Constraining warm dark matter candidates including sterile neutrinos and light gravitinos with WMAP and the Lyman-αforest

The matter power spectrum at comoving scales of $(1-40)h^{-1}\mathrm{Mpc}$ is very sensitive to the presence of Warm Dark Matter (WDM) particles with large free-streaming lengths. We present constraints on the mass of WDM particles from a combined analysis of the matter power spectrum inferred from the large samples of high-resolution high signal-to-noise Lyman-$\alpha$ forest data of Kim et al. (2004) and Croft et al. (2002) and the cosmic microwave background data of WMAP. We obtain a lower limit of $m_{\mathrm{WDM}}\gtrsim 550\mathrm{eV}$ ($2\sigma$) for early decoupled thermal relics and $m_{\mathrm{WDM}}\gtrsim 2.0\mathrm{keV}$ ($2\sigma$) for sterile neutrinos. We also investigate the case where in addition to cold dark matter a light thermal gravitino with fixed effective temperature contributes significantly to the matter density. In that case the gravitino density is proportional to its mass, and we find an upper limit $m_{3/2}\lesssim 16\mathrm{eV}$ ($2\sigma$). This translates into a bound on the scale of supersymmetry breaking, ${\Lambda }_{\mathrm{susy}}\lesssim 260\mathrm{TeV}$, for models of supersymmetric gauge mediation in which the gravitino is the lightest supersymmetric particle.