# THE 24TH CAPRA MEETING ON RADIATION REACTION IN GENERAL RELATIVITY

7-11 June 2021
America/Toronto timezone

## Contributed Talks - Abstracts

David Aruquipa, Centro Brasileiro de Pesquisas

Efforts on calculating the gravitational sef-force using the Green function method [work in progress]

In this talk we discuss current efforts to calculate the gravitational Green function and use it to calculate the self-force in Schwarzschild spacetime. We first calculate the Green function for the Regge-Wheeler equation. Then, using the Chandrasekhar transformation we are able to calculate the Green function for the spin-2 (radial) Teukolsky equation. By following the Chrzanowsky metric reconstruction procedure, we intent to obtain the Green function associated with the metric perturbation equation and, subsequently, the self-force.

Susanna Barsanti, University of Rome La Sapienza

Extreme Mass Ratio Inspirals, scalar fields and LISA

Extreme Mass Ratio Inspirals (EMRIs), binary systems in which a stellar mass compact object inspiral into a massive black hole (MBH), are among the primary targets for LISA, as they harbour the potential for precise gravity test. Although the description of these systems in modified theories of gravity can be dramatically complex, for a vast class of theories with additional scalar fields great simplifications occur. First, the MBH scalar charge is strongly suppressed, so that the background spacetime is simply described by the Kerr metric. Moreover, all information about the underlying gravity theory turns out to be encoded in the inspiralling body’s scalar charge. In this talk I will show how, for these theories, the surviving charge strongly affects the binary dynamics, accelerating its coalescence and leaving an imprint on the emitted gravitational wave. By analysing such singals, I will finally present the extremely promising results on the LISA’s detectability of the scalar charge, which render EMRIs encouraging probes probes of gravity and of new fundamental fields.

Yilber Fabian Bautista, Perimeter Institute

From Scattering in Kerr Backgrounds to Higher-Spin Amplitudes

We revisit the scattering of massless waves of helicity $|h|=0,\frac{1}{2},1,2$ in Schwarzschild and Kerr backgrounds, in the long-wavelength regime. The Newman-Penrose scattering amplitudes arising from the Black Hole Perturbation Theory (BHPT) framework are found in agreement with the classical limit of QFT amplitudes at finite values of the scattering angle and arbitrary spin orientation. The latter amplitudes are obtained from on-shell methods and describe the $2\to 2$ scattering of a massless particle of helicity $|h|$ with a massive particle of arbitrary spin $S$, where $S=0$ corresponds to the Schwarzschild case. The effect of the black hole spin in the polarization of the waves is found in agreement with previous analysis. Finally, unitarity constraints based on partial amplitudes and positive time delay are also discussed.

Donato Bini, Istituto per le Applicazioni del Calcolo, CNR

Hyperboliclike orbits in a two-body system: review of recent results

I will review some recent results obtained for the scattering angle in PN theory, including radiation reaction effects.

Ollie Burke, The University of Edinburgh & The Max Planck Institute for Gravitational Physics

Constraining the spin parameter of near-extremal black holes using LISA

We describe a model that generates first order adiabatic EMRI waveforms for quasi-circular equatorial inspirals of compact objects into rapidly rotating (near-extremal) black holes. Using our model, we show that LISA could measure the spin parameter of near-extremal black holes (for a0.9999) with extraordinary precision, 3-4 orders of magnitude better than for moderate spins, a0.9. Such spin measurements would be one of the tightest measurements of an astrophysical parameter within a gravitational wave context. Our results are primarily based off a Fisher matrix analysis, but are verified using both frequentest and Bayesian techniques. We present analytical arguments that explain these high spin precision measurements. The high precision arises from the spin dependence of the radial inspiral evolution, which is dominated by geodesic properties of the secondary orbit, rather than radiation reaction. High precision measurements are only possible if we observe the exponential damping of the signal that is characteristic of the near-horizon regime of near-extremal inspirals. Our results demonstrate that, if such black holes exist, LISA would be able to successfully identify rapidly rotating black holes up to a=1−1e−9 , far past the Thorne limit of a=0.998.

Horng Sheng Chia, Institute for Advanced Study

Tidal Deformation and Dissipation of Rotating Black Holes

Black holes are never isolated in realistic astrophysical environments; instead, they are often perturbed by complicated external tidal fields. How does a black hole respond to these tidal perturbations? In this talk, I will discuss both the conservative and dissipative responses of the Kerr black hole to a weak and adiabatic gravitational field. The former describes how the black hole would change its shape due to these tidal interactions, and is quantified by the so-called “Love numbers”. On the other hand, the latter describes how energy and angular momentum are exchanged between the black hole and its tidal environment due to the absorptive nature of the event horizon. In this talk, I will describe how the Love numbers of the Kerr black hole in a static tidal field vanish identically. I will also describe how the Kerr black hole's dissipative response implies that energy and angular momentum can either be lost to or extracted from the black hole, with the latter process commonly known as the black hole superradiance. I will end by discussing how these tidal responses leave distinct imprints on the gravitational waves emitted by binary black holes.

Kyriakos Destounis, Universität Tübingen

Gravitational-wave imprints of non-integrable extreme-mass-ratio inspirals

The detection of gravitational waves from extreme-mass-ratio inspirals (EMRIs) with upcoming space-borne detectors will allow for unprecedented tests of general relativity in the strong-field regime. Aside from assessing whether black holes are unequivocally described by the Kerr metric, they may place constraints on the degree of spacetime symmetry. Depending on exactly how a hypothetical departure from the Kerr metric manifests, the Carter symmetry, which implies the integrability of the geodesic equations, may be broken. In this talk, I will discuss the impact of non-integrability in EMRIs which involve a supermassive compact object with anomalous multipolar structure. After reviewing the features of chaotic phenomena in EMRIs, I will argue that non-integrability is precisely imprinted in the gravitational waveform. Explicit examples of non-integrable EMRIs will be discussed, as well as their role in LISA data analysis.

