Analytical Black-Hole Binary Merger Waveforms

We present a highly accurate, fully analytical model for the late inspiral, merger, and ringdown of black-hole binaries with arbitrary mass ratios and spin vectors and the associated gravitational radiation, including the contributions of harmonics beyond the fundamental mode. This model assumes only that nonlinear effects remain small throughout the entire coalescence and is developed based on a physical understanding of the dynamics of late stage binary evolution, in particular, on the tendency of the dynamical binary spacetime to behave like a linear perturbation of the stationary merger-remnant spacetime, even at times before the merger has occurred. We demonstrate that our model agrees with the most accurate numerical relativity results to within their own uncertainties throughout the merger-ringdown phase, and it does so for example cases spanning the full range of binary parameter space that is currently testable with numerical relativity. Furthermore, our model maintains accuracy back to the innermost stable circular orbit of the merger-remnant spacetime over much of the relevant parameter space, greatly decreasing the need to introduce phenomenological degrees of freedom to describe the late inspiral.

Figure 1

Historical comparison of waveform predictions for the dominant $\ell =m=2$ mode of strain for an equal mass, nonspinning merger, including the Lazarus project [6] [(green) dotted line], uncalibrated EOB attached to a ringdown [31] [(red) dash-dotted line], the first stable evolution of a binary merger in numerical relativity by Pretorius [1] [(magenta) solid line], and simulations by the SXS Collaboration [33]. We note that the times and extraction distances for all the waveforms are rescaled by their own estimates of the final mass and that the waveforms from Refs. [1, 31] have small nonzero initial spins. We estimated an uncertainty interval for the SXS waveform (SXS:BBH0001 from the online waveform catalog [32]) [(cyan) shaded region] by combining (in quadrature) the numerical error derived from the multiple available resolutions, the extrapolation error derived from the many available extraction radii, and systematics from residual eccentricity as estimated from a second available SXS simulation (SXS:BBH0002) that used different initial data. When we transition from the same uncalibrated EOB inspiral to the BOB model $5M$ after it reaches twice the ISCO orbital frequency of the merged remnant, so that we are replacing most of the EOB extrapolation to the light ring and subsequent ringdown attachment with BOB [(black) dashed line], the resulting waveform agrees with SXS to within SXS’s uncertainties throughout the merger ringdown and backwards in time beyond the ISCO. For reference, we also show the times (top) and frequencies (bottom) corresponding to the SXS waveform crossing twice the ISCO frequency and twice the light-ring (LR) frequency of the infalling perturber, as well as the frequency (bottom) for crossing twice the circular light-ring frequency (LR, null). All waveforms are aligned at the time of peak strain amplitude, $t_{p,h}$.