Gravitational wave constraints on Lorentz and parity violations in gravity: High-order spatial derivative cases

Abstract

High-order spatial derivatives are of crucial importance for constructing the low energy effective action of a Lorentz or parity violating theory of quantum gravity. One example is the Ho\ifmmode \check{r}\else \v{r}\fi{}ava-Lifshitz gravity, in which one has to consider at least the sixth-order spatial derivatives in the gravitational action in order to make the theory power-counting renormalizable. In this paper, we consider the Lorentz and parity violating effects on the propagation of GWs due to the fifth and sixth-order spatial derivatives, respectively. For this purpose we calculate the corresponding Lorentz and parity violating waveforms of GWs produced by the coalescence of compact binaries. By using these modified waveforms, we perform the full Bayesian inference with the help of the open source software bilby on the selected GW events of binary black hole (BBH) and binary neutron stars (BNS) merges in the LIGO-Virgo catalogs GWTC-1 and GWTC-2. Overall we do not find any significant evidence of Lorentz and parity violation due to the fifth and sixth-order spatial derivatives and thus place lower bounds on the energy scales ${M}_{\mathrm{LV}}>2.4\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}16}\text{ }\text{ }\mathrm{GeV}$ for Lorentz violation and ${M}_{\mathrm{PV}}>1.0\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}14}\text{ }\text{ }\mathrm{GeV}$ for parity violation at 90% confidence level. Both constraints represent the first constraints on the fifth- and sixth-order spatial derivative terms respectively in the framework of spatial covariant gravity by using the observational data of GWs.