Adrien Druart, Université Libre de Bruxelles

Complete set of quasi-conserved quantities for spinning particles around Kerr

I will revisit the conserved quantities of the Mathisson-Papapetrou-Tulczyjew equations describing the motion of spinning particles around a fixed background. Assuming Ricci-flatness and the existence of a Killing-Yano tensor, I obtain three non-trivial quasi-conserved quantities, i.e. conserved at linear order in the spin, thereby completing the two quasi-constants of motion found by Rüdiger with one new independent quasi-constant of motion. Finally, I will discuss the implications for the motion of spinning particles in the Kerr geometry.

Leanne Durkan, University College Dublin

Slow Time Derivatives of the Lorenz Gauge Metric Perturbation

One contribution to the second-order self-force calculations is the derivative of the first-order metric perturbation with respect to the slow inspiral time. Previous methods to compute this involve non-compact source terms which are challenging to work with. We employ the method of partial annihilators to obtain higher-order differential equations with a compact source, and solve these equations for the slowtime derivatives of the Regge-Wheeler and Zerilli master functions for circular orbits. We then use a gauge transformation to compute the slowtime derivative of the first-order Lorenz gauge metric perturbation.

Xuefeng Feng, Academy of Mathematics and Systems Science, Chinese Academy of Science

Hybrid waveform for neutron star binaries

"We consider the motion of nonspinning, compact objects orbiting around a Kerr black hole with tidal couplings. The tide-indcued quadrupole moment modifies both the orbital energy and out-going fluxes, so that over the inspiral timescale there is an accumulative shift in the orbital and gravitational wave phase. Previous studies on compact object tidal effects have been carried out in the Post-Newtonian (PN) and Effetive-One-Body (EOB) formalisms. In this work, within the black hole perturbation framework, we propose to characterize the tidal influence in the expansion of mass ratios, while higher-order PN corrections are naturally included. For the equatorial and circular orbit, we derive the leading order, frequency depedent tidal phase shift which agrees with the Post-Newtonian result at low frequencies but deviates at high frequencies. We also find that such phase shift has weak dependence (≤ 10%) on the spin of the primary black hole. Combining this black hole perturbation waveform with the Post-Newtonian waveform, we propose a frequency-domain, hybrid waveform that shows comparable accuracy as the EOB waveform in characterizing the tidal effects, as calibrated by numerical relativity simulations. Further improvement is expected as the next-leading order in mass ratio and the 2PN tidal corections are included. This hybrid approach is also applicable for generating binary black hole waveforms."

Nicola Franchini, SISSA

Detecting scalar field with extreme mass ratio inspirals

I will present extreme mass ratio inspirals (EMRIs), during which a small body spirals into a supermassive black hole, in gravity theories with additional scalar fields. No-hair theorems and properties of known theories that manage to circumvent them introduce a drastic simplification to the problem: the effects of the scalar on supermassive black holes, if any, are mostly negligible for EMRIs in vast classes of theories. I will show how to exploit this simplification to model the inspiral perturbatively and demonstrate that the scalar charge of the small body leaves a significant imprint on gravitational wave emission. This result is particularly appealing, as this imprint is observable with LISA, rendering EMRIs promising probes of scalar fields.

Davide Gerosa, University of Birmingham

Keep calm and mind the waveform

Gravitational-wave observations of binary black holes allow new tests of general relativity to be performed on strong, dynamical gravitational fields. These tests require accurate waveform models of the gravitational-wave signal, otherwise waveform errors can erroneously suggest evidence for new physics. Existing waveforms are generally thought to be accurate enough for current observations, and each of the events observed to date appears to be individually consistent with general relativity. In the near future, with larger gravitational-wave catalogs, it will be possible to perform more stringent tests of gravity by analyzing large numbers of events together. However, there is a danger that waveform errors can accumulate among events: even if the waveform model is accurate enough for each individual event, it can still yield erroneous evidence for new physics when applied to a large catalog. We presents a simple linearised analysis, in the style of a Fisher matrix calculation, that reveals the conditions under which the apparent evidence for new physics due to waveform errors grows as the catalog size increases. We estimate that, in the worst-case scenario, evidence for a deviation from general relativity might appear in some tests using a catalog containing as few as 10-30 events above a signal-to-noise ratio of 20. This is close to the size of current catalogs and highlights the need for caution when performing these sorts of experiments.

Lidia Gomes Da Silva, Queen Mary University of London

Conformal numerical method for self force applications in the time domain

"In 2034 LISA is due to be launched, which will provide the opportunity to extract physics from stellar objects and systems that would not otherwise be possible, among which are EMRIs. Unlike previous sources detected at LIGO, these sources can be simulated using an accurate computation of the gravitational self-force, resulting from the gravitational effects of the compact object orbiting around the massive BH. Whereas the field has seen outstanding progress in the frequency domain, metric reconstruction and self-force calculations are still an open challenge in the time domain. Such computations would not only further corroborate frequency domain calculations/models but also allow for full self-consistent evolution of the orbit under the effect of the self-force . Given we have a priori information about the local structure of the discontinuity at the particle, we will show how we can construct discontinuous spatial and temporal discretizations  by operating on discontinuous Lagrange and Hermite interpolation formulae and hence recover higher order accuracy. We will show how this technique in conjunction with well-suited conformal (hyperboloidal slicing) and numerical (discontinuous time symmetric ) methods can provide a relatively simple method of lines numerical recipe approach to the problem.  We will show, in particular, how this method can be applied to solve the Regge-Wheeler and Zerilli equations with a moving particle source in the time domain.

