Takahiro Tanaka, Yuko Urakawa

The gradient expansion and the separate universe approach provide an effective description of inflationary soft modes after coarse-graining shorter-wavelength degrees of freedom. We formulate a locality condition on the quantum state, requiring that the hard-mode state in each local universe depend on the soft modes only through the local soft-mode values in the same patch. When this condition is satisfied, the coarse-grained soft-mode dynamics remains local, and loop corrections from hard modes to superhorizon correlators of the adiabatic curvature perturbation are perturbatively suppressed. This provides a model-independent diagnosis of when enhanced corrections due to hard modes can invalidate the gradient expansion. We further show that the same locality condition implies a generalized soft theorem, from which the standard consistency relations follow under additional assumptions. This formulation clarifies the origin of possible deviations from the standard consistency relations in multi-field systems or in a non-attractor background. We also show that the locality condition guarantees the absence of infrared divergences for the correlators of operators invariant under a large gauge transformation. Thus, locality of the hard-mode state provides a unified criterion for the effective description of inflationary soft modes, generalized soft theorems, the suppression of hard-mode loop corrections, and the infrared regularity of observable correlators.