# Analysis of Birefringence and Dispersion Effects from Spacetime-Symmetry Breaking in Gravitational Waves

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## Abstract

**:**

## 1. Introduction

`LALSuite`[52], and we describe the statistical method used to infer the posterior probability of the coefficients for symmetry-breaking. In order to link the theoretical derivation to the analysis of astrophysical signals, we provide detailed explanations of the steps necessary to measure the coefficients for CPT and Lorentz violation, alongside simulations of the modified signals and studies of the sensitivity of current GW interferometers for parameter inference.

## 2. Theoretical Framework

#### 2.1. Background and General Relativity

#### 2.2. Spacetime-Symmetry Breaking Scenario

#### 2.2.1. Scalar Field Example

#### 2.2.2. Gravity Sector Case

#### 2.2.3. Gravitational Wave Signals

#### 2.3. Unit Changes and Dimension

## 3. Analysis Method

`LALSuite`modified for our purposes as described below [52].

#### 3.1. Implementation of the Modified Waveform

`LALSuite`. Such deformations can be anisotropic, as can be inferred from the appearance of $\widehat{n}$ in Equation (42) via $\beta $ and $\delta $. Here we focus on the simplest coefficients that produce dispersion and birefringence via Lorentz and CPT violating effects, i.e., those of mass dimension 5. These coefficients are contained within $\beta $ in (42) and obey the complex conjugate relation ${k}_{jm}^{\left(d\right)*}={(-1)}^{m}{k}_{j(-m)}^{\left(d\right)}$, for $j=0,1,2,3$, $-j\le m\le j$. There are a priori independent coefficients in this set of terms [26]. We display the first terms within $\beta $ in SI units:

`LALSuite`software.

#### 3.2. Bayesian Analysis

`LALInference`, the parameter estimation package of

`LALSuite`[88].

`LALInference`performs Bayesian inference of the posterior probability of the GW source parameters with the inclusion of the systematic uncertainties due to the detectors resolutions. The vector set of GR prior parameters, ${\overrightarrow{\theta}}_{GR}$, includes intrinsic parameters describing the binary system (e.g., the black holes masses and spins) as well as extrinsic parameters placing it in the astrophysical environment (e.g., the sky location, distance, and inclination). We add to the preexisting parameters the SME coefficients ${\left({k}_{\left(V\right)jm}^{\left(5\right)}\right)}_{eff}$ described in Section 3.1 for the mass dimension 5 case, contained within ${\overrightarrow{\theta}}_{SME}$.

## 4. Sensitivity Study

## 5. Conclusions and Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Notes

1 | |

2 | Alternatively one can choose $\mathrm{Js}$ to match classical mechanics. |

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**Figure 1.**The above waveforms with varying ${k}_{\left(V\right)00}^{\left(5\right)}$ values are for a simulated coalescence of a non-spinning binary system of black holes with ${m}_{1}={m}_{2}=50{M}_{\odot}$ located at a luminosity distance of 4 Gpc. GR denotes the case where ${k}_{\left(V\right)00}^{\left(5\right)}=0$, and Lorentz violation is the case where ${k}_{\left(V\right)00}^{\left(5\right)}$ has the value specified above the plot.

**Figure 2.**Posterior probability density on the ${k}_{\left(V\right)00}^{\left(5\right)}$ coefficient for a simulated coalescence of a non-spinning binary system of black holes with ${m}_{1}={m}_{2}=50{M}_{\odot}$ located at a luminosity distance of 5 Gpc. The left figure shows the 1$\sigma $ and 90% credible intervals in the ${D}_{L}-{k}_{\left(V\right)00}^{\left(d\right)}$ plane, and the right figure shows the posterior probability of ${k}_{\left(V\right)00}^{\left(5\right)}$ marginalizing the source and systematical uncertainty parameters.

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**MDPI and ACS Style**

O’Neal-Ault, K.; Bailey, Q.G.; Dumerchat, T.; Haegel, L.; Tasson, J.
Analysis of Birefringence and Dispersion Effects from Spacetime-Symmetry Breaking in Gravitational Waves. *Universe* **2021**, *7*, 380.
https://doi.org/10.3390/universe7100380

**AMA Style**

O’Neal-Ault K, Bailey QG, Dumerchat T, Haegel L, Tasson J.
Analysis of Birefringence and Dispersion Effects from Spacetime-Symmetry Breaking in Gravitational Waves. *Universe*. 2021; 7(10):380.
https://doi.org/10.3390/universe7100380

**Chicago/Turabian Style**

O’Neal-Ault, Kellie, Quentin G. Bailey, Tyann Dumerchat, Leïla Haegel, and Jay Tasson.
2021. "Analysis of Birefringence and Dispersion Effects from Spacetime-Symmetry Breaking in Gravitational Waves" *Universe* 7, no. 10: 380.
https://doi.org/10.3390/universe7100380