Detecting Accelerating Eccentric Binaries in the LISA Band Zeyuan Xuanand Smadar Naozy Department of Physics and Astronomy UCLA Los Angeles CA 90095 and

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Detecting Accelerating Eccentric Binaries in the LISA Band

Zeyuan Xuan∗and Smadar Naoz†

Department of Physics and Astronomy, UCLA, Los Angeles, CA 90095 and

Mani L. Bhaumik Institute for Theoretical Physics,

Department of Physics and Astronomy, UCLA, Los Angeles, CA 90095, USA

Xian Chen‡

Astronomy Department, School of Physics, Peking University, 100871 Beijing, China and

Kavli Institute for Astronomy and Astrophysics at Peking University, 100871 Beijing, China

(Dated: January 12, 2023)

Many gravitational wave (GW) sources in the LISA band are expected to have non-negligible

eccentricity. Furthermore, many of them can undergo acceleration because they reside in the pres-

ence of a tertiary. Here we develop analytical and numerical methods to quantify how the compact

binary’s eccentricity enhances the detection of its peculiar acceleration. We show that the general

relativistic precession pattern can disentangle the binary’s acceleration-induced frequency shift from

the chirp-mass-induced frequency shift in GW template ﬁtting, thus relaxing the signal-to-noise ra-

tio requirement for distinguishing the acceleration by a factor of 10 ∼100. Moreover, by adopting

the GW templates of the accelerating eccentric compact binaries, we can enhance the acceleration

measurement accuracy by a factor of ∼100, compared to the zero-eccentricity case, and detect

the source’s acceleration even if it does not change during the observational time. For example, a

stellar-mass binary black hole (BBH) with moderate eccentricity in the LISA band yields an error

of the acceleration measurement ∼10−7m·s−2for SNR = 20 and observational time of 4 yrs. In

this example, we can measure the BBHs’ peculiar acceleration even when it is ∼1pc away from a

4×106MSMBH. Our results highlight the importance of eccentricity to the LISA-band sources

and show the necessity of developing GW templates for accelerating eccentric compact binaries.

I. INTRODUCTION

The detection of gravitational wave (GW) by LIGO

and Virgo [1, 2] has greatly enhanced our understand-

ing of the properties of compact objects in the Uni-

verse. In the future, the Laser Interferometer Space An-

tenna (LISA) [3] would broaden the scope of GW as-

tronomy by detecting GW signals in a lower frequency

band (10−4−10−1Hz), where GW signal in the LISA

band may last for years [4], potentially having signatures

of compact binary’s long-term evolution long before they

merge. Therefore, LISA detection can shed light on com-

pact object binary’s formation channel and surrounding

environment.

Among all the parameters that can be measured us-

ing LISA, the GW source’s acceleration is of great im-

portance, because it carries the signature of the binary’s

environment. Several studies suggested that accelerating

compact object binaries may signiﬁcantly contribute to

the population observed by LISA [5–29]. For example,

observational endeavors shows that stars often reside in

binaries, and high mass stars reside in higher multiples

[5–10]. According to long term stability arguments, the

spatial separation of these multiples is likely to evolve

into one tight inner binary with tertiary in wide outer

orbit(s) [11–16]. Stellar evolution of such systems may

∗Corresponding author: zeyuan.xuan@physics.ucla.edu

†snaoz@astro.ucla.edu

‡xian.chen@pku.edu.cn

result in compact object binaries that host at least a

tertiary companion [17–19]. The observation of white

dwarfs (WDs) also directly suggests that many WDs bi-

naries reside in a triple conﬁguration [20–22]. These com-

pact object binaries can undergo non-negligible acceler-

ation in the triple system, thus leaving an imprint of

acceleration on the GW signal.

Additionally, many binaries and compact object bi-

naries are supposed to reside in the center of galaxies

[23–30]. Already a few stellar binaries were detected in

the inner ∼0.1 pc of our galactic center [31]. Moreover,

the number of compact binaries visible in the LISA band

within the inner parsec of our galactic center is estimated

to be 14 −150 WD-WD, 0 −2 neutron star (NS) - black

hole (BH), 0.2−4 NS-NS, 0.3−20 BH-BH, and the re-

sults are of the same order for other galaxies [29]. These

systems mentioned above can undergo non-negligible ac-

celeration caused by the gravitational potential of the

supermassive black hole (SMBH) in the galactic center.

