# Constraints on the Nieh-Yan modified teleparallel gravity with gravitational
waves

Qiang Wua,b Tao Zhua,b corresponding author<EMAIL_ADDRESS>Rui Niuc,d Wen
Zhaoc,d<EMAIL_ADDRESS>Anzhong Wange<EMAIL_ADDRESS>a Institute
for Theoretical Physics and Cosmology, Zhejiang University of Technology,
Hangzhou, 310032, China
b United Center for Gravitational Wave Physics (UCGWP), Zhejiang University of
Technology, Hangzhou, 310032, China
c CAS Key Laboratory for Research in Galaxies and Cosmology, Department of
Astronomy, University of Science and Technology of China, Hefei 230026, China;
d School of Astronomy and Space Sciences, University of Science and Technology
of China, Hefei, 230026, China;
e GCAP-CASPER, Physics Department, Baylor University, Waco, Texas 76798-7316,
USA

###### Abstract

The discovery of gravitational waves (GWs) from the compact binary components
by LIGO/Virgo Collaboration provides an unprecedented opportunity for testing
gravity in the strong and highly dynamical field regime of gravity. Currently
a lot of model-independent tests have been performed by LIGO/Virgo
Collaboration and no any significant derivation from general relativity has
been found. In this paper, we study the parity violating effects on the
propagation of GWs in the Nieh-Yan modified teleparallel gravity, a theory
which modifies general relativity by a parity violating Nieh-Yan term. We
calculate the corresponding parity violating waveform of GWs produced by the
coalescence of compact binaries. By comparing the two circular polarization
modes, we find the effects of the velocity birefringence of GWs in their
propagation caused by the parity violation due to the Nieh-Yan term, which are
explicitly presented in the GW waveforms by the phase modification. With such
phase modifications to the waveform, we perform the full Bayesian inference
with the help of the open source software Bilby on the GW events of binary
black hole merges (BBH) in the LIGO-Virgo catalogs GWTC-1 and GWTC-2. We do
not find any significant evidence of parity violation due to the parity
violating Nieh-Yan term and thus place an upper bound on the energy scale
$M_{\rm PV}<6.5\times 10^{-42}\;{\rm GeV}$ at 90% confidence level, which
represents the first constraint on the Nieh-Yan modified teleparallel gravity
so far.

##  I Introduction

The direct detections of gravitational waves (GWs) emitted from the compact
binary components (CBC) by LIGO/Virgo Collaboration open a new era for
exploring the nature of gravity in the strong and highly-dynamical field
regime of gravity gw150914 ; gw170817 ; gw-other ; LIGOScientific:2017ycc . To
data, there are 50 GW events that have been reported by Advanced Laser
Interferometer Gravitational-Wave Observatory (LIGO) and Advanced Virgo, which
are included in the Gravitational-Wave Transient Catalogs GWTC-1 gwtc1  and
GWTC-2 gwtc2 . In addition, two new GW events, GW200105 and GW200115, have
been released recently, which most likely, are signals from mergers of neutron
star and black hole binaries LIGOScientific:2021qlt . With these events,
various model-independent tests of general relativity (GR) have been performed
by LIGO/Virgo Collaboration and no any significant derivation from GR has been
found gw150914-testGR ; gw170817-testGR ; gw170817-speed ; testGR_GWTC1 ;
testGR_GWTC2 .

Although GR has been considered to be the most successful theory of gravity
since it was proposed, it faces difficulties both theoretically (e.g.,
singularity, quantization, etc), and observationally (e.g., dark matter, dark
energy, etc). Various modified gravities have been proposed to be one of the
effective ways to solve these anomalies MG1 ; MG2 ; MG3 ; MG4 . Therefore, the
tests of the modified gravities are essential to confirm the final theory of
gravity.

As is well known, symmetry permeates nature and is important to all laws of
physics. Thus, one important aspect for tests of gravity is to test its
symmetries. Parity symmetry is one of the fundamental symmetries of GR.
However, it is well known that nature is parity violating, since the first
discovery of parity violation in weak interactions parity_violation . On the
other hand, when one considers the quantization of gravity, such symmetry
could be also violated at high energy regimes. For examples, the parity
violations in gravity can in general arise from the gravitational anomaly of
the standard model of elementary particles, the Green-Schwarz anomaly
canceling mechanism in string theory, or the scalarization of the Barbero-
Immirzi parameter in the loop quantum gravity cs1 ; cs2 , see also cs_review
for a review. In this sense, the parity symmetry can only be treated as an
approximate symmetry, which emerges at low energies and is violated at higher
energies. With these thoughts, a lot of modified gravity theories or
phenomenological models with parity violation in the gravitational interaction
have been proposed, such as the Chern-Simons modified gravity cs_review ;
chern-simons1 ; chern-simons2 ; chern-simons3 ; chern-simons4 ; chern-simons4
; chern-simons5 , the symmetric teleparallel equivalence of GR theory Conroy ,
Horava-Lifshitz theories of quantum gravity horava1 ; horava2 ; horava3 ;
horava4 ; horava5 ; horava6 , chiral scalar-tensor theory chiral_ST ;
chiral_ST1 ; chiral_ST2 , and the standard model extension SME1 ; SME2 ; SME3
; SME4 ; SME5 .

Parity symmetry implies that a directional flipping to the left and right does
not change the laws of physics. The parity violation in gravity in general can
induce an asymmetry of the propagation speed and amplitude damping between
left- and right-hand polarizations of a GW, which leads to the velocity and
amplitude birefringence, respectively. In primordial cosmology, such
birefringence phenomenons can produce circularly polarized primordial GWs,
which leaves the significant imprints in the temperature and polarization
anisotropies of cosmic microwave background radiation (CMB) horava3 ; horava4
; horava5 ; PGW1 ; PGW2 ; PGW3 ; Fu:2020tlw . The detection of GW emitted from
the compact binary components by LIGO-Virgo provides a great opportunity to
test the parity violation in gravity as well. A lot of tests on both the
velocity and amplitude birefringence of GWs have been carried out by using the
observational data from GW events in LIGO-Virgo catalogs SME1 ; chiral_ST1 ;
CS_gb ; SME4 ; SME5 ; Okounkova:2021xjv ; Hu:2020rub ; sai_wang ; tanaka .
Recently, the parity violating effects which induces velocity birefringence
due to the leading-order higher derivative modification in the waveform has
been constrained through the Bayesian parameter estimation on the third Open
Gravitational-wave Catalog events yi-fan1 ; Wang:2021gqm . The parity
violating energy scale due to the leading-order higher derivative modification
has been constrained to be $M_{\rm PV}>0.14\;{\rm GeV}$ at 90% confidence
level, which represents the tightest bound on $M_{\rm PV}$ so far. Here the
$M_{\rm PV}$ denotes the parity violating energy scale from leading high
derivative corrections which corresponds to the case with $\beta_{\mu}=1$ as
defined in waveform . It is worth noting that the parity violating energy
scale $M_{\rm PV}$ in this paper corresponds to $\beta_{\mu}=-1$ waveform .

