Paper Explainer: Collapsed Dark Matter Structures
/This is a description of a paper I’ve written with my postdoc, Anthony DiFranzo. In our paper, we consider the possibility that dark matter could form gravitationally collapsed objects, evolving from an initial state of nearly uniform distribution across the Universe into one where it forms compact objects, analogous to have the regular matter that you and I are made of eventually formed stars and galaxies. Usually, we think this is not possible for dark matter, due to evidence that, on the largest scales, dark matter forms gravitationally bound structures that are much "fluffier" than the collapsed stars and galaxies.
However, as we show in the paper, there is a way for dark matter to evolve into compact objects on small scales (say, a thousandth the size of the Milky Way), while still satisfying the constraints we've observed at larger scales. In demonstrating that it is possible for dark matter to do this, I think our paper makes an important point about some open questions in the field of dark matter research.
To explain why I started thinking about this particular project, I want to motivate it with a somewhat whimsical question.
Can there be planets and stars made of dark matter?
This is a very fun question to ask, and one of those obvious extrapolations from scientific knowledge that are so popular in science fiction. After all, there is 5 times as much dark matter as regular matter (referred to generically as “baryons” amongst astrophysicists). We’re made of baryons. Baryons are interesting. Baryons have stars and galaxies and New York and so on. So maybe all that dark matter has something equally interesting in it. Who knows, right?
As I said, this is not a research paper in which I expound on the nature of the dark matter planets. But it is a fun question, if somewhat silly and wildly speculative, and it led me to think about a much more grounded question that I think is very interesting, very well-motivated, and very important to people like me who think about dark matter. But imagining ghostly planets and stars made of dark matter is one of those science fiction tropes that pops up again and again. And that got me thinking.
I love science fiction, I always have. It’s part of my journey that led me to become an actual physicist. As I learned more and more, I would often get a little twinge of — let’s call it professional annoyance when I read science fiction that relied on ideas I knew to be incorrect. Science fantasy a la Star Wars never bothers me, but when your Big Idea revolves around something like using quantum entanglement for faster-than-light communication, it’s sort of irksome, because that's not how it works. Call it magic and be done with it. (Before you contact me to tell me it’s all just fiction and just to relax, finding the logical holes in ideas is relaxing to me, and I usually don’t get personally affronted at the author for it. Unless they are implying that this is how the Universe really works. That’s just Dan Brown-level malpractice.)
So when I ran into dark matter in the form of planets and stars populating the pages of science fiction, my professional annoyance-sense got activated. “Surely,” I thought to myself “this can’t be. We’d have some evidence for the sort of physics even minimally necessary to get something as complicated as a planet in the dark sector. Right?”
And that’s in some way how I ended up writing one of my most highly cited papers, where we studied the possibility of a force like electromagnetism in the dark sector. After all, if you want planets made of dark matter, your dark matter better be more complicated than the neutralinos or axions that are the usual candidates among particle physicists (and before you contact me to say I’m lacking imagination, no, a bag of neutralinos is not going to give you anything a complicated as organism. There aren’t enough interactions). You need forces beyond gravity, and longer range than the weak nuclear force to get something interesting going on, so considering “dark electromagnetism” seemed like a good place to start. Also, I found the phrase “dark light” endlessly amusing. Of such things are good ideas born.
I’m telling this story, by the way, because I think it’s important to get across how the background of scientists influence what problems we find interesting and how we approach them. I’m very proud of that dark electromagnetism paper, and I’m proud other scientists found it important and interesting. It is a completely serious work. But the thoughts and conversations that led to that idea were deeply influenced on my part by the sort of thing I find fun and interesting, which is influenced by how I grew up. This is why having scientists from all walks of life and all backgrounds is critical. Even in something as hard and logical and rigorous as particle physics, there is always a need for the different viewpoints and ways of thinking.
After working on that idea of dark electromagnetism, I moved on to other “serious” ideas about self-scattering of dark matter, and what was and wasn’t possible from the data. Along the way, if you asked me about the possibility of dark matter planets, I would have said the following:
“To get a compact object made of dark matter of macroscopic size — say the size of a grain of sand, or a snowball, or a human, or a mountain, you need the dark matter to cool and stick together. We know that can’t happen. Because if you get dark matter to stick together on the size of a snowball, there is nothing to stop it from continuing to cool and collapse on the size of a star, or a thousand stars, or a billion stars. The dark matter in the Milky Way is not all collapsed and stuck together, therefore the dark matter can’t form a snowball. And sorry, that means it can’t form stars or planets either.”
