 So definitely, no worries, heavy weight. Yeah, OK, good, good, yeah. I'm just, yeah, just using the blackboard. I mean, at least the very last lecture, I might show some slides, but yeah, I'm planning to just do blackboard. Right, welcome back. With the second lecture of today, the first lecture on dark matter by Tracy Slater from MIT, expert of dark matter, can take advantage to ask everything you want to know about it. Welcome back, everyone. So as it was just introduced, my name's Tracy Slater. I'm a professor at MIT in the US. I'm going to be around all week, so feel free to come chat to me and ask questions, a few questions and I'll answer by these lectures. I work on dark matter from a range of different directions from a theoretical perspective and also from a detection perspective. I'm particularly interested in search of the dark matter using astrophysical and cosmological data, which I hope to get to in the last lecture here. So what I want you to know at the end of these lectures is the basic properties that we think dark matter has. We believe that it's about 80% of the matter in the universe, that it's fairly cold, that it's not relativistic in the present day, that it hasn't been for quite some time, that it's fairly non-interacting, at least that we can set fairly stringent upper limits on its interactions, both with other dark matter particles and with light and visible matter. So I want to take you through the evidence for those properties, as it was originally constituted, going back to the 1930s, and how we understand those similar probes today. I want to talk about various ideas that we have for dark matter and how it might be produced. And the zero thought statement here is that we really don't know what dark matter is. There's an enormous range of mass scales it could potentially inhabit. There's an enormous range of possible kinds of physics it could be connected to. I'm going to talk you through some general statements that we can make about how dark matter must behave depending on its mass scale and tell you about what kind of fairly model independent constraints we can put on different parts of that mass range and what kind of production mechanisms would allow you to get the observed amount of dark matter in the universe at those different scales. And then, so this is going to be mostly the focus of the first lecture today. Lectures two and three, I'm going to talk about dark matter models, ways to get the right production mechanism for illustrative models that populate sort of different parts of this big parameter space. And then in lectures three and four, I'm going to focus on how we actually look for this stuff. So both with terrestrial searches and colliders and in underground experiments and with my favorite kind of search, using telescopes to observe our universe and try to use that to tell us about the properties of dark matter, both what we know already, which is how it interacts gravitationally and going beyond that. So that's the outline of the general five lecture series. The goals for today, at the end of today, I'd like you to be able to explain what the evidence for dark matter is, what we think we know about it, and why we think we know that and how you would go about it. Okay, and please, as Matt already said this morning, please feel free to ask questions as I go. If you just wanted to sit here and let me recite everything that's in the lecture notes, there are lots of good books about dark matter. I mean, the things that make these summer schools particularly useful is that you get to ask questions, we get to talk about points of confusion. If you're confused about something there, there's a high probability that about 50% of the room is also confused about it, so you will be helping out your colleagues by erasing those points. Okay, so, where to begin? Well, let me begin by a bit of a historical overview, going back to the early evidence for dark matter, and this was first recognized as being an issue that we had to think about around in the 1920s and 1930s, and there were a number of people thinking about these problems, but I'm going to focus on one analysis in particular, which was work that Prince Wiki did starting in 1933, and there was a particular study that he did where his goal was to try to estimate the mass of a galaxy cluster, and in particular, the Kome galaxy cluster. Okay, so first, people at the back, is my writing large enough or does it need to be larger? Okay, if it's good? Also, people at the back, can you hear me okay? Okay, if at any point in the lecture, my writing starts to converge to a smaller value or my microphone switches off and I don't notice, I need you to wave urgently at me to let me know that this is a problem, because it's hard for me to notice from down here. Okay, so, Zwicky's goal was to measure the mass of this particular galaxy cluster. He was not looking for physics beyond the standard model, he was not trying to solve puzzles in particle physics, he just wanted to know how much mass there was in this cluster. But, you know, being a good scientist, he thought okay, we should, if I can get a number for the mass easily enough, but I should try to validate that number. So I should try to come up with two possible ways of measuring the mass and see if they give the same result. So, his two, he actually, I was rereading his paper earlier today and he actually suggested five methods, one of which like included looking at the gravitational lensing of light by the cluster, which is a sophisticated method that we actually do use today to look at dark matter masses, but the two that get a lot of attention because they were simple was method one was sort of a census method, was to look at the galaxy cluster, say okay, how many stars are there in this cluster, how much light is there coming from this cluster, how many galaxies are there in this cluster, say okay, I think I know roughly how much mass a galaxy should