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

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.

40 pages, 3 figures; v2: published in JCAP, references added, typos corrected, a minor revision in sec. 4.1

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• Unified approach to secondary effects on the CMB B-mode polarization
• Scalar induced gravitational waves review
• CMB anisotropies from patchy reionisation and diffuse Sunyaev-Zel'dovich effects
• CMB anisotropies at all orders: the non-linear Sachs-Wolfe formula
• Observable Gravitational Wave Strain at Second Order
• The intrinsic bispectrum of the CMB from isocurvature initial conditions