Universality and Falsifiability of Quantum Spacetime Decoherence: A Gauge-Invariant Framework for Gravitational-Wave Phase Diffusion

TL;DR

This work develops a gauge-invariant framework for gravitational-wave propagation through stochastic quantum spacetime, focusing on phase diffusion as the leading observable in the deep adiabatic regime. By projecting the Riemann tensor two-point function along null geodesics, the authors derive a master formula for the phase variance and prove a universality result: for short-range curvature fluctuations, the accumulated phase variance scales linearly with distance, characterized by $\langle \Delta\phi^2\rangle \propto D$ (i.e., $\beta=1$). The frequency dependence is captured by $\alpha = 2 + p$, where $p$ encodes microscopic energy dependence; as a worked example, stringy spacetime foam yields $\alpha = 2$ to $5$ with $\beta=1$, while long-range correlations lead to $\beta\neq 1$, signaling nonlocal spacetime structure. The framework enables a clean, falsifiable classification of quantum-spacetime models in the $(\alpha,\beta)$ plane and prescribes a hierarchical Bayesian strategy to combine multi-band GW data (LIGO/LISA/PTAs) to constrain or rule out exotic scenarios, thereby turning GW observations into precise tests of spacetime microstructure.

Abstract

We develop a fully gauge-invariant and rigorously derived framework for computing the cumulative decoherence of gravitational waves (GWs) propagating through a stochastic quantum spacetime. Working directly with the Riemann-tensor two-point function and exploiting the extreme adiabaticity of cosmological GW propagation, we show that phase diffusion, rather than amplitude attenuation or mode mixing, is the unique leading-order imprint of microscopic curvature fluctuations. Our main theoretical result is a universality theorem: for any quantum-gravity model whose curvature fluctuations possess a finite correlation length, the accumulated phase variance grows linearly with distance, independent of the underlying microphysics. This diffusive scaling contrasts sharply with coherent astrophysical effects and with nonlocal models. The frequency exponent therefore becomes a clean spectral discriminator, separating string-foam recoil, holographic or scale-invariant noise, and causal-set discreteness. We obtain these results from first principles by evaluating the projected Riemann correlator along null geodesics and determining the exact conditions under which deviations from universality can arise. Finally, we outline a hierarchical Bayesian strategy for measuring this effect with LIGO, LISA, and Pulsar Timing Arrays. Although standard Planck-scale fluctuations remain far below current sensitivity, this framework provides a sharp and falsifiable test of exotic quantum-spacetime scenarios, particularly those with macroscopic correlation lengths or strong energy dependence.

Figures (3)

• Figure 1: Schematic illustration of gravitational-wave propagation through a fluctuating spacetime background. A coherent gravitational wave (blue) samples microscopic fluctuations of the geometry (gray surface) along a cosmological trajectory. In the adiabatic regime, phase coherence is preserved and microscopic fluctuations contribute primarily through cumulative phase shifts.
• Figure 2: Classification of quantum spacetime models in the $(\bm{\alpha}, \bm{\beta})$ scaling plane. The vertical dashed line at $\beta=1$ represents the universality class derived in this work: all local, short-range quantum gravity fluctuations must exhibit linear distance scaling (diffusive random walk). Distinct microscopic theories populate specific "islands" along this line based on their spectral index $\alpha$: Causal Set discreteness ($\alpha \approx 0$), Holographic noise and Loop Quantum Gravity ($\alpha \approx 2$), and energy-dependent String Theory foam ($\alpha \approx 4\text{--}5$). The shaded region to the right ($\beta > 1$) corresponds to non-local or long-range correlations, physically associated with fractal spacetime geometry or coherent astrophysical foregrounds.
• Figure 3: Multi-band spectroscopy of quantum spacetime. The predicted cumulative phase error ($\Delta \phi_{\mathrm{rms}}$) as a function of gravitational-wave frequency for two representative "exotic" quantum gravity scenarios. The curves are normalized to a reference amplitude of $A \sim 10^{-5}$ at LISA frequencies to illustrate the distinct spectral slopes. Shaded rectangles indicate approximate phase-sensitivity floors for Pulsar Timing Arrays (PTA), LISA, and ground-based detectors (LIGO/ET). Stringy Spacetime Foam (dashed green line) predicts a steep frequency dependence ($\alpha \approx 4$, corresponding to $\Delta \phi_{\mathrm{rms}} \propto f^2$), making it most accessible to high-frequency ground-based detectors. Conversely, scale-invariant Holographic noise (solid blue line, $\alpha \approx 2$, corresponding to $\Delta \phi_{\mathrm{rms}} \propto f^1$) is effectively probed by the long baselines of LISA and PTAs. Simultaneous constraints across these bands can break degeneracies and uniquely determine the microphysical exponent $\alpha$.