Inspiral, merger and ringdown of unequal mass black hole binaries: a multipolar analysis

TL;DR

This work analyzes inspiral, merger, and ringdown of seven unequal-mass binary black holes using NR waveforms decomposed into multipoles, then contrasts these with Post-Newtonian quasi-circular (PNQC) predictions and perturbative quasinormal-mode (QNM) theory. It demonstrates that PNQC captures the amplitude–frequency relation for multiple multipoles and characterizes the non-monotonic PN convergence, while establishing robust scaling laws $E_{tot}\sim\eta^2$ and $j_{fin}\sim\eta$ and detailing the multipolar energy distribution as mass ratio grows. The ringdown analysis employs Prony-type methods and three ringdown-start definitions (including EMOP) to extract final BH mass and spin, showing cross-multipole consistency and agreement with energy–momentum balance; EMOP indicates a near-universal ringdown onset around the merger across mass ratios. The study also highlights practical considerations for data analysis, such as memory effects from waveform extraction, and lays the groundwork for hybrid PN/NR templates and improved parameter estimation in gravitational-wave astronomy.

Abstract

We study the inspiral, merger and ringdown of unequal mass black hole binaries by analyzing a catalogue of numerical simulations for seven different values of the mass ratio (from q=M2/M1=1 to q=4). We compare numerical and Post-Newtonian results by projecting the waveforms onto spin-weighted spherical harmonics, characterized by angular indices (l,m). We find that the Post-Newtonian equations predict remarkably well the relation between the wave amplitude and the orbital frequency for each (l,m), and that the convergence of the Post-Newtonian series to the numerical results is non-monotonic. To leading order the total energy emitted in the merger phase scales like eta^2 and the spin of the final black hole scales like eta, where eta=q/(1+q)^2 is the symmetric mass ratio. We study the multipolar distribution of the radiation, finding that odd-l multipoles are suppressed in the equal mass limit. Higher multipoles carry a larger fraction of the total energy as q increases. We introduce and compare three different definitions for the ringdown starting time. Applying linear estimation methods (the so-called Prony methods) to the ringdown phase, we find resolution-dependent time variations in the fitted parameters of the final black hole. By cross-correlating information from different multipoles we show that ringdown fits can be used to obtain precise estimates of the mass and spin of the final black hole, which are in remarkable agreement with energy and angular momentum balance calculations.

Figures (28)

• Figure 1: $|{\rm Re}(Mr\,\psi_{l\,,m})|$ for $q=1.0$ (left) and $q=2.0$ (right). For the equal mass ($q=1.0$) binary the $l=m=3$ component is strongly suppressed, and we do not show it.
• Figure 2: $|Mr\,\psi_{l\,,m}|$ for different mass ratios. Each plot shows only some of the dominant components: $l=m=2,~3,~4$ and $(l=2,~m=1)$. The initial burst of radiation is induced by the initial data, and the wiggles at late times are due to numerical noise.
• Figure 3: Convergence plots for $Mr\psi_{22}$ (left) and $Mr\psi_{44}$ (right). These plots show the differences between runs at different resolutions (as indicated in the inset), scaled to be consistent with second-order accuracy. They refer to run D7, mass ratio $q=4$ and $r_{\rm ext}=30M$.
• Figure 4: Effect of changing the extraction radius on the "memory effect" when we do not apply corrections to the integration constants. These plots refer to the dominant multipole ($l=m=2$) of run D10 with $q=1.0$.
• Figure 5: Effect of changing the extraction radius on the "memory effect" when we do not apply corrections to the integration constants. These plots refer to a small amplitude mode ($l=m=4$) of run D10 with $q=1.0$. The left panel shows the effect of changing the extraction radius at fixed resolution. In the right panel, we change the resolution at fixed extraction radius.