  The forest of Lyman-$\alpha$ absorption lines detected in the spectra of
distant quasars encodes information on the nature and properties of dark matter
and the thermodynamics of diffuse baryonic material. Its main observable -- the
1D flux power spectrum (FPS) -- should exhibit a suppression on small scales
and an enhancement on large scales in warm dark matter (WDM) cosmologies
compared to standard $\Lambda$CDM. Here, we present an unprecedented suite of
1080 high-resolution cosmological hydrodynamical simulations run with the
Graphics Processing Unit-accelerated code Cholla to study the evolution of the
Lyman-$\alpha$ forest under a wide range of physically-motivated gas thermal
histories along with different free-streaming lengths of WDM thermal relics in
the early Universe. A statistical comparison of synthetic data with the forest
FPS measured down to the smallest velocity scales ever probed at redshifts
$4.0\lesssim z\lesssim 5.2$ (Boera et al. 2019) yields a lower limit $m_{\rm
WDM}>3.1$ keV (95 percent CL) for the WDM particle mass and constrains the
amplitude and spectrum of the photoheating and photoionizing background
produced by star-forming galaxies and active galactic nuclei at these
redshifts. Interestingly, our Bayesian inference analysis appears to weakly
favor WDM models with a peak likelihood value at the thermal relic mass of
$m_{\rm WDM}=4.5$ keV. We find that the suppression of the FPS from
free-streaming saturates at $k\gtrsim 0.1\,\mathrm{s}\,\mathrm{km}^{-1}$
because of peculiar velocity smearing, and this saturated suppression combined
with a slightly lower gas temperature provides a moderately better fit to the
observed small-scale FPS for WDM cosmologies.

Cosmology and Nongalactic Astrophysics;Astrophysics of Galaxies