In this paper, we study the geometry and dynamics of ionization fronts during reionization using the CROC simulations. By constructing reionization timing maps, we extract front velocities and curvatures across the cosmic volume.

I find it most interesting that the curvature statistics in the simulations are well described by analytical expressions derived from a barrier-crossing model applied to a Gaussian random field (GRF). In particular, the mean Gaussian curvature correlates with the normalized density contrast $\nu = \delta / \sigma_0$, and agrees with the GRF prediction in overdense regions:

In underdense regions, the analytical model breaks down due to its assumption that ionization fronts follow isodensity contours.

The velocity distribution is also compared to predictions from the same barrier-crossing formalism. While the model reproduces the overall shape of the distribution at intermediate and high velocities, it fails to capture the low-velocity tail observed in simulations. This discrepancy is due to non-Gaussian features in the evolved density field, which are not included in the Gaussian-based model.

These results show that the topology of reionization, quantified through geometric measures such as curvature, can be connected to the statistical properties of the initial density field. This provides a compact and interpretable framework for linking simulation results to theory, with potential applications to large-scale observables such as the 21 cm signal and the thermal state of the intergalactic medium.