Generic Orbits Research Articles

Detecting the tidal heating with the generic extreme mass-ratio inspirals

The horizon of a classical black hole (BH), functioning as a one-way membrane, plays a vital role in the dynamic evolution of binary BHs, capable of absorbing fluxes entirely.Tidal heating, stemming from this phenomenon, exerts a notable influence on the production of gravitational waves (GWs). If at least one member of a binary is an exotic compact object (ECO) instead of a BH, the absorption of fluxes is expected to be incomplete and the tidal heating would be different. Thus, tidal heating can be utilized for model-independent investigations into the nature of compact object. In this paper, assuming that the extreme mass-ratio inspiral (EMRI) contains a stellar-mass compact object orbiting around a massive ECO with a reflective surface, we compute the GWs from the generic EMRI orbits. Using the accurate and analytic flux formulas in the black hole spacetime, we adapted these formulas in the vicinity of the ECO surface by incorporating a reflectivity parameter.Under the adiabatic approximation, we can evolve the orbital parameters and compute the EMRI waveforms. The effect of tidal heating for the spinning and non-spinning objects can be used to constrain the reflectivity of the surface at the level of 𝒪(10-6) by computing the mismatch and fisher information matrix.

The HHMP Decomposition of the Permutohedron and Degenerations of Torus Orbits in Flag Varieties

Abstract Let $Z\subset \operatorname{Fl}(n)$ be the closure of a generic torus orbit in the full flag variety. Anderson–Tymoczko express the cohomology class of $Z$ as a sum of classes of Richardson varieties. Harada–Horiguchi–Masuda–Park give a decomposition of the permutohedron, the moment map image of $Z$, into subpolytopes corresponding to the summands of the Anderson–Tymoczko formula. We construct an explicit toric degeneration inside $\operatorname{Fl}(n)$ of $Z$ into Richardson varieties, whose moment map images coincide with the HHMP decomposition, thereby obtaining a new proof of the Anderson–Tymoczko formula.

A Mañé-Manning formula for expanding measures for endomorphisms of ℙ^{𝕜}

Let k ≥ 1 k \ge 1 be an integer and f f a holomorphic endomorphism of P k ( C ) \mathbb {P}^k (\mathbb {C}) of algebraic degree d ≥ 2 d\geq 2 . We introduce a dynamical volume dimension for ergodic f f -invariant probability measures with strictly positive Lyapunov exponents. In particular, this class of measures includes all ergodic measures whose measure-theoretic entropy is strictly larger than ( k − 1 ) log ⁡ d (k-1)\log d , a natural generalization of the class of measures of positive measure-theoretic entropy in dimension 1. The volume dimension is equivalent to the Hausdorff dimension when k = 1 k=1 , but depends on the dynamics of f f to incorporate the absence of an analogue of Koebe’s theorem and the non-conformality of holomorphic endomorphisms for k ≥ 2 k\geq 2 . If ν \nu is an ergodic f f -invariant probability measure with strictly positive Lyapunov exponents, we prove a generalization of the Mañé-Manning formula relating the volume dimension, the measure-theoretic entropy, and the sum of the Lyapunov exponents of ν \nu . As a consequence, we give a characterization of the first zero of a natural pressure function for such expanding measures in terms of their volume dimensions. For hyperbolic maps, such zero also coincides with the volume dimension of the Julia set, and with the exponent of a natural (volume-)conformal measure. This generalizes results by Denker-Urbański [Nonlinearity 4 (1991), pp. 365–384; Trans. Amer. Math. Soc. 328 (1991), pp. 563–587] and McMullen [Comment. Math. Helv. 75 (2000), pp. 535–593] in dimension 1 to any dimension k ≥ 1 k\geq 1 . Our methods mainly rely on a theorem by Berteloot-Dupont-Molino [Ann. Inst. Fourier (Grenoble) 58 (2008), pp. 2137–2168], which gives a precise control on the distortion of inverse branches of endomorphisms along generic inverse orbits with respect to measures with strictly positive Lyapunov exponents.

