Cumulative Decoherence of Gravitational Waves

• Cumulative Decoherence of Gravitational Waves is the gradual loss of phase coherence due to interactions with quantum and classical environments, impacting observable GW signatures.
• Decoherence mechanisms are analyzed with gauge-invariant frameworks and environmental models, revealing linear and super-diffusive scaling of phase variance.
• The study links decoherence effects with observational limits in GW astronomy, offering constraints on quantum gravity and cosmological perturbations.

Cumulative decoherence of gravitational waves refers to the progressive loss of phase coherence, purity, or quantum correlations of gravitational wave (GW) modes during their propagation or interaction with quantum or classical environments. This phenomenon arises from various mechanisms: stochastic spacetime fluctuations (“quantum foam”), quantum or classical environmental interactions, graviton mixing, and inhomogeneities in the cosmological metric. Decoherence functions analogously to phase diffusion, damping, or the exponential decay of off-diagonal components of density matrices in open quantum systems. The cumulative effects, which may accrue over cosmological scales, have profound implications for GW astronomy, the classicality of cosmological perturbations, and tests of quantum gravity.

1. Gauge-Invariant Frameworks and Universality of GW Decoherence

The propagation of GWs in a stochastic or quantum spacetime is most rigorously described using gauge-invariant formalisms, typically via the projected Riemann-tensor two-point function along null geodesics. A GW with amplitude $A_{\mu\nu}$ and phase $\phi$,

$h_{\mu\nu}(x) = A_{\mu\nu}(x) e^{i \phi(x)},$

acquires a stochastic phase shift $\Delta \phi$ due to curvature fluctuations. The mean-square accumulated phase is given by

$$\langle \Delta\phi^2 \rangle = \frac{1}{4\omega^2} \int_0^D ds \int_0^D ds' C_R(s-s'),$$

where $C_R$ is the projected Riemann correlator. If $C_R$ decays on a finite correlation length $L_c$, the universality theorem asserts that

$\langle \Delta\phi^2 \rangle \propto D$

for $D \gg L_c$, i.e., phase variance grows linearly and is independent of microphysical details (Cang et al., 2 Dec 2025). The frequency scaling discriminates between models: holographic noise gives $\alpha=2$ (where $\langle \Delta\phi^2 \rangle \propto \omega^2$), string-foam recoil $\alpha=4$, and causal-set discreteness $0<\alpha<2$. Long-range correlations ($C_R \sim u^{-\gamma}$) produce super-diffusive scaling ($\langle \Delta\phi^2 \rangle \propto D^{2-\gamma}$) and could indicate nonlocal quantum-spacetime structure.

2. Decoherence by Quantum Environments: Inflationary and Scalar Field Interactions

Quantum decoherence of GWs in the early Universe—especially tensor modes of cosmological perturbations—generally arises from weak interactions with scalar fields during or immediately after inflation. For tensor fluctuations $h_{ij}$ coupled to a scalar "environment" $\phi$, decoherence is described by influence functional and Lindblad-type reduced master equations. The decoherence functional

$$\Gamma[h^\Delta] = \frac{1}{2\hbar^2} \int d\eta_1 d\eta_2 \int d^3p\, h^\Delta_{\vec{p}}(\eta_1) h^\Delta_{-\vec{p}}(\eta_2) \mathcal{N}(\eta_2-\eta_1,\vec{p})$$

encodes the decay of off-diagonal elements due to environmental noise kernel $\mathcal{N}$. The cumulative decoherence factor $D$ suppresses the quantum state as $e^{-D}$, with $D$ a functional of GW amplitude, frequencies, and the environmental correlation structure. Decoherence is maximized at high reheating temperatures and low frequencies, typically rendering primordial GW backgrounds classical except in restricted, high-frequency windows (e.g., $10^2$–$10^8$ Hz for standard reheating, above $10^7$ Hz for kinetic-dominated scenarios) (Takeda et al., 25 Feb 2025). Data-driven analyses constrain environmental interaction strengths and show observationally allowed regions where inflationary modes have not completed decoherence, thus potentially leaving quantum signatures in the stochastic background (Kruijf et al., 18 Nov 2025).

3. Decoherence from Propagation in Cosmological Metric Perturbations

Cosmological metric perturbations, primarily scalar potentials $\Phi$, $\Psi$, induce stochastic Shapiro delays in GW phases as they propagate through the perturbed FLRW metric. A GW of frequency $f$ acquires a random phase

$$\delta\phi(\hat{n}, f) = 2\pi f \int d\eta\, [\Phi + \Psi](x=\hat{n}(\eta_0-\eta), \eta),$$

or equivalently over comoving radial distance. The variance $\langle \delta\phi^2 \rangle$ integrates the cosmological power spectrum and transfer functions. For frequencies above $f \sim 10^{-12}$ Hz, phase decoherence is complete ($e^{-1/2\langle \delta\phi^2 \rangle} \to 0$), precluding phase-coherent mapping of stochastic GW backgrounds. Only intensity-based, quadratic estimators yield meaningful sky maps in practical GW astronomy (Margalit et al., 2020).

