PHYSICAL REVIEW D 99, 023010 (2019)
Gravitational waveforms, polarizations, response functions, and energy losses of triple systems in Einstein-aether theory
Kai Lin,1,2 Xiang Zhao,3,4 Chao Zhang,3,4 Tan Liu,5,6 Bin Wang,7,8 Shaojun Zhang,4 Xing Zhang,5,6 Wen Zhao,5,6 Tao Zhu,4 and Anzhong Wang3,4,* 1Hubei Subsurface Multi-scale Imaging Key Laboratory, Institute of Geophysics and Geomatics, China University of Geosciences, Wuhan, Hubei, 430074, China 2Escola de Engenharia de Lorena, Universidade de São Paulo, 12602-810, Lorena, São Paulo, Brazil 3GCAP-CASPER, Physics Department, Baylor University, Waco, Texas 76798-7316, USA 4Institute for Advanced Physics & Mathematics, Zhejiang University of Technology, Hangzhou 310032, China 5CAS Key Laboratory for Researches in Galaxies and Cosmology, Department of Astronomy, University of Science and Technology of China, Chinese Academy of Sciences, Hefei, Anhui 230026, China 6School of Astronomy and Space Science, University of Science and Technology of China, Hefei 230026, China 7Center for Gravitation and Cosmology, Yangzhou University, Yangzhou 225009, China 8School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
(Received 2 November 2018; published 14 January 2019)
Gravitationally bound hierarchies containing three or more components are very common in our Universe. In this paper we study the periodic gravitational wave (GW) form, its polarizations, the response function, the Fourier transform, and the energy loss rate of a triple system through three different channels of radiation, the scalar, vector, and tensor modes, in the Einstein-aether theory of gravity. The theory violates locally the Lorentz symmetry, and yet satisfies all the theoretical and observational constraints by properly choosing its four coupling constants ci’s. In particular, in the weak-field approximations and with the recently obtained constraints of the theory, we first analyze the energy loss rate of a binary system and find that the dipole contributions from the scalar and vector modes could be of the order of 2 −5 Oðc14ÞOðGNm=dÞ , where c14 ð≡c1 þ c4Þ is constrained to c14 ≲ Oð10 Þ by current observations, and GN, m, and d are, respectively, the Newtonian constant, mass, and size of the source. On the other hand, the “strong-field” effects for a binary system of neutron stars are about 6 orders lower than that of general relativity. So, in this paper we ignore these strong-field effects and first develop the general formulas to the lowest post-Newtonian order, by taking the coupling of the aether field with matter into account. Within this approximation, we find that the scalar breather mode and the scalar longitudinal mode are all suppressed by a factor of Oðc14Þ with respect to the transverse-traceless modes (hþ and h×), while the −15 vectorial modes (hX and hY ) are suppressed by a factor of c13 ≲ Oð10 Þ. Applying the general formulas to a triple system with periodic orbits, we find that the corresponding GW form, the response function, and its Fourier transform depend sensitively on the configuration of the triple system, their orientation with respect to the detectors, and the binding energies of the three compact bodies.
DOI: 10.1103/PhysRevD.99.023010
I. INTRODUCTION Advanced Virgo detector (aVirgo) [8] joined the second observation run of aLIGO on August 1, 2017, and jointly The detection of the gravitational wave from the coalesc- þ5 detected the last two GWs, GW170814 and GW170817 ing of two massive black holes (one with mass 36 M⨀ and −4 [5,6]. Except for GW170817, which was produced by the 29þ4 the other with mass −4 M⨀) by the advanced Laser merger of binary neutron stars (BNSs) [6], all the rest were Interferometer Gravitational-Wave Observatory (aLIGO) produced during the mergers of binary black holes (BBHs). marked the beginning of the era of the gravitational wave The detection of GW170817 is important, not only because it (GW) astronomy [1]. Following it, five more GWs were confirmed that BNSs are indeed one of the most promising detected [2–6], and one candidate was identified [7].The sources of GWs but also because it was companied by a short-duration gamma-ray burst [9], which enables a *Corresponding author. wealth of science unavailable from either messenger alone. [email protected] In particular, it allows us to measure simultaneously both
2470-0010=2019=99(2)=023010(23) 023010-1 © 2019 American Physical Society KAI LIN et al. PHYS. REV. D 99, 023010 (2019) distance and redshift of the source, with which we can study, binary and a third companion. The inner binary consists of for example, cosmology. a pulsar with mass m1 ¼ 1.44 M⨀ and a white dwarf with With the promise of increasing duration, observational mass m2 ¼ 0.20 M⨀ in a 1.6 day orbit. The outer binary sensitivity, and the number of detectors, many more events consists of the inner binary and a second dwarf with mass are expected to be detected. In particular, the Laser m3 ¼ 0.41 M⨀ in a 327 day orbit. The two orbits are very Interferometer Space Antenna (LISA) [10] is expected to −4 circular with its eccentricities eI ≃ 6.9 × 10 for the inner observe tens of thousands of compact galactic binaries −2 binary and eO ≃ 3.5 × 10 for the outer orbit. The two during its nominal four year mission lifetime [11].Asa orbital planes are remarkably coplanar with an inclination matter of fact, because of its high mass, GW150914 would ≲0.01°. A triple system is an ideal place to test the strong have been visible to LISA for several years prior to their equivalence principle [29]. Remarkably, after six-year coalescence [12]. observations, lately it was found that the accelerations of Despite significant investigations, the origins of these the pulsar and its nearby white-dwarf companion differ binary systems, particularly the heavier BBHs, remain an −6 fractionally by no more than 2.6 × 10 [30], which pro- open question (see, e.g., Ref. [13] and references therein). vides the severest constraint on the violation of the strong Not only from the point of view of theoretical simulations equivalence principle. but also from the observational estimates, it is found very In a triple system, the existence of the third companion difficult to have such heavy black holes (BHs). On the one can undertake the Lidov-Kozai oscillations [31,32], and hand, most of the BHs obtained from numerical simulations 10 cause the orbit of the inner binary to become nearly radial, have masses lower than M⨀, unless the stellar metal- whereby a rapid merger due to GW emissions can result licity is very low [14]. On the other hand, Özel et al. [33,34]. Such a system can emit GWs in the 10 Hz examined 16 low-mass x-ray binary systems containing frequency band [35–37], which are potential sources for BHs and found that the masses of BHs hardly exceeded 20 the current ground-based detectors, such as aLIGO, aVirgo, M⨀ and that there was a strongly peaked distribution at and KAGRA [38]. It can also emit GWs in the frequency 7 8 1 2 . . M⨀ [15]. Similar results were obtained by Farr bands to be detectable by LISA [39] and pulsar timing et al. [16]. arrays [40]. In particular, with such a high detectable event To reconcile the above mentioned problem, one of the rate, it is expected that LISA will detect many triple or mechanisms [13] considers a series merger of such binary higher multiple systems [41]. systems in a dense star cluster [17,18]. Assuming that such In this paper, we shall study the periodic gravitational a merger initially occurred with stellar mass (≃10 M⨀) waveforms, response functions, and energy losses of triple BHs, due to the presence of many massive stars/BHs in the systems in Einstein-aether theory [42]. This problem is dense cluster, the merger product could easily combine interesting particularly for the orbits in which two of the with a third massive companion to trigger another merger. bodies pass each other very closely and yet avoid their One of the interesting properties of this scenario is that such collisions, so they can produce periodic gravitational waves formed BHs usually have very high spins and can easily be with intension, which are the natural sources for the future identified with future aLIGO/aVirgo detections [19–22]. detections of GWs. Certainly, this problem is also very Recently, Rodriguez et al. [23] considered realistic models challenging, as even in Newtonian theory, the systems of globular clusters with fully post-Newtonian (PN) stellar allow chaotic and singular solutions, and only a few dynamics for three- and four-body encounters, and found periodic solutions are known [43,44].1 When one of the that nearly half of all binary BH mergers occur inside the 3-bodies is a test mass, it reduces to the restricted 3-body cluster and with about 10% of those mergers entering the problem, and a collinear solution was found by Euler [47]. aLIGO/aVirgo band with eccentricities greater than 0.1. In In 1772 Lagrange found a second class of periodic orbits particular, in-cluster mergers lead to the birth of a second for an equilateral triangle configuration [48] (a historic generation (2G) of BHs with larger masses and high spins. review of the subject can be found in [43]). These 2G BHs can reconcile the upper BH mass limit Gravitational waveforms of 3-body systems in general ð≲50 M⨀Þ created by the pair-instability supernovae [24]. relativity (GR) were calculated up to the 1PN approxima- Gravitationally bound hierarchies containing three or tions while the orbits of the 3-bodies are still Newtonian more components are very common in our Universe [25]. [49,50]. In GR, neither analytical nor numerical solutions Roughly speaking, about 13% of low-mass stellar systems contains three or more stars [26], and 96% of low-mass 1The three-body problem can be traced back to Newton in the binaries with periods shorter than 3 days are part of a larger 1680s. In the past 300 years, only three families of periodic hierarchy [27]. The simplest example is the 3-body system of solutions were found [43,44]. In 2013 a breakthrough was made, our Sun, Earth, and Moon. In fact, any star in the vicinity of a and 11 new families of the Newtonian planner 3-body problem with equal mass and totally zero-angular momentum were found supermassive BH binary naturally forms a triple system. numerically in [45]. In 2017, 695 families of such solutions (with Recently, a realistic triple system was observed, named equal mass and totally zero-angular momentum) were numeri- as PRS J0337 þ 1715 [28], which consists of an inner cally found in [46].
