Spontaneous collapse models lead to the emergence of classicality of the Universe

The paper addresses the quantum-to-classical transition of the Universe’s geometry under the assumption that quantum mechanics is universal. Standard decoherence relies on an external environment acting as a measurer, which is untenable for the Universe as a whole, yielding the cosmological measurement problem. The authors propose that spontaneous wavefunction collapse models can account for the emergence of a classical, well-defined space-time geometry from an initial quantum superposition of geometries. They focus on a flat FLRW Universe and set conventions for the metric and variables, aiming to demonstrate that collapse dynamics can select a single geometry (and, in a unimodular setting, a single value of the cosmological constant) prior to the CMB epoch, thereby bridging early quantum cosmology with later classical cosmology.

The work builds on foundational discussions of the measurement problem and decoherence, and on collapse models (e.g., CSL, DP) as objective modifications of quantum theory. In cosmology, prior approaches often studied quantum-to-classical transitions of matter-field perturbations on fixed classical backgrounds (e.g., inflationary perturbations, Mukhanov–Sasaki variable), sometimes invoking collapse to explain features of the CMB and constraining collapse parameters via the power spectrum. Other proposals introduced collapse terms to address the problem of time by modifying dynamics with respect to a foliation time coordinate. This paper differs by targeting the emergence of a classical space-time itself and by choosing a relational internal variable as a physical clock. The appendix reviews related works on unimodular gravity, choice of clock and boundary conditions, and alternative frameworks (e.g., de Broglie–Bohm) for cosmological perturbations.

• Models considered: (i) GR with a perfect fluid; (ii) Parametrised Unimodular gravity (PUM). Both lead to equivalent canonical structures after appropriate transformations.
• Classical setup: Starting from an action for GR plus a perfect fluid, and for PUM, the authors perform canonical transformations to variables (v, π_v) and (t, λ), with Poisson brackets {v, π_v} = 1 and {t, λ} = 1. The Hamiltonian constraint becomes C = −π_v^2 + λ = 0, with λ constant along classical trajectories (energy density in GR+fluid or the unimodular cosmological constant in PUM).
• Quantization: Promote the constraint to an operator acting on the wavefunction Ψ(v, t), leading to the Wheeler–DeWitt equation in the (v, t) representation: (∂^2/∂v^2 − i ∂/∂t) Ψ(v, t) = 0, upon identifying t as a relational clock. The effective Hamiltonian is Ĥ = −∂^2/∂v^2. Boundary conditions on the half-line v ≥ 0 are imposed to ensure self-adjoint dynamics (no probability flow at v = 0), characterized by a parameter γ; γ = ∞ is adopted for numerics.
• Collapse dynamics: Modify the WDW equation by adding stochastic, nonlinear terms characteristic of collapse models. Choose the collapse operator  proportional to the Hamiltonian,  = ε Ĥ, where ε is the collapse rate. Using the quantum form of the constraint (H = −λ^2/2), this choice induces localization in the λ basis (i.e., selection of a definite geometry). The resulting stochastic differential equation yields Itô equations for the mean and variance of λ: d⟨λ⟩t = −2 ε σ^2{λ,t} dW_t; dσ^2_{λ,t} = −4 ε^2 (σ^2_{λ,t})^2 dt − 2 Σ^{(3)}_{λ,t} dW_t, with W_t a Wiener process and Σ^{(3)} the third central moment.
• Gaussian approximation: For initial Gaussian superpositions over λ, Σ^{(3)} vanishes and the variance evolves analytically as σ^2_{λ,t} = σ^2_{λ,t0} / (1 + 4 ε^2 (t − t0) σ^2_{λ,t0}), exhibiting monotonic decay toward zero (localization). The mean follows ⟨λ⟩t = ⟨λ⟩{t0} − 2 ε ∫{t0}^t σ^2{λ,s} dW_s. Perturbative treatments beyond Gaussianity are outlined in the appendix.
• Interpretation in GR and PUM: In GR+fluid, collapse selects a definite fluid energy density (hence geometry). In PUM, collapse selects a definite value of the cosmological constant, offering a mechanism for fixing Λ.

• Incorporating spontaneous collapse terms into the Wheeler–DeWitt dynamics drives localization in λ, transforming an initial superposition of geometries into a single geometry without invoking external environments.
• For Gaussian initial states over λ, the variance decays as σ^2_{λ,t} = σ^2_{λ,t0} / (1 + 4 ε^2 (t − t0) σ^2_{λ,t0}), implying eventual localization (σ^2 → 0) and freezing of the mean ⟨λ⟩ once localized.
• The choice  = ε Ĥ ties localization to λ because H ∝ −λ^2/2; thus the collapse basis corresponds to distinct space-time geometries.
• In the PUM formulation, localization in λ corresponds to selecting a definite cosmological constant Λ, thereby providing a possible dynamical explanation for the observed Λ without invoking an anthropic multiverse.
• The model is general with respect to clock choice (as long as a suitable internal clock exists) and reduces to known collapse prescriptions in appropriate limits (e.g., nonrelativistic CSL-like behavior).
• Observationally, since cosmology affords a single realization, the post-collapse Universe is indistinguishable from a purely classical one; however, in principle, ε could be constrained by requiring localization before CMB photon decoupling.

The findings directly address the cosmological quantum measurement problem by providing an intrinsic, dynamics-driven mechanism for the emergence of a classical space-time. By selecting a physical internal clock, the modified WDW equation yields a nontrivial stochastic evolution that collapses superpositions in λ without external observers. This mechanism explains how an initially quantum Universe can become classical prior to the era described by classical GR and standard cosmology. The approach also differs from earlier proposals focused on time-reparametrization gauges or on field perturbations over fixed geometry; here the geometry itself becomes classical. In the unimodular setting, the selection of a single Λ offers a route to the cosmological constant problem that does not rely on anthropic arguments. While the model does not by itself solve the problem of time (the clock is chosen), it demonstrates that, given a relational clock, collapse dynamics can consistently produce a classical Universe and is compatible with various quantum cosmology frameworks.

The paper shows that augmenting the Wheeler–DeWitt equation with spontaneous collapse dynamics (using  ∝ Ĥ) generically localizes the Universe’s wavefunction in λ, selecting a unique space-time geometry. In GR+perfect fluid this fixes the fluid energy density; in PUM it selects a definite cosmological constant, offering a potential explanation of its observed value. The mechanism provides a coherent account of the quantum-to-classical transition of space-time without external environments and is adaptable to other quantum cosmology models with a definable internal clock. Future directions include: exploring different clock choices and their phenomenology; extending to matter fields and Loop Quantum Cosmology; analyzing non-Gaussian initial conditions and full perturbative regimes; and constraining the collapse rate ε by requiring classicality before CMB decoupling or via laboratory tests of collapse models.

• The choice of an internal clock variable is assumed and does not solve the problem of time; results depend on this clock choice.
• The Wheeler–DeWitt minisuperspace model is a simplified setting; full quantum gravity effects and more general spacetimes are not addressed.
• Gaussian assumptions are used for analytic tractability; beyond-Gaussian behavior is treated perturbatively and may affect localization timescales.
• The localization timescale can be long (variance decays as an inverse in time under Gaussian assumption), with quantitative predictions depending on the collapse rate ε, which is not fixed here.
• Cosmology affords only a single realization of the noise, limiting direct observational discriminability from classical cosmology; constraints on ε are only suggested indirectly.
• Boundary conditions (e.g., at v = 0 via parameter γ) and factor ordering choices may introduce additional ambiguities, though not expected to alter the qualitative conclusion of localization.