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

TL;DR

A space-based decihertz gravitational-wave detector like TianGO can significantly enhance astrophysical observations, improve source localization, and enable new tests of cosmology and stellar evolution, complementing ground-based detectors.

Contribution

This paper proposes the scientific potential of a decihertz GW detector, highlighting its unique capabilities for source localization, early warning, and testing astrophysical hypotheses.

Findings

Enhanced sky localization accuracy for GW sources.

Potential to constrain the Hubble constant using standard sirens.

Ability to test supernova progenitor models and detect intermediate-mass black holes.

Abstract

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.

Tables

Table 1. Table 1: Summary of TianGO science cases

Section	Scientific Objective	Target	Information to extract	Key references
II	Cosmography.	Binary BHs	Sky location	[19, 20, 21]
III	Multi-messenger astrophysics; NS physics.	Binary NSs	Sky location	[22]
IV	Structure formation; IMBHs.	Binaries involving IMBHs	Source population	[23, 24, 25]
V	Type-Ia SNe progenitors.	Binary WDs	Source population	[26, 27, 28]
VI	WD physics.	Binary WDs	Tidal dephasing	[29, 30, 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]
IX	Environment around BHs.	Binary BHs	Phase modulation	[36]

Table 2. Table 2: Comparison of sky localization for different networks of detectors. The neutron star system is a binary with ℳ c = 1.2 ​ M ⊙ subscript ℳ 𝑐 1.2 subscript 𝑀 direct-product \mathcal{M}_{c}=1.2\,M_{\odot} , q = 1.05 𝑞 1.05 q=1.05 , and D L = 50 ​ Mpc subscript 𝐷 𝐿 50 Mpc D_{L}=50\,\mathrm{Mpc} . The black hole system is a binary with ℳ c = 25 ​ M ⊙ subscript ℳ 𝑐 25 subscript 𝑀 direct-product \mathcal{M}_{c}=25\,M_{\odot} , q = 1.05 𝑞 1.05 q=1.05 , and D L = 600 ​ Mpc subscript 𝐷 𝐿 600 Mpc D_{L}=600\,\mathrm{Mpc} . The best and median sources are given in deg 2 superscript deg 2 \mathrm{deg}^{2} . The ground-based detectors have the Voyager design sensitivity.

	Neutron Star		Black Hole
Network	Best	Median	Best	Median
HLV	$7.9\times {10}^{-3}$	$4.1\times {10}^{-2}$	$1.1\times {10}^{-2}$	$5.4\times {10}^{-2}$
HLVKA	$2.0\times {10}^{-3}$	$5.6\times {10}^{-3}$	$3.1\times {10}^{-3}$	$8.5\times {10}^{-3}$
T	$3.5\times {10}^{-5}$	$5.4\times {10}^{-5}$	$4.4\times {10}^{-3}$	$1.1\times {10}^{-2}$
HLVKA + L2 T	$1.6\times {10}^{-5}$	$2.9\times {10}^{-5}$	$5.4\times {10}^{-4}$	$1.5\times {10}^{-3}$
HLVKA + $5^{\circ }$ T	$5.7\times {10}^{-6}$	$1.3\times {10}^{-5}$	$6.6\times {10}^{-5}$	$1.9\times {10}^{-4}$
HLVKA + ${20}^{\circ }$ T	$1.3\times {10}^{-6}$	$3.5\times {10}^{-6}$	$1.6\times {10}^{-5}$	$5.2\times {10}^{-5}$

Table 3. Table 3: Uncertainties in the sum of WDs’ moment of inertia for different initial GW frequencies at the start of a 5 year observation period.

$f_0$ [mHz]	5	10	20	30
$\frac{\Delta \left(I_1+I_2\right)}{\left(I_1+I_2\right)}$	1.1	$3.3\times {10}^{-3}$	$9.6\times {10}^{-6}$	$6.7\times {10}^{-7}$

Equations

$$t_{\mathrm{m}}(f)=5.4{\left(\frac{{\mathcal{M}}_c}{1.2M_{\odot }}\right)}^{-5/3}{\left(\frac{f}{1\mathrm{Hz}}\right)}^{-8/3}\mathrm{days}.$$

$$f_{\mathrm{QNM}}^{(\det )}\simeq 1.21\left(\frac{10}{1+z}\right)\left(\frac{10^3{M}_{\odot }}{M_1+M_2}\right)\mathrm{Hz}.$$

$$R_{\mathrm{wd}}(M_{\mathrm{wd}})=10^9{\left(\frac{M_{\mathrm{wd}}}{0.6M_{\odot }}\right)}^{-1/3}\mathrm{cm},$$

