Probability distribution and statistical properties of spherically
  compensated cosmic regions in $\Lambda$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 $\Lambda$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 $\delta_1$ while higher central
extrema are located in larger but smoother over/under massive regions.
