Detecting Kozai-Lidov imprints on the gravitational waves of intermediate-mass black holes in galactic nuclei

A third object in the vicinity of a binary system causes variations in the eccentricity and the inclination of the binary through the Kozai-Lidov effect. We examine if such variations leave a detectable imprint on the gravitational waves of a binary consisting of intermediate mass black holes and stellar mass objects. As a proof of concept, we present an example where LISA may detect the Kozai-Lidov modulated gravitational wave signals of such sources from at least a distance of 1Mpc if the perturbation is caused by a supermassive black hole tertiary. Although the quick pericenter precession induced by general relativity significantly reduces the appropriate parameter space for this effect by quenching the Kozai-Lidov oscillations, we still find reasonable parameters where the Kozai-Lidov effect may be detected with high signal-to-noise ratios.


INTRODUCTION
According to the current paradigm, nearly all galaxies, including our own, host a supermassive black hole (SMBH) at their centers (Kormendy & Ho 2013;Ghez et al. 2008;Genzel et al. 2010). Being the engine of galactic nuclear activity, they have a large influence both on their immediate environment (e.g., Inoue et al. 2020) and on more extended scales which leads to correlations between the SMBH and the host galaxy properties (King 2003). Observations such as periodic AGN variability show that some SMBHs are found in binaries (Kelley et al. 2019). They are the natural consequences of galaxy mergers predicted by the ΛCDM model (e.g., Di Matteo et al. 2005;Hopkins et al. 2006;Robertson et al. 2006). The SMBH at the center of the Milky Way may also have a massive binary companion (see Gualandris & Merritt 2009;Gualandris et al. 2010;Naoz et al. 2020, for observational constraints). SMBH binaries play a key role in galaxy evolution (e.g., Begelman et al. 1980;Blecha & Loeb 2008). They explain the mass deficit of stars observed in the centers of galaxies (Merritt 2006;Gualandris & Merritt 2012), they lead to the ejection of hyper velocity stars (e.g., Yu & Tremaine 2003;Luna et al. 2019;Rasskazov et al. 2019;Fragione & Gualandris 2019), affect tidal disruption and GW events (e.g., Ivanov et al. 2005;Chen et al. 2009Chen et al. , 2011Chen & Liu 2013;Wegg & Nate Bode 2011;Sesana et al. 2011;Li et al. 2015;Meiron & Laor 2013;Fragione et al. 2020), and lead to an electromagnetic signature from dark matter annihilation (Naoz & Silk 2014;Naoz et al. 2019). The inspiral of such SMBH binaries will be targets for the future space-E-mail: deme.barnabas@gmail.com borne gravitational wave (GW) observatory LISA 1 (e.g., Amaro-Seoane et al. 2017).
The mass spectrum of SMBHs arguably extends down to the regime of intermediate-mass black holes (IMBHs) (see Greene et al. 2019 andMezcua 2017 for recent reviews). An SMBH-IMBH binary may reside in the nucleus of some galaxies. The IMBHs may form in SMBH accretion disks (Goodman & Tan 2004;McKernan et al. 2012) or they may be transported to the galactic center region by infalling globular clusters that also help to form the nuclear star clusters around SMBHs (Portegies Zwart et al. 2006;Mastrobuono-Battisti et al. 2014). Gravitational wave (GW) astronomy, which has recently opened a new window on the Universe (Abbott et al. 2016), may directly test the existence of IMBHs in galactic nuclei.
There are multiple dynamical processes in the nuclear regions of galaxies which may affect the binaries' GWs. The pertubations associated with the SMBH in nuclear star clusters may be significant. The SMBH perturber leads to the acceleration of a binary's center of mass which may be detected by LISA (Yunes et al. 2011). Furthermore, variations caused by relativistic beaming, Doppler, and gravitational redshift associated with the SMBH companion may also lead to potentially detectable signatures (Meiron et al. 2017). In this paper, we examine if the Kozai-Lidov (KL) effect of the SMBH leads to detectable variations on binaries in nuclear star clusters.