Recently, a new parity violating gravity model, the Nieh-Yan modified
teleparallel gravity, was proposed in Li:2020xjt ; Li:2021wij . This model is
healthy and simple in form. The Nieh-Yan modified teleparallel gravity is
constructed based on the theory of GR equivalent teleparallel gravity (TEGR)
TEGR  which is formulated in flat spacetime with vanishing curvature and
vanishing nonmetricity (see Ref. Bahamonde:2021gfp  for a recent review). The
TEGR is equivalent to GR and the Nieh-Yan modified teleparallel gravity
modified TEGR by including an extra Nieh-Yan term into the gravitational
action, which breaks the parity symmetry in gravity. It is interesting to
mention here that such a coupling can also appear in the mechanisms NY-1 ;
NY-2  to regularize the infinities in theories of the Einstein-Cartan
manifold, similar to the QCD axion coupling in the Peccei-Quinn mechanism NY-3
for a solution to the strong CP problem. In contrast to other parity violating
gravities which break parity due to high-order derivative terms, the Nieh-Yan
modified teleparallel gravity has no higher derivatives and successfully
avoids the ghost mode. The cosmological perturbations and parametrized post-
Newtonian limit in this theory have also been explored recently in Li:2020xjt
; Li:2021wij ; PPN . Some other modified theories in the framework of
teleparallel gravity or with Nieh-Yan term and their implications in the GW
observations have also been considered in Bombacigno:2021bpk ;
Bahamonde:2017wwk ; Bahamonde:2015zma ; Hohmann:2020dgy ; Zhang:2021kqn .

Since no higher derivative is introduced in the Nieh-Yan modified teleparallel
gravity, the effects of parity violation due to the Nieh-Yan term on the
propagation of GWs is in the lower energy regime. This is in contrast to those
parity violating gravities due to high derivative terms, while the propagation
of GW is modified at high energy regime. This implies that the effect of Nieh-
Yan term is more sensitive to the low frequency GWs. Another important
property of GW in this model is that it only leads to velocity birefringence
and there is no amplitude birefringence. Similar properties of GW can also
arise from the low dimension modifications to GR in the symmetric teleparallel
gravity Li:2021mdp ; Conroy  and the linear gravity of standard model
extension SME1 ; SME3 ; SMExx  111In Refs. SME1 ; SME3 , only the operators
with dimension $d=5$ and $d=6$ are considered. However, as examples, the Nieh-
Yan term considered in this paper or lower dimension term in Li:2021mdp ;
Conroy  show that the inclusion of operators with lower dimension $d=3$ in the
linear gravity of the SME is also possible SMExx . It is worth noting that
such lower dimension operators have been also considered in the standard model
extension of electrodynamics SME-EM .. In this paper, we study in detail the
effects of the velocity birefringence due to the parity violating Nieh-Yan
term on the GWs waveform. Decomposing the GWs into the left-hand and right-
hand circular polarization modes, we find that the effects of velocity
birefringence can be explicitly presented by the modifications in the GW
phase. We also mapped such phase modification to the parametrized description
of 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 46 GW events of BBH in the LIGO-Virgo catalogs
GWTC-1 and GWTC-2. From our analysis, we do not find any signatures of parity
violation due to the parity violating Nieh-Yan term and then place an upper
bound on the energy scale $M_{\rm PV}$ to be $M_{\rm PV}<6.5\times
10^{-42}\;{\rm GeV}$ at 90% confidence level, which represents the first
constraint on the Nieh-Yan modified teleparallel gravity so far.

This paper is organized as follows. In the next section, we present a brief
introduction of the Nieh-Yan modified teleparallel gravity model and then
discuss the propagation of GWs in Sec. III. In Sec. IV, we discuss the
velocity birefringence effects of GWs, and then calculate the waveform of GWs
produced by the coalescence of compact binary systems and particularly focus
on the deviations from those in GR. In Sec. V, we present the basic
statistical framework of Bayesian analysis used in this work and report the
results of constraints on the Nieh-Yan modified teleparallel gravity from the
Bayesian analysis. We finish with concluding remarks and discussions in Sec.
VI.

Throughout this paper, the metric convention is chosen as $(-,+,+,+)$, greek
indices $(\mu,\nu,\cdot\cdot\cdot)$ and latin indices
$(a,b,c,\cdot\cdot\cdot)$ which run over $0,1,2,3$ denote the spacetime and
tangents space respectively, and the latin and latin indices
$(i,\;j,\;k,\;l,\cdot\cdot\cdot)$ which run over $1,2,3$ indicate the spatial
index. We choose the units $G=c=1$.

##  II  The Nieh-Yan modified teleparallel gravity

In this section, we present a brief introduction of the Nieh-Yan modified
teleparallel gravity, for details about this theory, see Li:2020xjt ;
Li:2021wij  and references therein.

The Nieh-Yan modified teleparallel gravity is constructed based on the theory
of teleparallel gravity (TEGR) TEGR  which is equivalent to GR but formulated
in flat spacetime with vanishing curvature and vanishing nonmetricity. In this
theory, the dynamical variable is the tetrad field $e_{a}^{\mu}$. The relation
between the metric $g_{\mu\nu}$ and the tetrad field $e^{a}_{\mu}$ read

$\displaystyle g_{\mu\nu}=e_{\mu}^{a}e_{\nu}^{a}\eta_{ab},$ (2.1)

where $\eta_{ab}=(-1,1,1,1)$ is the Minkowski metric. Therefore at each point
of the spacetime, the tetrad field follows an orthonormal basis for the
tangent field. In the construction of the TEGR, one normally starts from a
flat spacetime where the curvature of the spacetime vanishes and the gravity
is encoded by a nonzero torsion tensor. In this way, the torsion tensor
general depends on both the tetrad field and the spin connection,

