Linking gamma-ray spectra of supernova remnants to the cosmic ray injection properties in the aftermath of supernovae

The acceleration times of the highest-energy particles which emit gamma-rays in young and middle-age SNRs are comparable with SNR age. If the number of particles starting acceleration was varying during early times after the supernova explosion then this variation should be reflected in the shape of the gamma-ray spectrum. We use the solution of the non-stationary equation for particle acceleration in order to analyze this effect. As a test case, we apply our method to describe gamma-rays from IC443. As a proxy of the IC443 parent supernova we consider SN1987A. First, we infer the time dependence of injection efficiency from evolution of the radio spectral index in SN1987A. Then, we use the inferred injection behavior to fit the gamma-ray spectrum of IC443. We show that the break in the proton spectrum needed to explain the gamma-ray emission is a natural consequence of the early variation of the cosmic ray injection, and that the very-high energy gamma-rays originate from particles which began acceleration during the first months after the supernova explosion. We conclude that the shape of the gamma-ray spectrum observed today in SNRs critically depends on the time variation of the cosmic ray injection process in the immediate post explosion phases. With the same model, we estimate also the possibility in the future to detect gamma-rays from SN 1987A.