arXiv subjects

Barry Wardell

Publications and source records attributed to Barry Wardell.

At least 19 recordsLinked to original sources

arXiv2026

Kerr metric perturbations in Lorenz gauge via numerical relativity methods

Gravitational self-force calculations are needed to accurately describe the motion of a small body in a spinning (Kerr) black hole spacetime, which is vital to predict gravitational wave signals from asymmetric-mass compact binaries. Accurate calculations for the upcoming LISA mission require a minimum of second perturbative order in the mass ratio, which has thus far only been achieved for nonspinning (Schwarzschild) black holes due to nonlinear complexities of the Einstein equations. In this work, we calculate Kerr metric perturbations with a new computational approach based on numerical relativity methods, extending recent scalar self-force results to full general relativity. We formulate the perturbative Lorenz-gauge Einstein equations in terms of azimuthal $m$-modes in the frequency domain and solve the resulting elliptic partial differential equations using a spectral discontinuous Galerkin discretization, adaptive mesh refinement, and preconditioned iterative solvers. With our new method, we achieve exponentially convergent first-order gravitational self-force calculations for circular equatorial orbits in Kerr spacetime up to spin $|a|/M=0.998$ (the Thorne limit). Our framework is designed to carry over to second order by delivering accurate and well-behaved first-order data for nonlinear source construction, and as a solver for the second-order metric perturbations themselves.

gr-qc↗

arXiv2026

Multi-domain spectral method for Kerr spacetimes: a hyperboloidal Teukolsky $m$-mode solver for self-force computations

We solve the frequency-domain, $m$-mode Teukolsky equation for the spin-weight $s=-2$ Weyl scalar $ψ_4$ sourced by a point mass on a circular, equatorial orbit around a Kerr black hole. The equation is posed on hyperboloidal slices that reach the event horizon and future null infinity, so that the physical (retarded) solution is selected by regularity alone, without imposing radiation conditions. The particle is treated with the $m$-mode puncture/effective-source scheme, and the resulting two-dimensional elliptic problem is solved with a four-domain pseudo-spectral method in which the equatorial symmetry is broken and handled by an even/odd decomposition of the field. We test the method for orbital radius $r_0=10M$ and spins up to $|a|/M=0.9$, covering prograde and retrograde orbits. The energy fluxes to infinity agree with independent Black Hole Perturbation Toolkit results to $\sim10^{-7}$ for the dominant modes and to better than $2\times10^{-5}$ for all modes up to $m=20$ at moderate resolutions. The static $m=0$ mode requires no special treatment.

gr-qc↗

arXiv2026

Kerr metric perturbations for second-order self-force theory

We present a calculation of the first-order metric perturbation arising from a point mass moving on a circular, equatorial orbit in Kerr spacetime. The metric perturbation is specifically aimed at meeting the requirements of second-order self-force theory: it is highly accurate; incorporates all contributions, including those associated with perturbations to the black hole's mass and angular momentum; is in Lorenz gauge near the worldline and in (near) Bondi--Sachs gauge at large radius; and can be efficiently evaluated across the entire exterior spacetime. The Lorenz gauge gives a well-understood singular structure at the particle, while the (near) Bondi--Sachs gauge ensures the asymptotic fall-off needed for an asymptotically flat second-order source. Our formalism reconstructs the metric from separable Teukolsky-type equations, decomposed into spin-weighted spheroidal harmonic modes and solved spectrally on hyperboloidal slices. The resulting $(\ell,m)$-mode sums converge rapidly across most of the domain, but slowly near the point mass; we overcome this with a novel use of a puncture to extract accurate $m$-mode data, yielding output accurate to $\sim 10^{-10}$ per $m$-mode and suitable for use at second order. Our metric-perturbation results cover a range of Kerr spins $-0.9 \le a/M \le 0.9$ and orbital radii out to $50M$. In addition to its immediate application in second-order self-force calculations, our results can also be used as input to other calculations involving Kerr metric perturbations, including studies of environmental effects and quasinormal modes.

gr-qc↗

arXiv2026

Comparison of 4.5PN and 2SF gravitational energy fluxes from quasicircular compact binaries

Recent years have seen significant advances in models of gravitational waveforms emitted by quasicircular compact binaries in two regimes: the weak-field, post-Newtonian regime, in which the gravitational wave energy flux has now been calculated to fourth-and-a-half post-Newtonian order (4.5PN) [Phys. Rev. Lett. 131}, 121402 (2023)]; and the small-mass-ratio, gravitational self-force regime, in which the flux has now been calculated to second perturbative order in the mass ratio (2SF) [Phys. Rev. Lett. 127, 151102 (2021)]. We compare these results and find evidence of mutual consistency between the two (very distinct though both first-principle) perturbative calculations.

