Pseudo Non-Gaussianity: Origins & Observational Effects

• Pseudo non-Gaussianity is a phenomenon where observed deviations from Gaussian statistics arise due to biased subvolumes, foreground contamination, or nonlinear mappings rather than from the intrinsic properties of the field.
• Methodologies in finite-volume cosmology, CMB analysis, and quantum settings reveal that estimator bias and data processing can regenerate lower-order terms and obscure true statistical signals.
• Observational strategies, including masking and pseudo-likelihoods, demonstrate that apparent non-Gaussian signatures may be artifacts of sampling and projection, challenging the direct interpretation of primordial signals.

Pseudo non-Gaussianity denotes a class of situations in which an observed, inferred, or operationally defined non-Gaussian signal is not a faithful direct reflection of the underlying parent statistics. In the literature, the term is not used uniformly. In finite-universe cosmology it refers to non-Gaussianity generated by observing only a biased subvolume of a larger statistically homogeneous and isotropic space (Nelson et al., 2012). In CMB and large-scale-structure analysis it can refer to apparent non-Gaussianity induced by foregrounds, masking, estimator distributions, or Gaussian pseudo-likelihoods rather than by primordial physics (Saha, 2011). In quantum settings it can denote basis-dependent or superselection-dependent apparent non-Gaussianity, or the absence of a meaningful pseudorandom “pseudo-gap” for a particular non-Gaussianity resource measure (Moulonguet et al., 21 Mar 2026).

1. Finite-volume pseudo non-Gaussianity

The most explicit formulation appears in “Statistical Naturalness and non-Gaussianity in a Finite Universe” (Nelson et al., 2012). There the observed universe is treated as a subvolume $M$ of a larger volume $L$, with curvature perturbation defined through the local scale factor,

$$a(\mathbf{x})=\bar a_L\left(1+\zeta(\mathbf{x})\right), \qquad \mathbf{x}\in \mathrm{Vol}_L ,$$

and in the subvolume,

$$a(\mathbf{x})=\bar a_M\left(1+\zeta(\mathbf{x})\right), \qquad \mathbf{x}\in \mathrm{Vol}_M .$$

The relation between global and local perturbations is

$$1+\zeta_M(\mathbf{x}) = \left(1+\zeta_{I,M}\right)\left(1+\zeta_s(\mathbf{x})\right).$$

The central physical statement is that long-wavelength modes in the unobserved larger universe bias the statistics inside the observed patch. The parent field is written as a local non-Gaussian expansion,

$$\zeta(\mathbf{x})=\sum_{n=1}^{\infty}\frac{N_n}{n!} g(\mathbf{x})^n,$$

with $g$ Gaussian and $\langle \zeta\rangle=0$ in the large volume. Splitting $g=g_{I,M}+g_{s,M}$, the paper defines a bias parameter

$B=\frac{g_{I,M}}{\sigma_0},$

and shows that the local coefficients shift according to

$L$0

The consequence is a background-dependent renormalized local ansatz. A sufficiently biased small volume can therefore look like a different but still “natural” local model: coefficients are modified, lower-order terms can be regenerated, and the observed non-Gaussianity need not represent the global parent statistics. In this sense, pseudo non-Gaussianity is not the absence of non-Gaussianity, but a mismatch between global generation and local appearance.

2. Ordered moments, regenerated lower-order terms, and preserved local shape

The same finite-volume framework emphasizes that biased subvolumes tend toward weakly non-Gaussian, ordered moments even when the parent field is strongly non-Gaussian or fine-tuned (Nelson et al., 2012). The normalized connected moments are written as

$L$1

with hierarchical ordering expressed as

$L$2

for some small ratio $L$3.

For a large-volume model beginning only at order $L$4,

$L$5

the full-volume moments need not be nicely ordered. In a biased subvolume, however, long modes regenerate lower-order terms and the moments become ordered. For sufficiently biased subvolumes,

$L$6

The same mechanism also applies to a strongly non-Gaussian parent model with no linear term initially,

$L$7

for which a biased subvolume again regenerates missing lower-order pieces, including a linear term, and can look much more Gaussian.

