On the relation between sSFR and metallicity
  In this paper we present an exact general analytic expression
$Z(sSFR)=y/\Lambda(sSFR)+I(sSFR)$ linking the gas metallicity Z to the specific
star formation rate (sSFR), that validates and extends the approximate relation
put forward by Lilly et al. (2013, L13), where $y$ is the yield per stellar
generation, $\Lambda(sSFR)$ is the instantaneous ratio between inflow and star
formation rate expressed as a function of the sSFR, and $I$ is the integral of
the past enrichment history, respectively. We then demonstrate that the
instantaneous metallicity of a self-regulating system, such that its sSFR
decreases with decreasing redshift, can be well approximated by the first term
on the right-hand side in the above formula, which provides an upper bound to
the metallicity. The metallicity is well approximated also by the L13 ideal
regulator case, which provides a lower bound to the actual metallicity. We
compare these approximate analytic formulae to numerical results and infer a
discrepancy <0.1 dex in a range of metallicities and almost three orders of
magnitude in the sSFR. We explore the consequences of the L13 model on the
mass-weighted metallicity in the stellar component of the galaxies. We find
that the stellar average metallicity lags 0.1-0.2 dex behind the gas-phase
metallicity relation, in agreement with the data. (abridged)
