Cosmological information content of Betti curves and $k$-nearest neighbor distributions

We compare the cosmological constraints that can be obtained from halo clustering on non-linear scales ($2 h^{-1}$ Mpc < $r$ < $50 h^{-1}$ Mpc) using Betti curves, a topological summary statistic, and $k$-th nearest neighbor ($k$NN) distributions. We quantify the information content of each summary statistic through Fisher matrices computed from the Quijote simulations. Due to the use of simulation-based Fisher forecasts, we pay careful attention to the convergence of the Fisher matrices by looking at their eigendecompositions. We find that, in general, only two directions in the parameter space have constraints that are well converged given the number of Quijote simulations available. We then compare the information content of each summary statistic in the reduced parameter space $\{\Omega_m, \sigma_8\}$. We find that almost all of the information present in the Betti curves comes from the first two, $\beta_0$ and $\beta_1$, which track the number of connected components and one-dimensional loops respectively, and almost no constraining power comes from $\beta_2$ which tracks the number of topological voids. In comparison, we find that the $k$NNs provide very competitive constraints along with several potential advantages in regards to real data. Finally, we find that while the $k$NNs and Betti curves provide some complementary constraints, they are not fully independent, potentially indicating a connection between the two statistics.