arXiv subjects

Andrew Spiers

Publications and source records attributed to Andrew Spiers.

13 recordsLinked to original sources

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.

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arXiv2026

The significance of first post-adiabatic contributions for scalar charge measurements with intermediate and extreme mass ratio inspirals

We present the first self-force-based beyond-GR waveform model incorporating post-adiabatic orbital evolution for intermediate- and extreme-mass-ratio inspirals in theories of gravity with additional scalar fields. Focusing on quasi-circular inspirals into a non-spinning primary, we combine a first post-adiabatic (1PA) gravitational sector with leading-order scalar field effects and use Bayesian injection-recovery studies to assess the impact of waveform systematics on the inference of scalar charges with LISA. We find that neglecting 1PA effects in the gravitational sector can bias the inference of intrinsic binary parameters, while scalar-charge measurements remain robust across a wide range of mass ratios. In contrast, analysing signals from binaries in which the secondary carries a scalar charge using pure-GR templates leads to significant biases and underestimated uncertainties due to unmodelled correlations between the scalar charge and the binary parameters. We also investigate the role of secondary spin and find no significant correlation between the secondary spin and the scalar charge. Notably, up to a mass ratio of $10^{-4}$, the secondary spin itself remains unconstrained even in the pure-GR case, in contrast with previous claims in the literature. Finally, we show that modelling scalar emission with a leading-order dipolar post-Newtonian approximation -- for quasi-circular inspirals into a non-spinning primary -- introduces negligible systematic errors relative to fully relativistic scalar fluxes.

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arXiv2026

The Bondi--Sachs gauge, BMS frames, and memory in black hole perturbation theory

As LISA and other next-generation detectors demand increasingly accurate waveform models, there is a growing need for these models to precisely control gauge freedoms that had previously been inconsequential. One such intrinsic freedom is the choice of the asymptotic Bondi--Metzner--Sachs (BMS) frame. The need to control the BMS frame is particularly pronounced in black hole perturbation theory, where there has been little work to this end -- most glaringly in gravitational self-force calculations, which are in an unknown frame and encounter infrared, far-zone gauge singularities at second perturbative order. Here we present a framework for iteratively transforming to the Bondi--Sachs gauge and fixing the BMS frame on a Kerr background. This includes an extension of the Bondi--Sachs formalism to the multiscale expansions that underpin most self-force-based waveforms, introducing soft hair and a concept of ``forgetful gauges'' in the process. Our framework evades infrared divergences and naturally incorporates memory effects that had previously only ever been added ``after the fact'' in self-force waveforms, including the recently discovered ``memory distortion''. Our formalism could also be used for ringdown analysis, and we expect it to be vital for comparisons with numerical relativity and post-Newtonian theory.

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arXiv2025

Extreme-mass-ratio inspirals in relativistic accretion discs

We compute relativistic Lindblad torques for circular, equatorial extreme-mass-ratio inspirals (EMRIs) embedded in relativistic thin accretion discs, including spinning black hole configurations. We find that relativistic effects can amplify the magnitude of these torques by orders of magnitude in the strong-field regime, and that the torque can even reverse direction as the EMRI approaches the innermost stable circular orbit (ISCO). However, we show that the location of this reversal is highly spin-dependent, shifting progressively closer to the ISCO, where gravitational-wave emission completely dominates the inspiral, as the spin of the central black hole increases. Spin also modifies the radial dependence of the Lindblad torques. We investigate whether Lindblad torques can be approximated by parametrised power laws of the form T_LR = A(r_s / 10M)^n_r (or combinations thereof), and find significant spin- and disc-dependent variations in the slope parameter n_r. For instance, for spin a/M = 0.9, we find n_r = 3.6 in the strong-field regime, compared to the Newtonian value of n_r = 4.5. Given current forecasts of parameter recovery for ``golden'', loud EMRIs in accretion discs (\Delta n_r ~ 0.5), we predict LISA could distinguish between different disc configurations through their relativistic Lindblad torque signatures, providing the first direct probe of the midplane structure of the inner region of accretion discs, which is inaccessible to electromagnetic observations.

