Nearest Neighbor distributions: new statistical measures for
  cosmological clustering
  The use of summary statistics beyond the two-point correlation function to
analyze the non-Gaussian clustering on small scales is an active field of
research in cosmology. In this paper, we explore a set of new summary
statistics -- the $k$-Nearest Neighbor Cumulative Distribution Functions
($k{\rm NN}$-${\rm CDF}$). This is the empirical cumulative distribution
function of distances from a set of volume-filling, Poisson distributed random
points to the $k$-nearest data points, and is sensitive to all connected
$N$-point correlations in the data. The $k{\rm NN}$-${\rm CDF}$ can be used to
measure counts in cell, void probability distributions and higher $N$-point
correlation functions, all using the same formalism exploiting fast searches
with spatial tree data structures. We demonstrate how it can be computed
efficiently from various data sets - both discrete points, and the
generalization for continuous fields. We use data from a large suite of
$N$-body simulations to explore the sensitivity of this new statistic to
various cosmological parameters, compared to the two-point correlation
function, while using the same range of scales. We demonstrate that the use of
$k{\rm NN}$-${\rm CDF}$ improves the constraints on the cosmological parameters
by more than a factor of $2$ when applied to the clustering of dark matter in
the range of scales between $10h^{-1}{\rm Mpc}$ and $40h^{-1}{\rm Mpc}$. We
also show that relative improvement is even greater when applied on the same
scales to the clustering of halos in the simulations at a fixed number density,
both in real space, as well as in redshift space. Since the $k{\rm NN}$-${\rm
CDF}$ are sensitive to all higher order connected correlation functions in the
data, the gains over traditional two-point analyses are expected to grow as
progressively smaller scales are included in the analysis of cosmological data.