$$n(a)\propto \{\begin{array}{ll}{a}^3& \text{ for }\alpha \ge -1,\\ a^{\alpha +4}& \text{ for }\alpha <-1,\end{array}$$

$${\dot{\Omega }}_{\mathrm{s},1}\simeq {\dot{\Omega }}_{\mathrm{s},2}\simeq {\dot{\Omega }}_{\mathrm{orb}},$$

$$\frac{{\dot{E}}_{tide1(2)}}{{\dot{E}}_{\mathrm{pp}}}\simeq \frac{3}{2}\frac{I_{1(2)}{\Omega }_{\mathrm{orb}}^2}{E_{\mathrm{orb}}}\propto f^{4/3}.$$

$$I(M_{\mathrm{wd}})=3.1\times 10^{50}{\left(\frac{M_{\mathrm{wd}}}{0.6M_{\odot }}\right)}^{1/3}\mathrm{g}\mathrm{c}{\mathrm{m}}^2.$$

$$h(t)=\sum _{k=1}^{\infty }{h}_k(t),$$

$$f_k=k{\Omega }_{\mathrm{orb}}/2\pi +\dot{\gamma }/\pi .$$

$$h_{c,k}(f_k)=\frac{1}{\pi D_L}\sqrt{\frac{2G}{c^3}\frac{{\dot{E}}_k}{{\dot{f}}_k}},$$

$$⟨{\mathrm{SNR}}_k^2⟩=\int \frac{h_{c,k}^2(f)}{5f{S}_a(f)}\text{d}\ln f,$$

$${\varphi }_{\mathrm{bc}}(t)=2\pi \int f(t)\left[1-\frac{v_{\mathrm{mod}}}{c}\sin ({\Omega }_{\mathrm{mod}}t+{\varphi }_{\mathrm{mod}})\right]\text{d}t,$$

$$\frac{v_{\mathrm{mod}}}{c}=0.02{\left(\frac{M_3}{4\times 10^6{M}_{\odot }}\right)}^{1/2}{\left(\frac{r_3}{100\mathrm{AU}}\right)}^{-1/2},$$

$$\frac{{\Omega }_{\mathrm{mod}}}{2\pi }=\frac{1}{0.5\mathrm{yr}}{\left(\frac{M_3}{4\times 10^6{M}_{\odot }}\right)}^{1/2}{\left(\frac{r_3}{100\mathrm{AU}}\right)}^{-3/2},$$

$$s_a(f)=h_a(f,\mathbf{\theta })+n_a(f)$$

$$p(\Delta \theta )\propto {\mathrm{e}}^{-{\Gamma }_{ij}^a\Delta {\theta }_i\Delta {\theta }_j/2}$$

$${\Gamma }_{ij}^a={\left(\frac{\partial h}{\partial {\theta }_i}|\frac{\partial h}{\partial {\theta }_j}\right)}_a$$

$${\left(g|h\right)}_a=4\mathrm{Re}{\int }_0^{\infty }\frac{g^{\ast }(f)h(f)}{S_a(f)}\mathrm{d}f$$

$${\Gamma }_{ij}$$

$${\Sigma }_{ij}$$

$$\Delta \Omega =2\pi \left|\cos \delta \right|\sqrt{{\Sigma }_{\alpha \alpha }{\Sigma }_{\delta \delta }-{\left({\Sigma }_{\alpha \delta }\right)}^2},$$

$$u_{\mathrm{ph}}(f)=A_{\mathrm{ph}}(f){\mathrm{e}}^{\mathrm{i}{\Psi }_{\mathrm{ph}}(f)}$$

$$h_{+}(f)$$

$$h_{\times }(f)$$

$$\cos \iota$$

$$\tan \psi$$

$$h_a(f)={\mathrm{e}}^{\mathrm{i}\left[2\pi f(t_c+{\tau }_a)-{\varphi }_c\right]}\left[F_{+}^a{h}_{+}(f)+F_{\times }^a{h}_{\times }(f)\right].$$

$$\Lambda (f)=\sqrt{[1+(\hat{\mathbf{L}}\cdot \hat{\mathbf{n}}{)}^2{]}^2{F}_{+}^2(f)+4(\hat{\mathbf{L}}\cdot \hat{\mathbf{n}}{)}^2{F}_{\times }^2(f)}$$

$$\tan {\varphi }_p(f)=\frac{2(\hat{\mathbf{L}}\cdot \hat{\mathbf{n}})F_{\times }(f)}{[1+(\hat{\mathbf{L}}\cdot \hat{\mathbf{n}}{)}^2]F_{+}(f)}.$$

