Astrophysics and cosmology with a decihertz gravitational-wave detector: TianGO

Astrophysics and cosmology with a decihertz gravitational-wave detector:

TianGO

Kevin A. Kuns ,

1,2

Hang Yu ,

Yanbei Chen,

and Rana X. Adhikari

LIGO Laboratory, California Institute of Technology, Pasadena, California 91125, USA

LIGO Laboratory, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

Theoretical Astrophysics 350-17, California Institute of Technology, Pasadena, California 91125, USA

Bridge Laboratory of Physics, California Institute of Technology, Pasadena, California 91125, USA

(Received 25 June 2020; accepted 23 July 2020; published 3 August 2020)

We present the astrophysical science case for a space-based, decihertz gravitational-wave (GW) detector.

We particularly highlight an ability to infer a source

’

s sky location, both when combined with a network of

ground-based detectors to form a long triangulation baseline, and by itself for the early warning of merger

events. Such an accurate location measurement is the key for using GW signals as standard sirens for

constraining the Hubble constant. This kind of detector also opens up the possibility to test type Ia

supernovae progenitor hypotheses by constraining the merger rates of white dwarf binaries with both super-

and sub-Chandrasekhar masses separately. We will discuss other scientific outcomes that can be delivered,

including the constraint of structure formation in the early Universe, the search for intermediate-mass black

holes, the precise determination of black hole spins, the probe of binary systems

’

orbital eccentricity

evolution, and the detection of tertiary masses around merging binaries.

DOI:

10.1103/PhysRevD.102.043001

I. INTRODUCTION

The coming decades will be an exciting time for

gravitational-wave (GW) astronomy and astrophysics

throughout the frequency band ranging from nano- to

kilohertz. In the 10

–

10,000 Hz band, detectors including

Advanced LIGO (aLIGO)

[1]

, Advanced Virgo (aVirgo)

[2]

, and KAGRA

[3]

are steadily improving toward their

sensitivity goals. Meanwhile, various upgrades to current

facilities have been proposed, including the incremental

upgrade

[4]

and the Voyager design which aims to

reach the limits of the current infrastructure

[5]

. In the long

run, third-generation detectors, including the Einstein

Telescope

[6,7]

and Cosmic Explorer

[8]

, are expected

to push the audio-band reach of GW astronomy out to

cosmological distances. In the millihertz band, space-borne

laser interferometers such as LISA

[9]

and TianQin

[10,11]

would give us exquisitely sensitive probes for many

astrophysical signals

—

both are planned to be launched

around 2035. At even lower frequencies, pulsar timing

arrays are becoming evermore sensitive with more pulsars

being added to the network

[12

–

14]

. Nonetheless, gaps still

exist between these missions. This especially limits our

ability to have a coherent, multiband coverage of the same

source; even a relatively massive

⊙

–

⊙

black hole

(BH) binary at 0.01 Hz (where LISA is most sensitive) will

not enter a ground-based detector

’

s sensitive band until

20 years later.

Therefore, we propose a space-based detector, TianGO,

which is sensitive in the 0.01

–

10 Hz band and which fills

the gap between LISA and the ground-based detectors

[15]

A possible advanced TianGO (aTianGO) would have 10

times better sensitivity, but is not discussed further here. In

this paper, we expand on the pioneering work of Ref.

[16]

and explore the scientific promise of TianGO. Our work

also sheds light on other decihertz concepts such as the big

bang observer

[17]

, Pre-DECIGO

[18]

, and DECIGO

[19]

Figure

shows the sensitivity of TianGO and other

major detectors. For the rest of the paper, unless otherwise

stated, the ground-based detectors are assumed to have the

Voyager design sensitivity

[5]

and the ground-based net-

work consists of the three LIGO detectors at Hanford (H),

Livingston (L), the United States, and Aundha, India (A);

Virgo (V) in Italy; and KAGRA (K) in Japan. The

corresponding detection horizons for compact binary

sources of different total mass are shown in Fig.

.For

stellar-mass compact objects such as neutron stars (NSs)

and BHs, TianGO has a comparable range as the ground-

based detectors. Moreover, even a relatively light NS binary

starting at 0.12 Hz, where TianGO is most sensitive, will

evolve into the ground-based detectors

’

band and merge

within 5 years. This facilitates a multiband coverage of

astrophysical sources.

In particular, by placing TianGO in an orbit from

between a 5 and 170 s light travel time from the Earth,

K. A. K and H. Y. contributed equally to this work.

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the localization of astrophysical sources is significantly

improved over that possible with a ground-based network

alone: when combined with the ground-based network, this

long baseline allows a combined TianGO-ground-based

network to increase the angular resolution by a factor of

∼

over that of the ground-based network alone. This

exquisite ability to localize sources enables this hybrid

network to do precision cosmography. Furthermore, since a

binary of two NSs or of an NS and BH will stay in

TianGO

’

s sensitivity band for several years, TianGO will

provide an early warning for the ground-based GW

detectors and the electromagnetic telescopes.