Alexander Grant, University of Virginia

Flux-balance laws in the Kerr spacetime

The motion of a radiating point particle in the Kerr spacetime can be represented by a series of geodesics whose constants of motion change slowly over its motion.  In the case of energy and axial angular momentum, there are conserved currents, defined for the field, whose fluxes at infinity and the horizon directly determine the evolution of these constants of motion.  This relationship between the properties of point-particle motion and fluxes of conserved currents is known as a "flux-balance law".  Despite the flux-balance laws for energy and axial angular momentum, the third constant of motion in Kerr, the Carter constant, has no known flux-balance law.  While there are conserved currents that can be defined for the field that are, in certain senses, "associated with the Carter constant", the fluxes of these currents are not clearly related to the Carter constant for the particle.  In this talk, we present our recent efforts to find such flux-balance laws, in the case of a point particle in Kerr that is coupled to a scalar field.

Stephen Green, Albert Einstein Institute

Lorenz-gauge reconstruction for Teukolsky solutions with sources in electromagnetism

Reconstructing a metric or vector potential that corresponds to a given solution to the Teukolsky equation is an important problem for self-force calculations. Traditional reconstruction algorithms do not work in the presence of sources, and they give rise to solutions in a radiation gauge. In the electromagnetic case, however, Dolan (2019) and Wardell and Kavanagh (2020) very recently showed how to reconstruct a vector potential in Lorenz gauge, which is more convenient for self-force. Their algorithm is based on a new Hertz-potential 2-form. In this talk, I will first show that the electromagnetic Teukolsky formalism takes a simplified form when expressed in terms of differential forms and the exterior calculus. This formalism makes the new Lorenz-gauge construction much more transparent, and it enables an extension to nonzero sources. In particular, I will derive a corrector term, related to the charge current, which when added to the vector potential gives a solution to the Maxwell equations with nonzero source. I will conclude by discussing prospects for extending to the gravitational case.

Priti Gupta, Kyoto University

Importance of tidal resonances in EMRIs

In recent work, tidal resonances induced by the tidal field of nearby stars or black holes have been identified as potentially significant in the context of extreme mass-ratio inspirals (EMRIs). These resonances occur when the three orbital frequencies describing the orbit are commensurate. During the resonance, the orbital parameters of the small body experience a ‘jump’ leading to a shift in the phase of the gravitational waveform. We study how common and important such resonances are over the entire orbital parameter space. We find that a large proportion of inspirals encounter a low-order tidal resonance in the observationally important regime.

Abraham Harte, Dublin City University

Extended-body effects in general relativity: What is possible?

To a first approximation, objects in general relativity move along geodesics. Looked at more closely, a body's internal structure can affect its motion. This talk will explore some of the surprising possibilities which arise when such effects are taken into account. An object can, for example, control its orbit merely by manipulating its internal structure: unstable orbits can be stabilized, bound orbits can be made unbound, and more, all without a rocket.

Sk Jahanur Hoque, Charles University

Mass loss law for weak gravitational fields: With or without a positive cosmological constant

Bondi's celebrated mass loss formula measures the rate of change of energy carried away from an isolated system (in asymptotically flat space-time) by gravitational radiation. In this talk, we generalize this idea to the de Sitter setting. We derive a formula for the total canonical energy, and its flux, of weak gravitational waves on a de Sitter background. Based on arXiv:2003.09548 [gr-qc], arXiv: 2103.05982 [gr-qc].

Asad Hussain, University of Texas at Austin

A framework for the quasinormal mode shifts of arbitrary spin beyond-kerr black holes.

Gravitational wave detectors and their increasing precision have enabled more specific tests of general relativity, including spectroscopic tests of black holes by measuring the quasinormal modes within the ringdown signal. These tests ideally compare the QNM frequencies to predictions from theories beyond GR, where black holes may be described by deformations to the Kerr metric. I will present a framework to compute the first order QNM shifts of these deformed Kerr Black Holes at arbitrary spin and present some initial results for the spin-0 case. In addition, I will lay out some of the technical issues that come up when computing the shifts for the spin-2 modes, and explain how they are surmountable.

Tousif Islam, University of Massachusetts Dartmouth

A multi-mode time-domain surrogate model for gravitational wave signals from comparable to extreme mass-ratio black hole binaries

We present EMRISur1dq1e6, a reduced-order multi-mode time-domain surrogate model of gravitational waveforms for non-spinning black hole binary systems with comparable- to extreme mass-ratio configurations. This surrogate model is trained on waveform data generated by a point-particle black hole perturbation theory (ppBHPT) framework computed from a high-performance Teukolsky equation solver code.  In the comparable mass-ratio regime, the gravitational waveforms generated through ppBHPT agree surprisingly well with those from full numerical relativity after scaling of the ppBHPT’s total mass parameter.  This model extends the EMRISur1dq1e4 waveform model, which spans 13,500M in duration and includes modes only up to (l,m)=(5,5). EMRISur1dq1e6, on the other hand, covers mass ratios from 3 to 1,00,000, can generate waveforms of duration up to 350,000M, and includes several spherical harmonic modes up to (l,m)=(10,10).  The accuracy of training data is further improved by employing an updated plunge model in the ppBHPT framework. EMRISur1dq1e6 surrogate model has been extended to enable data analysis studies in the high-mass ratio regime, including potential intermediate mass-ratio signals from LIGO/Virgo and events of interest to the future observatories such as Einstein Telescope, Voyager and Cosmic Explorer.

Soichiro Isoyama, University of Southampton

Adiabatic waveform for extreme mass-ratio inspirals: an analytical approach

We will discuss an adiabatic waveform model for generic (eccentric, inclined) EMRI orbits in Kerr spacetime, based on a high-order PN expansion as well as an expansion in eccentricity to the (frequency-domain) Teukolsky equations.

