Numerical-relativity simulations of the quasi-circular inspiral and merger of non-spinning, charged black holes: methods and comparison with approximate approaches

Gabriele Bozzola1, ∗ and Vasileios Paschalidis1, 2, † 1Department of Astronomy, University of Arizona, Tucson, AZ, USA 2Department of Physics, University of Arizona, Tucson, AZ, USA (Dated: April 15, 2021) We present fully general relativistic simulations of the quasi-circular inspiral and merger of charged, non-spinning, binary black holes with charge-to-mass ratio λ 0.3. We discuss the key fea- tures that enabled long term and stable evolutions of these binaries. We≤ also present a formalism for computing the angular momentum carried away by electromagnetic waves, and the electromagnetic contribution to black-hole horizon properties. We implement our formalism and present the results for the first time in numerical-relativity simulations. In addition, we compare our full non-linear solutions with existing approximate models for the inspiral and ringdown phases. We show that Newtonian models based on the quadrupole approximation have errors of 20 %–100 % in key gauge- invariant quantities. On the other hand, for the systems considered, we find that estimates of the remnant black hole spin based on the motion of test particles in Kerr-Newman spacetimes agree with our non-linear calculations to within a few percent. Finally, we discuss the prospects for detecting black hole charge by future gravitational-wave detectors using either the inspiral-merger-ringdown signal or the ringdown signal alone.

I. INTRODUCTION detect these signals and perform the associated param- eter estimations. Therefore, it should come to no sur- In previous work [1,2], we initiated a systematic pro- prise that the scarcity of simulations of compact binary gram to study the interactions between charged black mergers in modified theories of gravity [9] severely lim- holes in full non-linear Einstein-Maxwell theory. These its out our ability to use gravitational-wave observations explorations are relevant for gravitational-wave astron- to place stronger constraints. The reason for this short- omy, exotic astrophysics, and fundamental physics (such age is that many models of modified gravity are “sick” as modified gravity and beyond-standard-model physics). (e.g., lack of well-posedness, ghosts, etc., see [10] for a We first review these applications in Sec.IA. We sum- discussion), thereby making these computations partic- marize the main goals and results of this work in Sec.IB, ularly challenging or impossible. Hence, with the ex- and outline the structure of the manuscript and conven- ception of a few cases [11–13], progress in this direction tions adopted in Sec.IC. Readers that are mainly inter- is usually made by means of order-reduced approaches, ested in our new results can skip Sec.IA. and not solutions of the full theory (see, e.g. [14–18]). In contrast to modified gravity, Einstein-Maxwell theory admits a well-posed initial-value problem, while sharing other non-trivial properties with modified gravity (e.g, A. Motivation for non-linear simulations in emission of dipole radiation1). Moreover, some modified Einstein-Maxwell theory gravity theories reduce effectively to Einstein-Maxwell in specific limits (e.g. [19]). Therefore, the inspiral and Over the past few years there has been a growing in- merger of charged black holes constitutes, in a sense, terest in modified theories of gravity to perform strong- the middle ground between traditional general relativ- arXiv:2104.06978v1 [gr-qc] 14 Apr 2021 field tests of general relativity. The data collected by ity and modified theories of gravity. Despite these facts, the Event Horizon Telescope [3] allowed for new tests the non-linear dynamics of charged binary black holes is of gravity around supermassive black holes, and the ob- uncharted territory, with [20–24] being the main works servation of gravitational waves by the LIGO-Virgo col- on the subject. laboration [4] enabled the first constraints on deviations Studying charged black holes in Einstein-Maxwell the- from Einstein’s theory in the highly dynamical strong- ory not only provides a way to capture some features field regime (see, e.g. [5–8]). In turn, this latter achieve- of specific modified theories in a controlled environment, ment was made possible by advancements in the field of but also simulations of such systems have direct astro- numerical relativity that has been able to produce the physical and fundamental physics applications. First, accurate gravitational-wave models that are needed to while black holes are expected to be electrically neu- tral [25–29], there is no definitive observational sup-

∗ [email protected] 1 † [email protected] We note that this is not gravitational dipole radiation. 2 port for this expectation. Therefore, this assumption binary black holes. In particular, we discuss how Kreiss- must be tested. Gravitational-wave observations offer Oliger dissipation [68] helps (or impedes) these evolu- a model-independent way to test this assumption. Sec- tions. We also describe the formalism that we adopt ond, “charge” is an umbrella term that applies to differ- for our evolutions detailing some features that have not ent models in exotic astrophysics and beyond-standard- been included in previous works (e.g., the computation of model physics, including dark matter with hidden charge the angular momentum carried away by electromagnetic and interacting with dark electromagnetism (e.g. [30–36] waves with the Newman-Penrose formalism, and the con- or dark matter with fractional charge [37–46]), as well as tribution of electromagnetic fields to the quasi-local spin modified theories of gravity with additional vector fields of a black hole). Using the gravitational waveforms gen- sourced by a “gravitational” charge [19]. Furthermore, erated by our simulations, we explore black hole charge through a duality transformation, “charge” can also be detectability by future ground- and space-based gravi- interpreted as magnetic charge. Recently, black holes tational wave detectors. The second thrust of this work with magnetic charge in astrophysics have received some consists of analyzing existing approximate models for the attention (e.g. [47–51]), with focus on primordial black inspiral of charged black hole binaries and their remnant holes in [47, 48]. Gravitational waves can be used to di- black holes, and comparing them with our non-linear so- rectly test whether black holes have any kind of charge. lutions. This was the goal of our previous paper on the sub- In [2], we presented the first simulations of the quasi- ject [2], where we found that charge-to-mass ratios of circular inspiral and merger of charged black holes in up to λ = 0.32 are compatible with GW150914 [52], as- full general relativity with valid initial data. Here, we suming that the role of black hole spins can be neglected. present more details about these computations. We fo- Upper bounds on black hole charge can be translated to cus on systems with mass ratio q = 29/36, as inferred constraints on the properties of the dark matter particles for GW150914 [52, 69], and restrict the charge-to-mass in the aforementioned models or on the parameters of ratio λ of the individual black holes to values |λ| ≤ 0.3. modified gravity theories [2]. The mass ratio is close to unity, so we expect that the Numerical-relativity simulations of charged black holes conclusions presented in this work will hold for equal- can be used to produce gravitational-wave templates mass binaries. We consider three systems: (1) binary that include charge (for example, by hybridizing analyt- black holes with same charge-to-mass ratio in magnitude ical waveforms with numerical ones, as done in [53–58]). and sign, (2) binary black holes with charge-to-mass ra- Since the detection of gravitational waves by the LIGO- tio equal in magnitude but oppositely charged, and (3) Virgo interferometers relies heavily on matched-filtering binary black holes in which only the primary is charged. techniques [59–61], extended gravitational-wave template In this first exploration, we do not study cases where the banks that encode additional physics are necessary for black holes have different (non-zero) charge-to-mass ratio the parameter estimation [55, 62–65]. A phenomenologi- λ. cal model based on numerical relativity with charge can For the comparison of our solutions with existing also be used in Bayesian analyses to directly constrain approximations for the inspiral phase, we consider a this parameter in LIGO-Virgo signals. model that is based on Newtonian physics coupled with Finally, charge provides a way for a black hole to reach the quadrupole formula to incorporate radiation reac- extremality (along with the spin). Thus, non-linear stud- tion. We will refer to this model with the letters “QA” ies of charged binary black holes offer new pathways to (quadrupole approximation). Given its simplicity, this investigate cosmic censorship in conditions where it has model has been routinely used to study the merger of never been probed before. For instance, it would be charged black holes (e.g. [36, 44, 47–49, 70, 71]). Prior of interest to tackle the question: “can black holes be work on head-on collisions of charged black hole reported overcharged?”, going beyond previous perturbative ap- good agreement in some quantities between these approx- proaches [66, 67]. imate calculations and full non-linear simulations [20, 21]. Therefore, a goal of this work is to determine the errors of the Newtonian approximation when applied to the quasi- B. Goals of this work circular inspiral of charged black holes. A key result of our study is that Newtonian models can be successfully In this paper, we continue our explorations of the non- applied to obtain order-of-magnitude estimates of observ- linear interaction of charged black holes by approaching ables or to build intuition, but they cannot be used for the problem of inspirals and mergers on two different precision studies of these mergers. thrusts. On one side, we present necessary ingredients A second focus of this paper is on the properties of for performing long-term and stable numerical relativ- the post-merger black hole and its quasi-normal modes. ity simulations of the quasi-circular inspiral of charged This is especially relevant for LISA, which will detect the ringdown signal arising from the merger of super- massive black holes with high signal-to-noise ratio [72]. The quasi-normal modes can be used to test general rel- 2 We are using geometrized units. See Sec.IC ativity [73], as their characteristic frequencies depend on 3 the spin, mass, and charge of the remnant black hole in II. METHODS AND FORMALISM a known way. Here, we consider the method described in [74] to estimate the remnant black hole spin using con- In this section, we describe the methods we adopt for servation arguments, and we compare it with our non- solving the full non-linear Einstein-Maxwell equations linear solutions. As we discuss later, we find that the (Sec.IIA). We discuss our approach to building quasi- quasi-normal-mode properties do not change much across circular initial data (Sec.IIB) and how to achieve long- the simulations considered here. This is due to the value term, stable evolutions (Sec.IIC). We also describe the of the final spin and to the relatively weak dependence formalism and implementation of two new features that of the quasi-normal-modes on the charge for the values have not appeared in previous simulations: the contri- of λ we consider. bution of the electromagnetic fields to black-hole horizon properties (Sec.IID), and the angular momentum car- ried away by electromagnetic waves (Sec.IIE).

C. Structure of the paper and conventions A. Equations and numerical setup

The structure of the remainder of the paper is the fol- In this paper, we study systems described by the lowing. In Sec.II, we describe the formalism that we source-free Einstein-Maxwell equations [77] (electrovac- adopt to perform simulations of the quasi-circular inspi- uum4) ral and merger of charged black holes. In particular, 1 we discuss how to obtain stable quasi-circular inspirals, R − g R = 8πT EM , (1a) and highlight some new features of the approach. Next ab 2 ab ab ab (Sec.III), we outline the simplest (Newtonian) model ∇aF = 0 , (1b) for the quasi-circular inspiral of non-spinning, charged ∇ ?F ab = 0 , (1c) binary black holes. Sec.IV describes what we can say a about the remnant black hole using results of relativistic where R is the Ricci tensor associated with the metric calculations and perturbation theory. In Sec.V, we follow ab g , R = Ra , F = 2 A is the Maxwell field-strength the black holes through their coalescence: first (Sec.VA) ab a ab [a,b] tensor, with A the electromagnetic four-vector poten- we study the inspiral and compare the non-linear solution a tial, and ?F is its Hodge dual, defined by with the Newtonian model, then (Sec.VB) we discuss re- ab sults from the full simulations, and finally (Sec.VC) we 1 ?F ab = abcd F , (2) report on the properties of the remnant black hole. Con- 2 cd clusions and future directions are collected in Sec.VI. abcd We adopt geometrized units with G = c = 1, with with  being the Levi-Civita tensor. The electromag- G being Newton’s constant and c the speed of light in netic stress-energy tensor is given by vacuum. We also adopt Gaussian units for the electro- EM cd 1 cd magnetic sector. Similarly, we denote the different sim- 4πTab = FacFbdg − gabFcdF . (3) ulations adding a superscript and a subscript when we 4 + + report physical quantities. For example, e+, e−, and We solve the coupled Einstein-Maxwell equations in a + e 0 indicate the eccentricity measured in the evolutions 3+1 decomposition of the spacetime (for more details, see where black holes have charges with the same signs, with Sec. II A in [1], or textbooks on the subject, e.g. [79–81]) opposite signs, and only one charged black hole, respec- and use the Einstein Toolkit [82–84] for the numerical tively.3 In geometrized units, quantities have units of integration. length. Here we report all the results in units in units The initial data are generated by of the Arnowitt-Deser-Misner (ADM) mass of the system TwoChargedPunctures, which solves the Hamil- M [75]. We use the letters a, b, c, d for spacetime indices, tonian constraint equation using a Bowen-York and i, j, k for spatial ones. For everything else, we follow approach [85, 86]. We developed and tested the same conventions as in [76]. TwoChargedPunctures in [1], starting from the widely used TwoPunctures code [87]. TwoChargedPunctures takes as input the locations, charges, bare masses, angular, and linear momenta of each of the two black

3 For figures reporting different charge-to-mass ratios, we use a consistent style: red dashed lines with circles are for systems with both black holes charged with the same sign, blue dotted 4 In all our discussion, we assume that the black holes are in vac- lines with squares are for oppositely charged ones, and green dash uum (see [78] for a discussion on the role of the environment). dotted lines with triangles are for evolutions in which only one We also ignore Schwinger pair-production and any other quan- black hole is charged. tum effects. 4 holes, and it can build arbitrary configurations. We B. Controlling the eccentricity start our simulations at a coordinate distance of 12.1 M, with the two black holes having fixed charge-to-mass In this work, we focus on quasi-circular inspirals. ratio λ and mass-ratio 29/36. In this first study, we only When considering charged black holes, the effect of the explore systems with the two black holes having the electromagnetic fields must be taken into account to same λ (up to the sign), or with only one charged black achieve a low-eccentricity coalescence. Here, we describe hole. In Sec.IIB, we discuss how to choose the initial a simple method to incorporate the effect of charge. This data parameters to achieve quasi-circular inspirals. approach successfully yields quasi-circular inspirals for the values of λ explored in this study. The evolution of the spacetime is performed with First, it is useful to summarize how quasi-circular in- the Lean code [88], which implements the Baumgarte- spirals are obtained in the case without charge. The sim- Shapiro-Shibata-Nakamura (BSSN) formulation of Ein- plest way is to start from Newtonian physics. Consider stein’s equations [89, 90]. Lean evolves the conformal two point particles with mass m1, m2, and assume that −1/6 factor χconf = γ , with γ determinant of the 3- they are in a circular Keplerian orbit. The orbital angular metric. We adopt as spacetime gauges the 1 + log velocity Ω of each particle is (restoring the gravitational and the Γ − driver conditions [91, 92]. The electro- constant G) magnetic fields are evolved with the massless version of r the ProcaEvolve [23] code. The code evolves the elec- G(m1 + m2) i Ω = , (4) tric field E , the scalar and vector potentials φ and Ai, d3 along with an auxiliary variable Z that is used to control the Maxwell constraints. The precise equations solved where mi is the mass of the i-th component, and d the or- by ProcaEvolve are presented in the Appendix of [23]. bital separation. If we denote the mass-ratio q = m1/m2, These codes are publicly available as part of the Canuda the linear velocity of two particles becomes suite [93, 94] and have been extensively tested and used Ωd Ωd throughout the years. We use sixth-order accurate finite v1 = and v2 = q . (5) differences for the spatial derivative, and we integrate in 1 + q 1 + q time with a fourth-order Runge-Kutta method. In numerical integrations, black holes are assumed to be- have like these point masses, so pi = mivi is the ini- Apparent horizons are located using tial linear momentum assigned to the i-th black hole. AHFinderDirect [95, 96], and their physical properties Evolutions initialized with such Newtonian values have are measured with QuasiLocalMeasuresEM, a version significant residual eccentricity [100], hence high-order of QuasiLocalMeasures [97] updated to implement post-Newtonian (PN) expansions are used to compute the isolated horizon formalism in full Einstein-Maxwell more accurately the linear momenta necessary for quasi- theory (see Sec. II C in [1]). This extension is neces- circularity. When going beyond the Newtonian approxi- sary when considering black holes in the presence of mation, radial contribution to the velocities appear. electromagnetic fields, as we discuss in Sec.IID. Now, let us endow the point particles with charges q1 = λ1m1, and q2 = λ2m2 (with λ being the charge- We extract waves via the Newman-Penrose formalism to-mass ratio). In Newtonian physics, both electromag- as implemented in NPScalars Proca (see, Sec.IIE) and netism and gravity are central forces, so, from the point recover the gravitational-wave strain with a time integra- of view of the dynamics, this system is indistinguish- tion using the fixed frequency integration method [98]. able from one with uncharged bodies but gravitational We consider a finite extraction radius of 110.69 M. Re- constant Ge = (1 − λ1λ2)G. For this reason, one can sults are approximately invariant if we consider different incorporate the effect of charge by rescaling G to Ge. extraction radii or if we extrapolate the waves to infinity We can use this fact to achieve low-eccentricity inspi- with the method described in [65]. rals for charged black holes. First, we compute the lin- ear momenta needed for a quasi-circular coalescence of We work with Cartesian grids with Berger-Oliger adap- the black holes without charges using the highest order tive mesh refinement as provided by Carpet [99]. We use post-Newtonian expansion available (in our simulations two sets of nine nested refinement levels that are centered √we used 2.5PN). Then, we rescale these momenta by on and track the centroid of the black hole apparent hori- 1 − λ1λ2 to introduce the effect of electromagnetism. zons. The resolution of our simulations is M/65, where This simple method is effective at keeping eccentricity M is the total ADM mass, with additional resolutions under control for the values of charge-to-mass ratio ex- to perform convergence study (which we reported in [2]). plored here, as we show next. The outer boundary is at 1033 M and we performed se- Following [100], we estimate the residual eccentricity lected simulations to verify that the location does not by fitting the time derivative of the coordinate separation affect the evolution. Some more details on evolution and of the two black holes d˙ with a function of the form grid parameters used in our simulations are reported in ˙ AppendixB. d(t) = A0 + A1t + B sin (ωt + ϕ) . (6) 5

