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Schematic diagram of the spin and orbital angular momentum vectors. The coordinate system is defined such that LN is along the z-axis and (θ1, θ2) are the respective angles between LN and (S1, S2). The projection of S1 onto the x-y plane is defined to be along the x-axis so the azimuthal spin angles are φ1 = 0 and φ2 = ∆φ.
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Starting with a post-Newtonian description of compact binary systems, we derive a set of equations that describes the evolution of the orbital angular momentum and both spin vectors during inspiral. We find regions of phase space that exhibit resonance behavior, characterized by small librations of the spin vectors around a fixed orientation. Due t...
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... the overall dynamics are preserved under rotation around L N so we can reduce the spin degrees of freedom by defining thê e x direction along φ 1 = 0, leaving four independent coordinates to define the orientation of the system: (L N , θ 1 , θ 2 , ∆φ). Figure 1 shows a schematic of the geometry used throughout this paper. Following the post-Newtonian formalism, all angles and vector magnitudes are defined in a Cartesian, flat space-time. ...
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... an initial probability distribution function f (θ 1 ), we can predict the final distribution of (θ 12 , ∆φ) by weighting the ensemble of distributions in Figures 7 and 8 by f (θ 1 ). This initial spin distribution function has been the focus of much recent work in compact binary systems [10,16]. ...
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... the moderate mass ratios (m 1 : m 2 ≈ 1 − 10) for which resonance behavior will be important, one can estimate roughly the range over which the spin locking occurs during the binary inspiral by examining equation (3.1). To leading order, we can write the equilibrium condition as ...
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... equation (5.1) suggests that the resonant locking effects might fall off for smaller values of S 1 and S 2 , in practice we find a relatively weak dependence on the magnitude of the spin parameter for a/m = S/m 2 > 0.5. Figure 10 shows the probability distributions of (θ 12 , ∆φ) near the end of inspiral for a variety of spin magnitudes. Each system has the same mass ratio m 1 : m 2 = 11 : 9 and initial spin-orbit angle θ 1 (t 0 ) = 10 • . ...
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... Each system has the same mass ratio m 1 : m 2 = 11 : 9 and initial spin-orbit angle θ 1 (t 0 ) = 10 • . Even for spins as small as a/m = 0.25 we see significant spin locking at the end of inspiral. Interestingly, the final distribution for a/m = 0.5 and m 1 : m 2 = 11 : 9 is almost identical to that of a/m = 1 and m 1 : m 2 = 3 : 1 (cf. Figs. 9 and 10), perhaps pointing to another relation in the evolution equations that could be used to reduce further the dimensionality of the total search space, analogous to the effective spins described in ...
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... In a frame that coprecesses with the binary [42][43][44], the mutual orientations of L, S 1 and S 2 are fully described by three angles [45]. These are often chosen to be the polar angles θ 1;2 ∈ ½0; π between the spin and orbital angular momentum: ...
... Physically, these are cases where the relative orientation of the spins and the orbital angular momentum is fixed on the precession timescale. These configurations are the so-called "spin-orbit resonances" first discovered by Schnittman [45] and later explored at length by several authors [3,4,11,32,54,[62][63][64][65][66][67][68][69]. In particular, Ref. [11] formally proved that there are always two spin-orbit resonances κ AE for each set of ðq; χ 1 ; χ 2 ; r; χ eff Þ, as previously suggested by extensive numerical explorations [4,45]. ...
... These configurations are the so-called "spin-orbit resonances" first discovered by Schnittman [45] and later explored at length by several authors [3,4,11,32,54,[62][63][64][65][66][67][68][69]. In particular, Ref. [11] formally proved that there are always two spin-orbit resonances κ AE for each set of ðq; χ 1 ; χ 2 ; r; χ eff Þ, as previously suggested by extensive numerical explorations [4,45]. ...
