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

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.