$\displaystyle\mathcal{T}^{\lambda}_{\mu\nu}=2e^{\lambda}_{a}(\partial_{[\mu}e_{\nu]}^{a}+\omega^{a}_{b[\mu}e^{b}_{\nu]}),$
(2.2)

where $\omega^{a}_{b\mu}$ is the spin connection which is related to the
Lorentz transformation matrix by

$\displaystyle\omega^{a}_{\;b\mu}=(\Lambda^{-1})^{a}_{\;c}\partial_{\mu}\Lambda^{c}_{\;b},$
(2.3)

and $\omega_{ab\mu}=-\omega_{ba\mu}$ with $\Lambda^{a}_{\;b}$ representing the
element of an arbitrary Lorentz transformation matrix. Here
$\Lambda^{a}_{\;b}$ is position dependent and satisfies the relation
$\eta_{ab}\Lambda^{a}_{\;c}\Lambda^{b}_{\;d}=\eta_{cd}$.  With the tetrad
field and the torsion tensor, the action of the TEGR can be written as

$\displaystyle S_{\rm TEGR}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\int
d^{4}x~{}e\mathcal{T}$ (2.4) $\displaystyle\equiv$ $\displaystyle\int
d^{4}x~{}e\Bigg{(}-\frac{1}{2}\mathcal{T}_{\mu}\mathcal{T}^{\mu}+\frac{1}{8}\mathcal{T}_{\alpha\beta\mu}\mathcal{T}^{\alpha\beta\mu}$
$\displaystyle~{}~{}~{}~{}+\frac{1}{4}\mathcal{T}_{\alpha\beta\mu}\mathcal{T}^{\beta\alpha\mu}\Bigg{)},$

where $e={\rm det}(e^{a}_{\mu})=\sqrt{-g}$ is the determinant of the tetrad,
$\mathcal{T}_{\mu}=\mathcal{T}^{\nu}_{\mu\nu}$ is the torsion vector, and
$\mathcal{T}$ is the torsion scalar. It is interesting to note that the
torsion scalar is related to the Ricci scalar by

$\displaystyle R=-\mathcal{T}+\frac{2}{e}\partial_{\mu}(e\mathcal{T}^{\mu}).$
(2.5)

Therefore, the action of the TEGR (2.4) is identical to the Einstein-Hilbert
action up to a surface term,

$\displaystyle S_{\rm TEGR}=\int
d^{4}x\sqrt{-g}\left[-\frac{1}{2}R(e)-\nabla_{\mu}\mathcal{T}^{\mu}\right].$
(2.6)

In the Nieh-Yan modified teleparallel gravity, the theory of TEGR is modified
by introducing a Nieh-Yan term into the TEGR action Li:2021wij ; Li:2020xjt ,
i.e,

$\displaystyle S_{\rm NY}=\frac{{\color[rgb]{0,0,0}c}}{4}\int
d^{4}x\sqrt{-g}\theta\,\mathcal{T}_{a\mu\nu}\widetilde{\mathcal{T}}^{a\mu\nu},$
(2.7)

where $c$ is the coupling constant,
$\widetilde{\mathcal{T}}^{a\mu\nu}=(1/2)\varepsilon^{\mu\nu\rho\sigma}\mathcal{T}^{a}_{~{}~{}\rho\sigma}$
is the dual of the torsion two form with
$\mathcal{T}^{a}_{\mu\nu}=2(\partial_{[\mu}e_{\nu]}^{a}+\omega^{a}_{b[\mu}e^{b}_{\nu]})$
and $\varepsilon^{\mu\nu\rho\sigma}$ being the Levi-Civita tensor. As
mentioned in Li:2020xjt , the Nieh-Yan term is itself a topological term and
thus it does not contribute to the gravitational dynamics. In general, the
Nieh-Yan term can be split into two individual parity violating terms
Hohmann:2020dgy . To incorporate the parity violation with this two terms in
the gravitational dynamics, only one of them can be added into the action. As
shown in PGG_pv , however, including either terms in the theory leads to a
propagating ghost mode so that such theory is not heathy. Another way to
incorporate the parity violating effects is to consider a coupling between the
Nieh-Yan term and a scalar field $\theta$. With such a scalar field, the
introduction of the Nieh-Yan term (2.7) into the action breaks the parity
symmetry of the gravitational interaction. It is shown in Li:2020xjt  through
perturbative analysis that such simple theory is ghost-free and healthy. As we
already mentioned in the Introduction, such a term can also appear in the
mechanisms NY-1 ; NY-2  to regularize the infinities in theories of the
Einstein-Cartan manifold, similar to the QCD axion coupling in the Peccei-
Quinn mechanism NY-3  for a solution to the strong CP problem.

By taking into account both the kinetic and potential terms of the scalar
field, the full action of the Nieh-Yan modified teleparallel gravity is

$\displaystyle S$ $\displaystyle=$ $\displaystyle S_{\rm TEGR}+S_{\rm
NY}+S_{\theta}+S_{\rm m}$ (2.8) $\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{-g}\Bigg{[}-\frac{R(e)}{2}+\frac{c}{4}\,\theta\,\mathcal{T}_{A\mu\nu}\widetilde{\mathcal{T}}^{A\mu\nu}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}+\frac{\mathfrak{b}}{2}\nabla_{\mu}\theta\nabla^{\mu}\theta-\mathfrak{b}V(\theta)\Bigg{]}+S_{\rm
m},$

where $\mathfrak{b}$ is a coupling constant, the curvature scalar $R(e)$ is
defined by the Levi-Civita connection and considered as being fully
constructed from the metric, and in turn from the tetrad. In writing the above
action, we have dropped all the surface terms arising in (2.6) and (2.7). At
classical level, surface terms do not contribute to the gravitational dynamics
and thus will not affect the analysis presented in this paper. It is worth
mention that the nondynamical surface terms in the gravitational action can
play essential roles in the phase integral approach to quantum gravity, the
calculation of black hole entropy, and the interpretation of energy or mass in
gravity Jimenez:2021nup , etc.