Mikhail Ivanov, New York University

Hidden Symmetry of Vanishing Love

We show that perturbations of massless fields in the Kerr black hole background enjoy a hidden infinite-dimensional ("Love") symmetry in the properly defined near zone approximation. Love symmetry mixes IR and UV modes. Still, this approximate symmetry allows us to derive exact results about static tidal responses (Love numbers) of static and spinning black holes. Generators of the Love symmetry are globally well defined and have a smooth Schwarzschild limit. The Love symmetry contains an SL(2,R)×U(1) subalgebra.  Generic regular solutions of the near zone Teukolsky equation form infinite-dimensional SL(2,R) representations. In some special cases  these are highest weight representations. This situation corresponds to vanishing Love numbers. In particular, static perturbations of four-dimensional Schwarzschild black holes belong to finite-dimensional representations. Other known facts about static Love numbers also acquire an elegant explanation in terms of the SL(2,R) representation theory.

Chris Kavanagh, Albert Einstein Institute

Progress in post-Newtonian self-force at second order in the mass-ratio

The lowered accuracy requirements at second order in the mass ratio greatly increases the utility of high-order post-Newtonian calculations at post-adiabatic order. In this talk I will present a methodology for solving the first- and second-order field equations using matched asymptotic expansions. As an application and test of the method, we solve the scalar wave equation sourced by a first order solution to the Regge-Wheeler equation with a particle moving in a circular orbit, and thus provide a construction of an analytic expansions of the Lorenz gauge metric perturbation.

Mohammed Khalil, Albert Einstein Institute & University of Maryland, College Park

New spin-orbit and spin-squared post-Newtonian results from first-order self-force

The scattering angle function exhibits a simple dependence on the mass ratio, which has been recently used to obtain new post-Newtonian (PN) results for arbitrary mass ratios from first-order self-force calculations. In this talk, I will present results for the spin-orbit coupling at fourth subleading PN order (5.5PN), including both local and nonlocal contributions, and the spin-squared coupling at third subleading PN order (5PN) for aligned spins. The spin-orbit results are missing one coefficient at second order in the mass ratio, and the spin-squared results are missing one coefficient at first order in the mass ratio. The latter could be determined from a self-force calculation of the spin-precession invariant for circular orbits in Schwarzschild to linear order in the spin of the small object. I will also discuss implications regarding the first law of binary mechanics with spin quadrupole and its relation to tidal invariants.

Lorenzo Küchler, Université Libre de Bruxelles - KU Leuven

Self-consistent adiabatic inspiral and transition motion

We describe the transition to plunge of a point particle around the last stable orbit of Kerr at leading order in the transition-timescale expansion. Taking systematically into account all self-force effects, we prove that the transition motion is still described by the Painlevé transcendent equation of the first kind. Using an asymptotically matched expansions scheme, we consistently match the quasi-circular adiabatic inspiral with the transition motion. The matching requires to take into account the secular change of angular velocity due to radiation-reaction during the adiabatic inspiral, which consistently leads to a leading-order radial self-force in the slow timescale expansion.

Michael LaHaye, University of Guelph

Improving Semi-Analytic Spin Precession with NITs

Semi-analytic solutions are useful because they are much faster than full numerical evolutions by virtue of the fact that they do not have to use as many points to achieve similar levels of accuracy. Currently there exists a semi-analytic solution for spin precessing binaries, which is implemented in LIGO and used to generate waveforms for comparison with gravitational waves. This solution comes with a caveat: it was calculated using precession averaged equations and thus has an oscillating error associated with the unaccounted for precession. This error is large enough that it can’t be overlooked for second and third generation detectors. By using near identity transforms (NITs) to reintroduce the effects of precession to the evolution, we can lower the error significantly. When implementing the NIT we found that the phase of oscillations we introduce (which coincides with the precession phase) does not line up with the precession in the numerical evolution, this causes the NIT to be less effective at reducing the error than it could be. To fix this issue we also introduce corrections to the evolution of the precession phase that were previously overlooked.

Benjamin Leather, University College Dublin

Calculating the second-order self-force for a Teukolsky formalism

The first iteration of the second-order self-force calculation considered the problem in the Lorenz gauge.  However, in the absence of a separable Lorenz gauge formalism for the astrophysical relevant scenario of Kerr spacetime, we are now pursuing a second-order calculation for the Teukolsky equation. In this talk I outline the numerical implementation to compute the second-order Weyl scalar.  In lieu of the second-order Teukolsky source I will demonstrate the code by presenting results for the slow-time derivative of the (first-order) Weyl-scalar as this alternative calculation is structurally similar to the second-order calculation.

Alexandre Le Tiec, Observatoire de Paris

Tidal Love numbers of Kerr black holes clarified

The open question of whether a black hole can become tidally deformed by an external gravitational field has profound implications for fundamental physics, astrophysics and gravitational-wave astronomy. Love tensors characterize the tidal deformability of compact objects such as astrophysical (Kerr) black holes under an external static tidal field. We prove that all Love tensors vanish identically for a Kerr black hole in the nonspinning limit or for an axisymmetric tidal perturbation. In contrast to this result, we show that Love tensors are generically nonzero for a spinning black hole. Specifically, to linear order in the Kerr black hole spin and the weak perturbing tidal field, we compute in closed form the Love tensors that couple the mass-type and current-type quadrupole moments to the electric-type and magnetic-type quadrupolar tidal fields. For a dimensionless spin ~ 0.1, the nonvanishing quadrupolar Love tensors are ~ 0.002, thus showing that black holes are particularly "rigid" compact objects. We also show that the induced quadrupole moments are closely related to the physical phenomenon of tidal torquing of a spinning body interacting with a tidal gravitational environment.

Kaye Li, University College London

Relativistic scattering of a fast spinning neutron star by a massive black hole

We consider the motion of a spinning neutron star with astrophysically relevant speed in the gravity field of a massive black hole. The orbital dynamic is described by the MPD equations which include up to quadrupole interaction. We compare the orbits of the neutron star under geodesic motion and under MPD equations and show that the difference in the orbital motion can translate into a variation of pulse-arrival-time. Such a difference is within the observational limit of the radio telescopes like SKA and FAST.

