1908.06004.pdf

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

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.

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 in-

cluding Advanced LIGO (aLIGO) [1], Advanced Virgo

(aVirgo) [2], and KAGRA [3] are steadily improving to-

wards their sensitivity goals. Meanwhile, various up-

grades to current facilities have been proposed, includ-

ing the incremental A+ upgrade [4] and the Voyager de-

sign which aims to reach the limits of the current in-

frastructure [5]. In the long run, third generation de-

tectors including the Einstein Telescope [6, 7] and Cos-

mic 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 fo many astrophysical signals

– both are planned to be launched around 2035. At even

lower frequencies, pulsar timing arrays are becoming ev-

ermore 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, multi-band coverage of the same source; even

a relatively massive 30

-30

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 dis-

cussed further here. In this paper we expand on the

∗

These two authors contributed equally

pioneering work of Ref. [16] and explore the scientific

promise of TianGO. Our work also sheds light on other

decihertz concepts [17, 18].

Fig. 1 shows the sensitivity of TianGO and other ma-

jor 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

network consists of the three LIGO detectors at Hanford

(H), Livingston (L), USA, and Aundha, India (A); Virgo

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

ing detection horizons for compact binary sources of dif-

ferent total mass are shown in Fig. 2. 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 multi-band coverage of

astrophysical sources.

In particular, by placing TianGO in an orbit from be-

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

localization of astrophysical sources is significantly im-

proved 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 fac-

tor of

∼

50 over that of the ground-based network alone.

This exquisite ability to localize sources enables this hy-

brid network to do precision cosmography. Furthermore,

since a binary of two NSs or of a 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 of IMBHs

and an IMBH with a stellar-mass compact companion.

arXiv:1908.06004v2 [gr-qc] 25 Jun 2020

TABLE I. Summary of TianGO science cases

Section

Scientific Objective

Target

Information to extract

Key references

Cosmography.

Binary BHs

Sky location

[19–21]

III

Multi-messenger astrophysics; NS physics.

Binary NSs

Sky location

[22]

Structure formation; IMBHs.

Binaries involving IMBHs

Source population

[23–25]

Type-Ia SNe progenitors.

Binary WDs

Source population

[26–28]

WD physics.

Binary WDs

Tidal dephasing

[29–31]

VII

Formation of binary BHs; Stellar evolution.

Binary BHs

Aligned and precession spin

[32, 33]

VIII

Formation of binary BHs.

Binary BHs

Orbital eccentricity

[34, 35]

Environment around BHs.

Binary BHs

Phase modulation

[36]

−

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 grav-

itational wave detectors. The sensitivities are averaged over

sky location and polarization. The LISA curve includes two

◦

interferometers and the ET curve includes three 60

◦

inter-

ferometers. The curve labeled “GW150914” is 2

√

, where

is the waveform of the first gravitational wave detected [37]

starting five years before merger.

Consequently, TianGO will be the ideal detector to ei-

ther solidly confirm the existence of IMBHs with a pos-

itive detection or strongly disfavor their existence with

a null-detection.

Meanwhile, mass transfer starts at

∼

30 mHz for a typical white dwarf (WD) binary. This

frequency will be higher for even more massive, super-

Chandrasekhar WD binaries. As LISA’s sensitivity starts

to degrade above 10 mHz, TianGO will be the most sensi-

tive instrument to study the interactions between double

WDs near the end of their binary evolution, which may

be the progenitors of type-Ia supernovae. Lastly, if a

system is formed with a high initial eccentricity, TianGO

will be able to capture the evolution history of the eccen-

tricity, which will in turn reveal the system’s formation

channel.

