We analyze branch decoherence in inflationary quantum cosmology by computing reduced density matrices and branch-overlap factors for long-wavelength perturbations. The Hartle-Hawking no-boundary state is real in the semiclassical regime and contains both expanding and contracting Wentzel-Kramers-Brillouin (WKB) components, whereas the tunneling state is selected as an outgoing complex WKB branch; expanding-contracting decoherence is therefore central for the former and mainly diagnostic for the latter. Using the influence-functional formalism, we derive the noise kernel for a light spectator environment and evaluate decoherence under horizon-based and effective-field-theory (EFT)-motivated coarse grainings. We then compute the single-mode branch overlap directly from the Bunch-Davies mode functions, obtaining $|{\mathcal{D}}_k(z)|=[z^2/(z^2+1){]}^{1/4}$ in the massless limit and $|{\mathcal{D}}_k(z)|\sim z^{\nu }$ on superhorizon scales for massive fields, where $z\equiv -k\eta$ is the dimensionless wave number with $\eta$ the conformal time. In the massless case the accumulated geometric branch functional is evaluated in closed form, with a leading cutoff-sensitive phase-space term and a universal subleading contribution. The calculation provides an explicit quantitative bridge between quantum-cosmological boundary conditions, inflationary squeezing, and the emergence of effectively classical cosmological histories.

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• Cosmology
• General relativity
• Gravitation
• Quantum cosmology
• Quantum gravity