arXiv:0704.0790v3  [hep-th]  15 Nov 2007
Dynamical Casimir effect for gravitons in bouncing braneworlds
Marcus Ruser∗ and Ruth Durrer†
Département de Physique Théorique, Université de Genève,
24 quai Ernest Ansermet, 1211 Genève 4, Switzerland.
We consider a two-brane system in five-dimensional anti-de Sitter space-time. We study particle
creation due to the motion of the physical brane which first approaches the second static brane
(contraction) and then recedes from it (expansion). The spectrum and the energy density of the
generated gravitons are calculated. We show that the massless gravitons have a blue spectrum
and that their energy density satisfies the nucleosynthesis bound with very mild constraints on the
parameters. We also show that the Kaluza-Klein modes cannot provide the dark matter in an anti-
de-Sitter braneworld. However, for natural choices of parameters, backreaction from the Kaluza-
Klein gravitons may well become important. The main findings of this work have been published
in form of a Letter [R.Durrer and M.Ruser, Phys. Rev. Lett. 99, 071601 (2007), arXiv:0704.0756].
PACS numbers: 04.50.+h, 11.10.Kk, 98.80.Cq
I. INTRODUCTION
In recent times, the possibility that our observed
Universe might represent a hypersurface in a higher-
dimensional space-time has received considerable
attention. The main motivation for this idea is the
fact, that string theory [1, 2], which is consistent only
in ten spac-etime dimensions (or 11 for M–theory)
allows for solutions where the standard model particles
(like fermions and gauge bosons) are confined to some
hypersurface, called the brane, and only the graviton can
propagate in the whole space-time, the bulk [2, 3]. Since
gravity is not well constrained at small distances, the
dimensions normal to the brane, the extra dimensions,
can be as large as 0.1mm.
Based on this feature, Arkani-Hamed, Dimopoulos
and Dvali (ADD) proposed a braneworld model where
the presence of two or more flat extra-dimensions can
provide a solution to the hierarchy problem, the problem
of the huge difference between the Planck scale and the
electroweak scale [4, 5].
In 1999 Randall and Sundrum (RS) introduced a model
with one extra dimension, where the bulk is a slice of
five-dimensional anti de-Sitter (AdS) space. Such curved
extra dimensions are also referred to as warped extra
dimensions. While in the RS I model [6] with two flat
branes of opposite tension at the edges of the bulk the
warping leads to an interesting solution of the hierarchy
problem, it localizes four-dimensional gravity on a single
positive tension brane in the RS II model [7].
Within the context of warped braneworlds, cosmological
evolution, i.e., the expansion of the Universe, can be
understood as the motion of the brane representing our
Universe through the AdS bulk. Thereby the Lanczos-
Sen-Darmois-Israel-junction conditions [8, 9, 10, 11],
relate the energy-momentum tensor on the brane to the
∗Electronic address: marcus.ruser@physics.unige.ch
†Electronic address: ruth.durrer@physics.unige.ch
extrinsic curvature and hence to the brane motion which
is described by a modified Friedmann equation. At low
energy, however, the usual Friedmann equations for the
expansion of the Universe are recovered [12, 13].
Since gravity probes the extra dimension, gravita-
tional perturbations on the brane, i.e. in our Universe,
carry five-dimensional effects in form of massive four-
dimensional gravitons, the so-called Kaluza-Klein (KK)
tower. Depending on the particular brane trajectory,
these perturbations may be significantly amplified lead-
ing to observable consequences, for example, a stochastic
gravitational wave background. (For a review of stochas-
tic gravitational waves see [14].) This amplification
mechanism is identical to the dynamical Casimir effect
for the electromagnetic field in cavities with dynamical
walls (moving mirrors); see [15, 16, 17] and references
therein. In the quantum field theoretical language,
such an amplification corresponds to the creation of
particles out of vacuum fluctuations. Hence, in the same
way a moving mirror leads to production of photons,
the brane moving through the bulk causes creation of
gravitons. Thereby, not only the usual four-dimensional
graviton might be produced, but also gravitons of the
KK tower can be excited. Those massive gravitons are of
particular interest, since their energy density could dom-
inate the energy density of the Universe and spoil the
phenomenology if their production is sufficiently copious.
The evolution of cosmological perturbations under
the influence of a moving brane has been the subject of
many studies during recent years. Since one has to deal
with partial differential equations and time-dependent
boundary conditions, the investigation of the evolution
of perturbations in the background of a moving brane is
quite complicated. Analytical progress has been made
based on approximations like the “near brane limit” and
a slowly moving brane [18, 19, 20, 21].
The case of de Sitter or quasi-de Sitter inflation
on the brane has been investigated analytically in
[22, 23, 24, 25, 26]. In [25] it is demonstrated that dur-
http://arxiv.org/abs/0704.0790v3
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yb(t)
FIG. 1: Two branes in an AdS5 spacetime, with y denot-
ing the fifth dimension and L the AdS curvature scale. The
physical brane is on the left at time dependent position yb(t).
While it is approaching the static brane its scale factor is
decreasing and when it moves away from the static brane it
is expanding [cf. Eq. (2.3)]. The value of the scale factor of
the brane metric as function of the extra dimension y is also
indicated.
ing slow-roll inflation (modeled as a period of quasi-de
Sitter expansion) the standard four-dimensional result
for the amplitude of perturbations is recovered at low
energies while it is enhanced at high energies.
However, most of the effort has gone into numerical
simulations [27, 28, 29, 30, 31, 32, 33, 34, 35, 36], in
particular in order to investigate the high-energy regime.
Thereby different coordinate systems have been used
for which the brane is at rest, and different numerical
evolution schemes have been employed in order to solve
the partial differential equation.
In this work we chose a different way of looking at
the problem. We shall apply a formalism used to
describe the dynamical Casimir effect to study the
production of gravitons in braneworld cosmology. This
approach and its numerical implementation offers many
advantages. The most important one is the fact that
this approach deals directly with the appearing mode
couplings by means of coupling matrices. (In [19] a
similar approach involving coupling matrices has been
used. However, perturbatively only, and not in the
complexity presented here.) Hence, the interaction
between the four-dimensional graviton and the KK
modes is not hidden within a numerical simulation but
can directly be investigated making it possible to reveal
the underlying physics in a very transparent way.
We consider a five-dimensional anti-de Sitter spacetime
with two branes in it; a moving positive tension brane
representing our Universe and a second brane which, for
definiteness, is kept at rest. This setup is depicted in
Fig. 1. For this model we have previously shown that
in a radiation dominated Universe, where the second,
fixed brane is arbitrarily far away, no gravitons are
produced [37].
The particular model which we shall consider is
strongly motivated by the ekpyrotic or cyclic Universe
and similar ideas [38, 39, 40, 41, 42, 43, 44, 45, 46].
In this model, roughly speaking, the hot big bang
corresponds to the collision of two branes; a moving
bulk brane which hits “our” brane, i.e. the observable
Universe. Within such a model, it seems to be possible to
address all major cosmological problems (homogeneity,
origin of density perturbations, monopole problem)
without invoking the paradigm of inflation. For more
details see [38] but also [39] for critical comments.
One important difference between the ekpyrotic model
and standard inflation is that in the latter one tensor
perturbations have a nearly scale invariant spectrum.
The ekpyrotic model, on the other hand, predicts a
strongly blue gravitational wave spectrum with spectral
tilt nT ≃ 2 [38]. This blue spectrum is a key test
for the ekpyrotic scenario since inflation always pre-
dicts a slightly red spectrum for gravitational waves.
One method to detect a background of primordial
gravitational waves of wavelengths comparable to the
Hubble horizon today is the polarization of the cosmic
microwave background. Since a strongly blue spectrum
of gravitational waves is unobservably small for large
length scales, the detection of gravitational waves in the
cosmic microwave background polarization would falsify
the ekpyrotic model [38].
Here we consider a simple specific model which is
generic enough to cover important main features of the
generation and evolution of gravitational waves in the
background of a moving brane whose trajectory involves
a bounce. First, the physical brane moves towards the
static brane, initially the motion is very slow. During
this phase our Universe is contracting, i.e. the scale
factor on the brane decreases, the energy density on
the brane increases and the motion becomes faster. We
suppose that the evolution of the brane is driven by a
radiation component on the brane, and that at some
more or less close encounter of the two branes which we
call the bounce, some high-energy mechanism which we
do not want to specify in any detail, turns around the
motion of the brane leading to an expanding Universe.
Modeling the transition from contraction to subsequent
expansion in any detail would require assumptions about
unknown physics. We shall therefore ignore results
which depend on the details of the transition. Finally
the physical brane moves away from the static brane
back towards the horizon with expansion first fast and
then becoming slower as the energy density drops. This
model is more similar to the pyrotechnic Universe of
Kallosh, Kofman and Linde [39] where the observable
Universe is also represented by a positive tension brane
rather than to the ekpyrotic model where our brane has
negative tension.
We address the following questions: What is the spec-
trum and energy density of the produced gravitons, the
massless zero mode and the KK modes? Can the gravi-
ton production in such a brane Universe lead to limits,
e.g. on the AdS curvature scale via the nucleosynthesis
bound? Can the KK modes provide the dark matter
or lead to stringent limits on these models? Similar
results could be obtained for the free gravi-photon and
gravi-scalar, i.e. when we neglect the perturbations of
the brane energy momentum tensor which also couple to
these gravity wave modes which have spin-1 respectively
spin-0 on the brane.
The reminder of the paper is organized as follows.
After reviewing the basic equations of braneworld cos-
mology and tensor perturbations in Sec. II, we discuss
the dynamical Casimir effect approach in Sec. III. In
Sec. IV we derive expressions for the energy density
and the power spectrum of gravitons. Thereby we show
that, very generically, KK gravitons cannot play the
role of dark matter in warped braneworlds. This is
explained by the localization of gravity on the moving
brane which we discuss in detail. Section V is devoted to
the presentation and discussion of our numerical results.
In Sec. VI we reproduce some of the numerical results
with analytical approximations and we derive fits for
the number of produced gravitons. We discuss our main
results and their implications for bouncing braneworlds
in Sec. VII and conclude in Sec. VIII. Some technical
aspects are collected in appendices.
The main and most important results of this rather long
and technical paper are published in the Letter [47].
II. GRAVITONS IN MOVING BRANEWORLDS
A. A moving brane in AdS5
We consider a AdS-5 spacetime. In Poincaré coordi-
nates, the bulk metric is given by
ds2 = gABdx
AdxB =
−dt2 + δijdxidxj + dy2
(2.1)
The physical brane (our Universe) is located at some time
dependent position y = yb(t), while the 2nd brane is at
fixed position y = ys (see Fig. 1). The induced metric on
the physical brane is given by
ds2 =
y2b (t)
dt2 + δijdx
= a2(η)
−dη2 + δijdxidxj
, (2.2)
where
a(η) =
yb(t)
(2.3)
is the scale factor and η denotes the conformal time of
an observer on the brane,
dt ≡ γ−1dt . (2.4)
We have introduced the brane velocity
v ≡ dyb
= − LH√
1 + L2H2
and (2.5)
1− v2
1 + L2H2 . (2.6)
Here H is the usual Hubble parameter,
H ≡ ȧ/a2 ≡ a−1H = −L−1γv , (2.7)
and an overdot denotes the derivative with respect to
conformal time η. The bulk cosmological constant Λ is
related to the curvature scale L by Λ = −6/L2. The
junction conditions on the brane lead to [37, 48]
(ρ+ T ) = 6
1 + L2H2
, (2.8)
(ρ+ P ) = − 2LḢ
1 + L2H2
. (2.9)
Here T is the brane tension and ρ and P denote the
energy density and pressure of the matter confined on
the brane. Combining (2.8) and (2.9) results in
ρ̇ = −3Ha(ρ+ P ) , (2.10)
while taking the square of (2.8) leads to
. (2.11)
These equations form the basis of brane cosmology and
have been discussed at length in the literature (for re-
views see [49, 50]). The last equation is called the mod-
ified Friedmann equation for brane cosmology [13]. For
usual matter with ρ+ P > 0, ρ decreases during expan-
sion and at sufficiently late time ρ ≪ T . The ordinary
four-dimensional Friedmann equation is then recovered if
and we set κ
= 8πG4 =
. (2.12)
Here we have neglected a possible four-dimensional cos-
mological constant. The first of these equations is the
RS fine tuning implying
κ5 = κ4 L . (2.13)
Defining the string and Planck scales by
= L3s , κ4 =
= L2Pl , (2.14)
respectively, the RS fine tuning condition leads to
. (2.15)
As outlined in the introduction, we shall be interested
mainly in a radiation dominated low-energy phase, hence
in the period where
ρ and |v| ≪ 1 so that γ ≃ 1 , dη ≃ dt .
(2.16)
In such a period, the solutions to the above equations are
of the form
a(t) =
|t|+ tb
, (2.17)
yb(t) =
|t|+ tb
, (2.18)
v(t) = − sgn(t)L
(|t|+ tb)2
≃ −HL . (2.19)
Negative times (t < 0) describe a contracting phase,
while positive times (t > 0) describe radiation dominated
expansion. At t = 0, the scale factor exhibits a kink and
the evolution equations are singular. This is the bounce
which we shall not model in detail, but we will have to in-
troduce a cutoff in order to avoid ultraviolet divergencies
in the total particle number and energy density which are
due to this unphysical kink. We shall show, that when
the kink is smoothed out at some length scale, the pro-
duction of particles (KK gravitons) of masses larger than
this scale is exponentially suppressed, as it is expected.
The (free) parameter tb > 0 determines the value of the
scale factor at the bounce ab, i.e. the minimal interbrane
distance, as well as the velocity at the bounce vb
ab = a(0) =
, |v(0)| ≡ vb =
. (2.20)
Apparently we have to demand tb > L which implies
yb(t) < L.
B. Tensor perturbations in AdS5
We now consider tensor perturbations on this back-
ground. Allowing for tensor perturbations hij(t,x, y) of
the spatial three-dimensional geometry at fixed y, the
bulk metric reads
ds2 =
−dt2 + (δij + 2hij)dxidxj + dy2
. (2.21)
Tensor modes satisfy the traceless and transverse condi-
tions, hii = ∂ih
j = 0. These conditions imply that hij has
only two independent degrees of freedom, the two polar-
ization states • = ×,+. We decompose hij into spatial
Fourier modes,
hij(t,x, y) =
(2π)3/2
•=+,×
eik·xe•ij(k)h•(t, y;k) ,
(2.22)
where e•ij(k) are unitary constant transverse-traceless po-
larization tensors which form a basis of the two polariza-
tion states • = ×,+. For hij to be real we require
h∗•(t, y;k) = h•(t, y;−k). (2.23)
The perturbed Einstein equations yield the equation of
motion for the mode functions h
, which obey the Klein-
Gordon equation for minimally coupled massless scalar
fields in AdS5 [25, 51, 52]
∂2t + k
2 − ∂2y +
(t, y;k) = 0 . (2.24)
In addition to the bulk equation of motion the modes also
satisfy a boundary condition at the brane coming from
the second junction condition,
LH∂th• −
1 + L2H2∂yh•
− γ (v∂t + ∂y)h•|yb =
aPΠ(T )
. (2.25)
Here Π(T )
denotes possible anisotropic stress perturba-
tions in the brane energy momentum tensor. We are in-
terested in the quantum production of free gravitons, not
in the coupling of gravitational waves to matter. There-
fore we shall set Π(T )
= 0 in the sequel, i.e. we make
the assumption that the Universe is filled with a perfect
fluid. Then, (2.25) reduces to 1
(v∂t + ∂y)h•|yb(t) = 0 . (2.26)
This is not entirely correct for the evolution of gravity
modes since at late times, when matter on the brane is
no longer a perfect fluid (e.g., free-streaming neutrinos)
and anisotropic stresses develop which slightly modify
the evolution of gravitational waves. We neglect this sub-
dominant effect in our treatment. (Some of the difficul-
ties which appear when Π(T )
6= 0 are discussed in [48].)
The wave equation (2.24) together with the boundary
condition (2.26) can also be obtained by variation of the
action
Sh = 2
yb(t)
|∂th•|2 − |∂yh•|2 − k2|h•|2
, (2.27)
which follows from the second order perturbation of the
gravitational Lagrangian. The factor 2 in the action is
1 In Equations (4) and (8) of our Letter [47] two sign mistakes
have creeped in.
due to Z2 symmetry. Indeed, Equation (2.26) is the only
boundary condition for the perturbation amplitude h•
which is compatible with the variational principle δSh =
0, except if h• is constant on the brane. Since this issue is
important in the following, it is discussed more detailed
in Appendix A.
C. Equations of motion in the late time/low energy
limit
In this work we restrict ourselves to relatively late
times, when
ρT ≫ ρ2 and therefore |v| ≪ 1. (2.28)
In this limit the conformal time on the brane agrees
roughly with the 5D time coordinate, dη ≃ dt and we
shall therefore not distinguish these times; we set t = η.
We want to study the quantum mechanical evolution
of tensor perturbations within a canonical formulation
similar to the dynamical Casimir effect for the electro-
magnetic field in dynamical cavities [15, 16, 17]. In or-
der to pave the way for canonical quantization, we have
to introduce a suitable set of functions allowing the ex-
pansion of the perturbation amplitude h• in canonical
variables. More precisely, we need a complete and or-
thonormal set of eigenfunctions φα of the spatial part
−∂2y + 3y∂y = −y
y−3∂y
of the differential opera-
tor (2.24). The existence of such a set depends on the
boundary conditions and is ensured if the problem is
of Sturm-Liouville type (see, e.g.,[53]). For the junc-
tion condition (2.26), such a set does unfortunately not
exist due to the time derivative. One way to proceed
would be to introduce other coordinates along the lines
of [54] for which the junction condition reduces to a sim-
ple Neumann boundary condition leading to a problem
of Sturm-Liouville type. This transformation is, however,
relatively complicated to implement without approxima-
tions and is the subject of future work.
Here we shall proceed otherwise, harnessing the fact that
we are interested in low energy effects only, i.e. in small
brane velocities. Assuming that one can neglect the
time derivative in the junction condition since |v| ≪ 1,
Eq. (2.25) reduces to a simple Neumann boundary con-
dition. We shall therefore work with the boundary con-
ditions
∂yh•|yb = ∂yh•|ys = 0 . (2.29)
Then, at any time t the eigenvalue problem for the spatial
part of the differential operator (2.24)
−∂2y +
φα(t, y) = −y3∂y
y−3∂yφα(t, y)
= m2α(t)φα(t, y) (2.30)
is of Sturm-Liouville type if we demand that the φα’s are
subject to the boundary conditions (2.29). Consequently,
the set of eigenfunctions {φα(t, y)}∞α=0 is complete,
φα(t, y)φα(t, ỹ) = δ(y − ỹ)y3 , (2.31)
and orthonormal with respect to the inner-product
(φα, φβ) = 2
yb(t)
φα(t, y)φβ(t, y) = δαβ . (2.32)
Note the factor 2 in front of both expressions which is
necessary in order to take the Z2 symmetry properly into
account.
The eigenvalues mα(t) are time-dependent and discrete
due to the time-dependent but finite distance between
the branes and the eigenfunctions φα(t, y) are time-
dependent in particular because of the time dependence
of the boundary conditions (2.29). The case α = 0
with m0 = 0 is the zero mode, i.e. the massless four-
dimensional graviton. Its general solution in accordance
with the boundary conditions is just a constant with re-
spect to the extra dimension, φ0(t, y) = φ0(t), and is fully
determined by the normalization condition (φ0, φ0) = 1:
φ0(t) =
ysyb(t)√
y2s − y2b (t)
. (2.33)
For α = i ∈ {1, 2, 3, · · · , } with eigenvalues mi > 0, the
general solution of (2.30) is a combination of the Bessel
functions J2 (mi(t) y) and Y2 (mi(t) y). Their particular
combination is determined by the boundary condition at
the moving brane. The remaining boundary condition at
the static brane selects the possible values for the eigen-
values mi(t), the KK masses. For any three-momentum
k these masses build up an entire tower of momenta in
the y-direction; the fifth dimension. Explicitely, the so-
lutions φi(t, y) for the KK modes read
φi(t, y) = Ni(t)y
2C2 (mi(t) y) (2.34)
Cν(miy) = Y1(miyb)Jν(miy)−J1(miyb)Yν(miy). (2.35)
The normalization reads
Ni(t, yb, ys) =
y2sC22(mi ys)− (2/(miπ))
(2.36)
where we have used that
C2(mi yb) =
πmi yb
. (2.37)
2 Note that we have changed the parameterization of the solutions
with respect to [37] for technical reasons. There, we also did not
take into account the factor 2 related to Z2 symmetry.
It can be simplified further by using
C2(mi ys) =
Y1(mi yb)
Y1(mi ys)
πmi ys
(2.38)
leading to
Y 21 (miys)
Y 21 (miyb)− Y 21 (miys)
. (2.39)
Note that it is possible to have Y 21 (mi ys)− Y 21 (mi yb) =
0. But then both Y 21 (miys) = Y
1 (miyb) = 0 and
Eq. (2.39) has to be understood as a limit. For that rea-
son, the expression (2.36) for the normalization is used
in the numerical simulations later on. Its denominator
remains always finite.
The time-dependent KK masses {mi(t)}∞i=1 are deter-
mined by the condition
C1 (mi(t)ys) = 0 . (2.40)
Because the zeros of the cross product of the Bessel func-
tions J1 and Y1 are not known analytically in closed form,
the KK-spectrum has to be determined by solving Eq.
(2.40) numerically 3. An important quantity which we
need below is the rate of change ṁi/mi of a KK mass
given by
m̂i ≡
= ŷb
m2i π
N2i (2.41)
where the rate of change of the brane motion ŷb is just
the Hubble parameter on the brane
ŷb(t) ≡
ẏb(t)
yb(t)
≃ −Ha = − ȧ
= −H . (2.42)
On account of the completeness of the eigenfunctions
φα(t, y) the gravitational wave amplitude h•(t, y;k)
subject to the boundary conditions (2.29) can now be
expanded as
h•(t, y;k) =
qα,k,•(t)φα(t, y) . (2.43)
The coefficients qα,k,•(t) are canonical variables describ-
ing the time evolution of the perturbations and the fac-
κ5/L3 has been introduced in order to render the
qα,k,•’s canonically normalized. In order to satisfy (2.23)
we have to impose the same condition for the canonical
variables, i.e.
q∗α,k,• = qα,−k,•. (2.44)
3 Approximate expressions for the zeros can be found in [55].
One could now insert the expansion (2.43) into the wave
equation (2.24), multiplying it by φβ(t, y) and integrating
out the y−dependence by using the orthonormality to de-
rive the equations of motion for the variables qα,k,•. How-
ever, as we explain in Appendix A, a Neumann boundary
condition at a moving brane is not compatible with a free
wave equation. The only consistent way to implement the
boundary conditions (2.29) is therefore to consider the
action (2.27) of the perturbations as the starting point
to derive the equations of motion for qα,k,•. Inserting
(2.43) into (2.27) leads to the canonical action
S = 1
|q̇α,k,•|2 − ω2α,k|qα,k,•|2
Mαβ (qα,k,•q̇β,−k,• + qα,−k,•q̇β,k,•)
+Nαβqα,k,•qβ,−k,•
. (2.45)
We have introduced the time-dependent frequency of a
graviton mode
ω2α,k =
k2 +m2α , k = |k| , (2.46)
and the time-dependent coupling matrices
Mαβ = (∂tφα, φβ) , (2.47)
Nαβ = (∂tφα, ∂tφβ) =
MαγMβγ (2.48)
which are given explicitely in Appendix B (see also [37]).
Consequently, the equations of motion for the canonical
variables are
q̈α,k,• + ω
α,kqα,k,• +
[Mβα −Mαβ ] q̇β,k,•
Ṁαβ −Nαβ
qβ,k,• = 0 . (2.49)
The motion of the brane through the bulk, i.e.
the expansion of the Universe, is encoded in the
time-dependent coupling matrices Mαβ , Nαβ. The
mode couplings are caused by the time-dependent
boundary condition ∂yh•(t, y)|yb = 0 which forces the
eigenfunctions φα(t, y) to be explicitly time-dependent.
In addition, the frequency of a KK mode ωα,k is also
time-dependent since the distance between the two
branes changes when the brane is in motion. Both time-
dependencies can lead to the amplification of tensor
perturbations and, within a quantum theory which is
developed in the next section, to graviton production
from vacuum.
Because of translation invariance with respect to the
directions parallel to the brane, modes with different k
do not couple in (2.49). The three-momentum k enters
the equation of motion for the perturbation only via the
frequency ωα,k, i.e. as a global quantity. Equation (2.49)
is similar to the equation describing the time-evolution
of electromagnetic field modes in a three-dimensional
dynamical cavity [16] and may effectively be described
by a massive scalar field on a time-dependent interval
[17]. For the electromagnetic field, the dynamics of the
cavity, or more precisely the motion of one of its walls,
leads to photon creation from vacuum fluctuations. This
phenomenon is usually referred to as dynamical Casimir
effect. Inspired by this, we shall call the production of
gravitons by the moving brane as dynamical Casimir
effect for gravitons.
D. Remarks and comments
In [37] we have already shown that in the limit where
the fixed brane is sent off to infinity, ys → ∞, only the
M00 matrix element survives with M00 = −H[1 + O(ǫ)]
and ǫ = yb/ys. M00 expresses the coupling of the zero
mode to the brane motion. Since all other couplings dis-
appear for ǫ → 0 all modes decouple from each other and,
in addition, the canonical variables for the KK modes de-
couple from the brane motion itself. This has led to the
result that at late times and in the limit ys ≫ yb, the KK
modes with non-vanishing mass evolve trivially, and only
the massless zero mode is coupled to the brane motion
q̈0,k,• +
k2 − Ḣ −H2
q0,k,• = 0 . (2.50)
Since φ0 ∝ 1/a [cf. Eqs. (4.2),(4.5)] we have found in [37]
that the gravitational zero mode on the brane h0,•(t;k) ≡√
κ5/L3q0,k,•φ0(t, yb) evolves according to
ḧ0,•(t;k) + 2Hḣ0,•(t;k) + k2h0,•(t;k) = 0 , (2.51)
which explicitely demonstrates that at low energies (late
times) the homogeneous tensor perturbation equation in
brane cosmology reduces to the four-dimensional tensor
perturbation equation.
An important comment is in order here concerning
the RS II model. In the limit ys → ∞ the fixed brane
is sent off to infinity and one ends up with a single
positive tension brane in AdS, i.e. the RS II model.
Even though we have shown that all couplings except
M00 vanish in this limit, that does not imply that this
is necessarily the case for the RS II setup. Strictly
speaking, the above arguments are only valid in a two
brane model with ys ≫ 1. Starting with the RS II model
from the beginning, the coupling matrices do in general
not vanish when calculated with the corresponding
eigenfunctions which can be found in, e.g., [22]. One
just has to be careful when taking those limits. But
what the above consideration demonstrates is that, if
the couplings of the zero mode to the KK modes vanish,
like in the ys ≫ 1 limit or in the low energy RS II model
as observed in numerical simulations (see below) the
standard evolution equation for the zero mode emerges
automatically from five-dimensional perturbation theory.
Starting from five-dimensional perturbation theory,
our formalism does imply the usual evolution equation
for the four-dimensional graviton in a FLRW-Universe
in the limit of vanishing couplings. This serves as a
very strong indication (but certainly not proof!) for
the fact that the approach based on the approximation
(2.29) and the expansion of the action in canonical
variables rather than the wave equation is consistent
and leads to results which should reflect the physics at
low energies. As already outlined, if one would expand
the wave equation (2.24) in the set of functions φα,
the resulting equation of motion for the corresponding
canonical variables is different from Eq. (2.49) and
cannot be derived from a Lagrangian or Hamiltonian
(see Appendix A). Moreover, in [30] the low energy RS
II scenario has been studied numerically including the
full junction condition (2.26) without approximations
(see also [27]). Those numerical results show that
the evolution of tensor perturbations on the brane is
four-dimensional, i.e. described by Eq. (2.51) derived
here analytically. Combining these observations gives
us confidence that the used approach based on the
Neumann boundary condition approximation and the
action as starting point for the canonical formulation is
adequate for the study of tensor perturbations in the
low energy limit. The many benefits this approach offers
will become visible in the following.
III. QUANTUM GENERATION OF TENSOR
PERTURBATIONS
A. Preliminary remarks
We now introduce a treatment of quantum generation
of tensor perturbations. This formalism is an advance-
ment of the method which is presented in [15, 16, 17]
for the dynamical Casimir effect for a scalar field and
the electromagnetic field to gravitational perturbations
in the braneworld scenario.
The following method is very general and not restricted
to a particular brane motion as long as it complies with
the low energy approach [cf. Eq. (2.28)]. We assume that
asymptotically, i.e. for t → ±∞, the physical brane
approaches the Cauchy horizon (yb → 0), moving very
slowly. Then, the coupling matrices vanish and the KK
masses are constant (for yb close to zero, Eq. (2.40) re-
duces to J1(miys) = 0):
Mαβ(t) = 0 , lim
mα(t) = const. ∀α, β .
(3.1)
In this limit, the system (2.49) reduces to an infinite set
of uncoupled harmonic oscillators. This allows to intro-
duce an unambiguous and meaningful particle concept,
i.e. notion of (massive) gravitons.
As a matter of fact, in the numerical simulations, the
brane motion has to be switched on and off at finite times.
These times are denoted by tin and tout, respectively. We
introduce vacuum states with respect to times t < tin < 0
and t > tout > 0. In order to avoid spurious effects influ-
encing the particle creation, we have to chose tin small,
respectively tout large enough such that the couplings are
effectively zero at these times. Checking the indepen-
dence of the numerical results on the choice of tin and
tout guarantees that these times correspond virtually to
the real asymptotic states of the brane configuration.
B. Quantization, initial and final state
Canonical quantization of the gravity wave amplitude
is performed by replacing the canonical variables qα,k,•
by the corresponding operators q̂α,k,•
ĥ•(t, y;k) =
q̂α,k,•(t)φα(t, y) . (3.2)
Adopting the Heisenberg picture to describe the quantum
time-evolution, it follows that q̂α,k,• satisfies the same
equation (2.49) as the canonical variable qα,k,•.
Under the assumptions outlined above, the operator
q̂α,k,• can be written for times t < tin as
q̂α,k,•(t < tin) = (3.3)
2ωinα,k
âinα,k,•e
−i ωinα,k t + âin†α,−k,•e
i ωinα,k t
where we have introduced the initial-state frequency
ωinα,k ≡ ωα,k(t < tin) . (3.4)
This expansion ensures that Eq. (2.44) is satisfied. The
set of annihilation and creation operators {âinα,k,•, â
α,k,•}
corresponding to the notion of gravitons for t < tin is
subject to the usual commutation relations
âinα,k,•, â
α′,k′,•′
= δαα′δ••′δ
(3)(k− k′) , (3.5)
âinα,k,•, â
α′,k′,•′
α,k,•, â
α′,k′,•′
= 0. (3.6)
For times t > tout, i.e. after the motion of the brane has
ceased, the operator q̂α,k,• can be expanded in a similar
manner,
q̂α,k,•(t > tout) = (3.7)
2ωoutα,k
âoutα,k,•e
−i ωout
t + â
out †
α,−k,•e
i ωout
with final state frequency
ωoutα,k ≡ ωα,k(t > tout) . (3.8)
The annihilation and creation operators {âoutα,k,•, â
out †
α,k,•}
correspond to a meaningful definition of final state gravi-
tons (they are associated with positive and negative fre-
quency solutions for t ≥ tout) and satisfy the same com-
mutation relations as the initial state operators.
Initial |0, in〉 ≡ |0, t < tin〉 and final |0, out〉 ≡ |0, t > tout〉
vacuum states are uniquely defined via 4
âinα,k,•|0, in〉 = 0 , âoutα,k,•|0, out〉 = 0 , ∀ α, k, • . (3.9)
The operators counting the number of particles defined
with respect to the initial and final vacuum state, respec-
tively, are
N̂ inα,k,• = â
α,k,•â
α,k,• , N̂
α,k,• = â
out †
α,k,•â
α,k,• . (3.10)
The number of gravitons created during the motion of
the brane for each momentum k, quantum number α and
polarization state • is given by the expectation value of
the number operator N̂outα,k,• of final-state gravitons with
respect to the initial vacuum state |0, in〉:
N outα,k,• = 〈0, in|N̂outα,k,•|0, in〉. (3.11)
If the brane undergoes a non-trivial dynamics between
tin < t < tout it is â
α,k,•|0, in〉 6= 0 in general, i.e.
graviton production from vacuum fluctuations takes
place.
From (2.22), the expansion (3.2) and Eqs.(3.3),
(3.7) it follows that the quantized tensor perturbation
with respect to the initial and final state can be written
ĥij(t < tin,x,y) =
(2π)3/2
âinα,k,• e
−i ωinα,k t
2ωinα,k
× u•ij,α(t < tin,x, y,k) + h.c. (3.12)
ĥij(t > tout,x,y) =
(2π)3/2
âoutα,k,• e
−i ωoutα,k t
2ωoutα,k
× u•ij,α(t > tout,x, y,k) + h.c. . (3.13)
We have introduced the basis functions
u•ij,α(t,x, y,k) = e
ik ·x e•ij(k)φα(t, y). (3.14)
which, on account of (e•ij(k))
∗ = e•ij(−k), satisfy
(u•ij,α(t,x, y,k))
∗ = u•ij,α(t,x, y,−k).
4 Note that the notations |0, t < tin〉 and |0, t > tout〉 do not mean
that the states are time-dependent; states do not evolve in the
Heisenberg picture.
C. Time evolution
During the motion of the brane the time evolution of
the field modes is described by the system of coupled
differential equations (2.49). To account for the inter-
mode couplings mediated by the coupling matrix Mαβ
the operator q̂α,k,• is decomposed as
q̂α,k,•(t) =
2ωinβ,k
âinβ,k,•ǫ
α,k(t) + â
β,−k,•ǫ
α,k (t)
(3.15)
The complex functions ǫ
α,k(t) also satisfy the system of
coupled differential equations (2.49). With the ansatz
(3.15) the quantized tensor perturbation at any time dur-
ing the brane motion reads
ĥij(t,x, y) = (3.16)
âinβ,k,•√
2ωinβ,k
α,k(t)u
ij,α(t,x, y,k) + h.c. .
Due to the time-dependence of the eigenfunctions φα,
the time-derivative of the gravity wave amplitude con-
tains additional mode coupling contributions. Using the
completeness and orthnormality of the φα’s it is readily
shown that
h•(t, y;k) =
p̂α,−k,•(t)φα(t, y) (3.17)
where
p̂α,−k,•(t) = ˙̂qα,k,•(t) +
Mβαq̂β,k,•(t). (3.18)
The coupling term arises from the time dependence of
the mode functions φα. Accordingly, the time derivative
hij reads
hij(t,x, y) =
âinβ,k,•√
2ωinβ,k
× (3.19)
× f (β)α,k(t)u
ij,α(t,x, y,k) + h.c.
where we have introduced the function
α,k(t) = ǫ̇
α,k(t) +
Mγα(t)ǫ
γ,k(t) . (3.20)
By comparing Eq. (3.12) and its time-derivative with
Eqs. (3.16) and (3.19) at t = tin one can read off the
initial conditions for the functions ǫ
α,k(tin) = δαβ Θ
α,k , (3.21)
α,k(tin) =
−iωinα,kδαβ −Mβα(tin)
Θinβ,k (3.22)
with phase
Θinα,k = e
−iωinα,k tin . (3.23)
The choice of this phase for the initial condition is in
principle arbitrary, we could as well set Θinα,k = 1. But
with this choice, ǫ
α,k(t) is independent of tin for t < tin
and therefore it is also at later times independent of tin
if only we choose tin sufficiently early. This is especially
useful for the numerical work.
D. Bogoliubov transformations
The two sets of annihilation and creation operators
{âinα,k,•, â
α,k,•} and {âoutα,k,•, â
out †
α,k,•} corresponding to the
notion of initial-state and final-state gravitons are re-
lated via a Bogoliubov transformation. Matching the
expression for the tensor perturbation Eq. (3.16) and
its time-derivative Eq. (3.19) with the final state expres-
sion Eq. (3.13) and its corresponding time-derivative at
t = tout one finds
âoutβ,k,• =
Aαβ,k(tout)âinα,k,• + B∗αβ,k(tout)â
α,−k,•
(3.24)
Aβα,k(tout) =
ωoutα,k
ωinβ,k
α,k(tout) +
ωoutα,k
α,k(tout)
(3.25)
Bβα,k(tout) =
Θoutα,k
ωoutα,k
ωinβ,k
α,k(tout)−
ωoutα,k
α,k(tout)
(3.26)
where we shall stick to the phase Θoutα,k defined like Θ
in (3.23) for completeness. Performing the matching at
tout = tin the Bogoliubov transformation should become
trivial, i.e. the Bogoliubov coefficients are subject to vac-
uum initial conditions
Aαβ,k(tin) = δαβ , Bαβ,k(tin) = 0. (3.27)
Evaluating the Bogoliubov coefficients (3.25) and (3.26)
for tout = tin by making use of the initial conditions
(3.21) and (3.22) shows the consistency. Note that the
Bogoliubov transformation (3.24) is not diagonal due
to the inter-mode coupling. If during the motion of
the brane the graviton field departs form its vacuum
state one has Bαβ,k(tout) 6= 0, i.e. gravitons have been
generated.
By means of Eq. (3.24) the number of generated
final state gravitons (3.11), which is the same for every
polarization state, is given by
N outα,k (t ≥ tout) =
•=+,×
〈0, in|N̂outα,k,•|0, in〉
|Bβα,k(tout)|2. (3.28)
Later we will sometimes interpret tout as a continuous
variable tout → t such that N outα,k → Nα,k(t), i.e. it
becomes a continuous function of time. We shall call
Nα,k(t) the instantaneous particle number [see Appendix
C 2], however, a physical interpretation should be made
with caution.
E. The first order system
From the solutions of the system of differential equa-
tions (2.49) for the complex functions ǫ
α,k, the Bogoli-
ubov coefficient Bαβ,k, and hence the number of cre-
ated final state gravitons (3.28), can now be calculated.
