Introduction

An extreme-mass-ratio inspiral (EMRI) normally consists of a compact stellar object, such as a stellar-mass black hole (BH), and a supermassive black hole (SMBH). It is an important target for future space-borne, milli-Hz gravitational-wave (GW) detectors, such as the laser interferometer space antenna (LISA1): an EMRI could dwell in the LISA band for years2,3,4, accumulating as many as 103–104 GW cycles in the data stream. Such a long waveform contains rich information about the space-time, as well as the astrophysical environment at the immediate exterior of a SMBH5,6,7. To decode this information, our model of an EMRI has to be highly accurate, both mathematically and physically.

In the canonical model of an EMRI, the stellar object is captured by the SMBH in two possible ways3: (i) it is scattered by other stars, a process known as relaxation, to such a small distance to the SMBH that the stellar object loses a significant amount of its orbital energy through GW radiation and becomes bound to the big BH8,9. The event rate of this type of EMRIs is difficult to estimate because it depends on factors that are poorly constrained by observations. The current estimation lies in a broad range between 10−9 and 10−6 per galaxy per year4,10,11,12,13,14,15; (ii) the small body could come from a binary which is also scattered to the vicinity of the SMBH. If the distance between the SMBH and the centre-of-mass of the binary becomes smaller than the tidal radius, Rt ≡ a(M3/m12)1/3, where a and m12 are the semi-major axis and total mass of the binary and M3 the mass of the SMBH, the interaction in general ejects the lighter member of the binary and leaves the other, more massive member on a bound orbit around the SMBH16,17,18. The corresponding event rate could be comparable to that produced by the aforementioned capture of individual BHs from eccentric orbits18.

In this article, we point to a third possibility. We show that a BH binary (BHB) could be tidally captured by a SMBH to a bound orbit. This could happen when the binary passes by the SMBH at such a close distance that the tidal interaction transforms a fraction of the kinetic energy into the internal potential energy of the binary17,19. We refer to the resulting triple system as the “binary EMRI” (b-EMRI) and investigate its long-term evolution using numerical simulations. We find that the captured binary, as a single unit, could survive around the SMBH for a long time. A significant fraction of the BHBs coalesce before their orbits around the SMBHs completely circularize. The result could be a burst of high-frequency (102 Hz) GWs happening at the same time and in the same sky position of a low-frequency (10−3 Hz) EMRI.

Results

Formation of a b-EMRI

Our BHB starts most likely around the influence radius of the SMBH, Rinf ≡ GM3/σ2, where σ is the one-dimensional velocity dispersion of the stars surrounding the SMBH (or “nuclear star cluster”, NSC20). Deeper inside this radius the BHB is susceptible to ionization by energetic encounters with interlopers21,22 and further outside the gravitational influence of the SMBH becomes negligible. Using the typical values M3 = 106M⊙ and σ = 60 km s−1 for NSCs, we find Rinf ≃ 1 pc.

Because the specific energy of the BHB initially is 3σ2/2 − GM3/Rinf = σ2/2, the orbit is a hyperbola with an asymptotic velocity of v0 = σ at infinity. Previous studies focus on parabolic orbits19 and have shown that tidal capture happens when the pericentre distance of the parabola, Rp, becomes comparable to the tidal radius Rt, i.e.,

$${R_{\mathrm{p}} \simeq \xi R_{\mathrm{t}} \simeq 74{\kern 1pt} R_{\mathrm{g}}\left( {\frac{\xi }{2}} \right)\left( {\frac{a}{{10^5{\kern 1pt} r_{\mathrm{g}}}}} \right) \times \left( {\frac{2}{{1 + q}}} \right)^{1/3}\left( {\frac{{m_1}}{{10{\kern 1pt} M_ \odot }}} \right)^{2/3}\left( {\frac{{M_3}}{{10^6{\kern 1pt} M_ \odot }}} \right)^{ - 2/3}},$$
(1)

