Constraining FIMP from the structure formation of the Universe: analytic
  mapping from $m_{\mathrm{WDM}}$
  A feebly interacting massive particle (FIMP), contrasting with a weakly
interacting massive particle (WIMP), is an intriguing dark matter (DM)
candidate. Light (keV-scale) FIMP DM is of particular interest: its radiative
decay leaves a line signal in x-ray spectra; and it is warm dark matter (WDM)
and alters the galactic-scale structure formation of the Universe from that
with WIMP DM. Once a possible x-ray line is reported (e.g., $3.5
\,\mathrm{keV}$ line and $7 \,\mathrm{keV}$ FIMP DM is inferred), one has to
check whether or not this FIMP DM is compatible with the structure formation.
Here is an issue: the structure formation constraint on WDM is often reported
in terms of the so-called thermal WDM mass $m_{\mathrm{WDM}}$, which cannot be
directly applied to FIMP parameters. In this paper, we introduce a benchmark
FIMP model that represents well a broad class of FIMP models. A big advantage
of this benchmark is that we can derive the analytic formula of the non-thermal
phase space distribution of FIMPs produced from freeze-in processes. By further
deriving a certain "warmness" quantity, we can analytically map
$m_{\mathrm{WDM}}$ to FIMP parameters. Our analytic map indicates that $7
\,\mathrm{keV}$ FIMP DM, without entropy production or a degenerate spectrum,
is in tension with the latest Lyman-$\alpha$ forest data. Our analytic map will
be very useful for future updates of observational constraints and reports of
x-ray lines. It is also very easy to incorporate our analytic formula into a
Boltzmann solver so that a linear matter power spectrum is readily accessible.
Our benchmark model will facilitate FIMP searches and particle physics
model-building.