gr-qc↗

arXiv2026

Second-order Teukolsky calculations for nonspinning, quasicircular binaries

Currently, the only second-order gravitational self-force calculations have been based on directly solving the perturbative Einstein equations in the Lorenz gauge. That method relied on the complete separability of the Einstein equations in a Schwarzschild background. In this paper, we present a new scheme based on the second-order Teukolsky equation. Crucially, this method promises to extend (reasonably straightforwardly) to the more realistic case of a Kerr background. Here we implement the scheme in the simplest setting of quasicircular orbits around a Schwarzschild black hole. In addition to working with the Teukolsky equation, our scheme incorporates several other advances over previous second-order self-force calculations: compactified hyperboloidal slicing, transformation to a Bondi-Sachs gauge, and a combination of spectral and variation-of-parameters methods. We also use these tools to re-examine the infrared divergences that arise in second-order Lorenz-gauge calculations, showing they are less pronounced in the Teukolsky case and completely eliminated in the Bondi-Sachs gauge. We conclude by calculating the asymptotic energy fluxes and benchmarking them against previous Lorenz-gauge calculations.

gr-qc↗

arXiv2026

Schwarzschild perturbations in Lorenz gauge via elliptic differential equations

Accurate predictions of gravitational wave signals from asymmetric compact binaries are accessible through black hole perturbation theory and self-force calculations. Faithful waveform models will require contributions from first- and second-order terms in the small mass-ratio expansion. The problem of second-order Kerr perturbations is exacerbated by non-separability of the metric perturbation equations and non-linear mode coupling, which motivates this $m$-mode approach. This work moves towards the eventual goal of second-order Kerr perturbations by calculating first-order Schwarzschild metric perturbations via $m$-modes in the frequency domain for the first time. We solve the Lorenz gauge field equations as a system of coupled elliptic partial differential equations that govern each $m$-mode. Our Mathematica code implements a second-order finite difference representation of the field equations, which we solve as a sparse linear algebra problem. Regularization near the small body is achieved through the effective source method, and our presentation introduces a new puncture expansion of the singular field for a point mass in Kerr spacetime. Issues related to problematic near-horizon behavior are explored and then mitigated by applying sophisticated near-horizon boundary conditions. Our results illustrate the features of each component and $m$-mode of the metric perturbation, and we are able to calculate gravitational wave energy fluxes with sufficient accuracy to enable future second-order self-force calculations.

gr-qc↗

arXiv2026

Enhancing the SEOBNRv5 effective-one-body waveform model with second-order gravitational self-force fluxes

We leverage recent breakthrough calculations using second-order gravitational self-force (2GSF) theory to improve both the gravitational-mode amplitudes and radiation-reaction force in effective-one-body~(EOB) waveform models. We achieve this by introducing new calibration parameters in the SEOBNRv5HM mode amplitudes, and matching them to the newly available 2GSF energy-flux multipolar data for quasicircular nonspinning binary black holes. We find that this significantly improves the SEOBNRv5HM energy flux, when compared to numerical-relativity (NR) simulations of binary black holes with mass ratios between 1:1 and 1:20. Moreover, we find that, once the conservative part of the SEOBNRv5 dynamics is calibrated, the SEOBNRv5HM waveform model with 2GSF information reproduces the binding energy of NR simulations more accurately, providing a powerful check of the consistency and naturalness of the EOB approach. While we only include nonspinning 2GSF information, the more accurate binding energy and energy flux carry over to the SEOBNRv5 waveform models for spinning binary black holes. Thus, our results improve the latest generation of SEOBNR waveform models (i.e., SEOBNRv5), which has been recently completed for use in the upcoming fourth observing (O4) run of the LIGO-Virgo-KAGRA Collaboration.

gr-qc↗

arXiv2026

Self-force calculations with numerical relativity methods

To model gravitational waveforms from extreme mass-ratio inspirals (EMRIs) for the upcoming LISA space mission, gravitational self-force calculations are needed to second order in perturbation theory. However, to date these calculations have only been attempted for the simplest case of circular orbits in Schwarzschild spacetime. In this work, we present a new computational method aimed at performing generic second-order self-force calculations in Kerr spacetime using methods from the adjacent field of numerical relativity. We perform an $m$-mode separation of variables, add null ("$vtu$") slicing in horizon-penetrating coordinates, and solve the resulting elliptic PDEs using high-order discontinuous Galerkin discretization, adaptive mesh-refinement, and an iterative Krylov-type linear solver with parallelizable multigrid-Schwarz preconditioning. We find that our method achieves exponential convergence for the self-force on a scalar point charge in Kerr spacetime up to spins of $a=0.998$ (Thorne limit) on circular equatorial orbits as close as the ISCO (prograde and retrograde), despite the non-smooth puncture on the grid. We solve for 20 $m$-modes in parallel in a few seconds and retain the flexibility to extend the method to gravitational self-force and more generic orbits in the future. The code to perform these calculations is publicly available in the open-source numerical relativity code SpECTRE.