In Fig.1, we show the coordinate separation d and its aration and let gravitational and electromagnetic waves derivative d˙ for the most extreme simulations that we are circularize the orbit. considering here. The scale of the amplitude of the oscil- lations in the bottom panel already provides an idea of the (small) amount of residual eccentricity. The very be- C. Achieving long-term stable evolutions ginning of the time series is noisy due to the relaxation of the initial data, so we exclude that part of the simulation. Performing long-term, stable evolutions of black holes We perform the fit with the Levenberg-Marquardt algo- in vacuum in 3+1 dimensions used to be the greatest rithm [101, 102] for non-linear least-square fitting as im- challenge in numerical relativity. Through substantial plemented in MINPACK. As in [100], we find that fits are developments over the past two decades, solving the Ein- not perfect, so the eccentricities reported should be con- stein equations in vacuum is considered a solved prob- sidered as estimates. In general, we find that the first or- lem, and there is considerable knowledge on the topic. bit is the one with the most eccentricity. Fitting the first However, simulations of black holes with electromagnetic + + + three orbits we find e+ ≈ 0.01, e− ≈ 0.02, and e 0 ≈ 0.01. fields are still in their infancy. It is not yet clear how The eccentricity is significantly reduced if we consider much of the technology developed for vacuum spacetimes + −3 the next three orbits after the first: e+ ≈ 2 × 10 , carries over to electrovacuum spacetimes. Being able to + −3 + −3 e− ≈ 4 × 10 , and e 0 ≈ 1 × 10 . perform long-term and stable evolutions of charged black holes is therefore not granted. Indeed, the simulations presented in this work are among the longest and most 12 sophisticated to date, and presented some challenges. We

] 11 found that adding artificial dissipation to all evolved vari- M

[ + ables in a specific way is critical for successful simula- λ+ = 0.3 d 10 + tions. We found that standard recipes for artificial dissi- λ = 0.3 − pation that work in the case of vacuum binary black hole 9 λ+ = 0.3 0 spacetimes lead to blowups in the case of electrovacuum 5 binary black holes. Artificial dissipation stabilizes evolutions by remov- ] 3

− 0 ing high-frequency unstable modes. This approach has

[10 proven necessary to achieve long-term simulations in ˙ d many cases (see, e.g. [105–111]). The most common fla- 5 − vor of artificial dissipation adopted in numerical relativ- ity is known as Kreiss-Oliger dissipation [68]. This tech- 0 200 400 600 800 1,000 nique consists of introducing artificial diffusion by adding t [M] a term to the evolution equation of a variable U as follows (schematically) FIG. 1. First few orbits for the simulations with charge-to-  ∂ U = ··· + (−1)(p+3)/2 ∆xp∂p+1U, (7) mass ratio of λ = 0.3. Top panel: (coordinate) separation t 2p+1 x d of the centroid of the two black holes as a function of the where ··· indicate the right-hand-side of the evolution (coordinate) time. Bottom panel: time derivative of d, this quantity can be used to estimate the residual eccentricity with equation of U, p is the order of the Kreiss-Oliger dissi- Eq. (6). The initial spike is due to the initial data relaxation, pation, ∆x is the grid spacing and the numerical fac- p p+1 and is not included in our analyses. Note that these quantities tor ∆x /2 is the diffusivity, which represents the are not gauge-invariant. strength of the dissipation. Note that although we only add a spatial derivative in the x direction in Eq. (7), the These values of eccentricity found are remarkably small actual operator has also corresponding y and z deriva- considering the simplicity of the method employed. Sim- tives. For simplicity of the presentation we do not write ulations with higher charge-to-mass ratio, especially in these extra terms here. In the infinite resolution limit, the case of opposite charges, may have a significant resid- this new term vanishes and the equations are the ones ual eccentricity. There are at least three ways to improve we started with. However, at finite resolution, it is im- our method and further remove the eccentricity. portant to ensure that the modification does not affect First, one can adapt iterative eccentricity-reduction the convergence order of the solution. Thus, p is typi- schemes, such as those described in [100, 103, 104] for cally chosen to be greater than the convergence order of uncharged black holes, to systems with charge. Alterna- the evolution operator. Moreover, p has to be odd, so tively, one can use post-Newtonian methods that include that this modification is an even-order parabolic opera- directly electromagnetic fields to estimate the linear mo- tor (since p+1 is even). In AppendixA, we present some menta. The two methods are not mutually exclusive, details on conditions that must be satisfied for numerical and for simulations with extreme charge both may be re- stability. quired to produce quasi-circular inspirals. Finally, one As in previous works, our simulations quickly crash can always start the evolution from a larger initial sep- (in the first 100 M) if we do not add artificial diffusion. 6

Since our evolutions are sixth-order accurate, we add a that there is unstable growth in some variables.5 We ver- seventh-order (p = 7) dissipation to all evolved variables ified that this instability is numerical and not physical, (i.e., including the spacetime and electromagnetic fields since its onset depends on the resolution of the simula- and the gauge variables). The strength of the dissipa- tion: the higher the resolution, the earlier the instability tion is determined by the coefficient  (see, AppendixA takes place. This behavior was not reported in previ- for details). We explored three prescriptions for setting  ous studies without charge, suggesting that the instabil- on different refinement levels: (1) constant , (2) propor- ity first arises in the electromagnetic sector, and then tional to the local Courant factor ∆t/∆x, and (3) “con- feeds the gravitational one. Indeed, we observe the nu- tinuous”  (described below). The rationale behind (1) is merical instability starts first in electromagnetic quanti- to have a form that respects the local Courant stability ties, such as the Gauss constraint. We also found that condition independently of ∆x as long as the Courant the unstable growth of the constraint first occurs near factor is the same everywhere. It also has increased ef- the outer boundary, suggesting that the approximate fective dissipation in regions with coarser resolution. Nu- outgoing-wave boundary conditions may be the trigger merical stability requirements often require that different of the instability. Finally, we noticed that a fourth-order refinement levels be evolved with different Courant fac- accurate finite-difference evolution (with fifth order dis- tors, so prescription (2) amends (1) by modifying  when sipation) and constant  does not lead to the same nu- the Courant factor is changed to ensure that Courant merical explosions at the same resolution and within the stability conditions are met (see AppendixA). While simulation times we considered. What is important to (1) and (2) are commonly used in numerical relativity, note regarding the goals of this paper is that prescription (3) is introduced in this paper and consists of setting (3) allows us to perform long evolutions with constraints −p i = n(∆xi/∆xn) , where i = 1, 2, . . . , n is the index converging to zero when the resolution is increased when indicating the refinement level (larger i indicating finer adopting sixth-order accurate finite differences. Study- level), with n the maximum number of refinement levels, ing the interplay between dissipation, convergence order and ∆xi the grid spacing of refinement level i. The above of the numerical scheme, and boundary conditions is left i−n p prescription can also be written as i = 0/(2 ) , since for future works. refinement level grid spacings differ by factors of 2, i.e., 6 ∆xi−1 = 2∆xi. Thus, the diffusivity entering Eq. (7) for 10− 2

−p p p+1 p k each level becomes n(∆xi/∆xn) ∆xi /2 = n∆xn/ 2p+1. In other words, this new prescription guarantees that the effective diffusivity is the same everywhere on the grid, ensuring that the same parabolic diffusion op- 7 10− erator is added to the set of equations on each refinement level. Moreover, the artificial diffusion again goes to 0 at order p, and thus does not affect the expected order of  = constant convergence of the finite difference scheme. We call this 8  ∆t/∆x continuous, because prescriptions (1) and (2) have jumps Hamiltonian constraint 10− ∝ k  continous in the effective diffusivity across refinement levels that 0 500 1,000 1,500 2,000 2,500 introduce discontinuities in the equations. Hence, the parabolic operator added to the equations depends on t [M] the refinement level, which results in effectively solving a different system of partial differential equations on differ- FIG. 2. Evolution of the L2 norm of the Hamiltonian con- ent refinement levels. This difference vanishes in the limit straint (outside the horizon) for the three different Kreiss- of infinite resolution. To our knowledge our approach (3) Oliger dissipation prescriptions. The “continuous” dissipa- for setting the Kreiss-Oliger diffusivity on adaptive-mesh- tion prescription introduced in this work is the only one that refinement grids has not been discussed before. We found allows long-term and stable simulations with seventh-order that this is a crucial ingredient for long-term and stable Kreiss-Oliger dissipation and sixth-order accurate finite dif- ferences. numerical evolutions of charged black holes.

For completeness, in AppendixC, we report two for- To demonstrate the performance of each of the three mulations that we experimented with to further improve prescriptions for setting , we consider a high-resolution simulation of a single charged, non-spinning black hole with mass M = 1 and charge Q = 0.5 M. In Fig.2, we show the L2 norm of the violation of the Hamilto- 5 nian constraint (excluding the domain interior to the ap- The jumps in the constraints at the beginning of the simulations are due to an initial pulse in the gauge variables propagating parent horizon) for each dissipation prescription. More outwards from the center of the simulation. Every time this details about our numerical setup are provided in Ap- pulse crosses a refinement boundary, it is turned in constraint- pendixB. The figure is representative of how our binary violating modes through refinement level interpolations. This charged black hole simulations evolve, and demonstrates behavior in the constraints is well-known [112]. 7 the stability of the evolutions but did not result in addi- ++ 2 tional improvements. 10− + − +0 final

/J 4 10− 2 final EM λ J final ∝ final /J 6 JEM D. Contribution of the electromagnetic fields to 10− the spin of a black hole 0.01 0.05 0.1 0.2 0.3 λ (charge-to-mass ratio) The quasi-isolated horizon formalism is widely used in numerical relativity to determine quasi-local physical FIG. 3. Contribution of the electromagnetic fields to the spin properties (such as mass or angular momentum) of black of the final black hole. The relative importance of the spin holes [113, 114]. However, the majority of previous simu- due to electromagnetic fields JEM scales as the charge-to-mass lations focused on the case with gravitational fields only, ratio squared. For details of the formalism, see Sec. II C in [1]. and hence, electromagnetic contributions to black hole quasi-local properties have not been considered. In [1], 0.6 we presented the code QuasiLocalMeasuresEM, an ex-

tended version of QuasiLocalMeasures that implements ] Q = 0.6 M 2 0.4 the formalism for the full Einstein-Maxwell theory [115]. M J The main differences with respect to the purely gravita- [ J JEM 0.2 JEM/J 11 % tional case are in the computation of the mass and the JGR ≈ angular momentum of a black hole. The full formalism also provides a quasi-local way to compute the charge 0.0 0 5 10 15 20 25 30 of a black hole. Here, we focus on spin (for a complete discussion, see Sec. II C in [1]). t [M]

While the expression for angular momentum at infin- FIG. 4. Various contributions to the angular momentum ity does not depend on electromagnetic fields [116], the of a rapidly spinning and highly charged black hole with quasi-local formula does [115]. So, to compute the an- J = 0.6 M 2 and Q = 0.6 M. As a result of the choice of the gular momentum J of a charged black hole, one must initial electromagnetic fields, TwoChargedPunctures [1] gen- calculate two terms: JGR and JEM (see, Sec. II C erates initial data with JEM = 0. During the evolution, the in [1] for more details). The computation of JGR is initial data relax to a non-zero JEM while J remains con- implemented in the QuasiLocalMeasures thorn of the served. Einstein Toolkit. QuasiLocalMeasuresEM computes JEM as well. We find that this contribution can be a E. Wave extraction significant fraction on the total spin. In Fig.3, we show the ratio between JEM and the total angular momentum of the remnant black hole forming in our binary simula- We adopt the Newman-Penrose formalism [117] with final final the same conventions as in [23] to extract gravitational tions JEM /J for the various simulations in our set. The maximum value in our simulations is with λ+ = 0.3, and electromagnetic waves from our simulations. In this + subsection, we review the formalism with focus on the where we find that J final/J final ≈ 3.8 %. Fig.3 shows EM electromagnetic sector. The Newman-Penrose formalism that this ratio depends quadratically on the charge-to- of the gravitational sector is a standard topic in the lit- mass ratio of the final black hole. A fit confirms quanti- erature (e.g. [81, 118, 119]). Details on the numerical tatively that this is the case. implementation of the calculation of the various quanti- TwoChargedPunctures always generates (binary) ties presented below can be found in [23] and [120]. a a a black hole initial data with JEM = 0, because of the We consider a null tetrad of complex vectors k , l , m a choice of the initial electromagnetic fields [1]. Soon af- and m∗ with null mutual inner products except for a a ∗ ter the evolution starts, the initial data relax to a non- −k la = 1 = m ma. We also define the Newman-Penrose zero JEM on a few light-crossing times while keeping the scalars constructed with the Faraday tensor [121] total angular momentum constant. In Fig.4, we show a b how the two contributions to the spin behave for the Φ0 = Fabl m , (8a) case of a highly charged, rapidly spinning black hole with 1 a Φ = F (lakb + m∗ mb) , (8b) Q = 0.6 M and J = 0.6 M 2. In this case, after the initial 1 2 ab final final ∗a b data relax JEM /J is approximately 11 %. Monitor- Φ2 = Fabm k . (8c) ing this quantity also provides a way to quantify when the initial data has relaxed. AppendixD connects Φ 0,Φ1 and Φ2 with the electric and 8 magnetic fields in flat spacetime and provides physical infinity is given by arguments to build intuition on what the different scalars 2 represent. d E 2 r = lim r T t , (14a) In asymptotically flat spacetimes, the Newman- dt dΩ r→+∞ 2 z Penrose scalars have a known fall-off behavior at large d L 2 r = lim r T ϕ . (14b) spatial distances (the so-called “peeling theorem” [122]) dt dΩ r→+∞ 1 1 1 Here we focus on the z component of the angular mo- Φ ∼ , Φ ∼ , Φ ∼ , (9) 0 r3 1 r2 2 r mentum. These quantities can be expressed in terms of the Newman-Penrose scalars by use of Eqs. (9) and (12). where the ∼ indicates the asymptotic behavior as r → The only non-zero contribution to the energy flux arises ∞. We will use these properties to find which terms are 2 from the term |Φ2| ltlr/2π. Hence, important for computations at infinity and which are not. 2 2 By use of Eqs. (8), we can express the electromagnetic d E 2 r r 2 = lim r T t = lim |Φ2| . (15) field tensor as dt dΩ r→+∞ r→+∞ 2π h i ∗ ∗ Considering the fall-off behavior of the Newman-Penrose Fab = 2 Φ1(k[alb] + m[amb]) + Φ2l[amb] + Φ0m[akb] scalars in Eqs. (9), the angular momentum flux becomes + complex conjugate , (10)

