Curvature perturbations from ekpyrotic collapse with
multiple fields
Kazuya Koyama† 1 Shuntaro Mizuno‡ 2 and David Wands† 3
† Institute of Cosmology and Gravitation, Mercantile House, University of Portsmouth,
Portsmouth PO1 2EG, United Kingdom
‡ Research Center for the Early Universe (RESCEU), School of Science, University of
Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan
Abstract
A scale-invariant spectrum of isocurvature perturbations is generated during col-
lapse in the ekpyrotic scaling solution in models where multiple fields have steep neg-
ative exponential potentials. The scale invariance of the spectrum is realized by a
tachyonic instability in the isocurvature field. This instability drives the scaling solu-
tion to the late time attractor that is the old ekpyrotic collapse dominated by a single
field. We show that the transition from the scaling solution to the single field domi-
nated ekpyrotic collapse automatically converts the initial isocurvature perturbations
about the scaling solution to comoving curvature perturbations about the late-time
attractor. The final amplitude of the comoving curvature perturbation is determined
by the Hubble scale at the transition.
1 Introduction
The existence of an almost scale-invariant spectrum of primordial curvature perturbations on
large scales is one of the most important observations that any model for the early universe
should explain. An inflationary expansion in the very early universe is most commonly
assumed to achieve this, but it is important to consider whether there is any alternative
model. In this paper we focus on the ekpyrotic scenario as an alternative [1] (see also [2, 3]).
In the old ekpyrotic scenario, the large scale perturbations are supposed to be generated
during a collapse driven by a single scalar field with a steep negative exponential potential.
It was shown that the Newtonian potential acquires a scale-invariant spectrum, but the
comoving curvature perturbation has a steep blue spectrum [4]. In this scenario we need a
mechanism to convert contraction to expansion, and for a regular four-dimensional bounce,
the scale-invariant Newtonian potential is matched to the decaying mode in an expanding
universe, and the growing mode of curvature perturbations acquires a steep blue spectrum
[5, 6]. It has been suggested that this conclusion might be altered by allowing a singular
matching between collapse and expanion [7], but the general rule that the comoving curvature
perturbation remains constant still holds for adiabatic perturbations on large scales [8]. In
1 E-mail: Kazuya.Koyama@port.ac.uk
2 E-mail: mizuno@resceu.s.u-tokyo.ac.jp
3E-mail: David.Wands@port.ac.uk
http://arxiv.org/abs/0704.1152v3
a braneworld context, the conversion from contraction to expansion might be accomplished
by a collision of two branes where one of extra-dimensions disappears [9]. It was argued
that the scale-invariant Newtonian potential can be transferred to the comoving curvature
perturbations by this singular bounce [10]. However, without having a concrete theory to
describe the singularity, it is difficult to have a definite conclusion on how perturbations pass
through the singularity.
Recently, there has been some progress in generating a scale invariant spectrum for
curvature perturbations in the ekpyrotic scenario [11, 12, 13], by considering non-adiabatic
perturbations which has been suggested previously by Ref. [15]. In this case we require two
or more fields. If these have steep exponential potentials then there exists a scaling solution
where the energy density of the fields grow at the same rate during collapse [16, 17]. The
isocurvature perturbations then have a scale-invariant spectrum [16]. These isocurvature
perturbations can be converted to curvature perturbations if there is a sharp turn in the
trajectory in field space [11, 12, 13]. For example Ref. [11] considered a situation where one
of the fields changes its direction in field space, which corresponds to a time when a negative
tension brane is reflected by a curvature singularity in the bulk, in the context of the heterotic
M-theory. Refs. [12, 13] considered a regular bounce realized by a ghost condensate. One
of the fields exits the ekpyrotic phase and hits the transition to the ghost condensate phase
that creates a sharp turn in the trajectory in field space and curvature perturbations can be
generated [12]. It is still necessary to match curvature perturbations in a contracting phase
to those in an expanding universe, but it is shown that the comoving curvature perturbation
is conserved on large scales resulting in an almost scale-invariant spectrum observed today
for a regular bounce like a ghost condensate model [12].
The isocurvature perturbations behave like δs ∝ H on large scales. As the Hubble
parameter is rapidly increasing in a collapsing universe, this signals an instability. In fact it
is easy to see that we always require an instability of this form in order to generate a scale-
invariant spectrum with a canonical scalar field during collapse4. As the amplitude of field
perturbations at Hubble exit are of order H , we require the super-Hubble perturbations to
grow at the same rate to maintain a scale-invariant spectrum. Ref. [19] studied this instability
in a phase space analysis. The multi-field scaling solution was shown to be a saddle point in
field space and the late-time attractor is the old ekpyrotic collapse dominated by a single-
field. A tachyonic instability drives the scaling solution towards the late-time attractor (see
also [11, 12]).
In Ref. [19], we pointed out that the natural turning point in the field space trajectory
due to the instability of the scaling solution might itself offer the possibility of converting the
scale invariant spectrum of isocurvature field perturbations into a scale invariant spectrum
of curvature perturbations. In this paper, we confirm this expectation by explicitly solving
the evolution equations for perturbations in a two field model. We find that the ratio
between curvature perturbations and isocurvature perturbations at the final old ekpyrotic
phase is solely determined by the ratio of exponents of the two exponential potentials, and
the amplitude is set by the Hubble rate at the transition time.
4The only way to produce a scale-invariant spectrum without the presence of an instability seems to
be due to non-canonical kinetic terms, as in the case of axion-type fields which can acquire scale-invariant
perturbation spectra while remaining massless [18].
2 Homogeneous field dynamics
We first review the background dynamics of the fields. During the ekpyrotic collapse the
contraction of the universe is assumed to be described by a 4D Friedmann equation in the
Einstein frame with scalar fields with negative exponential potentials
3H2 = V +
φ̇2i , (1)
where
V = −
−ciφi , (2)
and we take Vi > 0 and set 8πG equal to unity.
The authors of [11] found a scaling solution (previously studied in [16, 17]) in which both
fields roll down their potential as the universe approaches a big crunch singularity. In this
ekpyrotic scaling collapse we find a power-law solution for the scale factor
a ∝ (−t)p , where p =
, (3)
where
−Vie−ciφi
−Vje−cjφj
. (4)
As we will see in the next section, it is possible to generate scale-invariant isocurvature per-
turbations around this background. However, the ekpyrotic scaling solution (4) is unstable.
In addition to the scaling solution we have fixed points corresponding to any one of the
original fields φi dominating the energy density where the other fields have negligible energy
density. These correspond to the original ekpyrotic power-law solutions where
a ∝ (−t)pi , where pi =
, (5)
for c2i > 6. We find that any of these single field dominated solutions is a stable local
attractor at late times during collapse.