We summarize the major science targets we will be

considering in Table I and also in the text as follows. We

discuss the precision with which BBHs can be localized

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. 1. The maximum de-

tectable 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 max-

imum 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.

with a hybrid network and the application to cosmogra-

phy in Section II. We then examine TianGO’s ability to

localize coalescing binary NSs and to serve as an early

warning for ground-based and EM telescopes, the most

crucial component for multi-messenger astrophysics, in

Section III. In Section IV 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 progeni-

tor problem of type Ia supernovae in Section V. We then

discuss the detectability of tidal interactions in binary

WDs with TianGO in Section VI. In Section VII we an-

alyze 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 Section VIII we ex-

plore TianGO’s capability of measuring the orbital ec-

centricity evolution. In Section IX we discuss TianGO’s

ability to directly probe the existence of tertiary masses

around merging binaries.

II. GRAVITATIONAL-WAVE COSMOGRAPHY

The Hubble constant,

, quantifies the current ex-

pansion rate of the universe, and is one of the most fun-

damental parameters of the standard ΛCDM cosmolog-

ical model, yet the two traditional methods of measur-

ing it disagree at the 4

level [38]. The first method

relies on the physics of the early universe and our un-

derstanding of cosmology to fit observations of the CMB

to a cosmological model [39]. The second, local mea-

surement, relies on our understanding of astrophysics to

calibrate a cosmic distance ladder. This ladder relates

the redshifts of observed sources to their luminosity dis-

tances [38, 40, 41]. Gravitational wave astronomy adds

a third method of determining

and the prospect of

resolving this tension [19–21, 42, 43], a task for which a

combined TianGO-ground-based network is particularly

well suited.

To obtain the redshift-distance relationship necessary

to determine

, the local measurement first determines

the redshift of a galaxy. The luminosity distance can-

not be measured directly, however, and relies on the cal-

ibration of a cosmic distance ladder to provide “stan-

dard candles.” On the other-hand, the luminosity dis-

tance is measured directly from a GW observation re-

quiring no calibration and relying only on the assump-

tion that general relativity describes the source. This

makes gravitational waves ideal “standard sirens.” If

the host galaxy of a gravitational wave source is iden-

tified, 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 ob-

servatories [45]. Since the gravitational wave signal was

accompanied by an optical counterpart, the host galaxy

was identified and the first direct measurement of

us-

ing this method was made [46].

Identifying the host galaxy to make these measure-

ments requires precise sky localization from the GW de-

tector network. This ability is greatly enhanced when

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

TianGO will either be in a Earth-trailing orbit of up to

◦

or an orbit at the L2 Lagrange point [15] thereby

adding a baseline of between 1

km = 235

⊕

and

km = 8

⊕

to the network, where

⊕

The GW standard sirens can also be used to independently cali-

brate the EM standard candles forming the cosmic distance lad-

der [44].

−

∆Ω

[

deg

]

Best

50% detected

90% detected

Ground

TianGO

TianGO Only

Ground Only

−

∆

Redshift

∆

[

Mpc

]

Luminosity distance [Gpc]

−

Expected number of galaxies

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

calization precision as a function of redshift for a binary black

hole system with

= 25

= 1

05, and an inclination

= 30

◦

and TianGO in a 5

◦

Earth trailing orbit. A notional

density of 0

01 galaxies

Mpc

is used to convert ∆

to the

expected number of galaxies. Fig. 13 shows the same for face-

on binaries.

the radius of the Earth. Since the same source will be ob-

served by both TianGO and the ground-based network,

the timing accuracy formed by this large baseline signif-

icantly improves the sky localization ability over that of

the ground-based alone, as is illustrated in Figs. 3 and 13

and Table II.

The top panel of Fig. 3 shows the angular resolu-

tion ∆Ω 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

= 25

= 1

05, and

= 30

◦

. (The

probability of detecting binaries with a given inclination

TABLE II. Comparison of sky localization for different net-

works of detectors. The neutron star system is a binary with

= 1

05, and

= 50 Mpc. The black

hole system is a binary with

= 25

= 1

05, 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 + L2 T

−

HLVKA + 5

◦

−

HLVKA + 20

◦

−

peaks around

= 30

◦

[47]. The same figure for

= 0

◦

is shown in Fig. 13.) The extra long baseline formed

by TianGO and the ground-based network improves the

angular uncertainty by a factor of

∼

50.