Kunal Lobo, University of Arizona

Self-force effects in weak-field scattering

We revisit the old problem of the self-force on a particle moving in a weak-field spacetime in the context of renewed interest in gravitational two-body scattering.  We calculate the scalar, electromagnetic, and gravitational self-force on a particle moving on a straight-line trajectory in the spacetime of a Newtonian star and use these results to find the associated correction to the scattering angle in each case.  In the gravitational case we must also take into account the motion of the star via a matter-mediated'' force on the particle, which acts at the same perturbative order as the gravitational self-force.

Oliver Long, University of Southampton

Time-domain metric reconstruction for hyperbolic scattering

Hyperbolic-type scattering orbits are excellent probes of the strong-field regime around black holes, and their analysis can inform the construction of an accurate two-body Hamiltonian. In particular, it has been shown that knowledge of the scattering angle through linear order in the mass ratio completely determines the 4PM Hamiltonian. With this motivation in mind, we describe a technique for (numerical) self-force calculations that can efficiently tackle scatter orbits. The method is based on a time-domain metric reconstruction from a Hertz potential in a radiation gauge. The crucial ingredient in this formulation are certain jump conditions that (each multipole mode of) the Hertz potential must satisfy along the orbit, in a 1+1-dimensional multipole reduction of the problem. We show a closed-form expression for these jumps, for an arbitrary geodesic orbit in Schwarzschild spacetime, and present a full numerical implementation for a scatter orbit.

Philip Lynch, University College Dublin

Fast Self-Forced Inspirals into a Rotating Black Hole

Analysing the data for the upcoming LISA mission will require extreme mass ratio inpsiral (EMRI) waveforms waveforms that are not only accurate but also fast to compute and extensive in the parameter space. To this end, we present a method for rapidly calculating the inspiral trajectory of EMRIs with a spinning primary. We extend the work of van de Meent and Warburton (2018) by applying the technique of near-identity (averaging) transformations (NITs) to the osculating geodesic equations for a rotating (Kerr) black hole, resulting in equations of motion that do not explicitly depend upon the orbital phases. This allows us accurately to calculate the evolving constants of motion, orbital phases and waveform phase to within subradian accuracy, while dramatically reducing computational cost. We have implemented this scheme with an interpolated gravitational self-force model in both the equatorial and the spherical cases as a proof of concept, and present the first inspirals in Kerr spacetime to include all first order self-force effects.

Elisa Maggio, University of Rome La Sapienza

Extreme-mass ratio inspirals around a spinning horizonless compact object

"Extreme mass-ratio inspirals detectable by LISA are unique probes of the nature of supermassive compact objects. We compute the gravitational-wave signal emitted by a point particle in a circular equatorial orbit around a Kerr-like horizonless object defined by an effective radius and a reflectivity coefficient. Teukolsky equations are solved consistently with suitable boundary conditions, and the modified energy fluxes are used to evolve the orbital parameters adiabatically. We show that the gravitational fluxes have resonances corresponding to the low-frequency quasinormal modes of the central object, which can contribute significantly to the gravitational-wave phase. Overall, the absence of a classical event horizon in the central object affects the gravitational-wave signal dramatically, with deviations even larger than those previously estimated by a model-independent analysis of the tidal heating.

Talk based on: Elisa Maggio, Maarten van de Meent, Paolo Pani, in preparation"

Charalampos Markakis, Queen Mary University of London

Discontinuous collocation methods and self-force applications

Numerical simulations of extereme mass ratio inspirals face several computational challenges. We present a new approach to evolving partial differential equations occurring in black hole perturbation theory and calculations of the self-force acting on point particles orbiting supermassive black holes. Such equations are distributionally sourced, and standard numerical methods, such as finite-difference or spectral methods, face difficulties associated with approximating discontinuous functions. However, in the self-force problem we typically have access to full a-priori information about the local structure of the discontinuity at the particle. Using this information, we show that high-order accuracy can be recovered by adding to the Lagrange interpolation formula a linear combination of certain jump amplitudes. We construct discontinuous spatial and temporal discretizations by operating on the corrected Lagrange formula. In a method-of-lines framework, this provides a simple and efficient method of solving time-dependent partial differential equations, without loss of accuracy near moving singularities or discontinuities. This method is well-suited for the problem of time-domain reconstruction of the metric perturbation via the Teukolsky or Regge-Wheeler-Zerilli formalisms. Parallel implementations on modern CPU and GPU architectures are discussed.

Josh Mathews, University College Dublin

Gauge Invariant Self-Force Calculations with a Spinning Secondary

"We calculate the first order metric perturbation to a Schwarzschild background spacetime induced by a spinning secondary body in the Regge-Wheeler and Zerilli gauges. In particular we specialise to a secondary with spin (anti-)aligned to the total orbital angular momentum in a quasi-circular orbit. From the metric perturbation we can calculate gauge invariant self-force quantities such as Detweiler’s redshift invariant and compare with known PN results. In doing so we present the first strong field calculation of a conservative self-force quantity with a spinning secondary and emphasise:

1) The treatment of the additional spin term to the singular field by deriving additional tensor harmonic regularisation parameters.

2) Parametrising at fixed frequency and practically extracting the linear in spin contribution to the metric perturbation."

Zachary Nasipak, NASA Goddard Space Flight Center

Transient resonances in EMRIs

Sergi Navarro Albalat, University of Texas at Austin

Redshift factor from numerical relativity simulations

We present a method to extract the redshift factor in numerical relativity simulations by means of its connection to the surface gravity. We proceed to analyze the small mass ratio limit and extract 2nd and higher orders in the case of non-spinning, quasi-circular binaries. We also compare our results to analytic post-newtonian and self-force calculations.

