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.

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