Constrained probability distributions of correlation functions

Abstract

Context: Two-point correlation functions are used throughout cosmology as a measure for the statistics of random fields. When used in Bayesian parameter estimation, their likelihood function is usually replaced by a Gaussian approximation. However, this has been shown to be insufficient. Aims: For the case of Gaussian random fields, we search for an exact probability distribution of correlation functions, which could improve the accuracy of future data analyses. Methods: We use a fully analytic approach, first expanding the random field in its Fourier modes, and then calculating the characteristic function. Finally, we derive the probability distribution function using integration by residues. We use a numerical implementation of the full analytic formula to discuss the behaviour of this function. Results: We derive the univariate and bivariate probability distribution function of the correlation functions of a Gaussian random field, and outline how higher joint distributions could be calculated. We give the results in the form of mode expansions, but in one special case we also find a closed-form expression. We calculate the moments of the distribution and, in the univariate case, we discuss the Edgeworth expansion approximation. We also comment on the difficulties in a fast and exact numerical implementation of our results, and on possible future applications.

13 pages, 5 figures, updated to match version published in A&A (slightly expanded Sects. 5.3 and 6)

References in corpus (5)

• Very weak lensing in the CFHTLS Wide: Cosmology from cosmic shear in the linear regime
• Large-Scale Anisotropic Correlation Function of SDSS Luminous Red Galaxies
• The non-Gaussianity of the cosmic shear likelihood - or: How odd is the Chandra Deep Field South?
• Copula Cosmology: Constructing a Likelihood Function
• Constrained correlation functions

Cited by in corpus (10)

• CFHTLenS: Combined probe cosmological model comparison using 2D weak gravitational lensing
• Non-Gaussianity in the Weak Lensing Correlation Function Likelihood -- Implications for Cosmological Parameter Biases
• CFHTLenS: A Gaussian likelihood is a sufficient approximation for a cosmological analysis of third-order cosmic shear statistics
• Cosmological Inference from Galaxy-Clustering Power Spectrum: Gaussianization and Covariance Decomposition
• Sufficiency of a Gaussian power spectrum likelihood for accurate cosmology from upcoming weak lensing surveys
• Retrieving the three-dimensional matter power spectrum and galaxy biasing parameters from lensing tomography
• Likelihood of the Power Spectrum in Cosmological Parameter Estimation
• A quasi-Gaussian approximation for the probability distribution of correlation functions
• Constrained correlation functions from the Millennium Simulation
• Non-Gaussian likelihood of weak lensing power spectra