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Gravitational Waves from Hierarchical Triple Systems with Kozai-Lidov Oscillation

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Gravitational Waves from Hierarchical Triple Systems with Kozai-Lidov Oscillation Gravitational Waves from Hierarchical Triple Systems with Kozai-Lidov Oscillation Priti Gupta,1, ∗ Haruka Suzuki,2, y Hirotada Okawa,3, z and Kei-ichi Maeda3, 4, x 1 Department of Physics, Kyoto University, Kyoto 606-8502, Japan 2 Graduate School of Advanced Science and Engineering, Waseda University, Shinjuku, Tokyo 169-8555, Japan 3 Waseda Institute for Advanced Study (WIAS), Waseda University, Shinjuku, Tokyo 169-8050, Japan 4 Department of Physics, Waseda University, Shinjuku, Tokyo 169-8555, Japan (Dated: April 14, 2020) We study gravitational waves from a hierarchical three-body system up to first-order post- Newtonian approximation. Under certain conditions, the existence of a nearby third body can cause periodic exchange between eccentricity of an inner binary and relative inclination, known as Kozai-Lidov oscillations. We analyze features of the waveform from the inner binary system un- dergoing such oscillations. We find that variation caused due to the tertiary companion can be observed in the gravitational waveforms and energy spectra, which should be compared with those from isolated binaries and coplanar three-body system. The detections from future space-based interferometers will make possible the investigation of gravitational wave spectrum in mHz range and may fetch signals by sources addressed. PACS numbers: I. INTRODUCTION merger phase by incoming gravitational waves. The ob- servations so far, and the future GW detections may shed The LIGO-VIRGO collaboration observed eleven grav- light on binary formation channels, enable precision tests itational wave signals from compact binary mergers dur- of general relativity (GR) in the strong-field regime, and ing the first and second runs (O1 and O2)[1]. The initiate new avenues in astrophysics. third observation run (O3) by LIGO and VIRGO be- In this new stage of the beginning of gravitational wave gan in April 2019 and new detection alerts have started astronomy, we have to list up all possible gravitational following within months of operation[2]. With cur- wave sources and analyze what we expect from those rent sensitivity design, ground detectors are focusing on in observations. So far, we have seen intensive analy- the high-frequency range (10Hz to 1000Hz). Most ex- sis for binary systems composed of objects with various pected sources lying in this range are compact binaries masses[13{15, and references therein]. However, we may in the merger and ringdown phase. For ground-based expect more exotic sources in nature. One of the likely detectors, currently, we have five ground-based detec- sources is a three-body system, which we will study in tors with characteristic strain of order 10−22; advanced this paper. The environment near supermassive black LIGO (aLIGO) detectors in Hanford and Livingston[3], holes (SMBHs) in galactic nuclei comprises of a large advanced VIRGO (aVIRGO) in Italy[4], KAGRA in number of stars and compact objects. Triple systems Japan[5], and GEO600 in Germany[6]. could also emerge in these surroundings [16{21, and ref- Future space-based observatories like Laser Interferom- erences therein] with the likelihood of the presence of hi- eter Space Antenna (LISA) are expected to explore low erarchical structure. The potential astrophysical sources frequency range from 10−4 Hz to 10−1 Hz with character- of gravitational waves involving three-body system have istic strain of order 10−21[7] whereas DECi-hertz Inter- seen increasing interest over time [22{28]. All the detec- ferometer Gravitational wave Observatory (DECIGO) is arXiv:1911.11318v3 [gr-qc] 11 Apr 2020 tions so far (direct or indirect) came from binary inspirals aiming to fill the gap between LIGO and LISA with fre- and merger phase. It is good to contemplate how the quency band around 10−2 Hz to 10 Hz with characteristic waveform would look like if the binary system was ac- strain down to 10−24[8]. Several big projects have also companied by a tertiary companion in a hierarchical ar- been proposed in space; Big Bang Observer (BBO)[9], rangement. The hierarchical triple body approximation advanced LISA (aLISA)[10], and TianQin and TAIJI in has useful applications and can be applied to diverse set- China[11, 12]. With the launch of those space detec- tings, varying from planetary and stellar mass scales to tors, we can aim for low-frequency sources. It will be SMBHs. In this approximation, the energy of each orbit possible to map a source all the way from inspiral to a is separately conserved and therefore the two semi-major axes are constants. The Kozai-Lidov (KL) mechanism [29, 30], which is one of the most important phenomenon in a hierarchical triple system, is particularly exciting. ∗Electronic address: [email protected] yElectronic address: [email protected] KL mechanism occurs when the inner orbit inclined with zElectronic address: [email protected] respect