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 Ly \alpha 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 h ^ { -1 } { Mpc } 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 Ly \alpha forest data ; the ratio is different for an open universe ( \Lambda = 0 ) or a flat universe .
Because the Ly \alpha forest measurement only reaches comoving scales 2 \pi / k \sim 15 - 20 h ^ { -1 } { Mpc } , 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 \Omega _ { 0 } +0.2 \lambda _ { 0 } \approx 0.46 + 1.3 ( \Gamma - 0.2 ) , where \lambda _ { 0 } \equiv \Lambda / 3 H _ { 0 } ^ { 2 } .
Analysis of larger , existing samples of QSO spectra could greatly improve the measurement of P ( k ) from the Ly \alpha forest , allowing a determination of \Omega _ { 0 } by this method with a precision of \sim 15 \% , limited mainly by uncertainty in the cluster mass function .