4. Environmental Gravitational Decoherence and Tensor Noise Models

Emission of GWs by massive systems in superpositions leads to environmental decoherence, quantifiable via path-integral/Feynman–Vernon influence functionals and tensor noise models. The cumulative decoherence factor $D(t)$ for a system of mass $m$, separation $d$, and oscillation frequency $\omega$ exhibits exponential suppression:

$$D(t) = \exp\left[ -\frac{G m^2 d^2 \omega^6}{10 \pi} \coth\left(\frac{\beta\omega}{2} \right) t \right]$$

(at long times), with negligible values for laboratory-mass systems. Tensor noise models, such as Diόsi–Penrose-type collapses, add further contributions that scale with differences of mass distribution and Newtonian potential. For realistic interferometers and macroscopic superpositions, environmental gravitational decoherence remains several orders of magnitude below operational thresholds (Suzuki et al., 2015).

5. Decoherence via Graviton Emission, Memory Effects, and Soft Modes

Quantum superpositions subject to bursts of GW radiation decohere predominantly via soft-graviton emission linked to their time-dependent quadrupole moments. The expected number of emitted “which-path” gravitons $\langle N \rangle$ governs the decoherence factor:

$D = 1 - \exp(-\frac{1}{2} \langle N \rangle )$

with $\langle N \rangle$ computed from mode integrals over the Weyl tensor correlator. Decoherence splits into memory (linear in permanent displacement) and oscillatory (dependent on the phase when the burst is switched off) contributions. The cumulative effect for multiple bursts or a continuous background is additive, and formal analogues appear in the electromagnetic case (with electric dipole radiation replacing mass quadrupole) (Linton et al., 30 Jan 2025).

6. Decoherence Induced by Graviton Baths and Quantum Geometry Fluctuations

A background bath of long-wavelength gravitons induces pure dephasing of GW modes via quantum geometry fluctuations. The decoherence rate $\Gamma$ scales with the variance of metric fluctuations, and the off-diagonal components decay as $\exp(-\Gamma t^2)$,

$\Gamma = 2 (\Delta\omega)^2 \sigma^2,$

where $\sigma^2$ is the metric fluctuation variance. For a bath of gravitons at temperature $T$, phase decoherence times are inversely proportional to $T$ and frequency difference $\Delta\omega$. For cosmological GWs, cumulative decoherence over a path $L$ is $D(L) = \exp(-\Gamma L)$; for redshift-dependent parameters, an integral over cosmic history is required. For realistic current graviton backgrounds, $t_d$ is exceedingly long, making such decoherence negligible, but the framework allows estimation for arbitrary environmental models (Lorenci et al., 2014).

7. Bigravity, Wave-Packet Decoherence, and Phenomenological Implications

In bigravity theories with massive and massless graviton states, wave-packet decoherence leads to exponential suppression of oscillatory interference terms. The coherence length,

$$L_{\text{coh}} \simeq (2\sqrt{2} \sigma_x E^2) / \Delta m^2,$$

with $\Delta m^2$ the mass squared difference, sets the scale at which two packet components separate. At distances $D \gg L_{\text{coh}}$, the amplitude splits into distinct “primary” and “echo” pulses with amplitudes proportional to $\cos^2\theta$ and $\sin^2\theta$ respectively, where $\theta$ is the mixing angle. Observationally, non-detections imply bounds on $m_g$ and $\theta$; future searches may probe further via “echo” signatures and population rate mismatches (Max et al., 2017).

Summary Table: Key Decoherence Mechanisms

Mechanism	Scaling Relation	Applicability/Context
Riemann correlator phase diffusion (Cang et al., 2 Dec 2025)	$\langle \Delta\phi^2 \rangle \propto D, \alpha=f$	Quantum-spacetime foam, Planck-scale
GW–environment interactions (Takeda et al., 25 Feb 2025)	$D \sim (A^\Delta)^2(H_r/p)^2(T_r/p)^5$	Inflation, reheating, scalar fields
Cosmological metric perturbations (Margalit et al., 2020)	$e^{-1/2\langle \delta\phi^2 \rangle}$	Mapping stochastic GW backgrounds
Environmental GW emission (Suzuki et al., 2015)	$D(t) \sim \exp[-G m^2 d^2 \omega^6 t]$	Macroscopic superpositions, lab
Soft-graviton emission (Linton et al., 30 Jan 2025)	$D = 1 - \exp(-\langle N \rangle/2)$	GW bursts, quantum qubits
Graviton bath dephasing (Lorenci et al., 2014)	$D(t) \sim \exp[-\Gamma t^2]$	Bath temperature, GW propagation
Bigravity wave-packet splitting (Max et al., 2017)	$|I(D)| \sim \exp[-(D/L_{\text{coh}})^2]$	GW “echoes”, massive graviton

The cumulative decoherence of gravitational waves, through diverse quantum and classical mechanisms, is characterized by exponential or Gaussian suppression of coherence, often scaling linearly or super-diffusively with propagation distance or time. These effects, although negligible for many practical purposes, set hard limits on the observability of quantum features of GWs and serve as discriminators for quantum-gravity scenarios, cosmological origins, and fundamental physics beyond general relativity.