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The Laser Interferometer Space Antenna (LISA) will be a transformative experiment for gravitational wave astronomy, and, as such, it will offer unique opportunities to address many key astrophysical questions in a completely novel way. The synergy with ground-based and space-born instruments in the electromagnetic domain, by enabling multi-messenger observations, will add further to the discovery potential of LISA. The next decade is crucial to prepare the astrophysical community for LISA’s first observations. This review outlines the extensive landscape of astrophysical theory, numerical simulations, and astronomical observations that are instrumental for modeling and interpreting the upcoming LISA datastream. To this aim, the current knowledge in three main source classes for LISA is reviewed; ultra-compact stellar-mass binaries, massive black hole binaries, and extreme or interme-diate mass ratio inspirals. The relevant astrophysical processes and the established modeling techniques are summarized. Likewise, open issues and gaps in our understanding of these sources are highlighted, along with an indication of how LISA could help making progress in the different areas. New research avenues that LISA itself, or its joint exploitation with upcoming studies in the electromagnetic domain, will enable, are also illustrated. Improvements in modeling and analysis approaches, such as the combination of numerical simulations and modern data science techniques, are discussed. This review is intended to be a starting point for using LISA as a new discovery tool for understanding our Universe.

In general-relativistic perturbation theory, a point mass accelerates away from geodesic motion due to its gravitational self-force. Because the self-force is small, one can often approximate the motion as geodesic. However, it is well known that self-force effects accumulate over time, making the geodesic approximation fail on long time scales. It is less well known that this failure at large times translates to a failure at large distances as well. At second perturbative order, two large-distance pathologies arise: spurious secular growth and infrared-divergent retarded integrals. Both stand in the way of practical computations of second-order self-force effects. Utilizing a simple flat-space scalar toy model, I develop methods to overcome these obstacles. The secular growth is tamed with a multiscale expansion that captures the system's slow evolution. The divergent integrals are eliminated by matching to the correct retarded solution at large distances. I also show how to extract conservative self-force effects by taking local-in-time "snapshots" of the global solution. These methods are readily adaptable to the physically relevant case of a point mass orbiting a black hole. [ABSTRACT FROM AUTHOR]

Through second order in perturbative general relativity, a small compact object in an external vacuum spacetime obeys a generalized equivalence principle: although it is accelerated with respect to the external background geometry, it is in free fall with respect to a certain effective vacuum geometry. However, this single principle takes very different mathematical forms, with very different behaviors, depending on how one treats perturbed motion. Furthermore, any description of perturbed motion can be altered by a gauge transformation. In this paper, I clarify the relationship between two treatments of perturbed motion and the gauge freedom in each. I first show explicitly how one common treatment, called the Gralla-Wald approximation, can be derived from a second, called the self-consistent approximation. I next present a general treatment of smooth gauge transformations in both approximations, in which I emphasize that the approximations' governing equations can be formulated in an invariant manner. All of these analyses are carried through second perturbative order, but the methods are general enough to go to any order. Furthermore, the tools I develop, and many of the results, should have broad applicability to any description of perturbed motion, including osculating-geodesic and two-timescale descriptions. [ABSTRACT FROM AUTHOR]

Using black hole perturbation theory and arbitrary-precision computer algebra, we obtain the post- Newtonian expansions of the linear-in-mass-ratio corrections to the spin-precession angle and tidal invariants for a particle in circular orbit around a Schwarzschild black hole. We extract coefficients up to 20 post-Newtonian order from numerical results that are calculated with an accuracy greater than 1 part in 10500. These results can be used to calibrate parameters in effective-one-body models of compact binaries, specifically the spin-orbit part of the effective Hamiltonian and the dynamically significant tidal part of the main radial potential of the effective metric. Our calculations are performed in a radiation gauge, which is known to be singular away from the particle. To overcome this irregularity, we define suitable Detweiler-Whiting singular and regular fields in this gauge, and we compute the invariants using mode-sum regularization in combination with averaging from two sides of the particle. The detailed justification of this regularization procedure will be presented in a forthcoming companion paper. [ABSTRACT FROM AUTHOR]

