The observed Lyman-$alpha$ flux power spectrum (FPS) is suppressed on scales below $sim~ 30~{rm km~s}^{-1}$. This cutoff could be due to the high temperature, $T_0$, and pressure, $p_0$, of the absorbing gas or, alternatively, it could reflect the free streaming of dark matter particles in the early universe. We perform a set of very high resolution cosmological hydrodynamic simulations in which we vary $T_0$, $p_0$ and the amplitude of the dark matter free streaming, and compare the FPS of mock spectra to the data. We show that the location of the dark matter free-streaming cutoff scales differently with redshift than the cutoff produced by thermal effects and is more pronounced at higher redshift. We, therefore, focus on a comparison to the observed FPS at $z>5$. We demonstrate that the FPS cutoff can be fit assuming cold dark matter, but it can be equally well fit assuming that the dark matter consists of $sim 7$ keV sterile neutrinos in which case the cutoff is due primarily to the dark matter free streaming.
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