where 1 ≲ ξ ≲ 5. In the last equation, Rg = GM3/c2 is the gravitational radius of the SMBH, c is the speed of light, m1 and m2 are the masses of the two stellar BHs, q ≡ m2/m1 is the mass ratio assuming that m1 ≥ m2, and rg = Gm1/c2 is the gravitational radius of the bigger stellar BH. In the following we assume q ≃ 1 because star clusters produce, most likely, equal-mass binaries23,24,25. We also scale a with 105rg and the reason will become clear later. The lifetime of these BHBs can been calculated in the Keplerian approximation26,

$$t_{{\mathrm{gw}}}(a,e) = \frac{a}{{4\left| {\dot a} \right|}} = \frac{{5a^4F(e)}}{{256c{\kern 1pt} r_{\mathrm{g}}^3q(1 + q)}}$$
(2)
$$\simeq \frac{{3.1 \times 10^6}}{{q(1 + q)}}\left( {\frac{{m_1}}{{10{\kern 1pt} M_ \odot }}} \right)\left( {\frac{a}{{10^5{\kern 1pt} r_{\mathrm{g}}}}} \right)^4F(e){\kern 1pt} {\mathrm{years}},$$
(3)

where \(\dot a\) is the decay rate of the semi-major axis due to GW radiation, e is the orbital eccentricity, and F(e) = (1 − e2)7/2(1 + 73e2/24 + 37e4/96)−1.

To prove that the above criterion for tidal capture also applies to hyperbolic orbits, we conduct scattering experiments using the numerical tool Fewbody27. The default parameters are M3 = 106M⊙, m1 = m2 = 10M⊙, e = 0.1, and a0 = 5 × 104rg ≃ 0.005 AU. Initially we place the BHB at infinity with a velocity of v0 = 60 km s−1. The direction of the velocity is chosen in such a way that the hyperbolic orbit initially has a pericentre distance of ξRt. We then numerically integrate the triple system and, after the first pericentre passage, record the relative energy change of the BHB, η ≡ (E − E0)/|E0|, where E is the final energy of the BHB and E0 = −Gm1m2/(2a0) is the initial energy.

If η ≥ 1, the binary is tidally disrupted. Otherwise, the binary survives the pericentre passage. Since the BHB initially has a kinetic energy of \(K_0 = m_{12}v_0^2{\mathrm{/}}2\), we know that the binary becomes bound to the SMBH if E − E0 > K0, i.e., η ≳ \(a_0v_0^2{\mathrm{/}}(G\mu )\) ≃ 0.0041, where μ = m1m2/m12 is the reduced mass of the binary. If η < 0.0041, the BHB is re-ejected to infinity. Figure 1 shows the cumulative distribution of η for different ξ, where we have randomized the initial inclination of the BHB and repeated the experiment 103 times for each value of ξ. We find that the captured fraction is fcap = (10–30)% when ξ varies between 0.7 and 2.5. Therefore, b-EMRIs can form via capturing the BHBs on hyperbolic orbits.

Fig. 1
figure 1

Cumulative distribution function of the η parameter. This parameter characterizes the relative energy change of a black hole binary (BHB) during its first pericentre passage of a hyperbolic orbit. The curves with different thickness refer to simulations with different ξ factors. In our model, tidal captures fall in the interval of 0.0041 < η < 0.95

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The orbital elements of the b-EMRIs can be derived from two conservation laws: (i) because of the conservation of energy, the binding energy (relative to the SMBH) of a captured BHB is E3 = K0 − η|E0| ≃ ηGm1m2/(2a0), where the last approximation uses the previous result that η ~ 0.1. From E3 we derive a semi-major axis of R ≃ (a0/η)(M3/μ). (ii) The pericentre remains to be Rp because of the conservation of angular momentum. Consequently, the eccentricity e3 satisfies the condition

$${1 - e_3 = \frac{{R_{\mathrm{p}}}}{R} \simeq 9.3 \times 10^{ - 5}\frac{q}{{(1 + q)^{4/3}}}\left( {\frac{\xi }{2}} \right) \times \left( {\frac{\eta }{{0.1}}} \right)\left( {\frac{{m_1}}{{10{\kern 1pt} M_ \odot }}} \right)^{2/3}\left( {\frac{{M_3}}{{10^6{\kern 1pt} M_ \odot }}} \right)^{ - 2/3}}.$$
(4)

