Non-linear diffusion of cosmic rays escaping from supernova remnants: Cold partially neutral atomic and molecular phases

This paper deals with the Cosmic Ray (CR) propagation in the weakly ionized environments of supernovae remnants (SNRs) and is based the Cosmic Ray Cloud (CRC) model developed by Malkov et al. (2013) and Nava et al. (2016). It consists in solving two transport equation simultaneously: one for the CR pressure and one for the Alfven waves energy density where CR are initially confined in the SNR shock. CR trigger a streaming instability and produce slab-type resonant Alfven modes. The self-generated turbulence is damped by ion-neutral collision and by non-correlated interaction with Alfven modes generated at large scale. We show that CRs leaking in cold dense phases as Cold Neutral Medium (CNM) and Diffuse molecular Medium (DiM) can still be confined over distances of a few tens of parsecs from the CRC center for a few kyrs. At 10 TeV CR diffusion can be suppressed by two or three orders of magnitude. This effect results from a reduced ion-neutral collision damping in the decoupled regime. We calculate the grammage of CR in these environnements. We find both in single or multi-phase set ups that at 10 GeV CNM and DiM media can produce grammage in the range 10-20 g/cm$^2$ in the CNM and DiM phases. At 10 TeV because of non-linear propagation the grammage increases to values in the range 0.5-20 g/cm$^2$ in these two phases. We also present preliminary calculations in inhomogeneous ISM combining two or three different phases where we obtain the same trends.