# Observation of Dirac surface states in the hexagonal PtBi2, a possible
origin of the linear magnetoresistance

S. Thirupathaiah  Institute for Solid State Research, IFW Dresden, D-01171
Dresden, Germany.  Solid State and Structural Chemistry Unit, Indian Institute
of Science, Bangalore, Karnataka, 560012, India.  Y. Kushnirenko  Institute
for Solid State Research, IFW Dresden, D-01171 Dresden, Germany.  E. Haubold
Institute for Solid State Research, IFW Dresden, D-01171 Dresden, Germany.  A.
V. Fedorov  Institute for Solid State Research, IFW Dresden, D-01171 Dresden,
Germany.  E. D. L. Rienks  Institute for Solid State Research, IFW Dresden,
D-01171 Dresden, Germany.  T. K. Kim  Diamond Light Source, Harwell Campus,
Didcot, OX11 0DE, UK.  A. N. Yaresko  Max-Planck-institute for Solid State
Research, Heisenbergstrasse 1, 70569 Stuttgart, Germany.  C. G. F. Blum
Institute for Solid State Research, IFW Dresden, D-01171 Dresden, Germany.  S.
Aswartham  Institute for Solid State Research, IFW Dresden, D-01171 Dresden,
Germany.  B. Büchner  Institute for Solid State Research, IFW Dresden, D-01171
Dresden, Germany.  S. V. Borisenko  Institute for Solid State Research, IFW
Dresden, D-01171 Dresden, Germany.

(September 2, 2024)

###### Abstract

The nonmagnetic compounds showing extremely large magnetoresistance are
attracting a great deal of research interests due to their potential
applications in the field of spintronics. PtBi2 is one of such interesting
compounds showing large linear magnetoresistance (MR) in its both the
hexagonal and pyrite crystal structure. We use angle-resolved photoelectron
spectroscopy (ARPES) and density functional theory (DFT) calculations to
understand the mechanism of liner MR observed in the hexagonal PtBi2. Our
results uncover for the first time linear dispersive surface Dirac states at
the $\bar{\Gamma}$-point, crossing Fermi level with node at a binding energy
of $\approx$ 900 meV, in addition to the previously reported Dirac states at
the $\bar{M}$-point in the same compound. We further notice from our dichroic
measurements that these surface states show an asymmetric spectral intensity
when measured with left and right circularly polarized light, hinting at a
substantial spin polarization of the bands. Following these observations, we
suggest that the linear dispersive Dirac states at the $\bar{\Gamma}$ and
$\bar{M}$-points are likely to play a crucial role for the linear field
dependent magnetoresistance recorded in this compound.

The property of magnetoresistance in the nonmagnetic solids has majorly two
types of behaviour under the external magnetic fields. While the linear field
dependent magnetoresistance as observed in graphene Singh  _et al._ (2012),
topological insulators Bi2Te3 and Bi2Se3 Qu  _et al._ (2010); Wang  _et al._
(2012), Ag2+δTe/Se Xu  _et al._ (1997); Lee  _et al._ (2002), Dirac semimetals
Cd3As2 Liang  _et al._ (2014); Feng  _et al._ (2015) and Na3Bi Kushwaha  _et
al._ (2015), and type-I Weyl semimetals NbAs(P) Ghimire  _et al._ (2015);
Shekhar  _et al._ (2015); Wang  _et al._ (2016) and TaAs Huang  _et al._
(2015); Zhang  _et al._ (2015) is explained based on the linear dispersive
Dirac states present near the Fermi level Abrikosov (1998, 2003), the
quadratic field dependent MR as observed in the type-II Weyl semimetals WTe2
and MoTe2 Ali  _et al._ (2014); Zhou  _et al._ (2016), LaSb Zeng  _et al._
(2016), ZrSiS Lv  _et al._ (2016), and Bi Yang (1999) is explained based on
the charge compensation Ali  _et al._ (2014). However, the role of charge
compensation for the XMR of these compounds is still not clear as some reports
recently demonstrated quadratic field dependent MR without charge balance
Zandt  _et al._ (2007); Thirupathaiah  _et al._ (2017). Moreover, the theory
based on the impurity scattering or crystal disorder is inevitably brought
into the above two cases and as well to the subquadratic field dependent MR
Fatemi  _et al._ (2017); Parish and Littlewood (2003); Ping  _et al._ (2014).
The other existing mechanisms include metal-insulator transition Khveshchenko
(2001); Kopelevich  _et al._ (2006); Wang  _et al._ (2014); Zhao  _et al._
(2015); Xiang  _et al._ (2015) and strong spin-orbit interactions Jiang  _et
al._ (2015); Das  _et al._ (2016). Thus, there is no clear consensus yet on
the mechanism of magnetoresistance in the nonmagnetic solids.

PtBi2 is a very interesting compound as it shows MR in both the hexagonal Yang
_et al._ (2016) and pyrite structures Gao  _et al._ (2016). Astonishingly, the
pyrite PtBi2 has been found to show the highest MR observed among the known
nonmagnetic metals so far Xu  _et al._ (1997); Lee  _et al._ (2002); Liang
_et al._ (2014); Ali  _et al._ (2014); Feng  _et al._ (2015); Shekhar  _et
al._ (2015); Zhou  _et al._ (2016), which also has been predicted as a
3-dimensional Dirac semimetal Gibson  _et al._ (2015). Therefore, understating
the mechanism of extremely large MR in this compound is essential due to its
potential applications. While in the pyrite structure the extremely large MR
is attributed to its semimetalic nature Gao  _et al._ (2016), the large linear
MR observed in the hexagonal phase is explained based on the crystal disorder
theory Yang  _et al._ (2016); Yao  _et al._ (2016). Here, we report on the
electronic structure of hexagonal PtBi2 using the ARPES technique and first-
principles calculations. Our studies suggest that PtBi2 has two different
surface terminations as the experimental band structure looks different from
different cleavages. Our ARPES measurements demonstrate linear dispersive
surface Dirac states near the Fermi level with a node at $\approx$ 150 meV
below the Fermi level ($E_{F}$) at the $\bar{M}$-point with a 6-fold
rotational symmetry in the $\overline{\Gamma MK}$ plane. Thus, there exists 6
Dirac cones of this type in a Brillouin zone. This observation is consistent
with a previous ARPES report Yao  _et al._ (2016) on this compound.
Interestingly, in addition to this, we have uncovered for the first time an
another surface Dirac cone at the zone center with a node at $\approx$ 900 meV
below $E_{F}$. Much like in BiPd Thirupathaiah  _et al._ (2016), the Dirac
states at the zone center in PtBi2 show a gap opening as a function of photon
energy, possibly due to a surface-bulk hybridization. Based on these studies,
we suggest that the linear MR observed in the hexagonal PtBi2 could have
originated from the linear dispersive surface states in the quantum limit,
like in the case of topological insulators Qu  _et al._ (2010); Wang  _et al._
(2012) where the large linear MR is observed only from the surface band
structure.

Figure 1: Electronic structure of PtBi2. (a) Hexagonal crystal structure
projected on to the $ab$-plane. (b) 3D view of the Brillouin zone with the
high symmetry points. (c) Schematic representation of measuring geometry in
which the $s$\- and $p$-plane polarized lights are defined with respect to the
analyzer entrance slit (ES) and the scattering plane (SP). (d) Fermi surface
map extracted from the experiment using $p$-polarized light with a photon
energy of 100 eV. (e) Fermi surface map derived from the DFT calculations. (f)
Energy distribution map taken along the $A-L$ high symmetry line. (g) Energy-
momentum ($E-k$) plot from the DFT calculations along the $A-L$ high symmetry
line. In the figure, the red dashed-lines represent the Dirac surface states.

High quality single crystals of hexagonal PtBi2 were grown using floating zone
technique Cry . The crystals have a platelet-like shape with shiny surface.
The crystals were structurally characterized using powder X-ray diffraction to
confirm the bulk purity and the hexagonal crystal structure with $P3$ space
group. ARPES measurements were carried out in BESSY II (Helmholtz Zentrum
Berlin) synchrotron radiation facility at the UE112-PGM2b beam-line using the
”13-ARPES” Borisenko (2012); Borisenko  _et al._ (2012) end station equipped
with SIENTA R4000 spectrometer. The total energy resolution was set between 5
and 15 meV, depending on the incident photon energy. The measurements were
performed at T = 1 K. Another set of ARPES measurements were performed in the
Diamond light source at I05 beamline equipped with SIENTA R4000 spectrometer.
The sample temperature at Diamond was always below 10 K and the energy
resolution was set between 10 and 20 meV depending on the incident photon
energy.

Self-consistent band structure calculations were performed on the hexagonal
crystal structure of PtBi2 having the lattice constants $a$ = $b$ = 6.572 Å
and $c$ = 6.166 Å using the linear muffin-tin orbital (LMTO) method Andersen
(1975) in the atomic sphere approximation (ASA) as implemented in PY LMTO
computer code Antonov (2004). The Perdew-Wang parameterization Perdew and Wang
(1992) was used to construct the exchange correlation potential in the local
density approximation (LDA). Spin-orbit coupling was taken into account at the
variational step.

