Constrained correlation functions

Abstract

We show that correlation functions have to satisfy contraint relations, owing to the non-negativity of the power spectrum of the underlying random process. Specifically, for any statistically homogeneous and (for more than one spatial dimension) isotropic random field with correlation function $\xi (x)$, we derive inequalities for the correlation coefficients $r_n\equiv \xi (nx)/\xi (0)$ (for integer $n$) of the form $r_{n\mathrm{l}}\le r_n\le r_{n\mathrm{u}}$, where the lower and upper bounds on $r_n$ depend on the $r_j$, with $j<n$. Explicit expressions for the bounds are obtained for arbitrary $n$. These constraint equations very significantly limit the set of possible correlation functions. For one particular example of a fiducial cosmic shear survey, we show that the Gaussian likelihood ellipsoid has a significant spill-over into the forbidden region of correlation functions, rendering the resulting best-fitting model parameters and their error region questionable, and indicating the need for a better description of the likelihood function. We conduct some simple numerical experiments which explicitly demonstrate the failure of a Gaussian description for the likelihood of $\xi$. Instead, the shape of the likelihood function of the correlation coefficients appears to follow approximately that of the shape of the bounds on the $r_n$, even if the Gaussian ellipsoid lies well within the allowed region. For more than one spatial dimension of the random field, the explicit expressions of the bounds on the $r_n$ are not optimal. We outline a geometrical method how tighter bounds may be obtained in principle. We illustrate this method for a few simple cases; a more general treatment awaits future work.

18 pages, 9 figures, submitted to A&A

References in corpus (5)

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• Cosmological Constraints From the 100 Square Degree Weak Lensing Survey
• Large-Scale Anisotropic Correlation Function of SDSS Luminous Red Galaxies
• Analysis of two-point statistics of cosmic shear: III. Covariances of shear measures made easy
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• Dark Energy Survey Year 3 Results: Covariance Modelling and its Impact on Parameter Estimation and Quality of Fit
• A bias in cosmic shear from galaxy selection: results from ray-tracing simulations
• Cosmological information in Gaussianised weak lensing signals
• KiDS-SBI: Simulation-based inference analysis of KiDS-1000 cosmic shear
• Non-Gaussianity in the Weak Lensing Correlation Function Likelihood -- Implications for Cosmological Parameter Biases
• Cosmological Inference from Galaxy-Clustering Power Spectrum: Gaussianization and Covariance Decomposition
• A simulation-based inference pipeline for cosmic shear with the Kilo-Degree Survey
• Retrieving the three-dimensional matter power spectrum and galaxy biasing parameters from lensing tomography
• Constrained probability distributions of correlation functions
• Likelihood of the Power Spectrum in Cosmological Parameter Estimation
• A quasi-Gaussian approximation for the probability distribution of correlation functions
• Two-point statistics without bins: A continuous-function generalization of the correlation function estimator for large-scale structure
• Constrained correlation functions from the Millennium Simulation
• Non-Gaussian likelihood of weak lensing power spectra