The clustering of critical points in the evolving cosmic web

Focusing on both small separations and Baryonic Acoustic Oscillation scales, the cosmic evolution of clustering properties peak, void, wall, filament-type critical points is measured using two-point correlation functions in $\Lambda$CDM dark matter simulations as a function their relative rarity. A qualitative comparison to corresponding theory for Gaussian Random fields allows us understand following observed features: i) appearance an exclusion zone at separation, whose size depends rarity signature (\ie number negative eigenvalues) involved; ii) amplification bump with its reversal cross-correlations involving negatively biased points; iii) orientation-dependent small-separation divergence peaks filaments (voids walls) which reflects loci such filament's (wall's) eigenframe. The most significant features correlations are tabulated. (cross-) non-linear (peaks, voids) display variation redshift, while those less seem mostly insensitive redshift evolution, should prove advantageous model. distances maxima peak-to-wall peak-to-void over that peak-to-filament cross-correlation ratios $\sim\sqrt{2}$ $\sim\sqrt{3}$, respectively could be interpreted indication crystal being average close cubic lattice. insensitivity suggests absolute become topologically robust alternative standard techniques when analyzing upcoming large scale surveys Euclid or LSST.