Neutron Stars in the Effective Fly-By Framework: f-Mode Re-summation
NNeutron Stars in the Effective Fly-By Framework: f -Mode Re-summation J. Nijaid Arredondo
Department of Astrophysical Sciences, Princeton University, Princeton, NJ, 08544, USA
Nicholas Loutrel
Department of Physics, Princeton University, Princeton, NJ, 08544, USA (Dated: January 27, 2021)Eccentric compact binaries pose not only a challenge for gravitational wave detectors, but alsoprovide a probe into the nuclear equation of state if one of the objects is a neutron star. At theshort pericenter passage, tidal interactions excite f-modes on the star, which in turn emit theirown gravitational waves. We derive an analytic waveform for these stellar oscillations within theeffective fly-by framework, modeling the emission to leading post-Newtonian order. At this order,the f-mode response can be written in a Fourier decomposition in terms of orbital harmonics, withthe amplitudes of each harmonic depending on Hansen coefficients. Re-summing the harmonics ofthe f-mode results in a simple decaying harmonic oscillator, with the amplitude now determined by aHansen coefficient of complex harmonic number. We compute the match M between the re-summedf-mode and numerical integrations of the tidal response, and find M > .
97 for systems with highorbital eccentriciy ( e > .
9) and low semi-latus rectum ( p < M ). We further compare our modelto modes generated from multiple pericenter passages under the effect of radiation reaction, anddiscuss issues related to the timing of pericenter passages and its impact on the model. I. INTRODUCTION
It is now five years since the Laser InterferometerGravitational-wave Observatory (LIGO) [1] first detectedthe gravitational waves of merging binary black holes(BBHs) [2]. Since then, the LIGO and Virgo [3] Sci-entific Collaborations have in total detected the coales-cence of fifty pairs of compact objects, including blackholes (BHs) and neutron stars (NSs) [4]. LIGO initiatedan era of multi-messenger astronomy with the detectionof its first NS-NS binary [5], as it was accompanied by theelectromagnetic observation of its kilonova [6]. This hasallowed for quantitative tests of the structure of these NSsprobed by the strong gravity during merger [7], constrain-ing the equation of state (EOS) of degenerate matterwith measurement of the stars’ masses, radii, and tidaldeformabilities [8]. With this data and upcoming obser-vations, gravitational wave (GW) astronomy has openedup promising avenues towards probing strong gravity andthe equation of state of nuclear matter.GW astronomy is yet in its infancy, and its ability tocharacterize detections has already been pushed into un-charted territory. Only 4 cycles were captured of LIGOdetection GW190521, leading to uncertainties in its in-terpretation [9]. Besides its primary mass lying in thepair-instability mass gap, making it an outlier from pre-vious LIGO events, its short duration makes it difficult todetermine the precise interpretation of the signal, specif-ically quasi-circular spin precessing versus head-on colli-sion. Follow up studies performing model selection havefound that models with non-zero eccentricity ( e > . e ∼ .
7) precessing waveformsare even more favored [11]. However, the uncertain-ties between quasi-circular, precessing binary models and eccentric models hamper this interpretation from beingdefinitive [10, 12].A conclusive measurement of a merging binary withnon-zero eccentricity would be a strong indicator ofdynamical formation, tracing the formation channel ofthe merging components in dense stellar environments.LIGO’s detections have so far been consistent with therates expected from isolated binary evolution, althoughthis may be due to the large uncertainties in spin mea-surements and population models [13]. This formationchannel is expected to produce orbits in which the com-ponents’ spins are predominantly aligned with the orbitalangular momentum and are circularized by the time theyenter LIGO’s frequency band. Such assumptions willunfortunately exclude dynamical channels in which bi-naries merge with significant eccentricity or misalignedspins [14]. These channels are sourced by a diversity ofdynamical processes, such as GW captures between twounbound objects [15], binary-single interactions includ-ing Kozai-Lidov cycles [16, 17] and exchanges [18], andhierarchical mergers [19], which is a possible formationchannel for GW190521 [20]. These processes have thepotential to create eccentric binaries in dense stellar en-vironments whose GWs may be detectable with a vari-ety of detectors, including decihertz detectors, LISA, andLIGO [21–23].Dynamical formation channels are not only expected toproduce BBHs, but also BH-NS binaries and binary neu-tron stars (BNSs). The BH-NS merger rates in dense en-vironments have only been loosely constrained and varydepending on their formation channel. For the morepromising channels, the rate for binary-single interac-tions has been estimated to be as high as 0.25 Gpc − yr − in globular clusters [24]; for Kozai-Lidov to be ashigh as 0.33 Gpc − yr − in galactic nuclei [25, 26]; and a r X i v : . [ g r- q c ] J a n for the combination of binary evolution and dynamicalexchanges in young star clusters to be as high as ∼ − yr − [27], although this estimate assumes low na-tal NS kicks and high densities that may not be repre-sentative of all young star clusters (a more conservativeupper limit has been found to be ∼ − Gpc − yr − ;see [26]). The latter category is especially interestingas almost all of the dynamically assembled BH-NSs areejected from their cluster, thus merging in the field andpossibly mixing with binaries that did not form in clus-ters. These are all a fraction of LIGO’s empirical upperBH-NS merger rate of 610 Gpc − yr − [28], which is es-timated to be dominated by isolated binaries in the field.However, triple systems in the field may form eccentricbinaries within LIGO; the rate of BH-NS mergers formedfrom these triples is estimated to be 10 − −
19 Gpc − yr − [29]. Understanding these channels and being ableto identify their signals will be crucial as detections in-crease.The method of power stacking has been proposed as ameans of detecting the GWs from eccentric binaries [30].While quasi-circular orbits emit sinusoidal GWs, eccen-tric orbits emit bursts of radiation that are increasinglylocalized around pericenter the more eccentric the or-bit is. By searching for these bursts as excess powerin the detector data and stacking them, detectors canbe more sensitive to signals from eccentric binaries. Todistinguish these from other sources of uncorrelated ex-cess power such as glitches [31, 32], a timing model thatcan physically correlate a sequence of bursts would beneeded. Such a model has so far only been developedin the post-Newtonian (PN) formalism for point masses[33].However, to detect and reconstruct a signal, themethod of choice is match filtering with template wave-forms [4]. Complications that arise from non-circular ef-fects, such as orbital precession and the timescales in-volved in computing multiple eccentric orbits make quasi-circular waveforms simpler to produce and analyze. Someof these complications arise in numerical simulations thatmodel the evolution of the spacetime with full generalrelativity, as accurately capturing the GWs at pericenterpassage requires resolving timescales shorter than the pe-riod of an orbit. For eccentric orbits, these timescales aremuch more disparate, with the time spent at pericenterbeing orders of magnitude smaller. Yet numerical sim-ulations are key to accurately modeling the waveformsused in match filtering [34, 35].While LIGO has ruled out detections with significanteccentricity in its first two observing runs [36], it is recog-nized that quasi-circular waveforms lose sensitivity whenmatched against eccentric signals, in particular if the or-bit is of low mass [37]. Quantitatively, these models losesignificant sensitivity for e ∼ .
