Tree structure of the percolating Universe

We present a numerical study of topological descriptors of initially Gaussian and scale-free density perturbations evolving via gravitational instability in an expanding universe. We carefully evaluate and avoid numerical contamination in making accurate measurements on simulated fields on a grid in a finite box. Independent of extent of non linearity, the measured Euler number of the excursion set at the percolation threshold, $\delta_c$, is positive and nearly equal to the number of isolated components, suggesting that these structures are trees. Our study of critical point counts reconciles the clumpy appearance of the density field at $\delta_c$ with measured filamentary local curvature. In the Gaussian limit, we measure $|\delta_c|> \sigma$ in contrast to widely held belief that $|\delta_c| \sim \sigma$, where $\sigma^2$ is the variance of the density field.