Probing stellar binary black hole formation in galactic nuclei via the imprint of their center of mass acceleration on their gravitational wave signal

TL;DR

This paper explores how the center-of-mass acceleration of binary black holes in galactic nuclei affects their gravitational wave signals, enabling insights into their formation environments through multi-frequency GW observations.

Contribution

It introduces the idea that phase drifts in GW signals caused by acceleration can distinguish between different stellar-mass BBH formation channels in galactic nuclei.

Findings

Strong acceleration causes measurable phase drifts in GW signals.

LISA can identify host galaxies within the error volume.

Non-detection constrains nuclear formation scenarios.

Abstract

Multi-frequency gravitational wave (GW) observations are useful probes of the formation processes of coalescing stellar-mass binary black holes (BBHs). We discuss the phase drift in the GW inspiral waveform of the merging BBH caused by its center-of-mass acceleration. The acceleration strongly depends on the location where a BBH forms within a galaxy, allowing observations of the early inspiral phase of LIGO-like BBH mergers by the Laser Interferometer Space Antenna (LISA) to test the formation mechanism. In particular, BBHs formed in dense nuclear star clusters or via compact accretion disks around a nuclear supermassive black hole in active galactic nuclei would suffer strong acceleration, and produce large phase drifts measurable by LISA. The host galaxies of the coalescing BBHs in these scenarios can also be uniquely identified in the LISA error volume, without electromagnetic counterparts. A non-detection of phase drifts would rule out or constrain the contribution of the nuclear formation channels to the stellar-mass BBH population.

Figures

Tables

Table 1. Table 1: The first four columns show, in each BBH formation scenario, the expected center-of-mass orbital velocity ( v 𝑣 v ) and radius ( r 𝑟 r ), and the acceleration parameter ( ϵ italic-ϵ \epsilon ). Col. 5 shows the maximum distance ( d L , obs subscript 𝑑 𝐿 obs d_{L,{\rm obs}} ) at which the phase drift can be measured by LISA with an SNR of ρ ​ ( δ ​ h ) > 8 𝜌 𝛿 ℎ 8 \rho(\delta h)>8 , corresponding to Δ ​ Y / Y < 0.5 Δ 𝑌 𝑌 0.5 \Delta Y/Y<0.5 . In Col. 6, the number densities of the host objects are shown. Cols. 7 and 8 show the number of host objects ( n host subscript 𝑛 host n_{\rm host} ) and of GW events ( n m subscript 𝑛 m n_{\rm m} ) in the LISA band, within the cosmic volume of V eff = 4 ​ π ​ d eff 3 / 3 subscript 𝑉 eff 4 𝜋 superscript subscript 𝑑 eff 3 3 V_{\rm eff}=4\pi d_{\rm eff}^{3}/3 , where d eff = min ​ ( d L , obs , 640 ​ Mpc ) subscript 𝑑 eff min subscript 𝑑 𝐿 obs 640 Mpc d_{\rm eff}={\rm min}(d_{L,{\rm obs}},640~{}{\rm Mpc}) and n m = R ​ f / f ˙ ≃ 10 − 6 ​ Mpc 3 subscript 𝑛 m 𝑅 𝑓 ˙ 𝑓 similar-to-or-equals superscript 10 6 superscript Mpc 3 n_{\rm m}=Rf/\dot{f}\simeq 10^{-6}~{}{\rm Mpc}^{3} . Here we adopt our fiducial case: M cz = 40 ​ M ⊙ subscript 𝑀 cz 40 subscript M direct-product M_{\rm cz}=40~{}{\rm M}_{\odot} , f min = 0.014 subscript 𝑓 min 0.014 f_{\rm min}=0.014 Hz, and η = 0.25 𝜂 0.25 \eta=0.25 . In the three scenarios indicated by boldface, the acceleration is large and measurable.