07 [38] and may miss BNSwith e ≥ .
02 [39]. Significant efforts have gone into mod-eling the GW emission from eccentric binaries in recentyears, with PN models currently available up to 3PN or-der and up to e ∼ . ∼
5% of the NS binding en-ergy, given an initial separation on the order of the NSradius (see Fig. 3 of [57]). The phase shift of the GW canreach the tens of radians before merger, demonstratingthe effect of the f-modes on hastening the collision.The effects of eccentricity and EOS on waveforms havethus shown exciting results. Detectors that are more sen-sitive to them across different frequency bands are on thehorizon, such as the space-borne LISA [58] and the third-generation ground detectors Cosmic Explorer (CE) [59]and Einstein Telescope (ET) [60]. Upon realization, thethird-generation is expected to reduce LIGO uncertain-ties on the EOS by an order of magnitude [61]. Eccentricwaveforms will need to be available by then to accuratelycharacterize formation channels and the structure of NSs.
A. Executive Summary
In this present study, we seek to extend the PN ap-proximations introduced by [54, 55] to explicitly calcu-late the gravitational waveform of f-modes from highlyeccentric BH-NS binaries. The GWs from the orbitalmotion in this limit have already been calculated andanalyzed in [62, 63]. Here, we adapt their effective fly-by framework to NS f-modes. We work to leading or-der in the PN formalism, decomposing the elliptic orbitas a Fourier series of harmonics of the orbital period.The f-modes are excited by this orbit and can be writ-ten exactly as an integral of the Greens’ function of thedriving force, which can then be solved using the har-monic Fourier series of the orbit. However, such a seriesrequires multiple terms to accurately describe high ec-centricities and to capture the loudest harmonics. Wethus adapt the re-summation procedure of [62, 64] to in-tegrate the series. We find that the f-modes oscillate attheir natural frequency with an amplitude determined bythe stellar structure and the orbital parameters at clos-est approach, further demonstrating the effective fly-bydescription of the re-summation procedure. Mathemati-cally, this appears through the evaluation of the series atthe terms closest to resonating with the dominant f-modeharmonic. These coefficients determine the amplitude ofa simple sinusoidal tide. This is our main result, andwe compare it to the numerical integration of the oscil-lations. We find that our re-summations performs wellfor highly eccentric and close pericenter passages, reach-ing matches with the numerical integrations greater than0.97 for eccentricities e > . p < M .We also discuss how to account for subsequent modeexcitations with a radiation reaction model that deter-mines how the orbital parameters change between peri-center passages and the timing between these passages.However, we find our timing model to be insufficient fortwo reasons: 1) the timing model’s accuracy breaks downfor close pericenter passages, and 2) the timing modeldoes not incorporate the back-reaction of the tides onthe orbit, which from past studies have proven to be asessential as the radiation reaction. The first limitationcalls for a new, more accurate timing, that can perhapsbe obtained with a multi-scale analysis that accounts foreffects beyond the adiabatic approximation of radiationreaction. The second limitation requires further work tobe performed within the osculating orbit framework toaccount for how the orbital parameters evolve due to thetide within our fly-by formalism. Resolution of these lim-itations will then allow our model of the tidal excitationto be easily stitched into a sequence of tides, and thustake further steps towards accurate waveforms for matchfiltering eccentric binaries with NSs.In Sec. II we present the leading order dynamicaltides raised on the star and calculate its induced thequadrupole moment. Expanding the orbit in a Fourierseries, we then write it as a function of time and calcu-late its GW polarizations. We review the effective fly-byframework in Sec. III and apply it to the f-modes throughcontour integration, and introduce our timing model tocreate a sequence of tides. Sec. IV demonstrates theaccuracy of our re-summed modes and shows the matchbetween them and numerical waveforms within the sen- sitivities of LIGO and ET. We conclude in Sec. V bydiscussing the work needed to produce a complete ec-centric BH-NS model. All expressions are given in units G = c = 1. II. DYNAMICAL TIDES IN THEPOST-NEWTONIAN FORMALISM
At closest approach, the black hole’s tidal field stronglyaffects the neutron star, deforming it and exciting funda-mental f-modes on its surface [65]. In this section we re-view how this deformation perturbs the two-body pointmass problem into one with an additional quadrupolemoment induced by the raised tides. The excitation ofthese modes can be expressed in terms of the binary’sorbit which we then decompose into harmonics of themean anomaly allowing us to express the moment as afunction of time and thus obtain the GWs emitted by theperturbed star.The dynamic tide raised on a star has been analyzed indetail by linearizing the Newtonian fluid equations for anon-spinning body and finding the normal modes of thefluid displacement ξ ( x , t ) [44, 66]. The fluid displacementobeys the equation ∂ ξ ∂t + L ξ = ∇ U tidal , (1)where L is a self-adjoint operator representing the spa-tial derivatives of the perturbed Euler equations [65] and U tidal is the tidal potential. In this formalism, the dis-placement is decomposed into normal modes, ξ ( x , t ) = (cid:88) λ Q λ ( t ) ξ λ ( x ) , (2)where the eigenfunctions are normalized by an innerproduct over the star’s volume, (cid:90) ξ ∗ λ (cid:48) · ξ λ ρ ( x ) d x = δ λ (cid:48) λ , (3)with ρ being the body’s density and the asterisk denotingcomplex conjugation, and are constrained by the eigen-value equation L ξ λ − ω λ ξ λ = 0 . (4)By Fourier transformation, Eq. (1) yields a differentialequation for the mode amplitude ¨ Q λ + 2 γ λ ˙ Q λ + ω λ Q λ = U λ . (5) Note that here we have defined the damping coefficient couplingto ˙ Q λ to be 2 γ instead of γ as other references have. We do thisto avoid certain factors of two that appear in the calculation tosimplify expressions. The overall structure of the solutions doesnot change. To account for dissipative mechanisms, we’ve added thedamping rate γ λ = 1 /τ λ to account for a mode’s dissipa-tion within the fluid that occurs on a timescale τ λ . Thedriving force is U λ ( t ) = (cid:90) ξ ∗ λ · ∇ U tidal ρ d x. (6)Eq. (5) can be solved as the superposition of a homo-geneous and an inhomogeneous solution. Choosing theinitial values at some time t = 0, the first solution is Q h λ ( t ) = exp( − γ λ t ) (cid:34) Q λ (0) cos( ω (cid:48) λ t )+ ˙ Q λ (0) + γ λ Q λ (0) ω (cid:48) λ sin( ω (cid:48) λ t ) (cid:35) , (7)for the real frequency ω (cid:48) λ ≡ (cid:112) ω λ − γ λ . Using a Greens’function, the inhomogeneous solution driven by U λ is Q inh λ ( t ) = 1 ω (cid:48) λ (cid:90) t −∞ exp[ − γ λ ( t − t (cid:48) )] U λ ( t (cid:48) ) sin( ω (cid:48) λ ( t − t (cid:48) )) d t (cid:48) . (8)To begin integrating Eq (8), we use spherical harmon-ics to expand the tidal force of the black hole of mass M B [44], ∇ U tidal = M B (cid:88) lm W lm r l − R l +1 exp( − imv ) [ l e r + r ∇ ] Y lm (Ω) , (9)where the W lm coefficients are defined in Ref. [44]’s Eq.24, R is the distance between bodies, and ( r, Ω) describe apoint in space centered at the neutron star, with e r as theunit vector in the radial direction. As will be explainedin Sec. II B, v is the orbit’s true anomaly, making it afunction of time along with R . Identifying the modes asthe collection of integer indices { l, m } , m ∈ [ − l, l ], theeigenfunctions can be expanded similarly as ξ λ ( x ) = ξ lm ( r, Ω) = (cid:2) ξ Rlm ( r ) e r + ξ Slm ( r ) r ∇ (cid:3) Y lm (Ω) , (10)where the (real) functions ξ R and ξ S can be found by con-sidering hydrostatic equilibrium and the boundary con-ditions on the star’s surface [55], and the unperturbedstar is assumed to be spherically symmetric. Thanks tothe orthogonality relations (cid:90) Y lm (Ω) Y ∗ l (cid:48) m (cid:48) (Ω) dΩ = δ ll (cid:48) δ mm (cid:48) , (cid:90) r ∇ Y lm (Ω) · r ∇ Y ∗ l (cid:48) m (cid:48) (Ω) dΩ = l ( l + 1) δ ll (cid:48) δ mm (cid:48) , In wave theory, the indices are usually { n, l, m } , but f-modes arenot radial, so in our analysis the modes are independent of n .Furthermore, if the star is not spinning, the eigenfunctions andfrequencies are also independent of m [49, 55]. we can express Eq. (6) as U lm ( t ) = M B W lm K lm exp( − imv ) R l +1 , (11)where the overlap integral over the star’s radius r ∗ is K lm = l (cid:90) r ∗ r l +1 (cid:2) ξ Rlm ( r ) + ( l + 1) ξ Slm ( r ) (cid:3) ρ d r. (12)From these equations, the responses Q lm and frequencies ω lm can be constrained [55]. Solving for the response ofthe star to the gravitational potential, we can insert Eq.(11) into (8), Q lm ( t ) = M B W lm K lm ω (cid:48) lm × (cid:90) t −∞ exp[ − γ λ ( t − t (cid:48) )] exp( − imv ( t (cid:48) )) R l +1 ( t (cid:48) ) sin( ω (cid:48) lm ( t − t (cid:48) )) d t (cid:48) . (13)The inhomogeneous solution is the response of the starto the driving potential U tidal , which for a homogeneousstar excites only fundamental f-modes [66]. In general,not only are other perturbations, such as p- and g-modes,negligible compared to these, but the f-modes are primar-ily excited by the potential’s quadrupolar moment l = 2.Higher moments are suppressed by a factors of R , andas 1 /R ∼ V (the square of the orbital velocity), theyconstitute higher PN corrections. Thus, this quadrupo-lar moment is the leading PN order description of thetide. A. Quadrupole Moment
To obtain the waveform of the neutron star’squadrupole tide, we consider the symmetric trace-freequadrupole moment [65], I (cid:104) jk (cid:105) tidal ( t ) = 8 π (cid:88) m = − I m ( t ) Y ∗(cid:104) jk (cid:105) m , (14)where I m ( t ) = (cid:90) ρ ( t, x ) r Y ∗ m (Ω) d x (15)is the { l = 2 , m } multipole moment of the star, withthe asterisk indicating complex conjugation. We havealso used the spherical harmonic tensors Y (cid:104) jk (cid:105) m that aredefined as (for l = 2) Y (cid:104) jk (cid:105) m = 158 π (cid:90) n (cid:104) jk (cid:105) Y ∗ m (Ω) dΩ , (16)where n (cid:104) jk (cid:105) = n j n k − δ jk for the unit vector n j , and Y (cid:104) jk (cid:105) l, − m = ( − m Y ∗(cid:104) jk (cid:105) lm . The use of symmetric trace-freetensors (denoted by the bracketed indices) serves us toexpand the quadrupole moment in spherical harmonics,such as in Eq. 14.The utility of this formalism is more apparent when wefind the tidal contribution by perturbing the density ofthe neutron star as ρ → ρ + δρ , where in the Lagrangiandescription, δρ = − ∇ · ( ρ ξ ) (17)= − (cid:88) lm Q lm ( t ) Y lm (Ω) × (cid:20) ∂∂r (cid:0) ρξ Rlm ( r ) (cid:1) − ρr l ( l + 1) ξ Slm ( r ) (cid:21) , (18)recalling that r ∇ Y lm = − l ( l + 1) Y lm . The multipolemoment can be split into an integral over the unper-turbed star plus its deformation. As the star is spher-ically symmetric and stationary, its unperturbed density ρ does not yield a time varying quadrupole moment, andin the absence of spin can be neglected. We are then leftwith the moment of the density perturbation, I m ( t ) = (cid:90) δρ r Y ∗ m (Ω) d x = − ε m Q m ( t ) , (19)where ε lm = (cid:90) r ∗ (cid:20) ∂∂r (cid:0) ρξ Rlm ( r ) (cid:1) − ρr l ( l + 1) ξ Slm ( r ) (cid:21) r d r. (20)The quadrupole moment can now be found in termsof the tidal response. We henceforth suppress the l = 2index for convenience. As W ± = 0, the relevant coeffi-cients are W = (cid:114) π , W ± = (cid:114) π , (21)with the respective tensors Y (cid:104) jk (cid:105) = (cid:114) π diag( − , − , , (22) Y (cid:104) jk (cid:105) = Y ∗(cid:104) jk (cid:105)− = (cid:114) π − i − i − . (23)We thus obtain the quadrupole moment I (cid:104) jk (cid:105) tidal ( t ) = − πM B (cid:88) m = − ε m W m K m ω (cid:48) m Y ∗(cid:104) jk (cid:105) m × (cid:90) t −∞ exp[ − γ m ( t − t (cid:48) )] exp( − imv ) R sin( ω (cid:48) m ( t − t (cid:48) )) d t (cid:48) . (24)The amplitude of the oscillation is thus EOS-dependentthrough the constants ε m and K m . B. Harmonic Decomposition
