Recovering the initial Gaussian information content of the large-scale structure from weak-lensing observables: higher-order statistics, nonlinear transforms, super-sample modes, and survey design

Cosmic shear surveys measure a present-day, gravitationally evolved, highly non-Gaussian projection of the primordial near-Gaussian density field. Takada (2014) posed the underlying statistical question: can the initial Gaussian information content of the large-scale structure ever be fully recovered from present-day observables, or is part of it fundamentally lost to nonlinear mode coupling? For a Subaru-Hyper-Suprime-Cam-like survey, the power spectrum retains less than half of the initial Gaussian information at multipoles $\ell\gtrsim 1000$, and adding the bispectrum of all triangle configurations recovers only about 50% at $\ell_{\max}\sim 1000$, with most of the deficit caused by super-sample covariance from survey-scale modes. The question has three linked parts: whether higher-order correlations or nonlinear (log-normal-type) transforms of the field recover the remaining information; whether the super-survey background mode $\delta_b$ can be treated jointly with cosmological parameters to mitigate super-sample variance; and what survey design minimizes super-sample variance while maximizing recovered information. The problem is currently unresolved.

Background

Cosmological weak lensing (cosmic shear) is the coherent, parts-per-million distortion of the shapes of distant galaxies by the gravitational field of the intervening matter distribution. The observable is the convergence field $\kappa(\boldsymbol{\theta})$, a weighted line-of-sight projection of the three-dimensional matter density contrast. Cosmic shear is one of the main probes of the Stage-IV imaging surveys — the current generation of large-area, deep imaging surveys (Rubin Observatory LSST, Euclid, Roman). Because the primordial density fluctuations are Gaussian to high precision (as measured in the cosmic microwave background), the initial field is informationally simple: for a Gaussian random field the Fourier modes are statistically independent, so the total information content — whether quantified as the cumulative signal-to-noise of the field measurement or as the Fisher information (the curvature of the likelihood, giving the best-achievable parameter errors) on cosmological parameters — is just the sum of independent contributions from each mode, for a given survey window and noise.

Nonlinear gravitational clustering destroys this simplicity. As structure grows, Fourier modes couple, the field becomes highly non-Gaussian, and the covariance between band powers and between different n-point statistics reduces the information carried by any finite set of summary statistics below the Gaussian reference. Rimes & Hamilton (2005) and Hamilton, Rimes & Scoccimarro (2006) found that the cumulative signal-to-noise of the nonlinear matter power spectrum (the two-point function in Fourier space) stops growing with wavenumber on translinear scales: the trispectrum (the four-point function in Fourier space) correlates neighboring band powers, producing an information plateau. Neyrinck, Szapudi & Szalay (2009) and Seo et al. (2011) showed that a logarithmic (log-normal-type) transform of the density or convergence field nearly Gaussianizes it and restores much of this information to the power spectrum of the transformed field.

For a finite-area survey there is a second, distinct mechanism of information loss: super-sample covariance (SSC), also called halo sample variance. Density modes with wavelengths comparable to or larger than the survey footprint cannot be measured within the survey; their amplitude varies from one surveyed volume of the Universe to another and, through nonlinear mode coupling, coherently modulates all measurable sub-survey modes. To leading order this entire effect is parameterized by a single number, $\delta_b$, the mean density fluctuation over the survey volume (the background mode). Takada & Hu (2013) and Li, Hu & Takada (2014) identified SSC as the dominant contribution to the covariance of weak-lensing power spectrum measurements, including the correlations between the power spectra of different multipoles and the bispectra of different triangle configurations.

Against this background, Takada (2014), summarizing Kayo, Takada & Jain (2013) and Kayo & Takada (2013), quantified the problem for a survey like the Subaru Hyper Suprime-Cam: the weak-lensing power spectrum alone carries less than half of the initial Gaussian information at multipoles $\ell\gtrsim 1000$, and the joint power spectrum plus bispectrum information (the bispectrum being the three-point function in Fourier space, here summed over all triangle configurations of wavevectors, with the full non-Gaussian covariance, including SSC) recovers only about 50% of the Gaussian information at $\ell_{\max}\sim 1000$, improving the dark-energy figure of merit (the inverse of the confidence area in the plane of the dark-energy equation-of-state parameters $w_0$ and $w_a$) by about 60% over the power spectrum alone. Since most of this deficit is caused by super-sample covariance rather than intra-survey mode coupling, the question decomposes into whether higher-order statistics or nonlinear field transforms can recapture the mode-coupled information, whether the super-survey mode $\delta_b$ can be self-calibrated by treating it jointly with the cosmological parameters, and how the survey design itself should be optimized. This is the statistical question addressed here, in the concrete setting of cosmic-shear surveys, though the same question applies to any present-day tracer of the large-scale structure.

Problem Statement

Determine, quantitatively and for a specified cosmic-shear survey configuration (survey area, geometry, source-galaxy redshift distribution and density, and shape noise), whether the initial Gaussian information content of the large-scale structure — the information the field would carry if its Fourier modes remained statistically independent, for the same survey window and noise — can be fully recovered from present-day observables, and, over the closed admissible class of analyses defined here — the weak-lensing power spectrum, the bispectrum, higher-order ($n>3$) correlation functions of the field, and nonlinear transformations of the field such as log-normal (logarithmic-density) mappings, each evaluated with a stated covariance treatment — establish which member attains the largest recovered fraction of that information, the recovered fraction being that member's cumulative signal-to-noise divided by that of the Gaussian reference for the same survey window and noise (with the corresponding Fisher-information recovery reported alongside, as detailed below). The question, posed by Takada (2014) and still unresolved, has one objective with three linked deliverables:

1. Information accounting and recovery methods: quantify the fraction of the initial Gaussian information carried by the weak-lensing power spectrum and bispectrum, and determine whether higher-order correlation functions beyond the bispectrum, or nonlinear transformations of the field such as log-normal (logarithmic-density) mappings, recover the remaining information — or whether some of it is fundamentally lost through strong mode coupling in the deeply nonlinear regime. Any conclusion that part of the information is fundamentally lost must be stated as a bounded, checkable claim about the admissible class defined in the Problem Statement — for example, that no analysis in the class recovers more than a stated fraction of the Gaussian reference — and not as an assertion about all conceivable statistics.

2. Super-survey mode treatment: determine whether the survey-scale background density fluctuation $\delta_b$, which dominates the information deficit through super-sample covariance, can be treated jointly with the cosmological parameters (self-calibrated by fitting or marginalizing over it with response templates — the precomputed change of each observable under a uniform background density perturbation — rather than absorbed into the covariance), how much of the apparent information loss this recovers, and on which scales this treatment remains valid.

3. Survey design: identify the survey geometry and area strategy, for fixed observational cost, that minimizes super-sample variance while maximizing recovered information.

Deliverables 2 and 3 address the super-sample (survey-scale) part of the information deficit identified in the source, while deliverable 1 addresses the intra-survey nonlinear mode-coupling part; together they decide the single overarching question of whether and how the primordial information is recoverable. The answer should be stated in the same currencies the source uses — cumulative signal-to-noise of the field and Fisher information on cosmological parameters relative to the Gaussian reference — for the stated survey configuration, with the covariance treatment (including super-sample covariance) made explicit.

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