The middle panel of Fig. 3 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 degener-

acy. (This is especially true for face-on sources as can be

seen by comparing Figs. 3 and 13.) 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. 3 shows the uncertainty in

comoving volume localization ∆

If an optical coun-

terpart 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 num-

ber of galaxies contained in a comoving volume ∆

, the

value of 0

01 galaxies

Mpc

is assumed. The combined

network can localize a source to a single galaxy up to a

redshift of

∼

5 for the best face-on sources, and to

∼

35 for the median sources at

= 30

◦

Even if the host galaxy cannot be uniquely identified,

galaxy catalogs can be used to make a statistical infer-

ence about the location of the source [19, 48–51]. This

method has been used to reanalyze the measurement

from GW170817 to infer

without the unique galaxy

determination provided by the observation of the opti-

cal counterpart [52] and has been used to improve the

original analysis of Ref. [46] with further observations

of BBHs without optical counterparts [53]. Future work

will quantify the extent to which the TianGO-ground-

based network’s exquisite sky localization can improve

The Planck 2015 cosmology is assumed [39].

the reach of these methods.

III. EARLY WARNING OF BINARY NEUTRON

STAR COALESCENCE

The joint detection of a coalescing binary NS in

GW [45] and

-ray [54], and the follow-up observation

of the post-merger kilonova in electromagnetic radia-

tions [55] heralds the beginning of an exciting era of

multi-messenger 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 multi-messenger observations [22].

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

localization of coalescing compact binaries. As a typi-

cal 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 mo-

tion enables it to localize the source by itself with high

accuracy.

This is illustrated in detail in Fig. 4 and Table II. Fig. 4

shows the

cumulative angular uncertainty

for a typical NS

binary with (

, M

)=(1

). More specif-

ically, on the bottom of the frame we show the GW fre-

quency 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

(

) = 5

(

)

−

(

1 Hz

)

−

days

(1)

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,

= 50 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 fi-

nal merger. This provides sufficient time for the GW

network to process the data and inform the electromag-

netic observatories 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 bet-

ter than a network of 5 ground-based detectors each with

Voyager’s designed sensitivity (LHVKA; see Table II),

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

optical telescope of

(1) square-degree.

In addition to post-merger emissions, TianGO also sig-

nificantly enhances the possibility of capturing the poten-

tial precursor emissions during the inspiral phase (see,

e.g., Section 2.2 of Ref. [56]). One example is the en-

ergy release due to shattering of the NS crust [57], which

is suspected to be the source of short-

-ray burst pre-

cursors [58]. The timing when the precursor happens

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 (

, M

) = (1

) as a function of

GW frequency or time to coalescence.

is directly related to the equation of state of materials

near the crust-core interface. Additionally, if at least

one of the NS 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 [59]. In the radio band, the magnetospheric interac-

tion may also extract the orbital energy and give rise

to a short burst of coherent radio emission [60]. Such

an emission could be a mechanism leading to fast radio

bursts [61]. With TianGO’s ability to accurately pin-

point a source days prior to the merger, one can unam-

biguously 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 galax-

ies. Despite the fact that the mass of the central BH is

only

∼

1% of the total mass of the host galaxy, surpris-

ingly clear correlations between the massive BH’s mass

and the properties of the host galaxy have been observed

(e.g., Ref. [62]). This thus suggests a co-evolution of

the massive BH and its host galaxy [63], which is fur-

ther sensitive to the seed from which the massive BHs

grow (see Ref. [64] for a review). Broadly speaking, a

massive BH may grow from either a “heavy seed” with

mass

∼

−

at a relatively late cosmic time of

∼

−

10, or from a “light seed” with mass

100

−

600

at an earlier time of

20. Those “light seeds” may be

generated from the collapse of Pop III stars [65] and they

may merge with each other in the early Universe [66].

