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

Cheng Gong a,b,c Tao Zhub,c corresponding author<EMAIL_ADDRESS>Rui
Niud,e Qiang Wub,c Jing-Lei Cuia Xin Zhanga<EMAIL_ADDRESS>Wen
Zhaod,e<EMAIL_ADDRESS>Anzhong Wangf<EMAIL_ADDRESS>a
Department of Physics, College of Sciences, Northeastern University, Shenyang
110819, China
b Institute for Theoretical Physics and Cosmology, Zhejiang University of
Technology, Hangzhou, 310032, China
c United Center for Gravitational Wave Physics (UCGWP), Zhejiang University of
Technology, Hangzhou, 310032, China
d CAS Key Laboratory for Research in Galaxies and Cosmology, Department of
Astronomy, University of Science and Technology of China, Hefei 230026, China;
e School of Astronomy and Space Sciences, University of Science and Technology
of China, Hefei, 230026, China;
f GCAP-CASPER, Physics Department, Baylor University, Waco, Texas 76798-7316,
USA

###### 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 Hořava-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 $M_{\rm LV}>2.4\times
10^{-16}\;{\rm GeV}$ for Lorentz violation and $M_{\rm PV}>1.0\times
10^{-14}\;{\rm GeV}$ 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.

##  I Introduction

The detection of gravitational waves (GW) from individual compact binary
sources(CBC) by the LIGO/Virgo collaboration over the past few years has
started a new era in gravitational physics gw150914 ; gw-other ; gw170817 ;
LIGOScientific:2017ycc ; gwtc1 ; gwtc2 ; LIGOScientific:2021djp . These
signals are consistent with GWs produced by the coalescence of compact
binaries, as predicted by Einstein’s general relativity (GR). These GWs also
provide strong evidences in the strong field and highly dynamical regime in
which GR’s predictions are well tested. Up to now, GW have become an important
tool in testing GR, and it is a very important topic in the era of GW
astronomy gw150914-testGR ; gw170817-testGR ; gw170817-speed ; testGR_GWTC1 ;
testGR_GWTC2 .

Although GR is the most successful gravity theory at present, there are
difficulties in explaining some problems, including(e.g., singularity,
quantization, etc), and observationally (e.g., dark matter, dark energy, etc).
Therefore, in order to solve the above problems, various modified theories of
gravity have been proposed MG1 ; MG2 ; MG3 ; MG4 . Therefore, the tests of the
modified gravities are essential to confirm the final theory of gravity.

On the other hand, the Lorentz invariance is one of the fundamental principle
of GR. However, when one considers the quantization of the gravity, such
invariance could be violated at high energy regime. In this sense, the Lorentz
symmetry can be treated as a approximate symmetry, which emerges at low
energies and is violated at higher energies. With these thoughts, a lot of
modified gravity theories have been proposed, such as the Hořava-Lifshitz
theories of quantum gravity (see Refs. horava1 ; horava2 ; horava3 ; horava4 ;
horava5 ; horava6  for examples), Einstein-Æther theory davi ; d1 ;
jacobson_einstein-aether_2008 ; li ; bat , spatial covariant gravities
Gao:2020qxy ; Gao:2020yzr ; Gao:2019twq ; Gao:2019liu ; Joshi:2021azw , and
the standard model extension SME1 ; SME2 ; SME3 ; SME4 ; SME5 ; SME6 . When
the Lorentz symmetry is violated, the associated gauge symmetry of gravity can
be broken from the general covariance down to the three dimensional spatial
covariance. Then the gravitational action can be constructed from those terms
which respect the spatial diffeomorphisms. In such framework, 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.
For example, in Hořava-Lifshitz gravity horava1 ; horava6 , one has to
consider at least the sixth-order spatial derivatives in the gravitational
action, in order to make the theory power-counting renormalizable.

Breaking the Lorentz symmetry in gravitational interaction also provides a
natural way to incorporate the parity violation terms into the theory. With
spatial covariance, the parity violation can be achieved by including the odd-
order spatial derivatives into the gravitational action horava5 ; Gao:2019liu
. Several parity violating scalar-tensor theories can also be mapped to the
spatial covariant framework by imposing the unitary gauge Gao:2019liu ;
Gao:2020yzr ; cs1 ; chiral_ST . In these theories, the parity violating terms
can also be written in the form of high-order odd-order spatial derivatives.
Various implications of parity violations on the gravitational waves have been
extensively explored in cs_review ; cs3 ; cs4 ; cs5 ; chiral_ST1 ; PGW1 ; PGW2
; PGW3 ; Fu:2020tlw ; add1 ; add2 ; add3 . Note that in Conroy ; Li:2020xjt ;
Li:2021wij ; Li:2021mdp , parity-violating effects induced by lower dimension
operators have also been proposed and their phenomenological effects have been
explored.

Both the Lorentz and parity violations in gravity can introduce deviations
from GR in the propagation of GWs. When the Lorentz symmetry is violated, the
conventional linear dispersion relation of GWs can be modified into a
nonlinear one, which in turn changes the phase velocities of GWs at different
frequencies Mirshekari:2011yq . For parity violation, in general it causes the
polarization propagation velocity and amplitude damping of the left and right
hands of GW to be different, which leads to the velocity and amplitude
birefringence, respectively waveform .

Recently, a lot of tests on both the Lorentz symmetry and parity symmetry of
gravitational interaction have been carried out by using the observational
data from GW events in LIGO-Virgo catalogs Okounkova:2021xjv ; Hu:2020rub ;
CS_gb ; sai_wang ; tanaka ; yi-fan1 ; Wang:2021gqm . In all these works, only
the effects of Lorentz and parity violation due to the leading spatial
derivatives have been considered. In this paper, we study in detail the GW
effects of the Lorentz violation due to the sixth-order derivatives and parity
violation due to fifth-order derivatives in the gravitational action.
Decomposing the GWs into the left-hand and right-hand circular polarization
modes, we find that the GW effects of both the Lorentz and parity violations
can be explicitly presented by the modifications in the GW amplitude and
phase. We also mapped such amplitude and phase modification to the
parametrized description of Lorentz and parity violating waveforms proposed in
waveform . With the modified waveform, we perform the full Bayesian inference
with the help of the open source software BILBY on the selected GW events in
the LIGO-Virgo catalogs GWTC-1 and GWTC-2. From our analysis, we do not find
any signatures of Lorentz violation due to the sixth-order derivatives and
parity violation due to the fifth-order spatial derivatives in the
gravitational action in the GW data except GW190521. We thus place lower
bounds on the energy scale $M_{\rm LV}>2.4\times 10^{-16}\;{\rm GeV}$ for
Lorentz violation and $M_{\rm PV}>1.0\times 10^{-14}\;{\rm GeV}$ 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 spatial covariant framework by using the observational
data of GWs.

This paper is organized as follows. In the next section, we present a brief
introduction of GWs and waveform in Lorentz and parity violating gravities and
then discuss Lorentz violating dispersion relations and parity violating
birefringences. We also calculate in details the waveforms of GWs produced by
the coalescence of compact binary systems and particularly focus on the
deviations from those in GR. In Sec. III, we present two examples of modified
gravity which naturally involved high-order spatial derivatives in the spatial
covariant framework. In Sec. IV, we present the basic statistical framework of
Bayesian analysis used in this work and report the results of constraints on
the Lorentz and parity violation from the Bayesian analysis in Sec. V. We
finish with concluding remarks and discussion in Sec. VI.

Throughout this paper, the metric convention is chosen as $(-,+,+,+)$, and
greek indices $(\mu,\nu,\cdot\cdot\cdot)$ run over $0,1,2,3$ and latin indices
$(i,\;j,\;k)$ run over $1,2,3$. We choose the units to $G=c=1$.

##  II Waveform of GWs with Lorentz and parity violating effects

In this section, we present a brief review the waveform of GWs with Lorentz
and parity violations. Both the Lorentz and parity violations can modify the
damping rates or phase velocities of the two tensorial modes of GWs. For our
purpose we restrict our attention to the cases with Lorentz and parity
violations due to high-spatial derivative terms in gravitational actions and
adopt the general parametrized framework developed in waveform  for describing
the effects of Lorentz and parity violating effects in the propagation of GWs.

