The common approach to compute the cosmic-ray distribution in an starburst galaxy or region is equivalent to assume that at any point within that environment , there is an accelerator inputing cosmic rays at a reduced rate .
This rate should be compatible with the overall volume-average injection , given by the total number of accelerators that were active during the starburst age .
These assumptions seem reasonable , especially under the supposition of an homogeneous and isotropic distribution of accelerators .
However , in this approach the temporal evolution of the superposed spectrum is not explicitly derived ; rather , it is essentially assumed ab-initio .
Here , we test the validity of this approach by following the temporal evolution and spatial distribution of the superposed cosmic-ray spectrum and compare our results with those from theoretical models that treat the starburst region as a single source .
In the calorimetric limit ( with no cosmic-ray advection ) , homogeneity is reached ( typically within 20 % ) across most of the starburst region .
However , values of center-to-edge intensity ratios can amount to a factor of several .
Differences between the common homogeneous assumption for the cosmic-ray distribution and our models are larger in the case of two-zone geometries , such as a central nucleus with a surrounding disc .
We have also found that the decay of the cosmic-ray density following the duration of the starburst process is slow , and even approximately 1 Myr after the burst ends ( for a gas density of 35 cm ^ { -3 } ) it may still be within an order of magnitude of its peak value .
Based on our simulations , it seems that the detection of a relatively hard spectrum up to the highest gamma-ray energies from nearby starburst galaxies favors a relatively small diffusion coefficient ( i.e. , long diffusion time ) in the region where most of the emission originates .