scenario	$v\left(\mathrm{km}{s}^{-1}\right)$	$r\left(\mathrm{pc}\right)$	$ϵ$	$d_{L,\mathrm{obs}}\left(\mathrm{Mpc}\right)$	$n_{\mathrm{host}}\left({\mathrm{Mpc}}^{-3}\right)$	$n_{\mathrm{host}}{V}_{\mathrm{eff}}$	$n_m{V}_{\mathrm{eff}}$
Field binaries (A)
formed at $z\simeq 0$	$\sim 200$	$>5\times {10}^3$		$\sim 0.2$	$\sim 2\times {10}^{-2}$ Kopparapu et al. (2008)	$\ll 1$	$\ll 1$
formed at $z\simeq 3$	$\sim 300$	${10}^3-{10}^4$	$10-100$	$0.2-2$	$\sim 5\times {10}^{-4}$ Bell et al. (2003)	$\lesssim 0.02$	$\ll 1$
Dense stellar systems (B)
globular clusters	$\sim 200$	$\sim 5\times {10}^4$	$\sim 1$	$\sim 0.02$	$\sim 1$ Harris et al. (2013)	$\ll 1$	$\ll 1$
nuclear star clusters	$30-100$	$\sim 1$	${\mathrm{𝟏𝟎}}^{\mathrm{𝟑}}-{\mathrm{𝟏𝟎}}^{\mathrm{𝟒}}$	$20-200$	$\sim 0.01$ Miller and Lauburg (2009)	$\lesssim 3\times {10}^5$	$\lesssim 30$
AGN disks (C)
formed in disk	$\sim 200$	$\sim 1$	$\sim {\mathrm{𝟏𝟎}}^{\mathrm{𝟒}}$	$\sim 200$	$\sim {10}^{-5}$ Ueda et al. (2014)	$\sim 300$	$\sim 30$
captured or migrated in	$\sim 600$	$\sim 0.1$	$\sim {\mathrm{𝟏𝟎}}^{\mathrm{𝟓}}$	$\sim 950$	$\sim {10}^{-5}$	$\sim {10}^4$	$\sim {10}^3$
Very high-redshift (D)
Population III	$\sim 200$	$\lesssim {10}^3$	$10-100$	$0.2-2$	$\sim 2\times {10}^{-2}$ Kopparapu et al. (2008)	$\lesssim 0.7$	$\ll 1$

Equations

$$f\simeq 13.8M_{\mathrm{cz},40}^{-5/8}{\left(\frac{{\tau }_c}{4\mathrm{yr}}\right)}^{-3/8}\mathrm{mHz}.$$

$$\rho \simeq 51d_{L,100}^{-1}{M}_{\mathrm{cz},40}^{5/3}{f}_{14}^{2/3}\delta t_5^{1/2},$$

$$n_{\mathrm{m}}\equiv R\frac{f}{\dot{f}}=10^{-6}{f}_{14}^{-8/3}{M}_{\mathrm{cz},40}^{-5/3}{R}_{100}{\mathrm{Mpc}}^{-3},$$

$$\delta z_{\mathrm{acc}}\simeq \frac{a_{\mathrm{CoM}}\delta t}{c}\equiv 1.7\times 10^{-7}\delta t_5\left(\frac{ϵ}{10^4}\right),$$

$$ϵ\equiv 10^4{\left(\frac{v_{\mathrm{acc}}}{100{\mathrm{km}\mathrm{s}}^{-1}}\right)}^2{\left(\frac{r}{1\mathrm{pc}}\right)}^{-1}.$$

$$Y\equiv \frac{1}{2(1+z)}\cdot \frac{\delta z_{\mathrm{acc}}}{\delta t}\simeq 1.5\times 10^{-8}{ϵ}_{\mathrm{z},4}{\mathrm{yr}}^{-1},$$

$$\delta {\Phi }_{\mathrm{acc}}$$

$$\delta {\Psi }_{\mathrm{acc}}(f)\simeq -0.59{ϵ}_{\mathrm{z},4}{M}_{\mathrm{cz},40}^{-10/3}{f}_{14}^{-13/3}.$$

$$\rho (\delta h)\simeq 16{ϵ}_{\mathrm{z},4}{d}_{L,100}^{-1}\delta t_5^{1/2}{M}_{\mathrm{cz},40}^{-5/3}{f}_{14}^{-11/3}.$$

$$\Psi (f)=2\pi f{t}_c-\frac{\pi }{4}-{\Phi }_c+{\Psi }_{\mathrm{PN}}(f,\eta )+\delta {\Psi }_{\mathrm{acc}}(f).$$

Full text

I Introduction

Advanced LIGO has so far detected gravitational waves from three stellar binary black hole (BBH) mergers Abbott et al. (2016a, b, c, 2017). Several scenarios for the origin of such massive compact BBHs have been proposed Abbott et al. (2016d), through the evolution of isolated massive stellar binaries (Belczynski et al., 2004; Dominik et al., 2012; Kinugawa et al., 2014; Belczynski et al., 2016; Inayoshi et al., 2016), dynamical formation in dense stellar clusters Portegies Zwart and McMillan (2000); O’Leary et al. (2009); Rodriguez et al. (2016a); Antonini and Rasio (2016) or in active galactic nuclei (AGN) Bellovary et al. (2016); Bartos et al. (2016); Stone et al. (2017).

