The Cluster Mass Function and the $\sigma_8$-tension

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 $\mathtt{GalWCat19}$ 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 $\chi_{\rm min}^2$-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 ${\bar \sigma_8}=0.798\pm0.040$, 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.