have per unit light, just from looking at neighboring stellar systems, so use that measurement of the luminosity to convert that into an estimate of the mass. So this is, so method one is, okay, so this is, so that's method one. So we can, okay, so let's just go through that, the way that Zwicky did it, he said okay, I look at the total luminosity of the coma cluster, of course what I can actually see is it's luminosity diluted by the distance between in mass, and this was actually the biggest error in his early calculation that he got the distance wrong by a factor of several, but let's just take his numbers, see what he got, so he, back in 1933, said that this system has about 800 galaxies, and that the typical mass of that galaxy is about 10 to the nine solar masses, and then, so then you, so if you do this, if you do this estimate from the luminosity, and this was, and this was based on an estimate based on local measurements of systems with stars in them, that the total mass over the total luminosity should be about three times the ratio of the mass of the sun to the luminosity of the sun. Using this, he estimated that this was the total mass in the cluster just by counting up the galaxies that he can see, so this gives you an m tot of about 800 times 10 to the nine solar masses. Does anyone happen to know, it's totally fine if the answer to this is no, but does anyone happen to know roughly how much the sun weighs and grams or kilos? All right, I heard an answer up there, say it loud. 10 to the 30, great, excellent, so 10 to the 30 watt. Kilos, good, yeah, so it's about, so the number that I got and the Wiki used was about two times 10 to the 33 grams, but yeah, very good. So that is, okay, so this gives you a number of about, well, we don't really need this to several significant figures, but let's do it anyway. This gives you a number of about 1.6 times 10 to the 45 grams. Okay, great, we've measured the mass of this cluster. So okay, that's our method one, so let's try, that's sort of just count what we can see. So method two is to use the fact that the gravity depends on the mass of the cluster. So we can assume that this cluster is an equilibrium system and use the virial theorem which allows us to relate the kinetic energy of particles propagating in a gravitational potential to the overall potential energy. So this applies for systems that are sufficiently in equilibrium, sufficiently virialized, and the potential energy in this system, we can estimate as being M squared of R where M is the total mass of the system as the approximate radius of the system. Now this pre-factor depends on exactly how the system is configured. I mean, this is, I think this is assuming like just a spherical blob of matter. So is Wiki considered, okay, maybe this could be a different number, let's consider different distributions of matter. He said, look, you know, I've tried a lot of things and basically this number is always bigger than about one over five. So he actually took one over five as his calibration, but we'll use this three over five number. Okay, so if we can measure the potential energy, we can get the mass if we can estimate the radius, which we can do just by observations. And if we, and okay, so, and this tells us that to get the potential energy, all we need is the kinetic energy. So we can estimate the kinetic energy via looking at the typical velocities of particles in the cluster. So the kinetic energy, this other side of the equation, this is gonna be roughly, so this is gonna be roughly, this is gonna be a half times the mass of the cluster times the average velocity times the average V squared in the cluster. So putting this together, putting these two pieces together, what we end up with is that minus three fifths, gm squared over r equals minus two times the kinetic energy. So that's minus m times V squared. We can cancel out one factor of these masses. So now, great. So now we have m equals five thirds V squared r over g. So Zwicky estimated that the typical velocity, so he looked at eight galaxies in the cluster, he looked at the Doppler shift of the radiation coming from them, and he used that to estimate the typical velocity, value of about a thousand kilometers per second for the velocity, which is actually pretty good. The modern value, the velocity dispersion is measured to be about 1,082 kilometers per second. This is done by a paper from the 1990s. So this was actually, okay, it's not quite right within the error bars for the purposes of what we're doing. This is perfectly fine. So we can take this velocity dispersion, you take this to give us the characteristic scale of V squared. So we can talk the radius of coma to be about 10 to the six light years. It's about 10 to the 24 centimeters. And so then we can just plug these expressions into this result for the mass. And if you do this, I'm putting all my notes online, so yeah, question. How did he estimate the radius? It's a good question. I mean, I think it was big enough on the sky that he could just look at the radius. But of course, you need a distance estimate to get the radius, right? And I think this is actually one of, but yeah, so he got his, to get his distance estimate, he needed to know what the Hubble scale was, which he actually had wrong by a non-necological factor. But I think I could be wrong, but I believe it was just like, just the observation of how spread out the galaxies one. It's a good question. We can check it later that I was, the papers publicly, there's an English language version of the paper that's publicly available from 1937, which I was reading earlier, so we can check that. Right, so just to repeat the question, the question was how did, how did Wiki measure the radius? How did he get this 10 to the six light years number for the radius of the cluster? There was a question up the back there? Okay, so sorry. So you say you think by measuring the Doppler shift of the total system and of