- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Theoretical description of weak lensing critical points for power spectra in the mildly nonlinear regime

Zhengyangguang Gong*

Alexandre Barthelemy

Sandrine Codis

  • *Contact author: lgong@usm.lmu.de
PDF

Phys. Rev. D 112, 083519 – Published 9 October, 2025

DOI: https://doi.org/10.1103/c12h-8fpv

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 formulas for the power spectra and 2PCFs of two-dimensional (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. A generalized Gram-Charlier A series expansion is used to describe the probability density functional of the cosmic density field. We successfully derive the power spectrum of weak lensing critical points up to the next-to-next-to-leading order 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, which corresponds to the inclusion of bispectrum configurations, and transform them to the corresponding 2PCFs. An exact Monte Carlo integration is performed assuming a Gaussian distributed density field to validate our theoretical predictions. Overall, we find similar properties in two dimensions compared to the clustering of three-dimensional critical points previously measured from N-body simulations. Contrary to standard lensing power spectra analysis, we find distinct baryon acoustic oscillation 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 quasilinear scales, which could be important for current stage-IV surveys.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (72)

  1. V. Desjacques, D. Jeong, and F. Schmidt, Large-scale galaxy bias, Phys. Rep. 733, 1 (2018).
  2. J. M. Bardeen, J. R. Bond, N. Kaiser, and A. S. Szalay, The statistics of peaks of Gaussian random fields, Astrophys. J. 304, 15 (1986).
  3. S. L. Lumsden, A. F. Heavens, and J. A. Peacock, The clustering of peaks in a random Gaussian field, Mon. Not. R. Astron. Soc. 238, 293 (1989).
  4. J. A. Peacock and A. F. Heavens, Alternatives to the Press-Schechter cosmological mass function, Mon. Not. R. Astron. Soc. 243, 133 (1990).
  5. E. Regos and A. S. Szalay, Density and velocity correlations of peaks from random Gaussian fluctuations, Mon. Not. R. Astron. Soc. 272, 447 (1995).
  6. V. Desjacques, Baryon acoustic signature in the clustering of density maxima, Phys. Rev. D 78, 103503 (2008).
  7. V. Desjacques and R. K. Sheth, Redshift space correlations and scale-dependent stochastic biasing of density peaks, Phys. Rev. D 81, 023526 (2010).
  8. V. Desjacques, Local bias approach to the clustering of discrete density peaks, Phys. Rev. D 87, 043505 (2013).
  9. T. Matsubara and S. Codis, Large-separation expansion of peak clustering in Gaussian random fields, Phys. Rev. D 101, 063504 (2020).
  10. T. Baldauf, S. Codis, V. Desjacques, and C. Pichon, Nonperturbative halo clustering from cosmological density peaks, Phys. Rev. D 103, 083530 (2021).
  11. E. G. P. Bos, R. van de Weygaert, K. Dolag, and V. Pettorino, The darkness that shaped the void: Dark energy and cosmic voids, Mon. Not. R. Astron. Soc. 426, 440 (2012).
  12. A. Barreira, M. Cautun, B. Li, C. M. Baugh, and S. Pascoli, Weak lensing by voids in modified lensing potentials, J. Cosmol. Astropart. Phys. 08 (2015) 028.
  13. E. L. D. Perico, R. Voivodic, M. Lima, and D. F. Mota, Cosmic voids in modified gravity scenarios, Astron. Astrophys. 632, A52 (2019).