We present analytical and numerical progress on black-hole binary spin precession at second post-Newtonian order using multitimescale methods. In addition to the commonly used effective spin which acts as a constant of motion, we exploit the weighted spin difference and show that such reparametrization cures the coordinate singularity that affected the previous formulation for the case of equal-mass binaries. The dynamics on the precession timescale is written down in closed form in both coprecessing and inertial frames. Radiation reaction can then be introduced in a quasiadiabatic fashion such that, at least for binaries on quasicircular orbits, gravitational inspirals reduce to solving a single ordinary differential equation. We provide a broad review of the resulting phenomenology and rewrite the relevant physics in terms of the newly adopted parametrization. This includes the spin-orbit resonances, the up-down instability, spin propagation at past time infinity, and new precession estimators to be used in gravitational-wave astronomy. Our findings are implemented in version 2 of the public Python module precession. Performing a precession-averaged post-Newtonian evolution from/to arbitrarily large separation takes ≲0.1 s on a single off-the-shelf processor—a 50× speedup compared to our previous implementation. This allows for a wide variety of applications including propagating gravitational-wave posterior samples as well as population-synthesis predictions of astrophysical nature.
... The vast majority of these systems should be nearly circular (negligible eccentricity) [10,11] and they ought to include black holes with masses below the pair instability gap [12][13][14][15]. Binaries assembled dynamically in dense stellar environments are likely to contain black holes with isotropic random spin orientations [16][17][18][19][20][21]. Some fraction of these systems, perhaps ≈ 5%, may be measurably eccentric [22][23][24][25][26][27][28]. ...
The formation history of binary black hole systems is imprinted on the distribution of their masses, spins, and eccentricity. While much has been learned studying these parameters in turn, recent studies have explored the joint distribution of binary black hole parameters in two or more dimensions. Most notably, Callister et al. [T. A. Callister, C.-J. Haster, K. K. Y. Ng, S. Vitale and W. M. Farr, Astrophys. J. Lett. 922, L5 (2021)] find that binary black hole mass ratio and effective inspiral spin $\chi_\text{eff}$ are anti-correlated. We point out a previously overlooked subtlety in such two-dimensional population studies: in order to conduct a controlled test for correlation, one ought to fix the two marginal distributions -- lest the purported correlation be driven by improved fit in just one dimension. We address this subtlety using a tool from applied statistics: the copula density function. We use Callister et al. as a case study to demonstrate the power of copulas in gravitational-wave astronomy while scrutinising their astrophysical inferences. Our findings, however, affirm their conclusion that binary black holes with unequal component masses exhibit larger $\chi_\text{eff}$ (98.7% credibility). We conclude by discussing potential astrophysical implications of these findings as well as prospects for future studies using copulas.
... In addition to effects on the semimajor axis and eccentricity, Schnittman (2004) showed that the directions of the spins of the two black holes can change systematically during GW inspiral. In particular, if the more massive black hole is at least partially aligned with the orbital axis and the mass ratio is close to unity, then the spins of both holes tend to align with the orbital axis. ...
The next two decades are expected to open the door to the first coincident detections of electromagnetic (EM) and gravitational-wave (GW) signatures associated with massive black-hole (MBH) binaries heading for coalescence. These detections will launch a new era of multimessenger astrophysics by expanding this growing field to the low-frequency GW regime and will provide an unprecedented understanding of the evolution of MBHs and galaxies. They will also constitute fundamentally new probes of cosmology and would enable unique tests of gravity. The aim of this Living Review is to provide an introduction to this research topic by presenting a summary of key findings, physical processes and ideas pertaining to EM counterparts to MBH mergers as they are known at the time of this writing. We review current observational evidence for close MBH binaries, discuss relevant physical processes and timescales, and summarize the possible EM counterparts to GWs in the precursor, coalescence, and afterglow stages of a MBH merger. We also describe open questions and discuss future prospects in this dynamic and quick-paced research area.
... While the tilt angles θ 1 and θ 2 control precession, the orbital-plane spin angles ϕ 1 and ϕ 2 play a central role in binaries undergoing spin-orbit resonances (SORs) [6]. For these binaries, the χ 1 , χ 2 , and L vectors become locked into a common resonant plane such that Δϕ ¼ ϕ 1 − ϕ 2 is fixed at 0 or AEπ as the binary precesses. ...
... On the other hand, the preference for Δϕ ∼ AEπ in Fig. 2 is expected in some formation channels where SORs [6] are important. In particular, stellar binaries with significant supernova natal kicks and efficient stellar tides can be driven toward these resonances [13,14]. ...
