The spectrum of gravitational waves in Randall-Sundrum
braneworld cosmology
Tsutomu Kobayashi1
1Department of Physics, Kyoto University, Kyoto 606-8502, JapanAbstract
We study the generation and evolution of gravitational waves (tensor perturbations)in the context of Randall-Sundrum braneworld cosmology. We assume that the initial
and ﬁnal stages of the background cosmological model are given by de Sitter andMinkowski phases, respectively, and they are connected smoothly by a radiationdominated phase. Using the Wronskian formulation, we numerically compute thepower spectrum of gravitational waves, and ﬁnd that the eﬀect of initial vacuum
ﬂuctuations in the Kaluza-Klein modes is subdominant, contributing not more than
10% of the total power spectrum. Thus it is conﬁrmed that the damping due to theKaluza-Klein mode generation and the enhancement due to the modiﬁcation of thebackground Friedmann equation are the two dominant eﬀects, but they cancel eachother, leading to the same spectral tilt as the standard four-dimensional result.

1Introduction

Cosmological inﬂation predicts the gravitational wave background arising due to quantum ﬂuctuationsin the graviton ﬁeld. Gravitational wave ﬂuctuations are stretched beyond the horizon radius by rapid
expansion during inﬂation, and at a later stage they come back inside the horizon possibly with richinformation on the early universe and hence on high energy physics. Though yet undetected, gravitational
waves will provide us with a powerful tool to probe fundamental physics in near future. Motivated bystring theory, recently braneworld scenarios have attached much attention, in which our four-dimensional
universe is realized as a brane embedded in a higher dimensional bulk spacetime. In order to reveal
ﬁve-dimensional eﬀects particular to the braneworld scenario, we have to focus on the scales smaller thanthe bulk curvature scale . This is our motivation to study gravitational waves from inﬂation on thebrane.
In the Randall-Sundrum braneworld, at least in the low energy regime (H  1) corrections to theevolution of gravitational waves are shown to be small [1]. To understand the evolution of gravitational
waves in a much more general and interesting case, i.e., in the high energy (H  1) radiation-dominatedphase, numerical studies have been done by Hiramatsu et al. [2] and by Ichiki and Nakamura [3]. Theirresults give us a lot of implications, for example, on the damping nature of the gravitational wave amplitude due to the Kaluza-Klein mode generation, but the initial condition they adopt is naive, neglecting
initial quantum ﬂuctuations in the Kaluza-Klein modes. Hence, its validity is open to question.
The goal of the present paper is to clarify the late time power spectrum of gravitational waves in theRandall-Sundrum brane cosmology, evolving through the radiation-dominated stage after their generation
during inﬂation. We closely follow the same line as our previous work [4], in which, using the Wronskian
formulation, we have formulated a numerical scheme for the braneworld cosmological perturbations. The
initial condition in our analysis is imposed quantum-mechanically, and therefore we will be able to obtaina true picture of the generation and evolution of gravitational perturbations in the braneworld. A detailed
discussion is done in a much longer paper [5] in collaboration with Takahiro Tanaka.

1 E-mail:tsutomu@tap.scphys.kyoto-u.ac.jp12Preliminaries

2.1

The background modelNow we describe a model for the background. We shall work in the cosmological setting of the RandallSundrum braneworld, and so the bulk is given by a ﬁve-dimensional AdS spacetime. The AdS metric inthe Poincoré coordinates is ds2 = (/z)2 (−dt2 + δij dxi dxj + dz 2 ), where  is the bulk curvature scale.
A cosmological brane moves in this static bulk, the trajectory of which is given by z = z(t). The scale
factor of the universe is related to the position of the brane as a(t) = /z(t).
We consider the following cosmological model on the brane. The initial stage of the model is given by deSitter inﬂation with a constant Hubble parameter H = Hi , which is smoothly connected to the radiationdominated phase. In the radiation stage the scale factor evolves subject to the modiﬁed Friedmann22)(1 + ρr /2σ), where ρr is the radiation energy density and σ = 6MPl/2 is theequation H 2 = (ρr /3MPltension of the brane. After a period of time the energy scale of the universe becomes suﬃciently low, andthe radiation-dominated phase is then smoothly connected to the Minkowski phase.

