# Paper Explainer: Gravitational probes of dark matter physics

This is description of my recent paper with Annika Peter of OSU, where we consider how to use gravitational measurements of dark matter (that is: astronomy and cosmology) to understand the particle physics of dark matter. I have a separate post on a very fun section of the paper, where we think about what you could learn about the Standard Model if you were made of dark matter and could see dark matter (but couldn't see baryons). Beyond that motivating thought experiment, this monster of a paper (126 pages and over 600 citations) is trying to cover a lot of ground.

The central thesis of the paper is that there is a huge potential to learn about the properties of dark matter: things like mass, interactions, production, etc using measurements from astronomy. This is not a completely novel idea: we know a great deal about dark matter from astronomy and cosmology (for example: dark matter is "cold" and not "hot"). However, there is an immense opportunity in the near future to do far more, thanks to improvements in simulation and some powerful new astronomical surveys which will be occurring.

To take advantage of all this new astronomical data, we particle physicists need to start doing our homework. One of the issues with melding the results of astrophysics with the questions that particle physicists want to answer about dark matter is that the two groups speak different languages. For particle physicists, we're interested in the nitty gritty of how dark matter (and any other particle that interacts with dark matter) fits together at a really granular level. We don't really know as much about what big astronomical surveys do, and how they might impact our theories. Astronomers are much more big-picture, as you'd expect from people who look at objects that are, at minimum, some $10^{50}$ times larger than individual particles. They're not accustomed to throwing around models of particles where everything is a free parameter. Neither approach is worse or better than the other, it's just two different sets of training and problem-solving.

Adding to this is that particle physicists did entirely too good a job of selling a particular type of dark matter (the Weakly Interacting Massive Particle, as found in supersymmetry) to our astronomer colleagues. WIMPs won't change astronomical results that much, they're nearly the most vanilla of dark matter candidates, but they are just a candidate for what dark matter might be. The problem is that we seem to have convinced a non-negligible number of astronomers that all dark matter acts like a WIMP, so they're not looking at what interesting models of dark matter their results might constrain. So you have one group (particle physicists) who don't know what astronomers are capable of doing, and another group (astronomers) who don't know that what they're doing can be repurposed to measure dark matter properties in novel ways.

One of the major impetuses for the paper was trying to come up with a common language that both groups can use to communicate what they are talking about when they talk about dark matter. The issue boils down to the vast possibilities that could be lying in the physics of the dark sector. Dark matter might be something as "boring" as a WIMP or it might be an entire mirrored copy of the Standard Model, with all the complexity and baroqueness that would imply. But if you're trying to build bridges between fields, you need to be able to boil all of that down to one or two parameters, at least to start. Such a simplification will miss much of what is interesting about a particular model of dark matter, but it will hopefully allow both groups to find some common ground.

There are many ways to try to capture the physics of dark matter in a set of parameters. But when Prof. Peter and I surveyed the options, we found nothing really to our liking. So we made our own. We wanted two parameters: one particle physics and one astrophysics. Then we could imagine classifying dark matter theories according to these two parameters.

The two parameters we picked we call $\Lambda^{-1}$ and $M_{\rm halo}$.

The first, $\Lambda^{-1}$, is the particle physics parameter: it has units of inverse energy. Basically, it is a measure of the interaction between dark matter (whatever it is), and the Standard Model. If you imagine that dark matter exchanges some particle with the stuff we know about, that particle has some couplings (measuring "how strongly" it interacts with both dark matter and the Standard Model), and some mass. Very roughly, one expects any rate of interaction to be proportional to the combination

${\Large \frac{g^2}{M}}$

where $g^2$ is (very heuristically) the combination of the Standard Model and dark matter couplings, and $M$ is the mass of whatever particle is allowing the dark matter to "talk" to the Standard Model. The smaller this number is, the harder it is for the Standard Model to interact with dark matter. The couplings $g^2$ are just numbers, and mass $M$ can be written (by particle physicists) as an energy (via $E=mc^2$), so this combination has units of inverse energy. So we just call it $\Lambda^{-1}$, and viola, we have a parameter that captures some measure of one of the central questions particle physicists have about dark matter, namely "how hard is it to see dark matter in an experiment?"

Of course, the real answer to that question has to do with which Standard Model particles the dark matter interacts with, and lots of other issues besides. But hey, we're compressing everything down into two parameters, we have to simplify somewhere.

The second parameter, $M_{\rm halo}$ is the astrophysics parameter. It is defined as

The mass of a dark matter halo at which a particular theory of dark matter predicts deviations from a theory of pure cold dark matter.

What do we mean by that?

Well, first we have to define what we mean by "halo" and then what we mean by "pure cold dark matter." We actually spend a significant amount of time in the paper carefully describing what a dark matter halo is and how astronomers parametrize it. Very generally, we know that in the early Universe, the matter and energy was distributed very uniformly. But not perfectly so, and those regions where there was a slight overdensity grew (because gravity is attractive), and become more and more dense. Critically though, there was a distribution of these overdensities in physical extent: there were overdensities that were only a few parsecs across, overdensities of a few 10's of parsecs, hundreds, kiloparsecs, and so on. Larger halos contain smaller halos, and so on down the scale.