In Ref. [19], the ekpyrotic scaling solution (4) was shown to be a saddle point in the
phase space. We briefly review the phase space analysis. Introducing phase space variables
[20, 21, 17]
, (6)
Vie−ciφi√
. (7)
the first order evolution equations for the phase space variables are given by
= −3xi(1−
x2j)− ci
y2i , (8)
x2j − ci
 , (9)
where N = log a. The Friedmann equation gives a constraint
x2j −
y2j = 1. (10)
There are (n + 2) fixed points of the system where dxi/dN = dyi/dN = 0.
x2j = 1, yj = 0. (11)
Bi : xi =
, yi = −
− 1, xj = yj = 0, (for j 6= i), (12)
B : xj =
, yj = −
. (13)
In this paper, we focus on the fixed points B and Bi assuming c
i > 6 and
c−2i < 1/6. The
linearized analysis shows that the multi-field scaling solution, B, always has one unstable
mode. On the other hand, the single field dominated fixed points, Bi, are always stable.
From now on we concentrate our attention on two fields case. Then we have three fixed
points B,B1 and B2. It is interesting to note that in the (x1, x2) plane, the fixed points B,
B1 and B2 are connected by a straight line, which is given by
c2x1 + c1x2 =
c1c2√
. (14)
The eigenvector associated with the unstable mode around the scaling solution B lies in the
same direction as the line (14). Thus this is an attractor trajectory, which all solutions near
B approach. Fig. 1 shows numerical solutions for the evolution of x1. Initial positions in the
phase space are perturbed away from B along the line (14). The solutions go to B1 or B2,
depending on the initial position in the phase space.
An important observation is that, as we follow phase space trajectories during the tran-
sition from the scaling solution B to the attractor solutions B1 or B2, the solutions obey the
relation Eq. (14), even far away from the saddle point, B. Using the Friedmann equation
and the field equations, we can show that
H − φ̇1
− φ̇2
H − φ̇1
− φ̇2
= 0 , (15)
and hence
H − φ̇1
− φ̇2
, (16)
where C is an integration constant. In terms of φ1 and φ2, Eq. (14) can be rewritten as
= H. (17)
and hence we see that for trajectories starting from point B we have C = 0, which is the
late time attractor. Thus we see that Eq. (17) holds even during the transition caused by
the tachyonic instability from point B to B1 or B2 and in the final single-field dominated
phase. This fact will be important when we study perturbations.
We will study the behaviour of perturbations during the transition in the next section.
2.5 5 7.5 10 12.5 15
0.01 0.02 0.03 0.04 0.05
Figure 1: Left: Numerical solutions for x1(N). The horizontal axis is N = log a and we take
c1 = 40 and c2 = 30. The initial time is N = 0.05. Note that N decreases towards the future
in a collapsing universe. Right: The corresponding phase space trajectories in (x1, x2) plane.
3 Generation of quantum fluctuations
In this section, we consider inhomogeneous linear perturbations around the background
solution. We consider the scalar field perturbations on spatially flat hypersurfaces. Then
the scalar field perturbations are given by [22, 23, 24, 25]
δ̈φi + 3H
˙δφi +
δφi − c2iVi exp(−ciφi)δφi −
φ̇iφ̇j
δφj = 0. (18)
We can decompose the perturbations into the instantaneous adiabatic and entropy field
perturbations as follows [24]:
φ̇1δφ1 + φ̇2δφ2
φ̇21 + φ̇
, δs =
φ̇2δφ1 − φ̇1δφ2
φ̇21 + φ̇
. (19)
The adiabatic field perturbation δr is the component of the two-field perturbation along the
direction of the background fields’ evolution while the entropy perturbation δs represents
fluctuations orthogonal to the background classical trajectory. The adiabatic field perturba-
tion leads to a perturbation in the comoving curavture perturbation:
, (20)
whereas the entropy field perturbations correspond to isocurvature perturbations.
Their evolution equations are given by [24]
δ̈r + 3Hδ̇r +
V,rr − θ̇2 −
a3ṙ2
δr = 2θ̇δ̇s+ 2
δs, (21)
δ̈s+ 3Hδ̇s+
δs+ (V,ss − θ̇2)δs = −2
ṙδ̇r −
, (22)
where the angle θ is defined as
cos θ =
φ̇21 + φ̇
, sin θ =
φ̇21 + φ̇
, (23)
such that
ṙ = (cos θ)φ̇2 + (sin θ)φ̇1, (24)
θ̇ = −V,s
, (25)
V,r = (sin θ)c1V1 exp(−c1φ1) + (cos θ)c2V2 exp(−c2φ2), (26)
V,s = (cos θ)c1V1 exp(−c1φ1)− (sin θ)c2V2 exp(−c2φ2), (27)
V,rr = −(sin θ)2c21V1 exp(−c1φ1)− (cos θ)2c22V2 exp(−c2φ2), (28)
V,ss = −(sin θ)2c22V2 exp(−c2φ2)− (cos θ)2c21V1 exp(−c1φ1). (29)
For the multi-field scaling solution, B, we have
θ = arctan
, (30)
and for the single field scaling solutions we have
(B1) , θ = 0 (B2) . (31)
Thus we have θ =constant for the fixed points and the adiabatic and the entropy fields are
decoupled. This allows us to quantise the independent fluctuations in the two fields.
For the multi-field scaling solutionB, the spectrum of quantum fluctuations of the entropy
field is given on large scales (k ≪ aH) by
Pδs ≡
|δs2| = C2ν
(−kτ)1−2ν , (32)
where τ < 0 is conformal time, and
(ǫ− 1)2
, ǫ ≡ −Ḣ/H2 = 1/p, (33)
and Cν = 2
ν−3/2Γ(ν)/π3/2 [16]. The spectral tilt is given by
∆nδχ ≃
, (34)
to leading order in a fast-roll expansion (ǫ ≫ 1) [11, 12, 13]. In this limit, the spectrum (32)
can be written as
P1/2δs = ǫ
. (35)
Note that |H| is rapidly increasing and thus δs is also growing on super-Hubble scales due to
the tachyonic instability. This instability is essential in order to realize the scale invariance
of the spectrum (35). The amplitude of field perturbations at Hubble exit is of order H
and thus we require the super-Hubble perturbations to grow at the same rate in order to
maintain a scale-invariant spectrum.
Note that in the simplest model, the spectrum is slightly blue [11, 12, 13]. However, any
deviations from an exponential potential for adiabatic field can introduce the corrections to
the spectral tilt and thus it becomes model dependent [11, 12].
The spectrum of quantum fluctuations in the adiabatic field about the scaling solution
has the same power-law form on large scales
Pδr = C2µ
(−kτ)1−2µ , (36)
where to µ ≃ 1/2 to leading order in 1/ǫ. Thus the adiabatic field perturbations become
constant in the large scale limit and the spectral tilt is given by
∆nδr ≃ 2 . (37)
Thus we have
∝ (−kτ)2 , (38)
and hence in what follows we can neglect the adiabatic field fluctuations in the large-scale
limit.