Interestingly, it does not depend on our assumption of a0. The small value of 1 − e3 indicates that a newly formed b-EMRI, in general, is very eccentric.

The high eccentricity will affect the stability of a b-EMRI in two ways. On the one hand, the orbital motion of the BHB around the SMBH is more susceptible to perturbations by the surrounding stars because the angular momentum, \(\sqrt {GM_3R\left( {1 - e_3^2} \right)}\), is small. To be more quantitative, suppose Trlx characterizes the typical timescale for stellar relaxation processes to completely alter the orbital elements of a circular orbit3, the timescale to spoil an orbit of an eccentricity of e3 is only \(T_{{\mathrm{rlx}}}\left( {1 - e_3^2} \right)\). On the other hand, the orbit circularizes very fast due to GW radiation because the associated timescale Tgw (during which 1 − e3 increases) is proportional to \(R^4\left( {1 - e_3^2} \right)^{7/2}\). For our purpose, we use the relationship Rp = R(1 − e3) to rewrite Tgw and find that

$$T_{\mathrm{gw}} \simeq 15R_{\mathrm{p}}^{4} \, 256 \, c \, R_{\mathrm{g}}^{3} \left( {\frac{M_{3}} {m_{12}}} \right)\left( {1 - e_3} \right)^{ - 1/2} \\ \simeq {1.2 \times 10^6\left( {\frac{{1 + q}}{2}} \right)^{ - 7/3}\left( {\frac{\xi }{2}} \right)^4\left( {\frac{{1 - e_3}}{{10^{ - 4}}}} \right)^{ - 1/2}}$$
(5)
$$\times \left( {\frac{{m_1}}{{10{\kern 1pt} M_ \odot }}} \right)^{5/3}\left( {\frac{{M_3}}{{10^6{\kern 1pt} M_ \odot }}} \right)^{ - 2/3}\left( {\frac{a}{{10^5{\kern 1pt} r_{\mathrm{g}}}}} \right)^4{\mathrm{years}}.$$
(6)

Based on the understanding of the above two effects, we find that the b-EMRIs with \(\left( {1 - e_3^2} \right){\kern 1pt} T_{{\mathrm{rlx}}} > T_{{\mathrm{gw}}}\) are not any more affected by the relaxation of the background stars. Together with Eqs. (4) and (6), we find that these “clean” b-EMRIs satisfy

$${a < a_{{\mathrm{cri}}}} \simeq {4.2 \times 10^4{\kern 1pt} r_{\mathrm{g}}\frac{{q^{3/8}}}{{(1 + q)^{ - 1/12}}}\left( {\frac{{T_{{\mathrm{rlx}}}}}{{10^9{\kern 1pt} {\mathrm{years}}}}} \right)^{1/4}\left( {\frac{\xi }{2}} \right)^{ - 5/8}} \\ \times \left( {\frac{\eta }{{0.1}}} \right)^{3/8}\left( {\frac{{m_1}}{{10 M_ \odot }}} \right)^{ - 1/6}\left( {\frac{{M_3}}{{10^6{\kern 1pt} M_ \odot }}} \right)^{ - 1/12}.$$
(7)

This is the reason that we scaled a with 105rg in the previous equations. In the last equation we have adopted a typical value of 109 years for Trlx. We choose this value because our b-EMRI resides in the phase space of (1 − e3) ~ 10−4 and R = Rp/(1 − e3) ~ 0.02 pc (assuming a = acri), a region where two-body scattering dominates the relaxation process28,29.