The KL mechanism has been long recognized to be one of the important dynamical processes in galactic nuclei (see Naoz 2016 for a review). It describes the long-term dynamics of a hierarchical triple system in separation, i.e. when two of the bodies constitute a tight inner binary, which is orbited by a more distant tertiary (outer binary). This third object perturbs the inner binary in such a way that it exhibits eccentricity and inclination oscillations with nearly constant semi-major axis (Kozai 1962;Lidov 1962). It can be shown that the octupole-order perturbation by the third body can pump up the eccentricity to very high values close to unity (Lithwick & Naoz 2011) which leads to gravitational wave bursts during close periapsis encounters (O'Leary et al. 2009;Kocsis & Levin 2012). The KL torque from a SMBH may result in the merger of compact object binaries (e.g. Antonini et al. 2014;Hoang et al. 2018). 2 The eccentricity oscillations from KL effects are directly detectable in the inspiral phase long before the merger of the inner binary, which causes a periodic shift in the GW strain signal (Hoang et al. 2019;Randall & Xianyu 2019).
In this paper we examine how the mass and initial orbital parameters of the inner binary affect LISA's ability to identify the KL effect of the SMBH on binaries in galactic nuclei. We show that if the inner binary consists of an IMBH and a stellar-mass black hole (see Fig. 1), LISA may directly detect the KL oscillations from a distance of 1 Mpc. This paper is structured as follows. In Section 2 we introduce the timescales which have key role in the dynamics we investigate. In Section 3 we calculate the signal-to-noise ratio and in Section 4 we discuss our results.

TIMESCALES AND CONSTRAINTS
The relevant timescales for our study are the KL time (T KL ), the general relativistic (GR) apsidal precession time (T GR ), and the GW inspiral time (T GW ) (Naoz 2016;Peters 1964): where the 1 and 2 subscripts refer to the inner and outer binaries, respectively. Long-lived triples must satisfy the Hill stability criterion, i.e.
and we restrict attention to sufficiently hierarchical configurations so that we can neglect the terms beyond octupole in the expansion of the Hamiltonian Eqs. (5) and (6) show that Eq. (5) is always more strict if (m 1 + m 2 )/m 3 0.024. In particular, Hill-stable inner binaries with 10 and 10 5 M around a 10 8 M SMBH have (m 1 + m 2 )/m 3 ∼ 0.001 and automatically satisfy the hierarchy criterion.
It is useful to express Eqs.
(1)-(4) with distances normalized by the corresponding gravitational radii: and with the mass ratios µ 1 = m 1 /m 3 and µ 2 = m 2 /m 3 as (1 − e 2 1 ) 7/2 1 + 73 24 e 2 1 + 37 96 e 4 and the Hill stability criterion reads Fig. 2 shows the parameter space for different separations and mass ratios where these timescales are in the suitable range for the KL effect to play a role during LISA observations. We show cases where T KL < 10yr, T GR,1 > 10yr, T GW,1 > 10 8 yr, and T GW,2 > 10 8 yr for m 3 = 10 8 M . The values of the timescales are chosen arbitrarily, but in the case of GR and KL times (10 years) we took into account the operational time of LISA. We also note that once T GR,1 < T KL , the KL mechanism is quenched, i. e. the amplitude of the eccentricity oscillations is significantly damped, however, we will show that they are still detectable. The right panels show higher µ 1 (i.e. higher m 1 ), while the bottom one higher µ 2 . The initial outer eccentricity is set to e 2 = 10 −6 , which remains 10 3 10 4 approximately constant during the evolution since µ 2 1. This condition also implies that the evolution is well approximated by the quadrupole term of the Hamiltonian. Thus Eq. (1) is quite accurate and the outer argument of pericenter needs not be accounted for as the quadrupole Hamiltonian is independent of it (the so-called "happy coincidence" (Lidov & Ziglin 1976)). The ideal zone in the parameter space, where the triple is Hill stable and KL oscillations may occur in LISA observations, is the highlighted yellow area between the green and the blue curves. This region is larger in the case of the left panels. More specifically, in what follows we focus on the top left, where m 1 = 10 5 M and m 2 = 10 M and where the GR and GW timescales are slightly longer than in the bottom left. Further decreasing µ 2 would also decrease the GR timescale, allowing KL to pump the eccentricity higher, but µ 2 < 10 −8 with m 3 = 10 8 M would result in unphysically low compact object masses. Most of the yellow zone is of little use, though, because high a 1 gives weak GW signal for sources outside of the Milky Way. For this reason, in what follows we restrict the inner semi-major axis to the range a 1 ∈ [1; 10] AU, i.e. a 1 /R g1 between ∼ 10 3 and 10 4 . Fig. 3 demonstrates the time evolution of the system for a representative example shown with a star in Fig. 2 at a distance of 1 Mpc. We simulate the system using the secular OSPE code. The left panel shows the pericenter frequency evolution of the inner binary. Its oscillatory behavior at the beginning is due to the KL effect, which is later quenched by GR precession. The inset of the left panel shows the first year of the inner eccentricity evolution. One way to detect the KL effect in practice is to average the GW strain over time in two-month-long intervals, indicated by horizontal arrows. We calculate the strain spectra for these averaged intervals, which are shown in the right panel of Fig. 3. In order to detect the KL oscillations, both the GW spectral amplitude (black and red curves) and its variation (blue curve) are required to be above the LISA sensitivity curve (denoted by orange). Technically, by the difference of the strains we mean the strain of the difference of the GW signals obtained from two subsequent observational time segments.

