Cosmological evolution of warm dark matter fluctuations. I. Efficient computational framework with Volterra integral equations

We study the complete cosmological evolution of dark matter (DM) density fluctuations for DM particles that decoupled being ultrarelativistic during the radiation dominated era which is the case of keV scale warm DM (WDM). The new framework presented here can be applied to other types of DM and, in particular, we extend it to cold DM. The collisionless and linearized Boltzmann-Vlasov equations (B-V) for WDM and neutrinos in the presence of photons and coupled to the linearized Einstein equations are studied in detail in the presence of anisotropic stress with the Newtonian potential generically different from the spatial curvature perturbations. We recast this full system of B-V equations for DM and neutrinos into a system of coupled Volterra integral equations. These Volterra-type equations are valid both in the radiation dominated and matter dominated eras during which the WDM particles are ultrarelativistic and then nonrelativistic. This generalizes the so-called Gilbert integral equation only valid for nonrelativistic particles in the matter dominated era. We succeed to reduce the system of four Volterra integral equations for the density and anisotropic stress fluctuations of DM and neutrinos into a system of only two coupled Volterra equations. The kernels and inhomogeneities in these equations are explicitly given functions. Combining the Boltzmann-Vlasov equations and the linearized Einstein equations constrain the initial conditions on the distribution functions and gravitational potentials. In the absence of neutrinos the anisotropic stress vanishes and the Volterra-type equations reduce to a single integral equation. These Volterra integral equations provide a useful and precise framework to compute the primordial WDM fluctuations over a wide range of scales including small scales up to $k\sim 1/5\mathrm{kpc}$.

Figure 1

${log}_{10}{\mathcal{R}}_0(y)$ defined in Eq. (2.32) vs ${log}_{10}y$. The solid (red) line corresponds to fermions in thermal equilibrium; the dotted (green) line corresponds to sterile neutrinos out of thermal equilibrium in the $\chi$ model. Both freeze-out distributions give the same ${\mathcal{R}}_0(y)$ values for ${\xi }_{\mathrm{dm}}y\gtrsim 5$ [as in Eq. (2.33)] while ${\mathcal{R}}_0(y)$ does depend on the details of the freeze-out distribution for ${\xi }_{\mathrm{dm}}y\lesssim 5$. For ${\xi }_{\mathrm{dm}}y\lesssim 1$, ${\mathcal{R}}_0(y)$ takes the constant value given analytically in Eq. (2.33).