Energetics and phasing of nonprecessing spinning coalescing black hole binaries

We present an improved numerical relativity (NR) calibration of the new effective-one-body (EOB) model for coalescing non precessing spinning black hole binaries recently introduced by Damour and Nagar [Physical Review D 90, 044018 (2014)]. We do so by comparing the EOB predictions to both the phasing and the energetics provided by two independent sets of NR data covering mass ratios $1\leq q \leq 9.989$ and dimensionless spin range $-0.95\leq \chi\leq +0.994$. One set of data is a subset of the Simulating eXtreme Spacetimes (SXS) catalog of public waveforms; the other set consists of new simulations obtained with the Llama code plus Cauchy Characteristic Evolution. We present the first systematic computation of the gauge-invariant relation between the binding energy and the total angular momentum, $E_{b}(j)$, for a large sample of, spin-aligned, SXS and Llama data. The dynamics of the EOB model presented here involves only two free functional parameters, one ($a_6^c(\nu)$) entering the non spinning sector, as a 5PN effective correction to the interaction potential, and one ($c_3(\tilde{a}_1,\tilde{a}_2,\nu))$ in the spinning sector, as an effective next-to-next-to-next-to-leading order correction to the spin-orbit coupling. These parameters are determined (together with a third functional parameter $\Delta t_{\rm NQC}(\chi)$ entering the waveform) by comparing the EOB phasing with the SXS phasing, the consistency of the energetics being checked afterwards. The quality of the analytical model for gravitational wave data analysis purposes is assessed by computing the EOB/NR faithfulness. Over the NR data sample and when varying the total mass between 20 and 200~$M_\odot$ the EOB/NR unfaithfulness (integrated over the NR frequency range) is found to vary between $99.493\%$ and $99.984\%$ with a median value of $99.944\%$.Comment: 26 pages, 27 figures, results improved with respect to first versio

Geometry and Regularity of Moving Punctures

Significant advances in numerical simulations of black-hole binaries have recently been achieved using the puncture method. We examine how and why this method works by evolving a single black hole. The coordinate singularity and hence the geometry at the puncture are found to change during evolution, from representing an asymptotically flat end to being a cylinder. We construct an analytic solution for the stationary state of a black hole in spherical symmetry that matches the numerical result and demonstrates that the evolution is not dominated by artefacts at the puncture but indeed finds the analytical result.Comment: 4 pages, 2 figures. Replaced with version that matches the one published in PRL: one extra figure, and modified abstract and introductio

Vacuum Electromagnetic Counterparts of Binary Black-Hole Mergers

As one step towards a systematic modeling of the electromagnetic (EM) emission from an inspiralling black hole binary we consider a simple scenario in which the binary moves in a uniform magnetic field anchored to a distant circumbinary disc. We study this system by solving the Einstein-Maxwell equations in which the EM fields are chosen with astrophysically consistent strengths. We consider binaries with spins aligned or anti-aligned with the orbital angular momentum and study the dependence of gravitational and EM signals with these spin configurations. Overall we find that the EM radiation in the lowest l=2, m=2 multipole accurately reflects the gravitational one, with identical phase evolutions and amplitudes that differ only by a scaling factor. We also compute the efficiency of the energy emission in EM waves and find that it is given by E^rad_EM/M ~ 10^-15 (M/10^8 M_Sun)^2 (B/10^4 G)^2, hence 13 orders of magnitude smaller than the gravitational energy for realistic magnetic fields. The corresponding luminosity is much smaller than the accretion luminosity if the system is accreting at near the Eddington rate. Most importantly, this EM emission is at frequencies of 10^-4 (10^8 M_Sun/M) Hz, well outside those accessible to astronomical radio observations. As a result, it is unlikely that the EM emission discussed here can be detected directly and simultaneously with the gravitational-wave one. However, indirect processes, driven by changes in the EM fields behavior could yield observable events. In particular if the accretion rate of the circumbinary disc is small and sufficiently stable over the timescale of the final inspiral, then the EM emission may be observable indirectly as it will alter the accretion rate through the magnetic torques exerted by the distorted magnetic field lines