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arXiv2025

Quadratic quasinormal modes at null infinity on a Schwarzschild spacetime

The ringdown of perturbed black holes has been studied since the 1970s, but until recently, studies have focused on linear perturbations. There is now burgeoning interest in nonlinear perturbative effects during ringdown. Here, using a hyperboloidal framework, we provide a complete treatment of linear and quadratic quasinormal modes (QNMs and QQNMs) in second-order perturbation theory, in Schwarzschild spacetime. We include novel methods for extracting QNMs and QQNMs amplitudes using a Laplace transform treatment, allowing for the inclusion of arbitrary initial data. We produce both time- and frequency-domain codes. From these codes, we present new results further exploring the unforeseen dependence of QQNMs amplitudes on the parity of the progenitor system, as demonstrated in our letter [Phys. Rev. Lett. 134, 061401 (2025)]. Our numerical results are restricted to perturbations of a Schwarzschild black hole, but our methods extend straightforwardly to the astrophysically realistic case of a Kerr black hole.

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arXiv2024

Quadratic quasi-normal mode dependence on linear mode parity

Quasinormal modes (QNMs) uniquely describe the dominant piece of the gravitational-wave ringdown of postmerger black holes. While the linear QNM regime has been extensively studied, recent work has highlighted the importance of second-perturbative-order, quadratic QNMs (QQNMs) arising from the nonlinear coupling of linear QNMs. Previous attempts to quantify the magnitude of these QQNMs have shown discrepant results. Using a new hyperboloidal framework, we resolve the discrepancy by showing that the QQNM/QNM ratio is a function not only of the black hole parameters but also of the ratio between even- and odd-parity linear QNMs: the ratio QQNM/QNM depends on what created the ringing black hole, but only through this ratio of even- to odd-parity linear perturbations.

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arXiv2024

Analytically Separating the Source of the Teukolsky Equation

Recent gravitational wave detections from black hole mergers have underscored the critical role black hole perturbation theory and the Teukolsky equation play in understanding the behaviour of black holes. The separable nature of the Teukolsky equation has long been leveraged to study the vacuum linear Teukolsky equation; however, as theory and measurements advance, solving the sourced Teukolsky equation is becoming a frontier of research. In particular, second-order calculations, such as in quasi-normal mode and self-force problems, have extended sources. This paper presents a novel method for analytically separating the Teukolsky equation's source, aimed to improve efficiency. Separating the source is a non-trivial problem due to the angular and radial mixing of generic quantities in Kerr spacetime. We provide a proof-of-concept demonstration of our method and show that it is accurate, separating the Teukolsky source produced by the stress-energy tensor of an ideal gas cloud surrounding a Kerr black hole. The detailed application of our method is provided in an accompanying \textit{Mathematica} notebook. Our approach opens up a new avenue for accurate black hole perturbation theory calculations with sources in both the time and frequency domain.

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arXiv2023

Ringdowns for black holes with scalar hair: the large mass case

Deviations from General Relativity can alter the quasi-normal mode (QNM) ringdown of perturbed black holes. It is known that a shift-symmetric (hence massless) scalar can only introduce black hole hair if it couples to the Gauss-Bonnet invariant, in which case the scalar charge is fixed with respect to the black hole mass and controlled by the strength of that coupling. The charge per unit mass decreases with the mass and can, therefore, be used as a perturbative parameter for black holes that are sufficiently large with respect to the scale suppressing the deviation from General Relativity or the Standard model. We construct an effective field theory scheme for QNMs using this perturbative parameter to capture deviations from Kerr for both the background and the perturbations. We demonstrate that up to second order in the charge per unit mass, QNMs can be calculated by solving standard linearised perturbation equations for the Kerr metric with sources depending on solutions of the same equations up to first order. It follows that corrections to the QNM frequencies are heavily suppressed for sufficiently massive black holes, meaning that LISA is very unlikely to detect any evidence of scalar hair in ringdown signals.

