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 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 [282] and tensor [281] 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
):
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 (249
). Once
and
are determined or estimated, the numerical integration in
Equation (249
) 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.
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