 Primordial bispectrum

The primordial bispectrum is the Fourier-space connected correlation function of order three for an early-universe perturbation field. For a zero-mean real scalar field $\varphi$, statistical homogeneity impliesA common spatial translation multiplies the left side by $e^{i({\mathbf{k}}_1+{\mathbf{k}}_2+{\mathbf{k}}_3)\cdot \mathbf{a}}$, so translation invariance requires support on closed wavevector triangles. If the field has a nonzero mean, subtract the disconnected mean/two-point products before defining this connected statistic.

Statistical isotropy is the additional assumption that removes orientation dependence, allowing the shorter notation $B(k_1,k_2,k_3)$ with $k_i=|{\mathbf{k}}_i|$. The three positive magnitudes obey the triangle inequality. Statistical homogeneity alone does not justify replacing the vectors by their lengths. The usual primordial convention is illustrated by the Planck definition.^[ref]

A zero-mean Gaussian random field has vanishing three-point function. A measured nonzero connected primordial bispectrum therefore tests primordial non-Gaussianity beyond the power spectrum. The converse fails: a non-Gaussian field whose distribution is invariant under $\varphi \mapsto -\varphi$ has zero odd moments but can have a nonzero connected four-point function. Comparing predictions requires specifying the field, such as a gravitational potential or comoving curvature perturbation, because nonlinear field redefinitions change its primordial bispectrum.