Dynamical tides during the inspiral of rapidly spinning neutron stars: Solutions beyond mode resonance Hang Yu * eXtreme Gravity Institute, Department of Physics, Montana State University, Bozeman, Montana 59717, USA Phil Arras Department of Astronomy, University of Virginia, P.O. Box 400325, Charlottesville, Virginia 22904, USA Nevin N. Weinberg Department of Physics, University of Texas at Arlington, Arlington, Texas 76019, USA (Received 2 April 2024; accepted 14 June 2024; published 17 July 2024) We investigate the dynamical tide in a gravitational wave (GW)-driven coalescing binary involving at least one neutron star (NS). The deformed NS is assumed to spin rapidly, with its spin axis antialigned with the orbit. Such an NS may exist if the binary forms dynamically in a dense environment, and it can lead to a particularly strong tide because the NS f-mode can be resonantly excited during the inspiral. We present a new analytical solution for the f-mode resonance by decomposing the tide into a resummed equilibrium component varying at the tidal forcing frequency and a dynamical component varying at the f-mode eigenfrequency that is excited only around mode resonance. This solution simplifies numerical implementations by avoiding the subtraction of two diverging terms as was done in previous analyses. It also extends the solution’s validity to frequencies beyond mode resonance. When the dynamical tide back reacts on the orbit, we demonstrate that the commonly adopted effective Love number is insufficient because it does not capture the tidal torque on the orbit that dominates the back reaction during mode resonance. An additional dressing factor originating from the imaginary part of the Love number is therefore introduced to model the torque. The dissipative interaction between the NS and the orbital mass multipoles is computed including the dynamical tide and shown to be subdominant compared to the conservative energy transfer from the orbit to the NS modes. Our study shows that orbital phase shifts caused by the l ¼ 3 and l ¼ 2 f-modes can reach 0.5 and 10 radians at their respective resonances if the NS has a spin rate of 850 Hz and direction antialigned with the orbit. Because of the large impact of the l ¼ 2 dynamical tide, a linearized analytical description becomes insufficient, calling for future developments to incorporate higher-order corrections. After mode excitation, the orbit cannot remain quasicircular, and the eccentricity excited by the l ¼ 2 dynamical tide can approach nearly e ≃ 0.1, leading to nonmonotonic frequency evolution which breaks the stationary phase approximation commonly adopted by frequency domain phenomenological waveform constructions. Lastly, we demonstrate that the GW radiation from the resonantly excited f-mode alone can be detected with a signal-to-noise ratio exceeding unity at a distance of 50 Mpc with the next-generation GW detectors. DOI: 10.1103/PhysRevD.110.024039 I. INTRODUCTION The first observation of a binary neutron star (BNS) merger on August 17, 2017, demonstrated the future promise of detecting imprints of matter effects of a neutron star (NS) in gravitational wave (GW) signals [1–4]. Detecting an NS with a significant spin misaligned with the orbit can be especially interesting for at least two reasons. First, such a system can be highly informative about the formation history of the binary. A misaligned spin has been considered to be smoking gun evidence that the binary formed dynamically in a dense stellar environment [5]. While most NSs in the Galactic BNS population have low spins [6], nothing fundamental prevents a rapidly spinning NS in a coalescing binary. The maximum dimensionless spin an NS can have before it breaks up is about 0.67 [7], or about a spin rate of 1000 Hz (depending on the equation of state). The fastest-spinning pulsar known, J1748-2446ad, has a spin rate of 716 Hz. It resides in the globular cluster Terzan 5 and has a light companion [8], both conditions favorable for*Contact author: hang.yu2@montana.edu PHYSICAL REVIEW D 110, 024039 (2024) 2470-0010=2024=110(2)=024039(26) 024039-1 © 2024 American Physical Society https://orcid.org/0000-0002-6011-6190 https://ror.org/02w0trx84 https://orcid.org/0000-0001-5611-1349 https://ror.org/0153tk833 https://orcid.org/0000-0001-9194-2084 https://ror.org/019kgqr73 https://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevD.110.024039&domain=pdf&date_stamp=2024-07-17 https://doi.org/10.1103/PhysRevD.110.024039 https://doi.org/10.1103/PhysRevD.110.024039 https://doi.org/10.1103/PhysRevD.110.024039 https://doi.org/10.1103/PhysRevD.110.024039 the dynamical production of GW sources [9]. If an NS similar to J1748-2446ad is captured into a compact and/or highly eccentric orbit through, e.g., binary-single encoun- tering [10], it may merge quickly while retaining a significant spin when entering the sensitivity band of a ground-based GW detector. Second, matter effects can also lead to rich dynamics in the binary’s evolution when the spin is misaligned [11]. It can both modify the precession of the orbit [12,13] and cause extra dephasing of the GW signal through both Newtonian (via tidal interactions [14–20]) and post-Newtonian (or PN via, e.g., quadrupole-monopole interactions [21,22]) effects. Of particular interest to this work is the tidal interaction and we will focus on a special scenario in which the deformed NS has a spin antialigned with the orbit. It has been shown in both theoretical studies [23–27] and numeri- cal relativity simulations [28–30] that tidal deformation is significantly amplified when the spin is antialigned, because the antialigned spin shifts the eigenfrequency of the NS fundamental mode oscillation, or f-mode, down to a lower value in the inertial frame (or equivalently, it shifts the tidal forcing frequency up in the frame corotating with the NS), so that the NS f-mode can be resonantly excited during the inspiral stage. While leading to a strong signal, the resonant excitation of a mode also leads to theoretical challenges in modeling this effect. Some earlier milestones in this effort include Ref. [31] in which the author computed the asymptotic energy transfer of a resonantly excited mode, and Ref. [32] where the authors identified the key effect of a resonantly excited mode can be modeled as a jump in the orbit’s frequency evolution at the mode resonance. Other works discussing mode resonances in coalescing binaries include Refs. [23,24,26] for f-modes, Refs. [31,33–37] for gravity modes, Refs. [23,38–41] for Rossby/inertial modes, and Refs. [42–44] for interface/crustal modes. While a model as simple as a jump in the frequency evolution at the mode resonance may be sufficient in modeling other modes whose tidal coupling is small, for the f-mode that dominates the tidal interaction, more detailed modeling is desired when constructing faithful GW waveform templates for parameter estimation. In particular, such a model should smoothly connect the analysis of f-mode in the adiabatic (zero driving frequency) limit [14–20] to those that include corrections from finite- frequency effects, or even mode resonance. Analytical models for the f-mode’s evolution and its orbital back- reaction can be highly valuable for parameter estimation purposes because they can be generated in large quantities with manageable computational costs. The inference of tidal signatures in BNS signals further plays a central role in constraining the equation of state of NS matter [45–52]. An analytical model can also guide the empirical calibration against numerical simulations in the nonlinear regime [53,54]. Some major steps forward in integrating the f-mode resonance into GW waveform construction under the effective-one body (EOB) framework [55] include Refs. [56,57] for nonspinning NSs and Ref. [25] for spinning ones. The authors proposed to modify the tidal Love number with a frequency-dependent “dressing factor”, known as the “effective Love number” so that the tidal backreaction on the orbit takes the same form as in the adiabatic limit. The effective Love number description quickly gained popularity in the literature when modeling NS dynamical tides. See, e.g., Refs. [27,58–64]. However, there are a few aspects to be improved for the analysis in Ref. [25]. One is that its numerical implemen- tation may not be ideal when mode resonance does occur. The lowest-order finite-frequency correction of the tide has a diverging behavior at mode resonance, which is unphys- ical because the mode spends only a finite amount of time near resonance [31]. The authors of Ref. [25], following their earlier analyses in Refs. [56,57], introduced another diverging term with an opposite sign to cancel the diver- gence. A similar procedure to remove the divergence is also adopted in Ref. [24]. While theoretically accurate, sub- tracting two diverging terms to get a perfect cancellation can be challenging in numerical implementations. Moreover, the evolution of the tidally induced NS mass multipole in the formulation of Ref. [25] is accurate only up to resonance, and the effective Love number is adequate in describing the backreaction only when the tidal forcing frequency is below resonance. In particular, we will show explicitly that the effective Love number fails to describe the torque between the NS and the orbit, which dominates the tidal backreaction around mode resonance. It is actually what leads to the jump in the waveform’s frequency evolution shown in earlier analyses [32]. The effective Love number is also insufficient in describing the GW radiation due to the coherent interaction of NS and orbital mass multipoles when finite frequency corrections are included. Indeed, a recent analysis in Ref. [61] comparing waveforms generated following Ref. [25] found it insuffi- cient in describing systems where f-mode resonance hap- pens, and the effective Love number treatment does not improve the agreement with numerical relativity when compared with models without f-mode resonances [20,65]. We therefore aim to improve the analysis by Ref. [25] in describing the NS f-mode resonance including the leading- order correction due to NS spin. We will start by enhancing the treatment of the mode amplitude evolution in Sec. II so that all diverging terms are eliminated analytically for easy numerical implementation. More importantly, we extend the validity of the solution to frequencies beyond mode reso- nance. In Sec. III, we examine the relation between NSmass multipoles, the effective Love number, and tidal back- reactions on the orbit. We will show that the commonly adopted effective Love number captures only the radial interaction but misses the tidal torque, yet the torque is what dominates the backreaction around mode resonance. A new HANG YU, PHIL ARRAS, and NEVIN N. WEINBERG PHYS. REV. D 110, 024039 (2024) 024039-2 dressing factor is therefore introduced to describe the torque. It is followed by Sec. II B where we describe the system’s evolution throughout the inspiral, including both pre and postresonance regimes. We consider in Sec. IV B the tidal phase