Cluster Abundance and Large Scale Structure

We use the presently observed number density of large X-ray clusters and linear mass power spectra to constrain the shape parameter (Γ), the spectral index (n), the amplitude of matter density perturbations on the scale of 8h−1Mpc (σ8), and the redshift distortion parameter (β). The non-spherical-collapse model as an improvement to the Press-Schechter formula is used and yields significantly lower σ8 and β. An analytical formalism for the formation redshift of halos is also derived. One of the most important constraints on models of structure formation is the observed abundance of galaxy clusters. Because they are the largest virialized objects in the universe, their abundance can be simply predicted by and thus used to constrain the linear perturbation theory. In the light of the new observations and the improvement in modeling cluster evolution, we revisit this application [1], which has been extensively explored in the literature. Based on the maximum-likelihood analysis, we first use the observed linear mass power spectra P (k) by Peacock & Dodds [2] (PD, combination of galaxy surveys) and by Hamilton, Tegmark, & Padmanabhan [3] (HTP, based on PSCz [4]; see Figure 1), to estimate the spectral index n, the shape parameter Γ, and the amplitude of perturbations σ8 in the parameterization of the standard model P (k) ∝ σ 8kT (k/Γ) [1]. The results are shown in Figure 1 and Table 1, with σ8(I) = 0.78 ± 0.26 for IRAS galaxies. The degeneracy between Γ and n is clear, motivating us to find the theoretically expected ‘degenerated’ shape parameter Γ′ = 0.247Γ exp(1.4n) = 0.220 −0.031, which has a much more constrained likelihood. These results are consistent with the current constraints from CMB [5]. Following a similar formalism as in Ref. [6], we then derive the probability disTABLE 1. Best fits of different data sets (all errors at 95% confidence level). n Γ Γ′ χ/degrees of freedom (conf. level) HTP 0.91 −0.91 0.18 +0.74 −0.18 0.160 +0.085 −0.051 15.4/19 (70%) PD 0.99 −0.86 0.23 +0.55 −0.16 0.229 +0.042 −0.033 6.95/9 (64%) HTP+PD 0.84 −0.67 0.27 +0.42 −0.16 0.220 +0.036 −0.031 24.6/30 (74%) 10 −2 10 −1 10 3 10 4 10 5 k (h Mpc) P (k ) ( h − 3 M pc 3 ) HTP PD Best fit (HTP) Best fit (PD) Best fit (PD+HTP) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2