Cosmic-ray acceleration in supernova remnants: non-linear theory revised

A rapidly growing amount of evidences, mostly coming from the recent gamma-ray observations of Galactic supernova remnants (SNRs), is seriously challenging our understanding of how particles are accelerated at fast shocks. The cosmic-ray (CR) spectra required to account for the observed phenomenology are in fact as steep as E^-2.2-E^-2.4, i.e., steeper than the test-particle prediction of first-order Fermi acceleration, and significantly steeper than what expected in a more refined non-linear theory of diffusive shock acceleration. By accounting for the dynamical back-reaction of the non-thermal particles, such a theory in fact predicts that the more efficient the particle acceleration, the flatter the CR spectrum. In this work we put forward a self-consistent scenario in which the account for the magnetic field amplification induced by CR streaming produces the conditions for reversing such a trend, allowing — at the same time — for rather steep spectra and CR acceleration efficiencies (about 20%) consistent with the hypothesis that SNRs are the sources of Galactic CRs. In particular, we quantitatively work out the details of instantaneous and cumulative CR spectra during the evolution of a typical SNR, also stressing the implications of the observed levels of magnetization on both the expected maximum energy and the predicted CR acceleration efficiency. The latter naturally turns out to saturate around 10-30%, almost independently of the fraction of particles injected into the acceleration process as long as this fraction is larger than about 10^-4.