Figure 2: Photon energy dependent ARPES data. The data are measured using
$p$-polarized light. (a) $k_{z}$ Fermi surface map taken in the $\Gamma MLA$
plane. (b) Energy distribution map taken along the $\Gamma-A$ symmetry line at
$k_{y}$, $k_{x}$=0. (c) Energy distribution curves as a function of photon
energy. (d) $E-k$ plot derived from the DFT calculations in the $\Gamma$-$A$
high symmetry line. (e)-(h) Photon energy dependent EDMs taken along the
$\Gamma$-$M$ direction. (j)-(m) second derivatives of (e)-(h). (i) EDM
measured using $p$-polarized light along the $\Gamma$-$M$ symmetry line with a
photon energy of 80 eV from a different cleavage plane than the rest of the
data. (n) second derivative of (i). In the figure, the red and blue dashed
lines represent the surface states and the pink dashed lines represent the
bulk states. Dirac point (DP) of the surface states at the
$\bar{\Gamma}$-point is at around 900 meV and at the $\bar{M}$-point is at
around 150 meV below $E_{F}$.

The electronic structure of PtBi2 is shown in Fig. 1 obtained by means of
ARPES and DFT calculations. The ARPES data are measured using $p$-polarized
light with a photon energy of 100 eV. The data are taken at a sample
temperature of 1 K. We estimate that 100 eV photons detect the bands nearly
from the $k_{z}$=5.5 $\pi/c$ plane using an inner potential of Pt of 25 eV
Ichikawa  _et al._ (2003), i.e., from the $ALH$ plane. Fig. 1(a) shows the
hexagonal crystal structure of PtBi2 projected onto the $ab$-plane. Fig. 1(b)
shows 3D view of the Brillouin zone in which various high symmetry points are
located. In Fig. 1(c) we schematically show the measuring geometry where the
$s$\- and $p$-plane polarized lights are defined with respect to the analyzer
entrance slit (ES) and the scattering plane (SP). Fig. 1(d) shows the
experimental Fermi surface (FS) map in the $k_{x}-k_{y}$ plane extracted by
integrating over an energy window of 20 meV centered at $E_{F}$. Fig. 1(f)
shows the FS map derived from DFT calculations. Fermi surface obtained in our
studies [see Fig. 1 (d) and (e)] is inline with the hexagonal crystal
structure of PtBi2[see Fig. 1(a)]. Fermi surface obtained in our studies is
inline with the hexagonal crystal structure of PtBi2 [see Fig. 1(a)]. Energy
distribution map (EDMs) along the $A$-$L$ high symmetry lines is shown in Fig.
1 (h). In Fig. 1 (h), the red-dashed lines represent linearly dispersive Dirac
surface states with a node at $\approx$ 900 meV. Calculated band dispersions
along the $A$-$L$ high symmetry lines are shown in Fig. 1 (i). Note here that
the Fermi level of the calculated band structure is shifted about 200 meV
towards the lower-binding energy to match with the experiment. A similar
observation was also made in the previous ARPES report of this compound Yao
_et al._ (2016).

Figure 3: Circular dichroism. (a) EDM measured using left circular ($I^{-}$),
(b) right circular ($I^{+}$) and (c) the resultant EDM after the spectral
intensity subtraction ($I^{-}$-($I^{+}$) measured with a photon energy of 17
eV. (d)-(i) Similar data of (a)-(c) but measured with 40 and 100 eV photon
energies.

Next, we show photon energy dependent ARPES measurements performed to
understand the $k_{z}$ dependent band structure of this compound. Fig. 2(a)
shows the $k_{z}$ Fermi surface map measured using $p$-polarized light in the
$\Gamma MLA$ plane. Fig. 2(b) shows the EDM taken along the $\Gamma$-$A$ high
symmetry line at $k_{x},\leavevmode\nobreak\ k_{y}$=0. Energy distribution
curves (EDCs) as a function of incident photon energy are shown in Fig. 2(c).
Band dispersions in the $\Gamma$-$A$ symmetry line derived from the DFT
calculations are shown in Fig. 2(d). Figs. 2(e)-(h) depict EDMs measured using
various photon energies. Figs. 2(j)-(m) depict second derivatives of Figs.
2(e)-(h). In Fig. 2(i), we show EDM taken from a different cleavage (say
Surface B) surface using $p$-polarized light with a photon energy of 80 eV,
while the rest of the data are measured from another cleavage (say Surface A).
Fig. 2(n) depicts the second derivative of Fig. 2(i). In the figures, the red-
and blue-dashed lines represent the surface states (SS), while the pink-dashed
lines represent the bulk states (BS).

In the $k_{z}$ map shown in Fig. 2(a), we can clearly notice that the two
Fermi sheets shown by the red down-arrows have negligible $k_{z}$ warping. In
principle, surface states show $k_{z}$ independent band structure. On the
other hand, our bulk band structure calculations suggest a strong $k_{z}$
warping of the Fermi sheets in the $\Gamma MLA$ plane (not shown) . Therefore,
from these observations we suggest that these Fermi sheets are from the
surface contribution. This is further confirmed from the EDM taken along the
$\Gamma$-$A$ direction [see Fig. 2(b)] where one can notice a flat dispersion
for the Dirac point (DP) at $\approx$ 900 meV of the binding energy and no
such band is found from a similar EDM derived from the calculations [see Fig.
2(d)]. Interestingly, from the energy distribution curves (EDCs) extracted as
a function of photon energy we noticed that the Dirac states at the zone
center are gapped when measured with the photon energy $h\nu$ $<$ 55 eV. This
is further demonstrated from the photon energy dependent EDMs shown in Figs.
2(e)-(h) and from the corresponding second derivatives shown in Figs.
2(j)-(m). The estimated maximum energy gap is of 145 meV when measured with 16
eV of photon energy [see Fig. 2(j)]. A similar observation was made earlier in
the case of noncentrosymmetric BiPd Thirupathaiah  _et al._ (2016). There, it
was suggested that the gap opening is due to the hybridization between the
surface and the bulk states located in the same binding energy range. In the
present case as well it could be the similar scenario as the bulk bands
disperse in the $k_{z}$ direction over a window of 1 eV of binding energy. As
can be seen in Fig. 2(d), the binding energy of the band at $\Gamma$ is of
0.65 eV whereas at $A$ the binding energy of the same band is of 1.5 eV.
Therefore, as the surface states show a finite penetration depth into the
solid Autès  _et al._ (2015); Li  _et al._ , it is highly likely to have a
surface-bulk hybridization which may lead to the gap opening at the Dirac
point. On Fig. 2(m), we have shown schematically another set of Dirac surface
states by the blue-dashed lines located at the $\bar{M}$-point having a Dirac
point at $\approx$ 150 meV with a 6-fold rotational symmetry. Thus, there are
in total 6 Dirac cones of this type in a BZ. This observation is in good
agreement with previous ARPES report on this compound Yao  _et al._ (2016).
The electronic structure of PtBi2 seems to be cleavage dependent. As can be
seen in Figs. 2(i) and (n), the EDM taken from the cleavage surface B shows no
signatures of the Dirac-like surface states when compared to the data taken on
the cleavage surface A. Moreover, the band structure highlighted by pink-
dashed lines shown in Figs. 2(i) and (n) look much alike the bulk band
structure derived from the DFT calculations [see Fig. 1(i)].

Next we show the circular dichroic ARPES measurements. Energy distribution map
(I-) shown in Fig. 3(a) is measured using a left-circular polarized light with
a photon energy of 17 eV. Similarly, EDM (I+) shown in Fig. 3(b) is measured
using a right-circular polarized light. Fig. 3(c) is the resultant spectra
after the subtraction (I--I+). Figs. 3(d)-(i) are analogous data to Figs.
3(a)-(c) but measured with photon energies of 40 and 100 eV. Generally, in
spin-orbit coupled systems the helicity of light can couple to the electron
spin degree of freedom through the total angular momentum. Therefore, the
helicity dependent spectral intensity can provide indirectly information on
the spin polarization of the detected sates Wang  _et al._ (2011); Park  _et
al._ (2011); Wang and Gedik (2013). However, it is also argued that such
measurements suffer greatly from the final state effects and thus, reversing
the dichroic signal among the photon energies Scholz  _et al._ (2013).
Nevertheless, as can be seen from the intensity difference spectra [Figs.
3(c), (f) and (i)] taken from a wide range of photon energies (17, 40 and 100
eV), the dichroic signal is consistent at the zone center switching the
spectral intensity of the Dirac cone as $k\rightarrow-k$. Interestingly,
dichroic signal is noticed also for the Dirac states at the $\bar{M}$-point,
but with opposite sign.

