The matter power spectrum at small scales: An estimate from the Lyman-alpha forest optical depth

We measure the matter power spectrum from 31 Ly alpha spectra spanning the redshift range of 1.6-3.6. The optical depth, tau, for Ly alpha absorption of the intergalactic medium is obtained from the flux using the inversion method of Nusser & Haehnelt. The optical depth is converted to density by using a simple power-law relation, tau proportional to (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 one- and two-point statistics of the flux and compared against theoretical models with a likelihood analysis. A cold dark matter model with standard cosmological parameters fits the data well. The power-spectrum amplitude is measured to be (assuming a flat Universe), sigma(8) = (0.92 +/- 0.09) x (Omega(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 temperature-density relation and the thermal smoothing length of the gas. The inferred evolution is consistent with that expected if reionization of He-II occurred at z similar to 3.2. A joint analysis with the Wilkinson Microwave Anisotropy Probe results together with a prior on the Hubble constant as suggested by the Hubble Space Telescope key project data, yields values of Omega(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.