The Cluster Mass Function and the ${\sigma }_8$-tension

Abstract

We use a large set of halo mass function (HMF) models in order to investigate their ability to represent the observational Cluster Mass Function (CMF), derived from the $GalWCat19$ cluster catalogue, within the $\Lambda$CDM cosmology. We apply the ${\chi }^2$ minimization procedure to constrain the free parameters of the models, namely ${\Omega }_m$ and ${\sigma }_8$. We find that all HMF models fit well the observational CMF, while the Bocquet et. al. model provides the best fit, with the lowest ${\chi }^2$ value. Utilizing the {\em Index of Inconsistency} (IOI) measure, we further test the possible inconsistency of the models with respect to a variety of {\em Planck 2018} $\Lambda$CDM cosmologies, resulting from the combination of different probes (CMB - BAO or CMB - DES). We find that the HMF models that fitted well the observed CMF provide consistent cosmological parameters with those of the {\em Planck} CMB analysis, except for the Press $\&$ Schechter, Yahagi et. al., and Despali et. al. models which return large IOI values. The inverse ${\chi }_{\min }^2$-weighted average values of ${\Omega }_m$ and ${\sigma }_8$, over all 23 theoretical HMF models are: ${\bar{\Omega }}_{m,0}=0.313\pm 0.022$ and ${\bar{\sigma }}_8=0.798\pm 0.040$, which are clearly consistent with the results of {\em Planck}-CMB, providing $S_8={\sigma }_8{\left({\Omega }_m/0.3\right)}^{1/2}=0.815\pm 0.05$. Within the $\Lambda$CDM paradigm and independently of the selected HMF model in the analysis, we find that the current CMF shows no ${\sigma }_8$-tension with the corresponding {\em Planck}-CMB results.

Comments: 10 pages, 4 figures, Accepted for publication in Monthly Notices of the Royal Astronomical Society