Mixed Graviton and Scalar Bispectra in the EFT of Inflation: Soft Limits and Boostless Bootstrap

Boostless Bootstrap techniques have been applied by many in the literature to compute pure scalar and graviton correlators. In this paper, we focus primarily on mixed graviton and scalar correlators. We start by developing an EFT of Inflation (EFToI) with some general assumptions, clarifying various subtleties related to power counting. We verify explicitly the soft limits for mixed correlators, showing how they are satisfied for higher derivative operators beyond the Maldacena action. We clarify some confusion in the literature related to the soft limits for operators that modify the power spectra of gravitons or scalars. We then proceed to apply the boostless bootstrap rules to operators that do not modify the power spectra. Towards the end, we give a prescription that gives correlators for states that are Bogolyubov transforms of the Bunch-Davies vacuum, directly once we have the correlator for the Bunch-Davies vacuum. This enables us to bypass complicated in-in calculations for Bogolyubov states.