Recently, Ref. Sesana (2016) pointed out that the early inspiral of GW150914-like BBHs can be measured by the space-based detector LISA Audley et al. (2017). The BBH coalescence rate inferred from the LIGO detections implies that $\sim 10-100$ of such BBHs will be individually resolved by LISA and then merge in the LIGO band in $\leq 10$ yr. These LISA observations alone can determine the coalescence time with an accuracy of $\sim 10$ s and the sky position to within $<1$ deg2, allowing advance planning of electromagnetic (EM) observations of the merger.

Multi-frequency GW observations by LISA and LIGO are also useful to distinguish formation scenarios of stellar-mass BBHs (e.g., measurements of spin-orbit misalignments by LIGO Rodriguez et al. (2016b); Vitale et al. (2017); Stevenson et al. (2017)). LISA could detect non-zero eccentricities of the merging BBHs Seto (2016); Nishizawa et al. (2016); Breivik et al. (2016). Measurable eccentricities are expected in formation channels involving dynamical interactions in dense stellar clusters O’Leary et al. (2009), or as a result of the interaction with AGN accretion disks Bellovary et al. (2016); Bartos et al. (2016); Stone et al. (2017), but not if typical BBHs form via isolated binaries Belczynski et al. (2016). The predicted eccentricities are uncertain and alternative ways to distinguish formation channels remain useful.

Another important advantage of low-frequency GW observations by LISA (and/or by DECIGO Kawamura et al. (2006)) is that the acceleration of the merging BBHs can produce a measurable phase drift in their GW inspiral waveform. In the cosmological context, the apparent acceleration caused by the time evolution of the Hubble expansion is weak, but if detected, it would allow us to measure the accelerated expansion directly Seto et al. (2001); Nishizawa et al. (2012). However, the peculiar acceleration of the coalescing BBHs due to astrophysical processes could be much larger than that produced by the cosmic expansion Yunes et al. (2011a); Bonvin et al. (2016).

In this paper, we discuss the possibility to distinguish the formation channels of merging BBHs, focusing on the binary motion inside the host galaxy. We show that BBHs located in dense nuclear star clusters or in compact accretion disks around a nuclear supermassive BH (SMBH) suffer strong acceleration, and produce a large phase drift measurable by LISA 111For a stellar-mass BBH located exceptionally close ($\lesssim 10^{11}$cm) to an SMBH, the acceleration, together with other relativistic corrections to the GW waveform, could be measured by LIGO alone Meiron et al. (2017); see § IV.. Since the acceleration effect strongly depends on the location where BBHs form within a galaxy, observations by LISA of stellar-mass BBH mergers offer a test of proposed formation scenarios.

II Phase drift in gravitational waves

We consider a coalescing BBH with redshifted chirp mass $M_{\rm cz}$ ($M_{\rm cz,40}\equiv M_{\rm cz}/40M_{\odot}$) in the LISA band with a signal-to-noise ratio (SNR) of $\rho>8$ which will merge in the advanced LIGO/Virgo band in $\tau_{c}\lesssim 10$ yr Sesana (2016). The rest-frame GW frequency of the coalescing BBH is given by (see e.g. Peters (1964); Maggiore (2007))

[TABLE]

The SNR accumulated during the LISA observation time $\delta t$ can be approximated as Seto (2016)

[TABLE]

where $d_{L,100}\equiv d_{L}/(100~{}{\rm Mpc})$ is the luminosity distance to the GW source and $f_{14}\equiv f/(14~{}{\rm mHz})$ is a representative frequency. Throughout the paper we assume a LISA configuration with six links, 2 million km arm length and mission duration of $\delta t=5$ yr $(\delta t_{5}\equiv\delta t/5~{}{\rm yr})$. LISA’s noise curve $S_{\rm n}(f)$ has been taken from Klein et al. (2016). In Eq. (2), we have set LISA’s sensitivity to $\sqrt{S_{\rm n}(f)}=7\times 10^{-21}/\sqrt{\rm Hz}$ at $f\simeq 10$ mHz.

