An analytic solution for the minimal bathtub toy model: challenges in the star formation history of high-z galaxies

We study the minimal 'bathtub' toy model as a tool for capturing key processes of galaxy evolution and identifying robust successes and challenges in reproducing high-z observations. The source and sink terms of the continuity equations for gas and stars are expressed in simple terms from first principles. The assumed dependence of star formation rate (SFR) on gas mass self-regulates the system into a unique asymptotic behaviour, which is approximated by an analytic quasi-steady-state (QSS) solution. We address the validity of the QSS at different epochs independent of earlier conditions. At high z, where the accretion is gaseous, the specific SFR (sSFR) is predicted to be sSFR similar or equal to [(1 + z)/3](5/2) Gyr(-1), slightly above the cosmological specific accretion rate, as observed at z = 3-8. The gas fraction is expected to decline slowly, and the observations constrain the SFR efficiency per dynamical time to epsilon similar or equal to 0.02. The stellar-to-virial mass ratio f(sv) is predicted to be constant in time, and the observed value requires an outflow mass-loading factor eta similar or equal to 1-3, depending on the penetration efficiency of gas into the galaxy. However, at z similar to 2, where stars are also accreted through mergers, there is a conflict between model and observations. The model that maximizes the sSFR, with the outflows fully recycled, underestimates the sSFR by a factor of similar to 3 and overestimates f(sv). With strong outflows, the model can match the observed f(sv) but then it underestimates the sSFR by an order of magnitude. We discuss potential remedies including a bias due to the exclusion of quenched galaxies.