Warm Dark Matter via Ultra-Violet Freeze-In: Reheating Temperature and
  Non-Thermal Distribution for Fermionic Higgs Portal Dark Matter
  Warm dark matter (WDM) of order keV mass may be able to resolve the
disagreement between structure formation in cold dark matter simulations and
observations. The detailed properties of WDM will depend upon its energy
distribution, in particular how it deviates from the thermal distribution
usually assumed in WDM simulations. Here we focus on WDM production via the
Ultra-Violet (UV) freeze-in mechanism, for the case of fermionic Higgs portal
dark matter $\psi$ produced via the portal interaction $\bar{\psi}\psi
H^{\dagger}H/\Lambda$. We introduce a new method to simplify the computation of
the non-thermal energy distribution of dark matter from freeze-in. We show that
the non-thermal energy distribution from UV freeze-in is hotter than the
corresponding thermal distribution and has the form of a Bose-Einstein
distribution with a non-thermal normalization. The resulting range of dark
matter fermion mass consistent with observations is 5-7 keV. The reheating
temperature must satisfy $T_{R} \gtrsim 120 $ GeV in order to account for the
observed dark matter density when $m_{\psi} \approx 5 $ keV, where the lower
bound on $T_{R}$ corresponds to the limit where the fermion mass is entirely
due to electroweak symmetry breaking via the portal interaction. The
corresponding bound on the interaction scale is $\Lambda \gtrsim 6.0 \times
10^{9}$ GeV.
