7.1 Analytical approaches
The tensor perturbations are given by Equation (273), i.e., (for a flat background brane),
The transverse traceless
satisfies Equation (293), which implies, on splitting
into Fourier modes
with amplitude
,
By the transverse traceless part of Equation (279), the boundary condition is
where
is the tensor part of the anisotropic stress of matter-radiation on the brane.
The wave equation (355) 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 [272
, 176, 148
, 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 (186, 187, 188) with
where
.
The linearized wave equation (355) 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 (363) as a Schrödinger-type equation,
using the conformal coordinate
and defining
. The potential is given by (see Figure 13)
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],
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 [148
]
The massive modes decay during inflation, according to Equation (364), 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 [272]
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 [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
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 [281
]. Self-consistent low-energy approximations to compute this effect are developed
in [201
, 136
].
Update
At zero order, the low-energy approximation is based on the following [321
, 323
, 27
]. 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 [321, 323, 27]. 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 [136
]:
where
is a transfer matrix coefficient. The numerical integration of the equations [201
] confirms the
effect of massive mode generation and consequent damping of the zero-mode.