Constraining the mass of light bosonic dark matter using SDSS Lyman-$\alpha$ forest

If a significant fraction of the dark matter in the Universe is made of an ultra-light scalar field, named fuzzy dark matter (FDM) with a mass $m_a$ of the order of $10^{-22}-10^{-21}$ eV, then its de Broglie wavelength is large enough to impact the physics of large scale structure formation. In particular, the associated cut-off in the linear matter power spectrum modifies the structure of the intergalactic medium (IGM) at the scales probed by the Lyman-$\alpha$ forest of distant quasars. We study this effect by making use of dedicated cosmological simulations which take into account the hydrodynamics of the IGM. We explore heuristically the amplitude of quantum pressure for the FDM masses considered here and conclude that quantum effects should not modify significantly the non-linear evolution of matter density at the scales relevant to the measured Lyman-$\alpha$ flux power, and for $m_a \geq 10^{-22}$ eV. We derive a scaling law between $m_a$ and the mass of the well-studied thermal warm dark matter (WDM) model that is best adapted to the Lyman-$\alpha$ forest data, and differs significantly from the one infered by a simple linear extrapolation. By comparing FDM simulations with the Lyman-$\alpha$ flux power spectra determined from the BOSS survey, and marginalizing over relevant nuisance parameters, we exclude FDM masses in the range $10^{-22} \leq m_a < 2.3\times 10^{-21}$ eV at 95 % CL. Adding higher-resolution Lyman-$\alpha$ spectra extends the exclusion range up to $2.9\times 10^{-21}$ eV. This provides a significant constraint on FDM models tailored to solve the "small-scale problems" of $\Lambda$CDM.