  An extinction-free estimator of the star-formation rate (SFR) of galaxies is
critical for understanding the high-redshift universe. To this end, the nearly
linear, tight correlation of far-infrared (FIR) and radio luminosity of
star-forming galaxies is widely used. While the FIR is linked to massive star
formation, which also generates shock-accelerated cosmic ray (CR) electrons and
radio synchrotron emission, a detailed understanding of the underlying physics
is still lacking. Hence, we perform three-dimensional magneto-hydrodynamical
(MHD) simulations of isolated galaxies over a broad range of halo masses and
SFRs using the moving-mesh code AREPO, and evolve the CR proton energy density
self-consistently. In post-processing, we calculate the steady-state spectra of
primary, shock-accelerated and secondary CR electrons, which result from
hadronic CR proton interactions with the interstellar medium. The resulting
total radio luminosities correlate with the FIR luminosities as observed and
are dominated by primary CR electrons if we account for anisotropic CR
diffusion. The increasing contribution of secondary emission up to 30 per cent
in starbursts is compensated by the larger bremsstrahlung and Coulomb losses.
CR electrons are in the calorimetric limit and lose most of their energy
through inverse Compton interactions with star-light and cosmic microwave
background (CMB) photons while less energy is converted to synchrotron
emission. This implies steep steady-state synchrotron spectra in starbursts.
Interestingly, we find that thermal free-free emission hardens the total radio
spectra at high radio frequencies and reconciles calorimetric theory with
observations while free-free absorption explains the observed low-frequency
flattening towards the central regions of starbursts.