2 where “complex conjugate” refers to the term in square 2 r r ∗ ∗ ∗
lim r T ϕ = lim − Φ1Φ2lrmϕ + Φ1Φ2lrmϕ , brackets, and square brackets next to indices imply anti- r→+∞ r→+∞ 2π symmetrization. Considering the electromagnetic stress- (16) energy tensor which can be rewritten as r2 EM cd 1 cd 2 r ∗ ∗ 4πT = F F g − g F F , (11) lim r T ϕ = lim − (Φ1Φ2ir sin θ − Φ1Φ2ir sin θ) ab ac bd 4 ab cd r→+∞ r→+∞ 4π 3 ir sin θ ∗ ∗ we can express this in terms of the Newman-Penrose = lim − (Φ1Φ2 − Φ1Φ2) r→+∞ 4π scalars [121] r3 sin θ=[Φ∗Φ ] = lim 1 2 ,  r→+∞ EM ∗ ∗  ∗  2π 4πTab = Φ0Φ0kakb + 2Φ1Φ1 l(akb) + m(amb) where =[z] is the imaginary part of the complex number ∗ ∗ + Φ2Φ2lalb − 4Φ0Φ1k(amb) (12) z. Therefore, the flux of angular momentum that crosses  a sphere at large radius r is ∗ ∗ − 4Φ1Φ2l(amb) + 2Φ2Φ0mamb d2Lz 1 = r3 sin θ=[Φ Φ∗] . (17) + complex conjugate , dt dΩ 2π 1 2 where the complex conjugate is of the term in square Here we reach the same conclusion as in [123]: the flux of brackets. angular momentum does not depend only on the radia- Assuming asymptotically spherical coordinates tive degrees of freedom (encoded in Φ2), but there is also (r, θ, ϕ) centered on the Cartesian grid and oriented a Coulombic contribution (encoded in Φ1)[124]. This is along the z direction and t along the normal to the a striking difference compared to gravitational waves, for hypersurfaces, we can choose as null tetrad which the information contained in the radiative degrees of freedom, i.e. Ψ4, is sufficient for computing the flux of 1 ka = √ (ea − ea) , (13a) angular momentum [125]. 2 tˆ rˆ We perform a decomposition in spin-weighted spherical s a 1 a a harmonics Y`m with spin s = −2, −1, 0 l = √ (etˆ + erˆ) , (13b) 2 X X `m 0 Φ1(r, θ, ϕ) = φ1 Y`m , (18a) a 1 a a m = √ (e + ie ) , (13c) `≥0 −`≤m≤` 2 θˆ ϕˆ X X `m −1 1 Φ2(r, θ, ϕ) = φ Y , (18b) m∗a = √ (ea − iea ) , (13d) 2 `m 2 θˆ ϕˆ `≥1 −`≤m≤` X X Ψ (r, θ, ϕ) = ψ`mY −2 . (18c) with e , e , e , e orthonormal non-coordinate basis. 4 4 `m tˆ rˆ θˆ φˆ `≥2 −`≤m≤` Note that in the coordinate basis, it holds that ma ∼ r. The energy and angular momentum fluxes per solid Using the orthogonality relations between spin-weighted angle carried away by outgoing electromagnetic waves at spherical harmonics with the same spin, we can write [80] 9

is given by [47]

2 dip dip r dEEM r X `m 2 dJ dE  M = lim |φ2 | , (19a) EM EM dt r→+∞ 4π = (1 − λ1λ2) 3 , (22a) `,m dt dt d 2 Z t 2 quad quad r dEGW r X 0 `m dJGW dEGW M = lim dt ψ4 , (19b) = (1 − λ1λ2) . (22b) dt r→+∞ 16π dt dt d3 `,m −∞ 2 "Z t  Note that the denominator is the orbital angular velocity dLGW r X 0 `m = lim m= dt ψ4 (19c) of the binary. dt r→+∞ 16π `,m −∞ We can derive the equation of motion by taking the Z t Z t0 !# derivative of Eq. (20) and applying the chain rule × dt0 dt00ψ`,m∗ . 4 dE m m dd −∞ −∞ = (1 − λ λ ) 1 2 . (23) dt 1 2 2d2 dt The sums on m go from −l to l. In practice, we truncate the expansion to l = 8 and we use a finite extraction Therefore, radius. There is no simple relation between the angular dd 2d2 dE momentum lost by electromagnetic waves and the multi- = . (24) `m `m dt (1 − λ1λ2)m1m2 dt polar components φ2 and φ1 when decomposed with the respective spin-weight, so we implemented directly To respect energy conservation, dE/dt is given by the Eq. (17). We tested this new diagnostic for the flux of total energy loss via electromagnetic and gravitational angular momentum against the analytical Michel solu- radiation provided by Eqs. (21a), (21b), (21c). The re- tion [126, 127] (see, AppendixE). sulting equation of motion (24) cannot be solved ana- lytically in closed form for a non-zero charge. Here, we solve it numerically with the LSODA solver [128] of ODE- III. QUADRUPOLE APPROXIMATION MODEL PACK [129] through the SciPy interface [130]. The time integration is performed with a timestep that is propor- The simplest waveform model for charged binary black tional to the orbital separation. We continue the inte- holes is obtained from the Keplerian motion of charged gration up to d = 5 M, which is an average radius of the massive particles in Newtonian physics with the inclu- Innermost Stable Circular Orbit (ISCO) for neutral par- sion of radiation reaction. We indicate this model with ticles around Kerr-Newman black holes (more on this in the initials QA (quadrupole approximation). We will also Sec.IVA). refer to it as the “Newtonian model”. The solution of Eq. (24) provides the time evolution Let us consider two point particles with masses m1, of the orbital separation. Then, using Eqs. (21), (22), m2 and charges q1, q2 on a circular orbit at separation d. we can compute the energy and angular momentum lost The total energy (kinetic + gravitational + electrostatic) by gravitational and electromagnetic waves. We can also of the system E is compute the gravitational-wave strain in the quadrupole approximation [70, 131] measured at distance r from the m1m2 q1q2 m1m2 E = − − = −(1 − λ1λ2) , (20) binary 2d 2d 2d 5 m1m2 2 3 3 where λi = qi/mi is the charge-to-mass ratio of the i-th rh+ = 4(1 − λ1λ2) 1 (πfGW) cos φGW , (25a) particle. M 3 5 m1m2 2 3 3 The system loses energy by emission of gravitational rh× = 4(1 − λ1λ2) 1 (πfGW) sin φGW . (25b) and electromagnetic waves. In this work, we will re- M 3 strict to circular orbits and only consider dipole and The frequency of the gravitational waves fGW is twice quadrupole electromagnetic waves, and quadrupole grav- the orbital frequency, itational waves [36, 47] r 1 (1 − λ1λ2)M dip 2 2 f (d) = . (26) dE 2 m m GW 3 EM = (λ − λ )2(1 − λ λ )2 1 2 , (21a) π d dt 3 1 2 1 2 d4 quad 2 2 Defining ωGW(d) = 2πfGW(d), the phase of the gravita- dE 32 3 m m M GW = (1 − λ λ ) 1 2 , (21b) tional waves φGW is dt 5 1 2 d5 quad 2 quad Z d   ωGW(R) dEEM m2λ1 m1λ2 dEGW φ = φ + dR, (27) = + . (21c) GW 0 ˙ dt 2M 2M dt d0 d(R) ˙ This can be extended to eccentric orbits [47]. The angu- where φ0 is an arbitrary initial phase and d(R) is the time lar momentum carried away by gravitational and electro- derivative of d evaluated at orbital separation R, which magnetic waves in the dipole and quadrupolar channels can be computed from Eq. (24). 10

Finally, we note that the computational requirements at which point the two black holes plunge. During the of the QA model are negligible compared to numerical- plunge, little angular momentum is lost, so one can esti- relativity simulations (as Eq. (24) is a single ordinary mate the spin of the remnant black hole by studying the differential equation). In certain limits, the equations ISCO. can be even solved analytically [36]. Consider the merger of two non-spinning black holes with mass m1, m2 and charge q1, q2. The total mass and charge of the system are IV. THE REMNANT BLACK HOLE

M = m1 + m2 ,Q = q1 + q2 . (28) The properties of the remnant black hole that forms following a binary black hole merger are key to test- Given that the energy lost via gravitational waves is of ing general relativity [73, 132, 133]. Perturbation the- order of a few percent of M (smaller than the target ory applies to the post-merger black hole, and it is well accuracy of 10 %), the BKL method assumes that the established that perturbed black holes settle by under- total mass is conserved. Therefore, the final black hole going characteristic damped oscillations. This is known has mass M. Attempts to include energy loss in the un- as quasi-normal-mode ringing. In general relativity, the charged case were made [138], but we will not consider complex frequency of these oscillations is completely de- this here. Charge is exactly conserved, so the final black termined by the black hole mass, spin, and charge. hole must have charge Q. To determine the remnant In this section, we first present the details of an existing black hole spin parameter a, one invokes angular mo- method to estimate the spin of the remnant black hole mentum conservation. The BKL model postulates that forming following mergers of charged black holes. Next, the final angular momentum Ma is exactly the same as we discuss quasi-normal-modes. The two discussions are the orbital angular momentum at the onset of the plunge. closely related as the quasi-normal-modes depend on the This quantity is then estimated by considering the mo- properties of the black hole.6 tion in the spacetime of the remnant black hole of a test For the values of charge-to-mass ratios λ considered in particle with mass µ and charge q given by this work, the quasi-normal-modes are primarily deter- m m q q mined by the spin of the final black hole, because λ is not µ = 1 2 , q = 1 2 , (29) large enough to matter. However, the final spin depends M Q on the charge of the binary components. Even when the total charge is zero, e.g. the two black holes have equal where µ and q are the reduced mass and charge of the and opposite charges, the final spin is expected to be dif- binary system (namely, the mass and charge of the equiv- ferent from the uncharged case, allowing, in principle, to alent effective one-body problem). distinguish the two scenarios. Let LM,a,Q be the ISCO angular momentum of a test particle with mass µ and charge q in a Kerr-Newman spacetime with mass M, spin a, and charge Q, then the A. Remnant black hole spin model BKL approach sets

A simple way to estimate the spin of the remnant Ma = LM,a,Q(rISCO) . (30) black hole forming following the merger of two black holes on a quasi-circular orbit is to invoke conservation If l = LM,a,Q(rISCO)/µ is the test particle specific angu- arguments (this method is sometimes known with the lar momentum, then we have acronym BKL [137], from the names of the authors). The approach aims to be 10 % accurate [137], and while it a = νlM,a,Q(rISCO) , (31) was first applied to Kerr-Newman black holes in [74], the 2 performance of the approximation has not been tested where ν = µ/M = (m1m2)/M is the symmetric mass in the case of charged black hole inspirals. Therefore, it ratio. Equation (31) determines the spin a of the final is unknown if the accuracy goal is met for more generic black hole. It is straightforward to extend the method cases. Here we use our quasi-circular inspiral calculations to consider spinning binaries by adding the contribution to gauge the accuracy of the BKL approach. of the individual spins to the total angular momentum The basic assumption of the method is that during in Eq. (30), but we will not do this here, because our an inspiral the black-hole orbital separation shrinks due numerical relativity simulations of charged binaries do emission of gravitational waves until the ISCO is reached, not involve spin. To solve Eq. (31), we need to compute the specific angular momentum of the test particle with charge q in a 6 Another way in which the two discussions are linked is that the circular orbit at radius rISCO. We do this in the standard study of geodesic motion can be used to estimate the quasi- way by defining an effective radial potential Veff(r). normal-mode frequencies from the light-ring properties [134– In Boyer-Lindquist coordinates (t, r, θ, φ), the line el- 136]. ement ds2 of a Kerr-Newman black hole with mass M, 11 spin a, and charge Q is given by [139, 140] with l, m being the multipolar mode numbers and n the overtone number; A, τ, and ω are the characteristic am- ∆ − a2 sin2 θ ρ2 ds2 = − dt2 + dr2 + ρ2 dθ2 plitude, the decay time, and the frequencies of the quasi- ρ2 ∆ normal modes. These values depend on the mass, spin, (r2 + a2 − ∆) and charge of the black hole in a known way [73, 133]. In − 2a sin2 θ dt dφ (32) ρ2 this work, we are interested in exploring the charge in- formation contained in the ringdown waveforms. In par- (r2 + a2)2 − ∆a2 sin2 θ + sin2 θ dφ2 , ticular, we test whether it is possible to tell whether a ρ2 merging binary had charge by looking at the post-merger signal alone. with For Schwarzschild and Kerr black holes, the values of 2 2 2 2 ρ = r + a cos θ , (33a) ω`mn and τ`mn are tabulated [143, 144] or available in public codes, like the one we use here–qnm [145]. For ∆ = r2 − 2Mr + a2 + Q2 . (33b) generic Kerr-Newman solutions, while the problem has The electromagnetic vector potential is been solved [146], such tables are not publicly available. However, since the simulations in our set have relatively Qr A = − (dt − a sin2 θ dφ) . (34) small charge-to-mass ratio, we can work in the small ρ2 charge limit and we can use the formulas provided in [44] for the ` = 2, m = 2, n = 0 quasi-normal mode (which For Kerr-Newman black holes, V is given by [74, 141] eff typically dominates [147, 148]). For a Kerr-Newman ˜2 ˜ 2 black hole with charge-to-mass ratio λ and dimensionless gttl + 2gtφlε˜+ gφφε˜ − ∆ Veff = , (35) spin χ, the first correction to ω220 and τ220 with respect grr∆ λ=0 λ=0 to the uncharged values ω220 and τ220 is: with " # δω 0.3506 ˜l = l + qA , (36a) 220 2 φ λ=0 = λ −0.2812 − 0.0243χ + 0.505 , ω220 (1 − χ) ε˜ = ε − qAt , (36b) (39a) where ε and l are the specific energy and specific angular " δτ momentum, respectively. The properties of the ISCO are 220 2 2 λ=0 = −λ 0.1075 + 0.089 23χ + 0.023 14χ (39b) found solving the following equations simultaneously τ220 # V (r ) = 0 , (37a) 0.075 85 eff ISCO +0.094 43χ3 − , dV (1 − χ)1.2716 eff (r ) = 0 , (37b) dr ISCO λ=0 λ=0 d2V with δω220 = ω220 − ω220 and δτ220 = τ220 − τ220 . In eff λ=0 λ=0 (rISCO) = 0 . (37c) this work, we use qnm [145] to compute ω and τ . dr2 220 220 We can plug a representative remnant black hole spin for rISCO, ε(rISCO) and l(rISCO). value χ = 0.67 in Eq. (39) to gain some insight on the λ=0 2 For the estimate of the spin, we are particularly in- effect of charge, which yields δω220/ω220 ≈ λ /3, and λ=0 2 terested in l(rISCO). Once that is known, we use a root- δτ220/τ220 ≈ λ /10. For the values of λ treated in finding method to solve numerically Eq. (30) and find the our simulations, if we assumed that all black holes had spin of the final black hole. the same final mass and spin, then the deviations from the Kerr quasi-normal modes are at most at the percent level. The deviations are maximized for larger λ, so one B. Quasi-normal-modes + would expect that the simulation with λ+ is the easi- est to constrain. However, if one wanted to tackle the Following merger, the remnant black hole settles by un- question “Can we tell from the ringdown if the binary dergoing quasi-normal mode ringing, i.e., damped oscilla- was charged?” the problem is more complicated and one tions with specific frequencies ω and decay times τ [142].7 needs to consider the interplay between mass, spin, and During the ringdown phase, the Newman-Penrose scalar charge. We discuss this in Sec.VC2. Ψ4 looks like

