From haloes to Galaxies - I. The dynamics of the gas regulator model and
  the implied cosmic sSFR history
  We explore the basic parameters that drive the evolution of the fundamental
properties of star forming galaxies within the gas regulator model, or
bathtub-model. We derive the general analytic form of the evolution of the key
galaxy properties, i.e. gas mass, star formation rate (SFR), stellar mass,
specific SFR, gas fraction, gas phase metallicity and stellar metallicity,
without assuming that galaxies live in the equilibrium state. We find that the
timescale required to reach equilibrium, tau_eq, which is determined by the
product of star-formation efficiency and mass-loading factor, is the central
parameter that is essentially in control of the evolution of all key galaxy
properties. The scatters in most of the key scaling relations, such as the
mass-SFR relation and mass-metallicity relation, are primarily governed by
tau_eq. Most strikingly, the predicted sSFR evolution is controlled solely by
tau_eq (apart from the cosmic time). Although the precise evolution of the sSFR
depends on tau_eq, the sSFR history is largely insensitive to different values
of tau_eq. The difference between the minimum and maximum sSFR at any epoch is
less than a factor of four. The shape of the predicted sSFR history simply
mimics that of the specific mass increase rate of the dark matter halos
(sMIR_DM) with the typical value of the sSFR around 2*sMIR_DM, in good
agreement with the predictions from typical Semi-Analytic Models (SAMs), but
both are fundamentally different from the observed sSFR history. This clearly
implies that some key process is missing in both typical SAMs and gas regulator
model, and we hint at some possible culprit. We emphasize the critical role of
tau_eq in controlling the evolution of the galaxy population, especially for
gas rich low mass galaxies that are very unlikely to live around the
equilibrium state at any epoch and this has been largely ignored in many
similar studies.
