Massive Neutrinos in Cosmology: Analytic Solutions and Fluid Approximation

TL;DR

This paper evaluates the accuracy of the fluid approximation for modeling linear density fluctuations of massive neutrinos in cosmology, providing explicit solutions and quantifying its limitations at different scales and redshifts.

Contribution

It derives exact solutions of the Boltzmann equation for neutrinos and assesses the validity of the fluid approximation in a cosmological context.

Findings

Fluid approximation is accurate within 25% for certain neutrino masses and scales.

Including anisotropic stress improves the approximation at lower redshifts.

Fluid approximation accuracy depends on neutrino mass, scale, and redshift.

Abstract

We study the evolution of linear density fluctuations of free-streaming massive neutrinos at redshift of z<1000, with an explicit justification on the use of a fluid approximation. We solve the collisionless Boltzmann equation in an Einstein de-Sitter (EdS) universe, truncating the Boltzmann hierarchy at lmax=1 and 2, and compare the resulting density contrast of neutrinos, \delta_{\nu}^{fluid}, with that of the exact solutions of the Boltzmann equation that we derive in this paper. Roughly speaking, the fluid approximation is accurate if neutrinos were already non-relativistic when the neutrino density fluctuation of a given wavenumber entered the horizon. We find that the fluid approximation is accurate at few to 25% for massive neutrinos with 0.05<m_{\nu}<0.5eV at the scale of k<0.4~hMpc^{-1} and redshift of z<10. This result quantifies the limitation of the fluid approximation, for the massive neutrinos with m_{\nu}<0.5eV. We also find that the density contrast calculated from fluid equations (i.e., continuity and Euler equations) becomes a better approximation at a lower redshift, and the accuracy can be further improved by including an anisotropic stress term in the Euler equation. The anisotropic stress term effectively increases the pressure term by a factor of 9/5.