SIGNAL-TO-NOISE RATIOS
In order to estimate the detectability of the signal within an observation segment of time duration T obs , we calculate the signal-to-noise ratio (SNR) following Hoang et al. (2019) whereh is the Fourier transform of the GW strain signal and S n ≡ S( f n ) is the LISA spectral noise amplitude. For short time segments that satisfy T GW1 T obs , and that the orbital time around the SMBH is sufficiently long, i.e. T orb3 = 2π[a 3 3 /G(m 1 + m 2 + m 3 )] 1/2 T obs we may substitute the the strain for a fixed semsemimajor axis a 1 (Peters 1964).
The left panel of Fig. 4 shows the SNR for the initial parameters of the secular evolution (calculated with T obs = 2 months), i. e. the GW signal we would measure from the inner binary at the beginning. A red asterisk marks here the initial values used in the representative example shown in Fig. 3. The relevant timescales for the initial configuration are indicated with lines. However, note that these timescales change significantly during the evolution.  Fig. 2. It oscillates in the LISA frequency band due to the KL torque from the SMBH tertiary, but later becomes damped by the quick pericenter precession induced by GR. The in-set figure shows the first two KL-peaks in the eccentricity. The horizontal black and red arrows show the interval of averaging (2 months). We use these mean values for calculating the strain (shown in black and red in the right panel). We note that the GW inspiral time is only ∼50 years, even though Eq. (3) predicts much longer values from the initial parameters (T GW,1 ≈ 1.4 × 10 4 years), because the eccentricity is increased to very high values. Right: LISA sensitivity curve and the strain for the two 2-months intervals marked in the inset of the left panel. h 1 corresponds to a 1 = 5.999887 AU, e 1 = 0.989745 and SNR = 5.81 (D L /Mpc) −1 , while h 2 to a 1 = 5.968728 AU, e 1 = 0.996220 and SNR = 117 (D L /Mpc) −1 . The blue curve shows the difference of the strains, which is a useful measure of the time evolution of the orbital parameters in the inner binary, either by KL mechanism or GW.

Figure 4.
Left: SNR of the GW signal from the IMBH-SBH inner binary calculated from the initial orbital parameters e 1 and a 1 /R g1 for an observation time T obs = 2 months for sources at 1 Mpc. The red asterisk corresponds to the same initial position as the one marked in Fig. 2. We emphasize that the orbital parameters significantly change in time due to the KL oscillations, GR precession and GW radiation. By the same manner, timescales indicated by the lines reduce significantly, too. Right: The highest change in SNR during the evolution between two subsequent averaging time segments, maximized over both ω 1 and T obs . In order to demonstrate the underlying dynamics we show the evolution of the orbital elements from three different parts of the parameter space, denoted by A, B and C, in Fig. 5.
In order to calculate how the SNR changes during the KL evolution, we run ∼4000 simulations, each for 20 years and with a 2 = 400 AU, e 2 = 10 −6 . The right panel of Fig. 4 indicates the maximum ∆SNR, i.e. the highest change in the SNR during the evolution between two subsequent observational segments (T obs ) for the system initiated from that particular point of the (a 1 /R g1 , 1 − e 1 ) parameter space. Here ∆SNR is maximized over the argument of the inner pericenter ω 1 in a way that it is varied in a grid from 0 • to 360 • keeping the rest of the initial elements fixed, choosing the ω 1 that resulted in the highest |∆SNR|. We also optimize for the T obs observational time: we calculate the ∆SNR for T obs ∈ {10 −1 , 10 0 , 10 1 } years and choose whichever gives the highest change in SNR during the evolution. We note that it makes the predicitions of the right panel somewhat pessimistic: the ∆SNR values could be further in-creased if we chose such T obs that fits better to the eccentricity oscillation timescale.
