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

Abstract

Observations of the Universe reveal that matter within it clusters on a variety of scales. On scales between 10 and 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 that 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 discretized and closed triangulated surface, the MFs are determined using SURFGEN. Ratios of the MFs provide us with an excellent diagnostic of three-dimensional shapes of clusters, superclusters and voids. MFs 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 MFs of a triangulated isodensity 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 that are pancake-like, filamentary, ribbon-like and spherical. Remarkably, the first three MFs are computed to better than 1 per cent 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, ΛCDM, τCDM and SCDM, at the present epoch. Geometrical properties of the supercluster-void network are found to be sensitive to the underlying cosmological parameter set. For instance, the percolating supercluster in ΛCDM turns out to be more filamentary but topologically simpler than superclusters in τCDM and SCDM. It occupies just 0.6 per cent of the total simulation-box volume yet contains about 4 per cent of the total mass. Our results indicate that the surface modelling scheme to calculate MFs is accurate and robust and can successfully be used to quantify the topology and morphology of the supercluster-void network in the universe.