Clustering of the extreme: A theoretical description of weak lensing critical points power spectra in the mildly nonlinear regime

TL;DR

This work develops an analytical framework for the clustering of 2D weak-lensing critical points (peaks, voids, saddles) in mildly non-Gaussian fields. It extends a perturbative bias approach and employs Wiener-Hermite/Gram-Charlier expansions to derive a power-spectrum description up to NNLO, with NLO evaluated numerically and validated against Gaussian Monte Carlo integrations. The key results include a first-principles expression for all critical-point 2PCFs, enhanced BAO features in peak clustering, and non-Gaussian corrections at the 10% level on quasi-linear scales relevant for Stage-IV surveys. The framework enables fast, principled comparisons with simulations, offers a pathway for hybrid analytic-simulation analyses, and suggests using BAO features in critical-point clustering as a potential standard ruler in weak-lensing cosmology.

Abstract

In cosmic web analysis, complementary to traditional cosmological probes, the extrema (e.g. peaks and voids) two-point correlation functions (2PCFs) are of particular interest for the study of both astrophysical phenomena and cosmological structure formation. However most previous studies constructed those statistics via N-body simulations without a robust theoretical derivation from first principles. A strong motivation exists for analytically describing the 2PCFs of these local extrema, taking into account the nonlinear gravitational evolution in the late Universe. In this paper, we derive analytical formulae for the power spectra and 2PCFs of 2D critical points, including peaks (maxima), voids (minima) and saddle points, in mildly non-Gaussian weak gravitational lensing fields. We apply a perturbative bias expansion to model the clustering of 2D critical points. We successfully derive the power spectrum of weak lensing critical points up to the next-to-next-to-leading order (NNLO) in gravitational perturbation theory, where trispectrum configurations of the weak lensing field have to be included. We numerically evaluate those power spectra up to the next-to-leading order (NLO), which correspond to the inclusion of bispectrum configurations, and transform them to the corresponding 2PCFs. An exact Monte Carlo (MC) integration is performed assuming a Gaussian distributed density field to validate our theoretical predictions. Overall, we find similar properties in 2D compared to the clustering of 3D critical points previously measured from N-body simulations. Contrary to standard lensing power spectra analysis, we find distinct BAO features in the lensing peak 2PCFs due to the gradient and curvature constraints, and we quantify that non-Gaussianity makes for ~10% of the signal at quasi-linear scales which could be important for current stage-IV surveys.

Figures (11)

• Figure 1: A summary of 2PCFs of all pairs of critical points in weak lensing convergence fields above a threshold of $\nu=0.3$. The convergence is smoothed with a Gaussian kernel at scale $R=15^{\prime}$. Blue curves represent auto 2PCFs while orange curves display cross 2PCFs (between different types of critical points). Within each color, different curve configurations represent different types of critical points in a 2PCF. Note that the chosen convergence field is for sources located at $z_s = 1.5$.
• Figure 2: Illustration of the behaviour of the LO bias functions for extrema. We schematically decompose the field into its large and small-scales fluctuations. The critical point bias can at large $\nu$ be approximated by the value of the critical point amplitude plus its curvature.
• Figure 3: Auto and cross power spectrum of different critical points in 2D weak lensing fields above a given threshold $\nu=0.3$ and a smoothing scale $R=15'$. The subscript "p" represents peaks while "v" and "s" stand for voids and saddle points respectively. Blue curve represents the LO in the power spectrum, corresponding to the first line on the right hand side of Eq. (\ref{['eq:wl_power_spectrum_nlo']}). Orange curve is the sum of the LO and the 2nd-order Gaussian approximation ($\propto P(k)^2$ term) in the NLO, which is the second line term in Eq. (\ref{['eq:wl_power_spectrum_nlo']}). Green curve is the full NLO prediction including the bispectrum correction expressed by the third line term in Eq. (\ref{['eq:wl_power_spectrum_nlo']}). The color curves in all the other sub-panels have the same representation as that denoted in the top left subplot. The fluctuations on large $k$ scales are the residuals of the unphysical components from the perturbative bias expansion after smoothing. They will not impact the 2PCFs on intermediate and large angular separations after the Hankel transform as we will show in Sec. \ref{['sec:MC_integration']} with the peak 2PCF as an example.
• Figure 4: Auto and cross 2PCF of different critical points in 2D weak lensing fields above a given threshold $\nu=0.3$ and a smoothing scale $R=15'$. The subscript "p" represents peaks while "v" and "s" stand for voids and saddle points respectively. Blue curve represents the LO in the 2PCF, which is Hankel transformed from the first line on the right hand side of Eq. (\ref{['eq:wl_power_spectrum_nlo']}). The orange curve is the sum of the LO and the 2nd-order Gaussian approximation (whose Fourier counterpart is the $\propto P(k)^2$ term) in the NLO, which is the sum of the Hankel transform of the first two line terms in Eq. (\ref{['eq:wl_power_spectrum_nlo']}). The green curve is the full NLO prediction including the bispectrum correction, i.e. the Hankel transform of the complete expression of Eq. (\ref{['eq:wl_power_spectrum_nlo']}). The color curves in all the other sub-panels have the same representation as that denoted in the top left subplot.
• Figure 5: Left: The same peak 2PCF as in Fig. \ref{['fig:xi_critical_points_wl']} but multiplied by $\theta^2$. Right: Same as the left panel, but the underlying matter power spectrum is calculated without baryons while kept at the same total matter density parameter.