Go to previous page Go up Go to next page

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

( ) ρtot = ρ 1 + -ρ-+ ρℰ- , (360 ) 2λ ρ ρ-- ρℰ- ptot = p + 2λ(2p + ρ) + 3 , (361 ) tot ( ρ) ℰ qμ = qμ 1 + -- + qμ, (362 ) ( λ ) πtot= πμν 1 − ρ-+-3p- + πℰ , (363 ) μν 2λ μν
∑ ∑ ∑ (i) ρ = ρ(i), p = p(i), qμ = qμ (364 ) i i i
are the total matter-radiation density, pressure, and momentum density, respectively, and πμν 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 [201Jump To The Next Citation Point] and tensor [200] perturbations. But the systems are not closed, as discussed above, because of the presence of the KK anisotropic stress πℰμν, 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 Φ, defined by the electric part of the brane Weyl tensor (not to be confused with ℰ μν):

E = ⃗∇ ⃗∇ Φ. (365 ) μν ⟨μ ν⟩
In general relativity, the equation in Φ has no source term, but in the brane-world there is a source term made up of πℰ μν and its time-derivatives. At low energies (ρ ≪ λ), and for a flat background (K = 0), the equation is [201Jump To The Next Citation Point]
[ ( ) ] 3x Φ′′+ 12Φ ′ + xΦ = const. π ℰ′′− 1π ℰ′+ -2-− 3--+ 1- πℰ , (366 ) k k k λ k x k x3 x2 x k
where x = k∕(aH ), a prime denotes d∕dx, and Φk and ℰ π k are the Fourier modes of Φ and ℰ π μν, respectively. In general relativity the right hand side is zero, so that the equation may be solved for Φk, 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]:
729x2 Φ′k′′′+ 3888x Φ′k′′+ (1782 + 54x2)Φ ′′k + 144x Φ ′k + (90 + x2)Φk [ ( ℰ )′′′′ ( ℰ )′′′ 2 ( ℰ) ′′ = const.× 243 πk- − 810- πk- + 18(135-+-2x-)- πk- ρ x ρ x2 ρ ( )′ ( ) ] 30(162-+-x2-) πℰk- x4-+-30(162-+-x2)- πℰk- − x3 ρ + x4 ρ . (367 )
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 πℰ μν. 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 ℰ πμν. This solution will be of the form given in Equation (243View Equation). Once 𝒢 and Fk are determined or estimated, the numerical integration in Equation (243View Equation) 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.

 8.1 The low-energy approximation
 8.2 The simplest model

  Go to previous page Go up Go to next page