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 (CDM). 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 (RD) and matter dominated (MD) 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 MD 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 ~ 1/5 kpc.
