Propagation effects of Lorentz violation in gravitational waves

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 CPT-even dimension-four coefficient $$\mathring{k}^{(4)}_{(I)}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mover> <mml:mi>k</mml:mi> <mml:mo>˚</mml:mo> </mml:mover> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>I</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>4</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:msubsup> </mml:math> and the CPT-odd dimension-five coefficient $$\mathring{k}^{(5)}_{(V)}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mover> <mml:mi>k</mml:mi> <mml:mo>˚</mml:mo> </mml:mover> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>V</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>5</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:msubsup> </mml:math> on tensorial gravitational radiation. The modified dispersion relation induces both a rescaling of the propagation speed and helicity-dependent 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 an 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. Using GW170817/GRB 170817A, published GWTC-3 propagation tests, and conservative polarization consistency arguments, we translate existing observational constraints into bounds on $$\mathring{k}^{(4)}_{(I)}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mover> <mml:mi>k</mml:mi> <mml:mo>˚</mml:mo> </mml:mover> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>I</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>4</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:msubsup> </mml:math> and $$\mathring{k}^{(5)}_{(V)}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mover> <mml:mi>k</mml:mi> <mml:mo>˚</mml:mo> </mml:mover> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>V</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>5</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:msubsup> </mml:math> .