**Title: **Gravitational Wave Memory Imprints on the CMB from Populations of Massive Black Hole Mergers

**Authors: **Lorenz Zwick, David O’Neill, Kai Hendriks, Philip Kirkeberg, Miquel Miravet-Tenés

**First Author’s Institution: **Niels Bohr International Academy, Niels Bohr Institute, Copenhagen, Denmark

**Status:** Submitted to A&A letters [open access]

For a second, let’s pretend you’re me in the third year of college and you have just bought two hundred small rubber ducks online (true story). Instead of dumping them all into your roommate’s room while they’re gone on winter break as I did, let’s say you set some adrift on a nearby lake. You would notice that as they bobbed up and down on the gentle waves of the lake, the arrangement of the rubber ducks would not change after the waves passed them by. For example, if you started your ducks in a row, they would still be in the same row after the wave. Now let’s take those same ducks to interstellar space and plop them down in a circle. If a strong gravitational wave passes through, thanks to a nearby binary black hole merger for instance, you might be shocked to find that your ducks are now in an ellipse! This phenomenon is known as Gravitational Wave Memory because a system “remembers” (by being altered) that a gravitational wave has passed through it, see Figure 1. This phenomenon could have interesting implications all over cosmology and astrophysics, but the authors of today’s paper put forward an interesting proposal: could we measure the gravitational wave memories of cosmic microwave background (CMB) photons? Or, put another way, could the grand menagerie of cataclysmic mergers in the universe leave a gravitational wave memory imprint on the CMB? Today’s authors strive to find out.

## The Tensor is Killing Me!

When doing the general relativistic calculation to determine the effect of gravitational waves, one usually uses what is called the strain tensor. For now, a tensor is just what a physicist calls a matrix when they want to feel smarter than everyone. This strain tensor tells you how much and in what direction your test ducks will oscillate when a gravitational wave passes through. All we care about for our purpose is that the tensor tells us that gravitational waves get weaker with distance and their effect depends on the angle between the orbital axis of the merger and the line to the point in space you are curious about.

This tensor allows the authors to determine both how much photons are deflected by gravitational waves as well as how much their wavelengths are squashed or stretched. In this second case, since a photon’s wavelength is tied to its color or energy, we expect this to change as the wavelength is shortened or elongated by the gravitational wave. CMB maps are just maps of photon energy, so the effect of the memory of these gravitational waves could slightly change the CMB! We also said that gravitational waves could modify the paths of photons, but the authors choose to ignore this effect as the paths of photons are much more strongly affected by the gravitational lensing of the dark matter halo they are escaping.

To get an idea of what kind of impact on the CMB we might expect, the authors first focus on one singular gravitational wave event to determine its effect. One merger is defined by a handful of parameters: the time of the merger, its location, its redshift, the total mass of the two black holes, the angles between the merger and the viewer, and the angle between the merger and the test point. (Note: when we say angle, we mean the angle between the line of sight and the orbit of the binary, i.e., do we see it edge-on or face-on?) Another thing to consider is the speed of gravitational waves, which is the speed of light, so the authors model this as an outward propagating shell moving at the speed of light. Photons can only “know” about the merger once they have intersected this shell.

After crunching the numbers, the authors find some interesting results; see an example in Figure 2. It seems that the two angle parameters govern the shape of the pattern, while the mass, time, and distance to the merger all affect the overall scale of the effect. Even better, the authors are able to write down an empirical law governing the maximum size of the effect on the CMB based on the parameters mentioned above. This is important because it lets us determine the best-case scenario for real-world detection. Also, because gravitational waves stretch the wavelengths of photons just as often as they compress them, the overall average energy of photons remains the same; this will be important in a moment.

## Why have one when you can have them all?

Of course, there is more than one binary black hole pair in the universe (controversial, I know). So now that we’ve gotten one black hole merger under our belts, what does it look like when a photon has to brave a universe filled with mergers? When more mergers are added, the pattern gets more complicated (there is no Nobel prize for guessing that one). Still, the authors remind us that the pattern is just a superposition of the pattern we saw in Figure 2, just in different places and orientations, see Figure 3.

Because each of these individual mergers does not change the average energy of the photons around it (i.e., it lowers some photons’ energy and raises the same amount in others), the authors conclude that the effect of these mergers on a photon is statistically identical to a random walk. A random walk is a classic statistics problem that asks: if you start from a spot and take each step in a random direction, how far away will you get? This isn’t statbites, so I’ll spoil that the answer is essentially that the distance away you can get scales as the square root of your number of steps taken. Also, the standard deviation on how far away you end up from your initial position also scales as the square root of steps. The photons in this experiment are basically random walking in their energy. Some mergers give them a step up in energy, others a step down. The difference between their starting and final energy therefore scales as the square root of the number of mergers that affected the photon. The authors confirm this prediction via simulation! This result allows the authors to again make an empirical prediction of the maximum amount the CMB can be affected by this gravitational wave memory phenomenon and how much it should vary using the standard deviation. The authors then use this idea to show that the resulting pattern of energy changes has a specifically shaped power spectrum (a measurement of how strong the effect is at different size scales), which is very useful as most CMB experiments are concerned with the power spectrum.

## Observability

Could this power spectrum from gravitational wave memory be separated from the regular CMB power spectrum? Unfortunately, probably not yet. Based on the empirical formula for the maximum change in energy they found, the size of this effect is a few orders of magnitude weaker than what can be detected by modern all-sky CMB instruments. However, what about just focusing on one spot on the sky? What if a black hole “remembers” that it had a merger in the past? The authors pick the nearby ultra-massive black hole (name a scarier sequence of 5 words) Holmberg 15 as a test case. If one had a CMB interferometer that was sensitive to light of a typical CMB photon wavelength such as 2 mm, the telescope would require a baseline (distance between antennae in a radio telescope) of only about 150 meters. Since one can “cheat” and put their antennae on the other side of the planet (creatively named: very long baseline interferometry), a 150 meter baseline radio telescope is totally reasonable! The authors note that the merger would have to have happened about 10,000 years ago or older to have a large enough size on the sky to be hypothetically detectable, but higher resolution telescopes would allow this to be more recent. So, while detecting a whole sky CMB gravitational wave memory pattern might be a bit beyond the current capabilities of CMB instruments, a radio telescope could be powerful enough to ask nearby massive black holes to reminisce about their past.

*Astrobite edited by: Will Golay and Karthik Yadavalli*

*Featured image credit: Wikimedia Commons and NOIRLab*

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