The dark matter transfer function: free streaming, particle statistics and memory of gravitational clustering

TL;DR

This paper derives an exact transfer function for dark matter perturbations during matter domination, accounting for particle distribution functions and initial conditions, revealing effects on structure formation at various scales.

Contribution

It provides a novel exact expression for the dark matter transfer function applicable to arbitrary particle distributions and initial conditions, improving understanding of gravitational clustering effects.

Findings

Exact transfer function derived for arbitrary distributions

Approximate formula valid across all relevant scales

Quantitative estimates for free streaming scales of different dark matter models

Abstract

The transfer function $T(k)$ of dark matter (DM) perturbations during matter domination is obtained by solving the collisionless Boltzmann-Vlasov equation. We find an \emph{exact} expression for $T(k)$ for \emph{arbitrary} distribution functions of decoupled particles and initial conditions}. We find a remarkably accurate and simple approximation valid on all scales of cosmological relevance for structure formation in the linear regime. The natural scale of suppression is the free streaming wavevector at matter-radiation equality, $ k_{fs}(t_{eq}) = [{4\pi\rho_{0M}}/{[< \vec{V}^2> (1+z_{eq})]} ]^\frac12 $. An important ingredient is a non-local kernel determined by the distribution functions of the decoupled particles which describes the \emph{memory of the initial conditions and gravitational clustering} and yields a correction to the fluid description. Distribution functions that favor the small momentum region lead to an \emph{enhancement of power at small scales} $ k > k_{fs}(t_{eq}) $. For DM thermal relics that decoupled while ultrarelativistic we find $ k_{fs}(t_{eq}) \simeq 0.003 (g_d/2)^\frac13 (m/\mathrm{keV}) [\mathrm{kpc}]^{-1} $, where $ g_d $ is the number of degrees of freedom at decoupling. For WIMPS we obtain $ k_{fs}(t_{eq}) = 5.88 (g_d/2)^\frac13 (m/100 \mathrm{GeV})^\frac12 (T_d/10 \mathrm{MeV})^\frac12 [\mathrm{pc}]^{-1} $. For $k\ll k_{fs}(t_{eq})$, $T(k) \sim 1-\mathrm{C}[k/k_{fs}(t_{eq})]^2 $ where $C =\mathrm{O}(1)$ for all cases considered and simple and accurate fits for \emph{small} scales.