Long-term evolution

Although the long-term interaction between a SMBH and a binary has been studied previously21,30,31,32,33,34,35,36,–37, these earlier works focus on a binary that is far away from the SMBH, so that there is no energy exchange between the “inner binary”, i.e., the BHB, and the “outer binary”, i.e. the pair formed by the BHB and the SMBH. Such triple systems are called “hierarchical triples”. Moreover, because of the large distance, the GW radiation of the outer binary is also unimportant to the overall dynamics. Under these circumstances, the evolution reduces to a periodic oscillation of the inner binary in the eccentricity space, which is known as the “Lidov–Kozai” cycle38,39. In our problem, however, the BHB is much closer to the SMBH so that the energies of the inner and outer binaries are no longer conserved and the GW radiation from the outer binary is not negligible. These are the key differences between our problem and other studies.

The b-EMRIs, in general, do not satisfy the criterion of hierarchical triple. In our problem, since \(M{\mathrm{/}}m_{12} \gg 1\) and 1 + e3 ≃ 2, the criterion for hierarchical triple40 reduces to

$$\frac{{R_{\mathrm{p}}}}{{R_{\mathrm{t}}}}\, \gtrsim\, 23\left( {\frac{{1 - e_3}}{{10^{ - 4}}}} \right)^{ - 1/5}\left( {\frac{{M_3}}{{m_{12}}}} \right)^{1/15}.$$
(8)

Our b-EMRIs do not satisfy this condition because tidal capture requires that 1 ≲ Rp/Rt ≲ 2.5 (see Fig. 1). For this reason, we use again Fewbody to evolve the triple system numerically. Besides adopting the default parameters, we also choose ξ = 2 and 1 − e3 = 10−4 as the initial conditions and start the BHB at the apocentre with a random inclination.

Moreover, we notice that Tgw is of the same order of tgw. If the eccentricity of the inner binary gets excited by the tidal force of the SMBH, the BHB may coalesce. This would terminate the b-EMRI. To include this effect in the simulation, we implement post-Newtonian (PN) corrections to the equations of motion of the inner binary, as has been done by one of the authors in a previous work24. Figure 2 shows an example of a coalescence. It also shows that the energy of the inner binary is not conserved, confirming our prediction that the system is not a hierarchical triple.

Fig. 2
figure 2

An example showing the evolution of the semi-major axis and eccentricity of the inner black hole binary (BHB). The discontinuities coincide with the pericentre passages. The black dot at the end of each curve marks the final output of the code, where the BHB has passed the pericentre and is 103Rt away from the supermassive black hole (SMBH). At this moment tgw is only 35 days so that the BHB will coalesce shortly afterwards. The intersecting plot in each panel shows a close-up of the evolution of about one hour during the first pericentre passage

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With the above setup, we run 104 simulations to get reasonable statistics. Figure 3 summarizes the outcomes. Interestingly, in about 30% of the cases, the BHBs coalesce well before the time limit t = Tgw/2 (for circularization) is reached. Because coalescing BHBs generate high-frequency GWs (102 Hz) and their orbital motion around the SMBH produces low-frequency ones (10−3 Hz), we conclude that b-EMRIs, in principle, could be detected by both ground-based and space-based detectors. Figure 4 illustrates this idea and we will elaborate this point in Discussion.