By contrast, for the single field dominated scaling solutions, the adiabatic and entropy
field perturbations are both frozen on super-Hubble scales:
δs , δr = const. (39)
These perturbations both have a steep blue spectrum if they cross the horizon when the
background solutions are described by the single field dominated solution [19].
4 Generation of curvature perturbations
Now let us consider the evolution of perturbations in a situation where the classical solution
starts from near the saddle point, B. As emphasized in Ref. [19] this requires an additional
preceding mechanism that drives the classical background solution to the unstable saddle
point throughout our observable part of the universe. In this paper, we will not discuss the
mechanism required to bring the classical solution to the saddle point and we just assume
that the classical solution stays near the saddle point for long enough to ensure that a scale-
invariant spectrum of isocurvature perturbations is generated over the relevant scales for the
observed large scale structure of our Universe.
Then the initial conditions for the adiabatic and the entropy field perturbations can be
set from the amplitude of quantum fluctuation as described in the previous section:
δr = 0, δs = ǫ
, (40)
on sufficiently large scales and for 1/ǫ ≪ 1.
Unless the spatially homogeneous background solution is located exactly at the fixed
point, the tachyonic instability drives the background solution away from the multi-field
scaling solution, B, to one of the single field dominated solutions, B1 or B2, depending on
the initial conditions. During the transition θ is not constant and the adiabatic and entropy
field perturbations mix, so it is possible to generate perturbations in the adiabatic field, and
hence comoving curvature perturbations (20), from initial fluctuations in the entropy field.
We can solve the evolution equations (21) and (22) numerically for any given classical
background. Figure 2 shows the behaviour of δr and δs. Due to the coupling between δr
and δs during the transition, curvature perturbations are generated during the transition.
On the other hand, δs shows a tachyonic instability according to Eq. (40) close to the
scaling solution, but when the background solution goes to Bi, the entropy field perturbation
becomes constant. The final amplitude of δr depends on when the transition from B to Bi
occurs, but, interestingly, the final ratio between δr and δs does not depend on the details
of the transition. We find that the ratio is determined solely by the parameters c1 and c2 as
, at B1, (41)
= −c2
, at B2, (42)
as is shown in Fig. 2. We will explain later why such a simple result is found.
0.01 0.02 0.03 0.04 0.05
-0.002
0.002
0.004
0.006
0.008
0.01 0.02 0.03 0.04 0.05
Figure 2: Left: Solutions for δr(N), using the same parameters as in Figure 1. The corre-
sponding background solutions are shown in Figure 1. Right: The ratio between δr and δs.
The ratio approaches a constant given by Eqs. (41) and (42). In this case 1.3333 for B1 and
−0.75 for B2.
The resulting curvature perturbation on a comoving hypersurface in the final single-field
dominated phase is thus given by
δr, at B1, (43)
δr, at B2. (44)
The equation for the entropy field perturbation (22) can be rewritten as [24]
δ̈s+ 3Hδ̇s+
+ Vss + 3θ̇
δs = 4
Ψ , (45)
where Ψ is the curvature perturbations in Newtonian gauge. The change of the curvature
perturbations is determined by [24]
Ṙc =
θ̇δs. (46)
On large scales, we can neglect (k2/a2)Ψ and it is possible to reproduce the previous results
from Eq. (46).
Although the physical meaning of the instantaneous adiabatic and entropy field perturba-
tions is clear, the dynamics of the perturbations during the transition are rather complicated
in this basis. We find it is much easier to work in terms of new variables [19]
c2φ1 + c1φ2
c21 + c
, χ =
c1φ1 − c2φ2
c21 + c
, (47)
corresponding to a fixed rotation in field space. The potential Eq. (2) can then be simply
re-written as [29, 16, 19]
V = −U(χ) e−cϕ , (48)
where
, (49)
and the potential for the orthogonal field is given by
− U(χ) = −V1 e−(c1/c2)cχ − V2 e(c2/c1)cχ , (50)
which has a maximum at
χ = χ0 =
c21 + c
c21V1
c22V2
. (51)
The multi-field scaling solution corresponds to χ = χ0, while ϕ is rolling down the
exponential potential. The potential for χ has a negative mass-squared around χ = χ0, and
thus χ represents the instability direction. If the initial condition for χ is slightly different
from χ0 or χ̇ is not zero, then χ starts rolling down the potential and the solution approaches
a single-field dominated solution.
Note that perturbations δϕ and δχ coincide with the instantaeous adiabatic and entropy
field perturbations respectively, defined in Eq. (19), at the scaling solution, B. Thus we
can use the initial perturbations (40) due to vacuum fluctuations about the scaling solution
previously calculated. However as we follow the evolution away from this saddle point we
can no longer identify ϕ and χ with the adiabatic and entropy perturbations. Nevertheless,
the dynamics of perturbations turns out to be much simpler using these fields.
In terms of ϕ and χ the equations for perturbations are given by
δ̈ϕ+ 3H ˙δϕ+
δϕ+Mϕϕδϕ+Mϕχδχ = 0, (52)
δ̈χ+ 3H ˙δχ+
δχ+Mχχδχ+Mϕχδϕ = 0, (53)
where
Mϕϕ = V,ϕϕ −
, (54)
Mϕχ = V,ϕχ −
, (55)
Mχχ = V,χχ −
. (56)
A key observation is that the phase space trajectory of the background fields during the
transition, Eq. (17), can be re-written as
= c. (57)
Thus even away from the multi-field scaling solution, ϕ obeys a scaling relation. We can
then show that two of the effective mass terms become
Mϕϕ = V,ϕϕ + cV,ϕ = 0 , (58)
Mϕχ = V,ϕχ + cV,χ = 0 , (59)
independently of the form of U(χ) since V ∝ exp(−cϕ).
Thus on large scales δϕ is constant. As we take δϕ = 0 as our initial condition, this
remains so even during the transition and in the final single field dominated phase. Thus
from Eq. (47) we have a relation between δφ1 and δφ2,
c2δφ1 + c1δφ2 = 0, (60)
and δφ1 and δφ2 are given by
δφ1 =
c21 + c
δχ, (61)
δφ2 = −
c21 + c
δχ. (62)
In the single field dominated solutions, ϕ is no longer the adiabatic field. The adiabatic field
is simply φ1 (or φ2) in B1 (or B2) and the entropy field is −φ2 (or φ1) in B1 (or B2). Thus
at the late-time attractor we have
δr = δφ1, δs = −δφ2, at B1, (63)
δr = δφ2, δs = δφ1, at B2. (64)
Thus the ratio between δr and δs is determined by the ratio between δφ1 and δφ2 and we
can easily find the ratio δr/δs as given in Eqs.(41) and (42).
