Improved Constraints on Scalar Breathing Modes from Full GWTC-4 Catalog (O1-O4a): h_br/h_+ < 5.6 × 10⁻⁴

Version 2.0:  Critical Update: Full Catalog (O1-O4a) +  GPS Timing Correction + Adding Validation with Injection Recovery Tests + Tensor Component Verification.

Please download (pdf + code + data + methods), Published with complete reproducibility

Important Data Quality Issue Addressed

Version 1.0 Limitation: The initial release contained GPS timing inaccuracies in my custom-compiled event catalog that affected precise event selection and sky position mapping. These timing errors were due to my incorrect assembly of the event catalog from various sources, not issues with the official LIGO data. While the overall methodology and statistical approach were sound, these catalog timing errors introduced systematic uncertainties in the analysis.

Version 2.0 Correction: This release addresses the GPS timing issue through:

• Official LIGO Catalog Adoption: Now uses the verified LIGO Scientific Collaboration GWTC-4 catalog directly, eliminating my custom catalog compilation errors

• Proper Event Timing: Each event now uses the official GPS timestamps from the peer-reviewed LIGO catalog release

• Tensor Component Verification: Additional tests confirm proper tensor signal detection in each correctly-timed GPS window, ensuring events are accurately identified and processed

• Methodology Validation with Injection Recovery Tests. Added comprehensive injection recovery testing (Section 10) to validate the null-stream analysis pipeline

 --

Scientific Impact: The GPS timing corrections enable analysis of the complete GWTC-4 catalog (180 events vs. 104 in V1.0) with proper event timing. The improved constraint h_br/h_+ < 5.6 × 10⁻⁴ represents both the correction of my catalog compilation errors and the statistical benefit of the expanded, correctly-timed event sample.

 

Validation: The manuscript includes comprehensive validation tests demonstrating that tensor polarization components are correctly detected in each event's properly-timed GPS window, confirming the reliability of the null-stream scalar polarization analysis with the official LIGO event catalog.

Methodology Validation with Injection Tests

Added comprehensive injection recovery testing (Section 10) to validate the null-stream analysis pipeline:

• Synthetic signals with known tensor (A₊ = 1.0) and scalar breathing (Aᵦ = 0.01) components injected into simulated detector data with Gaussian noise (σₙ = 0.1)
• Null-stream optimization successfully cancels tensor mode, reducing residual energy from σ ~ 0.5 to σ = 0.205
• 24% of injected scalar energy survives null-stream construction (expected from antenna pattern geometry: Fᵦᴴ ≠ k·Fᵦᴸ)
• Recovered scalar signal (0.0024) buried beneath noise floor (0.167), yielding SNR = 0.014
• Validates that constraint is noise-limited, not methodology-limited

New content includes: 4 validation figures (Figures 6-9) showing optimization landscape, time series evolution, signal amplitude comparison, and conceptual diagram of noise-limited sensitivity.

The injection test confirms that further improvements require advanced noise reduction techniques (e.g., AI-based denoising) to push sensitivity into the 10⁻⁵–10⁻⁶ regime.

Recommendation

Users should use Version 2.0 for all scientific applications. Version 1.0 is retained for historical reference and methodological comparison, but Version 2.0 provides the definitive analysis with the corrected official LIGO event catalog and expanded coverage.

Scientific Context

General Relativity predicts gravitational waves have only two tensor polarizations (plus and cross modes). However, many alternative theories of gravity predict additional polarization modes, including scalar "breathing" modes that cause isotropic expansion and contraction of space. Detecting or constraining these additional polarizations provides crucial tests of fundamental physics beyond Einstein's theory.

 

This work establishes the most stringent independent constraint on scalar breathing mode polarization to date: h_br/h_+ < 5.6 × 10⁻⁴ (95% C.L.) from the complete GWTC-4 catalog analysis — representing a ~121× improvement over the previous best published constraint of A_S/A_T ≲ 0.068 from the single event GW170817 (Takeda et al. 2021).