The Asymptotic Falloff of Local Waveform Measurements in Numerical Relativity

We examine current numerical relativity computations of gravitational waves, which typically determine the asymptotic waves at infinity by extrapolation from finite (small) radii. Using simulations of a black hole binary with accurate wave extraction at $r=1000M$, we show that extrapolations from the near-zone are self-consistent in approximating measurements at this radius, although with a somewhat reduced accuracy. We verify that $\psi_4$ is the dominant asymptotic contribution to the gravitational energy (as required by the peeling theorem) but point out that gauge effects may complicate the interpretation of the other Weyl components

Spin Diagrams for Equal-Mass Black-Hole Binaries with Aligned Spins

Binary black-hole systems with spins aligned with the orbital angular momentum are of special interest as they may be the preferred end-state of the inspiral of generic supermassive binary black-hole systems. In view of this, we have computed the inspiral and merger of a large set of binary systems of equal-mass black holes with spins aligned with the orbital angular momentum but otherwise arbitrary. By least-square fitting the results of these simulations we have constructed two "spin diagrams" which provide straightforward information about the recoil velocity |v_kick| and the final black-hole spin a_fin in terms of the dimensionless spins a_1 and a_2 of the two initial black holes. Overall they suggest a maximum recoil velocity of |v_kick|=441.94 km/s, and minimum and maximum final spins a_fin=0.3471 and a_fin=0.9591, respectively.Comment: 4 pages, 3 figs; small changes matching published versio

The final spin from the coalescence of aligned-spin black-hole binaries

Determining the final spin of a black-hole (BH) binary is a question of key importance in astrophysics. Modelling this quantity in general is made difficult by the fact that it depends on the 7-dimensional space of parameters characterizing the two initial black holes. However, in special cases, when symmetries can be exploited, the description can become simpler. For black-hole binaries with unequal masses but with equal spins which are aligned with the orbital angular momentum, we show that the use of recent simulations and basic but exact constraints derived from the extreme mass-ratio limit allow to model this quantity with a simple analytic expression. Despite the simple dependence, the expression models very accurately all of the available estimates, with errors of a couple of percent at most. We also discuss how to use the fit to predict when a Schwarzschild BH is produced by the merger of two spinning BHs, when the total angular momentum of the spacetime ``flips'' sign, or under what conditions the final BH is ``spun-up'' by the merger. Finally, suggest an extension of the fit to include unequal-spin binaries, thus potentially providing a complete description of the final spin from the coalescence of generic black-hole binaries with spins aligned to the orbital angular momentum.Comment: Version matching the published one; small changes throughout to fit space constraints; corrects error in vii) about spin-up/dow

Accurate Evolution of Orbiting Binary Black Holes

We present a detailed analysis of binary black hole evolutions in the last orbit, and demonstrate consistent and convergent results for the trajectories of the individual bodies. The gauge choice can significantly affect the overall accuracy of the evolution. It is possible to reconcile certain gauge dependent discrepancies by examining the convergence limit. We illustrate these results using an initial data set recently evolved by Bruegmann (Phys. Rev. Lett. 92, 211101). For our highest resolution and most accurate gauge, we estimate the duration of this data set's last orbit to be approximately $59 M_{ADM}$.Comment: 4 pages, 3 figure

Horizon Pretracking

We introduce horizon pretracking as a method for analysing numerically generated spacetimes of merging black holes. Pretracking consists of following certain modified constant expansion surfaces during a simulation before a common apparent horizon has formed. The tracked surfaces exist at all times, and are defined so as to include the common apparent horizon if it exists. The method provides a way for finding this common apparent horizon in an efficient and reliable manner at the earliest possible time. We can distinguish inner and outer horizons by examining the distortion of the surface. Properties of the pretracking surface such as its expansion, location, shape, area, and angular momentum can also be used to predict when a common apparent horizon will appear, and its characteristics. The latter could also be used to feed back into the simulation by adapting e.g. boundary or gauge conditions even before the common apparent horizon has formed.Comment: 14 pages, 8 figures, minor change