###  II.1 GWs in parity and Lorentz violating gravities

We consider the GWs propagating on a homogeneous and isotropic background. The
spatial metric in the flat Friedmann-Robertson-Walker universe is written as

$\displaystyle g_{ij}=a(\tau)(\delta_{ij}+h_{ij}(\tau,x^{i})),$ (2.1)

where $\tau$ denotes the conformal time, which relates to the cosmic time $t$
by $dt=ad\tau$, and $a$ is the scale factor of the universe. Throughout this
paper, we set the present scale factor $a_{0}=1$. $h_{ij}$ denotes the GWs,
which represents the transverse and traceless metric perturbations, i.e,

$\displaystyle\partial^{i}h_{ij}=0=h^{i}_{i}.$ (2.2)

To study the Lorentz and parity violating effects on GWs, it is convenient to
decompose the GWs into the circular polarization modes. To study the evolution
of $h_{ij}$, we expand it over spatial Fourier harmonics,

$\displaystyle h_{ij}(\tau,x^{i})=\sum_{A={\rm
R,L}}\int\frac{d^{3}k}{(2\pi)^{3}}h_{A}(\tau,k^{i})e^{ik_{i}x^{i}}e_{ij}^{A}(k^{i}),$

where $e_{ij}^{A}$ denote the circular polarization tensors and satisfy the
relation

$\displaystyle\epsilon^{ijk}n_{i}e_{kl}^{A}=i\rho_{A}e^{jA}_{l},$ (2.4)

with $\rho_{\rm R}=1$ and $\rho_{\rm L}=-1$. We find that the propagation
equations of these two modes are decoupled, which can be casted into the
parametrized form waveform

$\displaystyle
h^{\prime\prime}_{A}+(2+\bar{\nu}+\nu_{A})\mathcal{H}h^{\prime}_{A}+(1+\bar{\mu}+\mu_{A})k^{2}h_{A}=0,$

where a prime denotes the derivative with respect to the conformal time $\tau$
and $\mathcal{H}=a^{\prime}/a$.

In the above parametrization, the new effects arising from theories beyond GR
are fully characterized by four parameters: $\bar{\nu}$, $\bar{\mu}$,
$\nu_{A}$ and $\mu_{A}$. The parameters $\nu_{A}$ and $\mu_{A}$ label the
effects of the parity violation, and $\bar{\nu}$ and $\bar{\mu}$ describe the
effects of other possible modifications which are not relevant to parity
violation. In these four parameters, $\mu_{A}$ and $\bar{\mu}$ determine the
speed of GWs, while $\nu_{A}$ and $\bar{\nu}$ determine the damping rate of
GWs during their propagation. As shown in waveform , such parametrization
provides an unifying description for the low-energy effective description of
GWs in generic modified gravities, including Chern-Simons modified gravity
cs_review , ghost-free parity violating scalar-tensor theory chiral_ST ;
chiral_ST1 ; Gao:2019liu , symmetric teleparallel equivalence of general
relativity Conroy , Hořava-Lifshitz gravities horava1 ; horava2 ; horava3 ;
horava4 ; horava5 ; horava6 , the Nieh-Yan modified teleparallel gravity
Li:2020xjt ; Li:2021wij .

With regard to the parity violation, the parameter $\mu_{A}$ leads to
different velocities of left- and right-hand circular polarizations of GWs, so
the arrival times of the two circular polarization modes could be different,
which is the phenomenon of velocity birefringence. The parameter $\nu_{A}$, on
the other hand, leads to different damping rates of left- and right-hand
circular polarizations of GWs, so the amplitude of left-hand circular
polarization of GWs will increase (or decrease) during the propagation, while
the amplitude for the right-hand modes will decrease (or increase). This is
the phenomenon of amplitude birefringence. It is also worth noting that for
large numbers of parity violating gravities, both parameters $\nu_{A}$ and
$\mu_{A}$ are frequency dependent.

For the effects which are not relevant to parity violation, the parameters
$\bar{\nu}$ and $\bar{\mu}$ can arise either from a modified gravity with
general covariance or a modified theories in the framework of spatial
covariant gravities. For the former case, both $\bar{\nu}$ and $\bar{\mu}$ can
be frequency-independent, as shown in Table. I of waveform  for a few of
modified gravities. For the later case, the theories only respect the spatial
covariance, not the usual general covariance of four dimensional spacetime. In
this case, there exists a preferred timelike direction which violates the
temporal diffeomorphism of the theories, thus breaks the Lorentz symmetry of
the theories as well. In these kind of theories, the parameters $\bar{\nu}$
and $\bar{\mu}$ can be frequency dependent. In this paper, in order to study
the Lorentz violating effects on GWs, we only focus on the later case.

As analyzed in the above, here we can assume all the four parameters are
frequency dependent, the two parameters $\nu_{A}$ and $\mu_{A}$ are effects
arising from parity violations and the other two parameters $\bar{\nu}$ and
$\bar{\mu}$ are effects from Lorentz violation. In this sense, we parametrize
these four parameters in the following forms waveform

$\displaystyle\mathcal{H}\bar{\nu}$ $\displaystyle=$
$\displaystyle\left[\alpha_{\bar{\nu}}(\tau)\left(k/aM_{\mathrm{LV}}\right)^{\beta_{\bar{\nu}}}\right]^{\prime},$
(2.6) $\displaystyle\bar{\mu}$ $\displaystyle=$
$\displaystyle\alpha_{\bar{\mu}}(\tau)\left(k/aM_{\mathrm{LV}}\right)^{\beta_{\bar{\mu}}},$
(2.7) $\displaystyle\mathcal{H}\nu_{\mathrm{A}}$ $\displaystyle=$
$\displaystyle\left[\rho_{\mathrm{A}}\alpha_{\nu}(\tau)\left(k/aM_{\mathrm{PV}}\right)^{\beta_{\nu}}\right]^{\prime},$
(2.8) $\displaystyle\mu_{\mathrm{A}}$ $\displaystyle=$
$\displaystyle\rho_{\mathrm{A}}\alpha_{\mu}(\tau)\left(k/aM_{\mathrm{PV}}\right)^{\beta_{\mu}},$
(2.9)

where $\beta_{\bar{\nu}}$ and $\beta_{\bar{\mu}}$ are arbitrary even numbers
and $\beta_{\nu}$ and $\beta_{\mu}$ are arbitrary odd numbers.
$\alpha_{\bar{\nu}}$, $\alpha_{\bar{\mu}}$, $\alpha_{\nu}$, and $\alpha_{\mu}$
are arbitrary functions of time. These parameters can only be determined given
a specific model of modified gravities. For the GW events at local Universe,
these two functions can be approximately treated as constant, i.e. ignoring
their time-dependence.

###  II.2 Waveforms of GWs with parity and Lorentz violations

With the above parametrization of Lorentz and parity violating effects on the
propagation of GWs, one can derive their explicit GW waveform by solving the
equation of motion (II.1). It is shown in waveform  that the amplitude and
phase modifications to the GR-based waveform due to both Lorentz and parity
violating effects can be written in the following parametrized form

$\displaystyle\tilde{h}_{A}(f)=\tilde{h}_{A}^{\rm GR}(1+\rho_{A}\delta
h_{1}+\delta h_{2})e^{i(\rho_{A}\delta\Psi_{1}+\delta\Psi_{2})},$ (2.10)

where $\tilde{h}_{A}^{\rm GR}$ is the corresponding waveform in GR, and its
explicit form can be found in the previous works waveform . The amplitude
corrections $\delta h_{1}$ and $\delta h_{2}$ are caused by the parameters
$\nu_{A}$ and $\bar{\nu}$ respectively, while the phase corrections
$\delta\Psi_{1}$ and $\delta\Psi_{2}$ are caused by the parameters $\mu_{A}$
and $\bar{\mu}$ respectively. The explicit forms of $\delta h_{1}$, $\delta
h_{2}$, $\delta\Psi_{1}$, and $\delta\Psi_{2}$ are given respectively by

$\displaystyle\delta h_{1}=-\frac{1}{2}\left(\frac{2\pi f}{M_{\rm
PV}}\right)^{\beta_{\nu}}\Big{[}\alpha_{\nu}(\tau_{0})-\alpha_{\nu}(\tau_{e})(1+z)^{\beta_{\nu}}\Big{]},$
(2.11) $\displaystyle\delta h_{2}=-\frac{1}{2}\left(\frac{2\pi f}{M_{\rm
LV}}\right)^{\beta_{\bar{\nu}}}\Big{[}\alpha_{\bar{\nu}}(\tau_{0})-\alpha_{\bar{\nu}}(\tau_{e})(1+z)^{\beta_{\bar{\nu}}}\Big{]},$
(2.12) $\displaystyle\delta\Psi_{1}=\begin{cases}\frac{(2/M_{\rm
PV})^{\beta_{\mu}}}{\beta_{\mu}+1}(\pi
f)^{\beta_{\mu}+1}\int_{t_{e}}^{t_{0}}\frac{\alpha_{\mu}}{a^{\beta_{\mu}+1}}dt,&\;\;\beta_{\mu}\neq-1,\\\
\frac{M_{\rm PV}}{2}\ln
u\int_{t_{e}}^{t_{0}}\alpha_{\bar{\mu}}dt,&\;\;\beta_{\mu}=-1,\end{cases}\;\;$

and

$\displaystyle\delta\Psi_{2}=\begin{cases}\frac{(2/M_{\rm
LV})^{\beta_{\bar{\mu}}}}{\beta_{\bar{\mu}}+1}(\pi
f)^{\beta_{\bar{\mu}}+1}\int_{t_{e}}^{t_{0}}\frac{\alpha_{\bar{\mu}}}{a^{\beta_{\bar{\mu}}+1}}dt,&\;\;\beta_{\bar{\mu}}\neq-1,\\\
\frac{M_{\rm LV}}{2}\ln
u\int_{t_{e}}^{t_{0}}\alpha_{\mu}dt,&\;\;\beta_{\bar{\mu}}=-1,\end{cases}\;\;$

where $t_{e}$ ($t_{0}$) denotes the emitted (arrival) time for a given GW
event, $z=1/a(t_{e})-1$ is the cosmological redshift, $f$ is the GW frequency
at the detector, and $u=\pi{\cal M}f$, where ${\cal M}$ is the measured chirp
mass of the binary system.

