Hironao Miyatake

We review recent advancements in cosmology with galaxy clusters. Galaxy clusters are the most massive objects in the Universe. Consequently the cluster number density as a function of cluster mass, or cluster abundance, is sensitive to cosmological parameters, particularly the matter density of the Universe $\Omega_{\rm m}$ and the amplitude of matter density fluctuation $\sigma_8$. In this review, we describe the methods used to detect galaxy clusters through optical near-infrared (O-NIR), X-ray, and CMB observations, outlining the advantages and disadvantages of cluster detection through different wavelengths. We describe methods for measuring cluster mass, with a particular focus on calibration by WL measurements. We then discuss how the connection between observables in different wavelengths and cluster abundance can be modeled through a cluster selection function and MOR, and quantify the impact of marginalization of nuisance parameters on cosmological constraints. Finally, we also walk through the recent results of cosmological constraints by cluster abundance with the O-NIR, X-ray, and CMB observations.