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Let $A_k=H/(2\sqrt{ϵ}{M}_{\mathrm{Pl}}{k}^{3/2})$ and $K=k_1+k_2+k_3>0$. For the unconjugated integrand on the lower contour, the De Sitter curvature mode functions giveHere the two derivative factors supply ${\tau }^2$, cancelling $a^2=1/(H^2{\tau }^2)$. The vacuum prescription for inflationary in-in integrals can be implemented by a factor $e^{\delta \tau }$ on the real negative axis, with $\delta >0$, followed by $\delta \downarrow 0$. In this notationThereforeAn undamped boundary evaluation on the real axis at $-\infty$ is not valid. On the upper contour the complex-conjugate integrand instead yields the conjugate value. Conjugating the modes without conjugating the contour would produce exponential growth.

Using all six connected Wick contractions from the preceding part and $U=A_{k_1}{A}_{k_2}{A}_{k_3}$ gives the literal-Hamiltonian resultThe connected correlator is $(2\pi {)}^3{\delta }^{(3)}(\sum {\mathbf{k}}_i)B_{\mathcal{C}}$. The three assignments constitute the curvature bispectrum from a zeta zeta-prime-squared interaction. The numerator contains six powers of $H$ from the six modes, two of which are cancelled by $a^2$; the six powers of $M_{\mathrm{Pl}}$ remain unless the vertex supplies two.

Consequently the printed final expression, with $M_{\mathrm{Pl}}^{-4}$, is obtained for $\boxed{\mathcal{C}=M_{\mathrm{Pl}}^2}$. With the literal printed Hamiltonian $\mathcal{C}=1$, the answer instead has $\boxed{M_{\mathrm{Pl}}^{-6}}$. The Planck normalization of a cubic curvature interaction requires a factor $M_{\mathrm{Pl}}^2$ in the Hamiltonian if its final bispectrum is intended. In addition, integrating the intermediate expression literally with only three assignments gives half the fully contracted amplitude. These are normalization defects, rather than changes in the momentum shape. For dimensionless comoving curvature perturbations, the usual interaction normalization is also required by the mass dimension of the action.

For the standard normalization, $P_{\zeta }(k)=H^2/(4ϵM_{\mathrm{Pl}}^2{k}^3)$, so $B/P_{\zeta }^2$ is of order $ϵ$. This slow-roll suppression means that the primordial non-Gaussianity from this interaction alone is not expected to be detectably large. For example, in the squeezed bispectrum configuration $k_1\ll k_2\simeq k_3$, matching this contribution to the local convention gives $f_{\mathrm{NL}}=5ϵ/24$. This is a single-vertex contribution, not the complete single-field slow-roll inflation prediction; other vertices and field redefinitions contribute at the same slow-roll order. A claim of detectability for enhanced interactions requires a model beyond this approximation.

For a statistically homogeneous comoving curvature perturbation, define the connected primordial bispectrum byStatistical isotropy makes $B$ depend only on the three magnitudes, which must form a triangle. A Gaussian random field has zero connected bispectrum. Canonical attractor single-field slow-roll inflation with the usual vacuum produces only slow-roll-sized primordial non-Gaussianity. A measurable signal can arise from additional light fields and nonlinear conversion of isocurvature perturbations, or from enhanced interactions such as a small sound speed, departures from an attractor, or suitable features/excited initial states. The resulting model must still reproduce the observed nearly scale-invariant power spectrum and remain under perturbative control, with acceptable backreaction and late-time conversion to the observed perturbations. Large non-Gaussianity is therefore possible but is not automatic in a viable model.

Put $c=3f_{\mathrm{NL}}/5$ and use the consistent Fourier transform convention with ${\zeta }_G(\mathbf{x})$, rather than the repeated momentum argument printed in the integrand. The quadratic part has transformThe subtraction sets the mean to zero and removes the internal contraction of a single quadratic factor. At first order in $f_{\mathrm{NL}}$, choose one of the three external factors to be quadratic. For example, the two connected Wick contractions at the third leg pair its two fields with the first two legs, givingSumming the three placements provesThis is the leading local primordial bispectrum. The factor $6/5$ is the quadratic coefficient $3/5$ multiplied by the two cross-pairings. Terms of order $f_{\mathrm{NL}}^2$ have an odd Gaussian moment and vanish.

The literal quadratic local model also has a connected cubic-in-$f_{\mathrm{NL}}$ contribution, from one quadratic factor at every leg:This is the loop correction to the local primordial bispectrum; regulators may be needed for idealized spectra. Thus the PDF's displayed formula is the tree/leading-order result, not an exact identity for arbitrary $f_{\mathrm{NL}}$. The weak-non-Gaussian expansion assumes these loop terms are small. The local shape is enhanced in the squeezed bispectrum configuration when a long-wavelength perturbation modulates small-scale power.

Write $K=k_1+k_2+k_3>0$ and $C=ϵ+1-c_s^2$. The inflationary scalar Fourier mode has real late-time value $A_k=H/\sqrt{4ϵM_{\mathrm{Pl}}^2{c}_s{k}^3}$ and $u_k^{\prime \ast }(\tau )=A_k{c}_s^2{k}^2\tau e^{ik{c}_s\tau }$. Thus the time integrand, before the factor $-2i$, isThe Bunch-Davies vacuum prescription supplies damping in the past. Equivalently, use a positive regulator $d$ and then take the limit:The real part of $-2i$ times this integral is $4/(c_s^3{K}^3)$. Including the six Wick contractions yields the cubic time-derivative curvature bispectrumThe resulting primordial bispectrum is a homogeneous function of degree $-6$ in the wavenumbers, as expected for scale-independent background parameters.

