Measure correlations beyond the primordial power spectrum

Take the positive-spatial-curvature variable ζ, agreeing with the adiabatic ℛ used above in its stated large-scale regime, and subtract its ensemble mean. Translation invariance makes a Fourier three-point function proportional to a delta function enforcing k1+k2+k3=0. The remaining bispectrum depends on the triangle’s side lengths for an isotropic scalar field. The vectors close; three arbitrary positive side lengths that violate the triangle inequality do not define an allowed configuration.

A mean-zero Gaussian field has vanishing connected three-point function, but a zero bispectrum does not prove the full distribution Gaussian. Higher connected moments can remain. The power has volume units in our Fourier convention, and the bispectrum has volume squared; an amplitude quoted without its template normalization can therefore be misleading. Permuting the three identical scalar fields leaves the primordial bispectrum symmetric, although observational estimators can apply nontrivial windows and weights.

A common local model writes ζ as a Gaussian field plus 3fNL/5 times its squared value minus its ensemble variance. The subtraction keeps the mean zero. Wick contraction gives a leading bispectrum proportional to the sum of three pairwise products of Gaussian powers. The coefficient is 6fNL/5 in this declared curvature convention. Reversing the sign of ζ reverses the corresponding odd-correlation sign, so the same printed fNL symbol is not enough to compare two different potential conventions.

For nearly scale-invariant power, the local template emphasizes squeezed triangles with one side much shorter than the other two. Equilateral or flattened templates describe different momentum patterns and need their own normalization. Multiple mechanisms can project onto similar templates, and a general bispectrum need not equal any one template exactly. The weak local expansion requires the quadratic correction to remain small for the relevant smoothed variance; a large coefficient cannot be assessed without the amplitude and range of the underlying fluctuations.

In a regular single-clock attractor with an appropriate adiabatic initial state, a sufficiently long curvature mode acts locally as a spatial dilation. This connects the squeezed correlation to how short-scale power changes with scale. Non-attractor evolution, additional light modes, or unsuitable initial-state behavior can invalidate a naive use of that standard argument. A measured squeezed signal must also be compared with projection and late-time effects before claiming a violation of a particular primordial model.

The probability lab illustrates a change in one-point distribution only. Reset and set success probability 0.5,trials 10; predict count mean 5 and variance 2.5, with symmetric bars. Reset and set probability 0.7,trials 10; predict mean 7 and variance 2.1, with asymmetric bars. These are exact binomial masses, not simulated observations. The lab has no spatial wave vectors, connected field correlator, primordial interaction, or bispectrum estimator; one-point asymmetry alone is not a measurement of cosmological triangle correlations.