Inflationary cross-correlations of a non-minimal spectator and their soft limits

Light spectator fields may not be dynamically relevant for the inflationary phase of the early universe, but they can still induce interesting imprints on cosmological observables. In this paper, we compute the cross-correlations of the inflationary perturbations, both scalar and tensor, with the fluctuations of a non-minimally interacting spectator field using the in-in formalism and investigate the consistency relations associated with such cross-correlations. In particular, the scalar consistency relation is derived semi-classically by generalizing the consistency relation obtained earlier for cosmic magnetic fields. Notably, we find that the direct coupling between the inflaton and the spectator solely determines the local non-linearity parameter associated with the scalar cross-correlation during slow-roll inflation, regardless of the specific form of the Lagrangian for the spectator field. Further, we calculate the tensor correlation with spectator fluctuations, explore the associated soft limits, and demonstrate the violation of the conventional tensor consistency relation with a non-minimal derivative coupling. Our analysis stresses that the violation of tensor consistency relations does not necessarily imply the superhorizon evolution of tensor modes. Instead, such violations can arise due to the non-minimal derivative coupling of the spectator field to gravity. Finally, we discuss the wider implications of our results in the context of cosmological soft theorems.