Previous transport Yang  _et al._ (2016) and ARPES studies Yao  _et al._
(2016) on this compound suggested crystal disorder for the observed linear
magnetoresistance. On the other hand, our present studies clearly show two
sets of linear dispersive Dirac states crossing the Fermi level with nodes
located at around 150 meV and 900 meV below $E_{F}$. Therefore, based on these
observations we suggest that the linear MR observed in this compound could be
originated from the linear dispersive surface states. This is argument is
inline with the theory based on the Landau levels of the linear dispersive
states in the presence of external magnetic field Abrikosov (1998, 2003),
where it was suggested that the linear dispersive states without a gap near
the Fermi level play a crucial role. In addition, following our circular
dichroic measurements we further suggest that the surface states are
substantially spin polarized. Therefore, spin polarization of the bands also
may play a vital role for the magnetoresistance Jiang  _et al._ (2015) of this
compound.

In conclusion, we studied the low-energy band structure of the hexagonal PtBi2
by means of ARPES technique and DFT calculations. Our studies uncover for the
first time linearly dispersive Dirac states near $\bar{\Gamma}$-point,
crossing the Fermi level with a node at $\approx$ 900 meV below $E_{F}$. In
addition, another surface Dirac cone is found at the $\bar{M}$-point-point
with a node at $\approx$ 150 meV below $E_{F}$. We further notice from the
circular dichroic measurements that these surface states show a significant
dichroic signal switching as $k\rightarrow-k$. Following these studies, we
suggest that the linear dispersive spin polarized Dirac states found at both
the $\bar{\Gamma}$-point and $\bar{M}$-point points could be a possible origin
for the large linear MR observed in this compound rather than the crystal
disorder that was suggested earlier for the same.

This work was supported under the DFG grant BO 1912/7-1. S.T. acknowledges
support by the Department of Science and Technology, India through the
INSPIRE-Faculty program (Grant number: IFA14 PH-86). We acknowledge Diamond
Light Source for time on Beamline/Lab I05 under Proposal SI15936-1.

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# Forecasts for Constraining Lorentz-violating Damping of Gravitational Waves
from Compact Binary Inspirals

Bo-Yang Zhanga,b,c<EMAIL_ADDRESS>Tao Zhub,c Corresponding author:
<EMAIL_ADDRESS>Jing-Fei Zhanga Corresponding author:
<EMAIL_ADDRESS>Xin Zhanga,d,e<EMAIL_ADDRESS>a Key
Laboratory of Cosmology and Astrophysics (Liaoning) & 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 Key Laboratory of Data Analytics and Optimization for Smart Industry
(Ministry of Education), Northeastern University, Shenyang 110819, China
e National Frontiers Science Center for Industrial Intelligence and Systems
Optimization, Northeastern University, Shenyang 110819, China

###### Abstract

Violation of Lorentz symmetry can result in two distinct effects in the
propagation of the gravitational waves (GWs). One is a modified dispersion
relation and another is a frequency-dependent damping of GWs. While the former
has been extensively studied in the literature, in this paper we concentrate
on the frequency-dependent damping effect that arises from several specific
Lorentz-violating theories, such as spatial covariant gravities, Hořava-
Lifshitz gravities, etc. This Lorentz-violating damping effect changes the
damping rate of GWs at different frequencies and leads to an amplitude
correction to the GW waveform of compact binary inspiral systems. With this
modified waveform, we then use the Fisher information matrix to investigate
the prospects of constraining the Lorentz-violating damping effect with GW
observations. We consider both ground-based and space-based GW detectors,
including the advanced LIGO, Einstein Telescope, Cosmic Explorer (CE), Taiji,
TianQin, and LISA. Our results indicate that the ground-based detectors in
general give tighter constraints than those from the space-based detectors.
Among the considered three ground-based detectors, CE can give the tightest
constraints on the Lorentz-violating damping effect, which improves the
current constraint from LIGO-Virgo-KAGRA events by about 8 times.

##  I INTRODUCTION

Since the landmark discovery of the first gravitational wave (GW) event,
GW150914, resulting from the coalescence of two massive black holes by the
LIGO-Virgo collaboration in 2015 LIGOScientific:2016aoc , the field of GW
astronomy has rapidly evolved. To date, approximately 90 events have been
meticulously identified by the LIGO-Virgo-KAGRA (LVK) scientific
collaborations LIGOScientific:2018mvr ; LIGOScientific:2019lzm ;
LIGOScientific:2020aai ; LIGOScientific:2021djp . On May 24, 2023, the
advanced LIGO (aLIGO) initiated the Observing Run 4 (O4) project. Following
LVK, the forthcoming third-generation ground-based GW detectors, such as
Einstein Telescope (ET) Branchesi:2023mws  and Cosmic Explorer (CE)
Evans:2021gyd , are presently in the design phase, with a specific emphasis on
detecting high-redshift GW events ($z>10$). Concurrently, a new cohort of
space-based detectors (Taiji Ruan:2018tsw ; Wu:2018clg ; Hu:2017mde , TianQin
Liu:2020eko ; Wang:2019ryf ; TianQin:2015yph ; Luo:2020bls ; Milyukov:2020kyg
, and LISA Robson:2018ifk ; LISACosmologyWorkingGroup:2022jok ) is designed to
explore the low-frequency GW signals ($f\sim 10^{-4}$ Hz). We anticipate that
these detectors will play a crucial role in the era of GW astronomy
LIGOScientific:2016lio ; LIGOScientific:2018dkp ; LIGOScientific:2017zic ;
LIGOScientific:2019fpa ; LIGOScientific:2020tif .

General relativity (GR) remains the preeminent theory for explaining
gravitational phenomena. Yet, it faces challenges in accounting for enigmatic
concepts such as dark matter and dark energy, and reconciling them with
quantum mechanics, particularly in the contexts of singularities and the
quantization of gravity. To address these issues, a plethora of experiments
have been devised to rigorously test GR’s predictions. Regrettably, the
majority of these experiments have been limited to investigating the weak-
field regime Hoyle:2000cv ; Everitt:2011hp ; Sabulsky:2018jma . GWs, one of
the fundamental predictions of GR, are produced in the tumultuous environments
of strong gravitational fields and interact only weakly with matter, making
them pristine messengers of the dynamics of space-time. The detection of GWs,
especially those originating from the coalescences of compact binary systems
LIGOScientific:2020iuh ; LIGOScientific:2017vwq ; LIGOScientific:2020ibl , has
thus heralded a new era for testing the robustness of GR under extreme
conditions. These observations offer a powerful tool for probing the strong-
field regime of gravity, potentially unlocking answers to the persistent
questions that challenge the current understanding within the framework of GR.

In the theoretical realm, various modified theories of gravity Cognola:2006eg
; Copeland:2006wr ; Frieman:2008sn  have been proposed to address challenges
within GR. A subset of these theories has garnered significant attention for
deviating from a fundamental principle of GR — the Lorentz invariance. At
high-energy levels, it is widely believed that this invariance will be broken
when gravity is quantized. Various modified gravity theories have been
proposed to explore the nature of Lorentz violation in gravity, including the
Einstein-Æther theory Jacobson:2000xp ; Eling:2004dk ; Jacobson:2007veq ;
Li:2007vz ; Battye:2017zvv ; Zhang:2023kzs ; Liu:2021yev ; Zhu:2019ura ,
Hořava-Lifshitz theories of quantum gravity Horava:2009uw ; Takahashi:2009wc ;
Wang:2012fi ; Zhu:2013fja , the standard model extension Kostelecky:2016kfm ;
Bailey:2006fd ; Mewes:2019dhj ; Shao:2020shv ; Wang:2021ctl ;
Kostelecky:2017zob ; Niu:2022yhr , and spatial covariant gravities Gao:2020qxy
; Gao:2020yzr ; Gao:2019twq ; Gao:2019liu ; Joshi:2021azw .

The violation of Lorentz symmetry in gravity can introduce deviations from GR
in the propagation of GWs. These deviations manifest in two distinct ways,
influencing the propagation behavior of GWs in the cosmological background.
First, with Lorentz violation, 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. This effect can arise from a large
number of Lorentz-violating theories. Second, Lorentz violation can introduce
frequency-dependent friction into the propagation equation of GWs, resulting
in varying damping rates for GWs of different frequencies during their
propagation. This effect normally arises from those theories with mixed
temporal and spatial derivatives of the spacetime metric in the modified
theories of gravity with spatial covariance, for example, the Hořava-Lifshitz
gravity Colombo:2014lta , the spatial covariant gravities Gao:2019liu ;
Zhu:2022uoq , etc.

Testing Lorentz symmetry of gravitational interaction by using the
observational data from GW events in LIGO-Virgo-KAGRA catalogs and future GW
detectors has been carried out in a lot of works, see Refs.
LIGOScientific:2019fpa ; LIGOScientific:2020tif ; Mirshekari:2011yq ;
Will:1997bb ; Zhu:2022uoq ; Gong:2021jgg  and references therein. In most of
these works, the effects due to the nonlinear dispersion relation have been
extensively considered. Recently, the constraint on the Lorentz-violating
damping effects from GW events in LIGO-Virgo-KAGRA catalogs was first obtained
Zhu:2023wci .

