Cosmological Constraints from Galaxy Clusters and Groups in the eROSITA Final Equatorial Depth Survey

We present the first cosmological study of a sample of $eROSITA$ clusters, which were identified in the $eROSITA$ Final Equatorial Depth Survey (eFEDS). In a joint selection on X-ray and optical observables, the sample contains $455$ clusters within a redshift range of $0.1<z<1.2$, of which $177$ systems are covered by the public data from the Hyper Suprime-Cam (HSC) survey that enables uniform weak-lensing cluster mass constraints. With minimal assumptions, at each cluster redshift $z$ we empirically model (1) the scaling relations between the cluster halo mass and the observables, which include the X-ray count rate, the optical richness, and the weak-lensing mass, and (2) the X-ray selection in terms of the completeness function $\mathtt{C}$. Using the richness distribution of the clusters, we directly measure the X-ray completeness and adopt those measurements as informative priors for the parameters of $\mathtt{C}$. In a blinded analysis, we obtain the cosmological constraints $\Omega_{\mathrm{m}} = 0.245^{+0.048}_{-0.058}$, $\sigma_{8} = 0.833^{+0.075}_{-0.063}$ and $S_{8} \equiv \sigma_{8}\left(\Omega_{\mathrm{m}}/0.3\right)^{0.3}= 0.791^{+0.028}_{-0.031}$ in a flat $\Lambda$CDM cosmology. Extending to a flat $w$CDM cosmology leads to the constraint on the equation of state parameter of the dark energy of $w = -1.25\pm 0.47$. The eFEDS constraints are in good agreement with the results from the $Planck$ mission, the galaxy-galaxy lensing and clustering analysis of the Dark Energy Survey, and the cluster abundance analysis of the SPT-SZ survey at a level of $\lesssim1\sigma$. With the empirical modelling, this work presents the first fully self-consistent cosmological constraints based on a synergy between wide-field X-ray and weak lensing surveys.