The SRG/eROSITA all-sky survey: Cosmology constraints from cluster abundances in the western Galactic hemisphere

The cluster mass function traces the growth of linear density perturbations and provides valuable insights into the growth of structures, the nature of dark matter, and the cosmological parameters governing the Universe. The primary science goal of eROSITA, on board the {\it Spectrum Roentgen Gamma (SRG)} mission, launched in 2019, is to constrain cosmology through the evolution of cluster mass function. In this paper, we present the cosmological constraints obtained from 5259 clusters of galaxies detected over an area of 12791~deg$^2$ in the Western Galactic Hemisphere of the eROSITA's first All-Sky Survey (eRASS1). The common footprint region between the eROSITA Survey and DES, KiDS, and HSC surveys is used for calibration of the scaling between X-ray count rate and their total mass through measurements of their weak gravitational lensing signal. eRASS1 cluster abundances constrain the $\Lambda$CDM parameters, which are the energy density of the total matter to $\Omega_{\mathrm{m}}=0.29^{+0.01}_{-0.02}$, and the normalization of the density fluctuations to $\sigma_8=0.88\pm0.02$ and their combination yields $S_8=\sigma_8 (\Omega_\mathrm{m} / 0.3)^{0.5}=0.86\pm0.01$, consistent and at a similar precision with the state-of-the-art CMB measurements. eRASS1 cosmological experiment places a most stringent upper limit on the summed masses of left-handed light neutrinos to $\sum m_\nu< 0.22\mathrm{~eV}$ (95\% confidence interval). Combining eRASS1 cluster abundance measurements with CMB and ground-based neutrino oscillation experiments, we measure the summed neutrino masses to be $\sum m_\nu=0.08_{-0.02}^{+0.03}\mathrm{~eV}$ or $\sum m_\nu=0.12_{-0.01}^{+0.03}\mathrm{~eV}$ depending on the mass hierarchy scenario for neutrino eigenstates. eRASS1 cluster abundances significantly improve the constraints on the dark energy equation of state parameter to $w=-1.12\pm0.12$. (ABRIDGED)