Binary black hole late inspiral: Simulations for gravitational wave observations

TL;DR

The paper tackles the challenge of predicting gravitational waves from the late inspiral through merger and ringdown of equal-mass, nonspinning BBHs, a regime previously inaccessible to analytic methods. It demonstrates stable, long-duration numerical relativity simulations using the moving-puncture method, achieving ~7 orbits and ~14 waveform cycles starting from the late inspiral, and shows strong agreement with PN waveforms when analyzed in frequency space. By stitching a 3.5PN/2.5PN inspiral to a high-accuracy NR merger–ringdown, the authors produce a full, mass-scalable waveform with phase errors well below a cycle, enabling robust SNR calculations for iLIGO, adLIGO, and LISA. The results underscore the importance of merger–ringdown for detectability and parameter estimation, quantify detector reach across masses and redshifts, and lay groundwork for extending analyses to unequal masses and spins in future work.

Abstract

Coalescing binary black hole mergers are expected to be the strongest gravitational wave sources for ground-based interferometers, such as the LIGO, VIRGO, and GEO600, as well as the space-based interferometer LISA. Until recently it has been impossible to reliably derive the predictions of General Relativity for the final merger stage, which takes place in the strong-field regime. Recent progress in numerical relativity simulations is, however, revolutionizing our understanding of these systems. We examine here the specific case of merging equal-mass Schwarzschild black holes in detail, presenting new simulations in which the black holes start in the late inspiral stage on orbits with very low eccentricity and evolve for ~1200M through ~7 orbits before merging. We study the accuracy and consistency of our simulations and the resulting gravitational waveforms, which encompass ~14 cycles before merger, and highlight the importance of using frequency (rather than time) to set the physical reference when comparing models. Matching our results to PN calculations for the earlier parts of the inspiral provides a combined waveform with less than half a cycle of accumulated phase error through the entire coalescence. Using this waveform, we calculate signal-to-noise ratios (SNRs) for iLIGO, adLIGO, and LISA, highlighting the contributions from the late-inspiral and merger-ringdown parts of the waveform which can now be simulated numerically. Contour plots of SNR as a function of z and M show that adLIGO can achieve SNR >~ 10 for some intermediate-mass binary black holes (IMBBHs) out to z ~ 1, and that LISA can see massive binary black holes (MBBHs) in the range 3x10^4 <~ M/M_Sun <~ 10^7 at SNR > 100 out to the earliest epochs of structure formation at z > 15.

Figures (20)

• Figure 1: Gravitational strain waveforms from the merger of equal-mass Schwarzschild black holes. The late part of the merger ($t \gtrsim -50M$) is robustly determined and relatively easily calculable, while simulations of the late inspiral (early part of the waveform) are rapidly approaching the phasing accuracy required for observational applications Baker:2006ha. The solid blue line shows our current "best" numerical waveform. The dashed red line shows a comparison waveform from a run starting with the same initial data as R4 in Ref. Baker:2006yw and the dash-dotted green curve shows the results from the highest resolution R1 run in Ref. Baker:2006yw. All waveforms have been extracted at $R_{\rm ext} = 40M$ and shifted in time so that the moment of maximum $\psi_4$ radiation amplitude occurs at time $t = 0$. The initial coordinate distance between the punctures, $d_i$, is indicated in all cases.
• Figure 2: The trajectory of one of the binary system's black holes through $\sim 7$ revolutions before coalescence for our high resolution case is shown by the dotted line. The solid line gives the trajectory of the moderately long comparison run. The initial coordinate distance between the punctures, $d_i$, is indicated in both cases.
• Figure 3: The coordinate separation between the puncture black holes is shown as a function of time. The solid line shows the results for the comparison run, which has relatively large eccentricity. The other curves show the three resolutions for our new simulations, all having noticeably less eccentricity. Note that equivalent gauge evolution equations were used in all four cases.
• Figure 4: Convergence plot for the Hamiltonian constraint $C_H$. The top panel shows results from the finest grid and has been scaled so that for 2.5 order convergence the curves should superpose. The bottom panel shows results from the second finest grid and has been scaled so that for fourth order convergence the curves should superpose; the curves indeed appear to be fourth order convergent.
• Figure 5: Convergence plot for the Momentum constraint $C_M$. The top panel shows results from the finest grid and has been scaled so that for 2.5 order convergence the curves should superpose. The bottom panel shows results from the second finest grid and has been scaled so that for fourth-order convergence the curves should superpose; the curves appear less than fourth-order convergent but better than second-order convergent.