X −t/τ`mn Ψ4(t, r) ∼ A`m(r)e sin(ω`mnt) , (38) V. SIMULATION RESULTS AND DISCUSSION `mn In this section, we describe the results from our black hole binary simulations through the inspiral, merger and 7 Here we are considering relatively small charge, so we will only ringdown phases. We start by analyzing the inspiral be focusing on the gravitational quasi-normal-modes. (Sec.VA). We compare the non-linear solutions with 12 the QA model, finding that the Newtonian approach al- We designate by tf the coordinate time at which the ways overestimates observable quantities by 20–100 %. gravitational-wave frequency is f, and using our non- Next, we present our numerical-relativity simulations up linear simulations we compute the error of the QA model to merger (Sec.VB), discussing properties of the emis- in gauge invariant quantities within the time intervals sion (Sec. VB1) and detectability of charge by future [tf0 , tf1 ] and [tf0 , tf2 ]. The quantities we consider are: gravitational wave observatories (Sec.VB2). Finally, we gravitational-wave phase, energy and angular momentum explore the ringdown phase (Sec.VC), testing the BKL lost through emission, number of gravitational-wave cy- approach (Sec.VC1) and discussing quasi-normal-modes cles NGW, and the signal-to-noise ratio (SNR) between (Sec.VC2). fmin and fmax (which are of particular interest to esti- mate charge detectability [44]). The latter two quantities are computed as (see e.g., [131, 149]) A. Inspiral Z fmax f N = GW df , (40a) GW ˙ GW Here, we focus on the inspiral part of our calculations fmin fGW and we assess the performance of the Newtonian model Z fmax ˜ 2 2 |h(f)| within the quadrupole approximation (QA). Since high- SNR = 4 df , (40b) Sn(f) order PN and/or effective-one-body waveforms including fmin electromagnetic fields are not yet available, this model is ˜ with h Fourier transform of the strain and Sn(f) the the most widely used to study inspirals of charged black power spectral noise density of the detector. In the rest holes (e.g. [36, 44, 47–49, 70, 71]). of the discussion, we focus on advanced LIGO at design To compare numerical-relativity simulations with the sensitivity. Newtonian model, we consider a set of quantities of in- terest. To keep our results gauge-independent, we an- QA alyze these quantities between reference gravitational- 0.2 NR wave frequencies f0, f1, and f2, which we choose as fol-

−3 −2 ] lows: Mf0 = 9.61 × 10 , Mf1 = 1.76 × 10 , and

−2 M Mf3 = 3.84 × 10 , where M is the detector-frame ADM [ 0.0 mass. The frequency Mf0 corresponds to approximately + rh 28 Hz for a binary with source-frame ADM mass 65 M (detector-frame mass of approximately 70.6 M ). These 0.2 + three frequencies are motivated by the LIGO sensitivity λ = 0.1 − − band with f0 corresponding to the onset of the latest 600 400 200 0 200 400 600 800 stage of the inspiral, f1 to the intermediate phase prior − − − t tf0 [M] to plunge, and f2 to the plunge. Previous studies used − the Newtonian model for LIGO-Virgo mergers, so our analysis here gauges how this approximate method per- FIG. 5. Plus polarization of the gravitational-wave strain pro- duced by oppositely charged binary black holes with λ+ = 0.1 forms and allows us to estimate the level of its accuracy. − Since our numerical-relativity simulations scale with the as extracted from the Newtonian (QA, blue, dash-dotted line) and full numerical-relativity simulation (NR, orange, solid total mass of the system M, we can target both stellar- line). The two waveforms are aligned a tf0 , which is when mass and supermassive black hole binaries. In TableI we 3 1 the gravitational wave frequency is f0 = 9.61 10− M − . show what frequencies correspond to our reference fre- The main difference is that the fully relativistic× simulations quencies f0, f1, f2 for different choices of M. The table predict a faster merger, as they include all non-linear terms. shows that this work is relevant to LIGO-Virgo as well as LISA sources [72]. In Sec. VB2, we discuss charge Before discussing the quantitative differences between detectability by LISA. the numerical relativity (NR) and QA models, we first provide a qualitative description. Figure5 shows the

3 plus polarization of the ` = 2, m = 2 mode obtained TABLE I. Reference frequencies Mf0 = 9.61 10− , Mf1 = 2 2 × with NR and the Newtonian model for a representative 1.76 10− , and Mf3 = 3.84 10− for different values of + the detector-frame× mass M. This× study targets both LIGO- charge black hole binary with λ− = 0.1. The two wave- Virgo and LISA sources. The choice of M = 70.6 M is in- forms shown in the figure are aligned at tf0 , when they spired by GW150914 [69]. The evolution from f0 to f1 is both have the same gravitational-wave frequency f0. As still not in the most relativistic part of the inspiral. On the the plot demonstrates, the QA model provides a decent other hand, f2 is reached in the latest stages of the merger, approximation to the NR signal up to t − tf0 ≈ 400 M. following which the black holes plunge. Shortly after that time, the black holes merge in the NR Freq. M = 30 M M = 70.6 M M = 104 M M = 107 M simulation. Merger is never captured in the QA model, f0 65 Hz 28 Hz 0.195 Hz 0.195 mHz and the Newtonian simulation is stopped when the sep- f1 120 Hz 51 Hz 0.357 Hz 0.357 mHz aration is of order of the ISCO (as in [70]). Also, there f2 260 Hz 111 Hz 0.780 Hz 0.780 mHz is substantial de-phasing before the frequency-alignment 13 time. A major difference between the two models is that the relativistic simulation predicts a faster merger, be- cause it includes all non-linear terms. As we will discuss later, this is the fundamental reason why the QA model B. Up to merger overestimates all the interesting physical quantities. Our simulations capture all non-linear effects that take 4 QA NR place during the late inspiral and merger. So, we can study interesting quantities that are not accessible with

] f2 1 approximate methods. In Fig.7, we show the coordinate − 3

M distance of the two black hole centroids as a function of 2

− the coordinate time for four representative NR simula-

[10 tions. This figure complements the top panel of Fig.1. 2 f1 It can be seen that our Newtonian expectations are met: GW

f the system that merges faster is the one with opposite f0 1 charge, due to the additional electrostatic attraction and λ+ = 0.1 − loss of energy due to dipole electromagnetic emission. 600 400 200 0 200 400 600 800 Next, we have the one with only one charged black hole, − − − due to additional loss of energy in the electromagnetic t tf0 [M] − emission. Finally, the system with black holes with the same charge is the last to merge, as it has to fight against FIG. 6. Gravitational-wave frequency evolution for the New- tonian model and for the fully general-relativistic simula- additional electrostatic repulsion. tion. The two simulations are aligned at tf0 , when the gravitational-wave frequency is f0. TableI reports the val- ues of f0, f1 and f2. 10

We further emphasize this point in Fig.6, where we ]

M 0 report the frequency of the gravitational waves in the [ λ 0 = 0

d 5 two models as a function of time. It is clear that the + λ+ = 0.2 frequency evolves faster in the non-linear calculation. As λ+ = 0.2 a result, the QA computations spend more time inspiral- +− λ 0 = 0.2 ing, so this model overestimates all relevant quantities, 0 500 1,000 1,500 2,000 as it overestimates the time from tf0 to tf1 and tf2 . This is exactly what happens at the quantitative level, too. t [M] TableII reports the relative error of the Newtonian cal- culations with respect to the NR simulations. For each FIG. 7. Coordinate distance of the two black holes as a func- quantity Υ, the error is computed as tion of the coordinate time for three representative cases with ΥQA − ΥNR charge-to-mass ratio of λ = 0.2, along with the case with no electromagnetic fields. Note that this is not a gauge- relative error = NR . (41) Υ independent plot. No absolute value is taken: a positive error means that the Newtonian approximation overestimates Υ. The An important first finding is that for a fixed binary ranges of error reported are across all NR simulations mass corresponding to GW150914, the SNR and the we performed for this work. As the values in the ta- number of in-band gravitational-wave cycles depend very ble demonstrate, the listed quantities are always overes- weakly on the charge: . 3 %. This is in spite of the differ- timated by order 20 % or more. However, we note that ent evolutions depicted in Fig.7. This result will likely the QA model always captures the correct order of mag- change if we consider low-mass binaries, which have a nitude in the amplitude up until a couple of cycles prior much longer inspiral that can be significantly affected by to peak gravitational wave amplitude in the non-linear the presence of charge earlier in the inspiral. calculations. Hence, for the values of λ considered here, the model can be used for rough estimates.8 1. Properties of the emission

8 In [150, 151] it was shown that when adopting the quadrupole In this subsection, we explore some properties of the formula to estimate gravitational waves, but with the fluid dis- electromagnetic and gravitational emission. We find that tribution computed based on general relativistic simulations, the errors are of order 20 % in the amplitude. Here, we show the per- in our simulations several quantities of interest scale with 2 formance of the quadrupole approximation when not coupled to λ . This scaling is expected to be exact in the limit where trajectories from numerical relativity simulations is significantly the charge has a negligible contribution to the dynamics, worse. and it will likely stop to be valid for larger values of λ, 14

TABLE II. Summary of relative errors for key quantities across all our simulations. The relative error for a quantity Υ is computed as (ΥQA ΥNR)/ΥNR, where ΥQA is the quantity in the quadrupole approximation and ΥNR in the simulations. The relative error is− with sign: positive error means that the Newtonian approximation overestimates the quantity. In all cases the quadrupole approximation overestimates the quantities. The values of the reference frequencies is reported in TableI. GW and EM indicate gravitational and electromagnetic waves respectively. When it comes to loss of angular momentum due to electromagnetic waves, the QA model only includes the dipolar channel, which is identically zero in the ++ case. Hence, we do not report this error for this quantity.

Relative error f0 f1 f0 f2 → → [(ΥQA ΥNR)/ΥNR] ++ + +0 ++ + +0 Time− 22 % 26 % 15 % −24 % 20 % 24 % 26 % 29 % 18 % −27 % 24 % 27 % Phase 20 % − 32 % 18 % − 30 % 17 % − 30 % 20 % − 32 % 18 % − 30 % 17 % − 30 % GW cycles 24 % − 26 % 15 % − 25 % 21 % − 25 % 24 % − 26 % 15 % − 25 % 21 % − 25 % GW energy 47 % − 48 % 34 % − 47 % 42 % − 47 % 56 % − 59 % 42 % − 56 % 52 % − 56 % EM energy 42 % − 44 % 59 % − 76 % 70 % − 76 % 47 % − 51 % 76 % − 99 % 91 % − 98 % GW angular momentum 45 % − 48 % 35 % − 45 % 41 % − 46 % 54 % − 57 % 42 % − 54 % 51 % − 55 % EM angular momentum − 57 % − 70 % 53 % − 58 % − 71 % − 87 % 65 % − 69 % GW150914-like SNR 13 % − 23 % 23 % − 26 % 21 % − 23 % 20 % − 28 % 65 % − 69 % 19 % − 21 % − − − − − −

where the self-gravity of the electromagnetic fields be- ] peak M

comes more important. Even if we do not expect these t 1.9 2 → relations to hold, they are still useful because that the − 0 f GW

t 1.8 charge-to-mass ratios explored here are up to 0.3, which [10 ++ E is a considerable amount of charge (corresponding to 30 % + ] −

of the maximum value allowed for a general-relativistic M +0 [ black hole). 4 10− peak

t 2 First, we notice that the energy and angular mo- λ → mentum lost in gravitational-wave emission depend only 0 EEM ∝ f EM t 6 10− weakly on the value of λ for the cases considered in our E set. This is shown in the top panel of Fig.8, where we 0.01 0.05 0.1 0.2 0.3 plot the total energy lost by gravitational waves for the λ (charge-to-mass ratio) various simulations we performed. Next, it is interesting to study which modes dominate FIG. 8. Top: total energy radiated away in gravitational the emission. We focus on the λ = 0.3 simulations, and waves (between frequencies f0 and fpeak) as a function of λ begin with the electromagnetic sector. Based on New- for the different cases we simulated. Bottom: total energy tonian arguments, we expect that in black hole binaries radiated away in electromagnetic waves EEM as a function of with opposite charge, most of the emission will be dipo- λ. We find that the energy scales as the charge squared for lar. By contrast, for simulations with like charge we ex- for the cases considered in this work. The simulation that pect no dipole, overall reduced emission, mostly in the radiates the most is the one with opposite charges because quadrupole. Finally, the case with only one charged black its electric dipole is the largest. In the same-charge case, the hole (which has a non-zero dipole) should lie in between. emission is dominated by the electric quadrupole and it is still The bottom panel of Fig.8 shows the total energy carried significant. The angular momentum lost by waves behaves away by electromagnetic waves starting at the reference similarly. frequency f0 until peak gravitational wave amplitude, and demonstrates that the aforementioned intuition is Figure8 demonstrates that the emitted electromag- correct. Moreover, the figure shows that the energy emit- netic energy scales as λ2. That is not the only quantity ted scales with the charge-to-mass ratio squared. When that does so. We find that several quantities scale the computing the mode-by-mode contributions to the elec- + same way, including EEM, JEM, EEM/EGW, and JEM/ tromagnetic emission, we find that for the λ− case, ` = 1 J . We observe an example of this in Fig.9 which dominates over all other modes and constitutes 98 % of GW + shows EEM/EGW, and JEM/JGW vs λ. As remarked at the energy lost. For λ+ the ` = 2 mode is the main chan- the beginning of this subsection, while we find the scaling nel through which energy is carried away, and 98 % is lost 2 + with λ for several quantities, it is possible that some of in this way. In the λ0 case both dipole and quadrupole this relationships will fail for larger values of the charge. are important, the first contributing 78 %, and the sec- In [44] it was estimated that for equal-mass mergers ond 20 %. Higher order modes constitute less than 2 % and black holes endowed with opposite charges, contribution. In all of our simulations the ` = 2 mode ac- counts for 98 % of the energy lost by gravitational waves. +− +− 2 EEM/EGW ≈ 2λ . (42) If we considered angular momentum instead of energy, we arrive at similar conclusions. Using our results, which strictly speaking do not apply 15

1 and is given by [63, 152–155] 10− peak t

→ 1 1

0 SNR(h , h ) = √ . (44) f 1 2

t p 3 2 mismatch(h , h )  10− 2 1 2 λ EM GW ∝ E E Note that the SNR threshold value of Eq. (44) corre-  5 sponds to a 68 % confidence level for distinguishability. 10− 1 ++ We compute the mismatch as [2, 155, 156] 10−

peak + t − → +0 mismatch(h1, h2) = 1 − max O(h1, h2) , (45) 0 f t 3  10− 2 λ with the maximum evaluated with respect to time-shifts, EM GW J J ∝ polarization angles, and mass shifts. We maximize over  5 mass shifts to include the possible degeneracy between 10− 0.01 0.05 0.1 0.2 0.3 mass and charge (see [2] for a detailed discussion). This λ (charge-to-mass ratio) operation is allowed because our simulations scale with the total mass M. In Eq. (45), O(h1, h2) is the overlap FIG. 9. Ratio between electromagnetic and gravitational en- between h1 and h2 ergy and angular momentum lost via emission of radiation (h , h ) from t to merger. The configuration with the strongest 1 2 f0 O(h1, h2) = p , (46) electromagnetic emission is the one with opposite charges, as (h1, h1)(h2, h2) the dipole is the most effective as emitting. with (h1, h2) being the noise-weighted inner product be- ˜ tween the two signals in the frequency domain h1(f) and ˜ to the equal mass case but should be close enough, we h2(f), which is given by [157] find Z fmax ˜ ˜? h1(f)h2(f) (h1, h2) = 4Re df , (47) +− +− 2 f Sn(f) EEM/EGW ≈ 1.6λ . (43) min

where Sn(f) is the power spectral density of the detec- Thus, in spite of the simple derivation in [44], the expres- tor noise, and the asterisk indicates complex conjuga- sion provided finds the correct order of magnitude in the tion. In the subsequent analysis we consider the fol- numerical coefficient. lowing detectors at design sensitivity: Advanced LIGO, If we consider charge in its traditional meaning of A+ [158], Voyager [159], Einstein Telescope [160], and electric charge, we can talk about electromagnetic lumi- Cosmic Explorer [161] and we set fmin = 25 Hz and nosity emitted. For the simulations with λ = 0.3, the f = 1024 Hz.9 −5 max peak electromagnetic luminosity varies from 2.7 × 10 Before we discuss the results, we highlight possible pit- −5 57 −1 to 8.6 × 10 , corresponding to ≈ 10 erg s (regard- falls in numerically evaluating Eq. (47). There are mul- less of the value of M). For M = 65 M , the maximum tiple steps to go from what the simulations output to 53 electromagnetic energy emitted is ≈ 10 erg , which is Eq. (47). In particular, Fourier transforms have to be much smaller than the energy emitted through gravita- computed and the time series have to be windowed and tional waves. This possibly explains the near perfect zero-padded to avoid aliasing and spectral leakage. Since 2 scaling with λ of several quantities we reported. How the simulations we are considering produce similar wave- this electromagnetic energy is converted to potentially forms (up to time, phase, and mass shifts), small differ- observable photons has to be modeled and it is highly ences in how the two waves are pre-processed can con- dependent on the environment. In standard astrophysi- tribute significantly to the value of the mismatch. First, cal conditions, the diluted plasma would not allow these it is important to trim the end of the two waveforms waves to propagate [78]. where the strain is practically zero and ensure that they have the same duration after the peak. If this is not done,

