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A feebly interacting massive particle (FIMP), contrasting with a weaklyinteracting massive particle (WIMP), is an intriguing dark matter (DM)candidate. Light (keV-scale) FIMP DM is of particular interest: its radiativedecay 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 thatwith 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 tocheck whether or not this FIMP DM is compatible with the structure formation.Here is an issue: the structure formation constraint on WDM is often reportedin terms of the so-called thermal WDM mass $m_{\mathrm{WDM}}$, which cannot bedirectly applied to FIMP parameters. In this paper, we introduce a benchmarkFIMP model that represents well a broad class of FIMP models. A big advantageof this benchmark is that we can derive the analytic formula of the non-thermalphase space distribution of FIMPs produced from freeze-in processes. By furtherderiving 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 willbe very useful for future updates of observational constraints and reports ofx-ray lines. It is also very easy to incorporate our analytic formula into aBoltzmann solver so that a linear matter power spectrum is readily accessible.Our benchmark model will facilitate FIMP searches and particle physicsmodel-building.

Constraining FIMP from the structure formation of the Universe: analytic mapping from $m_{\mathrm{WDM}}$