shift in the GW signal using both energy balancing arguments and osculating orbits. The osculating orbits further allow us to examine in Sec. IV C the deviation from the quasicircular inspiral in the postresonance regime. We also analyze the detectability of the GW from the resonantly excited f-mode in Sec. IV D. Lastly, we conclude and discuss in Sec. V. Throughout the paper, we adopt geometrical units with G ¼ c ¼ 1. We will use M1 to denote the mass of the tidally deformed NS and R1 its radius. Together they lead to natural units E1 ≡M2 1=R1 and ω1 ≡ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi M1=R3 1 p . The mass of the companion is denoted with M2. To describe the orbital dynamics, we further define Mt ¼ M1 þM2, μ ¼ M1M2=Mt, and η ¼ μ=Mt as the total mass, reduced mass, and symmetric mass ratio. As our focus is the tidal interaction, we simplify the analysis by modeling the orbital dynamics at Newtonian order while including the leading order quadrupole GW radiation. A more rigorous analysis should incorporate relativistic effects, as they are crucial in the late inspiral stage. This would entail replacing our Newtonian Hamiltonian to, e.g., that constructed under the EOB framework, as done in Refs. [25,56,57]. We defer such an upgrade to a future study. II. ANALYTICAL SOLUTION OF TIDAL AMPLITUDE We begin our discussion by first examining the evolution of the amplitude of a tidally driven NS mode throughout the inspiral stage. The mode amplitude is directly related to the tidally induced mass multipoles [see later Eq. (38)] and is what determines the backreaction on the orbit, a topic we will discuss in detail in Sec. III. The motion of a perturbed fluid parcel with Lagrangian displacement ξ can be written in the corotating frame to linear order in ξ as [66] ̈ξ þ 2Ω × ξ̇ þ Cξ ¼ aext; ð1Þ where Ω is the spin vector, C is a self-adjoint operator describing the hydrodynamic response of the star, and aext is an external acceleration acting on the star. We perform a phase-space decomposition of the fluid into eigenmodes as [66] � ξ ξ̇ � ¼ X a qa � ξa −iωaξa � ; ð2Þ where qa ¼ qaðtÞ is a mode’s amplitude whose temporal evolution is to be solved in this section, and ξa ¼ ξaðrnsÞ is the mode’s spatial eigenfunction. We use rns to denote a fluid element’s position inside the NS, which should be distinguished from the orbital separation denoted by r. The summation runs over all the radial and angular quantum numbers as well as the signs of the mode’s eigenfrequency ωa (in the corotating frame). When acting on an eigen- mode, the operator C satisfies Cξa ¼ ω2 aξa þ 2iωaΩ × ξa: ð3Þ The equation of motion for each mode in the corotating frame is given by [66] q̇a þ iωaqa ¼ fa; ð4Þ where fa ¼ i ϵa hξa; aexti; ð5Þ ϵa ¼ 2hξa;ωaξa þ iΩ × ξai: ð6Þ For this work, we will focus on the case where the binary moves in the x-y plane in an initially quasicircular orbit, and Ω is either aligned (Ω > 0) or antialigned (Ω < 0) with the z-axis, the direction of orbital angular momentum. We further consider the case where the fluid is perturbed by the companion’s tidal field so aext ¼ −∇U. We incorporate the spin as a small perturbation. To linear order, the eigenfrequency of a mode a in the corotating frame ωa is related to its nonrotating value ωa0 via [67,68] Δωa ≡ ωa − ωa0 ¼ −i hξa;Ω × ξai hξa; ξai ¼ −maΩ R drnsϱr2nsð2ξa;rξa;h þ ξ2a;hÞR drnsϱr2ns½ξ2a;r þ laðla þ 1Þξ2a;h� ; ≃ − ma la Ω: ð7Þ Here ξa;r and ξa;h are the radial and tangential components of ξa (see, e.g., Ref. [69]), which are evaluated for a nonrotating star. In the last line we have approximated the density ϱ as a constant and ξa;r ≃ lξa;h ∼ rl−1ns for the f-modes. Note that our result shows good agreement with, e.g., table I in Ref. [70] for different polytropic models. Higher-order spin effects are ignored in this study but see e.g., Refs. [63,71]. We also ignore the gravitational red- shift, relativistic frame dragging [25], post-Newtonian spin-tidal couplings [72], and nonlinear hydrodynamic corrections [73] to the mode frequency in this analysis for simplicity. We normalize the modes as ωa0ðωa0 þ ωb0Þhξa; ξbi ¼ E1δab ¼ ωa0ϵaδab; ð8Þ DYNAMICAL TIDES DURING THE INSPIRAL OF RAPIDLY … PHYS. REV. D 110, 024039 (2024) 024039-3 where E1 ≡M2 1=R1. We further define [69] Ua ¼ hξa;−∇Ui E1 ¼ WlmIa � M2 M1 �� R1 r � lþ1 e−imaðϕ−ΩtÞ; ð9Þ whereWlm ¼ 4πYlmðπ=2; 0Þ=ð2lþ 1Þ and Ia is the overlap integral given by Ia ¼ 1 MRl Z d3xϱξ�a · ∇ðrlnsYlmÞ: ð10Þ This allows us to write q̇a þ iωaqa ¼ iωa0Ua: ð11Þ We adopt the same reality condition as used in Ref. [66] and require that qωa;ma ¼ q�−ωa;−ma and ξωa;ma ¼ ξ�−ωa;−ma . In other words, for two modes with opposite signs of the frequency ωa and the azimuthal quantum number ma (but other quantum numbers are the same), their amplitudes and eigenfunctions are related by complex conjugation [66]. This condition further requires that Ima ¼ ð−1ÞmaI−ma for the overlap integral. While the mode expansion should include a spectrum of modes, in an NS the f-mode strongly dominates the tidal coupling (see, e.g., [58]). Therefore, in our numerical analysis, we will focus on the la ¼ 2 and la ¼ 3 f-modes and ignore all the other modes. Wewill nonetheless maintain the generality of our analytical expressions and keep summations over modes. The analytical nature of our study means our results apply to a generic choice of NS equation of state (EOS) as long as one chooses the appropriate values of ðR1;ωa; IaÞ for given M1. When presenting numerical results to validate our analytical expressions, a NS model consistent with the SLy [74] EOS is assumed by default, which has M1 ¼ 1.4M⊙ and R1 ¼ 11.7 km [45]. The companion is treated as a point particle (PP) with M2 ¼ M1. The initial orbit is assumed to be circular. We assume ωa0 ¼ 2π × 1.8 × 103 Hz (Mtωa0 ¼ 0.08) and Ia ¼ 0.18 for the la ¼ 2 f-modes, and ωa0 ¼ 2π × 2.5 × 103 Hz (Mtωa0 ¼ 0.11) and Ia ¼ 0.11 for the la ¼ 3 f-modes [25]. The overlap integrals Ia are related to the (adiabatic) Love numbers via Eq. (43) and the chosen values lead to k20 ¼ 0.08 and k30 ¼ 0.02. To have an extended postresonance region, we consider a rapidly spinning NS with Ω ¼ −2π × 850 Hz when pre- senting numerical results. The minus sign (Ω < 0) indicates that the spin is antialigned with the orbital angular momen- tum. This corresponds to a dimensionless spin of −0.45. While this value is higher than the spin rate of J1748- 2446ad [8], it is still less than half of the maximum spin rate a NS can in principle have. For the assumed SLy EOS, the maximum spin rate is estimated to be jΩmaxj ¼ 2π × 1.8 × 103 Hz [45]. A NS with a softer EOS will typically have higher mode frequencies, yet such a NS can also support a higher spin rate because both effects scale as the dynamical frequency of the NS ffiffiffiffiffiffiffiffiffiffiffiffiffiffi M1=R3 1 p . As a caveat, we emphasize that our treatment of the background NS is only accurate to the linear order inΩ=ω1 where ω2 1 ¼ M1=R3 1. This modifies the corotating frame mode frequencies by the Coriolis force according to Eq. (7). We do not account for the effect of rotation on other background quantities, and we also ignore rotational corrections to the eigenfunctions. For the modes to be resonantly excited before the merger, approximated by the location of the innermost stable circular orbit (ISCO) risco ¼ 6Mt (≃2.1R1 for the SLy EOS), or a GW frequency of fisco ¼ ωisco=π ¼ 1570 Hz, the critical spin rate is ΩðSLyÞ crit ≃ laωisco − ωa0 la − 1 ; ≃ �−2π × 260 Hz for la ¼ 2; −2π × 53 Hz for la ¼ 3: ð12Þ We have used Eq. (7) and set ma ¼ la for the estimation above. On the other hand, for a harder EOS like H4 [75], the binary contacts each other (r ¼ 2R1) outside the ISCO. The GW frequency when they contact is about fr¼2R1 ¼ 1345 Hz. In this case, the critical spin rates are ΩðH4Þ crit ≃ laωr¼2R1 − ωa0 la − 1 ; ≃ �−2π × 140 Hz for la ¼ 2; þ2π × 5 Hz for la ¼ 3: ð13Þ In this case, the la ≥ 3 f-modes can be resonantly excited even for a nonspinning NS. Motivated by the general solution of a driven harmonic oscillator, we solve the mode amplitude evolution analyti- cally by decomposing it as qa ¼ qðeqÞa þ qðdynÞa ¼ bðeqÞa e−imaðϕ−ΩtÞ þ cðdynÞa e−iðωat−ψrÞ: ð14Þ In other words, we decompose the mode into an equilib- rium component that varies with the driving force, and a dynamical tide component that varies at the oscillator’s eigenfrequency. We also introduce slow varying mode amplitudes bðeqÞa and cðdynÞa with the fast-evolving phase components factored out. We will in the following sections describe closed-form solutions to each component. Before proceeding, we emphasize that in our work the different meanings of “adiabatic,” “equilibrium,” and “dynamical.” We will use “adiabatic” to mean the zero- frequency limit where the orbital frequency is set to ω ¼ ϕ̇ ¼ 0 in the amplitude equation (while we still keep HANG YU, PHIL ARRAS, and NEVIN N. WEINBERG PHYS. REV. D 110, 024039 (2024) 024039-4 r and ϕ). In other words, in the adiabatic limit we set q̇a to ðimaΩqaÞ in Eq. (11). The word “equilibrium” is used when the mode amplitude can be approximated by Eq. (18) [or in many practical cases, Eq. (16)]. Therefore, the equilibrium solution in our language contains the finite- frequency (ω > 0) correction to the adiabatic solution. This is different than the equilibrium tide used in, e.g., Refs. [69,76], for the zero-frequency limit (which is our adiabatic tide). Our convention is also different from that used in, e.g., Refs. [25,56,57], where the finite-frequency correction was referred to as the dynamical tide. In our notation, however, the dynamical tide will mean strictly the resonantly excited component that oscillates at the mode’s eigenfrequency. A. Equilibrium tide via resummation Let ba ¼ qaeimðϕ−ΩtÞ and Va ¼ Uaeimðϕ−ΩtÞ, the equa- tion for ba is thus ḃa þ iðωa −maσÞba ¼ iωa0Va; ð15Þ where σ ¼ ω − Ω and ω ¼ ϕ̇. Prior to resonance, ba is slowly varying, allowing us to obtain an approximate equilibrium solution by ignoring the ḃa term [31], bð0Þa ¼ ωa0 ωa −maσ Va ¼ ωa0 Δa Va; ð16Þ where Δa ¼ ωa −maσ ¼ ðωa þmaΩÞ −maω. Note that while bð0Þa gives a good approximation to ba when maσ < ωa, it diverges when the mode becomes resonant with Δa ¼ 0 or maω ¼ ωa þmaΩ. However, the diver- gence is not real as the mode will only resonate with the orbit for a finite amount of time [31]. To prevent the divergence in the algebraic solution of the equilibrium tide, we proceed as follows. First, we write ba ¼ bð0Þa þ bð1Þa þ…, and obtain the next order approxi- mation to ba still under the equilibrium (i.e. away from resonance) assumption as bð1Þa ¼ i Δa ḃð0Þa ; ¼ ibð0Þa Δa � maω Δa ω̇ ω − ðla þ 1Þ ṙ r � : ð17Þ Inspired by Ref. [31], we now resum ba as bðeqÞa ¼ bð0Þa þ bð1Þa þ � � � ≃ bð0Þa 1 − bð1Þa =bð0Þa ¼ Δaωa0Va Δ2 a − i½maω̇ − Δaðla þ 1Þ ṙr� : ð18Þ Note that the equilibrium solution bðeqÞa is everywhere finite. The orbital decay acts as an effective damping term. When resonance happens, we have bðeqÞa ¼ 0. B. Dynamical tide After constructing an algebraic equation of the equilib- rium tide qðeqÞa ¼ bðeqÞa exp ½−imaðϕ −ΩtÞ�, we now solve for the