2. Charge detectability 9 When available, we use the sensitivity curves im- In this subsection, we use our simulation results to plemented in LALSimulation [162] via PyCBC [163]. estimate black hole charge detectability by future grav- The functions used are part of the pycbc.psd mod- ule and are: aLIGODesignSensitivityP1200087, itational wave detectors. To do this, we follow [2] and aLIGOAPlusDesignSensitivityT1800042, compute the mismatch between two signals h1 and h2. EinsteinTelescopeP1600143, and CosmicExplorerP1600143. This quantity determines the minimum SNR needed to For Voyager we obtained the sensitivity curve from LIGO Document T1500293-v11. distinguish h1 and h2, which we denote as SNR(h1, h2), 16 and the signals have different durations after merger, ap- our binaries to place them in the LISA band, which we plying a window has a different effect and introduces a assume ranges from 0.1 mHz to 1 Hz. Despite the scale systematic uncertainty. Second, in some cases the fixed freedom, there is one obstacle to doing a complete anal- frequency integration may leave small residual drifts at ysis using our simulation data as Fig. 11 demonstrates. the very end of the waveform. These depend on the sim- In the plot, we show the LISA sensitivity curve [164] ulation, so one must check that all the waves are well- and we schematically show with dashed lines the grav- behaved. In case they are not, one can adjust this by itational wave spectrum from our simulations when the cropping the signal or by changing the parameters of the mass is rescaled using two different values. In the case 7 window function or of the integration method. Improp- of a 10 M binary, our simulation signal is entirely in erly considering one of these effects may lead to system- the LISA band. On the other hand, at least part of the 4−6 10 atic errors. inspiral is missing for ∼ 10 M binaries. Despite We now discuss the results of our study by focusing on this obstacle, we know that for the case with opposite + the λ− case, which is relevant to constraining the dipole charge, the mismatch will increase if we included the in- emission. In Fig. 10, we report the value of the SNR spiral, due to the presence of dipole emission. Hence, if needed to distinguish uncharged binary waveforms from we use our data to find what is the minimum SNR needed + binary black holes with charge-to-mass ratio λ− = 0.1 to distinguish waveforms of charged binaries from those (circles), 0.2 (stars), or 0.3 (diamonds) for a GW150914- generated by uncharged binaries, we will find an upper + like event. For smaller values of λ−, the mismatch com- bound on that (if we included the entire in-band inspiral puted from our simulations is limited by their numerical waveform the minimum SNR for distinguishability would error, so higher resolution evolutions would be needed. only decrease). First, we note that there is not much variation of the SNR for distinguishability across the different detectors, 27 10− regardless of the significant variation in sensitivity. The Sims. cover entire signal ] reason for this result is that for the computation of the 1 Inspiral not available mismatch the overall noise curve does not matter: it is − [Hz 34 how the noise is distributed in different frequencies that 10− matters the most. The difference in sensitivity is reflected LISA n in how easy or not it is to achieve such SNRs. Second, the S values needed to detect λ+ ≥ 0.1 are already achievable M 107 M M 104 M − 41 ≈ ≈ today (GW150914 had a network-SNR of approximately 10− 5 4 3 2 1 0 1 2 25 [69], but its noise curve was not the one at deign sensi- 10− 10− 10− 10− 10− 10 10 10 tivity). Given their improved sensitivity, future detectors f [Hz] will immediately be able to detect this amount of charge. To estimate what limits on charge future detectors will FIG. 11. Schematic representation of how we can use our sim- place, one not only needs better simulations, but one ulations for LISA sources. We are free to vary the mass, but needs to include the effects of spin and eccentricity. in many cases our simulations do not cover the entirety of the signal, and part of the inspiral is missing. Since the inspiral 30 is the most constraining part of the signal, when we compute GW150914- λ+ = 0.1 λ+ = 0.2 λ+ = 0.3 the minimum SNR for distinguishability of charged binary λ like event − − − waveforms from uncharged binary ones, we are providing an 20 upper bound. Figure 12 shows the upper bound on the minimum 10 SNR needed to distinguish waveforms generated by

SNR to detect charged binaries from those generated by uncharged bi- naries for different charge configurations as a function of aLIGO A+ Voyager ET CE the detector-frame mass Mdetector. LISA is expected to detect binaries with SNR much higher than the one in FIG. 10. Signal-to-noise ratio required to distinguish wave- Fig. 12, so it will be able to detect or place constraints + forms from mergers of charged black holes from ones without on small values of λ− for multiple systems. These re- charge for different detectors assuming their design sensitiv- sults are not surprising. In [44, 71] it argued that LISA ity. The expected SNR for a GW150914-like event future will constrain the dipole moment at the level of the 10−4, detectors is significantly larger than the one needed to con- considering only the inspiral. strain λ+ < 0.1. See Footnote9 for details on the detector sensitivity− curves used.

10 One could extended our simulation data with some approximate Next, we discuss what charge-to-mass ratio LISA waveforms, like the one from the Newtonian model, and recom- would be able to detect for million solar mass binaries. pute the quantities with the entire signal. However, this goes As already mentioned, we are free to rescale the mass of beyond the scope of the current work. 17

80 in Fig. 13. For some mass ranges, LISA will essentially λ+ = 0.1 + distinguish a charged binary with λ− up to 0.1 every- +−

λ λ = 0.2 where it can detect black hole binaries. It is important 60 − λ+ = 0.3 to note that that plot provides only a lower limit on this − maximum distance, especially for lower masses, where we 40 do not include the long inspiral part of the waveform.

to constrain 20

L 20 + Upper bound on SNR D λ = 0.1 200 −

pc] + 5 6 7 λ = 0.2 16 G

10 10 10 [ − + z

λ λ = 0.3 Mdetector [M ] − 12 100 FIG. 12. Upper limit on the SNR needed to distinguish wave- 8 Redshift form from charged binaries with different λ+ from uncharted − 4 systems. This is an upper limit because most of the masses to constrain

do not include the entirety of the inspiral. Given that the Lower bound on max 0 0 inspiral is most constraining, including it would decrease the 105 106 107 SNR needed. Msource [M ]

We can understand the shape in Fig. 12 by considering FIG. 13. Lower limit on the maximum luminosity distance DL that the inspiral has the most important contribution to in gigaparsec (or max redshift z) at which LISA can detect a charged binary with λ of 0.1 as function of the source-frame the mismatch, so, it is the lower frequencies that mat- mass M . ter the most. Recall that Fig. 11 shows schematically source the power spectrum of the gravitational waves. Roughly We can ask the same question for the case λ+ = 0.3 speaking, there is more power in the lower frequencies + at the mass-scale for which we have the entire signal in than the higher ones because more time is spent there. LISA band (≈ 107 M ). We find that LISA will be able Increasing the binary mass from the minimum value con- to detect charge up to redshift of about 1, which would sidered here amounts to sliding the spectrum in Fig. 11 translate to a constraint on the α parameter of Mof- from the right to the left. When we are considering fat’s Scalar-Vector-Tensor gravity [19] of α 0.1 (see masses for which the signal lies to the right-hand-side of . also, [2]). the plot, we find that there is more noise in the higher fre- quencies of the signal than the lower ones. This is the op- timal condition to separate an uncharged waveform from a charged ones, because it is the lower frequencies that C. Properties of the remnant black hole and ringdown contain the most information. Hence, the signal to noise for the detection is the lowest. Now, when we consider masses for which the signal lies on the left-hand-side of Here we study the properties of the remnant black hole Fig. 11, initially not much changes in the signal-to-noise forming following the merger of charged binaries. We ratio. This is because the frequencies with the most infor- discuss the accuracy of the method presented in Sec.IVA mation are the ones with the lowest noise. The scenario to estimate the remnant black hole spin (Sec.VC1), and changes when we approach the minimum of the sensi- we explore the quasi-normal-modes from the ringdown tivity curve, at that point, the way noise is distributed phase (Sec.VC2). across frequency changes, and we see in Fig. 12 an in- crease in the signal-to-noise ratio, which reflects the fact that we are removing sensitivity in the lower frequencies 1. Remnant black hole properties to put it in the higher ones (that have less information). The trend continues when we increase further the mass In this subsection, we discuss the physical proper- scale and we climb up the sensitivity curve on the left ties of the charged binary black hole merger remnants side. Here, we are giving more weight to high frequen- in our numerical-relativity simulations as computed via cies, which cannot distinguish the two waves well, so we QuasiLocalMeasuresEM. Then, we compare the remnant need more signal-to-noise ratio for distinguishability of black hole spin found in the simulations with the expec- two signals. tations of the BKL approximation described in Sec.IVA. A natural question that arises next is the following: Before presenting our results, let us consider what one how far can LISA detect the minimum SNR required to might expect qualitatively. Following the discussion in distinguish charge? Computing the SNR with Eq. (40b), Sec.IIB, we know that, in Newtonian√ physics, the or- we can find what is the maximum distance at which the bital angular momentum scales as G. Next, consider SNR is larger than the threshold. This distance is plotted conservation of angular momentum, and assume (as in 18

the BKL method) that the spin of the final black hole is We now compare the method with our simulations. The determined by the angular momentum at the innermost- top panel of Fig. 15 shows that the method estimates stable circular orbit of the remnant black hole. We would the value of the final dimensionless spin with an error final expect the final spin χ √with respect to the uncharged of 3 % in the uncharged case. This method does not in- final case χ00 to increase as 1 − λ1λ2 for the case with op- clude energy loss. Including it would not improve accu- posite charge (where the ISCO of the remnant black hole racy, as the dimensionless spin would be overestimated 11 is the same√ as in the uncharged case ), and to decrease as opposed to underestimated. In the bottom panel of faster than 1 − λ1λ2 for the other two cases, where the Fig. 15 we normalize the final spin to the value of the un- ISCO is reduced compared to the uncharged case, result- charged case with simulations normalized with the values ing in additional emission of angular momentum. obtained from the simulations, and the BKL values are In Fig. 14 we show how these properties change with re- normalized with the value obtained applying the method spect to their values in the uncharged case. First, we find in absence of charge. This removes the normalization as that the final mass of the remnant is almost independent a possible parameter and allows us to test if the method of the charge configuration, with sub-percent variations. captures the correct trend (and only gets the normal- + In particular, the λ− remnant looks almost identical to ization wrong). In practice, this corresponds to testing the one from the uncharged simulation: spin, charge, whether removing trends arising from the normalization and mass are the same to within 1 %. Moreover, in that makes the black lines in the bottom panel of Fig. 15 over- case, the angular momentum is essentially independent lap with the colored ones. We observe that the method of λ, against the expectations from the Newtonian argu- indeed works well for the cases with like charge or with ment. In general, we find that the final properties of the only one charged black hole, but overestimates the final black hole depend weakly on the charge configuration, spin in the other cases. The approach does not capture + with the largest variation being about 4 % in the spin the fact that the spin of the final black hole in the λ− nu- of the λ+ = 0.3 case. Interestingly, the cases with like merical relativity simulations is essentially independent + √ charge follow the Newtonian scaling 1 − λ1λ2 as shown of charge. However, the method is still accurate to within in Fig. 14 with the black solid lines. a few per cent.

1.000 0.68 final 00

/M ++ 0.66 0.995 + final final − χ 0.64

M +0 ++ BKL ++ 1.050 0.62 + BKL + − − +0 +0 final 00 BKL 1.02

/a 1.000 final 00

final 1.00 a /χ 0.950 0.98 0.01 0.05 0.1 0.2 0.3 final χ 0.96 λ (charge-to-mass ratio) 0.01 0.05 0.1 0.2 0.3 FIG. 14. Mass (top panel) and spin (bottom panel) λ (charge-to-mass ratio) of the remnant black hole computed with the isolated horizon formalism [115, 165, 166] as implemented in QuasiLocalMeasuresEM [1], compared to the one in a merger FIG. 15. Dimensionless spin of the remnant black hole in the final final numerical relativity simulations (colored lines) and estimated with no charge (M00 0.96 M, a00 0.66 M). The solid black lines in the bottom≈ panel indicate the≈ Newtonian scaling with the method of Sec.IV (black lines with crossed markers). √1 λ1λ2. In the top panel, we compare the actual numerical values, in ± the bottom panel we rescale the values by the value measured In Sec.IVA, we described a simple way to estimate in the simulation without charge or the estimated value in ab- sence of charge (simulation data is normalized with the value the spin of the remnant using conservation arguments. obtained from the uncharged simulation, and BKL estimates are normalized with the value obtained assuming no charge).