To find the quadrupole moment explicitly, we mustintegrate Eq. (24). We investigate the time dependenceof the distance between the black hole and the neutronstar, R , and the true anomaly, v . In the Kepler problemof a binary with total mass M = M ∗ + M B , where M ∗ isthe neutron star’s mass, two other anomalies are known:the eccentric anomaly u that can be related to the orbitalphase as cos v = cos u − e − e cos u (25)and sin v = √ − e sin u − e cos u , (26)where e is the orbit’s eccentricity; and the mean anomaly (cid:96) = u − e sin u = n ( t − t p ) , (27)where n = (cid:112) M/a is the orbital frequency for a binaryof total mass M , semi-major axis a , and time of peri-center passage t p . Measured from the line of nodes, theorbit revolves by an angle φ = v + (cid:36) such that (cid:36) is thelongitude of pericenter, where the bodies are at closestapproach.These anomalies allow for useful representations of theorbit; for example, Eq. (25) allows for the equation ofthe orbit to be expressed as R = a (1 − e )1 + e cos v = a (1 − e cos u ) . (28)However, there is no simple function R ( t ) for non-circularorbits. To use v , the differential equationd φ d t = √ M pR (29)has to be integrated for v = φ − (cid:36) , or to use u , Eq. (27)has to be solved numerically. Here we have introducedthe semi-latus rectum p = a (1 − e ) of the orbital ellipse,which is a useful alternative to the semi-major axis foreccentric orbits.To calculate the orbit as an explicit function of time,we turn towards a more useful representation of the orbit:its harmonic decomposition in (cid:96) . Consider expanding afunction f of the anomalies as a Fourier series f = ∞ (cid:88) k = −∞ c k exp( ik(cid:96) ) , (30)where one can find the coefficients with c k = 12 π (cid:90) π − π f exp( − ik(cid:96) ) d (cid:96). (31)This expresses f as a sum of epicycles. With the integralrepresentation of the Bessel function J k ( x ) = 12 π (cid:90) π − π exp[ i ( ku − x sin u )] d u, (32)the following can be found:cos v = − e + 2 e (1 − e ) ∞ (cid:88) k =1 J k ( ke ) cos( k(cid:96) ) , (33)sin v = 2 (cid:112) − e ∞ (cid:88) k =1 J (cid:48) k ( ke ) sin( k(cid:96) ) , (34) aR = 1 + 2 ∞ (cid:88) k =1 J k ( ke ) cos( k(cid:96) ) , (35)where J (cid:48) k ( x ) = ddx J k ( x ). One can thus express the orbitexplicitly as a function of time, albeit in terms of infinitesums.
1. Hansen Coefficients
The orbital functions cos v , sin v , and a/r were ex-pressed as Fourier series of the orbit’s epicycles in theprevious section. However, in integrating Eq. (24), wehave the task of Fourier expanding the terms1 R , exp( ± i v ) R . We follow [55], considering the general expansion (cid:18) Ra (cid:19) q exp( imv ) = ∞ (cid:88) k = −∞ X q,mk ( e ) exp( ik(cid:96) ) . (36)We see that the quadrupole moment is composed of thecoefficients ( q, m ) = ( − , ±
2) and ( − , X q,mk areknown as the Hansen coefficients [67].With these coefficients, we can now integrate Eq. (24)to find the star’s response to the driving potential, (cid:90) t −∞ exp[ − γ m ( t − t (cid:48) )] exp( − imv ) R sin( ω (cid:48) m ( t − t (cid:48) )) d t (cid:48) = ω (cid:48) m a ∞ (cid:88) k = −∞ X − , − mk exp( ik(cid:96) ) ω (cid:48) m − ( kn ) + 2 iknγ m , (37)yielding the response Q m ( t ) = M B a W m K m ∞ (cid:88) k = −∞ X − , − mk exp( ik(cid:96) ) ω m − ( kn ) + 2 iknγ m . (38)Here we’ve assumed that the modes were quiet in thepast, such that this is the response for a single passage.Tides raised by a past passage can be added coherentlyusing the homogeneous solution (Eq. (7)). The initial conditions for a tide can be evaluated knowing the timeof each pericenter passage.It is expected that the series (36) will converge moreslowly as the eccentricity approaches 1. As one can alsosee from their definition in Eq. (31), the Hansen co-efficients can become difficult to evaluate numericallyfor large k as the integrand becomes highly oscillatory.These two facts complicate our analysis. As ω m lies in thekHz frequencies for NSs, one can see in Eq. (38) that thedominating terms of the series are of large index k suchthat the f-modes are close to resonance with the orbitalfrequency. The coefficients of interest cannot be found insimple analytic form, unless expanded in another seriesof their own [68, 69]. C. Gravitational Waves From f-Modes
In Sec. II A and II B, we calculated the f-mode exci-tation and induced quadrupole moment of a star. Eq.(24) yields this tidal response as a function of time. Itis then straightforward to find the leading order gravi-tational waves from these f-modes with the quadrupoleformula for the metric perturbation far from the source, h jk = 2 d L ¨ I jk , (39)where I jk is the quadrupole moment of the sources, and d L is the luminosity distance to origin of our coordinatesystem. Besides the moment from the motion of the bi-nary, I jk also includes the tidal response from the neu-tron star, I mode jk , so the total quadrupole moment is theorbit plus the tide [70]. However, we are interested pri-marily in the signal from the tidal excitations, and so theorbit’s radiation is excluded in this study. The gravita-tional waves from eccentric binaries at leading PN orderhave been studied in [62].To analyze the observable effects of the metric pertur-bation, we project it into the transverse traceless gauge.Setting our coordinate system on the orbital plane andcentered at its center of mass, the unit vector pointing toan observer is ˆ d = [sin ι cos ψ, sin ι sin ψ, cos ι ] , (40)where ι is the inclination angle to the binary’s orbitalangular momentum and ψ is an arbitrary polarizationangle. To describe the space orthogonal to ˆ d , we alsodefine Θ = [cos ι cos ψ, cos ι sin ψ, − sin ι ] , (41) Φ = [ − sin ψ, cos ψ, . (42)Although the position of the neutron star (the source of h jk ) is not at the center of our coordinates, the separa-tion between them is negligible compared to the distanceexpected from astrophysical sources. Thus, we can ap-proximate the line-of-sight from the star as the line-of-sight from the center of mass to leading order in 1 /d L , aswe show in Appendix A.
