Vacuum prescription for inflationary in-in integrals (${\tau }_i=-\infty (1-i\epsilon ),\epsilon >0$)

The Gaussian Bunch-Davies vacuum is selected by tilting the early endpoint to $-\infty (1-i\epsilon )$ for integrands $e^{iK\tau }$ with $K>0$. They then decay at the early boundary. The complex-conjugate integrand requires the opposite tilt. Ordered contractions and their contour must be conjugated together.

For $K=k_1+k_2+k_3$, $s_2=k_1{k}_2+k_1{k}_3+k_2{k}_3$ and $s_3=k_1{k}_2{k}_3$, the convergent in-in formalism integral is $I={\int }_{-\infty (1-i0)}^{0^{-}}{\tau }^{-2}{\prod }_i(1-i{c}_s{k}_i\tau )e^{i{c}_{s}K\tau }d\tau$. Its real endpoint divergence drops out of the expectation value, but the finite imaginary endpoint term $-c_{s}K$ must be kept. The remaining terms contribute $c_s{s}_2/K+c_s{s}_3/K^2$. Reversing the exponential requires conjugating both the contour and the outside prefactor.