TeV cosmic ray nuclei acceleration in shell-type supernova remnants with hard $\gamma$-ray spectra

The emission mechanism for hard $\gamma$-ray spectra from supernova remnants (SNRs) is still a matter of debate. Recent multi-wavelength observations of TeV source HESS J1912+101 show that it is associated with an SNR with an age of $\sim 100$ kyrs, making it unlikely produce the TeV $\gamma$-ray emission via leptonic processes. We analyzed Fermi observations of it and found an extended source with a hard spectrum. HESS J1912+101 may represent a peculiar stage of SNR evolution that dominates the acceleration of TeV cosmic rays. By fitting the multi-wavelength spectra of 13 SNRs with hard GeV $\gamma$-ray spectra with simple emission models with a density ratio of GeV electrons to protons of $\sim 10^{-2}$, we obtain reasonable mean densities and magnetic fields with a total energy of $\sim 10^{50}$ ergs for relativistic ions in each SNR. Among these sources, only two of them, namely SN 1006 and RCW 86, favor a leptonic origin for the $\gamma$-ray emission. The magnetic field energy is found to be comparable to that of the accelerated relativistic ions and their ratio has a tendency of increase with the age of SNRs. These results suggest that TeV cosmic rays mainly originate from SNRs with hard $\gamma$-ray spectra.