# Primordial Black Holes as Dark Matter?

I was recently asked by a family friend “have you heard about this new idea that primordial black holes could explain dark matter?”

Well I hadn’t, so I did a little investigating and it’s a pretty clever idea. Part of the backstory here is “what can we do with gravitational waves?”, so that’s where I’ll start.

One of the surprising things about the very first direct observations of gravitational waves by LIGO is the masses of the constituent black holes. The first pair was 36 and 29 solar masses, the second was 14 and 8, and the third was 31 and 19. What was immediately understood to be important about these sources is that they are generally more massive than the other stellar-mass black holes we’re found previously (from X-ray studies, usually. Max there is 18 solar masses). Significantly, the larger mass ones should also be *less* likely, from stellar formation scenarios. So while we are only talking about 6 new black holes, we clearly need to know if that will pose a problem for stellar formation models. (there are also some issues in regard to the spins of these black holes, but I won’t go down that particular rabbit hole).

So people started looking at it, and found that it was generally possible to get these kinds of higher-mass black holes, but it did put some constraints on the formation scenarios. Basically, the problem is you need to make giant stars, which generally need to have low metallicity to form. However, the conditions that generate those stars (high star formation rate in the past) generally turn out to produce higher overall metallicity quicker. If you tune the star formation rate a bit so there are actually fewer large-mass stars, you reduce the overall metalicity so you can effectively create massive black holes. So it’s constraining, but not overly so.

But that’s actually not what I want to talk about – what about other formation scenarios for these black holes? Specifically, what about primordial black holes (PBH)? These are black holes that formed as a result of density fluctuations in the early universe. It turns out it’s pretty easy to produce black holes of this mass in this manner (and the spin, which I skipped talking about above, is a little easier to produce as well). So, cool, we have at least two different ways the universe can give us the black holes found by LIGO.

But, are there any other implications of primordial mass black hole production at this rate? Well, without a stellar companion, there would typically not be an accretion disk and we would have no way to observe these black holes. But of course – that’s exactly the condition we need for dark matter!

So, in a recent paper, Juan Garcia-Bellido and his collaborators (who include Sebastien Clesse, Andre Linde, and David Wands) have worked this out in a bit of detail (and apparently there are others working on this as well, such as Alexander Kashlinsky).

The idea that black holes (or other compact objects) could be a model for dark matter is not new, actually. We’ve been looking for microlensing due to compact objects in the galactic halo for years (these objects are called MACHOS), but have essentially found nothing. What’s interesting about their new models is the mass distribution for primordial black holes in the 10-100 range sits right in the region of parameter space which was has not been covered by previous studies:

As you can see in the figure (which comes from the paper), the lower limits on PBH have a gap in between the lower mass MACHO/EROS observations and the higher mass WMAP3/FIRAS observations. It looks to me like that gap peaks around 0.01 of a solar mass and carries up to around 100. Which is broad range for black holes, but look at the range which we are talking about here (25 orders of magnitude!).

So there are lots of other interested details here, but what’s really fascinating about this new paper is that there are apparently a very large set of phenomenological signals we can use to test this hypothesis. It would affect the CMB, star formation in the early universe, X-ray transients, and a whole host of others. One particularly interesting idea is that rather then looking for lensing, we might try to look for the shift of the positions of stars over time. With the new plethora of data on stellar positions (like the GAIA satellite), it also might be the first time someone could actually attempt such a study. So there are a lot of interesting things to check.

As a sidenote, some of these black holes would of course develop an accretion disk through random interactions with stars or gas, and produce point sources that would emit in Gamma or X-ray range. Well, there actually is a large list of unidentified point sources in nearly all the X-ray catalogs. In fact, my undergraduate honors thesis was working on trying to identify unknown point sources in a Chandra X-ray image of the galactic center. The paper suggests that rather than looking at spectral characteristics, one should look for a correlation between the point sources and the expected dark matter distribution.

