Probability distribution and statistical properties of spherically compensated cosmic regions in $Λ$CDM cosmology

The statistical properties of cosmic structures are well known to be strong probes for cosmology. In particular, several studies tried to use the cosmic void counting number to obtain tight constrains on Dark Energy. In this paper we address this question by using the CoSphere model as introduced in de Fromont & Alimi (2017a). We derive their exact statistics in both primordial and non linearly evolved Universe for the standard $Λ$CDM model. We first compute the full joint Gaussian probability distribution for the various parameters describing these profiles in the Gaussian Random Field. We recover the results of Bardeen et al. (1986) only in the limit where the compensation radius becomes very large, i.e. when the central extremum decouples from its cosmic environment. We derive the probability distribution of the compensation size in this primordial field. We show that this distribution is redshift independent and can be used to model cosmic void size distribution. Interestingly, it can be used for central maximum such as DM haloes. We compute analytically the statistical distribution of the compensation density in both primordial and evolved Universe. We also derive the statistical distribution of the peak parameters already introduced by Bardeen et al. (1986) and discuss their correlation with the cosmic environment. We thus show that small central extrema with low density are associated with narrow compensation regions with a small $R_1$ and a deep compensation density $δ_1$ while higher central extrema are located in larger but smoother over/under massive regions.