11 Although the ISCO of the remnant black hole in the charged case with opposite charges is practically unaffected by charge (for the range of λ in this work), since the total charge is approximately 2. Quasi-normal-modes zero, the effective innermost stable orbit of the binary (the orbital separation at plunge) must be affected to some extend, especially for larger values of λ even when the component black holes have Here we discuss the ring-down phase and address how exactly opposite charges. challenging it is to tell if a binary black hole is charged 19 from the ringdown phase alone. 2 Since the frequency and decay time of the quasi- (M final, χfinal,Qfinal) normal-modes depend only on mass, spin, and charge, 0 we can use the values reported in Fig. 14 to compute ] ω and τ with Eqs. (39) and the qnm code [145]. The 2 ++ 220 220 − top panel of Fig. 16 shows the relative difference of the 2 + [10 − 00 00 00 2 − computed value (ω220 − ω220)/ω220, where ω220 is the +0 00 ring-down frequency computed with qnm using the mass 220 /ω and spin of the remnant black hole from the simulation ) 0 00 without charge, while ω is the ring-down frequency us- 220

220 ω final final final (M00 , χ00 ,Q ) ing Eqs. (39). The maximum difference we find is 0.9 % − 2 in the λ+ = 0.3 case. Therefore, to be able to distinguish − − 220 2 ω the ringdown from a merger of charged black holes from ( one with no charge, the parameters have to be estimated better than the percent level. For a given value of λ, even 0 higher accuracy is needed to be able to distinguish the final final final (M , χ ,Q00 = 0) three charged scenarios. 2 It is interesting to understand why the simulation with − 0.01 0.05 0.1 0.2 0.3 opposite charge, in which the remnant has the least λ (charge-to-mass ratio) amount of total charge for a fixed λ, is the one with largest difference with respect to the uncharged case: In FIG. 16. Relative difference between the analytical 220 quasi- Fig. 15, we see that for the case with opposite charges normal-mode frequency for the merger remnants in our evo- final final χ /χ00 ≈ 1 regardless of the value of λ. Combining lutions and the simulation with uncharged black holes. In the this information with the fact that the remnant black middle panel, we remove the effects of mass and spin and fix hole has Q/M  1, we deduce that the only difference the values to the ones they have in the uncharged case. In + the bottom panel, we remove the contribution of the charge between the uncharged case and the case with λ− = 0.3 must be in the mass of the final black hole. The difference described by Eqs. (39): the frequency becomes more differ- in mass alone is responsible for the observed difference in ent. The same happens with the decay time, but the variation with respect to the uncharged case is even smaller. the ringdown frequencies. It is also interesting to understand why the case with same charge is not the easiest to distinguish, despite that the remnant black hole mass in this case has λ ≈ 0.3, and charge, so it is natural that this parameter will contribute is the one with the most different spin (Fig. 15) compared the least if mass and angular momentum are the same. + On the other hand, the case with like charges is the most to the uncharged case. Interestingly, in the λ+ case, charge, angular momentum and mass conspire so that different from the uncharged ones, with relative increase the remnant black hole appears to have a quasi-normal in ω220 of 2.6 %. mode frequency that matches that from an uncharged bi- The bottom panel of Fig. 16 shows what happens when nary. To analyze this case, it is convenient to separate we completely ignore the contribution of charge to the the effects of charge, and mass and spin. The funda- quasi-normal modes. First, we find the confirmation that + mental properties of the quasi-normal-modes, ω220 and here is where the simulations λ− acquire the differences final final final τ220, depend on the triplet (M , χ ,Q ). We can of up to ≈ 1 % shown in the top panel of Fig. 16. Sec- hold some of these parameters fixed to the case with no ond, in the like charges case, we also find that charge charges to understand what is the dominant contribution introduces differences in the mass and angular momen- to the deviation of ω220 and τ220 from their corresponding tum of the remnant black hole that drive the change in values in the absence of charge. The results are reported the quasi-normal mode in the opposite direction com- in the middle and bottom panels of Fig. 16, where we pared to the ones shown in the middle panel. Therefore, report ω220. The same results hold for τ220. when we consider the complete triplet of mass, spin, and In the middle panel we vary only the charge, and fix the charge (“summing” middle and bottom panels), we find mass and spin to the value they obtain in the simulation that their effects almost cancel each other, so that the final final corresponding quasi-normal-modes are close to the un- with no charge M00 and χ00 . Here we find that bina- ries with opposite charge have remnant black holes that charged binary case. behave almost exactly like the uncharged one (there is no Hence, we conclude that for the λ explored here, to dis- difference in the quasi-normal-modes). This tells us that tinguish the quasi-normal modes of the final black hole for this specific configuration the deviation in ω220 is not from the ones of an uncharged black hole, one needs because of charge, but that the mass and spin are the pa- exquisite accuracy. The accuracy needed is also of the rameters that primarily control the quasi-normal-modes same order as that of the fitting functions in Eq. (39), properties. This is not unexpected: oppositely charged and for the highest values of λ adopted here, it is also binaries are those with the smallest remnant black hole of the same order as the errors due to the truncation of 20 the expansion in λ, which may affect the result. In our ACKNOWLEDGMENTS simulations, the ringdown signals are essentially indistin- guishable from one another, and we cannot identify the We used kuibit [169] for part of our analysis. We thank quasi-normal-mode parameters at the accuracy needed M. Zilh˜aofor help with ProcaEvolve. We are grateful to the to tell them apart. To sum up, values of charge up to developers and maintainers of the open-source codes that we λ ∼ 0.3 could be challenging to detect using the ring- used. This work was in part supported by NSF Grant PHY- down phase alone. 1912619 to the University of Arizona. G. B. is supported by a Texas Advanced Computing Center (TACC) Frontera Fellow- ship. Frontera is supported by NSF grant No. OAC-1818253. We acknowledge the hospitality of the Kavli Institute for The- oretical Physics (KITP), where part of the work was con- VI. CONCLUSIONS AND FUTURE ducted. KITP is partially supported by the NSF grant No. DIRECTIONS PHY-1748958. Computational resources were provided by the Extreme Science and Engineering Discovery Environment (XSEDE) under grant number TG-PHY190020. XSEDE is In this paper, we continued our program of explor- supported by the NSF grant No. ACI-1548562. Simulations ing the non-linear dynamics of black holes in Einstein- were performed on Comet, and Stampede2, which is funded by Maxwell theory. In Sec.II, we described our theoreti- the NSF through award ACI-1540931. cal and numerical approach, emphasizing new features and formalism that have not been treated before, includ- ing: how to prepare quasi-circular initial data for charged Appendix A: Courant stability of Kreiss-Oliger black hole binaries (Sec.IIB), how to perform long-term Dissipation and stable evolutions of quasi-circular inspirals of charged black holes (Sec.IIC), the electromagnetic contribution Kreiss-Oliger dissipation is described by Eq. (7), which to horizon properties of black holes (Sec.IID), and the we rewrite here for convenience computation of the angular momentum carried away by  ∂ U = ··· + (−1)(p+3)/2 ∆xp∂p+1U. (A1) electromagnetic waves with the Newman-Penrose formal- t 2p+1 x ism (Sec.IIE). This technique has a free a parameter  that controls We compared the results of our non-linear simulations the strength of the dissipation. This cannot be chosen with approximate approaches for the inspiral (Sec.VA) arbitrarily, as we show in this Appendix. and the ringdown (Sec.VC). For the systems consid- A finite-difference discretization transforms Eq. (7) to ered, our work shows that Newtonian models based on the quadrupole approximation find the correct order of  Dp+1U magnitude in a set of gauge-invariant quantities, but have ∂ U = ··· + (−1)(p+3)/2 ∆xp , (A2) errors O(20%) or larger, and hence they cannot be used t 2p+1 ∆xp+1 in precision studies of mergers of charged black holes or where Dp+1U is determined by the finite-difference sten- accurate parameter estimation. Similarly, estimates of cil. For an explicit time integration scheme, this operator the spin of the remnant black hole based on conservation alone leads to a Courant stability condition of the form of angular momentum and energy arguments are accu- rate up to few percent. Hence, a key result of this work ∆t  ≤ Λ , (A3) is extending what was found in [20]: these arguments can ∆x 2p+1 be used to build intuition and make order-of-magnitude where Λ is typically a fixed number that is determined estimates in the case of quasi-circular mergers, too. by the details of numerical integration method.12 Given Furthermore, we discussed properties of the emission that in our case we add dissipation to hyperbolic partial (Sec.VB) and estimated the detectability of charge by differential equations, condition (A3) can be recast to future gravitational wave observatories (Sec.VB2), fo- read cusing in particular on LISA. Finally, we studied the Λ quasi-normal-modes (Sec.VC2), finding that it may be  ≤ , (A4) challenging to extract charge information from the ring- µ down alone. where µ is the Courant factor ∆t/∆x. For the Einstein- There are multiple possible extensions of this work. Maxwell equations in standard finite-difference imple- On the non-linear side, including spin and increasing mentations, this quantity is typically chosen such that the charge-to-mass ratio would allow the exploration of a region of the parameter space never considered be- fore. On the side of approximate calculations, the next 12 The case with p = 1 using standard centered second-order accu- step in complexity after Newtonian physics is develop- rate finite differences is a textbook example of stability analysis ing 1PN models. Such waveforms are currently avail- for a standard diffusion equation with constant coefficients. In able [167, 168]. These models are important in the effort this case, and for a three-dimensional Cartesian grid with equal of generating gravitational-wave templates. grid spacing in the all spatial directions, Λ = 1/6. 21

a µ ≤ 0.5 for numerical stability. Since we are working potential, the Lorenz gauge is ∇aA = 0, while its gener- a a with an adaptive-mesh-refinement computational grid alized version is ∇aA = ξnaA , with ξ damping param- with sub-cycling in time, there are multiple values of ∆t eter. In [170], it was found that with a suitable choice of and of ∆x, so condition (A3) can be met in some parts ξ, this condition reduces spurious gauge modes that arise of the grid but not in others. Failure to satisfy the con- from interpolation at the refinement level boundaries. dition can result in numerical instabilities that spoil the While this is important for general-relativistic magneto- simulation. It should be noted that the Courant con- hydrodynamic simulations in which matter crosses refine- dition (A4) is necessary but not sufficient for stability. ment levels, in our evolutions we did not find significant This means that the scheme can be unstable even when improvements. Eq. (A4) is satisfied. For example, an ill-posed initial A second formulation we tested was motivated by the boundary value problem or other numerical instabilities parabolized Arnowitt-Deser-Misner formalism of general could cause simulations to crash. relativity [171, 172]. This consists of adding an extra parabolic term to the evolution of the electric field, which is 0 when the Gauss constraint is satisfied, Appendix B: Parameters of the numerical evolution i ij ∂tE = ··· + Eγ ∂jCE . (C1) In this Appendix, we report the parameters used for our simulations, including the test described in Sec.IIC. with CE is the Gauss constraint and E strength of the parabolic term. With this modification, the evolution of Our simulations are on a grid with outer boundary the constraint looks like at 1033 M and nine refinement levels with refinement boundaries located at 2iM, where i is the number of ∂ C = ··· +  γij∂ ∂ C , (C2) level. Our standard coarsest-level resolution is ≈ 4 M t E E i j E and the high-resolution runs are with 25 % smaller grid which is parabolic diffusion operator. The indented result spacing (≈ 3.2 M). Of the nine refinement levels, five is to further dissipate violations of the constraint from have the same timestep, the other levels had Courant perturbations with high wave number. In our tests, this factor fixed to ∆t/∆x of 0.4. This choice reduces the formulation did not result in noticeable improvements maximum timestep on the grid and prevents some nu- over our new method for setting the Kreiss-Oliger dissi- merical instability that would otherwise arise. We set κ pation parameter presented in Sec.IIC. (defined in Appendix A in [23]) to 9.9 M −1 and verified that changing this parameter does not produce signifi- cant differences. We set the η parameter in the evolu- Appendix D: Newman-Penrose scalars in flat tion of the shift vector (as defined in Eq. (13) in [88]) spacetime as function of electric and magnetic fields to 1.5 M −1. These parameters lead to instability unless κ 1.5/∆t (same for η), so we adjusted the time- . max In this Appendix, we provide expressions for the elec- stepping in our evolutions to ensure that this condition tromagnetic Newman-Penrose Φ ,Φ , and Φ in terms is met. We use fifth order prolongation in space and sec- 0 1 2 of the electric and magnetic fields in (asymptotically) flat ond in time. We tested selected cases with seventh order spacetime. We consider both coordinate and orthonor- spatial prolongation and found no significant differences. mal bases. These expressions can be used to quickly All our derivatives are obtained with sixth-order finite compute the Newman-Penrose scalars for a given electro- difference. magnetic field. (For a similar discussion, see Appendix The test described in Sec.IIC was performed with the A in [173] noting that a different convention is used for high-resolution grid and otherwise the same parameters the normalization of the tetrad.) described above. The tests showed that the instabilities In flat spacetime and given the spherical coordinates arise unless we choose the continuous prescription for the (r, θ, ϕ), consider the coordinate basis (∂ , ∂ , ∂ ), and dissipation. In the unstable cases, we found that sim- r θ φ an orthonormal basis (e , e , e ) so that for a vector v ulations with higher resolution become unstable earlier rˆ θˆ φˆ compared to the ones at lower resolution. we have that rˆ θˆ ϕˆ v = v erˆ + v eθˆ + v eϕˆ , (D1a) r θ ϕˆ Appendix C: Alternative formulations v = v ∂r + v ∂θ + v ∂ϕ , (D1b)

In our efforts to improve and stabilize our simulations, with we tested two additional formulations for the evolution of vrˆ = vr , v = v , (D2a) the electromagnetic fields. We found that improvements rˆ r arising from either of these formulations is subdominant θˆ θ 1 v = rv , vˆ = vθ , (D2b) compared to the role of the dissipation (see, Sec.IIC). θ r First, we followed [170] and implemented a general- ϕˆ ϕ 1 v = r sin θv , vϕˆ = vϕ . (D2c) ized Lorenz condition. If Aa is the electromagnetic four- r sin θ 22

The electromagnetic field strength can be written in Heaviside-Lorentz units are used, and many formulas dif- terms of the orthonormal tetrad components of the elec- fer by a factor of 4π with what is reported here). Our tric and magnetic fields as follows interest in Michel’s monopole is motivated by the fact that is a simple solution with stationary flow of energy   0 −Erˆ −rEθˆ −r sin θEϕˆ and angular momentum in flat spacetime that can be Erˆ 0 rBϕˆ −r sin θBθˆ used to test Eq. (17) and its numerical implementation. Fab =  2  .  rEθˆ −rBϕˆ 0 r sin θBrˆ  In an orthonormal basis, the non-zero components 2 r sin θEϕˆ r sin θBθˆ −r sin θBrˆ 0 of electric and magnetic fields of Michel’s monopole (D3) are [127] The Newman-Penrose scalars are, as computed by q Eqs. (8) and using Eq. (D3) E = B = − ω sin θ , (E1a) θˆ ϕˆ r 1
q Φ = −E + B − i(E + B ) , (D4a) Brˆ = . (E1b) 0 2 θˆ ϕˆ ϕˆ θˆ r2 1 Φ = (E + iB ) , (D4b) A four-vector potential that produces such configuration 1 2 rˆ rˆ is given by 1
Φ2 = Eθˆ + Bϕˆ − i(Eϕˆ − Bθˆ) . (D4c) 2  q θ  Aµˆ = qω cos θ, 0, − ω sin θ, q tan . (E2) Alternatively, expressing Eqs. (D4) in terms of the vector r 2 components in the coordinate basis: Notice that Aϕˆ → +∞ when θ → π, but the resulting 1  E B  E B  magnetic field is well-behaved, as shown in Eq. (E1b). Φ = − θ + ϕ − i ϕ + θ , (D5a) 0 2 r r sin θ r sin θ r We can compute the Newman-Penrose scalars Φ1 and Φ with Eqs. (D4): 1 2 Φ = (E + iB ) , (D5b) 1 2 r r q    Φ1 = − ω sin θ , (E3a) 1 Eθ Bϕ Eϕ Bθ r Φ = + − i − . (D5c) 2 2 r r sin θ r sin θ r i q Φ = . (E3b) 2 2 r2 For completeness, we also report the null tetrad of Eq. (13) is (in the coordinate basis) As expected, Φ1 and Φ2 follow the peeling behavior of Eqs. (9). Next, we can use Eq. (15) to compute the total 1 ka = √ (1, −1, 0, 0) , (D6a) power that crosses a sphere or radius r 2 dE 2 1 EM = q2ω2 . (E4) la = √ (1, 1, 0, 0) , (D6b) dt 3 2   We can also compute the flux of angular momentum a 1 1 i m = √ 0, 0, , , (D6c) along the z direction radiated with Eq. (17) 2 r r sin θ 1  1 i  dLz 2 m∗a = √ 0, 0, , − . (D6d) EM = q2ω . (E5) 2 r r sin θ dt 3 Hence, we find the relation

Appendix E: Michel’s solution dE dLz EM = ω EM . (E6) dt dt Michel’s rotating magnetic monopole solution [126] is a simple model for pulsar and black hole magneto- This is a well-known relation for pulsar and black hole spheres (for a rigorous description, see [127], noting that magnetospheres.