Measuring the acceleration will give us direct evidence

about the GW source’s dynamic environment. For ex-

ample, a binary black hole (BBH) orbiting the SMBH

on a close conﬁguration will undergo a time-dependent

Doppler shift induced by acceleration, which is poten-

tially detectable [32–38]. Additionally, an acceleration

signature on GW signal may help detect double WDs

(DWDs) accompanied by a star or a planet [34, 39–

43]. Further, eccentricity oscillations due to the Eccen-

tric Kozai Lidov mechanism [e.g., 44–46], can induce a

time-changing characteristic strain proﬁle in the LISA

band [47–50], thus imprinting a signature on a binary in

the presence of a tertiary. However, the eﬀect of accel-

arXiv:2210.03129v2 [astro-ph.HE] 11 Jan 2023

eration on the GW signal is not always distinguishable

and measurable. Sometimes it may degenerate with the

measurement of other parameters and yield a mislead-

ing interpretation of the physics behind the GW sources

[34, 38, 51, 52].

Consider a circular compact object binary as an exam-

ple. When its evolution is dominated by GW radiation

and the observer is in the rest frame of the source, the

intrinsic GW frequency fewill increase with time. The

frequency shift rate ˙

feis proportional to the “chirp mass”

as: ˙

fe∼ M5/3

c, where Mc≡(m1m2)3/5/(m1+m2)1/5,

and m1and m2are masses of the two compact objects

[53]. But if the GW source is accelerating, the peculiar

acceleration also leads to an extra frequency shift rate

facc by changing the peculiar velocity of the GW source

and inducing a time-dependent Doppler shift. Therefore,

the detected GW signal has two components that con-

tribute to the observed frequency shift rate, one from the

GW emission that shrinks and circularizes the orbit, and

the other from the acceleration of the system. In other

words, there is a degeneracy between the chirp mass and

peculiar acceleration since they both contribute to the

frequency shift rate in the GW signal for the leading or-

der.

Such kind of degeneracy makes it harder to measure

peculiar acceleration. Some studies suggested that GW

source with measurable acceleration may be limited to

the cases when the compact binary is very close to the

tertiary and the period of the outer orbit is short enough

[34, 38]. Further, for DWDs in the Milky Way, there is

a signiﬁcant parameter space where acceleration is large

enough to cause a non-negligible ˙

facc yet still not dis-

tinguishable (degenerate with other parameters) in GW

template ﬁtting [52].

However, there can be a diﬀerent story when the ec-

centricity of the GW source is detected. In fact, several

studies suggest that many eccentric compact binaries are

in the LISA band [23, 29, 47, 54–58]. And in some dy-

namical channels, the existence of a tertiary is supposed

to directly produce LISA-band sources by exciting the

eccentricity of inner orbit and accelerating the merger

[27–29, 57, 59–62]. Thus, we expect binaries in an accel-

erating environment to be eccentric.

If the binary has non-zero eccentricity, the general rel-

ativistic (GR) precession will induce a triplet waveform,

which in turn changes the frequency-peak position of each

harmonic. This signature can be used to extract the bi-

nary’s total mass independently of the frequency shift

rate [63, 64], making it possible to break the degener-

acy between peculiar acceleration and the chirp-mass-

induced frequency shift.

Here we propose a strategy aimed to break this degen-

eracy by considering the GR precession signature of the

accelerating eccentric binary. In Section II, we present

analytical methods yielding an overall understanding of

how eccentricity can disentangle the acceleration feature

from the chirp-mass-induced frequency shift. In Section

III, we introduce the numerical tools used to simulate

LISA event waveforms (§III A), analyze the waveform

(§III B), and estimating the error of parameter measure-

ments (§III C). Section IV shows the application of nu-

merical methods. In particular, in this section, we map

the parameter space where the GW signal from acceler-

ating sources can be distinguished from non-accelerating

GW templates and quantify the eﬀect of eccentricity on

the accuracy of peculiar acceleration measurement. Fi-

nally, in Section V, we oﬀer our discussion and conclude

that eccentric binaries have a clear signature on the GW

form when it is undergoing peculiar acceleration. Specif-

ically, the acceleration measurement enhanced by eccen-

tricity can shed light on the GW source’s environment.

