

7 Gravitational Wave
Perturbations in Brane-World Cosmology
The tensor perturbations are given by Equation (267), i.e., (for a flat
background brane),
The transverse traceless
satisfies Equation (287), which implies, on
splitting
into Fourier modes with amplitude
,
By the transverse traceless part of Equation (273), the boundary
condition is
where
is the tensor part of the anisotropic stress of
matter-radiation on the brane.
The wave equation (334) 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
. This includes the
RS case
already treated in Section 2.
The cosmologically relevant case is the de Sitter brane,
. We can calculate the spectrum of gravitational
waves generated during brane inflation [192
, 121, 102
, 167], if we approximate
slow-roll inflation by a succession of de Sitter phases. The metric
for a de Sitter brane
in
is given by Equations (180, 181, 182) with
where
.
The linearized wave equation (334) is separable. As
before, we separate the amplitude as
where
is the 4D mass, and this leads to:
The general solutions for
are
where
is a linear combination of Bessel functions,
is a linear combination of associated Legendre
functions, and
It is more useful to reformulate
Equation (342) as a Schrödinger-type
equation,
using the conformal coordinate
and defining
. The potential is given by
(see Figure 10)
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
brane [161, 296].)
The non-zero value of the Hubble parameter
implies the existence of a mass gap [104],
between the zero mode and the continuum of massive KK modes. This
result has been generalized: For
brane(s) with bulk
scalar field, a universal lower bound on the mass gap of the KK
tower is [102
]
The massive modes decay during inflation, according to
Equation (343), leaving only the
zero mode, which is effectively a 4D gravitational wave. The zero
mode, satisfying the boundary condition
is given by
where the normalization condition
implies that the function
is given by [192]
At low energies (
) we recover the general
relativity amplitude:
. At high energies, the
amplitude is considerably enhanced:
The factor
determines the modification of the
gravitational wave amplitude relative to the standard 4D result:
The modifying factor
can also be interpreted as a change in
the effective Planck mass [102].
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
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 [200
]. Self-consistent
low-energy approximations to compute this effect are developed
in [142
, 91
].
At zero order, the low-energy approximation is based
on the following [235
, 237
, 22
]. In the radiation
era, at low energy, the background metric functions obey
To lowest order, the wave equation therefore separates, and the
mode functions can be found analytically [235, 237, 22]. The massive modes in
the bulk,
, 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
where
marks the start of the low-energy regime (
), and
denotes a linear combination
of Bessel functions. The massive modes decay on super-Hubble
scales, unlike the zero-mode. Expanding the wave equation in
, one arrives at the first order, where mode-mixing
arises. The massive modes
on sub-Hubble
scales are sourced by the initial zero mode that is re-entering the
Hubble radius [91
]:
where
is a transfer matrix coefficient. The numerical
integration of the equations [142
] confirms the effect
of massive mode generation and consequent damping of the zero-mode,
as shown in Figure 11.