From the BBH merger rate inferred from the existing LIGO events, $R\simeq 9-240~{}{\rm Gpc}^{-3}~{}{\rm yr}^{-1}$ Abbott et al. (2016c), the space density of BBHs inspiralling near $f\simeq 10^{-2}$ Hz at a given time can be estimated as

[TABLE]

where $R_{100}=R/(100~{}{\rm Gpc}^{-3}~{}{\rm yr}^{-1})$. Thus, below we require $60~{}{\rm Mpc}<d_{L}<640~{}{\rm Mpc}$, in order to ensure that at least one event ($N_{\rm m}\equiv 4\pi n_{\rm m}d_{L}^{3}/3>1$, evaluated with $R_{100}=1$) occurs in the local cosmic volume with a total SNR $\rho>8$, during a $\delta t=$ 5yr LISA mission lifetime.

Next, we consider the impact of the center-of-mass (CoM) acceleration of a merging BBH. Over $\delta t$, the source will appear to change its redshift by an amount

[TABLE]

where we have expressed the acceleration $a_{\rm CoM}=v^{2}_{\rm acc}/r$ along the line of sight in terms of a characteristic velocity $v_{\rm acc}$ and distance $r$ (interpreted below as the orbital velocity and distance from the barycenter of the host galaxy), and defined the dimensionless acceleration parameter $\epsilon$

[TABLE]

We define a variable $Y$ which accounts for the CoM acceleration of a merging BBH by

[TABLE]

where $\epsilon/(1+z)\equiv 10^{4}\epsilon_{\rm z,4}$ 222 Note that the redshift drift due to the cosmic expansion is negligible for the range of $\epsilon$ we consider here.. The CoM acceleration causes a linear frequency drift $\delta f\propto Y\delta t$ and the corresponding phase drift in the GW inspiral waveform is expressed as Bonvin et al. (2016)

[TABLE]

The total number of GW cycles without the acceleration is $\mathcal{O}(10^{6})$ for stellar-mass binaries, and Eq. (7) gives about one extra cycle for the reference values of the parameters.

To detect the phase drift in the GW inspiral waveform, the strain perturbation $\delta h(f)=h(f)[1-e^{i\delta\Psi_{\rm acc}(f)}]$ must have a significant SNR Yunes et al. (2011b); Kocsis et al. (2011), where $\delta\Psi_{\rm acc}(f)$ is the phase drift in frequency space Cutler and Flanagan (1994)

[TABLE]

Fig. 1 shows the relative SNR of the perturbation $\rho(\delta h)/\rho=\left[\int^{f_{\rm max}}_{f_{\rm min}}df\,\frac{|\delta h(f)|^{2}}{S_{n}(f)}/\int^{f_{\rm max}}_{f_{\rm min}}df\,\frac{|h(f)|^{2}}{S_{n}(f)}\right]^{1/2}$ for different combinations of the redshifted chirp mass $M_{\rm cz}$ and the frequency $f_{\rm min}$ when the observation begins ($f_{\rm max}$ is the smaller between twice the inner-most-stable-circular-orbit frequency or the frequency reached in $\delta t=5$yr).

For small $\epsilon$, the SNR of the perturbation is proportional to $|\delta\Psi_{\rm acc}|$ Kocsis et al. (2011); Hayasaki et al. (2013) and is given by

[TABLE]

For larger $\epsilon(>\epsilon_{\rm crit})$, when the phase drift approaches a full cycle, the relative SNR of $\rho(\delta h)/\rho$ saturates at a roughly constant value ($\simeq 1.5$) because of de-phasing. Computing the relative SNR numerically, we found the critical accelerations to be $\epsilon_{\rm crit}/(1+z)\simeq 6.6\times 10^{4}~{}M_{\rm cz,40}^{10/3}~{}f_{14}^{13/3}$.

In order to estimate the LISA error on the acceleration parameter $Y$, including possible degeneracies with other system parameters, we perform a Fisher matrix analysis. We adopt the six parameters $M_{\rm cz}$, $\Phi_{c}$, $t_{c}$, $\eta$, $d_{L}$ and $Y$, where $\Phi_{c}$ is the phase at the coalescence time $t_{c}$, and $\eta$ is the symmetric mass ratio. We further follow Ref. Bonvin et al. (2016) and adopt the sky-averaged GW waveform of $h(f)=A(f)\,\exp[i\Psi(f)]$ with the amplitude $A(f)$ at Newtonian order, and the phase $\Psi(f)$ at 3.5PN order, plus the contribution of the CoM acceleration effect:

[TABLE]

The explicit form of $\Psi_{\rm PN}(f)$ is given in Mishra et al. (2010).