2.2

Minkowski and de Sitter braneworlds

Let us consider tensor perturbations in AdS spacetime bounded by a brane. We can decompose thegraviton ﬁeld into a zero mode and Kaluza-Klein (KK) modes without ambiguity when the brane ismaximally symmetric. This is the reason why the initial and ﬁnal stages of the background model are
given by the de Sitter and Minkowski phases, respectively.
√we decompose tensor perturbations hij into the spatial Fourier modes as hij = 2/(2πM5 )3/2 · 3As usuald k φk eik·x eij . Here M5 is the fundamental mass scale which is related to the four-dimensional Planck2.
mass MPl by (M5 )3 = MPl
For the analysis of perturbations from the Minkowski brane, the above Poincaré coordinate system
will be best suited. It is easy to ﬁnd mode solutions. Going to quantum theory, the graviton ﬁeld can beexpanded in terms of the zero mode and KK modes as
 ∞ 

(1)
dm Âm ϕm + Â†m ϕ∗m ,φ = Â0 ϕ0 + Â†0 ϕ∗0 +0

where Ân and Â†n (n = 0, m) are the annihilation and creation operators, respectively, of their corresponding modes. The mode functions form a complete orthonormal basis with respect to the Wronskian.
In the de Sitter braneworld we introduce another set of coordinates (η, ξ), which is related to (t, z) ast = η cosh ξ + t0 , and z = −η sinh ξ, where t0 is an arbitrary constant. In (η, ξ) frame the de Sitter brane
is located at a ﬁxed coordinate position ξ = ξb = constant, and the perturbation equation again has aseparable form. Treating φ as an operator, the graviton ﬁeld can be expanded as
 ∞


(2)
φ = â0 φ0 + â†0 φ∗0 +dν âν φν + â†ν φ∗ν ,0where ân and â†n (n = 0, ν) are the annihilation and creation operators of each mode. These mode
functions also form a complete orthonormal basis with respect to the Wronskian.

3Wronskian formulation

Due to the presence of an inﬁnite tower of Kaluza-Klein modes, cosmological perturbations in thebraneworld have inﬁnite degrees of freedom. Instead of solving an initial value problem for such a system,
it would be better to use the Wronskian formulation in order to take necessary degrees of freedom out of
inﬁnite information. In the present case, we would like to know the ﬁnal amplitude of the zero mode, andtherefore in fact what we need to do is solving the (backward) evolution of a single degree of freedom.
In the double null coordinates u = t − z and v = t + z, which will be convenient for numerical calculations, the trajectory of the brane can be speciﬁed arbitrarily by v = q(u). By a further coordinate transformation U = u and q(V ) = v, the bulk metric can be rewritten as ds2 =2[2/(q(V ) − U )]2 [−q  (V )dU dV + δij dxi dxj ]. Now in the new coordinates the position of the brane issimply given by U = V. We will use this coordinate system for actual numerical calculations.
The Klein-Gordon-type equation for a gravitational wave perturbation φ in the (U, V ) coordinates
reduces to62(∂V − q (V ) ∂U ) + q (V ) k φ = 0,(3)4∂U ∂V +q(V ) − U
supplemented by the boundary condition [∂U − ∂V ]φ|U =V = 0. The expression for the Wronskian evaluated on a constant V hypersurface is given by

3
 V
2
(X · Y ) = 2idU(X∂U Y ∗ − Y ∗ ∂U X),
(4)
q(V)
−U−∞
which is independent of the choice of the hypersurface.
In the initial de Sitter phase, the graviton ﬁeld can be expanded as Eq. (2), while in the ﬁnal Minkowski
phase it can be expanded as Eq. (1). We assume that initially the gravitons are in the de Sitter invariant
vacuum state annihilated by â0 and âν , â0 |0 = âν |0 = 0. The ﬁnal power spectrum is given byP(k):=4πk 3 2
1· Nf ,(2π)3 (M5 )3 k(5)where Nf := 0|Â†0 Â0 |0 is the number of created zero mode gravitons. The operator Â0 can be projectedout by making use of the Wronskian relations, and noting that the Wronskian is constant in time we haveÂ0 = (φ · ϕ0 )f = (φ · Φ) = (φ0 · Φ)i â0 + dν(φν · Φ)i âν + h.c.,(6)where Φ is a solution of the Klein-Gordon equation (3) whose ﬁnal conﬁguration is the zero mode function
ϕ0 in the Minkowski phase, and subscript f and i denote the quantities evaluated on the ﬁnal and initial
hypersurfaces, respectively. Thus we clearly see that ﬁnal zero mode gravitons are createdfrom thevacuum ﬂuctuations both in the initial zero mode and in the KK modes: Nf = |(φ∗0 ·Φ)i |2 + dν|(φ∗ν ·Φ)i |2 .
Correspondingly, the power spectrum (5) can be written as a sum of the two contributions:
P = P0 + PKK ,22where P0 := (k 2 /π 2 MPl)|(φ∗0 · Φ)i |2 and PKK := (k 2 /π 2 MPl)4(7)dν|(φ∗ν · Φ)i |2 .