  14. S. Contarini, F. Marulli, L. Moscardini, A. Veropalumbo, C. Giocoli, and M. Baldi, Cosmic voids in modified gravity models with massive neutrinos, Mon. Not. R. Astron. Soc. 504, 5021 (2021).
  15. C. M. Correa, Cosmic voids as cosmological laboratories, Bol. Asoc. Argent. Astron. Plata Argent. 64, 159 (2023), https://ui.adsabs.harvard.edu/abs/2023BAAA...64..159C/abstract.
  16. N. Hamaus, A. Pisani, J.-A. Choi, G. Lavaux, B. D. Wandelt, and J. Weller, Precision cosmology with voids in the final BOSS data, J. Cosmol. Astropart. Phys. 12 (2020) 023.
  17. S. Contarini, A. Pisani, N. Hamaus, F. Marulli, L. Moscardini, and M. Baldi, Cosmological constraints from the BOSS DR12 void size function, Astrophys. J. 953, 46 (2023).
  18. T. Sousbie, The persistent cosmic web and its filamentary structure—I. Theory and implementation, Mon. Not. R. Astron. Soc. 414, 350 (2011).
  19. M. Cautun, R. van de Weygaert, B. J. T. Jones, and C. S. Frenk, Evolution of the cosmic web, Mon. Not. R. Astron. Soc. 441, 2923 (2014).
  20. F. Sarron, C. Adami, F. Durret, and C. Laigle, Pre-processing of galaxies in cosmic filaments around AMASCFI clusters in the CFHTLS, Astron. Astrophys. 632, A49 (2019).
  21. J. Shim, S. Codis, C. Pichon, D. Pogosyan, and C. Cadiou, The clustering of critical points in the evolving cosmic web, Mon. Not. R. Astron. Soc. 502, 3885 (2021).
  22. J. Feldbrugge and R. van de Weygaert, What makes a cosmic filament? The dynamical origin and identity of filaments I. fundamentals in 2D, Mon. Not. R. Astron. Soc. 539, 873 (2025).
  23. D. Galárraga-Espinosa, C. Cadiou, C. Gouin, S. D. M. White, V. Springel, R. Pakmor, B. Hadzhiyska, S. Bose, F. Ferlito, L. Hernquist, R. Kannan, M. Barrera, A. Maria Delgado, and C. Hernández-Aguayo, Evolution of cosmic filaments in the MTNG simulation, Astron. Astrophys. 684, A63 (2024).
  24. L. Marian, R. E. Smith, S. Hilbert, and P. Schneider, The cosmological information of shear peaks: Beyond the abundance, Mon. Not. R. Astron. Soc. 432, 1338 (2013).
  25. H. Y. Shan, J.-P. Kneib, J. Comparat, E. Jullo, A. Charbonnier, T. Erben, M. Makler, B. Moraes, L. Van Waerbeke, F. Courbin, G. Meylan, C. Tao, and J. E. Taylor, Weak lensing mass map and peak statistics in Canada-France-Hawaii Telescope Stripe 82 survey, Mon. Not. R. Astron. Soc. 442, 2534 (2014).
  26. C. T. Davies, M. Cautun, and B. Li, The self-similarity of weak lensing peaks, Mon. Not. R. Astron. Soc. 488, 5833 (2019).
  27. C. T. Davies, M. Cautun, B. Giblin, B. Li, J. Harnois-Déraps, and Y.-C. Cai, Constraining cosmology with weak lensing voids, Mon. Not. R. Astron. Soc. 507, 2267 (2021).
  28. J. Harnois-Déraps, N. Martinet, T. Castro, K. Dolag, B. Giblin, C. Heymans, H. Hildebrandt, and Q. Xia, Cosmic shear cosmology beyond two-point statistics: A combined peak count and correlation function analysis of DES-Y1, Mon. Not. R. Astron. Soc. 506, 1623 (2021).
  29. C. T. Davies, M. Cautun, B. Giblin, B. Li, J. Harnois-Déraps, and Y.-C. Cai, Cosmological forecasts with the clustering of weak lensing peaks, Mon. Not. R. Astron. Soc. 513, 4729 (2022).
  30. D. Zürcher et al. (DES Collaboration), Dark energy survey year 3 results: Cosmology with peaks using an emulator approach, Mon. Not. R. Astron. Soc. 511, 2075 (2022).
  31. J. Harnois-Déraps, S. Heydenreich, B. Giblin, N. Martinet, T. Tröster, M. Asgari, P. Burger, T. Castro, K. Dolag, C. Heymans, H. Hildebrandt, B. Joachimi, and A. H. Wright, KiDS-1000 and DES-Y1 combined: Cosmology from peak count statistics, Mon. Not. R. Astron. Soc. 534, 3305 (2024).