Fig. 3
figure 3

Probabilities as a function of time for the following four outcomes. First, the black hole binary (BHB) in the binary extrme-mass-ratio inspiral (b-EMRI) remains bound to the supermassive black hole (SMBH) (black solid line). The grey shaded area correspond to the statistical Poisson error, calculated with the square root of the number of experiments. Second, the BHB is disrupted by the tidal force of the SMBH (thick dashed curve). Third, the BHB is ejected to infinity (thin dashed curve). Fourth, the BHB coalesces into a single black hole (blue solid curve). The vertical dashed line indicates half of the gravitational wave (GW) radiation timescale Tgw/2 of a typical b-EMRI with ξ = 2 and 1 − e3 = 10−4. Beyond this time, the BHB would circularize around the BHB due to GW radiation but our code cannot capture this feature

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Fig. 4
figure 4

Three evolutionary stages of a binary extrme-mass-ratio inspiral (b-EMRI). First, in (i), a compact black hole binary (BHB) is captured to a bound orbit around a supermassive black hole (SMBH) because the pericentre distance becomes comparable to the tidal-disruption radius, Rt. Next, in (ii), the outer binary circularizes due to gravitational wave (GW) radiation, and the GW frequency lies in the band of a space-borne GW detector. Finally, in (iii), the tidal force of the SMBH becomes strong enough to excite the eccentricity of the inner BHB and drive it to merge. The merger produces high-frequency GWs

Full size image

Furthermore, about 0.6% of our b-EMRIs survive till the time t = Tgw/2. In principle they could have significantly circularized, i.e., both the semi-major axis R and the eccentricity e3 of the outer binary could have significantly decreased. However, our code does not include PN corrections to the outer binary and this shortcoming prevents us from predicting the fate of these long-lived b-EMRIs. One way of circumventing this issue is to repeat our b-EMRI simulations with smaller e, i.e., e = 0.999 or 0.99, to mimic the process of circularization, but the integration time exceeds our current computational capacity. We plan to resolve this issue in a future work. We note that PN corrections would not significantly affect the previous results about the capturing process because during the first periapsis passage the periastron shift, which is of the order of 2Rg/Rp ~ 0.027, is small and the energy lost via GW radiation is negligible relative to the dynamical energy exchange |E − E0|.

Formation rate

The formation rate of b-EMRIs can be estimated with Γ = pfcapΓBHB. Here ΓBHB denotes the formation rate of the BHBs with a ≃ acri in NSCs, p is the probability that such a BHB is on a hyperbolic orbit with ξ ≲ 2, and fcap ≃ 0.3 is the probability for capture (from Fig. 1).

To quantify ΓBHB, we notice that the BHBs of our interest are short-lived (see Eq. (3)) relative to the age of a galaxy, which is about 1010 years. Therefore, ΓBHB is equal to the event rate of BHB mergers in a galaxy. Recent calculations22,41 suggest that ΓBHB ~ few × 10−7 galaxy−1 year−1. This rate dose not account for the enhancement due to the interaction between BHBs and SMBHs21. When this enhancement is considered, the rate could be as high as ΓBHB = (0.8 − 2) × 10−6 galaxy−1 year−1 according to the most optimistic estimations33,35.

To quantify p, we first recall that our BHBs start from a rather large distance, Rinf, from the SMBH, where the binaries have a maximum angular momentum of \(L_c \simeq \sqrt {GM_3R_{{\mathrm{inf}}}}\) (i.e., the angular momentum for a circular orbit of the same energy). To reach a pericentre distance of Rp, the BHBs would have to diffuse in the angular-momentum space (L-space) due to relaxation to reach a region where L ≲\(\,L_{\mathrm{p}}\sim \sqrt {2GM_3R_{\mathrm{p}}}\). If the relaxation process is incoherent42, like the aforementioned two-body relaxation, the probability that the BHB appears at L ≲ Lp is proportional to L2. With this consideration and assuming acri = 5 × 104rg, we derive p ≃ (Lp/Lc)2 ≃ 4 × 10−6. Such a small probability renders the b-EMRI rate formidably small, i.e. Γ ~ (10−12–10−11) galaxy−1 year−1.