The amplitude of the curvature perturbations can be estimated from δχ. In the initial
stage, δχ coincides with the entropy field perturbations δs, and thus its initial amplitude on
super-Hubble scales is given by
. (65)
where we used ǫ = c2/2 in Eq. (40) and c2 is given by Eq. (49). After the transition, δχ
becomes massless and the amplitude becomes frozen on large scales. In the single field
dominated solutions, the comoving curvature perturbation Rc is given by
|Rc| =
c21 + c
δχ. (66)
Assuming the transition occurs suddenly, the final amplitude of the comoving curvature
perturbation is given by
|Rc| =
c21 + c
, (67)
where the subscript T means that the quantity is evaluated at the transition time. On the
other hand the amplitude of the entropy field perturbation is given by
c21 + c
, at B1, (68)
c21 + c
, at B2. (69)
From the numerical solutions for δr and δs, we can reconstruct HT . We confirm that HT
constructed in this way agrees with the Hubble scale at the transition as is shown in Figure 3.
5 Conclusion
In this paper we have studied the generation of curvature perturbations during an ekpyrotic
collapse with multiple fields. We must assume that the classical background solution starts
from a state very close to a saddle point in the phase space that corresponds to the multi-
field ekpyrotic scaling solution. If the solution stays at this fixed point for long enough, a
scale-invariant spectrum of isocurvature perturbations is generated over the range of scales
that is relevant for large scale structure in the Universe. Even if the background solution
deviates only slightly from the multi-field scaling solution initially, the tachyonic instability
eventually drives the solution to the old ekpyrotic collapse dominated by a single field.
During this transition, the initial isocurvature field perturbations generate a scale-invariant
spectrum of comoving curvature perturbations.
First we studied the perturbations by decomposing them into the instantaneous adiabatic
and the entropy field perturbations. These fields are decoupled at the fixed points. The adia-
batic field perturbations are effectively massless around both the multi-field scaling solutions
0.01 0.02 0.03 0.04 0.05
0.00025
0.0005
0.00075
0.001
0.00125
0.0015
0.00175
0.005 0.01 0.015 0.02 0.025 0.03 0.035
Log|H|
Log|H   |T
Figure 3: Left: Solutions for δχ(N). We used the same parameters as Figures 1 and 2. Right:
Solutions for log |H| with three different background solutions. We also show log |HT | that
is determined from the numerical solutions for δχ.
and the single field dominated solution, so become constant on large scales. On the other
hand, the entropy field perturbations have a tachyonic mass around the multi-field scaling
solution, where they grow like δs ∝ H on large scales, but they are effectively massless
around the single field dominated solution. We set initial conditions for these perturbations
from quantum fluctuations about the multi-field scaling solution. As the adiabatic field
perturbations have a blue spectrum they can be neglected compared with the entropy field
perturbations on large scales. However, during the transition to the single-field attractor,
adiabatic field perturbations are generated from the entropy field perturbations.
It turns out to be more convenient to use new fields ϕ and χ defined in Eqs. (47) to
follow the perturbations through the transition. These fields coincide with the adiabatic and
entropy fields around the multi-field scaling solution but not during the transition or at the
final single field dominated phase. In terms of ϕ and χ, the potential is given by a product of
the potential for χ, U(χ), and an exponential potential for ϕ. U(χ) has an extremum which
corresponds to the multi-field scaling solution. A crucial observation is that even during the
transition and in the final phase, ϕ satisfies the scaling relation Eq. (57). We can then show
that the field perturbations for ϕ and χ are always decoupled. Since δϕ = 0 on large scales
during the ekpyrotic scaling solution, this remains so. This determines the ratio between the
adiabatic and entropy field perturbations at the final phase. On the other hand, δχ grows
during the scaling solution and then becomes constant during the single field dominated
solution.
Our final results are Eqs. (67), (68) and (69) for the amplitude of comoving curvature
and isocurvature field perturbations during the single field ekpyrotic collapse phase. The
amplitude of the comoving curvature perturbation is determined by the Hubble scale at the
transition.
We still need to convert the ekpyrotic collapse to expansion (see, for instance, [12, 13])
and see how this curvature perturbation is matched to that in an expanding universe. For
a regular bounce the comoving curvature perturbation is conserved for adiabatic pertur-
bations and thus Eq. (67) is directly related to the amplitude of the observed primordial
density perturbation. If the radiation and matter content in today’s universe comes solely
from the single field that dominates the final ekpyrotic phase, we will have no isocurvaure
perturbations in an expanding universe.
It is interesting to compare this multi-field model with a single field model that gives a
scale-invariant spectrum of comoving curvature perturbations during collapse [14, 26, 27].
The single-field model requires the correct exponent for a relatively flat and positive expo-
nential potential in order to obtain a ∝ |t|2/3, whereas the ekyprotic model only requires
sufficiently steep, negative exponential potentials to obtain a ∝ |t|1/ǫ with ǫ ≫ 1. On the
other hand both models require fine-tuned initial conditions as it is the existence of an insta-
bility that gives rise to the scale-invariant perturbation spectrum during collapse. In both
models the amplitude of tensor perturbations is determined by the Hubble scale when the
perturbations leave the horizon as tensor perturbations are then frozen on super-horizon
scales. In the single field model the tensor metric perturbations thus acquire the same al-
most scale-invariant spectrum [28] as the comoving curvature perturbation, with a similar
amplitude, giving rise to a dangerously large tensor-scalar ratio which severely constrains
the model [27]. But in the ekpyrotic scaling solution the tensor perturbations have a steep
blue spectrum and are completely negligible at scales relevant for the cosmic microwave
anisotropies.
In summary, a simple ekpyrotic model with two steep, negative exponential potentials
is capable of generating a scale-invariant spectrum of comoving curvature perturbations. A
key ingredient is the instability of the multi-field scaling solution [16, 11, 12, 19]. This insta-
bility generates a scale-invariant spectrum of isocurvature field perturbations from vacuum
fluctuations about the scaling solution, and converts them to a scale-invariant spectrum of
comoving curvature perturbations. We should emphasize that this conversion occurs auto-
matically due to the dynamics of the fields in our simple model and does not require any
change in the shape of the potential or any additional dynamics [11, 12, 13].
Thus ekpyrotic collapse with multiple fields can generate a scale-invariant spectrum for
curvature perturbations in several different ways (see also [30] for a different idea). In all
these models, we need some preceding phase that initially drives the classical background
solution to the unstable multi-field scaling solution throughout our observable region of
space. This problem cannot be solved within the simplest model with multiple exponential
potentials that we considered in this paper and we would need to appeal to a more ambitious
framework for the model such as the cyclic scenario [31, 32] to address this problem.
Acknowledgments
We would like to thank J-L. Lehners and P.J. Steinhardt for discussions. KK and DW are
supported by STFC. SM is grateful to the ICG, Portsmouth, for their hospitality when this
work was initiated. SM is supported in part by the Japan Society for Promotion of Science
(JSPS) Research Fellowship.
