  We present constraints on neutrino masses, the primordial fluctuation
spectrum from inflation, and other parameters of the $\Lambda$CDM model, using
the one-dimensional Ly$\alpha$-forest power spectrum measured by
Palanque-Delabrouille et al. (2013) from SDSS-III/BOSS, complemented by Planck
2015 cosmic microwave background (CMB) data and other cosmological probes. This
paper improves on the previous analysis by Palanque-Delabrouille et al. (2015)
by using a more powerful set of calibrating hydrodynamical simulations that
reduces uncertainties associated with resolution and box size, by adopting a
more flexible set of nuisance parameters for describing the evolution of the
intergalactic medium, by including additional freedom to account for systematic
uncertainties, and by using Planck 2015 constraints in place of Planck 2013.
  Fitting Ly$\alpha$ data alone leads to cosmological parameters in excellent
agreement with the values derived independently from CMB data, except for a
weak tension on the scalar index $n_s$. Combining BOSS Ly$\alpha$ with Planck
CMB constrains the sum of neutrino masses to $\sum m_\nu < 0.12$ eV (95\% C.L.)
including all identified systematic uncertainties, tighter than our previous
limit (0.15 eV) and more robust. Adding Ly$\alpha$ data to CMB data reduces the
uncertainties on the optical depth to reionization $\tau$, through the
correlation of $\tau$ with $\sigma_8$. Similarly, correlations between
cosmological parameters help in constraining the tensor-to-scalar ratio of
primordial fluctuations $r$. The tension on $n_s$ can be accommodated by
allowing for a running ${\mathrm d}n_s/{\mathrm d}\ln k$. Allowing running as a
free parameter in the fits does not change the limit on $\sum m_\nu$. We
discuss possible interpretations of these results in the context of slow-roll
inflation.