dynamical tide qðdynÞa ¼ qa − qðeqÞa . For this, it is convenient to define cðdynÞa ¼ qðdynÞa exp ½iðωat − ψa;rÞ�, where ψa;r ¼ ðωa þmaΩÞtr −maϕr. The subscript “r” under a quantity means the quantity is evaluated on resonance when Δa ¼ 0. The equation of motion for cðdynÞa is ċðdynÞa ¼ iωa0V ðdynÞ a ei½ωat−maðϕ−ΩtÞ−ψa;r�; ð19Þ with iωa0V ðdynÞ a ¼ iωa0Va − ḃðeqÞa − iΔab ðeqÞ a : ð20Þ Note that ca has net accumulations only when Δa ≃ 0, as the phase in the exponential is stationary and can be approximated as ωat −maðϕ − ΩtÞ − ψ r ≃ − ma 2 ω̇rτ 2: ð21Þ where we have defined a shifted time τ ¼ t − tr. The dominant driving term expanded around mode resonance (Δa ¼ 0) is given by iωa0V ðdynÞ a ≃ 2iωa0Va;r × ð1þ 2l−9 6 τ=tgw;rÞ ½1− 2 3 ðlþ 1Þτ=tgw;r þ imaω̇pp;rτ 2�2 : ð22Þ Since we solve the tidal perturbation to the first order, the PP orbit has been adopted to eliminate ṙ; ̈r; ω̈ in terms of ω̇pp, and tgw ¼ ω=ω̇pp [see Eq. (66)]. Note that the driving term vanishes as jτj increases, highlighting the fact that the dynamical tide’s excitation is localized near mode reso- nance in both phase and amplitude. We can evaluate the dynamical tide amplitude as cðdynÞa ≃ ωa0Va;rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sa ma 2 ω̇pp;r q FðuÞ ≃ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sa 2 ma 5 96η s WlmIa � M2 M1 �� R1 Mt �ðlþ1Þ × ðMtωrÞð4l−7Þ=6ðMtωa0ÞFðuÞ ð23Þ DYNAMICAL TIDES DURING THE INSPIRAL OF RAPIDLY … PHYS. REV. D 110, 024039 (2024) 024039-5 where in the second line we have used the PP orbit to express Va;r in terms of ωr. The dimensionless, order unity quantity FðuÞ describes the excitation of the dynamical tide and it reads FðuÞ ¼ Z u −∞ 2ið1þ a1uÞ ð1þ a2uþ 2isau2Þ2 e−isau 2 du; ð24Þ and u ¼ � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi samaω̇pp;r=2 q τ; ð25Þ τ ¼ t − tr ¼ 5 256 ηMt h ðMtωrÞ−8=3 − ðMtωÞ−8=3 i : ð26Þ Using Eq. (26), we effectively treated u ¼ uðωÞ because the instance of resonance (when ω ¼ ωr) is most straight- forwardly identified using frequency. Here we still use the PP orbit to convert time and frequency, which is accurate to the linear order in ðΔr=rÞ but loses accuracy when ðΔr=rÞ2 corrections become important. Such nonlinear corrections are left for future studies. Further, sa ¼ Sign½ma�. The constants a1 ¼ ½ð2l − 9Þ=6� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2sa=maÞ=ðωrtgw;rÞ p and a2 ¼ ½−2ðlþ 1Þ=3� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2sa=maÞ=ðωrtgw;rÞ p are both small quantities because ðωrtgw;rÞ ≫ 1. While FðuÞ can be numerically integrated easily, it is nonetheless instructive to derive its analytical expressions to know the asymptotic behaviors. For this, we decompose the integrand excluding the phase into two components that are respectively even and odd about u ¼ 0, with feðuÞ ¼ 2i ð1þ 2isau2Þ2 ; foðuÞ ¼ −4ia3u ð1þ 2isau2Þ3 ; ð27Þ wherea3¼a2−a1=2¼−½ð10l−1Þ=12� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2sa=maÞ=ðtgw;rωrÞ p . Since a3 ∝ ðωrtgw;rÞ−1=2 ≪ 1, we only keep a3 to the linear order. Note that only the even component has net accu- mulation across mode resonance. The odd component is a small correction important only near resonance and its significance decreases for modes excited earlier in the inspiral. The two components can be separately integrated, leading to FeðuÞ ¼ Z u −∞ feðuÞe−isau2du; ¼ ð4þ 16u4Þ−1 � sa � 4u sinðu2Þ þ 8u3 cosðu2Þ þ ffiffiffiffiffiffi 2π p ð1þ 4u4Þ � 1þ 2Fs � ffiffiffi 2 π r u ��� þi � 4u cosðu2Þ − 8u3 sinðu2Þ þ ffiffiffiffiffiffi 2π p ð1þ 4u4Þ � 1þ 2Fc � ffiffiffi 2 π r u ��� : ð28Þ FoðuÞ ¼ Z u −∞ foðuÞe−isau2du; ¼ a3 8 � −sa ffiffiffi e p Ei � − 1þ 2isau2 2 � þ 2isae−isau 2 1 − 2isau2 ð1þ 2isau2Þ2 − i ffiffiffi e p π ≃ 0.3654saa3 ð1þ 1.3685isau2Þ4 : ð29Þ and FðuÞ ¼ FeðuÞ þ FoðuÞ. In the function above, Fs and Fc are Fresnel integrals, and Ei is the exponential integral. Asymptotically, we have Feð−∞Þ ¼ Foð�∞Þ ¼ 0 and Feðþ∞Þ ¼ ð ffiffiffiffiffiffiffiffi π=2 p Þðiþ saÞ ≃ 1.25ðiþ saÞ. At mode res- onance, Feð0Þ¼Feð∞Þ=2¼ð ffiffiffiffiffiffiffiffi π=8 p ÞðiþsaÞ and Foð0Þ ¼ saa3ð2 − ffiffiffi e p Ei½−1=2�Þ=8 ≃ 0.3654saa3. Fig. 1 shows the real and imaginary parts of FðuÞ computed from the sum of FeðuÞ and FoðuÞ in Eqs. (28) and (29) and from numerical integration of Eq. (24). Going back to bðdynÞa with bðdynÞa ¼ cðdynÞa ei½maðϕ−ϕrÞ−ðωaþmaΩÞτ�; ð30Þ where we use the leading-order quadrupole formula to write ϕ − ϕr as functions of frequencies ðϕ − ϕrÞpp ¼ 1 32 Mt μ ½ðMωrÞ−5=3 − ðMωÞ−5=3�: ð31Þ The comparisons of the new analytical solution with numerical values and with previous results from Ref. [25] are presented in Fig. 2 for the mass multipole Q33 with l ¼ m ¼ 3. The relation between mass multipole and mode amplitude is given later in Eq. (38). We have normalized the result by its adiabatic limit obtained by replacing ba in Eq. (38) with ωa0Va=ðωa þmaΩsÞ. The gray line in the top HANG YU, PHIL ARRAS, and NEVIN N. WEINBERG PHYS. REV. D 110, 024039 (2024) 024039-6 panel is the numerical result from solving the coupled differential equations governing the evolution of the mode and the orbit [see Eqs. (52) and (53)]. In this process, we keep the tidal backreaction due to the l ¼ 3 tide but drop the backreactions from the l ¼ 2 tide. The l ¼ 2 tide can cause the orbit to deviate significantly from the PP limit, requiring higher-order corrections that are beyond the scope of this work (also Ref. [25]) that solves the tide at the linear order (in both backreaction and fluid displacement; see later the discussion about Fig. 5 and Sec. V). Our new solution agrees well with the numerical one throughout the inspiral as shown in the red dashed line in the top panel, and its decomposition into the equilibrium tide [Eq. (18)] and the dynamical tide [Eqs. (23) and (30)] is presented in the bottom panel. In comparison, the solution constructed following Ref. [25] is given in the olive dashed line.1 While it has excellent accuracy in the preresonance regime, it shows a spike at 510 Hz that is due to the imperfect numerical cancellation between the diverging term in Eq. (16) and the counterterm [second line in their Eq. (6.10)] at mode resonance. When extended to the postresonance regime, the solution from Ref. [25] loses accuracy in both amplitude and phase, which we will discuss quantitatively in the subsequent section (Sec. II C). As an aside, we also prove using Eq. (23) that the dynamical tide is a converging series as l increases. For this, we note first that ωr ≃ ωa0=lþ ½ðl − 1Þ=l�Ω for the f-modes that are most likely to be excited with ma ≃ l. Further, FIG. 2. Top: comparison of different analytical approximations for the l ¼ m ¼ 3 mass multipole. We have normalized the results by the adiabatic limit, and the result also corresponds to the Love number as k33=k ðeqÞ 33 ð0Þ ¼ Q33=Q33ð0Þ, see Eq. (42). The solution from Ref. [25] (olive dashed line) is corrected according to Footnote so that it applies to l ¼ 3. Because the Sþ 21 solution requires the subtraction of two diverging terms at mode resonance (around 510 Hz), it may cause numerical artifacts. Furthermore, the phase and amplitude of the multipole are inaccurate in the postresonance regime. The new solution (red-dashed line) however fixes these issues and matches the numerical solution throughout the entire evolution. Bottom: decomposition of the solution into an equilibrium tide component and a dynamical tide component. The equilibrium component vanishes smoothly at mode resonance, facilitating the numerical implementation. The dynamical tide is excited at mode resonance and dominates the postresonance solution. In this figure and all other figures for the l ¼ 3 tide, we include the tidal backreaction on the orbit only from the l ¼ 3 tide and exclude the l ¼ 2 tide. FIG. 1. Real and imaginary parts of FðuÞ. The FeðuÞ part (whose integrand is even with respect to u; red-dashed lines) dominates the result and it is only a function of u. The FoðuÞ component (included in the olive dashed lines) corrects the solution near mode resonance and vanishes when u → �∞. It depends on an overall scaling factor a3 ∝ ðωtgwÞ−1=2. In the example shown, a3 ¼ −0.18. 1In particular, we use the part after bl in Eq. (6.10) from Ref. [25]. The relation between the Love number and mass multipole is given by Eq. (42). Note that their Δω0l measures the frequency shift of a mode in the inertial frame, so it corresponds to our Δωa þmaΩ. DYNAMICAL TIDES DURING THE INSPIRAL OF RAPIDLY … PHYS. REV. D 110, 024039 (2024) 024039-7 Mtωa0 ∝ l1=2 � M1 Mt � 1=2 � R1 Mt � −3=2 : ð32Þ In the slow-rotating limit with jΩj ≪ jωa0j, we have cðdynÞa ∝ � 1 l �4l−13 12 � M1 Mt �4l−1 12 � R1 Mt �5 4 : ð33Þ This is clearly a converging series. In the case where jΩj dominates, we first note that only when Ω > 0 can there be infinitely many f-modes be resonantly excited all atωr ≃Ω. The maximum Ω a NS can have is ðM1=R3 1Þð1=2Þ, which leads to cðdynÞa ∝ l1=2 � M1 Mt �4l−1 12 � R1 Mt �5 4 : ð34Þ In the high-l limit, the ðM1=MtÞ < 1 term decreases exponentially while the l1=2 increases only as polynomial. Therefore, the series is also converging. For Ω < 0, only modes with ωa0 > −ðl − 1ÞΩ can experience resonance. Because this is a finite set of modes, divergence cannot happen. C. Comparison with previous analysis We elaborate further on the relation between our new solution and that obtained in previous analyses such as Ref. [56] and [25]. We note that Eq. (15) can be solved locally around mode resonance by expanding ωa −maσ ≃maω̇rτ ≃ma ω̇r ωr ϖ; ð35Þ where ϖ ¼ ðϕ − ϕrÞ. This leads to the solutions in the resonance region obtained in [56] and [25] (in particular, the last line in Eq. (6.10) of Ref. [25]). In this case, we have bðlocÞa ≃ iωa0Va;rei ma 2 ω̇rt2 Z t −∞ e−i ma 2 ω̇rt2dt; ≃ i ωa0Va;r ωr e ima 2 ω̇r ω2r ϖ2 Z ϖ −∞ e −ima 2 ω̇r ω2r ϖ2 dϖ; ¼ ωa0Va;rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sa ma 2 ω̇r p GðvÞ ð36Þ where2 v ¼ ð ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi samaω̇pp;r=2 p =ωrÞϖ ¼ ðω=ωrÞu and GðvÞ ¼ ieisav 2 Z v −∞ e−isav 2 dv ¼ ffiffiffi π 8 r ðsa þ iÞeisav2 � 1þ ð1þ isaÞFc � ffiffiffi 2 π r v � þ ð1 − isaÞFs � ffiffiffi 2 π r v � : ð37Þ Note that Gð0Þ ¼ Feð0Þ and limv→þ∞ GðvÞe−isav2 ¼ Fðþ∞Þ, indicating the consistency in both the mode amplitude at resonance and their asymptotic values. However, the phase of bðlocÞa is accurate only in the vicinity of mode resonance. It also overestimates the magnitude of oscillations in the postresonance region. Another way to obtain a local solution is to solve ca [cf. Eq. (23) without the “(dyn)” superscript] using a stationary phase approxi- mation, and then transfer the resultant ca to ba using Eq. (30) [15]. Such a solution gives the correct phase in the postresonance regime but not in the prior-resonance regime. Indeed, the general solution of a driven oscillator should contain two terms varying at distinct frequencies as in Eq. (14), and a local solution with a single term can thus be extended accurately only in one direction. The dis- cussion therefore highlights the necessity of decomposing the tide into the equilibrium and dynamical components in the presence of