Can we measure individual black-hole spins from gravitational-wave observations?

Measurements of black-hole spins from gravitational-wave observations of black-hole binaries with ground-based detectors are known to be hampered by partial degeneracies in the gravitational-wave phasing: between the two component spins, and between the spins and the binary’s mass ratio, at least for signals that are dominated by the binary’s inspiral. Through the merger and ringdown, however, a different set of degeneracies apply. This suggests the possibility that, if the inspiral, merger and ringdown are all within the sensitive frequency band of a detector, we may be able to break these degeneracies and more accurately measure both spins. In this work we investigate our ability to measure individual spins for nonprecessing binaries, for a range of configurations and signal strengths, and conclude that in general the spin of the larger black hole will be measurable (at best) with observations from Advanced LIGO and Virgo. This implies that in many applications waveform models parameterized by only one effective spin will be sufficient. Our work does not consider precessing binaries or subdominant harmonics, although we provide some arguments why we expect that these will not qualitatively change our conclusions.

Figure 1

Symmetric mass ratio $\eta$ versus rescaled reduced spin $\widehat{\chi }$ (left), and component spin ${\chi }_1$ versus ${\chi }_2$ (right) posteriors for configurations with mass ratios $q=1$, 4, 10 (top to bottom) at total mass $50M_{\odot }$ and SNR 30. Each panel shows configurations with equal aligned spins ${\chi }_1={\chi }_2=-0.9$, 0, 0.5, 0.9 (blue, green, orange, and magenta, respectively) with the true parameters indicated by a star symbol. The contours represent 95% credible regions. In the right column we show in addition to the ${\chi }_1\text{−}{\chi }_2$ samples colored bands that are delineated by two lines of constant reduced spin. These are obtained from the posteriors of the single-spin model at 95% credible level.

Figure 2

The 95% credible regions for unequal spins, but identical reduced spin $\chi$. The configurations have SNR 30, and $M_{\text{tot}}=50M_{\odot }$ and mass ratio $q=1$ (left) and $q=4$ (right). While the signal is symmetric under exchange of the masses for $q=1$, the prior is not.

Figure 3

The 95% credible regions for $\eta \text{−}\widehat{\chi }$ posteriors for mass ratio $q=4$ and a range of total masses. We give SNRs in the inspiral (up to the Schwarzschild ISCO) and merger-ringdown (from the ISCO onwards) for nonspinning configurations. The total SNR is 30 in all cases.

Figure 4

The 95% credible regions for ${\chi }_1\text{−}{\chi }_2$ posteriors for mass ratio $q=4$ and a range of total masses. We quote SNRs as in Fig. 3.

Figure 5

The 95% intervals for the rescaled reduced spin $\widehat{\chi }$ (left) and the symmetric mass ratio $\eta$ (right) as a function of total mass for a series of configurations at mass ratios $q=1$, 4, 10 (top to bottom).

Figure 6

The 95% joint credible regions of the component spins for a $q=4$ nonspinning system at a total mass of $12M_{\odot }$ (left) and $100M_{\odot }$ (right) as a function of SNR.

Figure 7

The 95% credible intervals for the component spins for nonspinning binaries at mass ratio $q=4$ as a function of total mass. The panels show the spin on the small black hole (blue) and on the large black hole (green) for SNR 30 (left), SNR 60 (middle), and SNR 100 (right).

Figure 8

The width of 95% credible intervals for chirp mass $M_c$, symmetric mass ratio $\eta$, rescaled reduced spin $\widehat{\chi }$, and the aligned-spin components ${\chi }_1$, ${\chi }_2$ against SNR. As in Fig. 6 results are shown for a nonspinning system at mass ratio $q=4$ at a total mass of $100M_{\odot }$.

Figure 9

The 68%, 90%, and 95% credible regions (solid, dashed, and dotted lines, respectively) for the spin tilt angles (left) and spin magnitudes (right) for two configurations at $q=3$ and $M_{\text{tot}}=12M_{\odot }$. We show credible regions for precessing spin on the small black hole (blue) or the large black hole (red). The true values of the spin magnitudes 0.9 and tilt angles $\pi /3$ are indicated by stars.

Figure 10

The 68%, 90%, and 95% credible regions (solid, dashed, and dotted lines, respectively) for aligned-spin SEOBNRv2-ROM configurations with $q=3$ and spin 0.9 on the small black hole (blue) or the large black hole (red). The left panel shows the spin magnitudes. The peculiar shape of the regions can be explained by the elongated regions in the component spins (right panel).