40 20 0 20 40 60 80time (M)0.0020.0010.0000.0010.0020.003 d L h ( t ) / m K m h mode+ h mode× FIG. 1. Plus and cross polarizations of a tidal excitation with ι = ψ = 0 (face-on binary) scaled by the luminosity distance,where the BH and NS are closest to each other at t = 0.The NS has an f-mode frequency f = ω/ π = 1 .
865 kHz anddampening timescale τ = 0 .
230 s, which is longer than theorbital period T orb = 2399 M ≈ .
133 s ( e = 0 . , p = 10 M ).. The plus and cross GW polarizations due to the tidecan then be found to be [65] h + = 12 (cid:0) Θ j Θ k − Φ j Φ k (cid:1) h jk , (43) h × = 12 (cid:0) Θ j Φ k + Φ j Θ k (cid:1) h jk . (44)Inserting Eq. (14) into Eq. (39) yields the polarizations h mode+ = 13 d L (cid:114) π ι ) (cid:104) ¨ I exp( i ψ ) + ¨ I − exp( − i ψ ) (cid:105) + 2 d L (cid:114) π ι ¨ I , (45) h mode × = i d L (cid:114) π ι (cid:104) ¨ I exp ( i ψ ) − ¨ I − exp ( − i ψ ) (cid:105) . (46)We show examples of these polarizations in Fig. 1 foran eccentric orbit ( e = 0 . , p = 10 M ) containing a 1.273M (cid:12) NS with the SLy4 equation of state (see Table XIin [71]) and a 10 M (cid:12)
BH. The ¨ I m are calculated fromEq. (19) by numerically integrating Q m ( t ) as describedin Sec. IV, where we compare the polarizations to theanalytic expressions developed in Sec. III. III. f -MODES IN EFFECTIVE FLY-BYS While in Section II we provided an analytic descriptionof the tides raised on a star in an arbitrary binary, wenow focus on eccentric binaries. We’ve mentioned thatthe series in orbital harmonics Eq. (36) converges moreslowly as e → e have sought to extend the range of circular models.Recently, a re-summation method that instead expandsthe harmonics in large e has been investigated for hered-itary fluxes [64] and resulted in waveforms for eccentricbinaries to leading order [62]. These waveforms replaceEqs. (33)-(34) with their asymptotic expansions about k (cid:29) k . This method yields accuraterepresentations of eccentric orbits, eliminating the needto sum over many terms. In this representation, the os-cillatory orbit is replaced by a post-parabolic orbit, thusmore accurately describing an effective fly-by. We nowapply this re-summation method to f-modes as follows. A. Re-summation of f -Modes Consider the tide we found in Eq. (38). As k rangesthrough all integers, there will be values at which it’llcome close to resonating with the f-mode, especially asthis frequency is expected to surpass the orbit and thedamping timescale. Therefore the dominant term in theseries is not at low values of k , but rather at high valueswhere the resonance dominates, as shown in Fig. 2. Withthe dominant terms being at large k , we can re-sum ourseries with an integral as ∞ (cid:88) k = −∞ → (cid:90) ∞−∞ d k, (47)since all epicycles contribute to the sum in the high ec-centricity limit.Applying this to the tide in Eq. (38), we now have thetask of integrating (cid:90) ∞−∞ X − , − mk exp( ik(cid:96) ) ω − ( kn ) + 2 iknγ d k. (48)Note that the integrand has poles at k ± ≡ n ( iγ ± ω (cid:48) ) , (49)which lie on the positive imaginary plane. We can thenmake the integral as part of a contour integral over thisplane, and is thus accompanied by an integral over asemi-circle on it, with our integral as its border on thereal axis. The integral over the semi-circle vanishes aslim k → + i ∞ exp( ikl ) → (cid:96) >
0; for (cid:96) <
0, we wouldplace the semi-circle on the negative imaginary planewhere the absence of poles would make the contour in-tegral vanish by Cauchy’s theorem. Only the integral onthe real axis is left, and we can cast the contour integral k | X , k / ( ( k n ) + i k n ) | FIG. 2. The absolute value of the Fourier coefficients of Q for the same system in Fig. 1. The dashed line marks where k = ω/n ≈ .
4. The bump in the tens of k corresponds tothe lower harmonics present in the series. as a sum of residues: (cid:90) ∞−∞ X − , − mk exp( ik(cid:96) ) ω − ( kn ) + 2 iknγ d k =2 πi (cid:88) k ± Res (cid:34) X − , − mk exp( ik(cid:96) ) ω − ( kn ) + 2 iknγ , k ± (cid:35) . (50)Note that this applies only for (cid:96) >
0, that is, after peri-center passage. In the high eccentricity limit, the tide israised at closest approach, with the black hole’s poten-tial ringing the neutron star like a bell. Evaluating theresiduesRes (cid:34) X − , − mk exp( ik(cid:96) ) ω − ( kn ) + 2 iknγ , k ± (cid:35) = ∓ X − , − mk ± exp (cid:2) − (cid:96)n ( γ ∓ iω (cid:48) ) (cid:3) nω (cid:48) , (51)we obtain the tide Q m ( t ) = M B (cid:18) − e p (cid:19) W m K m πinω (cid:48) exp( − γ ∆ t ) × (cid:16) X − , − mk − exp ( − iω (cid:48) ∆ t ) − X − , − mk + exp ( iω (cid:48) ∆ t ) (cid:17) , (52)where ∆ t ≡ t − t p . We refer to this as the re-summedf-mode.Naively, it appears the re-summed mode in Eq. (52)vanishes in the parabolic limit ( e → e ap-proaches unity. The divergence can be shown explicitlyin X − , , which can be calculated explicitly, X − , = (1 − e ) − / . (53)The remaining coefficients X − ,mk (cid:54) =0 do not admit exactclosed-form expressions, and we are forced to numeri-cally evaluate them. Generically, these coefficients areenhanced, and appear to diverge, as the eccentricity ap-proaches unity. We have numerically shown that multi-plying X − , − mk by (1 − e ) / for m = 0 , ± k up to 10,000. Thus, we postulate thatwe can write X − ,mk ( e ) = ˆ X − ,mk ( e ) / (1 − e ) / , whereˆ X − ,mk ( e ) is regular for all values of e . Further, one mustremember that n ∝ (1 − e ) / , and thus, it is now clearthat the product (1 − e ) X − , − mk /n in the amplitudeof the re-summed mode neither diverges nor vanishes inthe limit of high eccentricity. The star thus responds asa damped harmonic oscillator and is well behaved in theparabolic limit.An interesting effect of the re-summation is the pro-motion of k from integer to complex number, which alsocauses the Hansen coefficients to become complex-valued.The symmetry X − ,mk + = X − , − mk − appears, which aidsin evaluating these coefficients. Nonetheless, the lack ofan analytic form for X − ,mk keeps us from obtaining acompletely analytic model, and for now we continue nu-merically calculating the coefficients using the Pythonpackage SciPy’s quad module.Fig. 3 shows how well the re-summation Eq. (52)approximates the numerically integrated Q m for m = −
