Numerical solution of the 2+1 Teukolsky equation on a hyperboloidal and horizon penetrating foliation of Kerr and application to late-time decays

In this work we present a formulation of the Teukolsky equation for generic spin perturbations on the hyperboloidal and horizon penetrating foliation of Kerr recently proposed by Racz and Toth. An additional, spin-dependent rescaling of the field variable can be used to achieve stable, long-term, and accurate time-domain evolutions of generic spin perturbations. As an application (and a severe numerical test), we investigate the late-time decays of electromagnetic and gravitational perturbations at the horizon and future null infinity by means of 2+1 evolutions. As initial data we consider four combinations of (non-)stationary and (non-)compact-support initial data with a pure spin-weighted spherical harmonic profile. We present an extensive study of late time decays of axisymmetric perturbations. We verify the power-law decay rates predicted analytically, together with a certain "splitting" behaviour of the power-law exponent. We also present results for non-axisymmetric perturbations. In particular, our approach allows to study the behaviour of the late time decays of gravitational fields for nearly extremal and extremal black holes. For rapid rotation we observe a very prolonged, weakly damped, quasi-normal-mode phase. For extremal rotation the field at future null infinity shows an oscillatory behaviour decaying as the inverse power of time, while at the horizon it is amplified by several orders of magnitude over long time scales. This behaviour can be understood in terms of the superradiance cavity argument

A new gravitational wave generation algorithm for particle perturbations of the Kerr spacetime

We present a new approach to solve the 2+1 Teukolsky equation for gravitational perturbations of a Kerr black hole. Our approach relies on a new horizon penetrating, hyperboloidal foliation of Kerr spacetime and spatial compactification. In particular, we present a framework for waveform generation from point-particle perturbations. Extensive tests of a time domain implementation in the code {\it Teukode} are presented. The code can efficiently deliver waveforms at future null infinity. As a first application of the method, we compute the gravitational waveforms from inspiraling and coalescing black-hole binaries in the large-mass-ratio limit. The smaller mass black hole is modeled as a point particle whose dynamics is driven by an effective-one-body-resummed analytical radiation reaction force. We compare the analytical angular momentum loss to the gravitational wave angular momentum flux. We find that higher-order post-Newtonian corrections are needed to improve the consistency for rapidly spinning binaries. Close to merger, the subdominant multipolar amplitudes (notably the $m=0$ ones) are enhanced for retrograde orbits with respect to prograde ones. We argue that this effect mirrors nonnegligible deviations from circularity of the dynamics during the late-plunge and merger phase. We compute the gravitational wave energy flux flowing into the black hole during the inspiral using a time-domain formalism proposed by Poisson. Finally, a self-consistent, iterative method to compute the gravitational wave fluxes at leading-order in the mass of the particle is presented. For a specific case study with $\hat{a}$=0.9, a simulation that uses the consistent flux differs from one that uses the analytical flux by $\sim35$ gravitational wave cycles over a total of about $250$ cycles. In this case the horizon absorption accounts for about $+5$ gravitational wave cycles

Gravitational waves from black hole binaries in the point-particle limit

In this thesis we have developed a new numerical waveform generation algorithm for particle perturbations of rotating black hole spacetimes. The Teukolsky equation, describing the evolution of gravitational perturbations around such black holes, is rederived in horizon-penetrating and hyperboloidal coordinates using a rotated null-tetrad. By comparison with state-of-the-art literature results we prove that the reformulated equation is solvable numerically in the time-domain at excellent accuracy using standard numerical techniques. In this context it improves on the traditional time-domain Teukolsky algorithm presented by Krivan et al. in 1997, and should thus be viewed as the algorithm of choice for future researchers aiming at the numerical solution of the Teukolsky equation in the time domain. After severe sanity checks, the implementation of the algorithm, the teukode, is employed to study several aspects of the black hole binary problem in the particle limit; a multipolar analysis of merger waveforms, consistency checks of the radiation reaction, horizon absorbed gravitational wave fluxes during particle inspirals, kick and antikick velocties, and gravitational waves from spinning particles

The antikick strikes back: Recoil velocities for nearly extremal binary black hole mergers in the test-mass limit