Gravitational wave fluxes on generic orbits in near-extreme Kerr spacetime: Higher spin and large eccentricity

Gravitational wave fluxes on generic orbits in near-extreme Kerr spacetime: Higher spin and large eccentricity

Local in Time Conservative Binary Dynamics at Fourth Post-Minkowskian Order.

Leveraging scattering information to describe binary systems in generic orbits requires identifying local and nonlocal in time tail effects. We report here the derivation of the universal (nonspinning) local in time conservative dynamics at fourth post-Minkowskian order, i.e., O(G^{4}). This is achtieved by computing the nonlocal-in-time contribution to the deflection angle, and removing it from the full conservative value in [C. Dlapa et al., Phys. Rev. Lett. 128, 161104 (2022).PRLTAO0031-900710.1103/PhysRevLett.128.161104; C. Dlapa et al., Phys. Rev. Lett. 130, 101401 (2023).PRLTAO0031-900710.1103/PhysRevLett.130.101401]. Unlike the total result, the integration problem involves two scales-velocity and mass ratio-and features multiple polylogarithms, complete elliptic and iterated elliptic integrals, notably in the mass ratio. We reconstruct the local radial action, center-of-mass momentum and Hamiltonian, as well as the exact logarithmic-dependent part(s), all valid for generic orbits. We incorporate the remaining nonlocal terms for ellipticlike motion to sixth post-Newtonian order. The combined Hamiltonian is in perfect agreement in the overlap with the post-Newtonian state of the art. The results presented here provide the most accurate description of gravitationally bound binaries harnessing scattering data to date, readily applicable to waveform modeling.

Unveiling the Merger Structure of Black Hole Binaries in Generic Planar Orbits.

The precise modeling of binary black hole coalescences in generic planar orbits is a crucial step to disentangle dynamical and isolated binary formation channels through gravitational-wave observations. The merger regime of such coalescences exhibits a significantly higher complexity compared to the quasicircular case, and cannot be readily described through standard parametrizations in terms of eccentricity and anomaly. In the spirit of the effective one body formalism, we build on the study of the test-mass limit, and introduce a new modeling strategy to describe the general-relativistic dynamics of two-body systems in generic orbits. This is achieved through gauge-invariant combinations of the binary energy and angular momentum, such as a dynamical "impact parameter" at merger. These variables reveal simple "quasi-universal" structures of the pivotal merger parameters, allowing us to build an accurate analytical representation of generic (bounded and dynamically bounded) orbital configurations. We demonstrate the validity of these analytical relations using 311 numerical simulations of bounded noncircular binaries with progenitors from the RIT and SXS catalogs, together with a custom dataset of dynamical captures generated using the Einstein Toolkit, and test-mass data in bound orbits. Our modeling strategy lays the foundations of accurate and complete waveform models for systems in arbitrary orbits, bolstering observational explorations of dynamical formation scenarios and the discovery of new classes of gravitational wave sources.

Fast and Fourier: extreme mass ratio inspiral waveforms in the frequency domain

Extreme Mass Ratio Inspirals (EMRIs) are one of the key sources for future space-based gravitational wave interferometers. Measurements of EMRI gravitational waves are expected to determine the characteristics of their sources with sub-percent precision. However, their waveform generation is challenging due to the long duration of the signal and the high harmonic content. Here, we present the first ready-to-use Schwarzschild eccentric EMRI waveform implementation in the frequency domain for use with either graphics processing units (GPUs) or central processing units (CPUs). We present the overall waveform implementation and test the accuracy and performance of the frequency domain waveforms against the time domain implementation. On GPUs, the frequency domain waveform takes in median 0.044 s to generate and is twice as fast to compute as its time domain counterpart when considering massive black hole masses ≥2×106M⊙ and initial eccentricities e0 > 0.2. On CPUs, the median waveform evaluation time is 5 s, and it is five times faster in the frequency domain than in the time domain. Using a sparser frequency array can further speed up the waveform generation, reaching up to 0.3 s. This enables us to perform, for the first time, EMRI parameter inference with fully relativistic waveforms on CPUs. Future EMRI models, which encompass wider source characteristics (particularly black hole spin and generic orbit geometries), will require significantly more harmonics. Frequency domain models will be essential analysis tools for these astrophysically realistic and important signals.

