Robust inference of the primordial trispectrum from patch-to-patch variation of observed non-Gaussianity

A detectably large primordial trispectrum forces the bispectrum parameter $f_{\rm NL}$ to vary between horizon-sized patches of the inflated universe: with $g_{\rm NL}\gtrsim10^5$, the probability that a horizon-sized patch shows $f_{\rm NL}$ as small as Planck observes falls below 30% even for a single super-horizon e-folding of near scale-invariant perturbations, and below 10% for $\sim100$ such e-foldings, even if the global value of $f_{\rm NL}$ is zero (Byrnes, Nurmi, Tasinato and Wands, 2013). The same paper showed this relation cannot be inverted: posterior bounds on the global trispectrum amplitude $g^0_{\rm NL}$ from the observed $f^{\rm obs}_{\rm NL}$ depend strongly on the prior assumed for the variance, so no model-independent a posteriori bound exists, and the derivation rests on unquantified assumptions — the unknown number $N_{\rm in}$ of super-horizon e-foldings, the absence of large higher-order coefficients ($h_{\rm NL}$ and beyond), and squeezed-limit trispectrum shapes. The open problem is whether the mapping between patch-to-patch variation of observed non-Gaussianity and the global higher-order correlator content of the primordial curvature perturbation can be made robust and observationally usable: quantifying the $N_{\rm in}$ and shape dependence, controlling the higher-order truncation, constructing (or proving impossible) physically motivated model-independent priors, and realizing concrete patch-variation tests in CMB and large-scale-structure surveys.

Background

Inflation, the accelerated expansion believed to have preceded the hot big bang, stretched microscopic quantum fluctuations to cosmological scales, seeding the temperature anisotropies of the cosmic microwave background (CMB) and the large-scale distribution of galaxies. These perturbations are described by the curvature perturbation $\zeta$, a nearly scale-invariant random field with power spectrum amplitude $P_\zeta\simeq2\times10^{-9}$. Deviations from a Gaussian field are organized in a perturbative expansion; for the widely used local ansatz,

$$\zeta=\zeta_G+\frac{3}{5}f_{\rm NL}\left(\zeta_G^2-\langle\zeta_G^2\rangle\right)+\frac{9}{25}g_{\rm NL}\zeta_G^3+\frac{27}{125}h_{\rm NL}\zeta_G^4+\ldots$$

where $\zeta_G$ is Gaussian. The coefficients multiply the connected higher-order correlation functions: $f_{\rm NL}$ sets the bispectrum (connected three-point function), $g_{\rm NL}$ the trispectrum (connected four-point function, whose parameterization for local-type non-Gaussianity was introduced by Byrnes, Sasaki and Wands in 2006), and $h_{\rm NL}$ the connected five-point function. Planck constrains $f_{\rm NL}^{\rm local}$ to order unity; at the time of the source paper the local trispectrum amplitude $g_{\rm NL}$ was constrained only at the level of a few $\times10^5$ (from pre-Planck CMB data), roughly $10^5$ times weaker, so the second- and third-order terms of the expansion were comparably bounded — a detectable $g_{\rm NL}$ (threshold $|g_{\rm NL}|\gtrsim2\times10^5$) remained logically compatible with the small observed $f_{\rm NL}$. The key physical mechanism is mode coupling to unobservable longer wavelengths. In a universe much larger than our Hubble volume, long-wavelength modes of $\zeta$ (or of other light fields) act as a locally nearly homogeneous background that shifts the statistics measured in any smaller patch — the separate-universe picture. Byrnes, Nurmi, Tasinato and Wands (2012, Inhomogeneous non-Gaussianity) proposed exploiting this: the inhomogeneity of $f_{\rm NL}$ across patches probes higher-order correlators. The source paper (Byrnes, Nurmi, Tasinato and Wands 2013) made this quantitative. Splitting $\zeta_G=\zeta_{G,l}+\zeta_{G,s}$ at the horizon scale, with typical long mode $\zeta_{G,l}\simeq\sqrt{P_\zeta N_{\rm in}}$ for $N_{\rm in}$ e-foldings of super-horizon, near scale-invariant perturbations, the value measured in our patch obeys $f^{\rm obs}_{\rm NL}=f^0_{\rm NL}+\left(\frac{9}{5}g^0_{\rm NL}-\frac{12}{5}(f^0_{\rm NL})^2\right)\zeta_{G,l}+O(h^0_{\rm NL}\zeta^2_{G,l})$, so a large $g_{\rm NL}$ induces a patch-to-patch variance $\sigma^2_{f_{\rm NL}}\propto(g^0_{\rm NL})^2P_\zeta N_{\rm in}$. If $g_{\rm NL}\gtrsim10^5$, the probability that a horizon-sized patch displays $f_{\rm NL}$ as small as observed drops below 30% even for $N_{\rm in}\sim1$, and below 10% for $N_{\rm in}\sim100$, even when $f^0_{\rm NL}=0$. The paper also proved a shape-general inequality in the squeezed limit (one external momentum much smaller than the others), $\sigma^2_{f_{\rm NL}}\ge\left(\frac{5\beta_{\rm NL}}{12}\right)^2P_\zeta N_{\rm in}$, relating the variance of the generalized squeezed-limit $f_{\rm NL}$ to the squeezed limit of the four-point function through a new parameter $\beta_{\rm NL}$ — a higher-order analogue of the Suyama–Yamaguchi inequality, which relates the collapsed-limit trispectrum parameter $\tau_{\rm NL}$ to $f^2_{\rm NL}$ for any shape. Finally, Bayesian inversion — a posterior bound on the global $g^0_{\rm NL}$ from the observed small $f^{\rm obs}_{\rm NL}$ — proved strongly dependent on the prior assumed for the variance, and the authors concluded that no model-independent a posteriori bound on $g^0_{\rm NL}$ can be obtained from $f_{\rm NL}$ constraints alone. (Observationally, direct CMB trispectrum limits now exist — $g_{\rm NL}^{\rm local}=(-5.8\pm6.5)\times10^4$ from Planck 2018 — which changes the practical motivation but not the inference-theory questions below.) The paper is explicit that this framework rests on unverified assumptions and stops short of several natural completions: the value of $N_{\rm in}$ and possible departures from scale invariance on super-horizon scales are unknown; the truncation assuming the fourth-order coefficient $h^0_{\rm NL}$ (and higher) is not extremely large, which underpins $g^{\rm obs}_{\rm NL}\simeq g^0_{\rm NL}$, is unquantified; no physically motivated model-independent prior is constructed; the suggested observational test — measuring the variation of $f_{\rm NL}$ across sub-patches of our horizon in CMB or large-scale-structure surveys — is not developed; and the variance–trispectrum mapping is established only for squeezed-limit-dominated shapes.

