Cosmic voids are progressively emerging as a new viable cosmological probe. Their abundance and density profiles are sensitive to modifications of gravity, as well as to dark energy and neutrinos. The main goal of this work is to investigate the possibility of exploiting cosmic void statistics to disentangle the degeneracies resulting from a proper combination of $f(R)$ modified gravity and neutrino mass. We use N-body simulations to analyse the density profiles and size function of voids traced by both dark matter particles and haloes. We find clear evidence of the enhancement of gravity in $f(R)$ cosmologies in the void density profiles at $z=1$. However, these effects can be almost completely overridden by the presence of massive neutrinos because of their thermal free-streaming. Despite the limited volume of the analysed simulations does not allow us to achieve a statistically relevant abundance of voids larger than $40 mathrm{Mpc}/h$, we find that the void size function at high redshifts and for large voids is potentially an effective probe to disentangle these degenerate cosmological models, which is key in the prospective of the upcoming wide field redshift surveys.
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