标题： 集群质量函数和$σ_8$-张力 显示英文标题

标题： The Cluster Mass Function and the $σ_8$-tension

摘要： 我们使用一组大的晕质量函数（HMF）模型，以研究它们在$\mathtt{GalWCat19}$星系团目录中得出的观测星系团质量函数（CMF）的表示能力，处于$\Lambda$冷暗物质（CDM）宇宙学中。 我们将$\chi^2$最小化过程应用于约束模型的自由参数，即$\Omega_m$和$\sigma_8$。 我们发现所有HMF模型都能很好地拟合观测的CMF，而Bocquet等人的模型提供了最佳拟合，具有最低的$\chi^2$值。 利用{\em 不一致指数}（IOI）度量，我们进一步测试了模型在多种{\em 普朗克2018}$\Lambda$CDM 宇宙学模型下的可能不一致性，这些模型是通过不同探针（CMB - BAO 或 CMB - DES）的组合得到的。我们发现，拟合观测CMF良好的HMF模型，其宇宙学参数与{\em 普朗克}CMB 分析的结果一致，除了Press$\&$Schechter、Yahagi等和Despali等模型返回较大的IOI值。 所有23个理论HMF模型中，$\Omega_m$和$\sigma_8$的逆$\chi_{\rm min}^2$加权平均值为${\bar \Omega_{m,0}}=0.313\pm 0.022$和${\bar \sigma_8}=0.798\pm0.040$，这显然与{\em 普朗克}-CMB的结果一致，提供了$S_8=\sigma_8\left(\Omega_m/0.3\right)^{1/2}= 0.815\pm 0.05$。 在$\Lambda$CDM 框架下，且无论分析中选择哪种 HMF 模型，我们发现当前的 CMF 与相应的{\em 普朗克}-CMB 结果之间没有$\sigma_8$矛盾。 显示英文摘要

摘要： 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.