Closing In on ΩM: The Amplitude of Mass Fluctuations from Galaxy Clusters and the Lyα Forest

We estimate the present-day value of the matter density parameter Ω M by combining constraints from the galaxy cluster mass function with Croft et al.'s recent measurement of the mass power spectrum, P ( k ), from Lyα 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 Ω M , matching the observed cluster mass function then fixes the value of σ 8 , the rms amplitude of mass fluctuations in 8 h -1 Mpc spheres, and it thus determines the normalization of P ( k ) at z = 0. The value of Ω M also determines the ratio of P ( k ) at z = 0 to P ( k ) at z = 2.5, the central redshift of the Lyα forest data; the ratio is different for an open universe (Λ = 0) or a flat universe. Because the Lyα forest measurement only reaches comoving scales 2π/ k ~ 15-20 h -1 Mpc, the derived value of Ω M depends on the value of the power spectrum shape parameter Γ, which determines the relative contribution of larger scale modes to σ 8 . Adopting Γ = 0.2, a value favored by galaxy clustering data, we find Ω M = 0.46 +0.12 -0.10 for an open universe and Ω M = 0.34 +0.13 -0.09 for a flat universe (1 σ errors, not including the uncertainty in cluster normalization). Cluster-normalized models with Ω M = 1 predict too low an amplitude for P ( k ) at z = 2.5, while models with Ω M = 0.1 predict too high an amplitude. The more general best-fit parameter combination is Ω M + 0.2Ω Λ ≈ 0.46 + 1.3(Γ - 0.2), where Ω Λ ≡ Λ/3 H 2 0 . Analysis of larger, existing samples of QSO spectra could greatly improve the measurement of P ( k ) from the Lyα forest, allowing a determination of Ω M by this method with a precision of ~15%, limited mainly by uncertainty in the cluster mass function.