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Light spectator fields may not be dynamically relevant for the inflationaryphase of the early universe, but they can still induce interesting imprints oncosmological observables. In this paper, we compute the cross-correlations ofthe inflationary perturbations, both scalar and tensor, with the fluctuationsof a non-minimally interacting spectator field using the in-in formalism andinvestigate the consistency relations associated with such cross-correlations.In particular, the scalar consistency relation is derived semi-classically bygeneralizing the consistency relation obtained earlier for cosmic magneticfields. Notably, we find that the direct coupling between the inflaton and thespectator solely determines the local non-linearity parameter associated withthe scalar cross-correlation during slow-roll inflation, regardless of thespecific form of the Lagrangian for the spectator field. Further, we calculatethe tensor correlation with spectator fluctuations, explore the associated softlimits, and demonstrate the violation of the conventional tensor consistencyrelation with a non-minimal derivative coupling. Our analysis stresses that theviolation of tensor consistency relations does not necessarily imply thesuperhorizon evolution of tensor modes. Instead, such violations can arise dueto the non-minimal derivative coupling of the spectator field to gravity.Finally, we discuss the wider implications of our results in the context ofcosmological soft theorems.

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