We present new measurements of the free-streaming of warm dark matter (WDM) from Lyman-α flux-power spectra. We use data from the medium resolution, intermediate redshift XQ-100 sample observed with the X-shooter spectrograph (z = 3−4.2) and the high-resolution, high-redshift sample used in obtained with the HIRES/MIKE spectrographs (z = 4.2 − 5.4). Based on further improved modelling of the dependence of the Lyman-α flux-power spectrum on the freestreaming of dark matter, cosmological parameters, as well as the thermal history of the intergalactic medium (IGM) with hydrodynamical simulations, we obtain the following limits, expressed as the equivalent mass of thermal relic WDM particles. The XQ-100 flux power spectrum alone gives a lower limit of 1.4 keV, the re-analysis of the HIRES/MIKE sample gives 4.1 keV while the combined analysis gives our best and significantly strengthened lower limit of 5.3 keV (all 2σ C.L.). The further improvement in the joint analysis is partly due to the fact that the two data sets have different degeneracies between astrophysical and cosmological parameters that are broken when the data sets are combined, and more importantly on chosen priors on the thermal evolution. These results all assume that the temperature evolution of the IGM can be modelled as a power law in redshift. Allowing for a non-smooth evolution of the temperature of the IGM with sudden temperature changes of up to 5000K reduces the lower limit for the combined analysis to 3.5 keV. A WDM with smaller thermal relic masses would require, however, a sudden temperature jump of 5000 K or more in the narrow redshift interval z = 4.6 − 4.8, in disagreement with observations of the thermal history based on high-resolution resolution Lyman-α forest data and expectations for photo-heating and cooling in the low density IGM at these redshifts.

We present constraints on the masses of extremely light bosons dubbed fuzzy dark matter from Lyman-α forest data. Extremely light bosons with a De Broglie wavelength of ∼ 1 kpc have been suggested as dark matter candidates that may resolve some of the current small scale problems of the cold dark matter model. For the first time we use hydrodynamical simulations to model the Lyman-α flux power spectrum in these models and compare with the observed flux power spectrum from two different data sets: the XQ-100 and HIRES/MIKE quasar spectra samples. After marginalization over nuisance and physical parameters and with conservative assumptions for the thermal history of the IGM that allow for jumps in the temperature of up to 5000 K, XQ-100 provides a lower limit of 7.1×10 −22 eV, HIRES/MIKE returns a stronger limit of 14.3×10 −22 eV, while the combination of both data sets results in a limit of 20 ×10 −22 eV (2σ C.L.). The limits for the analysis of the combined data sets increases to 37.5×10 −22 eV (2σ C.L.) when a smoother thermal history is assumed where the temperature of the IGM evolves as a power-law in redshift. Light boson masses in the range 1 − 10 × 10 −22 eV are ruled out at high significance by our analysis, casting strong doubts that FDM helps solve the "small scale crisis" of the cold dark matter models.Introduction. Recently, there has been a growing interest in so-called Fuzzy Dark Matter (FDM) models where the dark matter is made of ultra-light bosons. Cosmological and astrophysical consequences have been comprehensively reviewed in [1-5] highlighting the particle physics motivation [6-9] for such models, as well as the importance of experimental searches [10]. A broad variety of astrophysical implications have been investigated in the literature: the halo mass function [11], the innermost structure of haloes [12][13][14], the dynamical properties of the smallest objects [15], the linear matter power spectrum [1], the development of non-linearities by using N-body simulations [16], the abundance of high redshift objects [17], the overall impact of FDM on galaxy formation and the reionization history of the Universe, the intergalactic medium [4,[18][19][20][21], pulsar timing and binary pulsars [22,23], and the properties of our galactic disk [24]. The general conclusion is that in order to have an