The Variance and Covariance of Counts-in-Cells Probabilities

TL;DR

This paper derives explicit formulas for the variance and covariance of counts-in-cells probabilities, enabling their use in cosmological parameter estimation by accurately modeling statistical uncertainties.

Contribution

It provides the first general expressions for the covariance matrix of CIC probabilities based on density field correlations, facilitating their application in cosmology.

Findings

Derived explicit formulas for variance and covariance of CIC probabilities.

Validated formulas against simulated galaxy catalog data.

Enabled more accurate cosmological parameter constraints using CIC measurements.

Abstract

Counts-in-cells (CIC) measurements contain a wealth of cosmological information yet are seldom used to constrain theories. Although we can predict the shape of the distribution for a given cosmology, to fit a model to the observed CIC probabilities requires the covariance matrix -- both the variance of counts in one probability bin and the covariance between counts in different bins. To date, there have been no general expressions for these variances. Here we show that correlations of particular levels, or "slices," of the density field determine the variance and covariance of CIC probabilities. We derive explicit formulae that accurately predict the variance and covariance among subvolumes of a simulated galaxy catalog, opening the door to the use of CIC measurements for cosmological parameter estimation.