  32. G. A. Marques, J. Liu, M. Shirasaki, L. Thiele, D. Grandón, K. M. Huffenberger, S. Cheng, J. Harnois-Déraps, K. Osato, and W. R. Coulton, Cosmology from weak lensing peaks and minima with Subaru Hyper Suprime-Cam Survey first-year data, Mon. Not. R. Astron. Soc. 528, 4513 (2024).
  33. C. T. Davies, J. Harnois-Déraps, B. Li, B. Giblin, C. Hernández-Aguayo, and E. Paillas, Constraining modified gravity with weak-lensing peaks, Mon. Not. R. Astron. Soc. 533, 3546 (2024).
  34. A. Halder, O. Friedrich, S. Seitz, and T. N. Varga, The integrated three-point correlation function of cosmic shear, Mon. Not. R. Astron. Soc. 506, 2780 (2021).
  35. M. Gatti et al. (DES Collaboration), Dark energy survey year 3 results: Cosmology with moments of weak lensing mass maps, Phys. Rev. D 106, 083509 (2022).
  36. A. Halder and A. Barreira, Response approach to the integrated shear 3-point correlation function: The impact of baryonic effects on small scales, Mon. Not. R. Astron. Soc. 515, 4639 (2022).
  37. Z. Gong, A. Halder, A. Barreira, S. Seitz, and O. Friedrich, Cosmology from the integrated shear 3-point correlation function: Simulated likelihood analyses with machine-learning emulators, J. Cosmol. Astropart. Phys. 07 (2023) 040.
  38. A. Halder, Z. Gong, A. Barreira, O. Friedrich, S. Seitz, and D. Gruen, Beyond 3×2-point cosmology: The integrated shear and galaxy 3-point correlation functions, J. Cosmol. Astropart. Phys. 10 (2023) 028.
  39. D. Anbajagane et al. (DES Collaboration), Beyond the 3rd moment: A practical study of using lensing convergence CDFs for cosmology with DES Y3, Mon. Not. R. Astron. Soc. 526, 5530 (2023).
  40. P. Cannon, D. Ward, and S. M. Schmon, Investigating the impact of model misspecification in neural simulation-based inference, arXiv:2209.01845.
  41. B. Horowitz and P. Melchior, Plausible adversarial attacks on direct parameter inference models in astrophysics, arXiv:2211.14788.
  42. C. Gay, C. Pichon, and D. Pogosyan, Non-Gaussian statistics of critical sets in 2D and 3D: Peaks, voids, saddles, genus, and skeleton, Phys. Rev. D 85, 023011 (2012).
  43. T. Matsubara, Statistics of peaks of weakly non-Gaussian random fields: Effects of bispectrum in two- and three-dimensions, Phys. Rev. D 101, 043532 (2020).
  44. N. Kaiser, On the spatial correlations of Abell clusters, Astrophys. J. Lett. 284, L9 (1984).
  45. M. Kac, On the average number of real roots of a random algebraic equation, Bull. Am. Math. Soc. 49, 314 (1943).
  46. S. O. Rice, Mathematical analysis of random noise-conclusion, Bell Syst. Tech. J. 24, 46 (1945).
  47. M. S. Longuet-Higgins, The statistical analysis of a random, moving surface, Phil. Trans. R. Soc. A 249, 321 (1957).
  48. D. Pogosyan, C. Gay, and C. Pichon, Invariant joint distribution of a stationary random field and its derivatives: Euler characteristic and critical point counts in 2 and 3D, Phys. Rev. D 80, 081301 (2009).
  49. P. J. E. Peebles, The Large-Scale Structure of the Universe (Princeton University Press, Princeton, NJ, 1980).
  50. S. Blinnikov and R. Moessner, Expansions for nearly Gaussian distributions, Astron. Astrophys. 130, 193 (1998).
  51. T. Matsubara, Diagrammatic methods in statistics and biasing in the large-scale structure of the Universe, Astrophys. J. Suppl. Ser. 101, 1 (1995).
  52. T. Matsubara, Nonlinear perturbation theory integrated with nonlocal bias, redshift-space distortions, and primordial non-Gaussianity, Phys. Rev. D 83, 083518 (2011).
