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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.

The comoving curvature perturbation is dimensionless. With spatial coordinates and time carrying inverse mass dimensions, the cubic interaction $\int dt{d}^{3}x{a}^3{ϵ}^2\zeta {\dot{\zeta }}^2$ needs a mass-squared coefficient. Its standard gravitational normalization is $M_{\mathrm{Pl}}^2$. Six De Sitter curvature mode functions supply $M_{\mathrm{Pl}}^{-6}$; this vertex then supplies $M_{\mathrm{Pl}}^2$, leaving the physical $M_{\mathrm{Pl}}^{-4}$ curvature primordial bispectrum. The Wick-pairing multiplicity for a zeta zeta-prime-squared vertex is independently six: three choices of undifferentiated external leg times two differentiated-leg assignments.