A further result concerns shape. Bias changes amplitudes and can induce extra logarithmic scale dependence, including factors such as $L$8 and $L$9, but it does not arbitrarily change the local-type shape of the $$a(\mathbf{x})=\bar a_L\left(1+\zeta(\mathbf{x})\right), \qquad \mathbf{x}\in \mathrm{Vol}_L ,$$0-point functions. The bispectrum retains the standard local form,

$$a(\mathbf{x})=\bar a_L\left(1+\zeta(\mathbf{x})\right), \qquad \mathbf{x}\in \mathrm{Vol}_L ,$$1

and in the squeezed limit,

$$a(\mathbf{x})=\bar a_L\left(1+\zeta(\mathbf{x})\right), \qquad \mathbf{x}\in \mathrm{Vol}_L ,$$2

Thus the local squeezed-limit behavior is protected under subsampling, even when amplitudes and effective coefficients are altered.

3. Observational and statistical pseudo-signals

A second major usage concerns apparent non-Gaussianity generated by data processing, contamination, masking, or inference methodology rather than by the primordial field itself. In WMAP foreground cleaning, kurtosis is used as a scalar measure of non-Gaussianity,

$$a(\mathbf{x})=\bar a_L\left(1+\zeta(\mathbf{x})\right), \qquad \mathbf{x}\in \mathrm{Vol}_L ,$$3

and a constrained linear combination of the five WMAP frequency maps is chosen to minimize it (Saha, 2011). The resulting “non-Gaussianity minimized map” is interpreted through the statement that the observed non-Gaussianity in the raw frequency maps is not primordial but mainly induced by foreground contamination. Residual foreground contamination remains near the inner Galactic plane, motivating the G20 mask, which retains about $$a(\mathbf{x})=\bar a_L\left(1+\zeta(\mathbf{x})\right), \qquad \mathbf{x}\in \mathrm{Vol}_L ,$$4 of the sky and yields $$a(\mathbf{x})=\bar a_L\left(1+\zeta(\mathbf{x})\right), \qquad \mathbf{x}\in \mathrm{Vol}_L ,$$5, consistent with zero outside the masked region.

Planck analyses sharpen the same point by showing that Gaussianity assessments are directional, patch-dependent, and mask-dependent (Bernui et al., 2014). The sky is divided into $$a(\mathbf{x})=\bar a_L\left(1+\zeta(\mathbf{x})\right), \qquad \mathbf{x}\in \mathrm{Vol}_L ,$$6 caps of aperture $$a(\mathbf{x})=\bar a_L\left(1+\zeta(\mathbf{x})\right), \qquad \mathbf{x}\in \mathrm{Vol}_L ,$$7, with local skewness and kurtosis defined by

$$a(\mathbf{x})=\bar a_L\left(1+\zeta(\mathbf{x})\right), \qquad \mathbf{x}\in \mathrm{Vol}_L ,$$8

For SMICA, NILC, and SEVEM, low-$$a(\mathbf{x})=\bar a_L\left(1+\zeta(\mathbf{x})\right), \qquad \mathbf{x}\in \mathrm{Vol}_L ,$$9 spectra under INPMASK and VALMASK show significant deviations from Gaussianity, while under the more conservative U73 mask all low-$$a(\mathbf{x})=\bar a_M\left(1+\zeta(\mathbf{x})\right), \qquad \mathbf{x}\in \mathrm{Vol}_M .$$0 values of $$a(\mathbf{x})=\bar a_M\left(1+\zeta(\mathbf{x})\right), \qquad \mathbf{x}\in \mathrm{Vol}_M .$$1 and $$a(\mathbf{x})=\bar a_M\left(1+\zeta(\mathbf{x})\right), \qquad \mathbf{x}\in \mathrm{Vol}_M .$$2 fall within the $$a(\mathbf{x})=\bar a_M\left(1+\zeta(\mathbf{x})\right), \qquad \mathbf{x}\in \mathrm{Vol}_M .$$3 bounds of Gaussian simulations. The quoted $$a(\mathbf{x})=\bar a_M\left(1+\zeta(\mathbf{x})\right), \qquad \mathbf{x}\in \mathrm{Vol}_M .$$4-probabilities for U73 include $$a(\mathbf{x})=\bar a_M\left(1+\zeta(\mathbf{x})\right), \qquad \mathbf{x}\in \mathrm{Vol}_M .$$5 for SMICA, $$a(\mathbf{x})=\bar a_M\left(1+\zeta(\mathbf{x})\right), \qquad \mathbf{x}\in \mathrm{Vol}_M .$$6 for NILC, and $$a(\mathbf{x})=\bar a_M\left(1+\zeta(\mathbf{x})\right), \qquad \mathbf{x}\in \mathrm{Vol}_M .$$7 for SEVEM. This identifies a mask-sensitive form of apparent non-Gaussianity likely tied to residual foregrounds or contaminated regions.