In this paper, we detail the Lorentz-violating damping effects in the
propagation of GW in a cosmological background. Decomposing the GWs into left-
hand and right-hand circular polarization modes, we observe that the Lorentz-
violating damping effects manifest through explicit modifications in the GW
amplitude. We derive corrections to the waveform of the compact binary
inspiral system accordingly. With this modified waveform, we use the Fisher
information matrix (FIM), which is widely used in cosmology and astrophysics
Yu:2023ico ; Jin:2023tou ; Shao:2023agv ; Jin:2023sfc ; Zhang:2023gaz ;
Wu:2022jkf ; Song:2022siz ; Jin:2022qnj ; Wu:2022dgy ; Wang:2022oou ;
Wu:2021vfz ; Jin:2021pcv ; Zhang:2021yof ; Yu:2021nvx ; Zhao:2019gyk ;
Zhang:2019loq ; Wang:2019tto ; Zhang:2019ple  to investigate the prospects of
constraining the Lorentz-violating damping effect with GW observations of
compact binary systems. We consider both ground-based and space-based GW
detectors, including aLIGO, CE, ET, Taiji, TianQin, and LISA. Our results
indicate that the ground-based detectors in general give tighter constraints
than those from the space-based detectors. Among the considered three ground-
based detectors, CE can give the tightest constraints on the Lorentz-violating
damping effect, which improves the current constraint from LIGO-Virgo-KAGRA
events by about 8 times Zhu:2023wci .

Our paper is organized as follows. In Sec. II, We present a very brief
introduction to the Lorentz-violating damping effect and calculate the
modified waveform of GWs of compact binary inspiral systems with the effect.
Sec. III summarizes the application of the FIM for constraining the modified
waveform parameters of GWs. The main results of our analysis are discussed in
Sec. IV. Finally, Sec. V provides a summary and further discussion of our work
in this paper.

##  II MODIFIED WAVEFORM OF GWS WITH LORENTZ-VIOLATING DAMPING EFFECT

In this section, we present a brief introduction to the modified waveform of
GWs with Lorentz-violating damping effect. As we mentioned, such Lorentz-
violating damping effect can modify the amplitude damping rates of the two
tensorial modes of GWs, which arise from several specific Lorentz-violating
theories of gravity, for instance, the spatial covariant gravities Gao:2019liu
; Zhu:2022uoq  and Hořava-Lifshitz gravity Colombo:2014lta .

Figure 1:  The noise spectral density of the six detectors considered in this
paper. Both ground-based and space-based detectors are included in the
picture.

###  II.1 Propagating equation of GWs with Lorentz-violating damping effects

Let us first investigate the propagation of GWs with Lorentz-violating damping
effect on a flat Friedmann-Robertson-Walker spacetime. Treating this spacetime
as a background, GWs can be described by the tensor perturbations of the
metric, where the metric is expressed in the form of

$\displaystyle
ds^{2}=a^{2}(\tau)\Big{[}-d\tau^{2}+(\delta_{ij}+h_{ij})dx^{i}dx^{j}\Big{]},$
(2.1)

where $a(\tau)$ is the scale factor of the expanding Universe and $\tau$
represents the conformal time. One can transform the conformal time $\tau$ to
the cosmic time $t$ by $dt=a(\tau)d\tau$. Throughout this paper, we set the
present expansion factor $a_{0}=1$. $h_{ij}$ denote GWs, which are transverse
and traceless, i.e.,

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

For later convenience, let us expand $h_{ij}$ over spatial Fourier harmonics,

$\displaystyle
h_{ij}(\tau,x^{i})=\sum_{A=R,L}\int\frac{d^{3}k}{(2\pi)^{3}}h_{A}(\tau,k^{i})e^{ik_{i}x^{i}}e_{ij}^{A}(k^{i}),$
(2.3)

where $e_{ij}^{A}$ is the circular polarization tensor and obeys the following
rules

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

with $\rho_{\rm R}=1$ and $\rho_{\rm L}=-1$.

To study the Lorentz-violating damping effect on the propagation of GWs, let
us first write the modified propagation equation of motions of the two GW
modes in the following parametrized form Zhu:2023wci ; Zhao:2019xmm ,

$\displaystyle
h_{A}^{\prime\prime}+(2+{\bar{\nu}}+\nu_{A})\mathcal{H}h^{\prime}_{A}+(1+\bar{\mu}+\mu_{A})k^{2}h_{A}=0,$
(2.5)

where a prime denotes the derivative concerning the conformal time $\tau$ and
${\cal H}=a^{\prime}/a$. The four parameters, ${\bar{\nu}}$, $\nu_{A}$,
${\bar{\mu}}$, and $\mu_{A}$ label the new effects on the propagation of GWs
arising from theories beyond GR. As mentioned in Ref. Zhu:2023wci , such
parametrization provides a general framework for exploring possible modified
GW propagations arising from a large number of modified theories of gravity.
Different parameters correspond to different effects on the propagation of
GWs. The parameters $\nu_{A}$ and $\mu_{A}$ represent the effects of parity
violations, while the parameters ${\bar{\nu}}$ and ${\bar{\mu}}$, if
frequency-dependent, can originate from other potential modifications
involving Lorentz violations. For ${\bar{\nu}}$ and ${\bar{\mu}}$, the former
provides an amplitude modulation of the GW waveform, while the latter one
${\bar{\mu}}$ determines the phase velocities of the GWs. In this paper, we
will only concentrate on the case of the Lorentz-violating damping effect, and
for this case one has

$\displaystyle{\bar{\nu}}\neq 0,\;\;\nu_{A}=0,\;\;{\bar{\mu}}=0=\mu_{A}.$
(2.6)

Thus the propagation equation now can be written in the form of

$\displaystyle
h^{\prime\prime}_{A}+(2+{\bar{\nu}})\mathcal{H}h^{\prime}_{A}+k^{2}h_{A}=0.$
(2.7)

With the Lorentz-violating damping effect, the only nonzero parameter
${\bar{\nu}}$ is generally frequency dependent. Thus, following Ref.
Zhu:2023wci , one can further parametrize ${\bar{\nu}}$ in the form of

$\displaystyle{\cal
H}{\bar{\nu}}=\left[\alpha_{\bar{\nu}}(\tau)\left(\frac{k}{aM_{\rm
LV}}\right)^{\beta_{\bar{\nu}}}\right]^{\prime},$ (2.8)

where $\beta_{\bar{\nu}}$ is an arbitrary number, $\alpha_{\bar{\nu}}$ is an
arbitrary function of time, and $M_{\rm LV}$ denotes the energy scale of the
Lorentz violation. The parameters $\alpha_{\bar{\nu}}$ and $\beta_{\bar{\nu}}$
depend on the specific modified theories of gravity.

In this paper, we consider the case with $\beta_{\bar{\nu}}=2$ and derive the
corresponding modified waveform of GWs. We then explore the potential
constraints on this modified waveform using proposed GW detectors such as
aLIGO, CE, ET, Taiji, TianQin, and LISA. It is worth noting that this case can
arise from the mixed temporal and spatial derivatives of the spacetime metric
in the modified theories of gravity with spatial covariance Gao:2019liu ;
Zhu:2022uoq ; Colombo:2014lta , for example, the mixed term in the form of

$\displaystyle\nabla_{k}K_{ij}\nabla^{k}K^{ij}$ (2.9)

can appear in both the spatial covariant gravity Gao:2019liu  and Hořava-
Lifshitz gravity Colombo:2014lta , where $\nabla_{k}$ denotes the covariant
derivative associated with the spatial metric $g_{ij}$ and $K_{ij}$ is the
extrinsic curvature tensor. It is also shown that by including mixed
derivative terms, the non-protectable Hořava-Lifshitz gravity could be power-
counting renormalizable and free of ghosts Colombo:2014lta . One can connect
the coupling coefficient $c_{1}^{(2,2)}$ of Eq. (2.9) in modified gravities to
the parameters $\alpha_{\bar{\nu}}$ and $M_{\rm LV}$ via

$\displaystyle\frac{\alpha_{\bar{\nu}}(\tau)}{M_{\rm
LV}^{2}}=\frac{2c_{1}^{(2,2)}}{M_{\rm Pl}^{2}},$ (2.10)

where $M_{\rm Pl}$ is the reduced Planck mass. Considering the coupling
coefficient as a constant, which is equivalent to treating
$\alpha_{\bar{\nu}}$ as constant, we can simply set $\alpha_{\bar{\nu}}=1$ by
absorbing its order of magnitude into $M_{\rm LV}$.

###  II.2 Amplitude modulation of GWs with Lorentz-violating damping effect

The nonzero parameter $\overline{\nu}$ provides a frequency-dependent damping
of GW amplitudes during propagation. This means that GWs with different
frequencies will experience different damping rates. This damping rate effect
induces an amplitude modification in the GWs.