the individual elements, you could get the, you can get a radial estimate. Yep. Oh, good. Oh, okay. Yeah, okay, okay. So the claim is that you can, you think they pull that from the frequency information? Yeah, okay, cause you know, okay, okay, okay, cool. So the claim is that, yeah, by measuring the frequency information of the overall system and of the sub galaxies, you can also get an estimate of the distance scatter. Okay, cool, I did not know that. So if you plug in these numbers, then what you end up with is a mass of a few times 10 to the 47 grams. Now the modern value is a little bit high, is actually a little bit higher than that, cause around 10 to the 48 grams. But this is a reasonably good estimate. This estimate's significantly closer than this one. But even like just taking Wiki's numbers at face value and certainly in the modern era, we've seen these numbers shift around by a factor of a few. So we keep noted, hang on a sec, this number and this number are different by about a factor of 200. You might say, okay, like this factor of three, maybe this is a little fishy, maybe it's different here. But I mean, this corresponds to a mass to light ratio for the Koma cluster of about three times 200, which is 600. So you're naive, the expectation was a mass to light ratio of three in these units and you got, I've written 600s where he got 500. So that's an interesting discrepancy. Okay, so the modern value, so the modern value finds that this discrepancy is actually more like a factor of 50. The mass to light ratio in Koma is about 160. Solar masses over solar luminosity. Swicky was off by a factor of a few, but he had the right general idea, which is, and this was one of the first clear observations that the mass of the system, and this holds generally for galaxies and for galaxy clusters and for all the large scale systems we see in the universe with a couple of rare exceptions, that the gravitationally bound mass of an object inferred from looking at the velocities of test particles is much larger, other might want to do orders of magnitude than the mass that you would infer from simply adding up the luminous matter in the system. Now this could be indicating a couple of different things. It could be indicating that maybe this Newtonian gravity calculation that we did here is not applicable to systems as large as this or it might be indicating that there's something important that we are missing when we do a census based only on the objects that we can see. Now in the 90 or so years since this paper we have gotten significant evidence for the extra matter that we can't see hypothesis, which is why most of the field now focuses on that hypothesis rather than on some modification to gravity, but this is in some sense where it started just by trying to measure the mass of these large systems. Okay, so nonetheless, that was in 1930, but people said, oh well okay, so there's stuff in this galaxy cluster that you haven't seen. Maybe it's stars that have burnt out, they're not shining brightly anymore, you don't see them anymore, but they're still just ordinary matter. Maybe this is just a puzzle that we won't resolve. The next major step forward in our understanding of what was going on happened in the 1970s when people started to look not just at total masses of systems like these, but at the velocity dispersions of test particles within galaxy scale objects and at how they varied with distance from the center of the galaxy. These were rotation curve studies. So the idea here is very, very similar to this Zwicky method too, which is to say okay, we can measure the velocities of objects by looking at their Doppler shifts relative to us. We're going to use those velocities as a tracer for the gravitational potential in which they're propagating. So let's think about the orbital velocity of objects, these could be stars, they could be clouds of gas that emit lion photons that we can measure their energies. Let's consider objects like this moving around the center of our galaxy. What does our galaxy look like? Well, it has a bulge of stars and gas at the center superimposed on a spiral disk. So let's suppose just as a first estimate that most of the mass in the galaxy is concentrated in some central region, that can be the bulge, it can be the inner part of the disk and that central region has a total mass capital M. This isn't a crazy approximation. I mean, you can do better, you can formally model the disk plus the bulge and try to work out exactly what this would look like, but to a first approximation, the stars in our galaxy are reasonably concentrated. When we go outside that region, we can just use the Newtonian mechanics that you learned back in high school, probably ask how does the velocity vary as a function of r? When you're orbiting a body of mass M, then we expect the centripetal force and the centripetal acceleration v squared over r to be given just by G capital M over r squared. So as a consequence, the velocity of a test particle should just scale like one over the square root of r. If we're looking at a test particle that is orbiting outside the central region. Okay, so what we'd expect is if we look at a range of different test particles, a range of different stars or gas clouds orbiting around our galaxy, a different radii, and we look at this velocity as a function of r, then we should see this falling, you should see this steadily, this typical velocity steadily falling off at large r. Again, this is just Newtonian physics. Now you can do a little bit better, so obviously once you get in close enough to the center, this curve is gonna turn around and fall off because then you're inside the central region, the enclosed mass will be smaller. So here's a particular example, doing a bit better than this dumb approximation. If we take an actual galaxy, ask what this should look like based on modeling of the halo, you expect to see a central peak as you move out of the region