Lorentz invariance violation and simultaneous emission of electromagnetic and gravitational waves

In this work, we compute some phenomenological bounds for the electromagnetic and massive gravitational high-derivative extensions supposing that it is possible to have an astrophysical process that generates simultaneously gravitational and electromagnetic waves. We present Lorentz invariance violating (LIV) higher-order derivative models, following the Myers-Pospelov approach, to electrodynamics and massive gravitational waves. We compute the corrected equation of motion of these models, their dispersion relations and the velocities. The LIV parameters for the gravitational and electromagnetic sectors, $\xi_{g}$ and $\xi_{\gamma}$, respectively, were also obtained for three different approaches: luminal photons, time delay of flight and the difference of graviton and photon velocities. These LIV parameters depend on the mass scales where the LIV-terms become relevant, $M$ for the electromagnetic sector and $M_{1}$ for the gravitational one. We obtain, using the values for $M$ and $M_{1}$ found in the literature, that $\xi_{g}\sim10^{-2}$, which is expected to be phenomenologically relevant and $\xi_{\gamma}\sim10^{3}$, which cannot be suitable for an effective LIV theory. However, we show that $\xi_{\gamma}$ can be interesting in a phenomenological point of view if $M\gg M_{1}$. Finally the relation between the variation of the velocities of the photon and the graviton in relation to the speed of light was calculated and resulted in $\Delta v_{g}/\Delta v_{\gamma}\lesssim1.82\times 10^{-3}$.