However, if the relaxation process is coherent, e.g., due to a massive perturber or an axisymmetric gravitational potential43,44,45,46, the probability p scales with L, so that p ≃ Lp/Lc ≃ 2 × 10−3. With this number, we find Γ ranges from 10−10 galaxy−1 year−1 to as large as few × 10−9 galaxy−1 year−1 in the most optimistic estimation. These numbers should be regarded as upper limits because the coherent relaxation processes normally affect only part of the angular-momentum space so that the real value of p could be significantly smaller than Lp/Lc.

Comparing these rates with the event rate of standard EMRIs4,10,11,12,13,14,15, which falls in the range of 10−9–10−6 yr−1 galaxy−1, we conclude that in the most optimistic case the b-EMRI rate could approach the lower boundary of the EMRI rate. Furthermore, since there are on average 2 × 107 galaxies per Gpc3 (earlier studies41 use the same galaxy density), we finally find that the above formation rate of b-EMRIs is equivalent to (10−5–10−4) Gpc3 year−1 in the pessimistic case and 0.1 Gpc3 year−1 in the most optimistic one.

Although the formation rate is relatively low, each b-EMRI has a non-negligible lifetime. The extended lifetime leads to a substantial duty cycle, Dbemri, for a galaxy to host a b-EMRI. To estimate Dbemri, we use (i) the formation rate Γ and (ii) the probability of surviving b-EMRIs as a function of time, fbemri(t), which is shown in Fig. 3 as the black solid curve, to derive Dbemri ≃ Γtfbemri(t). In the previous equation, the maximum value of tfbemri(t) appears at t ≃ 7600 years, where fbemri(t) ≃ 0.1. Therefore, we find that Dbemri ≃ (10−9–10−8) galaxy−1 in our pessimistic scenario and Dbemri ~ 10−6 galaxy−1 in the most optimistic one. This result means that within a volume of 1 Gpc3 (corresponding to a distance of about 600 Mpc), there are, on average, Nbemri ~ 0.02–20 b-EMRIs.

Discussion

We have seen that the BHBs in b-EMRIs, in general, have a ≲ acri. Interestingly, these BHBs are detectable by LISA. This is so because LISA is sensitive to the GWs with a frequency of f ~ 10−3 Hz and the strongest GW mode that a BHB emits47,48 is of a frequency of \(f \simeq \pi ^{ - 1}\sqrt {Gm_{12}{\mathrm{/}}[a(1 - e)]^3}\). As a result, those BHBs with a semi-major axis of

$$a_{{\mathrm{LISA}}}\sim 3.4 \times 10^4{\kern 1pt} r_{\mathrm{g}}{\kern 1pt} f_{ - 3}^{ - 2/3}\frac{{(1 + q)^{1/3}}}{{1 - e}}\left( {\frac{{m_1}}{{10{\kern 1pt} M_ \odot }}} \right)^{ - 2/3}$$
(9)

are inside the LISA band, where f−3 := f/(10−3 Hz). This critical semi-major axis is comparable to acri.

It is important to note that the BHBs in b-EMRIs are accelerating in the gravitational potential of the SMBHs. This condition could induce detectable shift to the phase of the GW inspiral waveform49,50. We have seen that within a distance of about 600 Mpc there are at most 20 accelerating BHBs for LISA to detect. In our default model, the BHB is at a typical distance of R ≃ 0.02 pc from the central SMBH. The corresponding acceleration is GM3/R2 and it could induce a phase shift as large as 103π per year according to the formula derived in the earlier studies49,50.

