Distinguishing black-hole spin-orbit resonances by their gravitational-wave signatures

If binary black holes form following the successive core collapses of sufficiently massive binary stars, precessional dynamics may align their spins, ${\mathbf{S}}_{\mathbf{1}}$ and ${\mathbf{S}}_{\mathbf{2}}$, and the orbital angular momentum $\mathbf{L}$ into a plane in which they jointly precess about the total angular momentum $\mathbf{J}$. These spin orientations are known as spin-orbit resonances since ${\mathbf{S}}_{\mathbf{1}}$, ${\mathbf{S}}_{\mathbf{2}}$, and $\mathbf{L}$ all precess at the same frequency to maintain their planar configuration. Two families of such spin-orbit resonances exist, differentiated by whether the components of the two spins in the orbital plane are either aligned or antialigned. The fraction of binary black holes in each family is determined by the stellar evolution of their progenitors, so if gravitational-wave detectors could measure this fraction they could provide important insights into astrophysical formation scenarios for binary black holes. In this paper, we show that even under the conservative assumption that binary black holes are observed along the direction of $\mathbf{J}$ (where precession-induced modulations to the gravitational waveforms are minimized), the waveforms of many members of each resonant family can be distinguished from all members of the other family in events with signal-to-noise ratios $\rho \simeq 10$, typical of those expected for the first detections with Advanced LIGO and Virgo. We hope that our preliminary findings inspire a greater appreciation of the capability of gravitational-wave detectors to constrain stellar astrophysics and stimulate further studies of the distinguishability of spin-orbit resonant families in more expanded regions of binary black-hole parameter space.

Figure 1

Conventions and definitions used in this paper. We work in the radiation frame, where the $z$ axis is oriented along the line of sight $\widehat{\mathbf{n}}$. The orbital angular momentum $\mathbf{L}$ lies in the $xz$-plane at $f_{\text{ref}}$ and is inclined by an angle $\iota$ with respect to the line of sight. The directions of the spins ${\mathbf{S}}_{\mathbf{1}}$ (blue) and ${\mathbf{S}}_{\mathbf{2}}$ (red) are specified using polar angles ${\theta }_i$ and azimuthal angles ${\mathrm{\Phi }}_i$ $(i=1,2)$ which are defined in a frame where the $z$ axis is aligned with the orbital angular momentum $\mathbf{L}$. As resonant binaries precess, their orbital angular momentum and spins remain coplanar, implying that the angle $\mathrm{\Delta }\mathrm{\Phi }={\mathrm{\Phi }}_2-{\mathrm{\Phi }}_1$ (green) remains fixed at either $0°$ or $\pm 180°$. In later sections of the paper, we will fix $\iota$ and ${\mathrm{\Phi }}_1$ by aligning the line of sight with the total angular momentum: cf. Eqs. (a4) and (a5).

Figure 2

One-parameter families of resonant binaries superimposed on contours of constant $\xi ={\mathbf{S}}_{\mathbf{0}}·\widehat{\mathbf{L}}/M^2$. Red (top-left) and green (bottom-right) curves show resonant configurations in the two coplanar families for our canonical choice of the parameters ($q=0.8$, $M=13.5M_{\odot }$, ${\chi }_1={\chi }_2=1$) at three different emitted frequencies: 20 Hz (dashed), 60 Hz (i.e., $f_{\text{ref}}$, solid), and 150 Hz (dotted). The value of $\xi \in [-1,1]$ is constant over the sloped dashed lines. Each of them always crosses the resonant curves exactly once, thus unambiguously identifying a single binary [i.e., a pair $({\theta }_1,{\theta }_2)$] in each family.

Figure 3

Inner product of unit vectors in the directions of the orbital angular momentum $\mathbf{L}$ and total angular momentum $\mathbf{J}$ for BBHs in spin-orbit resonances at the reference frequency $f_{\text{ref}}$. Solid and dashed curves correspond to the $\mathrm{\Delta }\mathrm{\Phi }=0°$ and $\mathrm{\Delta }\mathrm{\Phi }=\pm 180°$ families, respectively. If the line of sight points along $\widehat{\mathbf{J}}$, $\widehat{\mathbf{J}}·\widehat{\mathbf{L}}=\cos \iota$ as given by Eq. (a5).

Figure 4

Expected squared SNR per unit frequency [Eq. (7)] for binaries belonging to the two resonant families. The sources all have a projected effective spin $\xi =-0.5$, but they are viewed at different inclinations $\iota$. Waveforms from binaries in the $\mathrm{\Delta }\mathrm{\Phi }=0°$ resonance (top row) exhibit wider modulations due to greater precession of the orbital plane. On the other hand, in the $\mathrm{\Delta }\mathrm{\Phi }=180°$ family (bottom row), the components of the two spins in the orbital plane partially cancel each other, reducing the precession of $\mathbf{L}$. The expected modulation varies with $\iota$ and is minimized by the values of $\iota$ predicted by Eq. (a5), for which the line of sight $\widehat{\mathbf{n}}$ is parallel to the total angular momentum $\mathbf{J}$ (first column, red curves). With our canonical choice of the parameters and $\xi =-0.5$, the $\widehat{\mathbf{n}}=\widehat{\mathbf{J}}$ case corresponds to $\iota \simeq 27°$ for $\mathrm{\Delta }\mathrm{\Phi }=0°$, and $\iota \simeq 5°$ for $\mathrm{\Delta }\mathrm{\Phi }=180°$.

