Spatial Correlations of the Signature-Change Transition-Point Sensitivity in Robin-Boundary Quantum Cosmology

AbstractIn a companion paper [1], we computed the ensemble statistics of the shift in the exactEuclidean–Lorentzian signature-change surface of Robin-boundary quantum cosmology inducedby a light stochastic spectator field, treating the field as spatially homogeneous within eachHubble patch. Here we extend that calculation to retain the field’s spatial dependence, usingthe separate-universe (gradient-expansion) approximation together with the exact equal-timespatial correlation function of a light field in the de Sitter stochastic equilibrium state. Wederive, in closed form, the two-point correlation function of this counterfactual sensitivity andshow that it is exactly the ensemble variance of the companion paper multiplied by a power lawin physical separation, with exponent set by the same m2/H2 that controls every other suppression factor in this program. The associated correlation length is double-exponentially large inH2/m2; for any field light enough to satisfy the hierarchy already required for the homogeneouscalculation, this sensitivity is correlated only on scales vastly exceeding any observationally relevant patch– and we clarify, using the precise Multi-Domain-Universe/Multiverse distinction ofStoeger, Ellis and Kirchner, that our single-nucleation construction in fact produces an exactlyhomogeneous historical transition surface, so that this correlation function quantifies sensitivityto local field variations rather than realized spatial structure. We further show that this leadingorder (separate-universe) construction inherits the stability of the exact homogeneous solutionat the order considered, without introducing any instability of its own, adopt the same gaugeused in the standard curvaton literature, classify the possible couplings between the spectatorfield and the Robin boundary data into an exactly solvable class (shown to leave the transitionpoint unaffected) and a class that is not exactly solvable in closed form (shown, via the samelight-field hierarchy, to be perturbatively suppressed by an additional factor of m2/H2 relative to the already-negligible homogeneous effect), and identify the precise, named toolkit (thestochastic-δN formalism) that would extend the present equilibrium-regime treatment beyondit. The overall conclusion is, like that of the companion paper, a quantitative negative result:within the entire class of mechanisms considered here, the signature-change surface shows nostructure on any scale of physical interest.