Toward numerical-relativity informed effective-one-body waveforms for dynamical capture black hole binaries

Dynamical captures of black holes may take place in dense stellar media due to the emission of gravitational radiation during a close passage. Detection of such events requires detailed modeling, since their phenomenology qualitatively differs from that of quasicircular binaries. Very few models can deliver such waveforms, and none includes information from numerical relativity (NR) simulations of nonquasicircular coalescences. In this study we present a first step towards a fully NR-informed effective-one-body (EOB) model of dynamical captures. We perform 14 new simulations of single and double encounter mergers, and use this data to inform the merger-ringdown model of the TEOBResumS-Dalì approximant. We keep the initial energy approximately fixed to the binary mass, and vary the mass-rescaled, dimensionless angular momentum in the range (0.6, 1.1), the mass ratio in (1, 2.15), and aligned dimensionless spins in $(-0.5,0.5)$. We find that the model is able to match NR to 97%, improving previous performances, without the need of modifying the baseline template. Upon NR informing the model, this improves to 99% with the exception of one outlier corresponding to a direct plunge. The maximum $\mathrm{EOB}/\mathrm{NR}$ phase difference at merger for the uninformed model is of 0.15 radians, which is reduced to 0.1 radians after the NR information is introduced. We outline the steps towards a fully informed EOB model of dynamical captures, and discuss future improvements.

Figure 1

Schematic depiction of our initial data. The initial separation $D$ and the value of $P$ are kept fixed at $20M$ and $0.06175M$, respectively, and we vary the initial angle $\theta$ as well as the intrinsic spins of the BH components.

Figure 2

Example orbit, waveforms, frequency, and amplitude and phase difference, for a direct capture (ETK42q1s0, GRA42q1s0, left panel) and zoom whirl (ETK50q1s0, GRA50q1s0, right panel). The differences are defined in (6). We only show the trajectories of one of the black holes to ease visualization. Since these are equal-mass simulations, the other trajectory can be obtained by reflection symmetry. In the direct capture case, the black holes plunge almost immediately, without completing a full orbital cycle. Therefore, the waveform is dominated by merger and ringdown. Conversely, in the zoom-whirl scenario, the bodies undergo multiple encounters before merging, each close passage corresponds to a GW burst. In our code comparison, there is no alignment applied to the waveforms. We show the merger time as a vertical dashed on bottom panels.

Figure 3

Self-convergence for ETK50q1s0 (top row) and for GRA50q1s0 (middle row). Displayed are the differences in amplitude and phase for consecutive resolutions ($L-M$), ($M-H$). In addition, we overlay the curves for $SF(r)(M-H)$ which allows us to test for convergence of a given order $r$, see (16), which we show in each panel. Bottom row: comparison between ETK50q1s0 and GRA50q1s0 at fixed resolution. The simulations for $(L,M,H)$ resolutions have the same grid spacing at puncture location, see Table 2. For each case, we display the absolute value of the difference in normalized amplitude and phase for every available resolution.

Figure 4

Relation between NR initial data and EOB ones for two selected configurations. The procedure of tuning phenomenological corrections to $(E_0^{\mathrm{ADM}},J_0^{\mathrm{ADM}})$ using Eqs. (20) and (21) is globally more efficient (especially for the ETK48q1s0 dataset) than the straightforward transformation from ADM to EOB coordinates.

Figure 5

Equal-mass, nonspinning case. $\mathrm{EOB}/\mathrm{NR}$ waveform comparison. Top panels: $\ell =m=2$ real part and frequency evolution for ETK42q1s0 (left) and ETK50q1s0 (right). Bottom panels: $\mathrm{EOB}/\mathrm{NR}$ phase and (relative) amplitude differences obtained by increasing progressively the amount of NR input to describe the merger and ringdown part.

Figure 6

Unequal-mass ($q=2.15$), nonspinning case, dataset ETK42q215: $\mathrm{EOB}/\mathrm{NR}$ phasing comparison including the available higher harmonics. Note that the (2, 2) mode is completed by the NR-informed, noncircular, merger, and ringdown, while the higher harmonics still retain the standard quasicircular contributions. Despite this, the agreement is acceptable. Also note some unphysical features in the ringdown for (4, 4) and (5, 5) due to the calculation of the strain from ${\psi }_4$.

Figure 7

$\mathrm{EOB}/\mathrm{NR}$ unfaithfulness for the $\ell =m=2$ mode obtained in zero noise for all configurations considered. Note how $\overline{\mathcal{F}}$ decreases due to the NR improvement of the merger and ringdown part of the EOB waveform.

Figure 8

$\mathrm{EOB}/\mathrm{NR}$ unfaithfulness for the $\ell =m=2$ mode obtained with the zero-detuned, high-power noise spectral density of Advanced LIGO [77].

Figure 9

$\mathrm{EOB}/\mathrm{NR}$ comparisons for spin-aligned binaries. Top row: ETK42q1s* configurations with ${\chi }_1={\chi }_2=0$ (left) and ${\chi }_1=-{\chi }_2=0.5$ (right). The cancellation of the spin-orbit interaction as predicted by the EOB model is present also in the NR data. Bottom panels: ${\chi }_1={\chi }_2=-0.5$ (left) and ${\chi }_1={\chi }_2=+0.5$ (right). When spins are negative, the plunge occurs faster and the signal is shorter than the ${\chi }_1={\chi }_2=0$ case. When spins are positive, the repulsive character of the spin-orbit coupling yields a first encounter followed then by the plunge and merger. The phase differences are reported in Fig. 10.

Figure 10

$\mathrm{EOB}/\mathrm{NR}$ phase differences for the configurations of Fig. 9. The markers indicate the location of the waveform amplitude peak. Note the rather small differences (only during the precursor phase) between the ${\chi }_1={\chi }_2=0$ and ${\chi }_1=-{\chi }_2=0.5$ datasets.

Figure 11

Puncture tracks and ${\psi }_4^{(2,2)}$ for all einstein toolkit simulations.

Figure 12

Comparison between einstein toolkit and gr-athena++ simulations ETK50 and GRA50 at fixed resolution. We display the absolute value of the difference in normalized amplitude and phase of the nonextrapolated ${\psi }_4$ for every available resolution.

Figure 13

$\mathrm{EOB}/\mathrm{NR}$ unfaithfulness in zero noise for all configurations considered. We contrast the results for the strain computed from the leading mode only, and the strain including the modes $\{(2,2),(2,1),(3,3),(3,2),(4,4),(5,5)\}$. We consider an angular orientation with $\theta =\pi /3$, $\varphi =0$. The highest value for the unfaithfulness including subleading modes corresponds to the simulation ETK37q1s0.

Figure 14

Comparison between the NR configuration ETK50q1s0 and the corresponding EOB configuration initialized at $r_{\mathrm{EOB}}=300M$. We display the real parts of the 22 modes for each waveform, enlarging the region covered by the NR simulation on the right panel.

Figure 15

$\mathrm{EOB}/\mathrm{NR}$ comparison plots for the real parts of the waveforms for the leading (2, 2) modes, normalized by $\nu$. In the EOB case, we plot the fully NR-informed model, including ringdown and NQC parameters.

Figure 16

$\mathrm{EOB}/\mathrm{NR}$ comparison plots for the frequency of the leading (2, 2) modes. In the EOB case, we plot the fully NR-informed model, including ringdown and NQC parameters.

Figure 17

Binding energy versus dimensionless angular momentum curves for all simulations. The blue marker denotes the NR merger value for each dataset.