to the outer orbit. The main feature of KL mech- xElectronic address: [email protected] anism is the secular change (timescale much larger than 2 the orbital periods and referred as KL timescale) of the Indirect observation of GW from a triple system is also eccentricity of the inner orbit and the relative inclina- studied by analyzing the cumulative shift of periastron tion between inner and outer orbit. Both values oscil- time of a binary pulsar undergoing KL oscillations [53]. late inversely with respect to the other, that is, when In this paper, by integrating the equations of mo- the eccentricity decreases, the inclination increases, and tion directly, we analyze the features of the gravita- vice versa. The eccentricity of the orbit can reach ex- tional waves generated in a hierarchical triple system. treme values leading to various astrophysical phenomena. We discuss wider range of parameters, i.e., a triple sys- For instance, if the eccentricity becomes large enough be- tem of SBHs as well as a SBH binary around SMBH. We cause of KL oscillations, the merger rate of BHs can be also analyze models composed of an IMBH as well as a enhanced by large emission of gravitational waves [31{ SBH/SMBH. We show the waveform and spectra as well 34], the tidal disruptions rate in a stellar cusp containing as the observability of such sources by future space-based SMBH binary can be several orders larger in magnitude detectors. We also draw comparisons with the isolated than expected in case of single SMBH [35{40]. Further, binary systems. the formation of hot Jupiters or ultra-short-period plan- The paper is organized as follows: We review the ets is studied through the secular gravitational interac- characteristics of the hierarchical structure and KL- tions between two planets in eccentric and inclined orbits mechanism in xII A. In section xII B , we discuss the con- [41{44]. straints on KL oscillations due to relativistic effects. In Recently, there has been extensive work focusing on section xIII, we recall the fundamentals of quadrupole gravitational wave detections from such systems. The formalism and discuss GWs from highly eccentric orbits. probability of detection and frequency range from differ- In section xIV, we explore parameter space for stable KL ent sources has been discussed by many authors. The sec- oscillations, classifying the cases by the mass ratios. We ular evolution of stellar BH (SBH) binaries near SMBHs analyze seven models and show the properties of GWs for is studied, focusing on the tidal perturbation by the three typical models in xV. We also discuss the observ- SMBH [27, 45, 46]. The analysis shows that merging ability of those models. The paper is concluded in xVI. binaries will enter the LIGO observational window while In Appendix A, we provide the waveforms from isolated on orbits that are still very eccentric (e ∼ 0:5). The ef- binary systems and hierarchical triple coplanar systems ficient GW analysis for such systems would, therefore, for comparison. We present the characteristic strains for require the use of eccentric templates. The constraints all other models in Appendix B. are put on the importance of KL induced mergers for pro- ducing gravitational wave sources detectable by aLIGO, for triples with an inner pair of SBHs. Another interest- II. DYNAMICS OF TRIPLE SYSTEM ing model with triple SMBHs is studied [47]. Numeri- cal simulations of the dynamics of triples within galaxy In this section, we sequentially explain a hierarchical cores exhibit phases of very high eccentricity (as high as triple system, Kozai-Lidov mechanism, and their stability e ∼ 0:99) showing the likelihood of detection of GWs by and constraints. LISA. Some authors discuss variations in GWs in the pres- ence of the third body due to relativistic beaming, A. Hierarchical Triple System and Kozai-Lidov Doppler, and gravitational redshift [48{51]. The accel- Mechanism eration imparted by the hierarchical companions leads to potentially observable changes in the waveforms that We can have several stable configurations and shapes would reflect their gravitational interactions with the sur- when it comes to a three-body dynamical system. rounding matter. The GW signal would be distinguished Broadly addressing, it can be concentric, coplanar orbits by a net phase change or even a time-dependent Doppler or inclined orbits in a hierarchical arrangement. In this shift arising from the orbital motion. This could provide paper, we treat the so-called hierarchical triple system, direct information about the black hole binary environ- whose schematic picture is given in Fig. 1. ments and otherwise invisible ambient mass. As depicted in the figure, we can decouple a system into Recent work analyzes triple systems composed of the the two-body inner binary orbit and the outer orbit with SMBH near the galactic center and a pair of BHs with tertiary companion in motion around center of mass of different masses, from stellar to intermediate-mass BHs inner binary, if the distance between the first and second (IMBHs)[28, 52].
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