Testing the strong equivalence principle with multimessenger binary neutron star mergers

By developing a gravitational-wave waveform model that incorporates a slowly varying gravitational constant and applying it to a joint multi-messenger analysis of the binary neutron star merger GW170817, this study finds no evidence for temporal variation in $G$ and establishes the most stringent constraints to date on its fractional time derivative, thereby validating the strong equivalence principle in the relativistic regime.

The Gist

The Big Question: Is Gravity's "Volume Knob" Broken?

Imagine the universe is a giant radio station. For nearly a century, physicists have believed that the "volume knob" for gravity—known as Newton's Gravitational Constant ($G$)—is perfectly fixed. It never changes, no matter where you are or when you look. This idea is a cornerstone of Einstein's theory of General Relativity, which is like the rulebook for how the universe works.

However, some scientists wonder: What if the volume knob is actually slowly turning up or down over time? If $G$ changes, it would mean Einstein's rulebook has a typo, and a fundamental law of physics (called the Strong Equivalence Principle) is broken.

The Detective Work: A Cosmic Crash with a Flashlight

To test this, the authors of this paper acted like cosmic detectives. They looked at a specific event: GW170817.

Think of this event as a massive, violent crash between two neutron stars (the densest objects in the universe, like cosmic bowling balls made of pure matter).

1. The Sound: When they crashed, they sent out ripples in space-time called Gravitational Waves. We "heard" this crash with our detectors (LIGO and Virgo).
2. The Light: Just 1.7 seconds later, the crash flashed a burst of gamma rays (a "flashlight") that we saw with telescopes.

This was a Multimessenger event. Usually, we only get the "sound" (gravitational waves). But because we also got the "light" (electromagnetic signals), we knew exactly where the crash happened, how far away it was, and how it was tilted relative to us.

The Investigation: Listening for a "Drift"

The scientists built a super-advanced computer model to simulate what the sound of this crash should have sounded like if gravity was constant (Einstein's theory). Then, they built a second model that allowed gravity to slowly change over time.

They compared these models against the actual data from the crash.

The Analogy of the Echo:
Imagine you are shouting in a canyon.

• Scenario A (Constant Gravity): The echo returns exactly as expected based on the distance and the shape of the canyon.
• Scenario B (Changing Gravity): If the air in the canyon suddenly got thicker or thinner while your shout was traveling, the echo would sound slightly different. It might be louder, quieter, or arrive at a slightly different pitch.

The authors checked if the "echo" (the gravitational wave signal) from the neutron star crash showed any signs that the "air" (gravity) had changed while the signal was traveling across the universe.

The Results: The Volume Knob is Stuck

After running complex statistical analyses (basically asking the data millions of questions), the results were clear:

1. No Drift Found: The data matched Einstein's theory perfectly. There was no evidence that the gravitational constant was changing.
2. The Tightest Constraint Yet: They calculated that if gravity is changing, it is doing so so slowly that it's almost impossible to detect. They narrowed the possible change down to a tiny range: between -3.36 and +0.53 parts per billion per year.
   • Think of it this way: If you had a bank account with a billion dollars, and gravity was changing, the amount of money added or subtracted from your account every year would be less than a single penny.

Why This Matters

This is a huge deal for a few reasons:

• Strong-Field Test: Previous tests of gravity were done in "weak" environments, like our Solar System (where gravity is relatively gentle). This test happened in the "strong-field" regime, where gravity is incredibly intense, like in the heart of a neutron star. It's like testing a car's brakes on a gentle hill versus dropping it off a cliff; this paper tested the brakes on the cliff.
• The Power of Teamwork: The study highlights the power of Multimessenger Astronomy. Without the "flashlight" (the gamma-ray burst) to tell us exactly how far away the crash was and how it was tilted, the "sound" (gravitational waves) would have been too ambiguous to rule out a changing gravity. The light and the sound worked together to break the "degeneracy" (the confusion) that usually plagues these measurements.

The Bottom Line

The authors have used the most violent crash in the nearby universe to check if the fundamental rules of gravity are changing. The verdict? Gravity is stable. The "volume knob" is stuck, and Einstein's rulebook passes another rigorous test with flying colors. This gives us even more confidence that our understanding of the universe is on the right track.

Technical Summary

1. Problem Statement

The Strong Equivalence Principle (SEP) is a foundational pillar of General Relativity (GR), asserting that the laws of physics are independent of location and velocity in a freely falling frame. A direct consequence of the SEP is the constancy of Newton's gravitational constant, $G$. While the SEP has been rigorously tested in weak-field regimes (e.g., lunar laser ranging, pulsar timing), testing it in the strong-field, highly dynamical regime remains a significant open challenge.

The paper addresses the possibility of a temporal variation of the gravitational constant ($\dot{G} \neq 0$). Such a variation would violate the SEP and manifest in two distinct ways during a binary neutron star (BNS) merger:

1. Source Effects: A time-dependent $G$ modifies the internal structure (gravitational self-energy) and orbital dynamics of the compact binaries, altering the gravitational wave (GW) phase evolution.
2. Propagation Effects: A varying $G$ across cosmological distances modifies the amplitude of GWs as they travel through an expanding Friedmann–Lemaître–Robertson–Walker (FLRW) universe.

Previous studies often treated these effects separately, relied on Fisher matrix approximations, or lacked robust constraints from real multimessenger data. This work aims to perform a rigorous, joint Bayesian analysis using real data from the BNS merger GW170817 and its electromagnetic counterpart GRB 170817A to constrain $\dot{G}/G$.