To study the modified waveform of GWs with this frequency-dependent damping of
GW amplitudes, following the derivations in Refs. Zhao:2019xmm ; Qiao:2019wsh
, let us decompose $h_{A}$ in Eq. (2.7) as

$\displaystyle h_{A}=h_{A}^{\rm GR}e^{-i\theta(\tau)},\;\;h_{A}^{\rm GR}={\cal
A}_{A}^{\rm GR}e^{-i\Phi^{\rm GR}(\tau)}.$ (2.11)

Here $h_{A}^{\rm GR}$ is the solution of Eq. (2.7) when ${\bar{\nu}=0}$.
${\cal A}_{A}^{\rm GR}$ and $\Phi^{\rm GR}(\tau)$ are the amplitude and phase
of $h_{A}^{\rm GR}$ respectively. With this decomposition, $\theta(\tau)$
encodes the correction arising from ${\bar{\nu}}$ which characterizes the
Lorentz-violating damping effect.

Plugging the second equation of decomposition Eq. (2.11) into Eq. (2.7) with
${\bar{\nu}=0}$, one finds

$\displaystyle i\Phi^{\prime\prime}+\Phi^{\prime 2}+2i{\cal
H}\Phi^{\prime}-k^{2}=0.$ (2.12)

Similarly, plugging the first equation of the decomposition Eq. (2.11) into
Eq. (2.7), one gets

$\displaystyle
i(\theta^{\prime\prime}+\Phi^{\prime\prime})+(\Phi^{\prime}+\theta^{\prime})^{2}+i(2+{\bar{\nu}}){\cal
H}(\theta^{\prime}+\Phi^{\prime})-k^{2}=0.$

In GR, the time derivative of the phase $\Phi^{\prime}\sim k$. Here the
wavenumber $k$ is connected to the frequency of GWs by $k=2\pi f/a_{0}$. Since
the amplitude correction $\theta$ is induced by the expansion of the universe,
one has $\theta^{\prime}\sim{\cal H}$ and $\theta^{\prime\prime}\sim{\cal
H}^{2}$. Considering $k\gg{\cal H}$ and
$\theta^{\prime\prime}\ll\Phi^{\prime}\theta^{\prime}\sim k\theta^{\prime}$,
Eq. (II.2) can be simplified into

$\displaystyle 2\theta^{\prime}+i{\cal H}{\bar{\nu}}\simeq 0.$ (2.14)

Solving this equation gives

$\displaystyle\theta=-\frac{i}{2}\int_{\tau_{e}}^{\tau_{0}}{\cal
H}{\bar{\nu}}d\tau.$ (2.15)

The Lorentz-violating damping effect in the phase $\theta$ is purely
imaginary, indicating that it modifies the amplitude of the GWs during their
propagation. Considering ${\bar{\nu}}$ is also frequency-dependent, such
amplitude modulation depends on the frequency of GWs as well.

Specifically, with the above solution, one can write the waveform of GWs with
Lorentz-violating effect as

$\displaystyle h_{A}=h_{A}^{\rm GR}e^{\delta h_{2}},$ (2.16)

where

$\displaystyle\delta h_{2}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\int_{\tau_{e}}^{\tau_{0}}{\cal H}\bar{\nu}d\tau$
(2.17) $\displaystyle=$
$\displaystyle-\frac{1}{2}\left[\alpha_{\bar{\nu}}\left(\frac{k}{aM_{\rm
LV}}\right)^{\beta_{\bar{\nu}}}\right]\Bigg{|}^{a_{0}}_{a_{e}}.$

It can be further rewritten in the form

$\displaystyle\delta h_{2}=-\frac{1}{2}\left(\frac{2u}{M_{\rm LV}{\cal
M}}\right)^{\beta_{\bar{\nu}}}\Big{[}\alpha_{\bar{\nu}}(\tau_{0})-\alpha_{\bar{\nu}}(\tau_{e})(1+z)^{\beta_{\bar{\nu}}}\Big{]}.$

Here we define $u=\pi{\cal M}f$, where ${\cal M}=(1+z){\cal M}_{\rm c}$
represents the measured chirp mass, and ${\cal M}_{\rm
c}\equiv(m_{1}m_{2})^{3/5}/(m_{1}+m_{2})^{1/5}$ denotes the chirp mass of the
binary system with component masses $m_{1}$ and $m_{2}$.

###  II.3 Amplitude modification to the waveform of GWs

To derive the modified waveform of GWs, we consider the GWs produced during
the inspiral stage of the compact binaries. To directly contact with the
observations, it is convenient to analyze the GWs in the Fourier domain. In
this approach, under the stationary phase approximation, the responses of the
detectors to the GW signal ${\tilde{h}}(f)$ can be written in the form of

$\displaystyle{\tilde{h}}(f)=\Big{[}F_{+}h_{+}(f)+F_{\times}h_{\times}(f)\Big{]}e^{-2\pi
if\Delta t},$ (2.19)

where $F_{+}$ and $F_{\times}$ denote the beam pattern functions of GW
detectors, which depend on the GW source’s location and polarization angle
Jaranowski:1998qm ; Zhao:2017cbb . The two polarizations of GWs, $h_{+}(f)$
and $h_{\times}(f)$, are related to the left- and right-handed polarization
modes, $h_{\rm L}(f)$ and $h_{\rm R}(f)$, via

$\displaystyle h_{+}=\frac{h_{\rm L}+h_{\rm
R}}{\sqrt{2}},\;\;h_{\times}=\frac{h_{\rm L}-h_{\rm R}}{\sqrt{2}i}.$ (2.20)

Then using Eq. (2.16), after tedious calculations, one obtains the following
restricted form for the waveform of GWs in the Fourier domain as a function of
the GW frequency $f$, i.e.,

$\displaystyle\tilde{h}(f)=\mathcal{A}_{\rm
GR}f^{-7/6}e^{i\Psi_{\mathrm{GR}}(f)}e^{\delta h_{2}},$ (2.21)

where ${\cal A}_{\rm GR}$ and $\Psi_{\rm GR}$ represent the amplitude and
phase of GWs of a compact binary inspiral signal in GR, and $e^{\delta h_{2}}$
with $\delta h_{2}$ being given by Eq. (II.2) is the amplitude correction to
the waveform of GWs in GR. In the post-Newtonian approximation, the amplitude
and the phase of GWs in GR can be expressed as Wang:2022yxb

$\displaystyle{\cal A}_{\rm
GR}=\frac{2}{5}\times\sqrt{\frac{5}{24}}\pi^{-2/3}\frac{\mathcal{M}^{5/6}}{D_{\rm
L}},$ (2.22)

and

$\displaystyle\Psi_{\mathrm{GR}}(f)$ $\displaystyle=$ $\displaystyle 2\pi
ft_{\rm c}-\Phi_{\rm
c}-\frac{\pi}{4}+\frac{3}{128\eta}u^{-5/3}\left\\{1+\left(\frac{3715}{756}+\frac{55}{9}\eta\right)u^{2/3}-16\pi
u\right.$ (2.23)
$\displaystyle+\left(\frac{15293365}{508032}+\frac{27145}{504}\eta+\frac{3085}{72}\eta^{2}\right)u^{4/3}+\pi\left(\frac{38645}{756}-\frac{65}{9}\eta\right)(1+\ln
u)u^{5/3}$
$\displaystyle+\left[\frac{11583231236531}{4694215680}-\frac{640}{3}\pi^{2}-\frac{6848}{21}\gamma_{\mathrm{E}}-\frac{6848}{63}\ln(64u)+\left(-\frac{15737765635}{3048192}+\frac{2255}{12}\pi^{2}\right)\eta\right.$
$\displaystyle\left.\left.+\frac{76055}{1728}\eta^{2}-\frac{127825}{1296}\eta^{3}\right]u^{2}+\pi\left(\frac{77096675}{254016}+\frac{378515}{1512}\eta-\frac{74045}{756}\eta^{2}\right)u^{7/3}\right\\},$

where the luminosity distance $D_{\rm L}$ is expressed as

$\displaystyle D_{\rm
L}(z)=\frac{1+z}{H_{0}}\int_{0}^{z}\frac{dz^{\prime}}{\sqrt{\Omega_{\mathrm{M}}(1+z^{\prime})^{3}+\Omega_{\Lambda}}}.$
(2.24)

Here we adopt the $\Lambda$CDM model with Hubble constant $H_{0}\approx 67.4$
km $\mathrm{s}^{-1}$ $\mathrm{Mpc}^{-1}$, matter density fractoin
$\Omega_{\mathrm{m}}\approx 0.315$, and vacuum energy density fraction
$\Omega_{\mathrm{\Lambda}}\approx 0.685$ Planck:2018vyg ; Zhang:2019ylr .
$t_{c}$ and $\Phi_{c}$ are time and phase at coalescence, $\gamma_{\rm E}$ is
the Euler constant, and $\eta=m_{1}m_{2}/(m_{1}+m_{2})^{2}$ is the symmetric
mass ratio.

