Testing general relativity with TianQin the prospect of using the inspiral signals of black hole binaries Changfu Shi1Mujie Ji1 2Jian-dong Zhang1and Jianwei Mei1

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Testing general relativity with TianQin: the prospect of using the inspiral signals

of black hole binaries

Changfu Shi,1Mujie Ji,1, 2 Jian-dong Zhang,1and Jianwei Mei1, ∗

1MOE Key Laboratory of TianQin Mission, TianQin Research Center for Gravitational Physics

& School of Physics and Astronomy, Frontiers Science Center for TianQin, Gravitational Wave

Research Center of CNSA, Sun Yat-sen University (Zhuhai Campus), Zhuhai 519082, China.

2Shantou Jinshan Middle School, Shantou 515073, China

(Dated: October 25, 2022)

In this paper, we carry out a systematic study of the prospect of testing general relativity with

the inspiral signal of black hole binaries that could be detected with TianQin. The study is based

on the parameterized post-Einsteinian (ppE) waveform, so that many modiﬁed gravity theories

can be covered simultaneously. We consider black hole binaries with total masses ranging from

10 M∼107Mand ppE corrections at post-Newtonian (PN) orders ranging from −4PN to 2PN.

Compared to the current ground-based detectors, TianQin can improve the constraints on the ppE

phase parameter βby orders of magnitude. For example, the improvement at the −4PN and 2PN

orders can be about 13 and 3 orders of magnitude (compared to the results from GW150914),

respectively. Compared to future ground-based detectors, such as ET, TianQin is expected to

be superior below the −1PN order, and for corrections above the −0.5PN order, TianQin is still

competitive near the large mass end of the low mass range [10 M,103M] . Compared to the

future space-based detector LISA, TianQin can be competitive in the lower mass end as the PN order

is increased. For example, at the −4PN order, LISA is always superior for sources more massive

than about 30 M, while at the 2PN order, TianQin becomes competitive for sources less massive

than about 104M. We also study the scientiﬁc potentials of detector networks involving TianQin,

LISA and ET, and discuss the constraints on speciﬁc theories such as the dynamic Chern-Simons

theory and the Einstein-dilaton Gauss-Bonnet theory.

I. INTRODUCTION

Pushing the experimental limit on testing general

relativity (GR) is essential in helping ﬁnd out the

breaking point of the century-old theory and reveal-

ing the deeper nature of gravity. GR has been tested

under a variety of conditions, such as with solar sys-

tem experiments and astrophysical observations [1],

yet no sure sign of beyond GR eﬀect has been found

[2]. Gravitational waves (GWs) generated from the

very early universe or by extremely compact objects

such as black holes can help extend the realm of test-

ing GR to the genuinely strong ﬁeld regime. Since the

ﬁrst detection of GWs by LIGO [3,4], many tests of

GR have been carried out and all the GW data has

been found to be consistent with GR so far [5–14].

Space-based GW detection in the millihertz fre-

quency band enjoins rich types, large numbers, and

diverse spatial distributions of GW sources, and ex-

pects many GW signals that are large in magnitude

and/or long in duration [15–17]. These factors make

the millihertz frequency band the golden band in GW

detection, bearing great importance in fundamental

physics [18], astrophysics [19] and cosmology [20]. So

for a space-based detector, it is of great importance

to study its potential in testing GR [21].

A diﬃculty in assessing the capability of a space-

based GW detector in testing GR is the lack of a

unique direction for the task. The success of GR

against experimental tests has resulted in a lack of

∗Email: meijw@sysu.edu.cn (Corresponding author)

eﬀective guidance in the construction of modiﬁed

graivty theorys (MGs), leading to a rather diversiﬁed

literature that has to be navigated with the help of a

mathematical theorem (see, e.g. [22]). The many dif-

ferent types of GW signals expected for a space-based

detector only add to the complexity of the task.