Boundary to bound dictionary for generic Kerr orbits

Boundary to bound dictionary for generic Kerr orbits

Asymptotic gravitational-wave fluxes from a spinning test body on generic orbits around a Kerr black hole

Asymptotic gravitational-wave fluxes from a spinning test body on generic orbits around a Kerr black hole

One-skeleton posets of Bruhat interval polytopes

Introduced by Kodama and Williams, Bruhat interval polytopes are generalized permutohedra closely connected to the study of torus orbit closures and total positivity in Schubert varieties. We show that the 1-skeleton posets of these polytopes are lattices and classify when the polytopes are simple, thereby resolving open problems and conjectures of Fraser, of Lee–Masuda, and of Lee–Masuda–Park. In particular, we classify when generic torus orbit closures in Schubert varieties are smooth.

The Toda flow on Hessenberg elements of real, split simple Lie algebras

In this work, we consider the Toda flow associated with compact/Borel decompositions of real, split simple Lie algebras. Using the primitive invariant polynomials of Chevalley, we show how to construct integrals in involution which are invariants of the maximal compact subgroup, and moreover, we show that the number of such integrals is given by a formula involving only Lie-theoretic data. We then introduce the space of Hessenberg elements, characterize the generic Hessenberg coadjoint orbits, and show that the dimension of such orbits is precisely twice the number of nontrivial invariants which appeared earlier. For the class of classical, real split simple Lie algebras, we construct angle-type variables which in particular shows that the Toda flow is Liouville integrable on generic Hessenberg coadjoint orbits.

Geodesics and gravitational waves in chaotic extreme-mass-ratio inspirals: the curious case of Zipoy-Voorhees black-hole mimickers

Due to the growing capacity of gravitational-wave astronomy and black-hole imaging, we will soon be able to emphatically decide if astrophysical dark objects lurking in galactic centers are black holes. Sgr A*, one of the most prolific astronomical radio sources in our galaxy, is the focal point for tests of general relativity. Current mass and spin constraints predict that the central object of the Milky Way is supermassive and slowly rotating, thus can be conservatively modeled as a Schwarzschild black hole. Nevertheless, the well-established presence of accretion disks and astrophysical environments around supermassive compact objects can significantly deform their geometry and complicate their observational scientific yield. Here, we study extreme-mass-ratio binaries comprised of a minuscule secondary object inspiraling onto a supermassive Zipoy-Voorhees compact object; the simplest exact solution of general relativity that describes a static, spheroidal deformation of Schwarzschild spacetime. We examine geodesics of prolate and oblate deformations for generic orbits and reevaluate the non-integrability of Zipoy-Voorhees spacetime through the existence of resonant islands in the orbital phase space. By including radiation loss with post-Newtonian techniques, we evolve stellar-mass secondary objects around a supermassive Zipoy-Voorhees primary and find clear imprints of non-integrability in these systems. The peculiar structure of the primary, allows for, not only typical single crossings of transient resonant islands, that are well-known for non-Kerr objects, but also inspirals that transverse through several islands, in a brief period of time, that lead to multiple glitches in the gravitational-wave frequency evolution of the binary. The detectability of glitches with future spaceborne detectors can, therefore, narrow down the parameter space of exotic solutions that, otherwise, can cast identical shadows with black holes.

Universal corner symmetry and the orbit method for gravity

A universal symmetry algebra organizing the gravitational phase space has been recently found. It corresponds to the subset of diffeomorphisms that become physical at corners – codimension-2 surfaces supporting Noether charges. It applies to both finite distance and asymptotic corners. In this paper, we study this algebra and its representations, via the coadjoint orbit method. We show that generic orbits of the universal algebra split into sub-orbits spanned by finite distance and asymptotic corner symmetries, such that the full universal symmetry algebra gives rise to a unified treatment of corners in a manifold. We then identify the geometric structure that captures these algebraic properties on corners, which is the Atiyah Lie algebroid associated to a principal GL(2,R)⋉R2-bundle. This structure is suggestive of the existence of a novel quantum gravitational theory which would unitarily glue such geometric structures, with spacetime geometries appearing as semi-classical configurations.