Nami Nishimura, SUNY Geneseo

Kerr self-force via elliptic PDEs: Background and theory (part 1)

Our long-term goal is to calculate the Lorenz gauge gravitational self-force for an extreme mass-ratio binary system in Kerr spacetime. Past work in the time-domain has encountered time instabilities for the two lowest modes m=0 and m=1. In order to overcome this problem, we enter the frequency-domain, which introduces elliptic PDEs. To develop an appropriate scheme, we first investigate the scalar self-force in Kerr spacetime by separating the Φ and t variables. To calculate the self-force, we use the effective source method. This presentation discusses the background and theory, which will be followed by a discussion of numerical methods in a later presentation.

Conor O'Toole, University College Dublin

Progress in Green Function Methods for Extreme Mass Ratio Inspirals

We present an update on Green function methods for modelling Extreme Mass Ratio Inspirals. In particular, we present an accurate, efficient and robust procedure for computing the Green functions of the Regge-Wheeler and Zerilli equations, and show the application to the computation of self-force and energy flux results, considering a range of sample orbits from circular geodesics to unbound encounters. In addition, we demonstrate further possible applications and improvements to the method, including progress in extending it beyond the Regge-Wheeler-Zerilli formalism.

Thomas Osburn, SUNY Geneseo

Kerr self-force via elliptic PDEs: Numerical methods (part 2)

I will discuss the numerical methods we use to calculate the self-force on a scalar charge orbiting a Kerr black hole. We apply a 2nd-order finite difference scheme on a rectangular grid in the r*-θ plane. By working in the frequency domain and separating the ϕ variable (but not θ) we encounter elliptic PDEs, which present certain numerical challenges. One challenge is that every grid point is coupled to every other grid point so that a simultaneous solution requires solving a large linear system. Another related challenge involves how imperfect boundary conditions can introduce errors that would inevitably pollute the entire domain. We have applied various techniques to overcome these challenges, such as analyzing the boundary behavior to impose more sophisticated boundary conditions with improved accuracy (for a fixed outer boundary position). Various preliminary self-force results will be presented and discussed.

Zhen Pan, Perimeter Institute

Wet Extreme Mass Ratio Inspirals May Be More Common For Spaceborne Gravitational Wave Detection

Extreme Mass Ratio Inspirals (EMRIs) can be classified as dry EMRIs and wet EMRIs based on their formation mechanisms. Dry (or the" loss-cone") EMRIs, previsouly considered as the main EMRI sources for the Laser Interferometer Space Antenna, are primarily produced by multi-body scattering in the nuclear star cluster and gravitational capture. In this Letter, we highlight an alternative EMRI formation channel:(wet) EMRI formation assisted by the accretion flow around accreting galactic-center massive black holes (MBHs). In this channel, the accretion disk captures stellar-mass black holes that are intially moving on inclined orbits, and subsequently drives them to migrate towards the MBH-this process boosts the formation rate of EMRIs in such galaxies by orders of magnitude. Taking into account the fraction of active galactic nuclei where the MBHs are expected to be rapidly accreting, we forecast that wet EMRIs will contribute an important fraction of all EMRIs observed by spaceborne gravitational wave detectors and likely dominate for MBH hosts heavier than a few $10^5 M_\odot$.

Rodrigo Panosso Macedo, CENTRA, Universidade de Lisboa & Queen Mary University of London

Pseudospectrum and black hole quasi-normal mode (in)stability

We study the stability of quasi-normal modes (QNM) in asymptotically flat black hole spacetimes by means of a pseudospectrum analysis. The construction of the Schwarzschild QNM pseudospectrum reveals: i) the stability of the slowest decaying QNM under perturbations respecting the asymptotic structure, reassessing the instability of the fundamental QNM discussed by Nollert (1996); ii) the instability of all overtones under small scale perturbations of sufficiently high frequency, that migrate to a universal class of QNM branches along pseudospectra boundaries, shedding light on Nollert & Price's analysis (1996).

Gabriel Andres Piovano, University of Rome La Sapienza

Assessing the detectability of the secondary spin in extreme mass-ratio inspirals with fully-relativistic numerical waveforms

"Extreme mass-ratio inspirals (EMRIs) detectable by the Laser Inteferometric Space Antenna (LISA) are unique probes of astrophysics and fundamental physics.

Parameter estimation for these sources is challenging, especially because the waveforms are long, complicated, known only numerically, and slow to compute in the most relevant regime, where the dynamics is relativistic.

We perform a time-consuming Fisher-matrix error analysis of the EMRI parameters using fully-relativistic numerical waveforms to leading order in an adiabatic expansion on a Kerr background, taking into account the motion of the LISA constellation, higher harmonics, and also including the leading correction from the spin of the secondary in the post-adiabatic approximation.

We pay particular attention to the convergence of the numerical derivatives in the Fisher matrix and to the numerical stability of the covariance matrix, which for some systems requires computing the numerical waveforms with approximately 90-digit precision.

Our analysis confirms previous results (obtained with approximated but much more computationally efficient waveforms) for the measurement errors on the binary's parameters. We also show that the inclusion of higher harmonics improves the errors on the luminosity distance and on the orbital angular momentum angles by one order and two orders of magnitude, respectively, which might be useful to identify the environments where EMRIs live.

We particularly focus on the measurability of the spin of the secondary, confirming that it cannot be measured with sufficient accuracy. However, due to correlations, its inclusion in the waveform model can deteriorate the accuracy on the measurements of other parameters by orders of magnitude, unless a physically-motivated prior on the secondary spin is imposed.

This work is based on the pre-print arXiv:2105.07083 ."

LISA science will require EMRI waveforms that are accurate to first-post-adiabatic order, which in turn requires the calculation of second-order self-force effects. In this talk I describe a post-adiabatic waveform-generation framework and progress toward its implementation. This lays the groundwork for talks by Durkan, Warburton, Spiers, Leathers, Upton, and others.

Comparison of eccentric numerical relativity simulations to small mass-ratio perturbation theory

In this work we compare two approaches to modeling binary black holes (BBHs): 1) small mass-ratio (SMR) perturbation theory, and 2) numerical relativity (NR). We extend recent work on combining information from quasicircular nonspinning NR simulations of BBHs with results from SMR perturbation theory to nonspinning eccentric BBHs. We produce a dataset of long and accurate eccentric nonspinning NR simulations with the Spectral Einstein Code (SpEC) from mass ratios 1 to 10, and eccentricities up to 0.7. We analyze these NR simulations, compute gauge invariant quantities from the gravitational radiation, and develop tools to map points in parameter space between eccentric NR and SMR waveforms. Finally, we discuss discrepancies between SMR and NR predictions for the energy and angular momentum fluxes due to eccentricity, and limitations of such comparisons due to the limited parameter space in mass ratio covered by the NR simulations.

Hannes Rüter, Albert Einstein Institute

World tube excision method for intermediate-mass-ratio inspirals: scalar-field toy model

"We propose and explore a method for alleviating the scale disparity in numerical relativity simulations with mass ratios in the intermediate astrophysical range ($10^2 \lesssim q\lesssim 10^4$), where purely perturbative methods may not be adequate. A region around the smaller object considerably smaller than its horizon is excised from the numerical domain, and replaced with an analytical model approximating a tidally deformed black hole. We develop the basic idea and try it on the toy model of a scalar charge in a circular geodesic orbit around a Schwarzschild black hole, solving for the scalar field as a linear perturbation in a 1+1D framework.

Collaborators: Mekhi Dhesi, Adam Pound, Leor Brack, Harald Pfeiffer"

Zeyd, Sam, University of Southampton

Non-teleology and motion of a tidally perturbed Schwarzschild black hole

The prospect of gravitational wave astronomy with EMRIs has motivated increasingly accurate perturbative studies of binary black hole dynamics. Studying the apparent and event horizon of a perturbed Schwarzschild black hole, we find that the two horizons are identical at linear order regardless of the source of perturbation. This implies that the seemingly teleological behaviour of the linearly perturbed event horizon, previously observed in the literature, cannot be truly teleological in origin. The two horizons do generically differ at second order in some ways, but their Hawking masses remain identical. In the context of tidal distortion by a small companion, we also show how the perturbed event horizon in a small-mass-ratio binary is effectively localized in time, and we numerically visualize unexpected behaviour in the black hole’s motion around the binary’s center of mass.

Viktor Skoupý, Charles University; Astronomical Institute, Czech Academy of Sciences

Spinning test body orbiting around a Kerr black hole: eccentric equatorial orbits and their asymptotic gravitational-wave fluxes

We use the frequency and time domain Teukolsky formalism to calculate gravitational wave fluxes from a spinning body on a bound eccentric equatorial orbit around a Kerr black hole. The spinning body is represented as a point particle following the pole-dipole approximation of the Mathisson-Papapetrou-Dixon equations. Reformulating these equations we are not only able to find the trajectory of a spinning particle in terms of its constants of motion, but also to provide a method to calculate the azimuthal and the radial frequency of this trajectory. Using these orbital quantities, we introduce the machinery to calculate through the frequency domain Teukolsky formalism the energy and the angular momentum fluxes at infinity, and at the horizon, along with the gravitational strain at infinity. We crosscheck the results obtained from the frequency domain approach with the results obtained from a time domain Teukolsky equation solver called Teukode.

Mikhail Solon, University of California, Los Angeles

Binary Black Holes and Scattering Amplitudes

Future gravitational wave detectors will map out and characterize every binary merger in the history of the universe. The possibilities for new and unexpected scientific discoveries from this wealth of data is staggering, but hinges crucially on complementary advances in our theoretical understanding of the nature of gravitational wave sources. However, the path from Einstein’s equation to precision binary dynamics is notoriously difficult, and conventional methods do not scale to the demands of future detectors. I will describe our recent efforts in solving the relativistic two body problem using tools from quantum field theory.

Lorenzo Speri, Albert Einstein Institute

Assessing the impact of transient orbital resonances

"One of the primary sources for the future space-based gravitational wave detector, the Laser Interferometer Space Antenna, are the inspirals of small compact objects into massive black holes in the centres of galaxies.

The gravitational waveforms from such Extreme Mass Ratio Inspiral (EMRI) systems will provide measurements of their parameters with unprecedented precision, but only if the waveforms are accurately modeled.

Here we explore the impact of transient orbital resonances which occur when the ratio of radial and polar frequencies is a rational number. We introduce a new Effective Resonance Model, which is an extension of the numerical kludge EMRI waveform approximation to include the effect of resonances, and use it to explore the impact of resonances on EMRI parameter estimation.

For one-year inspirals, we find that the few cycle dephasings induced by 3:2 resonances can lead to systematic errors in parameter estimates, that are up to several times the typical measurement precision of the parameters. The bias is greatest for 3:2 resonances that occur closer to the central black hole. By regarding them as unknown model parameters, we further show that observations will be able to constrain the size of the changes in the orbital parameters across the resonance to a relative precision of $10\%$ for a typical one-year EMRI observation with signal-to-noise ratio of 20. Such measurements can be regarded as tests of fundamental physics, by comparing the measured changes to those predicted in general relativity."

Andrew Spiers, University of Southampton

The second-order Teukolsky source in Schwarzschild.

Precise parameter extraction from EMRI signals requires, among other things, the dissipative piece of the second-order self-force in a Kerr background. We have shown how a new form of the second-order Teukolsky equation has a well-defined source in a highly regular gauge, and how to construct gauge invariant second-order quantities using a gauge fixing method. For the current prospective second-order self-force methods in Kerr solving the second-order Teukolsky equation will be a crucial step. In this talk, I show our progress in calculating the source in the second-order Teukolsky equation for quasi-circular orbits in Schwarzschild, and discuss how the source can be made more regular at future null infinity by transforming to the Bondi-Sachs gauge.