Primer on dark matter halos from our paper. The "virial mass" is the mass of the dark matter contained in the halo (and maps to our astrophysical parameter). However, astronomers cannot measure mass directly, and so use proxies like orbital velocities of visible objects (galaxies, stars, and gas) to estimate mass. Presumably the hierarchical structure of halos continues down to ever-smaller scales, but below dwarf galaxies, no visible bodies are known to be embedded in the dark matter halos.

Naively, you'd expect the larger dark matter halos to contain more baryonic material, and this is more or less what we see. Groups of galaxies are found in dark matter halos massing 100 trillion to a quadrillion Suns ($10^{14-15}\,M_\odot$). These contain Milky Way-type galaxies, which mass a trillion Suns ($10^{12}\,M_\odot$). Galaxies like our own have satellite galaxies called "dwarfs" which also appear "in the field" (far from parent galaxies). The smallest known dwarfs might have a dark matter mass of a hundred million Suns ($10^8\,M_\odot$). Presumably, there are even smaller halos of dark matter, which contain baryonic gas. However, such small objects are not yet seen directly, because they do not contain many (or perhaps any) stars, and thus are not apparent in the surveys astronomers have performed. One of the characteristics of cold dark matter is that this hierarchy of clustered halos should continue down, to arbitrary small scales. There should be dark matter halos of a $100\,M_\odot$ or even the mass of a single Sun: $M_\odot$. We've never seen them though, so we don't know if they really exist. But that's the prediction from what we can infer about the distribution of these primordial density perturbations.

But if you start modifying the dark matter particle physics, you're going to erase or modify this hierarchy of halos. I called dark matter "cold" a couple of times here, and that's a good example. Cold dark matter just means dark matter that was not moving at relativistic speeds early in the history of the Universe (in particular, when the Universe was composed of approximately equal densities of matter and radiation). Material moving at or near the speed of light can't be trapped inside gravitational perturbations short of black holes, so if dark matter wasn't "cold" (i.e., if it was "hot"), then it would move out of these initial density perturbations, erasing them. However, they can only move so far before the Universe cools enough for them to become non-relativistic, so they only erase the density perturbations that are sufficiently small - that is, dark matter halos today that are very massive should be fine, but the smaller ones shouldn't exist. Thus, the presence of dwarf galaxies limits dark matter to being "cold" (or at most "lukewarm"). We can characterize a model of hot or warm dark matter by the minimum dark matter halo mass that survives: this would be the parameter $M_{\rm halo}$ for those models.

But other models of dark matter will also deviate from the predictions of cold dark matter at some scale. Even particles as "vanilla" as a WIMP would terminate the hierarchy of halos at some scale, it's just that this scale is expected to be tiny (often $10^{-6}\,M_\odot$, which is about the mass of the Earth). We can't detect such tiny halos yet (or maybe ever), because they contain next to no baryons, but if we could, that would provide powerful limitations on the type of dark matter particle physics that could be at play.

Hot dark matter (or WIMPs) alter the prediction of cold dark matter at very early times; that is, they set a $M_{\rm halo}$ primordially. But you could also imagine that dark matter particle physics could induce subtle changes over time in the structure of dark matter halos at some scale $M_{\rm halo}$ over time, that is, an evolutionary deviation. One example would be a model of self-interacting dark matter (SIDM), where dark matter particles can ricochet off each other at non-negligible rates. This would allow energy to move efficiently through a dark matter halo, heating some parts and cooling others. That wouldn't necessarily erase structure, but that would introduce some characteristic $M_{\rm halo}$ where you should see some difference from a model without self-interactions.

So these two parameters, $\Lambda^{-1}$ and $M_{\rm halo}$, allow you to capture a lot of interesting dark matter physics in a plot-able space. The larger $\Lambda^{-1}$ is, the easier it will be for a particle physicist to find dark matter in their experiments (all else being equal). The larger $M_{\rm halo}$ is, the easier it would be for astronomers (though we think a lot more work needs to be done to make this possible for general theories of dark matter, thus the paper). We made a plot estimating where many popular models of dark matter fall in our parameter space.

Estimates of particle physics and astrophysics parameters for a number of dark matter models. See paper for details.

Our take-away from all this is that there is a huge range of dark matter theories that have far too low of a $\Lambda^{-1}$ to be reasonable searched for in particle physics experiments. The only way such models would be apparent is in astrophysics. Further, even if a model has a measurable $\Lambda^{-1}$, that is just the dark matter-baryonic interaction; there may be internal interactions and physics at play that would be completely invisible to, say, the LHC. Again, the only hope to find such physics is by turning to astrophysics.

Having introduced what we hope is a useful way of thinking about dark matter for both particle physicists and astronomers, we turn in the paper to what we might hope to learn from astronomical surveys, and what work needs to be done for the power of these searches to be full realized. We start by discussing a set of "hints" of non-trivial dark matter physics that have been of great interest for the last 10 years. These hints are generally called the "Crisis in Small Scale Structure," and basically they boil down to a number of discrepancies between our expectations of cold dark matter and our observations in halos of dwarf galaxy mass and above. That is, the Crisis might be pointing towards an existing $M_{\rm halo}$ of $10^{8-11}\,M_\odot$.

