CMB and Random Flights: temperature and polarization in position space

TL;DR

This paper reformulates the CMB temperature and polarization fluctuations using a position-space approach involving random flight probabilities, providing new analytical tools and insights into the equilibrium behavior of the photon distribution.

Contribution

It introduces a position-space, causal description of CMB fluctuations using random flight probabilities and employs Fourier-Bessel decomposition for analytical calculations.

Findings

Random flight probabilities are key to describing CMB fluctuations.

The Fourier-Bessel decomposition facilitates calculation of probability densities.

The H-theorem implies temperature anisotropies and polarization vanish at equilibrium.

Abstract

The fluctuations in the temperature and polarization of the cosmic microwave background are described by a hierarchy of Boltzmann equations. In its integral form, this Boltzmann hierarchy can be converted from the usual Fourier-space base into a position-space and causal description. We show that probability densities for random flights play a key role in this description. The integral system can be treated as a perturbative series in the number of steps of the random flights, and the properties of random flight probabilities impose constraints on the domains of dependence. We show that, as a result of these domains, a Fourier-Bessel decomposition can be employed in order to calculate the random flight probability densities. We also illustrate how the H-theorem applies to the cosmic microwave background: by using analytical formulae for the asymptotic limits of these probability densities, we show that, as the photon distribution approaches a state of equilibrium, both the temperature anisotropies and the net polarization must vanish.