Supernova remnants are believed to be the major contributors to Galactic cosmic rays. In this paper, we explore how the non-thermal emission from young remnants can be used to probe the production of energetic particles at the shock (both protons and electrons). Our model couples hydrodynamic simulations of a supernova remnant with a kinetic treatment of particle acceleration. We include two important back-reaction loops upstream of the shock: energetic particles can (i) modify the flow structure and (ii) amplify the magnetic field. As the latter process is not fully understood, we use different limit cases that encompass a wide range of possibilities. We follow the history of the shock dynamics and of the particle transport downstream of the shock, which allows us to compute the non-thermal emission from the remnant at any given age. We do this in 3D, in order to generate projected maps that can be compared with observations. We observe that completely different recipes for the magnetic field can lead to similar modifications of the shock structure, although to very different configurations of the field and particles. We show how this affects the emission patterns in different energy bands, from radio to X-rays and $gamma$-rays. High magnetic fields ($>100 mu$G) directly impact the synchrotron emission from electrons, by restricting their emission to thin rims, and indirectly impact the inverse Compton emission from electrons and also the pion decay emission from protons, mostly by shifting their cut-off energies to respectively lower and higher energies.
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