A compact object moving on a quasicircular orbit about a Schwarzschild black hole gradually spirals inward due to the dissipative action of its gravitational self-force. But in addition to driving the inspirai, the self-force has a conservative piece. Within a second-order self-force formalism, I derive a second-order generalization of Detweiler's redshift variable, which provides a gauge-invariant measure of conservative effects on quasicircular orbits. I sketch a frequency-domain numerical scheme for calculating this quantity. Once this scheme has been implemented, its results may be used to determine high-order terms in post- Newtonian theory and parameters in effective-one-body theory. [ABSTRACT FROM AUTHOR]

Accurately modeling an extreme-mass-ratio inspirai requires knowledge of the second-order gravita-tional self-force on the inspiraling small object. Recently, numerical puncture schemes have been formulated to calculate this force, and their essential analytical ingredients have been derived from first principles. However, the "puncture," a local representation of the small object's self-field, in each of these schemes has been presented only in a local coordinate system centered on the small object, while a numerical implementation will require the puncture in coordinates covering the entire numerical domain. In this paper we provide an explicit covariant self-field as a local expansion in terms of Synge's world function. The self-field is written in the Lorenz gauge, in an arbitrary vacuum background, and in forms suitable for both self-consistent and Gralla-Wald-type representations of the object's trajectory. We illustrate the local expansion's utility by sketching the procedure of constructing from it a numerically practical puncture in any chosen coordinate system. [ABSTRACT FROM AUTHOR]

Calculations of the gravitational self-force (GSF) on a point mass in curved spacetime require as input the metric perturbation in a sufficiently regular gauge. A basic challenge in the program to compute the GSF for orbits around a Kerr black hole is that the standard procedure for reconstructing the metric perturbation is formulated in a class of "radiation" gauges, in which the particle singularity is non-isotropic and extends away from the particle's location. Here we present two practical schemes for calculating the GSF using a radiation-gauge reconstructed metric as input. The schemes are based on a detailed analysis of the local structure of the particle singularity in the radiation gauges. We show that three types of radiation gauge exist: two containing a radial stringlike singularity emanating from the particle, either in one direction ("half-string" gauges) or both directions ("full-string" gauges); and a third type containing no strings but with a jump discontinuity (and possibly a delta function) across a surface intersecting the particle. Based on a flat-space example, we argue that the standard mode-by-mode reconstruction procedure yields the "regular half" of a half-string solution, or (equivalently) either of the regular halves of a no-string solution. For the half-string case, we formulate the GSF in a locally deformed radiation gauge that removes the string singularity near the particle. We derive a mode-sum formula for the GSF in this gauge, which is analogous to the standard Lorenz-gauge formula but requires a correction to the values of the regularization parameters. For the no-string case, we formulate the GSF directly, without a local deformation, and we derive a mode-sum formula that requires no correction to the regularization parameters but involves a certain averaging procedure. We explain the consistency of our results with Gralla's invariance theorem for the regularization parameters, and we discuss the correspondence between our method and a related approach by Friedman et al. [ABSTRACT FROM AUTHOR]

A small extended body moving through an external spacetime gɑβ creates a metric perturbation hɑβ, which forces the body away from geodesic motion in gɑβ. The foundations of this effect, called the gravitational self-force, are now well established, but concrete results have mostly been limited to linear order. Accurately modeling the dynamics of compact binaries requires proceeding to nonlinear orders. To that end, I show how to obtain the metric perturbation outside the body at all orders in a class of generalized wave gauges. In a small buffer region surrounding the body, the form of the perturbation can be found analytically as an expansion for small distances r from a representative worldline. Given only a specification of the body's multipole moments, the field obtained in the buffer region suffices to find the metric everywhere outside the body via a numerical puncture scheme. Following this procedure at first and second order, I calculate the field in the buffer region around an arbitrarily structured compact body at sufficiently high order in r to numerically implement a second-order puncture scheme, including effects of the body's spin. I also define nth-order (local) generalizations of the Detweiler-Whiting singular and regular fields and show that in a certain sense, the body can be viewed as a skeleton of multipole moments. [ABSTRACT FROM AUTHOR]

