[Paper Review] Spherical Collapse Model And Dark Energy(I)

Ding-fang Zeng, Yi-hong Gao|arXiv (Cornell University)|May 9, 2005

Cosmology and Gravitation Theories2 references3 citations

TL;DR

This paper re-evaluates the spherical collapse model in flat QCDM and Phantom-CDM cosmologies by assuming dark energy clusters synchronously with matter, eliminating dark energy currents outside galaxy clusters. It finds that the equation of state parameter $ w $ of dark energy affects the number density of galaxy clusters and its evolution exponentially, suggesting that cluster abundance measurements could constrain $ w $, though degeneracy with $ \sigma_8 $ limits precision without additional priors.

ABSTRACT

In existing literatures about the top-hat spherical collapse model of galaxy clusters formation in cosmology containing dark energies, dark energies are usually assumed not to cluster on this scale. But all these literatures ignored the current describing the flowing of dark energies outside the clusters which should exist under this assumption, so the conclusions of these literatures are worth further explorations. In this paper we study this model in QCDM or Phantom-CDM cosmologies(flat) by assuming that dark energies will cluster synchronously with ordinary matters on the scale of galaxy clusters so the dark energy current flowing outside the clusters does not exist at all and find that in this case, the key parameters of the model exhibit rather non-trivial and remarkable dependence on the equation of state coefficients of dark energies. We then apply the results in Press-Scheter theory and calculate the number density of galaxy clusters and its evolutions. We find that this two quantities are both affected exponentially by the equation of state coefficients of dark energies. We leave the study of this model with the assumption that dark energies do not cluster on the scale of galaxy clusters at all as the topic of another paper where similar conclusions will be obtained also.

Motivation & Objective

• To address the inconsistency in existing spherical collapse models that assume dark energy does not cluster but neglects the resulting dark energy current in the energy-momentum tensor.
• To investigate the cosmological implications of assuming dark energy clusters synchronously with matter on the scale of galaxy clusters, thereby eliminating such currents.
• To assess how the equation of state parameter $ w $ of dark energy affects the number density of galaxy clusters and its redshift evolution under this assumption.
• To evaluate the potential of using observed galaxy cluster abundances to constrain $ w $, despite degeneracy with $ \sigma_8 $.

Proposed method

• Formulate the spherical collapse model in flat QCDM and Phantom-CDM cosmologies under the assumption that dark energy clusters synchronously with matter, thus removing the dark energy current from the energy-momentum tensor.
• Solve the Einstein equations under this assumption to derive the critical density contrast $ \delta_c $ and growth function $ D_1(a) $, which depend non-trivially on $ w $.
• Use the resulting model parameters in the Press-Schechter formalism to compute the number density of galaxy clusters as a function of mass, temperature, and redshift.
• Apply the derived expressions to calculate the number density vs. temperature function and its evolution with redshift, analyzing the dependence on $ w $.
• Perform a confidence level analysis using priors on $ \sigma_8 $, $ h $, $ n_s $, and $ \mu/\beta $ to estimate constraints on $ w $ and $ \Omega_{m0} $.

Experimental results

Research questions

• RQ1 How does assuming synchronous clustering of dark energy with matter affect the key parameters of the spherical collapse model?
• RQ2 What is the functional dependence of galaxy cluster number density on the equation of state parameter $ w $ of dark energy under this assumption?
• RQ3 How does the redshift evolution of massive galaxy cluster abundance depend on $ w $?
• RQ4 Can observational measurements of cluster abundance constrain $ w $, and what are the limitations due to degeneracy with other parameters?

Key findings

• The number density of galaxy clusters and its evolution with redshift are exponentially sensitive to the equation of state parameter $ w $ of dark energy.
• As $ w \to 0 $, the number density function increases; as $ w \to -\infty $, it decreases significantly.
• For $ w < -1 $, the number density of massive clusters decreases with lookback time; for $ w > -1 $, it increases.
• With priors $ \sigma_8 = 0.5 $, $ h = 0.71 $, $ n_s = 1.0 $, and $ \mu/\beta = 0.75 $, the 99% confidence level constraint yields $ w = -1.08 \pm 0.09 $.
• The model predicts an unusually low $ \sigma_8 \approx 0.5 $, which is inconsistent with WMAP and SDSS results, indicating that dark energy perturbations must be included in standard abundance analyses.
• The assumption of synchronous clustering simplifies the model and improves self-consistency, but the actual physics likely lies between synchronous clustering and no clustering, suggesting that dark energy clustering effects are crucial for accurate cluster abundance modeling.