Let me pick this apart a bit:
Pictured: Collapsed Structure built of baryons (Image Credit: NASA)
First, what do I mean by a collapsed object? A collapsed object is something like the Sun. Or the Earth. Or the Milky Way. Imagine you start with a cloud of particles. That cloud will self-gravitate, and the particles will start falling in towards the center. So eventually they’ll all end up in a ball in the center, right?
Wrong. Or, at least, wrong if the only force at play is gravity. If you let a bunch of particles that only interact via gravity attract each other, then as they fall to the center, they’ll pick up speed, potential energy being converted to kinetic. But, since gravity is the only force acting, when the particles reach the center of the mass, they’ll pass straight through (assuming we never form a black hole, which would actually be very difficult to do in this situation). There’s no other force to allow particles to hit each other, so there’s no way to lose potential energy and stop. So the particles zip straight through, and fly out to the other size of the conglomeration, losing speed until they stop, turn around, and start falling back in. The cloud you start with will remain a cloud.
There will be tides on the particles, which would cause the collection of particles to contract a little bit, but gravity alone will not let a cloud of particles collapse in on itself. Bits will fall in, but other bits will be “falling out” at the same time.
PictureD: Collapsed Structure Built of Baryons (Image Credit: NASA)
Baryons started as big clouds of particles, that self gravitated. So how did they go from those spherical clouds to the pancake-shaped Milky Way disk, or even smaller to a Sun, or the Earth? What had to happen was some of the kinetic energy of the infalling particles had to be lost. Then, when the particles zipped through the center of the cloud, they couldn’t make it out as far on the other side, and eventually the cloud would collapse. As it collapsed, it would fragment into smaller clouds, which themselves continued to collapse. So your big cloud forms the Milky Way, which fragmented down to form clouds of hydrogen gas that go on to form stars, and eventually planets.
So creatures like us exist because there was a cooling mechanism for baryons. I’ll get to the details of that cooling mechanism in a bit, but for now, just know that the secret is “photons.” Baryons feel the electromagnetic force, so as they go screaming through the center of a cloud of their sibling baryons, they interact via that force and radiate off extra energy in the form of light.
My reasoning for why there couldn’t be dark matter creatures wandering around the Universe comes from the fact that the collections of dark matter we know about don’t look like Milky Way spirals. They look like the blobs of blue shown in the picture here.
This is a map of dark matter, created using the lensing of photons by the gravitational potential of the dark matter. Dark matter forms big fuzzy spherical-ish clouds, surrounding the collapsed baryons in their center. Here’s another picture, this from simulation of dark matter. It’s not a “real” picture, but it’s clearer, and matches the known properties of dark matter well.
Again, you see the clouds of dark matter, connected by strands that are still much, much larger than the size of the baryonic collections we call galaxies. Dark matter appears very different from the “collapsed” structures we see: on the scale of galaxies dark matter doesn’t form collapsed objects. That means there is no cooling, and thus no force like electromagnetism that can trigger that cooling. No dark light for dark matter.
Simulation of dark matter. A Milky Way-type galaxy would be located in the center of the yellow blob in the center (Image Credit: MILLENNIUM Simulation)
If dark matter didn’t cool on the scale of galaxies (the Milky Way contains about a trillion — $10^{12}$ — solar masses worth of dark matter), then the lack of that cooling mechanism seemed to imply to me that dark matter couldn’t cool on the scale of a star. Or a planet. Or a snowball. If any of those objects could cool and collapse, then what would stop all of them from collapsing? Nothing, it seemed: the collapse mechanism would work for any collection of material. So big fuzzy clouds of dark matter around galaxies that didn't cool meant no collapsed objects like stars or planets or even clumps the size of snowballs, which meant no dark matter planets.
Now, I want to point out that there are models that allow some clumping of dark matter. But these fall into two categories. First, you can do your clumping up in the very early Universe, so by time things have settled down and dark matter and baryons started evolving from the nearly featureless smooth distribution they started in to the rich structure of galaxies we see today, dark matter was already made of objects that might be the size of a snowball, or a mountain, or an Earth. However, such objects can’t cool and collapse further, and that suggests the lack of additional forces that would make the dark sector “interesting.”