with most of the mass, and then a steady fall at larger radii. So in this example, the peak based on the stellar system, a detailed model gave you that you expected this peak to be at a velocity of about 130 kilometers per second, and that then out at large radii, you'd get down to velocities of about 50 kilometers per second. Now the scales here in this particular galaxy, if I look at the radius in KPC, the peak was expected to be about around 10 kiloparsecs, this is 20, this is a linear scale. Okay, so that's what we would expect, and even if there's a lot of extra mass in this galaxy, as we expect there would be from Coma, even if there are many, many old burnt out stars for every bright one that we see shining, you could, you might expect if that mass is coming from the same source as what we can see that it would still be distributed in more or less the same way, and thus we would still expect this one over square root R distribution to hold just maybe with a higher overall normalization, since the overall normalization is set by the enclosed mass. But what you actually see when you try to observe test particles at a range of radii for this system, and there are lots of stars and lots of gas clouds, so you can get pretty good data points. What you actually see is that this rides at smaller, that works, that happens fine as expected, but then at larger, instead of seeing this fall off, you see an essentially flat velocity distribution in this system, it's at a velocity of about 150 kilometers per second, so higher than this expected peak, that just keeps going for quite a long way. And this can be out to sort of, so the typical, this can continue out to the 100 KPC scale, just for calibration, the distance of the Earth from the center of the galaxy is about in a half kilofarsecs question there. So the way that we now define edges of galaxies, which is the virial radius of the galaxy, where the density, sorry, which is defined as the density within that radius is a certain fraction above the overall critical density, factor of 100, for the Milky Way, the virial radius is out at about 200 KPC. So on this linear scale, somewhere over here, the rotation curve doesn't typically stay flat all the way out to the virial radius. I'll tell you in a moment how we typically model those halos and what we actually think the density profile is doing. Yeah, so the luminous matter, yeah, the luminous matter is mostly pretty concentrated deep within this virial radius. So if you look at how far out the, how far out where the stellar density starts to significantly drop off, it's at order a few tens of KPC. Yeah, well, yeah, okay, I mean, what I said was, so the spiral arm, so the location of the, so if this was a Milky Way like galaxy, the location of the Earth would be right around here. As you go further out from there, the stellar density falls off in the Milky Way, you can assume that it falls off exponentially with a scale height of I think about 10 or 15 KPC. So the peak is here, but I mean, you can go out another 15 KPC and the density is not dramatically lower. But yeah, the fall off is, does that, so the question was just about what is the approximate radial extent of the galaxy as a whole and of the stellar disc of the galaxy, if the luminous matter? Yeah, so right, so this was the peak that you would expect in this galaxy just from modeling the observed disc stars. So yeah, so in this case, the reason that this peak is falling off is because once you get out to this point, you're outside the bulk of the stellar mass of the galaxy. But you can still see tracer particles quite a way out. I mean, it's not that hard to find gas clouds and stars that you are still fairly sure about to the galaxy even though you're outside the region when most of the mass is contained. There are still non-negligible stellar structures further out there. But of course, but yes, but what you empirically see is that there's not really much of a peak. In this galaxy, there's not really much of a peak at that radius, it just keeps going. Which tells you that either your theory of gravity is pretty wrong or there's a lot of extra mass contained in this region. So this is a paper from 1985. This is actually about a decade after large studies of the rotation curves of many systems were done particularly by Vera Rubin and her collaborators Ford and Thonard, but also by other groups. And they found that this was a pretty ubiquitous feature that in general, rotation curves were usually close to flat at large radii. Okay, so if the rotation curve is really asymptoting to a flat result. And again, let's assume spherical symmetry just for the convenience of doing this on the board. It's, of course, real galaxies are often not spherical symmetric, but we can make this approximation. Let's get just to get an estimate. So yeah, yep, let's go for it. Good question. So the question was, is it possible? Good question. So the question was, is it possible that this flatness is, that this flatness is coming from the presence of satellite galaxies that are orbiting the Milky Way? Because they can orbit a long way out. I mean, they, even the closest of these satellite galaxies can have orbits that come in with a few tens of KPC, within like 20 or 30 KPC of the galactic center, but the orbits can go out to 100 or 200 KPC. So yeah, so that's a good question. In general, the mass, so yeah, and to answer that question, you, I guess, need to know how the mass is distributed in the satellite galaxies. And that will tell you, is it possible for there to be enough mass contained in those objects to explain these flat rotation curves? The answer broadly is that it's not. You need most of the mass to be contained in a large halo, not in individually clumped objects within the inner galaxy. But yeah, that's a good question. So if we ask, okay, if the rotation curve is nearly flat, what does that mean in this picture? It tells us that, okay, so we want