  (abridged) The flux power spectrum of the Lyman-alpha forest in quasar (QSO)
absorption spectra is sensitive to a wide range of cosmological and
astrophysical parameters and instrumental effects. Modelling the flux power
spectrum in this large parameter space to an accuracy comparable to the
statistical uncertainty of large samples of QSO spectra is very challenging. We
use here a coarse grid of hydrodynamical simulations run with GADGET-2 to
obtain a ``best guess'' model around which we calculate a finer grid of flux
power spectra using a Taylor expansion of the flux power spectrum to first
order. We find that the SDSS flux power spectrum alone is able to constrain a
wide range of parameters including the amplitude of the matter power spectrum
sigma_8, the matter density Omega_m, the spectral index of primordial density
fluctuations n, the effective optical depth tau_eff and its evolution. The
thermal history of the Intergalactic Medium (IGM) is, however, poorly
constrained and the SDSS data favour either an unplausibly large temperature or
an unplausibly steep temperature-density relation. By enforcing a thermal
history of the IGM consistent with that inferred from high-resolution QSO
spectra, we find the following values for the best fitting model (assuming a
flat Universe with a cosmological constant and zero neutrino mass): Omega_
m=0.28 \pm 0.03, n=0.95\pm0.04, \sigma_8=0.91\pm0.07 (1\sigma error bars).We
argue that the major uncertainties in this measurement are still systematic
rather than statistical.
