Closing in on Omega₀: The Amplitude of Mass Fluctuations from Galaxy Clusters and the Lyman-alpha Forest

We estimate the value of the matter density parameter \Omega_0 by combining constraints from the galaxy cluster mass function with Croft et al.'s recent measurement of the mass power spectrum, P(k), from \lya forest data. The key assumption of the method is that cosmic structure formed by gravitational instability from Gaussian primordial fluctuations. For a specified value of \Omega_0, matching the observed cluster mass function then fixes the value of \sigma_8, the rms amplitude of mass fluctuations in 8\hmpc spheres, and it thus determines the normalization of P(k) at z=0. The value of \Omega_0 also determines the ratio of P(k) at z=0 to P(k) at z=2.5, the central redshift of the \lya forest data; the ratio is different for an open universe (\Lambda=0) or a flat universe. Because the \lya forest measurement only reaches comoving scales 2\pi/k ~ 15-20\hmpc, the derived value of \Omega_0 depends on the value of the power spectrum shape parameter \Gamma, which determines the relative contribution of larger scale modes to \sigma_8. Adopting \Gamma=0.2, a value favored by galaxy clustering data, we find \Omega_0 = 0.46^{+0.12}_{-0.10} for an open universe and \Omega_0=0.34^{+0.13}_{-0.09} for a flat universe (1\sigma errors, not including the uncertainty in cluster normalization). Cluster-normalized models with \Omega_0=1 predict too low an amplitude for P(k) at z=2.5, while models with \Omega_0=0.1 predict too high an amplitude. The more general best fit parameter combination is approximately \Omega_0 + 0.2\Lambda_0 = 0.46 + 1.3(\Gamma-0.2). Analysis of larger, existing samples of QSO spectra could greatly improve the measurement of P(k) from the \lya forest, allowing a determination of \Omega_0 by this method with a precision of ~15%, limited mainly by uncertainty in the cluster mass function.