Testing Cosmological Models With A $\lya$ Forest Statistic: The High End Of The Optical Depth Distribution

We pay particular attention to the high end of the $\lya$ optical depth distribution of a quasar spectrum. Based on the flux distribution (Miralda-Escud\'e et al 1996), a simple yet seemingly cosmological model -differentiating statistic, $\Delta_{\tau_0}$ -- the cumulative probability of a quasar spectrum with $\lya$ optical depth greater than a high value $\tau_0$ -- is emphasized. It is shown that two different models -- the cold dark matter model with a cosmological constant and the mixed hot and cold dark matter model, both normalized to COBE and local galaxy cluster abundance -- yield quite different values of $\Delta_{\tau_0}$: 0.13 of the former versus 0.058 of the latter for $\tau_0=3.0$ at $z=3$. Moreover, it is argued that $\Delta_{\tau_0}$ may be fairly robust to compute theoretically because it does not seem to depend sensitively on small variations of simulations parameters such as radiation field, cooling, feedback process, radiative transfer, resolution and simulation volume within the plausible ranges of the concerned quantities. Furthermore, it is illustrated that $\Delta_{\tau_0}$ can be obtained sufficiently accurately from currently available observed quasar spectra for $\tau_0\sim 3.0-4.0$, when observational noise is properly taken into account. We anticipate that analyses of observations of quasar $\lya$ absorption spectra over a range of redshift may be able to constrain the redshift evolution of the amplitude of the density fluctuations on small-to-intermediate scales, therefore providing an independent constraint on $\Omega_0$, $\Omega_{0,HDM}$ and $\Lambda_0$.