II. ANALYTICAL CONSIDERATION

We begin with establishing the eﬀects of acceleration

on the GW signal (§II A). This eﬀect causes a degen-

eracy between the peculiar acceleration and the chirp-

mass-induced frequency shift. We suggest that eccentric-

ity can be used to disentangle this degeneracy. The role

of eccentricity in detecting compact binaries’ peculiar ac-

celeration can be divided into two parts.

1. Ignoring the possible contribution of acceleration in

the signal. In this case, consider using GW tem-

plates, in the data analysis, that do not include

the binary’s peculiar acceleration. Below (§II B),

we show that eccentricity can help distinguish an

accelerating eccentric compact binary from non-

accelerating ones.

2. Including the possible contribution of acceleration

in the signal. In this case, the data analysis uses

the GW templates with accelerating features, mak-

ing it possible to measure the binary’s acceleration.

Below (§II C), we show how the measurement of

acceleration depends on the eccentricity of the bi-

nary. Later, using numerical analysis, we demon-

strate that the eccentricity can increase the accel-

eration measurement accuracy (§IV B).

A. Mass-Acceleration Degeneracy in GW Data

Analysis

Consider a compact object binary with a semi-major

axis a, eccentricity e, and the two components’ masses m1

and m2. Its energy is dissipated by the GW radiation,

which results in the decrease of the orbital period Pb.

The average changing rate of the semi-major axis, ais

[65]:

da

dt =−64

G3m1m2(m1+m2)

c5a3(1 −e2)7/21 + 73

24e2+37

96e4,

(1)

where Gis the gravitational constant and cis the speed

of light.

When the binary has a non-zero eccentricity, the GW

signal is made up of multiple harmonics. The frequency

of GW’s n-th harmonic is related to the binary’s orbital

period as fj=n

e≈n/Pb, where the subscript “e” denotes

quantities in the rest frame of the source. Note that

since the orbital period is shrinking, the frequency of the

GW signal will increase with time. This phenomenon is

known as the “chirp signal” of a GW source.

GW data analysis studies often focus on the second

harmonic of the GW signal, which is the dominant sig-

nal for circular and low eccentricities orbits. Thus, here

we will take fj=2

eas an example to demonstrate the

commonly used method for estimating the mass of the

GW source. It should be noted that GW data analysis

methodology often proposes to ﬁt the numerical template

for the parameter estimation of GW sources [53, 66],

while the analysis presented here aims to explain diﬀer-

ent factors’ contribution to the result of the numerical

ﬁtting and give an analytical estimation of the accelera-

tion measurement accuracy in diﬀerent cases.

The time derivative of the 2nd harmonic is uniquely

determined by the frequency of the GW signal, eccen-

tricity, and the mass of the GW source for the leading

order:

fj=2

e=96π8/3

5Mc

c35/3

fj=2

e11/3F(e),(2)

We emphasize that Mcis the intrinsic chirp mass of the

binary as a GW source, which only depends on the mass

of the compact binary’s components; The observed chirp

mass, Mo, can be diﬀerent from Mcbecause the ob-

served GW signal can be distorted. F(e) is a function

of the compact binary’s eccentricity (the enhancement

function)[67]:

F(e)≡1 + 73

24 e2+37

96 e4

(1 −e2)7/2.(3)

Thus, we can estimate the intrinsic chirp mass Mcof

compact binaries based on two observables, the frequency

feand its time derivative ˙

fe. For the circular binary

case, the details of this method can be found in Cutler

and Flanagan [53]. On the other hand, for an eccen-

tric binary, the amplitude proﬁle of diﬀerent harmonics

is related to the eccentricity [68–70]. It can enable us to

measure the eccentricity in template ﬁtting [71]. In this

way, the results of [53] can be generalized by plugging in

fe,˙

fe, and einto Eq. (2) and ﬁnding the intrinsic chirp

mass of the source :

Mc=5c5

96π8/3G5/33/5

(fj=2

e)−11/5(˙

fj=2

e)3/5F(e)−3/5.