Meanwhile, there are astrophysical sources that are

particularly well suited to be studied by a decihertz detector

like TianGO. For example, intermediate-mass black holes

(IMBHs) are one of such examples. TianGO is sensitive to

the mergers of both a binary IMBHs and an IMBH with a

stellar-mass compact companion. Consequently, TianGO

will be the ideal detector to either solidly confirm the

existence of IMBHs with a positive detection or strongly

disfavor their existence with a null detection. Meanwhile,

mass transfer starts at

∼

mHz for a typical white dwarf

(WD) binary. This frequency will be higher for even more

massive, super-Chandrasekhar WD binaries. As LISA

’

sensitivity starts to degrade above 10 mHz, TianGO will

be the most sensitive instrument to study the interactions

between double WDs near the end of their binary evolution,

whichmaybetheprogenitorsoftype-Iasupernovae.Last,ifa

system is formed with a high initial eccentricity, TianGO will

be able to capture the evolution history of the eccentricity,

which will in turn reveal the system

’

s formation channel.

We summarize the major science targets we will be

considering in Table

and also in the text as follows. We

discuss the precision with which binary black holes (BBHs)

can be localized with a hybrid network and the application

to cosmography in Sec.

. We then examine TianGO

’

ability to localize coalescing binary NSs and to serve as an

early warning for ground-based and EM telescopes, the

most crucial component for multimessenger astrophysics,

in Sec.

III

. In Sec.

, we discuss the possibility of using

TianGO to distinguish the cosmological structure formation

scenarios and to search for the existence of IMBHs. This is

followed by our study of the progenitor problem of type Ia

supernovae in Sec.

. We then discuss the detectability of

tidal interactions in binary WDs with TianGO in Sec.

.In

Sec.

VII

, we analyze TianGO

’

s ability to accurately

determine both the effective and the precession spin, and

how we may use it to constrain the formation channels of

stellar-mass BH binaries as well as the efficiency of angular

momentum transfer in the progenitor stars. In Sec.

VIII

,we

explore TianGO

’

s capability of measuring the orbital

eccentricity evolution. In Sec.

, we discuss TianGO

’

ability to directly probe the existence of tertiary masses

around merging binaries.

II. GRAVITATIONAL-WAVE COSMOGRAPHY

The Hubble constant,

, quantifies the current expan-

sion rate of the Universe, and is one of the most funda-

mental parameters of the standard

CDM cosmological

−

Frequency [Hz]

−

Strain

TianGO

aTianGO

LISA

DECIGO

aLIGO

Voyager

Cosmic Explorer 2

Einstein Telescope (D)

TianQin

GW150914

FIG. 1. Sensitivities of future ground-based and space gravi-

tational-wave detectors. The sensitivities are averaged over sky

location and polarization. The LISA curve includes two 60°

interferometers and the ET curve includes three 60° interferom-

eters. The curve labeled

“

GW150914

”

ffiffiffi

, where

is the

waveform of the first gravitational wave detected

[20]

starting

5 years before merger.

Total source frame mass [

]

−

Redshift

Horizon

50% detected

90% detected

Cosmic Explorer 2

Einstein Telescope (D)

LISA

TianGO

aTianGO

LIGO Voyager

aLIGO

TianQin

Luminosity distance [Mpc]

FIG. 2. Horizons for equal mass compact binaries oriented face

on for the detectors shown in Fig.

. The maximum detectable

distance, defined as the distance at which a source has an SNR of

8 in a given detector, is computed for 48 source locations

uniformly tiling the sky. The horizon is the maximum distance

at which the best source is detected; 50% of these sources are

detected within the dark shaded band, and 90% of the sources are

detected within the light shaded band. If a source stays in a space

detector

’

s sensitivity band for more than 5 years, the 5-year

portion of the system

’

s evolution that gives the best SNR in each

detector is used.

KUNS, YU, CHEN, and ADHIKARI

PHYS. REV. D

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model, yet the two traditional methods of measuring it

disagree at the

level

[39]

. The first method relies on

the physics of the early Universe and our understanding of

cosmology to fit observations of the CMB to a cosmological

model

[40]

. The second, local measurement, relies on our

understanding of astrophysics to calibrate a cosmic distance

ladder. This ladderrelatestheredshifts ofobserved sourcesto

their luminosity distances

[39,41,42]

. Gravitational-wave

astronomy adds a third method of determining

and the

prospect of resolving this tension

[21

–

23,43,44]

, a task for

which a combined TianGO-ground-based network is par-

ticularly well suited.

To obtain the redshift-distance relationship necessary to

determine

, the local measurement first determines the

redshift of a galaxy. The luminosity distance cannot be

measured directly, however, and relies on the calibration of

a cosmic distance ladder to provide

“

standard candles.

”

the other hand, the luminosity distance is measured directly

from a GWobservation requiring no calibration and relying

only on the assumption that general relativity describes the

source. This makes gravitational waves ideal

“

standard

sirens.

