Massive Neutrinos in Cosmology: Analytic Solutions and Fluid Approximation
arXiv:1003.0942 · doi:10.1103/PhysRevD.81.123516 10.1103/PhysRevD.82.089901
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.
18 pages, 7 figures. (v2) with up-dated reference, minor typos corrected. (v3) An error in the normalization on the scale factor has been corrected. We now find that the fluid approximation is accurate at a few~25% accuracy. This version supersedes the published version. The code for calculating exact solutions of Ψ_l is publicly available at http://gyudon.as.utexas.edu/~komatsu/CRL/index.html