bispectrum

The bispectrum is the Fourier-space three-point correlation function of a random field, defined as B(k1,k2,k3) = ⟨δ(k1) δ(k2) δ(k3)⟩, with a Dirac delta enforcing k1 + k2 + k3 = 0. It is the Fourier transform of the real-space three-point function ζ(x1,x2,x3) and generalizes the power spectrum, which is the two-point statistic.

For statistically isotropic fields, B depends only on the magnitudes |k1|, |k2|, |k3| and on the shape of the triangle formed by the wavevectors. In practice, the bispectrum is zero for Gaussian random fields, since all connected higher-order correlators vanish; a nonzero bispectrum indicates non-Gaussianity and phase correlations.

In cosmology, the bispectrum is used to probe non-Gaussianity of the primordial fluctuations and of the evolved large-scale structure, including the cosmic microwave background. It provides complementary information to the power spectrum and helps constrain inflationary models. The amplitude is often described by templates or the reduced bispectrum Q and by the non-Gaussianity parameter f_NL in local, equilateral, or orthogonal shapes.

Estimation from data involves computing Fourier modes of the observed field, averaging the product over closed triangles, and accounting for survey geometry, noise, and cosmic variance. Practical analyses use binning in triangle configurations and may employ estimators for the angular bispectrum in CMB or the 3D bispectrum for galaxy surveys.

In the CMB, the angular bispectrum B_l1l2l3, and its reduced form b_l1l2l3, are defined using spherical harmonics and Wigner 3j symbols. The bispectrum is a key diagnostic in tests of primordial non-Gaussianity and in modeling nonlinear evolution in the matter distribution.