It is however useful to introduce auxiliary functions
α,k(t), η
α,k(t) through
α,k(t) = ǫ
α,k(t) +
ωinα,k
α,k(t) (3.29)
α,k(t) = ǫ
α,k(t)−
ωinα,k
α,k(t) . (3.30)
These are related to the Bogoliubov coefficients via
Aβα,k(tout) = (3.31)
ωoutα,k
ωinβ,k
∆+α,k(tout)ξ
α,k(tout) + ∆
α,k(tout)η
α,k(tout)
Bβα,k(tout) = (3.32)
Θoutα,k
ωoutα,k
ωinβ,k
∆−α,k(tout)ξ
α,k(tout) + ∆
α,k(tout)η
α,k(tout)
where we have defined
∆±α,k(t) =
ωinα,k
ωα,k(t)
, (3.33)
Using the second order differential equation for ǫ
α,k, it is
readily shown that the functions ξ
α,k(t), η
α,k(t) satisfy
the following system of first order differential equations:
α,k(t) = −i
a+αα,k(t)ξ
α,k(t)− a
αα,k(t)η
α,k(t)
c−αγ,k(t)ξ
γ,k(t) + c
αγ,k(t)η
γ,k(t)
(3.34)
α,k(t) = −i
a−αα,k(t)ξ
α,k(t)− a
αα,k(t)η
α,k(t)
c+αγ,k(t)ξ
γ,k(t) + c
αγ,k(t)η
γ,k(t)
(3.35)
a±αα,k(t) =
ωinα,k
ωα,k(t)
ωinα,k
 , (3.36)
c±γα,k(t) =
Mαγ(t)±
ωinα,k
ωinγ,k
Mγα(t)
. (3.37)
The vacuum initial conditions (3.27) entail the initial
conditions
α,k(tin) = 2 δαβ Θ
α,k , η
α,k(tin) = 0. (3.38)
With the aid of Eq. (3.32), the coefficient Bαβ,k(tout),
and therefore the number of produced gravitons, can be
directly deduced from the solutions to this system of
coupled first order differential equations which can be
solved using standard numerics.
In the next section we will show how interesting
observables like the power spectrum and the energy den-
sity of the amplified gravitational waves are expressed in
terms of the number of created gravitons. The system
(3.34, 3.35) of coupled differential equations forms the
basis of our numerical simulations. Details of the applied
numerics are collected in Appendix D.
IV. POWER SPECTRUM, ENERGY DENSITY
AND LOCALIZATION OF GRAVITY
A. Perturbations on the brane
By solving the system of coupled differential equations
formed by Eqs. (3.34) and (3.35) the time evolution of
the quantized tensor perturbation ĥij(t,x, y) can be com-
pletely reconstructed at any position y in the bulk. Ac-
cessible to observations is the imprint which the pertur-
bations leave on the brane, i.e. in our Universe. Of par-
ticular interest is therefore the part of the tensor pertur-
bation which resides on the brane. It is given by eval-
uating Eq. (2.22) at the brane position y = yb (see also
[36])
ĥij(t,x, yb) =
(2π)3/2
•=+,×
eik·xe•ij(k)ĥ•(t, yb,k) .
(4.1)
The motion of the brane (expansion of the Universe) en-
ters this expression via the eigenfunctions φα(t, yb(t)).
We shall take (4.1) as the starting point to define ob-
servables on the brane.
The zero-mode function φ0(t) [cf. Eq. (2.33)] does not
depend on the extra dimension y. Using Eq. (2.37), one
reads off from Eq. (2.34) that the eigenfunctions on the
brane φα(t, yb) are
φα(t, yb) = yb Yα(yb) =
Yα(a) (4.2)
where we have defined
Y0(a) =
y2s − y2b
and (4.3)
Yn(a) =
Y 21 (mnys)
Y 21 (mnyb)− Y 21 (mnys)
, (4.4)
for the zero- and KK modes, respectively. One immedi-
ately is confronted with an interesting observation: the
function Yα(a) behaves differently with the expansion of
the Universe for the zero mode α = 0 and the KK modes
α = n. This is evident in particular in the asymptotic
regime ys ≫ yb, i.e. yb → 0 (|t|, a → ∞) where, exploit-
ing the asymptotics of Y1 (see [55]), one finds
Y0(a) ≃ 1 , Yn(a) ≃
|Y1(mnys)| ≃
(4.5)
Ergo, Y0 is constant while Yn decays with the expansion
of the Universe as 1/a. For large n one can approximate
mn ≃ nπ/ys and Y1(mnys) ≃ Y1(nπ) ≃ (1/π)
2/n [55],
so that
Yn(a) ≃
, Y2n(a) ≃
πL2mn
2 ysa2
. (4.6)
In summary, the amplitude of the KKmodes on the brane
decreases faster with the expansion of the Universe than
the amplitude of the zero mode. This leads to interest-
ing consequences for the observable power spectrum and
energy density and has a clear physical interpretation: It
manifest the localization of usual gravity on the brane.
As we shall show below, KK gravitons which are traces
of the five-dimensional nature of gravity escape rapidly
from the brane.
B. Power spectrum
We define the power spectrum P(k) of gravitational
waves on the brane as in four-dimensional cosmology by
using the restriction of the tensor amplitude to the brane
position (4.1):
(2π)3
P(k)δ(3)(k− k′) (4.7)
•=×,+
0, in
∣∣∣ĥ•(t, yb;k)ĥ†•(t, yb;k′)
∣∣∣0, in
i.e. we consider the expectation value of the field operator
ĥ• with respect to the initial vacuum state at the position
of the brane y = yb(t). In order to get a physically mean-
ingful power spectrum, averaging over several oscillations
of the gravitational wave amplitude has to be performed.
Equation (4.7) describes the observable power spectrum
imprinted in our Universe by the four-dimensional spin-2
graviton component of the five-dimensional tensor per-
turbation.
The explicit calculation of the expectation value involv-
ing a “renormalization” of a divergent contribution is car-
ried out in detail in Appendix C 2. The final result reads
P(k) = 1
(2π)3
Rα,k(t)Y2α(a). (4.8)
The function Rα,k(t) can be expressed in terms of the
Bogoliubov coefficients (3.25) and (3.26) if one considers
tout as a continuous variable t:
Rα,k(t) =
Nα,k(t) +ONα,k(t)
ωα,k(t)
. (4.9)
Nα,k(t) is the instantaneous particle number [cf. Ap-
pendix C 1] and the function ONα,k(t) is defined in
Eq. (C9).
It is important to recall thatNα,k(t) can in general not be
interpreted as a physical particle number. For example
zero modes with wave numbers such that kt < 1 can-
not be considered as particles. They have not performed
several oscillations and their energy density cannot be
defined in a meaningful way.
Equivalently, expressed in terms of the complex functions
α,k, one finds
Rα,k(t) =
|ǫ(β)α,k(t)|2
ωinβ,k
ωα,k(t)
+Oǫα,k(t), (4.10)
with Oǫα,k given in Eq. (C10). Equation (4.8) together
with (4.9) or (4.10) holds at all times.
If one is interested in the power spectrum at early times
kt ≪ 1, it is not sufficient to take only the instantaneous
particle number Nα,k(t) in Eq. (4.9) into account. This
is due to the fact that even if the mode functions ǫ
are already oscillating, the coupling matrix entering the
Bogoliubov coefficients might still undergo a non-trivial
time dependence [cf. Eq. (6.16)]. In the next section
we shall show explicitly, that in a radiation dominated
bounce particle creation, especially of the zero mode,
only stops on sub-Hubble times, kt > 1, even if the mode
functions are plane waves right after the bounce [cf, e.g.,
Figs. 6, 7, 9]. Therefore, in order to determine the per-
turbation spectrum of the zero mode, one has to make
use of the full expression expression (4.10) and may not
use (4.11), given below.
At late times, kt ≫ 1 (t ≥ tout) when the brane moves
slowly, the couplings Mαβ go to zero and particle cre-
ation has come to an end, both functions ONα,k and Oǫα,k
do not contribute to the observable power spectrum after
averaging over several oscillations. Furthermore, the in-
stantaneous particle number then equals the (physically
meaningful) number of created final state gravitons N outα,k
and the KK masses are constant. Consequently, the ob-
servable power spectrum at late times takes the form
P(k, t ≥ tout) =
(2π)3
N outα,k
ωoutα,k
Y2α(a) , (4.11)
where we have used that κ5/L = κ4. Its dependence on
the wave number k is completely determined by the spec-
tral behavior of the number of created gravitons N outα,k .
It is useful to decompose the power spectrum in its zero-
mode and KK-contributions:
P = P0 + PKK . (4.12)
In the late time regime, using Eqs. (4.11) and (4.5), the
zero-mode power spectrum reads
P0(k, t ≥ tout) =
(2π)3
N out0,k . (4.13)
As expected for a usual four-dimensional tensor perturba-
tion (massless graviton), on sub-Hubble scales the power
spectrum decreases with the expansion of the Universe
as 1/a2.
In contrast, the KK mode power spectrum for late times,
given by
PKK(k, t ≥ tout) =
N outn,k
ωoutn,k
Y 21 (mnys),
(4.14)
decreases as 1/a4, i.e. with a factor 1/a2 faster than
P0. The gravity wave power spectrum at late times is
therefore dominated by the zero-mode power spectrum
and looks four dimensional. Contributions to it arising
from five-dimensional effects are scaled away rapidly as
the Universe expands due to the 1/a4 behavior of PKK.
In the limit of large masses mnys ≫ 1, n ≫ 1 and for
wave lengths k ≪ mn such that ωn,k ≃ mn, the late-time
KK-mode power spectrum can be approximated by
PKK(k, t ≥ tout) =
16π2ys
N outn,k (4.15)
where we have inserted Eq. (4.6) for Y2n(a).
Note that the formal summations over the particle num-
ber might be ill defined if the brane trajectory contains
unphysical features like discontinuities in the velocity. An
appropriate regularization is then necessary, for example,
by introducing a physically motivated cutoff.
C. Energy density
For a usual four-dimensional tensor perturbation hµν
on a backgroundmetric gµν an associated effective energy
momentum tensor can be defined unambiguously by (see,
e.g., [14, 56])
Tµν =
〈hαβ‖µhαβ‖ν〉 , (4.16)
where the bracket stands for averaging over several pe-
riods of the wave and “‖” denotes the covariant deriva-
tive with respect to the unperturbed background metric.
The energy density of gravity waves is the 00-component
of the effective energy momentum tensor. We shall use
the same effective energy momentum tensor to calculate
the energy density corresponding to the four-dimensional
spin-2 graviton component of the five-dimensional ten-
sor perturbation on the brane, i.e. for the perturbation
hij(t,x, yb) given by Eq. (4.1). For this it is important
to remember that in our low energy approach, and in
particular at very late times for which we want to cal-
culate the energy density, the conformal time η on the
brane is identical to the conformal bulk time t. The en-
ergy density of four-dimensional spin-2 gravitons on the
brane produced during the brane motion is then given by
[see also [36]]
κ4 a2
0, in| ˙̂hij(t,x, yb) ˙̂hij(t,x, yb)|0, in
. (4.17)
Here the outer bracket denotes averaging over several os-
cillations, which (in contrast to the power spectrum) we
embrace from the very beginning. The factor 1/a2 comes
from the fact that an over-dot indicates the derivative
with respect t. A detailed calculation is carried out in
Appendix C 3 leading to
(2π)3
ωα,kNα,k(t)Y2α(a) (4.18)
where againNα,k(t) is the instantaneous particle number.
At late times t > tout after particle creation has ceased,
the energy density is therefore given by
(2π)3
ωoutα,k N outα,k Y2α(a). (4.19)
This expression looks at first sight very similar to a
“naive” definition of energy density as integration over
momentum space and summation over all quantum num-
bers α of the energy ωoutα,k N outα,k of created gravitons.
(Note that the graviton number N outα,k already contains
the contributions of both polarizations [see Eq. (3.28)].)
However, the important difference is the appearance of
the function Y2α(a) which exhibits a different dependence
on the scale factor for the zero mode compared to the
KK modes.
Let us decompose the energy density into zero-mode and
KK contributions
ρ = ρ0 + ρKK . (4.20)
For the energy density of the massless zero mode one
then obtains
(2π)3
kN out0,k . (4.21)
This is the expected behavior; the energy density of stan-
dard four-dimensional gravitons scales like radiation.
On contrast, the energy density of the KK modes at late
times is found to be
ρKK =
(2π)3
ωoutn,k N outn,k m2nY 21 (mnys),
(4.22)
which decays like 1/a6. As the Universe expands, the en-
ergy density of massive gravitons on the brane is there-
fore rapidly diluted. The total energy density of gravita-
tional waves in our Universe at late times is dominated
by the standard four-dimensional graviton (massless zero
mode). In the large mass limit mnys ≫ 1,n ≫ 1 the KK-
energy density can be approximated by
ρKK ≃
2a6ys
(2π)3
N outn,k ωoutn,kmn . (4.23)
Due to the factor mn coming from the function Y2n, i.e.
from the normalization of the functions φn(t, y), for the
summation over the KK-tower to converge, the number
of produced gravitons N outn,k has to decrease faster than
1/m3n for large masses and not just faster than 1/m
one might naively expect.
D. Escaping of massive gravitons and localization
of gravity
As we have shown, the power spectrum and energy
density of the KK modes scale, at late times when par-
ticle production has ceased, with the expansion of the
Universe like
PKK ∝ 1/a4 , ρKK ∝ 1/a6. (4.24)
Both quantities decay by a factor 1/a2 faster than the
corresponding expressions for the zero-mode graviton. In
particular, the energy density of the KK particles on the
brane behaves effectively like stiff matter. Mathemat-
ically, this difference arises from the distinct behavior
of the functions Y0(a) and Yn(a) [cf. Eq. (4.5)] and is a
direct consequence of the warping of the fifth dimension.
But what is the underlying physics? As we shall discuss
now, this scaling behavior for the KK particles has
indeed a very appealing physical interpretation which is
in the spirit of the RS model.
First, the mass mn is a comoving mass. The (in-
stantaneous) ’comoving’ frequency or energy of a KK
graviton is ωn,k =
k2 +m2n, with comoving wave
number k. The physical mass of a KK mode measured
by an observer on the brane with cosmic time dτ = adt
is therefore mn/a, i.e. the KK masses are redshifted
with the expansion of the Universe. This comes from
the fact that mn is the wave number corresponding to
the y-direction with respect to the bulk time t which
corresponds to conformal time η on the brane and not to
physical time. It implies that the energy of KK particles
on a moving AdS brane is redshifted like that of massless
particles. From this alone one would expect that the
energy density of KK modes on the brane decays like
1/a4 (see also Appendix D of [22]).
Now, let us define the “wave function” for a gravi-
Ψα(t, y) =
φα(t, y)
(4.25)
which, by virtue of (φα, φα) = 1, satisfies
dyΨ2α(t, y) = 1 (4.26)
From the expansion of the gravity wave amplitude
Eq. (2.43) and the normalization condition it is clear that
Ψ2α(t, y) gives the probability to find a graviton of mass
mα for a given (fixed) time t at position y in the Z2-
symmetric AdS-bulk. Since φα satisfies Equation (2.30),
the wave function Ψα satisfies the Schrödinger like equa-
− ∂2yΨα +
Ψα = m
αΨα (4.27)
and the junction conditions (2.29) translate into
Ψα|y={yb,ys} = 0. (4.28)
In Fig. 2 we plot the evolution of Ψ21(t, y) under the
influence of the brane motion Eq. (2.18) with vb = 0.1.
For this motion, the physical brane starting at yb → 0 for
t → −∞ moves towards the static brane, corresponding
to a contracting Universe. After a bounce, it moves
back to the Cauchy horizon, i.e. the Universe expands.
The second brane is placed at ys = 10L and y ranges
from yb(t) to ys. We set Ψ
1 ≡ 0 for y < yb(t) . The
time-dependent KK mass m1 is determined numerically
from Eq. (2.40). As it is evident from this Figure, Ψ21
is effectively localized close to the static brane, i.e. the
weight of the KK-mode wave function lies in the region
of less warping, far from the physical brane. Thus the
probability to find a KK mode is larger in the region
with less warping. Since the effect of the brane motion
on Ψ21 is hardly visible in Fig. 2, we show the behavior
of Ψ21 close to the physical brane in Fig. 3. This shows
that Ψ21 peaks also at the physical brane but with an
amplitude roughly ten times smaller than the amplitude
at the static brane. While the brane, coming from
t → −∞, approaches the point of closest encounter Ψ21
slightly increases and peaks at the bounce t = 0 where,
as we shall show in the next Section, the production
of KK particles takes place. Afterwards, for t → ∞,
when the brane is moving back towards the Cauchy
horizon, the amplitude Ψ21 decreases again and so does
the probability to find a KK particle at the position of
the physical brane, i.e. in our Universe. The parameter
settings used in Figures 2 and 3 are typical parameters
which we use in the numerical simulations described
later on. However, the effect is illustrated much better
if the second brane is closer to the moving brane. In
Figure 4 we show Ψ21 for the same parameters as in
Figures 2 and 3 but now with ys = L. In this case, the
probability to find a KK particle on the physical brane
is of the same order as in the region close to the second
brane during times close to the bounce. However, as the
Universe expands, Ψ21 rapidly decreases at the position
of the physical brane.
From Eqs. (4.2) and (4.5) it follows that Ψ2n(t, yb) ∝ 1/a.
The behavior of the KK-mode wave function suggests
the following interpretation: If KK gravitons are created
on the brane, or equivalently in our Universe, they
escape from the brane into the bulk as the brane moves
back to the Cauchy horizon, i.e. when the Universe
undergoes expansion. This is the reason why the power
spectrum and the energy density imprinted by the KK
modes on the brane decrease faster with the expansion
of the Universe than for the massless zero mode.
The zero mode, on the other hand, is localized at
the position of the moving brane. The profile of φ0 does
not depend on the extra dimension, but the zero-mode
wave function Ψ0 does. Its square is
Ψ20(t, y) =
y2s − y2b
if ys ≫ yb ,
(4.29)
such that on the brane (y = yb) it behaves as
Ψ20(t, yb) ≃
. (4.30)
Equation (4.29) shows that, at any time, the zero
mode is localized at the position of the moving brane.
For a better illustration we show Eq. (4.29) in Fig. 5
for the same parameters as in Fig. 4. This is the
“dynamical analog” of the localization mechanism for
four-dimensional gravity discussed in [7].
To establish contact with [7] and to obtain a intu-
itive physical description, we rewrite the boundary value
problem (4.27), (4.28) as a Schrödinger-like equation
− ∂2yΨα(t, y) + V (y, t)Ψα(y, t) = mα(t)Ψα(y, t) (4.31)
V (y, t) =
yb(t)
δ(|y| − yb(t))
− 3a(t)
δ(|y| − yb(t)) , (4.32)
where we have absorbed the boundary condition at the
moving brane into the (instantaneous) volcano potential
V (y, t) and made use of Z2 symmetry. Similar to the
static case [7], at any time the potential (4.32) supports
a single bound state, the four-dimensional graviton
(4.29), and acts as a barrier for the massive KK modes.
The potential, ensuring localization of four-dimensional
gravity on the brane and the repulsion of KK modes,
moves together with the brane through the fifth dimen-
sion. Note that with the expansion of the Universe, the
“depth of the delta-function” becomes larger, expressing
the fact that the localization of four-dimensional gravity
becomes stronger at late times [cf. Eq. (4.30), Fig. 5].
In summary, the different scaling behavior for the
zero- and KK modes on the brane is entirely a conse-
quence of the geometry of the bulk space-time, i.e. of
the warping L2/y2 of the metric (2.1) 5. It is simply a
manifestation of the localization of gravity on the brane:
as time evolves, the KK gravitons, which are traces of
the five-dimensional nature of gravity, escape into the
bulk and only the zero mode which corresponds to the
usual four-dimensional graviton remains on the brane.
This, and in particular the scaling behavior (4.24),
remains also true if the second brane is removed, i.e. in
the limit ys → ∞, leading to the original RS II model.
By looking at (4.15) and (4.23) one could at first think
that then the KK-power spectrum and energy density
vanish and no traces of the KK gravitons could be
observed on the brane since both expressions behave
as 1/ys. But this is not the case since the spectrum of
KK masses becomes continuous. In the continuum limit
ys → ∞ the summation over the discrete spectrum mn
has to be replaced by an integration over continuous
masses m in the following way:
f(mn) −→
dmf(m) . (4.33)
f is some function depending on the spectrum, for
example f(mn) = N outn,k . The pre-factor 1/ys in (4.15)
and (4.23) therefore ensures the existence of the proper
continuum limit of both expressions.
Another way of seeing this is to repeat the same
calculations but using the eigenfunctions for the case
with only one brane from the beginning. Those are
δ-function normalized and can be found in, e.g., [22].
They are basically the same as (2.34) except that the
normalization is different since it depends on whether
the fifth dimension is compact or not. In particular, on
the brane, they have the same scale factor dependence
as (4.2).
At the end, the behavior found for the KK modes
should not come as a surprise, since the RS II model
has attracted lots of attention because of exactly this;
it localizes usual four-dimensional gravity on the brane.
As we have shown here, localization of standard four-
dimensional gravity on a moving brane via a warped
geometry automatically ensures that the KK modes
escape into the bulk as the Universe expands because
their wave function has its weight in the region of less
warping, resulting in an KK-mode energy density on the
brane which scales like stiff matter.
An immediate consequence of this particular scaling
behavior is that KK gravitons in an AdS braneworld
5 Note that it does not depend on a particular type of brane motion
and is expected to be true also in the high energy case which we
do not consider here.
FIG. 2: Evolution of Ψ21(t, y) = φ
1(t, y)/y
3 corresponding to
the probability to find the first KK graviton at time t at the
position y in the AdS-bulk. The static brane is at ys = 10L
and the maximal brane velocity is given by vb = 0.1.
FIG. 3: Evolution of Ψ21(t, y) as in Fig. 2 but zoomed into
the bulk-region close to the moving brane.
cannot play the role of dark matter. Their energy density
in our Universe decays much faster with the expansion
than that of ordinary matter which is restricted to reside
on the brane.
V. NUMERICAL SIMULATIONS
A. Preliminary remarks
In this section we present results of numerical simula-
tions for the bouncing model described by the equations
(2.17)-(2.19).
In the numerical simulations we set L = 1, i.e. all
FIG. 4: Evolution of Ψ21(t, y) for ys = L and vb = 0.1.
FIG. 5: Localization of four-dimensional gravity on a moving
brane: Evolution of Ψ20(t, y) for ys = L = 1 and vb = 0.1
which should be compared with Ψ21(t, y) shown in Fig. 4.
dimensionful quantities are measured in units of the
AdS5 curvature scale. Starting at initial time tin ≪ 0
where the initial vacuum state |0, in〉 is defined, the
system (3.34,3.35) is evolved numerically up to final time
tout. Thereby we set tin = −2πNin/k with 1 ≤ Nin ∈ N,
such that Θin0,k = 1 [cf. Eq. (3.23)]. This implies
0 (tin) = 2, i.e. independent of the three-dimensional
momentum k a (plane wave) zero-mode solution always
performs a fixed number of oscillations between tin and
the bounce at t = 0 [cf. Eq. (3.38)]. The final graviton
spectrum at N outα,k is calculated at late times tout ≫ 1
when the brane approaches the Cauchy horizon and
graviton creation has ceased. This quantity is physically
well defined and leads to the late-time power spectrum
(4.11) and energy density (4.19) on the brane. For
illustrative purposes, we also plot the instantaneous
particle number Nα,k,•(t) which also determines the
power spectrum at all times [cf Eq.(4.9)]. In this section
we shall use the term particle number respectively
graviton number for both, the instantaneous particle
number Nα,k,•(t) as well as the final state graviton
number N outα,k,•, keeping in mind that only the latter one
is physically meaningful.
There are two physical input parameters for the
numerical simulation; the maximal brane velocity vb (i.e.
tb) and the position of the static brane ys. The latter
determines the number of KK modes which fall within a
particular mass range. On the numerical side one has to
specify Nin and tout, as well as the maximum number of
KK modes nmax which one takes into account, i.e. after
which KK mode the system of differential equations is
truncated. The independence of the numerical results
on the choice of the time parameters is checked and the
convergence of the particle spectrum with increasing
nmax is investigated. More detailed information on
numerical issues including accuracy considerations are
collected in Appendix D.
One strong feature of the brane motion (2.18) is its kink
at the bounce t = 0. In order to study how particle
production depends on the kink, we shall compare
the motion (2.18) with the following motion which
has a smooth transition from contraction to expansion
(L = 1):
yb(t) =
(|t|+ tb − ts)−1 if |t| > ts
a+ (b/2)t2 + (c/4)t4 if |t| ≤ ts
(5.1)
with the new parameter ts in the range 0 < ts < tb. This
motion is constructed such that its velocity at |t| = ts
is the same as the velocity of the kink motion at the
bounce. This will be the important quantity determin-
ing the number of produced gravitons. For ts → 0 the
motion with smooth transition approaches (2.18). The
parameters a, b and c are obtained by matching the mo-
tions and the first and second derivatives. Matching also
the second derivative guarantees that possible spurious
effects contributing to particle production are avoided.
The parameter ts has to be chosen small enough, ts ≪ 1,
such that the maximal velocity of the smooth motion is
not much larger than vb in order to have comparable sit-
uations.
For reasons which will become obvious in the next two
sections we shall discuss the cases of long k ≪ 1 and
short wavelengths k ≫ 1, separately.
B. Generic results and observations for long
wavelengths k ≪ 1
Figure 6 displays the results of a numerical simula-
tion for three-momentum k = 0.01, static brane position
ys = 10 and maximal brane velocity vb = 0.1. Depicted is
FIG. 6: Evolution of the graviton number Nα,k,•(t) for the
zero mode and the first ten KK modes for three-momentum
k = 0.01 and vb = 0.1, ys = 10.
FIG. 7: Nn,k,•(t) for the zero mode and the first ten KK
modes for the parameters of Fig. 6, but without coupling of
the zero mode to the KK modes, i.e. Mi0 ≡ 0.
the graviton number for one polarizationNα,k,•(t) for the
zero mode and the first ten KK modes as well as the evo-
lution of the scale factor a(t) and the position of the phys-
ical brane yb(t). Initial and final times are Nin = 5 and
tout = 2000, respectively. The KK-particle spectrum will
be discussed in detail below. One observes that the zero-
mode particle number increases slightly with the expan-
sion of the Universe towards the bounce at t = 0. Close
to the bounce N0,k,•(t) increases drastically, shows a lo-
cal peak at the bounce and, after a short decrease, grows
again until the mode is sub-horizon (kt ≫ 1). Inside the
horizon N0,k,•(t) is oscillating around a mean value with
diminishing amplitude. This mean value which is reached
asymptotically for t → ∞ corresponds to the number of
generated final state zero-mode gravitonsN out0,k,•. Produc-
tion of KK-mode gravitons takes effectively place only at
the bounce in a step-like manner and the graviton num-
ber remains constant right after the bounce.
In Fig. 7 we show the numerical results obtained for the
same parameters as in Fig. 6 but without coupling of the
zero mode to the KK modes, i.e. Mi0 = 0 (and thus
also Ni0 = N0i = 0). One observes that the production
of zero-mode gravitons is virtually not affected by the
artificial decoupling 6. Note that even if M0j ≡ 0 (see
Eqs. B2), which is in general true for Neumann bound-
ary conditions, the zero mode q0,k,• couples in Eq. (2.49)
to the KK modes via N0j = M00Mj0 and through the
anti-symmetric combination Mαβ −Mβα.
In contrast, the production of the first ten KK modes is
heavily suppressed if Mi0 = 0. The corresponding final-
state graviton numbers N outn,k,• are reduced by four orders
of magnitude. This shows that the coupling to the zero
mode is essential for the production of massive gravitons.
Later we will see that this is true for light KK gravitons
only. If the KK masses exceed mi ∼ 1, they evolve in-
dependently of the four-dimensional graviton and their
evolution is entirely driven by the intermode couplings
Mij . It will also turn out that the time-dependence of
the KK mass mi plays only an inferior role for the gen-
eration of massive KK modes. On the other hand, the
effective decoupling of the evolution of the zero mode
from the KK modes occurs in general as long as k ≪ 1
is satisfied, i.e. for long-wavelengths. We will see that it
is no longer true for short wavelengths k ≫ 1.
The effective decoupling of the zero-mode evolution from
the KK modes makes it possible to derive analytical ex-
pressions for the number of zero-mode gravitons, their
power spectrum and energy density. The calculations
are carried out in section VIA
In summary we emphasize the important observation
that for long wavelengths the amplification of the four
dimensional gravity wave amplitude during the bounce
is not affected by the evolution of the KK gravitons. We
can therefore study the zero mode separately from the
KK modes in this case.
C. Zero mode: long wavelengths k ≪ 1
In Figure 8 we show the numerical results for the num-
ber of generated zero-mode gravitons N0,k,•(t) and the
evolution of the corresponding power spectrum P0(k) on
the brane for momentum k = 0.01, position of the static
brane ys = 10 and maximal brane velocity vb = 0.1. The
results have been obtained by solving the equations for
the zero mode alone, i.e. without the couplings to the KK
modes, since, as we have just shown, the evolution of the
6 Quantitatively it is N0,k,•(t = 2000) = 965.01 with and
N0,k,•(t = 2000) = 965.06 without Mi0. Note that this differ-
ence lies indeed within the accuracy of our numerical simulations
(see Appendix D.)
four-dimensional graviton for long wavelengths is not in-
fluenced by the KK modes. Thereby the power spectrum
is shown before and after averaging over several oscilla-
tions, i.e. employing Eq. (4.9) with and without the term
ON0,k, respectively. Right after the bounce where the gen-
eration of gravitons is initiated and which is responsible
for the peak in N0,k,• at t = 0, the number of gravitons
first decreases again. AfterwardsN0,k,• grows further un-
til the mode enters the horizon at kt = 1. Once on sub-
horizon scales kt ≫ 1, the number of produced gravitons
oscillates with a diminishing amplitude and asymptoti-
cally approaches the final state graviton number N out0,k,•.
During the growth of N0,k,• after the bounce, the power
spectrum remains practically constant. Within the range
of validity it is in good agreement with the analytical pre-
diction (6.22) yielding (L2(2π)3/κ4)P0(k, t) = 4vb(kL)2.
When particle creation has ceased, the full power spec-
trum Eq.(4.8) starts to oscillate with an decreasing am-
plitude. The time-averaged power spectrum obtained
by using Eq. (4.9) without the ON0,k-term is perfectly
in agreement with the analytical expression Eq. (6.20)
which gives (L2(2π)3/κ4)P0(k, t) = 2vb/t2. Note that at
early times, the time-averaged power spectrum behaves
not in the same way as the full one, demonstrating the
importance of the term ON0,k.
Figure 9 shows a summary of numerical results for the
number of created zero-mode gravitons N0,k,•(t) for dif-
ferent values of the three-momentum k. The maximum
velocity at the bounce is vb = 0.1 and the second brane is
at ys = 10. These values are representative. Other values
in accordance with the considered low-energy regime do
not lead to a qualitatively different behavior. Note that
the evolution of the zero mode does virtually not depend
on the value of ys as long as ys ≫ yb(0) (see below). Ini-
tial and final integration times are given by Nin = 5 and
tout = 20000, respectively.
For sub-horizon modes we compare the final graviton
spectra with the analytical prediction (6.17). Both are in
perfect agreement. On super-horizon scales where parti-
cle creation has not ceased yet N0,k,• is independent of
k. The corresponding time-evolution of the power spec-
tra P0(k, t) is depicted in Fig. 10. For the sake of clarity,
only the results for t > 0, i.e. after the bounce, are shown
in both figures.
The numerical simulations and the calculations of sec-
tion VIA reveal that the power spectrum for the four-
dimensional graviton for long wavelengths is blue on
super-horizon scales, as expected for an ekpyrotic sce-
nario.
The analytical calculations performed in section VIA
rely on the assumption that yb ≪ ys and tin → −∞.
Figure 11 shows the behavior of the number of generated
zero-mode gravitons of momentum k = 0.01 in depen-
dence on the inter-brane distance and the initial integra-
tion time. The brane velocity at the bounce is vb = 0.1
which implies that at the bounce the moving brane is
at yb(0) =
vb ≃ 0.316 (L = 1). In case of a close en-
counter of the two branes as for ys = 0.35, the production
FIG. 8: Time evolution of the number of created zero-mode
gravitons N0,k,•(t) and of the zero-mode power spectrum
(4.8): (a) for the entire integration time; (b) for t > 0 only.
Parameters are k = 0.01, ys = 10 and vb = 0.1. Initial and fi-
nal time of integration are given by Nin = 10 and tout = 4000,
respectively. The power spectrum is shown with and without
the term ON0,k,•, i.e. before and after averaging, respectively,
and compared with the analytical results.
FIG. 9: Numerical results for the time evolution of the num-
ber of created zero-mode gravitons N0,k,•(t) after the bounce
t > 0 for different three-momenta k. The maximal brane
velocity at the bounce is vb = 0.1 and the second brane is
positioned at ys = 10. In the final particle spectrum the nu-
merical values are compared with the analytical prediction
Eq. (6.17). Initial and final time of integration are given by
Nin = 5 and tout = 20000, respectively.
of massless gravitons is strongly enhanced compared to
the analytical result. But as soon as ys ≥ 1, (i.e. ys ≥ L)
the numerical result is very well described by the analyt-
ical expression Eq. (6.16) derived under the assumption
ys ≫ yb. For ys ≥ 10 the agreement between both is
very good. From panels (b) and (c) one infers that the
numerical result becomes indeed independent of the ini-
FIG. 10: Evolution of the zero-mode power spectrum after the
bounce t > 0 corresponding to the values and parameters of
Fig. 9. The numerical results are compared to the analytical
predictions Eqs. (6.20) and (6.22).
0 1000 2000
time t
analytical
=0.35
10 100 1000
2nd brane pos. y
0.1 1 10 100 1000
2nd brane pos. y
analytical
(a) (b)
FIG. 11: Dependence of the zero-mode particle number on
inter-brane distance and initial integration time for momen-
tum k = 0.01, maximal brane velocity vb = 0.1 in comparison
with the analytical expression Eq. (6.16). (a) Evolution of the
instantaneous particle number N0,k,•(t) with initial integra-
tion time given by Nin = 5 for ys = 0.35, 0.5 and 1. (b) Final
zero-mode graviton spectrum N0,k,•(tout = 2000) for various
values of ys and Nin. (c) Close-up view of (b) for large ys.
tial integration time when increasing Nin. Note that in
the limit Nin ≫ 1 the numerical result is slightly larger
than the analytical prediction but the difference between
both is negligibly small. This confirms the correctness
and accuracy of the analytical expressions derived in Sec-
tion VIA for the evolution of the zero-mode graviton.
0.01 0.1 1 10
Kaluza-Klein mass m
t nmax=60
= 0.3
= 0.1
 = 0.5●
FIG. 12: Final state KK-graviton spectra for k = 0.001, ys =
100, different maximal brane velocities vb and Nin = 1, tout =
400. The numerical results are compared with the analytical
prediction Eq. (6.34) (dashed line).
D. Kaluza-Klein-modes: long wavelengths k ≪ 1
Because the creation of KK gravitons ceases right
after the bounce [cf Fig. 6] one can stop the numerical
simulation and read out the number of produced KK
gravitons N outn,k,• at times for which the zero mode is still
super-horizon.
Even though Eq. (2.40) cannot be solved analytically,
the KK masses can be approximated by mn ≃ nπ/ys.
This expression is the better the larger the mass.
Consequently, for the massive modes the position of the
second brane ys determines how many KK modes belong
to a particular mass range ∆m.
In Figure 12 we show the KK-graviton spectra N outn,k,• for
three-momentum k = 0.001 and second brane position
ys = 100 for maximal brane velocities vb = 0.1, 0.3
and 0.5. For any velocity vb two spectra obtained
with nmax = 60 and 80 KK modes taken into account
in the simulation are compared to each other. This
reveals that the numerical results are stable up to a
KK mass mn ≃ 1. One infers that first, N outn,k,• grows
with increasing mass until a maximum is reached. The
position of the maximum shifts slightly towards larger
masses with increasing brane velocity vb. Afterwards,
N outn,k,• declines with growing mass. Until the maximum
is reached, the numerical results for the KK-particle
spectrum are very stable. This already indicates that
the KK-intermode couplings mediated by Mij are not
very strong in this mass range. In Figure 13 we show
the final KK-particle spectrum for the same parameters
as in Fig. 12 but for three-momentum k = 0.01 and
0.01 0.1 1 10
Kaluza-Klein mass m
=60, M
=0 for all i,j
 = 0.1
 = 0.5
 = 0.3
 = 0.9
FIG. 13: Final state KK-graviton spectra for k = 0.01, ys =
100, different vb and Nin = 1, tout = 400. The numerical
results are compared with the analytical prediction Eq. (6.34)
(dashed line). For vb = 0.3, 0.5 the spectra obtained without
KK-intermode and self-couplings (Mij ≡ 0 ∀ i, j) are shown
as well.
the additional velocity vb = 0.9
7. We observe the same
qualitative behavior as in Fig. 12. In addition we show
numerical results obtained for vb = 0.3 and 0.5 without
the KK-intermode and self couplings, i.e. we have set
Mij ≡ 0 ∀ i, j by hand. One infers that for KK masses,
depending slightly on the velocity vb but at least up to
mn ≃ 1, the numerical results for the spectra do not
change when the KK-intermode coupling is switched off.
Consequently, the evolution of light, i.e. mn <∼ 1, KK
gravitons is virtually not affected by the KK-intermode
coupling.
In addition we find that also the time-dependence of the
KK masses is not important for the production of light
KK gravitons which is explicitly demonstrated below.
Thus, production of light KK gravitons is driven by
the zero-mode evolution only. This allows us to find
an analytical expression, Eq. (6.34), for the number of
produced light KK gravitons in terms of exponential
integrals. The calculations which are based on several
approximations are performed in Section VIC.
In Figs. 12 and 13 the analytical prediction (6.34) for
the spectrum of final state gravitons has already been
included (dashed lines). Within its range of validity
it is in excellent agreement with the numerical results
obtained by including the full KK-intermode coupling.
It perfectly describes the dependence of N outn,k,• on the
three-momentum k and the maximal velocity vb. For
small velocities vb <∼ 0.1 it is also able to reproduce the
position of the maximum. This reveals that the KK-
7 Such a high brane velocity is of course not consistent with a
Neumann boundary condition Eq. (2.29) at the position of the
moving brane.
intermode coupling is negligible for light KK gravitons
and that their production is entirely driven by their
coupling to the four-dimensional graviton.
The analytical prediction is very precious for testing the
goodness of the parameters used in the simulations, in
particular the initial time tin (respectively Nin). Since it
has been derived for real asymptotic initial conditions,
tin → −∞, its perfect agreement with the numerical
results demonstrates that the values for Nin used in the
numerical simulations are large enough. No spurious
initial effects contaminate the numerical results.