mode resonance. III. BACKREACTION ON THE ORBIT USING EFFECTIVE LOVE NUMBER In this section, we first convert the mode amplitudes found in the previous section to mass multipoles of the NS, and then use the mass multipoles to define effective Love numbers following the approach used in Ref. [57]. The tidal back reactions will then be written in terms of the effective Love number so that they take the same form as in the adiabatic limit, except for the Love number replaced by their frequency-dependent effective values. We highlight that a pure real effective Love number as introduced in Ref. [57] is insufficient to model the dynamics. In par- ticular, it captures only the tidal acceleration in the radial direction. To capture the acceleration in the tangential direction (i.e., the tidal torque), a new dressing factor, originating from the imaginary part of the effective Love number for each ðl; mÞ harmonic, will be introduced. We will demonstrate explicitly that it is this torque that dominates the impact on the orbit near mode resonance. We start by relating the multipole moment of the NS in the inertial frame to the amplitude of the associated mode with the same ðl; mÞ as [73], Qlm ¼ MRl Xla;ma a Iabae−imaϕ; ð38Þ 2Note that t̂ in Ref. [25] is related to our v as t̂ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2=maÞð8=3Þ p v and their ϵl ¼ ð3=8Þðω̇r=ω2 rÞ. Our analysis also indicates that a factor of ffiffiffiffiffiffiffiffiffiffiffi ma=2 p ¼ ffiffiffiffiffiffiffiffiffi la=2 p in the phases in Eq. (6.11) of Ref. [25] is missing. The magnitude of the last term in their Eq. (6.10) lacks a factor of ffiffiffiffiffiffiffiffiffi la=2 p . HANG YU, PHIL ARRAS, and NEVIN N. WEINBERG PHYS. REV. D 110, 024039 (2024) 024039-8 where we have used Pla;ma a as a short-hand notation ofPðla;maÞ¼ðl;mÞ a . Note further that the summation runs over modes with both signs of eigenfrequencies. This corre- sponds to a symmetric trace-free tensor (indicated by the brackets around the indices) in a Cartesian coordinate, Qhi1…ili ¼ Nl X m Yhi1…ili lm �Qlm ¼ NlM1Rl X m Yhi1…ili lm � Xla;ma a Iabae−imaϕ: ð39Þ where Nl ¼ 4πl!=ð2lþ 1Þ!! and the tensor Yhi1…ili lm is defined in Ref. [77]. At the same time, the tidal moment (also symmetric and trace-free) can be written as Ei1…il ≡ −M2∂i1…il 1 r ¼ −l! M2 rlþ1 X m Wlme−imϕYhi1…ili lm � ð40Þ We can define an effective Love number of each ðl; mÞ harmonic so that it maintains the same form as in the adiabatic limit as Qhi1…iliY hi1…ili lm ¼ − 2klm ð2l − 1Þ!!R 2lþ1Ei1…ilY hi1…ili lm ; ð41Þ which leads to klm ¼ 2π 2lþ 1 Xla;ma a I2a � ba Va � ¼ k�l;−m ∈C: ð42Þ The last equality follows the reality condition on the mode amplitude. Note that in the nonspinning adiabatic limit, klmjω¼Ω¼0 ¼ kl0 ¼ 4π 2lþ 1 I2a ð43Þ and the Love number is the same for all values of m. Using the leading-order equilibrium solution from Eq. (16), we have kðeq;0Þlm ðωÞ ¼ kðeq;0Þl;−m ðωÞ ¼ ω2 a0 ω2 a0 − ðmσ − ΔωaÞ2 kl0; ð44Þ where mode a has ðla; maÞ ¼ ðl; mÞ, and we have used the fact that Δωa ∝ ma [Eq. (7)], so Δma ¼ −Δ−ma . This result agrees with Ref. [70]. Because of rotation, we have kðeq;0Þlm ðω ¼ 0Þ ¼ ω2 a0 ω2 a0 − ðmΩþ ΔωaÞ2 kl0: ð45Þ Note that the difference between kðeq;0Þlm and kl0 are due to the finite frequency response of a mode as in Eq. (16). Their difference is on the order ðΩ=ωa0Þ2. As a caveat, the mode structure can be modified by rotation at order ðΩ=ωa0Þ2 (e.g., due to the centrifugal force), which is not accounted for in our current framework as Eq. (1) is only to linear order in rotation. But see Ref. [63] for more discussions on rotational corrections beyond Eq. (7). We can further define an effective dressing factor κl for each harmonic l as κl ¼ − ð2l − 1Þ!! 2kl0R2lþ1 Qhi1…iliE i1…il Ei1…ilE i1…il ¼ ð2lþ 1Þ 4π X m W2 lm klm kl0 ∈R; ð46Þ so that κlkl0 becomes the effective Love number in Ref. [57]. Note that P m W2 lm ¼ 4π=ð2lþ 1Þ, so when ω ¼ Ω ¼ 0, κl ¼ 1. The reality condition that klm ¼ k�l;−m further requires that κl is real. As we will see shortly, the effective Love number κlkl0 can only capture the radial acceleration of the tidal back- reaction. We hence introduce an additional factor γl to describe the torque, which we define as γl ¼ 2lþ 1 4π X m mW2 lmIm � klm kl0 � : ð47Þ Note that γl vanishes in the equilibrium limit but becomes finite when klm has imaginary component as mIm½klm� ¼ −mIm½kl;−m�. To derive the conservative part of the dynamics, we write the Hamiltonian of the system to linear order in the NS’s rotation [32,66,73] as, H ¼ Hpp þHmode þHint; ð48Þ where Hpp ¼ p2 r=ð2μÞ þ p2 ϕ=ð2μr2Þ − μMt=r; ð49Þ with pr ¼ μṙ and pϕ ¼ μr2ϕ̇, Hmode ¼ Xωa>0 a ϵaωaq�aqa ¼ E1 Xωa>0 a ωa ωa0 b�aba; ð50Þ and Hint ¼ Xωa>0 a ϵaðifaq�a − if�aqaÞ ¼ −E1 Xωa>0 a ðVab�a þV� abaÞ: ð51Þ DYNAMICAL TIDES DURING THE INSPIRAL OF RAPIDLY … PHYS. REV. D 110, 024039 (2024) 024039-9 The three pieces respectively represent the PP orbit, the eigenmodes inside the NS, and the mode-orbit interaction. While we use ba and Va in the above expressions, we emphasize that the generalized displacements we use are ðr;ϕ; qaÞ and their associated momenta are ½pr; pϕ; ðiE1=ωa0Þq�a�. The Hamiltonian depends explic- itly on time because Vab�a ∼ expðimaΩtÞ in the interaction. We will come back to this point later when discussing the energy conservation of the system [Eq. (60)]. The dynamics of the orbit is thus given by ̈r − rϕ̇2 þMt r2 ¼ gr; ð52Þ rϕ̈þ 2ṙ ϕ̇ ¼ gϕ ð53Þ Where gr and gϕ are accelerations along the radial and tangential directions and they include both a dissipative contribution due to GW radiation and a conservative piece due to tidal interaction, which we can derive from the interaction Hamiltonian as gðtÞr ¼− 1 μ ∂Hint ∂r ¼− 2E1 μr Xωa>0 a ðlaþ 1ÞReðVab�aÞ ¼− Mt r2 M2 M1 X l ðlþ 1Þð2lþ 1Þ 2π � R r � 2lþ1X m W2 lmRe½klm�; ¼−2 Mt r2 M2 M1 X l ðlþ 1Þκlkl0 � R r � 2lþ1 ; ð54Þ gðtÞϕ ¼ − 1 μr ∂Hint ∂ϕ ¼ 2E1 μr Xωa>0 a maIm½Vab�a�; ¼ − Mt r2 M2 M1 X l ð2lþ 1Þ 2π � R r � 2lþ1X m mW2 lmIm½klm�; ¼ −2 Mt r2 M2 M1 X l ðlþ 1Þγlkl0 � R r � 2lþ1 ; ð55Þ where we have used Eqs. (42), (46), and (47) and their (real and imaginary) dressing factors. Note that we can further write the interaction energy as Hint ¼ −2 M2 2 r X l κlkl0 � R r � 2lþ1 : ð56Þ In the limit where Ω → 0 and ω → 0 (i.e., κl ¼ 1), we have Hmodeðω¼Ω¼ 0Þ ¼− 1 2 Hint ¼ M2 2 r X l kl0 � R r � 2lþ1 : ð57Þ However, in general, there is no simple relation between Hmode and κlkl0. Note further that whenΩ ≠ 0,Hmode is the mode energy in the corotating frame. For a specific mode a in the inertial frame, its canonical energy is [78] EðinertialÞ mode;a ¼Hmode;aþΩJmode;a ¼ωaþmaΩ ωa Hmode;a ¼ E1 2 ωaþmaΩ ωa0 b�aba; ð58Þ where Jmode;a ¼ maHmode=ωa is the angular momentum of the mode. Using the asymptotic values of FeðuÞ, we have when u → ∞ (after summing over a mode a and its complex conjugate) EðinertialÞ mode ð∞Þ E1 ≃ 2π sama ωa0ðωaþmaΩÞ ω̇pp;r V2 a;r: ¼ 5ð2lþ1Þ 192η W2 lmkl0 � M2 M1 � 2 × � R1 Mt � 2ðlþ1Þ ðMtωa0ÞðMtωrÞ4ðl−1Þ=3: ð59Þ While Hmode does not relate to κl and γl in a simple manner, we note that its derivative is ĖðinertialÞ mode ¼ Ḣmode þ ∂Hint ∂t ¼ −2E1 Xωa>0 a ðωa þmaΩÞIm½Vab�a�: ð60Þ We remind readers that the ∂Hint=∂t term originates from the fact that the Hamiltonian depends on time explicitly. Together with the total time derivative of Hint, Ḣint¼2E1 Xωa>0 a � ΔaIm½Vab�a�þ ṙ r ðlaþ1ÞRe½Vaba� ; ð61Þ we have Ėtide ≡ ĖðinertialÞ mode þ Ḣint ¼ −μṙgðtÞr − ωμrgðtÞϕ ; ð62Þ which describes the expected energy transportation from the orbit to the tide. Figure 3 shows the integral of the Eq. (62). In the equilibrium limit with jωaj ≫ ω and jωaj ≫ jΩj, jImðbaÞ=ReðbaÞj ∼ jṙ=ðrΔaÞj, leading to ωrgðtÞϕ ṙgðtÞr ∼ ω Δa ≃ ω ωa ≪ 1 ðfor equilibrium tideÞ: ð63Þ Therefore, in the equilibrium limit, the torque term (or γl) can be ignored. However, when dynamical tide becomes important and jImðbaÞ=ReðbaÞj ∼ 1, we have HANG YU, PHIL ARRAS, and NEVIN N. WEINBERG PHYS. REV. D 110, 024039 (2024) 024039-10 ωrgðtÞϕ ṙgðtÞr ∼ ω ṙ=r ≫ 1 ðfor dynamical tideÞ: ð64Þ Furthermore, the torque term has a secular contribution when integrated across mode resonance,Z ∞ −∞ −ωμrgðtÞϕ dt ≃ 2E1maωrVa;rIm � cðdynÞa;r Z ei ma 2 ω̇ppt2dt � ¼ 2π sama maωωa0 ω̇pp;r V2 a;rE1 ≃ EðinertialÞ mode ð∞Þ; ð65Þ where we have used Eqs. (23), (30), and (55), and evaluated the integral with a stationary phase approxima- tion on the exponent in Eq. (30). Note that the resonance condition requires maωr ¼ ωa þmaΩ. The tidal torque leads to the postresonance mode energy, Eq. (59), and it is greater than the absolute value of the interaction energy by a factor of ∼ðωrtgw;rÞ1=2ðrr=rÞlþ1. Therefore, the torque term dominates the evolution near mode resonance, and missing the γl term will lead to an inaccurate description of the dynamical tide. IV. DYNAMICS THROUGHOUT THE INSPIRAL Using the tools developed above, we will discuss the coupled evolution of the orbit and the mode in this section. We will start by reviewing the orbital evolution under the PP limit in Sec. IVA. Tidal corrections to the orbit will then be derived using both an energy-balancing argument and the osculating orbit technique coupled with the tidal accelerations gðtÞr and gðtÞϕ derived above. In particular, Sec. IV B is devoted to deriving the phase shifts, the most dominant tidal signature, as functions of both frequency and time. The deviation from a quasicircular inspiral due to the dynamical tide will be discussed in Sec. IV C. Lastly, we consider additional GW signatures arising from the oscillating f-modes and orbital eccentricity in Sec. IV D. A. Point-particle (PP) evolution We begin the discussion of this section by briefly reviewing the orbital evolution of two PP masses inspiral- ing due to GW radiation. Results here are used in Sec. II to derive the linearized mode amplitude evolution. They will also serve as the baseline values onto which we add the tidal perturbation. The PP frequency evolution is given by ω̇pp ¼ 96 5 η M2 t ðMtωÞ11=3: ð66Þ We further define tgw ¼ ω=ω̇pp, which serves as a measure of the instantaneous GW decay timescale. The orbital energy and separation of the PP orbit changes according to Ėpp Epp ¼ − ṙpp rpp ¼ 2 3 1 tgw ; ð67Þ where Epp ¼ −M1M2=ð2rÞ. Note Ėpp < 0 and ṙpp < 0 in our convention. Expressing ω and r as functions of t leads to ðMtωppÞ−8=3 − ðMtω0Þ−8=3 ¼ � rpp Mt � 4 − � r0 Mt � 4 ¼ 256η 5 � t0 Mt − t Mt � : ð68Þ It is also useful to relate the phase ϕ to ω, which has been given earlier in Eq. (31). The PP orbital evolution can be modeled by adding the following Burke-Throne accelerations (e.g., Ref. [32]) to the orbital dynamics, Eqs. (52) and (53), gðgwÞr ¼ 16μMt 5r3 ṙ � ṙ2 þ 6r2ϕ̇2 þ 4Mt 3r � ; ð69Þ gðgwÞϕ ¼ 8μMt 5r2 ϕ̇ � 9ṙ2 − 6r2ϕ̇2 þ 2Mt r � ≃ − 32 5 μr3ϕ̇5 ð70Þ where we have written the ϕ component into a form that is valid even when the r − ϕ̇ relation is modified by the presence of the tide [73]. FIG. 3. Energy in the l ¼ 3 tide, Etide ¼ EðinertialÞ mode and Hint, shown in the gray line. Also shown in the dashed lines are the integrals of the two terms in Eq. (62). In the low-frequency, equilibrium limit, the radial term (olive line) dominates the change