Binary black hole spin measurements from gravitational wave observations can reveal the binary's evolutionary history. In particular, the spin orientations of the component black holes within the orbital plane, ϕ_{1} and ϕ_{2}, can be used to identify binaries caught in the so-called spin-orbit resonances. In a companion paper, we demonstrate that ϕ_{1} and ϕ_{2} are best measured near the merger of the two black holes. In this work, we use these spin measurements to provide the first constraints on the full six-dimensional spin distribution of merging binary black holes. In particular, we find that there is a preference for Δϕ=ϕ_{1}-ϕ_{2}∼±π in the population, which can be a signature of spin-orbit resonances. We also find a preference for ϕ_{1}∼-π/4 with respect to the line of separation near merger, which has not been predicted for any astrophysical formation channel. However, the strength of these preferences depends on our prior choices, and we are unable to constrain the widths of the ϕ_{1} and Δϕ distributions. Therefore, more observations are necessary to confirm the features we find. Finally, we derive constraints on the distribution of recoil kicks in the population and use this to estimate the fraction of merger remnants retained by globular and nuclear star clusters. We make our spin and kick population constraints publicly available.
... Precession and higher multipoles lead to very rich dynamics, which in turn is imprinted on the GW signal (see e.g. [65,67,210,243,244,249,[329][330][331][332][333][334][335][336][337][338]). Their measurements will be able to shed light on the formation mechanism of the observed systems, probe the astrophysical environment, break degeneracy among parameters, allowing more accurate measurements of cosmological parameters, masses and spins, and more sophisticated tests of GR. ...
In the last five years, gravitational-wave astronomy has gone from a purely theoretical field into a thriving experimental science. Many gravitational-wave signals, emitted by stellar-mass binary black holes and binary neutron stars, have been detected, and many more are expected in the future. The observation of the gravitational-wave signals from these systems, and the characterization of their sources, heavily relies on the precise models for the emitted gravitational waveforms. In this thesis, I present an updated version of the waveform models for spinning binary black holes within the effective-one-body formalism. The novelty of the waveform models presented in this work is the inclusion of beyond-quadupolar terms in the waveforms emitted by spinning binary black holes. I first construct the model in the simplified case of black holes with spins aligned with the orbital angular momentum of the binary, then I extend it to the case of generic spin orientations. The measurement of the source properties of a binary system emitting gravitational waves requires to compute $\mathcal{O}(10^7-10^9)$ different waveforms. Since the waveform models mentioned before can require $\mathcal{O}(1-10)$s to generate a single waveform, they can be difficult to use in data-analysis studies. To overcome this obstacle, I use the reduced-order-modeling technique to develop a faster version of the waveform model for black holes with spins aligned to the orbital angular momentum of the binary. The waveform models developed in this thesis have been used by the LIGO and Virgo collaborations for the inference of the source parameters of the gravitational-wave signals detected during the second and third observing runs (O2 and O3) of the LIGO and Virgo detectors. Here, I present a study on the source properties of the signals GW170729 and GW190412, for which I have been directly involved in the analysis.
... Precession and higher multipoles lead to very rich dynamics, which in turn is imprinted on the GW signal (see e.g. [65,67,210,243,244,249,[329][330][331][332][333][334][335][336][337][338]). Their measurements will be able to shed light on the formation mechanism of the observed systems, probe the astrophysical environment, break degeneracy among parameters, allowing more accurate measurements of cosmological parameters, masses and spins, and more sophisticated tests of GR. ...