Note, that the numerical values for N outn,k,• in the ex-
amples shown are all smaller than one. However, for
smaller values of k than the ones which we consider here
for purely numerical reasons, the number of generated
KK-mode particles is enhanced since N outn,k,• ∝ 1/k as
can be inferred from Eq. (6.34) in the limit k ≪ mn.
If we go to smaller values of ys, fewer KK modes
belong to a particular mass range. Hence, with the same
or similar number of KK modes as taken into account
in the simulations so far, we can study the behavior
of the final particle spectrum for larger masses. These
simulations shall reveal the asymptotical behavior of
N outn,k,• for mn → ∞ and therefore the behavior of the
total graviton number and energy density. Due to the
kink in the brane motion we cannot expect that the
energy density of produced KK-mode gravitons is finite
when summing over arbitrarily high frequency modes.
Eventually, we will have to introduce a cutoff setting the
scale at which the kink-approximation [cf. Eqs. (2.17)
- (2.19)] is no longer valid. This is the scale where the
effects of the underlying unspecified high-energy physics
which drive the transition from contraction to expansion
become important. The dependence of the final particle
spectrum on the kink will be studied later on in this
section in detail.
In Figures 14 and 15 we show final KK-graviton
spectra for ys = 10 and three-momentum k = 0.01 and
k = 0.1. The analytical expression Eq. (6.34) is depicted
as well and the spectra are always shown for at least
two values of nmax to indicate up to which KK mass
stability of the the numerical results is guaranteed.
Now, only two KK modes are lighter than m = 1. For
these modes the analytical expression Eq. (6.34) is valid
and in excellent agreement with the numerical results, in
particular for small brane velocities vb ∼ 0.1. As before,
the larger the velocity vb the more visible is the effect of
the truncation of the system of differential equations at
nmax.
For k = 0.01 the spectrum seems to follow a power law
decrease right after the maximum in the spectra. In
case of vb = 0.1 the spectrum is numerically stable up to
masses mn ≃ 20. In the region 5 <∼ mn <∼ 20 the spec-
trum is very well fitted by a power law N outn,k,• ∝ m−2.7n .
Also for larger velocities the decline of the spectrum
is given by the same power within the mass ranges
0.1 1 10
Kaluza-Klein mass m
 = 0.5
 = 0.3
 = 0.1
FIG. 14: Final state KK-graviton spectra for k = 0.01, ys =
10, different maximal brane velocities vb and Nin = 2, tout =
400. The numerical results are compared with the analytical
prediction Eq. (6.34) (dashed line).
0.1 1 10
Kaluza - Klein mass m
 = 0.5
 = 0.1
 = 0.3
FIG. 15: Final state KK-graviton spectra for k = 0.1, ys = 10,
different maximal brane velocities vb and Nin = 2, tout =
400. The numerical results are compared with the analytical
prediction Eq. (6.34) (dashed line).
where the spectrum is numerically stable. For k = 0.1,
however, the decreasing spectrum bends over at a mass
around mn ≃ 10 towards a less steep decline. This is
in particular visible in the two cases with vb = 0.1 and
0.3 where the first 100 KK modes have been taken into
account in the simulation. The behavior of the KK-mode
particle spectrum can therefore not be described by a
single power law decline for masses mn > 1. It shows
more complicated features instead, which depend on the
parameters. We shall demonstrate that this bending
over of the decline is related to the coupling properties of
the KK modes and to the kink in the brane motion. But
before we come to a detailed discussion of these issues,
let us briefly confront numerical results of different ys to
FIG. 16: Upper panel: Final state KK-particle spectra for
k = 0.01, vb = 0.1 and different ys = 3, 10, 30 and 100.
The analytical prediction Eq. (6.34) is shown as well (dashed
line). Lower panel: Energy ωoutn,kN
n,k,• of the produced fi-
nal state gravitons binned in mass intervals ∆m = 1 for
ys = 10, 30, 100.
demonstrate a scaling behavior.
In the upper panel of Figures 16 and 17 we com-
pare the final KK-spectra for several positions of the
second brane ys = 3, 10, 30 and 100 obtained for a
maximal brane velocity vb = 0.1 for k = 0.01 and
0.1, respectively. One observes that the shapes of the
spectra are identical. The bending over in the decline
of the spectrum at masses mn ∼ 1 is very well visible
for k = 0.1 and ys = 3, 10. For a given KK mode n
the number of particles produced in this mode is the
larger the smaller ys. But the smaller ys, the less KK
modes belong to a given mass interval ∆m. The energy
transferred into the system by the moving brane, which
is determined by the maximum brane velocity vb, is the
same in all cases. Therefore, the total energy of the pro-
duced final state KK gravitons of a given mass interval
∆m should also be the same, independent of how many
KK modes are contributing to it. This is demonstrated
in the lower panels of Figs. 16 and 17 where the energy
ωoutn,kN outn,k,•(in units of L) of the generated KK gravitons
binned in mass intervals ∆m = 1 is shown 8. One
observes that, as expected, the energy transferred into
the production of KK gravitons of a particular mass
range is the same (within the region where the numerical
results are stable), independent of the number of KK
modes lying in the interval. This is in particular evident
for ys = 30, 100. The discrepancy for ys = 10 is due
to the binning. As we shall discuss below in detail, the
8 The energy for the case ys = 3 is not shown because no KK mode
belongs to the first mass interval.
FIG. 17: Upper panel: Final state KK-particle spectra for
k = 0.1, vb = 0.1 and different ys = 3, 10, 30 and 100.
The analytical prediction Eq. (6.34) is shown as well (dashed
line). Lower panel: Energy ωoutn,kN
n,k,• of the produced fi-
nal state gravitons binned in mass intervals ∆m = 1 for
ys = 10, 30, 100.
particle spectrum can be split into two different parts.
The first part is dominated by the coupling of the zero
mode to the KK modes (as shown above), whereas the
second part is dominated by the KK-intermode couplings
and is virtually independent of the wave number k.
As long as the coupling of the zero mode to the KK
modes is the dominant contribution to KK-particle
production it is N outn,k,• ∝ 1/k [cf. Eq. (6.34)]. Hence,
Eoutn,k,• = ωoutn,kN outn,k,• ∝ 1/k if mn ≫ k. This explains why
the energy per mass interval ∆m is one order larger for
k = 0.01 (cf Fig. 16) than for k = 0.1 (cf Fig. 17) .
Let us now discuss the KK-spectrum for large masses.
The qualitative behavior of the spectrum N outn,k,• and the
mass at which the decline of the spectrum changes are
independent of ys. This is demonstrated in Figure 18
where KK-spectra for vb = 0.1, k = 0.1, ys = 10 [cf
Fig. 15] and ys = 3 [cf Fig. 17] are shown. The results
obtained by taking the full intermode coupling into
account are compared to results of simulations where
we have switched off the coupling of the KK modes to
each other as well as their self-coupling (Mij ≡ 0 ∀ i, j).
Furthermore we display the results for the KK-spectrum
obtained by taking only the KK-intermode couplings
into account, i.e. Mi0 = Mii = 0 ∀ i. One infers that
for the lowest masses the spectra obtained with all
couplings are identical to the ones obtained without
the KK-intermode (Mij = 0, i 6= j) and self-couplings
(Mii = 0). Hence, as already seen before, the primary
source for the production of light KK gravitons is their
coupling to the evolution of the four-dimensional gravi-
ton. In this mass range, the contribution to the particle
creation coming from the KK-intermode couplings is
very much suppressed and negligibly small.
1 10 100
Kaluza-Klein mass m
t full coupling
=0 for all i,j
=0, M
FIG. 18: KK-particle spectra for three-momentum k = 0.1,
maximum brane velocity vb = 0.1 and ys = 3 and 10 with
different couplings taken into account. The dashed lines indi-
cates again the analytical expression Eq. (6.34).
For masses mn ≃ 4 a change in the decline of the
spectrum sets in and the spectrum obtained without
the coupling of the KK modes to the zero mode starts
to diverge from the spectrum computed by taking all
the couplings into account. While the spectrum without
the KK-intermode couplings decreases roughly like a
power law N outn,k,• ∝ m−3n the spectrum corresponding
to the full coupling case changes its slope towards
a power law decline with less power. At this point
the KK-intermode couplings gain importance and the
coupling of the KK modes to the zero mode looses
influence. For a particular mass mc ≃ 9 the spectrum
obtained including the KK-intermode couplings only,
crosses the spectrum calculated by taking into account
exclusively the coupling of the KK modes to the zero
mode. After the crossing, the spectrum obtained by
using only the KK-intermode couplings approaches the
spectrum of the full coupling case. Both agree for large
masses. Thus for large masses mn > mc the production
of KK gravitons is dominated by the couplings of the
KK modes to each other and is not influenced anymore
by the evolution of the four-dimensional graviton. This
crossing defines the transition between the two regimes
mentioned before: for masses mn < mc the production
of KK gravitons takes place due to their coupling to
the zero mode Mi0, while it is entirely caused by the
intermode couplings Mij for masses mn > mc.
Decoupling of the evolution of the KK modes from
the dynamics of the four-dimensional graviton for
large masses implies that KK-spectra obtained for the
same maximal velocity are independent of the three-
momentum k. This is demonstrated in Fig. 19 where we
compare spectra obtained for vb = 0.1 and ys = 3 but
different k. As expected, all spectra converge towards
the same behavior for masses mn > mc.
1 2 3 4 5 10 20 30 1000.3
Kaluza-Klein masses m
t k=0.01
k=0.03
k=0.1
FIG. 19: Comparison of KK-particle spectra for ys = 3,
vb = 0.1 and three-momentum k = 0.01, 0.03, 0.1 and 1
demonstrating the independence of the spectrum on k for
large masses. nmax = 60 KK modes have been taken into
account in the simulations.
1 10 100
Kaluza-Klein mass m
t full coupling
=0 for all i, j
=0, i = j
=0, M
Mαβ = 0  for all α, β
FIG. 20: KK-particle spectra for three-momentum k = 0.1,
maximum brane velocities vb = 0.1 and ys = 3 for nmax = 40
obtained for different coupling combinations.
Figure. 20 shows KK-particle spectra for k = 0.1,vb = 0.1
and ys = 3 obtained for different couplings. This plot
visualizes how each particular coupling combination
contributes to the production of KK gravitons. It
shows, as already mentioned before but not shown
explicitly, that the Mii coupling which is the rate of
change of the corresponding KK mass [cf. Eqs. (2.41)
and (B4)] is not important for the production of KK
gravitons. Switching it off does not affect the final
graviton spectrum. We also show the result obtained
with all couplings but with α+ii(t) = ω
i,k and α
ii (t) = 0,
i.e. the time-dependence of the frequency [cf. Eq. (3.36)]
has been neglected. One observes that in this case the
spectrum for larger masses is quantitatively slightly
0.1 1 10 100
Kaluza-Klein mass m
t full coupling
=0 for all i, j
=0, M
k=0.01
k=0.1
k=0.1
FIG. 21: KK-particle spectra for ys = 10, vb = 0.1, nmax =
100 and three-momentum k = 0.01 and 0.1 with different
couplings taken into account. The thin dashed lines indicates
Eq. (6.34) and the thick dashed line Eq. (5.4).
different but has a identical qualitative behavior. If,
on the other hand, all the couplings are switched off
Mαβ ≡ 0 ∀α, β and only the time-dependence of the
frequency ωi,k is taken into account, the spectrum
changes drastically. Not only the number of produced
gravitons is now orders of magnitude smaller but also
the spectral tilt changes. For large masses it behaves as
Nn,k,• ∝ m−2n . Consequently, the time-dependence of
the graviton frequency itself plays only an inferior role
for production of KK gravitons.
The bottom line is that the main sources of the produc-
tion of KK gravitons is their coupling to the evolution of
the four-dimensional graviton (Mi0) and their couplings
to each other (Mij , i 6= j) for small and large masses,
respectively. Both are caused by the time-dependent
boundary condition. The time-dependence of the oscilla-
tor frequency ωj,k =
m2j(t) + k
2 is virtually irrelevant.
Note that this situation is very different from ordinary
inflation where there are no boundaries and particle
production is due entirely to the time dependence of the
frequency 9.
The behavior of the KK-spectrum, in particular
the mass mc at which the KK-intermode couplings start
to dominate over the coupling of the KK modes to
the zero mode depends only on the three-momentum
k = |k| and the maximal brane velocity vb. This is now
discussed. In Figure 21 we show KK-particle spectra
for ys = 10, vb = 0.1, nmax = 100 and three-momenta
k = 0.01 and 0.1. Again, the spectra obtained by taking
9 Note, however, that the time-dependent KK mass mj(t) enters
the intermode couplings.
0.1 1 10
Kaluza-Klein mass m
t full coupling
=0 for all i, j
=0, M
 = 0.1
 = 0.1
 = 0.3
 = 0.03
FIG. 22: KK-particle spectra for three-momentum k =
0.1,ys = 10 and maximum brane velocities vb = 0.03, 0.1 and
0.3 with nmax = 100. As in Fig. 21 different couplings
have been taken into account and thin dashed lines indicates
Eq. (6.34) and the thick dashed line Eq. (5.4).
all the couplings into account are compared to the case
where only the coupling to the zero mode is switched
on. One observes that for k = 0.01 the spectrum is
dominated by the coupling of the KK modes to the
zero mode up to larger masses than it is the case for
k = 0.1. For k = 0.01 the spectrum obtained taking
into account Mi0 only is identical to the spectrum
obtained with the full coupling up to mn ≃ 10. In case
of k = 0.1 instead, the spectrum is purely zero mode
dominated only up to mn ≃ 5. Hence, the smaller the
three-momentum k the larger is the mass range for
which the KK-intermode coupling is suppressed, and
the coupling of the zero mode to the KK modes is the
dominant source for the production of KK gravitons. As
long as the coupling to the zero mode is the primary
source of particle production, the spectrum declines
with a power law ∝ m−3n . Therefore, in the limiting case
k → 0 when the coupling of the zero mode to the KK
modes dominates particle production also for very large
masses it is N outn≫1,k→0,• ∝ 1/m3n.
Figure 22 shows KK-graviton spectra obtained for
the same parameters as in Fig. 21 but for fixed k = 0.1
and different maximal brane velocities vb. Again, the
spectra obtained by taking all the couplings into account
are compared with the spectra to which only the coupling
of the KK modes to the zero mode contributes. The
mass up to which the spectra obtained with different
couplings are identical changes only slightly with the
maximal brane velocity vb. Therefore, the dependence
of mc on the velocity is rather weak even if vb is changed
by an order of magnitude, but nevertheless evident.
This behavior of the spectrum can indeed be understood
qualitatively. In Section VIC we demonstrate that the
coupling strength of the KK modes to the zero mode
1 10 100
1 10 100
1 10 100
1 10 100
k=0.01 k=0.03
k=0.1 k=1
=25.54 mc=14.11
=8.24 mc=7.31
FIG. 23: KK-particle spectra for three-momentum k = 0.01,
0.03, 0.1 and 1 for ys = 3 and maximum brane velocity
vb = 0.1 with different couplings taken into account where
the notation is like in Fig. 22. From the crossing of the
Mii = Mij = 0- and Mii = Mi0 = 0 results we determine
the k-dependence of mc(k, vb). The thick dashed line indi-
cates Eq. (5.4).
at the bounce t = 0, where production of KK gravitons
takes place, is proportional to
. (5.2)
The larger this term the stronger is the coupling of
the KK modes to the zero mode, and thus the larger
is the mass up to which this coupling dominates over
the KK-intermode couplings. Consequently, the mass
at which the tilt of the KK-particle spectrum changes
depends strongly on the three-momentum k but only
weakly on the maximal brane velocity due to the square
root behavior of the coupling strength. This explains
qualitatively the behavior obtained from the numerical
simulations.
An approximate expression for mc(k, vb) can be
obtained from the numerical simulations. In Figure 23
we depict the KK-particle spectra for three-momentum
k = 0.01, 0.03, 0.1 and 1 for ys = 3 and maximum brane
velocity vb = 0.1 with different couplings taken into
account. The legend is as in Fig. 22. From the crossings
of the Mij = 0, i 6= j and Mii = Mi0 = 0 results one
can determine the k-dependence of mc. Note that the
spectra are not numerically stable for large masses, but
they are stable in the range where mc lies [cf., e.g.,
Fig. 25, for k = 0.1]. Using the data for k = 0.01, 0.03
and 0.1 one finds mc(k, vb) ∝ 1/
In Fig. 24 KK-graviton spectra are displayed for
k = 0.1, ys = 3 and maximal brane velocities
vb = 0.3, 0.2, 0.1, 0.08, 0.05 and 0.03 with different
couplings taken into account. It is in principle possible
to determine the vb-dependence of mc from the crossings
1 10 100
1 10 100
1 10 100
 = 0.3
=9.50
 = 0.2 v
 = 0.1
 = 0.08
=8.24
 = 0.05 v
 = 0.03
=9.04
=8.06 m
=7.72 m
=7.52
FIG. 24: KK-graviton spectra for three-momentum k =
0.1, ys = 3 and maximum brane velocities vb =
0.3, 0.2, 0.1, 0.08, 0.05 and 0.03 with different couplings taken
into account where the notation is like in Fig. 22. From the
crossing of the Mii = Mij = 0- and Mii = Mi0 = 0 results we
determine the vb-dependence of mc.
of the Mij = 0, i 6= j- and Mii = Mi0 = 0 results as
done for the k-dependence. However, the values for mc
displayed in the Figures indicate that the dependence
of mc on vb is very weak. From the given data it is not
possible to obtain a good fitting formula (as a simple
power law) for the vb-dependence of mc. (In the range
0.1 ≤ vb ≤ 0.3 a very good fit is mc = 1.12πv0.13b /
The reason is twofold. First of all, given the complicated
coupling structure, it is a priori not clear that a simple
power law dependence exists. Recall that also the ana-
lytical expression for the particle number Eq. (6.34) has
not a simple power law velocity dependence. Moreover,
for the number of modes taken into account (nmax = 40)
the numerical results are not stable enough to resolve
the weak dependence of mc on vb with a high enough
accuracy. (But it is good enough to perfectly resolve
the k-dependence.) The reason for the slow convergence
of the numerics will become clear below. As we shall
see, the corresponding energy density is dominated by
masses much larger than mc. Consequently the weak
dependence of mc on vb is not very important in that
respect and therefore does not need to be determined
more precisely. However, combining all the data we can
give as a fair approximation
mc(k, vb) ≃
π vαb
, with α ≃ 0.1. (5.3)
Taking α = 0.13 for 0.1 ≤ vb ≤ 0.3 and α = 0.08 for
0.03 ≤ vb ≤ 0.1 fits the given data reasonably well.
As we have seen, as long as the zero mode is the
dominant source of KK-particle production, the final
KK-graviton spectrum can be approximated by a power
law decrease m−3n . We can combine the presented
numerical results to obtain a fitting formula valid in this
regime:
N outn≫1,k≪1,• =
(Lmn)3
, for
< mn < mc. (5.4)
This fitting formula is shown in Figs. 21 22 and 23
and is in reasonable good agreement with the numerical
results. Since Eq. (5.4) together with (5.3) is an impor-
tant result, we have reintroduced dimensions, i.e. the
AdS scale L which is set to one in the simulations, in
both expressions.
Let us now investigate the slope of the KK-graviton
spectrum for masses mn → ∞ since it determines the
contribution of the heavy KK modes to the energy den-
sity. In Figure 25 we show KK-graviton spectra obtained
for three-momentum k = 0.1, second brane position
ys = 3 and maximal brane velocities vb = 0.01, 0.03
and 0.1. Up to nmax = 100 KK modes have been taken
into account in the simulations. One immediately is
confronted with the observation that the convergence
of the KK-graviton spectra for large mn is very slow.
This is since those modes, which are decoupled from the
evolution of the four-dimensional graviton, are strongly
affected by the kink in the brane motion. Recall that the
production of light KK gravitons with masses mn ≪ mc
is virtually driven entirely by the evolution of the
massless mode. Those light modes are not so sensitive
to the discontinuity in the velocity of the brane motion.
To be more precise, their primary source of excitation is
the evolution of the four-dimensional graviton but not
the kink which, as we shall discuss now, is responsible
for the production of heavy KK gravitons mn ≫ mc.
A discontinuity in the velocity will always lead to a di-
vergent total particle number. Arbitrary high frequency
modes are excited by the kink since the acceleration
diverges there. Due to the excitation of KK gravitons
of arbitrarily high masses, one cannot expect that the
numerical simulations show a satisfactory convergence
behavior which allows to determine the slope by fit-
ting the data. However, it is nevertheless possible to
give a quantitative expression for the behavior of the
KK-graviton spectrum for large masses. The studies of
the usual dynamical Casimir effect on a time-dependent
interval are very useful for this purpose.
For the usual dynamical Casimir effect it has been
shown analytically that a discontinuity in the velocity
will lead to a divergent particle number [57, 58]. In
Appendix E we discuss in detail the model of a massless
real scalar field on a time-dependent interval [0, y(t)] for
the boundary motion y(t) = y0 + v t with v = const,
and present numerical results for final particle spectra
(Fig. 34). For this motion it was shown in [58] that the
particle spectrum behaves as ∝ v2/ωn where ωn = nπ/y0
is the frequency of a massless scalar particle. This di-
vergent behavior is due to the discontinuities in the
velocity when the motion is switched on and off, and are
responsible for the slow convergence of the numerical
1 2 3 4 5 10 20 30 100
Kaluza-Klein mass m
t nmax = 40
 = 60
 = 80
=0.03
=0.01
FIG. 25: KK-particle spectra for k = 0.1, ys = 3 and max-
imal brane velocities vb = 0.01, 0.03, 0.1 up to KK masses
mn ≃ 100 compared with an 1/mn decline. The dashed lines
indicate the approximate expression (5.6) which describes the
asymptotic behavior of the final KK-particle spectra reason-
ably well, in particular for vb < 0.1.
results shown in Fig. 34 for this scenario.
At the kink in the brane-motion the total change of the
velocity is 2vb, similar to the case for the linear motion
where the discontinuous change of the velocity is 2v.
Consequently we may conclude that for large KK masses
mn ≫ mc for which the evolution of the KK modes is no
longer affected by their coupling to the four-dimensional
graviton the KK-graviton spectrum behaves as 10
N outn,k,• ∝
for mn ≫ mc . (5.5)
If we assume that the spectrum declines like 1/mn and
use that the numerical results for masses mn ≃ 20 are
virtually stable one finds N outn,k,• ∝ v2.08b /mn which de-
scribes the asymptotics of the numerical results well.
As for the dynamical Casimir effect for a uniform motion
discussed in Appendix E [cf. Fig. 34], the slow conver-
gence of the numerical results towards the 1/mn behav-
ior is well visible for large masses mn ≫ mc which do no
longer couple to the four-dimensional graviton. This is a
strong indication for the statement that the final gravi-
ton spectrum for large masses behaves indeed like (5.5).
It is therefore possible to give a single simple expression
for the final KK-particle spectrum for large masses which
10 Note that the discussion in Appendix E refers to Dirichlet bound-
ary conditions. For Neumann boundary conditions considered
here, the zero mode and its asymmetric coupling play certainly a
particular role. However, as we have shown, for large masses only
the KK-intermode couplings are important. Consequently, there
is no reason to expect that the qualitative behavior of the spec-
trum for large masses depends on the particular kind of boundary
condition.
-100 -50 0 50 100
time t
-100 -50 0 50 100
time t
1 10 50
KK mass m
1 10 50
KK mass m
=40 n
FIG. 26: Evolution of the zero-mode particle numberN0,k,•(t)
and final KK-graviton spectra N outn,k,• for ys = 3, maximal
brane velocity vb = 0.1 and three-momenta k = 10 and 30.
The dashed line in the upper plots indicate Eq. (6.17) (divided
by two) demonstrating the value of the number of produced
zero-mode gravitons without coupling to the KK modes.
comprises all the features of the spectrum even quantita-
tively reasonably well [cf. dashed lines in Fig. 25]
N outn,k,• ≃ 0.2
ωoutn,k ys
for mn ≫ mc . (5.6)
The 1/ys-dependence is compelling. It follows imme-
diately from the considerations on the energy and the
scaling behavior discussed above [cf. Figs. 16 and 17].
For completeness we now write 1/ωoutn,k instead of the KK
mass mn only, since what matters is the total energy
of a mode. Throughout this section this has not been
important since we considered only k ≪ 1 such that
ωoutn,k becomes independent of k for large masses mn ≫ k
[cf. Fig. 19].
E. Short wavelengths k ≫ 1
For short wave lengths k ≫ 1 (short compared to the
AdS-curvature scale L set to one in the simulations) a
completely new and very interesting effect appears. The
behavior of the four-dimensional graviton mode changes
drastically. We find that the zero mode now couples
to the KK gravitons and no longer evolves virtually
independently of the KK modes, in contrast to the
behavior for long wavelengths.
In Fig. 26 we show the evolution of the zero-mode
graviton number N0,k,•(t) and final KK-graviton spectra
N outn,k,• for ys = 3, maximal brane velocity vb = 0.1
and three-momenta k = 10 and 30. One observes that
the evolution of the four-dimensional graviton depends
on the number of KK modes nmax taken into account,
i.e. the zero mode couples to the KK gravitons. For
-100 -50 0 50 100
time t
0 10 20 30 40
0.02/(k-1.8)
analytical
FIG. 27: 4D-graviton number N0,k,•(t) for k = 3, 5, 10, 20 and
30 with ys = 3 and maximal brane velocity vb = 0.1. The
small plot shows the final graviton spectrum N out
0,k,• together
with a fit to the inverse law a/(k + b) [dashed line] and the
analytical fitting formula Eq. (6.23) [solid line]. For k = 10
and 30 the corresponding KK-graviton spectra are shown in
Fig. 26.
k = 10 the first 60 KK modes have to be included in the
simulation in order to obtain a numerically stable result
for the zero mode. In the case of k = 30 one already
needs nmax ≃ 100 in order to achieve numerical stability
for the zero mode.
Figure 27 displays the time-evolution of the number
of produced zero-mode gravitons N0,k,•(t) for ys = 3
and vb = 0.1. For large k the production of massless
gravitons takes place only at the bounce since these
short wavelength modes are sub-horizon right after
the bounce. Corresponding KK-particle spectra for
k = 10, 30 are depicted in Figs. 26 and 28. The insert
in Fig. 27 shows the resulting final four-dimensional
graviton spectrum N out0,k,•, which is very well fitted
by an inverse power law N out0,k,• = 0.02/(k − 1.8) 11.
Consequently, for k ≫ 1 the zero-mode particle number
N out0,k,• declines like 1/k only, in contrast to the 1/k2
behavior found for k ≪ 1.
The dependence of N out0,k,• on the maximal brane
velocity vb also changes. In Fig. 28 we show N0,k,•(t)
together with the corresponding KK-graviton spectra for
ys = 3, k = 5 and 10 in each case for different vb. Using
nmax = 60 KK modes in the simulations guarantees
numerical stability for the zero mode. The velocity
dependence of N out0,k,• is not given by a simple power law
as it is the case for k ≪ 1. This is not very surprising
11 The momenta k = 5, 10, 20, 30 and 40 have been used to obtain
the fit. Fitting the spectrum for k = 20, 30 and 40 to a power
law gives N out
0,k,•
∝ k−1.1.
-100 -50 0 50 100
time t
-100 -50 0 50 100
time t
1 10 100
KK mass m
1 10 100
KK mass m
 = 0.03
 = 0.05
 = 0.1
 = 0.03
 = 0.05
 = 0.1
 = 0.3
 = 0.1
 = 0.03
 = 0.03
 = 0.1
 = 0.3
FIG. 28: Zero-mode particle number N0,k,•(t) and corre-
sponding final KK-particle spectraN outn,k,• for ys = 3, k = 5, 10
and different maximal brane velocities vb. nmax = 60 guaran-
tees numerically stable solutions for the zero mode.
since now the zero mode couples strongly to the KK
modes [cf. Fig. 26]. For k = 10, for example, one finds
N out0,k,• ∝ v1.4b if vb <∼ 0.1.
As in the long wavelengths case, the zero-mode
particle number does not depend on the position of the
static brane ys even though the zero mode now couples
to the KK modes. This is demonstrated in Fig. 29 where
the evolution of the zero-mode particle number N0,k,•(t)
and the corresponding KK-graviton spectra with k = 10,
vb = 0.1 for the two values ys = 3 and 10 are shown.
One needs nmax = 60 for ys = 30 in order to obtain a
stable result for the zero mode which is not sufficient in
the case ys = 10. Only for nmax ≃ 120 the zero-mode
solution approaches the stable result which is identical
to the result obtained for ys = 3.
What is important is not the number of the KK modes
the four-dimensional graviton couples to, but rather
a particular mass mzm ≃ k. The zero mode couples
to all KK modes of masses below mzm no matter how
many KK modes are lighter. Recall that the value of ys
just determines how many KK modes belong to a given
mass interval ∆m since, roughly, mn ≃ nπ/ys. The
KK-spectra for k ≥ 1 show the same scaling behavior as
demonstrated for long wavelengths in Figs. 16 and 17.
The production of four-dimensional gravitons of
short wavelengths takes place on the expense of the
KK modes. In Fig. 30 we show the numerical results
for the final KK-particle spectra with vb = 0.1, ys = 3
and k = 3, 5, 10 and 30 obtained for different coupling
combinations. These spectra should be compared with
those shown in Fig. 23 for the long wavelengths case. For
k >∼ 10 the number of the produced lightest KK gravi-
tons is smaller in the full coupling case compared to the
situation where only the KK-intermode coupling is taken
-100 -50 0 50 100
time t
-100 -50 0 50 100
time t
1 10 50
KK mass m
1 10 50
KK mass m
FIG. 29: Zero-mode particle numberN0,k•(t) and correspond-
ing KK-graviton spectra for k = 10, vb = 0.1 and 2nd brane
positions ys = 3 and 10.
1 10 100
KK-mass m
1 10 100
KK-mass m
k=3 k=5
k=10 k=30
FIG. 30: Final KK-particle spectra N outn,k,• for vb = 0.1, ys = 3
and k = 3, 5, 10 and 30 and different couplings. Circles cor-
respond to the full coupling case, squares indicate the results
if Mij = Mii = 0, i.e. no KK-intermode couplings and dia-
monds correspond to Mi0 = 0, i.e. no coupling of KK modes
to the zero mode.
into account. In case k = 30, for instance, the numbers
of produced gravitons for the first four KK modes are
smaller for the full coupling case. This indicates that
the lightest KK modes couple strongly to the zero mode.
Their evolution is damped and graviton production in
those modes is suppressed. The production of zero-mode
gravitons on the other hand is enhanced compared to
the long wavelengths case. For short wavelengths, the
evolution of the KK modes therefore contributes to
the production of zero-mode gravitons. This may be
interpreted as creation of zero-mode gravitons out of
KK-mode vacuum fluctuations.
As in the long wavelengths case, the KK-particle
spectrum becomes independent of k if mn ≫ k and
10 20 100 200
frequency ω
k=40, n
k=30, n
k=20, n
k=10, n
k=5, n
FIG. 31: Final KK-particle spectra N outn,k,• for vb = 0.1,
ys = 3 and k = 5, 10, 20, 30 and 40. The dashed lines in-
dicate Eq. (6.35) for k = 10, 20, 30 and 40. For k ≥ 20, the
simple analytical expression (6.35) agrees quite well with the
numerical results.
the evolution of the KK modes is dominated by the
KK-intermode coupling. This is visible in Fig. 30 for
k = 3 and 5. Also the bend in the spectrum when the
KK-intermode coupling starts to dominate is observable.
For k = 10 and 30 this regime with mn ≫ k is not
reached.
As we have shown before, in the regime mn ≫ k
the KK-particle spectrum behaves as 1/ωoutn,k which will
dominate the energy density of produced KK gravitons.
If 1 ≪ mn <∼ k, however, the zero mode couples to
the KK modes and the KK-graviton spectrum does
not decay like 1/ωoutn,k. This is demonstrated in Fig. 31
where the number of produced final state gravitons
N outn,k,• is plotted as function of their frequency ωoutn,k for
parameters vb = 0.1, ys = 3 and k = 5, 10, 20, 30 and 40.
While for k = 5 the KK-intermode coupling dominates
for large masses [cf. Fig. 30] leading to a bending over
in the spectrum and eventually to an 1/ωoutn,k-decay, the
spectra for k = 20, 30 and 40 show a different behavior.
All the modes are still coupled to the zero mode leading
to a power-law decrease ∝ 1/(ωoutn,k)α with α ≃ 2. The
case k = 10 corresponds to an intermediate regime.
Also shown is the simple analytical expression given in
Eq. (6.35) which describes the spectra reasonably well
for large k (dashed line).
The KK-particle spectra in the region 1 ≪ mn <∼ k will
also contribute to energy density since the cutoff scale is
the same for the integration over k and the summation
over the KK-tower (see Section VID below).
1 2 3 4 5 10 20 30 100
Kaluza-Klein mass m
=0 (kink)
=0.005
=0.015
=0.05
 = 2.2 x 10
 exp(-0.1315 m
FIG. 32: KK-particle spectrum for ys = 3, vb = 0.1 and
k = 0.1 for the bouncing as well as smooth motions with
ts = 0.005, 0.015, and 0.05 to demonstrate the influence of the
bounce. nmax = 60 KK modes have been taken into account
in the simulations and the result for the kink motion is shown
as well.
F. A smooth transition
Let us finally investigate how the KK-graviton spec-
trum changes when the kink-motion (2.18) is replaced by
the smooth motion (5.1). In Fig. 32 we show the numeri-
cal results for the final KK-graviton spectrum for ys = 3,
vb = 0.1 and k = 0.1 for the smooth motion (5.1) with
ts = 0.05, 0.015 and 0.005. nmax = 60 modes have been
taken into account in the simulation and the results are
compared to the spectrum obtained with the kink-motion
(2.18). The parameter ts defines the scale Ls ≃ 2ts at
which the kink is smoothed, i.e. Ls corresponds to the
width of the transition from contraction to expansion.
The numerical results reveal that KK gravitons of masses
smaller than ms ≃ 1/Ls are not affected, but the pro-
duction of KK particles of masses larger than ms is
exponentially suppressed. This is in particular evident
for ts = 0.05 where the particle spectrum for masses
mn > 10 has been fitted to a exponential decrease. Going
to smaller values of ts, the suppression of KK-mode pro-
duction sets in for larger masses. For the example with
ts = 0.005 the KK-particle spectrum is identical to the
one obtained with the kink-motion within the depicted
mass range. In this case the exponential suppression of
particle production sets in only for masses mn > 100.
Note that the exponential decay of the spectrum for the
smooth transition from contraction to expansions also
shows that no additional spurious effects due to the dis-
continuities in the velocity when switching the brane dy-
namics on and off occur. Consequently, tin and tout are
appropriately chosen.
VI. ANALYTICAL CALCULATIONS AND
ESTIMATES
A. The zero mode: long wavelengths k ≪ 1/L
The numerical simulations show that the evolution of
the zero mode at large wavelengths is not affected by the
KK modes. To find an analytical approximation to the
numerical result for the zero mode, we neglect all the
couplings of the KK modes to the zero mode by setting
Mij = 0 ∀ i, j and keeping M00 only. Then only the evo-
lution equation for ǫ
0 ≡ δα0 ǫ is important; it decouples
and reduces to
ǫ̈+ [k2 + V(t)]ǫ = 0 , (6.1)
with “potential”
V = Ṁ00 −M200 . (6.2)
The corresponding vacuum initial conditions are
[cf. Eqs. (3.21), (3.22); here we do not consider the unim-
portant phase]
ǫ = 1 , lim
ǫ̇ = −ik. (6.3)
A brief calculation using the expression for M00 (cf. Ap-
pendix B) leads to
V = y
y2s − y2b
3y2b − 2y2s
y2s − y2b
(6.4)
= − y
y2s − y2b
y2s − y2b
. (6.5)
If one assumes that the static brane is much further away
from the Cauchy horizon than the physical brane, ys ≫
yb, it is simply
V = −H2 − Ḣ , (6.6)
and one recovers Eq. (2.50).
For the particular scale factor (2.17) one obtains
H = ȧ
sgn(t)
|t|+ tb
and (6.7)
Ḣ = 2δ(t)
(|t|+ tb)2
(6.8)
such that
Ḣ +H2 = 2δ(t)
. (6.9)
The δ-function in the last equation models the bounce.
Without the bounce, i.e. for an eternally radiation dom-
inated dynamics, one has V = 0 and the evolution equa-
tion for ǫ would be trivial. With the bounce, the potential
is just a delta-function potential with “height” propor-
tional to −2√vb/L
V = −
δ(t) , (6.10)
where vb is given in Eq (2.20). Equation (6.1) with poten-
tial (6.10) can be considered as a Schrödinger equation
with δ-function potential. Its solution is a classical text-
book problem.
Since the approximated potential V vanishes for all t < 0
one has, with the initial condition (6.3),
ǫ(t) = e−ikt , t < 0 . (6.11)
Assuming continuity of ǫ through t = 0 and integrating
the differential equation over a small interval t ∈ [0−, 0+]
around t = 0 gives
(6.12)
= ǫ̇(0+)− ǫ̇(0−)−
ǫ(0) . (6.13)
The jump of the derivative ǫ̇ at t = 0 leads to parti-
cle creation. Using ǫ(0+) = ǫ(0) = ǫ(0−) and ǫ̇(0+) =
ǫ̇(0−) +
ǫ(0) as initial conditions for the solution for
t > 0, one obtains
ǫ(t) = Ae−ikt +Beikt , t > 0 (6.14)
A = 1 + i
, B = −i
. (6.15)
The Bogoliubov coefficient B00 after the bounce is then
given by
B00(t ≥ 0) =
e−ikt
1 + i
ǫ(t)− i
ǫ̇(t)
(6.16)
where we have used that M00 = −H if ys ≫ yb. At
this point the importance of the coupling matrix M00
becomes obvious. Even though the solution ǫ to the dif-
ferential equation (6.1) is a plane wave right after the
bounce, |B00(t)|2 is not a constant due to the motion of
the brane itself. Only once the mode is inside the hori-
zon, i.e. H/k ≪ 1, |B00(t)|2 is constant and the number
of generated final state gravitons (for both polarizations)
is given by
N out0,k = 2|B00(kt ≫ 1)|2 = 2
|ǫ|2 + |ǫ̇|
(kL)2
(6.17)
where we have used that the Wronskian of ǫ, ǫ∗ is 2ik.
As illustrated in Fig. 9 the expression (6.17) is indeed in
excellent agreement with the (full) numerical results, not
only in its k-dependence but also the amplitude agrees
without any fudge factor. The evolution of the four-
dimensional graviton mode and the associated genera-
tion of massless gravitons with momentum k < 1/L can
therefore be understood analytically.