A summary of the hints for deviations from predictions of cold dark matter at particular halo mass scales (BTF is "baryonic Tully-Fisher relation" and "TBTF" is "too Big to Fail."), compared to the halo masses where baryonic effects are expected to exist and must be correctly accounted for.

The problem however, is that we can't be sure yet that we really understand both the prediction of cold dark matter (that is, the distribution of halos without any non-trivial physics), and we can't be sure that we're correctly mapping the observed properties of collections of stars and gas in to a mass for the halo. The former problem is that these class of galaxies are exactly where we expect baryons to start becoming important. Baryons can form stars. They can cool and form disks in galaxies. That allows energy to be injected into the halo (via supernova, for example), and increases the effects of gravitational tides. Without correctly modeling all those effects, which is computationally expensive, we can't be sure that what we're seeing isn't just normal gravitational physics in the dark sector combined with baryons.

Secondly, we need to remember that astronomers can't see dark matter directly. They only see stars and gas. They use the motion of those stars to infer the local gravitational field, and from that get a map of dark matter (this ignores gravitational lensing, which gets a more direct picture of the gravitational effects of dark matter). From the velocities of stars then, we must map to a particular halo mass. If that mapping is off for one reason or another, then the data will show a deviation from cold dark matter predictions where none exist.

After surveying the literature, this is basically where I come down, and hopefully the paper will make that clear to readers: there is room for new physics at small scales in dark matter. But the "Crisis" can probably be resolved without appealing to new physics: it most likely is a combination of baryonic effects and systematics in the measurements that can be corrected for. So we don't need new physics yet, but we need to keep looking. The Crisis in Small Scale Structure is both a perfect example of the sort of astrophysics-informs-particle-physics that we are interested in, and a cautionary tale.

So if the measurements resulting in the Crisis in Small Scale Structure aren't evidence of new physics, what are we going to do?

The rest of our paper is a long discussion of the many opportunities astrophysics will be affording to measure $M_{\rm halo}$ down to ever-smaller scales.

Our first point is that we really need to nail down the effects of baryons, which involves improving our simulations of dark matter structure formation. This is already in progress. There is a difficulty, in that these simulations are very computer-intensive: millions of hours of CPU time on large clusters. If you want to test the effect of new dark matter physics, you really should do that in conjunction with these baryonic simulations. However, since they're so expensive to run, simulators can't just throw in some crazy new model from a theorist to see what happens. So we need to figure out ways to get results about how new theories of dark matter would affect structure without using simulation that are at least approximately correct. If we have an interesting idea that these estimates suggest would have some very interesting result, then we can lobby our simulation-oriented colleagues to do a more expensive and accurate computation with.

Then we particle physicists need to start becoming much more aware of the opportunities coming our way from observations. We included a sketch of astronomical probes that are or will be available, and the scale of $M_{\rm halo}$ that we could hope they will be sensitive to.

Picking out some (since the full discussion is over 20 pages in the paper), at very large scales, we can look for novel effects of dark matter in the expansion history of the Universe. There are already some slight hints that the standard cosmology might not fit all the data, but these are not at a high enough statistical significance to take too seriously yet. But in the next few years, we'll get separate measurements of that cosmological history, that hopefully will resolve this question one way or the other. If discrepancies remain, then dark matter physics should be one of the first places to turn to for a resolution.

At smaller scale, recently astronomers have started identifying "ultradiffuse" galaxies that are difficult to fit into the current story of galaxy evolution. These are being discovered now as we map more of the sky in greater detail, and these objects might help us resolve the existing Crisis in Small Scale Structure more satisfactorily. And who knows, maybe reveal a new Crisis that requires new physics.

At smaller scales ($\sim 10^9\,M_\odot$, i.e., dwarf galaxies), the more accurate surveys are both discovering new galaxies and resolving their velocity structure (and thus their dark matter density) more accurately. Again, this will be important for the Crisis at Small Scales, as well as testing many models of non-trivial dark matter. There are enormous data sets relevant to this scale of $M_{\rm halo}$, and more will be on the way.

Below this mass scale are the halos we expect to exist from cold dark matter, but have never seen. We have some ideas to push down our measurements, but really, this is the final frontier and new innovations are required. My personal obsession in terms of dark matter astrophysics right now is the Gaia space satellite, which is mapping the position and location of billions of nearby stars. This will allow us to do many things, but my day-dream is that it would allow us to pick up evidence of small dark matter halos drifting through the nearest kiloparsec of the galaxy or so. Whether that happens, or if Gaia can detect it if it does, is so far an open question.

As is probably apparent from the length of the paper, we thought for a long time about these sorts of problems, and wrote for a long time too. Dark matter is one of the biggest open questions in physics today, and we think that astrophysics might contain the key. It might not, but if we don't look, we'll never know. There's lots to do, and lots of fun to be had. And that's why we do science, right? Because it's fun.

At least, that's why I'm here.