Robustness and operational characterization of inflationary branch decoherence beyond the idealized spectator environment

Environment-induced decoherence is a leading explanation for how the quantum fluctuations of inflation become the classical seeds of cosmic structure, and recent work (Nayeri, arXiv:2602.21263) has made this mechanism explicit and quantitative within an idealized setup: long-wavelength curvature perturbations during sustained de Sitter inflation, decohered by a single light spectator scalar in the Bunch–Davies vacuum treated through the influence-functional formalism, yielding a closed-form superhorizon branch-overlap law and a branch-relative arrow of time. The open problem is whether these conclusions survive the source's stated extensions: deriving the noise kernel and decoherence functional for the reheating epoch, treating the system–environment couplings for tensor perturbations and their B-mode consequences, and computing operational quantum-information measures of the decoherence — in particular the relative entropy or Bures fidelity between the two branch-conditioned Gaussian states. Whether the conclusions also survive non-Gaussian environmental statistics, non-Bunch–Davies initial states, and finite inflationary duration is a natural robustness question raised by the framework's assumptions.

Background

Inflation seeds all cosmic structure from quantum fluctuations: the curvature perturbation $\zeta$ (the gauge-invariant scalar variable describing spatial curvature on uniform-density slices) starts as a quantum field, yet the temperature and density patterns we observe behave classically. A foundational line of work (Polarski and Starobinsky, gr-qc/9504030) showed that superhorizon inflationary modes become highly squeezed states that are operationally equivalent to classical stochastic variables once the exponentially small decaying mode is dropped — but squeezing by itself does not remove quantum coherence between macroscopically distinct histories, so the quantum-to-classical transition requires an additional mechanism. Environment-induced decoherence supplies one: unobserved degrees of freedom (short modes, spectator fields, gravitational nonlinearities) interact with the long-wavelength system, and tracing them out suppresses the off-diagonal elements of the reduced density matrix in a pointer basis. The standard tool is the Feynman–Vernon influence functional: the effect of the traced-out environment on the system's reduced density matrix is encoded in an influence action, whose imaginary part defines a real noise kernel $N(x,x')$ governing both the stochastic noise felt by the system and the decoherence rate. The environment is conventionally taken in the Bunch–Davies vacuum — the de Sitter-invariant state corresponding to no particles in the remote past, with mode functions that behave like plane waves at short distances. Nayeri (arXiv:2602.21263, v1 2026, retitled and expanded in v3 2026 as "Inflationary branch decoherence and the cosmological arrow of time", published as Phys. Rev. D 113, 126030 (2026)) assembled this into an explicit quantum-cosmological calculation: starting from the Hartle–Hawking no-boundary state (a real standing-wave superposition of expanding and contracting semiclassical WKB branches of the universal wavefunction) and the tunneling state (a single outgoing complex WKB branch), and using the influence-functional formalism with a light massive environmental scalar $\sigma$ coupled linearly to $\zeta$ through $S_{\mathrm{int}}=g\int d\eta\, d^3x\, a^4\, \zeta_L \sigma^2$ , the work derives the noise kernel, evaluates the decoherence functional $\Gamma$ (the exponent suppressing interference between coarse-grained perturbation histories) under both horizon-based and effective-field-theory-motivated coarse grainings, and computes the single-mode branch overlap $|\mathcal{D}_k(z)|=[z^2/(z^2+1)]^{1/4}$ in the massless limit and $|\mathcal{D}_k(z)|\sim z^{\nu-1}$ on superhorizon scales for massive fields, where $z=-k\eta$ is the wavenumber in conformal time units and $\nu=\sqrt{9/4-m_\sigma^2/H^2}$ is the standard de Sitter mode index for an environmental field of mass $m_\sigma$ in a background with Hubble rate $H$ (so the exponent tends to $1/2$ as $m_\sigma\to0$ , reproducing the massless closed form). A nonzero field mass regulates the infrared accumulation, which is otherwise controlled only by the finite duration of inflation. The conclusions — efficient, coarse-graining-insensitive decoherence, and an emergent branch-relative arrow of time from environment-induced entanglement — rest on idealizations: the environment is a single weakly coupled spectator with Gaussian statistics in the exact Bunch–Davies vacuum, the background is sustained (effectively endless) de Sitter inflation, and the influence action is kept at leading (quadratic, Gaussian) order. Version 3 of the paper additionally proved a universality corollary: the same overlap law governs all massless de Sitter modes including tensor perturbations (gravitational-wave modes) — although the explicit system–environment couplings for tensors remain untreated. The author's stated future directions are the reheating-epoch extension, quantum-information-theoretic measures of the decoherence (specifically the relative entropy or Bures fidelity — standard distinguishability measures between quantum states, computable for Gaussian states from two-point functions — between the two branch-conditioned Gaussian states), and the tensor-to-B-mode connection; sensitivity to the Gaussian, Bunch–Davies, and infinite-duration assumptions is a natural robustness question raised but not analyzed by the framework.