Note that the approximation employed here is only valid
if y2s − yb(0)2 ≫ yb(0)2. In the opposite limit, if ∆y ≡
ys − yb(0) ≪ yb(0) one can also derive an analytical ap-
proximation along the same lines. For k ≤ 1/∆y one
obtains instead of Eq. (6.17)
N out0,k =
2(k∆y)2
, (6.18)
if ∆y ≡ ys − yb(0) ≪ yb(0) , k∆y <∼ 1 .
In order to calculate the energy density, we have to take
into account that the approximation of an exactly ra-
diation dominated Universe with an instant transition
breaks down on small scales. We assume this break
down to occur at the string scale Ls, much smaller than
L [cf. Eqs. (2.14),(2.15)]. Ls is the true width of the
transition from collapse to expansion, which we have set
to zero in the treatment. Modes with mode numbers
k ≫ (2π)/Ls will not ’feel’ the potential and are not
generated. We therefore choose kmax = (2π)/Ls as the
cutoff scale. Then, with Eq (4.21), one obtains for the
energy density
2 π2a4
∫ 2π/Ls
dkk3N0,k . (6.19)
For small wave numbers, k < 1/L, we can use the above
analytical result for the zero-mode particle number.
However, as the numerical simulations have revealed, as
soon as k >∼ 1/L, the coupling of the four-dimensional
graviton to the KK modes becomes important and for
large wave numbers N out0,k decays only like 1/k. Hence
the integral (6.19) is entirely dominated by the upper
cutoff. The contributions from long wavelengths to the
energy density are negligible.
For the power spectrum, on the other hand, we
are interested in cosmologically large scales, 1/k ≃
several Mpc or more, but not in short wavelengths
kL ≫ 1 dominating the energy density. Inserting the
expression for the number of produced long wavelength
gravitons (6.17) into (4.11), the gravity wave power
spectrum at late times becomes
P0(k) =
(2π)3
for kt ≫ 1. (6.20)
This is the asymptotic power spectrum, when ǫ starts
oscillating, hence inside the Hubble horizon, kt ≫ 1.
On super Hubble scales, kt ≪ 1 when the asymptotic
out-state of the zero mode is not yet reached, one may
use Eq. (4.10) with
R0,k(t) =
|ǫ(t)|2 − 1
≃ 4vba
. (6.21)
For the ≃ sign we assume t ≫ L and t ≫ tb so that
one may neglect terms of order t/L in comparison to√
vb(t/L)
2. We have also approximated a = (t+ tb)/L ≃
t/L. Inserting this in Eq. (4.8) yields
P0(k) =
2 , kt ≪ 1 . (6.22)
Both expressions (6.20) and (6.22) are in very good
agreement with the corresponding numerical results, see
Figs. 9, 10 and 11.
B. The zero mode: short wavelengths k ≫ 1/L
As we have demonstrated with the numerical analysis,
as soon as k >∼ 1/L, the coupling of the zero mode to
the KK modes becomes important, and for large wave
numbers N out0,k,• ∝ 1/k. We obtain a good asymptotic
behavior for the four-dimensional graviton spectrum if
we set
N out0,k,• ≃
5(kL)
. (6.23)
This function and Eq. (6.17) (divided by two for one po-
larization) meet at kL = 5. Even though the approxi-
mation is not good in the intermediate regime it is very
reasonable for large k [cf. Fig. 27].
Inserting this approximation into Eq (6.19) for the energy
density, one finds that the integral is dominated entirely
by the upper cutoff, i.e. by the blue, high energy modes:
. (6.24)
The power spectrum associated with the short wave-
lengths k ≫ 1/L is not of interest since the gravity wave
spectrum is measured on cosmologically large scales only,
k ≪ 1/L.
C. Light Kaluza-Klein modes and long wavelengths
k ≪ 1/L
The numerics indicates that light (mn < 1) long wave-
length KK modes become excited mainly due to their
coupling to the zero mode. Let us take only this cou-
pling into account and neglect also the time-dependence
of the frequency, setting ωn,k(t) ≡ ωoutn,k = ωinn,k since it
plays an inferior role as shown by the numerics.
The Bogoliubov coefficients are then determined by the
equations
ξ̇n,k + iω
n,kξn,k =
2ωoutn,k
Sn(t; k) (6.25)
η̇n,k − iωoutn,kηn,k = −
2ωoutn,k
Sn(t; k) (6.26)
with the “source”
Sn(t; k) = (ξ0 − η0)Mn0 . (6.27)
We have defined ξn,k ≡ ξ(0)n,k, ηn,k ≡ η
n,k, ξ0 ≡ ξ
0,k, and
η0 ≡ η(0)0,k. This source is known, since the evolution of
the four-dimensional graviton is know. From the result
for ǫ above and the definition of ξ0 and η0 in terms of ǫ
and ǫ̇ one obtains
ξ0 − η0 =
−ik + 1|t|+ tb
e−itk , t < 0 (6.28)
ξ0 − η0 = 2
1− iktb
k2tb(t+ tb)
e−itk
k2tb(t+ tb)
eitk , t > 0 .(6.29)
Furthermore, if ys ≫ yb, one has [cf. Eq. (B3)]
Mn0 = 2
Y1(mnys)2
Y1(mnyb)2 − Y1(mnys)2
. (6.30)
Assuming ysmn ≫ 1 and ybmn ≪ 1 one can expand the
Bessel functions and arrives at
Mn0 ≃
ẏb = −
πmnL2
L sgn(t)
(|t|+ tb)2
To determine the number of created final state gravi-
tons we only need to calculate ηn,k [cf. Eq. (3.32) with
∆+n,k(|t| → ∞) = 1 and ∆
n,k(|t| → ∞) = 0],
N outn,k,• = |B0n,k(tout)|2 =
ωoutn,k
|ηn,k|2 (6.31)
The vacuum initial conditions require limt→−∞ ηn,k = 0
so that ηn,k is given by the particular solution
ηn,k(t) =
ωoutn,k
′; k)e−it
′ωoutn,kdt′ , (6.32)
and therefore
N outn,k,• =
4ωoutn,k
Sn(t; k)e
−itωout
n,kdt
(6.33)
where the integration range has been extended from −∞
to +∞ since the source is very localized around the
bounce. This integral can be solved exactly. A some-
what lengthy but straight forward calculation gives
N outn,k,• =
πm5nL
2ωoutn,kkys
∣∣∣2iRe
+k)tbE1(i(ω
n,k + k)tb)
+(ktb)
−1ei(ω
n,k−k)tbE1(i(ω
n,k − k)tb)
−ei(ω
+k)tbE1(i(ω
n,k + k)tb)
. (6.34)
Here E1 is the exponential integral, E1(z) ≡∫∞
t−1e−tdt . This function is holomorphic in the com-
plex plane with a cut along the negative real axis, and
the above expression is therefore well defined. Note that
this expression does not give rise to a simple dependence
of N outn,k on the velocity vb = (L/tb)2. In the preceding
section we have seen that, within its range of validity,
Eq. (6.34) is in excellent agreement with the numerical
results (cf., for instance, Figs. 12 and 13).
As already mentioned before, this excellent agreement
between the numerics and the analytical approximation
demonstrates that the numerical results are not contam-
inated by any spurious effects.
D. Kaluza-Klein modes: asymptotic behavior and
energy density
The numerical simulations show that the asymptotic
KK-graviton spectra (i.e. for masses mn ≫ 1) decay
like 1/ωoutn,k if mn ≫ k and like
1/ωoutn,k
with α ≃ 2
if mn <∼ k. The corresponding energy density on the
brane is given by the summation of Eq. (4.23) over all
KK modes up to the cutoff. Since the mass mn is simply
the momentum into the extra dimension, it is plausible
to choose the same cutoff scale for both, the k-integral
and the summation over the KK modes, namely 2π/Ls.
The main contribution to the four-dimensional particle
density and energy density comes from mn ∼ 2π/Ls and
k ∼ 2π/Ls, i.e. the blue end of the spectrum.
The large-frequency behavior of the final KK-spectrum
can be approximated by
N outn,k,• ≃
0.2v2b


ωoutn,k
if 1/L <∼ k <∼ mn
2(α−1)/2
(ωoutn,k)
if mn <∼ k <∼ 2π/Ls
(6.35)
with α ≃ 2 which is particularly good for large k. Both
expression match at mn = k and are indicated in Figures
25 and 31 as dashed lines. Given the complicated
coupling structure of the problem and the multitude of
features visible in the particle spectra, these compact
expressions describe the numerical results reasonable
well for all parameters. The deviation from the numer-
ical results is at most a factor of two. This accuracy
is sufficient in order to obtain a useful expression for
the energy density from which bounds on the involved
energy scales can be derived.
The energy density on the brane associated with
the KK gravitons is given by [cf. Eq. (4.23)]
ρKK ≃
πa6ys
dkk2 N outn,k,• ωoutn,k mn . (6.36)
Splitting the momentum integration into two integrations
from 0 to mn and mn to the cutoff 2π/Ls, and replacing
the sum over the KK masses by an integral one obtains
ρKK ≃ C(α)
π5v2b
. (6.37)
The power α in Eq. (6.35) enters the final result for the
energy density only through the pre-factor C(α) which is
of order unity.
VII. DISCUSSION
The numerical simulations have revealed many inter-
esting effects related to the interplay between the evolu-
tion of the four-dimensional graviton and the KK modes.
All features observed in the numerical results have been
interpreted entirely on physical grounds and many of
them are supported by analytical calculations and ar-
guments. Having summarized the results for the power
spectrum and energy densities in the preceding section,
we are now in the position to discuss the significance of
these findings for brane cosmology.
A. The zero mode
For the zero-mode power-spectrum we have found that
P0(k) =
k2 if kt ≪ 1
(La)−2 if kt ≫ 1 . (7.1)
Therefore, the gravity wave spectrum on large, super
Hubble scales is blue with spectral tilt
nT = 2 , (7.2)
a common feature of ekpyrotic and pre-big-bang models.
The amplitude of perturbations on scales at which
fluctuations of the Cosmic Microwave Background
(CMB) are observed is of the order of (H0/mPl)
2, i.e.
very suppressed on scales relevant for the anisotropies
of the CMB. The fluctuations induced by these Casimir
gravitons are much too small to leave any observable
imprint on the CMB.
For the zero-mode energy density at late times,
kt ≫ 1, we have obtained [cf Eq. (6.24)]
ρh0 ≃
. (7.3)
In this section we denote the energy density of the
zero mode by ρh0 in order not to confuse it with the
12 Note that even the transition from the summation over the KK-
tower to an integration according to (4.33) “eats up” the 1/ys
term in (6.36), the final energy density (6.37) depends on ys since
it explicitly enters the particle number.
present density of the Universe. Recall that Ls is the
scale at which our kinky approximation (2.17) of the
scale factor breaks down, i.e. the width of the bounce.
If this width is taken to zero, the energy density of
gravitons is very blue and diverges. This is not so
surprising, since the kink in a(t) leads to the generation
of gravitons of arbitrary high energies. However, as the
numerical simulations have shown, when we smooth the
kink at some scale Ls, the production of modes with
energies larger than ≃ 1/Ls is exponentially suppressed
[cf. Fig. 32]. This justifies the introduction of Ls as a
cutoff scale.
In the following we shall determine the density pa-
rameter of the generated gravitons today and compare
it to the Nucleosynthesis bound. For this we need the
quantities ab given in Eq (2.20) and
Here ab is the minimal scale factor andHb is the maximal
Hubble parameter, i.e. the Hubble parameter right after
the bounce. (Recall that in the low energy approximation
t = η.) During the radiation era, curvature and/or a cos-
mological constant can be neglected so that the density
ρrad =
a−4 =
. (7.4)
In order to determine the density parameter of the gen-
erated gravitons today, i.e., at t = t0, we use
Ωh0 =
ρh0(t0)
ρcrit(t0)
ρh0(t0)
ρrad(t0)
ρrad(t0)
ρcrit(t0)
ρh0(t0)
ρrad(t0)
Ωrad.
(7.5)
The second factor Ωrad is the present radiation density
parameter. For the factor ρh0/ρrad, which is time inde-
pendent since both ρh0 and ρrad scale like 1/a
4, we insert
the above results and obtain
Ωh0 =
Ωrad =
Ωrad (7.6)
Ωrad . (7.7)
The nucleosynthesis bound [14] requests that
Ωh0 <∼ 0.1Ωrad , (7.8)
which translates into the relation
(LPl/Ls)
(L/Ls) <∼ 0.1 (7.9)
which, at first sight, relates the different scales involved.
But since we have chosen the cutoff scale Ls to be
the higher-dimensional fundamental scale (string scale),
Equation (7.9) reduces to
vb <∼ 0.2 (7.10)
by virtue of Equation (2.15). All one has to require to
be consistent with the nucleosynthesis bound is a small
brane velocity which justifies the low energy approach.
In all, we conclude that the model is not severely con-
strained by the zero mode. This result itself is remark-
able. If there would be no coupling of the zero mode
to the KK modes for small wavelengths the number of
produced high energy zero-mode gravitons would behave
as ∝ k−2 as it is the case for long wavelengths. The
production of high energy zero-mode gravitons from KK
gravitons enhances the total energy density by a factor of
about L/Ls. Without this enhancement, the nucleosyn-
thesis bound would not lead to any meaningful constraint
and would not even require vb < 1.
B. The KK modes
As derived above, the energy density of KK gravitons
on the brane is dominated by the high energy gravitons
and can be approximated by [cf. Eq. (6.37)]
ρKK ≃
π5v2b
. (7.11)
Let us evaluate the constraint induced from the require-
ment that the KK-energy density on the brane be smaller
than the radiation density ρKK(t) < ρrad(t) at all times.
If this is not satisfied, back-reaction cannot be neglected
and our results are no longer valid. Clearly, at early times
this condition is more stringent than at late times since
ρKK decays faster then ρrad. Inserting the value of the
scale factor directly after the bounce where the produc-
tion of KK gravitons takes place, a−2b = vb, one finds,
using again the RS fine tuning condition (2.15),
≃ 100 v3b
. (7.12)
If we use the largest value for the brane velocity vb ad-
mitted by the nucleosynthesis bound vb ≃ 0.2 and re-
quire that ρKK/ρrad be (much) smaller than one for back-
reaction effects to be negligible, we obtain the very strin-
gent condition
. (7.13)
Let us first discuss the largest allowed value for L ≃
0.1mm. The RS-fine tuning condition (2.15) then deter-
mines Ls = (LL
1/3 ≃ 10−22 mm ≃ 1/(106 TeV). In
this case the brane tension is T = 6κ4/κ25 = 6L2Pl/L6s =
6/(LL3s) ∼ (10TeV)4. Furthermore, we have (L/Ls)2 ≃
1042 so that ys > L(L/Ls)
2 ≃ 1041mm ≃ 3 × 1015Mpc,
which is about 12 orders of magnitude larger than the
present Hubble scale. Also, since yb(t) ≪ L in the low
energy regime, and ys ≫ L according to the inequality
(7.13), the physical brane and the static brane are very
far apart at all times. Note that the distance between
the physical and the static brane is
dy = L log(ys/yb) >∼ L ≫ Ls .
This situation is probably not very realistic. Some high
energy, stringy effects are needed to provoke the bounce
and one expects these to be relevant only when the
branes are sufficiently close, i.e. at a distance of order
Ls. But in this case the constraint (7.13) will be violated
which implies that back-reaction will be relevant.
On the other hand, if one wants that ys ≃ L and
back-reaction to be unimportant, then Eq. (7.12) implies
that the bounce velocity has to be exceedingly small,
vb <∼ 10−15.
A way out of this conclusion is to assume that the
brane distance at the bounce, ∆y = ys − yb(0), becomes
of the order of the cutoff Ls or smaller. Then the pro-
duction of KK gravitons is suppressed. However, then
the approximation (6.18) has to be used to determine
the energy density of zero-mode gravitons which then
becomes
ρh0 ≃
(Ls∆y)
Setting ∆y ≃ Ls, the nucleosynthesis bound,
ρh0 <∼ 0.1ρrad, then yields the much more stringent limit
on the brane velocity,
v2b <
. (7.14)
One might hope to find a way out of these conclusions
by allowing the bounce to happen in the high energy
regime. But then vb ≃ 1 and the nucleosynthesis bound
is violated since too many zero-mode gravitons are pro-
duced. Even if one disregards this limit for a moment,
saying that the calculation presented here only applies in
the low energy regime, vb ≪ 1, the modification coming
from the high energy regime are not expected to allevi-
ate the bounds. In the high energy regime one may of
course have yb(t) ≫ L and therefore the physical brane
can approach the static brane arbitrarily closely without
the latter having to violate (7.13). Those results suggest
that even in the scenario of a bounce at low energies, the
back reaction from KK gravitons has to be taken into
account. But this does not need to exclude the model.
VIII. CONCLUSIONS
We have studied the evolution of tensor perturbations
in braneworld cosmology using the techniques developed
for the standard dynamical Casimir effect. A model
consisting of a moving and a fixed 3-brane embedded in
a five-dimensional static AdS bulk has been considered.
Applying the dynamical Casimir effect formulation
to the study of tensor perturbations in braneworld
cosmology represents an interesting alternative to
other approaches existing in the literature so far and
provides a new perspective on the problem. The explicit
use of coupling matrices allows us to obtain detailed
information about the effects of the intermode couplings
generated by the time-dependent boundary conditions,
i.e. the brane motion.
Based on the expansion of the tensor perturbations
in instantaneous eigenfunctions, we have introduced a
consistent quantum mechanical formulation of graviton
production by a moving brane. Observable quantities
like the power spectrum and energy density can be
directly deduced from quantum mechanical expectation
values, in particular the number of gravitons created
from vacuum fluctuations. The most surprising and
at the same time most interesting fact which this
approach has revealed is that the energy density of the
massive gravitons decays like 1/a6 with the expansion
of the Universe. This is a direct consequence of the
localization of gravity: five-dimensional aspects of it,
like the KK gravitons, become less and less ’visible’
on the brane with the expansion of the Universe. The
1/a6-scaling behavior remains valid also when the fixed
brane is sent off to infinity and one ends up with
a single braneworld in AdS, like in the original RS II
scenario. Consequently, KK gravitons on a brane moving
through an AdS bulk cannot play the role of dark matter.
As an explicit example, we have studied graviton
production in a generic, ekpyrotic-inspired model of
two branes bouncing at low energies, assuming that
the energy density on the moving brane is dominated
by a radiation component. The numerical results have
revealed a multitude of interesting effects.
For long wavelengths kL ≪ 1 the zero mode evolves
virtually independently of the KK modes. zero-mode
gravitons are generated by the self coupling of the zero
mode to the moving brane. For the number of produced
massless gravitons we have found the simple analytical
expression 2vb/(kL). These long wavelength modes
are the once which are of interest for the gravitational
wave power spectrum. As one expects for an ekpyrotic
scenario, the power spectrum is blue on super-horizon
scales with spectral tilt nT = 2. Hence, the spectrum
of these Casimir gravitons has much too little power
on large scales to affect the fluctuations of the cosmic
microwave background.
The situation changes completely for short wavelengths
kL ≫ 1. In this wavelength range, the evolution of the
zero mode couples strongly to the KK modes. Produc-
tion of zero-mode gravitons takes place on the expense of
KK-graviton production. The numerical simulation have
revealed that the number of produced short-wavelength
massless gravitons is given by 2vb/(5kL). It decays
only like 1/k instead of the 1/k2-behavior found for
long wavelengths. These short wavelength gravitons
dominate the energy density. Comparing the energy
density with the nucleosynthesis bound and taking the
cutoff scale to be the string scale Ls, we have shown that
the model is not constrained by the zero mode. As long
as vb <∼ 0.2, i.e. a low energy bounce, the nucleosynthesis
bound is not violated.
More stringent bounds on the model come from
the KK modes. Their energy density is dominated by
the high energy modes which are produced due to the
kink which models the transition from contraction to
expansion. Imposing the reasonable requirement that
the energy density of the KK modes on the brane be
(much) smaller than the radiation density at all times in
order for back reaction effects to be negligible, has led to
two cases. On the one hand, allowing the largest values
for the AdS curvature scale L ≃ 0.1mm and the bounce
velocity vb ≃ 0.2, back reaction can only be neglected
if the fixed brane is very far away from the physical
brane ys ∼ 1041mm. As we have argued, this is not very
realistic since some high energy, stringy effects provoking
the bounce are expected to be relevant only when the
branes are sufficiently close, i.e. ys ∼ Ls. On the other
hand, by only requiring that ys ≃ L ≫ Ls, the bounce
velocity has already to be exceedingly small, vb <∼ 10−15,
for back reaction to be unimportant. Therefore, one of
the main conclusions to take away from this work is that
back reaction of massive gravitons has to be taken into
account for a realistic bounce.
Many of the results presented here are based on
numerical calculations. However, since the used ap-
proach provides the possibility to artificially switch
on and off the mode couplings, we were able identify
the primary sources driving the time evolution of the
perturbations in different wavelength and KK mass
ranges. This has allowed us to understand many of the
features observed in the numerical results on analytical
grounds.
On the other hand, it is fair to say that most of the
presented results rely on the low energy approach, i.e. on
the approximation of the junction condition (generalized
Neumann boundary condition) by a Neumann boundary
condition. Even though we have given arguments for
the goodness of this approximation, it has eventually
to be confirmed by calculations which take the exact
boundary condition into account. This is the subject of
future work.
Acknowledgment
We thank Cyril Cartier who participated in the early
stages of this work and Kazuya Koyama and David Lan-
glois for discussions. We are grateful for the use of the
’Myrinet’-cluster of Geneva University on which most of
the quite intensive numerical computations have been
performed. This work is supported by the Swiss National
Science Foundation.
APPENDIX A: VARIATION OF THE ACTION
Let us consider the variation of the action (2.27) with
respect to h•. It is sufficient to study the action for a
fixed wave number k and polarization •
Sh•(k) =
yb(t)
|∂th•|2 − |∂yh•|2 − k2|h•|2
and we omit the normalization factor L3/κ5 as well as
the factor two related to Z2 symmetry. The variation of
(A1) reads
δSh•(k) =
yb(t)
(∂th•)(∂tδh
•) (A2)
−(∂yh•)(∂yδh∗•)− k2h•δh∗•
+ h.c. .
Here, T denotes a time interval within the variation is
performed and it is assumed in the following that the
variation vanishes at the boundaries of the time interval
T . Performing partial integrations and demanding that
the variation of the action vanishes leads to
0 = (A3)
yb(t)
− ∂2t h• + y3
− k2h•
[(v∂t + ∂y)h•] δh
•|yb(t) −
(∂yh•)δh
with v = dyb(t)/dt. The first term in curly brackets is the
wave operator (2.24). In order for h• to satisfy the free
wave equation (perturbation equation) (2.24) the term
in curly brackets in the second integral has to vanish.
Allowing for an evolution of h• on the branes, i.e. in
general δh•|brane 6= 0, enforces the boundary conditions
(v∂t + ∂y)h•|yb(t) = 0 and ∂yh•|ys = 0 , (A4)
hence, the junction condition (2.26). Consequently, any
other boundary conditions than (A4) are not compati-
ble with the free perturbation equation (2.24) under the
influence of a moving brane (provided δh• 6= 0 at the
branes).
APPENDIX B: COUPLING MATRICES
The use of several identities of Bessel functions leads
M00 = ŷb
y2s − y2b
, (B1)
M0j = 0 , (B2)
Mi0 =
φ0 = ŷb
y2s − y2b
, (B3)
Mii = m̂i , (B4)
Mij = M
ij +M
ij (B5)
MAij = (ŷb + m̂i)yb
2m2iNiNj
m2j −m2i
× (B6)
× [ys C2(mjys)J1(miys)− yb C2(mjyb)J1(miyb)]
where
J1(mi y) = [J2(miyb)Y1(miy)− Y2(miyb)J1(miy)]
MNij = NiNjmim̂i
dyy2C1(miy)C2(mjy). (B8)
This integral has to be solved numerically. Note that,
because of the boundary conditions, one has the identity
dyy2C1(miy)C2(mjy) = −
dyy2C1(miy)C0(mjy).
Furthermore, one can simplify
J1(mi yb) =
πmiyb
, J1(mi ys) =
πmiyb
Y1(miys)
Y1(miyb)
(B10)
where the limiting value has to be taken for the last term
whenever Y1(miyb) = Y1(miys) = 0.
APPENDIX C: ON POWER SPECTRUM AND
ENERGY DENSITY CALCULATION
1. Instantaneous vacuum
In Section III the in - out state approach to particle
creation has been presented. The definitions of the in
- and out- vacuum states Eq. (3.9) are unique and the
particle concept is well defined and meaningful.
If we interpret tout as a continuous time variable t,
we can write the Bogoliubov transformation Eq. (3.24)
âα,k,•(t) =
Aβα,k(t)âinβ,k,• + B∗βα,k(t)â
β,−k,•
where at any time we have introduced a set of operators
{âα,k•(t), â†α,k,•(t)}. Vacuum states defined at any time
can be associated with these operators via
âα,k,•(t)|0, t〉 = 0 ∀ α,k • . (C2)
Similar to Eq. (3.11) a ”particle number” can be intro-
duced through
Nα,k(t) =
〈0, in|â†α,k•(t)âα,k,•(t)|0, in〉
|Bβα,k(t)|2 . (C3)
We shall denote |0, t〉 as the instantaneous vacuum state
and the quantity Nα,k(t) as instantaneous particle num-
ber 13. However, even if we call it ”particle number” and
plot it in section V for illustrative reasons, we consider
only the particle definitions for the initial and final state
(asymptotic regions) outlined in section III as physically
meaningful.
2. Power spectrum
In order to calculate the power spectrum Eq. (4.7) we
need to evaluate the expectation value
(t, yb,k)ĥ
(t, yb,k
′)〉in = (C4)
φα(t, yb)φα′ (t, yb)〈q̂α,k,•(t)q̂†α′,k′,•(t)〉in
where we have introduced the shortcut 〈...〉in =
〈0, in|...|0, in〉. Using the expansion (3.15) of q̂α′,k′,•(t)
in initial state operators and complex functions ǫ
α,k(t)
one finds
〈q̂α,k,•(t)q̂†α′,k′,•(t)〉in =
α,k(t) ǫ
α′,k (t)
2ωinβ,k
δ(3)(k− k′).
From the initial conditions (3.21) it follows that the sum
in (C4) diverges at t = tin. This divergence is related to
the usual normal ordering problem and can be removed
by a subtraction scheme. However, in order to obtain a
well defined power spectrum at all times, it is not suffi-
cient just to subtract the term (1/2)(δαα′/ω
α,k)δ
(3)(k −
′) which corresponds to 〈q̂α,k,•(tin)q̂†α′,k′,•(tin)〉in in the
above expression. In order to identify all terms contained
in the power spectrum we use the instantaneous particle
concept which allows us to treat the Bogoliubov coeffi-
cients (3.25) and (3.26) as continuous functions of time.
First we express the complex functions ǫ
α,k in (C5) in
terms of Aγα,k(t) and Bγα,k(t). This is of course equiv-
alent to calculating the expectation value (C5) using
[cf. Eq.(3.7)]
q̂α,k,•(t) =
2ωα,k(t)
âα,k,•(t)Θα,k(t)
α,−k,•(t)Θ
α,k(t)
and the Bogoliubov transformation Eq. (C1). The result
consists of terms involving the Bogoliubov coefficients
and the factor (1/2)(δαα′/ωα,k(t))δ
(3)(k − k′), leading
potentially to a divergence at all times. This term cor-
responds to 〈0, t|q̂α,k,•(t)q̂†α′,k′,•(t)|0, t〉, and is related to
13 It could be interpreted as the number of particles which would
have been created if the motion of the boundary (the brane)
stops at time t.
the normal ordering problem (zero-point energy) with re-
spect to the instantaneous vacuum state |0, t〉. It can be
removed by the subtraction scheme
〈q̂α,k,•(t)q̂†α′,k′,•(t)〉in,phys (C7)
= 〈q̂α,k,•(t)q̂†α′,k′,•(t)〉in − 〈0, t|q̂α,k,•(t)q̂
α′,k′,•(t)|0, t〉
where we use the subscript “phys” to denote the physi-
cally meaningful expectation value.
Inserting this expectation value into (C4), and using
Eq. (4.2), we find
〈ĥ•(t, yb,k)ĥ•(t, yb,k′)〉in (C8)
Rα,k(t)Y2α(a)δ(3)(k− k′)
with Rα,k(t) defined in Eq. (4.9). The function ONα,k
appearing in Eq. (4.9) is explicitely given by
ONα,k = 2ℜ
Θ2α,kAβα,kB∗βα,k +Θα,k
α′ 6=α
ωα′,k
Yα′ (a)
Yα(a)
Θ∗α′,kB∗βαBβα′ +Θα′,kAβαB∗βα′
and Oǫα,k appearing in Eq. (4.10) reads
Oǫα,k =
β,α′ 6=α
Yα′(a)
Yα(a)
ωinβ,k
. (C10)
3. Energy density
In order to calculate the energy density we need to
evaluate the expectation value 〈 ˙̂hij(t,x, yb) ˙̂hij(t,x, yb)〉in.
Using (2.22) and the relation e•ij(−k) = (e•ij(k))∗ we ob-
〈 ˙̂hij(t,x, yb) ˙̂hij(t,x, yb)〉in =
(2π)3/2
(2π)3/2
(C11)
× 〈 ˙̂h
(t, yb,k)
′(t, yb,k
′)〉inei(k−k
′)·xe•ij(k)
′ ij(k′)
By means of the expansion (3.17) the expectation value
〈 ˙̂h
(t, yb,k)
′(t, yb,k
′)〉in becomes
〈 ˙̂h
(t, yb,k)
′(t, yb,k
′)〉in (C12)
〈p̂α,k,•(t)p̂†α′,k′,•′(t)〉inφα(t, yb)φα′ (t, yb).
From the definition of p̂α,k,•(t) in Eq. (3.18) it is clear
that this expectation value will in general contain terms
proportional to the coupling matrix and its square when
expressed in terms of ǫ
α,k. However, we are interested in
the expectation value at late times only when the brane
moves very slowly such that the mode couplings go to
zero and a physical meaningful particle definition can be
given. In this case we can set
〈p̂α,k,•(t)p̂†α′,k′,•′(t)〉in =
˙̂qα,k,•(t) ˙̂q
α′,k′,•′(t)
.(C13)
Calculating this expectation value by using Eq. (3.15)
leads to an expression which, as for the power spec-
trum calculation before, has a divergent part related to
the zero-point energy of the instantaneous vacuum state
(normal ordering problem). We remove this part by a
subtraction scheme similar to Eq (C7). The final result
reads
〈 ˙̂qα,k,•(t) ˙̂q†α′,k′,•′(t)〉in,phys (C14)
α,k(t)ǫ̇
α′,k′(t)√
ωinβ,kω
− ωα,k(t)δαα′
′δ(3)(k− k′).
Inserting this result into Eq. (C12), splitting the summa-
tions in sums over α = α′ and α 6= α′ and neglecting the
oscillating α 6= α′ contributions (averaging over several
oscillations), leads to
〈 ˙̂h
(t, yb,k)
′(t, yb,k
′)〉in (C15)
Kα,k(t)Y2α(a)δ••′δ(3)(k− k′)
where the function Kα,k(t) is given by
Kα,k(t) =
|ǫ̇(β)α,k(t)|2
ωinβ,k
− ωα,k(t) = ωα,k(t)Nα,k(t) ,
(C16)
and we have made use of Eq. (4.2). The relation be-
tween
β |ǫ̇
α,k(t)|2/ωinβ,k and the number of created par-
ticles can easily be established. Using this expression in
Eq. (C11) leads eventually to
〈 ˙̂hij(t,x, yb) ˙̂hij(t,x, yb)〉in (C17)
(2π)3
Kα,k(t)Y2α(a)
where we have used that the polarization tensors satisfy
e•ij(k)
e• ij(k)
= 2. (C18)
The final expression for the energy density Eq. (4.18) is
then obtained by exploiting that κ5/L = κ4.
APPENDIX D: NUMERICS
In order to calculate the number of produced gravitons
the system of coupled differential equations (3.34) and
(3.35) is solved numerically. The complex functions ξ
α,k are decomposed into their real and imaginary parts:
α,k = u
α,k + iv
α,k , η
α,k = x
α,k + iy
α,k. (D1)
The system of coupled differential equations can then be
written in the form (cf. Eq. (A2) of [16])
k (t) = Wk(t)X
k (t) (D2)
where
0,k ...u
nmax,k
0,k ...x
nmax,k
0,k ...v
nmax,k
0,k ...y
nmax,k
The matrix Wk(t) is given by Eq. (A4) of [16] but
here indices start at zero. The number of produced
gravitons can be calculated directly from the solutions
to this system using Eqs. (3.28) and (3.32). Note
that for a given truncation parameter nmax the above
system of size 4(nmax + 1) × 4(nmax + 1) has to be
solved nmax + 1 - times, each time with different initial
conditions (3.38).
The main difficulty in the numerical simulations is that
most of the entries of the matrix Wk(t) [Eq. (A4) of
[16]] are not known analytically. This is due to the fact
that Eq. (2.40) which determines the time-dependent
KK masses mi(t) does not have an (exact) analytical
solution. Only the 00-component of the coupling matrix
Mαβ is known analytically. We therefore have to deter-
mine the time-dependent KK-spectrum {mi(t)}nmaxi=1 by
solving Eq. (2.40) numerically. In addition, also the part
MNij [Eq. (B8)] has to be calculated numerically since
the integral over the particular combination of Bessel
functions can not be found analytically.
We numerically evaluate the KK-spectrum and the
integral MNij for discrete time-values ti and use spline
routines to assemble Wk(t). The system (D2) can
then be solved using standard routines. We chose the
distribution of the ti’s in a non-uniform way. A more
dense mesh close to the bounce and a less dense mesh at
early and late times. The independence of the numerical
results on the distribution of the ti’s is checked. In order
to implement the bounce as realistic as possible, we do
not spline the KK-spectrum very close to the bounce
but re-calculate it numerically at every time t needed in
the differential equation solver. This minimizes possible
artificial effects caused by using a spline in the direct
vicinity of the bounce. The same was done for MNij but
we found that splining MNij when propagating through
the bounce does not affect the numerical results.
Routines provided by the GNU Scientific Library (GSL)
[59] have been employed. Different routines for root
finding and integration have been compared. The code
has been parallelized (MPI) in order to deal with the
1 10 100
KK-mass m
1 10 100
KK-mass m
FIG. 33: Comparison of the final KK-graviton spectrum
n,k,• with the expression dn,k(tout) describing to what accu-
racy the diagonal part of the Bogoliubov relation (D4) is sat-
isfied. Left panel: ys = 3, k = 0.1, vb = 0.03 and nmax = 100
[cf. Fig. 25]. Right panel: ys = 3, k = 30, vb = 0.1 and
nmax = 100 [cf. Fig. 26].
intensive numerical computations.
The accuracy of the numerical simulations can be
assessed by checking the validity of the Bogoliubov
relations
Aβα,k(t)A∗βγ,k(t)− B∗βα,k(t)Bβγ,k(t)
= δαγ (D4)
Aβα,k(t)B∗βγ,k(t)− B∗βα,k(t)Aβγ,k(t)
= 0. (D5)
In the following we demonstrate the accuracy of the nu-
merical simulations by considering the diagonal part of
(D4). The deviation of the quantity
dα,k(t) = 1−
|Aβα,k(t)|2 − |Bβα,k(t)|2
from zero gives a measure for the accuracy of the
numerical result. We consider this quantity at final
times tout and compare it with the corresponding final
particle spectrum. In Fig. 33 we compare the final KK-
graviton spectrum N outn,k,• with the expression dn,k(tout)
for two different cases. This shows that the accuracy
of the numerical simulations is very good. Even if the
expectation value for the particle number is only of order
10−7 to 10−6, the deviation of dn,k(tout) from zero is at
least one order of magnitude smaller. This demonstrates
the reliability of our numerical simulations and that we
can trust the numerical results presented in this work.
APPENDIX E: DYNAMICAL CASIMIR EFFECT
FOR A UNIFORM MOTION
We consider a real massless scalar field on a time-
dependent interval [0, y(t)]. The time evolution of its
mode functions are described by a system of differential
equations like (2.49) where the specific form of Mαβ de-
pends on the particular boundary condition the field is
subject to. In [15, 17] a method has been introduced to
study particle creation due to the motion of the boundary
y(t) (i.e. the dynamical Casimir effect) fully numerically.
We refer the reader to these publications for further de-
tails.
If the boundary undergoes a uniform motion y(t) = 1+vt
(in units of some reference length) it was shown in [57, 58]
that the total number of created scalar particles diverges,
caused by the discontinuities in the velocity at the begin-
ning and the end of the motion. In particular, for Dirich-
let boundary conditions (no zero mode), it was found in
[58] that 〈0, in|N̂outn |0, in〉 ∝ v2/n if n > 6 and v ≪ 1.
Thereby in- and out- vacuum states are defined like in
the present work and the frequency of a mode function
is given by ωn = π n , n = 1, 2, ... . In Figure 34 we
show spectra of created scalar particles obtained numer-
ically with the method of [17] for this particular case.
One observes that, as for our bouncing motion, the con-
vergence is very slow since the discontinuities in the ve-
locity lead to the excitation of arbitrary high frequency
modes. Nevertheless, it is evident from Fig. 34 that the
numerically calculated spectra approach the analytical
prediction. The linear motion discussed here and the
brane-motion (2.18) are very similar with respect to the
discontinuities in the velocity. In both cases, the total
discontinuous change of the velocity is 2v and 2vb, re-
spectively. The resulting divergence of the acceleration
is responsible for the excitation and therefore creation of
particles of all frequency modes. Consequently we ex-
pect the same ∝ v2/ωn behavior for the bouncing mo-
tion (2.18). Indeed, comparing the convergence behavior
of the final graviton spectrum for vb = 0.01 shown in
Fig. 25 with the one of the scalar particle spectrum for
v = 0.01 depicted in Fig. 34 shows that both are very
similar.
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  We consider a two-brane system in a five-dimensional anti-de Sitter
spacetime. We study particle creation due to the motion of the physical brane
which first approaches the second static brane (contraction) and then recedes
from it(expansion). The spectrum and the energy density of the generated
gravitons are calculated. We show that the massless gravitons have a blue
spectrum and that their energy density satisfies the nucleosynthesis bound with
very mild constraints on the parameters. We also show that the Kaluza-Klein
modes cannot provide the dark matter in an anti-de-Sitter braneworld. However,
for natural choices of parameters, backreaction from the Kaluza-Klein gravitons
may well become important. The main findings of this work have been published
in the form of a Letter [R. Durrer and M. Ruser, Phys. Rev. Lett. 99, 071601
(2007), arXiv:0704.0756].

