We discuss the properties of the effective-one-body (EOB) multipolar gravitational waveform emitted by nonspinning black-hole binaries of masses $mu$ and $M$ in the extreme-mass-ratio limit, $mu/M= ull 1$. We focus on the transition from quasicircula
r inspiral to plunge, merger and ringdown.We compare the EOB waveform to a Regge-Wheeler-Zerilli (RWZ) waveform computed using the hyperboloidal layer method and extracted at null infinity. Because the EOB waveform keeps track analytically of most phase differences in the early inspiral, we do not allow for any arbitrary time or phase shift between the waveforms. The dynamics of the particle, common to both wave-generation formalisms, is driven by leading-order ${cal O}( u)$ analytically--resummed radiation reaction. The EOB and the RWZ waveforms have an initial dephasing of about $5times 10^{-4}$ rad and maintain then a remarkably accurate phase coherence during the long inspiral ($sim 33$ orbits), accumulating only about $-2times 10^{-3}$ rad until the last stable orbit, i.e. $Deltaphi/phisim -5.95times 10^{-6}$. We obtain such accuracy without calibrating the analytically-resummed EOB waveform to numerical data, which indicates the aptitude of the EOB waveform for LISA-oriented studies. We then improve the behavior of the EOB waveform around merger by introducing and tuning next-to-quasi-circular corrections both in the gravitational wave amplitude and phase. For each multipole we tune only four next-to-quasi-circular parameters by requiring compatibility between EOB and RWZ waveforms at the light-ring. The resulting phase difference around merger time is as small as $pm 0.015$ rad, with a fractional amplitude agreement of 2.5%. This suggest that next-to-quasi-circular corrections to the phase can be a useful ingredient in comparisons between EOB and numerical relativity waveforms.
We show how to solve hyperbolic equations numerically on unbounded domains by compactification, thereby avoiding the introduction of an artificial outer boundary. The essential ingredient is a suitable transformation of the time coordinate in combina
tion with spatial compactification. We construct a new layer method based on this idea, called the hyperboloidal layer. The method is demonstrated on numerical tests including the one dimensional Maxwell equations using finite differences and the three dimensional wave equation with and without nonlinear source terms using spectral techniques.
We present the numerical implementation of a clean solution to the outer boundary and radiation extraction problems within the 3+1 formalism for hyperbolic partial differential equations on a given background. Our approach is based on compactificatio
n at null infinity in hyperboloidal scri fixing coordinates. We report numerical tests for the particular example of a scalar wave equation on Minkowski and Schwarzschild backgrounds. We address issues related to the implementation of the hyperboloidal approach for the Einstein equations, such as nonlinear source functions, matching, and evaluation of formally singular terms at null infinity.
We study linear gravitational perturbations of Schwarzschild spacetime by solving numerically Regge-Wheeler-Zerilli equations in time domain using hyperboloidal surfaces and a compactifying radial coordinate. We stress the importance of including the
asymptotic region in the computational domain in studies of gravitational radiation. The hyperboloidal approach should be helpful in a wide range of applications employing black hole perturbation theory.
We consider the initial value problem for the spherically symmetric, focusing cubic wave equation in three spatial dimensions. We give numerical and analytical evidence for the existence of a universal attractor which encompasses both global and blow
up solutions. As a byproduct we get an explicit description of the critical behavior at the threshold of blowup.
We present two methods to include the asymptotic domain of a background spacetime in null directions for numerical solutions of evolution equations so that both the radiation extraction problem and the outer boundary problem are solved. The first met
hod is based on the geometric conformal approach, the second is a coordinate based approach. We apply these methods to the case of a massless scalar wave equation on a Kerr spacetime. Our methods are designed to allow existing codes to reach the radiative zone by including future null infinity in the computational domain with minor modifications. We demonstrate the flexibility of the methods by considering both Boyer-Lindquist and ingoing Kerr coordinates near the black hole. We also confirm numerically predictions concerning tail decay rates for scalar fields at null infinity in Kerr spacetime due to Hod for the first time.
