Inspiral-merger-ringdown waveforms of spinning, precessing black-hole binaries in the effective-one-body formalism

We describe a general procedure to generate spinning, precessing waveforms that include inspiral, merger and ringdown stages in the effective-one-body (EOB) approach. The procedure uses a precessing frame in which precession-induced amplitude and phase modulations are minimized, and an inertial frame, aligned with the spin of the final black hole, in which we carry out the matching of the inspiral-plunge to merger-ringdown waveforms. As a first application, we build spinning, precessing EOB waveforms for the gravitational modes l=2 such that in the nonprecessing limit those waveforms agree with the EOB waveforms recently calibrated to numerical-relativity waveforms. Without recalibrating the EOB model, we then compare EOB and post-Newtonian precessing waveforms to two numerical-relativity waveforms produced by the Caltech-Cornell-CITA collaboration. The numerical waveforms are strongly precessing and have 35 and 65 gravitational-wave cycles. We find a remarkable agreement between EOB and numerical-relativity precessing waveforms and spins' evolutions. The phase difference is ~ 0.2 rad at merger, while the mismatches, computed using the advanced-LIGO noise spectral density, are below 2% when maximizing only on the time and phase at coalescence and on the polarization angle.Comment: 17 pages, 10 figure

Comparing Gravitational Waveform Extrapolation to Cauchy-Characteristic Extraction in Binary Black Hole Simulations

We extract gravitational waveforms from numerical simulations of black hole binaries computed using the Spectral Einstein Code. We compare two extraction methods: direct construction of the Newman-Penrose (NP) scalar $\Psi_4$ at a finite distance from the source and Cauchy-characteristic extraction (CCE). The direct NP approach is simpler than CCE, but NP waveforms can be contaminated by near-zone effects---unless the waves are extracted at several distances from the source and extrapolated to infinity. Even then, the resulting waveforms can in principle be contaminated by gauge effects. In contrast, CCE directly provides, by construction, gauge-invariant waveforms at future null infinity. We verify the gauge invariance of CCE by running the same physical simulation using two different gauge conditions. We find that these two gauge conditions produce the same CCE waveforms but show differences in extrapolated-$\Psi_4$ waveforms. We examine data from several different binary configurations and measure the dominant sources of error in the extrapolated-$\Psi_4$ and CCE waveforms. In some cases, we find that NP waveforms extrapolated to infinity agree with the corresponding CCE waveforms to within the estimated error bars. However, we find that in other cases extrapolated and CCE waveforms disagree, most notably for $m=0$ "memory" modes.Comment: 26 pages, 20 figure

"Kludge" gravitational waveforms for a test-body orbiting a Kerr black hole

One of the most exciting potential sources of gravitational waves for low-frequency, space-based gravitational wave (GW) detectors such as the proposed Laser Interferometer Space Antenna (LISA) is the inspiral of compact objects into massive black holes in the centers of galaxies. The detection of waves from such "extreme mass ratio inspiral" systems (EMRIs) and extraction of information from those waves require template waveforms. The systems' extreme mass ratio means that their waveforms can be determined accurately using black hole perturbation theory. Such calculations are computationally very expensive. There is a pressing need for families of approximate waveforms that may be generated cheaply and quickly but which still capture the main features of true waveforms. In this paper, we introduce a family of such "kludge" waveforms and describe ways to generate them. We assess performance of the introduced approximations by comparing "kludge" waveforms to accurate waveforms obtained by solving the Teukolsky equation in the adiabatic limit (neglecting GW backreaction). We find that the kludge waveforms do extremely well at approximating the true gravitational waveform, having overlaps with the Teukolsky waveforms of 95% or higher over most of the parameter space for which comparisons can currently be made. Indeed, we find these kludges to be of such high quality (despite their ease of calculation) that it is possible they may play some role in the final search of LISA data for EMRIs.Comment: 29 pages, 11 figures, requires subeqnarray; v2 contains minor changes for consistency with published versio

Statistical Gravitational Waveform Models: What to Simulate Next?

Models of gravitational waveforms play a critical role in detecting and characterizing the gravitational waves (GWs) from compact binary coalescences. Waveforms from numerical relativity (NR), while highly accurate, are too computationally expensive to produce to be directly used with Bayesian parameter estimation tools like Markov-chain-Monte-Carlo and nested sampling. We propose a Gaussian process regression (GPR) method to generate accurate reduced-order-model waveforms based only on existing accurate (e.g. NR) simulations. Using a training set of simulated waveforms, our GPR approach produces interpolated waveforms along with uncertainties across the parameter space. As a proof of concept, we use a training set of IMRPhenomD waveforms to build a GPR model in the 2-d parameter space of mass ratio $q$ and equal-and-aligned spin $\chi_1=\chi_2$. Using a regular, equally-spaced grid of 120 IMRPhenomD training waveforms in $q\in[1,3]$ and $\chi_1 \in [-0.5,0.5]$, the GPR mean approximates IMRPhenomD in this space to mismatches below $4.3\times 10^{-5}$. Our approach can alternatively use training waveforms directly from numerical relativity. Beyond interpolation of waveforms, we also present a greedy algorithm that utilizes the errors provided by our GPR model to optimize the placement of future simulations. In a fiducial test case we find that using the greedy algorithm to iteratively add simulations achieves GPR errors that are $\sim 1$ order of magnitude lower than the errors from using Latin-hypercube or square training grids