023010-2 GRAVITATIONAL WAVEFORMS, POLARIZATIONS, … PHYS. REV. D 99, 023010 (2019) of the 3-body problem of the full theory have been found, the ones presented in [73], subjected to some corrections of and most of the studies were restricted to PN approxima- typos. From the general expressions for the polarization tions; see, e.g., Refs. [25,51–55] and references therein. In modes hN given by Eq. (4.59), we can see that the scalar particular, the 1PN collinear solution was found in [56] and breather and the scalar longitudinal modes are always proved that it is unique in [57]. The 1PN triangular solution proportional to each other, so only five of the six polari- and stability were studied, respectively, in [58,59] and zation modes are independent. In addition, the two scalar −5 [60,61]. Lately, the existence and uniqueness of the 1PN modes are all suppressed by a factor of Oðc14Þ ≲ Oð10 Þ collinear solution in the scalar-tensor theory were studied with respect to the transverse-traceless modes (hþ and h×), in [62,63]. while the vectorial modes (hX and hY) are suppressed by a −15 In the framework of Einstein-aether theory, Foster [64] factor of Oðc13Þ ≲ Oð10 Þ. In Sec. V, we apply these and Yagi et al. [65] derived the metric and the equations of formulas to triple systems and obtain the GW forms, motion to the 1PN order for a N-body system. Recently, response functions, their Fourier transforms, and energy Will applied them to study the 3-body problem and losses for three representative cases. Our results show that obtained the accelerations of a 2-body system in the the GW forms sensitively depend on not only the configu- presence of the third body at the quasi-Newtonian order rations of the 3-body orbits but also their relative positions [66]. For nearly circular coplanar orbits, he also calculated to the detectors, sharply in contrast to the 2-body problem “ ” ηˆ the strong-field Nordtvedt parameter N. For the PRS [75,76]. Our paper ends with Sec. VI, in which we present J0337 þ 1715 system, ignoring the sensitivities of the two our main conclusions. white-dwarf companions, Will found that ηˆN is given by ηˆ ¼ s1=ð1 − s1Þ, where s1 denotes the sensitivity of the N II. EINSTEIN-AETHER THEORY pulsar. In this paper, we shall focus ourselves on periodic GWs In Einstein-aether (æ-) theory, the fundamental variables in the framework of Einstein-aether theory. The theory of the gravitational sector are [42] breaks locally the Lorentz symmetry by the presence of a μ globally timelike unit vector field—the aether—and allows ðgμν;u ; λÞ; ð2:1Þ three different types of gravitational modes—the scalar, μ ν ¼ 0 gμν vector and tensor [67]; all the modes in principle move at with the Greek indices , ,1,2,3,and is the four- different speeds [68]. Recently, it was found [69] that the dimensional metric of the spacetime with the signature ð−; þ; þ; þÞ uμ λ four independent coupling constants of the theory must [73,77], is the aether four-velocity, and is satisfy the constraints of Eq. (2.17) given below, after a Lagrangian multiplier, which guarantees that the aether several conditions are imposed. In the vacuum, GWs were four-velocity is always timelike. In this paper, we will adopt also studied in [67,70], while GW forms and angular the following conventions: all the repeated indices i; j; k; lði; j; k; l ¼ 1; 2; 3Þ will be summed over regardless momentum loss were studied for binary systems in [71,72], whether they are up or down, but repeated indices respectively. a; b; cða; b; c ¼ 1; 2; 3Þ The rest of the paper is organized as follows: in Sec. II will not be summed over unless we give a brief introduction to the Einstein-aether theory, the summation is explicitly indicated. In this paper, we c ¼ 1 and in Sec. III we first study the effects of the gravitational also adopt units so that the speed of light is one ( ). The general action of the theory is given by [68] radiations from the scalar and vector modes to the energy loss for a binary system to the lowest PN order [73], and S ¼ S þ S ; ð2:2Þ find that for a neutron star binary system their contributions æ m to the quadrupole, monopole and dipole are, respectively, where S denotes the action of matter and S is the −5 −5 −2 m æ the orders of Oð10 Þ, Oð10 Þ, and Oð10 Þ lower than gravitational action of the æ theory, given, respectively, by the quadrupole contributions of GR (in which the scalar and Z 1 pffiffiffiffiffiffi vector modes are absent) [cf. Eq. (3.13)], while the strong- 4 α S ¼ −gd x½RðgμνÞþL ðgμν;u ; λÞ ; Oð10−6Þ æ 16π æ field effects are, respectively, the orders of , Z Gæ Oð10−6Þ, and Oð10−7Þ lower. Additionally, the order for pffiffiffiffiffiffi ¼ − 4 ½L ð α; ψÞ ð Þ the cross term is of Oð10−6Þ lower [cf. Eq. (3.26)]. Similar Sm gd x m gμν;u : 2:3 conclusions can also be obtained by analyzing the ampli- tudes of polarization modes of a binary system with Here ψ collectively denotes the matter fields, R and g are, nonvanishing sensitivities given in [71]. Therefore, with respectively, the Ricci scalar and determinant of gμν, and the current (second) generation of GW detectors [74],we L ≡ − αβ ð μÞð νÞþλð α β þ 1Þ ð Þ can safely ignore these strong-field effects. Then, in Sec. IV æ M μν Dαu Dβu gαβu u ; 2:4 we consider the lowest PN approximations by taking the coupling of the aether with matter fields into account. where Dμ denotes the covariant derivative with respect to αβ When such couplings are turned off, our formulas reduce to gμν and M μν is defined as
023010-3 KAI LIN et al. PHYS. REV. D 99, 023010 (2019)
αβ αβ α β α β α β ð2 − Þ M μν ≡ c1g gμν þ c2δμδν þ c3δν δμ − c4u u gμν: ð2:5Þ 2 ¼ c123 c14 cS ; c14ð1 − c13Þð2 þ c13 þ 3c2Þ
Note that here we assume that matter fields couple not only 2c1 − c13ð2c1 − c13Þ μ 2 ¼ to gμν but also to the aether field u , in order to model cV ; 2c14ð1 − c13Þ effectively the radiation of a compact object, such as a 1 ’ 2 ¼ ð Þ neutron star [78]. The four coupling constants ci s are all cT ; 2:13 1 − c13 dimensionless, and Gæ is related to the Newtonian constant GN via the relation [79] where cijk ≡ ci þ cj þ ck. In addition, among the ten parametrized post-Newtonian ¼ Gæ ð Þ GN 1 ; 2:6 parameters [80], in the Einstein-aether theory the only two 1 − 2 c14 parameters that deviate from GR are α1 and α2, which measure the preferred frame effects. In terms of the four ≡ þ where cij ci cj. ’ μ dimensionless coupling constants ci s, they are given by [81] The variations of the total action with respect to gμν, u , and λ yield, respectively, the field equations 8ðc1c14 − c−c13Þ α1 ¼ − ; 2c1 − c−c13 1 μν − μν − μν ¼ 8π μν ð Þ 1 ð − 2 Þð3 þ þ Þ R g R S GæT ; 2:7 c14 c13 c2 c13 c14 2 α2 ¼ α1 þ ; ð2:14Þ 2 c123ð2 − c14Þ ¼ 8π ð Þ Æμ GæTμ; 2:8 where c− ≡ c1 − c3. In the weak-field regime, using lunar laser ranging and solar alignment with the ecliptic, Solar α β gαβu u ¼ −1; ð2:9Þ System observations constrain these parameters to very small values [80], where −4 −7 jα1j ≤ 10 ; jα2j ≤ 10 : ð2:15Þ μ μ μ Sαβ ≡ Dμ½J ðαuβÞ þ JðαβÞu − uðβJαÞ Recently, the combination of the gravitational wave μ μ þ c1½ðDαuμÞðDβu Þ − ðDμuαÞðD uβÞ event GW170817 [6], observed by the LIGO/Virgo 1 Collaboration, and the event of the gamma-ray burst þ þ λ − δ σ c4aαaβ uαuβ 2 gαβJ σDδu ; GRB 170817A [9] provides a remarkably stringent con- α α straint on the speed of the spin-2 mode, Æμ ≡ DαJ μ þ c4aαDμu þ λuμ; pffiffiffiffiffiffi 2 δð −gL Þ −3 × 10−15 023010-4 GRAVITATIONAL WAVEFORMS, POLARIZATIONS, … PHYS. REV. D 99, 023010 (2019) −5 −9 jαˆ 1j ≤ 10 ; jαˆ 2j ≤ 10 ðat 95% confidenceÞ; Here ρ ¼ T00 to the lowest order, and ti denotes the quadratic terms given in