”

If the host galaxy of a gravitational-wave source is

identified, optical telescopes can measure the redshift.

this way, both the redshift and the distance are measured

directly. The BNS GW170817 was the first GW source

observed by both gravitational and electromagnetic

observatories

[46]

. Since the gravitational-wave signal

was accompanied by an optical counterpart, the host galaxy

was identified and the first direct measurement of

using

this method was made

[47]

Identifying the host galaxy to make these measurements

requires precise sky localization from the GW detector

network. This ability is greatly enhanced when TianGO is

added to a network of ground-based detectors. TianGO will

be in either an Earth-trailing orbit of up to 20° or an orbit at

the L2 Lagrange point

[15]

, thereby adding a baseline of

between

235

⊕

and

⊕

to the network, where

⊕

is the radius of

the Earth. Since the same source will be observed by both

TianGO and the ground-based network, the timing accu-

racy formed by this large baseline significantly improves

the sky localization ability over that of the ground-based

alone, as is illustrated in Figs.

and

and Table

The top panel of Fig.

shows the angular resolution

ΔΩ

as a function of redshift as determined from the network of

ground-based detectors alone, TianGO alone, and the

combined network of the ground-based and TianGO in a

5° Earth-trailing orbit. The source is a BBH with

⊙

, and

°. (The probability

of detecting binaries with a given inclination peaks around

[48]

. The same figure for

° is shown in

Fig.

.) The extra long baseline formed by TianGO and

the ground-based network improves the angular uncertainty

by a factor of

∼

The middle panel of Fig.

shows the fractional uncer-

tainty

in measuring the luminosity distance. Note

that the inference accuracy for the ground-based network is

limited by the distance-inclination degeneracy. (This is

especially true for face-on sources as can be seen by

comparing Figs.

and

.) TianGO breaks this degeneracy

due to the time-dependent antenna pattern caused by its

tumbling orbit. The combined TianGO-ground-based

uncertainty is thus significantly better that of the ground-

based alone.

The bottom panel of Fig.

shows the uncertainty in

comoving volume localization

If an optical counter-

part is not observed, or does not exist as is likely for most of

the sources for which the TianGO-ground-based network

will be sensitive, the GW detector network must localize

the host to a single galaxy. To estimate the number of

galaxies contained in a comoving volume

, the value of

galaxies

Mpc

is assumed. The combined network

can localize a source to a single galaxy up to a redshift of

∼

for the best face-on sources and to

∼

for the

median sources at

°.

Even if the host galaxy cannot be uniquely identified,

galaxy catalogs can be used to make a statistical inference

about the location of the source

[21,49

–

52]

. This method

has been used to reanalyze the measurement from

TABLE I. Summary of TianGO science cases.

Section

Scientific objective

Target

Information to extract

Key references

Cosmography

Binary BHs

Sky location

[21

–

23]

III

Multimessenger astrophysics; NS physics Binary NSs

Sky location

[24]

Structure formation; IMBHs

Binaries involving IMBHs Source population

[25

–

27]

Type-Ia SNe progenitors

Binary WDs

Source population

[28

–

30]

WD physics

Binary WDs

Tidal dephasing

[31

–

33]

VII

Formation of binary BHs; stellar evolution Binary BHs

Aligned and precession spin

[34,35]

VIII

Formation of binary BHs

Binary BHs

Orbital eccentricity

[36,37]

Environment around BHs

Binary BHs

Phase modulation

[38]

The GW standard sirens can also be used to independently

calibrate the EM standard candles forming the cosmic distance

ladder

[45]

The Planck 2015 cosmology is assumed

[40]

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GW170817 to infer

without the unique galaxy deter-

mination provided by the observation of the optical

counterpart

[53]

and has been used to improve the original

analysis of Ref.

[47]

with further observations of BBHs

without optical counterparts

[54]

. Future work will quantify

the extent to which the TianGO-ground-based network

’

exquisite sky localization can improve the reach of these

methods.

III. EARLY WARNING OF BINARY NEUTRON

STAR COALESCENCE

The joint detection of a coalescing binary NS in GW

[46]

and

ray

[55]

, and the follow-up observation of the

postmerger kilonova in electromagnetic radiations

[56]

herald the beginning of an exciting era of multimessenger

astronomy. While the first detection has provided some

valuable insights on the nature of short

-ray bursts and

kilonovae, significantly more are expected to come from

future multimessenger observations

[24]

. The success of

such a joint observation relies critically on the GW

observatories to produce an accurate sky map of the

source

’

s location in a timely manner, and TianGO is an

ideal instrument to perform the early warning and locali-

zation of coalescing compact binaries. As a typical NS

binary will stay in TianGO

’

s band for a few years before the

final merger, the Doppler phase shift and time-dependent

antenna patterns due to TianGO

’

s orbital motion enable it to

localize the source by itself with high accuracy.

This is illustrated in detail in Fig.

and Table

. Figure

shows the

cumulative angular uncertainty

for a typical NS

binary with

Þ¼ð

⊙

;

⊙

. More spe-

cifically, on the bottom of the frame, we show the GW

frequency up to which we integrate the data, and on the top

of the frame we show the corresponding time to the final

merger, given by

Þ¼

⊙

−

days

−

ΔΩ

deg

Best

50% detected

90% detected

Ground + TianGO

TianGO Only

Ground Only

−

0.25

0.50

0.75

1.00

1.25

1.50

Redshift

Mpc

0.0

2.5

5.0

7.5

10.0

Luminosity distance [Gpc]

−

Expected number of galaxies

FIG. 3. Sky localization, luminosity distance, and volume

localization precision as a function of redshift for a binary black

hole system with

⊙

, and an inclination

° and TianGO in a 5° Earth-trailing orbit. A notional density

galaxies

Mpc

is used to convert

to the expected

number of galaxies. Figure

shows the same for face-on binaries.