Table 1: Characteristics of six GW detectors. Detector | Configuration |  $f_{\rm lower}$ [Hz] |  $f_{\rm upper}$ [Hz] | Reference
---|---|---|---|---
aLIGO | Rightangle | 10 | 5000 | Refs. Wang:2022yxb ; KAGRA:2023pio
ET | Rightangle | 1 | 10000 | Ref. Hild:2010id
CE | Rightangle | 5 | 4000 | Ref. Reitze:2019iox
Taiji | Triangle | 0.0001 | 0.1 | Ref. Liu:2023qap
TianQin | Triangle | 0.0001 | 1 | Ref. TianQin:2020hid
LISA | Triangle | 0.0001 | 0.1 | Refs. Liu:2023qap ; LISA:2017pwj

##  III ANALYSIS FRAMEWORK WITH FISHER INFORMATION MATRIX

###  III.1 General considerations

In this section, we provide a brief overview of the matched-filter analysis
with the FIM approach, which follows the method outlined for compact binary
inspiral in Refs. Cutler:1994ys ; Finn:1992xs ; Poisson:1995ef . We calculate
the noise-weighted inner product between the partial derivatives of each GW
waveform parameter and the one-sided power spectral density (PSD) of the
detector noises. This calculation yields the FIM. Inverting the FIM provides
the variance-covariance matrix, where the diagonal elements represent the
square root of the mean squared error for the estimated parameters of the
signal. Previous studies have showcased the precision and utility of the FIM
approach, particularly in situations with high signal-to-noise ratios (SNRs).

To be specific, we first give the noise-weighted inner product of two signals
$h_{1}$ and $h_{2}$

$\left({h}_{1}|{h}_{2}\right)=2\int_{f_{\mathrm{min}}}^{f_{\mathrm{max}}}\frac{\tilde{h}_{1}^{*}(f)\tilde{h}_{2}(f)+\tilde{h}_{2}^{*}(f)\tilde{h}_{1}(f)}{S_{\rm
n}(f)}df,$ (3.1)

where $\tilde{h}_{1}(f)$ and ${\tilde{h}}_{2}(f)$ are the Fourier
transformation of GW signal $h(t)$, $S_{\rm n}(f)$ is the PSD of the
detector’s noise, and the star superscript stands for complex conjugation. In
the above expression, $f_{\rm max}\ (f_{\rm min})$ represents the instrumental
maximum (minimum) threshold frequency.

For a given signal $h$, the SNR is defined as

$\rho=(h|h)^{1/2}.$ (3.2)

The modified waveform of GW from binary inspirals, influenced by the Lorentz-
violating damping effect as described in Eq. (2.21), are generally
characterized by a set of parameters $\theta_{i}$. In this context, one can
define the FIM as

$F_{ij}=\left(\left.\frac{\partial\tilde{h}}{\partial\theta_{i}}\right|\frac{\partial\tilde{h}}{\partial\theta_{j}}\right).$
(3.3)

Here $\theta_{i}$ and $\theta_{j}$ represent the elements in the set of
modified waveform parameters of GWs
$\\{\ln\mathcal{A},\ln\mathcal{M},\ln\mathcal{\eta},\phi_{\mathrm{c}},t_{\mathrm{c}},C_{\nu}\\}$,
where $C_{\nu}=M_{\mathrm{LV}}^{-2}$ characterizes the Lorentz-violating
damping effect in the waveform. Then one can calculate each element of the
FIM, which are given respectively in Appendix A.

In the large SNR approximation, if the noise is stationary and Gaussian, the
probability that the GW signal $h(t)$ can be characterized by a given set of
values of the parameters $\theta_{i}$,

$\displaystyle
p(\theta_{i}|h)=p^{(0)}(\theta_{i})\exp\left[-\frac{1}{2}F_{ij}\Delta\theta^{i}\Delta\theta^{j}\right],$
(3.4)

where $p^{(0)}(\theta_{i})$ represents the distribution of prior information.
Then, the standard deviations $\Delta\theta_{i}$ in measuring the parameter
$\theta_{i}$, which mean $1\sigma$ bounds on parameters, can be calculated in
the large SNR approximation. This can be obtained by taking the square root of
the corresponding diagonal elements in the inverse of FIM,

$\Delta\theta_{i}=\sqrt{(F^{-1})_{ii}},$ (3.5)

where $F^{-1}$ is the inverse of FIM.

Our main purpose in employing FIM analysis here is to gauge the potential of
future GW detectors in constraining the energy scale $M_{\rm{LV}}$ associated
with Lorentz violation, which induces the Lorentz-violating damping effect in
the propagation of GWs. We consider two types of GW detectors, ground-based GW
detectors including aLIGO, CE, and ET, and space-based GW detectors, including
Taiji, TianQin, and LISA. We summarize the information of all the six GW
detectors in Table 1.

To investigate the variation tendency of $M_{\rm LV}$ in GW events, we choose
the reasonable astrophysical horizon of each GW detector as the sources’
property and constrain $M_{\rm LV}$ by FIM. To maintain the effectiveness of
FIM, the ranges of redshift $z$ and total mass $M$ are suitable enough to
satisfy the SNR threshold ($\rho>8$ for ground-based detectors and $\rho>15$
for space-based detectors). For ground-based GW detectors, their detectable
frequency bands are several Hz to thousand Hz. As the third generation GW
detector, the frequency band of CE can reach $[5-4000]$ Hz. For space-based
detectors, with huge arm-lengths so that they can detect the GWs from compact
binary systems in a very low-frequency band $[10^{-4}-10^{-1}]$ Hz. The
compact binary systems in this low-frequency band usually consist of
supermassive black holes and their total mass range is $[10^{4}-10^{6}]\
\mathrm{M}_{\odot}$.

When calculating the integrals of the inner product, it is necessary to use
the appropriate limits of integration. The minimum frequency
$f_{\mathrm{min}}$ in Eq. (3.1) is taken as the instrumental minimum threshold
frequency of the GW detector as shown in Table 1. The upper cut-off frequency
$f_{\mathrm{max}}$ is chosen from
$\mathrm{min}\\{f_{\mathrm{upper}},f_{\mathrm{ISCO}}\\}$, where
$f_{\mathrm{upper}}$ is the upper frequency of detectors and
$f_{\mathrm{ISCO}}$ is usually estimated by the innermost stable circular
orbit (ISCO) Mirshekari:2011yq

$\displaystyle f_{\mathrm{ISCO}}=6^{-3/2}\pi^{-1}\eta^{3/5}\mathcal{M}^{-1}.$
(3.6)

When calculating the FIM, we set $C_{\nu}=0$ as the fiducial value.
Additionally, the values of $t_{\rm c}$ and $\phi_{\rm c}$ do not impact the
constraints on other parameters; hence, we set $t_{\rm c}=0$ and $\phi_{\rm
c}=0$.

###  III.2 Noise power spectral density of detector

In Fig. 1, we illustrate the noise PSD of both ground-based and space-based
detectors. The three ground-based detectors’ PSD are from the official data
files quoted in Table 1. The other three PSD data are from the theoretical
formula introduced below.

The sensitivity curve of Taiji can be obtained by the formula Liu:2023qap

$\displaystyle S_{\rm n}(f)$ $\displaystyle=\frac{10}{3L^{2}}\left(P_{\rm
dp}+2(1+\mathrm{cos}^{2}(f/f_{*}))\frac{P_{\rm acc}}{(2\pi f)^{4}}\right)$
$\displaystyle\times\left(1+0.6(\frac{f}{f_{*}})^{2}\right),$ (3.7)

where

$P_{\mathrm{dp}}=\left(8\times
10^{-12}\mathrm{~{}m}\right)^{2}\left(1+\left(\frac{2\,\mathrm{mHz}}{f}\right)^{4}\right)\mathrm{Hz}^{-1},$
(3.8) $\displaystyle P_{\mathrm{acc}}$ $\displaystyle=\left(3\times
10^{-15}\mathrm{~{}m}\mathrm{~{}s}^{-2}\right)^{2}\left(1+\left(\frac{0.4\,\mathrm{mHz}}{f}\right)^{2}\right)$
$\displaystyle\times\left(1+\left(\frac{f}{8\,\mathrm{mHz}}\right)^{4}\right)\mathrm{Hz}^{-1}.$
(3.9)

Here $P_{\rm dp}$ is the PSD of the displacement noise and $P_{\rm acc}$ is
the PSD of the acceleration noise. $f_{*}=1/(2\pi L)$ and $L$ is the arm-
length of the detector. For Taiji $L=3\times 10^{9}\ \rm m$.