Eq. (4.15). ð2:19Þ It should be noted that the scalar (monopole) perturba- tions have contributions to all three parts, quadrupole (Q⃛ ), obtained from the isolated millisecond pulsars PSR ij dipole (Σ_ ), and monopole (I⃛). The vector (dipole) pertur- B1937þ21 [84] and PSR J17441134 [85], where (αˆ 1, αˆ 2) i bations have contributions to both quadrupole and dipole denotes the strong-field generalization of (α1, α2) [86], σ terms, while the tensor perturbations have only contribu- because they depend on the sensitivity æ, which is not known for the new constraints of Eq. (2.18). In fact, in the tions to the quadrupoleR term. ThisR can be seen clearlyR from Ωϕ_ ϕ_ Ων_ ν_ Ω _ _ Einstein-æther theory, they are given by [65] the expressions for d ij ij, d i i, and d F F, given by Eqs. (102)–(104) in [73], where F ≡ Δf and ϕij, ð8 þ α Þσ c− 1 æ νi, and f are all defined in Eq. (4.3) below. αˆ 1 ¼ α1 þ ; 2c1 In the case of GR (ci ¼ 0), we have αˆ − α ð − 2Þðα − 2α Þσ 1 1 c14 1 2 æ αˆ 2 ¼ α2 þ − : ð2:20Þ AGR ¼ 1; BGR ¼ CGR ¼ 0: ð3:4Þ 2 2ðc14 − 2c13Þ To see clearly the effects of the aether field, in the rest of For details, we refer interested readers to [69]. this section let us restrict ourselves only to a binary system, for which Eq. (3.1) takes the form III. EFFECTS OF THE AETHER FIELD ON ENERGY LOSS RATE μ 2 8 E_ ¼− GN m Að12 2 −11_2Þþ4B_2 þCΣ2 GN 2 v r r ; Before proceeding further, let us pause here for a while to r 15 consider the effects of the aether field on energy loss of a ð3:5Þ given system, in order to have a better understanding of the approximations to be taken in this paper. Although in this where m ≡ m1 þ m2, μ ≡ m1m2=m, r ≡ r1 − r2, v ≡ r_, section we shall restrict ourselves only to binary systems, r ≡ jrj, and we believe that such estimations are also valid for other systems, as long as they are weak enough, so the PN 2 Ω1 Ω2 approximations are applicable. In particular, our studies of Σ ≡ α1 − α2 − ; ð3:6Þ 3 m1 m2 triple systems are consistent with such estimations as to be shown below. with Ωa denoting the binding energy of the ath compact To the lowest PN order (by setting the sensitivity body, and σ ¼ 0), Foster found that the energy loss rate in æ Einstein-Æther gravity is given by [73] (see also [65] for 10−3 Ωa GNma ; white dwarfs; the correction of a typo in the expression of Z given below) ≃ O ≃ ð3:7Þ m d 0.1 ∼ 0.3; pulsars; a a A E_ ¼ − ⃛ ⃛ þ B ⃛⃛þCΣ_ Σ_ ð Þ where d denotes the size of the ath body. GN 5 Qij Qij I I i i ; 3:1 a For double pulsars, we have v2 ≃ 10−6 − 10−5 [87].In addition, without loss of the generality, we assume that v2 where and r_2 are of the same order, Oðv2Þ ≃ Oðr_2Þ. Then, from Z −15 1 the constraint jc13j < 10 [cf. (2.17)], we find that ≡ − δ ≡ 3 ρ Qij Iij 3 I ij;Iij d x xixj; 1 2 1 2 Z c1 = c2 = 3 cT ≃ 1;cV ≃ O ;cS ≃ O : ð3:8Þ I ≡ Iii; Σi ≡ d xti; ð3:2Þ c14 c14 Hence, we obtain and A ¼ 1 þ Oð Þ½1 þ Oð −5Þ 1 2 2 3ð − 1Þ2 c14 cS ; A ¼ 1 − c14 þ c14c13 þ Z c14 2 2 2ð2 − Þ ; B ¼ Oð ÞOð −1Þ cT ð2c1 − c13c−Þ cV c14 cS c14 cS ; 2 −1 −3 −3 Z c14 ðα1 − 2α2Þð1 − c13Þ C ¼ Oðc14 Þ½Oðc ÞþOðc Þ ; ð3:9Þ B ¼ ;Z≡ ; V S 8c 3ð2c13 − c14Þ S where cI ≳ 1ðI ¼ S; V; TÞ because of the vacuum gravi- 2 2 − c14 1 C ¼ þ ð Þ Čerenkov effects [82]. Then, comparing Eqs. (3.4) and 3 3 3 : 3:3 c14 cV cS (3.9), we can see that the contributions of the aether field to 023010-5 KAI LIN et al. PHYS. REV. D 99, 023010 (2019) ˜ ½γ both of the quadrupole and monopole radiations are at most σ ≡ −d ln mA A Oð Þ ≲ 10−5 A ; of the order of c14 . To see its contributions to d ln γA γ ¼1 A the dipole radiation, we first note that 2 ˜ ½γ σ¯ ≡ d ln mA A ð Þ A γ 2 ; 3:15 2 d ln A γ ¼1 α − α ≲ Oð Þ A 1 3 2 c14 ; GNm which can be determined by considering asymptotic Σ ≲ Oðc14ÞO : ð3:10Þ d properties of perturbations of static stellar configurations. Indeed, σA was calculated for neutron stars in [65]. But, Thus, we find that unfortunately, the calculations were done by setting α1 ¼ α2 ¼ 0, which are no longer valid when the new 2 2 constraint (2.17) is taken into account. However, to our 2 GNm WC ≡ CΣ ≲ Oðc14ÞO current purpose, the exact values of σ are not important, d A and we can simply use the expression of the small σA limit 2 −5 GNm [64,65], ≲ 10 O : ð3:11Þ d 2 Ω 2 ¼ α − α A þ O GNm ð Þ On the other hand, the quadrupole and monopole radiations sA 1 3 2 ; 3:16 are given, respectively, by mA d 8 2 2 2 where s ≡ σ =ð1 þ σ Þ. W ≡ Að12v − 11r_ Þ ≲ ½1 þ Oðc Þ Oðv Þ; A A A A 15 14 After taking the strong-field effects into account, 2 2 WB ≡ 4Br_ ≲ Oðc14ÞOðv Þ: ð3:12Þ Eq. (3.5) has four different kinds of corrections, one for each of the three terms presented in Eq. (3.5) plus a crossing term. It is given explicitly by Eq. (116) in [65].3 In particular, for a binary system of neutron stars, taking In the following, let us consider these corrections term by Oðv2Þ ≃ 10−5 OðGN mÞ ≃ 10−1 and d , we find that term. First, to the first term of Eq. (3.5), the coefficient A is 2 replaced by ðA þ SA2 þ S A3Þ, where NS −5 −5 WA ≃ ½1 þ Oð10 Þ Oð10 Þ; WNS ≲ Oð10−10Þ; WNS ≲ Oð10−7Þ: ð3:13Þ B C S ≡ s1μ2 þ s2μ1; ð3:17Þ with μ ¼ m =m ≃ Oð1Þ. Thus, we have A. Strong-field effects A A Strong-field effects can be important in the vicinity of the compact bodies, such as neutron stars, as the fields inside GNm S ≃ Oðs Þ ≲ Oðc14ÞO : ð3:18Þ such bodies can be very strong. Following Eardley [78], A d these effects can be included by considering the action of one particle [64], In addition, we also have Z ¼ − τ ˜ ½γ 2 SA d AmA A Note that from Eq. (2.14) we can write c2 and c4 in terms of Z α1 and α2. Then, with the new constraint (2.17), to have a self- ð Þ ≡ ¼ − ˜ τ 1 þ σ ð1 − γ Þ consistent expansion of any given function F c1;c2;c3;c4 mA d A A A Fðc1; α1; α2;c13Þ in terms of the small quantities α1, α2, and c13, one must expand Fðc1; α1; α2;c13Þ at least to the third order of 2 1 α1, the second order of α2 (plus their mixed terms, such as α α2), þ ðσ þ σ2 þ σ¯ Þð1 − γ Þ2 þ : ð3:14Þ 1 2 A A A A and to the first order of c13, considering the fact that the constraints (2.15) and (2.17) are in different orders. If one naively sets α1, α2, c13 all to zero at the same time, then it will μ A A −15 Here γA ¼ −u vμ , where vμ is the four-velocity of the end up with c2 ¼ 0, c3 ¼ c4 ¼ −c1, and jc1j ≤ 10 , which are body, A labels the body, and τA is the proper time along the not only too strict, but also inconsistent, as this is equivalent to body’s curve. When the body is at rest with respect to assume that these three quantities were constrained all to the same order. the aether, we have γ j A¼ ¼ 1 and m˜ ¼ m˜ ½γ jγ ¼1. 3 A vμ uμ A A A A In Ref. [64], it is given by Eq. (89). However, some typos The sensitivities σA and σ¯ A are defined as appeared in this equation and were corrected in [65]. 