Constraining the gravitational-wave breathing mode with LVK antenna patterns and GWTC data

Technical Summary: Constraining the Gravitational-Wave Breathing Mode with LVK Antenna Patterns and GWTC Data

Problem Statement
Metric theories of gravity generally admit six polarization modes for gravitational waves (GWs): two tensor modes ($+$, $\times$), two vector modes ($x$, $y$), and two scalar modes (breathing $b$ and longitudinal $l$). While General Relativity (GR) predicts only the tensor modes, scalar-tensor theories (e.g., Brans–Dicke) and Gravitational Quantum Field Theory (GQFT) predict the existence of a massless breathing scalar mode. In standard scalar-tensor models, solar-system tests constrain the scalar coupling such that the breathing amplitude is negligible compared to the tensor sector. However, GQFT allows for a breathing amplitude ($h_b$) comparable to the tensor amplitude ($h_+$), parameterized phenomenologically as $h_b = \xi h_+$, where $|\xi| \sim \mathcal{O}(1)$.

Current GW tests using the LIGO–Virgo–KAGRA (LVK) network have largely focused on pure polarization hypotheses or null-stream searches for non-tensor content. However, these analyses often leave the mixed amplitude $\xi$ (where the breathing mode is tied to the tensor $+$ mode) unconstrained. Furthermore, the geometric response of terrestrial L-shaped interferometers to the isotropic breathing mode is highly dependent on sky location; specifically, the differential readout of a Michelson interferometer nearly cancels the breathing signal when the wave vector is orthogonal to the detector plane (near zenith), a suppression not always fully accounted for in catalog-wide constraints.

Methodology
The authors employ a semi-joint analysis framework using data from the Gravitational-wave Transient Catalog (GWTC) to constrain the parameter $\xi$. The methodology proceeds through several key steps:

1. Antenna Pattern Evaluation: The authors calculate the LVK network antenna patterns for the breathing mode ($F_b$) using the PyCBC toolkit. They explicitly model the differential arm-length response of L-shaped interferometers, which suppresses $F_b$ near the zenith while maximizing tensor sensitivity ($S_T$).
2. Geometric Ranking: They define the network breathing power as $S_b = \sum_d F_{b,d}^2$. Using 72 GWTC events (network SNR $\ge 8$, $\ge 2$ sites) with packaged strain data, they rank events by a geometric proxy $\text{SNR} \sqrt{S_b^*}$, where $S_b^*$ is the breathing power at the optimal inclined sky position. This identifies events where the network geometry is most favorable for detecting a breathing mode.
3. Parameter-Estimation (PE) Weighting: To account for localization uncertainty, the authors re-evaluate $S_b$ using posterior samples of sky location ($\alpha, \delta$) and polarization angle ($\psi$) from official LVK PE releases for 12 high-SNR events. This reveals that "geometrically attractive" sources (like GW170817) may actually lie near a breathing null in their true PE sky, reducing their sensitivity to $\xi$.
4. Bayesian Model Comparison: The study performs nested sampling using bilby to compare two hypotheses:
   • $H_{GR}$: Pure tensor modes ($\xi \equiv 0$).
   • $H_{GR+b}$: Tensor modes plus a breathing mode ($h_b = \xi h_+$, with $\xi$ free).
     The analysis is conducted in two modes:
   • Injection Grid: Controlled injections of $h_b = \xi h_+$ into Gaussian noise with design PSDs to calibrate the Bayes factor ($\ln B$) recovery under fixed masses, spins, and distance, varying only sky and $\xi$.
   • Semi-Joint Real Events: Analysis of 11 real GWTC events (GW170817 excluded due to its PE sky being on a breathing null). Masses and spins are fixed at preferred PE medians, while luminosity distance ($d_L$), inclination ($\theta_{JN}$), sky location, and $\xi$ are sampled within PE-informed priors.

Key Results

• Geometric Suppression: At the zenith of the H1 detector during the GW150914 epoch, the breathing antenna pattern is suppressed to $F_b \sim 5 \times 10^{-6}$, while tensor modes remain strong. For the 72-event sample, the median gain in breathing power ($S_b$) when moving from the local zenith to the optimal inclined sky is approximately $34\times$.
• Injection Recovery: On an inclined sky ($S_b \sim 0.18$), the analysis successfully recovers injected values of $\xi$. For $\xi=1$, the log Bayes factor is $\ln B \approx 13.2$, strongly favoring $H_{GR+b}$. For $\xi=0.5$, $\ln B \approx 3.8$. However, near the zenith ($S_b \sim 0.016$), even an injected $\xi=1$ yields an inconclusive $\ln B \sim \mathcal{O}(0)$, and the posterior on $\xi$ remains unconstrained. This demonstrates that geometric sensitivity ($S_b$), rather than SNR alone, dictates the reach in $\xi$.
• Real Event Constraints: None of the 11 analyzed real events favor the $H_{GR+b}$ hypothesis over GR.
  • The tightest constraints are found for GW150914 ($|\xi|_{90\%} \approx 0.15$), GW200129 065458 ($|\xi|_{90\%} \approx 0.18$), and GW200311 115853 ($|\xi|_{90\%} \approx 0.23$).
  • These bounds disfavor a phenomenological breathing amplitude of $|\xi| \sim \mathcal{O}(1)$ for these specific events.
  • Weaker bounds (e.g., GW190814 with $|\xi|_{90\%} \approx 0.61$) still allow for $|\xi| \sim \mathcal{O}(1)$.
• Degeneracy: The study identifies an approximate product degeneracy between the breathing amplitude and the relative projection strength ($I = \sqrt{S_b}/\sqrt{S_{b,med}}$), where $\xi I \sim \text{const}$. The semi-joint analysis marginalizes over sky-driven variations in $I$, ensuring the reported limits on $\xi$ are robust against localization uncertainty.

Significance and Claims
The paper claims to provide the first ground-based constraints on the phenomenological parameter $\xi$ (where $h_b = \xi h_+$) specifically accounting for the Michelson suppression of the breathing mode in terrestrial detectors. By integrating PyCBC antenna patterns with GWTC data and PE-informed sky distributions, the authors demonstrate that:

1. Geometric factors are critical: The ability to constrain $\xi$ is heavily dependent on the source's sky location relative to the detector network; sources near the zenith are effectively blind to the breathing mode regardless of SNR.
2. Current data favors GR: For the three events with the tightest constraints, the data is consistent with GR and disfavors a breathing mode amplitude comparable to the tensor sector ($|\xi| \sim \mathcal{O}(1)$), a scale motivated by GQFT.
3. Methodological advancement: The "semi-joint" approach (fixing masses/spins while sampling sky and $\xi$) effectively averages over projection uncertainties, providing a more realistic assessment of sensitivity than catalog rankings based solely on geometric proxies.

The authors note that while these results constrain the specific model $h_b = \xi h_+$, they do not rule out all scalar-tensor theories, particularly those where the breathing amplitude is naturally suppressed by other mechanisms. Future improvements will require freeing masses and spins, expanding the open-data sample, and addressing waveform systematics.