= Solution
Use the <comoving curvature perturbation> mode normalization. Take $H,\epsilon,c_s$ to be effectively constant at leading <slow-roll approximation> order, with $H,\epsilon,c_s>0$, and use the <Bunch-Davies vacuum> selected by the early-time $i\varepsilon$ contour. The regulator $\varepsilon$ is distinct from the <Hubble slow-roll parameter> $\epsilon$.
Let $C=\epsilon+1-c_s^2$ and use the <Fourier transform> convention $\zeta(\mathbf x)=\int d^3k\,\zeta(\mathbf k)e^{i\mathbf k\cdot\mathbf x}/(2\pi)^3$. Conversion to <conformal time> gives
$$
H_{\rm int}\,dt=-\frac{M_{\rm Pl}^2\epsilon C}{Hc_s^2}\int d^3x\,a\zeta'^3\,d\tau=\frac{M_{\rm Pl}^2\epsilon C}{H^2c_s^2\tau}\int d^3x\,\zeta'^3\,d\tau.
$$
The last sign follows from substituting $a=-1/(H\tau)$, which is positive for $\tau<0$; dropping the explicit minus sign in that substitution would reverse the final <primordial bispectrum>.
At the order needed by the <in-in formalism>, the unequal-time <Wick contraction> is
$$
\langle\zeta(\mathbf k,0)\zeta'(\mathbf p,\tau)\rangle=(2\pi)^3\delta^{(3)}(\mathbf k+\mathbf p)u_k(0)u_k'^*(\tau).
$$
It uses the Gaussian quantum vacuum two-point function with the displayed operator ordering; an arbitrary classical unequal-time <covariance> cannot replace it. For the connected three-point function, each external field contracts with one of the three fields at the vertex. There are $3!=6$ such <Wick contractions>, all equal. The spatial integral conserves total momentum and leaves one factor $(2\pi)^3\delta^{(3)}(\mathbf k_1+\mathbf k_2+\mathbf k_3)$. The other nine pairings are disconnected tadpole terms supported on a zero external momentum; they are excluded from the connected <primordial bispectrum>, equivalently by subtracting the one-point function or normal ordering the vertex.
The free Gaussian odd correlator is zero. Applying the stated first-order <in-in formalism> therefore gives
$$
\boxed{B(k_1,k_2,k_3)=6\operatorname{Re}\!\left[-2i\int_{-\infty(1-i0)}^0\!d\tau\,\frac{M_{\rm Pl}^2\epsilon C}{H^2c_s^2\tau}\prod_{r=1}^3u_{k_r}(0)u_{k_r}'{}^*(\tau)\right].}
$$
By definition, the full connected correlator is $(2\pi)^3\delta^{(3)}(\sum_r\mathbf k_r)B$. The assumptions are a Gaussian <Bunch-Davies vacuum>, the given leading cubic vertex, tree order, the late-time limit $\tau\to0^-$, effectively constant background parameters, and connected nonzero external modes. No symmetry factor $1/3!$ is inserted into the given Hamiltonian.
Back to article page