[1] G. Bozzola and V. Paschalidis, Phys. Rev. D 99, 104044 F. Acernese, K. Ackley, C. Adams, R. X. Adhikari, (2019), arXiv:1903.01036 [gr-qc]. V. B. Adya, C. Affeldt, and et al., Physical Review X [2] G. Bozzola and V. Paschalidis, Phys. Rev. Lett. 126, 9, 031040 (2019), arXiv:1811.12907 [astro-ph.HE]. 041103 (2021), arXiv:2006.15764 [gr-qc]. [5] B. P. Abbott, R. Abbott, T. D. Abbott, M. R. Aber- [3] Event Horizon Telescope Collaboration, Astrophys. J. nathy, F. Acernese, K. Ackley, C. Adams, T. Adams, Lett. 875, L1 (2019), arXiv:1906.11238 [astro-ph.GA]. P. Addesso, R. X. Adhikari, and et al., Physical Review [4] B. P. Abbott, R. Abbott, T. D. Abbott, S. Abraham, Letters 116, 221101 (2016), arXiv:1602.03841 [gr-qc]. 23

[6] K. Yagi and L. C. Stein, Classical and Quantum Gravity [32] R. Foot and S. Vagnozzi, Phys. Rev. D 91, 023512 33, 054001 (2016), arXiv:1602.02413 [gr-qc]. (2015), arXiv:1409.7174 [hep-ph]. [7] B. P. Abbott, R. Abbott, T. D. Abbott, S. Abraham, [33] R. Foot and S. Vagnozzi, Physics Letters B 748, 61 F. Acernese, K. Ackley, C. Adams, R. X. Adhikari, V. B. (2015), arXiv:1412.0762 [hep-ph]. Adya, C. Affeldt, and et al., Phys. Rev. D 100, 104036 [34] R. Foot and S. Vagnozzi, J. Cosm. Astropart. Phys. (2019), arXiv:1903.04467 [gr-qc]. 2016, 013 (2016), arXiv:1602.02467 [astro-ph.CO]. [8] B. P. Abbott, R. Abbott, T. D. Abbott, F. Acernese, [35] P. Agrawal, F.-Y. Cyr-Racine, L. Randall, and K. Ackley, C. Adams, T. Adams, P. Addesso, R. X. J. Scholtz, JCAP 05, 022, arXiv:1610.04611 [hep-ph]. Adhikari, V. B. Adya, and et al. (LIGO Scientific Col- [36] Ø. Christiansen, J. B. Jim´enez, and D. F. Mota, laboration and Virgo Collaboration), Phys. Rev. Lett. Classical Quant. Grav. 38, arXiv:2003.11452 (2021), 123, 011102 (2019). arXiv:2003.11452 [gr-qc]. [9] N. Yunes, K. Yagi, and F. Pretorius, Phys. Rev. D 94, [37] S. Davidson, B. Campbell, and D. Bailey, Phys. Rev. D 084002 (2016), arXiv:1603.08955 [gr-qc]. 43, 2314 (1991). [10] J. Cayuso, N. Ortiz, and L. Lehner, Phys. Rev. D 96, [38] M. L. Perl and E. R. Lee, American Journal of Physics 084043 (2017), arXiv:1706.07421 [gr-qc]. 65, 698 (1997). [11] E. W. Hirschmann, L. Lehner, S. L. Liebling, and [39] S. Davidson, S. Hannestad, and G. Raffelt, Journal of C. Palenzuela, Phys. Rev. D 97, 064032 (2018), High Energy Physics 5, 003 (2000), hep-ph/0001179. arXiv:1706.09875 [gr-qc]. [40] S. L. Dubovsky, D. S. Gorbunov, and G. I. Rubtsov, [12] J. L. Ripley and F. Pretorius, Class. Quant. Grav. 37, Soviet Journal of Experimental and Theoretical Physics 155003 (2020), arXiv:2005.05417 [gr-qc]. Letters 79, 1 (2004), arXiv:hep-ph/0311189 [astro-ph]. [13] W. E. East and J. L. Ripley, Phys. Rev. D 103, 044040 [41] S. Chatrchyan, V. Khachatryan, A. M. Sirunyan, A. Tu- (2021), arXiv:2011.03547 [gr-qc]. masyan, W. Adam, E. Aguilo, Bergauer, and et al., [14] M. Okounkova, L. C. Stein, M. A. Scheel, and Phys. Rev. D 87, 092008 (2013), arXiv:1210.2311 [hep- D. A. Hemberger, Phys. Rev. D 96, 044020 (2017), ex]. arXiv:1705.07924 [gr-qc]. [42] A. D. Dolgov, S. L. Dubovsky, G. I. Rubtsov, and [15] M. Okounkova, L. C. Stein, J. Moxon, M. A. Scheel, I. I. Tkachev, Phys. Rev. D 88, 117701 (2013), and S. A. Teukolsky, Phys. Rev. D 101, 104016 (2020), arXiv:1310.2376 [hep-ph]. arXiv:1911.02588 [gr-qc]. [43] H. Vogel and J. Redondo, J. Cosm. Astropart. Phys. [16] H. Witek, L. Gualtieri, P. Pani, and T. P. Sotiriou, Phys. 2014, 029 (2014), arXiv:1311.2600 [hep-ph]. Rev. D 99, 064035 (2019), arXiv:1810.05177 [gr-qc]. [44] V. Cardoso, C. F. B. Macedo, P. Pani, and V. Ferrari,J. [17] M. Okounkova, Phys. Rev. D 102, 084046 (2020), Cosm. Astropart. Phys. 5, 054 (2016), arXiv:1604.07845 arXiv:2001.03571 [gr-qc]. [hep-ph]. [18] H. Witek, L. Gualtieri, and P. Pani, Phys. Rev. D 101, [45] P. Gautham A. and S. Sethi, J. Cosm. Astropart. Phys. 124055 (2020), arXiv:2004.00009 [gr-qc]. 2020, 039 (2020), arXiv:1910.04779 [astro-ph.CO]. [19] J. W. Moffat, J. Cosm. Astropart. Phys. 2006, 004 [46] R. Plestid, V. Takhistov, Y.-D. Tsai, T. Bring- (2006), arXiv:gr-qc/0506021 [gr-qc]. mann, A. Kusenko, and M. Pospelov, arXiv e-prints , [20] M. Zilh˜ao, V. Cardoso, C. Herdeiro, L. Lehner, arXiv:2002.11732 (2020), arXiv:2002.11732 [hep-ph]. and U. Sperhake, Phys. Rev. D 85, 124062 (2012), [47] L. Liu, Z.-K. Guo, R.-G. Cai, and S. P. Kim, Phys. Rev. arXiv:1205.1063 [gr-qc]. D 102, 043508 (2020), arXiv:2001.02984 [astro-ph.CO]. [21] M. Zilh˜ao, V. Cardoso, C. Herdeiro, L. Lehner, [48] L. Liu, Ø. Christiansen, Z.-K. Guo, R.-G. Cai, and U. Sperhake, Phys. Rev. D 89, 044008 (2014), and S. P. Kim, Phys. Rev. D 102, 103520 (2020), arXiv:1311.6483 [gr-qc]. arXiv:2008.02326 [gr-qc]. [22] M. Zilh˜ao, V. Cardoso, C. Herdeiro, L. Lehner, [49] L. Liu, Ø. Christiansen, Z.-K. Guo, R.-G. Cai, and and U. Sperhake, Phys. Rev. D 90, 124088 (2014), S. P. Kim, arXiv e-prints , arXiv:2011.13586 (2020), arXiv:1410.0694 [gr-qc]. arXiv:2011.13586 [gr-qc]. [23] M. Zilh˜ao,H. Witek, and V. Cardoso, Classical Quant. [50] Y. Bai, J. Berger, M. Korwar, and N. Orlofsky, Grav. 32, 234003 (2015), arXiv:1505.00797 [gr-qc]. Journal of High Energy Physics 2020, 210 (2020), [24] S. L. Liebling and C. Palenzuela, Phys. Rev. D 94, arXiv:2007.03703 [hep-ph]. 064046 (2016), arXiv:1607.02140 [gr-qc]. [51] D. Ghosh, A. Thalapillil, and F. Ullah, Phys. Rev. D [25] R. M. Wald, Phys. Rev. D 10, 1680 (1974). 103, 023006 (2021), arXiv:2009.03363 [hep-ph]. [26] G. W. Gibbons, Communications in Mathematical [52] B. P. Abbott et al. (LIGO Scientific Collaboration and Physics 44, 245 (1975). Virgo Collaboration), Phys. Rev. Lett. 116, 061102 [27] D. M. Eardley and W. H. Press, Ann. Rev. Astron. As- (2016), 1602.03837. trophys. 13, 381 (1975). [53] M. Hannam, S. Husa, J. A. Gonz´alez, U. Sperhake, [28] Y. Gong, Z. Cao, H. Gao, and B. Zhang, Mon. Not. R. and B. Br¨ugmann, Phys. Rev. D 77, 044020 (2008), Astron. Soc. 488, 2722 (2019), arXiv:1907.05239 [gr-qc]. arXiv:0706.1305 [gr-qc]. [29] Z. Pan and H. Yang, Phys. Rev. D 100, 043025 (2019), [54] I. MacDonald, S. Nissanke, and H. P. Pfeiffer, arXiv:1905.04775 [astro-ph.HE]. Classical and Quantum Gravity 28, 134002 (2011), [30] J. L. Feng, M. Kaplinghat, H. Tu, and H.-B. arXiv:1102.5128 [gr-qc]. Yu, J. Cosm. Astropart. Phys. 2009, 004 (2009), [55] P. Ajith, M. Boyle, D. A. Brown, B. Br¨ugmann,L. T. arXiv:0905.3039 [hep-ph]. Buchman, L. Cadonati, M. Campanelli, T. Chu, Z. B. [31] L. Ackerman, M. R. Buckley, S. M. Carroll, and Etienne, S. Fairhurst, M. Hannam, J. Healy, I. Hinder, M. Kamionkowski, Phys. Rev. D 79, 023519 (2009), S. Husa, and others., Classical Quant. Grav. 29, 124001 arXiv:0810.5126 [hep-ph]. (2012), arXiv:1201.5319 [gr-qc]. 24

[56] J. Aasi, B. P. Abbott, R. Abbott, T. Abbott, M. R. lisneri, A. Vecchio, D. Vetrugno, S. Vitale, M. Volonteri, Abernathy, T. Accadia, F. Acernese, K. Ackley, G. Wanner, H. Ward, P. Wass, W. Weber, J. Ziemer, C. Adams, T. Adams, and et al., Classical and Quantum and P. Zweifel, arXiv e-prints , arXiv:1702.00786 (2017), Gravity 31, 115004 (2014), arXiv:1401.0939 [gr-qc]. arXiv:1702.00786 [astro-ph.IM]. [57] A. K. Mehta, C. K. Mishra, V. Varma, and P. Ajith, [73] E. Berti, V. Cardoso, and C. M. Will, Phys. Rev. D 73, Phys. Rev. D 96, 124010 (2017), arXiv:1708.03501 [gr- 064030 (2006), gr-qc/0512160. qc]. [74] P. Jaiakson, A. Chatrabhuti, O. Evnin, and L. Lehner, [58] V. Varma, S. E. Field, M. A. Scheel, J. Blackman, L. E. Phys. Rev. D 96, 044031 (2017), arXiv:1706.06519 [gr- Kidder, and H. P. Pfeiffer, Phys. Rev. D 99, 064045 qc]. (2019), arXiv:1812.07865 [gr-qc]. [75] R. Arnowitt, S. Deser, and C. W. Misner, General Rel- [59] B. S. Sathyaprakash and S. V. Dhurandhar, Phys. Rev. ativity and Gravitation 40, 1997 (2008). D 44, 3819 (1991). [76] C. W. Misner, K. S. Thorne, and J. A. Wheeler, General [60] S. V. Dhurandhar and B. S. Sathyaprakash, Phys. Rev. relativity (W.H. Freeman and Co., San Francisco, CA, D 49, 1707 (1994). 1973). [61] R. Balasubramanian, B. S. Sathyaprakash, and S. V. [77] R. M. Wald, General relativity (Chicago Univ. Press, Dhurandhar, Phys. Rev. D 53, 3033 (1996), arXiv:gr- Chicago, IL, 1984). qc/9508011 [gr-qc]. [78] V. Cardoso, W.-D. Guo, C. F. B. Macedo, and [62] E.´ E.´ Flanagan and S. A. Hughes, Phys. Rev. D 57, P. Pani, Mon. Not. R. Astron. Soc. 503, 563 (2021), 4535 (1998), gr-qc/9701039. arXiv:2009.07287 [gr-qc]. [63] E.´ E.´ Flanagan and S. A. Hughes, Phys. Rev. D 57, [79] M. Alcubierre, Introduction to 3+1 Numerical Relativity 4566 (1998), gr-qc/9710129. (Oxford University Press, UK, 2008). [64] B. Aylott, J. G. Baker, W. D. Boggs, M. Boyle, P. R. [80] T. W. Baumgarte and S. L. Shapiro, Numerical Rel- Brady, D. A. Brown, B. Br¨ugmann, L. T. Buchman, ativity: Solving Einstein’s Equations on the Computer A. Buonanno, L. Cadonati, J. Camp, M. Campanelli, (Cambridge University Press, 2010). J. Centrella, S. Chatterji, N. Christensen, T. Chu, P. Di- [81] M. Shibata, Numerical Relativity (World Scientific Pub- ener, N. Dorband, Z. B. Etienne, et al., Classical Quant. lishing Co, 2016). Grav. 26, 165008 (2009), arXiv:0901.4399 [gr-qc]. [82] F. L¨offler,J. Faber, E. Bentivegna, T. Bode, P. Diener, [65] I. Hinder et al., Class. Quant. Grav. 31, 025012 (2014), R. Haas, I. Hinder, B. C. Mundim, C. D. Ott, E. Schnet- arXiv:1307.5307 [gr-qc]. ter, G. Allen, M. Campanelli, and P. Laguna, Class. [66] V. E. Hubeny, Phys. Rev. D 59, 064013 (1999), Quantum Grav. 29, 115001 (2012), arXiv:1111.3344 [gr- arXiv:gr-qc/9808043 [gr-qc]. qc]. [67] R. M. Wald, International Journal of Modern Physics [83] Collaborative Effort, Einstein Toolkit for Relativistic D 27, 1843003 (2018). Astrophysics, Astrophysics Source Code Library (2011), [68] H. Kreiss and J. Oliger, Methods for the Approximate ascl:1102.014. Solution of Time Dependent Problems, Global Atmo- [84] EinsteinToolkit, Einstein Toolkit: Open software for rel- spheric Research Programme (GARP): GARP Publica- ativistic astrophysics (2019). tion Series, Vol. 10 (GARP Publication, 1973). [85] J. M. Bowen and J. York, James W., Phys. Rev. D 21, [69] B. P. Abbott, R. Abbott, T. D. Abbott, M. R. Aber- 2047 (1980). nathy, F. Acernese, K. Ackley, C. Adams, T. Adams, [86] J. M. Bowen, Annals of Physics 165, 17 (1985). P. Addesso, R. X. Adhikari, and et al., Physical Review [87] M. Ansorg, B. Br¨ugmann,and W. Tichy, Phys. Rev. D Letters 116, 241102 (2016), arXiv:1602.03840 [gr-qc]. 70, 064011 (2004), gr-qc/0404056. [70] H.-T. Wang, P.-C. Li, J.-L. Jiang, Y.-M. Hu, and [88] U. Sperhake, Phys. Rev. D 76, 104015 (2007), gr- Y.-Z. Fan, arXiv e-prints , arXiv:2004.12421 (2020), qc/0606079. arXiv:2004.12421 [gr-qc]. [89] M. Shibata and T. Nakamura, Phys. Rev. D 52, 5428 [71] V. Cardoso, C. F. B. Macedo, P. Pani, and V. Ferrari, (1995). J. Cosm. Astropart. Phys. 2020, E01 (2020). [90] T. W. Baumgarte and S. L. Shapiro, Phys. Rev. D 59, [72] P. Amaro-Seoane, H. Audley, S. Babak, J. Baker, E. Ba- 024007 (1998). rausse, P. Bender, E. Berti, P. Binetruy, M. Born, [91] M. Alcubierre, B. Br¨ugmann, T. Dramlitsch, J. A. D. Bortoluzzi, J. Camp, C. Caprini, V. Cardoso, Font, P. Papadopoulos, E. Seidel, N. Stergioulas, and M. Colpi, J. Conklin, N. Cornish, C. Cutler, K. Danz- R. Takahashi, Phys. Rev. D 62, 044034 (2000), arXiv:gr- mann, R. Dolesi, L. Ferraioli, V. Ferroni, E. Fitzsimons, qc/0003071. J. Gair, L. Gesa Bote, D. Giardini, F. Gibert, C. Gri- [92] M. Alcubierre, B. Br¨ugmann,P. Diener, M. Koppitz, mani, H. Halloin, G. Heinzel, T. Hertog, M. Hewit- D. Pollney, E. Seidel, and R. Takahashi, Phys. Rev. D son, K. Holley-Bockelmann, D. Hollington, M. Hueller, 67, 084023 (2003), arXiv:gr-qc/0206072. H. Inchauspe, P. Jetzer, N. Karnesis, C. Killow, [93] H. Witek, M. Zilhao, G. Ficarra, and M. Elley, Canuda: A. Klein, B. Klipstein, N. Korsakova, S. L. Larson, J. Li- a public numerical relativity library to probe fundamen- vas, I. Lloro, N. Man, D. Mance, J. Martino, I. Mateos, tal physics (2020). K. McKenzie, S. T. McWilliams, C. Miller, G. Mueller, [94] H. Witek and M. Zilh˜ao, Canuda code (2015), https: G. Nardini, G. Nelemans, M. Nofrarias, A. Petiteau, //bitbucket.org/canuda/. P. Pivato, E. Plagnol, E. Porter, J. Reiche, D. Robert- [95] J. Thornburg, Phys. Rev. D 54, 4899 (1996), arXiv:gr- son, N. Robertson, E. Rossi, G. Russano, B. Schutz, qc/9508014. A. Sesana, D. Shoemaker, J. Slutsky, C. F. Sopuerta, [96] J. Thornburg, Class. Quantum Grav. 21, 743 (2004), T. Sumner, N. Tamanini, I. Thorpe, M. Troebs, M. Val- arXiv:gr-qc/0306056. [97] O. Dreyer, B. Krishnan, D. Shoemaker, and E. Schnet- 25