arXiv:0704.0790v3  [hep-th]  15 Nov 2007
Dynamical Casimir effect for gravitons in bouncing braneworlds
Marcus Ruser∗ and Ruth Durrer†
Département de Physique Théorique, Université de Genève,
24 quai Ernest Ansermet, 1211 Genève 4, Switzerland.
We consider a two-brane system in five-dimensional anti-de Sitter space-time. We study particle
creation due to the motion of the physical brane which first approaches the second static brane
(contraction) and then recedes from it (expansion). The spectrum and the energy density of the
generated gravitons are calculated. We show that the massless gravitons have a blue spectrum
and that their energy density satisfies the nucleosynthesis bound with very mild constraints on the
parameters. We also show that the Kaluza-Klein modes cannot provide the dark matter in an anti-
de-Sitter braneworld. However, for natural choices of parameters, backreaction from the Kaluza-
Klein gravitons may well become important. The main findings of this work have been published
in form of a Letter [R.Durrer and M.Ruser, Phys. Rev. Lett. 99, 071601 (2007), arXiv:0704.0756].
PACS numbers: 04.50.+h, 11.10.Kk, 98.80.Cq
I. INTRODUCTION
In recent times, the possibility that our observed
Universe might represent a hypersurface in a higher-
dimensional space-time has received considerable
attention. The main motivation for this idea is the
fact, that string theory [1, 2], which is consistent only
in ten spac-etime dimensions (or 11 for M–theory)
allows for solutions where the standard model particles
(like fermions and gauge bosons) are confined to some
hypersurface, called the brane, and only the graviton can
propagate in the whole space-time, the bulk [2, 3]. Since
gravity is not well constrained at small distances, the
dimensions normal to the brane, the extra dimensions,
can be as large as 0.1mm.
Based on this feature, Arkani-Hamed, Dimopoulos
and Dvali (ADD) proposed a braneworld model where
the presence of two or more flat extra-dimensions can
provide a solution to the hierarchy problem, the problem
of the huge difference between the Planck scale and the
electroweak scale [4, 5].
In 1999 Randall and Sundrum (RS) introduced a model
with one extra dimension, where the bulk is a slice of
five-dimensional anti de-Sitter (AdS) space. Such curved
extra dimensions are also referred to as warped extra
dimensions. While in the RS I model [6] with two flat
branes of opposite tension at the edges of the bulk the
warping leads to an interesting solution of the hierarchy
problem, it localizes four-dimensional gravity on a single
positive tension brane in the RS II model [7].
Within the context of warped braneworlds, cosmological
evolution, i.e., the expansion of the Universe, can be
understood as the motion of the brane representing our
Universe through the AdS bulk. Thereby the Lanczos-
Sen-Darmois-Israel-junction conditions [8, 9, 10, 11],
relate the energy-momentum tensor on the brane to the
∗Electronic address: marcus.ruser@physics.unige.ch
†Electronic address: ruth.durrer@physics.unige.ch
extrinsic curvature and hence to the brane motion which
is described by a modified Friedmann equation. At low
energy, however, the usual Friedmann equations for the
expansion of the Universe are recovered [12, 13].
Since gravity probes the extra dimension, gravita-
tional perturbations on the brane, i.e. in our Universe,
carry five-dimensional effects in form of massive four-
dimensional gravitons, the so-called Kaluza-Klein (KK)
tower. Depending on the particular brane trajectory,
these perturbations may be significantly amplified lead-
ing to observable consequences, for example, a stochastic
gravitational wave background. (For a review of stochas-
tic gravitational waves see [14].) This amplification
mechanism is identical to the dynamical Casimir effect
for the electromagnetic field in cavities with dynamical
walls (moving mirrors); see [15, 16, 17] and references
therein. In the quantum field theoretical language,
such an amplification corresponds to the creation of
particles out of vacuum fluctuations. Hence, in the same
way a moving mirror leads to production of photons,
the brane moving through the bulk causes creation of
gravitons. Thereby, not only the usual four-dimensional
graviton might be produced, but also gravitons of the
KK tower can be excited. Those massive gravitons are of
particular interest, since their energy density could dom-
inate the energy density of the Universe and spoil the
phenomenology if their production is sufficiently copious.
The evolution of cosmological perturbations under
the influence of a moving brane has been the subject of
many studies during recent years. Since one has to deal
with partial differential equations and time-dependent
boundary conditions, the investigation of the evolution
of perturbations in the background of a moving brane is
quite complicated. Analytical progress has been made
based on approximations like the “near brane limit” and
a slowly moving brane [18, 19, 20, 21].
The case of de Sitter or quasi-de Sitter inflation
on the brane has been investigated analytically in
[22, 23, 24, 25, 26]. In [25] it is demonstrated that dur-
http://arxiv.org/abs/0704.0790v3
mailto:marcus.ruser@physics.unige.ch
mailto:ruth.durrer@physics.unige.ch
yb(t)
FIG. 1: Two branes in an AdS5 spacetime, with y denot-
ing the fifth dimension and L the AdS curvature scale. The
physical brane is on the left at time dependent position yb(t).
While it is approaching the static brane its scale factor is
decreasing and when it moves away from the static brane it
is expanding [cf. Eq. (2.3)]. The value of the scale factor of
the brane metric as function of the extra dimension y is also
indicated.
ing slow-roll inflation (modeled as a period of quasi-de
Sitter expansion) the standard four-dimensional result
for the amplitude of perturbations is recovered at low
energies while it is enhanced at high energies.
However, most of the effort has gone into numerical
simulations [27, 28, 29, 30, 31, 32, 33, 34, 35, 36], in
particular in order to investigate the high-energy regime.
Thereby different coordinate systems have been used
for which the brane is at rest, and different numerical
evolution schemes have been employed in order to solve
the partial differential equation.
In this work we chose a different way of looking at
the problem. We shall apply a formalism used to
describe the dynamical Casimir effect to study the
production of gravitons in braneworld cosmology. This
approach and its numerical implementation offers many
advantages. The most important one is the fact that
this approach deals directly with the appearing mode
couplings by means of coupling matrices. (In [19] a
similar approach involving coupling matrices has been
used. However, perturbatively only, and not in the
complexity presented here.) Hence, the interaction
between the four-dimensional graviton and the KK
modes is not hidden within a numerical simulation but
can directly be investigated making it possible to reveal
the underlying physics in a very transparent way.
We consider a five-dimensional anti-de Sitter spacetime
with two branes in it; a moving positive tension brane
representing our Universe and a second brane which, for
definiteness, is kept at rest. This setup is depicted in
Fig. 1. For this model we have previously shown that
in a radiation dominated Universe, where the second,
fixed brane is arbitrarily far away, no gravitons are
produced [37].
The particular model which we shall consider is
strongly motivated by the ekpyrotic or cyclic Universe
and similar ideas [38, 39, 40, 41, 42, 43, 44, 45, 46].
In this model, roughly speaking, the hot big bang
corresponds to the collision of two branes; a moving
bulk brane which hits “our” brane, i.e. the observable
Universe. Within such a model, it seems to be possible to
address all major cosmological problems (homogeneity,
origin of density perturbations, monopole problem)
without invoking the paradigm of inflation. For more
details see [38] but also [39] for critical comments.
One important difference between the ekpyrotic model
and standard inflation is that in the latter one tensor
perturbations have a nearly scale invariant spectrum.
The ekpyrotic model, on the other hand, predicts a
strongly blue gravitational wave spectrum with spectral
tilt nT ≃ 2 [38]. This blue spectrum is a key test
for the ekpyrotic scenario since inflation always pre-
dicts a slightly red spectrum for gravitational waves.
One method to detect a background of primordial
gravitational waves of wavelengths comparable to the
Hubble horizon today is the polarization of the cosmic
microwave background. Since a strongly blue spectrum
of gravitational waves is unobservably small for large
length scales, the detection of gravitational waves in the
cosmic microwave background polarization would falsify
the ekpyrotic model [38].
Here we consider a simple specific model which is
generic enough to cover important main features of the
generation and evolution of gravitational waves in the
background of a moving brane whose trajectory involves
a bounce. First, the physical brane moves towards the
static brane, initially the motion is very slow. During
this phase our Universe is contracting, i.e. the scale
factor on the brane decreases, the energy density on
the brane increases and the motion becomes faster. We
suppose that the evolution of the brane is driven by a
radiation component on the brane, and that at some
more or less close encounter of the two branes which we
call the bounce, some high-energy mechanism which we
do not want to specify in any detail, turns around the
motion of the brane leading to an expanding Universe.
Modeling the transition from contraction to subsequent
expansion in any detail would require assumptions about
unknown physics. We shall therefore ignore results
which depend on the details of the transition. Finally
the physical brane moves away from the static brane
back towards the horizon with expansion first fast and
then becoming slower as the energy density drops. This
model is more similar to the pyrotechnic Universe of
Kallosh, Kofman and Linde [39] where the observable
Universe is also represented by a positive tension brane
rather than to the ekpyrotic model where our brane has
negative tension.
We address the following questions: What is the spec-
trum and energy density of the produced gravitons, the
massless zero mode and the KK modes? Can the gravi-
ton production in such a brane Universe lead to limits,
e.g. on the AdS curvature scale via the nucleosynthesis
bound? Can the KK modes provide the dark matter
or lead to stringent limits on these models? Similar
results could be obtained for the free gravi-photon and
gravi-scalar, i.e. when we neglect the perturbations of
the brane energy momentum tensor which also couple to
these gravity wave modes which have spin-1 respectively
spin-0 on the brane.
The reminder of the paper is organized as follows.
After reviewing the basic equations of braneworld cos-
mology and tensor perturbations in Sec. II, we discuss
the dynamical Casimir effect approach in Sec. III. In
Sec. IV we derive expressions for the energy density
and the power spectrum of gravitons. Thereby we show
that, very generically, KK gravitons cannot play the
role of dark matter in warped braneworlds. This is
explained by the localization of gravity on the moving
brane which we discuss in detail. Section V is devoted to
the presentation and discussion of our numerical results.
In Sec. VI we reproduce some of the numerical results
with analytical approximations and we derive fits for
the number of produced gravitons. We discuss our main
results and their implications for bouncing braneworlds
in Sec. VII and conclude in Sec. VIII. Some technical
aspects are collected in appendices.
The main and most important results of this rather long
and technical paper are published in the Letter [47].
II. GRAVITONS IN MOVING BRANEWORLDS
A. A moving brane in AdS5
We consider a AdS-5 spacetime. In Poincaré coordi-
nates, the bulk metric is given by
ds2 = gABdx
AdxB =
−dt2 + δijdxidxj + dy2
(2.1)
The physical brane (our Universe) is located at some time
dependent position y = yb(t), while the 2nd brane is at
fixed position y = ys (see Fig. 1). The induced metric on
the physical brane is given by
ds2 =
y2b (t)
dt2 + δijdx
= a2(η)
−dη2 + δijdxidxj
, (2.2)
where
a(η) =
yb(t)
(2.3)
is the scale factor and η denotes the conformal time of
an observer on the brane,
dt ≡ γ−1dt . (2.4)
We have introduced the brane velocity
v ≡ dyb
= − LH√
1 + L2H2
and (2.5)
1− v2
1 + L2H2 . (2.6)
Here H is the usual Hubble parameter,
H ≡ ȧ/a2 ≡ a−1H = −L−1γv , (2.7)
and an overdot denotes the derivative with respect to
conformal time η. The bulk cosmological constant Λ is
related to the curvature scale L by Λ = −6/L2. The
junction conditions on the brane lead to [37, 48]
(ρ+ T ) = 6
1 + L2H2
, (2.8)
(ρ+ P ) = − 2LḢ
1 + L2H2
. (2.9)
Here T is the brane tension and ρ and P denote the
energy density and pressure of the matter confined on
the brane. Combining (2.8) and (2.9) results in
ρ̇ = −3Ha(ρ+ P ) , (2.10)
while taking the square of (2.8) leads to
. (2.11)
These equations form the basis of brane cosmology and
have been discussed at length in the literature (for re-
views see [49, 50]). The last equation is called the mod-
ified Friedmann equation for brane cosmology [13]. For
usual matter with ρ+ P > 0, ρ decreases during expan-
sion and at sufficiently late time ρ ≪ T . The ordinary
four-dimensional Friedmann equation is then recovered if
and we set κ
= 8πG4 =
. (2.12)
Here we have neglected a possible four-dimensional cos-
mological constant. The first of these equations is the
RS fine tuning implying
κ5 = κ4 L . (2.13)
Defining the string and Planck scales by
= L3s , κ4 =
= L2Pl , (2.14)
respectively, the RS fine tuning condition leads to
. (2.15)
As outlined in the introduction, we shall be interested
mainly in a radiation dominated low-energy phase, hence
in the period where
ρ and |v| ≪ 1 so that γ ≃ 1 , dη ≃ dt .
(2.16)
In such a period, the solutions to the above equations are
of the form
a(t) =
|t|+ tb
, (2.17)
yb(t) =
|t|+ tb
, (2.18)
v(t) = − sgn(t)L
(|t|+ tb)2
≃ −HL . (2.19)
Negative times (t < 0) describe a contracting phase,
while positive times (t > 0) describe radiation dominated
expansion. At t = 0, the scale factor exhibits a kink and
the evolution equations are singular. This is the bounce
which we shall not model in detail, but we will have to in-
troduce a cutoff in order to avoid ultraviolet divergencies
in the total particle number and energy density which are
due to this unphysical kink. We shall show, that when
the kink is smoothed out at some length scale, the pro-
duction of particles (KK gravitons) of masses larger than
this scale is exponentially suppressed, as it is expected.
The (free) parameter tb > 0 determines the value of the
scale factor at the bounce ab, i.e. the minimal interbrane
distance, as well as the velocity at the bounce vb
ab = a(0) =
, |v(0)| ≡ vb =
. (2.20)
Apparently we have to demand tb > L which implies
yb(t) < L.
B. Tensor perturbations in AdS5
We now consider tensor perturbations on this back-
ground. Allowing for tensor perturbations hij(t,x, y) of
the spatial three-dimensional geometry at fixed y, the
bulk metric reads
ds2 =
−dt2 + (δij + 2hij)dxidxj + dy2
. (2.21)
Tensor modes satisfy the traceless and transverse condi-
tions, hii = ∂ih
j = 0. These conditions imply that hij has
only two independent degrees of freedom, the two polar-
ization states • = ×,+. We decompose hij into spatial
Fourier modes,
hij(t,x, y) =
(2π)3/2
•=+,×
eik·xe•ij(k)h•(t, y;k) ,
(2.22)
where e•ij(k) are unitary constant transverse-traceless po-
larization tensors which form a basis of the two polariza-
tion states • = ×,+. For hij to be real we require
h∗•(t, y;k) = h•(t, y;−k). (2.23)
The perturbed Einstein equations yield the equation of
motion for the mode functions h
, which obey the Klein-
Gordon equation for minimally coupled massless scalar
fields in AdS5 [25, 51, 52]
∂2t + k
2 − ∂2y +
(t, y;k) = 0 . (2.24)
In addition to the bulk equation of motion the modes also
satisfy a boundary condition at the brane coming from
the second junction condition,
LH∂th• −
1 + L2H2∂yh•
− γ (v∂t + ∂y)h•|yb =
aPΠ(T )
. (2.25)
Here Π(T )
denotes possible anisotropic stress perturba-
tions in the brane energy momentum tensor. We are in-
terested in the quantum production of free gravitons, not
in the coupling of gravitational waves to matter. There-
fore we shall set Π(T )
= 0 in the sequel, i.e. we make
the assumption that the Universe is filled with a perfect
fluid. Then, (2.25) reduces to 1
(v∂t + ∂y)h•|yb(t) = 0 . (2.26)
This is not entirely correct for the evolution of gravity
modes since at late times, when matter on the brane is
no longer a perfect fluid (e.g., free-streaming neutrinos)
and anisotropic stresses develop which slightly modify
the evolution of gravitational waves. We neglect this sub-
dominant effect in our treatment. (Some of the difficul-
ties which appear when Π(T )
6= 0 are discussed in [48].)
The wave equation (2.24) together with the boundary
condition (2.26) can also be obtained by variation of the
action
Sh = 2
yb(t)
|∂th•|2 − |∂yh•|2 − k2|h•|2
, (2.27)
which follows from the second order perturbation of the
gravitational Lagrangian. The factor 2 in the action is
1 In Equations (4) and (8) of our Letter [47] two sign mistakes
have creeped in.
due to Z2 symmetry. Indeed, Equation (2.26) is the only
boundary condition for the perturbation amplitude h•
which is compatible with the variational principle δSh =
0, except if h• is constant on the brane. Since this issue is
important in the following, it is discussed more detailed
in Appendix A.
C. Equations of motion in the late time/low energy
limit
In this work we restrict ourselves to relatively late
times, when
ρT ≫ ρ2 and therefore |v| ≪ 1. (2.28)
In this limit the conformal time on the brane agrees
roughly with the 5D time coordinate, dη ≃ dt and we
shall therefore not distinguish these times; we set t = η.
We want to study the quantum mechanical evolution
of tensor perturbations within a canonical formulation
similar to the dynamical Casimir effect for the electro-
magnetic field in dynamical cavities [15, 16, 17]. In or-
der to pave the way for canonical quantization, we have
to introduce a suitable set of functions allowing the ex-
pansion of the perturbation amplitude h• in canonical
variables. More precisely, we need a complete and or-
thonormal set of eigenfunctions φα of the spatial part
−∂2y + 3y∂y = −y
y−3∂y
of the differential opera-
tor (2.24). The existence of such a set depends on the
boundary conditions and is ensured if the problem is
of Sturm-Liouville type (see, e.g.,[53]). For the junc-
tion condition (2.26), such a set does unfortunately not
exist due to the time derivative. One way to proceed
would be to introduce other coordinates along the lines
of [54] for which the junction condition reduces to a sim-
ple Neumann boundary condition leading to a problem
of Sturm-Liouville type. This transformation is, however,
relatively complicated to implement without approxima-
tions and is the subject of future work.
Here we shall proceed otherwise, harnessing the fact that
we are interested in low energy effects only, i.e. in small
brane velocities. Assuming that one can neglect the
time derivative in the junction condition since |v| ≪ 1,
Eq. (2.25) reduces to a simple Neumann boundary con-
dition. We shall therefore work with the boundary con-
ditions
∂yh•|yb = ∂yh•|ys = 0 . (2.29)
Then, at any time t the eigenvalue problem for the spatial
part of the differential operator (2.24)
−∂2y +
φα(t, y) = −y3∂y
y−3∂yφα(t, y)
= m2α(t)φα(t, y) (2.30)
is of Sturm-Liouville type if we demand that the φα’s are
subject to the boundary conditions (2.29). Consequently,
the set of eigenfunctions {φα(t, y)}∞α=0 is complete,
φα(t, y)φα(t, ỹ) = δ(y − ỹ)y3 , (2.31)
and orthonormal with respect to the inner-product
(φα, φβ) = 2
yb(t)
φα(t, y)φβ(t, y) = δαβ . (2.32)
Note the factor 2 in front of both expressions which is
necessary in order to take the Z2 symmetry properly into
account.
The eigenvalues mα(t) are time-dependent and discrete
due to the time-dependent but finite distance between
the branes and the eigenfunctions φα(t, y) are time-
dependent in particular because of the time dependence
of the boundary conditions (2.29). The case α = 0
with m0 = 0 is the zero mode, i.e. the massless four-
dimensional graviton. Its general solution in accordance
with the boundary conditions is just a constant with re-
spect to the extra dimension, φ0(t, y) = φ0(t), and is fully
determined by the normalization condition (φ0, φ0) = 1:
φ0(t) =
ysyb(t)√
y2s − y2b (t)
. (2.33)
For α = i ∈ {1, 2, 3, · · · , } with eigenvalues mi > 0, the
general solution of (2.30) is a combination of the Bessel
functions J2 (mi(t) y) and Y2 (mi(t) y). Their particular
combination is determined by the boundary condition at
the moving brane. The remaining boundary condition at
the static brane selects the possible values for the eigen-
values mi(t), the KK masses. For any three-momentum
k these masses build up an entire tower of momenta in
the y-direction; the fifth dimension. Explicitely, the so-
lutions φi(t, y) for the KK modes read
φi(t, y) = Ni(t)y
2C2 (mi(t) y) (2.34)
Cν(miy) = Y1(miyb)Jν(miy)−J1(miyb)Yν(miy). (2.35)
The normalization reads
Ni(t, yb, ys) =
y2sC22(mi ys)− (2/(miπ))
(2.36)
where we have used that
C2(mi yb) =
πmi yb
. (2.37)
2 Note that we have changed the parameterization of the solutions
with respect to [37] for technical reasons. There, we also did not
take into account the factor 2 related to Z2 symmetry.
It can be simplified further by using
C2(mi ys) =
Y1(mi yb)
Y1(mi ys)
πmi ys
(2.38)
leading to
Y 21 (miys)
Y 21 (miyb)− Y 21 (miys)
. (2.39)
Note that it is possible to have Y 21 (mi ys)− Y 21 (mi yb) =
0. But then both Y 21 (miys) = Y
1 (miyb) = 0 and
Eq. (2.39) has to be understood as a limit. For that rea-
son, the expression (2.36) for the normalization is used
in the numerical simulations later on. Its denominator
remains always finite.
The time-dependent KK masses {mi(t)}∞i=1 are deter-
mined by the condition
C1 (mi(t)ys) = 0 . (2.40)
Because the zeros of the cross product of the Bessel func-
tions J1 and Y1 are not known analytically in closed form,
the KK-spectrum has to be determined by solving Eq.
(2.40) numerically 3. An important quantity which we
need below is the rate of change ṁi/mi of a KK mass
given by
m̂i ≡
= ŷb
m2i π
N2i (2.41)
where the rate of change of the brane motion ŷb is just
the Hubble parameter on the brane
ŷb(t) ≡
ẏb(t)
yb(t)
≃ −Ha = − ȧ
= −H . (2.42)
On account of the completeness of the eigenfunctions
φα(t, y) the gravitational wave amplitude h•(t, y;k)
subject to the boundary conditions (2.29) can now be
expanded as
h•(t, y;k) =
qα,k,•(t)φα(t, y) . (2.43)
The coefficients qα,k,•(t) are canonical variables describ-
ing the time evolution of the perturbations and the fac-
κ5/L3 has been introduced in order to render the
qα,k,•’s canonically normalized. In order to satisfy (2.23)
we have to impose the same condition for the canonical
variables, i.e.
q∗α,k,• = qα,−k,•. (2.44)
3 Approximate expressions for the zeros can be found in [55].
One could now insert the expansion (2.43) into the wave
equation (2.24), multiplying it by φβ(t, y) and integrating
out the y−dependence by using the orthonormality to de-
rive the equations of motion for the variables qα,k,•. How-
ever, as we explain in Appendix A, a Neumann boundary
condition at a moving brane is not compatible with a free
wave equation. The only consistent way to implement the
boundary conditions (2.29) is therefore to consider the
action (2.27) of the perturbations as the starting point
to derive the equations of motion for qα,k,•. Inserting
(2.43) into (2.27) leads to the canonical action
S = 1
|q̇α,k,•|2 − ω2α,k|qα,k,•|2
Mαβ (qα,k,•q̇β,−k,• + qα,−k,•q̇β,k,•)
+Nαβqα,k,•qβ,−k,•
. (2.45)
We have introduced the time-dependent frequency of a
graviton mode
ω2α,k =
k2 +m2α , k = |k| , (2.46)
and the time-dependent coupling matrices
Mαβ = (∂tφα, φβ) , (2.47)
Nαβ = (∂tφα, ∂tφβ) =
MαγMβγ (2.48)
which are given explicitely in Appendix B (see also [37]).
Consequently, the equations of motion for the canonical
variables are
q̈α,k,• + ω
α,kqα,k,• +
[Mβα −Mαβ ] q̇β,k,•
Ṁαβ −Nαβ
qβ,k,• = 0 . (2.49)
The motion of the brane through the bulk, i.e.
the expansion of the Universe, is encoded in the
time-dependent coupling matrices Mαβ , Nαβ. The
mode couplings are caused by the time-dependent
boundary condition ∂yh•(t, y)|yb = 0 which forces the
eigenfunctions φα(t, y) to be explicitly time-dependent.
In addition, the frequency of a KK mode ωα,k is also
time-dependent since the distance between the two
branes changes when the brane is in motion. Both time-
dependencies can lead to the amplification of tensor
perturbations and, within a quantum theory which is
developed in the next section, to graviton production
from vacuum.
Because of translation invariance with respect to the
directions parallel to the brane, modes with different k
do not couple in (2.49). The three-momentum k enters
the equation of motion for the perturbation only via the
frequency ωα,k, i.e. as a global quantity. Equation (2.49)
is similar to the equation describing the time-evolution
of electromagnetic field modes in a three-dimensional
dynamical cavity [16] and may effectively be described
by a massive scalar field on a time-dependent interval
[17]. For the electromagnetic field, the dynamics of the
cavity, or more precisely the motion of one of its walls,
leads to photon creation from vacuum fluctuations. This
phenomenon is usually referred to as dynamical Casimir
effect. Inspired by this, we shall call the production of
gravitons by the moving brane as dynamical Casimir
effect for gravitons.
D. Remarks and comments
In [37] we have already shown that in the limit where
the fixed brane is sent off to infinity, ys → ∞, only the
M00 matrix element survives with M00 = −H[1 + O(ǫ)]
and ǫ = yb/ys. M00 expresses the coupling of the zero
mode to the brane motion. Since all other couplings dis-
appear for ǫ → 0 all modes decouple from each other and,
in addition, the canonical variables for the KK modes de-
couple from the brane motion itself. This has led to the
result that at late times and in the limit ys ≫ yb, the KK
modes with non-vanishing mass evolve trivially, and only
the massless zero mode is coupled to the brane motion
q̈0,k,• +
k2 − Ḣ −H2
q0,k,• = 0 . (2.50)
Since φ0 ∝ 1/a [cf. Eqs. (4.2),(4.5)] we have found in [37]
that the gravitational zero mode on the brane h0,•(t;k) ≡√
κ5/L3q0,k,•φ0(t, yb) evolves according to
ḧ0,•(t;k) + 2Hḣ0,•(t;k) + k2h0,•(t;k) = 0 , (2.51)
which explicitely demonstrates that at low energies (late
times) the homogeneous tensor perturbation equation in
brane cosmology reduces to the four-dimensional tensor
perturbation equation.
An important comment is in order here concerning
the RS II model. In the limit ys → ∞ the fixed brane
is sent off to infinity and one ends up with a single
positive tension brane in AdS, i.e. the RS II model.
Even though we have shown that all couplings except
M00 vanish in this limit, that does not imply that this
is necessarily the case for the RS II setup. Strictly
speaking, the above arguments are only valid in a two
brane model with ys ≫ 1. Starting with the RS II model
from the beginning, the coupling matrices do in general
not vanish when calculated with the corresponding
eigenfunctions which can be found in, e.g., [22]. One
just has to be careful when taking those limits. But
what the above consideration demonstrates is that, if
the couplings of the zero mode to the KK modes vanish,
like in the ys ≫ 1 limit or in the low energy RS II model
as observed in numerical simulations (see below) the
standard evolution equation for the zero mode emerges
automatically from five-dimensional perturbation theory.
Starting from five-dimensional perturbation theory,
our formalism does imply the usual evolution equation
for the four-dimensional graviton in a FLRW-Universe
in the limit of vanishing couplings. This serves as a
very strong indication (but certainly not proof!) for
the fact that the approach based on the approximation
(2.29) and the expansion of the action in canonical
variables rather than the wave equation is consistent
and leads to results which should reflect the physics at
low energies. As already outlined, if one would expand
the wave equation (2.24) in the set of functions φα,
the resulting equation of motion for the corresponding
canonical variables is different from Eq. (2.49) and
cannot be derived from a Lagrangian or Hamiltonian
(see Appendix A). Moreover, in [30] the low energy RS
II scenario has been studied numerically including the
full junction condition (2.26) without approximations
(see also [27]). Those numerical results show that
the evolution of tensor perturbations on the brane is
four-dimensional, i.e. described by Eq. (2.51) derived
here analytically. Combining these observations gives
us confidence that the used approach based on the
Neumann boundary condition approximation and the
action as starting point for the canonical formulation is
adequate for the study of tensor perturbations in the
low energy limit. The many benefits this approach offers
will become visible in the following.
III. QUANTUM GENERATION OF TENSOR
PERTURBATIONS
A. Preliminary remarks
We now introduce a treatment of quantum generation
of tensor perturbations. This formalism is an advance-
ment of the method which is presented in [15, 16, 17]
for the dynamical Casimir effect for a scalar field and
the electromagnetic field to gravitational perturbations
in the braneworld scenario.
The following method is very general and not restricted
to a particular brane motion as long as it complies with
the low energy approach [cf. Eq. (2.28)]. We assume that
asymptotically, i.e. for t → ±∞, the physical brane
approaches the Cauchy horizon (yb → 0), moving very
slowly. Then, the coupling matrices vanish and the KK
masses are constant (for yb close to zero, Eq. (2.40) re-
duces to J1(miys) = 0):
Mαβ(t) = 0 , lim
mα(t) = const. ∀α, β .
(3.1)
In this limit, the system (2.49) reduces to an infinite set
of uncoupled harmonic oscillators. This allows to intro-
duce an unambiguous and meaningful particle concept,
i.e. notion of (massive) gravitons.
As a matter of fact, in the numerical simulations, the
brane motion has to be switched on and off at finite times.
These times are denoted by tin and tout, respectively. We
introduce vacuum states with respect to times t < tin < 0
and t > tout > 0. In order to avoid spurious effects influ-
encing the particle creation, we have to chose tin small,
respectively tout large enough such that the couplings are
effectively zero at these times. Checking the indepen-
dence of the numerical results on the choice of tin and
tout guarantees that these times correspond virtually to
the real asymptotic states of the brane configuration.
B. Quantization, initial and final state
Canonical quantization of the gravity wave amplitude
is performed by replacing the canonical variables qα,k,•
by the corresponding operators q̂α,k,•
ĥ•(t, y;k) =
q̂α,k,•(t)φα(t, y) . (3.2)
Adopting the Heisenberg picture to describe the quantum
time-evolution, it follows that q̂α,k,• satisfies the same
equation (2.49) as the canonical variable qα,k,•.
Under the assumptions outlined above, the operator
q̂α,k,• can be written for times t < tin as
q̂α,k,•(t < tin) = (3.3)
2ωinα,k
âinα,k,•e
−i ωinα,k t + âin†α,−k,•e
i ωinα,k t
where we have introduced the initial-state frequency
ωinα,k ≡ ωα,k(t < tin) . (3.4)
This expansion ensures that Eq. (2.44) is satisfied. The
set of annihilation and creation operators {âinα,k,•, â
α,k,•}
corresponding to the notion of gravitons for t < tin is
subject to the usual commutation relations
âinα,k,•, â
α′,k′,•′
= δαα′δ••′δ
(3)(k− k′) , (3.5)
âinα,k,•, â
α′,k′,•′
α,k,•, â
α′,k′,•′
= 0. (3.6)
For times t > tout, i.e. after the motion of the brane has
ceased, the operator q̂α,k,• can be expanded in a similar
manner,
q̂α,k,•(t > tout) = (3.7)
2ωoutα,k
âoutα,k,•e
−i ωout
t + â
out †
α,−k,•e
i ωout
with final state frequency
ωoutα,k ≡ ωα,k(t > tout) . (3.8)
The annihilation and creation operators {âoutα,k,•, â
out †
α,k,•}
correspond to a meaningful definition of final state gravi-
tons (they are associated with positive and negative fre-
quency solutions for t ≥ tout) and satisfy the same com-
mutation relations as the initial state operators.
Initial |0, in〉 ≡ |0, t < tin〉 and final |0, out〉 ≡ |0, t > tout〉
vacuum states are uniquely defined via 4
âinα,k,•|0, in〉 = 0 , âoutα,k,•|0, out〉 = 0 , ∀ α, k, • . (3.9)
The operators counting the number of particles defined
with respect to the initial and final vacuum state, respec-
tively, are
N̂ inα,k,• = â
α,k,•â
α,k,• , N̂
α,k,• = â
out †
α,k,•â
α,k,• . (3.10)
The number of gravitons created during the motion of
the brane for each momentum k, quantum number α and
polarization state • is given by the expectation value of
the number operator N̂outα,k,• of final-state gravitons with
respect to the initial vacuum state |0, in〉:
N outα,k,• = 〈0, in|N̂outα,k,•|0, in〉. (3.11)
If the brane undergoes a non-trivial dynamics between
tin < t < tout it is â
α,k,•|0, in〉 6= 0 in general, i.e.
graviton production from vacuum fluctuations takes
place.
From (2.22), the expansion (3.2) and Eqs.(3.3),
(3.7) it follows that the quantized tensor perturbation
with respect to the initial and final state can be written
ĥij(t < tin,x,y) =
(2π)3/2
âinα,k,• e
−i ωinα,k t
2ωinα,k
× u•ij,α(t < tin,x, y,k) + h.c. (3.12)
ĥij(t > tout,x,y) =
(2π)3/2
âoutα,k,• e
−i ωoutα,k t
2ωoutα,k
× u•ij,α(t > tout,x, y,k) + h.c. . (3.13)
We have introduced the basis functions
u•ij,α(t,x, y,k) = e
ik ·x e•ij(k)φα(t, y). (3.14)
which, on account of (e•ij(k))
∗ = e•ij(−k), satisfy
(u•ij,α(t,x, y,k))
∗ = u•ij,α(t,x, y,−k).
4 Note that the notations |0, t < tin〉 and |0, t > tout〉 do not mean
that the states are time-dependent; states do not evolve in the
Heisenberg picture.
C. Time evolution
During the motion of the brane the time evolution of
the field modes is described by the system of coupled
differential equations (2.49). To account for the inter-
mode couplings mediated by the coupling matrix Mαβ
the operator q̂α,k,• is decomposed as
q̂α,k,•(t) =
2ωinβ,k
âinβ,k,•ǫ
α,k(t) + â
β,−k,•ǫ
α,k (t)
(3.15)
The complex functions ǫ
α,k(t) also satisfy the system of
coupled differential equations (2.49). With the ansatz
(3.15) the quantized tensor perturbation at any time dur-
ing the brane motion reads
ĥij(t,x, y) = (3.16)
âinβ,k,•√
2ωinβ,k
α,k(t)u
ij,α(t,x, y,k) + h.c. .
Due to the time-dependence of the eigenfunctions φα,
the time-derivative of the gravity wave amplitude con-
tains additional mode coupling contributions. Using the
completeness and orthnormality of the φα’s it is readily
shown that
h•(t, y;k) =
p̂α,−k,•(t)φα(t, y) (3.17)
where
p̂α,−k,•(t) = ˙̂qα,k,•(t) +
Mβαq̂β,k,•(t). (3.18)
The coupling term arises from the time dependence of
the mode functions φα. Accordingly, the time derivative
hij reads
hij(t,x, y) =
âinβ,k,•√
2ωinβ,k
× (3.19)
× f (β)α,k(t)u
ij,α(t,x, y,k) + h.c.
where we have introduced the function
α,k(t) = ǫ̇
α,k(t) +
Mγα(t)ǫ
γ,k(t) . (3.20)
By comparing Eq. (3.12) and its time-derivative with
Eqs. (3.16) and (3.19) at t = tin one can read off the
initial conditions for the functions ǫ
α,k(tin) = δαβ Θ
α,k , (3.21)
α,k(tin) =
−iωinα,kδαβ −Mβα(tin)
Θinβ,k (3.22)
with phase
Θinα,k = e
−iωinα,k tin . (3.23)
The choice of this phase for the initial condition is in
principle arbitrary, we could as well set Θinα,k = 1. But
with this choice, ǫ
α,k(t) is independent of tin for t < tin
and therefore it is also at later times independent of tin
if only we choose tin sufficiently early. This is especially
useful for the numerical work.
D. Bogoliubov transformations
The two sets of annihilation and creation operators
{âinα,k,•, â
α,k,•} and {âoutα,k,•, â
out †
α,k,•} corresponding to the
notion of initial-state and final-state gravitons are re-
lated via a Bogoliubov transformation. Matching the
expression for the tensor perturbation Eq. (3.16) and
its time-derivative Eq. (3.19) with the final state expres-
sion Eq. (3.13) and its corresponding time-derivative at
t = tout one finds
âoutβ,k,• =
Aαβ,k(tout)âinα,k,• + B∗αβ,k(tout)â
α,−k,•
(3.24)
Aβα,k(tout) =
ωoutα,k
ωinβ,k
α,k(tout) +
ωoutα,k
α,k(tout)
(3.25)
Bβα,k(tout) =
Θoutα,k
ωoutα,k
ωinβ,k
α,k(tout)−
ωoutα,k
α,k(tout)
(3.26)
where we shall stick to the phase Θoutα,k defined like Θ
in (3.23) for completeness. Performing the matching at
tout = tin the Bogoliubov transformation should become
trivial, i.e. the Bogoliubov coefficients are subject to vac-
uum initial conditions
Aαβ,k(tin) = δαβ , Bαβ,k(tin) = 0. (3.27)
Evaluating the Bogoliubov coefficients (3.25) and (3.26)
for tout = tin by making use of the initial conditions
(3.21) and (3.22) shows the consistency. Note that the
Bogoliubov transformation (3.24) is not diagonal due
to the inter-mode coupling. If during the motion of
the brane the graviton field departs form its vacuum
state one has Bαβ,k(tout) 6= 0, i.e. gravitons have been
generated.
By means of Eq. (3.24) the number of generated
final state gravitons (3.11), which is the same for every
polarization state, is given by
N outα,k (t ≥ tout) =
•=+,×
〈0, in|N̂outα,k,•|0, in〉
|Bβα,k(tout)|2. (3.28)
Later we will sometimes interpret tout as a continuous
variable tout → t such that N outα,k → Nα,k(t), i.e. it
becomes a continuous function of time. We shall call
Nα,k(t) the instantaneous particle number [see Appendix
C 2], however, a physical interpretation should be made
with caution.
E. The first order system
From the solutions of the system of differential equa-
tions (2.49) for the complex functions ǫ
α,k, the Bogoli-
ubov coefficient Bαβ,k, and hence the number of cre-
ated final state gravitons (3.28), can now be calculated.