and it is negative (the change of the total equilibrium energy is still positive when including ΔEpp). Near mode resonance, the tangential torque (red line; positive) dominates the change of tidal energy. DYNAMICAL TIDES DURING THE INSPIRAL OF RAPIDLY … PHYS. REV. D 110, 024039 (2024) 024039-11 B. Tidal phase shift The dominant impact of the tide on the orbital evolution is that it causes a phase shift relative to the PP limit. This section is dedicated to deriving such a phase shift. We will start by deriving first the orbital phase shift as a function of frequency (that is, comparing the phases of two systems at the same frequency). While the phase shift at a given frequency offers a straightforward description of the tide, physically what is measurable is the phase shift measured at the same time. Therefore, in the second half of this section, we will also derive the phase shift directly as a function of time using the technique of osculating orbits. The osculating orbits also allow us to extract the post- resonance eccentricity excited by the dynamical tide which we will discuss further in Sec. IV C. 1. Energy balancing We use an energy-balancing argument to derive the dynamics as a function of GW frequency f ¼ ω=π. The duration of coalescence t and the orbital phase ϕ can be computed as t ¼ Z dE=df Ė df; ð71Þ ϕ ¼ π Z f dE=df Ė df; ð72Þ Therefore, the tidal corrections to the time and phase can be computed as [73] Δt ≃ Z 1 Ėpp � dΔEeq dfgw − dEeq dfgw ΔĖ Ėpp � df; ð73Þ ΔϕðfÞ ≃ π Z f Ėpp � dΔEeq dfgw − dEeq dfgw ΔĖ Ėpp � df; ð74Þ where the first term in the parenthesis describes a conservative effect that at a given frequency, the equilibrium energy of the system is modified relative to the PP orbit. We have ΔEeq ¼ ΔEpp þ Etide and Etide ¼ Hint þ EðinertialÞ mode . The ΔEpp describes the change of the orbital energy because both the location and the value of the radial effective potential’s minimum change due to the presence of the tide [73], Δr r ≃ − gðt;eqÞr 3rω2 ; ð75Þ ΔEpp Epp ¼ −4 Δr r : ð76Þ Note that we have used gðt;eqÞr , or the equilibrium component of the radial acceleration [using bðeqÞa from Eq. (18) in Eq. (54)], as it describes the component that is phase coherent with the orbit. When the dynamical tide is excited after mode resonance, the effective potential changes on a timescale shorter than the orbital decay timescale and excites eccentricity of the orbit, a point we will discuss further in Sec. IV C. TheEtide term describes energies due to the tidal degrees of freedom, including both its interaction with the orbit [Eq. (56)] and the inertial-frame mode energy from Eq. (58). The tide also affects how much energy is radiated through GW [second term in Eq. (74)]. Here we restrict our discussion to the lowest-order quadrupole radiation that is affected by the tide in two ways. The first one is the r − ω relation is modified compared to the PP orbit, and the second equality in Eq. (70) captures this modification as discussed in Ref. [73]. The second effect is that the tidally induced NS mass quadrupole can couple with the orbital quadrupole, allowing additional GW radiation. In our numerical calculations, we do not include the second effect. Instead, we present here an analytical calculation of it. The impact of this term in the adiabatic limit has been well established, see, e.g., Ref. [73] and references therein. Thus, we focus on the finite frequency correction (which can still be considered the equilibrium tide in our defi- nition) and the dynamical tide (only excited on resonance) to the NS quadrupole. Our calculation complements the EOB framework which computes the dissipative effect only under the adiabatic (i.e., the zero-frequency) limit; see, e.g., Ref. [57]. From [73], ΔĖns−orb¼− 2 5 hQ…ij nsQ …ij orbi¼− 4 15 X m W2mhQ …ns 2mQ …orb 2m �i: ð77Þ For the orbital mass quadrupole, we have Q …orb 2m ≃ ð−imωÞ3μr2e−imϕ; ð78Þ the ignored term is smaller by a factor of 1=ðωtgwÞ. We note that the NS quadrupole can be decomposed into an equilibrium and a dynamical component because Qns 2m ∝ ½bðeqÞa þ bðdynÞa �. The time variation of each compo- nent also mainly comes from the phase evolution (respec- tively goes as 2ω and 2ωr for the l ¼ jmj ¼ 2 components), while the rate of change on the amplitude of ba is smaller by a factor of ∼u̇=ωr ∼ ðωrtgw;rÞ−1=2. Furthermore, ḃa þ ḃ�a ¼ 2ΔaIm½ba�, which is small both before reso- nance (as Im½ba� is small) and during mode resonance as Δa ≃ 0. The equilibrium component is coherent with the orbit throughout the evolution because, in our definition, it is the component that varies at e−imϕ in the inertial frame. HANG YU, PHIL ARRAS, and NEVIN N. WEINBERG PHYS. REV. D 110, 024039 (2024) 024039-12 The derivation presented in Ref. [73] can thus be readily extended to include the equilibrium tide as ΔĖðeqÞ ns−orb ¼ − 8 15 μM1R2 1M 2=3 t ω14=3 × X m¼�2 m6W2m Xma¼m a;ωa>0 IaRe½bðeqÞa �: ð79Þ Note that ΔĖns−orbðωÞ ΔĖns−orbðω ¼ 0Þ ¼ Q22ðωÞ Q22ðω ¼ 0Þ ≠ κl; ð80Þ as m ¼ 0 modes do not contribute to the radiation but they affect the radial acceleration. Therefore, an effective Love number κl is inaccurate in modeling the radiation from the beat of the NS and orbital mass quadrupoles even in the equilibrium limit. We approximate the third derivative of the mass quadru- pole due to the dynamical tide as Q …ns;dyn 22 ≃ ð−imaωrÞ3MR2Iac ðdynÞ a e−imaϕr−imaωrτ: ð81Þ where mode a in the above equation stands specifically for the la ¼ ma ¼ 2 f-mode with ωa > 0. We also have Q …ns;dyn 2;−2 ¼ ðQ…ns;dyn 22 Þ�. The energy dissipation rate with the phase expanded near mode resonance reads ΔĖðdynÞ ns−orb ≃ − 8 15 ð2ωrÞ6μr2rM1R2 1 ×W22IaRe � cðdynÞa eiω̇rτ 2� ; ð82Þ The phase shift due to this term can be evaluated as ΔϕðdynÞ ns−orb ¼ − Z ω ΔĖðdynÞ ns−orb Ėpp dt; ≃ 5πk20 192 � M2 η2Mt �� ωa0 ωr � ðMtωrÞ5=3 � R1 Mt � 5 × � Fs � ffiffiffi π 2 r u � − Fc � ffiffiffi π 2 r u �� ; ⟶ u→þ∞ 0; ð83Þ where u ¼ ffiffiffiffiffi ω̇r p τ and we have used a stationary phase approximation in the second line. The mode amplitude is approximated using Eq. (23) at resonance with Fðu ¼ 0Þ ≃ Feð0Þ [see the discussion below Eq. (28)]. This is justified because eiω̇rτ 2 is even about τ ¼ u ¼ 0 while ½FeðuÞ − Feð0Þ� is odd. Note that under this approximation, ΔϕðdynÞ ns−orbðu → ∞Þ ¼ 0. While the Foð0Þ contribution may survive, it is nonetheless a small correction because Fo ∝ a3 ∝ ðωrtgw;rÞ−1=2. Therefore, at least at the leading order, the interaction between the NS mass quadrupole due to the dynamical tide and the orbital quadrupole does not lead to net time or phase shift when integrated across mode resonance. In the vicinity of mode resonance with juj≲ 1 and Fs ≠ Fc, the magnitude of tidal phase shift due to this dissipative effect compared to that due to the conservative effect [first term in Eq. (74)] can be estimated from the ratio ΔϕðdynÞ ns−orb ΔϕðconÞ ≃ ΔĖ ðdynÞ ns−orb=Ėpp EðinertialÞ mode =Eorb ∼ ωa0Var maωrjbaj ∼ ðωrtgw;rÞ−1=2: ð84Þ The equilibrium component has a similar magnitude because the equilibrium and dynamical mass quadrupoles have comparable magnitude near mode resonance (see the lower panel of Fig. 2). Therefore, the conservative effect (energy transfer from the orbit into the excited mode) dominates the phase shift. The phase shift at a given frequency is shown in the top panel of Fig. 4 for the l ¼ 3 tide. The l ¼ 2 tide is excluded in the computation here. We use the gray and red lines to respectively represent the numerical and analytical results. It is interesting to note that the main effect can be captured by a steplike jump at mode resonance. The amount of jump can be calculated as Δta ¼ EðinertialÞ mode ð∞Þ Ėpp;r ; and Δϕa ¼ ωrΔta; ð85Þ be the asymptotic time and phase shifts of a mode a (and its complex conjugate) integrated to infinity. More explicitly, Δϕa ¼ − 25ð2lþ 1Þ 6144η2 W2 lmkl0 � M2 M1 �� R1 Mt � 2lþ1 × ðMtωa0ÞðMtωrÞð4l−11Þ=3: ð86Þ The numerical values for the la ¼ 3 and la ¼ 2 f-modes are Δϕajla¼3 ¼ −0.48 rad × ð4ηÞ−2 � M2 M1 �� k30 0.02 � × � R1 11.7 km � 7 � ωa0=2π 2.5× 103 Hz �� ωr=π 510 Hz � 1=3 : ð87Þ Δϕajla¼2 ¼ −17 rad × ð4ηÞ−2 � M2 M1 �� k20 0.08 � × � R1 11.7 km � 5 � ωa0=2π 1.8 × 103 Hz �� ωr=π 950 Hz � −1 : ð88Þ DYNAMICAL TIDES DURING THE INSPIRAL OF RAPIDLY … PHYS. REV. D 110, 024039 (2024) 024039-13 With the jump magnitudes Δta and Δϕa, we can then write the approximate frequency evolutions of the time and phase shifts as Δt̄ðfÞ ≃ Δta 1þ tanh uðfÞ 2 ; ð89Þ Δϕ̄ðfÞ ≃ Δϕa 1þ tanhuðfÞ 2 ; ð90Þ where the tanh function is used to smoothly transition the phase shift from nearly 0 to its asymptotic value after mode resonance. The numerical constant in front of u is set to unity for simplicity, yet it provides sufficient accuracy for the approximation, as shown in the olive dashed line in the top panel of Fig. 4. After a mode’s resonance, there appears to be a large fluctuation in the phase shift. Our analytical solution does not fully capture this oscillation because the orbit is no longer quasicircular and Eq. (75) can capture only the averaged change of separation (see Sec. IV C). Nonetheless, a large portion of the oscillation is not physically observable and therefore does not impact the waveform analysis. For this, we note that phase shift should be measured at a fixed time t instead of frequency f (or ω). They are related through ΔtðfÞ ≃ − ΔωðtÞ ω̇ ; ð91Þ ΔϕðfÞ ≃ ΔϕðtÞ þ ωΔt ≃ ΔϕðtÞ − ω ω̇ ΔωðtÞ; ð92Þ where ΔωðtÞ and ΔϕðtÞ are the shifts of orbital frequency and of orbital phase measured at a fixed time. Since ω2=ω̇ ≫ 1, the second term in the last equality in Eq. (92) can appear large. It causes a large fluctuation in ΔϕðfÞ, though the true observable ΔϕðtÞ varies much less. Equivalently, we can consider the phase of the frequency domain GW waveform, which under the stationary phase is [79] ΨðfÞ ¼ 2πftðfÞ − 2ϕðfÞ − π 4 ; ð93Þ so the tidal shift is given by3 ΔΨðfÞ ¼ 2πfΔtðfÞ − 2ΔϕðfÞ ≃ −2ΔϕðtÞ; ð94Þ and the Δt contribution cancels out. While ΔϕðfÞ is not directly measurable, it is nonetheless a useful quantity to compute, because its asymptotic behavior can be captured by a particularly simple form, Eq. (90). Using Eq. (92) and replacing Δt with Δϕ=ωr [Eq. (85)], we have ΔΨðfÞ ≃ 2 � ω ωr − 1 � 1þ tanh u 2 Δϕa; ð95Þ FIG. 4. Tidal phase shift due to the l ¼ 3 f-modes. The top panel shows the orbital phase shift at a given frequency and the bottom one at a given time (tc is the time when r ¼ 2R). The middle panel is the GW phase shift of the frequency domain waveform. The gray curves are results extracted numerically while the red curves are from analytical calculations [Eqs. (74), (94), and (120)]. Also shown in the olive dashed lines are approximate estimations [Eqs. (90), (95), and (97)] that estimate the secular phase shifts. 