$${\varphi }_D(f)=\frac{2\pi f}{c}R\cos \delta \cos \left[{\varphi }_{\mathrm{T}}(f)-\alpha \right],$$

$$h_{\mathrm{T}}(f)=\Lambda (f)A_{\mathrm{ph}}(f){\mathrm{e}}^{\mathrm{i}\left[{\Psi }_{\mathrm{ph}}(f)-{\varphi }_D(f)-{\varphi }_p(f)-{\varphi }_c\right]}.$$

$$x={\left(\frac{G}{c^3}\pi M_{\odot }\right)}^{2/3}\frac{{\mathcal{M}}_c}{\mu }{\left[\left(\frac{{\mathcal{M}}_c}{M_{\odot }}\right)\left(\frac{f}{1\mathrm{Hz}}\right)\right]}^{2/3}$$

Full text

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 towards their sensitivity goals. Meanwhile, various upgrades to current facilities have been proposed, including the incremental A+ 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 fo 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, 13, 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 $M_{\odot}$-30 $M_{\odot}$ 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 [17, 18].

Fig. 1 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 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 corresponding detection horizons for compact binary sources of different 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 between a 5 and 170 s light travel time from the Earth, 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 ${\sim}50$ 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 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. 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 $\sim$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 sensitive 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 eccentricity, which will in turn reveal the system’s formation channel.

We summarize the major science targets we will be considering in Table 1 and also in the text as follows. We discuss the precision with which BBHs can be localized with a hybrid network and the application to cosmography 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 progenitor 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 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 Section VIII we explore TianGO’s capability of measuring the orbital eccentricity 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, $H_{0}$, quantifies the current expansion rate of the universe, and is one of the most fundamental parameters of the standard $\Lambda$CDM cosmological model, yet the two traditional methods of measuring it disagree at the $4.4\sigma$ level [38]. 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 [39]. The second, local measurement, relies on our understanding of astrophysics to calibrate a cosmic distance ladder. This ladder relates the redshifts of observed sources to their luminosity distances [40, 41, 38]. Gravitational wave astronomy adds a third method of determining $H_{0}$ and the prospect of resolving this tension [19, 42, 43, 20, 21], a task for which a combined TianGO-ground-based network is particularly well suited.

To obtain the redshift-distance relationship necessary to determine $H_{0}$, 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.” On the other-hand, the luminosity distance is measured directly from a GW observation 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.111The GW standard sirens can also be used to independently calibrate the EM standard candles forming the cosmic distance ladder [44]. In 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 [45]. Since the gravitational wave signal was accompanied by an optical counterpart, the host galaxy was identified and the first direct measurement of $H_{0}$ using this method was made [46].

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 either be in a Earth-trailing orbit of up to $20^{\circ}$ or an orbit at the L2 Lagrange point [15] thereby adding a baseline of between $1.5\times 10^{6}\,\mathrm{km}=235R_{\oplus}$ and $5.2\times 10^{7}\,\mathrm{km}=8.2\times 10^{3}R_{\oplus}$ to the network, where $R_{\oplus}$ is the radius of the Earth. Since the same source will be observed by both TianGO and the ground-based network, the timing accuracy formed by this large baseline significantly improves the sky localization ability over that of the ground-based alone, as is illustrated in Figs. 3 and 13 and Section II.

The top panel of Fig. 3 shows the angular resolution $\Delta\Omega$ 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 $5^{\circ}$ Earth-trailing orbit. The source is a BBH with $\mathcal{M}_{c}=25\,M_{\odot}$, $q=1.05$, and $\iota=30^{\circ}$. (The probability of detecting binaries with a given inclination peaks around $\iota=30^{\circ}$ [47]. The same figure for $\iota=0^{\circ}$ 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 $\sim 50$.

The middle panel of Fig. 3 shows the fractional uncertainty $\Delta D_{L}/D_{L}$ 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. 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 $\Delta V_{C}$.222The Planck 2015 cosmology is assumed [39]. If an optical counterpart 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 $\Delta V_{C}$, the value of $0.01\,\mathrm{galaxies}/\mathrm{Mpc}^{3}$ is assumed. The combined network can localize a source to a single galaxy up to a redshift of $z\sim 0.5$ for the best face-on sources, and to $z\sim 0.35$ for the median sources at $\iota=30^{\circ}$.

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 [19, 48, 49, 50, 51]. This method has been used to reanalyze the measurement from GW170817 to infer $H_{0}$ without the unique galaxy determination provided by the observation of the optical 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 reach of these methods.