Then variation of the action with respect to the tetrad field $e^{a}_{\mu}$
and Lorentz matrix element $\Lambda^{a}_{\;\;b}$, one obtains,

$\displaystyle G^{\mu\nu}+N^{\mu\nu}$ $\displaystyle=$ $\displaystyle
T^{\mu\nu}+T^{\mu\nu}_{\theta}~{},$ (2.9) $\displaystyle N^{[\mu\nu]}$
$\displaystyle=$ $\displaystyle 0,$ (2.10)

where $G^{\mu\nu}$ is the Einstein tensor, $T^{\mu\nu}=-(2/\sqrt{-g})(\delta
S_{m}/\delta g_{\mu\nu})$ and
$T^{\mu\nu}_{\theta}=\mathfrak{b}[V(\theta)-\nabla_{\alpha}\theta\nabla^{\alpha}\theta/2]g^{\mu\nu}+\nabla^{\mu}\theta\nabla^{\nu}\theta$
are the energy-momentum tensors for the matter and the scalar field $\theta$
respectively, and
$N^{\mu\nu}=c\,e_{a}^{~{}\,\nu}\partial_{\rho}\theta\,\widetilde{\mathcal{T}}^{a\mu\rho}$.
Variation of the action with respect to the scalar field $\theta$ leads to the
equation of motion for the scalar field, which is

$\displaystyle\mathfrak{b}\Big{[}\nabla_{\mu}\nabla^{\mu}\theta+V^{\prime}(\theta)\Big{]}-\frac{c}{4}\mathcal{T}_{a\mu\nu}\tilde{T}^{a\mu\nu}=0.$
(2.11)

Here $V^{\prime}(\theta)=dV(\theta)/d\theta$.

It is mentioned in Li:2020xjt ; Li:2021wij  that Eq. (2.10) is the
antisymmetric part of Eq. (2.9). From (2.10), it is evident that the
antisymmetric part of the field equations simply vanishes thus there is no
antisymmetric degrees of freedom in this theory. This leads to six strong
constraints on the theory. The theory contains 16+6 basic variables, 16 in the
tetrad fields $e_{\mu}^{a}$ and 6 in the Lorentz matrix $\Lambda^{a}_{\;b}$.
In the meantime, the theory also has 4 spacetime diffeomorphisms and 6 local
Lorentz symmetries. Considering the 6 additional constraints given by (2.10),
the theory can have the same physical degrees of freedom as that in GR. At the
linear perturbative level, it is shown Li:2021wij  that both the scalar and
vector modes are not dynamical degrees of freedom and only two dynamical
tensorial modes exist. It is worth mentioning that the degrees of freedom of
the theory may also be hidden in certain backgrounds. This is also known as
the strong coupling problem if such hidden modes do exist strong . It implies
that the perturbative analysis around specific background geometries cannot be
fully trusted strong . The phenomenon of degrees of freedom being hidden under
special backgrounds also appears in $f(T)$ models Ong:2013qja , the models of
massive gravity DeFelice:2012mx , and the Einsteinian cubic gravity
Jimenez:2020gbw .

Similar to the Chern-Simons modified gravity, there are two different versions
of the Nieh-Yan modified teleparallel gravity, the dynamical version and
nondynamical version. The dynamical version corresponds to $\mathfrak{b}\neq
0$, while the non-dynamical version corresponds to $\mathfrak{b}=0$. When
$\mathfrak{b}=0$, the equations of motion now reduce to

$\displaystyle G^{\mu\nu}+N^{\mu\nu}$ $\displaystyle=$ $\displaystyle
T^{\mu\nu},$ (2.12) $\displaystyle N^{[\mu\nu]}$ $\displaystyle=$
$\displaystyle 0~{},$ (2.13)

and the equation of motion for the scalar field now becomes a constraint

$\displaystyle\mathcal{T}_{a\mu\nu}\tilde{T}^{a\mu\nu}=0.$ (2.14)

As shown in Li:2020xjt ; Li:2021wij , the propagation of GWs in both theories
follows the same propagating equation, thus in this paper we will not
distinguish this two versions and take $\mathfrak{b}=1$ hereafter for
simplification.

##  III GWs in the Nieh-Yan modified teleparallel gravity

Let us investigate the propagation of GW in the Nieh-Yan modified teleparallel
gravity with the action given by (2.8). According to Eq. (2.10), the theory
does not contain the antisymmetric perturbations thus we only focus on the
symmetric tensorial perturbations, i.e., two independent modes of GW. We
consider the GWs propagating on a homogeneous and isotropic background. The
spatial metric in the flat Friedmann- Robertson-Walker universe is written as

$\displaystyle g_{ij}=a(\tau)(\delta_{ij}+h_{ij}(\tau,x^{i})),$ (3.1)

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.,

$\displaystyle\partial^{i}h_{ij}=0=h^{i}_{i}.$ (3.2)

The above spatial metric in (3.1) corresponds to the perturbations of the
tetrad fields as

$\displaystyle e_{0}^{0}=a,e_{i}^{0}=0,e^{a}_{0}=0,$ (3.3) $\displaystyle
e_{i}^{a}=a\left(\gamma_{i}^{a}+\frac{1}{2}\gamma^{aj}h_{ij}\right),$ (3.4)

where $\gamma_{i}^{a}$ can be regarded as the spatial tetrad on three-
dimensional spatial hypersurface. For a flat universe one has
$\delta_{ij}=\delta_{ab}\gamma^{a}_{i}\gamma^{b}_{j}$. Here we would like to
mention that the tensor perturbations only come from the tetrad field, and the
spin connection or the local Lorentz matrices do not contribute to the tensor
perturbations.

To proceed further one can substitute the above tetrad fields in to the action
(2.8) and expand the second order in $h_{ij}$. After tedious calculations, one
finds Li:2020xjt ; Li:2021wij ,

$\displaystyle S^{(2)}=\int
d^{4}x\frac{a^{2}}{8}\left(h^{\prime}_{ij}h^{\prime}_{ij}-\partial_{k}h_{ij}\partial^{k}h^{ij}-c\theta^{\prime}\epsilon_{ijk}h_{il}\partial_{j}h_{kl}\right),$

where $\epsilon_{ijk}$ is the antisymmetric symbol and a prime denotes the
derivative with respect to the conformal time $\tau$. We consider the GWs
propagating in the vacuum, and ignore the source term. Varying the action with
respect to $h_{ij}$, we obtain

$\displaystyle
h^{\prime\prime}_{ij}+2\mathcal{H}h^{\prime}_{ij}-\partial^{2}h_{ij}+\frac{1}{2}c\theta^{\prime}(\epsilon_{lki}\partial_{l}h_{jk}+\epsilon_{lkj}\partial_{l}h_{ik})=0,$

where $\mathcal{H}\equiv a^{\prime}/a$.

