CMB Anisotropies in Brane-World Cosmology

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

where

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 [201 ] 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 $ℰ μν$ ):

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 [201

]

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 (243 ). Once $𝒢$ and $Fk$ 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.

8.1 The low-energy approximation
8.2 The simplest model