gr-qc↗

arXiv2026

Post-adiabatic self-force waveforms: slowly spinning primary and precessing secondary

Recent progress in gravitational self-force theory has led to the development of a first post-adiabatic (1PA) waveform model for nonspinning, quasicircular compact binaries [Phys. Rev. Lett. 130, 241402 (2023)]. In this paper, we extend that model to allow for a slowly spinning primary black hole and a generic, precessing spin on the secondary object, restricting to the case of small misalignment between the primary spin and the orbital angular momentum. We demonstrate excellent agreement between our waveforms and fully nonlinear numerical relativity simulations for mass ratios $q\gtrsim 5$ and primary spins $|χ_1|\lesssim 0.1$ and arbitrary secondary spin $χ_2 \lesssim 1$. In particular we present the re-summed 1PAT1R waveform model, which significantly improves the accuracy of the original 1PAT1 waveforms for comparable masses and increasing primary spin. Our models are publicly available in the WaSABI package.

gr-qc↗

arXiv2026

Effective source for second-order self-force calculations: quasicircular orbits in Schwarzschild spacetime

Recent years have seen the first production of "post-adiabatic" gravitational-waveform models based on second-order gravitational self-force theory. These models rely on calculations of an effective source in the perturbative second-order Einstein equation. Here, for the first time, we detail the calculation of the effective source in a Schwarzschild background, which underlies the second-order self-force results in [Phys. Rev. Lett. 127, 151102 (2021); ibid. 128, 231101 (2022); ibid. 130, 241402 (2023)]. The source is designed for use in the multiscale form of the Lorenz-gauge Einstein equation, decomposed in tensor spherical harmonics, or in the analogous second-order Teukolsky equation. It involves, among other things, contributions from (i) quadratic coupling of first-order field modes, (ii) the slow evolution of first-order fields, (iii) quadratic products of a first-order puncture field, and (iv) the second-order puncture field. We validate each of these pieces through numerical and analytical tests.

gr-qc↗

arXiv2025

Post-Newtonian expansion of fluxes from a scalar charge on an inclined-spherical orbit about a Kerr black hole

Efforts are underway to accurately model extreme-mass-ratio inspirals for binaries with a spinning (Kerr) primary. At lowest order the adiabatic evolution depends on the radiation fluxes. Fluxes and other self-force quantities can be expanded analytically in post-Newtionian (PN) series allowing the early evolutionary phase to be understood. When it comes to more complicated background geodesic orbits, it proves useful to use the scalar field model problem to guide development and testing of techniques. In this paper, we present analytical expressions for the scalar fluxes from a scalar point-charge in inclined-spherical orbit about a Kerr black hole up to 12PN relative order, with expressions that are exact in terms of the inclination parameter $x$ and black hole spin $a$. The expressions are constructed using the Mano, Suzuki, and Takasugi method of solving the scalar wave equation in a Kerr background. We compare the numerical evaluation of these flux expressions to full numerical ($s=0$) Teukolsky code results, examining their degree of utility as the strong-field region is approached.

gr-qc↗

arXiv2025

Spin-aligned inspiral waveforms from self-force and post-Newtonian theory

We present the state-of-the-art waveform model WaSABI-C for quasicircular inspirals of spinning black hole binaries with aligned or anti-aligned spins. Our model synthesizes the most up-to-date first- and second-order gravitational self-force results with high-order post-Newtonian expansions through a systematic hybridization procedure. This approach captures both strong-field and weak-field dynamics with high fidelity, enabling accurate modeling of spin-(anti)aligned inspirals across a wide parameter space. The resulting waveforms mark a significant advance in the precision of self-force-based templates, providing critical input for the detection and interpretation of gravitational waves from compact binaries with future observatories such as LISA and ET. We accompany this work with the release of WaSABI (Waveform Simulations of Asymmetric Binary Inspirals), a public package implementing our model for community use and further development.

gr-qc↗

arXiv2025

Sourced metric perturbations of Kerr spacetime in Lorenz gauge

We derive a formalism for solving the Lorenz gauge equations for metric perturbations of Kerr spacetime sourced by an arbitrary stress-energy tensor. The metric perturbation is obtained as a sum of differential operators acting on a set of six scalars, with two of spin-weight $\pm2$, two of spin-weight $\pm1$, and two of spin-weight $0$. We derive the sourced Teukolsky equations satisfied by these scalars, with the sources given in terms of differential operators acting on the stress-energy tensor. The method can be used to obtain both linear and higher order nonlinear metric perturbations, and it fully determines the metric perturbation up to a time integral, omitting only static contributions which must be handled separately.

