A Complete Treatment of CMB Anisotropies in a FRW Universe

TL;DR

This work extends the total angular momentum method to FRW universes with constant curvature $K$, enabling a unified treatment of CMB temperature and polarization from scalar, vector, and tensor perturbations. It introduces eigenmodes and normal modes based on curved-space geometry, recasts the Boltzmann equation into line-of-sight integral solutions, and yields efficient linear-theory predictions suitable for open and closed cosmologies. The formalism is implemented in CMBFAST with LOS, and is used to compare open and critical-density models, highlighting curvature effects on angular features and polarization, as well as the ISW contribution. The framework also accommodates open-inflation spectra and provides a robust platform for tensor perturbations in open geometries.

Abstract

We generalize the total angular momentum method for computing Cosmic Microwave Background anisotropies to Friedman-Robertson-Walker (FRW) spaces with arbitrary geometries. This unifies the treatment of temperature and polarization anisotropies generated by scalar, vector and tensor perturbations of the fluid, seed, or a scalar field, in a universe with constant comoving curvature. The resulting formalism generalizes and simplifies the calculation of anisotropies and, in its integral form, allows for a fast calculation of model predictions in linear theory for any FRW metric.

Figures (2)

• Figure 1: The scalar (left) and tensor (right) angular power spectra for anisotropies in a critical density model (thick lines) and an open model (thin lines) with $\Omega_0=0.4$. Solid lines are $C_l^{\Theta\Theta}$, dashed $C_l^{EE}$ and dotted $C_l^{BB}$.
• Figure 2: The scalar (left) and tensor (right) temperature-polarization cross correlation $C_l^{\Theta E}$ with the same parameters and notation as Fig. \ref{['fig:temp']} (thick: flat; thin open). Dotted lines represent negative correlation.