023010-6 GRAVITATIONAL WAVEFORMS, POLARIZATIONS, … PHYS. REV. D 99, 023010 (2019) Z ≃ Oð1Þ; for other systems, they require V2 ≲ Oð100Þv2. In any case, without loss of the generality, we assume that 2ðZ − 1Þ 2c13 A ≡ þ Oð i iÞ2 ≃ Oð 2Þ 2 ð − 2Þ 3 ð2 − Þ 3 V n V . In addition, c14 cS c1 c−c13 cV ≃ Oð −5Þ 1 cS ; C ≡ ≲ Oð −1ÞOð −5Þ ð Þ 2 5 c14 cV : 3:23 1 2 6c14c A ≡ þ V 3 5 5 2c14c 3c14ð2 − c14Þc V S Thus, we find that ≃ Oðc−1Þ½Oðc−5ÞþOðc−5Þ : ð : Þ 14 V S 3 19 18 δW ≡ ð − Þ2 A þ 2C 2 Thus, we find that the correction to the quadrupole term is C s1 s2 5 3 2 V given by 6 þ A þ 36B − 2C ð i iÞ2 2 2 2 3 3 2 V n δWA ≡ ðSA2 þ S A3Þð12v − 11r_ Þ 5 2 −3 −3 ≲ Oðc14Þ½Oðc ÞþOðc Þ ≃ Oð Þ½Oð −5ÞþOð −5Þ O GNm Oð 2Þ S V c14 cV cS v d G m 2 × O N OðV2Þ −5 GNm 2 d þ Oðc14ÞOðc ÞO Oðv Þ S 2 d GNm 2 ≲ Oðc14ÞO OðV Þ GNm 2 d ≲ Oðc14ÞO Oðv Þ d G m 2 ≲ 10−5O N OðV2Þ: ð3:24Þ G m d ≲ 10−5O N Oðv2Þ: ð3:20Þ d The fourth kind of corrections is involved with the Second, the correction to the coefficient B is crossing terms, ðVinivjnjÞ and ðviViÞ. Again, without loss 2 i i j j i i ðSB2 þ S B3Þ, where of the generality, we assume that OðV n v n Þ ≃ OðV v Þ. Then, the fourth kind of corrections is given by Z −3 B2 ≡ ≃ Oðc Þ; 3ð − 2Þ 3 S c14 cS i i j j δWD ≡ ðs1 − s2Þ 12ðB2 þ 2SB3ÞðV n v n Þ 1 −1 −5 B3 ≡ ≃ Oðc ÞOðc Þ: ð3:21Þ 9 ð2 − Þ 5 14 S 8 c14 c14 cS þ ðA þ 2SA Þ ið3 i − 2 i j jÞ 5 2 3 V v n v n Therefore, we have GNm i i −3 ≲ Oðc14ÞO Oðv V Þ Oðc Þ 2 2 S δWB ≡ 4ðSB2 þ S B3Þr_ d GNm −5 −5 −3 GNm þ Oðc14ÞO ½Oðc ÞþOðc Þ ≲ Oðc14ÞOðc ÞO S V S d d 2 GNm i i −5 GNm 2 ≲ Oðc14ÞO Oðv V Þ þ Oðc14ÞOðc ÞO Oðv Þ S d d −5 GNm GNm 2 ≲ 10 O OðviViÞ: ð3:25Þ ≲ Oðc14ÞO Oðv Þ d d G m i i ≲ 10−5O N Oðv2Þ; ð3:22Þ For the binary system of neutron stars, taking Oðv V Þ ≃ d 2 2 −5 −1 Oðv Þ ≃ OðV Þ ≃ Oð10 Þ and OðGNm=dÞ ≃ 10 ,we find that which has the same order as that of δWA, as shown by Eq. (3.20). δWNS ≲ Oð10−11Þ; δWNS ≲ Oð10−11Þ; For the third kind of corrections, they are involved with A B i i i i NS −12 NS −11 the velocity V of the center of mass and V n , where n is δWC ≲ Oð10 Þ; δWD ≲ Oð10 Þ; ð3:26Þ the unity vector, defined by ni ≡ ri=r, where r is the module of the vector ri. As pointed out by Foster in [64], Vi which are all smaller than the quadrupole, monopole, and NS NS NS is not directly measurable, but the validity of leading PN dipole terms WA , WB , and WC . With the current order for the double pulsar [88] requires V2 ≲ 3v2, while accuracy of GW observations, it is hard to detect these 023010-7 KAI LIN et al. PHYS. REV. D 99, 023010 (2019) effects [74,76]. This justifies our current studies of triple vectors, γi, νi, and ϕi; and one transverse-traceless tensor, systems only up to the lowest PN order. ϕij. It should be noted that similar conclusions can also be Note that, to the zeroth order, Tμν and Tμ all vanish obtained by analyzing the polarization modes of a binary identically, while to the first order, we can decompose system with nonvanishing sensitivities given in [71]. them as ¼ Ψ þ Ψ ¼ Θ þ Θ IV. GRAVITATIONAL RADIATION IN T0i ;i i;Ti ;i i; EINSTEIN-AETHER THEORY 1 Tij ¼ Π;ij þ Pij½ϒ þ2Πði;jÞ þ Πij; ð4:5Þ As shown in the last section, the strong-field effects are 2 very small with the current constraints of Eq. (2.17) [69] and can safely be ignored for the current generation of where detectors. So, in the rest of this paper, we shall consider i i i gravitational radiation in Einstein-aether theory to the ∂ Ψi ¼ ∂ Θi ¼ ∂ Πi ¼ 0; lowest PN order, similar to what was already done in ∂jΠ ¼ 0 Πi ¼ 0 ð Þ ij ; i : 4:6 [73], but without setting Tμ ¼ 0 for our future studies. Recall that Tμ originates from the direct coupling between matter and aether. Therefore, in the following we shall try Therefore, in the matter sector, in general we have six scalars, T00, T0, Ψ, Θ, Π, and Υ; three vectors, Ψi, Θi, and to provide some detailed derivations of the formulas with Π Π the risk of repeating some materials already presented i; and one tensor, ij. previously in [73], although we shall try to limit these to Under the following coordinate transformations: their minimum.4 0 As mentioned in the previous section, the Minkowski t˜ ¼ t þ ξ ; x˜i ¼ xi þ ξi þ ∂iξ; ð4:7Þ spacetime is a vacuum solution of the Einstein-aether i theory with the aether being along the time direction, i.e., where ∂iξ ¼ 0, we find α ¯ ¯ ¯ α ðg¯μν; u¯ ; λ; Tμν; TμÞ¼ðημν; δ0; 0; 0; 0Þ; ð4:1Þ ˜ _0 0 0 _0 h00 ¼ h00 þ 2ξ ; w˜ ¼ w þ ξ ; 0 _ _ where ημν ≡ diagð−1; 1; 1; 1Þ represents the Minkowski γ˜ ¼ γ þ ξ − ξ; ν˜ ¼ ν þ ξ; spacetime metric in the Cartesian coordinates xμ ¼ðt;xiÞ. f˜ ¼ f; ϕ˜ ¼ ϕ − 2ξ; ð4:8Þ Then, let us consider the linear perturbations, ¼ − η 0 ¼ 0 − 1 i ¼ i ð Þ _ _ hμν gμν μν;wu ;wu ; 4:2 γ˜i ¼ γi − ξi; ν˜i ¼ νi þ ξi; ϕ˜ ¼ ϕ − ξ ð Þ 0 i i i i; 4:9 where the perturbations hμν, w , and w are decomposed to the forms [73] ˜ ϕij ¼ ϕij: ð4:10Þ h0i ¼ γi þ γ;i;wi ¼ νi þ ν;i; 1 Thus, out of the six scalar fields, we can construct four ¼ ϕ þ ½ þ2ϕ þ ϕ ð Þ hij ;ij 2 Pij f ði;jÞ ij; 4:3 gauge-invariant quantities, while out of the three vector fields, two gauge-invariant quantities can be constructed, with i, j ¼ 1,2,3,ða; bÞ ≡ ðab þ baÞ=2, and which can be chosen, respectively, by5 ∂iγ ¼ ∂iν ¼ ∂iϕ ¼ 0 ∂jϕ ¼ 0 ϕ i ¼ 0 1 i i i ; ij ; i ; ΦI ≡ − 2 0 ΦII ≡ Δ h00 w ; 2 f; P ½f ≡ δ Δf − f ; Δ ≡ δij∂ ∂ : ð4:4Þ ij ij ;ij i j 1 1 ΦIII ≡ ν þ ϕ_ − f;_ All spatial indices are raised or lowered by δij and δ , 2 4 ij 1 1 ∂i ≡ δij∂ 0 ̈ ̈ respectively. For example, j, and so on. Therefore, ΦIV ≡ γ_ − w − ϕ þ f; ð4:11Þ 0 2 4 we have six scalars, h00, w , γ, ν, f, and ϕ; three transverse and 4This also allows us to correct some typos. Reference [73] has been published in [89], but in Ref. [73] some corrections of typos were made and also with the same signature as used in the current 5Since they are all gauge invariant, any function of them is also paper, while in [89] the signature ðþ1; −1; −1; −1Þ was used. gauge invariant. 023010-8 GRAVITATIONAL WAVEFORMS, POLARIZATIONS, … PHYS. REV. D 99, 023010 (2019) ΨI ≡ γ þ ν ΨII ≡ γ − ϕ_ ð Þ i i i; i i i: 4:12 ¼ 8π ð þ Þ ð Þ Æμ Gæ Tμ tμ ; 4:15 Clearly, the tensor mode ϕij is already gauge invariant. 0 On the other hand, since Tμν ¼ OðϵÞ and Tμ ¼ OðϵÞ, h00 ¼ 2w ; ð4:16Þ where ϵ ¼ OðξμÞ ≪ 1, we find that where, to simplify the notations and without causing any ∂ α ∂ β ˜ x x 2 confusions, except tμν and tμ, we use the same notations of Tμνðx˜Þ¼TαβðxÞ μ ν ¼ TμνðxÞþOðϵ Þ; ∂x˜ ∂x˜ Eqs. (2.7)–(2.9) to denote the linearized ones, as they all ∂ α ˜ x 2 vanish for the Minkowski background. The quantities tμν Tμðx˜Þ¼TαðxÞ μ ¼ TμðxÞþOðϵ Þ; ð4:13Þ 6 ∂x˜ and tμ represent the nonlinear source terms [73]. In particular, to the linear order, we have that is, to the first order of ϵ, all the quantities of the matter sector remain the same and are gauge invariant. λ ¼ ∂ α − 8π With the above analysis in mind, to the leading order, we αJ 0 GæT0; – ¼ ∂ α − ð∂ α − 8π Þδ0 ð Þ find that Eqs. (2.7) (2.9) reduce to Æμ αJ μ αJ 0 GæT0 μ; 4:17 − ¼ 8π ð þ Þ ð Þ Gμν Sμν Gæ Tμν tμν ; 4:14 and 1 ¼ − Δ2 G00 2 f; 1 1 ¼ − Δðγ − ϕ_ Þ − Δ_ G0i 2 i i 2 f;i; 1 1 1 ¼ − ðΔϕ − ϕ̈ Þþðϕ̈ − γ_ Þ − ̈ þ ½Δ − 4 0 þ 4_γ − ̈− 2ϕ̈ Gij 2 ij ij ði;jÞ ði;jÞ 2 f;ij 4 Pij f w f ; 0 S00 ¼ c14Δðν_ þ γ_ − w Þ − Æ0; ¼ ðν̈ þ γ̈Þ − c− Δðν þ γ Þþ ∂ ðν̈þ ̈γ − _ 0Þ S0i c14 i i 2 i i c14 i w 1 ¼ Δ½ ðν þ ϕ_ Þþ ð2ν þ ϕ_ Þ þ _ − 2 c13 i i c123 ;i c2f;i Æi; 1 1 ¼ c13 ϕ̈ þ ðν_ þ ϕ̈ Þþ ð2ν_ þ ϕ̈þ ̈Þþc13 ̈ þ ½ ðϕ̈þ 2ν_Þþ ̈ ð Þ Sij 2 ij c13 ði;jÞ ði;jÞ 2 Pij c2 f 2 f 2 c123 c2f ;ij: 4:18 Setting which leads to the following conservation laws: 0 0 ν τμν ≡ Tμν − Tμδν þðtμν − tμδνÞ; ð4:19Þ ∂ τμν ¼ ∂iτμi − τ_μ0 ¼ 0: ð4:22Þ we find that Eq. (4.14) can be cast in the form When Tμ ¼ 0, it can be shown that the above equations reduce to the ones given in [73].7 − − δ0 ¼ 8π τ ð Þ Gμν Sμν Æμ ν Gæ μν: 4:20 Similar to the linearized terms Tμν and Tμ, we can also decompose the nonlinear terms tμν and tμ in the forms of It should be noted that τμν defined by Eq. (4.19) in general Eqs. (4.5) and (4.6), is not symmetric. In particular, we have τ0i ≠ τi0. Then, it can be shown that the left-hand side of the above equation t0i ¼ ψ ;i þ ψ i;ti ¼ θ;i þ θi; satisfies 1 ¼ π þ ½υ þ2π þ π ð Þ tij ;ij 2 Pij ði;jÞ ij; 4:23 ν 0 ∂ ðGμν − Sμν − ÆμδνÞ¼0; ð4:21Þ where 6 Note that here we use tμ to denote the nonlinear source term of 7 ν0 0 the aether field, instead of σμ [73], as we shall reserve the latter A term ∂νJ δμ in the left-hand side of Eq. (23) in [73] is for other uses. missing. 