Gravitational waves emitted from a generic binary black-hole merger carry away linear momentum anisotropically, resulting in a gravitational recoil, or "kick", of the center of mass. For certain merger configurations the time evolution of the magnitude of the kick velocity has a local maximum followed by a sudden drop. Perturbative studies of this "antikick" in a limited range of black hole spins have found that the antikick decreases for retrograde orbits as a function of negative spin. We analyze this problem using a recently developed code to evolve gravitational perturbations from a point-particle in Kerr spacetime driven by an effective-one-body resummed radiation reaction force at linear order in the mass ratio $\nu\ll 1$. Extending previous studies to nearly-extremal negative spins, we find that the well-known decrease of the antikick is overturned and, instead of approaching zero, the antikick increases again to reach $\Delta v/(c\nu^{2})=3.37\times10^{-3}$ for dimensionless spin $\hat{a}=-0.9999$. The corresponding final kick velocity is $v_{end}/(c\nu^{2})=0.076$. This result is connected to the nonadiabatic character of the emission of linear momentum during the plunge. We interpret it analytically by means of the quality factor of the flux to capture quantitatively the main properties of the kick velocity. The use of such quality factor of the flux does not require trajectories nor horizon curvature distributions and should therefore be useful both in perturbation theory and numerical relativity.Comment: 9 pages, 6 figures, submitted to Phys. Rev.

The most powerful astrophysical events: Gravitational-wave peak luminosity of binary black holes as predicted by numerical relativity

For a brief moment, a binary black hole (BBH) merger can be the most powerful astrophysical event in the visible Universe. Here we present a model fit for this gravitational-wave peak luminosity of nonprecessing quasicircular BBH systems as a function of the masses and spins of the component black holes, based on numerical relativity (NR) simulations and the hierarchical fitting approach introduced by X. Jiménez-Forteza et al. [Phys. Rev. D 95, 064024 (2017).]. This fit improves over previous results in accuracy and parameter-space coverage and can be used to infer posterior distributions for the peak luminosity of future astrophysical signals like GW150914 and GW151226. The model is calibrated to the ℓ≤6 modes of 378 nonprecessing NR simulations up to mass ratios of 18 and dimensionless spin magnitudes up to 0.995, and includes unequal-spin effects. We also constrain the fit to perturbative numerical results for large mass ratios. Studies of key contributions to the uncertainty in NR peak luminosities, such as (i) mode selection, (ii) finite resolution, (iii) finite extraction radius, and (iv) different methods for converting NR waveforms to luminosity, allow us to use NR simulations from four different codes as a homogeneous calibration set. This study of systematic fits to combined NR and large-mass-ratio data, including higher modes, also paves the way for improved inspiral-merger-ringdown waveform models

Asymptotic gravitational wave fluxes from a spinning particle in circular equatorial orbits around a rotating black hole

We present a new computation of the asymptotic gravitational wave energy fluxes emitted by a spinning particle in circular equatorial orbits about a Kerr black hole. The particle dynamics is computed in the pole-dipole approximation, solving the Mathisson-Papapetrou equations with the Tulczyjew spin-supplementary-condition. The fluxes are computed, for the first time, by solving the 2+1 Teukolsky equation in the time-domain using hyperboloidal and horizon-penetrating coordinates. Denoting by M the black hole mass and by μ the particle mass, we cover dimensionless background spins a/M=(0,±0.9) and dimensionless particle spins -0.9≤S/μ2≤+0.9. Our results span orbits of Boyer-Lindquist coordinate radii 4≤r/M≤30; notably, we investigate the strong-field regime, in some cases even beyond the last-stable-orbit. We compare our numerical results for the gravitational wave fluxes with the 2.5th order accurate post-Newtonian (PN) prediction obtained analytically by Tanaka et al. [Phys. Rev. D 54, 3762 (1996)]: we find an unambiguous trend of the PN-prediction toward the numerical results when r is large. At r/M=30 the fractional agreement between the full numerical flux, approximated as the sum over the modes m=1, 2, 3, and the PN prediction is 0.5% in all cases tested. This is close to our fractional numerical accuracy (∼0.2%). For smaller radii, the agreement between the 2.5PN prediction and the numerical result progressively deteriorates, as expected. Our numerical data will be essential to develop suitably resummed expressions of PN-analytical fluxes in order to improve their accuracy in the strong-field regime

Spinning test-body orbiting around Schwarzschild black hole: circular dynamics and gravitational-wave fluxes

We release gravitational wave fluxes at null-infinity from a spinning test-body in circular equatorial orbits around a Schwarzschild black hole. Four different prescriptions are used for the dynamics: the Mathisson-Papapetrou formalism under the Tulczyjew (TUL) spin-supplementary-condition (SSC), the Pirani (PIR) SSC and the Ohashi-Kyrian-Semerak (OKS) SSC, and the spinning particle limit of the effective-one-body Hamiltonian (HAM) of [Phys.~Rev.~D.90,~044018(2014)]. For more details see xxxx . The multipolar fluxes are given for l=2,3 m=1,2,3 at the Boyer-Lindquist radii r = 4 5 6 7 8 10 12 15 20 30 , in cases they were not computed the data contains a "42". Note that the fluxes in these data files are assumed to contain both the +m and -m contributions, since they are identical for equatorial orbits and aligned spins. Additionally, the data files contain the key numbers describing the circular dynamics (see paper). Units c=G=1