There have been a few strategies to deal with the

problem. For example, one can focus on testing if

the detected GW signals are consistent with the pre-

dictions of GR, such as residual test, inspiral-merge-

ringdown coincidence test, polarization test, and so

on. One can also employ waveform models that use

a set of purely phenomenological parameters to sig-

nify possible deviation from GR and use the observed

data to place constraints on these parameters. Both

schemes have been used by the LIGO-Virgo-KAGRA

collaboration [5–8]. For more focused treatment, one

can use phenomenological waveforms that are tai-

lored to a chosen set of MGs, then one not only can

place constraints on several MGs simultaneously, but

can also translate the results to an individual MG if

needed. A good example here is the parameterized

post-Einsteinian (ppE) waveform [23], which is based

on the post-Newtonion (PN) approximation and is

most suitable for binary systems in their early inspiral

stage and with comparable component masses.

In this paper, we use the ppE waveform to carry

out a systematic study of the prospect of using Tian-

Qin to test GR. TianQin is a planned space-based

GW detector expected around 2035 [24–27]. The tar-

get frequency band of TianQin is between 10−4Hz

and 1 Hz [28,29], and the expected sources include

galactic ultra-compact binary (GCB) [30], massive

black hole binary (MBHB) [31,32], intermediate-mass

arXiv:2210.13006v1 [gr-qc] 24 Oct 2022

black hole binary (IMBHB) [33], extreme mass ra-

tio inspiral (EMRI) [34], stellar-mass black hole bi-

nary (SBHB) [35], and stochastic gravitational wave

background (SGWB) [36]. There might also be unex-

pected sources [25,37]. A series of work has been car-

ried out to assess the scientiﬁc potential of TianQin,

such as on studying the astrophysical history of galax-

ies and black holes [30,31], the dynamics of dense star

clusters and galactic centers [34], the nature of gravity

and black holes [38–42], the expansion history of the

universe [43,44], and the fundamental physics related

to the very early universe [45–47]. This work is part

of the eﬀort.

Apart from doing a broad test of GR by using the

ppE waveform, we study how the results look like for

individual MGs. For this purpose we use two theories

as examples: the dynamic Chern-Simons (dCS) the-

ory and the Einstein-dilaton Gauss-Bonnet (EdGB)

theory. There is no particular reason why these two

theories are chosen, apart from the fact that the ppE

waveforms are known in these theories.

We also carry out a parallel study of some other

detectors as a comparison and to ﬁgure out the scien-

tiﬁc potential of detector networks made of these de-

tectors. Important examples include the third gener-

ation ground-based detectors, Cosmic Explorer (CE)

[48] and Einstein Telescope (ET) [49], and the space-

based detector, LISA [50]. Since there have been re-

sults on the joint detection of TianQin and CE [51],

we focus on ET and LISA in this paper.

The paper is organised as following. In section II,

we summarise the main existing works that are related

to this one. In section III, we recall the basic results

on the ppE waveform. In Section IV, we present the

methods and key assumptions used in the calculations.

In sections Vand VI, we present our main ﬁndings.

The paper concludes with a summary in section VII.

Throughout this paper, we use the natural units in

which GN=~=c= 1 .

II. SUMMARY OF EXISTING RESULTS

A lot of works have already been done on using the

inspiral signals detected by the space-based detector

LISA to test GR. Early works included using sig-

nals from extreme mass ratio inspiral systems to test

the no-hair theorem [52,53] and using signals from

neutron stars inspiraling into intermediate-mass black

holes to test the Scalar-Tensor theory [54].

For systems with comparable component masses,

Berti et al. have considered using inspiral signals to

constrain the massive Brans-Dicke theory by introduc-

ing leading order corrections to the PN waveform [55].

Arun et al. have used a set of phenomenological phase

parameters (one for each PN order) to characterize

the deviation of an MG from GR [56] and placed con-

straints on these phenomenological phase parameters

[56,57]. This is the precursor to the ppE method [23],

which uses a new set of phenomenological parameters

to replace the phenomenological phase parameters, by

dividing out the corresponding velocity factor at each

PN order.

Connish et al. have studied how the ppE param-

eters can be constrained by future detectors, such as

aLIGO/aVirgo and LISA [58]. Huwyler et al. have in-

vestigated the potential of using LISA to constrain the

ppE phase parameter β, as to be deﬁned in (2), with

MBHB [59]. The ppE formalism has also been used to

place constraints on speciﬁc MGs, such as Brans-Dicke

theory [60], Lorentz-Violating Gravity [61], G(t) the-

ory [62], and theories with massive gravitons, modiﬁed

dispersion relations or dipole radiation [63–67].

