Genus statistics using the Delaunay tessellation field estimation method: (I) tests with the Millennium Simulation and the SDSS DR7

TL;DR

This paper introduces a novel Delaunay tessellation method for analyzing the topology of cosmic large-scale structures, outperforming traditional smoothing techniques in extracting topological information from galaxy distributions.

Contribution

The paper presents a new Delaunay tessellation field estimation method for genus statistics, enabling precise topology measurement without free parameters and improving discrimination between galaxy catalogues.

Findings

Delaunay-based method captures more topological information.

It distinguishes between different galaxy catalogues effectively.

Detects discrepancies between models and SDSS data.

Abstract

We study the topology of cosmic large-scale structure through the genus statistics, using galaxy catalogues generated from the Millennium Simulation and observational data from the latest Sloan Digital Sky Survey Data Release (SDSS DR7). We introduce a new method for constructing galaxy density fields and for measuring the genus statistics of its isodensity surfaces. It is based on a Delaunay tessellation field estimation (DTFE) technique that allows the definition of a piece-wise continuous density field and the exact computation of the topology of its polygonal isodensity contours, without introducing any free numerical parameter. Besides this new approach, we also employ the traditional approaches of smoothing the galaxy distribution with a Gaussian of fixed width, or by adaptively smoothing with a kernel that encloses a constant number of neighboring galaxies. Our results show that the Delaunay-based method extracts the largest amount of topological information. Unlike the traditional approach for genus statistics, it is able to discriminate between the different theoretical galaxy catalogues analyzed here, both in real space and in redshift space, even though they are based on the same underlying simulation model. In particular, the DTFE approach detects with high confidence a discrepancy of one of the semi-analytic models studied here compared with the SDSS data, while the other models are found to be consistent.