Gradient Flows, Adjoint Orbits, and the Topology of Totally Nonnegative Flag Varieties

One can view a partial flag variety in \({\mathbb {C}}^n\) as an adjoint orbit \({\mathcal {O}}_{\lambda }\) inside the Lie algebra of \(n\times n\) skew-Hermitian matrices. We use the orbit context to study the totally nonnegative part of a partial flag variety from an algebraic, geometric, and dynamical perspective. The paper has three main parts: (1) We introduce the totally nonnegative part of \({\mathcal {O}}_{\lambda }\), and describe it explicitly in several cases. We define a twist map on it, which generalizes (in type A) a map of Bloch, Flaschka, and Ratiu (Duke Math. J. 61(1): 41–65, 1990) on an isospectral manifold of Jacobi matrices. (2) We study gradient flows on \({\mathcal {O}}_{\lambda }\) which preserve positivity, working in three natural Riemannian metrics. In the Kähler metric, positivity is preserved in many cases of interest, extending results of Galashin, Karp, and Lam (Adv. Math. 397: Paper No. 108123, 1–23, 2022; Adv. Math. 351: 614–620, 2019). In the normal metric, positivity is essentially never preserved on a generic orbit. In the induced metric, whether positivity is preserved appears to depends on the spacing of the eigenvalues defining the orbit. (3) We present two applications. First, we discuss the topology of totally nonnegative flag varieties and amplituhedra. Galashin, Karp, and Lam (2022, 2019) showed that the former are homeomorphic to closed balls, and we interpret their argument in the orbit framework. We also show that a new family of amplituhedra, which we call twisted Vandermonde amplituhedra, are homeomorphic to closed balls. Second, we discuss the symmetric Toda flow on \({\mathcal {O}}_{\lambda }\). We show that it preserves positivity, and that on the totally nonnegative part, it is a gradient flow in the Kähler metric up to applying the twist map. This extends a result of Bloch, Flaschka, and Ratiu (1990).

Effective-action model for dynamical scalarization beyond the adiabatic approximation

In certain scalar-field extensions to general relativity, scalar charges can develop on compact objects in an inspiraling binary -- an effect known as dynamical scalarization. This effect can be modeled using effective-field-theory methods applied to the binary within the post-Newtonian approximation. Past analytic investigations focused on the adiabatic (or quasi-stationary) case for quasi-circular orbits. In this work, we explore the full dynamical evolution around the phase transition to the scalarized regime. This allows for generic (eccentric) orbits and to quantify nonadiabatic (e.g., oscillatory) behavior during the phase transition. We also find that even in the circular-orbit case, the onset of scalarization can only be predicted reliably when taking the full dynamics into account, i.e., the adiabatic approximation is not appropriate. Our results pave the way for accurate post-Newtonian predictions for dynamical scalarization effects in gravitational waves from compact binaries.

Assessing the data-analysis impact of LISA orbit approximations using a GPU-accelerated response model

The analysis of gravitational wave (GW) datasets is based on the comparison of measured time series with theoretical templates of the detector's response to a variety of source parameters. For LISA, the main scientific observables will be the so-called time-delay interferometry (TDI) combinations, which suppress the otherwise overwhelming laser noise. Computing the TDI response to GW involves projecting the GW polarizations onto the LISA constellation arms, and then combining projections delayed by a multiple of the light propagation time along the arms. Both computations are difficult to perform efficiently for generic LISA orbits and GW signals. Various approximations are currently used in practice, e.g., assuming constant and equal armlengths, which yields analytical TDI expressions. In this article, we present 'fastlisaresponse', a new efficient GPU-accelerated code that implements the generic TDI response to GWs in the time domain. We use it to characterize the parameter-estimation bias incurred by analyzing loud Galactic-binary signals using the equal-armlength approximation. We conclude that equal-armlength parameter-estimation codes should be upgraded to the generic response if they are to achieve optimal accuracy for high (but reasonable) SNR sources within the actual LISA data.

