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 ( 1999 ) .
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 \alpha of the relation between gas and neutral hydrogen ( HI ) density are degenerate .
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 } / \alpha ^ { 2 } ] } -1 } , with 1.56 < \alpha < 2 for plausible reionization histories .
This corresponds to { { { { { { { { 0.9 \mathrel { \mathchoice { \vbox { \offinterlineskip \halign { \cr } $ \displaystyle < $ \cr% $ \displaystyle \sim$ } } } { \vbox { \offinterlineskip \halign { \cr } $ \textstyle < $ \cr$% \textstyle \sim$ } } } { \vbox { \offinterlineskip \halign { \cr } $ \scriptstyle < $ \cr$% \scriptstyle \sim$ } } } { \vbox { \offinterlineskip \halign { \cr } $ \scriptscriptstyle < $% \cr$ \scriptscriptstyle \sim$ } } } } \sigma \mathrel { \mathchoice { \vbox { % \offinterlineskip \halign { \cr } $ \displaystyle < $ \cr$ \displaystyle \sim$ } } } { \vbox { % \offinterlineskip \halign { \cr } $ \textstyle < $ \cr$ \textstyle \sim$ } } } { \vbox { % \offinterlineskip \halign { \cr } $ \scriptstyle < $ \cr$ \scriptstyle \sim$ } } } { \vbox { % \offinterlineskip \halign { \cr } $ \scriptscriptstyle < $ \cr$ \scriptscriptstyle \sim$ } % } } } 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 \Omega .
For power spectra normalized to reproduce the space density of present-day clusters and with a slope parameter of \Gamma = 0.21 consistent with the observed galaxy power spectrum , the inferred \Omega can be expressed as : \Omega = 0.61 \left ( \alpha / 1.7 \right ) ^ { 1.3 } \left ( x _ { { } _ { J } } / 0.62 \right ) ^ { -0.6 } for a flat universe , and \Omega = 0.91 \left ( \alpha / 1.7 \right ) ^ { 1.3 } \left ( x _ { { } _ { J } } / 0.62 \right ) ^ { -0.7 } for a \lambda = 0 universe . x _ { { } _ { J } } is the effective Jeans scale in ( comoving ) h ^ { -1 } { Mpc } .
Based on a suit of detailed mock spectra the 1- \sigma error is \approx 25 \% .
The estimates increase with increasing \Gamma .
For the second spectrum we obtain 15 \% lower values .