gr-qc↗

arXiv2025

Post-Newtonian expansion of gravitational energy and angular momentum fluxes: inclined spherical orbits about a Kerr black hole

We present analytical expressions for the fluxes of energy and angular momentum from a point mass on an inclined spherical orbit about a Kerr black hole. The expressions are obtained using the method of Mano, Suzuki and Takasugi to construct analytical solutions of the Teukolsky equation, and are given as post-Newtonian expansions valid through 12PN, with arbitrary values for the inclination parameter $x$ and black hole spin $a$. We characterize the structure of the PN expansions in terms of their dependence on $x$ and $a$, and we validate our results against numerical calculations.

gr-qc↗

arXiv2024

Implementation of a GHZ-Teukolsky puncture scheme for gravitational self-force calculations

Post-adiabatic models of extreme- and intermediate-mass-ratio inspirals will require calculations of second-order gravitational self-force effects in the spacetime of a spinning, Kerr black hole. We take a step toward such calculations by implementing the recently formulated Teukolsky puncture scheme with Green-Hollands-Zimmerman metric reconstruction [CQG 39, 015019 (2022)]. This scheme eliminates the critical obstacle of gauge singularities that arise in the standard no-string metric reconstruction. Our first proof-of-principle implementation is limited to the simple case of circular orbits in Schwarzschild spacetime, but the method also applies to generic orbits on a Kerr background. We conclude with a discussion of various approaches to the second-order self-force problem in Kerr.

gr-qc↗

arXiv2024

Second-order perturbations of the Schwarzschild spacetime: practical, covariant and gauge-invariant formalisms

High-accuracy gravitational-wave modeling demands going beyond linear, first-order perturbation theory. Particularly motivated by the need for second-order perturbative models of extreme-mass-ratio inspirals and black hole ringdowns, we present practical spherical-harmonic decompositions of the Einstein equation, Regge-Wheeler-Zerilli equations, and Teukolsky equation at second perturbative order in a Schwarzschild background. Our formulations are covariant on the $t$--$r$ plane and on the two-sphere, and we express the field equations in terms of gauge-invariant metric perturbations. In a companion Mathematica package, PerturbationEquations, we provide these invariant formulas as well as the analogous formulas in terms of raw, gauge-dependent metric perturbations. Our decomposition of the second-order Einstein equation, when specialized to the Lorenz gauge, was a key ingredient in recent second-order self-force calculations [Phys. Rev. Lett. 124, 021101 (2020); ibid. 127, 151102 (2021); ibid. 130, 241402 (2023)].

gr-qc↗

arXiv2024

Metric perturbations of Kerr spacetime in Lorenz gauge: Circular equatorial orbits

We construct the metric perturbation in Lorenz gauge for a compact body on a circular equatorial orbit of a rotating black hole (Kerr) spacetime, using a newly-developed method of separation of variables. The metric perturbation is formed from a linear sum of differential operators acting on Teukolsky mode functions, and certain auxiliary scalars, which are solutions to ordinary differential equations in the frequency domain. For radiative modes, the solution is uniquely determined by the $s=\pm2$ Weyl scalars, the $s=0$ trace, and $s=0,1$ gauge scalars whose amplitudes are determined by imposing continuity conditions on the metric perturbation at the orbital radius. The static (zero-frequency) part of the metric perturbation, which is handled separately, also includes mass and angular momentum completion pieces. The metric perturbation is validated against the independent results of a 2+1D time domain code, and we demonstrate agreement at the expected level in all components, and the absence of gauge discontinuities. In principle, the new method can be used to determine the Lorenz-gauge metric perturbation at a sufficiently high precision to enable accurate second-order self-force calculations on Kerr spacetime in future. We conclude with a discussion of extensions of the method to eccentric and non-equatorial orbits.

gr-qc↗

arXiv2024

Accuracy Requirements: Assessing the Importance of First Post-Adiabatic Terms for Small-Mass-Ratio Binaries

We investigate the impact of post-adiabatic (1PA) terms on parameter estimation for extreme and intermediate mass-ratio inspirals using state-of-the-art waveform models. Our analysis is the first to employ Bayesian inference to assess systematic errors for 1PA waveforms. We find that neglecting 1PA terms introduces significant biases for the (small) mass ratio $ε\gtrsim 10^{-5}$ for quasi circular orbits in Schwarzschild spacetime, which can be mitigated with resummed 3PN expressions at 1PA order. Moreover, we show that the secondary spin is strongly correlated with the other intrinsic parameters, and it can not be constrained for $ε\lesssim 10^{-5}$. Finally, we highlight the need for addressing eccentric waveform systematics in the small-mass-ratio regime, as they yield stronger biases than the circular limit in both intrinsic and extrinsic parameters.

gr-qc↗