2. For the numerical integration, we start the bi-nary at apocenter half an orbital period away from clos-est approach and assume that there is no pre-existingmode driving on the star, specifically Q m ( − T orb /
2) =0 = ˙ Q m ( − T orb / − and is phase-accurate. Our approxima-tion, however, does miss the large amplitude excitationat closest passage, which is caused by the low k values inthe sum in Eq. (38). Similar results are obtained for theother m modes.If no f-modes are present in the star before a passage,only the inhomogeneous solution to the wave equation inEq. (5) is excited. On subsequent passages, however, pastexcitations must be taken into account if they haven’tbeen dissipated away due to GW emission or viscosityin the star. At the close separations that the star andBH must be for a significant tide to be excited, the timebetween passages may not be enough for this dissipationto have dampened the modes away. Therefore to con-struct a sequence of tidal excitations we must add themodes from previous passages. This can be done by us-ing the homogeneous solution (7), evaluating the initial R e { Q / K } numericre-summation
100 50 0 50 100 150time (M)10 a b s . r e s i d u a l s I m { Q / K } numericre-summation
100 50 0 50 100 150time (M)10 a b s . r e s i d u a l s FIG. 3. The real (left) and imaginary (right) amplitudes of the m = − R e { Q / K } numericre-summation a b s . r e s i d u a l s FIG. 4. The real amplitude of the m = − conditions Q ( t = t p ,N +1 ) and ˙ Q ( t = t p ,N +1 ) at the N + 1pericenter passage, and adding it to the first excited tide described by Eq. (52). This requires knowing each t p ,N and the value of n at that time. We demonstrate theresults of this procedure in Fig. 4, where n is a constantwithout radiation or tidal back-reaction. The expecta-tion is that with a model detailing the evolution of n ,one can evaluate the initial conditions in the same wayand obtain a sequence of tides that remain as accurate. B. Timing Model
The dynamical tide we obtained in Sec. III A respondsto the BH’s tidal potential, its amplitude depending onthe value of n at t p ,N +1 (see Eq. (49)). Were this asimple Keplerian orbit, this would be a known constant,as in Fig 4; however, the orbit is radiating away energyand angular momentum, necessitating a radiation reac-tion model that informs the evolution of the orbit. Wenow detail our model.To leading order, the radiation reaction of an or-bit appears at 2.5PN. The orbit loses energy and an-gular momentum to gravitational radiation, changingits Newtonian description over time. We parametrizethe osculating orbit with p and e , which then defines n = (cid:112) M (1 − e ) /p and the energy and angular mo-mentum as E = − M η (1 − e ) / p and L = η (cid:112) M p , re-spectively, where η ≡ M B M ∗ /M is the symmetric massratio. In the harmonic gauge [65], the orbital evolutionis described byd p d t = − η (cid:18) Mp (cid:19) (1 + e cos( v )) (4 + e + 5 e cos( v )) , (54)d e d t = − ηM (cid:18) Mp (cid:19) (1 + e cos( v )) (cid:2) (96 + 109 e ) cos( v ) + e (104 + 6 e + 2(56 + 9 e ) cos(2 v ) + 35 e cos(3 v )) (cid:3) , (55)0d (cid:36) d t = − ηeM (cid:18) Mp (cid:19) (1 + e cos( v )) (cid:2) e ) + e (56 + 9 e ) cos( v ) + 35 e cos( v ) (cid:3) sin( v ) , (56)which also changes the evolution of v in Eq. (29) tod v d t = √ M pR − d (cid:36) d t . (57)Averaging these equations over an orbit yields the adia-batic approximation, first calculated by Peters [72], (cid:28) d p d t (cid:29) = − η (cid:18) Mp (cid:19) (1 − e ) / (cid:18) e (cid:19) , (58) (cid:28) d e d t (cid:29) = − eηM (cid:18) Mp (cid:19) (1 − e ) / (cid:18) e (cid:19) . (59)Further, the secular change in (cid:36) vanishes. However, thisapproximation can lose accuracy for small p and large e ,affecting the timing model.The accuracy of the timing model is key to calcu-lating the time of pericenter passage and their Hansencoefficients. One can obtain the orbital parameters { p N +1 , e N +1 } at the N + 1 passage by considering thatthey change primarily at pericenter, where almost all ofthe radiation occurs. A timing model can then be ob-tained from the orbit-averaged evolution [33, 62] p N +1 = p N (cid:34) − π η (cid:18) Mp N (cid:19) / (cid:18) e N (cid:19)(cid:35) , (60) e N +1 = e N (cid:34) − π η (cid:18) Mp N (cid:19) / (cid:18) e N (cid:19)(cid:35) . (61)This yields a sequence of orbital parameters at each peri-center passage. To obtain the time at which these pas-sages occur, the time from one passage to the next is theorbital period T orb = 2 π/n of the intermediate osculatingorbit, t p ,N +1 = t p ,N + 2 π (cid:118)(cid:117)(cid:117)(cid:116) p N +1 / M (1 − e N +1 / ) , (62)where { p N +1 / , e N +1 / } are the orbital parameters atapocenter between the N and N + 1 passage. This isdue to the fact that for the duration of an orbit exceptat pericenter, the orbital parameters for an eccentric bi-nary don’t change significantly. Thus, the time in be-tween passages is going to be determined by the periodof that orbit, aptly described by the parameters furthestfrom pericenter. These can be determined by substitut-ing N → N − / N passageas p N +1 / = p N + p N +1 − p N , (63) e N +1 / = e N + e N +1 − e N . (64)This timing model yields highly accurate results on or-bital timescales when comparing the orbital parametersto the values found with numerical integration. This thusbuilds a sequence of pericenter passages, from which wecan evaluate the tides raised on the star. We explore theaccuracy of this model for the tides in the next section. IV. ACCURACY OF THE RE-SUMMED MODES
With the re-summation of the f-modes presented inSec. III A and timing model from Sec. III B, we now havean analytic model for the leading-order GWs of theseoscillations, save for the Hansen coefficients, which wecalculate numerically in this study for the reasons statedin Sec II B 1.In this section we compare the re-summation to nu-merical integration of the orbit and the f-modes. Theintegration of the latter is performed by solving Eq. (5)with the driving force Eq. (11) using the 4(5)th orderRunge-Kutta method from SciPy’s solve_ivp . Simulta-neously, the orbit is integrated with Eq. (57) while evolv-ing p , e , and (cid:36) with Eqs. 54 - 56. This provides a 2.5PNnumerical model of the orbit that excludes tidal effects.While it has been shown that the tide also causes theorbit to decay faster than from radiation reaction alone[55, 57], it is generally sub-dominant to the gravitationalradiation. The combined effects of radiation reaction andthe tide will be investigated in a future work. A. Comparison to Numerics
