A hyperboloidal study of tail decay rates for scalar and Yang-Mills fields

We investigate the asymptotic behavior of spherically symmetric solutions to scalar wave and Yang-Mills equations on a Schwarzschild background. The studies demonstrate the astrophysical relevance of null infinity in predicting radiation signals for gravitational wave detectors and show how test fields on unbounded domains in black hole spacetimes can be simulated conveniently by numerically solving hyperboloidal initial value problems.Comment: 8 pages, 7 figure

Caustic echoes from a Schwarzschild black hole

We present the first numerical construction of the scalar Schwarzschild Green function in the time-domain, which reveals several universal features of wave propagation in black hole spacetimes. We demonstrate the trapping of energy near the photon sphere and confirm its exponential decay. The trapped wavefront propagates through caustics resulting in echoes that propagate to infinity. The arrival times and the decay rate of these caustic echoes are consistent with propagation along null geodesics and the large l-limit of quasinormal modes. We show that the four-fold singularity structure of the retarded Green function is due to the well-known action of a Hilbert transform on the trapped wavefront at caustics. A two-fold cycle is obtained for degenerate source-observer configurations along the caustic line, where the energy amplification increases with an inverse power of the scale of the source. Finally, we discuss the tail piece of the solution due to propagation within the light cone, up to and including null infinity, and argue that, even with ideal instruments, only a finite number of echoes can be observed. Putting these pieces together, we provide a heuristic expression that approximates the Green function with a few free parameters. Accurate calculations and approximations of the Green function are the most general way of solving for wave propagation in curved spacetimes and should be useful in a variety of studies such as the computation of the self-force on a particle.Comment: 18 pages, 23 figure

A geometric framework for black hole perturbations

Black hole perturbation theory is typically studied on time surfaces that extend between the bifurcation sphere and spatial infinity. From a physical point of view, however, it may be favorable to employ time surfaces that extend between the future event horizon and future null infinity. This framework resolves problems regarding the representation of quasinormal mode eigenfunctions and the construction of short-ranged potentials for the perturbation equations in frequency domain.Comment: 4 pages, 1 figur

Binary black hole coalescence in the extreme-mass-ratio limit: Testing and improving the effective-one-body multipolar waveform

We discuss the properties of the effective-one-body (EOB) multipolar gravitational waveform emitted by nonspinning black-hole binaries of masses and M in the extreme-mass-ratio limit µ/M = v « 1. We focus on the transition from quasicircular inspiral to plunge, merger, and ringdown. We compare the EOB waveform to a Regge-Wheeler-Zerilli 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 a leading-order O(v) analytically resummed radiation reaction. The EOB and the Regge-Wheeler-Zerilli waveforms have an initial dephasing of about 5 X 10^(-4) rad and maintain then a remarkably accurate phase coherence during the long inspiral (~33 orbits), accumulating only about -2 X 10^(-3) rad until the last stable orbit, i.e. ΔØ/Ø~-5.95 X 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 studies concerning the Laser Interferometer Space Antenna. We then improve the behavior of the EOB waveform around merger by introducing and tuning next-to-quasicircular corrections in both the gravitational wave amplitude and phase. For each multipole we tune only four next-to-quasicircular parameters by requiring compatibility between EOB and Regge-Wheeler-Zerilli waveforms at the light ring. The resulting phase difference around the merger time is as small as ±0.015 rad, with a fractional amplitude agreement of 2.5%. This suggest that next-to-quasicircular corrections to the phase can be a useful ingredient in comparisons between EOB and numerical-relativity waveforms

Horizon-absorption effects in coalescing black-hole binaries: An effective-one-body study of the non-spinning case

We study the horizon absorption of gravitational waves in coalescing, circularized, nonspinning black hole binaries. The horizon absorbed fluxes of a binary with a large mass ratio (q=1000) obtained by numerical perturbative simulations are compared with an analytical, effective-one-body (EOB) resummed expression recently proposed. The perturbative method employs an analytical, linear in the mass ratio, effective-one-body (EOB) resummed radiation reaction, and the Regge-Wheeler-Zerilli (RWZ) formalism for wave extraction. Hyperboloidal (transmitting) layers are employed for the numerical solution of the RWZ equations to accurately compute horizon fluxes up to the late plunge phase. The horizon fluxes from perturbative simulations and the EOB-resummed expression agree at the level of a few percent down to the late plunge. An upgrade of the EOB model for nonspinning binaries that includes horizon absorption of angular momentum as an additional term in the resummed radiation reaction is then discussed. The effect of this term on the waveform phasing for binaries with mass ratios spanning 1 to 1000 is investigated. We confirm that for comparable and intermediate-mass-ratio binaries horizon absorbtion is practically negligible for detection with advanced LIGO and the Einstein Telescope (faithfulness greater than or equal to 0.997)

Hyperboloidal layers for hyperbolic equations on unbounded domains

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 combination 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.Comment: 23 pages, 23 figure

Saddle-point dynamics of a Yang-Mills field on the exterior Schwarzschild spacetime

We consider the Cauchy problem for a spherically symmetric SU(2) Yang-Mills field propagating outside the Schwarzschild black hole. Although solutions starting from smooth finite energy initial data remain smooth for all times, not all of them scatter since there are non-generic solutions which asymptotically tend to unstable static solutions. We show that a static solution with one unstable mode appears as an intermediate attractor in the evolution of initial data near a border between basins of attraction of two different vacuum states. We study the saddle-point dynamics near this attractor, in particular we identify the universal phases of evolution: the ringdown approach, the exponential departure, and the eventual decay to one of the vacuum states.Comment: 15 pages, 5 figure

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.

Universality of global dynamics for the cubic wave equation

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 blowup solutions. As a byproduct we get an explicit description of the critical behavior at the threshold of blowup.Comment: 13 pages, 15 figures. Uses IOP-style. Updated to conform with published versio