023010-9 KAI LIN et al. PHYS. REV. D 99, 023010 (2019) i i i □ ≡ Δ − −2∂2 ∂ ψ i ¼ ∂ θi ¼ ∂ πi ¼ 0; where I cI t and j i ∂ πij ¼ 0; π i ¼ 0: ð4:24Þ τ0 ≡ T00 þ t00; Γ0 ≡ T0 þ t0; Then, we find that τμν has the following nonvanishing Γ ≡ Θ þ θ; Γi ≡ Θi þ θi; ð4:31Þ components: with cI’s being given by Eq. (2.13). τ00 ¼ T00 − T0 þðt00 − t0Þ; The second group is given by τ0i ¼ T0i þ t0i ¼ Ψ þ Ψ þðψ þ ψ Þ Δ½ ðν þ ϕ_ Þþγ − ϕ_ ¼−16π ðτ − Γ Þ ð Þ ;i i ;i i c13 i i i i Gæ i i ; 4:32 ≡ τ;i þ τi; Δ½ − 2 0 þ 2 ðγ_ þ ν_Þ ¼ −16π ðτ − Γ Þ ð Þ τi0 ¼ T0i − Ti þðt0i − tiÞ F c14w c14 Gæ 0 0 ; 4:33 ¼ Ψ − Θ þ Ψ − Θ þ½ðψ − θÞ þðψ − θ Þ ;i ;i i i ;i i i Δ½ð1 þ Þ_ þ ðϕ_ þ 2νÞ ¼ −16π ðτ − ΓÞ ð Þ ≡ χ;i þ χi; c2 f c123 ;i Gæ ;i; 4:34 τij ¼ Tij þ tij 1 Δ½ð1 þ Þ̈þ ðϕ̈þ 2ν_Þ ¼ −16π ΔϜ ð Þ ¼ Ϝ þ ½ϱ þ2Ϝ þ Ϝ ð Þ c2 f c123 Gæ ; 4:35 ;ij 2 Pij ði;jÞ ij; 4:25 which all take the form where Δψ ¼ −4πτ ð Þ τ ≡ Ψ þ ψ; τi ≡ Ψi þ ψ i; : 4:36 χ ≡ τ − Θ − θ; χi ≡ τi − Θi − θi; When far away from the source, the above Poisson Ϝ ≡ Π þ π ϱ ≡ ϒ þ υ ; ; equations have solutions of the form Ϝ ≡ Π þ π Ϝ ≡ Π þ π ð Þ i i i; ij ij ij: 4:26 Z 1 1 ψð ⃗Þ¼ 3 0τð 0ÞþO ð Þ Clearly, such defined three vectors τ , χ , and Ϝ are t; x d x t; x 2 ; 4:37 i i i R V0 R transverse, and the tensor Ϝij is traceless and transverse, i.e., where R ≡ jx⃗j ≫ d, with d denoting the size of the source. As argued in [73], the contributions of this part to the ∂iτ ¼ ∂iχ ¼ ∂iϜ ¼ 0; i i i waveforms are negligible, and without loss of the general- i i ∂ Ϝij ¼ 0; Ϝ i ¼ 0: ð4:27Þ ity, we can safely set it to zero, With the above decompositions, it can be shown that the ψðt; x⃗Þ ≃ 0 ðR ≫ dÞ; ð4:38Þ field equations can be divided into two groups; one represents the propagation equations for the scalar, vector, and tensor modes, and the other represents the type of in the wave zone. Poisson equations. In terms of the gauge-invariant quan- tities defined in Eqs. (4.11) and (4.12), the first group is A. Polarizations of gravitational waves given by in Einstein-aether theory 16π 1 1 þ To consider the polarizations of gravitational waves in II Gæc14 c2 □SΦ ¼ − Δϱ − ΔϜ Einstein-aether theory, let us consider the timelike geodesic 2 − c14 2 c123 deviations. In the spacetime described by the metric, 1 gμν ¼ ημν þ hμν, the spatial part, ζi, takes the form [70] þ ðτ0 − Γ0Þ ; ð4:28Þ c14 1 ζ̈ ¼ − ζj ≡ P̈ ζj ð Þ 16πG i R0i0j ij ; 4:39 □ ΨI ¼ − æ ð τ − Γ Þ ð Þ 2 V i c13 i i ; 4:29 2c1 − c−c13 where ζμ describes the deviation vector between two nearby □ ϕ ¼ −16π Ϝ ð Þ T ij Gæ ij; 4:30 trajectories of test particles and 023010-10 GRAVITATIONAL WAVEFORMS, POLARIZATIONS, … PHYS. REV. D 99, 023010 (2019) 1 2 − c14 ≃ ð þ − − Þ Φ_ III ¼ − ΦII ð Þ R0i0j 2 h0j;0i h0i;0j hij;00 h00;ij 2 ð1 − Þ : 4:48 c14 c13 1 ¼ 2_γ þ 2_γ − 2 0 − ϕ̈ 2 ði;jÞ ;ij w;ij ij Note that in writing the above expressions, we had used □ ΦII ¼ 0 1 1 S in the wave zone. The combination of − 2ϕ̈ − ϕ̈ − δ Δ̈þ ̈ Eqs. (4.46) and (4.48) yields ði;jÞ ;ij 2 ij f 2 f;ij 1 1 c14 − 2c13 c14 − 2c13 ¼ − ϕ̈ þ Ψ_ II þ ΦIV − δ Φ̈II ð Þ ΦIV ¼ Φ_ III ¼ ΦII ð Þ ij ði;jÞ ij : 4:40 : 4:49 2 ;ij 2 2 − c14 2c14ðc13 − 1Þ When deriving the last expression of the above equation, In the wave zone, we also have [73] we had used Eqs. (4.11) and (4.12). ≫ – 1 In the wave zone (R d), Eqs. (4.38) and (4.32) (4.35) ΨI ¼ − Ψ_ I i;j iNj; imply that cV 1 _ _ ΦII ¼ − Φ_ IIN ; ð4:50Þ c13ðνi þ ϕiÞþγi − ϕi ¼ 0; ð4:41Þ ;i i cS − 2 0 þ 2 ðγ_ þ ν_Þ¼0 ð Þ F c14w c14 ; 4:42 where Nk denotes the unit vector along the direction between the source and the observer. Then, inserting the _ _ ð1 þ c2Þf þ c123ðϕ þ 2νÞ¼0: ð4:43Þ above expressions into Eq. (4.39) we obtain 2c13 From Eq. (4.41), we find that P ¼ ϕ − ΨI ij ij ðiNjÞ ð1 − c13ÞcV γ þ ν ϕ_ ¼ i c13 i ð Þ c14 − 2c13 i : 4:44 − ΦIIN N þ δ ΦII: ð4:51Þ 1 − c13 ð − 1Þ 2 i j ij c14 c13 cS Inserting it into Eq. (4.12), we obtain Assuming that (eX, eY, eZ) are three unity vectors and form an orthogonal basis with e ≡ N, so that (e , e ) lay c Z X Y ΨII ¼ − 13 ΨI ð Þ on the plane orthogonal to the propagation direction N of i 1 − i: 4:45 c13 the gravitational wave, we find that, in the coordinates xμ ¼ðt; xiÞ, these three vectors can be specified by two On the other hand, the combination of Eqs. (4.11) angles, ϑ and φ, via the relations [76] and (4.42) yields e ¼ð ϑ φ ϑ φ − ϑÞ II IV _ III X cos cos ; cos sin ; sin ; Φ ¼ −c14ðΦ þ Φ Þ; ð4:46Þ eY ¼ð− sin φ; cos φ; 0Þ; while from Eq. (4.43) we obtain eZ ¼ðsin ϑ cos φ; sin ϑ sin φ; cos ϑÞ: ð4:52Þ c123 2 ΦIII þ 1 þ þ _ ¼ 0 ð Þ Then, we can define the six polarization h ’sby c123 c2 2 f : 4:47 N 1 1 ≡ ðP − P Þ ≡ ðP þ P Þ Then, combining it with Eq. (4.11) we have hþ 2 XX YY ;h× 2 XY YX ; 1 h ≡ ðP þ P Þ;h≡ P ; 2 ΔΦ_ III ¼ − 1 þ þ c123 Δ̈ b 2 XX YY L ZZ c123 c2 2 f 1 1 hX ≡ ðPXZ þ PZXÞ;hY ≡ ðPYZ þ PZYÞ; ð4:53Þ c123 2 2 ¼ −2 1 þ þ Φ̈II c2 2 i j where PXY ≡ Pije e , and so on. However, in Einstein- c123 X Y ¼ −2 1 þ þ 2ΔΦII c2 2 cS aether theory, only five of them are independent: two from each of the vector and tensor modes, and one from the c123ð2 − c14Þ scalar mode. ¼ − ΔΦII; c14ð1 − c13Þ In order to calculate the waveforms, let us first assume that the detector is located at x with R ≡ jxj ≫ d, where d that is, denotes the size of the source. Then, under the gauge, 023010-11 KAI LIN et al. PHYS. REV. D 99, 023010 (2019) ϕi ¼ 0; ν ¼ γ ¼ 0; ð4:54Þ B. Response function To study the GW forms, an important quantity is the we find response function hðtÞ, with respect to a specific detector, which, for the sake of simplicity, is assumed to have two 2G orthogonal arms, such as aLIGO, aVirgo, or KAGRA. ϕ ¼ æ ðQ̈ ÞTT; ð4:55Þ ij R ij Assume that the two arms of a detector are along, respectively, the e1 and e2 directions. Then, from them e 2 ̈ T we can construct another unity (spacelike) vector 3 that Gæ c13QijNj ðe e νi ¼ − − 2Σi ; forms an orthogonal basis together with 1; 2), that is, ð2c1 − c13c−ÞR ð1 − c13Þc V ei · ej ¼ δij. The choice of this frame is independent of the e e e γi ¼ −c13νi; ð4:56Þ one ( X, Y, Z) that we just introduced in the last subsection. But, we can always rotate properly from one frame to get the other with three independent angles, θ; ϕ, Gæ c14 ̈ ψ – F ¼ 6ðZ − 1ÞQijNiNj and [cf. Fig. 11.5 and Eqs. (11.319a) (11.319c) of [76]], R 2 − c14 given by 8 ̈ − ΣiNi þ 2ZI ; c14cS e1 ¼ðcos θ cos ϕ cos ψ − sin ϕ sin ψÞeX 1 1 þ 0 c2 þð θ ϕ ψ þ ϕ ψÞe h00 ¼ 2ω ¼ F; ϕ ¼ − f; ð4:57Þ cos cos sin sin cos Y c14 c123 − sin θ cos ϕeZ; 1 e2 ¼ðcos θ sin ϕ cos ψ þ cos ϕ sin ψÞeX ΨI ¼ð1 − Þν ΦII ¼ ð Þ i c13 i; 2 F; 4:58 þðcos θ sin ϕ sin ψ − cos ϕ cos ψÞeY − θ ϕe TT ¼ sin sin Z; where, for any given symmetric tensor Sij,wehaveSij Λ T ¼ Λ e3 ¼ − sin θ cos ψe − sin θ sin ψe − cos θe : ð4:60Þ ij;klSkl and Si PijSj, where ij;kl and Pij are the X Y Z projection operators defined, respectively, by Eqs. (1.35) and (1.39) in [75]. Then, the response function hðtÞ is defined as Inserting Eq. (4.51) into Eq. (4.53) and using the above equations, we find that 1 ð Þ ≡ ð i j − i j ÞPij h t 2 e1e1 e2e2 G G ¼ æ ̈ kl ¼ æ ̈ kl ¼ þ þ þ hþ Qkleþ;h× Qkle× ; Fþhþ F×h× Fbhb FLhL R R þ F h þ F h ; ð4:61Þ c14G 4 X X Y Y ¼ æ 3ð − 1Þ ̈ i j − Σ i þ ̈ hb Z QijeZeZ ieZ ZI ; Rð2 − c14Þ c14c S − 2 where ¼ 1 − c14 c13 hL 2 hb; c14ðc13 − 1Þc 1 S 2 ̈ k Fþ ≡ ð1 þ cos θÞ cos 2ϕ cos 2ψ − cos θ sin 2ϕ sin 2ψ; 2 c13Q e 2 ¼ c13Gæ jk Z − 2Σ j hX ð2 − Þ ð1 − Þ j eX; 1 c1 c13c− cVR c13 cV ≡ ð1 þ 2 θÞ 2ϕ 2ψ þ θ 2ϕ 2ψ F× cos cos sin cos sin cos ; ̈ k 2 2 c13Q e ¼ c13Gæ jk Z − 2Σ j ð Þ hY j eY; 4:59 1 2 ð2c1 − c13c−Þc R ð1 − c13Þc ≡ − θ 2ϕ V V Fb 2 sin cos ; 1 where ekl ≡ ek el − ek el and ekl ≡ ek el þ ek el . From ≡ 2 θ 2ϕ þ X X Y Y × X Y Y X FL 2 sin cos ; the above expressions we can see