This paper asks when the effective description of very long-wavelength fluctuations during cosmic inflation remains valid after we average over shorter-wavelength behavior. The authors propose a simple physical criterion: the quantum state of the short-wavelength or “hard” degrees of freedom inside each local patch should depend on the long-wavelength or “soft” fields only through the soft-field values inside the same patch. When this locality condition holds, the coarse-grained dynamics of the soft modes stays local and familiar approximation schemes remain trustworthy.

The work is framed in the language used for early-universe calculations: the gradient expansion and the separate-universe approach. These methods treat regions much larger than the causal scale (the Hubble scale) as nearly independent “local universes.” The authors define coarse-grained soft fields by averaging over fixed comoving patches of size L and define the hard modes as the leftover short-scale pieces. They then formulate a precise locality condition on the hard-mode quantum state (referred to in the paper as Eq. (28)): in each patch the hard-mode state must depend on the soft fields only through the coarse-grained values in that patch.

When the locality condition is satisfied, several useful results follow. Coarse-graining does not introduce non-local terms that would break the separate-universe picture. Loop corrections from hard modes to superhorizon correlations of the adiabatic curvature perturbation ζ (the quantity that encodes the observable, large-scale curvature produced during inflation) are only perturbatively small. The condition therefore provides a model-independent way to diagnose when short-scale quantum effects might otherwise grow large enough to invalidate the gradient expansion.

The authors also show that the same locality requirement leads to a generalized soft theorem. A soft theorem relates a correlation function that includes very long-wavelength modes to a simpler correlation function without them. Under extra, stated assumptions this generalized theorem reduces to the standard consistency relations used in inflationary cosmology. This clarifies why those relations can fail in multi-field models or in non-attractor backgrounds (situations where the background evolution does not settle into the usual attractor trajectory).