The right panel of Fig. 4 shows that the high ∆SNR values are found at large initial a 1 independently of e 1 and at small a 1 and 1 − e 1 . This is not unexpected because for the former the KL time is shortest at high a 1 while the GR precession and inspiral time are longer there (see Eq. (1)), therefore KL oscillations are less damped there. Interestingly, in this region the binary would not be detected without the KL oscillations, which push the binary to high eccentricities. For small a 1 and high e 1 , the orbital parameters change rapidly due to the GW inspiral independently of the KL effect, which explains the lower left peak of ∆SNR in Fig. 4.
To better understand this behavior, we select three representative points (denoted by A, B and C in Fig. 4) and plot the time evolution of their orbital elements in Fig. 5  C Figure 5. The orbital element evolution of the three systems sampled from the right panel of Fig. 4. Evolution A and B are both highly damped by GR. The high ∆SNR value at B is the result of the strong GW decay. amplitude in Fig. 6. The first row of panels shows the pericenter frequency calculated as The figures show that Case C exhibits multiple prominent KL oscillation cycles, which leads to a high ∆SNR. However, in Case B, KL oscillations are quenched by the rapid GR precession. We note that even in such a quenched case there are some small oscillations (∆e ∼ 10 −4 , T GR ∼ 10 −2 years (e.g., Naoz et al. 2013)), but they do not produce significant ∆SNR because of the small change in the eccentricity. The high ∆SNR is obtained with T obs = 10yr: the reason for this is that most of the variation of the orbital parameters is caused by the GW inspiral (not by the SMBH), so we need to have a T obs that is comparable to the inspiral time, T GW,1 ≈ 57yr. The high ∆SNR in thus mostly independent of the KL effect. In case A, the orbital parameters are almost constant as the system is neither inspiraling nor does it exhibit KL oscillations.

DISCUSSION AND CONCLUSION
We have shown that the dynamical imprint of SMBHs may be significantly detected with LISA from 1 Mpc for compact objects orbiting IMBHs in galactic nuclei. Fig. 4 showed the initial orbital parameters where this identification is possible. We found that the imprint of KL oscillations are most prominent for IMBH sources orbited by a stellar mass compact object which orbit around a SMBH. A binary of two stellar mass compact objects also exhibit similar oscillations in the vicinity of a SMBH, but in this case either the GW strain amplitude is much smaller or the GR precession rate is higher which decreases the KL oscillation amplitude. Further, KL oscillations are also less prominent in hierarchical SMBH triples since in this case the KL timescale is typically much longer than the observation time.
To demonstrate the detectability of KL oscillations in a robust way, we calculated the variations of the SNR during the observation in fixed T obs duration segments of the total observation period, and marginalized over the value of T obs . This analysis showed that the variations due to the KL effect can be highly significant and detectable with LISA to at least 1 Mpc.
While we have highlighted cases where the full KL oscillations may be detected with LISA with very high significance, the true parameter space where the KL effect may be detected is certainly much larger. Since the number of GW cycles is of order N GW ∼ T obs f orb ∼ T obs f p (1 − e 1 ) 3/2 (Eq. 14), a very small variation of eccentricity of order 4 × 10 −4 T obs 4yr −2/3 f p 1mHz −2/3 (15) may cause order unity change in the number of detected cycles during a 4 year observation. Thus, the KL effect may be significant even if only a 10 −3 fraction of a full KL cycle is observed. Furthermore, Figure 6. The GW spectra of the systems sampled from the right panel of Fig. 4. The black and red curves correspond to the signals which give the highest ∆SNR during the evolution. The blue curve is the difference of them. In case A the black and red curves are so close to each other that they are not separable by eye.
KL oscillations may push the binary to so high eccentricities that the binary merges during the observation. For merging binaries, the number of GW cycles is proportional to the inverse GW timescale, which for asymptotically high e 1 close to unity is proportional to (1 − e 2 1 ) −7/2 (Eq. 3), implying an even higher sensitivity to eccentricity. Thus, the KL effect of inspiraling GW sources may be highly significant even in cases where the observation time and/or the GW inspiral time is much shorter than the KL timescale.
While we leave the detailed GW data analysis exploration of KL imprints to a future study, these arguments suggest that the detection prospects of the KL effect may be possible even beyond the case of IMBH-stellar mass compact object triples around SMBHs.