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 \simeq [ ( 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 \simeq 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 \simeq 1 - 3 , depending on the penetration efficiency of gas into the galaxy .
However , at z \sim 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 \sim 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 .