Non-Gaussian statistics and tests of primordial models

Equal two-point statistics can hide different distributions

A power spectrum summarizes two-point statistical information. Processes can share that information while differing in asymmetry, higher correlations and the probability of particular combinations of fluctuations.

For a Gaussian random field, its mean and covariance determine its probability distribution. Departures from Gaussianity introduce information beyond that specification. A nonzero third central moment is one possible departure, but its absence does not prove Gaussianity.

Cosmological higher-correlation tests can therefore help distinguish models that make similar two-point predictions. Their interpretation also requires the scale and configuration dependence, transfer to observables and non-primordial contributions.

The example below is an exact finite probability construction. It is not a physical model of the early Universe, an observed CMB distribution, a random simulated sky or a claim about the topology of space.

Separate the variance normalization from the asymmetry

At eight independent teaching sites, let G_j be independent standard Gaussian variables: each has mean zero and variance one. For a chosen dimensionless coefficient f, define Y_j = [G_j + f(G_j² − 1)]/√(1 + 2f²).

The subtraction of one gives zero mean because ⟨G_j²⟩ = 1. The numerator variance is 1 + 2f²: the cross term uses a vanishing odd Gaussian moment, while ⟨(G_j² − 1)²⟩ = 2.

Dividing by √(1 + 2f²) therefore makes the variance exactly one. Since each output depends only on its own independent G_j, different sites remain independent. Their two-point covariance is one at the same site and zero at different sites.

Choose f = −0.1, 0 and 0.1. This coefficient belongs to the stated teaching normalization. It is not an empirical primordial f_NL parameter and has no supplied conversion to a curvature-perturbation convention.

The second moment is even in f while the third is odd

With unit variance, the skewness equals the third central moment S₃ = (6f + 8f³)/(1 + 2f²)^(3/2). The denominator is unchanged by reversing f, while the numerator changes sign.

At f = 0.1, S₃ ≈ 0.590206. At f = −0.1, it is approximately −0.590206. Both examples still have mean zero and variance one, so the opposite asymmetries are absent from their shared two-point description.

Their fourth moment is (3 + 60f² + 60f⁴)/(1 + 2f²)², approximately 3.465975 for either nonzero chosen coefficient. The Gaussian f = 0 example instead has fourth moment three. This gives another distinction even when the sign of the skewness is reversed.

These values are exact ensemble moments evaluated by Gaussian-moment algebra, not estimates from a finite random sample. No sampling uncertainty, observed significance or measurement detection threshold is assigned.

Equal mode power can coexist with unequal three-mode correlations

For an optional finite Fourier account, define Ỹ_m = 8⁻¹ᐟ²∑ⱼY_j exp(−2πimj/8), with j and m running from zero to seven. This unitary discrete transform is a mathematical representation of the eight variables, not a model of cosmic periodicity.

The independent unit-variance sites give ⟨Ỹ_mỸ_n*⟩ = δ_mn. Every mode has unit expected squared magnitude in all three examples. Their common white power is another expression of the same two-point covariance.

The three-mode expectation instead equals S₃/√8 when m + n + p = 0 modulo eight, and zero otherwise. This follows because independent zero-mean sites leave only the terms in which all three site indices coincide, followed by the finite Fourier sum.

Modes (1,2,5) close modulo eight and give about +0.208669 for f = 0.1, zero for f = 0 and −0.208669 for f = −0.1. Modes (1,2,4) do not close and give zero in every case. The configuration matters as well as the overall asymmetry.

Higher statistics test predictions under specified conditions

Actual primordial non-Gaussianity tests compare predictions for correlations and their configurations with observables. Late evolution, foregrounds, instrument effects and estimator behavior can also contribute, so those alternatives enter the analysis.

Theoretical relations can be restrictive when their assumptions are restrictive. For example, a canonical single-field, standard-vacuum slow-roll calculation gives a particular three-point structure and a relation involving a squeezed long-short configuration. That result should retain those assumptions.

More general inflationary proposals and competing initial-condition mechanisms need their own predictions and conversion to observations. A single nonzero skewness number does not uniquely select one mechanism, and a constraint on one template does not automatically exclude every alternative.

The finite construction establishes a narrower lesson: matching the power spectrum need not match higher statistics. Using that extra information scientifically requires a defined physical model, the relevant configurations and scales, and a tested inference procedure.

Worked reasoning

Try the reasoning

1. Does zero skewness alone prove that a distribution is Gaussian?

   Show the answer and reasoning

   No.

   A distribution can have zero third central moment while differing from a Gaussian in other higher moments or features.

2. Which supplied triad can have a nonzero expectation: (1,2,5) or (1,2,4)?

   Show the answer and reasoning

   (1,2,5).

   Its indices sum to eight, which is zero modulo eight; the other sum is seven and its Fourier sum vanishes.

3. Is f = 0.1 a measured local primordial f_NL or a unique origin model?

   Show the answer and reasoning

   Neither.

   It is a chosen coefficient in a finite independent-site probability model, with no physical normalization or observational inference supplied.

What to carry forward

• Equal power spectra can conceal unequal higher-order statistics.
• An explicit normalized transformation separates fixed variance from signed skewness.
• Cosmological interpretation requires model assumptions, configuration dependence and an observation model beyond a teaching example.