Cluster Abundance Constraints on Quintessence Models

TL;DR

This work extends cluster-abundance cosmological constraints to quintessence (QCDM) models by deriving a generalized dependence $σ_8 Ω_m^{γ}$ that incorporates the spectral index $n$, Hubble parameter $h$, and quintessence equation of state $w$. It integrates a virial-based mass–temperature relation, density-perturbation growth, and Press–Schechter theory to obtain a robust $σ_8$–$Ω_m$ calibration across ΛCDM and QCDM, tying it to COBE normalization. The authors show how the growth of structure and the evolution of cluster abundance with redshift can help break degeneracies between dark energy models that produce similar CMB spectra. This framework provides a practical pathway to discriminate between ΛCDM and quintessence using current and future cluster and CMB observations, with implications for estimating $Ω_m$, $h$, and $w$.

Abstract

The abundance of rich clusters is a strong constraint on the mass power spectrum. The current constraint can be expressed in the form $σ_8 Ω_m^γ = 0.5 \pm 0.1$ where $σ_8$ is the $rms$ mass fluctuation on 8 $h^{-1}$ Mpc scales, $Ω_m$ is the ratio of matter density to the critical density, and $γ$ is model-dependent. In this paper, we determine a general expression for $γ$ that applies to any models with a mixture of cold dark matter plus cosmological constant or quintessence (a time-evolving, spatially-inhomogeneous component with negative pressure) including dependence on the spectral index $n$, the Hubble constant $h$, and the equation-of-state of the quintessence component $w$. The cluster constraint is combined with COBE measurements to identify a spectrum of best-fitting models. The constraint from the evolution of rich clusters is also discussed.

Figures (4)

• Figure 1: The evolution index of cluster abundance for six models selected because they are the best-fit to the combination of COBE and cluster abundance (at $z=0$) constraints. The model parameters are (from bottom to top): (1) $w=-1$, $\Omega_m=0.35$, $n_s=1$, $h=0.65$, $\Omega_b=0.047$; (2) $w=-5/6$, $\Omega_m=0.34$, $n_s=1$, $h=0.66$, $\Omega_b=0.046$; (3) $w=-2/3$, $\Omega_m=0.35$, $n_s=1$, $h=0.66$, $\Omega_b=0.046$; (4) $w=-1/2$, $\Omega_m=0.36$, $n_s=1$, $h=0.68$, $\Omega_b=0.043$; (5) $w=-1/3$, $\Omega_m=0.44$, $n_s=1$, $h=0.67$, $\Omega_b=0.045$; (6) $w=-1/6$, $\Omega_m=0.49$, $n_s=1.1$, $h=0.70$, $\Omega_b=0.042$.
• Figure 2: COBE-normalized $\sigma_8$ as a function of $\Omega_Q$ for six constant $w$ models with $n_s=1$ and $h=0.65$. They are (from top to bottom): (1) $w=-1$; (2) $w=-2/3$; (3) $w=-1/2$; (4) $w=-1/3$; (5) $w=-1/6$; (6) $w=0$. The highlighted regions indicate where the x-ray cluster abundance constraints imposed by Eq. (\ref{['sigma8']}) overlap the COBE constraint. Best-fit models correspond to the overlap region.
• Figure 3: Cosmic microwave background (CMB) anisotropy measurements, combined with other constraints on cosmological parameters, may be unable to distinguish among a family of $\Lambda$ and quintessence models, as illustrated here. All models along the degeneracy curve shown in the figure produce a temperature anisotropy power spectrum that is indistinguishable, given cosmic variance uncertainty. In addition, the CMB can determine other parameters: for this illustration, we have assumed $\Omega_m h^2=1.5$, $\Omega_b h^2=0.02$ and $n_s=1$ (reasonable values). Even if these parameters are determined precisely and combined with other observational constraints, there remains substantial uncertainty (shaded region) that may not do much to discriminate among the degenerate models, as illustrated here.
• Figure 4: The evolution of cluster abundance may break down the degeneracy between $\Lambda$ and quintessence models illustrated in Fig. 3. $A(M_{1.5})$, the slope of log-abundance vs. redshift $z$, for the models along the degeneracy curve shown in Figure 3.