Key Results

Complete GWTC Analysis (180 events):

• Final constraint: h_br/h_+ < 5.6 × 10⁻⁴ (95% C.L.)

• 14.3× improvement from inverse-variance weighting

• Top contributor: GW170817 (5.2% of total constraining power)
  
  We perform polarization angle optimization over 181 values in [0, π), following standard LIGO methodology. This optimization yields k values that minimize the null stream RMS, providing optimal tensor mode cancellation. For comparison, analysis with fixed ψ=0 gives h_br/h_+ < 1.2×10⁻³, while the optimized approach yields h_br/h_+ < 5.6×10⁻⁴, demonstrating a 2.1× improvement. This confirms that polarization angle optimization is essential for achieving the best sensitivity to scalar modes

 

O4a-Only Analysis (113 events):

• Final constraint: h_br/h_+ < 8.2 × 10⁻⁴ (95% C.L.)

• 13.5× improvement from inverse-variance weighting

• Top contributor: GW231226_101520 (8.9% of total constraining power)

Methodology

We analyze gravitational wave events from GWTC-4 using a model-independent null-stream approach with comprehensive quality controls and three key innovations:

 

1. Polarization angle optimization: Grid search over 181 angles per event to minimize residual tensor signal, achieving optimal detector geometry factors |k| typically < 0.05

 

2. Inverse-variance weighted stacking: Optimal statistical combination following Allen & Romano (1999): w_i ∝ ρ²_i/σ²_off,i, maximizing constraining power from high-SNR events

 

3. Dual analysis capability: Complete pipeline supporting both O4a-only (113 events) and full GWTC catalog (180 events) analysis modes

 

4. Verified GPS timing: Version 2.0 addresses timing precision issues identified in initial analysis, ensuring accurate event selection and sky position mapping using official LIGO catalog data

Technical Improvements in Version 2.0

This release represents a significant methodological advancement over the initial implementation:

 

Enhanced Data Quality: Refined event selection and GPS time handling using verified LIGO Scientific Collaboration catalogs ensures precise timing alignment and sky position accuracy.

 

Expanded Analysis Scope: Added capability to analyze the complete GWTC-4 catalog (180 events from O1-O4a) alongside the O4a-only subset (113 events), providing both focused recent-data analysis and comprehensive multi-observing-run constraints.

 

Improved Statistical Power: The complete catalog analysis achieves h_br/h_+ < 5.6 × 10⁻⁴, approaching the 10⁻⁴ sensitivity regime where many alternative theories of gravity make testable predictions.

 

Rigorous Validation: Comprehensive bootstrap uncertainty analysis and cross-checks between analysis modes confirm robust constraint methodology and statistical significance.

Archive Contents

This package contains everything needed for complete reproducibility of both analysis modes:

 

Analysis Pipeline:

 

• gwtc4_scalar_analysis_final_find_angle.py - Main analysis script with polarization optimization

• weighted_stacking_methodology.py - Statistical combination and constraint calculation

• gwtc4_scalar_analysis_final_fixed_zero_angle.py - Reference implementation (ψ=0°)

 

Input Data:

 

• data/all/gwtc4_event_list_all.csv - Complete GWTC-4 catalog (219 events, O1-O4a)

• data/o4/gwtc4_event_list_o4.csv - O4a subset (128 events, May 2023 - Jan 2024)

• data/*/event_catalog_ra_dec.json - Sky positions and event metadata

 

Verified Results:

 

• results/all/ - Complete GWTC analysis results and constraints

• results/o4/ - O4a-only analysis results and constraints

• logs/ - Complete execution logs from verified production runs

 

Documentation:

 

• README.md - Comprehensive setup and execution instructions

• Analysis methodology, statistical approach, and interpretation guidelines

Data Processing and Validation

Input Data Sources: Official LIGO Scientific Collaboration GWTC-4 catalog with verified GPS times, sky positions, and network SNR values from peer-reviewed release.