The circular polarization modes $\tilde{h}_{\rm R}$ and $\tilde{h}_{\rm L}$
relate to the modes $\tilde{h}_{+}$ and $h_{\times}$ via

$\displaystyle\tilde{h}_{+}=\frac{\tilde{h}_{\rm L}+\tilde{h}_{\rm
R}}{\sqrt{2}},$ (2.15) $\displaystyle\tilde{h}_{\times}=\frac{\tilde{h}_{\rm
L}-\tilde{h}_{\rm R}}{\sqrt{2}i}.$ (2.16)

Thus the waveforms for the plus and cross modes become

$\displaystyle\tilde{h}_{+}=\tilde{h}_{+}^{\mathrm{GR}}\left(1+\delta
h_{2}+i\delta\Psi_{2}\right)-\tilde{h}_{\times}^{\mathrm{GR}}\left(i\delta
h_{1}-\delta\Psi_{1}\right),$
$\displaystyle\tilde{h}_{\times}=\tilde{h}_{\times}^{\mathrm{GR}}\left(1+\delta
h_{2}+i\delta\Psi_{2}\right)+\tilde{h}_{+}^{\mathrm{GR}}\left(i\delta
h_{1}-\delta\Psi_{1}\right).$

As we mentioned, the coefficients $\alpha_{\nu}$($\beta_{\nu}$),
$\alpha_{\bar{\nu}}$($\beta_{\bar{\nu}}$), $\alpha_{\mu}$ ($\beta_{\mu}$), and
$\alpha_{\bar{\mu}}$ ($\beta_{\bar{\mu}}$) depend on specific modified
gravities. Different numbers of $\beta_{\nu,\bar{\nu},\mu,\bar{\mu}}$
correspond to different effects. In this paper, we consider two distinct
effects, one from Lorenz violation and the other from parity violation.

####  II.2.1 Lorentz violating dispersion relations

Except $\beta_{\bar{\mu}}=0=\beta_{\bar{\nu}}$ and
$\beta_{\bar{\mu}}=-2=\beta_{\bar{\nu}}$, the coefficients $\alpha_{\nu}$
($\beta_{\nu}$) and $\alpha_{\mu}$ ($\beta_{\mu}$) represent the Lorentz
violating effects on the propagation of GWs. Among these parameters,
$\alpha_{\nu}$ ($\beta_{\nu}$) label the amplitude corrections to the waveform
of the GWs, while $\alpha_{\mu}$ ($\beta_{\mu}$) is related to the phase
corrections. In general, the amplitude and phase corrections can be related to
each other. Since the GW detectors are more sensitive to the phase, rather
than the amplitude of GWs from all the events detected by LIGO and Virgo, in
this paper let us only focus on those effects that can lead to the phase
corrections to the GW waveform. Thus hereafter we can set $\alpha_{\nu}=0$.
Then the Lorentz violating effects associated with parameters $\alpha_{\mu}$
($\beta_{\mu}$) can be related to a Lorentz-violating dispersion relation,

$\displaystyle\omega_{k}^{2}=k^{2}\Big{[}1+\alpha_{\bar{\mu}}\left(\frac{k}{aM_{\rm
LV}}\right)^{\beta_{\bar{\mu}}}\Big{]},$ (2.19)

where $\omega_{k}^{2}$ is the energy of the GWs. The waveform of GWs with the
above Lorentz-violating dispersion relation has been also derived in
Mirshekari:2011yq  and then their possible constraints with ground-based and
space-based detectors has been analyzed in the framework of Fisher matrix.
With the GW events detected by LIGO-Virgo, the gravitational wave constraints
on the Lorenz-violating dispersion relations for
$\beta_{\bar{\mu}}=-2,\;-1.5,\;-1,\;-0.5,\;0.5,\;1,\;1.5,\;2$ have been
obtained by comparing the modified waveforms with GW data testGR_GWTC2 ;
testGR_GWTC1 . With these constraints, one is able to infer the GW constraints
on the corresponding Lorentz violating gravities. In this paper, we shall
consider the case with $\beta_{\bar{\mu}}=4$ and derive the corresponding
constraints by using the signals of GW events detected by LIGO-Virgo. As we
will show later, the Lorentz-violating dispersion relation with
$\beta_{\bar{\mu}}=4$ is related to the sixth-order spatial derivatives in
modified gravities with spatial covariance and of crucial importance in the
construction for a power-counting renormalizable theory of quantum gravity
horava1 ; horava6 .

####  II.2.2 Parity violating birefringences

The coefficients $\alpha_{\nu}$ ($\beta_{\nu}$) and $\alpha_{\mu}$
($\beta_{\mu}$) represent the parity violating effects on the propagation of
the GWs. Among these parameters, $\alpha_{\nu}$ ($\beta_{\nu}$) label the
effects of amplitude birefringence, while $\alpha_{\mu}$ ($\beta_{\mu}$) is
related to the velocity birefringence. In this paper we will consider the
velocity birefringence effects only. With the GW events detected by LIGO and
Virgo, the GW constraints on the parity-violating birefringence for
$\beta_{\mu}=1$ yi-fan1 ; Wang:2021gqm  and $\beta_{\mu}=-1$ ZHUTAO  have been
obtained by comparing the modified waveforms with GW data. With these
constraints, one is able to infer the gravitational wave constraints on the
corresponding parity-violating gravities. In this paper, we shall consider the
case with $\beta_{\mu}=3$ and derive the corresponding constraints by using
the signals of GW events detected by LIGO-Virgo. This case corresponds to the
fifth-order spatial derivatives terms in modified gravities with spatial
covariance, which can arise naturally from a specific spatial covariant
gravity Gao:2019liu  and Hořava-Lifshitz gravity horava5 .

##  III Specific theories with high-order spatial derivatives

In this section we present two specific theories that contain the high-order
spatial derivatives in the framework of spatial covariant gravity.

####  III.0.1 Hořava-Lifshitz gravity

The Hořava-Lifshitz (HL) gravity is based on the view that Lorentz symmetry is
a symmetry only at low energy, but may not exist at high energy horava1 .
Because of this feature, one can establish a high-energy gravity without
Lorentz symmetry in the UV. This gravity theory contains higher-order spatial
derivatives, and also keep the time derivative operators to the second-order.
In this paper, we are going to focus on an extension of the HL gravity by
abandoning the projectability condition but imposing an extra local U(1)
symmetry that was proposed in Zhu:2011yu , in which the gravitational sector
has the same degree of freedom as that in GR, i.e., only spin-2 massless
gravitons exist.

The action of the HL gravity consists kinetic and potential parts. The kinetic
part only includes at most second time derivatives, which then is given by
horava1 ; horava6

$\displaystyle S_{\rm k}=\frac{M_{\rm Pl}^{2}}{2}\int
dtd^{3}x\sqrt{g}N(K_{ij}K^{ij}-\lambda K^{2}),$ (3.1)

where $K_{ij}$ is the extrinsic curvature of the constant time hypersurface
defined by $K_{ij}=(\dot{g}_{ij}-\nabla_{i}N_{j}-\nabla_{j}N_{i})/2N$ with $N$
and $N_{i}$ being the lapse function and shift vector respectively, $K$ is the
trace of $K_{ij}$. In construction of the potential part, it is of crucial
importance to consider the spatial derivative up to the sixth order, in order
to make the theory to be power-counting renormalizable. In different versions
of the HL theory, the potential part has different forms, but in general it
can be cast into the sum of different orders of spatial derivatives, i.e.,

$\displaystyle S_{\rm V}=\frac{M_{\rm Pl}^{2}}{2}\int
dtd^{3}x\sqrt{g}N\mathcal{L}_{\rm V},$ (3.2)

where

$\displaystyle\mathcal{L}_{\rm
V}=\sum_{n=2}\mathcal{L}^{(n)}(N,N^{i},g_{ij}),$ (3.3)

with $\mathcal{L}^{(n)}$ denoting the part of Lagrangian that contains
operators of the $n$th-order spatial derivative only. In the above, $n$ is an
even number for parity-conserving terms and is odd for parity violating terms.

By abandoning the Lorentz symmetry, the HL theory also provides a natural way
to incorporate the parity violation terms into the theory. For our current
purpose, we consider the third- and/or fifth-order spatial derivative
operators to the potential term ${\cal{L}}$ of the total action in horava4 ;
horava5 ; Zhu:2011yu ,

$\displaystyle\Delta{\cal{L}}$ $\displaystyle=$ $\displaystyle\frac{1}{M_{\rm
PV}^{3}}\left(\alpha_{0}K_{ij}R_{ij}+\alpha_{2}\varepsilon^{ijk}R_{il}\Delta_{j}R^{l}_{k}\right)$
(3.4) $\displaystyle+\frac{\alpha_{1}\omega_{3}(\Gamma)}{M_{\rm
PV}}+``\dots",$

here $M_{\rm PV}$ is the energy scale above which the high-order derivative
operators become important. The coupling constant
$\alpha_{0},\;\alpha_{1},\;\alpha_{2}$ are dimensionless and arbitrary, and
$\omega_{3}(\Gamma)$ the 3-dimensional gravitational CS term. “…” denotes the
rest of the fifth-order operators given in Eq.(2.6) of Zhu:2011yu . Since they
have no contributions to tensor perturbations, in this paper we shall not
write them out explicitly.