TABLE II. Comparison of sky localization for different networks of detectors. The neutron star system is a binary

with

⊙

, and

Mpc. The black hole system is a binary with

⊙

and

600

Mpc. The best and median sources are given in deg

. The ground-based detectors have the Voyager

design sensitivity.

Neutron star

Black hole

Network

Best

Median

Best

Median

HLV

−

HLVKA

−

HLVKA

−

HLVKA

°T

−

HLVKA

°T

−

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We assume that the source has a face-on orientation, yet we

vary its right ascension and declination to cover the entire

sky. Two representative distances,

Mpc and

100

Mpc, are shown in the plot. With TianGO alone,

we can localize the majority of sources to within a few ×

−

deg

approximately 10 days before the final merger.

This provides sufficient time for the GW network to

process the data and inform the electromagnetic observa-

tories to prepare the telescopes for the final merger.

Furthermore, the localization accuracy for NS binaries

obtained by TianGO alone is in fact nearly 100 times better

than a network of five ground-based detectors each with

Voyager

’

s designed sensitivity (LHVKA; see Table

) and

is much smaller than the typical field of view of an optical

telescope of

square degree.

In addition to postmerger emissions, TianGO also

significantly enhances the possibility of capturing the

potential precursor emissions during the inspiral phase

(see, e.g., Sec. 2.2 of Ref.

[57]

). One example is the energy

release due to shattering of the NS crust

[58]

, which is

suspected to be the source of short

-ray burst precursors

[59]

. The timing when the precursor happens is directly

related to the equation of state of materials near the crust-

core interface. Additionally, if at least one of the NSs is

highly magnetized, the orbital motion during the inspiral

may also trigger an electron-positron pair fireball that will

likely emerge in hard x ray/gamma ray

[60]

. In the radio

band, the magnetospheric interaction may also extract the

orbital energy and give rise to a short burst of coherent

radio emission

[61]

. Such an emission could be a mecha-

nism leading to fast radio bursts

[62]

. With TianGO

’

ability to accurately pinpoint a source day prior to the

merger, one can unambiguously associate a precursor

emission at the right time and location to coalescing binary

NSs.

IV. COSMOLOGICAL STRUCTURE FORMATION

AND INTERMEDIATE-MASS BLACK HOLES

Massive BHs reside in the center of most local galaxies.

Despite the fact that the mass of the central BH is only

∼

of the total mass of the host galaxy, surprisingly

clear correlations between the massive BH

’

s mass and the

properties of the host galaxy have been observed (e.g.,

Ref.

[63]

). This thus suggests a coevolution of the massive

BH and its host galaxy

[64]

, which is further sensitive to the

seed from which the massive BHs grow (see Ref.

[65]

for a

review). Broadly speaking, a massive BH may grow from

either a

“

heavy seed

”

with mass

∼

–

⊙

at a

relatively late cosmic time of

∼

–

, or from a

“

light

seed

”

with mass

≃

100

–

600

⊙

at an earlier time of

≃

Those light seeds may be generated from the collapse of

Pop III stars

[66]

, and they may merge with each other in

the early Universe

[67]

The characteristic frequency of such a merger is given by

the system

’

s quasinormal mode frequency. For a

Schwarzschild BH, the fundamental, axially symmetric,

quadrupolar mode oscillates at a frequency of

[68]

det

QNM

≃

⊙

We have used the superscript

“

(det)

”

to represent quantities

measured in the detector frame. While it is a frequency too

low for ground-based detectors and too high for LISA, it

falls right into TianGO

’

s most sensitive band. Indeed, as

shown in Fig.

, TianGO is especially sensitive to systems

with masses in the range of

100

–

1000

⊙

and can detect

them up to a redshift of

∼

100

. Consequently, if massive

BHs grow from light seeds, TianGO will be able to map out

the entire growth history throughout the Universe. On the

other hand, a null detection of such mergers by TianGO can

then rule out the light seed scenario. It will also constrain

our models of Pop III stars that will be otherwise chal-

lenging to detect even with the

James Webb Space

Telescope

[69]

. In either case, TianGO will provide

indispensable insights in our understanding of cosmologi-

cal structure formation (see also Refs.

[25

–

27]

for relevant

discussions for LISA and the third-generation ground-

based GW observatories).