Alternatively, you can postulate that only some of the dark matter has the ability to cool. Maybe there are baryons, then dark matter Type 1 which cools, and dark matter Type 2, which doesn’t. Maybe there are 10 different types of dark matter, or 100, and some cool and others can’t. We don’t know, but it’s possible. Lisa Randall and her collaborators have been exploring these ideas, and found that 10% or so of the dark matter cooling and collapsing isn’t obviously ruled out (see here, here, and here). I want to point these ideas out, since thinking about them helped me come up with the idea my paper is about.
So, this was my thought process for nearly a decade. No collapsed objects at human (or stellar) scales in dark matter, because dark matter at the galaxy scale doesn’t collapse.
Then I realize I was just wrong.
Let’s go back and look at baryons. Baryons on the scale of the Earth are collapsed: all the mass in a region is swept up to a central object.
Same with baryons on the scale of a star.
Baryons on the scale of a dwarf galaxy also are collapsed: they’ve fragmented down to stars, but the baryons are much smaller than the cloud of dark matter around them.
Same with the spiral galaxies.
Virgo Cluster of Galaxies (Image Source Rogelio Bernal Andreo via Astronomy Picture of the Day)
But now look at larger collections of matter: galaxy clusters. Galaxy clusters are… clusters of galaxies, just as galaxies are clusters of stars. But while galaxies have their component objects tightly packed into a central region, the galaxies in a cluster just sort of zip around in a big spherical mess. In fact, 90% of the baryons in a cluster aren’t in galaxies: they’re in uncollapsed gas in between the galaxies. Baryons don’t collapse at the largest scales.
Why? Because the more massive the object is, the more energy the baryons would have to lose in order to collapse (because more mass means more gravitational potential energy and thus more kinetic energy). But if the mechanisms that allow energy to be radiated away don’t scale up fast enough, then the baryons can’t cool efficiently. Whereas a baryon falling in towards a Milky Way-mass galaxy might lose 95% of its energy as it falls, in a galaxy cluster, maybe it only loses 1%. That’s not enough to get the baryons to collapse.
This behavior: where the cooling mechanism scales slower than the mass of the cloud, is pretty generic. It happens with electromagnetism, and is why there aren’t galaxies bigger than $\sim 10^{12}$ solar masses or so. Bigger clouds just can collapse.
So, maybe there is a dark electromagnetism. Maybe it does allow dark matter to collapse on scales smaller than we can discern yet — after all, our gravitational lensing measurements only extend down to objects on the order of $10^{8-9}$ solar masses. There would be a natural reason why those objects would collapse, yet the dark matter in the Milky Way remains nice and spherical and non-clumped. Our Milky Way dark matter would be like the baryons in a galaxy cluster: some little cores of cooled matter, but the vast majority just cruising along driven by gravitational forces.
This would be super interesting, even without the whole dark matter planets thing. It’d be interesting because we know so little about how dark matter works: what it’s structure is, how it moves around us. We have an idea at the biggest scales, but not smaller. If I had to bet, I’d say that this idea of dark matter cooling is probably wrong, but I don’t know it is wrong. I’ve convinced myself it is possible, so now my job is to prove that it can’t be done. Vie some new measurement, or simulation, or theoretical argument. Because, hey, maybe I’ll be wrong again, and it turns out dark matter does form interesting collapsed objects. That’d be fun, right?
So, can we get this idea to work?
We wanted a proof-of-principle, so rather than construct all possible versions of physics in the dark sector that allowed cooling, we just picked the simplest idea. We copied the baryons: very imaginative of us, I know. But hey, it worked once.
We postulate dark matter made of two particles, a heavy particle $H$ (the dark analogue of a proton), and a light $L$ (the dark version of an electron). The masses of these two particles are free parameters, and will turn out to be much heavier than the baryonic ones we know and love. We then add in dark electromagnetism, with a free parameter of its coupling strength — the dark fine structure constant $\alpha_D$.
We then consider this dark matter after the early Universe. At this point, there will be regions that, due to quantum fluctuations in the very earliest moments of the Universe’s life, there is more matter than elsewhere. Those overdensities are going to grow and collect more matter and become the galaxy clusters, galaxies, and dwarf galaxies all around us today. This is completely standard dark matter physics, and there’s tons of evidence to back this up.
