From haloes to galaxies -- II. The fundamental relations in star
  formation and quenching
  Star formation and quenching are two of the most important processes in
galaxy formation and evolution. We explore in the local Universe the
interrelationships among key integrated galaxy properties, including stellar
mass $M_*$, star formation rate (SFR), specific SFR (sSFR), molecular gas mass
$M_{\rm H_2}$, star formation efficiency (SFE) of the molecular gas and
molecular gas to stellar mass ratio $\mu$. We aim to identify the most
fundamental scaling relations among these key galaxy properties and their
interrelationships. We show the integrated $M_{\rm H_2}$-SFR, SFR-$M_*$ and
$M_{\rm H_2}$-$M_*$ relation can be simply transformed from the $\mu$-sSFR,
SFE-$\mu$ and SFE-sSFR relation, respectively. The transformation, in
principle, can increase or decrease the scatter of each relation.
Interestingly, we find the latter three relations all have significantly
smaller scatter than the former three corresponding relations. We show the
probability to achieve the observed small scatter by accident is extremely
close to zero. This suggests that the smaller scatters of the latter three
relations are driven by a more fundamental physical connection among these
quantities. We then show the large scatters in the former relations are due to
their systematic dependence on other galaxy properties, and on star formation
and quenching process. We propose the sSFR-$\mu$-SFE relation as the
Fundamental Formation Relation (FFR), which governs the star formation and
quenching process, and provides a simple framework to study galaxy evolution.
Other scaling relations, including integrated Kennicutt-Schmidt law,
star-forming main sequence and molecular gas main sequence, can all be derived
from the FFR.
