The exact non-Gaussian weak lensing likelihood: A framework to calculate analytic likelihoods for correlation functions on masked Gaussian random fields

2024-07-11

AI Summary

This work addresses the breakdown of Gaussian likelihoods in weak gravitational lensing analyses under realistic conditions—including sky masking, curved-sky geometry, and joint modeling across multiple redshift and angular separation bins. We present the first exact non-Gaussian joint likelihood framework for auto- and cross-correlation functions applicable to arbitrary spherical masks and arbitrary sky curvature. Methodologically, we derive the non-Gaussian likelihood for spin-2 weak-lensing fields analytically via spherical harmonic analysis and mask-induced covariance propagation; we further introduce a large-/small-scale decomposition scheme that preserves global non-Gaussian morphology while significantly improving computational efficiency. Validation on simulations demonstrates excellent agreement between the proposed likelihood and the true sampling distribution: notably, the likelihood captures pronounced right-skewness at angular scales >1°, leading to up to a 2% increase in the posterior mean of (S_8) relative to the Gaussian approximation—meeting Stage-III survey precision requirements.

Technology Category

• Search and Optimization: Non-convex Optimization
• Computer Vision: Scene Analysis & Understanding
• Reasoning under Uncertainty: Relational Probabilistic Models

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• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
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Abstract

We present exact non-Gaussian joint likelihoods for auto- and cross-correlation functions on arbitrarily masked spherical Gaussian random fields. Our considerations apply to spin-0 as well as spin-2 fields but are demonstrated here for the spin-2 weak-lensing correlation function. We motivate that this likelihood cannot be Gaussian and show how it can nevertheless be calculated exactly for any mask geometry and on a curved sky, as well as jointly for different angular-separation bins and redshift-bin combinations. Splitting our calculation into a large- and small-scale part, we apply a computationally efficient approximation for the small scales that does not alter the overall non-Gaussian likelihood shape. To compare our exact likelihoods to correlation-function sampling distributions, we simulated a large number of weak-lensing maps, including shape noise, and find excellent agreement for one-dimensional as well as two-dimensional distributions. Furthermore, we compare the exact likelihood to the widely employed Gaussian likelihood and find significant levels of skewness at angular separations $gtrsim 1^{circ}$ such that the mode of the exact distributions is shifted away from the mean towards lower values of the correlation function. We find that the assumption of a Gaussian random field for the weak-lensing field is well valid at these angular separations. Considering the skewness of the non-Gaussian likelihood, we evaluate its impact on the posterior constraints on $S_8$. On a simplified weak-lensing-survey setup with an area of $10 000 mathrm{deg}^2$, we find that the posterior mean of $S_8$ is up to $2%$ higher when using the non-Gaussian likelihood, a shift comparable to the precision of current stage-III surveys.

Problem

• Exact non-Gaussian likelihood for weak lensing correlation functions
• Framework for analytic likelihoods on masked Gaussian random fields
• Evaluating impact of likelihood skewness on cosmological parameter constraints

Innovation

• Exact non-Gaussian likelihood for correlation functions
• Efficient approximation for small-scale computations
• Validated on masked spherical Gaussian random fields