It is however useful to introduce auxiliary functions
α,k(t), η
α,k(t) through
α,k(t) = ǫ
α,k(t) +
ωinα,k
α,k(t) (3.29)
α,k(t) = ǫ
α,k(t)−
ωinα,k
α,k(t) . (3.30)
These are related to the Bogoliubov coefficients via
Aβα,k(tout) = (3.31)
ωoutα,k
ωinβ,k
∆+α,k(tout)ξ
α,k(tout) + ∆
α,k(tout)η
α,k(tout)
Bβα,k(tout) = (3.32)
Θoutα,k
ωoutα,k
ωinβ,k
∆−α,k(tout)ξ
α,k(tout) + ∆
α,k(tout)η
α,k(tout)
where we have defined
∆±α,k(t) =
ωinα,k
ωα,k(t)
, (3.33)
Using the second order differential equation for ǫ
α,k, it is
readily shown that the functions ξ
α,k(t), η
α,k(t) satisfy
the following system of first order differential equations:
α,k(t) = −i
a+αα,k(t)ξ
α,k(t)− a
αα,k(t)η
α,k(t)
c−αγ,k(t)ξ
γ,k(t) + c
αγ,k(t)η
γ,k(t)
(3.34)
α,k(t) = −i
a−αα,k(t)ξ
α,k(t)− a
αα,k(t)η
α,k(t)
c+αγ,k(t)ξ
γ,k(t) + c
αγ,k(t)η
γ,k(t)
(3.35)
a±αα,k(t) =
ωinα,k
ωα,k(t)
ωinα,k
 , (3.36)
c±γα,k(t) =
Mαγ(t)±
ωinα,k
ωinγ,k
Mγα(t)
. (3.37)
The vacuum initial conditions (3.27) entail the initial
conditions
α,k(tin) = 2 δαβ Θ
α,k , η
α,k(tin) = 0. (3.38)
With the aid of Eq. (3.32), the coefficient Bαβ,k(tout),
and therefore the number of produced gravitons, can be
directly deduced from the solutions to this system of
coupled first order differential equations which can be
solved using standard numerics.
In the next section we will show how interesting
observables like the power spectrum and the energy den-
sity of the amplified gravitational waves are expressed in
terms of the number of created gravitons. The system
(3.34, 3.35) of coupled differential equations forms the
basis of our numerical simulations. Details of the applied
numerics are collected in Appendix D.
IV. POWER SPECTRUM, ENERGY DENSITY
AND LOCALIZATION OF GRAVITY
A. Perturbations on the brane
By solving the system of coupled differential equations
formed by Eqs. (3.34) and (3.35) the time evolution of
the quantized tensor perturbation ĥij(t,x, y) can be com-
pletely reconstructed at any position y in the bulk. Ac-
cessible to observations is the imprint which the pertur-
bations leave on the brane, i.e. in our Universe. Of par-
ticular interest is therefore the part of the tensor pertur-
bation which resides on the brane. It is given by eval-
uating Eq. (2.22) at the brane position y = yb (see also
[36])
ĥij(t,x, yb) =
(2π)3/2
•=+,×
eik·xe•ij(k)ĥ•(t, yb,k) .
(4.1)
The motion of the brane (expansion of the Universe) en-
ters this expression via the eigenfunctions φα(t, yb(t)).
We shall take (4.1) as the starting point to define ob-
servables on the brane.
The zero-mode function φ0(t) [cf. Eq. (2.33)] does not
depend on the extra dimension y. Using Eq. (2.37), one
reads off from Eq. (2.34) that the eigenfunctions on the
brane φα(t, yb) are
φα(t, yb) = yb Yα(yb) =
Yα(a) (4.2)
where we have defined
Y0(a) =
y2s − y2b
and (4.3)
Yn(a) =
Y 21 (mnys)
Y 21 (mnyb)− Y 21 (mnys)
, (4.4)
for the zero- and KK modes, respectively. One immedi-
ately is confronted with an interesting observation: the
function Yα(a) behaves differently with the expansion of
the Universe for the zero mode α = 0 and the KK modes
α = n. This is evident in particular in the asymptotic
regime ys ≫ yb, i.e. yb → 0 (|t|, a → ∞) where, exploit-
ing the asymptotics of Y1 (see [55]), one finds
Y0(a) ≃ 1 , Yn(a) ≃
|Y1(mnys)| ≃
(4.5)
Ergo, Y0 is constant while Yn decays with the expansion
of the Universe as 1/a. For large n one can approximate
mn ≃ nπ/ys and Y1(mnys) ≃ Y1(nπ) ≃ (1/π)
2/n [55],
so that
Yn(a) ≃
, Y2n(a) ≃
πL2mn
2 ysa2
. (4.6)
In summary, the amplitude of the KKmodes on the brane
decreases faster with the expansion of the Universe than
the amplitude of the zero mode. This leads to interest-
ing consequences for the observable power spectrum and
energy density and has a clear physical interpretation: It
manifest the localization of usual gravity on the brane.
As we shall show below, KK gravitons which are traces
of the five-dimensional nature of gravity escape rapidly
from the brane.
B. Power spectrum
We define the power spectrum P(k) of gravitational
waves on the brane as in four-dimensional cosmology by
using the restriction of the tensor amplitude to the brane
position (4.1):
(2π)3
P(k)δ(3)(k− k′) (4.7)
•=×,+
0, in
∣∣∣ĥ•(t, yb;k)ĥ†•(t, yb;k′)
∣∣∣0, in
i.e. we consider the expectation value of the field operator
ĥ• with respect to the initial vacuum state at the position
of the brane y = yb(t). In order to get a physically mean-
ingful power spectrum, averaging over several oscillations
of the gravitational wave amplitude has to be performed.
Equation (4.7) describes the observable power spectrum
imprinted in our Universe by the four-dimensional spin-2
graviton component of the five-dimensional tensor per-
turbation.
The explicit calculation of the expectation value involv-
ing a “renormalization” of a divergent contribution is car-
ried out in detail in Appendix C 2. The final result reads
P(k) = 1
(2π)3
Rα,k(t)Y2α(a). (4.8)
The function Rα,k(t) can be expressed in terms of the
Bogoliubov coefficients (3.25) and (3.26) if one considers
tout as a continuous variable t:
Rα,k(t) =
Nα,k(t) +ONα,k(t)
ωα,k(t)
. (4.9)
Nα,k(t) is the instantaneous particle number [cf. Ap-
pendix C 1] and the function ONα,k(t) is defined in
Eq. (C9).
It is important to recall thatNα,k(t) can in general not be
interpreted as a physical particle number. For example
zero modes with wave numbers such that kt < 1 can-
not be considered as particles. They have not performed
several oscillations and their energy density cannot be
defined in a meaningful way.
Equivalently, expressed in terms of the complex functions
α,k, one finds
Rα,k(t) =
|ǫ(β)α,k(t)|2
ωinβ,k
ωα,k(t)
+Oǫα,k(t), (4.10)
with Oǫα,k given in Eq. (C10). Equation (4.8) together
with (4.9) or (4.10) holds at all times.
If one is interested in the power spectrum at early times
kt ≪ 1, it is not sufficient to take only the instantaneous
particle number Nα,k(t) in Eq. (4.9) into account. This
is due to the fact that even if the mode functions ǫ
are already oscillating, the coupling matrix entering the
Bogoliubov coefficients might still undergo a non-trivial
time dependence [cf. Eq. (6.16)]. In the next section
we shall show explicitly, that in a radiation dominated
bounce particle creation, especially of the zero mode,
only stops on sub-Hubble times, kt > 1, even if the mode
functions are plane waves right after the bounce [cf, e.g.,
Figs. 6, 7, 9]. Therefore, in order to determine the per-
turbation spectrum of the zero mode, one has to make
use of the full expression expression (4.10) and may not
use (4.11), given below.
At late times, kt ≫ 1 (t ≥ tout) when the brane moves
slowly, the couplings Mαβ go to zero and particle cre-
ation has come to an end, both functions ONα,k and Oǫα,k
do not contribute to the observable power spectrum after
averaging over several oscillations. Furthermore, the in-
stantaneous particle number then equals the (physically
meaningful) number of created final state gravitons N outα,k
and the KK masses are constant. Consequently, the ob-
servable power spectrum at late times takes the form
P(k, t ≥ tout) =
(2π)3
N outα,k
ωoutα,k
Y2α(a) , (4.11)
where we have used that κ5/L = κ4. Its dependence on
the wave number k is completely determined by the spec-
tral behavior of the number of created gravitons N outα,k .
It is useful to decompose the power spectrum in its zero-
mode and KK-contributions:
P = P0 + PKK . (4.12)
In the late time regime, using Eqs. (4.11) and (4.5), the
zero-mode power spectrum reads
P0(k, t ≥ tout) =
(2π)3
N out0,k . (4.13)
As expected for a usual four-dimensional tensor perturba-
tion (massless graviton), on sub-Hubble scales the power
spectrum decreases with the expansion of the Universe
as 1/a2.
In contrast, the KK mode power spectrum for late times,
given by
PKK(k, t ≥ tout) =
N outn,k
ωoutn,k
Y 21 (mnys),
(4.14)
decreases as 1/a4, i.e. with a factor 1/a2 faster than
P0. The gravity wave power spectrum at late times is
therefore dominated by the zero-mode power spectrum
and looks four dimensional. Contributions to it arising
from five-dimensional effects are scaled away rapidly as
the Universe expands due to the 1/a4 behavior of PKK.
In the limit of large masses mnys ≫ 1, n ≫ 1 and for
wave lengths k ≪ mn such that ωn,k ≃ mn, the late-time
KK-mode power spectrum can be approximated by
PKK(k, t ≥ tout) =
16π2ys
N outn,k (4.15)
where we have inserted Eq. (4.6) for Y2n(a).
Note that the formal summations over the particle num-
ber might be ill defined if the brane trajectory contains
unphysical features like discontinuities in the velocity. An
appropriate regularization is then necessary, for example,
by introducing a physically motivated cutoff.
C. Energy density
For a usual four-dimensional tensor perturbation hµν
on a backgroundmetric gµν an associated effective energy
momentum tensor can be defined unambiguously by (see,
e.g., [14, 56])
Tµν =
〈hαβ‖µhαβ‖ν〉 , (4.16)
where the bracket stands for averaging over several pe-
riods of the wave and “‖” denotes the covariant deriva-
tive with respect to the unperturbed background metric.
The energy density of gravity waves is the 00-component
of the effective energy momentum tensor. We shall use
the same effective energy momentum tensor to calculate
the energy density corresponding to the four-dimensional
spin-2 graviton component of the five-dimensional ten-
sor perturbation on the brane, i.e. for the perturbation
hij(t,x, yb) given by Eq. (4.1). For this it is important
to remember that in our low energy approach, and in
particular at very late times for which we want to cal-
culate the energy density, the conformal time η on the
brane is identical to the conformal bulk time t. The en-
ergy density of four-dimensional spin-2 gravitons on the
brane produced during the brane motion is then given by
[see also [36]]
κ4 a2
0, in| ˙̂hij(t,x, yb) ˙̂hij(t,x, yb)|0, in
. (4.17)
Here the outer bracket denotes averaging over several os-
cillations, which (in contrast to the power spectrum) we
embrace from the very beginning. The factor 1/a2 comes
from the fact that an over-dot indicates the derivative
with respect t. A detailed calculation is carried out in
Appendix C 3 leading to
(2π)3
ωα,kNα,k(t)Y2α(a) (4.18)
where againNα,k(t) is the instantaneous particle number.
At late times t > tout after particle creation has ceased,
the energy density is therefore given by
(2π)3
ωoutα,k N outα,k Y2α(a). (4.19)
This expression looks at first sight very similar to a
“naive” definition of energy density as integration over
momentum space and summation over all quantum num-
bers α of the energy ωoutα,k N outα,k of created gravitons.
(Note that the graviton number N outα,k already contains
the contributions of both polarizations [see Eq. (3.28)].)
However, the important difference is the appearance of
the function Y2α(a) which exhibits a different dependence
on the scale factor for the zero mode compared to the
KK modes.
Let us decompose the energy density into zero-mode and
KK contributions
ρ = ρ0 + ρKK . (4.20)
For the energy density of the massless zero mode one
then obtains
(2π)3
kN out0,k . (4.21)
This is the expected behavior; the energy density of stan-
dard four-dimensional gravitons scales like radiation.
On contrast, the energy density of the KK modes at late
times is found to be
ρKK =
(2π)3
ωoutn,k N outn,k m2nY 21 (mnys),
(4.22)
which decays like 1/a6. As the Universe expands, the en-
ergy density of massive gravitons on the brane is there-
fore rapidly diluted. The total energy density of gravita-
tional waves in our Universe at late times is dominated
by the standard four-dimensional graviton (massless zero
mode). In the large mass limit mnys ≫ 1,n ≫ 1 the KK-
energy density can be approximated by
ρKK ≃
2a6ys
(2π)3
N outn,k ωoutn,kmn . (4.23)
Due to the factor mn coming from the function Y2n, i.e.
from the normalization of the functions φn(t, y), for the
summation over the KK-tower to converge, the number
of produced gravitons N outn,k has to decrease faster than
1/m3n for large masses and not just faster than 1/m
one might naively expect.
D. Escaping of massive gravitons and localization
of gravity
As we have shown, the power spectrum and energy
density of the KK modes scale, at late times when par-
ticle production has ceased, with the expansion of the
Universe like
PKK ∝ 1/a4 , ρKK ∝ 1/a6. (4.24)
Both quantities decay by a factor 1/a2 faster than the
corresponding expressions for the zero-mode graviton. In
particular, the energy density of the KK particles on the
brane behaves effectively like stiff matter. Mathemat-
ically, this difference arises from the distinct behavior
of the functions Y0(a) and Yn(a) [cf. Eq. (4.5)] and is a
direct consequence of the warping of the fifth dimension.
But what is the underlying physics? As we shall discuss
now, this scaling behavior for the KK particles has
indeed a very appealing physical interpretation which is
in the spirit of the RS model.
First, the mass mn is a comoving mass. The (in-
stantaneous) ’comoving’ frequency or energy of a KK
graviton is ωn,k =
k2 +m2n, with comoving wave
number k. The physical mass of a KK mode measured
by an observer on the brane with cosmic time dτ = adt
is therefore mn/a, i.e. the KK masses are redshifted
with the expansion of the Universe. This comes from
the fact that mn is the wave number corresponding to
the y-direction with respect to the bulk time t which
corresponds to conformal time η on the brane and not to
physical time. It implies that the energy of KK particles
on a moving AdS brane is redshifted like that of massless
particles. From this alone one would expect that the
energy density of KK modes on the brane decays like
1/a4 (see also Appendix D of [22]).
Now, let us define the “wave function” for a gravi-
Ψα(t, y) =
φα(t, y)
(4.25)
which, by virtue of (φα, φα) = 1, satisfies
dyΨ2α(t, y) = 1 (4.26)
From the expansion of the gravity wave amplitude
Eq. (2.43) and the normalization condition it is clear that
Ψ2α(t, y) gives the probability to find a graviton of mass
mα for a given (fixed) time t at position y in the Z2-
symmetric AdS-bulk. Since φα satisfies Equation (2.30),
the wave function Ψα satisfies the Schrödinger like equa-
− ∂2yΨα +
Ψα = m
αΨα (4.27)
and the junction conditions (2.29) translate into
Ψα|y={yb,ys} = 0. (4.28)
In Fig. 2 we plot the evolution of Ψ21(t, y) under the
influence of the brane motion Eq. (2.18) with vb = 0.1.
For this motion, the physical brane starting at yb → 0 for
t → −∞ moves towards the static brane, corresponding
to a contracting Universe. After a bounce, it moves
back to the Cauchy horizon, i.e. the Universe expands.
The second brane is placed at ys = 10L and y ranges
from yb(t) to ys. We set Ψ
1 ≡ 0 for y < yb(t) . The
time-dependent KK mass m1 is determined numerically
from Eq. (2.40). As it is evident from this Figure, Ψ21
is effectively localized close to the static brane, i.e. the
weight of the KK-mode wave function lies in the region
of less warping, far from the physical brane. Thus the
probability to find a KK mode is larger in the region
with less warping. Since the effect of the brane motion
on Ψ21 is hardly visible in Fig. 2, we show the behavior
of Ψ21 close to the physical brane in Fig. 3. This shows
that Ψ21 peaks also at the physical brane but with an
amplitude roughly ten times smaller than the amplitude
at the static brane. While the brane, coming from
t → −∞, approaches the point of closest encounter Ψ21
slightly increases and peaks at the bounce t = 0 where,
as we shall show in the next Section, the production
of KK particles takes place. Afterwards, for t → ∞,
when the brane is moving back towards the Cauchy
horizon, the amplitude Ψ21 decreases again and so does
the probability to find a KK particle at the position of
the physical brane, i.e. in our Universe. The parameter
settings used in Figures 2 and 3 are typical parameters
which we use in the numerical simulations described
later on. However, the effect is illustrated much better
if the second brane is closer to the moving brane. In
Figure 4 we show Ψ21 for the same parameters as in
Figures 2 and 3 but now with ys = L. In this case, the
probability to find a KK particle on the physical brane
is of the same order as in the region close to the second
brane during times close to the bounce. However, as the
Universe expands, Ψ21 rapidly decreases at the position
of the physical brane.
From Eqs. (4.2) and (4.5) it follows that Ψ2n(t, yb) ∝ 1/a.
The behavior of the KK-mode wave function suggests
the following interpretation: If KK gravitons are created
on the brane, or equivalently in our Universe, they
escape from the brane into the bulk as the brane moves
back to the Cauchy horizon, i.e. when the Universe
undergoes expansion. This is the reason why the power
spectrum and the energy density imprinted by the KK
modes on the brane decrease faster with the expansion
of the Universe than for the massless zero mode.
The zero mode, on the other hand, is localized at
the position of the moving brane. The profile of φ0 does
not depend on the extra dimension, but the zero-mode
wave function Ψ0 does. Its square is
Ψ20(t, y) =
y2s − y2b
if ys ≫ yb ,
(4.29)
such that on the brane (y = yb) it behaves as
Ψ20(t, yb) ≃
. (4.30)
Equation (4.29) shows that, at any time, the zero
mode is localized at the position of the moving brane.
For a better illustration we show Eq. (4.29) in Fig. 5
for the same parameters as in Fig. 4. This is the
“dynamical analog” of the localization mechanism for
four-dimensional gravity discussed in [7].
To establish contact with [7] and to obtain a intu-
itive physical description, we rewrite the boundary value
problem (4.27), (4.28) as a Schrödinger-like equation
− ∂2yΨα(t, y) + V (y, t)Ψα(y, t) = mα(t)Ψα(y, t) (4.31)
V (y, t) =
yb(t)
δ(|y| − yb(t))
− 3a(t)
δ(|y| − yb(t)) , (4.32)
where we have absorbed the boundary condition at the
moving brane into the (instantaneous) volcano potential
V (y, t) and made use of Z2 symmetry. Similar to the
static case [7], at any time the potential (4.32) supports
a single bound state, the four-dimensional graviton
(4.29), and acts as a barrier for the massive KK modes.
The potential, ensuring localization of four-dimensional
gravity on the brane and the repulsion of KK modes,
moves together with the brane through the fifth dimen-
sion. Note that with the expansion of the Universe, the
“depth of the delta-function” becomes larger, expressing
the fact that the localization of four-dimensional gravity
becomes stronger at late times [cf. Eq. (4.30), Fig. 5].
In summary, the different scaling behavior for the
zero- and KK modes on the brane is entirely a conse-
quence of the geometry of the bulk space-time, i.e. of
the warping L2/y2 of the metric (2.1) 5. It is simply a
manifestation of the localization of gravity on the brane:
as time evolves, the KK gravitons, which are traces of
the five-dimensional nature of gravity, escape into the
bulk and only the zero mode which corresponds to the
usual four-dimensional graviton remains on the brane.
This, and in particular the scaling behavior (4.24),
remains also true if the second brane is removed, i.e. in
the limit ys → ∞, leading to the original RS II model.
By looking at (4.15) and (4.23) one could at first think
that then the KK-power spectrum and energy density
vanish and no traces of the KK gravitons could be
observed on the brane since both expressions behave
as 1/ys. But this is not the case since the spectrum of
KK masses becomes continuous. In the continuum limit
ys → ∞ the summation over the discrete spectrum mn
has to be replaced by an integration over continuous
masses m in the following way:
f(mn) −→
dmf(m) . (4.33)
f is some function depending on the spectrum, for
example f(mn) = N outn,k . The pre-factor 1/ys in (4.15)
and (4.23) therefore ensures the existence of the proper
continuum limit of both expressions.
Another way of seeing this is to repeat the same
calculations but using the eigenfunctions for the case
with only one brane from the beginning. Those are
δ-function normalized and can be found in, e.g., [22].
They are basically the same as (2.34) except that the
normalization is different since it depends on whether
the fifth dimension is compact or not. In particular, on
the brane, they have the same scale factor dependence
as (4.2).
At the end, the behavior found for the KK modes
should not come as a surprise, since the RS II model
has attracted lots of attention because of exactly this;
it localizes usual four-dimensional gravity on the brane.
As we have shown here, localization of standard four-
dimensional gravity on a moving brane via a warped
geometry automatically ensures that the KK modes
escape into the bulk as the Universe expands because
their wave function has its weight in the region of less
warping, resulting in an KK-mode energy density on the
brane which scales like stiff matter.
An immediate consequence of this particular scaling
behavior is that KK gravitons in an AdS braneworld
5 Note that it does not depend on a particular type of brane motion
and is expected to be true also in the high energy case which we
do not consider here.
FIG. 2: Evolution of Ψ21(t, y) = φ
1(t, y)/y
3 corresponding to
the probability to find the first KK graviton at time t at the
position y in the AdS-bulk. The static brane is at ys = 10L
and the maximal brane velocity is given by vb = 0.1.
FIG. 3: Evolution of Ψ21(t, y) as in Fig. 2 but zoomed into
the bulk-region close to the moving brane.
cannot play the role of dark matter. Their energy density
in our Universe decays much faster with the expansion
than that of ordinary matter which is restricted to reside
on the brane.
V. NUMERICAL SIMULATIONS
A. Preliminary remarks
In this section we present results of numerical simula-
tions for the bouncing model described by the equations
(2.17)-(2.19).
In the numerical simulations we set L = 1, i.e. all
FIG. 4: Evolution of Ψ21(t, y) for ys = L and vb = 0.1.
FIG. 5: Localization of four-dimensional gravity on a moving
brane: Evolution of Ψ20(t, y) for ys = L = 1 and vb = 0.1
which should be compared with Ψ21(t, y) shown in Fig. 4.
dimensionful quantities are measured in units of the
AdS5 curvature scale. Starting at initial time tin ≪ 0
where the initial vacuum state |0, in〉 is defined, the
system (3.34,3.35) is evolved numerically up to final time
tout. Thereby we set tin = −2πNin/k with 1 ≤ Nin ∈ N,
such that Θin0,k = 1 [cf. Eq. (3.23)]. This implies
0 (tin) = 2, i.e. independent of the three-dimensional
momentum k a (plane wave) zero-mode solution always
performs a fixed number of oscillations between tin and
the bounce at t = 0 [cf. Eq. (3.38)]. The final graviton
spectrum at N outα,k is calculated at late times tout ≫ 1
when the brane approaches the Cauchy horizon and
graviton creation has ceased. This quantity is physically
well defined and leads to the late-time power spectrum
(4.11) and energy density (4.19) on the brane. For
illustrative purposes, we also plot the instantaneous
particle number Nα,k,•(t) which also determines the
power spectrum at all times [cf Eq.(4.9)]. In this section
we shall use the term particle number respectively
graviton number for both, the instantaneous particle
number Nα,k,•(t) as well as the final state graviton
number N outα,k,•, keeping in mind that only the latter one
is physically meaningful.
There are two physical input parameters for the
numerical simulation; the maximal brane velocity vb (i.e.
tb) and the position of the static brane ys. The latter
determines the number of KK modes which fall within a
particular mass range. On the numerical side one has to
specify Nin and tout, as well as the maximum number of
KK modes nmax which one takes into account, i.e. after
which KK mode the system of differential equations is
truncated. The independence of the numerical results
on the choice of the time parameters is checked and the
convergence of the particle spectrum with increasing
nmax is investigated. More detailed information on
numerical issues including accuracy considerations are
collected in Appendix D.
One strong feature of the brane motion (2.18) is its kink
at the bounce t = 0. In order to study how particle
production depends on the kink, we shall compare
the motion (2.18) with the following motion which
has a smooth transition from contraction to expansion
(L = 1):
yb(t) =
(|t|+ tb − ts)−1 if |t| > ts
a+ (b/2)t2 + (c/4)t4 if |t| ≤ ts
(5.1)
with the new parameter ts in the range 0 < ts < tb. This
motion is constructed such that its velocity at |t| = ts
is the same as the velocity of the kink motion at the
bounce. This will be the important quantity determin-
ing the number of produced gravitons. For ts → 0 the
motion with smooth transition approaches (2.18). The
parameters a, b and c are obtained by matching the mo-
tions and the first and second derivatives. Matching also
the second derivative guarantees that possible spurious
effects contributing to particle production are avoided.
The parameter ts has to be chosen small enough, ts ≪ 1,
such that the maximal velocity of the smooth motion is
not much larger than vb in order to have comparable sit-
uations.
For reasons which will become obvious in the next two
sections we shall discuss the cases of long k ≪ 1 and
short wavelengths k ≫ 1, separately.
B. Generic results and observations for long
wavelengths k ≪ 1
Figure 6 displays the results of a numerical simula-
tion for three-momentum k = 0.01, static brane position
ys = 10 and maximal brane velocity vb = 0.1. Depicted is
FIG. 6: Evolution of the graviton number Nα,k,•(t) for the
zero mode and the first ten KK modes for three-momentum
k = 0.01 and vb = 0.1, ys = 10.
FIG. 7: Nn,k,•(t) for the zero mode and the first ten KK
modes for the parameters of Fig. 6, but without coupling of
the zero mode to the KK modes, i.e. Mi0 ≡ 0.
the graviton number for one polarizationNα,k,•(t) for the
zero mode and the first ten KK modes as well as the evo-
lution of the scale factor a(t) and the position of the phys-
ical brane yb(t). Initial and final times are Nin = 5 and
tout = 2000, respectively. The KK-particle spectrum will
be discussed in detail below. One observes that the zero-
mode particle number increases slightly with the expan-
sion of the Universe towards the bounce at t = 0. Close
to the bounce N0,k,•(t) increases drastically, shows a lo-
cal peak at the bounce and, after a short decrease, grows
again until the mode is sub-horizon (kt ≫ 1). Inside the
horizon N0,k,•(t) is oscillating around a mean value with
diminishing amplitude. This mean value which is reached
asymptotically for t → ∞ corresponds to the number of
generated final state zero-mode gravitonsN out0,k,•. Produc-
tion of KK-mode gravitons takes effectively place only at
the bounce in a step-like manner and the graviton num-
ber remains constant right after the bounce.
In Fig. 7 we show the numerical results obtained for the
same parameters as in Fig. 6 but without coupling of the
zero mode to the KK modes, i.e. Mi0 = 0 (and thus
also Ni0 = N0i = 0). One observes that the production
of zero-mode gravitons is virtually not affected by the
artificial decoupling 6. Note that even if M0j ≡ 0 (see
Eqs. B2), which is in general true for Neumann bound-
ary conditions, the zero mode q0,k,• couples in Eq. (2.49)
to the KK modes via N0j = M00Mj0 and through the
anti-symmetric combination Mαβ −Mβα.
In contrast, the production of the first ten KK modes is
heavily suppressed if Mi0 = 0. The corresponding final-
state graviton numbers N outn,k,• are reduced by four orders
of magnitude. This shows that the coupling to the zero
mode is essential for the production of massive gravitons.
Later we will see that this is true for light KK gravitons
only. If the KK masses exceed mi ∼ 1, they evolve in-
dependently of the four-dimensional graviton and their
evolution is entirely driven by the intermode couplings
Mij . It will also turn out that the time-dependence of
the KK mass mi plays only an inferior role for the gen-
eration of massive KK modes. On the other hand, the
effective decoupling of the evolution of the zero mode
from the KK modes occurs in general as long as k ≪ 1
is satisfied, i.e. for long-wavelengths. We will see that it
is no longer true for short wavelengths k ≫ 1.
The effective decoupling of the zero-mode evolution from
the KK modes makes it possible to derive analytical ex-
pressions for the number of zero-mode gravitons, their
power spectrum and energy density. The calculations
are carried out in section VIA
In summary we emphasize the important observation
that for long wavelengths the amplification of the four
dimensional gravity wave amplitude during the bounce
is not affected by the evolution of the KK gravitons. We
can therefore study the zero mode separately from the
KK modes in this case.
C. Zero mode: long wavelengths k ≪ 1
In Figure 8 we show the numerical results for the num-
ber of generated zero-mode gravitons N0,k,•(t) and the
evolution of the corresponding power spectrum P0(k) on
the brane for momentum k = 0.01, position of the static
brane ys = 10 and maximal brane velocity vb = 0.1. The
results have been obtained by solving the equations for
the zero mode alone, i.e. without the couplings to the KK
modes, since, as we have just shown, the evolution of the
6 Quantitatively it is N0,k,•(t = 2000) = 965.01 with and
N0,k,•(t = 2000) = 965.06 without Mi0. Note that this differ-
ence lies indeed within the accuracy of our numerical simulations
(see Appendix D.)
four-dimensional graviton for long wavelengths is not in-
fluenced by the KK modes. Thereby the power spectrum
is shown before and after averaging over several oscilla-
tions, i.e. employing Eq. (4.9) with and without the term
ON0,k, respectively. Right after the bounce where the gen-
eration of gravitons is initiated and which is responsible
for the peak in N0,k,• at t = 0, the number of gravitons
first decreases again. AfterwardsN0,k,• grows further un-
til the mode enters the horizon at kt = 1. Once on sub-
horizon scales kt ≫ 1, the number of produced gravitons
oscillates with a diminishing amplitude and asymptoti-
cally approaches the final state graviton number N out0,k,•.
During the growth of N0,k,• after the bounce, the power
spectrum remains practically constant. Within the range
of validity it is in good agreement with the analytical pre-
diction (6.22) yielding (L2(2π)3/κ4)P0(k, t) = 4vb(kL)2.
When particle creation has ceased, the full power spec-
trum Eq.(4.8) starts to oscillate with an decreasing am-
plitude. The time-averaged power spectrum obtained
by using Eq. (4.9) without the ON0,k-term is perfectly
in agreement with the analytical expression Eq. (6.20)
which gives (L2(2π)3/κ4)P0(k, t) = 2vb/t2. Note that at
early times, the time-averaged power spectrum behaves
not in the same way as the full one, demonstrating the
importance of the term ON0,k.
Figure 9 shows a summary of numerical results for the
number of created zero-mode gravitons N0,k,•(t) for dif-
ferent values of the three-momentum k. The maximum
velocity at the bounce is vb = 0.1 and the second brane is
at ys = 10. These values are representative. Other values
in accordance with the considered low-energy regime do
not lead to a qualitatively different behavior. Note that
the evolution of the zero mode does virtually not depend
on the value of ys as long as ys ≫ yb(0) (see below). Ini-
tial and final integration times are given by Nin = 5 and
tout = 20000, respectively.
For sub-horizon modes we compare the final graviton
spectra with the analytical prediction (6.17). Both are in
perfect agreement. On super-horizon scales where parti-
cle creation has not ceased yet N0,k,• is independent of
k. The corresponding time-evolution of the power spec-
tra P0(k, t) is depicted in Fig. 10. For the sake of clarity,
only the results for t > 0, i.e. after the bounce, are shown
in both figures.
The numerical simulations and the calculations of sec-
tion VIA reveal that the power spectrum for the four-
dimensional graviton for long wavelengths is blue on
super-horizon scales, as expected for an ekpyrotic sce-
nario.
The analytical calculations performed in section VIA
rely on the assumption that yb ≪ ys and tin → −∞.
Figure 11 shows the behavior of the number of generated
zero-mode gravitons of momentum k = 0.01 in depen-
dence on the inter-brane distance and the initial integra-
tion time. The brane velocity at the bounce is vb = 0.1
which implies that at the bounce the moving brane is
at yb(0) =
vb ≃ 0.316 (L = 1). In case of a close en-
counter of the two branes as for ys = 0.35, the production
FIG. 8: Time evolution of the number of created zero-mode
gravitons N0,k,•(t) and of the zero-mode power spectrum
(4.8): (a) for the entire integration time; (b) for t > 0 only.
Parameters are k = 0.01, ys = 10 and vb = 0.1. Initial and fi-
nal time of integration are given by Nin = 10 and tout = 4000,
respectively. The power spectrum is shown with and without
the term ON0,k,•, i.e. before and after averaging, respectively,
and compared with the analytical results.
FIG. 9: Numerical results for the time evolution of the num-
ber of created zero-mode gravitons N0,k,•(t) after the bounce
t > 0 for different three-momenta k. The maximal brane
velocity at the bounce is vb = 0.1 and the second brane is
positioned at ys = 10. In the final particle spectrum the nu-
merical values are compared with the analytical prediction
Eq. (6.17). Initial and final time of integration are given by
Nin = 5 and tout = 20000, respectively.
of massless gravitons is strongly enhanced compared to
the analytical result. But as soon as ys ≥ 1, (i.e. ys ≥ L)
the numerical result is very well described by the analyt-
ical expression Eq. (6.16) derived under the assumption
ys ≫ yb. For ys ≥ 10 the agreement between both is
very good. From panels (b) and (c) one infers that the
numerical result becomes indeed independent of the ini-
FIG. 10: Evolution of the zero-mode power spectrum after the
bounce t > 0 corresponding to the values and parameters of
Fig. 9. The numerical results are compared to the analytical
predictions Eqs. (6.20) and (6.22).
0 1000 2000
time t
analytical
=0.35
10 100 1000
2nd brane pos. y
0.1 1 10 100 1000
2nd brane pos. y
analytical
(a) (b)
FIG. 11: Dependence of the zero-mode particle number on
inter-brane distance and initial integration time for momen-
tum k = 0.01, maximal brane velocity vb = 0.1 in comparison
with the analytical expression Eq. (6.16). (a) Evolution of the
instantaneous particle number N0,k,•(t) with initial integra-
tion time given by Nin = 5 for ys = 0.35, 0.5 and 1. (b) Final
zero-mode graviton spectrum N0,k,•(tout = 2000) for various
values of ys and Nin. (c) Close-up view of (b) for large ys.
tial integration time when increasing Nin. Note that in
the limit Nin ≫ 1 the numerical result is slightly larger
than the analytical prediction but the difference between
both is negligibly small. This confirms the correctness
and accuracy of the analytical expressions derived in Sec-
tion VIA for the evolution of the zero-mode graviton.
0.01 0.1 1 10
Kaluza-Klein mass m
t nmax=60
= 0.3
= 0.1
 = 0.5●
FIG. 12: Final state KK-graviton spectra for k = 0.001, ys =
100, different maximal brane velocities vb and Nin = 1, tout =
400. The numerical results are compared with the analytical
prediction Eq. (6.34) (dashed line).
D. Kaluza-Klein-modes: long wavelengths k ≪ 1
Because the creation of KK gravitons ceases right
after the bounce [cf Fig. 6] one can stop the numerical
simulation and read out the number of produced KK
gravitons N outn,k,• at times for which the zero mode is still
super-horizon.
Even though Eq. (2.40) cannot be solved analytically,
the KK masses can be approximated by mn ≃ nπ/ys.
This expression is the better the larger the mass.
Consequently, for the massive modes the position of the
second brane ys determines how many KK modes belong
to a particular mass range ∆m.
In Figure 12 we show the KK-graviton spectra N outn,k,• for
three-momentum k = 0.001 and second brane position
ys = 100 for maximal brane velocities vb = 0.1, 0.3
and 0.5. For any velocity vb two spectra obtained
with nmax = 60 and 80 KK modes taken into account
in the simulation are compared to each other. This
reveals that the numerical results are stable up to a
KK mass mn ≃ 1. One infers that first, N outn,k,• grows
with increasing mass until a maximum is reached. The
position of the maximum shifts slightly towards larger
masses with increasing brane velocity vb. Afterwards,
N outn,k,• declines with growing mass. Until the maximum
is reached, the numerical results for the KK-particle
spectrum are very stable. This already indicates that
the KK-intermode couplings mediated by Mij are not
very strong in this mass range. In Figure 13 we show
the final KK-particle spectrum for the same parameters
as in Fig. 12 but for three-momentum k = 0.01 and
0.01 0.1 1 10
Kaluza-Klein mass m
=60, M
=0 for all i,j
 = 0.1
 = 0.5
 = 0.3
 = 0.9
FIG. 13: Final state KK-graviton spectra for k = 0.01, ys =
100, different vb and Nin = 1, tout = 400. The numerical
results are compared with the analytical prediction Eq. (6.34)
(dashed line). For vb = 0.3, 0.5 the spectra obtained without
KK-intermode and self-couplings (Mij ≡ 0 ∀ i, j) are shown
as well.
the additional velocity vb = 0.9
7. We observe the same
qualitative behavior as in Fig. 12. In addition we show
numerical results obtained for vb = 0.3 and 0.5 without
the KK-intermode and self couplings, i.e. we have set
Mij ≡ 0 ∀ i, j by hand. One infers that for KK masses,
depending slightly on the velocity vb but at least up to
mn ≃ 1, the numerical results for the spectra do not
change when the KK-intermode coupling is switched off.
Consequently, the evolution of light, i.e. mn <∼ 1, KK
gravitons is virtually not affected by the KK-intermode
coupling.
In addition we find that also the time-dependence of the
KK masses is not important for the production of light
KK gravitons which is explicitly demonstrated below.
Thus, production of light KK gravitons is driven by
the zero-mode evolution only. This allows us to find
an analytical expression, Eq. (6.34), for the number of
produced light KK gravitons in terms of exponential
integrals. The calculations which are based on several
approximations are performed in Section VIC.
In Figs. 12 and 13 the analytical prediction (6.34) for
the spectrum of final state gravitons has already been
included (dashed lines). Within its range of validity
it is in excellent agreement with the numerical results
obtained by including the full KK-intermode coupling.
It perfectly describes the dependence of N outn,k,• on the
three-momentum k and the maximal velocity vb. For
small velocities vb <∼ 0.1 it is also able to reproduce the
position of the maximum. This reveals that the KK-
7 Such a high brane velocity is of course not consistent with a
Neumann boundary condition Eq. (2.29) at the position of the
moving brane.
intermode coupling is negligible for light KK gravitons
and that their production is entirely driven by their
coupling to the four-dimensional graviton.