For LISA, $L=2.5\times 10^{9}$ m and the displacement noise can be written as
Liu:2023qap ; Berti:2004bd

$P_{\mathrm{dpL}}=\left(15\times
10^{-12}\mathrm{~{}m}\right)^{2}\left(1+\left(\frac{2\,\mathrm{mHz}}{f}\right)^{4}\right)\mathrm{Hz}^{-1}.$
(3.10)

The sensitivity curve for TianQin can be modeled by the following equation
TianQin:2020hid

$\displaystyle S_{\rm n}(f)$ $\displaystyle=\frac{10}{3L^{2}}\left[S_{\rm
x}+\frac{4S_{\rm a}}{(2\pi
f)^{4}}\left(1+\frac{10^{-4}\mathrm{~{}Hz}}{f}\right)\right]$
$\displaystyle\times\left[1+0.6\left(\frac{f}{f_{*}}\right)^{2}\right],$
(3.11)

where displacement measurement noise $S^{1/2}_{\rm x}$ and residual
acceleration noise $S^{1/2}_{\rm a}$ are defined as

$\displaystyle S^{1/2}_{\rm x}=1\times 10^{-12}\
\mathrm{m}/\mathrm{Hz}^{1/2},$ (3.12)

and

$\displaystyle S^{1/2}_{\rm a}=1\times 10^{-15}\ \mathrm{m}\
\mathrm{s}^{-2}/\mathrm{Hz}^{1/2}.$ (3.13)

$f_{*}=c/(2\pi L)$ is the transfer frequency and $L=\sqrt{3}\times 10^{8}\
\mathrm{m}$.

##  IV RESULTS AND DISCUSSION

Table 2: The best constraints of the simulated single GW events and the combination of joint events from each GW detector. The results from the single event are the best constraint which are from Fig. 3. The redshift ranges and the total mass ranges are only for the joint events. The number of joint events contains 100 (for both ground-based and space-based detectors) simulated GW events. We choose $\rho>8$ and $\rho>15$ as the threshold for ground-based detectors and space-based detectors respectively. Detector | Single event [Gev] | Joint redshift range | Joint total mass range [$M_{\odot}$] | Joint number | Joint event [Gev]
---|---|---|---|---|---
aLIGO | $8.54\times 10^{-22}$ | [0.01-0.5] | [3-10] | 100 | $2.39\times 10^{-21}$
CE | $3.02\times 10^{-21}$ | [0.01-0.5] | [3-10] | 100 | $8.84\times 10^{-21}$
ET | $2.39\times 10^{-21}$ | [0.01-0.5] | [3-10] | 100 | $7.18\times 10^{-21}$
Taiji | $2.10\times 10^{-24}$ | [5-10] | $[4.2-9.2]\times 10^{3}$ | 100 | $5.86\times 10^{-24}$
TianQin | $1.89\times 10^{-24}$ | [0.01-5] | $[1.5-2]\times 10^{4}$ | 100 | $4.56\times 10^{-24}$
LISA | $1.52\times 10^{-24}$ | [5-10] | $[8.3-13.3]\times 10^{3}$ | 100 | $4.37\times 10^{-24}$

Figure 2: The results of $M_{\rm LV}$ and SNR of GW150914-like and
GW170817-like events in three ground-based detectors. GW170817-like event is
the compact binary system of two neutron stars and GW150914-like event is a
system containing two black holes.

In this section, we present the results of the potential constraints on the
parameters of the modified waveform of GWs with the Lorentz-violating damping
effect. Among these parameters, we focus on the energy scale $M_{\rm LV}$ of
Lorentz violation which characterizes the frequency-dependent damping of GWs
during their propagations. To conduct a comprehensive and reliable analysis of
GW parameter estimation, we consider the frequency bands of both ground-based
and space-based detectors. We employ simulated GW data to constrain the
Lorentz-violating damping effect through the FIM approach. For the simulated
GW data, we refer to the astrophysical horizons of the detectors from Ref.
Evans:2021gyd  (for aLIGO, ET), Refs. Srivastava:2022slt ; Evans:2021gyd  (for
CE), Ref. Liu:2023qap  (for Taiji), Ref. TianQin:2020hid  (for TianQin), and
Ref. LISA:2017pwj  (for LISA). We set $\rho>8$ and $\rho>15$ as the thresholds
for ground-based detectors and space-based detectors, respectively. To fully
exploit the high event rates of future GW detectors, we also decide to conduct
a multi-event joint constraint analysis. The results of constraining both
individual GW events and their combinations are detailed in Table 2. The
constraint results and SNR from GW150914-like and GW170817-like events are
shown in Fig. 2. We depict the dependence of the lower bound of
$M_{\mathrm{LV}}$ in Fig. 3.

Figure 3: Dependence of the lower bound of $M_{\rm LV}$ on the total mass and
redshift of the binary inspiral systems for different detectors, aLIGO (top
left), CE (top middle), ET (top and right), Taiji (bottom left), TianQin
(bottom middle), and LISA (bottom right).

###  IV.1 $M_{\rm LV}$ from ground-based detectors

First, let us analyze the constraints on $M_{\rm LV}$ from the three ground-
based detectors. In Fig. 2, we illustrate the constraints on $M_{\rm LV}$
(right panel) and SNRs (left panel) of two examples of GW events, the
GW150914-like and GW170817-like events with three different ground-based
detectors. As observed in the left panel of Fig. 2, the lower bounds on
$M_{\rm LV}$ from both CE and ET extend to $\gtrsim 10^{-21}$ GeV. CE also
gives the best constraints on $M_{\rm LV}$ for both GW150914-like and
GW170817-like events. Notably, under similar redshift conditions, the
GW170817-like event, characterized by a smaller total mass, attains a higher
lower bound of $M_{\rm LV}$. Moving to the right panel of Fig. 2, CE exhibits
a distinct advantage in SNR, suggesting its effectiveness in ensuring the
detection of such GW events. There appears to be an indicative trend implying
that GW events with a greater total mass could potentially yield higher SNRs.

In Table 2, it is evident that ground-based detectors, capable of observing
high-frequency GWs, exhibit superior performance compared to their space-based
counterparts. Among ground-based detectors, the strongest constraint for
$M_{\rm LV}$ is achieved by the third-generation detector CE at $3.02\times
10^{-21}$ GeV. The result from the ET is marginally smaller than that of CE.
For aLIGO, the constraint can reach $8.54\times 10^{-22}$ Gev.

Considering the prospect of observing a large number of GW events in the
future, we also do research on the joint analysis for individual detectors. As
anticipated, the outcomes from each detector exhibit a marked enhancement. The
most notable improvement comes from CE, yielding the optimal result with
$M_{\mathrm{LV}}>8.84\times 10^{-21}$ GeV. We combine 100 simulated GW events
within the redshift range [0.01-0.5] Hz and total mass range [3-10]
$M_{\odot}$ of sources. aLIGO attains a result of $M_{\mathrm{LV}}>2.39\times
10^{-21}\mathrm{GeV}$. This value is approximately twice as high as the result
reported in Ref. Zhu:2023wci , which means that aLIGO continues to be a
powerful tool in testing the Lorentz-violating damping effect. It is crucial
to highlight that the arm-length of ET is 10 km. Meanwhile, CE is designed
with arm-lengths of 20 km and 40 km. Due to their long arm-lengths and
enhanced sensitivity, CE and ET can provide values of $M_{\mathrm{LV}}$ that
are roughly more than double the value obtained by aLIGO. The expected
detection rate, as reported in Refs. Li:2023gtu ; Han:2023exn , is
conservatively estimated at 100.

###  IV.2 $M_{\rm LV}$ from space-based detectors

For the space-based detectors, including Taiji, TianQin, and LISA, the
constraints on $M_{\rm LV}$ are about three orders of magnitude weaker than
those from the ground-based detectors. As shown in Table 2, the constraints on
$M_{\rm LV}$ is roughly at $\gtrsim 2\times 10^{-24}\ {\rm GeV}$ for single GW
event. When considering a joint analysis of 100 simulated GW events, the
constraints from all three detectors are roughly at $\gtrsim 5\times
10^{-24}\;{\rm GeV}$. Among the three detectors, Taiji achieves the best
result, with $M_{\mathrm{LV}}>5.86\times 10^{-24}$ GeV. The reason why the
constraints from the space-based detectors are weaker than those from the
ground-based detectors is easy to understand. As one can see from Eq. (II.2),
the amplitude correction to the waveform due to the Lorentz-violating damping
effect is proportional to the square of the GW frequencies, which implies this
effect is more sensitive to the higher GW frequencies, and thus the ground-
based detectors can give stronger constraints than space-based detectors. We
note that to estimate the number of events within our joint redshift and total
mass range, we employ the data and methods from Refs. Wang:2021srv ;
Klein:2015hvg .

###  IV.3 Trends of $M_{\rm LV}$

In Fig. 3, we illustrate how the lower bound of the Lorentz-violating
parameter, $M_{\rm LV}$, varies with the total mass and redshift of binary
inspiral systems across a selection of detectors. Specifically, for aLIGO, CE,
and ET, we focus on a redshift range of $[0.01,0.5]$ Hz and a total mass range
of $[3,50]\ M_{\odot}$. For Taiji, TianQin, and LISA, we extend the redshift
range to $[0.01,10]$ Hz. The rationale behind selecting different total mass
ranges for these latter three detectors is linked to ensuring that the SNR,
$\rho$, exceeds 15 when the redshift value is maximized. This approach is
designed to sharpen the visibility of the $M_{\mathrm{LV}}$ trend,
highlighting the impact of high redshift values. As illustrated in Fig. 3, the
highest values of the Lorentz violation energy scale, $M_{\mathrm{LV}}$, are
achieved for sources that are both nearest and of the lowest mass, as observed
in the cases of aLIGO, CE, ET, and TianQin. Conversely, for Taiji and LISA,
the trend deviates. Here, $M_{\rm LV}$ does not attain its maximum in regions
characterized by lower total masses and smaller redshifts. This discrepancy
arises because the innermost stable circular orbit frequency,
$f_{\mathrm{ISCO}}$, in this domain, surpasses the upper-frequency limit of
these detectors. This is indicated by Eq. (3.6) and illustrated in Fig. 4.
Consequently, this suggests that Taiji and LISA might be less efficient in
detecting the final inspiral phase in compact binary systems that have lower
redshifts and smaller total masses.

