Momentum distribution of dark matter produced in inflaton decay: effect
  of inflaton mediated scatterings
  Post-inflationary reheating is a widely discussed mechanism for non-thermal
production of dark matter (DM). In this scenario the momentum distribution of
the produced DM particles is usually taken to be the one obtained at reheating,
red-shifted at later times due to the expansion of the Universe. However, since
in such a scenario both the DM and the standard model (SM) fields couple to the
inflaton, the DM particles necessarily undergo self-scatterings, as well as
elastic and inelastic scattering reactions with the SM bath, all of which
proceed through $s-$channel or $t-$channel inflaton exchange. We compute the
momentum distribution of the DM particles including the effect of these
scatterings, and find that the distributions can be significantly altered, even
though DM remains non-thermal throughout the cosmological evolution. We observe
that if the inflaton dominantly couples to the SM Higgs boson through a
renormalizable interaction, then reheating temperatures and inflaton masses at
the TeV scale lead to a large effect from the scattering processes, with the
DM-inflaton coupling constrained by the DM density. The scattering effects are
found to be sensitive to the duration of the reheating process -- larger the
duration, more momentum modes are filled at reheating, leading to an enhanced
scattering probability. We also obtain the free-streaming length of such DM
using the resulting non-thermal momentum distribution, which can be used to
estimate the implications of the Lyman-$\alpha$ constraints on the DM mass. It
is observed that in the scenarios considered, including the scattering effects
can reduce the DM average velocity at matter-radiation equality, and its
free-streaming length, by upto a factor of $40$, thereby making the constraints
on light DM produced in inflaton decay significantly weaker.