In the last five years, gravitational-wave astronomy has gone from a purerly theoretical field into a thriving experimental science. Several gravitational- wave signals, emitted by stellar-mass binary black holes and binary neutron stars, have been detected, and many more are expected in the future as consequence of the planned upgrades in the gravitational-wave detectors. The observation of the gravitational-wave signals from these systems, and the characterization of their sources, heavily relies on the precise models for the emitted gravitational waveforms. To take full advantage of the increased detector sensitivity, it is then necessary to also improve the accuracy of the gravitational-waveform models. In this work, I present an updated version of the waveform models for spinning binary black holes within the effective-one-body formalism. This formalism is based on the notion that the solution to the relativistic two- body problem varies smoothly with the mass ratio of the binary system, from the equal-mass regime to the test-particle limit. For this reason, it provides an elegant method to combine, under a unique framework, the solution to the relativistic two-body problem in different regimes. The main two regimes that are combined under the effective-one-body formalism are the slow-motion, weak field limit (accessible through the post-Newtonian theory), and the extreme mass-ratio regime (described using the black-hole- perturbation theory). This formalism is nevertheless flexible enough to integrate information about the solution to the relativistic two-body problem obtained using other techniques, such as numerical relativity. The novelty of the waveform models presented in this work is the inclusion of beyond-quadupolar terms in the waveforms emitted by spinning binary black holes. In fact, while the time variation of the source quadupole moment is the leading contribution to the waveforms emitted by binary black holes observable by LIGO and Virgo detectors, beyond-quadupolar terms can be important for binary systems with asymmetric masses, large total mass, or observed with large inclination angle with respect to the orbital angular momentum of the binary. For this purpose, I combine the approximate analytic expressions of these beyond-quadupolar terms, with their calculations from numerical relativity, to develop an accurate waveform model including inspiral, merger and ringdown for spinning binary black holes. I first construct this model in the simplified case of black holes with spins aligned with the orbital angular momentum of the binary, then I extend it to the case of generic spin orientations. Finally, I test the accuracy of both these models against a large number of waveforms obtained from numerical relativity. The waveform models I present in this work are the state of the art for spinning binary black holes, without restrictions in the allowed values for the masses and the spins of the system. The measurement of the source properties of a binary system emitting gravitational waves requires to compute O(107 − 109) different waveforms. Since the waveform models mentioned before can require O(1 − 10)s to generate a single waveform, they can be difficult to use in data-analysis studies given the increasing number of sources observed by the LIGO and Virgo detectors. To overcome this obstacle, I use the reduced-order-modeling technique to develop a faster version of the waveform model for black holes with spins aligned to the orbital angular momentum of the binary. This version of the model is as accurate as the original and reduces the time for evaluating a waveform by two orders of magnitude. The waveform models developed in this thesis have been used by the LIGO and Virgo collaborations in the inference of the source parameters of the gravitational-wave signals detected during the second observing run (O2), and first half of the third observing run (O3a) of LIGO and Virgo detectors. Here, I present a study on the source properties of the signals GW170729 and GW190412, for which I have been directly involved in the analysis. In addition, these models have been used by the LIGO and Virgo collaborations to perform tests on General Relativity employing the gravitational-wave signals detected during O3a, and to analyze the population of the observed binary black holes.
... [24,70] and model the distribution as the sum of two populations motivated by the most popular BBH formation channels (see, e.g., Refs. [6,8,12,71]): an isotropic component and a preferentially aligned component, where the hyperparameters σ 1=A and σ 2=B control the spread in the possible tilt angles around θ 1=A ¼ θ 2=B ¼ 0. A nonzero value for σ indicates that not all tilts are aligned. We conduct hierarchical Bayesian inference using this model for the tilt angles under both the mass and spin sorting. ...
Gravitational waves from binary black holes have the potential to yield information on both of the intrinsic parameters that characterize the compact objects: their masses and spins. While the component masses are usually resolvable, the component spins have proven difficult to measure. This limitation stems in great part from our choice to inquire about the spins of the most and least massive objects in each binary, a question that becomes ill defined when the masses are equal. In this Letter, we show that one can ask a different question of the data: what are the spins of the objects with the highest and lowest dimensionless spins in the binary? We show that this can significantly improve estimates of the individual spins, especially for binary systems with comparable masses. When applying this parametrization to the first 13 gravitational-wave events detected by the LIGO-Virgo Collaboration (LVC), we find that the highest-spinning object is constrained to have nonzero spin for most sources and to have significant support at the Kerr limit for GW151226 and GW170729. A joint analysis of all the confident binary black hole detections by the LVC finds that, unlike with the traditional parametrization, the distribution of spin magnitude for the highest-spinning object has negligible support at zero spin. Regardless of the parametrization used, the configuration where all of the spins in the population are aligned with the orbital angular momentum is excluded from the 90% credible interval for the first ten events and from the 99% credible interval for all current confident detections.
... An alternative Weyl-invariant (WI) relativistic theory-fourth-order Weyl gravity-was later proposed to account for the anomalous rotation curves in galaxies with no recourse to DM [63][64][65][66]. The proposed remedy is based on an exact spherically symmetric static solution of the field equations. ...
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... The study of chaos and integrability in the spinning, eccentric BBH system has an interesting history [18][19][20][21][22][23][24][25][26][27][28][29][30][31]. We will recap some of the highlights below. ...