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	Introduction
	Homogeneous field dynamics
	Generation of quantum fluctuations
	Generation of curvature perturbations
	Conclusion

  A scale-invariant spectrum of isocurvature perturbations is generated during
collapse in the ekpyrotic scaling solution in models where multiple fields have
steep negative exponential potentials. The scale invariance of the spectrum is
realized by a tachyonic instability in the isocurvature field. This instability
drives the scaling solution to the late time attractor that is the old
ekpyrotic collapse dominated by a single field. We show that the transition
from the scaling solution to the single field dominated ekpyrotic collapse
automatically converts the initial isocurvature perturbations about the scaling
solution to comoving curvature perturbations about the late-time attractor. The
final amplitude of the comoving curvature perturbation is determined by the
Hubble scale at the transition.

Introduction
The existence of an almost scale-invariant spectrum of primordial curvature perturbations on
large scales is one of the most important observations that any model for the early universe
should explain. An inflationary expansion in the very early universe is most commonly
assumed to achieve this, but it is important to consider whether there is any alternative
model. In this paper we focus on the ekpyrotic scenario as an alternative [1] (see also [2, 3]).
In the old ekpyrotic scenario, the large scale perturbations are supposed to be generated
during a collapse driven by a single scalar field with a steep negative exponential potential.
It was shown that the Newtonian potential acquires a scale-invariant spectrum, but the
comoving curvature perturbation has a steep blue spectrum [4]. In this scenario we need a
mechanism to convert contraction to expansion, and for a regular four-dimensional bounce,
the scale-invariant Newtonian potential is matched to the decaying mode in an expanding
universe, and the growing mode of curvature perturbations acquires a steep blue spectrum
[5, 6]. It has been suggested that this conclusion might be altered by allowing a singular
matching between collapse and expanion [7], but the general rule that the comoving curvature
perturbation remains constant still holds for adiabatic perturbations on large scales [8]. In
1 E-mail: Kazuya.Koyama@port.ac.uk
2 E-mail: mizuno@resceu.s.u-tokyo.ac.jp
3E-mail: David.Wands@port.ac.uk
http://arxiv.org/abs/0704.1152v3
a braneworld context, the conversion from contraction to expansion might be accomplished
by a collision of two branes where one of extra-dimensions disappears [9]. It was argued
that the scale-invariant Newtonian potential can be transferred to the comoving curvature
perturbations by this singular bounce [10]. However, without having a concrete theory to
describe the singularity, it is difficult to have a definite conclusion on how perturbations pass
through the singularity.
Recently, there has been some progress in generating a scale invariant spectrum for
curvature perturbations in the ekpyrotic scenario [11, 12, 13], by considering non-adiabatic
perturbations which has been suggested previously by Ref. [15]. In this case we require two
or more fields. If these have steep exponential potentials then there exists a scaling solution
where the energy density of the fields grow at the same rate during collapse [16, 17]. The
isocurvature perturbations then have a scale-invariant spectrum [16]. These isocurvature
perturbations can be converted to curvature perturbations if there is a sharp turn in the
trajectory in field space [11, 12, 13]. For example Ref. [11] considered a situation where one
of the fields changes its direction in field space, which corresponds to a time when a negative
tension brane is reflected by a curvature singularity in the bulk, in the context of the heterotic
M-theory. Refs. [12, 13] considered a regular bounce realized by a ghost condensate. One
of the fields exits the ekpyrotic phase and hits the transition to the ghost condensate phase
that creates a sharp turn in the trajectory in field space and curvature perturbations can be
generated [12]. It is still necessary to match curvature perturbations in a contracting phase
to those in an expanding universe, but it is shown that the comoving curvature perturbation
is conserved on large scales resulting in an almost scale-invariant spectrum observed today
for a regular bounce like a ghost condensate model [12].
The isocurvature perturbations behave like δs ∝ H on large scales. As the Hubble
parameter is rapidly increasing in a collapsing universe, this signals an instability. In fact it
is easy to see that we always require an instability of this form in order to generate a scale-
invariant spectrum with a canonical scalar field during collapse4. As the amplitude of field
perturbations at Hubble exit are of order H , we require the super-Hubble perturbations to
grow at the same rate to maintain a scale-invariant spectrum. Ref. [19] studied this instability
in a phase space analysis. The multi-field scaling solution was shown to be a saddle point in
field space and the late-time attractor is the old ekpyrotic collapse dominated by a single-
field. A tachyonic instability drives the scaling solution towards the late-time attractor (see
also [11, 12]).
In Ref. [19], we pointed out that the natural turning point in the field space trajectory
due to the instability of the scaling solution might itself offer the possibility of converting the
scale invariant spectrum of isocurvature field perturbations into a scale invariant spectrum
of curvature perturbations. In this paper, we confirm this expectation by explicitly solving
the evolution equations for perturbations in a two field model. We find that the ratio
between curvature perturbations and isocurvature perturbations at the final old ekpyrotic
phase is solely determined by the ratio of exponents of the two exponential potentials, and
the amplitude is set by the Hubble rate at the transition time.
4The only way to produce a scale-invariant spectrum without the presence of an instability seems to
be due to non-canonical kinetic terms, as in the case of axion-type fields which can acquire scale-invariant
perturbation spectra while remaining massless [18].
2 Homogeneous field dynamics
We first review the background dynamics of the fields. During the ekpyrotic collapse the
contraction of the universe is assumed to be described by a 4D Friedmann equation in the
Einstein frame with scalar fields with negative exponential potentials
3H2 = V +
φ̇2i , (1)
where
V = −
−ciφi , (2)
and we take Vi > 0 and set 8πG equal to unity.
The authors of [11] found a scaling solution (previously studied in [16, 17]) in which both
fields roll down their potential as the universe approaches a big crunch singularity. In this
ekpyrotic scaling collapse we find a power-law solution for the scale factor
a ∝ (−t)p , where p =
, (3)
where
−Vie−ciφi
−Vje−cjφj
. (4)
As we will see in the next section, it is possible to generate scale-invariant isocurvature per-
turbations around this background. However, the ekpyrotic scaling solution (4) is unstable.
In addition to the scaling solution we have fixed points corresponding to any one of the
original fields φi dominating the energy density where the other fields have negligible energy
density. These correspond to the original ekpyrotic power-law solutions where
a ∝ (−t)pi , where pi =
, (5)
for c2i > 6. We find that any of these single field dominated solutions is a stable local
attractor at late times during collapse.
