Deriving the Dark Matter-Dark Energy Interaction Term in the Continuity Equation from the Boltzmann Equation

TL;DR

This paper derives an analytical expression for the dark matter-dark energy interaction term from the Boltzmann equation, considering a quantum scalar field for dark energy, and discusses implications for observational constraints.

Contribution

It provides a novel analytical derivation of the interaction kernel in the continuity equation from fundamental quantum field assumptions.

Findings

Interaction kernel $Q$ is very small, consistent with expectations.

A finite distribution function for dark energy with negative equation of state is obtained.

Method can be adapted for other dark sector couplings.

Abstract

Dark energy and dark matter are two of the biggest mysteries of modern cosmology, and our understanding of their fundamental nature is incomplete. Many parameterizations of couplings between the two in the continuity equation have been studied in the literature, and observational data from the growth of perturbations can constrain these parameterizations. Assuming standard general relativity with a simple Yukawa-type coupling between dark energy and dark matter fields in the Lagrangian, we use the Boltzmann equation to analytically express and calculate the interaction kernel $Q$ in the continuity equation and compare it to that of a typical parametrization. We arrive at a comparably very small result, as expected. Since the interaction is a function of the dark matter mass, other observational data sets can be used to constrain the mass. This calculation can be modified to account for other couplings of the dark energy and dark matter fields. This calculation required obtaining a distribution function for dark energy that leads to an equation of state parameter that is negative, which neither Bose-Einstein nor Fermi-Dirac statistics can supply, and this is the main result of this paper. Treating dark energy as a quantum scalar field, we use adiabatic subtraction to obtain a finite analytic approximation for its distribution function that assumes the FLRW metric and nothing more.