Topological approach to void finding applied to the SDSS galaxy map

The structure of the low redshift Universe is dominated by a multi-scale void distribution delineated by filaments and walls of galaxies. The characteristics of voids; such as morphology, average density profile, and correlation function, can be used as cosmological probes. However, their physical properties are difficult to infer due to shot noise and the general lack of tracer particles used to define them. In this work, we construct a robust, topology-based void finding algorithm that utilizes Persistent Homology (PH) to detect persistent features in the data. We apply this approach to a volume limited sub-sample of galaxies in the SDSS I/II Main Galaxy catalog with the $r$-band absolute magnitude brighter than $M_r=-20.19$, and a set of mock catalogs constructed using the Horizon Run 4 cosmological $N$-body simulation. We measure the size distribution of voids, their averaged radial profile, sphericity, and the centroid nearest neighbor separation, using conservative values for the threshold and persistence. We find $32$ topologically robust voids in the SDSS data over the redshift range $0.02 \leq z \leq 0.116$, with effective radii in the range $21 - 56 \, h^{-1} \, {\rm Mpc}$. The median nearest neighbor void separation is found to be $\sim 57 \, h^{-1} \, {\rm Mpc}$, and the median radial void profile is consistent with the expected shape from the mock data.