Furthermore, the orbital motion of the BHBs around the SMBHs also generates GWs. It is important to understand whether this radiation is detectable. Similar to the previous analysis for BHBs, we calculate the frequency of the strongest GW mode as \(f = \pi ^{ - 1}\sqrt {GM_3{\mathrm{/}}R_{\mathrm{p}}^3}\), so that a b-EMRI is inside the LISA band if its periapsis is

$$R_{\mathrm{p}}\sim R_{{\mathrm{LISA}}} \simeq 16{\kern 1pt} R_{\mathrm{g}}{\kern 1pt} f_{ - 3}^{ - 2/3}{\kern 1pt} M_6^{ - 2/3}.$$
(10)

From Eqs. (1) and (7), we find that our b-EMRIs satisfy the condition

$${R_{\mathrm{p}} \,\lesssim \,24{\kern 1pt} R_{\mathrm{g}}\frac{{(q{\kern 1pt} \xi )^{3/8}}}{{(1 + q)^{1/4}}}\left( {\frac{{T_{{\mathrm{rlx}}}}}{{10^9{\kern 1pt} {\mathrm{yr}}}}} \right)^{1/4} \times \left( {\frac{\eta }{{0.1}}} \right)^{3/8}\left( {\frac{{m_1}}{{10{\kern 1pt} M_ \odot }}} \right)^{1/2}\left( {\frac{{M_3}}{{10^6{\kern 1pt} M_ \odot }}} \right)^{ - 3/4}}.$$
(11)

Therefore, they are indeed inside the LISA band: Each pericentre passage will generate a burst of GWs detectable by LISA51,52,53,54.

These b-EMRIs are particularly interesting from the observational point of view because they generate two types of GWs, i.e., the continuous waves from the BHBs and bursts from the orbital motion of the BHBs around the SMBHs. Moreover, the two types of GWs are emitted at the same time in the same band. LISA can detect a single burst of the second type out to a distance of about 200(m12/20 M⊙) Mpc52. If the BHB has a total mass of about 60M⊙, like what the Laser Interferometer Gravitational-wave Observatory (LIGO) first detected23, the detection horizon would become as far as 600 Mpc. We already know that there are at most Nbemri ≃ 20 b-EMRIs within this distance. Now we estimate the chance of catching a GW burst at the moment of the pericentre passage. Since the majority of the b-EMRIs cannot circularize (see Fig. 3), the successive pericentre passages are separated by a long orbital period of the outer binary, which is about P3 ≃ 200 years in our fiducial model. Given that LISA is designed to have a mission duration of Δt = 5 years1, the chance of catching a b-EMRI at its pericentre passage is about (Δt/P)Nbemri ~ 50% in our most optimistic scenario.

Probably the most interesting feature about b-EMRIs is that the BHBs have a 30% chance to coalesce. The coalescence gives rise to a LIGO/Virgo event. The event rate is 0.3Γ, which is 0.03 Gpc3 year−1 in the most optimistic scenario and two orders of magnitude smaller in the most pessimistic one. If detected, the LIGO/Virgo event is likely separated from a LISA event, i.e., the GW burst generated during the pericentre passage, by about half of the orbital period of the outer binary, or 100 years in our fiducial model, because the coalescence happens most likely at the apocentre where the passing time is the longest. For this reason, it would be difficult to identify the LISA counterpart to the LIGO/Virgo event.

However, exceptions might exist. If the BHB in a b-EMRI could circularize around the SMBH, the orbital period P3 would be much shorter and hence the LISA and LIGO/Virgo events would appear much closer in time. According to our calculation, the event rate of such an exceptional case is at most Γ fbemri(t = Tgw/2) ~ 6 × 10−4 Gpc3 year−1.

In summary, we have shown that the small bodies in EMRIs could, in fact, be BH binaries. Such b-EMRIs could form via tidal capture and a significant fraction of the captured BHBs could merge due to the perturbation by the SMBHs. Although rare, they are ideal targets for future multi-band GW observations55,56. First, the merger generates high-frequency GWs, i.e. a LIGO/Virgo event that is in the same sky location of a LISA EMRI event. Second, the high-frequency GWs could be redshifted because they are generated very close to a SMBH57, providing a rare opportunity of studying the propagation of GWs in the regime of strong gravity. Third, the merger also induces a kick to the BH remnant58. This kick causes a glitch in the EMRI waveform, which, through a careful analysis, is discernible in the data stream59.