Two analytic relations connecting the hot gas astrophysics with the cold dark matter model for galaxy clusters

Galaxy clusters are good targets for examining our understanding of cosmology. Apart from numerical simulations and gravitational lensing, X-ray observation is the most common and conventional way to analyze the gravitational structures of galaxy clusters. Therefore, it is valuable to have simple analytical relations that can connect the observed distribution of the hot, X-ray emitting gas to the structure of the dark matter in the clusters as derived from simulations. In this article, we apply a simple framework that can analytically connect the hot gas empirical parameters with the standard parameters in the cosmological cold dark matter model. We have theoretically derived two important analytic relations, $r_s \approx \sqrt{3}r_c$ and $\rho_s \approx 9\beta kT/8 \pi Gm_gr_c^2$, which can easily relate the dark matter properties in galaxy clusters with the hot gas properties. This can give a consistent picture describing gravitational astrophysics for galaxy clusters by the hot gas and cold dark matter models.