Persistent topology of the reionization bubble network - I. Formalism and phenomenology

We present a new formalism for studying the topology of H II regions during the Epoch of Reionization, based on persistent homology theory. With persistent homology, it is possible to follow the evolution of topological features over time. We introduce the notion of a persistence field as a statistical summary of persistence data and we show how these fields can be used to identify different stages of reionization. We identify two new stages common to all bubble ionization scenarios. Following an initial pre-overlap and subsequent overlap stage, the topology is first dominated by neutral filaments (filament stage) and then by enclosed patches of neutral hydrogen undergoing outside-in ionization (patch stage). We study how these stages are affected by the degree of galaxy clustering. We also show how persistence fields can be used to study other properties of the ionization topology, such as the bubble size distribution and the fractal-like topology of the largest ionized region.