(4)

When the GW source is moving relative to the observer

with a velocity v, the frequency of GW’s each harmonic

will be shifted because of the Doppler eﬀect. In the ob-

server frame, the frequency, fo, is then [32–34, 72]:

fj=n

o=fj=n

ep1−β2

1 + βcos θ=fj=n

e(1 + zdop)−1,(5)

where β=v/c,θis the angle between the velocity vector

and the line of sight, and zdop is the Doppler coeﬃcient.

We are interested in the case when the GW source is

accelerating, which means that the peculiar velocity and

Doppler shift factor are changing with time. Thus, from

Eq. (5) it is straightforward to ﬁnd the time derivative

of the observed frequency [73], i.e.,

dtfj=n

o=˙

fj=n

(1 + zdop)2+fj=n

dt p1−β2

1 + βcos θ!,(6)

where the power 2 factor on the Doppler coeﬃcient comes

due to the transformation from the observed frames to

the source frame. The last term of Eq. (6), represents

the line of sight acceleration a///c. In other words :

a//

c=d

dt p1−β2

1 + βcos θ!.(7)

Equivalently, from Eq. (6), this acceleration relates to

the diﬀerence between the time derivative of frequency

in the observer’s frame and the rest frame of the GW

source. Such diﬀerence is directly caused by the change

of peculiar velocity, thus can be used to constrain the line

of sight acceleration a//, i.e.,

a//

c=˙

fo−˙

fe(1 + zdop)−2

.(8)

However, we cannot directly measure the frequency

(fe) and its time derivative ( ˙

fe) in the source frame.

Instead, the observed quantities are foand ˙

foin the ob-

server frame. As mentioned above, even when a// = 0,

constant peculiar velocity and cosmological redshift can

cause a signiﬁcant diﬀerence between foand feand

distorts the parameters of the GW source. This phe-

nomenon is known as “mass-redshift degeneracy,” [e.g.,

72, 74–76]. Because of the scale-free property of gravity,

the eﬀect of mass-redshift degeneracy can be exactly can-

celed out by making the measured chirp mass the “red-

shifted chirp mass” and the measured distance of the

source the luminosity distance [75] while keeping other

parameters the same. Thus, for simplicity, we will focus

on the eﬀect of acceleration and limit the discussion to

the cases when cosmological and gravitational redshift

can be neglected, i.e., z1. In particular, we assume

zdop a//fe/2c˙

fe, and simplify Eq. (8) to be:

fo≈˙

fe+fe

a//

c.(9)

The complete result for arbitrarily large redshift can be

recovered by replacing all the Mcin the following dis-

cussion to be the redshifted chirp-mass Mc(1 + z).

Most of the proposed LISA mission strategies are to

search for a signal using non-accelerating compact bi-

nary’s GW templates. [e.g. 71, 77] However, as discussed

above we expect a degeneracy between Mcand a// since

they both contribute to the frequency shift rate ˙

foin the

leading order. For the case of a compact object binary

with no or negligible eccentricity, only the 2nd harmonic

of the GW signal dominates. In the LISA band, such

kind of signal is a nearly-monochromatic sinusoidal wave

with slowly increasing frequency. Since the waveform

does not have other signiﬁcant features except for foand

fo, neglecting the acceleration’s contribution can lead to

a biased estimation of the chirp mass. In particular, if one

assumes ˙

fo≈˙

fe+fea///c (see Eq. (9)) as the intrinsic

frequency shift rate ˙

fein template ﬁtting, the resultant

chirp mass will diﬀer from the “true” chirp mass. This

can be seen by combining Eq. (4) and (9), which yields

an observed chirp mass with the following expression:

Mo=5c5

96π8/3G5/3F(e)3/5

(fj=2

o)−11/5(˙

fj=2

o)3/5

=5c5

96π8/3G5/3F(e)3/5

(fj=2

o)−11/5(˙

fj=2

e+a///c)3/5

6=Mc.(10)

Thus, there is a degeneracy between the binary’s acceler-

ation and mass in GW data analysis. In other words, the

acceleration can signiﬁcantly bias the chirp mass estima-

tion if the compact binary has small or zero eccentricity

and only the 2nd harmonic is detected.

B. Disentangling the Signatures of Accelerating

Eccentric GW Sources - Analytical Approach

For a GW source with peculiar acceleration, the chirp

mass measurement without considering an accelerating

template can lead to erroneous results (see examples

above). However, when the source has non-negligible

eccentricity and detectable multiple harmonics, the GR

precession creates a unique signature in the GW signal

and disentangles the acceleration from the bias on the

chirp mass. As a result, the accelerating GW signal will

be diﬀerent from any of the non-accelerating GW tem-

plates, thus can be identiﬁed.

