Dust coagulation feedback on magnetohydrodynamic resistivities in protostellar collapse

The degree of coupling between the gas and the magnetic field during the collapse of a core and the subsequent formation of a disk depends on the assumed dust size distribution. We study the impact of grain-grain coagulation on the evolution of magnetohydrodynamic (MHD) resistivities during the collapse of a prestellar core. We use a 1-D model to follow the evolution of the dust size distribution, out-of-equilibrium ionization state and gas chemistry during the collapse of a prestellar core. To compute the grain-grain collisional rate, we consider models for both random and systematic, size-dependent, velocities. We include grain growth through grain-grain coagulation and ice accretion, but ignore grain fragmentation. Starting with a MRN (Mathis et al. 1977) size distribution, we find that coagulation in grain-grain collisions generated by hydrodynamical turbulence is not efficient at removing the smallest grains, and as a consequence does not affect much the evolution of the Hall and ambipolar diffusion MHD resistivities which still severly drop during the collapse like in models without coagulation. The inclusion of systematic velocities, possibly induced by the presence of ambipolar diffusion, increases the coagulation rate between small and large grains, removing small grains earlier in the collapse and therefore limiting the drop in the Hall and ambipolar diffusion resistivities. At intermediate densities ($n_{\rm H} \sim 10^8\,$cm$^{-3}$), the Hall and ambipolar diffusion resistivities are found to be higher by 1 to 2 orders of magnitude in models with coagulation than in models where coagulation is ignored, and also higher than in a toy model without coagulation where all grains smaller than $0.1\,{\mu}$m would have been removed in the parent cloud before the collapse. When grain drift velocities induced by ambipolar diffusion are included, dust coagulation ... (abridged)