2. Methodology

A. Theoretical Framework

The authors develop a physically consistent GW waveform model incorporating a slowly varying $G$ within the Parametrized Post-Einsteinian (ppE) framework:

• Source Dynamics: They derive the orbital evolution of a binary system where the total mass $M$ and angular momentum $j$ evolve due to $\dot{G}$. This introduces corrections to the GW phase and amplitude characterized by ppE parameters $\alpha$ (amplitude) and $\beta$ (phase).
  • The phase correction scales as $u^{-13}$ (where $u = (\pi M_c f)^{1/3}$).
  • The amplitude correction scales as $u^{-8}$.
• Propagation Effects: They demonstrate that in a FLRW universe with varying $G$, the GW amplitude acquires a correction factor of $\sqrt{G_0/G_*}$, where $G_*$ is the gravitational constant at the source and $G_0$ at the detector.
• Sensitivities ($s$): The magnitude of the source effects depends on the "sensitivity" of the compact objects ($s = -\partial \ln m / \partial \ln G$).
  • For Neutron Stars (NS): Sensitivities are computed numerically by solving the Tolman–Oppenheimer–Volkoff (TOV) equations for various Equations of State (EOS). The authors show that the empirical linear approximation often used in literature is valid only at low masses; accurate sensitivities require EOS-dependent numerical solutions.
  • For Black Holes: They provide a model-independent proof that the sensitivity is universally $s=1/2$ for spherically symmetric black holes in varying-$G$ theories.

B. Waveform Construction

The authors construct a custom GW waveform template ($\tilde{h}_{VG}$) that combines:

1. Standard GR waveforms (generated via PyCBC using models TaylorF2 and IMRPhenomXAS_NRTidalv2).
2. A phase correction term $\exp(i \beta_{VG} u^{b_{VG}})$ derived from the varying-$G$ dynamics.
3. An amplitude correction factor $1/\sqrt{\kappa}$, where $\kappa \equiv G_*/G_0$.
4. Physical Constraints: The model enforces that the "EOS mass" (a rescaled mass dependent on $\kappa$) must lie within the physically allowed range for neutron stars (0.08 to 2.01 $M_\odot$) based on the H4 EOS.

C. Bayesian Analysis

• Data: GW strain data from LIGO Hanford and Livingston (GW170817) with a frequency cutoff of 20 Hz.
• Multimessenger Constraints: To break parameter degeneracies inherent in GW-only analyses (specifically between $\kappa$, luminosity distance $D_L$, and inclination $\iota$), the authors incorporate independent electromagnetic constraints:
  • Luminosity Distance ($D_L$): Fixed to 40.7 Mpc (from Hubble Space Telescope surface brightness fluctuations).
  • Sky Location: Fixed based on Dark Energy Camera observations.
  • Inclination ($\iota$): Constrained to the 90% credible interval $[2.70, 2.81]$ rad using Very Long Baseline Interferometry (VLBI) of the afterglow.
• Inference: A joint Bayesian analysis is performed to infer the posterior distributions of standard GR parameters and the non-GR parameter $\kappa$. The fractional time derivative is derived as $\dot{G}/G = (1-\kappa)/T$, where $T$ is the lookback time ($1.31 \times 10^8$ years).

3. Key Contributions

1. Unified Framework: Developed a comprehensive GW waveform model that simultaneously accounts for source-generation effects (orbital dynamics, sensitivities) and propagation effects (amplitude damping) of a varying gravitational constant.
2. EOS-Dependent Sensitivities: Moved beyond empirical approximations by numerically calculating neutron star sensitivities for multiple realistic Equations of State (AkmalPR, AP4, SLy4, etc.), ensuring the physical consistency of the mass bounds used in the analysis.
3. Multimessenger Degeneracy Breaking: Demonstrated the critical power of combining GW data with independent electromagnetic constraints (specifically inclination and distance) to disentangle varying-$G$ effects from standard astrophysical parameters.
4. Rigorous Bayesian Inference: Performed a full Bayesian parameter estimation using real data (GW170817) rather than relying on Fisher matrix projections or Monte Carlo simulations, utilizing two distinct waveform models to ensure robustness.

4. Results

• No Evidence for Variation: The analysis found no statistically significant evidence for a temporal variation of the gravitational constant. The posterior distributions for the varying-$G$ parameter $\kappa$ are consistent with the GR prediction ($\kappa = 1$).
• Parameter Constraints:
  • Using the TaylorF2 model: $\kappa = 1.17^{+0.27}_{-0.19}$.
  • Using the IMRPhenomXAS_NRTidalv2 model: $\kappa = 1.11^{+0.25}_{-0.18}$.
• Final Bound on $\dot{G}/G$:
  By converting the constraints on $\kappa$ using the lookback time, the authors established a conservative 3$\sigma$ bound:
  $$\frac{\dot{G}}{G} \in [-3.36 \times 10^{-9}, 5.34 \times 10^{-10}] \, \text{yr}^{-1}$$
  This represents the most stringent constraint to date derived from real gravitational wave observations.

5. Significance

• Precision Test of SEP: This work provides the tightest strong-field test of the Strong Equivalence Principle using compact binary mergers, significantly improving upon previous GW-based limits (which were often orders of magnitude weaker or based on projected data).
• Validation of GR: The results reinforce the validity of General Relativity in the strong-field, dynamical regime, confirming that $G$ is constant to within $\sim 10^{-9}$ per year over cosmological timescales.
• Multimessenger Synergy: The study highlights that the full potential of multimessenger astronomy (combining GWs with EM counterparts) is essential for precision fundamental physics, as it allows for the breaking of degeneracies that would otherwise render such tests impossible with GW data alone.
• Methodological Benchmark: The developed framework and numerical sensitivity calculations serve as a robust template for future tests of fundamental physics with upcoming GW detectors (e.g., Einstein Telescope, Cosmic Explorer) and more precise multimessenger events.