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arXiv2023

Waveform Modelling for the Laser Interferometer Space Antenna

LISA, the Laser Interferometer Space Antenna, will usher in a new era in gravitational-wave astronomy. As the first anticipated space-based gravitational-wave detector, it will expand our view to the millihertz gravitational-wave sky, where a spectacular variety of interesting new sources abound: from millions of ultra-compact binaries in our Galaxy, to mergers of massive black holes at cosmological distances; from the beginnings of inspirals that will venture into the ground-based detectors' view to the death spiral of compact objects into massive black holes, and many sources in between. Central to realising LISA's discovery potential are waveform models, the theoretical and phenomenological predictions of the pattern of gravitational waves that these sources emit. This white paper is presented on behalf of the Waveform Working Group for the LISA Consortium. It provides a review of the current state of waveform models for LISA sources, and describes the significant challenges that must yet be overcome.

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arXiv2023

Measuring scalar charge with compact binaries: High accuracy modelling with self-force

Using the self-force approach, we present the premier first-post-adiabatic accuracy formalism for modelling compact binaries in theories with a massless scalar field non-minimally coupled to gravity. We limit the binary secondary to being a non-spinning compact body with no scalar dipole (we will address the spinning and scalar dipole cases in an upcoming paper). By producing an ansatz for the scalar charged point particle action, we derive first- and second-order perturbative field equations and equations of motion for the secondary compact object. Under our assumptions, implementing this formalism will produce sufficiently accurate waveform templates for precision measurements of the scalar charge of the secondary with LISA data on extreme-mass-ratio inspirals. Our formalism is consistent with almost general scalar-tensor theories of gravity. Implementing our formalism builds on self-force models in General Relativity; we show the incorporation into the two-timescale formalism is straightforward. Excitingly, implementation poses no significantly more challenging barriers than computing first-post adiabatic waveforms in General Relativity.

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arXiv2023

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)].

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arXiv2023

Second-order Teukolsky formalism in Kerr spacetime: formulation and nonlinear source

To fully exploit the capabilities of next-generation gravitational wave detectors, we need to significantly improve the accuracy of our models of gravitational-wave-emitting systems. This paper focuses on one way of doing so: by taking black hole perturbation theory to second perturbative order. Such calculations are critical for the development of nonlinear ringdown models and of gravitational self-force models of extreme-mass-ratio inspirals. In the most astrophysically realistic case of a Kerr background, a second-order Teukolsky equation presents the most viable avenue for calculating second-order perturbations. Motivated by this, we analyse two second-order Teukolsky formalisms and advocate for the one that is well-behaved for gravitational self-force calculations and which meshes naturally with recent metric reconstruction methods due to Green, Hollands, and Zimmerman [CQG 37, 075001 (2020)] and others. Our main result is an expression for the nonlinear source term in the second-order field equation; we make this available, along with other useful tools, in an accompanying Mathematica notebook. Using our expression for the source, we also show that infrared divergences at second order can be evaded by adopting a Bondi--Sachs gauge.

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arXiv2021

New metric reconstruction scheme for gravitational self-force calculations

Inspirals of stellar-mass objects into massive black holes will be important sources for the space-based gravitational-wave detector LISA. Modelling these systems requires calculating the metric perturbation due to a point particle orbiting a Kerr black hole. Currently, the linear perturbation is obtained with a metric reconstruction procedure that puts it in a "no-string" radiation gauge which is singular on a surface surrounding the central black hole. Calculating dynamical quantities in this gauge involves a subtle procedure of "gauge completion" as well as cancellations of very large numbers. The singularities in the gauge also lead to pathological field equations at second perturbative order. In this paper we re-analyze the point-particle problem in Kerr using the corrector-field reconstruction formalism of Green, Hollands, and Zimmerman (GHZ). We clarify the relationship between the GHZ formalism and previous reconstruction methods, showing that it provides a simple formula for the "gauge completion". We then use it to develop a new method of computing the metric in a more regular gauge: a Teukolsky puncture scheme. This scheme should ameliorate the problem of large cancellations, and by constructing the linear metric perturbation in a sufficiently regular gauge, it should provide a first step toward second-order self-force calculations in Kerr. Our methods are developed in generality in Kerr, but we illustrate some key ideas and demonstrate our puncture scheme in the simple setting of a static particle in Minkowski spacetime.

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