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 per cent bias for the hydrostatic mass enclosed within r(500).