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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.

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.

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.

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.

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.

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.

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.

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].

"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"

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.