The general formulas of the linearized tensor perturbations were given in
horava5 , so in the rest of this subsection we give a very brief summary of
the main results obtained there. Consider a flat FRW universe and assuming
that matter fields have no contributions to tensor perturbations, the
quadratic part of the total action can be cast in the form,

$\displaystyle S^{(2)}_{\text{ g}}$ $\displaystyle=$
$\displaystyle\frac{M_{\rm Pl}^{2}}{2}\int d\eta
d^{3}x\Bigg{\\{}\frac{a^{2}}{4}(h_{ij}^{\prime})^{2}-\frac{1}{4}a^{2}(\partial_{k}h_{ij})^{2}$
$\displaystyle\;\;\;-\frac{\hat{\gamma_{3}}}{4M_{\rm
LV}^{2}}(\partial^{2}h_{ij})^{2}-\frac{\hat{\gamma}_{5}}{4M_{\rm
LV}^{4}a^{2}}(\partial^{2}\partial_{k}h_{ij})^{2}$
$\displaystyle\;\;\;-\frac{\alpha_{1}a\epsilon^{ijk}}{2M_{\rm
PV}}(\partial_{l}h_{i}^{m}\partial_{m}\partial_{j}h_{k}^{l}-\partial_{l}h_{im}\partial^{l}\partial_{j}h^{m}_{k})$
$\displaystyle\;\;\;-\frac{\alpha_{2}\epsilon^{ijk}}{4M_{\rm
PV}^{3}a}\partial^{2}h_{il}(\partial^{2}h^{l}_{k})_{,j}-\frac{3\alpha_{0}{\cal{H}}}{8M_{\rm
PV}a}(\partial_{k}h_{ij})^{2}\Bigg{\\}},$

where $\hat{\gamma}_{3}\equiv({2M_{\rm LV}}/{M_{\rm Pl}})^{2}{\gamma}_{3}$ and
$\hat{\gamma}_{5}\equiv({2M_{\rm LV}}/{M_{\rm Pl}})^{4}{\gamma}_{5}$, and
$\gamma_{3}$ and $\gamma_{5}$ are the dimensionless coupling constants of the
theory. To avoid fine-tuning, ${{\alpha}}_{n}$ and $\hat{\gamma}_{n}$ are
expected to be of the same order. Then, the field equations for $h_{ij}$ read,

$\displaystyle
h^{\prime\prime}_{ij}+2\mathcal{H}h^{\prime}_{ij}-\alpha^{2}\partial^{2}h_{ij}+\frac{\hat{\gamma}_{3}}{a^{2}M_{\rm
LV}^{2}}\partial^{4}h_{ij}-\frac{\hat{\gamma}_{5}}{a^{4}M_{\rm
LV}^{4}}\partial^{6}h_{ij}$
$\displaystyle\;\;\;+\epsilon_{i}^{\;\;lk}\left(\frac{2\alpha_{1}}{aM_{\rm
PV}}+\frac{\alpha_{2}}{a^{3}M_{\rm
PV}^{3}}\partial^{2}\right)\left(\partial^{2}h_{jk}\right)_{,l}=0,$ (3.6)

where $\alpha^{2}\equiv 1+{3\alpha_{0}{\cal{H}}}/{(2M_{\rm PV}a)}$. In the
late universe, $a\sim 1$, and $\mathcal{H}\ll M_{\rm PV}$, so we find
$\alpha^{2}\rightarrow 1$.

To study the evolution of $h_{ij}$, we expand it over spatial Fourier
harmonics. For each circular polarization mode, the equation of motion of GW
is given by

$\displaystyle h_{\rm A}^{\prime\prime}+2\mathcal{H}h_{\rm
A}^{\prime}+\omega_{\rm A}^{2}h_{\rm A}=0,$ (3.7)

with

$\displaystyle\omega^{2}_{\rm A}(k,\eta)$ $\displaystyle\equiv$
$\displaystyle\alpha^{2}k^{2}\Bigg{[}1+\delta_{1}\rho_{\rm A}\frac{\alpha
k}{M_{\rm PV}a}+\delta_{2}\left(\frac{\alpha k}{M_{\rm LV}a}\right)^{2}$
$\displaystyle-\delta_{3}\rho_{\rm A}\left(\frac{\alpha k}{M_{\rm
PV}a}\right)^{3}+\delta_{4}\left(\frac{\alpha k}{M_{\rm
LV}a}\right)^{4}\Bigg{]},$

where $\delta_{1}\equiv{2\alpha_{1}}/{\alpha^{3}}$,
$\delta_{2}\equiv{\hat{\gamma_{3}}}/{\alpha^{4}}$,
$\delta_{3}\equiv{\alpha_{2}}/{\alpha^{5}}$,
$\delta_{4}\equiv{\hat{\gamma_{5}}}/{\alpha^{6}}$. In comparison with the
formula in Eq.(II.1), we find the coefficients in HL gravity,

$\displaystyle\mathcal{H}\bar{\nu}$ $\displaystyle=$ $\displaystyle 0,$ (3.9)
$\displaystyle\bar{\mu}$ $\displaystyle=$
$\displaystyle\delta_{2}\left(\frac{k}{aM_{\rm
LV}}\right)^{2}+\delta_{4}\left(\frac{k}{aM_{\rm LV}}\right)^{4}$ (3.10)
$\displaystyle+\frac{3\alpha_{0}{\cal{H}}}{(2M_{\rm PV}a)},$
$\displaystyle\mathcal{H}\nu_{\rm A}$ $\displaystyle=$ $\displaystyle 0,$
(3.11) $\displaystyle\mu_{\rm A}$ $\displaystyle=$
$\displaystyle\delta_{1}\rho_{\rm A}\left(\frac{k}{aM_{\rm
PV}}\right)-\delta_{3}\rho_{\rm A}\left(\frac{k}{aM_{\rm PV}}\right)^{3},$
(3.12)

where we have considered the relation $\alpha^{2}\rightarrow 1$.

For the Lorentz violating effects, labeled by $\bar{\mu}$ and $\bar{\nu}$, the
observational constraint on the effect of the first term (c.f. $\delta_{2}$
term) in (3.10) from GWTC-1 and GWTC-2 has been derived in testGR_GWTC1 ;
testGR_GWTC2 . For parity violating effects, labeled by $\mu_{A}$, the
observational constraint on the effect of the first term in (3.12) has been
considered in sai_wang ; Wang:2021gqm ; yi-fan1 . In this paper, we will not
consider $\delta_{1}$ and $\delta_{2}$ terms, and concentrate on the effects
of $\delta_{3}$ and $\delta_{4}$ on GWs and their constriants from GW data.

####  III.0.2 Spatial covariant gravity

We first start with the general action of the spatial covariant gravity
Gao:2020yzr ; Gao:2019liu ,

$\displaystyle S=\int dtd^{3}xN\sqrt{g}{\cal
L}(N,g_{ij},K_{ij},R_{ij},\nabla_{i},\varepsilon_{ijk}),$

where $R_{ij}$ is the intrinsic curvature tensor, $\nabla_{i}$ denotes the
spatial covariant derivative with respect to $g_{ij}$, and
$\varepsilon_{ijk}=\sqrt{g}\epsilon_{ijk}$ is the spatial Levi-Civita tensor
with $\epsilon_{ijk}$ being the total antisymmetric tensor. The most important
feature of the spatial covariant gravity is that it is only invariant under
the 3-dimensional spatial diffeomorphism, which breaks the time
diffeomorphism. Normally, the violation of the time diffeomorphism can lead to
extra degree of freedom, in addition to the two tensorial degree of freedom in
GR. Indeed, it has been verified that the spatial covariant gravity described
by the action (III.0.2) can propagate up to 3 dynamical degrees of freedom.

