Local-type primordial non-Gaussianity ($f_{\mathrm{NL}}$)

A local quadratic departure from a Gaussian random field isThe subtraction removes the homogeneous mean. At leading order, Wick theorem gives the primordial bispectrum $B_{\phi }=2f_{\mathrm{NL}}[P_{\phi }(k_1)P_{\phi }(k_2)+\text{cyclic terms}]$. The numerical convention depends on which potential or comoving curvature perturbation is used.

If the linear density contrast obeys $\delta (\mathbf{k})=\alpha (k)\phi (\mathbf{k})$, then local-type primordial non-Gaussianity contributesIn a squeezed bispectrum configuration and for a nearly scale-invariant potential, the leading term is $4f_{\mathrm{NL}}{P}_m(k_L)P_m(k_S)/\alpha (k_L)$. This excludes the standard perturbation theory in cosmology contribution from nonlinear gravity.

For a long potential approximately constant over the short-scale region, local-type primordial non-Gaussianity gives ${\phi }_s=(1+2f_{\mathrm{NL}}{\phi }_{G,L}){\phi }_{G,s}$, to first order in the long mode and ignoring short-mode squares. A linear cosmological Poisson equation therefore givesto the same order. The variance is conditional on the long mode.