Too hot to handle? Analytic solutions for massive neutrino or warm dark matter cosmologies

TL;DR

This paper derives exact solutions for cosmological models with a 'hot matter' component, modeling neutrinos and warm dark matter, and analyzes their impact on cosmic evolution and perturbations.

Contribution

It introduces a novel closed-form solution for the Friedmann equation with a smoothly transitioning hot matter component, accurately modeling neutrinos and warm dark matter.

Findings

Exact solutions match neutrino evolution within 1.5% accuracy

Closed-form solutions for super-horizon perturbations in various cosmologies

Generalization to arbitrary momentum distributions achieved

Abstract

We obtain novel closed form solutions to the Friedmann equation for cosmological models containing a component whose equation of state is that of radiation $(w=1/3)$ at early times and that of cold pressureless matter $(w=0)$ at late times. The equation of state smoothly transitions from the early to late-time behavior and exactly describes the evolution of a species with a Dirac Delta function distribution in momentum magnitudes $|\vec{p}_0|$ (i.e. all particles have the same $|\vec{p}_0|$). Such a component, here termed "hot matter", is an approximate model for both neutrinos and warm dark matter. We consider it alone and in combination with cold matter and with radiation, also obtaining closed-form solutions for the growth of super-horizon perturbations in each case. The idealized model recovers $t(a)$ to better than $1.5\%$ accuracy for all $a$ relative to a Fermi-Dirac distribution (as describes neutrinos). We conclude by adding the second moment of the distribution to our exact solution and then generalizing to include all moments of an arbitrary momentum distribution in a closed form solution.