The characteristic frequency of such a merger is given

by the system’s quasi-normal mode frequency. For a

Schwarzschild BH, the fundamental, axially symmetric,

quadrupolar mode oscillates at a frequency of [67],

(det)

QNM

(

1 +

)(

)

(2)

We have used the superscript “(det)” to represent quan-

tities measured in the detector-frame. While it is a fre-

quency too low for ground-based detectors and too high

for LISA, it falls right into TianGO’s most sensitive band.

Indeed, as shown in Figure 2, TianGO is especially sensi-

tive to systems with masses in the range of 100

−

1000

and can detect them up to a redshift of

∼

100. Conse-

quently, if massive BHs grow from light seeds, TianGO

will be able to map out the entire growth history through-

out 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 challenging to detect even

with the

James Webb Space Telescope

[68]. In either

case, TianGO will provide indispensable insights in our

understanding of cosmological structure formation (see

also Refs. [23–25] 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 [69, 70]. While a few IMBH

candidates have been reported (see, e.g., [71–73]), 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 IMBHs (similar to the mergers of light BH seeds dis-

cussed 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 [74, 75].

TianGO will detect a typical IMRI source with

(

)=(1000

) at

=1 with an SNR of 10

after averaging over both orientation and sky location.

If the event rate for such a merger is about 1 per Gpc

per year as argued in Ref. [16], we would be able to de-

tect 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 [76].

V. BINARY WHITE DWARVES AS

PROGENITORS OF TYPE IA SUPERNOVAE

Type Ia supernovae are one of the most powerful fam-

ily of standard candles for determining the cosmologi-

cal distance [77] and they have led to the discovery of

[mHz]

(

) [

]

100

SNR

FIG. 5. Upper panel: the GW frequency

for WD bina-

ries with different total masses (

) and mass ratios

≡

≥

1 at the onset of Roche-lobe overflow. Bot-

tom panel: angle-averaged SNR seen by TianGO, assuming a

source distance of 10 kpc and an observation period of 5 years.

the accelerating expansion of the Universe [40]. How-

ever, the identity of their progenitors remains an unre-

solved 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 Chan-

drasekhar limit is a necessary condition for a supernova

explosion (for recent reviews, see Refs. [27, 28]). In this

section we show how TianGO can help to improve our

understanding of the problem (see Ref. [26] for a similar

discussion for LISA).

The key is that TianGO is capable of individually re-

solve essentially

all

the Galactic WD binaries when they

are close to starting or have just started mass transfer.

This is illustrated in Figure 5. 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

(

) = 10

(

)

−

(3)

and we find the orbital separation such that the donor

star’s radius is equal to the volume-equivalent radius of

its Roche lobe [78]. For such systems, the SNR (averag-

ing 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. [79, 80]). Then, comparing the merger rate of dou-

ble 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

α <

−

(4)

where

is the power-law index of the population’s initial

separation distribution. This scaling is valid for binaries

with a current separation of

01 AU and prior to

Roche-lobe overflow. Once we determine the constant of

proportionality with TianGO, we can then predict the

merger rate as

(

[27].

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 sen-

sitive band of 1-20 mHz will evolve in frequency by so

little over a

∼

5-year observation that it either is un-

resolvable or can only be used to measure the system’s

chirp mass. In the case of the type Ia supernovae progen-

itor problem, 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

(

20 mHz) and therefore will see a greater amount of

frequency evolution. Moreover, those systems will expe-

rience a stronger tidal effect which depends on the masses

in a different way than the chirp mass, allowing for a de-

termination of the component masses (see Section VI for

more details). Consequently, with TianGO we can de-

termine the distributions for double WD systems with

different total masses. This is critical for examining the

possibility of sub-Chandrasekhar progenitors [81–84].

At the same time, TianGO will also be able to iden-

tify the deviation of the power-law distribution for dif-

ferent WD binaries due to the onset of mass transfer.