  53. A. Eggemeier, R. Scoccimarro, and R. E. Smith, Bias loop corrections to the galaxy bispectrum, Phys. Rev. D 99, 123514 (2019).
  54. M. Bartelmann and P. Schneider, Weak gravitational lensing, Phys. Rep. 340, 291 (2001).
  55. P. Schneider, C. S. Kochanek, and J. Wambganss, Saas-Fee Advanced Course 33: Gravitational Lensing: Strong, Weak and Micro (Springer Science & Business Media, New York, 2006).
  56. M. Kilbinger, Cosmology with cosmic shear observations: A review, Rep. Prog. Phys. 78, 086901 (2015).
  57. D. N. Limber, The analysis of counts of the extragalactic nebulae in terms of a fluctuating density field, Astrophys. J. 117, 134 (1953).
  58. N. Kaiser, Weak lensing and cosmology, Astrophys. J. 498, 26 (1998).
  59. R. Scoccimarro, S. Colombi, J. N. Fry, J. A. Frieman, E. Hivon, and A. Melott, Nonlinear evolution of the bispectrum of cosmological perturbations, Astrophys. J. 496, 586 (1998).
  60. R. E. Smith, J. A. Peacock, A. Jenkins, S. D. M. White, C. S. Frenk, F. R. Pearce, P. A. Thomas, G. Efstathiou, and H. M. P. Couchman, Stable clustering, the halo model and non-linear cosmological power spectra, Mon. Not. R. Astron. Soc. 341, 1311 (2003).
  61. R. Takahashi, M. Sato, T. Nishimichi, A. Taruya, and M. Oguri, Revising the halofit model for the nonlinear matter power spectrum, Astrophys. J. 761, 152 (2012).
  62. D. Blas, J. Lesgourgues, and T. Tram, The Cosmic Linear Anisotropy Solving System (class). Part II: Approximation schemes, J. Cosmol. Astropart. Phys. 07 (2011) 034.
  63. N. Aghanim et al. (Planck Collaboration), Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys. 641, A6 (2020).
  64. T. Baldauf, S. Codis, V. Desjacques, and C. Pichon, Peak exclusion, stochasticity and convergence of perturbative bias expansions in 1+1 gravity, Mon. Not. R. Astron. Soc. 456, 3985 (2016).
  65. S. Codis, D. Pogosyan, and C. Pichon, On the connectivity of the cosmic web: Theory and implications for cosmology and galaxy formation, Mon. Not. R. Astron. Soc. 479, 973 (2018).
  66. T. Baldauf, U. Seljak, R. E. Smith, N. Hamaus, and V. Desjacques, Halo stochasticity from exclusion and nonlinear clustering, Phys. Rev. D 88, 083507 (2013).
  67. V. Desjacques, M. Crocce, R. Scoccimarro, and R. K. Sheth, Modeling scale-dependent bias on the baryonic acoustic scale with the statistics of peaks of Gaussian random fields, Phys. Rev. D 82, 103529 (2010).
  68. M. Jarvis, G. Bernstein, and B. Jain, The skewness of the aperture mass statistic, Mon. Not. R. Astron. Soc. 352, 338 (2004).
  69. K. M. Górski, E. Hivon, A. J. Banday, B. D. Wandelt, F. K. Hansen, M. Reinecke, and M. Bartelmann, healpix: A framework for high-resolution discretization and fast analysis of data distributed on the sphere, Astrophys. J. 622, 759 (2005).
  70. A. Zonca, L. Singer, D. Lenz, M. Reinecke, C. Rosset, E. Hivon, and K. Gorski, healpy: Equal area pixelization and spherical harmonics transforms for data on the sphere in python, J. Open Source Softwaare 4, 1298 (2019).
  71. Y. Mellier et al.(Euclid Collaboration), Euclid: I. Overview of the Euclid mission, Astron. Astrophys. 697, A1 (2025).
  72. S. Ma, Statistical Mechanics (World Scientific, Singapore, 1985).

Outline

Information

Phys. Rev. D 112, 083519– Published 9 October, 2025
  • Received 10 February 2025
  • Accepted 5 August 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation