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For approximately constant $H$ and $ϵ$, a vacuum-normalized curvature mode is $u_k(\tau )=H(1+ik\tau )e^{-ik\tau }/\sqrt{4ϵM_{\mathrm{Pl}}^2{k}^3}$. Its derivative is $u_k^{\prime }=H{k}^2\tau e^{-ik\tau }/\sqrt{4ϵM_{\mathrm{Pl}}^2{k}^3}$. Ordered unequal-time Wick contractions use the complex conjugate at the earlier vertex, consistently with the vacuum prescription for inflationary in-in integrals.

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.

Distinguish the slow-roll approximation parameter $ϵ$ from the infinitesimal positive contour regulator $\epsilon$. In the interaction picture, the ordered unequal-time Wick contraction isA differentiated vertex field instead supplies $u_k^{\ast \prime }(\tau )$. These conjugates are fixed by the annihilation operators and creation operators and cannot be dropped while retaining the same contour.

Convert the interaction time integral to conformal time. Since $dt=ad\tau$ and each cosmic-time derivative is $a^{-1}{\partial }_{\tau }$, the integrated cubic vertex isThe in-in formalism supplies $-2i$ multiplying the expectation of the Hamiltonian, so its negative sign gives $+2i$ multiplying this vertex. At nonzero external momenta, the connected contractions join each of the three external fields to one vertex field. There are six bijections, not three. Choosing which external field meets the undifferentiated vertex gives three cyclic choices; swapping the two differentiated fields gives a further factor of two. This is the Wick-pairing multiplicity for a zeta zeta-prime-squared vertex.

For example, before the internal momenta are integrated, one cyclic contribution includesThe factors of $2\pi$ cancel to leave one overall $(2\pi {)}^3$ and the external momentum delta. Thus the properly normalized reduced expression isThe vacuum prescription for inflationary in-in integrals damps the early-time oscillations and selects the Bunch-Davies vacuum. Internal self-contractions correspond to disconnected zero-momentum tadpole terms and are excluded from this connected three-point function.

The printed intermediate expression has unconjugated modes and only three literal cyclic terms with coefficient $-2i$. The conjugate form of the result above would use $-4i$, together with the conjugated contour. A literal $-2i$ with only three terms gives half the final printed answer. The six-contraction expression above is the consistent reduction of the supplied Hamiltonian and leads to that final answer.