Massive Neutrinos in Cosmology: Analytic Solutions and Fluid Approximation

TL;DR

This work assesses the validity of treating free-streaming massive neutrinos as a fluid in linear cosmological perturbation theory. By solving the collisionless Boltzmann equation exactly in an Einstein–de Sitter universe and comparing to fluid-limit solutions truncated at $l_{\rm max}=1$ and $2$, the authors quantify the accuracy of the fluid approximation across neutrino masses, wavenumbers, and redshifts. They find the fluid approach is typically accurate to a few to 25 percent for $k\lesssim 0.4\,h\mathrm{Mpc}^{-1}$ and $z\lesssim 10$ when $0.05\lesssim m_\nu\lesssim 0.5$ eV, with improvements when neutrinos become non-relativistic before horizon entry; including anisotropic stress effectively increases pressure by a factor of $9/5$. The results provide concrete guidance on when a fluid treatment is adequate and how to refine it (e.g., higher $l_{\rm max}$ or anisotropic-stress ansatz) for incorporating neutrino physics into perturbation theory and mildly non-linear modeling of large-scale structure.

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, δ_ν^{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_ν<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_ν<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.

Figures (7)

• Figure 1: Free-streaming scale of a massive neutrino, $k_{\rm FS,i}$, (black line), comoving horizon scale, $aH(a)$, (thick black line) and an approximation to the free-streaming scale in the non-relativistic limit given by Eq.(\ref{['eq:kfs']}), (dotted line) as functions of the scale factor, $a$. We use $m_{\rm\nu,i}=0.13~{\rm eV}$. The horizontal lines show (1) large, (2) small, and (3) intermediate scale modes as described in § \ref{['sec:freestreaming']}.
• Figure 2: We show $\tilde{\Psi}_0(k,q,x)$ as functions of $x\equiv k\tau$ with two different scales ($k/C=10$ and $100$, where $C=0.0092~h~{\rm Mpc^{-1}}$) and two different momenta ($m/q=10$ and $100$). $\tilde{\Psi}_0(k,q,x)$ is calculated from the exact solution, and the fractional difference is given as $\Delta\tilde{\Psi}_0/\tilde{\Psi}_0\equiv\tilde{\Psi}_0^{\rm fluid}/\tilde{\Psi}_0^{\rm exact}-1$, where the solid line is for $l_{\rm max}=1$, the dotted line is for $l_{\rm max}=2$ and the dashed line is for $l_{\rm max}=3$.
• Figure 3: Same as Figure \ref{['psi0']} for $\tilde{\Psi}_1(k,q,x)$.
• Figure 4: We show $\tilde{\Psi}_2(k,q,x)/\tilde{\Psi}_0(k,q,x)$ as functions of $x\equiv k\tau$ with two different scales ($k/C=10$ and $100$) and two different momenta ($m/q=10$ and $100$). Both $\tilde{\Psi}_0(k,q,x)$ and $\tilde{\Psi}_2(k,q,x)$ are calculated from the exact solution (solid line), or fluid approximation with $l_{\rm max}=2$ (dotted line) and $3$ (dashed line).
• Figure 5: