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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.

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.

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.

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.

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.

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.

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.

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.

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.

"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. /************ Note to organizers: if both my talk and my supervisor, Charalampos Markakis, are selected could this please be after his ? Thank you for your consideration *************/"

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.