Figure 5

Overlap between resonant binaries. A source in either the $\mathrm{\Delta }\mathrm{\Phi }=0°$ resonance (left) or the $\mathrm{\Delta }\mathrm{\Phi }=180°$ resonance (right), is compared with members of the other family parametrized by ${\xi }_{\text{template}}$. Each one-parameter family is built varying over the spin direction through $\xi$, while all the remaining parameters are fixed. Five different sources are considered, but the same trend holds for every value of ${\xi }_{\text{source}}$. Each curve possesses a clear unique maximum, pairing the source binary with a best matching template in the other family.

Figure 6

Pairing between binaries in different resonant families. Each source ${\xi }_{\text{source}}$ is paired with the best matching template ${\xi }_{\text{template}}^{\text{BM}}$ from the other resonant family, i.e., the template that maximizes the overlap $\mathcal{O}$ of Eq. (8). Within our numerical precision, each member of the pair is the other’s best match; the two curves are symmetric about the diagonal ${\xi }_{\text{source}}={\xi }_{\text{template}}^{\text{BM}}$. Maximized overlaps (the tips of the peaks shown in Fig. 5) are shown on the color scale.

Figure 7

Distributions of $\xi$ and $\mathrm{\Delta }\mathrm{\Phi }$ at $f_{\text{ref}}=60\mathrm{Hz}$ in the astrophysical models we developed in Ref. [14]. All scenarios shown here assume isotropic supernova kicks. Binaries for which tides align the spin of the secondary with the orbital angular momentum prior to the second supernova are typically locked into resonances by the end of the inspiral. When mass transfer prior to the first supernova causes the secondary to form the more massive BH (“Tides RMR,” green circles), the BBHs tend to be attracted to the $\mathrm{\Delta }\mathrm{\Phi }=0°$ resonances. If this mass-ratio reversal does not occur (“Tides SMR,” red triangles), binaries will instead fall into the $\mathrm{\Delta }\mathrm{\Phi }=180°$ resonances. Without this tidal alignment (“No Tides”, blue squares), BBHs will show no preference for either resonant family. BBHs inside the dashed boxes are within $\pm 50°$ degrees of either $\mathrm{\Delta }\mathrm{\Phi }=0°$ or $\mathrm{\Delta }\mathrm{\Phi }=180°$ at $f_{\text{ref}}$ and have maximum overlaps below 0.99 with the other family.

Figure 8

Fraction of binaries from the astrophysical distributions shown in Fig. 7 that can be identified as belonging to one of the resonant families as a function of their maximum allowed overlap ${\mathcal{O}}_{\text{max}}=1-{\rho }^{-2}$ with their match in the opposite family. As the SNR $\rho$ increases, the range of values of ${\xi }_{\text{source}}$ with $\mathcal{O}<{\mathcal{O}}_{\text{max}}$ increases as seen in Fig. 6. This range determines the heights of the dashed boxes shown in Fig. 7; as the areas of these boxes increase so too does the fraction of the points contained within them. The solid (dashed) curves show the fraction of binaries contained within the box centered on $\mathrm{\Delta }\mathrm{\Phi }=0°(180°)$. The green, red, and blue curves correspond to the “Tides RMR,” “Tides SMR,” and “No Tides” scenarios, respectively. We see that virtually all identified binaries belong to the $\mathrm{\Delta }\mathrm{\Phi }=0°(180°)$ family in the “Tides RMR” (SMR) scenario, while a comparable fraction of identified binaries belong to each family in the “No Tides” scenario.

Figure 9

Phasing of best matching BH binaries using the single-spin approximation, for a single source in the $\mathrm{\Delta }\mathrm{\Phi }=0°$ family (left) and in the $\mathrm{\Delta }\mathrm{\Phi }=180°$ family (right). We denote by $\mathrm{\Delta }X$ (left axis in each panel) the difference between the quantity $X$ for the source and for the best matching template, as a function of the GW frequency $f$. The solid blue line shows the GW phase offset between the source and the best matching template computed within the single-spin approximation (15). The dashed red and dot-dashed green lines show the orbital and the precessional contributions, respectively. The dotted black lines show the source squared SNR per unit frequency assuming a luminosity distance $D=1\text{Mpc}$, as reported on the right axis in each panel.

Figure 10

Predictions for the best matching binaries using the single-spin approximation. Solid blue lines show the overlap between a fixed source (with $\mathrm{\Delta }\mathrm{\Phi }=0°$ on the left and $\mathrm{\Delta }\mathrm{\Phi }=180°$ on the right) and different templates from the other family. The dashed red curves show the offset $|\mathrm{\Delta }\mathcal{N}|$ of the effective number of cycles (21) between the fixed source and each template. The dash-dotted green curves show the approximate dephasing ${\varphi }_{\text{rms}}^2$ (27); both $|\mathrm{\Delta }\mathcal{N}|$ and ${\varphi }_{\text{rms}}^2$ are shown in arbitrary units. Vertical dotted lines show the best matching template and the predictions for this template using the criteria $\min |\mathrm{\Delta }\mathcal{N}|$ and $\min {\varphi }_{\text{rms}}^2$ described in Sec. 4b.

Figure 11

Performance of the two predictors, $\min |\mathrm{\Delta }\mathcal{N}|$ and $\min {\varphi }_{\text{rms}}^2$. For various sources in both families, we show the difference between the highest-overlap template ${\xi }_{\text{template}}^{\text{BM}}$ and the predicted value ${\xi }_{\text{template}}^{\mathrm{P}}$.