In Ref. [19], the ekpyrotic scaling solution (4) was shown to be a saddle point in the
phase space. We briefly review the phase space analysis. Introducing phase space variables
[20, 21, 17]
, (6)
Vie−ciφi√
. (7)
the first order evolution equations for the phase space variables are given by
= −3xi(1−
x2j)− ci
y2i , (8)
x2j − ci
 , (9)
where N = log a. The Friedmann equation gives a constraint
x2j −
y2j = 1. (10)
There are (n + 2) fixed points of the system where dxi/dN = dyi/dN = 0.
x2j = 1, yj = 0. (11)
Bi : xi =
, yi = −
− 1, xj = yj = 0, (for j 6= i), (12)
B : xj =
, yj = −
. (13)
In this paper, we focus on the fixed points B and Bi assuming c
i > 6 and
c−2i < 1/6. The
linearized analysis shows that the multi-field scaling solution, B, always has one unstable
mode. On the other hand, the single field dominated fixed points, Bi, are always stable.
From now on we concentrate our attention on two fields case. Then we have three fixed
points B,B1 and B2. It is interesting to note that in the (x1, x2) plane, the fixed points B,
B1 and B2 are connected by a straight line, which is given by
c2x1 + c1x2 =
c1c2√
. (14)
The eigenvector associated with the unstable mode around the scaling solution B lies in the
same direction as the line (14). Thus this is an attractor trajectory, which all solutions near
B approach. Fig. 1 shows numerical solutions for the evolution of x1. Initial positions in the
phase space are perturbed away from B along the line (14). The solutions go to B1 or B2,
depending on the initial position in the phase space.
An important observation is that, as we follow phase space trajectories during the tran-
sition from the scaling solution B to the attractor solutions B1 or B2, the solutions obey the
relation Eq. (14), even far away from the saddle point, B. Using the Friedmann equation
and the field equations, we can show that
H − φ̇1
− φ̇2
H − φ̇1
− φ̇2
= 0 , (15)
and hence
H − φ̇1
− φ̇2
, (16)
where C is an integration constant. In terms of φ1 and φ2, Eq. (14) can be rewritten as
= H. (17)
and hence we see that for trajectories starting from point B we have C = 0, which is the
late time attractor. Thus we see that Eq. (17) holds even during the transition caused by
the tachyonic instability from point B to B1 or B2 and in the final single-field dominated
phase. This fact will be important when we study perturbations.
We will study the behaviour of perturbations during the transition in the next section.
2.5 5 7.5 10 12.5 15
0.01 0.02 0.03 0.04 0.05
Figure 1: Left: Numerical solutions for x1(N). The horizontal axis is N = log a and we take
c1 = 40 and c2 = 30. The initial time is N = 0.05. Note that N decreases towards the future
in a collapsing universe. Right: The corresponding phase space trajectories in (x1, x2) plane.
3 Generation of quantum fluctuations
In this section, we consider inhomogeneous linear perturbations around the background
solution. We consider the scalar field perturbations on spatially flat hypersurfaces. Then
the scalar field perturbations are given by [22, 23, 24, 25]
δ̈φi + 3H
˙δφi +
δφi − c2iVi exp(−ciφi)δφi −
φ̇iφ̇j
δφj = 0. (18)
We can decompose the perturbations into the instantaneous adiabatic and entropy field
perturbations as follows [24]:
φ̇1δφ1 + φ̇2δφ2
φ̇21 + φ̇
, δs =
φ̇2δφ1 − φ̇1δφ2
φ̇21 + φ̇
. (19)
The adiabatic field perturbation δr is the component of the two-field perturbation along the
direction of the background fields’ evolution while the entropy perturbation δs represents
fluctuations orthogonal to the background classical trajectory. The adiabatic field perturba-
tion leads to a perturbation in the comoving curavture perturbation:
, (20)
whereas the entropy field perturbations correspond to isocurvature perturbations.
Their evolution equations are given by [24]
δ̈r + 3Hδ̇r +
V,rr − θ̇2 −
a3ṙ2
δr = 2θ̇δ̇s+ 2
δs, (21)
δ̈s+ 3Hδ̇s+
δs+ (V,ss − θ̇2)δs = −2
ṙδ̇r −
, (22)
where the angle θ is defined as
cos θ =
φ̇21 + φ̇
, sin θ =
φ̇21 + φ̇
, (23)
such that
ṙ = (cos θ)φ̇2 + (sin θ)φ̇1, (24)
θ̇ = −V,s
, (25)
V,r = (sin θ)c1V1 exp(−c1φ1) + (cos θ)c2V2 exp(−c2φ2), (26)
V,s = (cos θ)c1V1 exp(−c1φ1)− (sin θ)c2V2 exp(−c2φ2), (27)
V,rr = −(sin θ)2c21V1 exp(−c1φ1)− (cos θ)2c22V2 exp(−c2φ2), (28)
V,ss = −(sin θ)2c22V2 exp(−c2φ2)− (cos θ)2c21V1 exp(−c1φ1). (29)
For the multi-field scaling solution, B, we have
θ = arctan
, (30)
and for the single field scaling solutions we have
(B1) , θ = 0 (B2) . (31)
Thus we have θ =constant for the fixed points and the adiabatic and the entropy fields are
decoupled. This allows us to quantise the independent fluctuations in the two fields.
For the multi-field scaling solutionB, the spectrum of quantum fluctuations of the entropy
field is given on large scales (k ≪ aH) by
Pδs ≡
|δs2| = C2ν
(−kτ)1−2ν , (32)
where τ < 0 is conformal time, and
(ǫ− 1)2
, ǫ ≡ −Ḣ/H2 = 1/p, (33)
and Cν = 2
ν−3/2Γ(ν)/π3/2 [16]. The spectral tilt is given by
∆nδχ ≃
, (34)
to leading order in a fast-roll expansion (ǫ ≫ 1) [11, 12, 13]. In this limit, the spectrum (32)
can be written as
P1/2δs = ǫ
. (35)
Note that |H| is rapidly increasing and thus δs is also growing on super-Hubble scales due to
the tachyonic instability. This instability is essential in order to realize the scale invariance
of the spectrum (35). The amplitude of field perturbations at Hubble exit is of order H
and thus we require the super-Hubble perturbations to grow at the same rate in order to
maintain a scale-invariant spectrum.
Note that in the simplest model, the spectrum is slightly blue [11, 12, 13]. However, any
deviations from an exponential potential for adiabatic field can introduce the corrections to
the spectral tilt and thus it becomes model dependent [11, 12].
The spectrum of quantum fluctuations in the adiabatic field about the scaling solution
has the same power-law form on large scales
Pδr = C2µ
(−kτ)1−2µ , (36)
where to µ ≃ 1/2 to leading order in 1/ǫ. Thus the adiabatic field perturbations become
constant in the large scale limit and the spectral tilt is given by
∆nδr ≃ 2 . (37)
Thus we have
∝ (−kτ)2 , (38)
and hence in what follows we can neglect the adiabatic field fluctuations in the large-scale
limit.
