| Name: | Kwanit Gangopadhyay |
| Affiliation: | Indian Institute of Science Education and Research, Pune |
| Conference ID: | ASI2025_474 |
| Title: | Geometric and Topological Interpretations of the Nearest Neighbour Measurements |
| Authors: | Arka Banerjee 1, Tom Abel 2,3,4 |
| Authors Affiliation: | 1 Department of Physics, Indian Institute of Science Education and Research, Homi Bhabha Road, Pashan, Pune 411008, India
2 Kavli Institute for Particle Astrophysics and Cosmology, Stanford University, 452 Lomita Mall, Stanford, CA 94305, USA
3 Department of Physics, Stanford University, 382 Via Pueblo Mall, Stanford, CA 94305, USA
4 SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, CA 94025, USA |
| Mode of Presentation: | Poster |
| Abstract Category: | Galaxies and Cosmology |
| Abstract: | The matter field in the universe is a Gaussian Random Field on large scales, summarized by the power spectrum or two-point correlation function. However, on smaller scales, non-linear gravitational evolution results in non-Gaussian clustering, necessitating improved summary statistics. k-Nearest Neighbour Cumulative Distribution Functions (kNN CDFs) are sensitive to all connected N-point correlation functions and related to the Void Probability Function and Counts-in-Cell statistic. kNN CDFs can tighten cosmological parameter constraints by over a factor of 2 compared to the two-point function and are faster to compute than higher-order statistics.
Our work provides geometric and topological interpretations of kNN CDFs. The 1NN CDF at radius r reflects the volume fraction within spheres of radius r centered on tracers. Its derivatives relate to the geometry of sphere intersections: the first derivative to area, the second to angles/arcs at two-sphere intersections, and the third to angles, arcs, and solid angles at three-sphere intersections. These interpretations align with the geometric insights provided by Minkowski functionals, known for measuring volume, area, mean curvature, and Euler characteristic. We demonstrate that the third derivative of the 1NN CDF links to the Euler characteristic, showing that kNN CDFs contain topological information. We have conducted Fisher analyses to compare the constraining power of kNN CDFs and Minkowski Functionals, demonstrating that both effectively contain the same information. We also provide the geometric interpretations of higher k-NN CDFs in terms of intersections of multiple spheres.
While kNN CDFs are less sensitive to topological statistics like Betti Numbers or Persistent Homology, kNN distance maps may retain more spatial information, offering new analysis opportunities. We have used these maps for Voronoi tessellation and Delaunay triangulation of tracers. Overall, kNN CDFs are a fast, robust method for detecting non-Gaussian clustering and can reveal geometric and some topological insights into tracer distributions. |