Meanwhile, those seed BHs that failed to grow into

massive and supermassive BHs may be left to become

IMBHs in the local Universe

[70,71]

. While a few IMBH

candidates have been reported (see, e.g.,

[72

–

74]

), a solid

confirmation is still lacking from electromagnetic obser-

vation. This makes the potential GW detection of an IMBH

particularly exciting. In addition to the merger of two

0.3

GW frequency [Hz]

−

Angular uncertainty deg

Best

50% detected

90% detected

= 50 Mpc

= 200 Mpc

−

Time to coalescence [days]

FIG. 4. Angular uncertainty as determined by TianGO alone for

a face-on BNS (at 12 source locations uniformly tiling the sky)

with

Þ¼ð

⊙

;

⊙

as a function of GW

frequency or time to coalescence.

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IMBHs (similar to the mergers of light BH seeds discussed

above), another potential GW source involving an IMBH is

the intermediate-mass-ratio inspirals (IMRIs): a stellar-

mass object (BH, NS, or WD) merges with an IMBH.

IMRIs may be found in the dense cores of globular

clusters

[75,76]

TianGO will detect a typical IMRI source with

Þ¼ð

1000

⊙

;

⊙

with an SNR

of 10 after averaging over both orientation and sky location.

If the event rate for such a merger is about

per Gpc

per

year as argued in Ref.

[16]

, we would be able to detect

nearly 1000 IMRI mergers over a 5-year observation

period. The numerous detections would thus allow us to

both place constraints on the dynamics in globular clusters

and perform potential tests of general relativity in a way

similar to those using the extreme-mass-ratio inspirals

[77]

V. BINARY WHITE DWARVES AS PROGENITORS

OF TYPE IA SUPERNOVAE

Type Ia supernovae are one of the most powerful family

of standard candles for determining the cosmological

distance

[78]

and they have led to the discovery of the

accelerating expansion of the Universe

[41]

. However, the

identity of their progenitors remains an unresolved problem

in modern astrophysics despite decades of research. Among

all possibilities, the merger of two WDs (also known as the

double-degenerate progenitor) is an increasingly favored

formation channel, yet it is still unclear if the system

’

s total

mass exceeding the Chandrasekhar limit is a necessary

condition for a supernova explosion (for recent reviews, see

Refs.

[29,30]

). In this section, we show how TianGO can

help to improve our understanding of the problem (see

Ref.

[28]

for a similar discussion for LISA).

The key is that TianGO is capable of individually resolve

essentially

all

the galactic WD binaries when they are close

to starting or have just started mass transfer. This is

illustrated in Fig.

. In the upper panel, we show the

GW frequency for WD binaries at the onset of the Roche-

lobe overflow. Here we assume a simple mass-radius

relation for WDs as

Þ¼

⊙

−

;

and we find the orbital separation such that the donor star

’

radius is equal to the volume-equivalent radius of its Roche

lobe

[79]

. For such systems, the SNR (averaging over both

sky location and source orientation) seen by TianGO over a

5-year observation period is shown in the lower panel. The

source distance is fixed at 10 kpc. TianGO thus allows us to

construct thorough statistics on the WD population which

can further be used to calibrate theoretical population

synthesis models (e.g., Refs.

[80,81]

). Then, comparing

the merger rate of double WDs predicted in the model to the

observed type Ia supernovae rate allows a test of the

double-degenerate progenitor hypothesis.

Specifically, for a population of WDs driven by GW

radiation only, the number density per orbital separation

should scale with the orbital separation

∝

for

≥−

;

for

−

;

where

is the power-law index of the population

’

s initial

separation distribution. This scaling is valid for binaries

with a current separation of

≪

AU and prior to

Roche-lobe overflow. Once we determine the constant of

proportionality with TianGO, we can then predict the

merger rate as

[29]

While LISA is expected to detect a similar number of

WD binaries as TianGO, there are nonetheless unique

advantages of TianGO in constraining the binary WD

population. Note that a WD binary in LISA

’

s more sensitive

band of 1

–

20 mHz will evolve in frequency by so little over

∼

-year observation that it either is unresolvable or can

only be used to measure the system

’

s chirp mass. In the

case of the type Ia supernovae progenitor problem,

[mHz]

=1.0

=1.5

=2.0

0.9

1.2

1.5

1.8

2.1

(

)[

]

100

SNR

FIG. 5. Upper panel: the GW frequency

for WD binaries

with different total masses

and mass ratios

≡

≥

at the onset of Roche-lobe overflow. Bottom panel:

angle-averaged SNR seen by TianGO, assuming a source dis-

tance of 10 kpc and an observation period of 5 years.

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however, it is the system

’

s total mass and mass ratio that are

of interest. TianGO, on the other hand, is more sensitive to

systems at higher frequencies (

≳

mHz) and therefore

will see a greater amount of frequency evolution. Moreover,

those systems will experience a stronger tidal effect which

depends on the masses in a different way than the chirp

mass, allowing for a determination of the component

masses (see Sec.

for more details). Consequently, with

TianGO, we can determine the distributions for double WD

systems with different total masses. This is critical for

examining the possibility of sub-Chandrasekhar progeni-

tors

[82

–

85]

At the same time, TianGO will also be able to identify

the deviation of the power-law distribution for different

WD binaries due to the onset of mass transfer. The stability

of the mass transfer is a complicated problem that depends

on factors like the system

’

s mass ratio, the nature of the

accretion, and the efficiency of tidal coupling

[86

–

90]

TianGO will provide insights on this problem by both

locating the cutoffs in the distribution that marks the onset

of unstable mass transfer and measuring directly the

waveforms of the surviving systems that may evolve into

AM CVn stars

[91]

. TianGO also has the potential of

resolving the current tension between the observed spatial

density of AM CVn stars and that predicted by population

synthesis models

[92]

VI. DETECTING WHITE DWARF TIDAL

INTERACTIONS

When a WD binary

’

s orbit decays due to GW radiation,

tidal interaction starts to play an increasingly significant

role in its evolution. In this section, we discuss the

prospects of detecting tides in WDs with TianGO.

