How warm is too warm? Towards robust Lyman-$\alpha$ forest bounds on
  warm dark matter
  The Lyman-$\alpha$ forest is a powerful tool to constrain warm dark matter
models (WDM). Its main observable -- flux power spectrum -- should exhibit a
suppression at small scales in WDM models. This suppression, however, can be
mimicked by a number of thermal effects related to the instantaneous
temperature of the intergalactic medium (IGM), and to the history of
reionization and of the IGM heating ("pressure effects"). Therefore, to put
robust bounds on WDM one needs to disentangle the effect of free-streaming of
dark matter particles from the influence of all astrophysical effects. This
task cannot be brute-forced due to the complexity of the IGM modelling. In this
work, we model the sample of high-resolution and high-redshift quasar spectra
(Boera et al 2018) assuming a thermal history that leads to the smallest
pressure effects while still being broadly compatible with observations. We
explicitly marginalize over observationally allowed values of IGM temperature
and find that (thermal) WDM models with masses above 1.9 keV (at 95% CL) are
consistent with the spatial shape of the observed flux power spectrum at
$z=4-5$. Even warmer models would produce a suppression at scales that are
larger than observed, independently of assumptions about thermal effects. This
bound is significantly lower than previously claimed bounds, demonstrating the
importance of the knowledge about the reionization history and of the proper
marginalization over unknowns.
