Comparison Between Poisson and Telegraph Models

We investigate the impact of finite-speed gravitational interactions on the formation of cosmic structure using N-body simulations modified to include telegraph-type corrections to the standard Poisson equation. By applying persistent homology to particle distributions, we extract Betti numbers and persistence diagrams to quantify the emergence and evolution of topological features such as voids and filaments. Our analysis reveals that simula- tions incorporating finite-speed gravity suppress the early formation of H (void) features compared to traditional Poisson-based simulations. This suggests that delayed gravitational response alters the morphological evolution of cosmic structure in measurable ways. Our results highlight the utility of topological data analysis in probing non-standard gravitational dynamics in cosmological models