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Ringdown bounds and spectral density limits from GWTC-3

Christian Balfagon

Abstract

We establish the first observational bounds on causal nonlocal extensions of gravity characterized by retarded Stieltjes-type kernels with positive spectral density rho(mu) >= 0, using two complementary gravitational-wave channels. From a Bayesian ringdown analysis of 17 binary black hole events in the LIGO-Virgo GWTC-3 catalogue, we set an observational ceiling on universal fractional quasi-normal mode deformations of |epsilon_Omega| < 0.05 (90% C.L.), with a cumulative log Bayes factor ln B = -0.46 +/- 0.77. By mapping published GWTC-3 modified dispersion relation bounds together with the GW170817 propagation speed constraint onto the Stieltjes spectral parameter space (mu_char, M0), we exclude a broad class of infrared-extended spectral densities with mu <= 10^{-6} m^{-2}, thereby ruling out non-trivial regions of the nonlocal kernel parameter space for the first time. The theoretically motivated fiducial range mu_char ~ M_*^2 ~ 10^8-10^10 m^{-2} satisfies all current bounds. We further show that sub-millimetre gravity experiments, which already operate at the predicted causal scale l_* ~ 10^{-4} m, provide the most promising path toward a direct test. These results define quantitative benchmarks against which future observations across the gravitational-wave, short-range gravity, and cosmological sectors can be compared.

Ringdown bounds and spectral density limits from GWTC-3

Abstract

We establish the first observational bounds on causal nonlocal extensions of gravity characterized by retarded Stieltjes-type kernels with positive spectral density rho(mu) >= 0, using two complementary gravitational-wave channels. From a Bayesian ringdown analysis of 17 binary black hole events in the LIGO-Virgo GWTC-3 catalogue, we set an observational ceiling on universal fractional quasi-normal mode deformations of |epsilon_Omega| < 0.05 (90% C.L.), with a cumulative log Bayes factor ln B = -0.46 +/- 0.77. By mapping published GWTC-3 modified dispersion relation bounds together with the GW170817 propagation speed constraint onto the Stieltjes spectral parameter space (mu_char, M0), we exclude a broad class of infrared-extended spectral densities with mu <= 10^{-6} m^{-2}, thereby ruling out non-trivial regions of the nonlocal kernel parameter space for the first time. The theoretically motivated fiducial range mu_char ~ M_*^2 ~ 10^8-10^10 m^{-2} satisfies all current bounds. We further show that sub-millimetre gravity experiments, which already operate at the predicted causal scale l_* ~ 10^{-4} m, provide the most promising path toward a direct test. These results define quantitative benchmarks against which future observations across the gravitational-wave, short-range gravity, and cosmological sectors can be compared.
Paper Structure (18 sections, 2 theorems, 21 equations, 6 figures, 3 tables)

This paper contains 18 sections, 2 theorems, 21 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Ringdown analysis
  3. Method
  4. Results
  5. Dispersion and propagation constraints
  6. Stieltjes transfer function and modified dispersion
  7. Mapping LVK bounds to Stieltjes parameters
  8. Excluded spectral density shapes
  9. Sensitivity projections and alternative probes
  10. Future GW detectors
  11. Short-range gravity experiments
  12. Conclusions
  13. Properties of the Stieltjes kernel
  14. Complete monotonicity
  15. Asymptotic expansion and moment hierarchy
  16. ...and 3 more sections

Key Result

Proposition 1

Let $\rho(\mu)\geq 0$ with $M_0 = \int_0^\infty\rho\,d\mu < \infty$. Then $m(\omega^2)$ defined by (eq:transfer) is completely monotone on $(0,\infty)$:

Figures (6)

  • Figure 1: Left: Joint posterior for spin $a$ and deformation parameter $\varepsilon_\Omega$ from GW150914 (real data). The horizontal dashed line marks the Kerr value $\varepsilon_\Omega=0$. Contours enclose the 68% and 95% credible regions. Right: Cumulative log Bayes factor $\ln B_{\rm stack}$ as a function of the number of events included (chronological order, 17 events). The shaded bands show the $\pm 1\sigma$ and $\pm 2\sigma$ uncertainties propagated from the individual nested-sampling evidence errors.
  • Figure 2: Individual log Bayes factors $\ln B_{\rm CET/Kerr}$ for each of the 17 GWTC-3 BBH events. Error bars show the $\pm 1\sigma$ nested-sampling uncertainty. No event deviates significantly from the Kerr hypothesis ($\ln B = 0$, dashed line).
  • Figure 3: Left: Individual-event posteriors on $\varepsilon_\Omega$. Each curve represents one event; all are broad and uninformative individually. Right: Population-level constraint obtained by multiplying the individual posteriors. The 90% credible interval $\varepsilon_\Omega\in[-0.047,+0.032]$ (orange band) is consistent with Kerr ($\varepsilon_\Omega=0$, dashed line).
  • Figure 4: Leave-one-out (jackknife) analysis. Each bar shows the cumulative $\ln B$ when the corresponding event is excluded from the stack. The red line marks the full-stack value. The result is stable under removal of any single event; GW190521 produces the largest shift.
  • Figure 5: Exclusion regions in the Stieltjes spectral parameter space $(\mu_{\rm char}, M_0)$ from LVK data. Current bounds from the GW170817 speed measurement (blue solid), graviton mass limit (maroon dashed), and GWTC-3 MDR analyses (green, orange) exclude the shaded region. Projected bounds from the Einstein Telescope and LISA (dashed lines) will extend the excluded region downward. The fiducial CET $\Omega$ range $\mu_{\rm char}\sim 10^{8}$--$10^{10}\, {\rm m}^{-2}$ (orange band) satisfies all current and projected GW bounds. The Eöt-Wash torsion-balance constraint (thick red segment) provides the only direct probe at the fiducial scale. The upper axis shows the corresponding effective correlation length $\ell_{\rm eff}=1/\sqrt{\mu}$.
  • ...and 1 more figures

Theorems & Definitions (2)

  • Proposition 1
  • Proposition 2