First Higher-Multipole Model of Gravitational Waves from Spinning and Coalescing Black-Hole Binaries

Gravitational-wave observations of binary black holes currently rely on theoretical models that predict the dominant multipoles $(\ell =2,|m|=2)$ of the radiation during inspiral, merger, and ringdown. We introduce a simple method to include the subdominant multipoles to binary black hole gravitational waveforms, given a frequency-domain model for the dominant multipoles. The amplitude and phase of the original model are appropriately stretched and rescaled using post-Newtonian results (for the inspiral), perturbation theory (for the ringdown), and a smooth transition between the two. No additional tuning to numerical-relativity simulations is required. We apply a variant of this method to the nonprecessing PhenomD model. The result, PhenomHM, constitutes the first higher-multipole model of spinning and coalescing black-hole binaries, and currently includes the $(\ell ,|m|)=(2,2),(3,3),(4,4),(2,1),(3,2),(4,3)$ radiative moments. Comparisons with numerical-relativity waveforms demonstrate that PhenomHM is more accurate than dominant-multipole-only models for all binary configurations, and typically improves the measurement of binary properties.

Figure 1

A GW signal decomposed into its multipolar contributions, for a system with mass ratio $1:8$ and spin on the larger BH of ${\chi }_1={\mathbf{S}}_1/m_1^2=(0,0,-0.5)$. Our model (PhenomHM) is included as solid ($m=\ell$) and dashed ($m=\ell -1$) black lines. Numerical Relativity (NR)multipoles are displayed in gray, thick lines. Axes are in dimensionless units, where $M$ is the total system mass and $d_L$ is the source’s luminosity distance.

Figure 2

The system considered in Fig. 1, with an inclination of $\iota =\pi /3$, a total mass of $90M_{\odot }$, and a distance of 500 Mpc. In each panel, the NR data are displayed in gray, thick lines. The PhenomHM and PhenomD models are shown in thin black lines that are continuous and dashed, respectively. Top panel: a time domain comparisons of plus polarizations. Bottom panel: a comparison of frequency domain amplitudes. Modeled aLIGO and Einstein Telescope noise spectral densities [20, 21] are displayed in dashed-dotted and dotted black lines, respectively.

Figure 3

Matches between models and NR. All curves are symmetric about $\iota =pi/2$. The NR waveforms contain all multipoles up to $\ell =5$, while PhenomHM contains multipoles with $\ell =|m|\le 4$ and $|m|=\ell -1$. Each curve corresponds to a NR simulation within the PhenomD calibration region [19] scaled to $100M_{\odot }$ with a minimum frequency of 30 Hz. Higher multipoles are not significant for configurations with $M_1/M_2=1$ (green curves with diamond markers), as opposed to cases with $M_1/M_2=4$ (purple curves with circles), and especially $M_1/M_2=8$ (orange curves with squares) and $M_1/M_2=18$ (blue curves with triangles). Each mass-ratio set contains systems with varying nonprecessing spins [19]. Left panel: average matches between NR and a model with only $l=|m|=2$ multipoles. While we used PhenomD, these results are common to all models that lack higher multipoles. Right panel: matches between NR and PhenomHM, which shows significant improvement.

Figure 4

Parameter recovery for a $100M_{\odot }$ mass-ratio $1:4$ binary, with aligned spins ${\chi }_1={\chi }_2=0.5$, optimally oriented to the detector at a distance of 671 Mpc. The higher-multipole PhenomHM model allows us to correctly identify the source orientation and to reduce the uncertainty in distance by approximately 40%.