Binary black hole spectroscopy

We study parameter estimation with post-Newtonian (PN) gravitational waveforms for the quasi-circular, adiabatic inspiral of spinning binary compact objects. The performance of amplitude-corrected waveforms is compared with that of the more commonly used restricted waveforms, in Advanced LIGO and EGO. With restricted waveforms, the properties of the source can only be extracted from the phasing. For amplitude-corrected waveforms, the spectrum encodes a wealth of additional information, which leads to dramatic improvements in parameter estimation. At distances of $\sim 100$ Mpc, the full PN waveforms allow for high-accuracy parameter extraction for total mass up to several hundred solar masses, while with the restricted ones the errors are steep functions of mass, and accurate parameter estimation is only possible for relatively light stellar mass binaries. At the low-mass end, the inclusion of amplitude corrections reduces the error on the time of coalescence by an order of magnitude in Advanced LIGO and a factor of 5 in EGO compared to the restricted waveforms; at higher masses these differences are much larger. The individual component masses, which are very poorly determined with restricted waveforms, become measurable with high accuracy if amplitude-corrected waveforms are used, with errors as low as a few percent in Advanced LIGO and a few tenths of a percent in EGO. The usual spin-orbit parameter $\beta$ is also poorly determined with restricted waveforms (except for low-mass systems in EGO), but the full waveforms give errors that are small compared to the largest possible value consistent with the Kerr bound. This suggests a way of finding out if one or both of the component objects violate this bound. We also briefly discuss the effect of amplitude corrections on parameter estimation in Initial LIGO.Comment: 28 pages, many figures. Final version accepted by CQG. More in-depth treatment of component mass errors and detectability of Kerr bound violations; improved presentatio

Length requirements for numerical-relativity waveforms

One way to produce complete inspiral-merger-ringdown gravitational waveforms from black-hole-binary systems is to connect post-Newtonian (PN) and numerical-relativity (NR) results to create "hybrid" waveforms. Hybrid waveforms are central to the construction of some phenomenological models for GW search templates, and for tests of GW search pipelines. The dominant error source in hybrid waveforms arises from the PN contribution, and can be reduced by increasing the number of NR GW cycles that are included in the hybrid. Hybrid waveforms are considered sufficiently accurate for GW detection if their mismatch error is below 3% (i.e., a fitting factor about 0.97). We address the question of the length requirements of NR waveforms such that the final hybrid waveforms meet this requirement, considering nonspinning binaries with q = M_2/M_1 \in [1,4] and equal-mass binaries with \chi = S_i/M_i^2 \in [-0.5,0.5]. We conclude that for the cases we study simulations must contain between three (in the equal-mass nonspinning case) and ten (the \chi = 0.5 case) orbits before merger, but there is also evidence that these are the regions of parameter space for which the least number of cycles will be needed.Comment: Corrected some typo

Matching post-Newtonian and numerical relativity waveforms: systematic errors and a new phenomenological model for non-precessing black hole binaries

We present a new phenomenological gravitational waveform model for the inspiral and coalescence of non-precessing spinning black hole binaries. Our approach is based on a frequency domain matching of post-Newtonian inspiral waveforms with numerical relativity based binary black hole coalescence waveforms. We quantify the various possible sources of systematic errors that arise in matching post-Newtonian and numerical relativity waveforms, and we use a matching criteria based on minimizing these errors; we find that the dominant source of errors are those in the post-Newtonian waveforms near the merger. An analytical formula for the dominant mode of the gravitational radiation of non-precessing black hole binaries is presented that captures the phenomenology of the hybrid waveforms. Its implementation in the current searches for gravitational waves should allow cross-checks of other inspiral-merger-ringdown waveform families and improve the reach of gravitational wave searches.Comment: 22 pages, 11 figure

Supermassive Black Hole Tests of General Relativity with eLISA

Motivated by the parameterized post-Einsteinian (ppE) scheme devised by Yunes and Pretorius, which introduces corrections to the post-Newtonian coefficients of the frequency domain gravitational waveform in order to emulate alternative theories of gravity, we compute analytical time domain waveforms that, after a numerical Fourier transform, aim to represent (phase corrected only) ppE waveforms. In this formalism, alternative theories manifest themselves via corrections to the phase and frequency, as predicted by General Relativity (GR), at different post-Newtonian (PN) orders. In order to present a generic test of alternative theories of gravity, we assume that the coupling constant of each alternative theory is manifestly positive, allowing corrections to the GR waveforms to be either positive or negative. By exploring the capabilities of massive black hole binary GR waveforms in the detection and parameter estimation of corrected time domain ppE signals, using the current eLISA configuration (as presented for the ESA Cosmic Vision L3 mission), we demonstrate that for corrections arising at higher than 1PN order in phase and frequency, GR waveforms are sufficient for both detecting and estimating the parameters of alternative theory signals. However, for theories introducing corrections at the 0 and 0.5 PN order, GR waveforms are not capable of covering the entire parameter space, requiring the use of non-GR waveforms for detection and parameter estimation.Comment: 13 pages, 5 figure