The amplitude of mass density fluctuations at $z\approx 3.25$ from the Ly-alpha 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. The first two moments of the truncated optical depth distribution are used to constrain the density fluctuation amplitude of the intergalactic medium (IGM). The $rms$ of the IGM density at $z\approx 3.25$ estimated from the first spectrum is $\sigma =\sqrt{\exp [(1.8\pm 0.27{)}^2\mathrm{/}{\alpha }^2]-1}$, with $1.56 <\alpha <2$forplausiblereionizationhistories\mathrm{.}Thiscorrespondsto$0.9 \la \sigma \la 2.1$with$\sigma(\alpha =1.7)= 1.44\pm 0.3 $. The values obtained from the second spectrum are higher by $\approx 20$. 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 about a hundred to two hundred (comoving) kpc. For CDM-like power spectra the amplitude of dark matter fluctuations on these small scales depends on the cosmological density parameter $\mathrm{\Omega }$. For power spectra normalized to reproduce the space density of present-day clusters and with a slope parameter of $\mathrm{\Gamma }=0.21$ consistent with the observed galaxy power spectrum, the inferred $\mathrm{\Omega }$ can be expressed as: $\Omega= 0.61 (\alpha/1.7)^{1.3}(x_{{\rm J}} /0.62)^{-0.6}$foraflatuniverse,and$\Omega= 0.91(\alpha/1.7)^{1.3} (x{{\rm J}}/0.62)^{-0.7}$fora$\lambda=0$universe\mathrm{.}$x{_{\rm J}}$istheeffectiveJeansscalein(comoving)$\mpc$. Based on a suit of detailed mock spectra the 1-$\sigma$ error is $\approx 25 %$\mathrm{.}Theestimatesincreasewithincreasing$\Gamma$. For the second spectrum we obtain 15% lower values.