ter, Phys. Rev. D 67, 024018 (2003), arXiv:gr- [124] B. Bonga, E. Poisson, and H. Yang, American Journal of qc/0206008. Physics 86, 839 (2018), arXiv:1805.01372 [physics.class- [98] C. Reisswig and D. Pollney, Classical Quant. Grav. 28, ph]. 195015 (2011), arXiv:1006.1632 [gr-qc]. [125] B. Bonga and E. Poisson, Phys. Rev. D 99, 064024 [99] E. Schnetter, S. H. Hawley, and I. Hawke, Class. Quan- (2019), arXiv:1808.01288 [gr-qc]. tum Grav. 21, 1465 (2004), arXiv:gr-qc/0310042. [126] F. C. Michel, Astrophys. J. 180, 207 (1973). [100] H. P. Pfeiffer, D. A. Brown, L. E. Kidder, L. Lindblom, [127] S. E. Gralla and T. Jacobson, Mon. Not. R. Astron. Soc. G. Lovelace, and M. A. Scheel, Class. Quantum Grav. 445, 2500 (2014), arXiv:1401.6159 [astro-ph.HE]. 24, S59 (2007), arXiv:gr-qc/0702106. [128] L. Petzold, SIAM Journal on Scientific and Statistical [101] K. Levenberg, Quarterly of Applied Mathematics 2, 164 Computing 4 (1983). (1944). [129] A. Hindmarsh and L. L. Laboratory, ODEPACK, a Sys- [102] D. W. Marquardt, Journal of the Society for Industrial tematized Collection of ODE Solvers (Lawrence Liver- and Applied Mathematics 11, 431 (1963). more National Laboratory, 1982). [103] M. P¨urrer,S. Husa, and M. Hannam, Phys. Rev. D 85, [130] P. Virtfanen, R. Gommers, T. E. Oliphant, M. Haber- 124051 (2012), arXiv:1203.4258 [gr-qc]. land, T. Reddy, D. Cournapeau, E. Burovski, P. Pe- [104] A. Ramos-Buades, S. Husa, and G. Pratten, Phys. Rev. terson, W. Weckesser, J. Bright, S. J. van der Walt, D 99, 023003 (2019), arXiv:1810.00036 [gr-qc]. M. Brett, J. Wilson, K. Jarrod Millman, N. Mayorov, [105] F. Pretorius, Phys. Rev. Lett. 95, 121101 (2005), A. R. J. Nelson, E. Jones, R. Kern, E. Larson, C. Carey, arXiv:gr-qc/0507014. I.˙ Polat, Y. Feng, E. W. Moore, J. Vand erPlas, D. Lax- [106] J. G. Baker, J. Centrella, D.-I. Choi, M. Koppitz, alde, J. Perktold, R. Cimrman, I. Henriksen, E. A. Quin- and J. van Meter, Phys. Rev. Lett. 96, 111102 (2006), tero, C. R. Harris, A. M. Archibald, A. H. Ribeiro, arXiv:gr-qc/0511103. F. Pedregosa, P. van Mulbregt, and S. . . Contributors, [107] M. Campanelli, C. O. Lousto, P. Marronetti, and Y. Zlo- Nature Methods 17, 261 (2020). chower, Phys. Rev. Lett. 96, 111101 (2006), arXiv:gr- [131] M. Maggiore, Gravitational Waves. Vol. 1: Theory and qc/0511048. Experiments, Oxford Master Series in Physics (Oxford [108] Y. Zlochower, J. G. Baker, M. Campanelli, and C. O. University Press, 2007). Lousto, Phys. Rev. D 72, 024021 (2005), arXiv:gr- [132] H. Yang, K. Yagi, J. Blackman, L. Lehner, V. Pascha- qc/0505055 [gr-qc]. lidis, F. Pretorius, and N. Yunes, Phys. Rev. Lett. 118, [109] G. Calabrese, I. Hinder, and S. Husa, Journal of Compu- 161101 (2017), arXiv:1701.05808 [gr-qc]. tational Physics 218, 607 (2006), arXiv:gr-qc/0503056 [133] E. Berti, K. Yagi, H. Yang, and N. Yunes, General Rel- [gr-qc]. ativity and Gravitation 50, 49 (2018), arXiv:1801.03587 [110] S. Husa, J. A. Gonz´alez,M. Hannam, B. Br¨ugmann, [gr-qc]. and U. Sperhake, Classical and Quantum Gravity 25, [134] E. Berti and K. D. Kokkotas, Phys. Rev. D 71, 124008 105006 (2008), arXiv:0706.0740 [gr-qc]. (2005), arXiv:gr-qc/0502065 [gr-qc]. [111] M. C. Babiuc, S. Husa, D. Alic, I. Hinder, C. Lechner, [135] V. Cardoso, A. S. Miranda, E. Berti, H. Witek, and E. Schnetter, B. Szil´agyi,Y. Zlochower, N. Dorband, V. T. Zanchin, Phys. Rev. D 79, 064016 (2009), D. Pollney, and J. Winicour, Classical and Quantum arXiv:0812.1806 [hep-th]. Gravity 25, 125012 (2008), arXiv:0709.3559 [gr-qc]. [136] G. Khanna and R. H. Price, Phys. Rev. D 95, 081501 [112] Z. B. Etienne, J. G. Baker, V. Paschalidis, B. J. Kelly, (2017), arXiv:1609.00083 [gr-qc]. and S. L. Shapiro, Phys. Rev. D 90, 064032 (2014), [137] A. Buonanno, L. E. Kidder, and L. Lehner, Phys. Rev. arXiv:1404.6523 [astro-ph.HE]. D 77, 026004 (2008), arXiv:0709.3839 [astro-ph]. [113] B. Krishnan, Isolated Horizons in Numerical Relativity, [138] M. Kesden, Phys. Rev. D 78, 084030 (2008), Ph.D. thesis, The Pennsylvania State University (2002). arXiv:0807.3043 [astro-ph]. [114] O. Dreyer, B. Krishnan, D. Shoemaker, and E. Schnet- [139] R. P. Kerr, Phys. Rev. Lett. 11, 237 (1963). ter, Phys. Rev. D 67, 024018 (2003), gr-qc/0206008. [140] E. T. Newman, E. Couch, K. Chinnapared, A. Exton, [115] A. Ashtekar, C. Beetle, and J. Lewandowski, Phys. Rev. A. Prakash, and R. Torrence, Journal of Mathematical D 64, 044016 (2001), gr-qc/0103026. Physics 6, 918 (1965). [116] A. Ashtekar and R. O. Hansen, Journal of Mathematical [141] H. M. Siahaan, Phys. Rev. D 101, 064036 (2020), Physics 19, 1542 (1978). arXiv:1907.02158 [gr-qc]. [117] E. Newman and R. Penrose, Journal of Mathematical [142] E. Berti, V. Cardoso, and A. O. Starinets, Classical Physics 3, 566 (1962). Quant. Grav. 26, 163001 (2009), arXiv:0905.2975 [gr- [118] M. Ruiz, M. Alcubierre, D. N´u˜nez,and R. Takahashi, qc]. General Relativity and Gravitation 40, 1705 (2008), [143] Ringdown data. arXiv:0707.4654 [gr-qc]. [144] Ringdown data. [119] N. T. Bishop and L. Rezzolla, Living Reviews in Rela- [145] L. C. Stein, J. Open Source Softw. 4, 1683 (2019), tivity 19, 2 (2016), arXiv:1606.02532 [gr-qc]. arXiv:1908.10377 [gr-qc]. [120] H. Witek, V. Cardoso, C. Herdeiro, A. Nerozzi, U. Sper- [146] O.´ J. C. Dias, M. Godazgar, and J. E. Santos, Phys. Rev. hake, and M. Zilh˜ao, Phys. Rev. D 82, 104037 (2010), Lett. 114, 151101 (2015), arXiv:1501.04625 [gr-qc]. arXiv:1004.4633 [hep-th]. [147] E. Berti, V. Cardoso, J. A. Gonzalez, U. Sperhake, [121] S. A. Teukolsky, Astrophys. J. 185, 635 (1973). M. Hannam, S. Husa, and B. Br¨ugmann, Phys. Rev. [122] E. T. Newman and R. Penrose, Proceedings of the Royal D 76, 064034 (2007), arXiv:gr-qc/0703053 [gr-qc]. Society of London Series A 305, 175 (1968). [148] E. Barausse, A. Buonanno, S. A. Hughes, G. Khanna, [123] A. Ashtekar and B. Bonga, General Relativity and S. O’Sullivan, and Y. Pan, Phys. Rev. D 85, 024046 Gravitation 49, 122 (2017), arXiv:1707.09914 [gr-qc]. 26

(2012), arXiv:1110.3081 [gr-qc]. S. Vitale, R. Weiss, C. Wipf, and M. Zucker, in Bulletin [149] C. J. Moore, R. H. Cole, and C. P. L. Berry, of the American Astronomical Society, Vol. 51 (2019) Classical and Quantum Gravity 32, 015014 (2015), p. 35, arXiv:1907.04833 [astro-ph.IM]. arXiv:1408.0740 [gr-qc]. [162] LIGO Scientific Collaboration, LIGO Algorithm Li- [150] M. Shibata and Y.-I. Sekiguchi, Phys. Rev. D 68, brary - LALSuite, free software (GPL) (2018). 104020 (2003), arXiv:astro-ph/0402184 [astro-ph]. [163] A. Nitz, I. Harry, D. Brown, C. M. Biwer, J. Willis, [151] C. Reisswig, C. D. Ott, U. Sperhake, and E. Schnetter, T. D. Canton, C. Capano, L. Pekowsky, T. Dent, Phys. Rev. D 83, 064008 (2011), arXiv:1012.0595 [gr- A. R. Williamson, M. Cabero, S. De, G. Davies, qc]. D. Macleod, B. Machenschalk, P. Kumar, S. Reyes, [152] L. Lindblom, B. J. Owen, and D. A. Brown, Phys. Rev. T. Massinger, F. Pannarale, M. T´apai, dfinstad, D 78, 124020 (2008), arXiv:0809.3844 [gr-qc]. S. Fairhurst, S. Khan, A. Nielsen, shasvath, S. Kumar, [153] S. T. McWilliams, B. J. Kelly, and J. G. Baker, Phys. idorrington92, L. Singer, H. Gabbard, and B. U. V. Rev. D 82, 024014 (2010), arXiv:1004.0961 [gr-qc]. Gadre, gwastro/pycbc: Pycbc release v1.14.2 (2019). [154] E. Baird, S. Fairhurst, M. Hannam, and P. Murphy, [164] T. Robson, N. J. Cornish, and C. Liu, Classical and Phys. Rev. D 87, 024035 (2013), arXiv:1211.0546 [gr- Quantum Gravity 36, 105011 (2019), arXiv:1803.01944 qc]. [astro-ph.HE]. [155] B. P. Abbott, R. Abbott, T. D. Abbott, M. R. Aber- [165] A. Ashtekar, C. Beetle, O. Dreyer, S. Fairhurst, B. Kr- nathy, F. Acernese, K. Ackley, C. Adams, T. Adams, ishnan, J. Lewandowski, and J. Wi´sniewski, Phys. Rev. P. Addesso, R. X. Adhikari, and et al., Classical and Lett. 85, 3564 (2000), gr-qc/0006006. Quantum Gravity 34, 104002 (2017), arXiv:1611.07531 [166] A. Ashtekar and B. Krishnan, Living Reviews in Rela- [gr-qc]. tivity 7, 10 (2004), gr-qc/0407042. [156] T. Damour, B. R. Iyer, and B. S. Sathyaprakash, Phys. [167] F.-L. Juli´e, J. Cosm. Astropart. Phys. 2018, 033 (2018), Rev. D 57, 885 (1998), gr-qc/9708034. arXiv:1809.05041 [gr-qc]. [157] I. W. Harry and S. Fairhurst, Phys. Rev. D 83, 084002 [168] M. Khalil, N. Sennett, J. Steinhoff, J. Vines, and (2011), arXiv:1012.4939 [gr-qc]. A. Buonanno, Phys. Rev. D 98, 104010 (2018), [158] J. Miller, L. Barsotti, S. Vitale, P. Fritschel, M. Evans, arXiv:1809.03109 [gr-qc]. and D. Sigg, Phys. Rev. D 91, 062005 (2015), [169] G. Bozzola, The Journal of Open Source Software 6, arXiv:1410.5882 [gr-qc]. 3099 (2021). [159] L. Collaboration, Instrument science white paper, ligo [170] B. D. Farris, R. Gold, V. Paschalidis, Z. B. Etienne, document ligo-t1400316-v5. and S. L. Shapiro, Physical Review Letters 109, 221102 [160] M. Punturo, M. Abernathy, F. Acernese, B. Allen, (2012), arXiv:1207.3354 [astro-ph.HE]. N. Andersson, K. Arun, F. Barone, B. Barr, M. Bar- [171] V. Paschalidis, Phys. Rev. D 78, 024002 (2008), suglia, and e. a. Beker, Classical and Quantum Gravity arXiv:0704.2861 [gr-qc]. 27, 194002 (2010). [172] V. Paschalidis, J. Hansen, and A. Khokhlov, Phys. Rev. [161] D. Reitze, R. X. Adhikari, S. Ballmer, B. Barish, D 78, 064048 (2008), arXiv:0712.1258 [gr-qc]. L. Barsotti, G. Billingsley, D. A. Brown, Y. Chen, [173] T. D. Brennan, S. E. Gralla, and T. Jacobson, Classical D. Coyne, R. Eisenstein, M. Evans, P. Fritschel, Quant. Grav. 30, 195012 (2013), arXiv:1305.6890 [gr- E. D. Hall, A. Lazzarini, G. Lovelace, J. Read, B. S. qc]. Sathyaprakash, D. Shoemaker, J. Smith, C. Torrie,