Precise calculation of the EFT likelihood with primordial non-Gaussianities

We perform a precise calculation of the effective field theory (EFT) conditional likelihood for large-scale structure (LSS) using the saddle-point expansion method in the presence of primordial non-Gaussianities (PNG). The precision is manifested at two levels: one corresponding to the consideration of higher-order noise terms, and the other to the inclusion of contributions around the saddle points. In computing the latter, we encounter the same issue of the negative modes as in the context of false vacuum decay, which necessitates deforming the original integration contour into a combination of the steepest descent contours to ensure a convergent and real result. We demonstrate through detailed calculations that, upon incorporating leading-order PNG, both types of extensions introduce irreducible field-dependent contributions to the conditional likelihood. This insight motivates the systematic inclusion of additional effective terms within the forward modeling framework. Our work facilitates Bayesian forward modeling under non-Gaussian initial conditions, thereby enabling more stringent constraints on the parameters describing PNG.