that the scalar longi- ≡ − θð θ 2ϕ ψ − 2ϕ ψÞ tudinal mode hL is proportional to the scalar breather mode FX sin cos cos cos sin sin ; hb. So, out of these six components, only five of them are F ≡ − sin θðcos θ cos 2ϕ sin ψ þ sin 2ϕ cos ψÞ: ð4:62Þ independent. Y It is remarkable to note that both of the scalar breather and the scalar longitudinal modes are suppressed by a Hence, its Fourier transformation takes the form −5 factor c14 ≲ Oð10 Þ with respect to the transverse-trace- Z 1 less modes hþ and h×, while the vectorial modes hX and hY h˜ðfÞ¼ hðtÞe−i2πftdt: ð4:63Þ −15 2π are suppressed by a factor of c13 ≲ Oð10 Þ. 023010-12 GRAVITATIONAL WAVEFORMS, POLARIZATIONS, … PHYS. REV. D 99, 023010 (2019) V. GRAVITATIONAL WAVEFORMS AND RADIATIONS OF TRIPLE SYSTEMS Consider a triple system with masses ma and positions xaðtÞ, where a specifies the three bodies, a ¼ 1,2,3. Defining r1 ≡ x2 − x3, r2 ≡ x3 − x1, r3 ≡ x1 − x2, and ra ≡ jraj, we find that FIG. 1. Trajectory of Simo’s figure-eight 3-body system [93]. ¼ 1 2 X In this plot, we set m . Q⃛ ¼ G Pa ; ij 3 N ij a 2 2 D2 2D D _ 2 2 1 2 3 i i _ _ _ ðΣ Þ ¼ α1 − α2 G þ r r ⃛¼ −2 r1 þ r2 þ r3 i 3 N ð Þ4 ð Þ3ð Þ3 2 3 I GN m2m3 2 m1m3 2 m1m2 2 ; r1 r2 r3 ðr1Þ ðr2Þ ðr3Þ D2 2D D 2 X þ 2 þ 1 3 i i i 4 3 3 r1r3 Σ ¼ α1 − α2 ðv Ω Þ; ðr2Þ ðr1Þ ðr3Þ i 3 a a a 2 D3 2D1D2 X i þ þ i i ð Þ 2 ra 4 3 3 r1r2 ; 5:5 Σ_ ¼ − α − α D ð Þ ðr3Þ ðr1Þ ðr2Þ i 1 3 2 GN a ð Þ3 ; 5:1 a ra where where Ω is the binding energy of the ath body, as 36 ð i j Þ ð i j Þ a 1 2 ¼ 2 d r1r1 d r2r2 mentioned previously, and PijPij m1m2m3 3 3 ðr1Þ ðr2Þ dt dt X 54 i j _ ð i j Þ 54 i j _ ð i j Þ − r1r1r1 d r2r2 − r2r2r2 d r1r1 D ≡ ϵabcΩ 4 3 3 4 c amb; ðr1Þ ðr2Þ dt ðr1Þ ðr2Þ dt a;b i j i j 81r r r r r_1r_2 3r_1r_2 X jϵabcj ð i j Þ þ 1 1 2 2 − ð Þ c mamb 2 d rcrc i j 4 4 2 2 ; 5:6 P ≡ ðr Þ r_ δ − 6r þ 9r_ r r : ðr1Þ ðr2Þ ðr1Þ ðr2Þ ij 2ð Þ4 c c ij c c c c a;b rc dt ð5:2Þ and so on. Then, from Eq. (4.59) we can see that hNðx; xaðtÞÞ, where xaðtÞ’s are the trajectories of the three bodies. So, abc i i 2 x ð Þ’ Here ϵ is the Levi-Civit`a symbol. Setting rara ≡ ðraÞ , once a t s are known, from Eqs. (4.59) and (5.1) we can i i i i 2 study the polarizations of GWs emitted by this triple rar_a ≡ rar_a, and r_ar_a ≡ ðvaÞ , we obtain system. On the other hand, inserting Eqs. (5.3)–(5.6) into _ Eq. (3.1) we obtain the energy loss rate EðtÞ. 8 2 2 1 ð ⃛ Þ2 ¼ 2 m2m3 ð12 2 − 11ð_ Þ2Þþ 2 3 However, in the framework of Einstein-aether theory the Qij GN 4 v1 r1 PijPij 3 ðr1Þ 3 trajectories of a triple system have not yet been studied. So, 2 2 1 in this paper we shall use the Newtonian trajectories of the þ m1m2 ð12 2 − 11ð_ Þ2Þþ 1 2 8 4 v3 r3 PijPij triple systems, which have been intensively studied in the ðr3Þ 3 past 300 years, and various periodic solutions have been 2 2 1 found; see, e.g., [46] and references therein. Some of them þ m1m3 ð12 2 − 11ð_ Þ2Þþ 1 3 ð Þ 4 v2 r2 PijPij ; 5:3 ðr2Þ 3 have also been used to study the GW forms in the framework of GR. In particular, in [50] it was shown that the quadrupole GW form of a figure-eight trajectory discovered by Moore in 1993 [90] is indistinguishable ð_ Þ2 _ _ from that of a binary system. In addition, Dmitrasinovic, ð ⃛Þ2 ¼ 4 2 2 2 r1 þ 2 2 r2r3 I GN m2m3 4 m2m3m1 2 2 Suvakov, and Hodomal calculated the quadrupole wave- ðr1Þ ðr2Þ ðr3Þ forms and the corresponding luminosities for the 13 þ 11 ð_ Þ2 _ _ þ 2 2 r2 þ 2 2 r1r3 periodic orbits of three-body problems in Newtonian m1m3 4 m1m3m2 2 2 ðr2Þ ðr1Þ ðr3Þ ð_ Þ2 _ _ 8 þ 2 2 r3 þ 2 2 r1r2 ð Þ Corrections due to the aether effects are expected to be small, m1m2 4 m1m2m3 2 2 ; 5:4 and they should be consistent with the lowest PN order ðr3Þ ðr1Þ ðr2Þ approximations adopted in this paper. 023010-13 KAI LIN et al. PHYS. REV. D 99, 023010 (2019) FIG. 3. The polarization modes hb and hL defined in Eq. (4.53) for Simo’s figure-eight 3-body system in æ theory, where the modes are propagating along the positive z direction., and −6 −5 ðΩ1; Ω2; Ω3Þ¼ð−0.1; −2.76 × 10 ; −2.9 × 10 Þ. of Oð10−5Þ lower than that of GR. This can be seen clearly from this figure, in which the lines are almost identical in both of the theories. Note that when plotting these figures, we assumed that the binding energies of the three −6 bodies are, respectively, Ω1 ¼ −0.1, Ω2 ¼ −2.76 × 10 , −5 and Ω3 ¼ −2.9 × 10 , which are the same as those given for the PRS J0337 þ 1715 [28], although here the three masses are assumed to be equal. In Fig. 3 we plot out the polarization modes hb and hL in FIG. 2. The polarization modes hþ and h× defined in Eq. (4.53) ’ æ theory, which all vanish in GR. Comparing with Fig. 2 it for Simo s figure-eight 3-body system in both GR and æ theory, can be seen that the amplitudes of these modes are about 5 where the modes are propagating along the positive z direction, −6 −5 orders lower than hþ and h×, which is again consistent with and ðΩ1; Ω2; Ω3Þ¼ð−0.1; −2.76 × 10 ; −2.9 × 10 Þ. our analysis given in Sec. III. – gravity [49], discovered, respectively, in [45] and [91]. In Figs. 4 6, we plot out the corresponding response hðtÞ h˜ðfÞ Among other things, they found that all these 13 þ 11 function , its Fourier transform , and the radiation ð≡−E_Þ ’ orbits produce different waveforms and their luminosities power P for Simo s figure-eight 3-body system. vary by up to 13 orders of magnitude in the mean, and up to From Fig. 6 we can see that both of the dipole and 20 orders for the peak values. monopole contributions are suppressed with the orders In this paper, we shall consider some of the trajectories given in Sec. III. provided in [92].9 Before doing so, let us first consider the GW form of Simo’s figure-eight trajectory [93], studied in [49].In Fig. 1, the trajectory of the 3-body problem is plotted out in the (x, y) plane for many periods, in order to make sure that our numerical codes converge well after a sufficiently long run. Assuming that the detector is along the z axis, we plot out the polarization modes hþ and h× in Fig. 2. In this figure, we plot these modes given in GR as well as in Einstein-aether theory. As we noted previously, the contributions from the aether field is of the order 9In the configurations provided in this site, the three bodies are assumed all to have equal masses, m1 ¼ m2 ¼ m3 ¼ m, and also ð Þ ’ the units were chosen so that GN ¼ 1, m ¼ 1. Therefore, in our FIG. 4. The response function h t for Simo s figure-eight numerical simulations presented in this paper, we adopt the same 3-body system in GR and æ theory, where the modes are units so that m ¼ 1 ¼ GN. However, restoring the physical units, propagating along the positive z direction, and ðΩ1; Ω2; Ω3Þ¼ 10 −6 −5 this is equivalent to set mi ¼ 1=GN ≃ 1.5 × 10 kg ≪ M⨀. ð−0.1; −2.76 × 10 ; −2.9 × 10 Þ. 023010-14 GRAVITATIONAL WAVEFORMS, POLARIZATIONS, … PHYS. REV. D 99, 023010 (2019) FIG. 5. The Fourier transform h˜ðfÞ of the response function hðtÞ for Simo’s figure-eight 3-body system in GR and æ theory, where the modes are propagating along the positive z direction, FIG. 7. The polarization modes hþ and h defined in Eq. (4.53) ðΩ Ω Ω Þ¼ð−0 1 −2 76 10−6 −2 9 10−5Þ × and 1; 2; 3 . ; . × ; . × . for Simo’s figure-eight 3-body system in both GR and æ theory, where the modes are propagating along the direction specified by ðϑ; φ; θ; ϕ; ψÞ¼ð0.6; 5.2; 1.3; 1.2; 1.8Þ with ðΩ1; Ω2; Ω3Þ¼ Note that in drawing the above figures, we had set −6 −5 −5 −5 −5 ð−0.1; −2.76 × 10 ; −2.9 × 10 Þ. c1 ¼ 4×10 , c2 ¼ 9×10 , c3 ¼ −c1, and c4 ¼ −2×10 , a condition that will be adopted for the rest of this paper. In addition, with these choices, we have c13 ¼ 0, and For such choices, the