This review is concerned with the motion of a point scalar charge, a point electric charge, and a point mass in a specified background spacetime. In each of the three cases the particle produces a field that behaves as outgoing radiation in the wave zone, and therefore removes energy from the particle. In the near zone the field acts on the particle and gives rise to a self-force that prevents the particle from moving on a geodesic of the background spacetime. The self-force contains both conservative and dissipative terms, and the latter are responsible for the radiation reaction. The work done by the self-force matches the energy radiated away by the particle. The field’s action on the particle is difficult to calculate because of its singular nature: the field diverges at the position of the particle. But it is possible to isolate the field’s singular part and show that it exerts no force on the particle — its only effect is to contribute to the particle’s inertia. What remains after subtraction is a regular field that is fully responsible for the self-force. Because this field satisfies a homogeneous wave equation, it can be thought of as a free field that interacts with the particle; it is this interaction that gives rise to the self-force. The mathematical tools required to derive the equations of motion of a point scalar charge, a point electric charge, and a point mass in a specified background spacetime are developed here from scratch. The review begins with a discussion of the basic theory of bitensors (Part I). It then applies the theory to the construction of convenient coordinate systems to chart a neighbourhood of the particle’s word line (Part II). It continues with a thorough discussion of Green’s functions in curved spacetime (Part III). The review presents a detailed derivation of each of the three equations of motion (Part IV). Because the notion of a point mass is problematic in general relativity, the review concludes (Part V) with an alternative derivation of the equations of motion that applies to a small body of arbitrary internal structure.

In recent years, gravitational-wave astronomy has motivated increasingly accurate perturbative studies of gravitational dynamics in compact binaries. This in turn has enabled more detailed analyses of the dynamical black holes in these systems. For example, Pound et al. [Phys. Rev. Lett. 124, 021101 (2020)] recently computed the surface area of a Schwarzschild black hole's apparent horizon, perturbed by an orbiting body, to second order in the binary's mass ratio. In this paper, we take that as the starting point for a comprehensive study of a perturbed Schwarzschild black hole's apparent and event horizon at second perturbative order, deriving generic formulas for the first- and second-order corrections to the horizons' radial profiles, surface areas, Hawking masses, and intrinsic curvatures. We find that the two horizons are remarkably similar, and that any teleological behavior of the event horizon is suppressed in several ways. Critically, we establish that at all orders, the perturbed event horizon in a small-mass-ratio binary is effectively localized in time. Even more pointedly, the event horizon is identical to the apparent horizon at linear order regardless of the source of perturbation, implying that the seemingly teleological "tidal lead," previously observed in linearly perturbed event horizons, is not genuinely teleological in origin. The two horizons do generically differ at second order, but their Hawking masses remain identical, implying that the event horizon obeys the same energy-flux balance law as the apparent horizon. At least in the case of a binary system, the difference between their surface areas remains extremely small even in the late stages of inspiral. In the course of our analysis, we also numerically illustrate puzzling behavior in the black hole's motion around the binary's center of mass. [ABSTRACT FROM AUTHOR]

Extreme-mass-ratio inspirals (EMRIs) will be key sources for LISA. However, accurately extracting system parameters from a detected EMRI waveform will require self-force calculations at second order in perturbation theory, which are still in a nascent stage. One major obstacle in these calculations is the strong divergences that are encountered on the worldline of the small object. Previously, it was shown by one of us [A. Pound, Nonlinear gravitational self-force: Second-order equation of motion, Phys. Rev. D 95, 104056 (2017)] that a class of "highly regular" gauges exist in which the singularities have a qualitatively milder form, promising to enable more efficient numerical calculations. Here we derive expressions for the metric perturbation in this class of gauges, in a local expansion in powers of distance r from the worldline, to sufficient order in r for numerical implementation in a puncture scheme. Additionally, we use the highly regular class to rigorously derive a distributional source for the second-order field and a pointlike second-order stress-energy tensor (the Detweiler stress energy) for the small object. This makes it possible to calculate the second-order self-force using mode-sum regularization rather than the more cumbersome puncture schemes that have been necessary previously. Although motivated by EMRIs, our calculations are valid in an arbitrary vacuum background, and they may help clarify the interpretation of point masses and skeleton sources in general relativity more broadly. [ABSTRACT FROM AUTHOR]

Extreme-mass-ratio inspirals, in which a stellar-mass compact object spirals into a supermassive black hole in a galactic core, are expected to be key sources for LISA. Modeling these systems with sufficient accuracy for LISA science requires going to second (or postadiabatic) order in gravitational self-force theory. Here we present a practical two-timescale framework for achieving this and generating postadiabatic waveforms. The framework comprises a set of frequency-domain field equations that apply on the fast, orbital timescale, together with a set of ordinary differential equations that determine the evolution on the slow, inspiral timescale. Our analysis is restricted to the special case of quasicircular orbits around a Schwarzschild black hole, but its general structure carries over to the realistic case of generic (inclined and eccentric) orbits in Kerr spacetime. In our restricted context, we also develop a tool that will be useful in all cases: a formulation of the frequency-domain field equations using hyperboloidal slicing, which significantly improves the behavior of the sources near the boundaries. We give special attention to the slow evolution of the central black hole, examining its impact on both the two-timescale evolution and the earlier self-consistent evolution scheme. [ABSTRACT FROM AUTHOR]