The stability of the mass transfer is a complicated prob-

lem that depends on factors like the system’s mass ratio,

the nature of the accretion, and the efficiency of tidal

coupling [85–89]. TianGO will provide insights on this

problem by both locating the cutoffs in the distribution

that marks the onset of unstable mass transfer, and mea-

suring directly the waveforms of the surviving systems

that may evolve into AM CVn stars [90]. 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 [91].

VI. DETECTING WHITE DWARF TIDAL

INTERACTIONS

When a WD binary’s orbit decays due to GW radia-

tion, tidal interaction starts to play an increasingly sig-

nificant 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

gravitational equipotential instantaneously. In most sit-

uations, this already captures the large-scale distortion

of the star. The dynamical tide, on the other hand,

accounts for the star’s 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 [92–94], for WDs in bina-

ries, it is the dynamical tide that has the most significant

effect.

As shown in Refs. [29–31, 95–97], when a WD binary

enters TianGO’s band, the dynamical tide can keep the

WD’s spin nearly synchronized with the orbit. Conse-

quently,

Ω

orb

(5)

where Ω

1(2)

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

In terms of energy, we have

tide1(2)

1(2)

Ω

orb

∝

(6)

Here

tide1(2)

is the amount of energy transferred per

unit time from the orbit to the interior of mass 1(2) and

being dissipated there,

1(2)

is the moment of inertia of

WD 1(2), and

is the point-particle GW power.

In the top panel of Figure 6, we show the energy dis-

sipation rate via different channels as a function of the

system’s GW frequency. Here we focus on a (

, M

) =

) WD binary. We compute the radii

using Eq. (3) and assume

1(2)

= 0

1(2)

. When

the system enters TianGO’s most sensitive band of

f >

10 mHz, the dynamical tide accounts for more than 10%

of the orbital energy loss. As a comparison, the energy

transferred into the equilibrium tide (as computed fol-

lowing Ref. [30]) is only a minor 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. 6. 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

Here we ignore the rotational modification of the WD structure,

as the Coriolis force only mildly modifies the tidal dissipation in

subsynchronously rotating WDs [98].

[erg s

−

]

GW Radiation

Dynamical Tide

Equilibrium Tide

[mHz]

−

∆

[

Hz]

GW Only

Tide

Resolution

FIG. 6. Tidal interactions for a 0

-0

WD binary.

Upper panel: orbital energy dissipation/transfer rates

different 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.

as a function of the initial frequency

at the start of the

observation. Note that

tide1(2)

∝

1(2)

. Therefore, mea-

suring the excess frequency shift will allow us to directly

constrain the moment of inertia of WDs.

To quantify the detectability of

1(2)

, we construct GW

waveforms taking into account the tidal interactions (see

Appendix B for details) and then use the Fisher matrix

to estimate the parameter estimation error. We focus on

the same (

, M

) = (0

) WD binary as

before and fix its distance to be 10 kpc but randomize

its orientation and sky location. The median uncertainty

in WD’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

f >

10 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 detec-

tions. We can then use this tidal effect to improve the

measurability 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 or-

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

−

bital velocity — it is challenging to measure parameters

such as the mass ratio that come from high-order post-

Newtonian corrections using the point-particle GW wave-

form 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 super-

novae (Section V). 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 Figure 7. Here we compare the

parameter estimation uncertainty on the mass ratio for

systems with different total masses. We set each system’s

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’s moment of inertia and mass as

(

) = 3

(

)

g cm

(7)

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 three

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 [99, 100] and dynamical formation in dense

star clusters [101]. A potentially powerful discriminator

of a system’s progenitor is the spin orientation (see, e.g.,

Refs. [32, 102–105]). Isolated field binaries will prefer-

entially 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

ef-

fective aligned spin parameter

eff

(the mass-weighted

sum of two BHs’ dimensionless spins along the direction

of orbital angular momentum [106]), the determination

(

) [

]

−

∆

Best

50% detected

90% detected

GW Only

Tide

FIG. 7. Uncertainties in inferring the mass ratio, ∆

, for WD

binaries with different total masses.

of spin components that lie in the orbital plane, often

parameterized as the

effective precession spin parameter

[107], will be challenging due to the limited sub-10 Hz

sensitivities for ground-based detectors [108]. 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-mass BH binaries that cannot be con-

structed with ground-based detectors alone, and conse-

quently provide valuable insights into the formation his-

tory of binaries.

