Geodesic "curve"-of-sight formulae for the cosmic microwave background: a unified treatment of redshift, time delay, and lensing

TL;DR

This paper presents a unified, second-order line-of-sight formula for gravitational effects on CMB anisotropies, integrating the Boltzmann equation along perturbed geodesics to distinguish gravitational and intrinsic contributions.

Contribution

It introduces a novel second-order line-of-sight approach based on the Boltzmann equation, unifying redshift, time delay, and lensing effects on the CMB.

Findings

Derivation of a second-order line-of-sight formula for CMB anisotropies.

Unified treatment of gravitational effects on the CMB.

Clear separation of gravitational and intrinsic effects.

Abstract

In this paper, we introduce a new approach to a treatment of the gravitational effects (redshift, time delay and lensing) on the observed cosmic microwave background (CMB) anisotropies based on the Boltzmann equation. From the Liouville's theorem in curved spacetime, the intensity of photons is conserved along a photon geodesic when non-gravitational scatterings are absent. Motivated by this fact, we derive a second-order line-of-sight formula by integrating the Boltzmann equation along a perturbed geodesic (curve) instead of a background geodesic (line). In this approach, the separation of the gravitational and intrinsic effects are manifest. This approach can be considered as a generalization of the remapping approach of CMB lensing, where all the gravitational effects can be treated on the same footing.