On My Soapbox

I've been applying to a bunch of faculty positions recently, and one requested a recorded teaching sample. I made this, giving an introduction to special relativity. Questions & feedback are welcome!

Pulsar-Teacher Association

On Thursday, the NANOGrav project, along with international partners, made the announcement that they had detected a stochastic gravitational-wave background! This week, I thought I'd talk a bit about the news, and how the discovery was made.

First though, we should talk about what a stochastic gravitational-wave background is. Gravitational waves are produced whenever large amounts of mass move around in an asymmetric way. In the case of (still undetected) continuous waves, a bump on the neutron star, or for CBCs a pair of black holes or neutron stars. In the case of stochastic waves, we're talking about galaxies colliding, which is a much slower process. Since the movement is slower, that means the frequency is lower, on the order of nanohertz, or about 1/(32 years). That range of frequencies is far below what LIGO, or even LISA can detect:

Wikipedia

The orange region on the left is the background signals we're talking about, and the type of detectors used are called Pulsar Timing Arrays (PTAs). Pulsars are rapidly-spinning neutron stars, which produce pulses of radio-frequency signals at extremely regular intervals. They were initially referred to (jokingly) as LGMs, or "little green men", since it seemed like regular radio bursts would be a hallmark of an intelligent species.

The strength of a gravitational wave depends in part on the size of the masses that are moving. Since this background signal is due to entire galaxies moving, the gravitational waves are a million times stronger than those detected by LIGO! You might wonder then, why they were not detected before the CBCs that LIGO found. While I was thinking about this myself, an analogy occurred to me: Shifts in the Earth's tectonic plates are responsible for both earthquakes and continental drift. Even though the drift is on a significantly higher scale than the earthquakes, it's much harder to detect, due to the long periods (low frequency) involved, while earthquakes are picked up every day.

Since the first detection by Jocelyn Bell in 1968, many more pulsars have been found. The regular signals from these pulsars can be thought of as distant clocks ticking, from which the idea of pulsar timing arrays was conceived. A passing gravitational wave will cause a change in the signal's arrival time on Earth, but that change will depend on the direction of the pulsar, and the direction and polarization of the gravitational wave.

An isotropic signal means it should be the same in all directions. In 1983, Hellings and Downs suggested a method to detect such a signal: If two pulsars are affected by the same gravitational wave background, then the measurement of those pulse deviations on Earth should depend on the strength of that background, the noise in our measurements, and the orientation of the pulsar relative to Earth. By averaging the correlation between two pulsars over a long period, we can reduce the noise (which should be uncorrelated) and increase the background signal. Hellings and Downs derived a specific curve that that correlation should follow, according to the angle between the pairs of pulsars. After 15 years of collecting data from 67 pulsars, the collaboration presented this comparison to the expected curve:

Figure 1c

The points clearly deviate from the straight line that would result from no stochastic background signal, and instead follow the predicted curve, indicating a background signal is present. It's exciting to have another part of the gravitational wave spectrum filled in, and I look forward to more results from PTA groups!

Dirty, Disgusting, Filthy, Lice-Ridden Boids

[Title from The Producers]

Long ago I had a screen saver with a simulation of bird flocking behavior – A group of 2000s-era 3D blocks would fly in formation, land, and take off. I was recently reminded of it, and grew curious how it was made. My best guess is that it used a model developed in 1987 called Boids. The model consists of a group of agents (boids) that each act according to a set of simple rules. In this case, the rules are

• Separation: Avoid flying into nearby flockmates
• Alignment: Fly in the same average direction as nearby flockmates
• Cohesion: Fly toward the average position of nearby flockmates

Along with these, I also included a target location that all boids want to get to. There's a lot of details unspecified here, like what "nearby" means, and how these various rules are weighted, e.g. cohesion and separation can be directly opposed at times. I looked at a couple implementations I found online, and tweaked my own model until I got reasonable looking results.

In the simulation below, the boids are all heading for the center of the map, and "nearby" covers about 1/4 of the map. I used "toroidal boundary conditions" which means if a boid goes off one edge, it wraps around to the opposite side. You can see that a few boids sometimes break out of the "nearby" region and head off on their own before rejoining:

I was pretty happy with how this worked out, but I felt like I should give it more of a physics spin. It occurred to me: What if these were relativistic boids? If they're moving at a significant fraction of the speed of light, then the observations they use to follow the rules above are based on the light arrival time, which may not reflect the current motion of a particular boid. What this amounts to is that based on the distance between two boids, their observations are delayed by a certain time. Here's what happens in that situation:

I was really surprised by the results – I expected that they would have a much harder time sticking together, since their actions would be too imprecise, but instead they show better clustering. I think this is because by staying closer, they get more up-to-date measurements, so it ends up being a positive-reinforcement.

I figured I could get a more qualitative comparison by computing some summary statistics. First I looked at the average distance between the boids at each time:

This shows the relativistic boids bunching up quickly, but then oscillating sharply around a higher value than then lowest the non-relativistic boids are able to hit. We can also look at the deviations in the boid's headings for the two cases:

This shows the relativistic boids frequently going in different directions, which we can see in the animation above: The tight grouping requires quickly changing direction to stay in one place.

I'm not sure whether any of the space organizations have plans for high-speed probe swarms, but if they do, I hope this will serve as my grant proposal!