No-truncation approach to cosmic microwave background anisotropies

We offer a method of calculating the source term in the line-of-sight integral for cosmic microwave background anisotropies without using a truncated partial-wave expansion in the Boltzmann hierarchy.

Figure 1

Iterative Solution of Eq. (29) for the Source Function $\Psi$. Short dashes indicate the zeroth iteration (31); longer dashes indicate the first iteration; and further iterations are indistinguishable from the solid curve, which therefore represents the solution. Here the source function is calculated for a gravitational perturbation ${\mathcal{D}}_q(t)$ whose value ${\mathcal{D}}_q^o$ before horizon entry is unity; for any other initial condition, $\Psi$ should be multiplied by the value of ${\mathcal{D}}_q^o$. The quantity $\kappa$ is the wave number $q$ in units of the wave number that just comes into the horizon at matter-radiation equality, and $y$ is the Robertson-Walker scale factor, in units of the scale factor at matter-radiation equality. The calculation was done by R. Flauger using the electron number density calculated with the program Recfast [6] and taking ${\Omega }_M{h}^2=0.133$, ${\Omega }_B{h}^2=0.02238$, $h=0.72$, $T_0=2.725\mathrm{K}$, and $Y_{\mathrm{He}}=0.24$.