Dark matter transfer function: Free streaming, particle statistics, and memory of gravitational clustering

Abstract

The transfer function $T(k)$ of dark matter (DM) perturbations during matter domination is obtained by solving the linearized collisionless Boltzmann-Vlasov equation. We provide an exact expression for $T(k)$ for arbitrary distribution functions of decoupled particles and initial conditions, which can be systematically expanded in a Fredholm series. An exhaustive numerical study of thermal relics for different initial conditions reveals that the first two terms in the expansion of $T(k)$ provide 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 wave vector at matter-radiation equality, ${k}_{fs}({t}_{\mathrm{eq}})=[4\ensuremath{\pi}{\ensuremath{\rho}}_{0M}/[⟨{\stackrel{\ensuremath{\rightarrow}}{V}}^{2}⟩(1+{z}_{\mathrm{eq}})]{]}^{1/2}$. An important ingredient is a nonlocal kernel determined by the distribution functions of the decoupled particles which describes the memory of the initial conditions and gravitational clustering and yields a correction to the fluid description. This correction is negligible at large scales $k\ensuremath{\ll}{k}_{fs}({t}_{\mathrm{eq}})$ but it becomes important at small scales $k\ensuremath{\ge}{k}_{fs}({t}_{\mathrm{eq}})$. Distribution functions that favor the small momentum region yield longer-range memory kernels and lead to an enhancement of power at small scales $k>{k}_{fs}({t}_{\mathrm{eq}})$. Fermi-Dirac and Bose-Einstein statistics lead to long-range memory kernels, with longer-range for bosons, both resulting in enhancement of $T(k)$ at small scales. For DM thermal relics that decoupled while ultrarelativistic we find ${k}_{fs}({t}_{\mathrm{eq}})\ensuremath{\simeq}0.003({g}_{d}/2{)}^{1/3}\text{ }\text{ }(m/\mathrm{keV})\text{ }[\mathrm{kpc}{]}^{\ensuremath{-}1}$, where ${g}_{d}$ is the number of degrees of freedom at decoupling. For WIMPS we obtain ${k}_{fs}({t}_{\mathrm{eq}})=5.88({g}_{d}/2{)}^{1/3}\text{ }\text{ }(m/100\text{ }\text{ }\mathrm{GeV}{)}^{1/2}\text{ }({T}_{d}/10\text{ }\text{ }\mathrm{MeV}{)}^{1/2}\text{ }[\mathrm{pc}{]}^{\ensuremath{-}1}$. For $k\ensuremath{\ll}{k}_{fs}({t}_{\mathrm{eq}})$, $T(k)\ensuremath{\sim}1\ensuremath{-}\mathrm{C}[k/{k}_{fs}({t}_{\mathrm{eq}}){]}^{2}$ where $C=\mathrm{O}(1)$ and independent of statistics for thermal relics. We provide simple and accurate fits for $T(k)$ in a wide range of small scales $k>{k}_{fs}({t}_{\mathrm{eq}})$ for thermal relics and different initial conditions. The numerical and analytic results for arbitrary distribution functions and initial conditions allow an assessment of DM candidates through their impact on structure formation.