Problem Statement

Determine whether environment-induced branch decoherence of long-wavelength inflationary perturbations — as quantified by the influence-functional decoherence functional $\Gamma$ and the branch-overlap law $|\mathcal{D}_k(z)|=[z^2/(z^2+1)]^{1/4}$ (massless) and $\sim z^{\nu-1}$ (massive, with $\nu$ the de Sitter mass index) derived for a single light spectator scalar in the Bunch–Davies vacuum during sustained de Sitter inflation — persists under the source's stated extensions. Concretely, within the influence-functional (Feynman–Vernon) open-system treatment, and following the future directions stated in the source: (a) derive the noise kernel and decoherence functional for the reheating epoch, where the background is time-dependent and the mode structure differs from eternal de Sitter; (b) treat the system–environment couplings for tensor (gravitational-wave) perturbations and establish what the overlap-law universality implies for primordial B-mode observables via the tensor-to-scalar ratio $r$ ; and (c) compute operational quantum-information-theoretic measures of the decoherence — in particular the relative entropy or Bures fidelity between the two branch-conditioned Gaussian states, directly computable from the overlap factor — going beyond the scalar decoherence functional $\Gamma$ , including whether such measures confirm or qualify the claimed classicality. All three components (a)–(c) are mandatory and jointly settle the genericity question: delivering all three resolves the target, whereas delivering any proper subset does not. These components jointly address one question: is the environment-induced quantum-to-classical transition of inflationary perturbations — and the arrow-of-time conclusion built on it — generic, or an artifact of the idealized spectator-scalar environment? In addition, the framework's assumptions raise natural robustness questions that the source does not pose as future work: the sensitivity of the decoherence conclusions to non-Gaussian environmental statistics (higher-order influence-action terms), to departures from the exact Bunch–Davies environmental initial state, and to finite-duration inflationary histories, particularly for the infrared accumulation that controls the accumulated decoherence and the branch-relative arrow of time. Results on these sensitivities bear on the genericity question but are not required components of the target.

References

Ali Nayeri, "Inflationary branch decoherence and the cosmological arrow of time" (v1 titled "Quantum Cosmology, Decoherence, and the Emergence of Classical Spacetime", 2026-02-24; v3 2026-06-10), arXiv:2602.21263, DOI 10.48550/arxiv.2602.21263, published as Phys. Rev. D 113, 126030 (2026), journal DOI 10.1103/1k1w-c7vd, https://doi.org/10.1103/1k1w-c7vd (preprint: https://arxiv.org/abs/2602.21263)
S. Shajidul Haque and Bret Underwood, "A Landscape of Cosmological Decoherence", arXiv:2606.07663, https://arxiv.org/abs/2606.07663
Mattia Cielo, Simone Scarlatella, Gianpiero Mangano, Ofelia Pisanti and Louis Hamaide, "Quantum Recoherence in Presence of Excited States in the Early Universe", arXiv:2512.01932, https://arxiv.org/abs/2512.01932
Jessie de Kruijf and Nicola Bartolo, "The effect of quantum decoherence on inflationary gravitational waves", arXiv:2408.02563, published as J. Cosmol. Astropart. Phys. 11 (2024) 041, DOI 10.1088/1475-7516/2024/11/041, https://doi.org/10.1088/1475-7516/2024/11/041 (preprint: https://arxiv.org/abs/2408.02563)
Jessie de Kruijf, Giacomo Galloni and Nicola Bartolo, "The first data-driven bounds on the quantum decoherence of inflationary gravitational waves", arXiv:2511.14727, https://arxiv.org/abs/2511.14727
David Polarski and Alexei A. Starobinsky, "Semiclassicality and Decoherence of Cosmological Perturbations", arXiv gr-qc/9504030, published as Class. Quantum Grav. 13 (1996) 377–392, DOI 10.1088/0264-9381/13/3/006, https://doi.org/10.1088/0264-9381/13/3/006 (preprint: https://arxiv.org/abs/gr-qc/9504030)