The printed limit $c_s\ll 0$ is not a physical sound-speed limit for these modes. The normalization and positive-frequency choice assume $c_s>0$; a negative value is outside the positive sound-speed convention used by these modes. The likely intended limit is $0<c_s\ll 1$, which is analyzed explicitly here rather than silently substituted for the PDF.

To compare primordial non-Gaussianity, normalize by the power spectrum, not only by the raw three-point amplitude. HereAt an equilateral bispectrum configuration, the exact normalized amplitude isThus $c_s\to 1$ gives an amplitude of order $ϵ$, consistent with weak non-Gaussianity in single-field slow-roll inflation. For $0<c_s\ll 1$, this vertex gives an amplitude of order one, enhanced relative to the slow-roll limit but not diverging as $c_s^{-2}$ after power-spectrum normalization. At fixed $H,ϵ,M_{\mathrm{Pl}}$ the raw bispectrum does scale as $c_s^{-2}$, but $P_{\zeta }^2$ has the same scaling. The often-quoted larger small-sound-speed amplitudes of more general inflationary actions require their actual interaction coefficients and other vertices; they do not follow from the lone supplied vertex. These conclusions concern its tree contribution within the stated weak-perturbation approximation.

The bispectrum shape function isAt fixed perimeter, the arithmetic-geometric mean inequality bounds this by $1/27$, attained only at an equilateral bispectrum configuration. In a squeezed bispectrum configuration with $k_1=q\ll k_2\simeq k_3=k$, it behaves as $q/(8k)\to 0$. At a flattened bispectrum configuration $(k,k,2k)$ it is finite, $1/32$, with no boundary singularity. Thus the shape prefers equilateral triangles and suppresses squeezed triangles, with a broad finite flattened contribution. This is a shape preference, not a complete signal-to-noise calculation: actual observational weighting also depends on cosmological transfer functions, covariance and accessible triangles. If $C=0$, this vertex produces no bispectrum at any configuration; if $C<0$, the signed shape reverses while its absolute triangle preference is unchanged.

For an attractor with one adiabatic clock and a regular Bunch-Davies vacuum, a long comoving curvature perturbation ${\zeta }_L$ acts at leading order as a spatial dilation. The single-field inflation squeezed-limit consistency relation therefore fixesFor the local convention $\zeta ={\zeta }_G+(3/5)f_{\mathrm{NL}}({\zeta }_G^2-⟨{\zeta }_G^2⟩)$, comparison with its squeezed primordial bispectrum givesThe matter-era negative gravitational potential is $\phi =3\zeta /5$, so this is also the conventional $f_{\mathrm{NL}}$ of part (i). In single-field slow-roll inflation, the spectral tilt is small and the effective local coupling is of slow-roll size, typically of order $10^{-2}$ for a tilt of a few percent. This is a statement about the normalized coupling: the squeezed primordial bispectrum still contains the large long-mode power spectrum.

Formally putting this small coupling into the local template of part (ii) predicts only a tiny correction on accessible scales, rather than an order-unity local signal. For a physical halo abundance there is a stronger qualification. The consistency-relation modulation at fixed coordinate wavelength is a dilation; the coordinate size of a fixed physical halo must be dilated too. Those responses cancel. This is the absence of primordial scale-dependent halo bias in single-clock inflation: the leading consistency-relation term produces no physical $k^{-2}$ halo bias. The first physical long-mode response contains two spatial derivatives, so dividing it by ${\delta }_L\propto k_L^2{\phi }_L$ does not produce the local-template enhancement. Thus the leading primordial large-scale galaxy power spectrum remains the Gaussian-bias result, apart from ordinary transfer, bias, and projection effects.

The cancellation can be seen with a top-hat filter. At fixed local time, a constant long curvature rescales physical lengths by $e^{{\zeta }_L}$, so the coordinate radius of a fixed physical halo is $R_c=e^{-{\zeta }_L}{R}_0$, and the coordinate short density contrast is ${\delta }_s(\mathbf{x})={\delta }_s^{(0)}(e^{{\zeta }_L}\mathbf{x})$. Changing variables $\mathbf{y}=e^{{\zeta }_L}\mathbf{x}$ givesThus the smoothed field and its variance at fixed physical mass are unchanged by the constant long mode.

The assumptions matter: a non-attractor such as ultra-slow-roll inflation, additional fluctuating fields, or an excited initial state need not obey this squeezed-limit conclusion. A sizable physical local modulation would test the stated single-clock attractor assumptions, rather than every possible one-field model.

The operator ${\dot{\varphi }}^2\varphi$ is not invariant under a scalar-field shift symmetry because one field is undifferentiated. An arbitrary large coefficient would therefore radiatively generate equally unsuppressed nonderivative operators, including a mass and a steep scalar potential, and would spoil the approximate shift symmetry and flat potential needed for single-field slow-roll inflation. In a technically natural slow-roll model its coefficient must consequently be small, so it does not produce parametrically large primordial non-Gaussianity. Treated merely as an effective spectator-field interaction, it does generate the tree-level primordial bispectrum computed below, whose size is controlled by the dimensionless interaction strength at the Hubble scale; taking that strength large abandons the controlled slow-roll premise.