Fig. 2 shows the marginalized $1\sigma$ error $\Delta Y/Y$ provided by LISA alone (solid blue) for our fiducial values of $\epsilon=3\times 10^{4}$, $f_{\rm min}=0.014\,{\rm Hz}$, $z=0.05$ ($d_{L}\simeq 200~{}{\rm Mpc}$; well inside the horizon of both LIGO and LISA) and $\eta=0.25$. The error is small enough to detect non-zero $Y$ (i.e., $\Delta Y/Y<1$) at $35\lesssim M_{\rm cz}/{\rm M}_{\odot}\lesssim 63$ because the GW chirping helps break the degeneracies among the waveform parameters. When the binary frequency hardly evolves during the LISA observation, i.e., for lower $M_{\rm cz}$ (and/or $f_{\rm min}$), strong degeneracies remain and render the acceleration undetectable. For higher masses (and/or $f_{\rm min}$), the binary exits the LISA band more rapidly, diminishing the SNR.

LIGO observations during/after the LISA operation time can reduce parameter degeneracies, by detecting the merger event and fixing the coalescence time $t_{c}$. As shown by the dashed blue curve in Fig. 2, for low masses the error $\Delta Y/Y$ provided by LISA + LIGO (i.e. $t_{c}$ fixed) is reduced by a factor of $3-10$ from that by LISA alone. As a result, the best error estimate in this case is given by $\Delta Y/Y\simeq 0.3$ at $M_{\rm cz}=40~{}{\rm M}_{\odot}$.

In Fig. 3, we show the measurement error $\Delta Y/Y$ as a function of $M_{\rm cz}$ and $f_{\rm min}$. In this case we also present the results for a LISA mission lasting 10 yr to show how they could improve: the nominal mission duration is 4 yr but a duration in flight up to 10 yr is conceivable Audley et al. (2017). In a suitable range of values for $f_{\rm min}$, for a 5 yr LISA mission without any input from LIGO (top left), the acceleration effect can be detected with an error of $\Delta Y/Y<0.5(1)$ only for relatively massive binaries with $M_{\rm cz}\gtrsim 50(40)~{}{\rm M}_{\odot}$. Fixing $t_{c}$ with LIGO (top right) extends the same limits down to $M_{\rm cz}\gtrsim 35(25)~{}{\rm M}_{\odot}$. A 10 yr LISA mission instead can provide errors of $\Delta Y/Y<0.5$ for masses $M_{\rm cz}\gtrsim 25~{}{\rm M}_{\odot}$, even without detecting the merger with LIGO (bottom left), while for BBHs with higher masses the acceleration effect can be measured with more accuracy: $\Delta Y/Y<0.2$ for $M_{\rm cz}\gtrsim 50~{}{\rm M}_{\odot}$. Fixing the merger time with LIGO (bottom right) improves these results yielding the possibility of detecting the acceleration effect with $\Delta Y/Y<0.5$ for $M_{\rm cz}$ below $20~{}{\rm M}_{\odot}$, while $\Delta Y/Y<0.2$ can be reached for $M_{\rm cz}\gtrsim 40~{}{\rm M}_{\odot}$. Long-lived binaries with $\tau_{c}>$ 5 (10) yr do not chirp rapidly enough to break parameter degeneracies, whereas short-lived binaries with $\tau_{c}<2$ yr do not spend sufficient time in the LISA band to accumulate SNR. We conclude that merging BBHs with 2 yr $\lesssim\tau_{c}\lesssim$ 5 (10) yr provide the best combinations of $f_{\rm min}$ and $M_{\rm cz}$ to probe CoM acceleration.

Fig. 4 presents the maximum distances out to which phase drifts can be measured with errors of $\Delta Y/Y<0.1-1.0$. We consider equal-mass BBH mergers with $M_{cz}=40~{}{\rm M}_{\odot}$ and $f_{\rm min}=0.014$ Hz. The horizontal dashed lines show the conditions $60~{}{\rm Mpc}<d_{L}<640~{}{\rm Mpc}$ discussed below Eq. (3). For $\epsilon>\epsilon_{\rm crit}$, the maximum distance does not increase linearly with $\epsilon$ because of the saturation of the relative SNR discussed below Eq. (9) and shown in Fig. 1. For merging BBHs located in the shaded region, the phase drifts in the GW inspiral waveform can be observed.