Pseudo non-Gaussianity also arises at the level of estimators. The standard CMB bispectrum estimator for $$a(\mathbf{x})=\bar a_M\left(1+\zeta(\mathbf{x})\right), \qquad \mathbf{x}\in \mathrm{Vol}_M .$$8 is a sum of products of temperatures in three different pixels, with $$a(\mathbf{x})=\bar a_M\left(1+\zeta(\mathbf{x})\right), \qquad \mathbf{x}\in \mathrm{Vol}_M .$$9 terms built from only $$1+\zeta_M(\mathbf{x}) = \left(1+\zeta_{I,M}\right)\left(1+\zeta_s(\mathbf{x})\right).$$0 measurements. The central-limit theorem therefore does not necessarily apply, and the estimator PDF need not be Gaussian even when the field is Gaussian or nearly Gaussian (Smith et al., 2011). The null-hypothesis minimum-variance estimator is very nearly Gaussian only when the true $$1+\zeta_M(\mathbf{x}) = \left(1+\zeta_{I,M}\right)\left(1+\zeta_s(\mathbf{x})\right).$$1; for $$1+\zeta_M(\mathbf{x}) = \left(1+\zeta_{I,M}\right)\left(1+\zeta_s(\mathbf{x})\right).$$2 its PDF becomes skewed, with a long tail on the side $$1+\zeta_M(\mathbf{x}) = \left(1+\zeta_{I,M}\right)\left(1+\zeta_s(\mathbf{x})\right).$$3. By contrast, the improved CSZ estimator is nearly Gaussian for observationally allowed values of $$1+\zeta_M(\mathbf{x}) = \left(1+\zeta_{I,M}\right)\left(1+\zeta_s(\mathbf{x})\right).$$4.

An analogous issue appears in large-scale-structure inference, where the standard likelihood is often only a Gaussian pseudo-likelihood rather than the true sampling distribution of the summary statistic (Hahn et al., 2018). Significant non-Gaussianity is found in both the $$1+\zeta_M(\mathbf{x}) = \left(1+\zeta_{I,M}\right)\left(1+\zeta_s(\mathbf{x})\right).$$5 and $$1+\zeta_M(\mathbf{x}) = \left(1+\zeta_{I,M}\right)\left(1+\zeta_s(\mathbf{x})\right).$$6 likelihoods. For Beutler et al. (2017), likelihood non-Gaussianity shifts the $$1+\zeta_M(\mathbf{x}) = \left(1+\zeta_{I,M}\right)\left(1+\zeta_s(\mathbf{x})\right).$$7 constraint by $$1+\zeta_M(\mathbf{x}) = \left(1+\zeta_{I,M}\right)\left(1+\zeta_s(\mathbf{x})\right).$$8. For Sinha et al. (2017), the Gaussian pseudo-likelihood underestimates uncertainties and biases halo-occupation constraints: $$1+\zeta_M(\mathbf{x}) = \left(1+\zeta_{I,M}\right)\left(1+\zeta_s(\mathbf{x})\right).$$9 and $$\zeta(\mathbf{x})=\sum_{n=1}^{\infty}\frac{N_n}{n!} g(\mathbf{x})^n,$$0 are shifted by $$\zeta(\mathbf{x})=\sum_{n=1}^{\infty}\frac{N_n}{n!} g(\mathbf{x})^n,$$1 and $$\zeta(\mathbf{x})=\sum_{n=1}^{\infty}\frac{N_n}{n!} g(\mathbf{x})^n,$$2, and broadened by $$\zeta(\mathbf{x})=\sum_{n=1}^{\infty}\frac{N_n}{n!} g(\mathbf{x})^n,$$3 and $$\zeta(\mathbf{x})=\sum_{n=1}^{\infty}\frac{N_n}{n!} g(\mathbf{x})^n,$$4, respectively.

