Stochastic Backreaction on the Signature-Change Surface in Robin Boundary Quantum Cosmology

The Lorentzian gravitational path integral of Einstein–Hilbert gravity, supplemented by aRobin boundary condition at the initial hypersurface, admits an exact solution in the closedFriedmann–Lemaitre–Robertson–Walker (FLRW) minisuperspace in which the Universe undergoes a signature change from Euclidean to Lorentzian at a fixed value of the scale factor,a = ℓ = 3k/Λ, independently of the Robin parameter. We revisit this exact result andask how the location of this signature-change surface responds when a light, stochastically fluctuating spectator scalar field is present in addition to the bare cosmological constant Λ. Workingwithin an explicit semiclassical (Born–Oppenheimer-type) scheme, we derive the field’s de Sitter equilibrium distribution directly from the stochastic-inflation Fokker–Planck equation, andfurther show, by explicit solution of that equation, that this equilibrium is reached from anarbitrary normalizable initial distribution with finite variance– so that the (currently unsolved)problem of deriving the field’s Robin-induced initial condition need not be solved for our resultto hold, provided sufficiently many e-folds separate the transition from the epoch of interest.We show that the induced shift in the effective cosmological constant, Λeff = Λ0 + 8πGV(ϕ),is distributed as a scaled χ21 variable, and we derive closed-form expressions for the resultingmean and variance of the shift in the transition point. The shift is universal (∝ H4/M2Pl, independent of the spectator mass) and is suppressed by H2/M2Pl relative to the unperturbedtransition point for any sub-Planckian expansion rate. We show explicitly that this suppressionmakes the effect observationally inaccessible for both inflationary and present-day values of H.The present work therefore establishes the homogeneous equilibrium limit of stochastic backreaction in Robin-boundary quantum cosmology, providing a controlled starting point for futureinhomogeneous and jointly-quantized extensions