III Formation scenarios of LIGO BBHs and corresponding acceleration

In this section, we review proposed stellar-mass BBH formation scenarios, from field binaries (A), dynamical formation in dense stellar systems (B) and in AGN accretion disks (C) and massive high-redshift binaries (D). We discuss the typical value of the acceleration parameter $\epsilon$ expected in each case, summarized in Table 1.

A. Field binaries — A compact ($\lesssim 0.1$ AU) massive stellar binary could form a BH remnant coalescing due to GW emission in a Hubble time. Such BBHs are formed in low-metallicity star forming regions Abbott et al. (2016d), possibly over an extended range of redshifts ($0\lesssim z\lesssim 3$; e.g. Ref. Dominik et al. (2012)).

In the nearby universe, most star-formation occurs in disks of spiral galaxies, within their half-light radii of $\sim 5$ kpc van der Kruit and Searle (1982). Assuming that the stars are orbiting around the center of the galaxy at the circular velocity $\sim 200~{}{\rm km~{}s}^{-1}$ of a typical disk galaxy, the acceleration parameter is $\epsilon\simeq 8$ 333The line-of-sight acceleration will be reduced by $\cos\theta$ if the instantaneous acceleration vector is inclined by an angle $\theta$ to the LOS; see discussion.. However, LIGO BBHs are expected to arise from massive stellar binaries with $Z<0.1~{}{\rm Z}_{\odot}$ Abbott et al. (2016d). Since metallicities decrease farther out in the disk Pilkington et al. (2012), BBH formation could occur preferentially at these larger radii, where the acceleration parameter is reduced to $\epsilon\sim O(1)$.

A large fraction of low-metallicity massive (binary) stars could form in high-redshift star-forming galaxies. Their host galaxies will undergo several mergers and most of these binaries may end up in the cores of massive elliptical galaxies. These old remnant BBHs would be located in the core with a typical size of a few kpc Faber et al. (1997); Yu (2002) and with the circular velocity of $v\sim 300~{}{\rm km~{}s}^{-1}$ (Kormendy and Ho, 2013), resulting in somewhat larger accelerations of $\epsilon\simeq 10-100$.

B. Dynamical formation in dense stellar systems — Two single BHs can be paired when they interact and form a bound binary in a dense stellar system, either through a chance close fly-by, or involving a third object. These processes likely occur in globular clusters (GCs), nuclear star clusters (NSCs) and around SMBHs in galactic nuclei.

Most GCs are in orbit inside dark matter (DM) halos with $M\simeq 10^{12}~{}{\rm M}_{\odot}$, because galaxies in such a mass range contain most of the present-day stellar mass, and the number of GCs scales with their host galaxy mass Harris et al. (2013). The acceleration parameter is as low as $\epsilon\lesssim 1$, for the circular velocity of $v\simeq 200~{}{\rm km~{}s}^{-1}$ at $r\sim 50~{}{\rm kpc}$ ($\sim$ half of the virial radius). On the other hand, the BBHs also orbit inside GCs, where the velocity dispersion is at most $\sim 10~{}{\rm km~{}s}^{-1}$ and half-light radii are $\sim 2-3$ pc van den Bergh (2010). Since massive BBHs should have sunk to the center due to dynamical friction, the acceleration parameter could increase to $\epsilon\approx 100~{}(r/{\rm pc})^{-1}$. However, many BBHs would be ejected from the GCs, reducing their acceleration back to values for orbits in the halo ($\epsilon\lesssim 1$).

LIGO binaries could also be formed in NSCs and/or in galactic nuclei due to mass segregation through dynamical friction O’Leary et al. (2009); Antonini and Rasio (2016). Since the escape velocity from these systems is higher, a larger number of BBHs can remain within smaller radii of $r\sim 1$ pc with velocities of $\sim 30-100~{}{\rm km~{}s}^{-1}$. The acceleration parameter for these binaries would be larger, $\epsilon\simeq 10^{3}~{}(v/30~{}{\rm km~{}s}^{-1})^{2}(r/{\rm pc})^{-1}$.

