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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 an analytic tool for capturing key processes of galaxy evolution and identifying robust successes and challenges in reproducing observations at high redshift. 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 behavior, which is approximated by an analytic quasi-steady-state solution (QSS). We address the validity of the QSS at different epochs independent of earlier conditions. At high z, where the accretion is assumed to consist of gas only, the specific SFR is robustly predicted to be sSFR = [(1+z)/3]^{5/2} Gyr^{-1}, slightly higher than the cosmological specific accretion rate, in agreement with observations at z=3-8. The gas fraction is expected to decline slowly, and the observations constrain the SFR efficiency per dynamical time to epsilon=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-lading factor of eta=1-3, depending on the penetration efficiency of gas into the galaxy. However, at z=2, where stars are also accreted through mergers, the simplest model has an apparent difficulty in matching observations. The model that maximizes the sSFR, with the outflows fully recycled, falls short by a factor 3 in sSFR, and overestimates f_sv. With strong outflows, the model can reproduce the observed f_sv but at the expense of underestimating the sSFR by an order of magnitude. We discuss potential remedies including a bias due to the exclusion of quenched galaxies.