coupling constants ci’s clearly satisfy then from Eq. (4.59) we find that the vectorial modes hX the theoretical and observational constraints of the æ theory and h are identically zero, [69], given by Eq. (2.18). Y hX ¼ hY ðc13 ¼ 0Þ: ð5:7Þ FIG. 6. The radiation power Pð≡ − E_Þ of Simo’s figure-eight 3-body system in æ theory, where the modes are all pro- FIG. 8. The polarization modes hb and hL defined in pagating along the positive z direction, and ðΩ1;Ω2;Ω3Þ¼ Eq. (4.53) for Simo’s figure-eight 3-body system in æ theory, ð−0.1;−2.76×10−6;−2.9×10−5Þ. The dotted (blue), dash-dotted where the modes are propagating along the direction specified (red), and solid (green) lines denote, respectively, the parts of by ðϑ; φ; θ; ϕ; ψÞ¼ð0.6; 5.2; 1.3; 1.2; 1.8Þ with ðΩ1; Ω2; Ω3Þ¼ quadrupole, monopole, and dipole radiations given in Eq. (3.1). ð−0.1; −2.76 × 10−6; −2.9 × 10−5Þ. 023010-15 KAI LIN et al. PHYS. REV. D 99, 023010 (2019) FIG. 11. The radiation power Pð≡ − E_Þ of Simo’s figure-eight FIG. 9. The response function hðtÞ for Simo’s figure-eight 3-body system in æ theory, where the modes are propaga- 3-body system in GR and æ theory, where the modes are ting along the direction specified by ðϑ; φ; θ; ϕ; ψÞ¼ ðϑ φ; θ ϕ ψÞ¼ −6 propagating along the direction specified by ; ; ; ð0.6; 5.2; 1.3; 1.2; 1.8Þ with ðΩ1;Ω2;Ω3Þ¼ð−0.1;−2.76×10 ; −6 ð0.6; 5.2; 1.3; 1.2; 1.8Þ with ðΩ1;Ω2;Ω3Þ¼ð−0.1;−2.76×10 ; −2.9×10−5Þ. The dotted (blue), dash-dotted (red), and solid −5 −2.9×10 Þ. (green) lines denote, respectively, the parts of quadrupole, monopole, and dipole radiations given in Eq. (3.1). ˜ð Þ FIG. 10. The Fourier transform h f of the response function FIG. 12. The polarization modes hþ and h× defined in hðtÞ for Simo’s figure-eight 3-body system in GR and æ theory, Eq. (4.53) for Simo’s figure-eight 3-body system in both GR where the modes are propagating along the direction specified and æ theory, where the modes are propagating along the by ðϑ; φ; θ; ϕ; ψÞ¼ð0.6; 5.2; 1.3; 1.2; 1.8Þ with ðΩ1; Ω2; Ω3Þ¼ direction specified by ðϑ; φ; θ; ϕ; ψÞ¼ð0.6; 5.2; 1.3; 1.2; 1.8Þ, −6 −5 −2 ð−0.1; −2.76 × 10 ; −2.9 × 10 Þ. with Ω1 ¼ Ω2 ¼ Ω3 ¼ −10 . 023010-16 GRAVITATIONAL WAVEFORMS, POLARIZATIONS, … PHYS. REV. D 99, 023010 (2019) FIG. 13. The polarization modes hb and hL defined in Eq. (4.53) for Simo’s figure-eight 3-body system in æ theory, where the modes are propagating along the direction specified by ðϑ; φ; θ; ϕ; ψÞ¼ð0.6; 5.2; 1.3; 1.2; 1.8Þ, with Ω1 ¼ Ω2 ¼ −2 Ω3 ¼ −10 . So, in the rest of this paper we only need to consider the hþ, h×, hb, and hL modes. Yet, we also plot the above figures by assuming that the orbit is in the (x, y) plane, while the detector is along the z axis. The same conditions were also assumed in [49]. However, as we mentioned in the last section, the locations and orientations of the sources as well as the detectors are FIG. 15. The Fourier transform h˜ðfÞ of the response function all independent, which are specified by the five angles hðtÞ for Simo’s figure-eight 3-body system in GR and æ theory, ðϑ; φ; θ; ϕ; ψÞ, defined in Eqs. (4.52) and (4.60).For where the modes are propagating along the direction specified ðϑ φ; θ ϕ ψÞ¼ð0 6 5 2; 1 3 1 2 1 8Þ Ω ¼ Ω ¼ different choices of these parameters, the waveforms and by ; ; ; . ; . . ; . ; . , with 1 2 Ω ¼ −10−2 response functions will also be different. In Figs. 7–11 we 3 . plot the mode functions hN, response function hðtÞ, its ˜ Fourier transform hðfÞ, and the radiation powers PA, PB, transform are all different from the case in which the 3- and PC, respectively, for ðϑ; φ; θ; ϕ; ψÞ¼ð0.6; 5.2; 1.3; bodies are in the ðx; yÞ plane, while the detector is localized 1.2; 1.8Þ, while still choosing ðΩ1; Ω2; Ω3Þ¼ð−0.1; along the z axis. However, the radiation powers are the −2.76 × 10−6; −2.9 × 10−5Þ. Clearly, the corresponding same and are independent of the choice of these five mode functions, response function, and its Fourier angular parameters, as can be seen from Figs. 6 and 11. To study the effects of the binding energies of the three bodies on the waveforms and energy losses, let us consider the same case as shown by Figs. 7–11 but now −2 with the same binding energy, Ωa ¼ −10 . With this choice, the dipole contributions are identically zero, as one can see from Eqs. (5.2) and (5.5), since now we have _ 2 Dc ¼ 0ðc ¼ 1; 2; 3Þ and Σi ¼ 0. This does not contradict with the results given by Eq. (3.13), as there we assumed that Ω1=m1 −Ω2=m2 ≃OðΩ=mÞ [cf. Eqs. (3.6) and (3.10)]. But, here due to our choice of Ωa and ma,wehave Ω1=m1 −Ω2=m2 ¼ 0 and so on. In Figs. 12–16 we plot the mode functions hN, response function hðtÞ, its Fourier ˜ transform hðfÞ, and the radiation powers PA and PB, respectively, for ðϑ; φ; θ; ϕ; ψÞ¼ð0.6; 5.2; 1.3; 1.2; 1.8Þ, −2 −2 −2 while setting ðΩ1; Ω2; Ω3Þ¼ð−10 ; −10 ; −10 Þ. FIG. 14. The response function hðtÞ for Simo’s figure-eight Clearly, the corresponding mode functions, response func- 3-body system in GR and æ theory, where the modes are tions, Fourier transforms, and radiation powers are all propagating along the direction specified by ðϑ; φ; θ; ϕ; ψÞ¼ different from the case in which the 3-bodies are in the −2 ð0.6; 5.2; 1.3; 1.2; 1.8Þ, with Ω1 ¼ Ω2 ¼ Ω3 ¼ −10 . ðx; yÞ plane, while the detector is localized along the z axis. 023010-17 KAI LIN et al. PHYS. REV. D 99, 023010 (2019) FIG. 16. The radiation power Pð≡ − E_Þ of Simo’s figure-eight trajectory of the 3-body system in GR and æ theory, where the modes are propagating along the direction specified by ðϑ; φ; θ; ϕ; ψÞ¼ð0.6; 5.2; 1.3; 1.2; 1.8Þ,withΩ1 ¼ Ω2 ¼ Ω3 ¼ −10−2. The dotted (blue) and solid (red) lines denote, respec- tively, the parts of quadrupole and monopole radiations given in Eq. (3.1). Because the choices of the binding energies Ωa and masses ma are all the same for the three compact objects, the dipole contributions, denoted by the C part in Eq. (3.1), are identically zero. In particular, the dipole contributions vanish now, as explained above. In Figs. 17–21, we plot out, respectively, the trajectory of the Broucke R7 3-body system provided in [92], the polarization modes hN, the response function hðtÞ, its ˜ Fourier transform hðfÞ, and the radiation powers PA, PB, and PC, for ðϑ; φ; θ; ϕ; ψÞ¼ð0.6; 5.2; 1.3; 1.2; 1.8Þ, −6 −5 and ðΩ1; Ω2; Ω3Þ¼ð−0.1; −2.76 × 10 ; −2.9 × 10 Þ. FIG. 18. The polarization modes hN defined in Eq. (4.53) for the Broucke R7 3-body system with the choice, ðϑ; φ; θ; ϕ; ψÞ¼ −6 ð0.6; 5.2; 1.3; 1.2; 1.8Þ and ðΩ1; Ω2; Ω3Þ¼ð−0.1; −2.76 × 10 ; −2.9 × 10−5Þ. In Fig. 22 we plot out the trajectory of the Broucke A16 3-body system provided in [92], while in Figs. 23–26 we plot out the corresponding physical quantities for the same choice of the five angular parameters as selected in the case for the Broucke R7 3-body system in both GR and æ theory −6 −5 with ðΩ1; Ω2; Ω3Þ¼ð−0.1; −2.76 × 10 ; −2.9 × 10 Þ. From these figures we can see clearly that the GW forms and radiation powers not only depend on the relative positions, orientations between the source and detector, but also depend on the configurations of the orbits of the 3- FIG. 17. Trajectory of the 3-body system for the Broucke R7 body system. In addition, they also depend on their binding figure provided in [92]. energies of the three compact bodies. 