The void probability function (VPF) is the likelihood of finding no tracers (galaxies or dark-matter haloes) within a given volume, and is sensitive to the large-scale matter distribution in the Universe. In this work, we investigate the VPF of dark matter haloes and galaxies using the quijote N-body simulations across a wide range of cosmological models, including both the standard $\mathrm{\Lambda }\mathrm{CDM}$ scenario and several well-motivated extensions. We examine in detail how the VPF responds to variations in cosmology, as well as to redshift ($z$), mass, morphology, and sample dilution, and compared the results with theoretical predictions from the geometric hierarchical (GH) and negative binomial (NB) models. We find that the VPF of haloes is sensitive to cosmology and halo mass, and follows the GH model at $z=0$, with the $\mathrm{\Lambda }\mathrm{CDM}$ model showing the smallest deviation, while the cosmology with massive neutrinos shows the largest deviation, likely due to neutrino free-streaming suppressing small-scale clustering and producing a smoother matter distribution. At higher $z$ and larger sample dilution, the VPF gradually departs from the GH model and trends toward the NB model. The VPF exhibits a dependency on the halo mass, with low-mass haloes showing the largest deviations from the GH model in the $\mathrm{\Lambda }\mathrm{CDM}$ cosmology, while intermediate- and high-mass haloes show the strongest deviations in the modified gravity scenario. In redshift space, the VPF shifts toward the NB model, with primordial non-Gaussianity producing the largest deviations, possibly due to enhanced redshift-space distortions. Furthermore, for galaxies, the VPF is dependent on the morphological type and underlying cosmology, with ellipticals following the GH model, while spirals follow the NB model. Lastly, we investigate the impact of cosmic variance using multiple $\mathrm{\Lambda }\mathrm{CDM}$ realizations and find that it introduces a non-negligible scatter in the VPF measurements. Overall, these results suggest that the VPF could be useful as a computationally efficient probe to distinguish extensions of the standard $\mathrm{\Lambda }\mathrm{CDM}$ model in upcoming large-volume galaxy surveys.

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• Dark energy
• Dark matter
• Evolution of the Universe
• Large scale structure of the Universe