The shape distribution of superclusters in SDSS DR 12

Superclusters are the largest structures formed due to the gravitational collapse and thus their geometrical information can shed light on the structure formation process on cosmological scales, hence on the fundamental properties of gravity itself. In this work we study the distributions of the shape, topology and morphology of the superclusters, defined in two different ways -- using constant and individual density threshold on a luminosity weighted density field constructed from SDSS DR 12 main galaxy sample. For this purpose, we employ Minkowski functionals and Shapefinders, precisely calculated by the shape diagnostic tool SURFGEN2. We notice a striking similarity in the shape distributions of the supercluster samples those are obtained following these two distinct techniques. Not surprisingly, most of them are spherical in shape with trivial topology. But there are a significant number of superclusters that are quite filamentary and statistically tend to be large in size with non-zero genus values. The shape distribution results (catalogues) along with the modified SURFGEN2 code have been made publicly available.