In Figure 8, we show the sky-location-averaged un-

certainty in

for sources located at a redshift of

= 2 (

16 Gpc). To capture the precession ef-

fect, we use the

IMRPhenomPv2

waveform model [106]

and assume all sources to have a moderate spin rate of

(

eff

, χ

)=(

−

These values are chosen for illus-

trative purposes, yet the conclusions we draw are generic.

The source-frame chirp mass

and mass ratio

are al-

lowed to vary. As shown in the figure, for TianGO (left

panel),

is measurable (∆

) in almost the en-

tire 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

Specifically, here we set the components of the spins as

eff

, and

= 0. The (non-

unique) way of choosing the components does not significantly

affect the final results, as these components only enter the inspi-

ral part of the waveform via the combination (

eff

, χ

) in the

IMRPhenomPv2

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.

(TianGo)

(Ground)

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

, χ

)=(

−

6). The source is assumed to be at

= 2 and the sky location is marginalized over. Note that

the color scales are different in the two panels.

space (

and

q >

4). This demonstrates

TianGO’s unparalleled ability to determine

One caveat though is that the above analysis assumes

binary BHs have a broad range of spins with 0

a/M <

1 as in the case of X-ray binaries [109]. How-

ever, the BBHs detected by aLIGO and aVirgo during

the first and second observing runs [110] suggest that

most BHs may have only low spins of

a/M <

[112],

which may be the consequence of an efficient angular mo-

mentum transfer in the progenitor stars [33]. In this case,

a moderate

would be an indication of the merger event

involving a second-generation BH [113].

As for the majority of the slowly spinning BHs,

TianGO can still deliver valuable information ground-

based detectors cannot access. This is illustrated in Fig-

ure 9 where we present the uncertainty in

eff

. This

time we assume the system to only have a slow spin

rate of (

eff

, χ

)=(0

0) while the other parameters

are the same as in Figure 8. The Voyager network can-

not constrain

eff

for systems spinning at such a slow

rate. TianGO, on the other hand, can still achieve an

accuracy of ∆

eff

/χ

eff

3 over most of the parame-

ter space. This opens up the possibility of discriminat-

ing different angular momentum transfer models that all

predict the majority of BHs having spins in the range of

a/M

∼

−

1 [33, 114–118].

Ref. [111] 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 homoge-

neous formation [100].

(TianGo)

eff

× 100

(Ground)

eff

× 100

FIG. 9. Similar to Figure 8 but showing the uncertainties

eff

, for binaries with (

eff

, χ

)=(0

0). 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.

VIII. REVEALING ORBITAL ECCENTRICITY

EVOLUTION

So far our discussions have focused on systems with

circular orbits. This is a good assumption for signals

f >

10 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 ini-

tial eccentricity is mild (see, e.g., Refs. [119, 120]), it

will likely miss those formed with very high initial ec-

centricities of (1

−

)

01 [35]. Such a high initial ec-

centricity can be produced if the binary is formed via

binary-single scattering [121, 122], hierarchical triple in-

teractions [105, 123–125], or gravitational braking [126];

see Ref. [35] 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. [34] 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

(

) =

∞

∑

(

)

(8)

where each harmonic varies at a frequency

given by

Ω

orb

+ ̇

γ/π.

(9)

The angle

represents the direction of the pericenter,

and we have defined

as the average between the radial