Evolution of cosmological dark matter perturbations

TL;DR

This work addresses how dark matter perturbations with non-zero velocity dispersion, particularly massive neutrinos, influence CMB anisotropies and structure formation. It introduces a covariant two-dimensional momentum-integrated multipole hierarchy derived from a PSTF expansion of the distribution function, with velocity-weight and mass-based expansions to cover both relativistic and non-relativistic regimes. An efficient approximate scheme, plus a truncated hierarchy for late times, yields ~1% accuracy for CMB power spectra and substantial computational speedups over traditional methods, while the full distribution-function approach remains available for high-precision matter power spectra. The framework and accompanying scalar-mode code integrate smoothly with existing cosmological perturbation tools (e.g., CAMB), enabling rapid inclusion of massive neutrino effects in parameter estimation and broad applicability to other warm/hot dark matter distributions.

Abstract

We discuss the propagation of dark matter perturbations with non-zero velocity dispersion in cosmological models. In particular a non-zero massive neutrino component may well have a significant effect on the matter power spectrum and cosmic microwave background anisotropy. We present a covariant analysis of the evolution of a dark matter distribution via a two-dimensional momentum-integrated hierarchy of multipole equations. This can be expanded in the velocity weight to provide accurate approximate equations if the matter is non-relativistic, and we also perform an expansion in the mass to study the propagation of relativistic matter perturbations. We suggest an approximation to the exact hierarchy that can be used to calculate efficiently the effect of the massive neutrinos on the CMB power spectra. We implement the corresponding scalar mode equations numerically achieving a considerable reduction in computation time compared with previous approaches.

Figures (2)

• Figure 1: The change in the scalar CMB temperature (solid lines) and E polarization (dashed line) power spectra compared to a CDM model, with $\Omega_\nu h^2 = 6.7\times 10^{-3}$, $N_\nu=3$ (top panel) and $\Omega_\nu h^2 = 2\times 10^{-3}$, $N_\nu=1$ (bottom panel) computed by integrating the distribution function perturbations. The dotted lines show the results from using the approximate Eq. \ref{['crude']}, which agree very closely almost everywhere. We used $h=0.69$, $\Omega_\Lambda=0.7$, $\Omega_K=0$, $\Omega_b h^2 = 0.022$.
• Figure 2: The evolution of $k=0.005 \text{Mpc}^{-1}$(top panel) and $k=0.01\text{Mpc}^{-1}$ (bottom panel) scalar modes, with three degenerate massive neutrinos giving $\Omega_\nu h^2 = 6.7\times 10^{-3}$. The blue and black dashed lines respectively show the background ionization fraction and $3 p_\nu/\rho_\nu$. The solid black and red lines show the massive neutrino density perturbation and momentum density, the dotted lines show the result of using Eq. \ref{['crude']} rather than integrating the distribution function. The green and cyan lines show the massless neutrino and photon perturbations. The perturbations are scaled corresponding to an initial curvature perturbation of $0.2$ and are evaluated in the zero acceleration frame.