C. Binary BH formation in AGN disks — It is possible to form BBHs, detectable by LIGO, with the help of AGN disks. They could form either from massive stellar binaries in the disk itself, at a few pc from the central SMBHs Stone et al. (2017) or at migration traps located closer in Bellovary et al. (2016); pre-existing binaries in the 3D bulge can also be captured in the inner regions ($<1~{}{\rm pc}$) of the disk Bartos et al. (2016). In these scenarios, SMBHs with masses of $10^{6-7}~{}{\rm M}_{\odot}$ likely dominate, since they are the most numerous and most efficiently accreting SMBHs with the densest disks in the local universe Greene and Ho (2007, 2009). At the location of the birth of the BBHs ($\sim 1~{}{\rm pc}$), their orbital velocity around the SMBH would be $\sim 200~{}{\rm km~{}s}^{-1}$. The acceleration is already as high as $\epsilon\simeq 4\times 10^{3}(M_{\bullet}/10^{7}~{}{\rm M}_{\odot})(r/1~{}{\rm pc})^{-2}$. However, these binaries are expected to migrate inward through the accretion disk, and many of them may be located closer to the center when they enter the LISA band Bellovary et al. (2016); Bartos et al. (2016); Stone et al. (2017). At a distance of $0.1~{}{\rm pc}$ from the center, the Keplerian velocity increases and the acceleration parameter is $\epsilon\sim 10^{5}$.

D. Very high-z binaries — Finally, another scenario is massive BBH formation in extremely metal-poor environments at high redshift. The first generation of stars in the universe at $z>10-20$, the so-called Population III (PopIII) stars, are typically as massive as $\sim 10-300~{}{\rm M}_{\odot}$ Hirano et al. (2014). PopIII binaries form coalescing BBHs efficiently, which can contribute to the rate of detectable LIGO events Kinugawa et al. (2014), including the existing O1 detections Inayoshi et al. (2017). PopIII remnants are expected to be located inside spiral galaxies like Milky-Way in the current universe Tumlinson (2010); Ishiyama et al. (2016); Griffen et al. (2016). Cosmological N-body simulations have suggested that PopIII remnants are distributed in the bulge, with $\sim 0.1-1\%$ of the remnants concentrated inside $r\lesssim 1$ kpc. In this case, the acceleration parameter is $\epsilon\simeq 10-100$.

IV Discussion and implications

Different formation scenarios of stellar-mass BBHs predict a wide range of typical acceleration parameters (see Table 1). As pointed out in Bonvin et al. (2016) and confirmed by our analysis, in the BBH formation scenarios with low values of $\epsilon<100$, the effect of the CoM acceleration of merging BBHs is difficult to observe by LISA in the operation time of $\delta t\simeq 5$ yr. On the other hand, BBHs produced in NSCs and in AGN disks are expected to have large and measurable accelerations, with $\epsilon>10^{3}$. Moreover, the number density of the NSCs ($n_{\rm NSC}\simeq 10^{-2}~{}{\rm Mpc}^{-3}$ Miller and Lauburg (2009)) and AGN ($n_{\rm AGN}\simeq 10^{-5}~{}{\rm Mpc}^{-3}$ Ueda et al. (2014)) are higher than the number density of merging BBHs at $f\simeq 14$ mHz, $n_{\rm m}(=Rf/\dot{f})\simeq 10^{-6}~{}{\rm Mpc}^{-3}$, inferred from the existing LIGO detections. Thus, LISA will likely observe phase drifts in the GW inspiral waveform if these scenarios contribute significantly to the total event rate.

Conversely, if no acceleration is detected among a total of $N_{\rm tot}$ events in the region of measurable parameter space, then this requires that other formation channels, not involving SMBHs are dominant. To illustrate this quantitatively, let us consider BBHs with intrinsic acceleration $\epsilon$, and suppose that line-of-sight accelerations $\epsilon_{\rm los}=|\cos\theta|\epsilon\geq\epsilon_{\rm obs}$ are detectable. Assuming that $\theta$, the angle between the line of sight to the BBH and the BBH’s instantaneous acceleration vector, has an isotropic distribution, we would expect $N_{\rm det}=f(\epsilon)N_{\rm tot}[1-\epsilon_{\rm obs}/\epsilon]$ events with measurable acceleration (i.e., within double cones with $|\cos\theta|\geq\epsilon_{\rm obs}/\epsilon$), where $f(\epsilon)$ is the fraction of events with a 3D acceleration above $\epsilon$. Setting $N_{\rm det}<1$ yields the upper limit $f(\epsilon)<N_{\rm tot}^{-1}[1-\epsilon_{\rm obs}/\epsilon]^{-1}$. For example, if we have $N_{\rm tot}=100$ events and the sensitivity was $\epsilon_{\rm obs}=10^{4}$, then at most $3\%$ of BBHs could have $\epsilon>1.5\times 10^{4}$, or be located within $\lesssim 0.5~{}{\rm pc}~{}(M_{\bullet}/10^{7}{\rm M_{\odot}})^{1/2}$ from SMBHs.