There are a lot approaches to construct the gravitational theories with
spatial covariance. In this paper, for simplicity, we will not write out the
action explicitly and only give a brief summary of GWs in the spatial
covariant gravity. In general, the equation of motion of $h_{A}$ in a spatial
covariant gravity can be written in the form Gao:2019liu ,

$\displaystyle h_{A}^{\prime\prime}+(2+\Gamma_{A}){\cal
H}h_{A}^{\prime}+\omega_{A}^{2}h_{A}=0,$ (3.14)

where

$\displaystyle\Gamma_{A}=\frac{1}{\cal H}\frac{\partial_{\tau}{\cal
G}^{(A)}(\tau,k)}{{\cal G}^{(A)}(\tau,k)},$ (3.15)
$\displaystyle\frac{\omega_{A}^{2}}{k^{2}}=\frac{{\cal W}^{(A)}(\tau,k)}{{\cal
G}^{(A)}(\tau,k)},$ (3.16)

where ${\cal G}^{(A)}$ and ${\cal W}^{(A)}$ are functions of time and wave
number $k$. Depending specific terms in the spatial covariant gravity, ${\cal
G}^{(A)}$ and ${\cal W}^{(A)}$ can have different forms. In above expressions,
$\Gamma_{A}$ can lead to amplitude modifications, while $\omega_{A}^{2}$
modifies phases of GWs. In this paper, we only focus on the phase modification
on GWs thus we will not consider the effect of $\Gamma_{A}$ and their GW
constraints. For $\omega_{A}^{2}$, the general form can be expressed as
Gao:2019liu

$\displaystyle\frac{\omega_{A}^{2}}{k^{2}}=\frac{{\cal W}_{0}+\rho_{A}{\cal
W}_{1}\tilde{k}+{\cal W}_{2}\tilde{k}^{2}+\rho_{A}{\cal
W}_{3}\tilde{k}^{3}+{\cal W}_{4}\tilde{k}^{4}+\cdots}{{\cal
G}_{0}+\rho_{A}{\cal G}_{1}\tilde{k}+{\cal G}_{2}\tilde{k}^{2}+\rho_{A}{\cal
G}_{3}\tilde{k}^{3}+{\cal G}_{4}\tilde{k}^{4}+\cdots},$

where ${\cal G}_{n}$ and ${\cal W}_{n}$ are functions of time. In the above
expression, ${\cal G}_{n}$ and ${\cal W}_{n}$ for even number $n$ denote the
Lorentz violating effects, while for odd number are effects from parity
violating operators.

The observational constraints on the terms with coefficients ${\cal W}_{1}$,
${\cal W}_{2}$, and ${\cal G}_{1}$ by using GW data can be inferred directly
from the analysis in testGR_GWTC1 ; testGR_GWTC2 ; sai_wang ; tanaka ; yi-fan1
; Wang:2021gqm . In this paper, we will focus on the observational constraints
on ${\cal W}_{3}$ and ${\cal W}_{4}$, which can arise from the fifth-order and
sixth-order spatial derivative terms in the action of gravity with spatial
covariance.

##  IV Bayesian inference for GW data

###  IV.1 Bayesian inference for GW data

In this paper we perform the Bayesian inference to constraint the parity and
Lorentz violations with selected GW events from the LIGO-Virgo catalogs GWTC-1
and GWTC-2. Bayesian analysis is also often used in GW parameter estimation
such as chirp mass, redshift, right ascension and luminosity distance etc. of
GW source. When we have GW data $d$, we compare the GW data with the parameter
$\vec{\theta}$ inferred by the instrument with the data of LIGO-Virgo catalogs
to infer the distribution of the parameter $\vec{\theta}$. Next, we write the
Bayesian theorem in the context of GW astronomy:

$\displaystyle
P({\vec{\theta}}|d,H)=\frac{P(d|\vec{\theta},H)P(\vec{\theta}|H)}{P(d|H)},$
(4.1)

where $H$ is the waveform model, $P(\vec{\theta}|H)$ is the prior distribution
for model parameters $\vec{\theta}$ and $P(d|\vec{\theta},H)$ is the
likelihood given a specific set of model parameters, $P(d|H)$ is normalization
factor called the ‘evidence’,

$\displaystyle P(d|H)\equiv\int
d\vec{\theta}P(d|\vec{\theta},H)P(\vec{\theta}|H),$ (4.2)

and $P({\vec{\theta}}|d,H)$ denotes the posterior probability distributions
for physical parameters ${\vec{\theta}}$ which describe the observational
data.

In most cases, we use the matched filtering method to extract the signal from
the noise (we assume that the noise is Gaussian white noise) because the GW
signal from compact binary combination is very weak compared with most other
signals, such as electromagnetic wave signals. The likelihood function of the
matched filtering method could be defined in the following form:

$\displaystyle
P(d|{\vec{\theta}},H)\propto\prod_{i=1}^{n}e^{-\frac{1}{2}\langle
d_{i}-h({\vec{\theta}})|d_{i}-h({\vec{\theta}})\rangle},$ (4.3)

where $h(\vec{\theta})$ is the GW waveform template for model $H$ and $i$
represents the $i$ th GW detector. The noise weighted inner product $\langle
A|B\rangle$ is defined as

$\displaystyle\langle A|B\rangle=4\;{\rm
Re}\left[\int_{0}^{\infty}\frac{A(f)B(f)^{*}}{S(f)}df\right],$ (4.4)

where $\;{}^{*}$ denotes complex conjugation and $S(f)$ is the power spectral
density (PSD) function of the detector.

Next, we restrict our attention to the cases with Lorentz and parity
violations due to high-spatial derivative terms in gravitational actions. We
utilize the Python package BILBY to perform the Bayesian inference by
analyzing the GW data from selected 47 events of BBH and BNS mergers in the
LIGO-Virgo catalogs GWTC-1 and GWTC-2. We use the waveform template given in
(II.2) and (II.2) with $\delta\Psi_{2}$ for the Lorentz violating effect and
$\delta\Psi_{1}$ for parity violating effect respectively. We employ template
IMRPhenomPv2 for the GR waveform $h^{\rm GR}_{+,\;\times}(f)$ for BBH events
except the event GW190814 and GW190521, and IMRPhenomPv2_NRTidal for BNS
events. For GW190814 and GW190521, we employ the waveform IMRPhenomXPHM which
includes the subdominant harmonic modes of GW and accounts for spin-precession
effects for a quasicircular-orbit binary black hole coalescence. The
explanation of parameter vector $\vec{\theta}$ for these waveform templates
can be found in ZHUTAO ; gwtc1 .

We have constructed modified GW waveform through Eq. (II.2) and Eq. (II.2) for
$\beta_{\mu}=3$ and $\beta_{\bar{\mu}}=4$ with respect to parity violation and
Lorentz violation respectively. We ignore the contribution of $\delta h_{1}$
and $\delta h_{2}$ because their contributions to the GW strain are negligibly
small compared to phase modifications from $\delta\Psi_{1}$ and
$\delta\Psi_{2}$. For our purpose we consider the Lorentz violation and parity
violation effects separately, which means when we consider the Lorentz
violating effects in the waveform, we have set the contributions of parity
violation to vanish, and vice-versa. When $\beta_{\mu}$ =3 and
$\beta_{\bar{\mu}}$ =4, the phase modification $\delta\Psi_{1}$ and
$\delta\Psi_{2}$ are written respectively in the following forms,

$\displaystyle\delta\Psi_{1}$ $\displaystyle=$ $\displaystyle A_{\mu}{(\pi
f)}^{4},$ (4.5) $\displaystyle\delta\Psi_{2}$ $\displaystyle=$ $\displaystyle
A_{\bar{\mu}}(\pi f)^{5},$ (4.6)

where

$\displaystyle A_{\mu}$ $\displaystyle=$ $\displaystyle\frac{2}{(M_{\rm
PV})^{3}}\int_{t_{e}}^{t_{0}}\frac{\alpha_{\mu}}{a^{4}}dt,$ (4.7)

and

$\displaystyle A_{\bar{\mu}}$ $\displaystyle=$
$\displaystyle\frac{16}{5(M_{\rm
LV})^{4}}\int_{t_{e}}^{t_{0}}\frac{\alpha_{\bar{\mu}}}{a^{5}}dt,$ (4.8)

Then we consider a series of GW events composed of data $\\{d_{i}\\}$,
described by parameters $\\{{\vec{\theta}_{i}}\\}$, then for each event,
posterior for all the parameters describing the event can be written as

$\displaystyle P(A_{\mu},{\vec{\theta}_{i}}|d_{i})=\frac{P(M_{\rm
PV},\vec{\theta}_{i})P(d_{i}|M_{\rm PV},\vec{\theta}_{i})}{P(d_{i})},$ (4.9)

for testing the parity violating waveform, and

$\displaystyle P(A_{\bar{\mu}},{\vec{\theta}_{i}}|d_{i})=\frac{P(M_{\rm
LV},\vec{\theta}_{i})P(d_{i}|M_{\rm LV},\vec{\theta}_{i})}{P(d_{i})},$ (4.10)

for testing Lorentz violating waveforms.

###  IV.2 Selection of data samples

Up to now, the LIGO-Virgo catalogs GWTC-1 and GWTC-2 have found and reported a
total of 50 gravitational wave events. However, it is mentioned in
LIGOScientific:2021usb  that three of GWTC-2 events, the events
GW190424_180648, GW190426_152155, and GW190909_114149, have a probability of
astrophysical origin $p_{\rm astro}<0.5$, so we carefully exclude them and
then select 47 BBH and BNS events in our analysis as presented in Table. 1
(see also in Table. 2. The data of these 47 GW events are downloaded from
Gravitational Wave Open Science Center data_GW . For these events, we perform
parameter estimation using Bayesian analysis by selecting $4\;{\rm s}$ or
$8\;{\rm s}$ or $16\;{\rm s}$ data for BBH and $128\;{\rm s}$ data for BNS
over the selected GW parameters $\vec{\theta}$ and the parity violating
parameter $A_{\mu}$ (the Lorentz violating parameter $A_{\bar{\mu}}$). The
prior for the standard GW parameters ${\vec{\theta}}$ are consistent with
those used in gwtc1 ; gwtc2 . The prior for $A_{\mu}$ and $A_{\bar{\mu}}$ are
chosen to be uniformly distributed. We use the package BILBY bilby  to perform
the Bayesian analysis and the posterior distribution is sampled by the nest
sampling method dynesty over the fiducial BBH and BNS data and the parity
violating parameter $A_{\mu}$ and the Lorentz violating parameter
$A_{\bar{\mu}}$. We report our main results in the next section.

