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Let $G={\Phi }_G$ and $f=f_{NL}^{\mathrm{local}}$. The Gaussian random field covariance in Fourier space isThe Fourier transform of the centered quadratic term isThe Gaussian three-point function vanishes. At first order in $f$, the only terms are $f⟨Q_1{G}_2{G}_3⟩$ and its two cyclic placements. For the first, Wick theorem pairs four Gaussian factors in three ways. Pairing the two factors inside $Q_1$ together is canceled by the mean subtraction. Pairing one of those factors with $G_2$ and the other with $G_3$ gives $P(k_2)P(k_3)$ and the overall momentum delta; exchanging the two factors gives the same contribution. ThusAdding the three placements givesThere is no second-order term because it would contain five centered Gaussian factors. The two-point function receives its first correction at $f^2$, whereas the displayed local-type primordial non-Gaussianity gives a three-point correlation already at $f$. For $P\propto k^{-3}$ the squeezed limit is $B_{\Phi }(q,k,k)\simeq 4fP(q)P(k)$; this explains the strong correlation between a long potential and short-scale power. A finite variance or regulator is needed to define the subtracted $⟨G^2⟩$, as noted in part (b).

Use the Fourier transform convention $\zeta (\mathbf{x})=\int d^{3}k{e}^{i\mathbf{k}\cdot \mathbf{x}}\zeta (\mathbf{k})/(2\pi {)}^3$. Statistical homogeneity and isotropy define the primordial bispectrum byThe Dirac delta function enforces momentum conservation; the three magnitudes describe a closed triangle. Set $\alpha =3f_{\mathrm{NL}}/5$. For the centered quadratic Gaussian transformation, its quadratic Fourier term isAt first order in $\alpha$, choose the quadratic field at one of the three external positions. For the first position, Wick contractions pair the two internal fields with the two remaining external fields in two ways. The self-pairing is precisely removed by the subtracted mean. The surviving contribution is $2\alpha P(k_2)P(k_3)$ times the momentum delta. Adding the other positions gives the tree-level local-type primordial non-Gaussianity resultThis expression is at leading order in $f_{\mathrm{NL}}$. The exact quadratic map also has a loop correction to the local primordial bispectrum, from three quadratic vertices:There is no quadratic-in-$\alpha$ bispectrum term, because it would involve five centered Gaussian fields. A regulator may be needed for this higher-order integral. The displayed leading formula uses the Gaussian power $P$, consistently neglecting its order-$f_{\mathrm{NL}}^2$ correction.

At late time the Gaussian curvature power spectrum is $P(k)=H^2/(4ϵM_{\mathrm{Pl}}^2{k}^3)$. Consider the squeezed bispectrum configuration $k_1=q\ll k_2\simeq k_3=k$. The local-type primordial non-Gaussianity result is dominated byIn the curvature bispectrum from a zeta zeta-prime-squared interaction, the leading cyclic term has the long mode in the undifferentiated slot. Its momentum numerator tends to $k^4/(2k)=k^3/2$, while the other two cyclic terms contain $q^2$. HenceThis is the slow-roll amplitude of the squeezed zeta zeta-prime-squared bispectrum. It retains the local-like $q^{-3}$ enhancement: assigning the soft momentum only to differentiated legs would incorrectly miss the dominant permutation. Comparing squeezed amplitudes givesFor comparison, in the equilateral configuration $B^{\mathrm{sf}}(k,k,k)=(4ϵ/3)P(k{)}^2$, so the signal is slow-roll suppressed away from the squeezed limit as well.

Use the quoted observational bound as the sensitivity benchmark supplied by this problem, rather than as a new current measurement. A local-template sensitivity at the order-ten level is far above this order-$ϵ$ signal; detection of this interaction with comparable CMB measurements is consequently very unlikely. The entire shape is not identical to the local template, so its bound is not an exact constraint on every single-field shape, but the squeezed amplitude already exposes the large suppression. The obstacle is the slow-roll amplitude, even though this specified interaction has a squeezed enhancement.

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.

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.