Quantum mechanics

Gaussian

Cosmology

Statistics

Poisson distribution

Astrophysics

Cumulative distribution function

Statistical physics

Statistic

Physics

Mathematics

Cluster analysis

Probability density function

Dark matter

Monthly Notices of the Royal Astronomical Society

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.

Nearest neighbour distributions: New statistical measures for cosmological clustering