Interpreting the Relation between the Gamma-Ray and Infrared Luminosities of Star-forming Galaxies

It has been found that there is a quasi-linear scaling relationship between the gamma-ray luminosity in GeV energies and the total infrared luminosity of star-forming galaxies, i.e. $L_{\gamma}\propto L_{\rm IR}^{\alpha}$ with $\alpha\simeq 1$. However, the origin of this linear slope is not well understood. Although extreme starburst galaxies can be regarded as calorimeters for hadronic cosmic ray interaction and thus a quasi-linear scaling may hold, it may not be the case for low star-formation-rate (SFR) galaxies, as the majority of cosmic rays in these galaxies are expected to escape. We calculate the gamma-ray production efficiency in star-forming galaxies by considering realistic galaxy properties, such as the gas density and galactic wind velocity in star-forming galaxies. We find that the slope for the relation between gamma-ray luminosity and the infrared luminosity gets steeper for low infrared luminosity galaxies, i.e. $\alpha\rightarrow 1.6$, due to increasingly lower efficiency for the production of gamma-ray emission. We further find that the measured data of the gamma-ray luminosity is compatible with such a steepening. The steepening in the slope suggests that cosmic-ray escape is very important in low-SFR galaxies.