The effect of the gravitational constant variation on the phase of gravitational waves (2508.21746v1)

Abstract: We have previously investigated the effect of the gravitational constant variation on the gravitational wave propagation. Pure theoretical analysis indicates that the leading order effect of the gravitational constant variation corrects the amplitude of gravitational waves, and the second order effect corrects the phase of gravitational waves. As the matched filtering technique is used by gravitational wave data analysis, the phase is more important than the amplitude. In the current paper we use LIGO-VIRGO-KAGRA data to constrain the gravitational constant variation. Our findings indicate that we need to wait until the distance of the detected gravitational wave events and/or the signal-to-noise ratio increases by $2n$ orders compared to the current detection state, and then we can use phase correction to get constraint $|\frac{G'}{G}|<10^{-n}$/yr for gravitational wave events without electromagnetic counterparts. For gravitational wave events with electromagnetic counterparts which provide the information of source's luminosity distance, the phase correction can almost always be neglected to constrain $\frac{G'}{G}$.

• The paper investigates the impact of a time-varying gravitational constant on GW phase, demonstrating that phase corrections remain subdominant with current LVK data.
• Using Fisher matrix analysis and LVK data, the study shows that amplitude corrections, degenerate with luminosity distance, constrain |G'/G| to ≲10⁻⁷/yr.
• The work underscores that improved detector sensitivity and multi-messenger observations are required to meaningfully probe gravitational constant variations through GW phase analysis.

The Effect of Gravitational Constant Variation on Gravitational Wave Phase

Introduction

This paper rigorously investigates the impact of temporal variation in the gravitational constant $G$ on the phase of gravitational waves (GWs), with a focus on constraints derived from LIGO-VIRGO-KAGRA (LVK) data. The motivation stems from both theoretical and observational perspectives: Dirac's large number hypothesis, potential explanations for dark matter, dark energy, and Hubble tension, and the prevalence of varying-$G$ scenarios in alternative gravity theories. The authors adopt a phenomenological approach, treating $G$ as a pre-fixed field rather than a dynamical one, and analyze the propagation effects on GW signals, particularly the phase corrections relevant for matched filtering in GW data analysis.

Theoretical Framework

The propagation of GWs in a spacetime with a varying gravitational constant introduces corrections to both the amplitude and phase of the waveform. The leading-order amplitude correction is $O(G'/G)$, while the phase correction is $O((G'/G)^2)$, where $G'$ denotes the derivative of $G$ with respect to the affine parameter along the GW trajectory. The corrected frequency-domain waveform for compact binary coalescences (CBCs) is given by:

$$\tilde{h}(f) \approx \sqrt{\frac{G_{\text{d}}}{G_{\text{s}}}} e^{-i\frac{\Xi D_L}{4\pi f}} \tilde{h}_0(f)$$

where $G_{\text{d}}$ and $G_{\text{s}}$ are the values of $G$ at the detector and source, respectively, $D_L$ is the luminosity distance, $\Xi = \frac{3}{4}\left(\frac{G'}{G}\right)^2$, and $\tilde{h}_0(f)$ is the standard GR waveform. The amplitude correction is degenerate with $D_L$ unless the luminosity distance is independently measured (e.g., via electromagnetic counterparts), while the phase correction is subdominant and only becomes relevant at higher-order sensitivity.

Data Analysis and Constraints

The authors utilize LVK data, specifically GWTC-3 events, to constrain $G'/G$. For binary black hole events without electromagnetic counterparts, the amplitude correction is unobservable due to degeneracy with $D_L$, and the phase correction is undetectable at current sensitivity. For events with electromagnetic counterparts (e.g., GW170817), where $D_L$ is known, both amplitude and phase corrections can, in principle, be constrained. However, the analysis demonstrates that the inclusion of phase correction negligibly improves the constraint on $G'/G$, with posterior distributions remaining consistent whether or not phase corrections are included.

The current constraint from GW data is $|\frac{G'}{G}| \lesssim 10^{-7}/\text{yr}$, which is orders of magnitude weaker than constraints from other astrophysical and laboratory measurements. The Fisher matrix analysis reveals that to achieve a constraint of $|\frac{G'}{G}| < 10^{-n}/\text{yr}$ via phase corrections, the product of signal-to-noise ratio ($\rho$) and luminosity distance ($D_L$) must increase by $2n$ orders of magnitude relative to present capabilities.

Fisher Matrix Analysis and Scaling

The Fisher information matrix formalism is employed to quantify the detectability of $G'/G$ via phase corrections. The measurement uncertainty scales as:

$$\Delta\left(\frac{G'}{G}\right) \propto \frac{1}{\rho D_L |\frac{G'}{G}|}$$

For realistic data, where noise induces a Gaussian posterior centered away from zero, the scaling becomes:

$$\Delta\left(\frac{G'}{G}\right) \propto \frac{1}{\sqrt{\rho D_L}}$$

This implies that only with significant improvements in detector sensitivity and/or detection of much more distant GW events can phase corrections provide meaningful constraints on $G'/G$. For events with known $D_L$, phase corrections are negligible unless the GW frequency is extremely low ($f_{\rm GW} < 10^{-16}$ Hz), a regime inaccessible to current detectors.

Implications and Future Directions

The results indicate that, under the assumption of a pre-fixed $G$ field, GW phase corrections due to $G$ variation are currently undetectable and do not improve constraints beyond those from amplitude corrections when $D_L$ is known. This conclusion is sensitive to the adopted phenomenology; if $G$ is treated as a dynamical field, additional effects such as dipole radiation and $-4$PN phase corrections arise, potentially leading to stronger constraints.

From a practical standpoint, the findings suggest that GW observations are presently limited in their ability to probe $G$ variation via phase effects. Future improvements in detector sensitivity, increased event rates, and multi-messenger observations could enhance the constraining power. Theoretically, the work underscores the importance of model assumptions regarding the nature of $G$ and motivates further paper of GW generation and propagation in alternative gravity theories.

Conclusion

This paper provides a comprehensive analysis of the effect of gravitational constant variation on GW phase, demonstrating that phase corrections are subdominant and currently undetectable with LVK data. The amplitude correction remains degenerate with luminosity distance unless electromagnetic counterparts are available. The results highlight the necessity of significant advances in GW detector sensitivity and event characterization to improve constraints on $G$ variation via GW phase analysis. The conclusions are contingent on the assumption of a pre-fixed $G$ field; alternative models may yield different observational signatures and constraints.