Methodology Validation with Injection Tests

Added comprehensive injection recovery testing (Section 10) to validate the null-stream analysis pipeline:

• Synthetic signals with known tensor (A₊ = 1.0) and scalar breathing (Aᵦ = 0.01) components injected into simulated detector data with Gaussian noise (σₙ = 0.1)
• Null-stream optimization successfully cancels tensor mode, reducing residual energy from σ ~ 0.5 to σ = 0.205
• 24% of injected scalar energy survives null-stream construction (expected from antenna pattern geometry: Fᵦᴴ ≠ k·Fᵦᴸ)
• Recovered scalar signal (0.0024) buried beneath noise floor (0.167), yielding SNR = 0.014
• Validates that constraint is noise-limited, not methodology-limited

 

Quality Controls: Systematic cuts on detector geometry (|k| < 2.0), noise stationarity (σ_on/σ_off < 1.2), and antenna response patterns ensure analysis robustness.

 

Statistical Validation: Bootstrap uncertainty analysis with 10,000 resamples provides rigorous confidence intervals. Multiple cross-checks between weighted/unweighted methods and O4a/complete catalog results demonstrate consistency.

 

Reproducibility: All analysis code provided with detailed instructions. Pipeline automatically downloads and caches LIGO strain data from public GWOSC archive. Users can independently reproduce both O4a and complete catalog constraints.

Physical Interpretation

These constraints approach theoretical sensitivity thresholds where scalar-tensor theories diverge significantly from General Relativity predictions. The complete GWTC analysis provides the most comprehensive test of gravitational wave polarization content to date, leveraging the full statistical power of the current gravitational wave catalog.

 

The results demonstrate that current gravitational wave observations are consistent with Einstein's prediction of purely tensor polarizations, with no evidence for scalar breathing modes at the level of sensitivity achieved by this analysis.

Software and Data Availability

Analysis Code: Complete Python pipeline with all dependencies specified in requirements.txt

 

Computational Requirements: 150 GB disk space, 8 GB RAM, stable network connection for initial LIGO data download

 

Strain Data: Automatically downloaded from public LIGO Open Science Center (GWOSC) archive

 

Platform Compatibility: Tested on Linux/macOS systems with Python 3.8+

Citation

Primary Reference:

 

Barodkin, A. (2025). "Constraints on Scalar Gravitational Wave Polarizations 

from GWTC-4: Complete Catalog Analysis with Optimized Null-Stream Methodology." 

[In preparation]

 

Data Sources:

 

LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration. 

"GWTC-4.0: Updating the Gravitational-Wave Transient Catalog with Observations 

from the First Part of the Fourth LIGO-Virgo-KAGRA Observing Run." 

arXiv:2508.18082 (2025).

 

LIGO Open Science Center. "Gravitational Wave Strain Data Release." 

https://gwosc.org/

Author Information

Aliaksei Barodkin
Independent Researcher in Gravitational Wave Physics
Warsaw, Poland
Email: aborodkin@gmail.com

Disclaimer

The author conducts this research independently, without affiliation to or support from any employer or institution. The views, methodology, and results presented here are solely those of the author. This work utilizes publicly available LIGO data and follows established gravitational wave analysis techniques as documented in the scientific literature.

License and Usage

Code: MIT License - freely available for scientific use and modification
Documentation: Creative Commons CC BY 4.0
Data: Original LIGO data subject to LIGO Open Science Center usage policies

 

 

 

 

Archive Version: 2.0
Release Date: November 2025
Analysis Period: GWTC-4 commulative catalog (O1-O4a observing runs)

Version History

Version 2.0 (November 2025):

• Fixed incorrect getting gps time by using correct catalog of GWTC events gps times.

• Added complete GWTC-4 analysis capability (180 events(which passed quality bar from 228 events)

• Improved constraint: h_br/h_+ < 5.6 × 10⁻⁴ (95% C.L.)

• Comprehensive dual-mode analysis pipeline

• Extensive validation and cross-check procedures

 

Version 1.0 (November 2025):

 

• Initial O4a-only analysis implementation

• Polarization angle optimization methodology

• Inverse-variance weighted stacking approach

• Proof-of-concept constraint: h_br/h_+ < 8.5 × 10⁻⁴ (95% C.L.)