Measuring the geometry and topology of large‐scale structure using SURFGEN: methodology and preliminary results

Observations of the universe reveal that matter within it clusters on a variety of scales. On scales between 10 - 100 Mpc, the universe is spanned by a percolating network of superclusters interspersed with large and almost empty regions – voids. This paper, the first in a series, presents a new ansatz which can successfully be used to determine the morphological properties of the supercluster-void network. The ansatz is based on a surface modelling scheme (SURFGEN), developed explicitly for the purpose, which generates a triangulated surface from a discrete data set representing (say) the distribution of galaxies in real (or redshift) space. The triangulated surface describes, at progressively lower density thresholds, clusters of galaxies, superclusters of galaxies and voids. Four Minkowski functionals (MFs) – surface area, volume, extrinsic curvature and genus – describe the geometry and topology of the supercluster-void network. On a discretised and closed triangulated surface the MFs are determined using SURFGEN. Ratio’s of the Minkowski functionals provide us with an excellent diagnostic of three dimensional shapes of clusters, superclusters and voids. Minkowski functionals can be studied at different levels of the density contrast and therefore probe the morphology of large scale structure on a variety of length scales. Our method for determining the Minkowski functionals of a triangulated iso-density surface is tested against both simply and multiply connected eikonal surfaces such as triaxial ellipsoids and tori. The performance of our code is thereby evaluated using density distributions which are pancake-like, filamentary, ribbon-like and spherical. Remarkably, the first three Minkowski functionals are computed to better than 1% accuracy while the fourth (genus) is known exactly. SURFGEN also gives very accurate results when applied to Gaussian random fields. We apply SURFGEN to study morphology in three cosmological models, �CDM, �CDM and SCDM, at the present epoch. Geometrical properties of the supercluster-void network are found to be sensitive to the underlying

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