The tidal response of a fluid can be decomposed into an

equilibrium component and a dynamical component. In the

equilibrium tide, the fluid distribution follows the gravita-

tional equipotential instantaneously. In most situations, this

already captures the large-scale distortion of the star. The

dynamical tide, on the other hand, accounts for the star

’

dynamical response to the tidal forcing and represents the

excitation of waves. Whereas for NSs in coalescing

binaries, the equilibrium component dominates the tidal

interaction

[93

–

95]

; for WDs in binaries, it is the dynamical

tide that has the most significant effect.

As shown in Refs.

[31

–

33,96

–

98]

, when a WD binary

enters TianGO

’

s band, the dynamical tide can keep the WD

’

spin nearly synchronized with the orbit. Consequently,

;

≃

;

≃

orb

;

where

;

is the angular spin velocity of mass 1 (2). In

terms of energy, we have

tide

≃

orb

∝

Here

tide

isthe amount ofenergytransferred per unittime

fromtheorbittotheinteriorofmass1(2)and beingdissipated

there,

is the moment ofinertia of WD 1(2), and

is the

point-particle GW power.

In the top panel of Fig.

, we show the energy dissipation

rate via different channels as a function of the system

’

sGW

frequency. Here we focus on a

Þ¼ð

⊙

;

⊙

WD binary. We compute the radii using Eq.

(3)

and assume

. When the system enters

TianGO

’

s most sensitive band of

mHz, the dynami-

cal tideaccountsformorethan 10% ofthe orbitalenergy loss.

As a comparison, the energy transferred into the equilibrium

tide (as computed following Ref.

[32]

)isonlyaminor

amount.

The tidal interaction accelerates the orbital decay and

thus increases the amount of frequency chirping during a

given period, as is illustrated in the bottom panel of Fig.

[erg s

−1

]

GW Radiation

Dynamical Tide

Equilibrium Tide

[mHz]

−2

−1

[ Hz]

GW Only

GW + Tide

Resolution

FIG. 6. Tidal interactions for a

⊙

–

⊙

WD binary.

Upper panel: orbital energy dissipation/transfer rates

in differ-

ent channels. Lower panel: total GW frequency shift of the binary

over a 5-year observation period,

, as a function of the initial

GW frequency

. Frequency shifts greater than

obs

are

resolvable.

Here we ignore the rotational modification of the WD

structure, as the Coriolis force only mildly modifies the tidal

dissipation in subsynchronously rotating WDs

[99]

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In the plot, we show the increase in system

’

s GW frequency

over an observation period of 5 years with (the orange

trace) and without (the blue trace) the tidal effect as a

function of the initial frequency

at the start of the

observation. Note that

tide

∝

. Therefore, measur-

ing the excess frequency shift will allow us to directly

constrain the moment of inertia of WDs.

To quantify the detectability of

, we construct GW

waveforms taking into account the tidal interactions (see

Appendix

for details) and then use the Fisher matrix to

estimate the parameter estimation error. We focus on the

same

Þ¼ð

⊙

;

⊙

WD binary as

before and fix its distance to be 10 kpc but randomize

its orientation and sky location. The median uncertainty in

’

s moment of inertia over a 5-year observation is

summarized in Table

III

for different initial GW frequen-

cies. Due to the way the moment of inertia enters the

waveform, we are most sensitive to the sum

and it

can be constrained to a level of better than 1% for sources at

a gravitational-wave frequency of

mHz.

With such a high level of statistical accuracy, we can

imagine that a precise relation between WD

’

s mass and

moment of inertia can be established after a few detections.

We can then use this tidal effect to improve the measur-

ability of other parameters. For example, due to a WD

binary

’

s slow orbital motion

—

orb

−

even at the

onset of Roche-lobe overflow, where

orb

is orbital velocity

—

it is challenging to measure parameters such as the mass

ratio that come from high-order post-Newtonian correc-

tions using the point-particle GW waveform alone.

However, it is critical to know not only the chirp mass

but also the component masses when tackling problems

like identifying progenitors of type Ia supernovae (Sec.

Nonetheless, if we assume

, the tide will then

introduce a mass dependence that is different from the chirp

mass and has a more prominent effect on the orbital

evolution than the post-Newtonian terms. It is thus a

promising way to help constrain a WD binary

’

s component

masses.

This is illustrated in Fig.

