Measuring the Spin of Black Holes in Binary Systems Using Gravitational Waves

Compact binary coalescences are the most promising sources of gravitational waves (GWs) for ground-based detectors. Binary systems containing one or two spinning black holes are particularly interesting due to spin-orbit (and eventual spin-spin) interactions and the opportunity of measuring spins directly through GW observations. In this Letter, we analyze simulated signals emitted by spinning binaries with several values of masses, spins, orientations, and signal-to-noise ratios, as detected by an advanced LIGO-Virgo network. We find that for moderate or high signal-to-noise ratio the spin magnitudes can be estimated with errors of a few percent (5%–30%) for neutron star–black hole (black hole–black hole) systems. Spins’ tilt angle can be estimated with errors of 0.04 rad in the best cases, but typical values will be above 0.1 rad. Errors will be larger for signals barely above the threshold for detection. The difference in the azimuth angles of the spins, which may be used to check if spins are locked into resonant configurations, cannot be constrained. We observe that the best performances are obtained when the line of sight is perpendicular to the system’s total angular momentum and that a sudden change of behavior occurs when a system is observed from angles such that the plane of the orbit can be seen both from above and below during the time the signal is in band. This study suggests that direct measurement of black hole spin by means of GWs can be as precise as what can be obtained from x-ray binaries.

Figure 1

1-sigma error in the estimation of the spin magnitude (top, %) and tilt angle (bottom, rad) of the black hole for NSBH systems. The color represents ${\theta }_{\overrightarrow{J}\overrightarrow{N}}$ in radians. Small symbols are systems for which both objects have a tilt angle of 60°; large symbols have ${\tau }_1=45°$ and ${\tau }_2=135°$.

Figure 2

1-sigma % error in the estimation of the most massive black hole spin magnitude in $(10,5)M_{\odot }$ BBHs. The color represents ${\theta }_{\overrightarrow{J}\overrightarrow{N}}$ in radians. Small symbols are systems for which both objects have a tilt angle of 60°, and large symbols have ${\tau }_1=45°$ and ${\tau }_2=135°$. Circles are systems where both objects have spin magnitude of 0.9; squares have a spin of 0.9 in the $10M_{\odot }$ black hole and 0.1 in the lighter one. Results for $s_1=s_2=0.1$ systems are not shown, as the errors in that case are above 100%.

Figure 3

Correlation between $1\text{−}\sigma$ errors for the spin magnitude and the tilt angle of the $10M_{\odot }$ black hole. Triangles are systems with reduced spin magnitude of 0.9 for both black holes. Stars are 0.9–0.1 systems, and circles are 0.1–0.1 systems. The size is proportional to the SNR (12, 17, or 30).

Figure 4

$1\text{−}\sigma$ errors for the spin magnitude (circles), tilt angle (pentagons, rad), and ${\theta }_{\overrightarrow{J}\overrightarrow{N}}$ (triangles, rad) against the injected value of ${\theta }_{\overrightarrow{J}\overrightarrow{N}}$ (in units of $\pi$). A change of behavior in the region of ${\theta }_{\overrightarrow{J}\overrightarrow{N}}$ [1.2–1.8] rad is clearly visible.