The extragalactic $\gamma$-ray background: imprints from the physical
  properties and evolution of star-forming galaxy populations
  Star-forming galaxies (SFGs) harbour an abundant reservoir of cosmic rays
(CRs). At GeV energies, these CRs undergo interactions with their environment
to produce $\gamma$-rays, and the unresolved $\gamma$-ray emission from
populations of SFGs forms a component of the isotropic extragalactic
$\gamma$-ray background (EGB). In this work, we investigate the contribution to
the 0.01 - 50 GeV EGB from SFG populations located up to redshift $z=3$. We
find this is dominated by starbursts, while the contribution from main sequence
SFGs is marginal at all energies. We also demonstrate that most of the
$\gamma$-ray contribution from SFGs emanates from low mass galaxies, with over
80 per cent of the emission originating from galaxies with stellar masses below
$10^8 \;\!{\rm M}_{\odot}$. Many of these galaxies are located at relatively
high redshift, with their peak EGB contribution arising $\sim$ 700 Myr before
the noon of cosmic star-formation. We find that the precise redshift
distributions of EGB sources at different energies imprint intensity signatures
at different angular scales, which may allow their contribution to be
distinguished using analyses of small-scale EGB intensity anisotropies,
particularly if the diffuse EGB is dominated by hadronic CR-driven $\gamma$-ray
emission from SFGs. We show that the EGB is sensitive to the evolution of low
mass populations of galaxies, particularly around $z\sim2.5$, and that it
provides a new means to probe the engagement of CRs in these galaxies before
the high noon of cosmic star-formation.
