[Paper Review] Cluster Abundance Constraints on Quintessence Models

Li-Min Wang, Paul J. Steinhardt|Apr 2, 1998

Cosmology and Gravitation TheoriesPhysics and Astronomy473 citations

TL;DR

This paper derives a general expression for the exponent γ in the cluster abundance constraint σ₈Ωₘ^γ = 0.5 ± 0.1, extending it to quintessence models with time-evolving dark energy (w ≠ -1), cold dark matter, and varying spectral index n and Hubble parameter h. The key result is that cluster abundance and evolution data can break degeneracies between ΛCDM and QCDM models that are indistinguishable via CMB anisotropy alone, especially when combined with future high-precision cluster evolution measurements.

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.

Motivation & Objective

• To extend the cluster abundance constraint σ₈Ωₘ^γ = 0.5 ± 0.1 to models with time-evolving quintessence dark energy, not just cosmological constant (Λ) models.
• To derive a general expression for γ that depends on spectral index n, Hubble parameter h, and equation of state w of quintessence.
• To demonstrate that cluster abundance and evolution data can break degeneracies between ΛCDM and QCDM models that are indistinguishable via CMB anisotropy alone.
• To identify best-fitting ΛCDM and QCDM models by combining cluster abundance constraints with COBE normalization of the power spectrum.

Proposed method

• Derives the mass-temperature relation for virialized clusters in models with quintessence, incorporating the time-evolving vacuum energy contribution to the virial theorem.
• Uses the Press-Schechter formalism to relate observed cluster abundance to σ₈, with γ derived as a function of n, h, and w.
• Incorporates the COBE-DMR normalization of the power spectrum to constrain σ₈ independently of cluster abundance.
• Analyzes the evolution of cluster abundance from z=0 to z≈1 to distinguish models with degenerate CMB anisotropy.
• Constructs degeneracy curves in w–Ωₘ parameter space where CMB and cluster data are consistent, and evaluates the spread in A(M₁.₅) to assess discriminability.
• Uses the redshift evolution of cluster abundance (quantified by A(M₁.₅)) as a discriminant, showing a 2-order-of-magnitude variation in abundance at z=0.5 across degenerate models.

Experimental results

Research questions

• RQ1 How does the exponent γ in the cluster abundance constraint σ₈Ωₘ^γ depend on the spectral index n, Hubble parameter h, and equation of state w in quintessence models?
• RQ2 Can cluster abundance and evolution data break the degeneracy between ΛCDM and QCDM models that produce identical CMB anisotropy power spectra?
• RQ3 What range of σ₈ values are consistent with both COBE normalization and cluster abundance constraints across a variety of quintessence models?
• RQ4 How sensitive is the predicted cluster abundance evolution (A(M₁.₅)) to variations in w, h, and Ωₘ along degeneracy curves?
• RQ5 To what extent can future high-precision cluster evolution measurements (e.g., from MAP or Planck) distinguish between ΛCDM and QCDM models?

Key findings

• The paper derives a general expression for γ in the cluster abundance constraint σ₈Ωₘ^γ = 0.5 ± 0.1 that depends on n, h, and w, extending it beyond ΛCDM to quintessence models.
• For models with degenerate CMB anisotropy, the predicted cluster abundance evolution (A(M₁.₅)) varies by nearly two orders of magnitude at z=0.5, indicating strong discriminative power.
• The range of A(M₁.₅) spans from -3.5 to -5.5 for models along degeneracy curves, showing that cluster evolution can distinguish between otherwise indistinguishable models.
• When combined with COBE normalization, the cluster abundance constraint selects a narrow band of best-fitting ΛCDM and QCDM models, with σ₈ values constrained to the range 0.5–1.0.
• Future measurements of A(M₁.₅) with precision better than ±0.5 could resolve the degeneracy between ΛCDM and QCDM models, even when CMB data alone cannot.
• The study identifies that cluster evolution is a critical probe for distinguishing time-evolving quintessence from a cosmological constant, especially when combined with CMB and other cosmological constraints.