So, we’ve got LIGO finding a new class of black holes, which could be created in the early universe, and a new model for dark matter. Given how much trouble the particle model for dark matter is having (sorry LHC!), we should be taking these new ideas seriously. And what’s great about this is there are *bunch* of great ways to look for this primordial black hole signal. Of course, maybe that means it won’t last long as an explanation for dark matter, but it’s something new to look at that doesn’t require any exotic new physics.

And, not to belabor the point, but all of this wouldn’t have been possible with LIGO. Thanks LIGO!

# Cosmic Strings and Spatial Topology

Some of my recent work has focused on foliations of spacetime – which are essentially 1- or 2-dimensional parametrizations of 3- or 4-manifolds. I am mostly interested in them because my work often requires 1- and 2-dimensional embedded spaces, and you can use these initial spaces to construct entire foliations of the manifold. If you have a 2-dimensional space (a surface), you can always smooth it out by squishing all the curvature to a single point, called a conical point (you can try this – next time you get an ice cream cone, take the paper off the cone and try to flatten it. You will be able to do it everywhere except for one point, at the tip of the cone). Mathematically, a surface with a conical point looks exactly like a surface intersecting a cosmic string, so this is a long winded way of explaining how my interest in foliations has lead to an interest in cosmic strings!

The latest part of this story is using some of the interesting properties of cosmic strings, I’ve been able to use the lack of observational evidence for them to constrain the nature of the global topology of the universe. This is a pretty interesting idea because there are very few ways that we can study the overall shape of the universe (shape here means topology, so does the surface looks like a plane, a sphere, a donut, or what?). There are lots of ways we can study the local details of the universe (the geometry), because we can look for the gravitational effects from massive objects like stars, galaxies, black holes, etc. However, we have essentially no access to the topological structure, because gravity is actually only a local theory, not a global theory (I could write forever about this, but I’ll just leave it for a later post maybe…). We have zero theoretical understanding of the global topology, and our only observational understanding comes from studying patterns in the CMB. The trick with cosmic strings is that they actually serve to connect the local gravitational field (the geometry) to the global structure of the universe (the topology).

The game is this – take a spacetime with cosmic strings running around everywhere, and take a flat surface which intersects some of them. This surface can always be taken as flat, so the intersections are conical points. If you measure an angular coordinate around each point, you won’t get $2\pi$, you’ll get something a bit smaller or a bit larger since the surfaces are twisted up around the points. It turns out that if you add up all the twists, you had better get an integer – the genus of the surface. The genus is essentially the number of holes in the surface. A sphere has $g=0$, a torus $g=1$, two tori attached to each other have $g=2$, and so on.

Now we consider what kind of observational evidence there is for cosmic strings. The short answer, none! People have been looking for them in the CMB, but so far they’ve only been able to say “if cosmic strings exist, they must be in such-and-such numbers and have energies of such-and-such.” If we use these limits, we find that to a very good approximation, if cosmic strings exist, a surface passing through them must have genus 1, and therefore be a torus (the surface of a doughnut)!

Ok big deal – but here’s where the foliations come in. For example, if we parametrize our spatial (3-dimensional) manifold with tori, the result is a 3-torus. So this actually implies that space is not a sphere, but is a solid torus (like a doughnut). The mathematics behind this statement are actually quite profound, and were worked out in the early days of foliation theory by the likes of Reeb, Thurston, and Novikov. But the idea is that such foliations of 3-manifolds are very stable, and a single closed surface greatly restricts the kinds of foliations allowed for the manifold as a whole.

The archive paper where I discuss this in more detail can be found here. This idea that space is not a sphere is not new, and there is actually some evidence for it in the CMB, in the form of a repeating pattern (or a preferred direction) in space. But my primary interest is pointing out that this is an independent way of measuring the topology of the universe, since it’s based on local observations of strings in the CMB rather than overall patterns. If strings don’t actually exist, it can still be used to study the presence of the conical singularities, but I expect the restrictions on the topology are much less strict. Perhaps I’ll look into that further into that, but for the moment I’m happy with this. It’s a new way to determine information about the global topology of the universe, and it’s a great combination of pure mathematics, theoretical physics, and observational cosmology.