A self-similar model of galaxy formation and dark halo relaxation

Abstract We develop a spherical self-similar model for the formation of a galaxy through gas collapsing in an isolated self-gravitating dark matter halo. As is well known, the self-similarity assumption makes the problem eminently tractable by reducing it to a system of ordinary differential equations. We improve upon the existing literature on self-similar collapse in two ways. First, we include the effects of radiative cooling and the formation of a pseudo-disk at the center of collapse, in a parametrised manner. More importantly, we solve for the evolution of gas and dark matter simultaneously and self-consistently using a novel iterative approach. As a result, our model produces shell trajectories of both gas and dark matter that qualitatively agree with the results of full hydrodynamical simulations of self-gravitating systems. We discuss the impact of various ingredients such as the accretion rate, gas equation of state, disk radius and cooling rate amplitude on the evolution of the gas shells, although we leave the inclusion of stellar and black hole activity to future work. The self-consistent evolution of gas and dark matter allows us to study the response (or `quasi-adiabatic relaxation') of the dark matter trajectories to the presence of collapsing gas, an effect that has gained increasing importance recently in the context of precision estimates of small-scale statistics like the matter power spectrum. Our default configuration produces a relaxation relation in qualitative agreement with that seen in cosmological hydrodynamical simulations, and further allows us to easily study the impact of the model ingredients mentioned above. As an initial application, we vary one ingredient at a time and find that the accretion rate and gas equation of state have the largest impact on the relaxation relation, while the cooling amplitude plays only a minor role. Our model thus provides a convenient framework to rapidly explore the coupled nonlinear impact of multiple astrophysical processes on the mass and velocity profiles of dark matter in galactic halos, and consequently on observables such as rotation curves and gravitational lensing signals.