By contrast, for the single field dominated scaling solutions, the adiabatic and entropy
field perturbations are both frozen on super-Hubble scales:
δs , δr = const. (39)
These perturbations both have a steep blue spectrum if they cross the horizon when the
background solutions are described by the single field dominated solution [19].
4 Generation of curvature perturbations
Now let us consider the evolution of perturbations in a situation where the classical solution
starts from near the saddle point, B. As emphasized in Ref. [19] this requires an additional
preceding mechanism that drives the classical background solution to the unstable saddle
point throughout our observable part of the universe. In this paper, we will not discuss the
mechanism required to bring the classical solution to the saddle point and we just assume
that the classical solution stays near the saddle point for long enough to ensure that a scale-
invariant spectrum of isocurvature perturbations is generated over the relevant scales for the
observed large scale structure of our Universe.
Then the initial conditions for the adiabatic and the entropy field perturbations can be
set from the amplitude of quantum fluctuation as described in the previous section:
δr = 0, δs = ǫ
, (40)
on sufficiently large scales and for 1/ǫ ≪ 1.
Unless the spatially homogeneous background solution is located exactly at the fixed
point, the tachyonic instability drives the background solution away from the multi-field
scaling solution, B, to one of the single field dominated solutions, B1 or B2, depending on
the initial conditions. During the transition θ is not constant and the adiabatic and entropy
field perturbations mix, so it is possible to generate perturbations in the adiabatic field, and
hence comoving curvature perturbations (20), from initial fluctuations in the entropy field.
We can solve the evolution equations (21) and (22) numerically for any given classical
background. Figure 2 shows the behaviour of δr and δs. Due to the coupling between δr
and δs during the transition, curvature perturbations are generated during the transition.
On the other hand, δs shows a tachyonic instability according to Eq. (40) close to the
scaling solution, but when the background solution goes to Bi, the entropy field perturbation
becomes constant. The final amplitude of δr depends on when the transition from B to Bi
occurs, but, interestingly, the final ratio between δr and δs does not depend on the details
of the transition. We find that the ratio is determined solely by the parameters c1 and c2 as
, at B1, (41)
= −c2
, at B2, (42)
as is shown in Fig. 2. We will explain later why such a simple result is found.
0.01 0.02 0.03 0.04 0.05
-0.002
0.002
0.004
0.006
0.008
0.01 0.02 0.03 0.04 0.05
Figure 2: Left: Solutions for δr(N), using the same parameters as in Figure 1. The corre-
sponding background solutions are shown in Figure 1. Right: The ratio between δr and δs.
The ratio approaches a constant given by Eqs. (41) and (42). In this case 1.3333 for B1 and
−0.75 for B2.
The resulting curvature perturbation on a comoving hypersurface in the final single-field
dominated phase is thus given by
δr, at B1, (43)
δr, at B2. (44)
The equation for the entropy field perturbation (22) can be rewritten as [24]
δ̈s+ 3Hδ̇s+
+ Vss + 3θ̇
δs = 4
Ψ , (45)
where Ψ is the curvature perturbations in Newtonian gauge. The change of the curvature
perturbations is determined by [24]
Ṙc =
θ̇δs. (46)
On large scales, we can neglect (k2/a2)Ψ and it is possible to reproduce the previous results
from Eq. (46).
Although the physical meaning of the instantaneous adiabatic and entropy field perturba-
tions is clear, the dynamics of the perturbations during the transition are rather complicated
in this basis. We find it is much easier to work in terms of new variables [19]
c2φ1 + c1φ2
c21 + c
, χ =
c1φ1 − c2φ2
c21 + c
, (47)
corresponding to a fixed rotation in field space. The potential Eq. (2) can then be simply
re-written as [29, 16, 19]
V = −U(χ) e−cϕ , (48)
where
, (49)
and the potential for the orthogonal field is given by
− U(χ) = −V1 e−(c1/c2)cχ − V2 e(c2/c1)cχ , (50)
which has a maximum at
χ = χ0 =
c21 + c
c21V1
c22V2
. (51)
The multi-field scaling solution corresponds to χ = χ0, while ϕ is rolling down the
exponential potential. The potential for χ has a negative mass-squared around χ = χ0, and
thus χ represents the instability direction. If the initial condition for χ is slightly different
from χ0 or χ̇ is not zero, then χ starts rolling down the potential and the solution approaches
a single-field dominated solution.
Note that perturbations δϕ and δχ coincide with the instantaeous adiabatic and entropy
field perturbations respectively, defined in Eq. (19), at the scaling solution, B. Thus we
can use the initial perturbations (40) due to vacuum fluctuations about the scaling solution
previously calculated. However as we follow the evolution away from this saddle point we
can no longer identify ϕ and χ with the adiabatic and entropy perturbations. Nevertheless,
the dynamics of perturbations turns out to be much simpler using these fields.
In terms of ϕ and χ the equations for perturbations are given by
δ̈ϕ+ 3H ˙δϕ+
δϕ+Mϕϕδϕ+Mϕχδχ = 0, (52)
δ̈χ+ 3H ˙δχ+
δχ+Mχχδχ+Mϕχδϕ = 0, (53)
where
Mϕϕ = V,ϕϕ −
, (54)
Mϕχ = V,ϕχ −
, (55)
Mχχ = V,χχ −
. (56)
A key observation is that the phase space trajectory of the background fields during the
transition, Eq. (17), can be re-written as
= c. (57)
Thus even away from the multi-field scaling solution, ϕ obeys a scaling relation. We can
then show that two of the effective mass terms become
Mϕϕ = V,ϕϕ + cV,ϕ = 0 , (58)
Mϕχ = V,ϕχ + cV,χ = 0 , (59)
independently of the form of U(χ) since V ∝ exp(−cϕ).
Thus on large scales δϕ is constant. As we take δϕ = 0 as our initial condition, this
remains so even during the transition and in the final single field dominated phase. Thus
from Eq. (47) we have a relation between δφ1 and δφ2,
c2δφ1 + c1δφ2 = 0, (60)
and δφ1 and δφ2 are given by
δφ1 =
c21 + c
δχ, (61)
δφ2 = −
c21 + c
δχ. (62)
In the single field dominated solutions, ϕ is no longer the adiabatic field. The adiabatic field
is simply φ1 (or φ2) in B1 (or B2) and the entropy field is −φ2 (or φ1) in B1 (or B2). Thus
at the late-time attractor we have
δr = δφ1, δs = −δφ2, at B1, (63)
δr = δφ2, δs = δφ1, at B2. (64)
Thus the ratio between δr and δs is determined by the ratio between δφ1 and δφ2 and we
can easily find the ratio δr/δs as given in Eqs.(41) and (42).
