A fully covariant description of CMB anisotropies

TL;DR

This paper derives a comprehensive covariant and gauge-invariant formula for CMB temperature anisotropies, incorporating non-linear effects and extending previous first-order approaches, with implications for understanding large-scale anisotropies.

Contribution

It provides a fully covariant, gauge-invariant framework for CMB anisotropies based on the Weyl tensor, extending earlier linear work to include non-linear effects and detailed physical contributions.

Findings

Derived a covariant formula for CMB anisotropies using Weyl tensor components.

Reproduced the Sachs-Wolfe effect and included Rees-Sciama, entropy, and velocity contributions.

Validated the interpretation of large-scale anisotropies as matter motion relative to the last scattering surface.

Abstract

Starting from the exact non-linear description of matter and radiation, a fully covariant and gauge-invariant formula for the observed temperature anisotropy of the cosmic microwave background (CBR) radiation, expressed in terms of the electric ($E_{ab}$) and magnetic ($H_{ab}$) parts of the Weyl tensor, is obtained by integrating photon geodesics from last scattering to the point of observation today. This improves and extends earlier work by Russ et al where a similar formula was obtained by taking first order variations of the redshift. In the case of scalar (density) perturbations, $E_{ab}$ is related to the harmonic components of the gravitational potential $\Phi_k$ and the usual dominant Sachs-Wolfe contribution $\delta T_R/\bar{T}_R\sim\Phi_k$ to the temperature anisotropy is recovered, together with contributions due to the time variation of the potential (Rees-Sciama effect), entropy and velocity perturbations at last scattering and a pressure suppression term important in low density universes. We also explicitly demonstrate the validity of assuming that the perturbations are adiabatic at decoupling and show that if the surface of last scattering is correctly placed and the background universe model is taken to be a flat dust dominated Friedmann-Robertson-Walker model (FRW), then the large scale temperature anisotropy can be interpreted as being due to the motion of the matter relative to the surface of constant temperature which defines the surface of last scattering on those scales.