After the direct detection of GWs, Yunes et al.

have analyzed the constraints on the ppE phase pa-

rameters using the GW190514 and GW151226 sig-

nals, and have translated the results to some spe-

ciﬁc MGs [14]. Chamberlain et al. have studied how

some future detectors (four possible conﬁgurations of

LISA, aLIGO, A+, Voyager, CE, and ET-D) can con-

strain the ppE phase parameter βand some MGs (in-

cluding dipole radiation, extra dimensions, G(t) the-

ory, Einstein-Æther theory, Khronometric gravity and

Massive graviton theory), by using some example GW

signals [68].

After the multiband work on SBHB by Sesana [69],

Barausse et al. have employed the ppE formalism to

show that the multiband observation with aLIGO and

LISA can improve the expected constraints on the

GW dipole radiation by 6 orders of magnitude [70],

Carson et al. have studied constraints on the ppE pa-

rameters with multiband observation using CE and

several space-based detectors (LISA, TianQin, DE-

CIGO and B- DECIGO) [51], and they have also ana-

lyzed the multiband enhancement on constraining the

EdGB theory and the IMR consistency test [71].

Comparing to these existing works, we will do a

more thorough exploration on how the constraints on

the ppE parameters will depend on diﬀerent source

parameters, diﬀerent detectors, diﬀerent detection

schemes, and possibly, also diﬀerent detector net-

works.

III. THE PARAMETERIZED

POST-EINSTEINIAN WAVEFORM

Black holes binaries are ideal systems for testing

GR, for the strong ﬁeld condition they can provide and

for the less of environmental contamination that often

aﬀects other astrophysical systems. The evolution of

a black hole binary can be divided into three phases:

inspiral, merger, and ringdown. During the inspiral

phase, the two components of the system start widely

separated and their velocities are relatively small. The

corresponding waveforms can be well modeled through

the PN approximation for systems with comparable

component masses. In GR, the frequency domain

waveform is given by

hGR(f) = A(f)eiψ(f),

ψ(f)=2πtc+φc+ Σ∞

k=0φPN

ku(k−5)/3,(1)

where fis frequency, A(f) is the amplitude, tcand φc

are the coalescence time and phase, respectively, u=

(πMf)1/3is a characteristic velocity, M=η3/5M

is the chirp mass, M=m1+m2is the total mass,

η=m1m2/(m1+m2)2is the symmetrical mass ratio,

and φPN

kis the phase coeﬃcient at the (k/2)PN order.

Note φPN

kis completely determined by the source pa-

rameters for a binary black hole system [72].

The ppE waveform has been proposed [23] to study

MGs whose inspiral waveform has the same PN struc-

ture as (1). The diﬀerence between a given MG and

GR resides in how the amplitude and the coeﬃcients

φPN

kdepend on the source parameters. Suppose the

MG correction only happens at a particular PN or-

der or keeping only the leading order correction, the

waveform is given by

hppE(f) = hGR(f)(1 + αua)eiβub,(2)

where αand βare the ppE parameters, and aand b

are the ppE order parameters, satisfying

b= 2 PN −5, a =b+ 5 .(3)

GR is recovered with α=β= 0 .

The original work of [23] has only considered the

two GW polarization modes found in GR and has fo-

cused on quasi-circular orbits for the black hole bi-

naries. Extensions have been made to include extra

polarization modes [73], time domain waveforms [74],

eccentricity [75] and environmental eﬀect [76]. For any

particular MG, the relation between the theory and

the ppE parameter can be established by calculating

corrections to the evolution of the binary orbits [77].

In this way, the ppE parameters have been calculated

for a series of theories, such as Brans-Dicke gravity

[60], screened modiﬁed gravity [78], parity-violating

gravity [79], Lorentz-violating gravity [80], noncom-

mutative gravity [81], and quadratic modiﬁed gravity

[82]. For the EdGB and dCS theories that will be con-

sidered in this paper, the ppE parameters have also

been calculated [82].

