Characterising the signatures of star-forming galaxies in the
  extra-galactic $\gamma$-ray background
  Galaxies experiencing intense star-formation episodes are expected to be rich
in energetic cosmic rays (CRs). These CRs undergo hadronic interactions with
the interstellar gases of their host to drive $\gamma$-ray emission, which has
already been detected from several nearby starbursts. Unresolved $\gamma$-ray
emission from more distant star-forming galaxies (SFGs) is expected to
contribute to the extra-galactic $\gamma$-ray background (EGB). However,
despite the wealth of high-quality all-sky data from the Fermi-LAT $\gamma$-ray
space telescope collected over more than a decade of operation, the exact
contribution of such SFGs to the EGB remains unsettled. We investigate the
high-energy $\gamma$-ray emission from SFGs up to redshift $z=3$ above a GeV,
and assess the contribution they can make to the EGB. We show the $\gamma$-ray
emission spectrum from a SFG population can be determined from just a small
number of key parameters, from which we model a range of possible EGB
realisations. We demonstrate that populations of SFGs leave anisotropic
signatures in the EGB, and that these can be accessed using the spatial power
spectrum. Moreover, we show that such signatures will be accessible with
ongoing operation of current $\gamma$-ray instruments, and detection prospects
will be greatly improved by the next generation of $\gamma$-ray observatories,
in particular the Cherenkov Telescope Array.