4. Nonlinear mappings, hidden structure, and approximate initial states

A further class of pseudo non-Gaussianity arises when a Gaussian or nearly Gaussian underlying field is mapped nonlinearly into a manifestly non-Gaussian observable. Tensor-induced scalar density perturbations provide a clean example (Abdelaziz et al., 9 Apr 2025). The second-order density contrast obeys

$$\zeta(\mathbf{x})=\sum_{n=1}^{\infty}\frac{N_n}{n!} g(\mathbf{x})^n,$$5

so the source is quadratic in the tensor perturbations. Even if the primordial tensor modes are Gaussian, the induced density contrast follows a chi-squared–type distribution and therefore exhibits significant non-Gaussianity. The bispectrum shape is sensitive to the underlying gravitational-wave spectrum by construction, with Gaussian-bump and monochromatic sources producing a strong signal peaking in the equilateral configuration.

An older cosmological version of the same logic appears in isocurvature perturbations (0808.0009). Observable CMB non-Gaussianity need not originate in the inflaton-generated adiabatic curvature mode; it can be induced indirectly by a non-Gaussian isocurvature perturbation $$\zeta(\mathbf{x})=\sum_{n=1}^{\infty}\frac{N_n}{n!} g(\mathbf{x})^n,$$6, with

$$\zeta(\mathbf{x})=\sum_{n=1}^{\infty}\frac{N_n}{n!} g(\mathbf{x})^n,$$7

In the Sachs–Wolfe regime,

$$\zeta(\mathbf{x})=\sum_{n=1}^{\infty}\frac{N_n}{n!} g(\mathbf{x})^n,$$8

The resulting CMB signal is distinct from ordinary local adiabatic non-Gaussianity and can be strongly enhanced on large angular scales. In the QCD axion example, sizable non-Gaussianity is possible for $$\zeta(\mathbf{x})=\sum_{n=1}^{\infty}\frac{N_n}{n!} g(\mathbf{x})^n,$$9.

The term also appears in initial-state physics through the notion of a pseudo-$g$0-vacuum (Kanno et al., 2022). In exact de Sitter space there exists an infinite family of de Sitter-invariant $g$1-vacua, with $g$2 giving the Bunch–Davies vacuum. In slow-roll inflation, exact de Sitter invariance is broken, so one can only consider pseudo-$g$3-vacua: states with weakly $g$4-dependent $g$5, an appropriate UV cutoff, and approximate validity over a finite range of modes. In this setting the graviton bispectrum may be exponentially enhanced to be detectable by observation even if the spectrum is too small to be detected.

High-dimensional random fields introduce yet another mechanism: real non-Gaussianity can be hidden in low-dimensional projections (Braspenning et al., 2021). The full joint distribution of all pixels can be strongly non-Gaussian, while one- and two-pixel marginals retain an almost generic Gauss-like appearance. For a $g$6 field, $g$7, so a two-pixel marginal integrates over $g$8 dimensions. This suggests a form of pseudo non-Gaussianity in which the field is globally non-Gaussian but appears Gaussian in simple diagnostics.

5. Quantum, superselection, and pseudorandom-state formulations

In quantum-information settings, pseudo non-Gaussianity is tied less to cosmic observables than to the meaning and detectability of non-Gaussian resources. A superselection-rule analysis argues that what is traditionally called quadrature non-Gaussianity is partly a representation-dependent artifact unless the phase reference is treated as a quantum system and photon-number superselection rules are respected (Moulonguet et al., 21 Mar 2026). In that formulation, quadrature non-Gaussianity and nonzero stellar rank act as witnesses of particle entanglement in the enlarged SSR-compliant description. “Pseudo-non-Gaussianity” then refers to apparent non-Gaussianity produced by the choice of phase reference or basis, rather than to an intrinsic single-mode resource.

