Propagation effects of Lorentz violation in gravitational waves

TL;DR

The paper investigates how isotropic Lorentz- and diffeomorphism-violating operators in the linearized SME affect gravitational-wave propagation. By deriving the modified dispersion relation and retarded Green function, it separates the effects of the nondispersive CPT-even coefficient $\mathring{k}^{(4)}_{(I)}$ (speed rescaling) from the CPT-odd dimension-five coefficient $\mathring{k}^{(5)}_{(V)}$ (helicity-dependent dispersion and birefringence), and demonstrates these modify the observed waveform without adding new propagating degrees of freedom. Focusing on a binary black hole source, the study shows the standard quadrupole radiation structure persists, but with propagation-induced corrections including higher derivatives and polarization mixing, interpreted as energy exchange with the Lorentz-violating background. Using LVK propagation and polarization constraints, along with strain-based consistency checks, the work translates current observations into bounds $|\mathring{k}^{(4)}_{(I)}| \lesssim 3 \times 10^{-15}$ and $|\mathring{k}^{(5)}_{(V)}| \lesssim (1.5-3.8) \times 10^{-18}$ s (with complementary birefringence limits), highlighting the complementary roles of phase, amplitude, and timing analyses in testing gravitational Lorentz violation.

Abstract

We investigate the propagation of gravitational waves in the presence of Lorentz- and diffeomorphism-violating operators within the linearized gravitational sector of the Standard Model Extension. Focusing on isotropic contributions, we analyze the combined effects of the nondispersive CPT-even dimension-four coefficient $\mathring{k}^{(4)}_{(I)}$ and the CPT-odd dimension-five coefficient $\mathring{k}^{(5)}_{(V)}$ on tensorial gravitational radiation. The modified dispersion relation induces both a rescaling of the propagation speed and helicity-dependent dispersive corrections, leading to birefringence and polarization mixing without introducing additional propagating degrees of freedom. We derive the retarded Green function associated with the modified wave operator and obtain explicit expressions for the gravitational waveform generated by matter sources. As a concrete application, we examine a binary black hole system and show how Lorentz violation alters the observed strain through shifted retarded times, amplitude rescaling, and higher derivative corrections to the quadrupole formula. The CPT-odd term produces characteristic attenuation and distortions in the waveform, which can be interpreted as energy exchange between the gravitational wave and the Lorentz-violating background rather than a violation of energy conservation. Using published LIGO-Virgo-KAGRA propagation tests and polarization consistency arguments, we translate current observational constraints into bounds on $\mathring{k}^{(4)}_{(I)}$ and $\mathring{k}^{(5)}_{(V)}$.

Figures (5)

• Figure 1: Illustration of a two-body black-hole system viewed in the barycentric frame. The compact constituents, labeled by masses $m_1$ and $m_2$, undergo orbital motion restricted to the $xy$ plane, with their trajectories specified by radii $r_1$ and $r_2$.
• Figure 2: Waveform $h^{+}_{xx}(t,r)$ as a function of time $t$ for a representative configuration with $\omega=0.5$, $r=20$, $\mu=1$, and $l_0=1$. The signal incorporates the combined effects of the Lorentz--violating coefficients $\mathring{k}^{(4)}_{(I)}$ and $\mathring{k}^{(5)}_{(V)}$, and displays a clear attenuation characterized by a gradual decrease of the amplitude as time evolves.
• Figure 3: Impact of the Lorentz--violating operators $\mathring{k}^{(4)}_{(I)}$ and $\mathring{k}^{(5)}_{(V)}$ on the attenuation of gravitational waves, as illustrated by the corresponding waveform behavior.
• Figure 4: Parametric plot of the helicity $(+)$ waveform in the $(h^{+}_{xx},\,h^{+}_{yx})$ plane, tracing the polarization state over one observation interval. The reference case with $\mathring{k}^{(5)}_{(V)}=0$ and $v=0.99$ yields an almost circular trajectory, characteristic of a monochromatic signal with a fixed phase relation between components. When the CPT--odd coefficient $\mathring{k}^{(5)}_{(V)}$ is included ($\mathring{k}^{(5)}_{(V)}=0.1$, $v=1$), the trajectory becomes distorted and non--closed, with a noticeable drift and loop deformation that signal time--dependent phase and amplitude modifications associated with altered wave propagation.
• Figure 5: Transverse deformation of a ring of freely falling test particles induced by a gravitational wave. Snapshots are shown at fixed retarded phase $\phi = 2\omega t_r$, with $t_r = t - r/v$, over one oscillation period. The dashed circle denotes the undeformed ring, while solid curves show the instantaneous displacement from the transverse--traceless mapping with nonzero components $h_{xx}$ and $h_{yx}$. Markers indicate selected particles along the principal axes (white: Lorentz--violating case; black: reference). Parameters are $\mu=1$, $l_0=1$, $\omega=0.5$, and $r=20$. The black curve corresponds to $\mathring{k}^{(5)}_{(V)}=0$, $v=0.99$, and the wine--colored curve to $\mathring{k}^{(5)}_{(V)}=0.1$, $v=1$, illustrating birefringent distortions over the cycle.