

8 CMB Anisotropies in
Brane-World Cosmology
For the CMB anisotropies, one needs to consider a multi-component
source. Linearizing the general nonlinear expressions for the total
effective energy-momentum tensor, we obtain
where
are the total matter-radiation density, pressure, and momentum
density, respectively, and
is the photon anisotropic
stress (neglecting that of neutrinos, baryons, and CDM).
The perturbation equations in the previous
Section 7 form the basis for an analysis of
scalar and tensor CMB anisotropies in the brane-world. The full
system of equations on the brane, including the Boltzmann equation
for photons, has been given for scalar [201
] and
tensor [200] perturbations. But the systems are not
closed, as discussed above, because of the presence of the KK
anisotropic stress
, which acts a source
term.
In the tight-coupling radiation era, the scalar
perturbation equations may be decoupled to give an equation for the
gravitational potential
, defined by the electric
part of the brane Weyl tensor (not to be confused with
):
In general relativity, the equation in
has no source term,
but in the brane-world there is a source term made up of
and its time-derivatives. At low energies (
), and for a flat background (
), the equation is [201
]
where
, a prime denotes
, and
and
are the Fourier
modes of
and
, respectively.
In general relativity the right hand side is zero, so that the
equation may be solved for
, and then for the remaining
perturbative variables, which gives the basis for initializing CMB
numerical integrations. At high energies, earlier in the radiation
era, the decoupled equation is fourth order [201]:
The formalism and machinery are ready to compute the temperature
and polarization anisotropies in brane-world cosmology, once a
solution, or at least an approximation, is given for
. The resulting power spectra will reveal the nature
of the brane-world imprint on CMB anisotropies, and would in
principle provide a means of constraining or possibly falsifying
the brane-world models. Once this is achieved, the implications for
the fundamental underlying theory, i.e., M theory, would need
to be explored.
However, the first step required is the solution
for
. This solution will be of the form given in
Equation (243). Once
and
are determined or estimated, the
numerical integration in Equation (243) can in principle be
incorporated into a modified version of a CMB numerical code. The
full solution in this form represents a formidable problem, and one
is led to look for approximations.