In the parity-violating gravities, 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

$\displaystyle\epsilon^{ijk}n_{i}e_{kl}^{A}=i\rho_{A}e^{jA}_{l},$ (3.8)

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
standard parametrized form proposed in waveform ,

$\displaystyle
h^{\prime\prime}_{A}+(2+\nu_{A})\mathcal{H}h^{\prime}_{A}+(1+\mu_{A})k^{2}h_{A}=0,$
(3.9)

where

$\displaystyle\nu_{A}=0,\;\;\;\mu_{A}=\frac{\rho_{A}c\theta^{\prime}}{k}.$
(3.10)

In the above parametrization, the effects of the parity violation are fully
characterized by two parameters: $\nu_{A}$ and $\mu_{A}$. As shown in refs.
waveform , such parametrization provides an unifying description for the low-
energy effective description of GWs in generic parity violating gravities,
including Chern-Simons modified gravity, ghost-free parity violating scalar-
tensor theory, symmetric teleparallel equivalence of general relativity,
Hořava-Lifshitz gravities, and the Nieh-Yan modified teleparallel gravity. The
parameter $\mu_{A}$ leads to different velocities of left-hand and right- hand
circular polarizations of GWs, so that the arrival times of the two circular
polarization modes could be different. The parameter $\nu_{A}$, on the other
hand, leads to different damping rates of left-hand and right- hand circular
polarizations of GWs, so that the amplitude of left-hand circular polarization
of gravitational waves will increase (or decrease) during the propagation,
while the amplitude for the right-hand modes will decrease (or increase). In
the Nieh-Yan modified teleparallel gravity, we have $\nu_{A}=0$ and
$\mu_{A}=\rho_{A}c\theta^{\prime}/k$, therefore there is no modification on
the damping rate of GWs and the parity violation due to the Nieh-Yan term can
only affect the velocities of GWs. This is the phenomenon of velocity
birefringence. It is worth noting here that similar corrections on
$\mu_{A}\propto 1/k$ can also arise from the lower dimension operators in the
parity violating symmetric teleparallel equivalence of GR theory Conroy ;
Li:2021mdp .

##  IV Velocity birefringence and phase modifications to the waveform of GWs

In this section, we study the velocity birefringence effects during the
propagation of GWs. As shown in waveform , for each circular polarization mode
$h_{A}$, the velocity birefringence effect induces the phase corrections to
the waveform of GWs. In order to derive the phase modification, similar to our
previous works waveform , let us first define
$u_{A}(\tau)=\frac{1}{2}a(\tau)M_{\rm Pl}h_{A}(\tau)$, and then the equation
of motion (3.9) can be casted into the form

$\displaystyle\frac{d^{2}u_{A}}{d\tau^{2}}+\left(\omega_{A}^{2}-\frac{a^{\prime\prime}}{a}\right)h_{A}=0,$
(4.1)

where

$\displaystyle\omega_{A}=k^{2}\left(1+\rho_{A}\frac{c\theta^{\prime}}{k}\right),$
(4.2)

is the modified dispersion relation. Then, one can find that GWs with
different helicities will have different phase velocities

$\displaystyle v_{A}\simeq 1+\rho_{A}\frac{c\theta^{\prime}}{2k}.$ (4.3)

Since $\rho_{A}$ has the opposite signs for left-hand and right-hand
polarization modes, it is straightforward to see that the phase velocities for
these two modes are different. For later convenience, one can introduce a
characteristic energy scale $M_{\rm PV}=c\theta^{\prime}/a=c\dot{\theta}$, and
then one has

$\displaystyle v_{A}=1+\rho_{A}\frac{aM_{\rm PV}}{2k}.$ (4.4)

Now considering a graviton emitted radially at $r=r_{e}$ and received at
$r=0$, we have,

$\displaystyle\frac{dr}{dt}=-\frac{1}{a}\left[1+\rho_{A}\frac{aM_{\rm
PV}}{2k}\right].$ (4.5)

Note that in the above we have assumed $c\dot{\theta}$ to be a constant.
Integrating this equation from the emission time ($r=r_{e}$) to arrival time
($r=0$), one obtains

$\displaystyle r_{e}=\int_{t_{e}}^{t_{0}}\frac{dt}{a(t)}+\rho_{A}\frac{M_{\rm
PV}}{2k}\int_{t_{e}}^{t_{0}}dt.$ (4.6)

Consider gravitons emitted at two different times $t_{e}$ and
$t_{e}^{\prime}$, with wave numbers $k$ and $k^{\prime}$, and received at
corresponding arrival times $t_{0}$ and $t_{0}^{\prime}$ ($r_{e}$ is the same
for both). Assuming $\Delta t_{e}\equiv t_{e}-t_{e}^{\prime}\ll a/\dot{a}$,
then the difference of their arrival times is given by

$\displaystyle\Delta t_{0}=(1+z)\Delta
t_{e}+\frac{\rho_{A}}{2}\left(\frac{M_{\rm PV}}{k^{\prime}}-\frac{M_{\rm
PV}}{k}\right)\int_{t_{e}}^{t_{0}}dt,$ (4.7)

where $z=1/a(t_{e})-1$ is the cosmological redshift.

Let us turn to consider the GW signal emitted from the nonspinning,
quasicircular inspiral of compact binary system in the post-Newtonian
approximation. Relative to the GW in GR, the parity violation due to the Nieh-
Yan term modifies the phase of GWs. In the Fourier domain, $h_{A}$ can be
calculated analytically in the stationary phase approximation, which is given
by

$\displaystyle h_{A}(f)=\frac{{\cal A}_{A}(f)}{\sqrt{df/dt}}e^{i\Psi(f)},$
(4.8)

where $f$ is the frequency at the detector and $\Psi$ denotes the phase of the
GWs. As shown in waveform , the difference of arrival times induces the
modification of GW phase $\Psi$ as follows,

$\displaystyle\Psi_{A}(f)=\Psi^{\rm GR}_{A}(f)+\rho_{A}\delta\Psi_{1}(f),$
(4.9)

where

$\displaystyle\delta\Psi_{1}(f)=A_{\mu}\ln u$ (4.10)

with

$\displaystyle A_{\mu}=\frac{M_{\rm
PV}}{2H_{0}}\int_{0}^{z}\frac{dz}{(1+z)\sqrt{(1+z)^{3}\Omega_{m}+\Omega_{\Lambda}}}.$
(4.11)

Here $u=\pi{\cal M}f$ with $f=k/2\pi$ being the frequency of GWs and ${\cal
M}=(1+z){\cal M}_{\rm c}$, where ${\cal
M}_{c}\equiv(m_{1}m_{2})^{3/5}/(m_{1}+m_{2})^{1/5}$ is the chirp mass of the
binary system with component masses $m_{1}$ and $m_{2}$. We adopt a Planck
cosmology with $\Omega_{m}=0.315$, $\Omega_{\Lambda}=0.685$, and
$H_{0}=67.4\;{\rm km}\;{\rm s}^{-1}\;{\rm Mpc}^{-1}$. With the above phase
correction, one can write the waveform of GWs with the effects of the Nieh-Yan
term in the form