The analytical prediction is very precious for testing the
goodness of the parameters used in the simulations, in
particular the initial time tin (respectively Nin). Since it
has been derived for real asymptotic initial conditions,
tin → −∞, its perfect agreement with the numerical
results demonstrates that the values for Nin used in the
numerical simulations are large enough. No spurious
initial effects contaminate the numerical results.
Note, that the numerical values for N outn,k,• in the ex-
amples shown are all smaller than one. However, for
smaller values of k than the ones which we consider here
for purely numerical reasons, the number of generated
KK-mode particles is enhanced since N outn,k,• ∝ 1/k as
can be inferred from Eq. (6.34) in the limit k ≪ mn.
If we go to smaller values of ys, fewer KK modes
belong to a particular mass range. Hence, with the same
or similar number of KK modes as taken into account
in the simulations so far, we can study the behavior
of the final particle spectrum for larger masses. These
simulations shall reveal the asymptotical behavior of
N outn,k,• for mn → ∞ and therefore the behavior of the
total graviton number and energy density. Due to the
kink in the brane motion we cannot expect that the
energy density of produced KK-mode gravitons is finite
when summing over arbitrarily high frequency modes.
Eventually, we will have to introduce a cutoff setting the
scale at which the kink-approximation [cf. Eqs. (2.17)
- (2.19)] is no longer valid. This is the scale where the
effects of the underlying unspecified high-energy physics
which drive the transition from contraction to expansion
become important. The dependence of the final particle
spectrum on the kink will be studied later on in this
section in detail.
In Figures 14 and 15 we show final KK-graviton
spectra for ys = 10 and three-momentum k = 0.01 and
k = 0.1. The analytical expression Eq. (6.34) is depicted
as well and the spectra are always shown for at least
two values of nmax to indicate up to which KK mass
stability of the the numerical results is guaranteed.
Now, only two KK modes are lighter than m = 1. For
these modes the analytical expression Eq. (6.34) is valid
and in excellent agreement with the numerical results, in
particular for small brane velocities vb ∼ 0.1. As before,
the larger the velocity vb the more visible is the effect of
the truncation of the system of differential equations at
nmax.
For k = 0.01 the spectrum seems to follow a power law
decrease right after the maximum in the spectra. In
case of vb = 0.1 the spectrum is numerically stable up to
masses mn ≃ 20. In the region 5 <∼ mn <∼ 20 the spec-
trum is very well fitted by a power law N outn,k,• ∝ m−2.7n .
Also for larger velocities the decline of the spectrum
is given by the same power within the mass ranges
0.1 1 10
Kaluza-Klein mass m
 = 0.5
 = 0.3
 = 0.1
FIG. 14: Final state KK-graviton spectra for k = 0.01, ys =
10, different maximal brane velocities vb and Nin = 2, tout =
400. The numerical results are compared with the analytical
prediction Eq. (6.34) (dashed line).
0.1 1 10
Kaluza - Klein mass m
 = 0.5
 = 0.1
 = 0.3
FIG. 15: Final state KK-graviton spectra for k = 0.1, ys = 10,
different maximal brane velocities vb and Nin = 2, tout =
400. The numerical results are compared with the analytical
prediction Eq. (6.34) (dashed line).
where the spectrum is numerically stable. For k = 0.1,
however, the decreasing spectrum bends over at a mass
around mn ≃ 10 towards a less steep decline. This is
in particular visible in the two cases with vb = 0.1 and
0.3 where the first 100 KK modes have been taken into
account in the simulation. The behavior of the KK-mode
particle spectrum can therefore not be described by a
single power law decline for masses mn > 1. It shows
more complicated features instead, which depend on the
parameters. We shall demonstrate that this bending
over of the decline is related to the coupling properties of
the KK modes and to the kink in the brane motion. But
before we come to a detailed discussion of these issues,
let us briefly confront numerical results of different ys to
FIG. 16: Upper panel: Final state KK-particle spectra for
k = 0.01, vb = 0.1 and different ys = 3, 10, 30 and 100.
The analytical prediction Eq. (6.34) is shown as well (dashed
line). Lower panel: Energy ωoutn,kN
n,k,• of the produced fi-
nal state gravitons binned in mass intervals ∆m = 1 for
ys = 10, 30, 100.
demonstrate a scaling behavior.
In the upper panel of Figures 16 and 17 we com-
pare the final KK-spectra for several positions of the
second brane ys = 3, 10, 30 and 100 obtained for a
maximal brane velocity vb = 0.1 for k = 0.01 and
0.1, respectively. One observes that the shapes of the
spectra are identical. The bending over in the decline
of the spectrum at masses mn ∼ 1 is very well visible
for k = 0.1 and ys = 3, 10. For a given KK mode n
the number of particles produced in this mode is the
larger the smaller ys. But the smaller ys, the less KK
modes belong to a given mass interval ∆m. The energy
transferred into the system by the moving brane, which
is determined by the maximum brane velocity vb, is the
same in all cases. Therefore, the total energy of the pro-
duced final state KK gravitons of a given mass interval
∆m should also be the same, independent of how many
KK modes are contributing to it. This is demonstrated
in the lower panels of Figs. 16 and 17 where the energy
ωoutn,kN outn,k,•(in units of L) of the generated KK gravitons
binned in mass intervals ∆m = 1 is shown 8. One
observes that, as expected, the energy transferred into
the production of KK gravitons of a particular mass
range is the same (within the region where the numerical
results are stable), independent of the number of KK
modes lying in the interval. This is in particular evident
for ys = 30, 100. The discrepancy for ys = 10 is due
to the binning. As we shall discuss below in detail, the
8 The energy for the case ys = 3 is not shown because no KK mode
belongs to the first mass interval.
FIG. 17: Upper panel: Final state KK-particle spectra for
k = 0.1, vb = 0.1 and different ys = 3, 10, 30 and 100.
The analytical prediction Eq. (6.34) is shown as well (dashed
line). Lower panel: Energy ωoutn,kN
n,k,• of the produced fi-
nal state gravitons binned in mass intervals ∆m = 1 for
ys = 10, 30, 100.
particle spectrum can be split into two different parts.
The first part is dominated by the coupling of the zero
mode to the KK modes (as shown above), whereas the
second part is dominated by the KK-intermode couplings
and is virtually independent of the wave number k.
As long as the coupling of the zero mode to the KK
modes is the dominant contribution to KK-particle
production it is N outn,k,• ∝ 1/k [cf. Eq. (6.34)]. Hence,
Eoutn,k,• = ωoutn,kN outn,k,• ∝ 1/k if mn ≫ k. This explains why
the energy per mass interval ∆m is one order larger for
k = 0.01 (cf Fig. 16) than for k = 0.1 (cf Fig. 17) .
Let us now discuss the KK-spectrum for large masses.
The qualitative behavior of the spectrum N outn,k,• and the
mass at which the decline of the spectrum changes are
independent of ys. This is demonstrated in Figure 18
where KK-spectra for vb = 0.1, k = 0.1, ys = 10 [cf
Fig. 15] and ys = 3 [cf Fig. 17] are shown. The results
obtained by taking the full intermode coupling into
account are compared to results of simulations where
we have switched off the coupling of the KK modes to
each other as well as their self-coupling (Mij ≡ 0 ∀ i, j).
Furthermore we display the results for the KK-spectrum
obtained by taking only the KK-intermode couplings
into account, i.e. Mi0 = Mii = 0 ∀ i. One infers that
for the lowest masses the spectra obtained with all
couplings are identical to the ones obtained without
the KK-intermode (Mij = 0, i 6= j) and self-couplings
(Mii = 0). Hence, as already seen before, the primary
source for the production of light KK gravitons is their
coupling to the evolution of the four-dimensional gravi-
ton. In this mass range, the contribution to the particle
creation coming from the KK-intermode couplings is
very much suppressed and negligibly small.
1 10 100
Kaluza-Klein mass m
t full coupling
=0 for all i,j
=0, M
FIG. 18: KK-particle spectra for three-momentum k = 0.1,
maximum brane velocity vb = 0.1 and ys = 3 and 10 with
different couplings taken into account. The dashed lines indi-
cates again the analytical expression Eq. (6.34).
For masses mn ≃ 4 a change in the decline of the
spectrum sets in and the spectrum obtained without
the coupling of the KK modes to the zero mode starts
to diverge from the spectrum computed by taking all
the couplings into account. While the spectrum without
the KK-intermode couplings decreases roughly like a
power law N outn,k,• ∝ m−3n the spectrum corresponding
to the full coupling case changes its slope towards
a power law decline with less power. At this point
the KK-intermode couplings gain importance and the
coupling of the KK modes to the zero mode looses
influence. For a particular mass mc ≃ 9 the spectrum
obtained including the KK-intermode couplings only,
crosses the spectrum calculated by taking into account
exclusively the coupling of the KK modes to the zero
mode. After the crossing, the spectrum obtained by
using only the KK-intermode couplings approaches the
spectrum of the full coupling case. Both agree for large
masses. Thus for large masses mn > mc the production
of KK gravitons is dominated by the couplings of the
KK modes to each other and is not influenced anymore
by the evolution of the four-dimensional graviton. This
crossing defines the transition between the two regimes
mentioned before: for masses mn < mc the production
of KK gravitons takes place due to their coupling to
the zero mode Mi0, while it is entirely caused by the
intermode couplings Mij for masses mn > mc.
Decoupling of the evolution of the KK modes from
the dynamics of the four-dimensional graviton for
large masses implies that KK-spectra obtained for the
same maximal velocity are independent of the three-
momentum k. This is demonstrated in Fig. 19 where we
compare spectra obtained for vb = 0.1 and ys = 3 but
different k. As expected, all spectra converge towards
the same behavior for masses mn > mc.
1 2 3 4 5 10 20 30 1000.3
Kaluza-Klein masses m
t k=0.01
k=0.03
k=0.1
FIG. 19: Comparison of KK-particle spectra for ys = 3,
vb = 0.1 and three-momentum k = 0.01, 0.03, 0.1 and 1
demonstrating the independence of the spectrum on k for
large masses. nmax = 60 KK modes have been taken into
account in the simulations.
1 10 100
Kaluza-Klein mass m
t full coupling
=0 for all i, j
=0, i = j
=0, M
Mαβ = 0  for all α, β
FIG. 20: KK-particle spectra for three-momentum k = 0.1,
maximum brane velocities vb = 0.1 and ys = 3 for nmax = 40
obtained for different coupling combinations.
Figure. 20 shows KK-particle spectra for k = 0.1,vb = 0.1
and ys = 3 obtained for different couplings. This plot
visualizes how each particular coupling combination
contributes to the production of KK gravitons. It
shows, as already mentioned before but not shown
explicitly, that the Mii coupling which is the rate of
change of the corresponding KK mass [cf. Eqs. (2.41)
and (B4)] is not important for the production of KK
gravitons. Switching it off does not affect the final
graviton spectrum. We also show the result obtained
with all couplings but with α+ii(t) = ω
i,k and α
ii (t) = 0,
i.e. the time-dependence of the frequency [cf. Eq. (3.36)]
has been neglected. One observes that in this case the
spectrum for larger masses is quantitatively slightly
0.1 1 10 100
Kaluza-Klein mass m
t full coupling
=0 for all i, j
=0, M
k=0.01
k=0.1
k=0.1
FIG. 21: KK-particle spectra for ys = 10, vb = 0.1, nmax =
100 and three-momentum k = 0.01 and 0.1 with different
couplings taken into account. The thin dashed lines indicates
Eq. (6.34) and the thick dashed line Eq. (5.4).
different but has a identical qualitative behavior. If,
on the other hand, all the couplings are switched off
Mαβ ≡ 0 ∀α, β and only the time-dependence of the
frequency ωi,k is taken into account, the spectrum
changes drastically. Not only the number of produced
gravitons is now orders of magnitude smaller but also
the spectral tilt changes. For large masses it behaves as
Nn,k,• ∝ m−2n . Consequently, the time-dependence of
the graviton frequency itself plays only an inferior role
for production of KK gravitons.
The bottom line is that the main sources of the produc-
tion of KK gravitons is their coupling to the evolution of
the four-dimensional graviton (Mi0) and their couplings
to each other (Mij , i 6= j) for small and large masses,
respectively. Both are caused by the time-dependent
boundary condition. The time-dependence of the oscilla-
tor frequency ωj,k =
m2j(t) + k
2 is virtually irrelevant.
Note that this situation is very different from ordinary
inflation where there are no boundaries and particle
production is due entirely to the time dependence of the
frequency 9.
The behavior of the KK-spectrum, in particular
the mass mc at which the KK-intermode couplings start
to dominate over the coupling of the KK modes to
the zero mode depends only on the three-momentum
k = |k| and the maximal brane velocity vb. This is now
discussed. In Figure 21 we show KK-particle spectra
for ys = 10, vb = 0.1, nmax = 100 and three-momenta
k = 0.01 and 0.1. Again, the spectra obtained by taking
9 Note, however, that the time-dependent KK mass mj(t) enters
the intermode couplings.
0.1 1 10
Kaluza-Klein mass m
t full coupling
=0 for all i, j
=0, M
 = 0.1
 = 0.1
 = 0.3
 = 0.03
FIG. 22: KK-particle spectra for three-momentum k =
0.1,ys = 10 and maximum brane velocities vb = 0.03, 0.1 and
0.3 with nmax = 100. As in Fig. 21 different couplings
have been taken into account and thin dashed lines indicates
Eq. (6.34) and the thick dashed line Eq. (5.4).
all the couplings into account are compared to the case
where only the coupling to the zero mode is switched
on. One observes that for k = 0.01 the spectrum is
dominated by the coupling of the KK modes to the
zero mode up to larger masses than it is the case for
k = 0.1. For k = 0.01 the spectrum obtained taking
into account Mi0 only is identical to the spectrum
obtained with the full coupling up to mn ≃ 10. In case
of k = 0.1 instead, the spectrum is purely zero mode
dominated only up to mn ≃ 5. Hence, the smaller the
three-momentum k the larger is the mass range for
which the KK-intermode coupling is suppressed, and
the coupling of the zero mode to the KK modes is the
dominant source for the production of KK gravitons. As
long as the coupling to the zero mode is the primary
source of particle production, the spectrum declines
with a power law ∝ m−3n . Therefore, in the limiting case
k → 0 when the coupling of the zero mode to the KK
modes dominates particle production also for very large
masses it is N outn≫1,k→0,• ∝ 1/m3n.
Figure 22 shows KK-graviton spectra obtained for
the same parameters as in Fig. 21 but for fixed k = 0.1
and different maximal brane velocities vb. Again, the
spectra obtained by taking all the couplings into account
are compared with the spectra to which only the coupling
of the KK modes to the zero mode contributes. The
mass up to which the spectra obtained with different
couplings are identical changes only slightly with the
maximal brane velocity vb. Therefore, the dependence
of mc on the velocity is rather weak even if vb is changed
by an order of magnitude, but nevertheless evident.
This behavior of the spectrum can indeed be understood
qualitatively. In Section VIC we demonstrate that the
coupling strength of the KK modes to the zero mode
1 10 100
1 10 100
1 10 100
1 10 100
k=0.01 k=0.03
k=0.1 k=1
=25.54 mc=14.11
=8.24 mc=7.31
FIG. 23: KK-particle spectra for three-momentum k = 0.01,
0.03, 0.1 and 1 for ys = 3 and maximum brane velocity
vb = 0.1 with different couplings taken into account where
the notation is like in Fig. 22. From the crossing of the
Mii = Mij = 0- and Mii = Mi0 = 0 results we determine
the k-dependence of mc(k, vb). The thick dashed line indi-
cates Eq. (5.4).
at the bounce t = 0, where production of KK gravitons
takes place, is proportional to
. (5.2)
The larger this term the stronger is the coupling of
the KK modes to the zero mode, and thus the larger
is the mass up to which this coupling dominates over
the KK-intermode couplings. Consequently, the mass
at which the tilt of the KK-particle spectrum changes
depends strongly on the three-momentum k but only
weakly on the maximal brane velocity due to the square
root behavior of the coupling strength. This explains
qualitatively the behavior obtained from the numerical
simulations.
An approximate expression for mc(k, vb) can be
obtained from the numerical simulations. In Figure 23
we depict the KK-particle spectra for three-momentum
k = 0.01, 0.03, 0.1 and 1 for ys = 3 and maximum brane
velocity vb = 0.1 with different couplings taken into
account. The legend is as in Fig. 22. From the crossings
of the Mij = 0, i 6= j and Mii = Mi0 = 0 results one
can determine the k-dependence of mc. Note that the
spectra are not numerically stable for large masses, but
they are stable in the range where mc lies [cf., e.g.,
Fig. 25, for k = 0.1]. Using the data for k = 0.01, 0.03
and 0.1 one finds mc(k, vb) ∝ 1/
In Fig. 24 KK-graviton spectra are displayed for
k = 0.1, ys = 3 and maximal brane velocities
vb = 0.3, 0.2, 0.1, 0.08, 0.05 and 0.03 with different
couplings taken into account. It is in principle possible
to determine the vb-dependence of mc from the crossings
1 10 100
1 10 100
1 10 100
 = 0.3
=9.50
 = 0.2 v
 = 0.1
 = 0.08
=8.24
 = 0.05 v
 = 0.03
=9.04
=8.06 m
=7.72 m
=7.52
FIG. 24: KK-graviton spectra for three-momentum k =
0.1, ys = 3 and maximum brane velocities vb =
0.3, 0.2, 0.1, 0.08, 0.05 and 0.03 with different couplings taken
into account where the notation is like in Fig. 22. From the
crossing of the Mii = Mij = 0- and Mii = Mi0 = 0 results we
determine the vb-dependence of mc.
of the Mij = 0, i 6= j- and Mii = Mi0 = 0 results as
done for the k-dependence. However, the values for mc
displayed in the Figures indicate that the dependence
of mc on vb is very weak. From the given data it is not
possible to obtain a good fitting formula (as a simple
power law) for the vb-dependence of mc. (In the range
0.1 ≤ vb ≤ 0.3 a very good fit is mc = 1.12πv0.13b /
The reason is twofold. First of all, given the complicated
coupling structure, it is a priori not clear that a simple
power law dependence exists. Recall that also the ana-
lytical expression for the particle number Eq. (6.34) has
not a simple power law velocity dependence. Moreover,
for the number of modes taken into account (nmax = 40)
the numerical results are not stable enough to resolve
the weak dependence of mc on vb with a high enough
accuracy. (But it is good enough to perfectly resolve
the k-dependence.) The reason for the slow convergence
of the numerics will become clear below. As we shall
see, the corresponding energy density is dominated by
masses much larger than mc. Consequently the weak
dependence of mc on vb is not very important in that
respect and therefore does not need to be determined
more precisely. However, combining all the data we can
give as a fair approximation
mc(k, vb) ≃
π vαb
, with α ≃ 0.1. (5.3)
Taking α = 0.13 for 0.1 ≤ vb ≤ 0.3 and α = 0.08 for
0.03 ≤ vb ≤ 0.1 fits the given data reasonably well.
As we have seen, as long as the zero mode is the
dominant source of KK-particle production, the final
KK-graviton spectrum can be approximated by a power
law decrease m−3n . We can combine the presented
numerical results to obtain a fitting formula valid in this
regime:
N outn≫1,k≪1,• =
(Lmn)3
, for
< mn < mc. (5.4)
This fitting formula is shown in Figs. 21 22 and 23
and is in reasonable good agreement with the numerical
results. Since Eq. (5.4) together with (5.3) is an impor-
tant result, we have reintroduced dimensions, i.e. the
AdS scale L which is set to one in the simulations, in
both expressions.
Let us now investigate the slope of the KK-graviton
spectrum for masses mn → ∞ since it determines the
contribution of the heavy KK modes to the energy den-
sity. In Figure 25 we show KK-graviton spectra obtained
for three-momentum k = 0.1, second brane position
ys = 3 and maximal brane velocities vb = 0.01, 0.03
and 0.1. Up to nmax = 100 KK modes have been taken
into account in the simulations. One immediately is
confronted with the observation that the convergence
of the KK-graviton spectra for large mn is very slow.
This is since those modes, which are decoupled from the
evolution of the four-dimensional graviton, are strongly
affected by the kink in the brane motion. Recall that the
production of light KK gravitons with masses mn ≪ mc
is virtually driven entirely by the evolution of the
massless mode. Those light modes are not so sensitive
to the discontinuity in the velocity of the brane motion.
To be more precise, their primary source of excitation is
the evolution of the four-dimensional graviton but not
the kink which, as we shall discuss now, is responsible
for the production of heavy KK gravitons mn ≫ mc.
A discontinuity in the velocity will always lead to a di-
vergent total particle number. Arbitrary high frequency
modes are excited by the kink since the acceleration
diverges there. Due to the excitation of KK gravitons
of arbitrarily high masses, one cannot expect that the
numerical simulations show a satisfactory convergence
behavior which allows to determine the slope by fit-
ting the data. However, it is nevertheless possible to
give a quantitative expression for the behavior of the
KK-graviton spectrum for large masses. The studies of
the usual dynamical Casimir effect on a time-dependent
interval are very useful for this purpose.
For the usual dynamical Casimir effect it has been
shown analytically that a discontinuity in the velocity
will lead to a divergent particle number [57, 58]. In
Appendix E we discuss in detail the model of a massless
real scalar field on a time-dependent interval [0, y(t)] for
the boundary motion y(t) = y0 + v t with v = const,
and present numerical results for final particle spectra
(Fig. 34). For this motion it was shown in [58] that the
particle spectrum behaves as ∝ v2/ωn where ωn = nπ/y0
is the frequency of a massless scalar particle. This di-
vergent behavior is due to the discontinuities in the
velocity when the motion is switched on and off, and are
responsible for the slow convergence of the numerical
1 2 3 4 5 10 20 30 100
Kaluza-Klein mass m
t nmax = 40
 = 60
 = 80
=0.03
=0.01
FIG. 25: KK-particle spectra for k = 0.1, ys = 3 and max-
imal brane velocities vb = 0.01, 0.03, 0.1 up to KK masses
mn ≃ 100 compared with an 1/mn decline. The dashed lines
indicate the approximate expression (5.6) which describes the
asymptotic behavior of the final KK-particle spectra reason-
ably well, in particular for vb < 0.1.
results shown in Fig. 34 for this scenario.
At the kink in the brane-motion the total change of the
velocity is 2vb, similar to the case for the linear motion
where the discontinuous change of the velocity is 2v.
Consequently we may conclude that for large KK masses
mn ≫ mc for which the evolution of the KK modes is no
longer affected by their coupling to the four-dimensional
graviton the KK-graviton spectrum behaves as 10
N outn,k,• ∝
for mn ≫ mc . (5.5)
If we assume that the spectrum declines like 1/mn and
use that the numerical results for masses mn ≃ 20 are
virtually stable one finds N outn,k,• ∝ v2.08b /mn which de-
scribes the asymptotics of the numerical results well.
As for the dynamical Casimir effect for a uniform motion
discussed in Appendix E [cf. Fig. 34], the slow conver-
gence of the numerical results towards the 1/mn behav-
ior is well visible for large masses mn ≫ mc which do no
longer couple to the four-dimensional graviton. This is a
strong indication for the statement that the final gravi-
ton spectrum for large masses behaves indeed like (5.5).
It is therefore possible to give a single simple expression
for the final KK-particle spectrum for large masses which
10 Note that the discussion in Appendix E refers to Dirichlet bound-
ary conditions. For Neumann boundary conditions considered
here, the zero mode and its asymmetric coupling play certainly a
particular role. However, as we have shown, for large masses only
the KK-intermode couplings are important. Consequently, there
is no reason to expect that the qualitative behavior of the spec-
trum for large masses depends on the particular kind of boundary
condition.
-100 -50 0 50 100
time t
-100 -50 0 50 100
time t
1 10 50
KK mass m
1 10 50
KK mass m
=40 n
FIG. 26: Evolution of the zero-mode particle numberN0,k,•(t)
and final KK-graviton spectra N outn,k,• for ys = 3, maximal
brane velocity vb = 0.1 and three-momenta k = 10 and 30.
The dashed line in the upper plots indicate Eq. (6.17) (divided
by two) demonstrating the value of the number of produced
zero-mode gravitons without coupling to the KK modes.
comprises all the features of the spectrum even quantita-
tively reasonably well [cf. dashed lines in Fig. 25]
N outn,k,• ≃ 0.2
ωoutn,k ys
for mn ≫ mc . (5.6)
The 1/ys-dependence is compelling. It follows imme-
diately from the considerations on the energy and the
scaling behavior discussed above [cf. Figs. 16 and 17].
For completeness we now write 1/ωoutn,k instead of the KK
mass mn only, since what matters is the total energy
of a mode. Throughout this section this has not been
important since we considered only k ≪ 1 such that
ωoutn,k becomes independent of k for large masses mn ≫ k
[cf. Fig. 19].
E. Short wavelengths k ≫ 1
For short wave lengths k ≫ 1 (short compared to the
AdS-curvature scale L set to one in the simulations) a
completely new and very interesting effect appears. The
behavior of the four-dimensional graviton mode changes
drastically. We find that the zero mode now couples
to the KK gravitons and no longer evolves virtually
independently of the KK modes, in contrast to the
behavior for long wavelengths.
In Fig. 26 we show the evolution of the zero-mode
graviton number N0,k,•(t) and final KK-graviton spectra
N outn,k,• for ys = 3, maximal brane velocity vb = 0.1
and three-momenta k = 10 and 30. One observes that
the evolution of the four-dimensional graviton depends
on the number of KK modes nmax taken into account,
i.e. the zero mode couples to the KK gravitons. For
-100 -50 0 50 100
time t
0 10 20 30 40
0.02/(k-1.8)
analytical
FIG. 27: 4D-graviton number N0,k,•(t) for k = 3, 5, 10, 20 and
30 with ys = 3 and maximal brane velocity vb = 0.1. The
small plot shows the final graviton spectrum N out
0,k,• together
with a fit to the inverse law a/(k + b) [dashed line] and the
analytical fitting formula Eq. (6.23) [solid line]. For k = 10
and 30 the corresponding KK-graviton spectra are shown in
Fig. 26.
k = 10 the first 60 KK modes have to be included in the
simulation in order to obtain a numerically stable result
for the zero mode. In the case of k = 30 one already
needs nmax ≃ 100 in order to achieve numerical stability
for the zero mode.
Figure 27 displays the time-evolution of the number
of produced zero-mode gravitons N0,k,•(t) for ys = 3
and vb = 0.1. For large k the production of massless
gravitons takes place only at the bounce since these
short wavelength modes are sub-horizon right after
the bounce. Corresponding KK-particle spectra for
k = 10, 30 are depicted in Figs. 26 and 28. The insert
in Fig. 27 shows the resulting final four-dimensional
graviton spectrum N out0,k,•, which is very well fitted
by an inverse power law N out0,k,• = 0.02/(k − 1.8) 11.
Consequently, for k ≫ 1 the zero-mode particle number
N out0,k,• declines like 1/k only, in contrast to the 1/k2
behavior found for k ≪ 1.
The dependence of N out0,k,• on the maximal brane
velocity vb also changes. In Fig. 28 we show N0,k,•(t)
together with the corresponding KK-graviton spectra for
ys = 3, k = 5 and 10 in each case for different vb. Using
nmax = 60 KK modes in the simulations guarantees
numerical stability for the zero mode. The velocity
dependence of N out0,k,• is not given by a simple power law
as it is the case for k ≪ 1. This is not very surprising
11 The momenta k = 5, 10, 20, 30 and 40 have been used to obtain
the fit. Fitting the spectrum for k = 20, 30 and 40 to a power
law gives N out
0,k,•
∝ k−1.1.
-100 -50 0 50 100
time t
-100 -50 0 50 100
time t
1 10 100
KK mass m
1 10 100
KK mass m
 = 0.03
 = 0.05
 = 0.1
 = 0.03
 = 0.05
 = 0.1
 = 0.3
 = 0.1
 = 0.03
 = 0.03
 = 0.1
 = 0.3
FIG. 28: Zero-mode particle number N0,k,•(t) and corre-
sponding final KK-particle spectraN outn,k,• for ys = 3, k = 5, 10
and different maximal brane velocities vb. nmax = 60 guaran-
tees numerically stable solutions for the zero mode.
since now the zero mode couples strongly to the KK
modes [cf. Fig. 26]. For k = 10, for example, one finds
N out0,k,• ∝ v1.4b if vb <∼ 0.1.
As in the long wavelengths case, the zero-mode
particle number does not depend on the position of the
static brane ys even though the zero mode now couples
to the KK modes. This is demonstrated in Fig. 29 where
the evolution of the zero-mode particle number N0,k,•(t)
and the corresponding KK-graviton spectra with k = 10,
vb = 0.1 for the two values ys = 3 and 10 are shown.
One needs nmax = 60 for ys = 30 in order to obtain a
stable result for the zero mode which is not sufficient in
the case ys = 10. Only for nmax ≃ 120 the zero-mode
solution approaches the stable result which is identical
to the result obtained for ys = 3.
What is important is not the number of the KK modes
the four-dimensional graviton couples to, but rather
a particular mass mzm ≃ k. The zero mode couples
to all KK modes of masses below mzm no matter how
many KK modes are lighter. Recall that the value of ys
just determines how many KK modes belong to a given
mass interval ∆m since, roughly, mn ≃ nπ/ys. The
KK-spectra for k ≥ 1 show the same scaling behavior as
demonstrated for long wavelengths in Figs. 16 and 17.
The production of four-dimensional gravitons of
short wavelengths takes place on the expense of the
KK modes. In Fig. 30 we show the numerical results
for the final KK-particle spectra with vb = 0.1, ys = 3
and k = 3, 5, 10 and 30 obtained for different coupling
combinations. These spectra should be compared with
those shown in Fig. 23 for the long wavelengths case. For
k >∼ 10 the number of the produced lightest KK gravi-
tons is smaller in the full coupling case compared to the
situation where only the KK-intermode coupling is taken
-100 -50 0 50 100
time t
-100 -50 0 50 100
time t
1 10 50
KK mass m
1 10 50
KK mass m
FIG. 29: Zero-mode particle numberN0,k•(t) and correspond-
ing KK-graviton spectra for k = 10, vb = 0.1 and 2nd brane
positions ys = 3 and 10.
1 10 100
KK-mass m
1 10 100
KK-mass m
k=3 k=5
k=10 k=30
FIG. 30: Final KK-particle spectra N outn,k,• for vb = 0.1, ys = 3
and k = 3, 5, 10 and 30 and different couplings. Circles cor-
respond to the full coupling case, squares indicate the results
if Mij = Mii = 0, i.e. no KK-intermode couplings and dia-
monds correspond to Mi0 = 0, i.e. no coupling of KK modes
to the zero mode.
into account. In case k = 30, for instance, the numbers
of produced gravitons for the first four KK modes are
smaller for the full coupling case. This indicates that
the lightest KK modes couple strongly to the zero mode.
Their evolution is damped and graviton production in
those modes is suppressed. The production of zero-mode
gravitons on the other hand is enhanced compared to
the long wavelengths case. For short wavelengths, the
evolution of the KK modes therefore contributes to
the production of zero-mode gravitons. This may be
interpreted as creation of zero-mode gravitons out of
KK-mode vacuum fluctuations.
As in the long wavelengths case, the KK-particle
spectrum becomes independent of k if mn ≫ k and
10 20 100 200
frequency ω
k=40, n
k=30, n
k=20, n
k=10, n
k=5, n
FIG. 31: Final KK-particle spectra N outn,k,• for vb = 0.1,
ys = 3 and k = 5, 10, 20, 30 and 40. The dashed lines in-
dicate Eq. (6.35) for k = 10, 20, 30 and 40. For k ≥ 20, the
simple analytical expression (6.35) agrees quite well with the
numerical results.
the evolution of the KK modes is dominated by the
KK-intermode coupling. This is visible in Fig. 30 for
k = 3 and 5. Also the bend in the spectrum when the
KK-intermode coupling starts to dominate is observable.
For k = 10 and 30 this regime with mn ≫ k is not
reached.
As we have shown before, in the regime mn ≫ k
the KK-particle spectrum behaves as 1/ωoutn,k which will
dominate the energy density of produced KK gravitons.
If 1 ≪ mn <∼ k, however, the zero mode couples to
the KK modes and the KK-graviton spectrum does
not decay like 1/ωoutn,k. This is demonstrated in Fig. 31
where the number of produced final state gravitons
N outn,k,• is plotted as function of their frequency ωoutn,k for
parameters vb = 0.1, ys = 3 and k = 5, 10, 20, 30 and 40.
While for k = 5 the KK-intermode coupling dominates
for large masses [cf. Fig. 30] leading to a bending over
in the spectrum and eventually to an 1/ωoutn,k-decay, the
spectra for k = 20, 30 and 40 show a different behavior.
All the modes are still coupled to the zero mode leading
to a power-law decrease ∝ 1/(ωoutn,k)α with α ≃ 2. The
case k = 10 corresponds to an intermediate regime.
Also shown is the simple analytical expression given in
Eq. (6.35) which describes the spectra reasonably well
for large k (dashed line).
The KK-particle spectra in the region 1 ≪ mn <∼ k will
also contribute to energy density since the cutoff scale is
the same for the integration over k and the summation
over the KK-tower (see Section VID below).
1 2 3 4 5 10 20 30 100
Kaluza-Klein mass m
=0 (kink)
=0.005
=0.015
=0.05
 = 2.2 x 10
 exp(-0.1315 m
FIG. 32: KK-particle spectrum for ys = 3, vb = 0.1 and
k = 0.1 for the bouncing as well as smooth motions with
ts = 0.005, 0.015, and 0.05 to demonstrate the influence of the
bounce. nmax = 60 KK modes have been taken into account
in the simulations and the result for the kink motion is shown
as well.
F. A smooth transition
Let us finally investigate how the KK-graviton spec-
trum changes when the kink-motion (2.18) is replaced by
the smooth motion (5.1). In Fig. 32 we show the numeri-
cal results for the final KK-graviton spectrum for ys = 3,
vb = 0.1 and k = 0.1 for the smooth motion (5.1) with
ts = 0.05, 0.015 and 0.005. nmax = 60 modes have been
taken into account in the simulation and the results are
compared to the spectrum obtained with the kink-motion
(2.18). The parameter ts defines the scale Ls ≃ 2ts at
which the kink is smoothed, i.e. Ls corresponds to the
width of the transition from contraction to expansion.
The numerical results reveal that KK gravitons of masses
smaller than ms ≃ 1/Ls are not affected, but the pro-
duction of KK particles of masses larger than ms is
exponentially suppressed. This is in particular evident
for ts = 0.05 where the particle spectrum for masses
mn > 10 has been fitted to a exponential decrease. Going
to smaller values of ts, the suppression of KK-mode pro-
duction sets in for larger masses. For the example with
ts = 0.005 the KK-particle spectrum is identical to the
one obtained with the kink-motion within the depicted
mass range. In this case the exponential suppression of
particle production sets in only for masses mn > 100.
Note that the exponential decay of the spectrum for the
smooth transition from contraction to expansions also
shows that no additional spurious effects due to the dis-
continuities in the velocity when switching the brane dy-
namics on and off occur. Consequently, tin and tout are
appropriately chosen.
VI. ANALYTICAL CALCULATIONS AND
ESTIMATES
A. The zero mode: long wavelengths k ≪ 1/L
The numerical simulations show that the evolution of
the zero mode at large wavelengths is not affected by the
KK modes. To find an analytical approximation to the
numerical result for the zero mode, we neglect all the
couplings of the KK modes to the zero mode by setting
Mij = 0 ∀ i, j and keeping M00 only. Then only the evo-
lution equation for ǫ
0 ≡ δα0 ǫ is important; it decouples
and reduces to
ǫ̈+ [k2 + V(t)]ǫ = 0 , (6.1)
with “potential”
V = Ṁ00 −M200 . (6.2)
The corresponding vacuum initial conditions are
[cf. Eqs. (3.21), (3.22); here we do not consider the unim-
portant phase]
ǫ = 1 , lim
ǫ̇ = −ik. (6.3)
A brief calculation using the expression for M00 (cf. Ap-
pendix B) leads to
V = y
y2s − y2b
3y2b − 2y2s
y2s − y2b
(6.4)
= − y
y2s − y2b
y2s − y2b
. (6.5)
If one assumes that the static brane is much further away
from the Cauchy horizon than the physical brane, ys ≫
yb, it is simply
V = −H2 − Ḣ , (6.6)
and one recovers Eq. (2.50).
For the particular scale factor (2.17) one obtains
H = ȧ
sgn(t)
|t|+ tb
and (6.7)
Ḣ = 2δ(t)
(|t|+ tb)2
(6.8)
such that
Ḣ +H2 = 2δ(t)
. (6.9)
The δ-function in the last equation models the bounce.
Without the bounce, i.e. for an eternally radiation dom-
inated dynamics, one has V = 0 and the evolution equa-
tion for ǫ would be trivial. With the bounce, the potential
is just a delta-function potential with “height” propor-
tional to −2√vb/L
V = −
δ(t) , (6.10)
where vb is given in Eq (2.20). Equation (6.1) with poten-
tial (6.10) can be considered as a Schrödinger equation
with δ-function potential. Its solution is a classical text-
book problem.
Since the approximated potential V vanishes for all t < 0
one has, with the initial condition (6.3),
ǫ(t) = e−ikt , t < 0 . (6.11)
Assuming continuity of ǫ through t = 0 and integrating
the differential equation over a small interval t ∈ [0−, 0+]
around t = 0 gives
(6.12)
= ǫ̇(0+)− ǫ̇(0−)−
ǫ(0) . (6.13)
The jump of the derivative ǫ̇ at t = 0 leads to parti-
cle creation. Using ǫ(0+) = ǫ(0) = ǫ(0−) and ǫ̇(0+) =
ǫ̇(0−) +
ǫ(0) as initial conditions for the solution for
t > 0, one obtains
ǫ(t) = Ae−ikt +Beikt , t > 0 (6.14)
A = 1 + i
, B = −i
. (6.15)
The Bogoliubov coefficient B00 after the bounce is then
given by
B00(t ≥ 0) =
e−ikt
1 + i
ǫ(t)− i
ǫ̇(t)
(6.16)
where we have used that M00 = −H if ys ≫ yb. At
this point the importance of the coupling matrix M00
becomes obvious. Even though the solution ǫ to the dif-
ferential equation (6.1) is a plane wave right after the
bounce, |B00(t)|2 is not a constant due to the motion of
the brane itself. Only once the mode is inside the hori-
zon, i.e. H/k ≪ 1, |B00(t)|2 is constant and the number
of generated final state gravitons (for both polarizations)
is given by
N out0,k = 2|B00(kt ≫ 1)|2 = 2
|ǫ|2 + |ǫ̇|
(kL)2
(6.17)
where we have used that the Wronskian of ǫ, ǫ∗ is 2ik.
