Toy Models for Galaxy Formation versus Simulations

We describe simple and useful toy models for key processes of galaxy formation in its most active phase, at z>1, and test the approximate expressions against a suite of high-resolution hydro-cosmological simulations of massive galaxies at z=4-1. We address in particular, as a function of redshift and mass, the evolution of (a) the total mass inflow rate from the cosmic web into galactic haloes based on the EPS approximation, (b) the penetration of baryonic streams into the inner galaxy, (c) the disc size, (d) the implied steady-state gas content and star-formation rate in the galaxy subject to mass conservation and a universal star-formation law, (e) the inflow rate within the disc to a central bulge and black hole as derived using energy conservation and self-regulated Q ~1 violent disc instability (VDI), and (f) the implied steady state in the disc and bulge. The toy models are found to provide useful approximations for the average behaviour of the simulated galaxies. We find that (a) the inflow rate is proportional to mass and to (1+z)^5/2, (b) the penetration to the inner halo is ~60% at z=4-2, (c) the disc radius is about 5% of the virial radius, (d) the galaxies reach a steady state with the star-formation rate (SFR) following the accretion rate into the galaxy, (e) there is an intense inflow through the disc, comparable to the SFR, following the predictions of VDI, and (f) the galaxies approach a steady state with the bulge mass comparable to the disc mass, where the draining of gas by SFR, outflows and inflows in the disc is replenished by fresh accretion. Given the agreement with simulations, these toy models are useful for understanding the complex phenomena in simple terms and for back-of-the-envelope predictions.