$\displaystyle h_{A}(f)=h_{A}^{\rm GR}(f)e^{i\rho_{A}\delta\Psi_{1}}.$ (4.12)

In order to test the Nieh-Yan modified teleparallel gravity with observations
of GWs, it is convenient to analyze the GWs in the Fourier domain. The
responses of detectors for the GW signals $h(f)$ in the Fourier domain can be
written in terms of waveforms of $h_{+}$ and $h_{\times}$ as

$\displaystyle h(f)=[F_{+}h_{+}(f)+F_{\times}h_{\times}(f)]e^{-2i\pi
f\triangle t},$ (4.13)

where $F_{+}$ and $F_{\times}$ are the beam pattern functions of GW detectors,
depending on the source location and polarization angle GR_wave . $\triangle
t$ is the arrival time difference between the detector and the geocenter. In
GR, the waveform of the two polarizations $h_{+}(f)$ and $h_{\times}(f)$ are
given respectively by GR_wave2

$\displaystyle h^{\rm GR}_{+}=(1+\chi^{2})\mathcal{A}e^{i\Psi},~{}~{}h^{\rm
GR}_{\times}=2\chi\mathcal{A}e^{i(\Psi+\pi/2)},$ (4.14)

where $\mathcal{A}$ and $\Psi$ denote the amplitude and phase of the waveforms
$h^{\rm GR}_{+,\times}$, and $\chi=\cos\iota$ with $\iota$ being the
inclination angle of the binary system. In GR, the explicit forms of
$\mathcal{A}$ and $\Psi$ have been calculated in the high-order PN
approximation (see for instance GR_wave2  and references therein).

Now using the relationship between $h_{+,\times}$ and $h_{\rm R,L}$, i.e.,

$\displaystyle h_{+}=\frac{h_{\rm L}+h_{\rm
R}}{\sqrt{2}},\;\;h_{\times}=\frac{h_{\rm L}-h_{\rm R}}{\sqrt{2}i},$ (4.15)

one can obtain the effects of the Nieh-Yan term on the above waveform for the
plus and cross modes. It is not difficult to get

$\displaystyle h_{+}=h_{+}^{\rm GR}\cos(\delta\Psi_{1})+h_{\times}^{\rm
GR}\sin(\delta\Psi_{1}),$ (4.16) $\displaystyle h_{\times}=h_{\times}^{\rm
GR}\cos(\delta\Psi_{1})-h_{+}\sin(\delta\Psi_{1}).$ (4.17)

Therefore, the Fourier waveform $h(f)$ becomes

$\displaystyle h(f)=\mathcal{A}\delta\mathcal{A}e^{i(\Psi+\delta\Psi)},$
(4.18)

where

$\displaystyle\delta\mathcal{A}$ $\displaystyle=$
$\displaystyle\sqrt{(1+\chi^{2})^{2}F^{2}_{+}+4\chi^{2}F^{2}_{\times}}$
$\displaystyle\times\left[1-\frac{(1-\chi^{2})^{2}F_{+}F_{\times}}{(1+\chi^{2})^{2}F^{2}_{+}+4\chi^{2}F^{2}_{\times}}\delta\Psi_{1}\right],$
$\displaystyle\delta\Psi$ $\displaystyle=$
$\displaystyle\tan^{-1}\left[\frac{2\chi
F_{\times}}{(1+\chi^{2})F_{+}}\right]$ (4.19)
$\displaystyle+\frac{2\chi(1+\chi^{2})(F^{2}_{+}+F^{2}_{\times})}{(1+\chi^{2})^{2}F^{2}_{+}+4\chi^{2}F^{2}_{\times}}\delta\Psi_{1}.$

##  V Constraints on Nieh-Yan term

###  V.1 Bayesian inference for GW data

In this paper we use the Bayesian inference to obtain the constraints on the
Nieh-Yan term with selected GW events from the LIGO-Virgo catalogs GWTC-1 and
GWTC-2. The Bayesian inference framework has been broadly employing in the
inference of scientific conclusions from GW data. Given a set of compact
binary merger observations from GW detectors with the data $d$ of GW signals,
one in general can infer the distribution of parameters $\vec{\theta}$ by
comparing the predicted GW strain in each instrument with the data. One can
write the Bayes’ theorem in the context of GW astronomy as:

$\displaystyle
P({\vec{\theta}}|d,H)=\frac{P(d|\vec{\theta},H)P(\vec{\theta}|H)}{P(d|H)},$
(5.1)

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 for obtaining the data given a specific set of model parameters,
$P(d|H)$ is the marginalized likelihood or evidence for the model or
“hypothesis” $H$, and $P({\vec{\theta}}|d,H)$ denotes the posterior
probability distributions for physical parameters ${\vec{\theta}}$ which
describe the observational data.

In general, the GW signals from the compact binary mergers are extremely weak
and the matched filtering method has been used to extract these signals from
the noises. Considering Gaussian and stationary noise from GW detectors, the
matched filtering method allows to define the likelihood function in the form
of

$\displaystyle\ln P(d|{\vec{\theta}},H)=-\frac{1}{2}\sum_{j=1}^{N}\langle
d_{j}-h({\vec{\theta}})|d_{j}-h({\vec{\theta}})\rangle,$ (5.2)

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

$\displaystyle\langle A|B\rangle=4\;{\rm
Re}\left[\int_{0}^{\infty}\frac{A(f)B(f)^{*}}{S(f)}df\right],$ (5.3)

where $\;{}^{*}$ denotes complex conjugation and $S(f)$ is the power spectral
density (PSD) function of the detector. In Bayes’ theorem, the Bayes’ evidence
$P(d|H)$ in (5.1) can be computed by integrating the likelihood
$P(d|{\vec{\theta}},H)$ of parameters ${\vec{\theta}}$ times the prior
probabilities $P({\vec{\theta}}|H)$ for these parameters within the waveform
model $H$,

$\displaystyle P(d|H)=\int
d{\vec{\theta}}P(d|{\vec{\theta}},H)P({\vec{\theta}}|H).$ (5.4)