As illustrated in Fig. 9 the expression (6.17) is indeed in
excellent agreement with the (full) numerical results, not
only in its k-dependence but also the amplitude agrees
without any fudge factor. The evolution of the four-
dimensional graviton mode and the associated genera-
tion of massless gravitons with momentum k < 1/L can
therefore be understood analytically.
Note that the approximation employed here is only valid
if y2s − yb(0)2 ≫ yb(0)2. In the opposite limit, if ∆y ≡
ys − yb(0) ≪ yb(0) one can also derive an analytical ap-
proximation along the same lines. For k ≤ 1/∆y one
obtains instead of Eq. (6.17)
N out0,k =
2(k∆y)2
, (6.18)
if ∆y ≡ ys − yb(0) ≪ yb(0) , k∆y <∼ 1 .
In order to calculate the energy density, we have to take
into account that the approximation of an exactly ra-
diation dominated Universe with an instant transition
breaks down on small scales. We assume this break
down to occur at the string scale Ls, much smaller than
L [cf. Eqs. (2.14),(2.15)]. Ls is the true width of the
transition from collapse to expansion, which we have set
to zero in the treatment. Modes with mode numbers
k ≫ (2π)/Ls will not ’feel’ the potential and are not
generated. We therefore choose kmax = (2π)/Ls as the
cutoff scale. Then, with Eq (4.21), one obtains for the
energy density
2 π2a4
∫ 2π/Ls
dkk3N0,k . (6.19)
For small wave numbers, k < 1/L, we can use the above
analytical result for the zero-mode particle number.
However, as the numerical simulations have revealed, as
soon as k >∼ 1/L, the coupling of the four-dimensional
graviton to the KK modes becomes important and for
large wave numbers N out0,k decays only like 1/k. Hence
the integral (6.19) is entirely dominated by the upper
cutoff. The contributions from long wavelengths to the
energy density are negligible.
For the power spectrum, on the other hand, we
are interested in cosmologically large scales, 1/k ≃
several Mpc or more, but not in short wavelengths
kL ≫ 1 dominating the energy density. Inserting the
expression for the number of produced long wavelength
gravitons (6.17) into (4.11), the gravity wave power
spectrum at late times becomes
P0(k) =
(2π)3
for kt ≫ 1. (6.20)
This is the asymptotic power spectrum, when ǫ starts
oscillating, hence inside the Hubble horizon, kt ≫ 1.
On super Hubble scales, kt ≪ 1 when the asymptotic
out-state of the zero mode is not yet reached, one may
use Eq. (4.10) with
R0,k(t) =
|ǫ(t)|2 − 1
≃ 4vba
. (6.21)
For the ≃ sign we assume t ≫ L and t ≫ tb so that
one may neglect terms of order t/L in comparison to√
vb(t/L)
2. We have also approximated a = (t+ tb)/L ≃
t/L. Inserting this in Eq. (4.8) yields
P0(k) =
2 , kt ≪ 1 . (6.22)
Both expressions (6.20) and (6.22) are in very good
agreement with the corresponding numerical results, see
Figs. 9, 10 and 11.
B. The zero mode: short wavelengths k ≫ 1/L
As we have demonstrated with the numerical analysis,
as soon as k >∼ 1/L, the coupling of the zero mode to
the KK modes becomes important, and for large wave
numbers N out0,k,• ∝ 1/k. We obtain a good asymptotic
behavior for the four-dimensional graviton spectrum if
we set
N out0,k,• ≃
5(kL)
. (6.23)
This function and Eq. (6.17) (divided by two for one po-
larization) meet at kL = 5. Even though the approxi-
mation is not good in the intermediate regime it is very
reasonable for large k [cf. Fig. 27].
Inserting this approximation into Eq (6.19) for the energy
density, one finds that the integral is dominated entirely
by the upper cutoff, i.e. by the blue, high energy modes:
. (6.24)
The power spectrum associated with the short wave-
lengths k ≫ 1/L is not of interest since the gravity wave
spectrum is measured on cosmologically large scales only,
k ≪ 1/L.
C. Light Kaluza-Klein modes and long wavelengths
k ≪ 1/L
The numerics indicates that light (mn < 1) long wave-
length KK modes become excited mainly due to their
coupling to the zero mode. Let us take only this cou-
pling into account and neglect also the time-dependence
of the frequency, setting ωn,k(t) ≡ ωoutn,k = ωinn,k since it
plays an inferior role as shown by the numerics.
The Bogoliubov coefficients are then determined by the
equations
ξ̇n,k + iω
n,kξn,k =
2ωoutn,k
Sn(t; k) (6.25)
η̇n,k − iωoutn,kηn,k = −
2ωoutn,k
Sn(t; k) (6.26)
with the “source”
Sn(t; k) = (ξ0 − η0)Mn0 . (6.27)
We have defined ξn,k ≡ ξ(0)n,k, ηn,k ≡ η
n,k, ξ0 ≡ ξ
0,k, and
η0 ≡ η(0)0,k. This source is known, since the evolution of
the four-dimensional graviton is know. From the result
for ǫ above and the definition of ξ0 and η0 in terms of ǫ
and ǫ̇ one obtains
ξ0 − η0 =
−ik + 1|t|+ tb
e−itk , t < 0 (6.28)
ξ0 − η0 = 2
1− iktb
k2tb(t+ tb)
e−itk
k2tb(t+ tb)
eitk , t > 0 .(6.29)
Furthermore, if ys ≫ yb, one has [cf. Eq. (B3)]
Mn0 = 2
Y1(mnys)2
Y1(mnyb)2 − Y1(mnys)2
. (6.30)
Assuming ysmn ≫ 1 and ybmn ≪ 1 one can expand the
Bessel functions and arrives at
Mn0 ≃
ẏb = −
πmnL2
L sgn(t)
(|t|+ tb)2
To determine the number of created final state gravi-
tons we only need to calculate ηn,k [cf. Eq. (3.32) with
∆+n,k(|t| → ∞) = 1 and ∆
n,k(|t| → ∞) = 0],
N outn,k,• = |B0n,k(tout)|2 =
ωoutn,k
|ηn,k|2 (6.31)
The vacuum initial conditions require limt→−∞ ηn,k = 0
so that ηn,k is given by the particular solution
ηn,k(t) =
ωoutn,k
′; k)e−it
′ωoutn,kdt′ , (6.32)
and therefore
N outn,k,• =
4ωoutn,k
Sn(t; k)e
−itωout
n,kdt
(6.33)
where the integration range has been extended from −∞
to +∞ since the source is very localized around the
bounce. This integral can be solved exactly. A some-
what lengthy but straight forward calculation gives
N outn,k,• =
πm5nL
2ωoutn,kkys
∣∣∣2iRe
+k)tbE1(i(ω
n,k + k)tb)
+(ktb)
−1ei(ω
n,k−k)tbE1(i(ω
n,k − k)tb)
−ei(ω
+k)tbE1(i(ω
n,k + k)tb)
. (6.34)
Here E1 is the exponential integral, E1(z) ≡∫∞
t−1e−tdt . This function is holomorphic in the com-
plex plane with a cut along the negative real axis, and
the above expression is therefore well defined. Note that
this expression does not give rise to a simple dependence
of N outn,k on the velocity vb = (L/tb)2. In the preceding
section we have seen that, within its range of validity,
Eq. (6.34) is in excellent agreement with the numerical
results (cf., for instance, Figs. 12 and 13).
As already mentioned before, this excellent agreement
between the numerics and the analytical approximation
demonstrates that the numerical results are not contam-
inated by any spurious effects.
D. Kaluza-Klein modes: asymptotic behavior and
energy density
The numerical simulations show that the asymptotic
KK-graviton spectra (i.e. for masses mn ≫ 1) decay
like 1/ωoutn,k if mn ≫ k and like
1/ωoutn,k
with α ≃ 2
if mn <∼ k. The corresponding energy density on the
brane is given by the summation of Eq. (4.23) over all
KK modes up to the cutoff. Since the mass mn is simply
the momentum into the extra dimension, it is plausible
to choose the same cutoff scale for both, the k-integral
and the summation over the KK modes, namely 2π/Ls.
The main contribution to the four-dimensional particle
density and energy density comes from mn ∼ 2π/Ls and
k ∼ 2π/Ls, i.e. the blue end of the spectrum.
The large-frequency behavior of the final KK-spectrum
can be approximated by
N outn,k,• ≃
0.2v2b


ωoutn,k
if 1/L <∼ k <∼ mn
2(α−1)/2
(ωoutn,k)
if mn <∼ k <∼ 2π/Ls
(6.35)
with α ≃ 2 which is particularly good for large k. Both
expression match at mn = k and are indicated in Figures
25 and 31 as dashed lines. Given the complicated
coupling structure of the problem and the multitude of
features visible in the particle spectra, these compact
expressions describe the numerical results reasonable
well for all parameters. The deviation from the numer-
ical results is at most a factor of two. This accuracy
is sufficient in order to obtain a useful expression for
the energy density from which bounds on the involved
energy scales can be derived.
The energy density on the brane associated with
the KK gravitons is given by [cf. Eq. (4.23)]
ρKK ≃
πa6ys
dkk2 N outn,k,• ωoutn,k mn . (6.36)
Splitting the momentum integration into two integrations
from 0 to mn and mn to the cutoff 2π/Ls, and replacing
the sum over the KK masses by an integral one obtains
ρKK ≃ C(α)
π5v2b
. (6.37)
The power α in Eq. (6.35) enters the final result for the
energy density only through the pre-factor C(α) which is
of order unity.
VII. DISCUSSION
The numerical simulations have revealed many inter-
esting effects related to the interplay between the evolu-
tion of the four-dimensional graviton and the KK modes.
All features observed in the numerical results have been
interpreted entirely on physical grounds and many of
them are supported by analytical calculations and ar-
guments. Having summarized the results for the power
spectrum and energy densities in the preceding section,
we are now in the position to discuss the significance of
these findings for brane cosmology.
A. The zero mode
For the zero-mode power-spectrum we have found that
P0(k) =
k2 if kt ≪ 1
(La)−2 if kt ≫ 1 . (7.1)
Therefore, the gravity wave spectrum on large, super
Hubble scales is blue with spectral tilt
nT = 2 , (7.2)
a common feature of ekpyrotic and pre-big-bang models.
The amplitude of perturbations on scales at which
fluctuations of the Cosmic Microwave Background
(CMB) are observed is of the order of (H0/mPl)
2, i.e.
very suppressed on scales relevant for the anisotropies
of the CMB. The fluctuations induced by these Casimir
gravitons are much too small to leave any observable
imprint on the CMB.
For the zero-mode energy density at late times,
kt ≫ 1, we have obtained [cf Eq. (6.24)]
ρh0 ≃
. (7.3)
In this section we denote the energy density of the
zero mode by ρh0 in order not to confuse it with the
12 Note that even the transition from the summation over the KK-
tower to an integration according to (4.33) “eats up” the 1/ys
term in (6.36), the final energy density (6.37) depends on ys since
it explicitly enters the particle number.
present density of the Universe. Recall that Ls is the
scale at which our kinky approximation (2.17) of the
scale factor breaks down, i.e. the width of the bounce.
If this width is taken to zero, the energy density of
gravitons is very blue and diverges. This is not so
surprising, since the kink in a(t) leads to the generation
of gravitons of arbitrary high energies. However, as the
numerical simulations have shown, when we smooth the
kink at some scale Ls, the production of modes with
energies larger than ≃ 1/Ls is exponentially suppressed
[cf. Fig. 32]. This justifies the introduction of Ls as a
cutoff scale.
In the following we shall determine the density pa-
rameter of the generated gravitons today and compare
it to the Nucleosynthesis bound. For this we need the
quantities ab given in Eq (2.20) and
Here ab is the minimal scale factor andHb is the maximal
Hubble parameter, i.e. the Hubble parameter right after
the bounce. (Recall that in the low energy approximation
t = η.) During the radiation era, curvature and/or a cos-
mological constant can be neglected so that the density
ρrad =
a−4 =
. (7.4)
In order to determine the density parameter of the gen-
erated gravitons today, i.e., at t = t0, we use
Ωh0 =
ρh0(t0)
ρcrit(t0)
ρh0(t0)
ρrad(t0)
ρrad(t0)
ρcrit(t0)
ρh0(t0)
ρrad(t0)
Ωrad.
(7.5)
The second factor Ωrad is the present radiation density
parameter. For the factor ρh0/ρrad, which is time inde-
pendent since both ρh0 and ρrad scale like 1/a
4, we insert
the above results and obtain
Ωh0 =
Ωrad =
Ωrad (7.6)
Ωrad . (7.7)
The nucleosynthesis bound [14] requests that
Ωh0 <∼ 0.1Ωrad , (7.8)
which translates into the relation
(LPl/Ls)
(L/Ls) <∼ 0.1 (7.9)
which, at first sight, relates the different scales involved.
But since we have chosen the cutoff scale Ls to be
the higher-dimensional fundamental scale (string scale),
Equation (7.9) reduces to
vb <∼ 0.2 (7.10)
by virtue of Equation (2.15). All one has to require to
be consistent with the nucleosynthesis bound is a small
brane velocity which justifies the low energy approach.
In all, we conclude that the model is not severely con-
strained by the zero mode. This result itself is remark-
able. If there would be no coupling of the zero mode
to the KK modes for small wavelengths the number of
produced high energy zero-mode gravitons would behave
as ∝ k−2 as it is the case for long wavelengths. The
production of high energy zero-mode gravitons from KK
gravitons enhances the total energy density by a factor of
about L/Ls. Without this enhancement, the nucleosyn-
thesis bound would not lead to any meaningful constraint
and would not even require vb < 1.
B. The KK modes
As derived above, the energy density of KK gravitons
on the brane is dominated by the high energy gravitons
and can be approximated by [cf. Eq. (6.37)]
ρKK ≃
π5v2b
. (7.11)
Let us evaluate the constraint induced from the require-
ment that the KK-energy density on the brane be smaller
than the radiation density ρKK(t) < ρrad(t) at all times.
If this is not satisfied, back-reaction cannot be neglected
and our results are no longer valid. Clearly, at early times
this condition is more stringent than at late times since
ρKK decays faster then ρrad. Inserting the value of the
scale factor directly after the bounce where the produc-
tion of KK gravitons takes place, a−2b = vb, one finds,
using again the RS fine tuning condition (2.15),
≃ 100 v3b
. (7.12)
If we use the largest value for the brane velocity vb ad-
mitted by the nucleosynthesis bound vb ≃ 0.2 and re-
quire that ρKK/ρrad be (much) smaller than one for back-
reaction effects to be negligible, we obtain the very strin-
gent condition
. (7.13)
Let us first discuss the largest allowed value for L ≃
0.1mm. The RS-fine tuning condition (2.15) then deter-
mines Ls = (LL
1/3 ≃ 10−22 mm ≃ 1/(106 TeV). In
this case the brane tension is T = 6κ4/κ25 = 6L2Pl/L6s =
6/(LL3s) ∼ (10TeV)4. Furthermore, we have (L/Ls)2 ≃
1042 so that ys > L(L/Ls)
2 ≃ 1041mm ≃ 3 × 1015Mpc,
which is about 12 orders of magnitude larger than the
present Hubble scale. Also, since yb(t) ≪ L in the low
energy regime, and ys ≫ L according to the inequality
(7.13), the physical brane and the static brane are very
far apart at all times. Note that the distance between
the physical and the static brane is
dy = L log(ys/yb) >∼ L ≫ Ls .
This situation is probably not very realistic. Some high
energy, stringy effects are needed to provoke the bounce
and one expects these to be relevant only when the
branes are sufficiently close, i.e. at a distance of order
Ls. But in this case the constraint (7.13) will be violated
which implies that back-reaction will be relevant.
On the other hand, if one wants that ys ≃ L and
back-reaction to be unimportant, then Eq. (7.12) implies
that the bounce velocity has to be exceedingly small,
vb <∼ 10−15.
A way out of this conclusion is to assume that the
brane distance at the bounce, ∆y = ys − yb(0), becomes
of the order of the cutoff Ls or smaller. Then the pro-
duction of KK gravitons is suppressed. However, then
the approximation (6.18) has to be used to determine
the energy density of zero-mode gravitons which then
becomes
ρh0 ≃
(Ls∆y)
Setting ∆y ≃ Ls, the nucleosynthesis bound,
ρh0 <∼ 0.1ρrad, then yields the much more stringent limit
on the brane velocity,
v2b <
. (7.14)
One might hope to find a way out of these conclusions
by allowing the bounce to happen in the high energy
regime. But then vb ≃ 1 and the nucleosynthesis bound
is violated since too many zero-mode gravitons are pro-
duced. Even if one disregards this limit for a moment,
saying that the calculation presented here only applies in
the low energy regime, vb ≪ 1, the modification coming
from the high energy regime are not expected to allevi-
ate the bounds. In the high energy regime one may of
course have yb(t) ≫ L and therefore the physical brane
can approach the static brane arbitrarily closely without
the latter having to violate (7.13). Those results suggest
that even in the scenario of a bounce at low energies, the
back reaction from KK gravitons has to be taken into
account. But this does not need to exclude the model.
VIII. CONCLUSIONS
We have studied the evolution of tensor perturbations
in braneworld cosmology using the techniques developed
for the standard dynamical Casimir effect. A model
consisting of a moving and a fixed 3-brane embedded in
a five-dimensional static AdS bulk has been considered.
Applying the dynamical Casimir effect formulation
to the study of tensor perturbations in braneworld
cosmology represents an interesting alternative to
other approaches existing in the literature so far and
provides a new perspective on the problem. The explicit
use of coupling matrices allows us to obtain detailed
information about the effects of the intermode couplings
generated by the time-dependent boundary conditions,
i.e. the brane motion.
Based on the expansion of the tensor perturbations
in instantaneous eigenfunctions, we have introduced a
consistent quantum mechanical formulation of graviton
production by a moving brane. Observable quantities
like the power spectrum and energy density can be
directly deduced from quantum mechanical expectation
values, in particular the number of gravitons created
from vacuum fluctuations. The most surprising and
at the same time most interesting fact which this
approach has revealed is that the energy density of the
massive gravitons decays like 1/a6 with the expansion
of the Universe. This is a direct consequence of the
localization of gravity: five-dimensional aspects of it,
like the KK gravitons, become less and less ’visible’
on the brane with the expansion of the Universe. The
1/a6-scaling behavior remains valid also when the fixed
brane is sent off to infinity and one ends up with
a single braneworld in AdS, like in the original RS II
scenario. Consequently, KK gravitons on a brane moving
through an AdS bulk cannot play the role of dark matter.
As an explicit example, we have studied graviton
production in a generic, ekpyrotic-inspired model of
two branes bouncing at low energies, assuming that
the energy density on the moving brane is dominated
by a radiation component. The numerical results have
revealed a multitude of interesting effects.
For long wavelengths kL ≪ 1 the zero mode evolves
virtually independently of the KK modes. zero-mode
gravitons are generated by the self coupling of the zero
mode to the moving brane. For the number of produced
massless gravitons we have found the simple analytical
expression 2vb/(kL). These long wavelength modes
are the once which are of interest for the gravitational
wave power spectrum. As one expects for an ekpyrotic
scenario, the power spectrum is blue on super-horizon
scales with spectral tilt nT = 2. Hence, the spectrum
of these Casimir gravitons has much too little power
on large scales to affect the fluctuations of the cosmic
microwave background.
The situation changes completely for short wavelengths
kL ≫ 1. In this wavelength range, the evolution of the
zero mode couples strongly to the KK modes. Produc-
tion of zero-mode gravitons takes place on the expense of
KK-graviton production. The numerical simulation have
revealed that the number of produced short-wavelength
massless gravitons is given by 2vb/(5kL). It decays
only like 1/k instead of the 1/k2-behavior found for
long wavelengths. These short wavelength gravitons
dominate the energy density. Comparing the energy
density with the nucleosynthesis bound and taking the
cutoff scale to be the string scale Ls, we have shown that
the model is not constrained by the zero mode. As long
as vb <∼ 0.2, i.e. a low energy bounce, the nucleosynthesis
bound is not violated.
More stringent bounds on the model come from
the KK modes. Their energy density is dominated by
the high energy modes which are produced due to the
kink which models the transition from contraction to
expansion. Imposing the reasonable requirement that
the energy density of the KK modes on the brane be
(much) smaller than the radiation density at all times in
order for back reaction effects to be negligible, has led to
two cases. On the one hand, allowing the largest values
for the AdS curvature scale L ≃ 0.1mm and the bounce
velocity vb ≃ 0.2, back reaction can only be neglected
if the fixed brane is very far away from the physical
brane ys ∼ 1041mm. As we have argued, this is not very
realistic since some high energy, stringy effects provoking
the bounce are expected to be relevant only when the
branes are sufficiently close, i.e. ys ∼ Ls. On the other
hand, by only requiring that ys ≃ L ≫ Ls, the bounce
velocity has already to be exceedingly small, vb <∼ 10−15,
for back reaction to be unimportant. Therefore, one of
the main conclusions to take away from this work is that
back reaction of massive gravitons has to be taken into
account for a realistic bounce.
Many of the results presented here are based on
numerical calculations. However, since the used ap-
proach provides the possibility to artificially switch
on and off the mode couplings, we were able identify
the primary sources driving the time evolution of the
perturbations in different wavelength and KK mass
ranges. This has allowed us to understand many of the
features observed in the numerical results on analytical
grounds.
On the other hand, it is fair to say that most of the
presented results rely on the low energy approach, i.e. on
the approximation of the junction condition (generalized
Neumann boundary condition) by a Neumann boundary
condition. Even though we have given arguments for
the goodness of this approximation, it has eventually
to be confirmed by calculations which take the exact
boundary condition into account. This is the subject of
future work.
Acknowledgment
We thank Cyril Cartier who participated in the early
stages of this work and Kazuya Koyama and David Lan-
glois for discussions. We are grateful for the use of the
’Myrinet’-cluster of Geneva University on which most of
the quite intensive numerical computations have been
performed. This work is supported by the Swiss National
Science Foundation.
APPENDIX A: VARIATION OF THE ACTION
Let us consider the variation of the action (2.27) with
respect to h•. It is sufficient to study the action for a
fixed wave number k and polarization •
Sh•(k) =
yb(t)
|∂th•|2 − |∂yh•|2 − k2|h•|2
and we omit the normalization factor L3/κ5 as well as
the factor two related to Z2 symmetry. The variation of
(A1) reads
δSh•(k) =
yb(t)
(∂th•)(∂tδh
•) (A2)
−(∂yh•)(∂yδh∗•)− k2h•δh∗•
+ h.c. .
Here, T denotes a time interval within the variation is
performed and it is assumed in the following that the
variation vanishes at the boundaries of the time interval
T . Performing partial integrations and demanding that
the variation of the action vanishes leads to
0 = (A3)
yb(t)
− ∂2t h• + y3
− k2h•
[(v∂t + ∂y)h•] δh
•|yb(t) −
(∂yh•)δh
with v = dyb(t)/dt. The first term in curly brackets is the
wave operator (2.24). In order for h• to satisfy the free
wave equation (perturbation equation) (2.24) the term
in curly brackets in the second integral has to vanish.
Allowing for an evolution of h• on the branes, i.e. in
general δh•|brane 6= 0, enforces the boundary conditions
(v∂t + ∂y)h•|yb(t) = 0 and ∂yh•|ys = 0 , (A4)
hence, the junction condition (2.26). Consequently, any
other boundary conditions than (A4) are not compati-
ble with the free perturbation equation (2.24) under the
influence of a moving brane (provided δh• 6= 0 at the
branes).
APPENDIX B: COUPLING MATRICES
The use of several identities of Bessel functions leads
M00 = ŷb
y2s − y2b
, (B1)
M0j = 0 , (B2)
Mi0 =
φ0 = ŷb
y2s − y2b
, (B3)
Mii = m̂i , (B4)
Mij = M
ij +M
ij (B5)
MAij = (ŷb + m̂i)yb
2m2iNiNj
m2j −m2i
× (B6)
× [ys C2(mjys)J1(miys)− yb C2(mjyb)J1(miyb)]
where
J1(mi y) = [J2(miyb)Y1(miy)− Y2(miyb)J1(miy)]
MNij = NiNjmim̂i
dyy2C1(miy)C2(mjy). (B8)
This integral has to be solved numerically. Note that,
because of the boundary conditions, one has the identity
dyy2C1(miy)C2(mjy) = −
dyy2C1(miy)C0(mjy).
Furthermore, one can simplify
J1(mi yb) =
πmiyb
, J1(mi ys) =
πmiyb
Y1(miys)
Y1(miyb)
(B10)
where the limiting value has to be taken for the last term
whenever Y1(miyb) = Y1(miys) = 0.
APPENDIX C: ON POWER SPECTRUM AND
ENERGY DENSITY CALCULATION
1. Instantaneous vacuum
In Section III the in - out state approach to particle
creation has been presented. The definitions of the in
- and out- vacuum states Eq. (3.9) are unique and the
particle concept is well defined and meaningful.
If we interpret tout as a continuous time variable t,
we can write the Bogoliubov transformation Eq. (3.24)
âα,k,•(t) =
Aβα,k(t)âinβ,k,• + B∗βα,k(t)â
β,−k,•
where at any time we have introduced a set of operators
{âα,k•(t), â†α,k,•(t)}. Vacuum states defined at any time
can be associated with these operators via
âα,k,•(t)|0, t〉 = 0 ∀ α,k • . (C2)
Similar to Eq. (3.11) a ”particle number” can be intro-
duced through
Nα,k(t) =
〈0, in|â†α,k•(t)âα,k,•(t)|0, in〉
|Bβα,k(t)|2 . (C3)
We shall denote |0, t〉 as the instantaneous vacuum state
and the quantity Nα,k(t) as instantaneous particle num-
ber 13. However, even if we call it ”particle number” and
plot it in section V for illustrative reasons, we consider
only the particle definitions for the initial and final state
(asymptotic regions) outlined in section III as physically
meaningful.
2. Power spectrum
In order to calculate the power spectrum Eq. (4.7) we
need to evaluate the expectation value
(t, yb,k)ĥ
(t, yb,k
′)〉in = (C4)
φα(t, yb)φα′ (t, yb)〈q̂α,k,•(t)q̂†α′,k′,•(t)〉in
where we have introduced the shortcut 〈...〉in =
〈0, in|...|0, in〉. Using the expansion (3.15) of q̂α′,k′,•(t)
in initial state operators and complex functions ǫ
α,k(t)
one finds
〈q̂α,k,•(t)q̂†α′,k′,•(t)〉in =
α,k(t) ǫ
α′,k (t)
2ωinβ,k
δ(3)(k− k′).
From the initial conditions (3.21) it follows that the sum
in (C4) diverges at t = tin. This divergence is related to
the usual normal ordering problem and can be removed
by a subtraction scheme. However, in order to obtain a
well defined power spectrum at all times, it is not suffi-
cient just to subtract the term (1/2)(δαα′/ω
α,k)δ
(3)(k −
′) which corresponds to 〈q̂α,k,•(tin)q̂†α′,k′,•(tin)〉in in the
above expression. In order to identify all terms contained
in the power spectrum we use the instantaneous particle
concept which allows us to treat the Bogoliubov coeffi-
cients (3.25) and (3.26) as continuous functions of time.
First we express the complex functions ǫ
α,k in (C5) in
terms of Aγα,k(t) and Bγα,k(t). This is of course equiv-
alent to calculating the expectation value (C5) using
[cf. Eq.(3.7)]
q̂α,k,•(t) =
2ωα,k(t)
âα,k,•(t)Θα,k(t)
α,−k,•(t)Θ
α,k(t)
and the Bogoliubov transformation Eq. (C1). The result
consists of terms involving the Bogoliubov coefficients
and the factor (1/2)(δαα′/ωα,k(t))δ
(3)(k − k′), leading
potentially to a divergence at all times. This term cor-
responds to 〈0, t|q̂α,k,•(t)q̂†α′,k′,•(t)|0, t〉, and is related to
13 It could be interpreted as the number of particles which would
have been created if the motion of the boundary (the brane)
stops at time t.
the normal ordering problem (zero-point energy) with re-
spect to the instantaneous vacuum state |0, t〉. It can be
removed by the subtraction scheme
〈q̂α,k,•(t)q̂†α′,k′,•(t)〉in,phys (C7)
= 〈q̂α,k,•(t)q̂†α′,k′,•(t)〉in − 〈0, t|q̂α,k,•(t)q̂
α′,k′,•(t)|0, t〉
where we use the subscript “phys” to denote the physi-
cally meaningful expectation value.
Inserting this expectation value into (C4), and using
Eq. (4.2), we find
〈ĥ•(t, yb,k)ĥ•(t, yb,k′)〉in (C8)
Rα,k(t)Y2α(a)δ(3)(k− k′)
with Rα,k(t) defined in Eq. (4.9). The function ONα,k
appearing in Eq. (4.9) is explicitely given by
ONα,k = 2ℜ
Θ2α,kAβα,kB∗βα,k +Θα,k
α′ 6=α
ωα′,k
Yα′ (a)
Yα(a)
Θ∗α′,kB∗βαBβα′ +Θα′,kAβαB∗βα′
and Oǫα,k appearing in Eq. (4.10) reads
Oǫα,k =
β,α′ 6=α
Yα′(a)
Yα(a)
ωinβ,k
. (C10)
3. Energy density
In order to calculate the energy density we need to
evaluate the expectation value 〈 ˙̂hij(t,x, yb) ˙̂hij(t,x, yb)〉in.
Using (2.22) and the relation e•ij(−k) = (e•ij(k))∗ we ob-
〈 ˙̂hij(t,x, yb) ˙̂hij(t,x, yb)〉in =
(2π)3/2
(2π)3/2
(C11)
× 〈 ˙̂h
(t, yb,k)
′(t, yb,k
′)〉inei(k−k
′)·xe•ij(k)
′ ij(k′)
By means of the expansion (3.17) the expectation value
〈 ˙̂h
(t, yb,k)
′(t, yb,k
′)〉in becomes
〈 ˙̂h
(t, yb,k)
′(t, yb,k
′)〉in (C12)
〈p̂α,k,•(t)p̂†α′,k′,•′(t)〉inφα(t, yb)φα′ (t, yb).
From the definition of p̂α,k,•(t) in Eq. (3.18) it is clear
that this expectation value will in general contain terms
proportional to the coupling matrix and its square when
expressed in terms of ǫ
α,k. However, we are interested in
the expectation value at late times only when the brane
moves very slowly such that the mode couplings go to
zero and a physical meaningful particle definition can be
given. In this case we can set
〈p̂α,k,•(t)p̂†α′,k′,•′(t)〉in =
˙̂qα,k,•(t) ˙̂q
α′,k′,•′(t)
.(C13)
Calculating this expectation value by using Eq. (3.15)
leads to an expression which, as for the power spec-
trum calculation before, has a divergent part related to
the zero-point energy of the instantaneous vacuum state
(normal ordering problem). We remove this part by a
subtraction scheme similar to Eq (C7). The final result
reads
〈 ˙̂qα,k,•(t) ˙̂q†α′,k′,•′(t)〉in,phys (C14)
α,k(t)ǫ̇
α′,k′(t)√
ωinβ,kω
− ωα,k(t)δαα′
′δ(3)(k− k′).
Inserting this result into Eq. (C12), splitting the summa-
tions in sums over α = α′ and α 6= α′ and neglecting the
oscillating α 6= α′ contributions (averaging over several
oscillations), leads to
〈 ˙̂h
(t, yb,k)
′(t, yb,k
′)〉in (C15)
Kα,k(t)Y2α(a)δ••′δ(3)(k− k′)
where the function Kα,k(t) is given by
Kα,k(t) =
|ǫ̇(β)α,k(t)|2
ωinβ,k
− ωα,k(t) = ωα,k(t)Nα,k(t) ,
(C16)
and we have made use of Eq. (4.2). The relation be-
tween
β |ǫ̇
α,k(t)|2/ωinβ,k and the number of created par-
ticles can easily be established. Using this expression in
Eq. (C11) leads eventually to
〈 ˙̂hij(t,x, yb) ˙̂hij(t,x, yb)〉in (C17)
(2π)3
Kα,k(t)Y2α(a)
where we have used that the polarization tensors satisfy
e•ij(k)
e• ij(k)
= 2. (C18)
The final expression for the energy density Eq. (4.18) is
then obtained by exploiting that κ5/L = κ4.
APPENDIX D: NUMERICS
In order to calculate the number of produced gravitons
the system of coupled differential equations (3.34) and
(3.35) is solved numerically. The complex functions ξ
α,k are decomposed into their real and imaginary parts:
α,k = u
α,k + iv
α,k , η
α,k = x
α,k + iy
α,k. (D1)
The system of coupled differential equations can then be
written in the form (cf. Eq. (A2) of [16])
k (t) = Wk(t)X
k (t) (D2)
where
0,k ...u
nmax,k
0,k ...x
nmax,k
0,k ...v
nmax,k
0,k ...y
nmax,k
The matrix Wk(t) is given by Eq. (A4) of [16] but
here indices start at zero. The number of produced
gravitons can be calculated directly from the solutions
to this system using Eqs. (3.28) and (3.32). Note
that for a given truncation parameter nmax the above
system of size 4(nmax + 1) × 4(nmax + 1) has to be
solved nmax + 1 - times, each time with different initial
conditions (3.38).
The main difficulty in the numerical simulations is that
most of the entries of the matrix Wk(t) [Eq. (A4) of
[16]] are not known analytically. This is due to the fact
that Eq. (2.40) which determines the time-dependent
KK masses mi(t) does not have an (exact) analytical
solution. Only the 00-component of the coupling matrix
Mαβ is known analytically. We therefore have to deter-
mine the time-dependent KK-spectrum {mi(t)}nmaxi=1 by
solving Eq. (2.40) numerically. In addition, also the part
MNij [Eq. (B8)] has to be calculated numerically since
the integral over the particular combination of Bessel
functions can not be found analytically.
We numerically evaluate the KK-spectrum and the
integral MNij for discrete time-values ti and use spline
routines to assemble Wk(t). The system (D2) can
then be solved using standard routines. We chose the
distribution of the ti’s in a non-uniform way. A more
dense mesh close to the bounce and a less dense mesh at
early and late times. The independence of the numerical
results on the distribution of the ti’s is checked. In order
to implement the bounce as realistic as possible, we do
not spline the KK-spectrum very close to the bounce
but re-calculate it numerically at every time t needed in
the differential equation solver. This minimizes possible
artificial effects caused by using a spline in the direct
vicinity of the bounce. The same was done for MNij but
we found that splining MNij when propagating through
the bounce does not affect the numerical results.
Routines provided by the GNU Scientific Library (GSL)
[59] have been employed. Different routines for root
finding and integration have been compared. The code
has been parallelized (MPI) in order to deal with the
1 10 100
KK-mass m
1 10 100
KK-mass m
FIG. 33: Comparison of the final KK-graviton spectrum
n,k,• with the expression dn,k(tout) describing to what accu-
racy the diagonal part of the Bogoliubov relation (D4) is sat-
isfied. Left panel: ys = 3, k = 0.1, vb = 0.03 and nmax = 100
[cf. Fig. 25]. Right panel: ys = 3, k = 30, vb = 0.1 and
nmax = 100 [cf. Fig. 26].
intensive numerical computations.
The accuracy of the numerical simulations can be
assessed by checking the validity of the Bogoliubov
relations
Aβα,k(t)A∗βγ,k(t)− B∗βα,k(t)Bβγ,k(t)
= δαγ (D4)
Aβα,k(t)B∗βγ,k(t)− B∗βα,k(t)Aβγ,k(t)
= 0. (D5)
In the following we demonstrate the accuracy of the nu-
merical simulations by considering the diagonal part of
(D4). The deviation of the quantity
dα,k(t) = 1−
|Aβα,k(t)|2 − |Bβα,k(t)|2
from zero gives a measure for the accuracy of the
numerical result. We consider this quantity at final
times tout and compare it with the corresponding final
particle spectrum. In Fig. 33 we compare the final KK-
graviton spectrum N outn,k,• with the expression dn,k(tout)
for two different cases. This shows that the accuracy
of the numerical simulations is very good. Even if the
expectation value for the particle number is only of order
10−7 to 10−6, the deviation of dn,k(tout) from zero is at
least one order of magnitude smaller. This demonstrates
the reliability of our numerical simulations and that we
can trust the numerical results presented in this work.
APPENDIX E: DYNAMICAL CASIMIR EFFECT
FOR A UNIFORM MOTION
We consider a real massless scalar field on a time-
dependent interval [0, y(t)]. The time evolution of its
mode functions are described by a system of differential
equations like (2.49) where the specific form of Mαβ de-
pends on the particular boundary condition the field is
subject to. In [15, 17] a method has been introduced to
study particle creation due to the motion of the boundary
y(t) (i.e. the dynamical Casimir effect) fully numerically.
We refer the reader to these publications for further de-
tails.
If the boundary undergoes a uniform motion y(t) = 1+vt
(in units of some reference length) it was shown in [57, 58]
that the total number of created scalar particles diverges,
caused by the discontinuities in the velocity at the begin-
ning and the end of the motion. In particular, for Dirich-
let boundary conditions (no zero mode), it was found in
[58] that 〈0, in|N̂outn |0, in〉 ∝ v2/n if n > 6 and v ≪ 1.
Thereby in- and out- vacuum states are defined like in
the present work and the frequency of a mode function
is given by ωn = π n , n = 1, 2, ... . In Figure 34 we
show spectra of created scalar particles obtained numer-
ically with the method of [17] for this particular case.
One observes that, as for our bouncing motion, the con-
vergence is very slow since the discontinuities in the ve-
locity lead to the excitation of arbitrary high frequency
modes. Nevertheless, it is evident from Fig. 34 that the
numerically calculated spectra approach the analytical
prediction. The linear motion discussed here and the
brane-motion (2.18) are very similar with respect to the
discontinuities in the velocity. In both cases, the total
discontinuous change of the velocity is 2v and 2vb, re-
spectively. The resulting divergence of the acceleration
is responsible for the excitation and therefore creation of
particles of all frequency modes. Consequently we ex-
pect the same ∝ v2/ωn behavior for the bouncing mo-
tion (2.18). Indeed, comparing the convergence behavior
of the final graviton spectrum for vb = 0.01 shown in
Fig. 25 with the one of the scalar particle spectrum for
v = 0.01 depicted in Fig. 34 shows that both are very
similar.
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