... Along independent lines, a large body of literature has been developed by taking advantage of orbit-averaging and precession-averaging. The principle at work is that there is a large separation of timescales, t orb t prec t rad ; so the orbital variables' influence on precession dynamics may be approximated by averaging, and similarly for precession-averaging. Early post-Newtonian works invoking orbit-averaging to study spin effects include [23,37,38], and precession-averaging followed in [39,40]. An important milestone was Racine's discovery that a quantity L · S 0 (to be introduced later) is constant under the Newtonian-orbit-average of the 2PN equations of motion (EOMs), despite not being constant under the full 2PN equations. ...
... This is relevant to the method of torus-averaging, which is used in canonical perturbation theory [34,50]. Since the actions depends on spin, it is easy to see that torus-averaging will differ from orbit-averaging (over Newtonian orbits) which has been used extensively in the literature [23,[37][38][39][40]. We expect torus-averaging to be more accurate at 1PN and higher orders. ...
Accurate and efficient modeling of the dynamics of binary black holes (BBHs) is crucial to their detection and parameter estimation through gravitational waves, with both LIGO/Virgo and LISA. General BBH configurations will have misaligned spins and eccentric orbits, eccentricity being particularly relevant at early times. Modeling these systems is both analytically and numerically challenging. Even though the 1.5 post-Newtonian (PN) order is Liouville integrable, numerical work has demonstrated chaos at 2PN order, which impedes the existence of an analytic solution. In this article we revisit integrability at both 1.5PN and 2PN orders. At 1.5PN, we construct four (out of five) action integrals. At 2PN, we show that the system is indeed integrable—but in a perturbative sense—by explicitly constructing five mutually commuting constants of motion. Because of the KAM theorem, this is consistent with the past numerical demonstration of chaos. Our method extends to higher PN orders, opening the door for a fully analytical solution to the generic eccentric, spinning BBH problem.
... [34,35,39] all report a dynamical attractor that drives each component's spin into the orbital plane at the end of the LK evolution. Consequently, the effective spin parameter [the mass-weighted sum of the component spins along the direction of the orbital AM; see Eq. (44)] of the inner binary is attracted toward zero. However, Refs. ...
... Previous studies suggest that the post-Newtonian (PN) spin evolution may play a significant role in shaping the final distribution of this angle (e.g., Refs. [44][45][46][47][48][49][50][51]). While this is not a leadingorder post-Newtonian (PN) effect in the inspiral stage, the spin-spin alignment nonetheless affects the GW radiation during the final merger-ringdown stage, and plays a crucial role in determining the GW recoil (also known as the GW kick; [45,46,52]). ...
... To proceed, we note that the key foundation of the derivation in Ref. [67] is that the effective spin parameter χ eff [Eq. (44)] is preserved to at least the 2.5 PN order. 7) and (10)] at the end of the LK evolution a i ¼ a ð0Þ i =10 of our simulations. ...
We study the spin and eccentricity evolution of black-hole (BH) binaries that are perturbed by tertiary masses and experience the Lidov-Kozai (LK) excitation. We focus on three aspects. First, we study the spin-orbit alignment of the inner binary following the approach outlined by Antonini et al. [Mon. Not. R. Astron. Soc. 480, L58 (2018)] and Liu and Lai [Astrophys. J. 863, 68 (2018)], yet allowing the spins to have random initial orientations. We confirm the existence of a dynamical attractor that drives the spin-orbit angle at the end of the LK evolution to a value given by the initial angle between the spin and the outer orbital angular momentum (instead of to a specific value of the effective spin). Second, we follow the (inner) binary’s evolution further to the merger to study the final spin-spin alignment. We generalize the effective potential theory to include orbital eccentricity, which allows us to efficiently evolve the system in the early inspiral stages. We further find that the spin-spin and spin-orbit alignments are correlated and the correlation is determined by the initial spin-orbit angle. For systems with the spin vectors initially in the orbital plane, the final spins strongly disfavor an aligned configuration and could thus lead to a greater value of the GW recoil than a uniform spin-spin alignment would predict. Lastly, we study the maximum eccentricity excitation that can be achieved during the LK process, including the effects of gravitational-wave radiation. We find that when the tertiary mass is a supermassive BH and the inner binary is massive, then even with the maximum LK excitation, the residual eccentricity is typically less than 0.1 when the binary’s orbital frequency reaches 10 Hz, and a decihertz detector would be necessary to follow such a system’s orbital evolution.