In this work, in order to constrain the Nieh-Yan modified teleparallel
gravity, we employ the open-source package Bilby to perform the Bayesian
inference by analyzing the GW data from selected events of binary black hole
mergers in the LIGO-Vrigo catalogs GWTC-1 and GWTC-2. For the GR waveform
$h^{\rm GR}_{+,\;\times}(f)$, we use the spin precessing waveform IMRPhenomPv2
Schmidt:2014iyl ; Hannam:2013oca  for all the BBH events except GW190521 with
the parameter vector ${\vec{\theta}}=\\{\alpha,\delta,\psi,\phi,t_{\rm
c},d_{L},{\cal
M},\eta,a_{1},a_{2},\cos\theta_{1},\cos\theta_{2},\phi_{12},\phi_{\rm
JL},\theta_{JN}\\}$, where $\alpha$ and $\delta$ are the right ascension and
declination of the binary system in the sky, $\psi$ is the polarization angle
of the source defined with respect to the Earth centered coordinates, $\phi$
is the binary phase at a reference frequency, $t_{\rm c}$ is the time of
coalescence, $d_{L}$ is the luminosity distance to the source, ${\cal M}$ is
the detector-frame chirp mass of the binary, $\eta$ is the symmetric mass
ratio, $a_{1}(a_{2})$ is the dimensionless spin magnitude of the larger
(smaller) black hole, $\theta_{1}(\theta_{2})$ is the angle between the spin
direction and the orbital angular momentum of the binary for the larger
(smaller) BH, $\phi_{12}$ is the difference between total and orbital angular
momentum azimuthal angles, $\phi_{\rm JL}$ denotes the difference between the
azimuthal angles of the individual spin direction projections onto the orbital
plane, and $\theta_{JN}$ represents the angle between the total angular
momentum and the line of sight. For GW190521, we use the state-of-the-art
approximant IMRPhenomXPHM which includes the subdominant harmonic modes of GW
and accounts for spin-precession effects for a quasicircular-orbit binary
black hole coalescence Pratten:2020ceb .

In order to constrain the parity violating Nieh-Yan term, we construct the
parity-violating waveform based on the above template through (4.16) and
(4.17) with $\delta\Psi_{1}$ being given by (4.10). For this purpose we append
one additional parameter $A_{\mu}$, which represents the effects of the parity
violation due to the Nieh-Yan modified teleparallel gravity, in addition to
the parameter vector ${\vec{\theta}}$. Then we consider a series of GW events
comprised of data $\\{d_{i}\\}$, described by parameters
$\\{{\vec{\theta}_{i}}\\}$, where $i$ runs from 1 to $N$ with $N$ being the
number of the analyzed GW events in the Bayesian inference. Then for each
event, posterior for all the parameters describing the event can be written as

$\displaystyle P(A_{\mu},{\vec{\theta}_{i}}|d_{i})=\frac{P(M_{\rm
PV},\vec{\theta}_{i})P(d_{i}|M_{\rm PV},\vec{\theta}_{i})}{P(d_{i})}.$ (5.5)

To infer the posterior of the parameter $A_{\mu}$, one can marginalize over
all parameters $\vec{\theta}_{i}$ for the individual GW events. This procedure
gives the marginal posterior distribution on $A_{\mu}$ for the $i$th GW event,

$\displaystyle P(A_{\mu}|d_{i})$ $\displaystyle=$
$\displaystyle\frac{P(A_{\mu})}{P(d_{i})}\int
P(\vec{\theta}_{i})P((d_{i}|A_{\mu},\vec{\theta}_{i})d\vec{\theta_{i}}.$ (5.6)

###  V.2 Results of constraints

To data, a total of 50 GW events have been reported by the LIGO-Virgo catalogs
GWTC-1 and GWTC-2. Among these events, not all of them are of our interest for
the Bayesian analysis. For our purpose, we consider all 46 GW events of BBH as
presented in Table. 1. We exclude the GW events of binary neutron star or
possible binary neutron star-black hole merges, since these events are not
expected to improve the constraint on $M_{\rm PV}$ drastically. The data of
these 46 GW events are downloaded from the Gravitational Wave Open Science
Center data_GW . Besides strain data, power spectral densities (PSDs) are also
needed for parameter estimation. Instead of directly estimating PSDs from
strain data by the Welch method, we use the event-specific PSDs which are
encapsulated in LVC posterior sample releases for specific events data_GW2 ;
data_GW1 . These PSDs are expected to lead to more stable and reliable
parameter estimation PSD1 ; PSD2 .

Table 1: 90% credible level upper bounds on $M_{\rm PV}$ from the Bayesian inference by analyzing 46 GW events of BBH in the LIGO-Virgo catalogs GWTC-1 and GWTC-2. catalogs | GW events | Constraints [$10^{-41}\;{\rm GeV}$]
---|---|---
GWTC-1 | GW150914 | 6.5
GW151012 | 6.9
GW151226 | 23.5
GW170104 | 16.2
GW170608 | 17.2
GW170729 | 2.9
GW170809 | 6.4
GW170814 | 7.2
GW170818 | 4.5
GW170823 | 3.4
GWTC-2 | GW190408_181802 | 3.6
GW190412 | 7.0
GW190413_052954 | 6.6
GW190413_134308 | 3.2
GW190421_213856 | 6.9
GW190424_180648 | 2.5
GW190503_185404 | 3.7
GW190512_180714 | 4.2
GW190513_205428 | 3.8
GW190514_065416 | 3.9
GW190517_055101 | 9.1
GW190519_153544 | 5.0
GW190521 | 4.5
GW190521_074359 | 3.9
GW190527_092055 | 5.1
GW190602_175927 | 3.1
GW190620_030421 | 3.4
GW190630_185205 | 6.3
GW190701_203306 | 8.7
GW190706_222641 | 5.7
GW190707_093326 | 13.2
GW190708_232457 | 11.5
GW190719_215514 | 7.8
GW190720_000836 | 4.4
GW190727_060333 | 3.3
GW190728_064510 | 9.1
GW190731_140936 | 4.4
GW190803_022701 | 4.4
GW190828_063405 | 3.0
GW190828_065509 | 9.6
GW190909_114149 | 10.3
GW190910_112807 | 2.6
GW190915_235702 | 3.0
GW190924_021846 | 36.7
GW190929_012149 | 11.4
GW190930_133541 | 21.6
| Combined | 0.65

For these events, we perform parameter estimations using Bayesian analysis by
selecting $4\;{\rm s}$ or $8\;{\rm s}$ data over all GW parameters
$\vec{\theta}$ and the parity violating parameter $A_{\mu}$. The prior for the
standard GW parameters ${\vec{\theta}}$ are consistent with those used in
gwtc1 ; gwtc2 . The prior for $A_{\mu}$ is chosen to be uniformly distributed.
We use the package BILBY bilby  to perform the analysis and the posterior
distribution is sampled by the nest sampling method dynesty over the fiducial
BBH and the parity violating parameter $A_{\mu}$. We report our main results
in the next subsection.

