Stellar velocity dispersion and initial mass function gradients in dissipationless galaxy mergers

The stellar initial mass function (IMF) is believed to be non-universal among early-type galaxies (ETGs). Parameterizing the IMF with the so-called IMF mismatch parameter $α_{\rm IMF}$, which is a measure of the stellar mass-to-light ratio of an ensemble of stars and thus of the 'heaviness' of its IMF, one finds that for ETGs $α_{\rm e}$ (i.e. $α_{\rm IMF}$ integrated within the effective radius $R_{\rm e}$) increases with $σ_{\rm e}$ (the line-of-sight velocity dispersion $σ_{\rm los}$ integrated within $R_{\rm e}$) and that, within the same ETG, $α_{\rm IMF}$ tends to decrease outwards. We study the effect of dissipationless (dry) mergers on the distribution of the IMF mismatch parameter $α_{\rm IMF}$ in ETGs using the results of binary major and minor merging simulations. We find that dry mergers tend to make the $α_{\rm IMF}$ profiles of ETGs shallower, but do not alter significantly the shape of the distributions in the spatially resolved $σ_{\rm los}α_{\rm IMF}$ space. Individual galaxies undergoing dry mergers tend to decrease their $α_{\rm e}$, due to erosion of $α_{\rm IMF}$ gradients and mixing with stellar populations with lighter IMF. Their $σ_{\rm e}$ can either decrease or increase, depending on the merging orbital parameters and mass ratio, but tends to decrease for cosmologically motivated merging histories. The $α_{\rm e}$-$σ_{\rm e}$ relation can vary with redshift as a consequence of the evolution of individual ETGs: based on a simple dry-merging model, ETGs of given $σ_{\rm e}$ are expected to have higher $α_{\rm e}$ at higher redshift, unless the accreted satellites are so diffuse that they contribute negligibly to the inner stellar distribution of the merger remnant.