. Here we compare the

parameter estimation uncertainty on the mass ratio for

systems with different total masses. We set each system

’

GW frequency to be the one right before the Roche-lobe

overflow and fix the true mass ratio to be 1.2. When tides

are included, we assume a fixed relation between a WD

’

moment of inertia and mass as

Þ¼

⊙

gcm

Compared to the point-particle results (blue traces), the

ones including the tidal effect (orange traces) can reduce

the statistical error on mass ratio,

, by nearly 3 orders of

magnitude over a large portion of parameter space.

VII. CONSTRAINING PROGENITORS OF BLACK

HOLE BINARIES BY MEASURING SPINS

The detections by aLIGO and aVirgo have confirmed the

existence of stellar-mass BH binaries. A question to ask

next is then what is the astrophysical process that gives

birth to these systems. Currently, the two most compelling

channels are isolated binary evolution in galactic fields

[100,101]

and dynamical formation in dense star clusters

[102]

. A potentially powerful discriminator of a system

’

progenitor is the spin orientation (see, e.g., Refs.

[34,103

–

106]

). Isolated field binaries will preferentially have the

spin aligned with the orbital angular momentum, whereas

in the case of dynamical formation, the orientation is more

likely to be isotropic.

While ground-based detectors are sensitive to the effec-

tive aligned spin parameter

eff

(the mass-weighted sum of

two BHs

’

dimensionless spins along the direction of orbital

angular momentum

[107]

), the determination of spin

components that lie in the orbital plane, often parametrized

as the effective precession spin parameter

[108]

, will be

challenging due to the limited sub-10 Hz sensitivities for

ground-based detectors

[109]

. TianGO, on the other hand,

is sensitive down to 10 mHz and can thus measure the

modulations due to the precession spin

with much higher

accuracy. TianGO thus allows us to construct a

two-

dimensional

spin distribution (in

eff

and

) of stellar-

TABLE III. Uncertainties in the sum of WDs

’

moment of

inertia for different initial GW frequencies at the start of a 5-year

observation period.

[mHz]

1.1

−

0.9

1.2

1.5

1.8

2.1

(

)[

]

−

Best

50% detected

90% detected

GW Only

GW + Tide

FIG. 7. Uncertainties in inferring the mass ratio,

, for WD

binaries with different total masses.

KUNS, YU, CHEN, and ADHIKARI

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043001-8

mass BH binaries that cannot be constructed with ground-

based detectors alone, and consequently provide valuable

insights into the formation history of binaries.

In Fig.

, we show the sky-location-averaged uncertainty

for sources located at a redshift of

(

≃

Gpc). To capture the precession effect, we use

the

IMRP

henom

v2 waveform model

[107]

and assume all

sources to have a moderate spin rate of

eff

;

Þ¼

−

;

These values are chosen for illustrative pur-

poses, yet the conclusions we draw are generic. The source-

framechirpmass

andmassratio

areallowedtovary.As

shown in the figure, for TianGO (left panel),

is measurable

(

) in almost the entire parameter space as long as

the mass ratio is slightly greater than 1. As a comparison, a

network of ground-based detectors consisting of HLVKA

(right panel) can only detect

over a small portion of the

parameter space (

⊙

and

). This demon-

strates TianGO

’

s unparalleled ability to determine

One caveat though is that the above analysis assumes

binary BHs have a broad range of spins with

≲

a=M<

as in the case of x-ray binaries

[110]

. However, the BBHs

detected by aLIGO and aVirgo during the first and second

observing runs

[111]

suggest that most BHs may have only

low spins of

a=M <

[113]

, which may be the

consequence of an efficient angular momentum transfer

in the progenitor stars

[35]

. In this case, a moderate

would be an indication of the merger event involving a

second-generation BH

[114]

As for the majority of the slowly spinning BHs, TianGO

can still deliver valuable information ground-based detec-

tors cannot access. This is illustrated in Fig.

where we

present the uncertainty in

eff

. This time we assume the

system to only have a slow spin rate of

eff

;

Þ¼

;

while the other parameters are the same as in

Fig.

. The Voyager network cannot constrain

eff

for

systems spinning at such a slow rate. TianGO, on the other

hand, can still achieve an accuracy of

eff

≲

over

most of the parameter space. This opens up the possibility

of discriminating different angular momentum transfer

models that all predict the majority of BHs having spins

in the range of

a=M

∼

–

[35,115

–

119]

VIII. REVEALING ORBITAL ECCENTRICITY

EVOLUTION

So far, our discussions have focused on systems with

circular orbits. This is a good assumption for signals at

Hz as the GW radiation may have efficiently

dissipated away the initial eccentricity. Nonetheless, at

lower frequencies, the residual eccentricity left from the

binary

’

s formation may leave a detectable imprint on the

GW waveform. While LISA can detect a fraction of

the eccentric systems at a few tens of millihertz if the

initial eccentricity is mild (see, e.g., Refs.

[120,121]

), it will

1.25

1.50

1.75

2.00

2.25

0.2

0.4

0.6

0.8

1.0

1.25

1.50

1.75

2.00

2.25

FIG. 9. Similar to Fig.

but showing the uncertainties in

eff

for binaries with

eff

;

Þ¼ð

;

. Note that the uncertainties

eff

are amplified by a factor of 100 in the plots, and that the

color scales are different in the two panels.