The amplitude of the curvature perturbations can be estimated from δχ. In the initial
stage, δχ coincides with the entropy field perturbations δs, and thus its initial amplitude on
super-Hubble scales is given by
. (65)
where we used ǫ = c2/2 in Eq. (40) and c2 is given by Eq. (49). After the transition, δχ
becomes massless and the amplitude becomes frozen on large scales. In the single field
dominated solutions, the comoving curvature perturbation Rc is given by
|Rc| =
c21 + c
δχ. (66)
Assuming the transition occurs suddenly, the final amplitude of the comoving curvature
perturbation is given by
|Rc| =
c21 + c
, (67)
where the subscript T means that the quantity is evaluated at the transition time. On the
other hand the amplitude of the entropy field perturbation is given by
c21 + c
, at B1, (68)
c21 + c
, at B2. (69)
From the numerical solutions for δr and δs, we can reconstruct HT . We confirm that HT
constructed in this way agrees with the Hubble scale at the transition as is shown in Figure 3.
5 Conclusion
In this paper we have studied the generation of curvature perturbations during an ekpyrotic
collapse with multiple fields. We must assume that the classical background solution starts
from a state very close to a saddle point in the phase space that corresponds to the multi-
field ekpyrotic scaling solution. If the solution stays at this fixed point for long enough, a
scale-invariant spectrum of isocurvature perturbations is generated over the range of scales
that is relevant for large scale structure in the Universe. Even if the background solution
deviates only slightly from the multi-field scaling solution initially, the tachyonic instability
eventually drives the solution to the old ekpyrotic collapse dominated by a single field.
During this transition, the initial isocurvature field perturbations generate a scale-invariant
spectrum of comoving curvature perturbations.
First we studied the perturbations by decomposing them into the instantaneous adiabatic
and the entropy field perturbations. These fields are decoupled at the fixed points. The adia-
batic field perturbations are effectively massless around both the multi-field scaling solutions
0.01 0.02 0.03 0.04 0.05
0.00025
0.0005
0.00075
0.001
0.00125
0.0015
0.00175
0.005 0.01 0.015 0.02 0.025 0.03 0.035
Log|H|
Log|H   |T
Figure 3: Left: Solutions for δχ(N). We used the same parameters as Figures 1 and 2. Right:
Solutions for log |H| with three different background solutions. We also show log |HT | that
is determined from the numerical solutions for δχ.
and the single field dominated solution, so become constant on large scales. On the other
hand, the entropy field perturbations have a tachyonic mass around the multi-field scaling
solution, where they grow like δs ∝ H on large scales, but they are effectively massless
around the single field dominated solution. We set initial conditions for these perturbations
from quantum fluctuations about the multi-field scaling solution. As the adiabatic field
perturbations have a blue spectrum they can be neglected compared with the entropy field
perturbations on large scales. However, during the transition to the single-field attractor,
adiabatic field perturbations are generated from the entropy field perturbations.
It turns out to be more convenient to use new fields ϕ and χ defined in Eqs. (47) to
follow the perturbations through the transition. These fields coincide with the adiabatic and
entropy fields around the multi-field scaling solution but not during the transition or at the
final single field dominated phase. In terms of ϕ and χ, the potential is given by a product of
the potential for χ, U(χ), and an exponential potential for ϕ. U(χ) has an extremum which
corresponds to the multi-field scaling solution. A crucial observation is that even during the
transition and in the final phase, ϕ satisfies the scaling relation Eq. (57). We can then show
that the field perturbations for ϕ and χ are always decoupled. Since δϕ = 0 on large scales
during the ekpyrotic scaling solution, this remains so. This determines the ratio between the
adiabatic and entropy field perturbations at the final phase. On the other hand, δχ grows
during the scaling solution and then becomes constant during the single field dominated
solution.
Our final results are Eqs. (67), (68) and (69) for the amplitude of comoving curvature
and isocurvature field perturbations during the single field ekpyrotic collapse phase. The
amplitude of the comoving curvature perturbation is determined by the Hubble scale at the
transition.
We still need to convert the ekpyrotic collapse to expansion (see, for instance, [12, 13])
and see how this curvature perturbation is matched to that in an expanding universe. For
a regular bounce the comoving curvature perturbation is conserved for adiabatic pertur-
bations and thus Eq. (67) is directly related to the amplitude of the observed primordial
density perturbation. If the radiation and matter content in today’s universe comes solely
from the single field that dominates the final ekpyrotic phase, we will have no isocurvaure
perturbations in an expanding universe.
It is interesting to compare this multi-field model with a single field model that gives a
scale-invariant spectrum of comoving curvature perturbations during collapse [14, 26, 27].
The single-field model requires the correct exponent for a relatively flat and positive expo-
nential potential in order to obtain a ∝ |t|2/3, whereas the ekyprotic model only requires
sufficiently steep, negative exponential potentials to obtain a ∝ |t|1/ǫ with ǫ ≫ 1. On the
other hand both models require fine-tuned initial conditions as it is the existence of an insta-
bility that gives rise to the scale-invariant perturbation spectrum during collapse. In both
models the amplitude of tensor perturbations is determined by the Hubble scale when the
perturbations leave the horizon as tensor perturbations are then frozen on super-horizon
scales. In the single field model the tensor metric perturbations thus acquire the same al-
most scale-invariant spectrum [28] as the comoving curvature perturbation, with a similar
amplitude, giving rise to a dangerously large tensor-scalar ratio which severely constrains
the model [27]. But in the ekpyrotic scaling solution the tensor perturbations have a steep
blue spectrum and are completely negligible at scales relevant for the cosmic microwave
anisotropies.
In summary, a simple ekpyrotic model with two steep, negative exponential potentials
is capable of generating a scale-invariant spectrum of comoving curvature perturbations. A
key ingredient is the instability of the multi-field scaling solution [16, 11, 12, 19]. This insta-
bility generates a scale-invariant spectrum of isocurvature field perturbations from vacuum
fluctuations about the scaling solution, and converts them to a scale-invariant spectrum of
comoving curvature perturbations. We should emphasize that this conversion occurs auto-
matically due to the dynamics of the fields in our simple model and does not require any
change in the shape of the potential or any additional dynamics [11, 12, 13].
Thus ekpyrotic collapse with multiple fields can generate a scale-invariant spectrum for
curvature perturbations in several different ways (see also [30] for a different idea). In all
these models, we need some preceding phase that initially drives the classical background
solution to the unstable multi-field scaling solution throughout our observable region of
space. This problem cannot be solved within the simplest model with multiple exponential
potentials that we considered in this paper and we would need to appeal to a more ambitious
framework for the model such as the cyclic scenario [31, 32] to address this problem.
Acknowledgments
We would like to thank J-L. Lehners and P.J. Steinhardt for discussions. KK and DW are
supported by STFC. SM is grateful to the ICG, Portsmouth, for their hospitality when this
work was initiated. SM is supported in part by the Japan Society for Promotion of Science
(JSPS) Research Fellowship.
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	Introduction
	Homogeneous field dynamics
	Generation of quantum fluctuations
	Generation of curvature perturbations
	Conclusion