For all the 46 GW events we analyzed, we find that the posterior distribution
of $A_{\mu}$ are all consistent with its GR value $A_{\mu}=0$, which means we
do not find any signatures of parity violation of Nieh-Yan modified
teleparallel gravity in the data of these events. To illustrate the results of
$A_{\mu}$ from each individual GW event, we plot Fig. 1 to show the
marginalized posterior distribution of $A_{\mu}$. In this figure, the region
in the posterior between the upper and lower bar denotes the $90\%$ credible
interval, and the bar at the middle denotes the median value. It is shown that
the GR value $A_{\mu}=0$ is well within the 90% confidence level for each GW
event.

From the posterior distributions of $A_{\mu}$ and redshift $z$ calculated from
the 46 events, one can convert $A_{\mu}$ and $z$ into $M_{\rm PV}$ through Eq.
(4.11). In Fig. 2 we plot the marginalized posterior distribution of $M_{\rm
PV}$. Then the upper bounds on $M_{\rm PV}$ for each individual event can be
calculated from the corresponding posterior distribution of $M_{\rm PV}$. In
Table. 1, we present the 90% credible level upper bounds on $M_{\rm PV}$ from
the Bayesian inference of each event. From Table. 1, one can see that the best
constraints on $M_{\rm PV}$ are all from those events with ${\cal M}_{\rm
c}\gtrsim 30M_{\odot}$.

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

$\displaystyle P(M_{\rm PV}|\\{d_{i}\\},H)\propto\prod_{i=1}^{N}P(M_{\rm
PV}|d_{i},H),$ (5.7)

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

$\displaystyle M_{\rm PV}<6.5\times 10^{-42}\;{\rm GeV}$ (5.8)

at 90% confidence level. We thus conclude that we do not find any significant
evidence of parity violation due to the Nieh-Yan modified teleparallel gravity
at $M_{\rm PV}>6.5\times 10^{-42}\;{\rm GeV}$. This constraint in turn can
convert into a constraint on parameter $c\dot{\theta}$ as

$\displaystyle c\dot{\theta}<6.5\times 10^{-42}\;{\rm GeV}.$ (5.9)

So far, this upper bound represents the only observational constraint on the
Nieh-Yan modified teleparallel gravity.

Figure 1: Violin plots of the posteriors of the parameter $A_{\mu}$. The
results are obtained by analyzing the 46 GW events of BBH in the catalogs
GWTC-1 and GWTC-2. The region in the posterior between the upper and lower bar
denotes the $90\%$ credible interval.
Figure 2: The posterior distributions for $M_{\rm PV}$ from 46 GW events in
the LIGO-Virgo catalogs GWTC-1 and GWTC-2. The solid black curve represents
the combined posterior distribution of $M_{\rm PV}$ and the vertical dash line
denotes the 90% upper limits for $M_{\rm PV}$ from combined results.

##  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. Therefore, the studies of GWs in the alternative
theories of gravity and inference of their constraints from data of GW events
detected by LIGO/Virgo Collaboration are of crucial importance for
understanding gravity under extreme conditions. In this paper, we focus on a
new parity-violating gravity model, the Nieh-Yan modified teleparallel gravity
Li:2020xjt ; Li:2021wij . This model is healthy and simple in form which
modifies the TEGR by a parity violating Nieh-Yan term. Such a term can arise
from the mechanisms to regularize the infinities in theories of the Einstein-
Cartan manifold as mentioned in NY-1 ; NY-2 . In contrast to other parity
violating gravities which break parity due to high-order derivative terms, the
Nieh-Yan modified teleparallel gravity has no higher derivatives and
successfully avoids the ghost mode.

In order to test this model with GWs, we study in detail the effects of the
velocity birefringence due to the parity violating Nieh-Yan term on the GW
waveforms. Decomposing the GWs into the left-hand and right-hand circular
polarization modes, we find that the effects of velocity birefringence can be
explicitly presented by the modifications in the GW phase. We also mapped such
phase modification to the parametrized description of 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
46 GW events of BBH in the LIGO-Virgo catalogs GWTC-1 and GWTC-2. From our
analysis, we do not find any signatures of parity violation due to the parity
violating Nieh-Yan term and then place an upper bound on the energy scale
$M_{\rm PV}$ to be $M_{\rm PV}<6.5\times 10^{-42}\;{\rm GeV}$ at 90%
confidence level, which represents the first constraints on the Nieh-Yan
modified teleparallel gravity so far.

The above constraint on $M_{\rm PV}$ can be straightforwardly mapped to bound
on the lower dimensional parity violating terms in the symmetric teleparallel
gravity or possible operators with dimension $d=3$ in the linear gravity of
the standard model extension. In the framework of the symmetric teleparallel
gravity, one can modify the GR equivalent symmetric teleparallel gravity by a
parity violating interaction, $c\phi Q\tilde{Q}$, here
$Q\tilde{Q}=\epsilon^{\mu\nu\rho\sigma}Q_{\mu\nu\alpha}Q_{\rho\sigma}^{\alpha}$
with $Q_{\mu\nu\sigma}$ being the nonmetricity tensor Li:2021mdp ; Conroy . It
is worth noting that such a parity violating interaction can lead to the ghost
problem in the vector perturbations Li:2021mdp . The propagating equation of
GWs in the symmetric teleparallel gravity with the parity violating term
$c\phi Q\tilde{Q}$ shares exactly the same form of (3.9) with $\nu_{A}=0$ and
$\mu_{A}=\rho_{A}c\dot{\phi}/(ak)$ Li:2021mdp . This corresponds to $M_{\rm
PV}=c\dot{\phi}$, thus one obtains

$\displaystyle|c\dot{\phi}|<6.5\times 10^{-42}\;{\rm GeV}.$ (6.1)

With the constant upgrading of the advanced LIGO and Virgo detectors, we
expect more GW events, especially those with heavier chirp mass and higher
redshifts, will be detected in the future. With more data, we are expected to
improve significantly the constraint on $M_{\rm PV}$ and the corresponding
modified gravities 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 Grants 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.
W.Z. and R.N. 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.

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