  We introduce a simple analytic model of galaxy formation that links the
growth of dark matter haloes in a cosmological background to the build-up of
stellar mass within them. The model aims to identify the physical processes
that drive the galaxy-halo co-evolution through cosmic time. The model
restricts the role of baryonic astrophysics to setting the relation between
galaxies and their haloes. Using this approach, galaxy properties can be
directly predicted from the growth of their host dark matter haloes. We explore
models in which the effective star formation efficiency within haloes is a
function of mass (or virial temperature) and independent of time. Despite its
simplicity, the model reproduces self-consistently the shape and evolution of
the cosmic star formation rate density, the specific star formation rate of
galaxies, and the galaxy stellar mass function, both at the present time and at
high redshifts. By systematically varying the effective star formation
efficiency in the model, we explore the emergence of the characteristic shape
of the galaxy stellar mass function. The origin of the observed double
Schechter function at low redshifts is naturally explained by two efficiency
regimes in the stellar to halo mass relation, namely, a stellar feedback
regulated stage, and a supermassive black hole regulated stage. By providing a
set of analytic differential equations, the model can be easily extended and
inverted, allowing the roles and impact of astrophysics and cosmology to be
explored and understood.
