7.1 Analytical approaches

The tensor perturbations are given by Equation (273View Equation), i.e., (for a flat background brane),
(5)ds2 = − N 2(t,y)dt2 + A2(t,y)[δij + fij]dxidxj + dy2. (354 )
The transverse traceless fij satisfies Equation (293View Equation), which implies, on splitting fij into Fourier modes with amplitude f(t,y),
[ ( ) ] 1 ˙A ˙N k2 ( A ′ N ′) --- f¨+ 3 --− --- f˙ + ---f = f′′ + 3---+ --- f ′. (355 ) N 2 A N A2 A N
By the transverse traceless part of Equation (279View Equation), the boundary condition is
| f′ij|brane = ¯πij, (356 )
where ¯πij is the tensor part of the anisotropic stress of matter-radiation on the brane.

The wave equation (355View Equation) cannot be solved analytically except if the background metric functions are separable, and this only happens for maximally symmetric branes, i.e., branes with constant Hubble rate H0. This includes the RS case H0 = 0 already treated in Section 2. The cosmologically relevant case is the de Sitter brane, H0 > 0. We can calculate the spectrum of gravitational waves generated during brane inflation [272Jump To The Next Citation Point, 176, 148Jump To The Next Citation Point, 232], if we approximate slow-roll inflation by a succession of de Sitter phases. The metric for a de Sitter brane dS4 in AdS5 is given by Equations (186View Equation, 187View Equation, 188View Equation) with

N (t,y) = n (y ), (357 ) A (t,y) = a(t)n(y), (358 ) ( ρ0) n(y) = cosh μy − 1 + λ sinh μ|y|, (359 ) a(t) = a expH (t − t), (360 ) 02 ( 0 ) 0 H2 = κ--ρ 1 + ρ0- , (361 ) 0 3 0 2λ
where μ = ℓ−1.
View Image

Figure 13: Graviton “volcano” potential around the dS4 brane, showing the mass gap. (Figure taken from [270].)

The linearized wave equation (355View Equation) is separable. As before, we separate the amplitude as ∑ f = φm (t)fm (y ) where m is the 4D mass, and this leads to:

[ k2] φ¨m + 3H0 φ˙m + m2 + -2- φm = 0, (362 ) a ′′ n ′ ′ m2 fm + 4--f m + --2 fm = 0. (363 ) n n
The general solutions for m > 0 are
( ) ( ) 3- -k- − H0t φm (t) = exp − 2 H0t B ν H0 e , (364 ) ( ∘ -------------) −3∕2 ν μ2 fm (y) = n(y) L3∕2 1 + H2- n(y)2 , (365 ) 0
where Bν is a linear combination of Bessel functions, L ν 3∕2 is a linear combination of associated Legendre functions, and
∘ -------- 2 ν = i m--− 9. (366 ) H20 4

It is more useful to reformulate Equation (363View Equation) as a Schrödinger-type equation,

2 d-Ψm--− V (z)Ψm = − m2 Ψm, (367 ) dz2
using the conformal coordinate
∫ y ( ) z = zb + -d&tidle;y-, zb = -1-sinh− 1 H0- , (368 ) 0 n(&tidle;y) H0 μ
and defining 3∕2 Ψm ≡ n fm. The potential is given by (see Figure 13View Image)
15H2 9 ( ρ ) V (z ) = -----2-0-----+ -H20 − 3μ 1 + --0 δ (z − zb), (369 ) 4sinh (H0z ) 4 λ
where the last term incorporates the boundary condition at the brane. The “volcano” shape of the potential shows how the 5D graviton is localized at the brane at low energies. (Note that localization fails for an dS4 brane [225, 396].)

The non-zero value of the Hubble parameter implies the existence of a mass gap [154],

3- Δm = 2H0, (370 )
between the zero mode and the continuum of massive KK modes. This result has been generalized: For dS4 brane(s) with bulk scalar field, a universal lower bound on the mass gap of the KK tower is [148Jump To The Next Citation Point]
∘ -- 3- Δm ≥ 2 H0. (371 )
The massive modes decay during inflation, according to Equation (364View Equation), leaving only the zero mode, which is effectively a 4D gravitational wave. The zero mode, satisfying the boundary condition
f′(x,0) = 0, (372 ) 0
is given by
√ -- f0 = μF (H0 ∕μ), (373 )
where the normalization condition
∫ ∞ 2 |Ψ20|dz = 1 (374 ) zb
implies that the function F is given by [272]
{ [ ∘ ------]} − 1∕2 √ ------- 2 1 1 F (x) = 1 + x2 − x ln --+ 1 + -2- . (375 ) x x
At low energies (H ≪ μ 0) we recover the general relativity amplitude: F → 1. At high energies, the amplitude is considerably enhanced:
∘ ----- 3H0 H0 ≫ μ ⇒ F ≈ ----. (376 ) 2μ
The factor F determines the modification of the gravitational wave amplitude relative to the standard 4D result:
[ ] 8 ( H0 )2 A2t = --2- --- F 2(H0∕μ ). (377 ) M p 2π
The modifying factor F can also be interpreted as a change in the effective Planck mass [148].

This enhanced zero mode produced by brane inflation remains frozen outside the Hubble radius, as in general relativity, but when it re-enters the Hubble radius during radiation or matter domination, it will no longer be separated from the massive modes, since H will not be constant. Instead, massive modes will be excited during re-entry. In other words, energy will be lost from the zero mode as 5D gravitons are emitted into the bulk, i.e., as massive modes are produced on the brane. A phenomenological model of the damping of the zero mode due to 5D graviton emission is given in [281Jump To The Next Citation Point]. Self-consistent low-energy approximations to compute this effect are developed in [201Jump To The Next Citation Point, 136Jump To The Next Citation Point]. UpdateJump To The Next Update Information

At zero order, the low-energy approximation is based on the following [321Jump To The Next Citation Point, 323Jump To The Next Citation Point, 27Jump To The Next Citation Point]. In the radiation era, at low energy, the background metric functions obey

A(t,y) → a(t)e− μy, N (t,y) → e− μy. (378 )
To lowest order, the wave equation therefore separates, and the mode functions can be found analytically [321, 323, 27]. The massive modes in the bulk, fm(y), are the same as for a Minkowski brane. On large scales, or at late times, the mode functions on the brane are given in conformal time by
( ma2 μ ) φ (0m)(η) = η−1∕2B1∕4 -√-h--η2 , (379 ) 2
where ah marks the start of the low-energy regime (ρh = λ), and B ν denotes a linear combination of Bessel functions. The massive modes decay on super-Hubble scales, unlike the zero-mode. Expanding the wave equation in ρ0∕ λ, one arrives at the first order, where mode-mixing arises. The massive modes (1) φ m (η) on sub-Hubble scales are sourced by the initial zero mode that is re-entering the Hubble radius [136Jump To The Next Citation Point]:
( 2 2 ) -∂--− ∂ηa- a φ(1) + k2aφ (1)+ m2a3 φ(1) = − 4ρ0-Im0k2aφ (0), (380 ) ∂η2 a m m m λ 0
where Im0 is a transfer matrix coefficient. The numerical integration of the equations [201Jump To The Next Citation Point] confirms the effect of massive mode generation and consequent damping of the zero-mode.
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