When a small, uncharged, compact object is immersed in an external background spacetime, at zeroth order in its mass, it moves as a test particle in the background. At linear order, its own gravitational field alters the geometry around it, and it moves instead as a test particle in a certain effective metric satisfying the linearized vacuum Einstein equation. In the letter [Phys. Rev. Lett. 109, 051101 (2012)], using a method of matched asymptotic expansions, I showed that the same statement holds true at second order: if the object's leading-order spin and quadrupole moment vanish, then through second order in its mass, it moves on a geodesic of a certain smooth, locally causal vacuum metric defined in its local neighborhood. Here I present the complete details of the derivation of that result. In addition, I extend the result, which had previously been derived in gauges smoothly related to Lorenz, to a class of highly regular gauges that should be optimal for numerical self-force computations. [ABSTRACT FROM AUTHOR]

The computational cost of inspiral and merger simulations for black hole binaries increases in inverse proportion to the square of the mass ratio q≔m2/m1≤1. One factor of q comes from the number of orbital cycles, which is proportional to 1/q, and another is associated with the required number of time steps per orbit, constrained (via the Courant-Friedrichs-Lewy condition) by the need to resolve the two disparate length scales. This problematic scaling makes simulations progressively less tractable at smaller q. Here we propose and explore a method for alleviating the scale disparity in simulations with mass ratios in the intermediate astrophysical range (10-4≲q≲10-2), where purely perturbative methods may not be adequate. A region of radius much larger than m2 around the smaller object is excised from the numerical domain, and replaced with an analytical model approximating a tidally deformed black hole. The analytical model involves certain a priori unknown parameters, associated with unknown bits of physics together with gauge-adjustment terms; these are dynamically determined by matching them to the numerical solution outside the excision region. In this paper we develop the basic idea and apply it to a toy model of a scalar charge in a circular-geodesic orbit around a Schwarzschild black hole, solving for the massless Klein-Gordon field in a (1+1)D framework. Our main goal here is to explore the utility and properties of different matching strategies, and to this end we develop two independent implementations, a finite-difference one and a spectral one. We discuss the extension of our method to a full 3D numerical evolution and to gravity. [ABSTRACT FROM AUTHOR]

Second-order self-force computations, which will be essential in modeling extreme-mass-ratio inspirals, involve two major new difficulties that were not present at first order. One is the problem of large scales, discussed in Pound [Phys. Rev. D 92, 104047 (2015)]. Here we discuss the second difficulty, which occurs instead on small scales: if we expand the field equations in spherical harmonics, then because the first-order field contains a singularity, we require an arbitrarily large number of first-order modes to accurately compute even a single second-order mode. This is a generic feature of nonlinear field equations containing singularities, allowing us to study it in the simple context of a scalar toy model in flat space. Using that model, we illustrate the problem and demonstrate a robust strategy for overcoming it. [ABSTRACT FROM AUTHOR]

Vacuum perturbations of the Kerr metric can be reconstructed from the corresponding perturbation in either of the two Weyl scalars ψ0 or ψ4, using a procedure described by Chrzanowski and others in the 1970s. More recent work, motivated within the context of self-force physics, extends the procedure to metric perturbations sourced by a particle in a bound geodesic orbit. However, the existing procedure leaves undetermined a certain stationary, axially symmetric piece of the metric perturbation. In the vacuum region away from the particle, this "completion" piece corresponds simply to mass and angular-momentum perturbations of the Kerr background, with amplitudes that are, however, a priori unknown. Here, we present and implement a rigorous method for finding the completion piece. The key idea is to impose continuity, off the particle, of certain gauge-invariant fields constructed from the full (completed) perturbation, in order to determine the unknown amplitude parameters of the completion piece. We implement this method in full for bound (eccentric) geodesic orbits in the equatorial plane of the Kerr black hole. Our results provide a rigorous underpinning of recent results by Friedman et al. for circular orbits and extend them to noncircular orbits. [ABSTRACT FROM AUTHOR]