Table 1: 90% credible level lower bounds on $M_{\rm PV}$ from the Bayesian inference by analyzing the selected 46 GW events in the LIGO-Virgo catalogs GWTC-1 and GWTC-2. Note that result from GW190521 is excluded. catalogs | GW events | Lower limit [$10^{-15}\;{\rm GeV}$]
---|---|---
GWTC-1 | GW150914 | 1.9
GW151012 | 3.3
GW151226 | 5.9
| GW170104 | 2.0
| GW170608 | 5.8
| GW170729 | 1.6
| GW170809 | 2.1
| GW170814 | 2.8
| GW170817 | 7.5
| GW170818 | 2.2
| GW170823 | 2.0
GWTC-2 | GW190408_181802 | 3.2
GW190412 | 3.1
GW190413_052954 | 2.0
GW190413_134308 | 0.9
GW190421_213856 | 1.5
GW190425 | 5.1
GW190503_185404 | 1.7
GW190512_180714 | 3.7
GW190513_205428 | 2.1
GW190514_065416 | 1.4
GW190517_055101 | 2.6
GW190519_153544 | 1.4
GW190521_074359 | 2.4
GW190527_092055 | 2.1
GW190602_175927 | 0.9
GW190620_030421 | 1.1
| GW190630_185205 | 2.4
| GW190701_203306 | 1.2
| GW190706_222641 | 0.6
| GW190707_093326 | 6.3
| GW190708_232457 | 3.9
| GW190719_215514 | 0.6
| GW190720_000836 | 5.4
| GW190727_060333 | 2.3
| GW190728_064510 | 7.3
| GW190731_140936 | 1.4
| GW190803_022701 | 1.4
| GW190814 | 4.5
| GW190828_063405 | 3.0
| GW190828_065509 | 3.6
| GW190910_112807 | 1.5
| GW190915_235702 | 2.1
| GW190924_021846 | 8.0
| GW190929_012149 | 0.8
| GW190930_133541 | 6.0
combined |  | 10.2

##  V Results of constraints on Lorentz and parity violations

For 46 GW events in all the 47 events we analyzed, we do not find any
significant signatures of Lorentz and parity violation due to the fifth and
sixth-order spatial derivatives. To illustrate our results, we present briefly
the marginalized posterior distributions of $A_{\mu}$ (upper panel) and
$A_{\bar{\mu}}$ (bottom panel) in Fig. 1 for those GW events with chirp mass
${\cal M}\lesssim 25M_{\odot}$. In this figure, the region in the posterior
between the upper and lower bar denotes the 90% credible interval for all
these events. From Fig.1, it is also shown that the relatively low-mass BBH
and BNS events, such as GW170817, GW190425, GW190728_064510, GW190924_021846
give tighter constraints on both $A_{\mu}$ and $A_{\bar{\mu}}$. This is
exactly what we expected, as the low-mass event can lead to larger phase
corrections, as shown in (4.5) and (4.6). For all the 46 events among the 47
events, our results show that the GR values $A_{\mu}=0$ and $A_{\bar{\mu}}=0$
is well within the 90% confidence level.

The only one exception is the GW events GW190521. In Fig. 2, we present the
posterior distributions of $A_{\mu}$ and $A_{\bar{\mu}}$ for GW190521, which
seems favor nonzero values for $A_{\mu}$ and $A_{\bar{\mu}}$. Similar results
for constraining parity violation with event GW190521 has also been reported
in Wang:2021gqm . It is mentioned in Wang:2021gqm  that such result may be
caused by the limitations of the existing waveform approximants, such as
systematic errors during merger phase of the waveform, or by the existence of
physical effects such as eccentricity which are not taken into account by the
current waveform approximants. For this reason, we exclude this event in our
analysis in the rest of this paper.

From the posterior distributions of $A_{\mu}$($A_{\bar{\mu}}$) and redshift
$z$ calculated from the selected 46 events, one can convert
$A_{\mu}$($A_{\bar{\mu}}$) and $z$ into $M_{\rm PV}$ ($M_{\rm LV}$). In Fig. 3
we separately plot the marginalized posterior distribution of $M_{\rm
PV}^{-3}$ and $M_{\rm LV}^{-4}$. Then the lower bounds on $M_{\rm PV}$ and
$M_{\rm LV}$ for each individual event can be calculated from the
corresponding posterior distributions. In Table-1, we present the 90% credible
level lower bounds on $M_{\rm PV}$ from the Bayesian inference of individual
event and in Table-2, we present the 90% credible level lower bounds on
$M_{\rm LV}$ from the Bayesian inference of individual event.

Figure 1: Violin plots of the posteriors of the parameter $A_{\mu}$ and
$A_{\bar{\mu}}$. The results are obtained by analyzing the selected 25 GW
events. The region in the posterior between the upper and lower bar denotes
the $90\%$ credible interval.

Figure 2: The distribution of posteriors of the parameter $A_{\mu}$ and
$A_{\bar{\mu}}$ for GW150921.

The parameter $M_{\rm PV}$($M_{\rm LV}$) are universal quantities for all GW
events. One can combine all the individual posterior of $M_{\rm PV}$ ($M_{\rm
LV}$) for each event to get the overall constraint. This can be done by
multiplying the posterior distributions of all these events together through

$\displaystyle P(M_{\rm PV}|\\{d_{i}\\},H)\propto\prod_{i=1}^{N}P(M_{\rm
PV}|d_{i},H),$ (5.1) $\displaystyle P(M_{\rm
LV}|\\{d_{i}\\},H)\propto\prod_{i=1}^{N}P(M_{\rm LV}|d_{i},H),$ (5.2)

where $d_{i}$ denotes data of the $i$ th GW event and N denotes selected
quantities of data. We find that $M_{\rm PV}$ and $M_{\rm LV}$ can be
constrained to be

$\displaystyle M_{\rm PV}>1.0\times 10^{-14}\;{\rm GeV},$ (5.3)

and

$\displaystyle M_{\rm LV}>2.4\times 10^{-16}\;{\rm GeV},$ (5.4)

at 90% confidential level.

##  VI conclusion and outlook

With the discovery of GWs from the coalescence of compact binary systems by
LIGO/Virgo Collaboration, the testing of gravity in the strong gravitational
fields becomes possible. In this paper, we consider the tests of Lorentz and
parity violations arising from high-order spatial derivatives of gravity with
spatial covariance by using the selected GW events in the LIGO-Virgo catalogs
GWTC-1 and GWTC-2. For this purpose we calculate in detail the effects of the
modified dispersion relation and velocity birefringence due to the Lorentz and
parity violations on the GW waveforms. Decomposing the GWs into the left-hand
and right-hand circular polarization modes, we find that the GW effects of
both the Lorentz and parity violation can be explicitly presented by the
modifications in the GW amplitude and phase. We also mapped such amplitude and
phase modification to the parametrized description of Lorentz and parity
violating waveforms proposed in waveform .

With the modified waveforms, we perform the full Bayesian inference with the
help of the open source software BILBY on the selected 47 GW events in the
LIGO-Virgo catalogs GWTC-1 and GWTC-2. From our analysis, we do not find any
signatures of Lorentz violation due to the sixth-order derivatives and parity
violation due to the fifth-order spatial derivatives in the gravitational
action with spatial covariance. We thus place lower bounds on the energy scale
$M_{\rm LV}>2.4\times 10^{-16}\;{\rm GeV}$ for Lorentz violation and $M_{\rm
PV}>1.0\times 10^{-14}\;{\rm GeV}$ 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 spatial covariant
framework by using the observational data of GWs. Note that in deriving the
above conclusions, we have excluded the result of GW190521 since its result
may be significantly biased by waveform templates used in the analysis
Wang:2021gqm ; testGR_GWTC2 .

We observe that the relatively low-mass BBH and BNS events, such as GW170817,
GW190425, GW190728_064510, GW190924_021846 give tighter constraints. Since the
next generation of GW detectors can detect lighter and more distant BBH and
BNS events, it is expected such system can lead to more tighter constraints on
$M_{\rm PV}$ and $M_{\rm LV}$ in the future.

## Acknowledgments

T.Z., Q.W., and A.W. are supported in part by the National Key Research and
Development Program of China Grant No. 2020YFC2201503, and the Zhejiang
Provincial Natural Science Foundation of China under Grant No. LR21A050001 and
No. LY20A050002, the National Natural Science Foundation of China under Grant
No. 11675143 and No. 11975203, and the Fundamental Research Funds for the
Provincial Universities of Zhejiang in China under Grant No. RF-A2019015. R.N.
and W.Z. are supported by NSFC Grants No. 11773028, No. 11633001, No.
11653002, No. 11421303, No. 11903030, the Fundamental Research Funds for the
Central Universities, and the Strategic Priority Research Program of the
Chinese Academy of Sciences Grant No. XDB23010200. J.C is supported by the
Fundamental Research Funds for the Central Universities of China (No.
N180503018). X.Z. is supported by the National Natural Science Foundation of
China (Grants No. 11975072, No. 11835009, and No. 11690021), the Liaoning
Revitalization Talents Program (Grant No. XLYC1905011), and the National 111
Project of China (Grant No. B16009).

