Matthew Mould
Career Stage
Student (postgraduate)
Poster Abstract
Binary black holes inspiral and eventually merge due to the emission of gravitational waves. Black holes also rotate, and in binaries the two black hole spins interact with each other and their orbit in a very complex way. But if the spins point exactly orthogonal to the orbital plane this complex motion vanishes, much like a spinning top that spins perfectly upright. We prove, however, that in one such configuration the black hole spins begin an unstable runaway, just as a spinning top begins to tilt and wobble. We use numerical evolutions of inspiraling black holes to verify this instability. Surprisingly, the unstable binaries evolve towards a precise, predictable endpoint before merger. We simulate a large astrophysical population of black-hole binaries and show that the instability leaves a significant imprint on the population, but improvements to detection techniques may be required to measure the effect with current gravitational-wave observatories.
Plain text summary
Introduction.
Black holes are warps in spacetime so strong that they consume ALL things that get too close. In black-hole binary systems, two black holes orbit each other. The emission of gravitational waves leads the black holes to inspiral and eventually merge. The gravitational-wave signal from a merging binary black hole was first detected by the LIGO in 2015. Pictured: Figure 1. The signal GW150914, the first ever observation of gravitational waves. Noisy signal with increasing amplitude and frequency, peaking at the black hole merger.
Precession.
Consider a spinning top. Unless spun perfect upright, it wobbles in a cone as it rotates - this is called spin precession. Black holes precess if the spins are not orthogonal to the orbit, significantly altering any gravitational-wave signal. There are four of these aligned spin configurations. Pictured: Figure 2. The four aligned spin configurations. The two black hole spins can either both point (up-up), both downwards (down-down), or one up and the other down (down-up and up-down). Are these systems stable? Or does spin precession change the configuration as the black holes evolve?
The maths.
1. Definitions.
Mass ratio = mass of smaller hole divided by mass of larger hole. Total mass = sum of the masses. Kerr parameters = how fast the black holes spin. The spin vectors and their relative orientations are defined in the following figure and the evolution of binary black hole spins is governed by the following equations. Pictured: Figure 3. The relative orientations of the spin vectors of the black holes with the orbital angular momentum, and the angles between them. Pictured: Equation. A coupled set of first order ordinary differential equations describing the evolution of the black hole spins and orbital angular momentum.
2. Perturbations.
To test the stability of the aligned spin configurations, we introduce small perturbations to the spin directions. This means the two spin vectors are slightly misaligned with the orbital angular momentum. The perturbation evolves to leading order as a simple harmonic oscillator. Pictured: Equation. The second order ordinary differential equation of a simple harmonic oscillator. The evolution of the perturbation depends on the square of the oscillation frequency: if greater than zero the spins are stable, if less than zero the spins are unstable and begin to precess, and if equal to zero there is a transition from stability to instability.
Results.
From the square of the oscillation frequency, we find that only the up-down configuration is unstable to precession. Those binaries become unstable when the distance between the black holes shrinks to value between two critical separations. Pictured: Figure 4. The evolution of the two black hole spins after an up-down binary encounters the instability. Before instability, the spins undergo tight oscillations. After the instability, the spin vectors spiral outwards, leading to large misalignments. Pictured: Equation. The critical separation at which an up-down binary becomes unstable, in terms of the mass ratio and the Kerr parameters. The endpoint of the unstable evolution is given by expressions for the angles between the black hole spins and the orbital angular momentum. Pictured: Equation. The angles between the black hole spins and the orbital angular momentum, in terms of the mass ratio and the Kerr parameters. The two misalignments equal each other. Pictured: Figure 5. Astrophysical population of up-down black-hole binaries. Unstable binaries make up most of the population (left three panels) and undergo the instability before reaching the LIGO sensitivity window (frequency > 50 Hz, right panel). The first three panels show the distributions of the mass ratio and Kerr parameters, and the right panel shows the distribution of the instability onset separation.
Black holes are warps in spacetime so strong that they consume ALL things that get too close. In black-hole binary systems, two black holes orbit each other. The emission of gravitational waves leads the black holes to inspiral and eventually merge. The gravitational-wave signal from a merging binary black hole was first detected by the LIGO in 2015. Pictured: Figure 1. The signal GW150914, the first ever observation of gravitational waves. Noisy signal with increasing amplitude and frequency, peaking at the black hole merger.
Precession.
Consider a spinning top. Unless spun perfect upright, it wobbles in a cone as it rotates - this is called spin precession. Black holes precess if the spins are not orthogonal to the orbit, significantly altering any gravitational-wave signal. There are four of these aligned spin configurations. Pictured: Figure 2. The four aligned spin configurations. The two black hole spins can either both point (up-up), both downwards (down-down), or one up and the other down (down-up and up-down). Are these systems stable? Or does spin precession change the configuration as the black holes evolve?
The maths.
1. Definitions.
Mass ratio = mass of smaller hole divided by mass of larger hole. Total mass = sum of the masses. Kerr parameters = how fast the black holes spin. The spin vectors and their relative orientations are defined in the following figure and the evolution of binary black hole spins is governed by the following equations. Pictured: Figure 3. The relative orientations of the spin vectors of the black holes with the orbital angular momentum, and the angles between them. Pictured: Equation. A coupled set of first order ordinary differential equations describing the evolution of the black hole spins and orbital angular momentum.
2. Perturbations.
To test the stability of the aligned spin configurations, we introduce small perturbations to the spin directions. This means the two spin vectors are slightly misaligned with the orbital angular momentum. The perturbation evolves to leading order as a simple harmonic oscillator. Pictured: Equation. The second order ordinary differential equation of a simple harmonic oscillator. The evolution of the perturbation depends on the square of the oscillation frequency: if greater than zero the spins are stable, if less than zero the spins are unstable and begin to precess, and if equal to zero there is a transition from stability to instability.
Results.
From the square of the oscillation frequency, we find that only the up-down configuration is unstable to precession. Those binaries become unstable when the distance between the black holes shrinks to value between two critical separations. Pictured: Figure 4. The evolution of the two black hole spins after an up-down binary encounters the instability. Before instability, the spins undergo tight oscillations. After the instability, the spin vectors spiral outwards, leading to large misalignments. Pictured: Equation. The critical separation at which an up-down binary becomes unstable, in terms of the mass ratio and the Kerr parameters. The endpoint of the unstable evolution is given by expressions for the angles between the black hole spins and the orbital angular momentum. Pictured: Equation. The angles between the black hole spins and the orbital angular momentum, in terms of the mass ratio and the Kerr parameters. The two misalignments equal each other. Pictured: Figure 5. Astrophysical population of up-down black-hole binaries. Unstable binaries make up most of the population (left three panels) and undergo the instability before reaching the LIGO sensitivity window (frequency > 50 Hz, right panel). The first three panels show the distributions of the mass ratio and Kerr parameters, and the right panel shows the distribution of the instability onset separation.
Poster file
Poster Title
Runaway spins in binary black holes
Url
mmould@star.sr.bham.ac.uk