023010-18 GRAVITATIONAL WAVEFORMS, POLARIZATIONS, … PHYS. REV. D 99, 023010 (2019) ð≡ − E_Þ ð Þ FIG. 21. The radiation power P of the 3-body system FIG. 19. The response function h t for the Broucke R7 3-body ðϑ φ; θ ϕ ψÞ¼ ðϑ φ; θ ϕ ψÞ¼ð0 6 5 2; 1 3 1 2 1 8Þ of the Broucke R7 figure with the choice, ; ; ; system with the choice, ; ; ; . ; . . ; . ; . −6 −6 −5 ð0.6; 5.2; 1.3; 1.2; 1.8Þ and ðΩ1; Ω2; Ω3Þ¼ð−0.1; −2.76 × 10 ; and ðΩ1; Ω2; Ω3Þ¼ð−0.1; −2.76 × 10 ; −2.9 × 10 Þ. −2.9 × 10−5Þ. The dotted (blue), dashdotted (red), and solid (green) lines denote, respectively, the parts of quadrupole, monopole, and dipole radiations given in Eq. (3.1). FIG. 20. The Fourier transform h˜ðfÞ of the response function hðtÞ for the Broucke R7 3-body system with the choice, FIG. 22. Trajectory of the 3-body system for the Broucke A16 ðϑ; φ; θ; ϕ; ψÞ¼ð0.6; 5.2; 1.3; 1.2; 1.8Þ and ðΩ1; Ω2; Ω3Þ¼ figure provided in [92]. ð−0.1; −2.76 × 10−6; −2.9 × 10−5Þ. 023010-19 KAI LIN et al. PHYS. REV. D 99, 023010 (2019) FIG. 24. The response function hðtÞ for the Broucke A16 3-body system with the choice, ðϑ; φ; θ; ϕ; ψÞ¼ð0.6; 5.2; 1.3; −6 −5 1.2; 1.8Þ and ðΩ1;Ω2;Ω3Þ¼ð−0.1;−2.76×10 ;−2.9×10 Þ. In this paper, we have studied the lowest PN order of 3- body problems in the framework of Einstein-aether theory [68], a theory that violates locally the Lorentz symmetry and yet passes all the theoretical and observational tests carried out so far [69]. Although these tests were mainly in the weak-field limits, strong-field effects of binary neutron stars systems have also been investigated [64,65,71,72]. FIG. 23. The polarization modes hN defined in Eq. (4.52) for the Broucke A16 3-body system with the choice, ðϑ; φ; θ; ϕ; ψÞ¼ð0.6; 5.2; 1.3; 1.2; 1.8Þ and ðΩ1; Ω2; Ω3Þ¼ ð−0.1; −2.76 × 10−6; −2.9 × 10−5Þ. VI. CONCLUSIONS Three-body systems have been attracting more and more attention recently [25], especially after the detections of GWs from binary systems [1–6], as they are common in our Universe [26] and can be ideal sources for periodic GWs. In particular, we are very much interested in the cases in which the orbits of two bodies pass each other very closely and yet FIG. 25. The Fourier transform h˜ðfÞ of the response function avoid collisions, so they can produce intense periodic hðtÞ for the Broucke A16 3-body system with the choice, gravitational waves and provide natural sources for the ðϑ; φ; θ;ϕ; ψÞ¼ð0.6;5.2; 1.3; 1.2; 1.8Þ and ðΩ1;Ω2;Ω3Þ¼ð−0.1; future detection of GWs. −2.76×10−6;−2.9×10−5Þ. 023010-20 GRAVITATIONAL WAVEFORMS, POLARIZATIONS, … PHYS. REV. D 99, 023010 (2019) NS NS for a binary neutron star system, where δWA , δWB , and NS δWC represent the contributions of the strong-field effects to the quadrupole, monopole, and dipole parts of Eq. (3.1), NS while δWD denotes a cross term due to the motion of the center of mass of the system [65]. Clearly, these effects are much smaller than the ones mentioned above and are out of the detectability of the current generation of detectors. So, in this paper we have ignored these effects, and simply set the sensitivities sA of the compact bodies to zero. However, in the development of the general formulas, we have kept Tμ not zero to its first order of perturbations in Sec. IV, for our later applications of the formulas. Here Tμ FIG. 26. The radiation power Pð≡−E_Þ of the 3-body system of represents the coupling between aether and matter fields ¼ 0 the Broucke A16 figure with the choice, ðϑ; φ; θ; ϕ; ψÞ¼ [cf. Eq. (2.10)]. Setting Tμ , our results presented in −6 ð0.6; 5.2; 1.3; 1.2; 1.8Þ and ðΩ1; Ω2; Ω3Þ¼ð−0.1; −2.76 × 10 ; Sec. IV reduce to the ones of Ref. [73], subjected to some −2.9 × 10−5Þ. The dotted (blue), dashdotted (red), and solid corrections of typos. From the expressions of the six (green) lines denote, respectively, the parts of quadrupole, polarization modes of Eq. (4.59) we can see that the scalar monopole, and dipole radiations given in Eq. (3.1). longitudinal mode hL is proportional to the scalar breather mode hb. Therefore, out of these six components, only five of them are independent. In addition, the scalar breather In particular, the accelerations and strong-field Nordtvedt and the scalar longitudinal modes are all suppressed by a parameter were calculated recently for a triple system to the −5 factor of Oðc14Þ ≲ Oð10 Þ with respect to the transverse- quasi-Newtonian order [66]. traceless modes hþ and h×, while the vectorial modes hX In this paper, we have first shown that the contributions −15 and h are suppressed by a factor of Oðc13Þ ≲ Oð10 Þ. of the presence of the aether field to the quadruple part of Y These conclusions should also be valid for general cases the energy loss rate of a binary system is the order of −5 and consistent with the analysis of triple systems presented Oðc14Þ ≲ Oð10 Þ lower than that of GR when only the in Sec. V. lowest PN order is taken into account [cf. Eq. (3.12)]. Applying the general formulas developed in Sec. IV to a Because of the presence of two additional modes, the scalar triple system, in Sec. V we have studied the GW forms, the and vector, in Einstein-aether theory, two additional parts response function, its Fourier transform, and the energy also appear in the energy loss rate of a binary system [73], loss rates due to each part of the GW radiations given in given, respectively, by the second and third terms in Eq. (3.1) for three different kinds of periodic orbits of three- Eq. (3.1), representing the monopole and dipole contribu- body problems: one is Simo’s figure-eight configuration tions. In comparison with the quadruple contributions of [93], given by Fig. 1, and the other two are, respectively, the GR, which is the order of Oðv2Þ, the monopole contribu- 2 Broucke R7 and A16 configurations provided in [92] and tions are only of the order of Oðc14ÞOðv Þ; that is, it is illustrated by Figs. 17 and 22 in the current paper. In the −5 about Oðc14Þ ≲ Oð10 Þ order lower than that of GR. Here case of Simo’s figure-eight configuration, we have studied v is the relative velocity of the two compact objects. the effects of the relative orientations between the source However, the dipole contributions can be much larger than and detector, as well as the effects of binding energies of the those of monopole. In particular, for a binary system with three compact bodies. Through this case, we have shown large differences between their binding energies, the dipole explicitly that the GW form, the response function, and its part can be as large as Oðc14ÞOðGNm=dÞ, where m denotes Fourier transform all depend on the relative orientations the mass of a binary system and d the distance between and binding energies, as they are expected from Eq. (3.1) the two stars. For a realistic neutron star, we have [cf. Figs. 7 and 26]. In the cases of the Broucke R7 and A16 OðGNm=dÞ ≃ 0.1–0.3, so that Oðc14ÞOðGNm=dÞ ≃ configurations, the five angles are chosen to be the same as 10−2Oðv2Þ. It should be noted that the scalar mode has in the second case of Simo’s figure-eight configuration, and contributions to all three parts, quadrupole, dipole, and the corresponding GW form, the response function, its monopole, while the vector mode has contributions only to Fourier transform, and the powers of radiation are given, the quadrupole and dipole parts, as can be seen clearly from respectively, by Figs. 18–21 and 23–26. From these figures Eqs. (102)–(104) of Ref. [73]. On the other hand, the we find that all these physical quantities are different. strong-field contributions are only of the orders of Therefore, the GW form, the response function, and its Fourier transform of a triple system depend not only on NS −11 NS −11 their configuration of orbits but also on their orientation δWA ≲ Oð10 Þ; δWB ≲ Oð10 Þ; with respect to the detector and binding energies of the δWNS ≲ Oð10−12Þ δWNS ≲ Oð10−11Þ ð Þ C ; D ; 6:1 three compact bodies. 023010-21 KAI LIN et al. PHYS. REV. 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