The amplitude of mass density fluctuations at z ≈ 3.25 from the Ly α forest of Q1422+231

The real‐space optical‐depth distribution along the line of sight to the QSO Q1422+231 is recovered from two HIRES spectra using a modified version of the inversion method proposed by Nusser & Haehnelt. The first two moments of the truncated optical‐depth distribution are used to constrain the density‐fluctuation amplitude of the intergalactic medium (IGM) assuming that the IGM is photoionized by a metagalactic UV background and obeys a temperature–density relation. The fluctuation amplitude and the power‐law index α of the relation between gas and neutral hydrogen (H  i ) density are degenerate. The rms of the IGM density at z ≈3.25 estimated from the first spectrum is with 1.56< α <2 for plausible reionization histories. This corresponds to 0.9≲ σ ≲2.1 with σ ( α =1.7)=1.44±0.3. The values obtained from the second spectrum are higher by ≈20 per cent. If the IGM density traces the dark matter (DM) as suggested by numerical simulations we have measured the fluctuation amplitude of the DM density at an effective Jeans scale of a few 100 kpc. For cold dark matter (CDM)‐like power spectra the amplitude of dark matter fluctuations on these small scales depends on the cosmological density parameter Ω. For power spectra normalized to reproduce the space density of present‐day clusters and with a slope parameter of Γ=0.21 consistent with the observed galaxy power spectrum, the inferred Ω can be expressed as: Ω=0.61( α /1.7) 1.3 ( x J /0.62) −0.6 for a flat universe, and Ω=0.91( α /1.7) 1.3 ( x J /0.62) −0.7 for a λ =0 universe. x J is the effective Jeans scale in (comoving) h −1  Mpc. Based on a suite of detailed mock spectra the 1 σ error is ≈25 per cent. The estimates increase with increasing Γ. For the second spectrum we obtain 15 per cent lower values.