Analytical model for non-thermal pressure in galaxy clusters

Non-thermal pressure in the intracluster gas has been found ubiquitously in numerical simulations, and observed indirectly. In this paper we develop an analytical model for intracluster non-thermal pressure in the virial region of relaxed clusters. We write down and solve a first-order differential equation describing the evolution of non-thermal velocity dispersion. This equation is based on insights gained from observations, numerical simulations, and theory of turbulence. The non-thermal energy is sourced, in a self-similar fashion, by the mass growth of clusters via mergers and accretion, and dissipates with a time-scale determined by the turnover time of the largest turbulence eddies. Our model predicts a radial profile of non-thermal pressure for relaxed clusters. The non-thermal fraction increases with radius, redshift, and cluster mass, in agreement with numerical simulations. The radial dependence is due to a rapid increase of the dissipation time-scale with radii, and the mass and redshift dependence comes from the mass growth history. Combing our model for the non-thermal fraction with the Komatsu-Seljak model for the total pressure, we obtain thermal pressure profiles, and compute the hydrostatic mass bias. We find typically 10% bias for the hydrostatic mass enclosed within $r_{500}$.