The leading order modiﬁcation from EdGB starts

at the −1PN order, corresponding to b=−7 and

a=−2. The ppE parameters are [82]:

αEdGB =−5ζEdGB

192

(m2

1˜s2−m2

2˜s1)2

M4η18/5,

βEdGB =−5ζEdGB

7168

(m2

1˜s2−m2

2˜s1)2

M4η18/5,(4)

where ζEdGB ≡16π¯α2

EdGB/M 4, ¯αEdGB is the coupling

between the scalar ﬁeld and quadratic curvature term

in the theory [83], and ˜sn≡2(p1−χ2

n−1 + χ2

n)/χ2

n= 1,2, is the spin-dependent scalar charge of the n-

th component, with χnbeing the eﬀective spin. The

current best constraint on the theory comes from the

observation of GW200115, giving p|¯αEdGB|<1.3 km

[84].

The leading order modiﬁcation from dCS starts at

the 2PN order, corresponding to b=−1 and a= 4.

The ppE parameters are [77,82]:

αdCS =57713η−14/5ξdCS

344064 h1−14976η

57713 χ2

+1−215876η

57713 χ2

s−2δmχaχsi,

βdCS =−1549225η−14/5ξdCS

11812864 h1−16068η

61969 χ2

+1−231808η

61969 χ2

s−2δmχaχsi,(5)

where δm≡(m1−m2)/M ,χs= (χ1+χ2)/2 ,

χa= (χ1−χ2)/2 , ξdCS ≡16π¯α2

dCS/M 4, and ¯αdCS

is the coupling constant of the Chern-Simons correc-

tion [85]. The current best constraint on the theory

comes from the observation of neutron star systems,

giving √¯αdCS <8.5 km [86]. So far one is unable to

place a meaningful constraint on the dCS theory using

GW data directly, due to a lack of viable waveform.

IV. METHODS AND ASSUMPTIONS

We use the Fisher information matrix (FIM)

method to estimate the constraints on the ppE pa-

rameters αand β, and on the theory speciﬁc couplings

¯αEdGB and ¯αdCS . The whole parameter space is given

θ={M, η, DL, tc, φc, χ1, χ2, θnonGR},(6)

where DLis the luminosity distance, and θnonGR

stands for the non-GR parameters such as α,β,

¯αEdGB and ¯αdCS .

Assuming large signal-to-noise ratio (SNR) and

Gaussian noise, the uncertainties in the waveform pa-

rameters are characterized by

∆θa≡ph∆θa∆θai ≈ p(Γ−1)aa ,(7)

where h. . . istands for statistical average and Γ−1, the

covariance matrix, is the inverses of FIM [87,88],

Γab =∂h

∂θa



∂h

∂θb.(8)

When a signal is observed by multiple detectors si-

multaneously, the combined FIM is

Γtotal

ab = Γ(1)

ab + Γ(2)

ab +. . . , (9)

where 1,2, . . . denote diﬀerent detectors.

The inner product in (8) is deﬁned as

(p|q)≡2Zfhigh

flow

p∗(f)q(f) + p(f)q∗(f)

Sn(f)df , (10)

where Sn(f) is the sensitivity of the detector. The

low- and high-frequency cutoﬀs are taken to be:

flow = max hfPN

low , fD

lowi,

fhigh = min hfISCO , f D

highi,(11)

where fD

low and fD

high mark the end points of the sen-

sitivity band of the detector, fISCO = (63/2πM)−1

is the frequency at the innermost stable circular or-

bit (ISCO), and

fPN

low = (8πη3/5M)−1(5η3/5M/Tob)3/8(12)

is determined by the total observation time Tob . In

this paper, we will take Tob to be the length of time

from the beginning of the observation to the moment

when the binaries reach ISCO.

For the detectors we consider TianQin [28], LISA

[89], ET [49], and the twin constellation conﬁguration

of TianQin [31,38]. The sky averaged Michelson sen-

sitivity of TianQin can be modeled as [28,31],

Sn(f) = 10

3h1 + 2fL0

0.41 2iSN(f),(13)

where we use the following noise model [28,30,36]:

SN(f) = 1

0h4Sa

(2πf)41 + 10−4Hz

f+Sxi.(14)

Here L0=√3×108m is the arm length, √Sa=

1×10−15 m/s2/Hz1/2is the residual acceleration on

each test mass, and √Sx= 1 ×10−12 m/Hz1/2is

the displacement measurement noise in each laser link.

