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We present constraints on the parameters of the $\Lambda$CDM cosmologicalmodel in the presence of massive neutrinos, using the one-dimensionalLy$\alpha$ forest power spectrum obtained with the Baryon OscillationSpectroscopic Survey (BOSS) of the Sloan Digital Sky Survey (SDSS) byPalanque-Delabrouille et al. (2013), complemented by additional cosmologicalprobes. The interpretation of the measured Ly$\alpha$ spectrum is done using asecond-order Taylor expansion of the simulated power spectrum. BOSS  Ly$\alpha$ data alone provide better bounds than previous Ly$\alpha$ results,but are still poorly constraining, especially for the sum of neutrino masses$\sum m_\nu$, for which we obtain an upper bound of 1.1~eV (95\% CL), includingsystematics for both data and simulations. Ly$\alpha$ constraints on$\Lambda$CDM parameters and neutrino masses are compatible with CMB bounds fromthe Planck collaboration. Interestingly, the combination of Ly$\alpha$ with CMBdata reduces the uncertainties significantly, due to very different directionsof degeneracy in parameter space, leading to the strongest cosmological boundto date on the total neutrino mass, $\sum m_\nu < 0.15$~eV at 95\% CL (with abest-fit in zero). Adding recent BAO results further tightens this constraintto $\sum m_\nu < 0.14$~eV at 95\% CL. This bound is nearly independent of thestatistical approach used, and of the different combinations of CMB and BAOdata sets considered in this paper in addition to Ly$\alpha$. Given themeasured values of the two squared mass differences $\Delta m^2$, this resulttends to favor the normal hierarchy scenario against the inverted hierarchyscenario for the masses of the active neutrino species.

Constraint on neutrino masses from SDSS-III/BOSS Ly$\alpha$ forest and other cosmological probes