Spectrum of gravitational waves

2De Sitter inﬂation on the brane predicts the ﬂat primordial spectrum δT2 := [2C 2 (Hi )/MPl](Hi /2π)2 [6].
During inﬂation the gravitational wave perturbations stretched to super-horizon scales, and then theystays constant until horizon reentry, with their amplitude given by δT .
For long wavelength modes with k  k∗ , where k∗ := a∗ H∗ = a∗ / labels the mode that reenters thehorizon when H = 1, the amplitude will decay as hµν ∝ a−1 after horizon reentry, because gravity on
the brane is basically described by four-dimensional general relativity in the low energy regime (H  1).
In fact, it is explicitly shown that leading order corrections to the cosmological evolution of gravitational
waves are suppressed by 2 and 2 ln  at low energies [1]. In particular, modes with k < k0 , wherek0 := a0 H0 = H0 , and H0 is the Hubble parameter evaluated at the end of the radiation-dominatedphase (i.e., just before the Minkowski phase), reenter the “horizon” in the ﬁnal Minkowski phase, andthen they begin to oscillate (but their amplitude will not decay). As a result, the mean-square vacuum
ﬂuctuations of such modes become half of the initial value, leading to P = δT2 /2 for k < k0 . For thereason mentioned above, modes with k0 < k  k∗ have the standard spectrum P = (δT2 /2)(k/k0 )−2 .
Something nontrivial may happen to gravitational waves with k >∼ k∗ . If the eﬀects of mode mixing
are neglected, only the modiﬁcation of the background expansion rate alters the spectrum for theseshort wavelength modes to P̃  (δT2 /2)(k/k∗ )−2/3 (k∗ /k0 )−2 . However, this evaluation will not be correct
because mode mixing is expected to be eﬃcient at high energies.

30.1

-2

0.01

-2.5

rKKlog( spectrum )-1.5-3-3.5

0.0001-4-4.5

0.001

00.2

0.4

0.6

0.8

11.2

1.4

00.2log(k=k )

0.4

0.6

0.8

1

1.2

1.4log(k=k ã)Figure 1: (left) Power spectrum of gravitational waves from inﬂation with H = 100, normalized by δT2 /2.
The total power spectrum is shown by red circles, while blue crosses represent the contribution only fromthe initial zero mode. Green squires indicate results from basically four-dimensional calculations, only
including the eﬀect of the modiﬁcation of the background expansion rate. (right) Contribution frominitial Kaluza-Klein ﬂuctuations.
Our procedure to obtain a correct power spectrum is as follows. We solve the perturbation equation (3)with its boundary condition set to be Φ(U, Vf ) = ϕ0 on the ﬁnal hypersurface. The numerical backward
evolution scheme we use here is the same as that used in our previous work [4], and the detailed descriptionof the scheme is found there. After obtaining the conﬁguration on the initial hypersurface we evaluate
the Wronskian to get the power spectrum.
We performed numerical calculations for the inﬂationary Hubble parameter Hi = 100. The radiationdominated phase is terminated when the Hubble parameter decreases down to H  0.03/ =: H0 , andthen it is connected smoothly to the Minkowski phase, so that we can see the amplitude of the well-deﬁned
zero mode. The power spectra of gravitational waves are shown in the left ﬁgure of Fig. 1. We ﬁnd thatirrespective of the inﬂationary energy scale, the power spectrum is approximately given byP

δT22kk0−2,(8)Namely, we have the same spectrum as the standard one even for short wavelength modes with k >∼ k∗ .
As is shown in the right ﬁgure of Fig. 1, the contribution of the vacuum ﬂuctuations in the initial KK
modes to the ﬁnal spectrum, rKK (k) := PKK /(P0 + PKK ), never exceeds 10% so far as the present
calculations are concerned, and hence it gives a subdominant eﬀect. On the other hand, the excitationof KK modes suppress the amplitude of the gravitational waves relative to P̃, and our result implies
that the eﬀect of the modiﬁcation of the background Friedmann equation compensate this suppression,leading to approximately the same spectral tilt as that in conventional four-dimensional cosmology.

References[1] T. Kobayashi and T. Tanaka, JCAP 0410, 015 (2004).
[2] T. Hiramatsu, K. Koyama and A. Taruya, Phys. Lett. B 609, 133 (2005).
[3] K. Ichiki and K. Nakamura, arXiv:astro-ph/0406606.
[4] T. Kobayashi and T. Tanaka, Phys. Rev. D 71, 124028 (2005).
[5] T. Kobayashi and T. Tanaka, arXiv:hep-th/0511186.
[6] D. Langlois, R. Maartens and D. Wands, Phys. Lett. B 489, 259 (2000).

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