Exploring the geometry, topology and morphology of large scale structure using Minkowski functionals

Modern redshift surveys such as the 2 degree field Galaxy Redshift Survey (2dFGRS) and the Sloan Digital Sky Survey (SDSS) reveal fully 3-dimensional distribution of 105-6 galaxies over a large cosmological volume ⋍ 0.1-1 [h−1 Gpc]3. It is well established that the galaxies in these surveys show strong clustering. To the eye, the galaxies are distributed along sheet-like and/ or filamentary superclusters. The CfA Great Wall, Southern Great Wall and the recently discovered SDSS Great Wall are the most popular superclusters of this kind. The superclusters are interwoven with one another, leaving ≥ 70% of volume devoid of any visible matter. This volume is occupied by voids. One is motivated to test the theoretical predictions for the clustering of galaxies against rich datasets resulting from these redshift surveys. To this end, several workers have recently proposed and developed an approach to quantify the large scale structure (LSS) by studying the geometry and topology of the superclusters and voids. Concretely, this can be achieved by evaluating the Minkowski functionals (MFs) for LSS-datasets. The MF-based approach further provides an unbiased description of the shapes and sizes of the elements of LSS, i.e. the superclusters and voids. This eventually leads to a framework within which to quantify LSS and compare outputs from simulations with redshift surveys. In this review we give a summary of the progress made in this direction. After reviewing the status of observations and of numerical simulations, we comment upon the nature of bias which itself serves as a link between theoretical predictions and observations. Next we introduce MFs and give sufficient motivation for employing them in cosmology. We summarize the methods developed for efficient numerical estimation of MFs for cosmological datasets. Next we list out several important results obtained using these methods. Specifically, we stress the discriminatory power of MFs and of the derived morphological statistics, the Shapefinders. Shapefinders are specifically important to study the shapes and sizes of the superclusters and voids. Several successes of Shapefinders are highlighted here. We further note some of the important effects of scaledependent bias which are brought out by an MF-based study of the mock catalogues of galaxies. Such effects, we note, should be accounted for before comparing theoretical models with observations