Learning about compact binary merger: the interplay between numerical relativity and gravitational-wave astronomy

TL;DR

The paper tackles the challenge of incorporating numerical-relativity waveforms into gravitational-wave data analysis by developing a framework to quantify waveform accuracy for detection in initial LIGO. It introduces a quadrupole-based re-parametrization to simplify template banks and uses NR results to assess detectability and cross-simulation consistency. The authors estimate that roughly 100 non-spinning BBH templates over 100–400 M⊙ are needed for detection, with about 10 simulations across mass ratios to survey parameter space, and they discuss systematic extraction errors and the practicality of NR waveform archives. They further discuss how NR can enhance detection strategies and what GW observations could reveal about BBHs, BNSs, and BH–NS binaries, highlighting the interplay between strong-field gravity, dense matter, and future detector capabilities.

Abstract

Activities in data analysis and numerical simulation of gravitational waves have to date largely proceeded independently. In this work we study how waveforms obtained from numerical simulations could be effectively used within the data analysis effort to search for gravitational waves from black hole binaries. We propose measures to quantify the accuracy of numerical waveforms for the purpose of data analysis and study how sensitive the analysis is to errors in the waveforms. We estimate that ~100 templates (and ~10 simulations with different mass ratios) are needed to detect waves from non-spinning binary black holes with total masses in the range 100 Msun < M < 400 Msun using initial LIGO. Of course, many more simulation runs will be needed to confirm that the correct physics is captured in the numerical evolutions. From this perspective, we also discuss sources of systematic errors in numerical waveform extraction and provide order of magnitude estimates for the computational cost of simulations that could be used to estimate the cost of parameter space surveys. Finally, we discuss what information from near-future numerical simulations of compact binary systems would be most useful for enhancing the detectability of such events with contemporary gravitational wave detectors and emphasize the role of numerical simulations for the interpretation of eventual gravitational-wave observations.

Figures (11)

• Figure 1: The noise sensitivity curves for the LIGO interferometers published in June 2006 ligo-noise. The blue and red curves are the 4km interferometers at Hanford and Livingston, respectively. The green curve is the 2km Hanford interferometer. The LIGO-I noise curve used for the sample calculations in this paper is the solid purple line. Compact binaries generate gravitational-waves which sweep upward in frequency as they inspiral and merge. The frequency ($\approx 40 \textrm{ Hz}$) below which the noise curve rises sharply determines the longest dynamical time-scale of the sources to which the LIGO instruments are sensitive; this, in turn, translates to a largest mass compact binary system to which LIGO is sensitive.
• Figure 2: Samples of $\Psi_4$ from the binary black hole merger simulations discussed here. Evolutions from three different initial conditions are shown: Cook-Pfeiffer $d=16$ (CP d=16), and two scalar field collapse binaries (SFCB), one equal mass, the other with a mass ratio of $1.5:1$. The top plot shows the real part of $\Psi_4$ evaluated along the axis $\theta=0$ orthogonal to the orbital plane (and azimuthal angle $\phi=0$); for brevity we do not show the imaginary part as it looks almost identical modulo a phase shift. The figures below show the real and imaginary parts of $\Psi_4$ evaluated at $\theta=3\pi/8$ (note the different vertical scale). Here we show both components as there are noticeable differences between the two polarizations. In all cases the waveform was extracted at a coordinate radius of $r=50m$, where $m$ is the sum of initial apparent horizon masses; also, the time has been shifted so that $t=0$ corresponds to the peak in wave amplitude, and $\Psi_4$ has been multiplied by a constant complex phase angle to aid comparison.
• Figure 3: A plot demonstrating the dependence on numerical resolution of Cook-Pfeiffer $d=16$ initial data evolutions. The lowest characteristic resolution (dashed line) has a characteristic mesh spacing of $h$, the next lowest one of $3h/4$ (dotted) while the finest resolution has a mesh spacing of $h/2$ (solid). The dominant component of the numerical error is in the phase evolution of the inspiral portion of the wave. See buonanno_et_al for a detailed discussion of the numerical errors in this set of evolutions.
• Figure 4: The amplitude of the Fourier transform of the gravitational waveform, from the evolution of Cook-Pfeiffer initial data, shown in Fig. \ref{['f:waveforms']}. The vertical dashed lines are the estimated frequency of the inner-most stable circular orbit given in Eq. (\ref{['e:fisco']}) and the frequency of the dominant quasi-normal mode, assuming $a=0.7$, given in Eq. (\ref{['e:fqnr']}). The gray shaded region indicates variations in this frequency due to 10% changes in the mass used. Notice how the power in these waves is predominantly emitted between these two frequencies; the initial data is such that the binary is orbiting at or near the ISCO frequency. In addition, the dashed Gray line which follows the amplitude is proportional to $f^{-5/6}$. While this is a convincing fit to the amplitude, we note that there is weak evidence for two power laws $f^{-7/6}$, as given by post-Newtonian approximations Blanchet:1996pi, below $\sim 1.5 f_{\mathrm{isco}}$ and $f^{-5/6}$ above that frequency. These simulations do not cover the inspiral phase well enough to confirm this result.
• Figure 5: The amplitude of the Fourier transform of the gravitational waveforms shown in Fig. \ref{['f:waveforms']}: the top panel is an equal mass SFCB, and the lower panel is a mass ratio $1.5:1$ SFCB. Note the differences between these spectra and that shown in Fig. \ref{['f:spectrum']}. The bump in the equal mass spectrum arises from the hangup of the binary at roughly constant separation the a brief whirl phase prior to merger. While these waveforms are unlikely to be realized in astrophysical scenarios, these spectra provide an example of the rich phenomenology that may be realized in binary mergers.