3When comparing the phase shift of two waveforms, it is important to keep in mind that one has the freedom to shift one waveform’s time and phase at a reference point relative to the other by arbitrary constants. This is because we do not know a priori of a signal’s time and phase of arrival. As a theoretical study, we fix ΔtðfrefÞ ¼ 0 and ΔϕðfrefÞ ¼ 0 at fref ¼ 100 Hz for simplicity, which approximately corresponds to Δtðfref ¼ 0Þ ≃ Δϕðfref ¼ 0Þ ≃ 0 as the tide is important only when f ≳ 500 Hz. When computing, e.g., the mismatch between two waveforms [80], however, the freedom in ΔtðfrefÞ and ΔϕðfrefÞ needs to be marginalized over. HANG YU, PHIL ARRAS, and NEVIN N. WEINBERG PHYS. REV. D 110, 024039 (2024) 024039-14 Δϕ̄ðtÞ ≃ � 1 − ωðtÞ ωr � 1þ tanh u 2 Δϕa; ð96Þ ¼ � 1 − 256η 5 τ Mt ðMtωrÞ8=3 � −3=8 1þ tanhu 2 Δϕa; ð97Þ as estimations of the secular shift of the orbital phase in the frequency and time domain, respectively. Note the form is consistent with previous results obtained in, e.g., Refs. [32,35] except for the minor difference that we use here a tanh instead of Heaviside used in Ref. [35] to smoothly connect the pre- and postresonance values. These approximate expressions are shown in the mid and bottom panels of Fig. 4 in the olive dashed lines, which show decent agreements with both the numerical (gray lines) and the full analytical (red lines) results. 2. Osculating orbits For the completeness of the study, we also present a direct derivation of the dynamics as a function of time t. We use the method of osculating orbits and change ðr; ṙ;ϕ; ϕ̇Þ to ðp; e;ϕ;ϕ0Þ, the instantaneous semilatus rectum, eccen- tricity, orbital phase, and argument of pericenter, according to [32] r ¼ p 1þ ecϕ ; ð98Þ ṙ ¼ ffiffiffiffiffiffi Mt p s esϕ; ð99Þ ϕ̇ ¼ ffiffiffiffiffiffi Mt p3 s ½1þ ecϕ�2; ð100Þ where cϕ ¼ cosðϕ − ϕ0Þ, sϕ ¼ sinðϕ − ϕ0Þ. The inverse relations are p ¼ r4ϕ̇2 Mt ; ð101Þ e2 ¼ � p r − 1 � 2 þ pṙ2 Mt ; ð102Þ esϕ ¼ ṙ ffiffiffiffiffiffi p Mt r ; ð103Þ ecϕ ¼ p r − 1: ð104Þ The evolution of osculating variables is given by ṗ ¼ 2 ffiffiffiffiffi p3 p ffiffiffiffiffiffi Mt p ð1þ ecϕÞ gϕ; ð105Þ ė ¼ ffiffiffiffi p p 2 ffiffiffiffiffiffi Mt p ð1þ ecϕÞ ½ð3eþ 4cϕ þ ec2ϕÞgϕ þ ð2sϕ þ es2ϕÞgr�; ð106Þ eϕ̇0 ¼ − ffiffiffiffi p p 2 ffiffiffiffiffiffi Mt p ð1þ ecϕÞ ½ðeþ 2cϕ þ ec2ϕÞgr − ð4sϕ þ es2ϕÞgϕ� ð107Þ and Eq. (100). The zeroth order solution of the set of equations can be obtained from the PP orbit under GW decay ppp ¼ rpp; ðecϕÞpp ¼ 0; ðesϕÞpp ¼− 2 3 1 ωtgw : ð108Þ To find the tidal correction, we first ignore GW decay and consider a system interacting just due to the equilib- rium tide with a radial acceleration gðt;eqÞr . Such a conservative system permits a solution with ṗ ¼ ė ¼ 0 and cϕ ¼ 1, which indicates eϕ̇0 ¼ eϕ̇. From Eqs. (100) and (107), we have ecϕ ≃ − gðt;eqÞr Mt=r2pp : ð109Þ The same result can also be obtained by first combining Eqs. (98) and (100) to get ϕ̇2 ¼ ðMt=r3Þð1þ ecϕÞ, and then compare it with Eq. (75) or Eq. (52) with ̈r set to zero. Note again the above equation is accurate only before mode resonance; an additional oscillatory component in ecϕ will be excited by the dynamical tide in the post- resonance regime (Sec. IV C). Nonetheless, Eq. (109) is sufficient for us to find the deviation in p caused by tide, which follows from Eq. (105), Δṗ ṙpp ¼ 3 2 Δp rpp þ Δgϕ gϕ − ecϕ: ð110Þ The Δgϕ term contains two contributions, Δgϕ ¼ gðtÞϕ þ ΔgðgwÞϕ : ð111Þ The first piece is due to the tidal torque [mainly from the dynamical tide; Eq. (55)]. The second piece is because the GW acceleration is modified by the tide. By replacing ðr; ϕ̇Þ in the second equality in Eq. (70) with osculating variables, we have ΔgðgwÞϕ gðgwÞϕ ≃ − 9 2 Δp rpp þ 7ecϕ ð112Þ DYNAMICAL TIDES DURING THE INSPIRAL OF RAPIDLY … PHYS. REV. D 110, 024039 (2024) 024039-15 Consequently, we can rewrite the equation governing the evolution of Δp as Δṗ ṙpp ¼ � −3 Δp rpp þ gðtÞϕ gðgwÞϕ − 6 gðt;eqÞr Mt=p2 0 � : ð113Þ The solution of this equation is given by ΔpðtÞ ¼ 1 r3pp Z r3ppṙpp gðtÞϕ gϕ0 − 6 gðt;eqÞr Mt=r2pp ! dt; ¼ 1 r3pp Z r3pp gðtÞϕ gϕ0 − 6 gðt;eqÞr Mt=r2pp ! drpp dω dω; ð114Þ Note that while we change the variable to ω in the second line for easier evaluation of the integral [as the mode amplitudes are given in u and u is treated as a function of ω; Eqs. (23), (30), (26)], here ΔpðtÞ is the change in p measured at a fixed time (assuming the tidal and PP orbits are aligned at past infinity). After evaluating the integral to a certain frequency ω, we map it back to the desired time according to tðωÞ ≃ tppðωÞ þ ΔtðωÞ ≃ tppðω − ΔωÞ; ð115Þ where tpp is the PP mapping from ω to t, Δt is from Eq. (73), andΔω is computed from Eq. (119) which wewill introduce shortly. We can compare the sizes of the two terms in the parenthesis in Eq. (114). First, we note that gϕ0 Mt=p2 0 ∼ ṙ ωr : ð116Þ Therefore, gðtÞϕ =gϕ0 gðtÞr =ðMt=p2 0Þ ∼ ωrgϕ ṙgr ; ð117Þ which becomes the comparison shown in Eqs. (63) and (64). In particular, whereas in the equilibrium limit, the radial acceleration plays the major role, near mode reso- nance it is the torque that dominates the orbit evolution. If we ignore the ecϕ term and adopt the same stationary phase approximation that leads to Eq. (65), we can approximately carry out the integral in Eq. (114) as ΔpðtÞ ≃ − � rpp;r rpp � 3 EðinertialÞ mode ð∞Þ μω2 rrr ½1þ tanhuðtÞ�; ≃ −ṙppðtÞΔta 1þ tanhuðtÞ 2 : ð118Þ The tanh function is introduced to smoothly connect the pre and postresonance values. Alternatively, the same result can be obtained by noticing the change in p at a given frequency f ¼ ω=π is nearly zero as ΔpðfÞ ≃ ð4=3Þecϕ ≃ 0 based on Eq. (100). Eq. (118) then follows the same argument that leads to Eq. (92). Once we have Δp calculated, we can compute the frequency and phase shifts as functions of t. Δω ω ¼ − 3 2 Δp rpp þ 2ecϕ ≃ − 3 2 Δp rpp − 2gðt;eqÞr Mt=r2pp ; ð119Þ ΔϕðtÞ ¼ Z Δωdt ≃ Z Δω ω̇pp dω: ð120Þ The resultant Δϕ is shown in the red line in the bottom panel of Fig. 4 for the l ¼ 3 tide. We use ½− lgðtc=Mt − t=Mt þ 1Þ� as a modified time coordinate with tc the time when r ¼ 2R. Note that when computed as a function of time, the amount of oscillations in the postresonance part is much milder and it will be quantified in the following section. As our computation is at the linear order in Δp=p, it is accurate only when the tide is not too strong. This is why in the figures, we have focused on the l ¼ 3 tide when comparing the numerical and analytical calcula- tions. The backreactions are included for the l ¼ 3 tide and the l ¼ 2 contributions have been excluded. This does not impact much the calculation of Δϕa for the l ¼ 3 f-mode because the l ¼ 3 is excited at a lower frequency (ωr=ð2πÞ ¼ 510 Hz), or an earlier time, compared to the l ¼ 2 mode (ωr=ð2πÞ ¼ 950 Hz). For completeness, we show in Fig. 5 the phase shift induced by the l ¼ 2 tide (with the tidal back reactions from the l ¼ 3 components turned off). The analytical estimations [Eq. (88)] start to lose accuracy, emphasizing the need to incorporate higher order effects [which includes both higher order terms in Δp=r that come in at ðR=rÞ10 and higher order terms in ξ=R that come in at ðR=rÞ8]. Interestingly, frequency is no longer a monotonic function of time when the l ¼ 2 tide is included and this can be seen from the backward turning of the gray line in the top panel of Fig. 5. This is due to the excitation of orbital eccentricities by the dynamical tide, which we will discuss in Sec. IV C below (see also Fig. 7). C. Eccentricities excited by the dynamical tide In this section, we discuss the orbital dynamics in the postresonance part, focusing on the eccentricity excited by the dynamical tide. While a nonzero eccentricity e appears in the osculating variables even in the pre-resonance regime, caution is required when interpreting this result. While both the GW decay and the equilibrium tide can cause a finite “eccentricity” in the osculating equations [see Eqs. (108) HANG YU, PHIL ARRAS, and NEVIN N. WEINBERG PHYS. REV. D 110, 024039 (2024) 024039-16 and (109)], these effects do not break the quasicircular approximation of the orbit. This is because the argument of pericenter ϕ0 proceeds at the same rate as ϕ itself. More specifically, we have tanðϕ − ϕ0Þ ≃ 2 3 1 ωtgw Mt=r2pp gðt;eqÞr ; ðequilibrium tide onlyÞ ð121Þ which evolves only on the GW decay timescale, much longer than the orbital decay timescale. This means the separation r evolves also slowly over ∼tgw. See also Ref. [81] on related discussions. In contrast, once a mode is excited, the interaction energy changes on a timescale ½mðω − ωrÞ�−1, which can be much faster than the GW decay timescale. Therefore, the orbit cannot settle at the bottom of the effective potential and therefore cannot remain quasicircular due to the interaction energy being oscillatory. Instead, a forced eccentricity is excited [82]. The results are numerically confirmed in Figs. 6 and 7. To quantify this, we note that the binary separation r and its time derivative ṙ do not depend on e alone but instead depend on ecϕ and esϕ. Our goal is therefore to find approximate expressions for the oscillatory parts in ecϕ and esϕ as they are the terms causing deviations to the quasicircular approximation. To proceed, we start with the assumption that ecϕ ≃ Δp=rpp is small. This assumption will be justified later. To estimate esϕ then, we note from Eq. (99) that esϕ ∝ ṙ ≃ ṗ, and from Eq. (105) we have ṗ ∝ gϕ. Therefore, we arrive at ðesϕÞðoscÞ ≃ − gðtÞϕ gðgwÞϕ ðesϕÞpp ¼ 2 gðtÞϕ Mt=r2pp ; ð122Þ where we have used the superscript “(osc)” to emphasize that it is the oscillatory component, in contrast to the secular component in Eq. (108). To estimate ðecϕÞðoscÞ, we note that ðϕ − ϕ0Þ ≃ −3π=2, so d dt ðecϕÞðoscÞ ≃ eppðω− ϕ̇0Þ; ≃ω � −ðesϕÞppþ 2 gϕ Mt=r2pp � ≃ 2 ωrpp gðtÞϕ : ð123Þ Integrating the equation and approximate quantities vary- ing on the GW decay timescale as constants [or equiv- alently, doing integration by parts and ignoring terms suppressed by 1=ðωtgwÞ], we have ðecϕÞðoscÞ ≃ 2 ωrpp Z gðtÞϕ dt; ð124Þ Further, FIG. 6. Oscillatory eccentricities in excited by the l ¼ 3 dynamical tide. FIG. 5. Similar to Fig. 4 but for the l ¼ 2 tide. The analytical calculations [e.g., Eq. (88)] become inaccurate because the tidal perturbation is so large and keeping only terms linear in Δp=rpp is no longer sufficient. DYNAMICAL TIDES DURING THE INSPIRAL OF RAPIDLY … PHYS. REV. D 110, 024039 (2024) 024039-17 gðtÞϕ ≃ − 1 ðla þ 1Þ ġðtÞr ω − ωr ; ð125Þ gðtÞr ≃ la þ 1 m2 a ġðtÞϕ ω − ωr ð126Þ in the postresonance regime. Using Eq. (125) to evaluateR gðtÞϕ dt, we have ðecϕÞðoscÞ ≃ − 2 lþ 1 gðtÞr Mt=r2pp ; ð127Þ where we have replaced the ðω − ωrÞ part in Eq. (125) by ω to avoid divergence near resonance and it does not affect much the postresonant estimations. Note that ðecϕÞðt;oscÞ=ðesϕÞðt;oscÞ ≃ l=ðlþ 1Þ2 < 1, which is consis- tent with the assumptions we made at the beginning. The comparison between the numerical and analytical estimations of the oscillatory eccentricity is shown in Fig. 6 for the l ¼ 3 tide (with l ¼ 2 tide excluded). To numeri- cally extract the oscillatory component, we use Eqs. (103) and (104) to compute the total eccentricities from the numerical ðr; ṙ;ϕ; ϕ̇Þ and then remove the PP values given in Eq. (108). The equilibrium tide can also cause a secular contribution to ecϕ [Eq. (109)], yet numerically it is sufficiently small to be ignored. The analytical estimations are accurate for a small range of frequencies after mode resonance and lose accuracy