Iason Timogiannis, National and Kapodistrian University of Athens

Spinning test body orbiting around a Schwarzschild black hole: Comparing Spin Supplementary Conditions for Circular Equatorial Orbits

The Mathisson-Papapetrou-Dixon (MPD) equations  describe the motion of an extended test body in general relativity. This system of equations, though, is underdetermined and has to be accompanied by constraining supplementary conditions, even in its simplest version, which is the pole-dipole approximation corresponding to a spinning test body. In particular, imposing a spin supplementary condition (SSC) fixes the center of the mass of the spinning body, i.e. the centroid of the body. In the present study, we examine whether characteristic features of the centroid of a spinning test body, moving in a circular equatorial orbit around a massive black hole, are preserved under the transition to another centroid of the same physical body, governed by a different SSC. For this purpose, we establish an analytical algorithm for deriving the orbital frequency of a spinning body, moving in the background of an arbitrary, stationary, axisymmetric spacetime with reflection symmetry, for the Tulczyjew-Dixon, the Mathisson-Pirani and the Ohashi-Kyrian-Semerak SSCs. Then, we focus on the Schwarzschild black hole background and a power series expansion method is developed, in order to investigate the discrepancies in the orbital frequencies expanded in power series of the spin among the different SSCs. Lastly, by employing the fact that the position of the centroid and the measure of the spin alters under the centroid's transition, we impose proper corrections to the power expansion of the orbital frequencies, which allows to improve the convergence between the SSCs. Our concluding argument is that when we shift from one circular equatorial orbit to another in the Schwarzschild background, under the change of a SSC, the convergence between the SSCs holds only up to certain powers in the spin expansion, and it cannot be achieved for the whole power series.

Vahid Toomani, Leipzig Univesity

Metric perturbations to Kerr blackhole in suitable gauges

I outline an algorithm for obtaining the first order metric perturbation solving the linearized Einstein equations sourced by a point mass in (a) a no-string gauge and (b) the Lorentz gauge via a variant of the Teukolsky formalism. The aim is to suitably combine the attractive features of this formalism with the milder singularity structure and better localization properties of the solutions in the latter gauges. This part of our work is intended as a starting point for the second order self force problem and relies in an essential way on the recently introduced corrector formalism by Green, Hollands and Zimmerman.

Samuel Upton, University of Southampton

Transforming to the highly regular gauge for use in second-order self-force calculations

With the publication of the first second-order self-force results, it has become even more clear of the need for fast and efficient calculations to avoid the computational expense encountered when using current methods in the Lorenz gauge. One ingredient for efficient calculation of second-order self-force data will be the use of the highly regular gauge (1703.02836 and 2101.11409) with its weaker divergences along the worldline of the small object. In this talk, we will present steps towards transforming the current Lorenz gauge data into the highly regular gauge to be used for quasicircular orbits in Schwarzschild spacetime. The end result will be a source that can be used as an input into the second-order Einstein equations (see talks by Andrew Spiers and Benjamin Leather). In particular, this will allow us to solve the second-order Teukolsky equation using a point particle source instead of requiring the use of a puncture scheme.

Niels Warburton, University College Dublin

Gravitational wave flux for compact binaries through second-order in the mass-ratio

Within the framework of self force theory we compute the gravitational wave flux through second-order in the mass ratio for quasi-circular compact binaries. Our results are consistent with post-Newtonian calculations in the weak field and we find they agree remarkably well with numerical relativity simulations of comparable mass binaries in the strong field. We also find good agreement for binaries with a spinning secondary or a slowly spinning primary.

Barry Wardell, University College Dublin

Separable electromagnetic perturbations of rotating black holes

We identify a set of Hertz potentials for solutions to the vector wave equation on black hole spacetimes. The Hertz potentials yield Lorenz gauge electromagnetic vector potentials that represent physical solutions to the Maxwell equations, satisfy the Teukolsky equation, and are related to the Maxwell scalars by straightforward and separable inversion relations. Our construction, based on the GHP formalism, avoids the need for a mode ansatz and leads to potentials that represent both static and non-static solutions. As an explicit example, we specialise the procedure to mode-decomposed perturbations of Kerr spacetime and in the process make connections with previous results.

Christopher Whittall, University of Southampton

Frequency domain approach to self-force in hyperbolic scattering

Hyperbolic scattering orbits, able to penetrate deep into the sub-ISCO region even at relatively low energies, provide an excellent probe of the strong-field regime outside black holes. Self-force calculations of the scatter angle can greatly advance the development of post-Minkowskian theory and of the EOB model of binary dynamics. We develop a frequency-domain method for calculating the 1st order scalar self-force acting on a charge moving along a hyperbolic Schwarzschild geodesic, outlining the formulation of the problem, the challenges faced and our attempted solutions. Particular attention will be paid to issues faced by the usual method of extended homogeneous solutions (EHS) used to circumvent the Gibbs phenomenon.

Vojtech Witzany, University College Dublin

Are we already observing inspirals into massive black holes?

Everybody talks about EMRIs and IMRIs in connection with LISA and the 2030s. However, inspirals into massive black holes are happening at this very moment. Would we be able to recognize them with our current electromagnetic observations? Even more, are we maybe observing these inspirals at the very moment without realising it? We simulated accretion-disks  perturbed by light perturbers and deduced that the orbital periods show in the disk variability. I present a list of candidate sources that, based on their variability periods and period derivatives, may contain ongoing inspirals into massive black holes.

Peter Zimmerman, Albert Einstein Institute

Gauge and effective source regularization of reconstructed metric perturbations

Understanding extended sources is an important aspect of the second-order problem in Kerr. Metric reconstruction procedures based on the Teukolsky equation bring challenges at second order due to singularities in the radiation gauge. One approach would be to soften these singularities using an effective source derived from a puncture. I will describe how the corrector-tensor reconstruction algorithm may be applied in a puncture scheme with a spatially compact effective source. To illustrate the method I will use an example in flat spacetime which provides some insights into the structure of the problem in Kerr.