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Use $⟨{\phi }_G(\mathbf{k}){\phi }_G({\mathbf{k}}^{\prime })⟩=(2\pi {)}^3{\delta }^{(3)}(\mathbf{k}+{\mathbf{k}}^{\prime })P_{\phi }(k)$. For nonzero external momenta, the quadratic term in local-type primordial non-Gaussianity isInsert it in one of the three external fields. Wick theorem supplies two connected Wick contractions at each insertion, whereas the mean subtraction removes the disconnected zero-mode contribution. Thus the leading primordial bispectrum isLinear propagation by the cosmological Poisson equation gives $P_m(k)=\alpha (k{)}^2{P}_{\phi }(k)$ and multiplies the primordial bispectrum by ${\alpha }_1{\alpha }_2{\alpha }_3$. Defining the matter three-point function by its momentum-conserving Dirac delta function times $B_m$, the primordial contribution to the matter bispectrum isThe nonlinear gravitational contribution from standard perturbation theory in cosmology is excluded here.

In a squeezed bispectrum configuration, $k_1=k_L\ll k_2\simeq k_3=k_S$, soFor a nearly scale-invariant potential, $P_{\phi }(k_L)\gg P_{\phi }(k_S)$, so the second term is subleading andRelative to the product of matter power spectra, the squeezed response is enhanced by $k_L^{-2}$. Its sign is the sign of $f_{\mathrm{NL}}$. This is the three-point counterpart of the conditional short-scale variance modulation by local non-Gaussianity.

Write ${\phi }_G={\phi }_{G,L}+{\phi }_{G,s}$. In the local-type primordial non-Gaussianity expansion, the term linear in the short field is ${\phi }_{G,s}+2f_{\mathrm{NL}}{\phi }_{G,L}{\phi }_{G,s}$. The long-field square contributes to the long background, the mean subtraction is spatially homogeneous, and the short-field square is excluded at the order requested. HenceThis is the long-short coupling measured by a squeezed bispectrum configuration.

Because the long potential is constant over the short-scale region, multiplication by $1+2f_{\mathrm{NL}}{\phi }_{G,L}$ commutes with the linear cosmological Poisson equation and any fixed linear smoothing filter. Thus the short density contrast satisfies ${\delta }_s=(1+2f_{\mathrm{NL}}{\phi }_{G,L}){\delta }_{G,s}$. Taking the conditional expected value over the short Gaussian random field givesThe short-scale variance modulation by local non-Gaussianity is conditional on the long mode; averaging over that mode as well would erase the term linear in its zero mean.