Cosmic rays and non-thermal emission in simulated galaxies. II. $\gamma$-ray maps, spectra and the far infrared-$\gamma$-ray relation

The $\gamma$-ray emission of star-forming (SF) galaxies is attributed to hadronic interactions of cosmic ray (CR) protons with the interstellar gas and contributions from CR electrons via bremsstrahlung and inverse Compton (IC) scattering. The relative importance of these processes in different galaxy types is still unclear. We model these processes in three-dimensional magneto-hydrodynamical (MHD) simulations of the formation of isolated galactic discs using the moving-mesh code AREPO, including dynamically coupled CR protons and adopting different CR transport models. We calculate steady-state CR spectra and also account for the emergence of secondary electrons and positrons. This allows us to produce detailed $\gamma$-ray maps, luminosities and spectra of our simulated galaxies at different evolutionary stages. Our simulations with anisotropic CR diffusion and a low CR injection efficiency at supernovae (SNe, $\zeta_{\mathrm{SN}}=0.05$) can successfully reproduce the observed far infrared (FIR)-$\gamma$-ray relation. Starburst galaxies are close to the calorimetric limit, where CR protons lose most of their energy due to hadronic interactions and hence, their $\gamma$-ray emission is dominated by neutral pion decay. However, in low SF galaxies, the increasing diffusive losses soften the CR proton spectra due to energy-dependent diffusion, and likewise steepen the pionic $\gamma$-ray spectra. In turn, IC emission hardens the total spectra and can contribute up to $\sim40$ per cent of the total luminosity in low SF galaxies. Furthermore, in order to match the observed $\gamma$-ray spectra of starburst galaxies, we require a weaker energy dependence of the CR diffusion coefficient, $D\propto{E}^{0.3}$, in comparison to Milky Way-like galaxies.