Table 2: 90% credible level lower bounds on $M_{\rm LV}$ from the Bayesian inference by analyzing the selected 46 GW events in the LIGO-Virgo catalogs GWTC-1 and GWTC-2. Note that result from GW190521 is excluded. catalogs | GW events | Lower limits [$10^{-17}\;{\rm GeV}$]
---|---|---
GWTC-1 | GW150914 | 4.9
GW151012 | 8.1
GW151226 | 15.7
| GW170104 | 5.1
| GW170608 | 15.6
| GW170729 | 3.9
| GW170809 | 5.1
| GW170814 | 6.9
| GW170817 | 21.1
| GW170818 | 5.2
| GW170823 | 4.8
GWTC-2 | GW190408_181802 | 7.6
GW190412 | 8.5
GW190413_052954 | 4.4
GW190413_134308 | 2.3
GW190421_213856 | 3.5
GW190425 | 11.1
GW190503_185404 | 3.9
GW190512_180714 | 9.0
GW190513_205428 | 5.2
GW190514_065416 | 3.8
GW190517_055101 | 6.5
GW190519_153544 | 3.5
GW190521_074359 | 5.6
GW190527_092055 | 5.3
GW190602_175927 | 2.1
GW190620_030421 | 2.6
| GW190630_185205 | 6.8
| GW190701_203306 | 2.9
| GW190706_222641 | 2.9
| GW190707_093326 | 15.6
| GW190708_232457 | 9.8
| GW190719_215514 | 2.2
| GW190720_000836 | 13.3
| GW190727_060333 | 5.2
| GW190728_064510 | 19.7
| GW190731_140936 | 3.5
| GW190803_022701 | 3.4
| GW190814 | 12.9
| GW190828_063405 | 7.1
| GW190828_065509 | 9.7
| GW190910_112807 | 3.8
| GW190915_235702 | 6.1
| GW190924_021846 | 20.5
| GW190929_012149 | 1.8
| GW190930_133541 | 17.3
combined |  | 24.0

Figure 3: The posterior distributions for $M_{\rm PV}$ and $M_{\rm LV}$ from
selected 46 GW events in the LIGO-Virgo catalogs GWTC-1 and GWTC-2. The
vertical dash line denotes the 90% upper limits for $M_{\rm PV}$ ($M_{\rm
LV}$) from combined results. Note that result from GW190521 is excluded.

