How to constrain warm dark matter with the Lyman $\alpha$ forest

The flux power spectrum of the high resolution Lyman-$\alpha$ forest data exhibits suppression at small scales. The origin of this suppression can be due to long-sought warm dark matter (WDM) or to thermal effects, related to the largely unknown reionization history of the Universe. Previous works explored a specific class of reionization histories that exhibit sufficiently strong thermal supression and leave little room for warm dark matter interpretation. In this work we choose a different class of reionization histories, fully compatible with available data on evolution of reionization, but much colder then the reionization histories used by previous authors in determining the nature of dark matter, thus leaving the broadest room for the WDM interpretation of the suppression in the flux power spectrum. We find that WDM thermal relics with masses below 1.9 keV (95% CL) would produce a suppression at scales that are larger than observed maximum of the flux power spectrum, independently of assumptions about thermal effects. This WDM mass is significantly lower than previously claimed bounds, demonstrating the level of systematic uncertainty of the Lyman-$\alpha$ forest method, due to the previous modelling. We also discuss how this uncertainty may affect also data at large scales measured by eBOSS.