We start our numerical integrations at apocenter t = − T orb , /
2, where T orb , is the period of the orbit at t .As the orbit progresses, e and p decrease due to radiationreaction. We integrate the orbit until either t = T orb , or p < M (3+ e ), the latter corresponding to the last stableorbit for Schwarzschild geodesics. With v , p , e , Q m ( t ),and ˙ Q m ( t ) calculated, we can evaluate Eq. (5) to find¨ I m and their GWs.We set the initial conditions Q m ( t ) = ˙ Q m ( t ) = 0for the f-modes. Physically, these initial conditions areonly valid when the binary is infinitely far apart on anunbound orbit, such that the tidal force can be neglected.If starting the evolution at a finite separation as we aredoing here, one would expect the modes to not be quietfor any e <
1. Essentially, the binary will have to haveevolved from a dynamical capture to an orbit with thespecificy e , and thus modes would have been excitedfrom previous pericenter passages. However, if we simply1 d L h m o d e + ( t ) / m K m numerictiming modelre-summation a b s . r e s i d u a l s FIG. 5. The plus polarization around the second passage of anorbit with e = 0 . p = 10 M . Three models of the tide areshown: the numeric (solid black), the timed (blue dashed),and re-summed (orange dashed). The latter two use the re-summation, but the pericenter passages for the first are timedusing the model of Sec. III B while those for the second arefound numerically. The bottom panel shows the residuals ofthese two models to the numeric. The black vertical dashedline marks the time of true pericenter passage, while the bluemarks the timed passage. desire to study the accuracy of the re-summed mode,it suffices to compare to numerical evolutions with theinitial conditions Q m ( t ) = 0 = ˙ Q m ( t ). It is worthnoting that using such data at a finite orbital separationresults in a low amplitude mode being activated beforethe initial pericenter passage, especially when e < . t p , ,and let the timing model (Eqs. 60 - 62) calculate them atsubsequent passages. This model appears in blue, dashedlines. The black vertical line marks the numerical t p , ,and it is evident that it is offset from its timed coun-terpart in blue. Compared to orbital timescales, this 4M difference is only a 0.35% relative error of the truepericenter passage time. However, the f-mode is fasterthan the orbit, making this difference 5% of the f-modewavelength. Thus, while the timing model provides greataccuracy on orbital timescales, it fails for timing f-modes,and the error increases by an order of magnitude after the passage. Yet if we give the re-summation the true time ofpericenter passage while still using Eqs. (60)-(61) for theevolution of p and e , we find that the model retains itsorder of accuracy across the passage. This is illustratedby the model plotted in the dotted line.The timing model (Eqs. (60) - (62)) was originallyderived by assuming any changes in the orbital parame-ters occur instantaneously at pericenter passage, in accor-dance with the burst-like nature of the gravitational ra-diation. Thus, the orbital parameters are assumed to beconstant between subsequent pericenter passages, whichresults in the timing algorithm in Eq. (62). However,when the orbital is sufficiently compact, i.e. p is small,this model breaks down because the orbital parameterscan change significantly on an orbital timescale. To prop-erly handle such effects, one would have to solve the exactradiation reaction equations using a multiple scale analy-sis, which would properly take into account secular effectsbeyond the adiabatic limit. In such a problem, there aretwo timescales, namely the orbital timescale T orb and theradiation reaction timescale T rr = p/ | dp/dt | . However,because we are also dealing with NSs, a timing modelwould also need to incorporate the tidal effects of thestars, and thus there is one additional timescale, namely T mode ∼ /ω . Multiple scale analysis becomes more com-plicated the more scales of relevance to the problem. Dueto the difficulty of this calculation, we leave the derivationof an accurate multiple scale timing model that capturesboth radiation reaction and tidal effects to future work. B. Waveform Match
In Sec. IV A, we compared the time-domain waveformsevaluated from numerical integration of the f-modes andfrom their re-summation for a specific e and p . To es-timate how well the re-summation would perform in de-tecting an f-mode GW, however, one should vary theseinitial parameters to understand where our fly-by approx-imation applies. In this section, we do this across a rangeof parameters to calculate the match of a single f-modeexcitation between the re-summation and the numericalwaveforms.The match between the waveforms h A and h B is de-fined as M = max ( δt,δφ ) (cid:0) h A (cid:12)(cid:12) h B exp[2 πif δt + iδφ ] (cid:1)(cid:113)(cid:0) h A (cid:12)(cid:12) h A (cid:1) (cid:0) h B (cid:12)(cid:12) h B (cid:1) (65)where δt and δφ are overall time and phase shifts, andthe noise-weighted inner product is (cid:0) h A (cid:12)(cid:12) h B (cid:1) = 4 (cid:60) (cid:90) ˜ h A ( f )˜ h ∗ B ( f ) S n ( f ) d f (66)where ˜ h ( f ) is the Fourier transform of h + ( t ) and S n is thespectral noise density of the detector under consideration.The match generally ranges from 0 (A and B are perfectly2
10 11 12 13 14 15 p (M)0.8000.8250.8500.8750.9000.9250.9500.975 e . . . p (M)0.8000.8250.8500.8750.9000.9250.9500.975 e . . . FIG. 6. Match between re-summed modes and numerical waveforms for BH-NS binaries as a function of e and p with M B = 10M (cid:12) and the SLy4 NS from Fig. 1. The match on the left was calculated with the aLIGO sensitivity curve while the match onthe right was calculated with ET. Contour lines for M > .
95 are shown. out of phase) to 1 (A and B are in perfect agreement)[40]. Thus, the closer M is to 1, the more accurate ourre-summation models the numerical waveform.Similarly as in Sec. IV A, we generate the numerical(A) and re-summed (B) plus polarizations for orbits witha range of parameters e ∈ [0 . , .