in the later evolution. Nonetheless, they provide a decent estimation of the order of magnitude for the deviation from the quasicircular approximation. Interestingly, the magnitude of the eccen- tricities increases with frequency as the magnitude of gðtÞr;ðϕÞ increases. The numerically extracted eccentricity due to the l ¼ 2 tide is shown in Fig. 7. Here in the top panel, we present also the instantaneous GW frequency (still defined through f ¼ ϕ̇=π) as a function of time, which is no longer a monotonically increasing function as in the PP case. This confirms the eccentricity excitation as shown in the middle and bottom panels. The magnitude of this eccentricity can exceed 0.05 near the end of the inspiral. An estimation of its detectability will be presented later in Sec. IV D. Lastly, we estimate the magnitude of the oscillatory eccentricity as a function of the phase shift Δϕa [Eq. (85)]. We have ΔEeq ≃ ωrμrrg ðtÞ ϕ tres; ð128Þ Δϕ ≃ ωr ΔEeq Ėpp ; ð129Þ where tres ¼ ffiffiffiffiffiffiffiffiffiffi 1=ω̇r p is the duration of resonance. We thus have esϕ ∼ jΔϕj � rr r � l ðωrtgw;rÞ−3=2: ð130Þ While the magnitude of the forced eccentricity increases as 1=rl ∝ ω2l=3, its overall magnitude is suppressed by the large factor of ðωrtgw;rÞ3=2 ∝ ω−5=2 r . The postresonance eccentricity is therefore less significant compared to the phase shift for modes excited earlier during the inspiral (e.g., gravity and inertial modes). D. GW from the dynamical tide While the dominant impact of the tide is the phase shift discussed in Figs. 4 and 5, we also consider additional features caused by the dynamical tide. In particular, once FIG. 7. The top panel shows the frequency-time relation when the l ¼ 2 tide is included. The frequency is computed f ¼ ϕ̇=π ¼ ω=π. The middle and bottom panels are similar to Fig. 6 and it shows the eccentricity excited by the l ¼ 2 dynamical tide. All results shown in this figure are extracted numerically. HANG YU, PHIL ARRAS, and NEVIN N. WEINBERG PHYS. REV. D 110, 024039 (2024) 024039-18 the f-mode is resonantly excited, it will ring at its natural frequency (¼ maωr in the inertial frame) and emit GW accordingly. This is similar to the postmerger oscillation of two nonspinning NSs [45,83–90]. For the oscillation to be excited during the inspiral stage, it requires either a highly eccentric orbit or a rapid, antialigned spin of the NS. Numerical simulations [91,92] have confirmed the former case, and our study aims to provide a detailed discussion of the latter scenario when the NS is rapidly spinning. Further, the deviation from the quasicircular approximation (Fig. 7) allows GW to be emitted at frequencies other than ϕ̇=π for the quadrupole GW. Both effects are examined in this section for the l ¼ 2 tide that dominates the back reaction. To compute the GW strain from the oscillating f-mode, we use the quadrupole formula [77], hij ¼ 2 DL Q̈hiji: ð131Þ From Eq. (38), we have Q̈ij ns ≃ −2N2ð2ωrÞ2M1R2 1Ia × Re½ðYij 22Þ�cae−2iωrtþiψ r �; ð132Þ where we have expressed the mode amplitude in terms of ca ¼ qa exp½iðωat − ψ rÞ� which stays nearly constant after resonance. The subscript a stands for the f mode with ðla; jmajÞ ¼ ð2; 2Þ and the factor of 2 comes from summing over ma ¼ �2 contributions. The strain tensor can be projected to the two polar- izations as hþ ¼ 1 2 ðeiXejX − eiYe j YÞhij; ð133Þ h× ¼ 1 2 ðeiXejY þ eiYe j XÞhij; ð134Þ where eX¼ðcosψ ;−sinψ ;0Þ, eY ¼ ðcos ι sinψ ; cos ι cosψ ; − sin ιÞ. The angle ι is the line of sight and ψ specifies the x-axis of the detector relative to the orbit. We have hnsþ ¼ − ð1þ cos2 ιÞ 2 � 2ωr ω1 � 2 hns0 Re½cae−i2ωrtþiψ0 �; ð135Þ hns× ¼ cos ι � 2ωr ω1 � 2 hns0 Im½cae−i2ωrtþiψ0 �; ð136Þ where ω2 1 ¼ M1=R3 1, ψ0 ¼ ψ r − 2ψ , and hns0 ¼ ffiffiffiffiffiffiffiffi 32π 15 r M2 1 R1DL Ia: ð137Þ For the rest of the discussions, we will assume that the detector’s antenna response is such that the observed strain is given by hþ. To gain some analytical insights, we can replace ca with the asymptotic mode energy, Eq. (59), leading to hnsþ ≃ ð1þ cos2 ιÞ 2 Ans cosð2ωrtþ ψ 0 0Þ; ð138Þ where Ans ≃ ffiffiffiffiffiffiffiffiffiffiffiffiffi ωrtgw;r 4π r � M2 Mt �� ωa0 ω1 �� 2ωr ω1 � 3 k20M2 1 R1DL ∝ ω13=6 r : ð139Þ While for the l ¼ 2 tide, Eq. (59) loses accuracy because of the neglecting of higher-order effects, it nonetheless pro- vides the scaling of the strain amplitude with different parameters, in particular the frequency, or ωr, at which the mode is resonantly excited, hence the spin rate Ωs as maωr ¼ ωa þmaΩs. The Fourier transform is given by (with f > 0) jh̃nsþðfÞj ¼ j Z hnsþei2πftdtj ≃ ð1þ cos2ιÞ 2 AnsT 2 sinc � ð2πf − 2ωrÞ T 2 � ; ð140Þ where T ≃ ð3=8Þtgw;r is the approximate duration for the f-mode to ring, so jh̃nsþðfÞj ∝ ω−1=2 r . This allows us to compute the signal-to-noise ratio (SNR ρ) of the f-mode, ρ2 ¼ Z 4fh̃�ðfÞh̃ðfÞ SðfÞ d log f; ð141Þ which varies slowly with respect to ωr when the noise power spectral density (PSD) SðfÞ is nearly flat. Damping on the excited f-mode has been ignored in the calculations above. This is justified by noticing that the energy loss from the excited f-mode can be estimated as [assuming it is dominated by GW radiation and other mechanisms, such as Urca reactions [93] or nonlinear fluid instabilities [76], are subdominant; cf. Eq. (77)] Ėmode ¼ − 8 15 W22hQ …ns 22Q …ns 22 �i: ð142Þ The energy-damping timescale of the excited f-mode is thus tmode ¼ EðinertialÞ mode ð∞Þ jĖmodej ¼ 3π 64 1 W22k20 � ω1 ωr � 5 � ω1 ωa0 � ðM1ω1Þ−5=3ω−1 1 : ð143Þ DYNAMICAL TIDES DURING THE INSPIRAL OF RAPIDLY … PHYS. REV. D 110, 024039 (2024) 024039-19 The ratio of this timescale to the remaining time of the inspiral is thus tmode 3 8 tgw;r ¼ 12πη 5W22k20 � Mt M1 � 5=3 � ω1 ωr � 7=3 � ω1 ωa0 � : ð144Þ Numerically, this ratio is about 1.4 × 103 for the system we consider. Therefore, damping of the f-mode can be safely ignored. In Fig. 8 we show the strain from the la ¼ jmaj ¼ 2 NS f-mode excited by the dynamical tide. The red trace is computed by plugging into Eq. (135) the numerical mode amplitude ca with the equilibrium component [Eq. (18)] removed. The source is placed at a luminosity distance of DL ¼ 50 Mpc. The time domain waveform is sampled at a rate of 8192 Hz and then transferred to the frequency domain. To avoid boundary effects in the Fourier trans- form, we let the system evolve to r ¼ R1 and then window out the portion where r < 2.3R1 with half of a Hann window (i.e., the window function varies from 1 at time when r ¼ 2.3R1 to 0 when r ¼ R1). The GW frequency corresponding to the truncation is 1390 Hz ≃ 0.89fisco. The SNR estimate is insensitive to the truncation location because the amplitude of the signal in the frequency domain is proportional to the duration of the signal, and T ∼ tgw ∼ f−8=3 is dominated by the low-frequency end (i.e., the location of resonance). If we instead more conservatively window out the portion where r < 2.6R1, or f > 1160 Hz, the SNR is reduced only by 7%. As a reference, the olive dotted line shows the analytical estimation based on Eq. (140), which overestimates (underestimates) the amplitude (width) the width of the peak. We nonetheless verified its accuracy by artificially reducing the Love number by a factor of 100 to eliminate higher-order ðΔr=rÞ effects. Also plotted in the gray line is the sensitivity of Cosmic Explorer (CE) [94–96]. Interestingly, the GW from the f-mode can be detected with an SNR ρ ¼ 1.3 for a source at 50 Mpc from the numerical result (red curve in Fig. 8). Because CE’s sensitivity is nearly flat up to 1000 Hz, the SNR does not vary much concerning the spin rate of the NS as long as it is more negative than the critical value estimated in Eq. (12), consistent with the scaling discussed below Eq. (140). At Ω ¼ −2π × 300 Hz, we can still recover the signal with ρ ¼ 0.8. If the NS is described by the harder H4 EOS, then the SNR becomes ρ ¼ 0.9 when Ω ¼ −2π × 200 Hz, and ρ ¼ 2 when −Ω≳ 2π × 400 Hz. This indicates GW from the NS f-mode can be a signature to look for in addition to GW from the orbit with the next generation of detectors (including CE and the Einstein Telescope [97,98], as well as the Neutron Star ExtremeMatter Observatory [99] that has a sensitivity comparable to CE with f ∼ a few × 1000 Hz). The peaklike feature of this signal should allow it to be detected with little degeneracy with other effects from the orbital inspiral, and identifying the peak frequency directly constrains the combination of NS f-mode frequency and its spin rate, while the height provides additional information on the Love number. The deviation of a quasicircular inspiral is another effect caused by the dynamical tide that we want to examine. For this, we decompose the strain tensor into modes as hþ − ih× ¼ X m h2m−2Y2m; ð145Þ where −2Y2m is spin weighted spherical harmonic with spin −2. When computing the derivatives of the orbital quadru- pole Qorb ij ¼ μrirj, we replace ̈r and ϕ̈ using Eqs. (52) and (53), leading to h22 ¼ 4 ffiffiffi π 5 r e−2iϕ ηMt DL � Mt r þ r2ϕ̇2 −rgr − ṙ2 þ irðgϕ þ 2ṙ ϕ̇Þ � : ð146Þ In terms of the osculating variables, we have h22¼ h0e−2iϕ � 1þ � 3 2 ecϕþ iesϕ � þð−grþ igϕÞ 2Mt=p2 � ; ð147Þ where FIG. 8. GW strain from the la ¼ jmaj ¼ 2 NS f-mode excited by the dynamical tide, assuming the source is located at DL ¼ 50 Mpc and the detector’s antenna response is such that the observed strain is hþ. The red line is computed using the numerical mode amplitude and the olive dotted line is from the analytical estimation Eq. (140). The sensitivity of the next generation of GW detector Cosmic Explore is shown in the gray line. The numerically computed strain can be detected with a matched-filtering SNR of ρ ¼ 1.3. Varying the spin rate of the NS does not affect the SNR much. HANG YU, PHIL ARRAS, and NEVIN N. WEINBERG PHYS. REV. D 110, 024039 (2024) 024039-20 h0 ¼ 8 ffiffiffi π 5 r μMt DLp : ð148Þ Because gðtÞrðϕÞ=ð2Mt=p2Þ ∼ ðesϕÞðoscÞ [Eq. (122)], the non- circular component of the GW can be estimated as δh22 ∼ ðesϕÞðoscÞh0e−2iϕ: ð149Þ Because ðesϕÞðoscÞ ∼ cos ½maðϕ − ωrtÞ�, we see that δh22 has a component that varies at 2ωr (together with a high- frequency component at 4ω − 2ωr which is outside the sensitivity band of a GW detector). Therefore, this effect should be added coherently with the GW from the NS f-mode [Eq. (136) and (135)]. It is thus interesting to compare DLδh22 and hðnsÞ22 . The GW strain from the oscillatory orbital eccentricity is DLδh22 ∼DLðesϕÞðoscÞh0 ∼ E1Vaba; ð150Þ where we have replaced ðesϕÞðoscÞ in terms of gtϕ using Eq. (122) and then used Eq. (55). On the other hand, the GW strain from excited NS mode is DLh ðnsÞ 22 ∼ ω2 abaIaM1R2 1 ≃ E1Iaba: ð151Þ Therefore,DLδh22 ∼ ðR1=rÞ3DLh ðnsÞ 22 , and the GW from the orbital eccentricity is subdominant compared to the GW from NS f-mode itself. V. CONCLUSION AND DISCUSSION The key conclusion of the paper is the following. (1) We showed a new approach to analytically solve the time evolution of mode amplitude, or equivalently NS mass multipoles [Eq. (38)], in the presence of resonance, by decomposing the amplitude into a resummed equilibrium component [including finite