The choice to initiate the redshift ($z$) analysis from 0.01 stems from the
observation that the lowest redshift value recorded in the LVK event catalog
is 0.01. Considering that the mass of a neutron star is typically around 1.5
$M_{\odot}$, we adopt this figure as the minimum mass threshold for our total
mass range. We cap the redshift at $z=0.5$ and set the maximum total mass at
50 $M_{\odot}$, which corresponds to the median GW source mass reported in the
LVK events. To streamline our analysis, we concentrate on binary systems
comprising either two black holes (BBH) or two neutron stars (BNS), assuming
equal mass for both components.

##  V CONCLUSION

Figure 4: The distribution of $f_{\mathrm{ISCO}}$ for Taiji and LISA is
depicted within the same astrophysical horizon, as illustrated in Fig. 3.

With the advent of future detectors, GWs are poised to play a pivotal role in
testing gravity in the strong field regime. Both ground-based and space-based
detectors are designed to capture GWs across different frequency bands,
spanning from $10^{-4}$ to $10^{4}$ Hz. In this study, we delve into the
investigation of the Lorentz-violating damping effect, which influences the
propagation of GWs. We aim to evaluate the capability of both ground-based
(aLIGO, CE, and ET) and space-based (Taiji, TianQin, and LISA) detectors in
constraining this effect. We begin by formulating the modified equations of
motion for the two polarizations of GWs. We then proceed to derive the altered
GW waveform in the Fourier domain, incorporating the Lorentz-violating damping
effect. Utilizing the FIM, we set out to quantify the constraints on the
energy scale $M_{\mathrm{LV}}$, showcasing its projected sensitivity for each
detector. For the FIM analysis, we establish detection thresholds for ground-
based detectors and space-based detectors at $\rho>8$ and $\rho>15$,
respectively. Additionally, we conduct a joint analysis of $M_{\mathrm{LV}}$
using simulated GW events to further our understanding of the constraints
achievable with future GW observations.

For ground-based detectors, the tightest constraint from a single event is set
by CE, with $M_{\rm LV}>3.02\times 10^{-21}$ GeV. When conducting a joint
analysis of 100 GW events, this constraint improves to $M_{\rm LV}>8.84\times
10^{-21}$ GeV. Regarding space-based detectors, the results are in line with
our expectations. For a single event, the constraints on $M_{\rm LV}$ from
these detectors are approximately $\gtrsim 2\times 10^{-24}\;{\rm GeV}$, which
is roughly three orders of magnitude less stringent than those obtained from
ground-based detectors. Upon performing a joint analysis of 100 simulated GW
events, the constraints from the space-based detectors converge to
approximately $\gtrsim 5\times 10^{-24}\;{\rm GeV}$.

In our conservative estimation of event numbers, we note an improvement of
more than twofold over the constraints obtained from individual events. We
posit that, with the ongoing accumulation of observational data, even more
stringent constraints on this effect will be achievable. Our analysis
indicates that targeting the high-frequency band offers a more efficacious
approach for constraining the Lorentz-violating damping effect. This suggests
that ground-based detectors are more adept at imposing rigorous constraints on
the effect in comparison to space-based detectors. Within this group of
detectors, CE distinguishes itself by offering the most stringent lower bound
on $M_{\mathrm{LV}}$.

Our analysis of the data distribution reveals that an effective strategy for
aLIGO, CE, ET, and TianQin to constrain the Lorentz-violating damping effect
involves concentrating on compact binary systems characterized by both a small
total mass and low redshift. Conversely, for Taiji and LISA, targeting GW
sources that have a small total mass but are situated at higher redshifts
proves to be more appropriate. This strategic focus is informed by the
differential sensitivity of these detectors to the frequency and amplitude of
GW signals, which in turn affects their capability to place constraints on the
Lorentz-violating effect.

## ACKNOWLEDGMENTS

We thank Hong-Chao Zhang, Chao Zhang, and Peng-Ju Wu for useful discussions.
Tao Zhu and Bo-Yang Zhang are supported by the National Key Research and
Development Program of China under Grant No.2020YFC2201503, the National
Natural Science Foundation of China under Grants No.12275238 and No. 11675143,
the Zhejiang Provincial Natural Science Foundation of China under Grants
No.LR21A050001 and No. LY20A050002, and the Fundamental Research Funds for the
Provincial Universities of Zhejiang in China under Grant No. RF-A2019015.
Jing-Fei Zhang and Xin Zhang are supported by the National Natural Science
Foundation of China (Grants Nos. 11975072, 11875102, and 11835009), the
National SKA Program of China (Grants Nos. 2022SKA0110200 and 2022SKA0110203),
and the National 111 Project (Grant No. B16009).

## APPENDIX A: PARTIAL DERIVATIVES OF THE WAVEFORM OF BINARY INSPIRAL

In this appendix, we present the partial derivatives of the GW waveform
parameters
$\\{\ln\mathcal{A},\ln\mathcal{M},\ln\mathcal{\eta},\phi_{\mathrm{c}},t_{\mathrm{c}},C_{\nu}\\}$
as follows,

$\displaystyle\frac{\partial\tilde{h}(f)}{\partial\ln{\cal{A}}}$
$\displaystyle=$ $\displaystyle\tilde{h}(f),$ (A.1)
$\displaystyle\frac{\partial\tilde{h}(f)}{\partial\ln\cal{M}}$
$\displaystyle=$ $\displaystyle\frac{5}{6}\tilde{h}(f),$ (A.2)
$\displaystyle\frac{\partial\tilde{h}(f)}{\partial\ln{\eta}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}+i\eta\Bigg{\\{}-\frac{3}{128}\eta^{-2}u^{-5/3}C_{\Psi}+\frac{3}{128}\eta^{-1}u^{-5/3}\Bigg{[}\frac{55}{9}u^{2/3}+\left(\frac{27145}{504}+\frac{3085}{36}\eta\right)u^{4/3}-\frac{65\pi}{9}\left(1+\rm\ln{u}\right)u^{5/3}$
(A.3)
$\displaystyle+\left(-\frac{15737765635}{3048192}+\frac{2255}{12}\pi^{2}+\frac{76055}{864}\eta-\frac{127825}{432}\eta^{2}\right)u^{2}+\pi\left(\frac{378515}{1512}-\frac{74045}{378}\eta\right)u^{7/3}\Bigg{]}\Bigg{\\}},$
$\displaystyle\frac{\partial\tilde{h}(f)}{\partial{\phi_{\rm c}}}$
$\displaystyle=$ $\displaystyle-i\tilde{h}(f),$ (A.4)
$\displaystyle\frac{\partial\tilde{h}(f)}{\partial{t_{\rm c}}}$
$\displaystyle=$ $\displaystyle(2\pi if)\tilde{h}(f),$ (A.5)
$\displaystyle\frac{\partial\tilde{h}(f)}{\partial{C_{\nu}}}$ $\displaystyle=$
$\displaystyle\Big{[}\frac{1}{2}(2\pi f)^{2}((1+z)^{2}-1)\Big{]}\tilde{h}(f),$
(A.6)

where $C_{\Psi}$ in Eq. (A.3) is given by

$\displaystyle C_{\Psi}$ $\displaystyle=$
$\displaystyle\left\\{1+\left(\frac{3715}{756}+\frac{55}{9}\eta\right)u^{2/3}-16\pi
u\right.$ (A.7)
$\displaystyle+\left(\frac{15293365}{508032}+\frac{27145}{504}\eta+\frac{3085}{72}\eta^{2}\right)u^{4/3}+\pi\left(\frac{38645}{756}-\frac{65}{9}\eta\right)(1+\ln
u)u^{5/3}$
$\displaystyle+\left[\frac{11583231236531}{4694215680}-\frac{640}{3}\pi^{2}-\frac{6848}{21}\gamma_{\mathrm{E}}-\frac{6848}{63}\ln(64u)+\left(-\frac{15737765635}{3048192}+\frac{2255}{12}\pi^{2}\right)\eta\right.$
$\displaystyle\left.\left.+\frac{76055}{1728}\eta^{2}-\frac{127825}{1296}\eta^{3}\right]u^{2}+\pi\left(\frac{77096675}{254016}+\frac{378515}{1512}\eta-\frac{74045}{756}\eta^{2}\right)u^{7/3}\right\\}.$

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