The sky position and the distance to merging BBHs for $\delta t>2$ yr with a high SNR ($\rho\gtrsim 10$) can be estimated by LISA alone to a statistical accuracy of $\Delta\Omega_{\rm s}\simeq 1.2~{}f_{14}^{-2}(\rho/10)^{-2}~{}{\rm deg}^{2}$ and $\Delta d_{L}/d_{L}\simeq 0.2(\rho/10)^{-1}$ Takahashi and Seto (2002); Kyutoku and Seto (2016). The corresponding error volume is given by $\Delta V=d_{L}^{2}\Delta d_{L}\Delta\Omega\simeq 9.6\times 10^{3}~{}f_{14}^{-2}(\rho/10)^{-6}~{}{\rm Mpc}^{3}$. Note that $\rho\simeq 10(d_{L}/510~{}{\rm Mpc})^{-1}$. Since AGN are rare objects, with abundance of a few $\times 10^{-5}\,{\rm Mpc}^{3}$ Greene and Ho (2007, 2009), the number of random interloping AGN within the error volume is well below unity even for $\rho=8$. This means that the AGN hosts can be identified uniquely from LISA observations alone, without EM counterparts. By comparison, the advanced LIGO-Virgo O3 observing run can achieve a 3D error volume of $\sim 10^{5}~{}{\rm Mpc}^{3}$ or better only for $<10\%$ of merging BBHs with $30+30~{}{\rm M}_{\odot}$ (Chen and Holz, 2016). This still allows a secure identification of the connection with AGN hosts, but only statistically Bartos et al. (2017). In the NSC scenario, the LISA error volume contains several candidate host galaxies even for relatively high SNR, $\rho>15$ ($d_{L}<340~{}{\rm Mpc}$), so that one would have to resort to a statistical correlation between LISA events and NSCs.

The LISA data predict the coalescence time of BBHs with an error of $<10$ s Sesana (2016). However, this prediction would be biased due to the CoM acceleration of the BBHs Bonvin et al. (2016). This bias has to be taken into account for any advance planning of follow-up EM observations of the merging BBHs. The phase drifts caused by the acceleration could be partially mimicked by a slight change in the mass ratio and time of coalescence. However, our Fisher analysis indicates that for sources with $35~{}{\rm M}_{\odot}\lesssim M_{\rm cz}\lesssim 63~{}{\rm M}_{\odot}$ which chirp inside the LISA band for 2-5 yrs, and especially for those whose eventual merger is detected by LIGO, these degeneracies are mitigated, and a measurement of the acceleration remains viable (see Figs. 3 and 4). Such a measurement will robustly test formation channels of coalescing stellar-mass BBHs involving an SMBH in a galactic nucleus.

The possibility of measuring the CoM acceleration of a merging BBH due to a nearby SMBH has been previously discussed for extreme mass ratio inspirals ($10^{5-6}~{}{\rm M}_{\odot}+10~{}{\rm M}_{\odot}$) in the LISA Yunes et al. (2011a) band, and for stellar-mass BBHs in the LIGO band Meiron et al. (2017). In the latter case, detection of the phase drift of the BBH during the handful of orbits executed inside the LIGO band requires an extremely close separation between the BBH and the SMBH ($\sim 10^{11}\,{\rm cm}$); these rare cases of extremely close-in binaries would however provide the opportunity to measure several other relativistic effects.

Acknowledgements.

We thank Enrico Barausse, Camille Bonvin, Tomoaki Ishiyama, Antoine Klein, Nicholas Stone, Riccardo Sturani and Kent Yagi for useful discussions. This work is partially supported by the Simons Foundation through the Simons Society of Fellows (KI) and by a Simons Fellowship in Theoretical Physics (ZH), and by NASA grant NNX15AB19G (to ZH). NT acknowledge support from the Labex P2IO and an Enhanced Eurotalents Fellowship.