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.