Detection of spatial clustering in the 1000 richest SDSS DR8 redMaPPer clusters with Nearest Neighbor distributions

Distances to the $k$-nearest-neighbor ($k$NN) data points from volume-filling query points are a sensitive probe of spatial clustering. Here we present the first application of $k$NN summary statistics to observational clustering measurement, using the 1000 richest redMaPPer clusters ($0.1\leqslant z\leqslant 0.3$) from the SDSS DR8 catalog. A clustering signal is defined as a difference in the cumulative distribution functions (CDFs) of $k$NN distances from fixed query points to the observed clusters versus a set of unclustered random points. We find that the $k=1,2$-NN CDFs of redMaPPer deviate significantly from the randoms' across scales of 35 to 155 Mpc, which is a robust signature of clustering. In addition to $k$NN, we also measure the two-point correlation function for the same set of redMaPPer clusters versus random points, which shows a noisier and less significant clustering signal within the same radial scales. Quantitatively, the $\chi^2$ distribution for both the $k$NN-CDFs and the two-point correlation function measured on the randoms peak at $\chi^2\sim 50$ (null hypothesis), whereas the $k$NN-CDFs ($\chi^2\sim 300$, $p = 1.54\times 10^{-36}$) pick up a much more significant clustering signal than the two-point function ($\chi^2\sim 100$, $p = 1.16\times 10^{-6}$) when measured on redMaPPer. Finally, the measured 3NN and 4NN CDFs deviate from the predicted $k=3, 4$-NN CDFs assuming an ideal Gaussian field, indicating a non-Gaussian clustering signal for redMaPPer clusters, although its origin might not be cosmological due to observational systematics. Therefore, $k$NN serves as a more sensitive probe of clustering complementary to the two point correlation function, providing a novel approach for constraining cosmology and galaxy-halo connection.