The fermionic case leads to a different conclusion. Using fermionic antiflatness,

$g$9

the paper “Practical Tests and Witnesses of Fermionic non-Gaussianity” introduces pseudo non-Gaussianity as an analogue of pseudo-entanglement, pseudomagic, and pseudocoherence, but argues that for $\langle \zeta\rangle=0$0 there is essentially no meaningful pseudo-resource gap (Haug et al., 25 May 2026). Because $\langle \zeta\rangle=0$1 is efficiently estimable, any inverse-polynomial gap from Haar-random behavior would be detectable and would contradict pseudorandomness. Haar-random states satisfy

$\langle \zeta\rangle=0$2

so asymptotically $\langle \zeta\rangle=0$3, and pseudorandom states must match this up to negligible corrections. In the Gaussian-plus-local-non-Gaussian-gate model, this yields the circuit lower bound $\langle \zeta\rangle=0$4 for constant-size non-Gaussian gates.

This should be distinguished from standard resource theories of non-Gaussianity. In continuous-variable systems, one recent framework defines ordinary non-Gaussianity operationally through the statement that two identical copies of a state become correlated at a $\langle \zeta\rangle=0$5 beam splitter if and only if the state is non-Gaussian, with the resulting measures monotonic under Gaussian channels (Hahn et al., 27 Aug 2025). That work explicitly concerns non-Gaussianity in the standard resource-theoretic sense, not pseudo non-Gaussianity as a separate label.

6. Conceptual boundaries and recurrent misunderstandings

Several recurrent ambiguities follow from this heterogeneous usage. First, pseudo non-Gaussianity is not synonymous with a false or spurious signal. In finite-volume cosmology, the observed non-Gaussianity can be physically real within the subvolume while still failing to represent the global parent field (Nelson et al., 2012). In tensor-induced density perturbations, the induced field is genuinely non-Gaussian even when the primordial tensor source is Gaussian (Abdelaziz et al., 9 Apr 2025). In LSS inference, the non-Gaussianity may reside in the true likelihood rather than in the assumed Gaussian pseudo-likelihood (Hahn et al., 2018).

Second, the term should not be confused with papers in which “pseudo” modifies the field rather than the statistics. Hilltop non-Gaussianity from a pseudo Nambu-Goldstone boson, pseudo-NG curvatons, and non-minimal inflation with a pseudo-Goldstone angular direction all describe mechanisms for generating genuine local-type primordial non-Gaussianity; the adjective “pseudo” refers there to the bosonic degree of freedom, not to an apparent or artifactual statistical signal (0810.1585). The same distinction applies to generic pseudo-NG curvaton dynamics, where non-quadratic potentials and a non-uniform onset of oscillation can produce large $\langle \zeta\rangle=0$6 with either sign and, in hilltop pseudo-NG curvatons, typically $\langle \zeta\rangle=0$7 (Kawasaki et al., 2011). It also applies to non-minimal two-field inflation in which the angular direction becomes a pseudo-Goldstone boson and large local non-Gaussianity arises from isocurvature-to-curvature conversion during slow roll (Gong et al., 2011).

Third, pseudo non-Gaussianity is not a standardized bispectrum template alongside local, equilateral, and orthogonal shapes. Standard reviews continue to organize primordial non-Gaussianity primarily by those shape classes, together with feature, folded, and scale-dependent variants (Byrnes, 2014). A plausible implication is that “pseudo non-Gaussianity” functions less as a single model class than as an interpretive label for mismatches between origin, appearance, and operational diagnosis.

In that broader encyclopedic sense, pseudo non-Gaussianity designates a family of phenomena in which non-Gaussianity is generated, revealed, hidden, or reinterpreted by finite sampling, observational masking, nonlinear projection, estimator behavior, likelihood modeling, basis choice, or pseudorandomness constraints. The unifying theme is not a common formula, but a common warning: the appearance of non-Gaussian statistics need not identify their microscopic origin without additional structural information.