Problem Statement

Can the relation between the primordial trispectrum and the patch-to-patch variance of observed non-Gaussianity be developed into a robust, model-independent channel for inferring the global higher-order correlator content of the primordial curvature perturbation from observations in a finite sub-volume of the universe (our Hubble horizon, or patches within it)? The source framework establishes $\sigma^2_{f_{\rm NL}}\propto(g^0_{\rm NL})^2P_\zeta N_{\rm in}$ for local non-Gaussianity and the squeezed-limit inequality $\sigma^2_{f_{\rm NL}}\ge\left(\frac{5\beta_{\rm NL}}{12}\right)^2P_\zeta N_{\rm in}$ for general shapes, but cannot be inverted without priors. The linked components are: (1) scale-range robustness — how the variance and probability bounds change when the assumption of $N_{\rm in}$ e-foldings of exactly scale-invariant perturbations beyond our horizon is relaxed (finite field mass, running of the spectral index, i.e. slow scale-drift of the spectral index $n_s$, other departures from scale invariance on super-horizon scales), and what this implies for the numerical probabilities and bounds; (2) higher-order truncation — under what conditions large higher-order coefficients ($h^0_{\rm NL}$ and beyond) invalidate $g^{\rm obs}_{\rm NL}\simeq g^0_{\rm NL}$ and the derived variance formulas, established either by general analysis or across explicit model classes; (3) prior structure — whether physically motivated, model-independent priors on the variance (equivalently upper cutoffs on $|g^0_{\rm NL}|$) can be constructed under which a posteriori bounds on $g^0_{\rm NL}$ from $f^{\rm obs}_{\rm NL}$ become meaningful, or a demonstration that none exist within specified model classes; (4) observational realization — concrete strategies and estimators for measuring the variation of $f_{\rm NL}$ or patch-wise bispectra across sub-patches of the sky in CMB or large-scale-structure surveys, with quantified sensitivity, covariance and systematic limitations for constraining the trispectrum and higher correlators this way; and (5) shape generality — the quantitative mapping between shape-specific trispectrum features beyond the squeezed limit and the variance of measurable non-Gaussianity in concrete non-local models, including any sharpening or extension of the squeezed-limit $\beta_{\rm NL}$ inequality. A resolving answer must quantitatively address at least the core components (1) scale-range robustness and (3) prior structure, and must explicitly state which of the remaining components it resolves and which it leaves open, and why; all components serve the single objective of determining whether, and how far, non-Gaussianity measured in our patch can be trusted to represent the global primordial statistics.

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