Toward faithful templates for non-spinning binary black holes using the effective-one-body approach

TL;DR

This paper enhances the effective-one-body (EOB) framework for non-spinning binary black holes by introducing a pseudo-4PN correction to the radial potential and calibrating it against numerical-relativity (NR) waveforms for mass ratios $m_1/m_2 \in \{1, 3/2, 2, 4\}$. It constructs complete inspiral–merger–ringdown waveforms by evolving the EOB dynamics through plunge and attaching a three-mode quasi-normal-mode ringdown using NR-informed final mass $M_f$ and spin $a_f$, with key parameters such as $M_f/M$ and $a_f/M_f$ fitted as functions of the symmetric mass ratio $\eta$. The resulting p4PN-EOB waveforms achieve phase differences $<0.08$ cycles relative to NR across tested cases and display high overlap (minmax FFs $\gtrsim 0.98$) for multiple modes, validating their use for detection and initial parameter estimation in ground-based detectors. The work also identifies amplitude differences and mode-mixing as current limitations, outlining paths for future refinements and extensions to spinning or precessing binaries. Overall, the study shows that NR-guided analytic templates based on the EOB approach can faithfully capture the full coalescence waveform of non-spinning binary black holes.

Abstract

We present an accurate approximation of the full gravitational radiation waveforms generated in the merger of non-eccentric systems of two non-spinning black holes. Utilizing information from recent numerical relativity simulations and the natural flexibility of the effective-one-body (EOB) model, we extend the latter so that it can successfully match the numerical relativity waveforms during the last stages of inspiral, merger and ringdown. By ``successfully'' here, we mean with phase differences < 8% of a gravitational-wave cycle accumulated by the end of the ringdown phase, maximizing only over time of arrival and initial phase. We obtain this result by simply adding a 4-post-Newtonian order correction in the EOB radial potential and determining the (constant) coefficient by imposing high-matching performances with numerical waveforms of mass ratios m1/m2 = 1, 3/2, 2 and 4, m1 and m2 being the individual black-hole masses. The final black-hole mass and spin predicted by the numerical simulations are used to determine the ringdown frequency and decay time of three quasi-normal-mode damped sinusoids that are attached to the EOB inspiral-(plunge) waveform at the EOB light-ring. The EOB waveforms might be tested and further improved in the future by comparison with extremely long and accurate inspiral numerical-relativity waveforms. They may already be employed for coherent searches and parameter estimation of gravitational waves emitted by non-spinning coalescing binary black holes with ground-based laser-interferometer detectors.

Figures (9)

• Figure 1: In the left panel we show the position of the LSO and light-ring as function of the parameter $\lambda$, for different mass ratios: 4:1 (dotted line), 2:1 (dot-dashed line), 3:2 (dashed line) and 1:1 (continuous line). In the right panel we show: (top part) the energy for circular orbits as a function of the frequency evaluated from the EOB Hamiltonian, (bottom part) the radial potential as function of the radial coordinate for a massless particle in the EOB model. The various curves refer to different PN orders.
• Figure 2: In the left panel we show the (minmax) FF between the high and medium resolution runs, $\langle h^{\rm NR,h}, h^{\rm NR,m} \rangle$, as a function of the binary total-mass $M$. The ${\rm FF}$s are evaluated using LIGO's PSD. If we use white noise we find 0.9922 and 0.9920 for mass ratios $1:1$ and $4:1$, respectively. In the right panel, for different mass ratios, we show how the (minmax) FF $\langle h^{\rm NR}, h^{\rm EOB} \rangle$ (computed using white noise) depends on the parameter $\lambda$. For mass ratios $1:1$, $2:1$, and $3:2$ we compute $\langle h_{22}^{\rm NR}, h_{22}^{\rm EOB} \rangle$, while for $4:1$ we show also results for $\langle h_{33}^{\rm NR}, h_{33}^{\rm EOB} \rangle$ and $\langle h_{44}^{\rm NR}, h_{44}^{\rm EOB} \rangle$. The vertical line refers to the value $\lambda=60$ which we employ in all subsequent analyses.
• Figure 3: We show the differences in the orbital frequency between the 3.5PN-Tt3 model and 3.5PN-T, 3.5PN-Tt1, 3.5PN-EOB, p4PN-EOB models for mass ratios 1:1 (left panel) and 4:1 (right panel). The PN frequencies coincide at $\omega M = 0.017$ at $t=0$, and end at $\omega M = 0.035$.
• Figure 4: Equal-mass binary. In the left panel we plot the NR and p4PN-EOB frequencies and amplitudes, and the phase difference between the EOB and NR $h_{22}$. In the right panel we compare the EOB and NR $h_{22}$. We maximize only on the initial phase and time of arrival. Note that we show only the last few cycles. The complete inspiral waveform has 14 GW cycles.
• Figure 5: Binary with mass ratio $4:1$. In the left panel we plot the NR and p4PN-EOB frequencies and amplitudes, and the phase difference between the EOB and NR $h_{22}$. In the right panel we compare the EOB and NR $h_{22}$. We maximize only over the initial phase and time of arrival. Note that we show only the last few cycles. The complete inspiral waveform has 9 GW cycles.