The sensitivity band of the detectors are chosen as

low = 10−4Hz , fD

high = 1 Hz ,for TianQin ,

low = 10−6Hz , fD

high = 1 Hz ,for LISA ,

low = 1 Hz , fD

high = 104Hz ,for ET .(15)

All detectors are limited to one year of operation,

except in part of subsection V A, when the eﬀect of

Tob is discussed. For TianQin, all binaries used in the

calculation are assumed to reach their ISCOs right

when TianQin ﬁnishes a 3 month observation (except

in subsection V B, which is dedicated to cases when

ISCOs are reached when TianQin is in between obser-

vation time windows). So only the last 0 ∼3 months

and 6 ∼9 months data will be used for TianQin. The

frequency bounds in the integrals are modiﬁed accord-

ingly.

The GR waveform hGR in (2) is generated using

IMRPhenomD [90,91]. We take tc= 0, φc= 0,

χ1= 0.4 and χ2= 0.2 in all the calculations, and we

only consider ppE corrections starting from the PN

orders in {−4PN, −3.5PN, ···, 2PN}, corresponding

to b∈[−13,−1] , and black hole binaries with total

masses in the range [10 M,107M] . Only sources

in the lower mass end will be observable by ET, so we

roughly divide the mass range into two sectors: the

low mass range, M∈[10 M, 103M] , and the high

mass range, M∈[103M, 107M] . All plots in

this paper will be made separately for these two mass

ranges.

A laser interferometer type detector is more sensi-

tive to the phase of a GW signal, so the ppE parameter

βis more severely constrained, while only in limited

cases that the eﬀect of the parameter αis not negli-

gible [92]. So for most part of the discussion, we will

focus on the constraints on β, while only in subsection

VI B that we will discuss the eﬀect of the amplitude

correction parameter α.

V. PROJECTED CONSTRAINTS ON β

In this section, we discuss the expected constraints

on the ppE parameter β. Our main ﬁndings are the

following.

A. What kind of sources are the best for

constraining β?

Although all components of the FIM contribute to

Eq.(7), the dominant contribution comes from

Γ(b)

ββ =∂h

∂β 



∂h

∂β = 4 Zfhigh

flow

u2bhGRh∗

Sn(f)df

'5π2b−4/3

24D2

η(5+2b)/5M(5+2b)/3

×Zfhigh

flow

f(2b−7)/3

Sn(f)df . (16)

One can see that M,η,DLand Tob are the main pa-

rameters aﬀecting the constraints on β, and the eﬀect

may diﬀer for diﬀerent PN orders. The luminosity

distance DLcontributes rather trivially through an

overall scaling,

∆β=q(Γ−1)ββ ∝DL,(17)

and so we will not consider it any longer.

The total mass contributes to ∆βthrough two

places. One is through the factor,

∆β(b)≈q(Γ−1

(b))ββ ∼M−(5+2b)/6,(18)

which improves with Mmonotonically for b=−2,−1

and worsens with Mmonotonically for b < −2 .1The

other is through the bounds in the integral,

∆β(b)∼Zfhigh

flow

f(2b−7)/3df

Sn(f)−1/2

,(19)

which does not have a clear trend and is diﬀerent for

diﬀerent detectors.

The dependences of ∆βon Mat diﬀerent PN orders

are shown in FIG. 1and FIG. 2. For all the plots, we

take DL= 500 Mpc, η= 0.22 for sources in the low

mass range, and DL= 15 Gpc, η= 0.22 for sources

in the high mass range.

One can see that the impact of the total mass on

the constraint can be signiﬁcant. For example, at the

1Here “improve” means that the value of ∆β(b)is decreasing,

and “worsen” means that the value of ∆β(b)is increasing.

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TestinggeneralrelativitywithTianQin:theprospectofusingtheinspiralsignalsofblackholebinariesChangfuShi,1MujieJi,1,2Jian-dongZhang,1andJianweiMei1,1MOEKeyLaboratoryofTianQinMission,TianQinResearchCenterforGravitationalPhysics&SchoolofPhysicsandAstronomy,FrontiersScienceCenterforTianQin,GravitationalW...

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Testing general relativity with TianQin the prospect of using the inspiral signals of black hole binaries Changfu Shi1Mujie Ji1 2Jian-dong Zhang1and Jianwei Mei1

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