Here we detailed the steps of this strategy. In particu-

lar, Section II B 1 describes how the GR precession pat-

tern of the eccentric compact binary leaves a signature on

the GW waveform, thus diﬀerentiating the accelerating

GW signal from the non-accelerating one. Section II B 2

estimates the critical acceleration for distinguishing ac-

celerating GW sources in data analysis.

1. Eccentricity as Means to Disentangle the

Mass-Acceleration Degeneracy

The GR precession of an eccentric orbit shifts the pe-

riastron by an angle δφ0for each period [78]. Thus, the

frequency of GW’s n-th harmonic will split into a triplet

0.000 0.002 0.004 0.006 0.008

0 . 0

5.0×10- 2 2

1.0×10- 2 1

1.5×10- 2 1

2.0×10- 2 1

h ( f )

o v e r a l l s p e c t r u m

0.001501 0.001502 0.001503 0.001504

0 . 0

2.0×10- 2 2

4.0×10- 2 2

6.0×10- 2 2

8.0×10- 2 2 1 s t h a r m o n i c

0.003000 0.003003 0.003006

0 . 0

5.0×10- 2 2

1.0×10- 2 1

1.5×10- 2 1

2.0×10- 2 1

h ( f )

f [ H z ]

2 n d h a r m o n i c

0.004501 0.004504 0.004507

0 . 0

5.0×10- 2 2

1.0×10- 2 1

1.5×10- 2 1

f [ H z ]

3 r d h a r m o n i c

FIG. 1. The frequency spectrum of GW signals from

two eccentric BBHs systems with the same chirp mass

but diﬀerent total mass. For the ﬁrst system (blue line)

m1= 8 M, m2= 40 M, e = 0.3; For the second system

(red line) m1=m2= 16.867 M, e = 0.3. Both of these

systems have the same chirp mass Mc∼14.684 M, but

as depicted the diﬀerence in total mass results in a diﬀerent

position of the harmonics. The initial period of radial motion

for the ﬁrst systems is 1.5mHz, but its GW frequency of each

harmonic is not the integer multiple of 1.5mHz because of the

shift induced by GR precession (∆f). For the second system,

its GR precession shifts the position of each harmonic by a

diﬀerent value. For illustration purposes, we adjust the initial

radial frequency of the second system to be slightly diﬀerent

from 1.5mHz, so that we compensate for the diﬀerence in

∆fand make its GW’s 2nd harmonic having the same initial

frequency as the ﬁrst system’s. The observation duration is

set to be two years.

(nf −∆f, nf, nf + ∆f) [63, 68], in which

∆f=6(2πG)2/3

(1 −e2)c2M2/3

total(Pb)−5/3,(11)

where Mtotal is the total mass of compact binary sys-

tem, Pbis the orbital period, and fequals 1/Pb. For

small eccentricity, coeﬃcients of some terms in the triplet

waveform are negligible, and the dominant component is

nf +∆f. Therefore, the position of each GW’s harmonic

will be shifted by ∆fin the frequency domain.

∆fis independent of the harmonic number because it

is a feature of the GR dynamic of the orbit. Thus, all

the harmonics will be shifted by the same value ∆f. This

feature allows us to disentangle ∆fand extract the total

mass of the system, Mtotal [63].

Figure 1 demonstrates this feature of the GR preces-

sion pattern. In this ﬁgure, we adopt the x-model [79] to

generate the GW signal from eccentric compact object

binaries (see Section III A for detailed information) and

show the frequency spectrum of the GW signals from two

diﬀerent compact binary systems. They have the same

chirp mass Mc, which means that for each harmonic the

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DetectingAcceleratingEccentricBinariesintheLISABandZeyuanXuanandSmadarNaozyDepartmentofPhysicsandAstronomy,UCLA,LosAngeles,CA90095andManiL.BhaumikInstituteforTheoreticalPhysics,DepartmentofPhysicsandAstronomy,UCLA,LosAngeles,CA90095,USAXianChenzAstronomyDepartment,SchoolofPhysics,PekingUniversity,1...

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