Now, with our new idea of dark matter, the $H$ and $L$ will combine to form “dark atoms”, bound states of $H$ and $L$. In overdensities that go on to form very massive galaxies today, those dark atoms will be moving very fast as they fall in, their “virial temperature” is high. They will scatter off of the few remaining $L$ particles that are unbound, and in that scattering, knock the bound $L$ up from the ground state of the dark atom to an excited state. The dark atom will de-excite, emitting a dark photon, and cool. For the most massive halos, this cooling will not radiate enough energy fast enough to appreciably cool the halo. There is a maximum mass that can cool (there is also a minimum, it turns out, since at low masses there aren’t enough free $L$’s around). This is called collisional excitation, and is how the earliest baryonic halos cooled to form the first protogalaxies.
Plot of mass of dark matter halos that can collapse, as function of light dark matter particle mass, assuming heavy dark matter mass of 1.2 TeV. Image from Buckley & DiFranzo 2017.
By dialing the masses of $H$ and $L$, and the coupling $\alpha_D$, we can make that maximum and minimum masses whatever we want. Here are some choices that give interesting results.
You can see we can make dark matter halos much bigger than the Milky Way collapse, down to halos the size of the Sun, depending on what I set as the mass of the lighter dark matter particle $L$. Obviously, the former is ruled out by the existence of the Milky Way, which is definitely not made of collapsed, cooled dark matter. But somewhere around $10^8$ solar masses, I can’t say for certain if dark matter could collapse. A lot more detailed work would need to be done to figure out whether such a collapsed object is in tension with observations. So there’s work to be done. Which is nice.
There are other limits. If $\alpha_D$ is too big for low $H$ masses, then dark matter scatters against itself too much (this is what my original paper on dark electromagnetism actually was interested in). But the parameter point I show here, with 1.2 TeV $H$ particles, is ok.
If dark matter was too “hot” in the early Universe, than the presence of this new dark electromagnetism would erase the very primordial overdensities that would go on to form the halos that I want to collapse. We show a (conservative) estimate of the minimum halo size that would exist, assuming that the dark photons in the early Universe were a factor $\xi$ times cooler than the real photons we can see. We need the collapsing regions to be above the $\xi$ lines. $\xi$ of 0.5 is trivial to achieve. $0.1$ is harder, and $0.02$ requires some more exotic physics. But this is a conservative estimate, exotic physics is totally plausible, and there are some workable parameter points regardless.
So what would this look like? Let’s take the parameter point that results in a $10^8$ solar mass dark matter cloud collapsing. Today, we estimate that about 2% of the Milky Way would be made of such clouds, so maybe 100 total. But that is the result of a cloud of that mass falling into the Milky Way, and most of such clouds are torn apart by tidal forces. If collapse happens, tidal forces are not as effective. Perhaps 10% of the Milky Way could be bound up in these objects, $\sim1000 $ of collapsed “mini-galaxies” of dark matter. Smaller collapsed objects would also exist, an order of magnitude more for each order of magnitude their mass is below $10^8$ solar masses.
What do these collapsed objects "look" like? We can't specify that in our model. We’d need to start making assumptions about the rest of the dark matter physics, and we didn’t do that in this proof of concept paper because it was extraneous to our idea of cooling the dark matter halos. If something like fusion exists, then the cooling and collapse would be stopped by the same process that stopped the baryons from collapsing down to nothing. Stellar fusion adds energy back to the baryons, and you could imagine something similar would happen to dark matter. If no extra physics is around, then these collapsing objects would fragment down to many black holes, though we’d have to do some more detailed estimates to know how many.
As I said, this idea, like most theoretical physics ideas, is probably wrong. But we can’t prove it is wrong with the information we have on hand, and that’s interesting. It gives us a sense of what is possible in the physics of dark matter, and so we can construct tests to rule these ideas out. That’s the job of a theoretical physicist: come up with a new idea, and come up with a way to shoot it down. Most of the time, your idea gets killed by the harsh light of reality.
But if you work hard enough and the Universe cooperates, sometimes you get lucky. It's likely there are no cooling mechanisms that can allow dark matter to form interesting objects like stars and planets, much less the things that wander around on top of planets. But we won't know for sure until we do a lot more work evaluating what the dark matter is doing on smaller scales than we have previously. Even if this idea is wrong, the knowledge we'll gain by proving it is wrong will teach us about how dark matter distributes itself around the Universe, and that will help point our way to the end goal of discovering what the hell this mysterious stuff is.