The matter power spectrum at small scales: an estimate from the Lyα forest optical depth

We measure the matter power spectrum on small scales from 31 Lyα spectra spanning the redshift range of 1.6-3.6. The optical depth, τ , for Lyα 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, τ ∝ (1 + δ)α. 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), σ8 = (0.92 ± 0.06)× (Ωm/0.3) , with α 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 Heii occurred at z ∼ 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 Ωm and σ8 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.