0) and p /M ∈ [10 , Hz. These waveforms are cut offat a time t f = t p , + 0 . t p , − t p , ) to exclude any f-modeexcitations from subsequent pericenter passages. Theyare then padded on either side to have a total lengthof 2 , sufficient to cover the longest generated orbit.We finally perform their Fourier transformation with theSciPy module fft .To calculate the match, we use the publicly availableAdvanced LIGO [73] and ET-D high frequency config-uration [74] sensitivity curves. While both have a sim-ilar sensitivity in the decahertz range, ET improves onLIGO’s strain by an order of magnitude in the kilohertz.This is the range in which f-modes oscillate, and thus ofimportance for their detection. While ET has yet to startconstruction, our estimates here provide an outlook forthe future of GW science.Fig. 6 shows the values of Eq. (65) for our rangeof orbital parameters. At pericenter passage, where thetide is excited, e and p will remain close to their initialvalue. The re-summation performs best at larger e andsmaller p , reaching matches of 0.97 above e = 0 . p = 12 M . The more eccentric the orbit is, thesharper the tidal excitation is at pericenter; the closerthe bodies are at that point, the stronger the excitationis. For lower e /larger p orbits, the tidal excitation of thestar also generates an adiabatic “spike” during pericen-ter passage, which contributes to the gravitational wavesignal at frequencies lower than the f-mode frequency ω .This can be seen from Fig. 2, which has power at values k ∼ − e /small p where tidal effects are the strongest, and wehave found through the match calculation that the re-summed mode is capable of accurately capturing theseeffects. It is worth noting that difference between thedetectors is small in our parameter space. In the upperleft corner ( M > . ∼ − of the LIGO match. V. DISCUSSION
As GW observatories continue to detect binary merg-ers from space, the likelihood increases of detecting a bi-nary with signatures of dynamical formation, making itimperative to have waveform models of eccentric binariesat hand. Here we have developed the first analytic wave-forms for the f-modes from highly eccentric BH-NS bina-ries using the effective fly-by framework. At leading PNorder, we solved the f-modes in harmonics of the Keple-rian orbit and re-summed them to obtain a damped har-monic oscillator excited at closest approach. Comparingthe re-summed modes to their numerical integration, wefind that they are an accurate representation for highlyeccentric and close pericenter passages, where the tidesare excited sharply. We have also outlined and shown thefeasability of timing these excitations and adding themcoherently in sequence. However, this model is incom-plete, and further work remains to be done to provide amore comprehensive analysis.In this study, we have primarily focused on BH-NSbinaries. However, the results presented here are alsoapplicable to BNSs in the following way. In such a case,f-modes would be generated on both NSs, being sourcedby the tidal potential of the companion. To leading PN3order, this tidal potential is given by the monopole termsof the gravitational potentials of the compact objects.Thus, the total f-mode response and their gravitationalwaves would simply be the sum of the f-modes from bothcomponents. At higher PN orders, the f-modes will con-tribute to the tidal potential, but the response will besuppressed by V . We thus expect the non-linear inter-action between modes to be subdominant, and one cantreat BNSs using the superposition of modes modeled aswe have done so here.The primary concern with the current model is the tim-ing of pericenter passages, which we showed in Sec. IV Ato be inaccurate at close passages, returning tides offsetfrom their true value and phase. Yet if given the correcttime, the re-summation yields accurate modes that canbe stitched together. Thus, work is needed on improvingthe timing model’s accuracy beyond the adiabatic ap-proximation of the 2.5 PN radiation reaction. Anotherconcern is the inclusion of tidal effects on the orbital evo-lution, which hastens the decay of an orbit and introducesnew physics into the timing model. We will pursue thesetasks in a future study to further the model presentedhere.A source of contention in the applicability of our modelare the Hansen coefficients, for which an analytic formhas eluded us. As the indices of interest k ± ∼ ω m /n are of large magnitude, one might expect the stationaryphase approximation (SPA) to be useful in evaluatingEq. (31), but the SPA does not account for k ± havingan imaginary component. Resorting to evaluating thecoefficients with numerical quadrature has shown to takea large portion of the time spent calculating re-summedmodes, which makes this procedure less attractive forobtaining models on the fly. Tabulating X − ,mk acrossvalues of k may alleviate this concern, but nonethelessholds back the model from being fully analytic. Analysisof these coefficients and a closed form for them wouldgreatly improve application of the f-mode re-summation.In conclusion, we have only begun development of an-alytic waveforms for highly eccentric binaries that in-clude the effects of dynamical tides. While the modelcan be improved by much, we have shown how wellour re-summation can match with numerical waveformsfor eccentric and close pericenter passages. Such orbitscan form in dense stellar environments, and if their pa-rameters are within our region of high match, our re- summation would be a candidate for detecting them andthus characterizing their formation channel. ACKNOWLEDGMENTS
N.L. acknowledges support from NSF grant PHY-1912171, the Simons Foundation, and the Canadian In-stitute for Advanced Research (CIFAR). We would liketo thank Frans Pretorius for useful discussions.
Appendix A: Line-of-Sight Vector to NSs
To project the metric perturbation due to the NS to thetransverse traceless gauge, we establish a tetrad definedby a line-of-sight vector to an observer from the NS’scenter of mass, ˆ d ∗ = D ∗ / | D ∗ | , where D ∗ = d L ˆ d − R ∗ , (A1)and ˆ d is defined in Eq. (40) along with the luminositydistance d L . R ∗ is the vector from the binary center ofmass to the NS, R ∗ = M B M R [cos φ, sin φ, , (A2)where R and φ describe the binary orbit (see Sec II B).We define two vectors perpendicular to this line-of-sight, y ∗ = ˆ z × ˆ d ∗ , x ∗ = y ∗ × ˆ d ∗ (A3)to complete the tetrad with the transverse unit vectors Θ ∗ = x ∗ / | x ∗ | , Φ ∗ = y ∗ / | y ∗ | . (A4)Unlike ˆ d for a stationary orbit (with respect to the ob-server), these vectors are time-dependent, describing theNS’s revolution around the binary’s center. This mod-ulates the f-mode’s waveform on an orbital timescale.However, the separations between the NS and the cen-ter of mass we consider for tidal effects to be significantdo not exceed the hundreds of kilometers, while the dis-tances to binaries are expected to be at least in the kilo-parsecs if in our galaxy. Thus, expanding this tetradin 1 /d L , we find that the leading order terms give Eqs.(40)-(42), the same tetrad for the binary’s center of mass. [1] J. Aasi et al. (LIGO Scientific), Advanced LIGO, Class.Quant. Grav. , 074001 (2015), arXiv:1411.4547 [gr-qc].[2] B. Abbott et al. (LIGO Scientific, Virgo), Observation ofGravitational Waves from a Binary Black Hole Merger,Phys. Rev. Lett. , 061102 (2016), arXiv:1602.03837[gr-qc].[3] B. Caron, A. Dominjon, C. Drezen, R. Flaminio,X. Grave, et al. , The Virgo interferometer,Class.Quant.Grav. , 1461 (1997). [4] R. Abbott et al. 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