1.25

1.50

1.75

2.00

2.25

0.1

0.2

0.3

0.4

0.5

1.25

1.50

1.75

2.00

2.25

0.3

0.5

0.7

0.9

ΔΔ

FIG. 8. Uncertainties in the precession spin parameter

for

TianGO (left) and a network of five Voyager-like detectors

(right). We vary the source

’

s chirp mass and mass ratio, while

fixing

eff

;

Þ¼ð

−

;

. The source is assumed to be at

and the sky location is marginalized over. Note that the

color scales are different in the two panels.

Specifically, here we set the components of the spins as

eff

, and

. The (non-

unique) way of choosing the components does not significantly

affect the final results, as these components only enter the inspiral

part of the waveform via the combination

eff

;

in the

IMRP

henom

waveform. The initial frequency we choose to

set the spin components is fixed at 0.01 Hz, consequently fixing

the orbital and spin precession phases.

Reference

[112]

reported a highly spinning BBH, yet this

event has lower detection significance compared to the others. If

the event is indeed astrophysical, it might hint at a chemically

homogeneous formation

[101]

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likely miss those formed with very high initial eccentricities

−

≲

[37]

. Such a high initial eccentricity can

be produced if the binary is formed via binary-

single scattering

[122,123]

, hierarchical triple interactions

[106,124

–

126]

, or gravitational braking

[127]

;seeRef.

[37]

for a comprehensive summary. A decihertz detector like

TianGO will then be the only way to detect the evolution of

such systems. We elaborate on this point further in this

section.

The detectability of the orbital eccentricity has been

studied in detail in Ref.

[36]

whose key components are

summarized in the following. The GW strain from an

eccentric binary can be decomposed into a superposition of

different orbital harmonics as

Þ¼

∞

;

where each harmonic varies at a frequency

and is

given by

orb

The angle

represents the direction of the pericenter, and

we have defined

as the average between the radial and

azimuthal frequencies. Note that for a circular orbit, all the

GW power is radiated through the

harmonic in the

leading-order quadrupole approximation, and hence there

exists a one-to-one mapping between time and GW

frequency [cf. Eq.

(1)

]. When the orbit is eccentric,

however, at a given instant the GW strain contains multiple

frequency components.

In the frequency domain, the characteristic strain ampli-

tude

c;k

of harmonic

[36]

c;k

Þ¼

ffiffiffiffiffiffiffiffiffiffiffiffi

;

where

is the GW power radiated at frequency

(see

Ref.

[128]

). The corresponding SNR for each harmonic can

then be evaluated as

SNR

i¼

c;k

dln

where

is the power spectral density of the noise in

detector

, and the factor of 5 in the denominator accounts

for the averaging over sky location. To evaluate Eq.

(10)

first integrate the Keplerian elements

orb

;e;

(i.e., the

orbital frequency, eccentricity, and argument of pericenter,

respectively) from a set of initial values to the final plunge.

We then evaluate

and

at each instant

with the corresponding Keplerian elements at the same

moment under a post-Newtonian approximation as was

done in Ref.

[36]

. The spin has been neglected throughout

this section.

In Fig.

, we show the evolution of the first four

harmonics for a system with

⊙

at a

redshift of

. We consider two initial conditions.

The solid lines correspond to a system formed with an

initial semimajor axis and eccentricity of

;

−

Þ¼

;

−

and the dashed lines correspond to

;

−

Þ¼ð

;

−

. Such systems can form via

triple interactions in dense stellar environments. On each

curve, we also mark the times corresponding to (5 years,

1 day, 1 hour) prior to the final plunge with the (plus, dot,

cross) symbols. As shown in the plot, systems with such high

initial eccentricities are likely to be missed by LISA because

when the system has an orbital frequency of

orb

a few × mHz, a significant amount of the total GW power is

radiated through the third and forth (and even higher)

harmonics whose GW frequencies

≃

orb

are higher

than LISA

’

s most sensitive band. Meanwhile, the amplitude

of the

≥

harmonics decays quickly as the eccentricity is

damped by the GW radiation and becomes negligible in the

−

Frequency [Hz]

−

Characteristic strain

LISA

TianGO

Voyager

5 years

1 year

1 hour

years

year

hour

5 years

1 year

1 hour

FIG. 10. Evolution of the characteristic strain amplitude

for

the first four orbital harmonics of a BH binary with

⊙

. The solid (dashed) curves represent systems

with an initial eccentricity of

−

and

AU. The x axis corresponds to the frequency of each

harmonic [Eq.

(9)

]. The thin black traces represent the sky-

averaged sensitivity (in

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

) for LISA, TianGO, and (a

single) Voyager, respectively. The pluses, dots, and crosses

correspond, respectively, to instants 5 years, 1 day, and 1 hour

prior to the final merger.

Note that

c;k

is a dimensionless quantity.

This is the same system as the highly eccentric binary

considered in Fig. 1 of Ref.

[37]

except for that we place the

system at a further distance.

KUNS, YU, CHEN, and ADHIKARI

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