The matter power spectrum from the Lyman-alpha forest: an optical depth estimate

We measure the matter power spectrum from the Lyman-alpha forest of 31 quasar spectra spanning the redshift range of 1.6-3.6. The optical depth, $\tau$, for absorption of the intergalactic medium is obtained from the flux using the inversion method of Nusser & Haehnelt (1999). The optical depth is converted to density by using a simple power law relation, $\tau \propto (1+\delta)^\alpha$. The non-linear 1D power spectrum of the gas density is then inferred with a method that makes simultaneous use of the 1 and 2 point statistics of the flux and compared against theoretical models with a likelihood analysis. A Cold Dark Matter (CDM) model with standard cosmological parameters fits the data well. The power spectrum amplitude is measured to be (assuming a flat Universe), $\sigma_8 = (0.9 \pm 0.09)\times (\Omega_{\rm m}/0.3)^{-0.3}$, with $\alpha$ varying in the range of $1.56-1.8$ with redshift. Enforcing the same cosmological parameters in all four redshift bins, the likelihood analysis suggests some evolution in the density-temperature relation and the thermal smoothing length of the gas. The inferred evolution is consistent with that expected if reionization of \hep occurred at $z\sim 3.2$. A joint analysis with the WMAP results together with a prior on the Hubble constant as suggested by the HST key project data, yields values of $\Omega_{\rm m}$ and $\sigma_8$ that are consistent with the cosmological concordance model. We also perform a further inversion to obtain the linear 3D power spectrum of the matter density fluctuations.