## References

  * (1) B. P. Abbott et al. (LIGO Scientific and Virgo Collaborations), Observation of Gravitational Waves from a Binary Black Hole Merger, Phys. Rev. Lett. 116, 061102 (2016) ; GW150914: First results from the search for binary black hole coalescence with Advanced LIGO, Phys. Rev. D 93, 122003 (2016); Properties of the Binary Black Hole Merger GW150914, Phys. Rev. Lett. 116, 241102 (2016); GW150914: The Advanced LIGO Detectors in the Era of First Discoveries, Phys. Rev. Lett. 116, 131103 (2016).
  * (2) B. P. Abbott et al. (LIGO Scientific and Virgo Collaborations), GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral, Phys. Rev. Lett. 119, 161101 (2017).
  * (3) B. P. Abbott et al. (LIGO Scientific and Virgo Collaborations), GW151226: Observation of Gravitational Waves from a 22-Solar-Mass Binary Black Hole Coalescence, Phys. Rev. Lett. 116, 241103 (2016); GW170104: Observation of a 50-Solar-Mass Binary Black Hole Coalescence at Redshift 0.2, Phys. Rev. Lett. 118, 221101 (2017); GW170608: Observation of a 19-solar-mass Binary Black Hole Coalescence, Astrophys. J. Lett. 851, L35 (2017).
  * (4) B. P. Abbott et al. (LIGO Scientific and Virgo Collaborations), GW170814: A Three-Detector Observation of Gravitational Waves from a Binary Black Hole Coalescence, Phys. Rev. Lett. 119, 141101 (2017).
  * (5) B. P. Abbott et al. (LIGO Scientific and Virgo Collaborations), GWTC-1: A Gravitational-Wave Transient Catalog of Compact Binary Mergers Observed by LIGO and Virgo during the First and Second Observing Runs, Phys. Rev. X 9, 031040 (2019).
  * (6) R. Abbott et al. (LIGO Scientific and Virgo Collaborations), GWTC-2: Compact Binary Coalescences Observed by LIGO and Virgo During the First Half of the Third Observing Run, Phys. Rev. X 11, 021053 (2021).
  * (7) R. Abbott et al. (LIGO Scientific, VIRGO and KAGRA Collaborations), GWTC-3: Compact binary coalescences observed by LIGO and Virgo during the second part of the third observing run, arXiv:2111.03606.
  * (8) B. P. Abbott et al. (LIGO Scientific and Virgo Collaborations), Tests of general relativity with GW150914, Phys. Rev. Lett. 116, 221101 (2016); Phys. Rev. Lett. 121, 129902(E) (2018)].
  * (9) B. P. Abbott et al. (LIGO Scientific and Virgo Collaborations), Tests of General Relativity with GW170817, Phys. Rev. Lett. 123, 011102 (2019).
  * (10) B. P. Abbott et al. (LIGO Scientific, Virgo, Fermi-GBM and INTEGRAL Collaborations), Gravitational waves and gamma-rays from a binary neutron star merger: GW170817 and GRB 170817A, Astrophys. J. Lett. 848, L13 (2017).
  * (11) B. P. Abbott et al. (LIGO Scientific and Virgo Collaborations), Tests of general relativity with the binary black hole signals from the LIGO-Virgo catalog GWTC-1, Phys. Rev. D 100, 104036 (2019).
  * (12) R. Abbott et al. (LIGO Scientific and Virgo Collaborations), Tests of general relativity with binary black holes from the second LIGO-Virgo gravitational-wave transient catalog, Phys. Rev. D 103, 122002 (2021).
  * (13) G. Cognola, E. Elizalde, S. Nojiri, S. D. Odintsov, and S. Zerbini, Dark energy in modified Gauss-Bonnet gravity: Late-time acceleration and the hierarchy problem, Phys. Rev. D 73, 084007 (2006).
  * (14) E. J. Copeland, M. Sami, and S. Tsujikawa, Dynamics of dark energy, Int. J. Mod. Phys. D 15, 1753 (2006).
  * (15) J. Frieman, M. Turner, and D. Huterer, Dark energy and the accelerating universe, Annu. Rev. Astron. Astrophys. 46, 385 (2008).
  * (16) M. Li, X. D. Li, S. Wang, and Y. Wang, Dark energy, Commun. Theor. Phys. 56, 525 (2011).
  * (17) P. Horava, Quantum gravity at a Lifshitz point, Phys. Rev. D 79, 084008 (2009).
  * (18) T. Takahashi and J. Soda, Chiral Primordial Gravitational Waves from a Lifshitz Point, Phys. Rev. Lett. 102, 231301 (2009).
  * (19) D. Yoshida and J. Soda, Exploring the string axiverse and parity violation in gravity with gravitational waves, Int. J. Mod. Phys. D 27, 1850096 (2018); Higher curvature corrections to primordial fluctuations in slow-roll inflation, J. Cosmol. Astropart. Phys. 09, (2008) 019; M. Satoh, S. Kanno, and J. Soda, Circular polarization of primordial gravitational waves in string-inspired inflationary cosmology, Phys. Rev. D 77, 023526 (2008); J. Soda, H. Kodama, and M. Nozawa, Parity violation in graviton non-Gaussianity, J. High Energy Phys. 08 (2011) 067.
  * (20) A. Wang, Q. Wu, W. Zhao, and T. Zhu, Polarizing primordial gravitational waves by parity violation, Phys. Rev. D 87, 103512 (2013).
  * (21) T. Zhu, W. Zhao, Y. Huang, A. Wang, and Q. Wu, Effects of parity violation on non-gaussianity of primordial gravitational waves in Hořava-Lifshitz gravity, Phys. Rev. D 88, 063508 (2013).
  * (22) A. Wang, Hořava gravity at a Lifshitz point: A progress report, Int. J. Mod. Phys. D 26, 1730014 (2017).
  * (23) T. Jacobson and D. Mattingly, Gravity with a dynamical preferred frame, Phys. Rev. D 64, 024028 (2001).
  * (24) C. Eling, T. Jacobson, and D. Mattingly, Einstein-Aether theory, arXiv:gr-qc/0410001.
  * (25) T. Jacobson, Einstein-aether gravity: A status report, Proc. Sci. QG-PH2007 2007 020 [arXiv:0801.1547].
  * (26) B. Li, D. F. Mota, and J. D. Barrow, Detecting a lorentz-violating field in cosmology, Phys. Rev. D 77, 024032 (2008).
  * (27) R. A. Battye, F. Pace, and D. Trinh, Cosmological perturbation theory in generalized Einstein-Aether models, Phys. Rev. D 96, 064041 (2017).
  * (28) X. Gao, Higher derivative scalar-tensor theory from the spatially covariant gravity: A linear algebraic analysis, J. Cosmol. Astropart. Phys. 11 (2020) 004.
  * (29) X. Gao and Y. M. Hu, Higher derivative scalar-tensor theory and spatially covariant gravity: The correspondence, Phys. Rev. D 102, 084006 (2020).
  * (30) X. Gao and Z. B. Yao, Spatially covariant gravity theories with two tensorial degrees of freedom: The formalism, Phys. Rev. D 101, 064018 (2020).
  * (31) X. Gao and X. Y. Hong, Propagation of gravitational waves in a cosmological background, Phys. Rev. D 101, 064057 (2020).
  * (32) P. Joshi and S. Panda, Higher derivative scalar tensor theory in unitary gauge, arXiv:2111.11791.
  * (33) V. A. Kostelecký and M. Mewes, Testing local Lorentz invariance with gravitational waves, Phys. Lett. B 757, 510 (2016).
  * (34) Q. G. Bailey and V. A. Kostelecky, Signals for Lorentz violation in post-Newtonian gravity, Phys. Rev. D 74, 045001 (2006).
  * (35) M. Mewes, Signals for Lorentz violation in gravitational waves, Phys. Rev. D 99, 104062 (2019).
  * (36) L. Shao, Combined search for anisotropic birefringence in the gravitational-wave transient catalog GWTC-1, Phys. Rev. D 101, 104019 (2020).
  * (37) Z. Wang, L. Shao, and C. Liu, New limits on the Lorentz/CPT symmetry through fifty gravitational-wave events, Astrophys. J. 921, 158 (2021).
  * (38) V. A. Kostelecký and M. Mewes, Lorentz and diffeomorphism violations in linearized gravity, Phys. Lett. B 779, 136 (2018).
  * (39) R. Jackiw and S. Y. Pi, Chern-Simons modification of general relativity, Phys. Rev. D 68, 104012 (2003).
  * (40) M. Crisostomi, K. Noui, C. Charmousis, and D. Langlois, Beyond Lovelock gravity: Higher derivative metric theories, Phys. Rev. D 97, 044034 (2018).
  * (41) S. Alexander and N. Yunes, Chern-Simons modified general relativity, Phys. Rep. 480, 1 (2009).
  * (42) K. Yagi, N. Yunes, and T. Tanaka, Gravitational Waves from Quasi-Circular Black Hole Binaries in Dynamical Chern-Simons Gravity, Phys. Rev. Lett. 109, 251105 (2012); Phys. Rev. Lett. 116, 169902(E) (2016); Phys. Rev. Lett. 124, 029901(E) (2020).
  * (43) S. H. Alexander and N. Yunes, Gravitational wave probes of parity violation in compact binary coalescences, Phys. Rev. D 97, 064033 (2018).
  * (44) K. Yagi and H. Yang, Probing gravitational parity violation with gravitational waves from stellar-mass black hole binaries, Phys. Rev. D 97, 104018 (2018).
  * (45) A. Nishizawa and T. Kobayashi, Parity-violating gravity and GW170817, Phys. Rev. D 98, 124018 (2018).
  * (46) A. Lue, L. M. Wang, and M. Kamionkowski, Cosmological Signature of New Parity Violating Interactions, Phys. Rev. Lett. 83, 1506 (1999).
  * (47) S. Alexander and J. Martin, Birefringent gravitational waves and the consistency check of inflation, Phys. Rev. D 71, 063526 (2005).
  * (48) N. Bartolo, L. Caloni, G. Orlando, and A. Ricciardone, Tensor non-Gaussianity in chiral scalar-tensor theories of gravity, J. Cosmol. Astropart. Phys. 03 (2021) 073.
  * (49) C. Fu, J. Liu, T. Zhu, H. Yu, and P. Wu, Resonance instability of primordial gravitational waves during inflation in Chern–Simons gravity, Eur. Phys. J. C 81, 204 (2021).
  * (50) S. H. S. Alexander, M. E. Peskin, and M. M. Sheikh-Jabbari, Leptogenesis from gravity waves in models of inflation, Phys. Rev. Lett. 96, 081301 (2006).
  * (51) S. H. S. Alexander, M. E. Peskin, and M. M. Sheikh-Jabbari, Gravi-Leptogenesis: Leptogenesis from Gravity Waves in Pseudo-scalar Driven Inflation Models, arXiv:hep-ph/0701139.
  * (52) S. Hou, T. Zhu, and Z. H. Zhu, Asymptotic analysis of Chern-Simons modified gravity and its memory effects, Phys. Rev. D 105, 024025 (2022).
  * (53) A. Conroy and T. Koivisto, Parity-violating gravity and GW170817 in non-Riemannian cosmology, J. Cosmol. Astropart. Phys. 12 (2019) 016.
  * (54) M. Li, H. Rao, and D. Zhao, A simple parity violating gravity model without ghost instability, J. Cosmol. Astropart. Phys. 11 (2020) 023.
  * (55) M. Li, H. Rao, and Y. Tong, Revisiting a parity violating gravity model without ghost instability: Local Lorentz covariance, Phys. Rev. D 104, 084077 (2021).
  * (56) M. Li and D. Zhao, A simple parity violating model in the symmetric teleparallel gravity and its cosmological perturbations, arXiv:2108.01337.
  * (57) S. Mirshekari, N. Yunes, and C. M. Will, Constraining generic lorentz violation and the speed of the graviton with gravitational waves, Phys. Rev. D 85, 024041 (2012).
  * (58) W. Zhao, T. Zhu, J. Qiao, and A. Wang, Waveform of gravitational waves in the general parity-violating gravities, Phys. Rev. D 101, 024002 (2020); Waveform of gravitational waves in the ghost-free parity-violating gravities, Phys. Rev. D 100, 124058 (2019).
  * (59) M. Okounkova, W. M. Farr, M. Isi, and L. C. Stein, Constraining gravitational wave amplitude birefringence and Chern-Simons gravity with GWTC-2, arXiv:2101.11153.
  * (60) Q. Hu, M. Li, R. Niu, and W. Zhao, Joint observations of space-based gravitational-wave detectors: Source localization and implication for parity-violating gravity, Phys. Rev. D 103, 064057 (2021).
  * (61) R. Nair, S. Perkins, H. O. Silva, and N. Yunes, Fundamental Physics Implications for Higher-Curvature Theories from Binary Black Hole Signals in the LIGO-Virgo Catalog GWTC-1, Phys. Rev. Lett. 123, 191101 (2019).
  * (62) S. Wang and Z. C. Zhao, Tests of CPT invariance in gravitational waves with LIGO-Virgo catalog GWTC-1, Eur. Phys. J. C 80, 1032 (2020).
  * (63) K. Yamada and T. Tanaka, Parametrized test of parity-violating gravity using GWTC-1 events, Prog. Theor. Exp. Phys. 2020, 093E01 (2020).
  * (64) Y. F. Wang, R. Niu, T. Zhu, and W. Zhao, Gravitational wave implications for the parity symmetry of gravity in the high energy region, Astrophys. J. 908, 58 (2021).
  * (65) Y. F. Wang, S. M. Brown, L. Shao, and W. Zhao, Tests of gravitational-wave birefringence with the open gravitational-wave catalog, arXiv:2109.09718.
  * (66) Q. Wu, T. Zhu, R. Niu, W. Zhao, and A. Wang, Constraints on the Nieh-Yan modified teleparallel gravity with gravitational waves, Phys. Rev. D 105, 024035 (2022).
  * (67) R. Abbott et al. (LIGO Scientific and VIRGO Collaborations), GWTC-2.1: Deep extended catalog of compact binary coalescences observed by LIGO and Virgo during the first half of the third observing run, arXiv:2108.01045.
  * (68) T. Zhu, F. W. Shu, Q. Wu, and A. Wang, General covariant Horava-Lifshitz gravity gravity without projectability condition and its applications to cosmology, Phys. Rev. D 85, 044053 (2012).
  * (69) GWTC-2 Data Release Documentation, see the link https://doi.org/10.7935/99gf-ax93.
  * (70) G. Ashton, M. Hübner, P. D. Lasky, C. Talbot, K. Ackley, S. Biscoveanu, Q. Chu, A. Divakarla, P. J. Easter, B. Goncharov et al., BILBY: A user-friendly Bayesian inference library for gravitational-wave astronomy, Astrophys. J. Suppl. Ser. 241, 27 (2019); I. M. Romero-Shaw, C. Talbot, S. Biscoveanu, V. D’Emilio, G. Ashton, C. P. L. Berry, S. Coughlin, S. Galaudage, C. Hoy, and M. Hübner et al., Bayesian inference for compact binary coalescences with BILBY: Validation and application to the first LIGO–Virgo gravitational-wave transient catalogue, Mon. Not. R. Astron. Soc. 499, 3295 (2020).