frequency corrections; Eq. (18)] and a dynamical component that is excited only around mode reso- nance [Eq. (23)]. The new formalism simplifies the numerical implementation as both components re- main finite throughout the evolution. This avoids the need to subtract two diverging terms at mode reso- nance as required in previous analysis [25,56,57]. Furthermore, the solution can be extended in the postresonance regime whereas previous works are accurate only up to the resonant frequency (Fig. 2). (2) The effective Love number proposed in, e.g., Refs. [25,57] (κl in our notation) can capture the radial backreaction [Eq. (54)] but it does not account for the tidal torque [Eq. (55)]. To capture the torque, an additional dressing factor γl origi- nating from the imaginary part of the Love number of each ðl; mÞ harmonic is necessary [Eq. (47)]. Near mode resonance, it is the tidal torque that dominates the impact on the orbit (Fig. 3). The effective Love number is also insufficient to de- scribe the finite-frequency correction to the GW radiation from the interaction of NS and orbital mass multipoles [Eq. (80)], yet this effect is subdominant compared to the conservative tidal torque near mode resonance [Eq. (84)]. (3) We computed the tidal phase shift as functions of both frequency and time using both energy-balancing arguments and osculating orbits (Sec. IV B). The dominant effect of mode resonance is a conservative energy transfer from orbit to the NS mode [Eq. (65)]. Consistent with the previous analysis, this effect can be approximated by a sudden change in the wave- form frequency [Eqs. (90) and (97)]. The effect scales linearly with the amount of phase shift at mode resonance, which can be 0.5 and 10 radians for the l ¼ 3 and l ¼ 2 f-modes, respectively (top panels in Figs. 4 and 5). While our analytical approxima- tions of the phase shifts have good accuracy for the l ¼ 3 f-mode (Fig. 4), for the l ¼ 2 f-mode linear theory is insufficient, as illustrated in Fig. 5; we discuss this point below. We further considered the dissipative effect due to the interaction of NS and orbital mass quadrupoles and showed the finite frequency correction cannot be captured with the effective Love number [Eq. (80)]. The dynamical tide has no net contribution to this effect [Eq. (83)]. (4) Further signatures associated with the dynamical tide were examined in Secs. IV C and IV D. In particular, the orbit cannot remain quasicircular after a mode’s resonance. The eccentricity excited by the dynamical tide can further cause the frequency to be nonmonotonic with time (Fig. 7). The excited f-mode can also emit GW on its own. Such a signal can be detected with an SNR greater than unity at 50 Mpc with the next generation of GW detectors for a wide range of NS spin (Fig. 8). While we focused on the resonant excitation of f-modes, our approach is general and can be applied to the resonant excitation of other NS modes [23,31,33–44]. For example, the equilibrium component of each mode would allow us to compute the correction to the effective Love number from other NS modes (similar to, e.g., Ref. [59]). Our Eq. (18) enables the incorporation of finite-frequency corrections while avoiding divergence near resonance. On the other hand, our analysis in Sec. IV C proves that the phase shift near mode resonance [Eqs. (74) and (120)] is the dominant dynamical tide effect of low-frequency modes (such as gravity modes) whereas the postresonance eccentricity excited by such a mode is subdominant [Eq. (130)]. As a caveat, our analysis adopted a few simplifying assumptions. In particular, our treatment of the orbital dynamics is at the lowest order (i.e., Newtonian). Since the DYNAMICAL TIDES DURING THE INSPIRAL OF RAPIDLY … PHYS. REV. D 110, 024039 (2024) 024039-21 f-mode excitation mainly happens during the late inspiral stage, relativistic corrections are crucial and we plan to upgrade our study to the EOB framework [25,56,57] to generate waveforms to be used for data analysis. Of particular significance to the dynamical tide are the corrections to the resonance frequency produced by redshift and frame dragging (see e.g., Fig. 1 of Ref. [25]). These effects further determine the critical spin rate required for the f-modes to experience resonance. The amplitude of the mode further depends on the duration of resonance through the ffiffiffiffiffiffiffiffiffiffi 1=ω̇r p factor [e.g., Eq. (23)], and in the late inspiral stage the orbital decay rate can see significant corrections from the higher PN terms. The path forward to upgrading our analysis to the EOB formulation is well-defined. Note that our tidal HamiltonianHmode þHint plays the same role as HDT in Ref. [25], which also accounts for relativistic redshift and frame dragging. The total Hamiltonian, the central part of the EOB formulation, is constructed when we further sum the tidal Hamiltonian to the upgrade PP Hamiltonian, which can be found in, e.g., Ref. [100]. More specifically, our κl can directly replace the effective Love number in Ref. [25], while the γl [Eq. (47)] term describes a backreaction torque missing in previous studies that should be further integrated into the EOB evolution equations. Besides the conservative dynamics, our Eqs. (80) and (82) further provide the finite-frequency and dynamical tide corrections to the dissipative GW radiation from the interaction between NS and orbital quadrupoles, which improves the existing EOB models that compute the dissipative effects under the adiabatic limit [see, e.g., the discussion below eq. (5) of Ref. [57] ]. At the same time, higher-order spin corrections of the NS should be incorporated for the rapidly spinning NSs considered here. We note that spin corrects the effective Love number [Eq. (45)] starting at Ω2 due to the finite frequency response of the modes. The modification to the background structure also enters at the same order [101], yet such an effect is ignored in the current study. Further investigations are hence required. Our Fig. 5 highlights the need for developing tidal theories in spinning NSs beyond the linear order in both ðΔr=rÞ and (ξ=R). The higher-order effect in ðΔr=rÞ, which corresponds to higher-order back reactions formally entering at ðR=rÞ10, can be captured if we do not use a closed-form expression for the tidal phase shift but instead numerically solve the coupled differential equations gov- erning the evolution of tide and orbit. For computational efficiency, a hybrid approach may be taken where analyti- cal expressions can be when u ≲ −1 [Eq. (25)], and only the evolution with u≳ −1 is tracked numerically. To capture higher-order hydrodynamical effects [i.e., higher-order effects in (ξ=R)], new physical ingredients must be included. At the lowest order, this includes adding a three-wave interaction in the Hamiltonian [Eq. (48)] and a nonlinear correction to the mass quadrupole [Eq. (38)]; see Ref. [73] for details. The nonlinear hydrodynamic effects enter at ðR=rÞ8 and therefore can be more significant than the higher-order backreaction effects. Indeed, as shown in Ref. [73], the nonlinear hydrodynamic effect effectively lowers the eigenfrequency of a mode, enabling f-mode resonance to happen in (more realistic) NSs with milder spins. Moreover, the anharmonicity of the f-mode [102], while being a higher order effect in (R=r), can become import as it is amplified by mode resonance. The la ¼ 2 f-mode can reach an energy nearly 10% of the binding energy with resonance, which suggests the effective natural frequency of the f-modewould be lower than the linear value by ∼10% [73]. Such a correction can become comparable or even exceed various relativistic corrections considered in Ref. [25]. The peak frequency of the GW signal from a resonantly excited f-mode (Fig. 8) will also shift due to the nonlinear hydrodynamic corrections (cf. the anharmonicity of an oscillator [102]). Another type of nonlinear hydro- dynamic effect is the excitation of small-scale fluid insta- bilities such as the p-g instability [4,76,103–105]. As the tide deviates more from the adiabatic limit and reaches higher energy in a spinning NS, cancellations in the instabilities growth rate may be reduced, potentially amplifying their impact on the GW signal. Future investigations along these directions are therefore crucial. The strong l ¼ 2 tide in the presence of f-mode reso- nance also means we need to be cautious when building frequency-domain phenomenological models including tides (e.g., Ref. [54]). As shown in Fig. 7, the quasicircular approximation is no longer accurate and the frequency evolution is no longer monotonic after the l ¼ 2 f-mode’s excitation. In the example shown in Fig. 7, the orbit hits f ¼ 1450 Hz at 4 different instants of time. As a result, the orbit is not uniquely defined at a given frequency, violating the underlying assumption of many frequency-domain phenomenological models. Nonetheless, approaches sim- ilar to that proposed in, e.g., Ref. [106] may be applied to handle eccentricity in the frequency domain. We do not consider the impact of NS crust. Reference [44] showed that the presence of a crust does not affect the f-mode we consider here. We nonetheless note that the energy stored in the f-mode is large enough to potentially break the crust [42,43]. If this does happen, its impact on the GW signal (both the tidal phase shift and the GWemission from the f-mode itself) is unclear and requires further investigation. Besides the tidal phase shift considered in this work, a rapidly spinning NS produces other matter effects in the inspiral stage including dephasing and precession (in generic spin configurations) due to spin-induced quadrupole, see, e.g., Ref. [11] and references therein. The postmerger oscillation is yet another signal due to matter effects with promising detection prospects. A complete waveform model should coherently integrate all these components to maxi- mize the information that can be extracted. HANG YU, PHIL ARRAS, and NEVIN N. WEINBERG PHYS. REV. D 110, 024039 (2024) 024039-22 Suppose a misaligned spin is produced because of the dynamical formation of the binary. In that case, there is also a possibility that the binarywill have some residual eccentricity when it enters the sensitivity band of a ground-based GW detector [9,107–111]. Discussions of tides in eccentric BNS systems with varying initial eccentricities can be found in Refs. [112–118]. Interestingly, Ref. [61] showed that f-mode excitations observed in numerical relativity simulations are not captured with the effective Love number prescription. This is consistent with our finding because the solutions constructed following Refs. [25,56,57] do not correctly capture the oscillation frequency of the NS mass multipoles in the postresonance regime (Fig. 2). A decomposition of the mass multipoles into an equilibrium (following the orbital phase) and a dynamical component (corresponding to the excited oscillator) similar to our Eq. (14) (see also Ref. [119]) is instead a promising way of modeling tides in eccentric systems. 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