Dynamical perturbations around an extreme mass ratio inspiral near resonance

Extreme mass ratio inspirals (EMRIs) -- systems with a compact object orbiting a much more massive (e.g., galactic center) black hole -- are of interest both as a new probe of the environments of galactic nuclei, and their waveforms are a precision test of the Kerr metric. This work focuses on the effects of an external perturbation due to a third body around an EMRI system. This perturbation will affect the orbit most significantly when the inner body crosses a resonance with the outer body, and result in a change of the conserved quantities (energy, angular momentum, and Carter constant) or equivalently of the actions, which results in a subsequent phase shift of the waveform that builds up over time. We present a general method for calculating the changes in action during a resonance crossing, valid for generic orbits in the Kerr spacetime. We show that these changes are related to the gravitational waveforms emitted by the two bodies (quantified by the amplitudes of the Weyl scalar $\psi_4$ at the horizon and at $\infty$) at the frequency corresponding to the resonance. This allows us to compute changes in the action variables for each body, without directly computing the explicit metric perturbations, and therefore we can carry out the computation by calling an existing black hole perturbation theory code. We show that our calculation can probe resonant interactions in both the static and dynamical limit. We plan to use this technique for future investigations of third-body effects in EMRIs and their potential impact on waveforms for LISA.

Regularization of a scalar charged particle for generic orbits in Kerr spacetime

A scalar charged particle moving in a curved background spacetime will emit a field affecting its own motion; the resolving of this resulting motion is often referred to as the self-force problem. This also serves as a toy model for the astrophysically interesting compact body binaries, extreme mass ratio inspirals, targets for the future space-based gravitational wave detector, LISA. In the modelling of such systems, a point-particle assumption leads to problematic singularities which need to be safely removed to solve for the motion of the particle regardless of the scenario: scalar, electromagnetic or gravitational. Here, we concentrate on a scalar charged particle and calculate the next order of the Detweiler-Whiting singular field and its resulting regularisation parameter when employing the mode-sum method of regularisation. This enables sufficiently faster self-force calculations giving the same level of accuracy with significantly less $\ell$ modes. Due to the similarity of the governing equations, this also lays the groundwork for similar calculations for an electromagnetic or mass charged particle in Kerr spacetime and has applications in other regularisation schemes like the effective source and matched expansion.

Quadratic-in-spin interactions at fifth post-Newtonian order probe new physics

We obtain for the first time all quadratic-in-spin interactions in spinning binaries at the third subleading order in post-Newtonian (PN) gravity, and provide their observable binding energies and their gauge-invariant relations to the angular momentum. Our results are valid for generic compact objects, orbits, and spin orientations, and enter at the fifth PN order for maximally-rotating objects, thus pushing the state of the art. This is accomplished through an extension of the effective field theory of spinning gravitating objects, and of its computational application. We also discover a new finite-size effect which is unique to spinning objects, with a new “Spin Love number” as its characteristic coefficient, that is a new probe for gravity and QCD.

Precisely computing bound orbits of spinning bodies around black holes. II. Generic orbits

In this paper, we continue our study of the motion of spinning test bodies orbiting Kerr black holes. Non-spinning test bodies follow geodesics of the spacetime in which they move. A test body's spin couples to the curvature of that spacetime, introducing a "spin-curvature force" which pushes the body's worldline away from a geodesic trajectory. The spin-curvature force is an important example of a post-geodesic effect which must be modeled carefully in order to accurately characterize the motion of bodies orbiting black holes. One motivation for this work is to understand how to include such effects in models of gravitational waves produced from the inspiral of stellar mass bodies into massive black holes. In this paper's predecessor, we describe a technique for computing bound orbits of spinning bodies around black holes with a frequency-domain description which can be solved very precisely. In that paper, we present an overview of our methods, as well as present results for orbits which are eccentric and nearly equatorial (i.e., the orbit's motion is no more than $\mathcal{O}(S)$ out of the equatorial plane). In this paper, we apply this formulation to the fully generic case -- orbits which are inclined and eccentric, with the small body's spin arbitrarily oriented. We compute the trajectories which such orbits follow, and compute how the small body's spin affects important quantities such as the observable orbital frequencies $\Omega_r$, $\Omega_\theta$ and $\Omega_\phi$.