V squared over R. We want this to be G times some radial distribution of mass now divided by R. If we want V to be approximately constant, so this is the enclosed mass within some radius R, then this tells us that the enclosed mass within a radius R should be roughly proportional to the radius R itself. So if I'm looking out at this sum, so if I'm looking out at this 40 KPC or 50 KPC radius and this rotation curve extends flat out to here, then the bulk of the mass that I'm looking at actually needs to be between 10 KPC and 40 or 50 KPC, well outside the region where the stellar distribution is peaked. So this suggests that not only is there extra mass in our galaxy beyond what that we can see, but it's distributed very differently to the visible matter. Now, we still don't know at this stage what it is. It could be a lot of, for some reason, there's a lot of old non-luminous, there's a lot of non-luminous, non-radiating matter orbiting out beyond the distribution of the stars. Maybe it's black holes, maybe it's burnt out old stars. Then this could explain these observations if there's a lot of mass stored in smaller bound objects orbiting the galaxy at large radii, that could potentially also explain this observation, but the distribution is different, question up there. That's right, in galaxies as a whole, most of the mass is in the gas, not in the stars. Yeah, good, so that's a really good question first. Maybe the stars just aren't a good tracer, maybe the galaxy as a whole really is a much bigger blob of gas and only in the central region was the density high enough to have star formation. So again, you can try to observe where the gas density is. So my memory is it's more extended than the stellar distribution, it's still not as extended as this. It's again in that same general ballpark that the scale radius for extension of the gas is in the ballpark of 10 KPC or 10 or 20 KPC, question? Good, so okay, so the question was what's the history of this, like at what point did people really become convinced that what we were looking at was not just interstellar gas, but actually had to be something different? Okay, so I should say I'm, the history of this period is not my area of primary expertise, so I'll tell you what I can, but you absolutely should go and read up on this. So first off, just the question of where is all the baryonic gas in the universe and how is it distributed is a question that's, I mean, that's still in some senses an open problem. And even in recent years, there's been a lot of work on observations finding reservoirs of gas at large radii in galaxies and in galaxy clusters that was not previously known to be there. So this is not, so like that part of the story, just okay, where's the baryonic, where is all the gas? Was a question that is still, maybe not fully resolved today, and that there's been a lot of progress on since the 1970s. That's, so yeah, so okay, so that's one thing. Second thing is, back in the 1970s, there was a lot of skepticism, like just about the rotation curve measurements, were they correct? My understanding, looking back at this from here, is that once people understood that, yeah, these rotation curve measurements are really believable, then I don't think that, well, to my knowledge, which is limited, there was not a ton of pushback in the sense of, oh, maybe it's just all gas, because I mean, you can get at least a decent handle on where the bulk of the gas is just by looking at lion emission signatures, which people had been doing in our galaxy and in other galaxies for decades. But I don't know exactly what the time scale was for getting those sorted out. So I think I'd say, you know, I think it was already reasonably clear by the 1970s that the dominant component of the gas did not extend out to hundreds of KPC, which is what you would need for this. But the question, and also like the gas isn't, yeah, another way to say that is, the gas is not non-luminous. You can do a census of the gas, looking at how it's illuminated by the starlight, looking at emission and absorption lines. So I think that problem was already pretty solved by the 70s, but the question of where are subdominant components of the gas located? Even if the bulk distribution is an exponential falloff with a scaled radius of 10 or 20 KPC, out at 200 KPC, there still may be streamers of gas, well, that's a question that is still being investigated today. No, it's a really good question. This is good. Keep on coming. Okay, so this tells us, okay, so we believe that there's something out there. At this point, we, so the other thing is that the real evidence that there was some dark component of the universe that was not baryonic and was much more abundant in the dark matter came in the 1990s from a different set of observations. But, okay, but I'll get to that shortly. Okay, so in the 1970s, we had this evidence that there was something, whether it be baryonic or not, distributed out beyond the radius where we see the dominant components of the stars and the gas clouds. So what's the modern version? So the modern version of this statement based on improved observations and also improved simulations is that every galaxy and our universe is embedded in an extended halo of dark matter which can extend out to hundreds of KPC beyond the visible matter. And that the densities of dark matter within these halos approximately follow at least at large distances characteristic density profiles which are extracted from simulations of collisionless dark matter. We're gonna use these in later lectures. So we'll just write down the forms for you here. So you can see how they relate to this simple estimate that the enclosed mass increases roughly with the radius. So these are profiles that provide reasonably good fits to the bulk of galactic rotation curves. So NFW, this is Navarro, Frank, and white is, so these are both just analytic profiles that have been fitted to simulation data. They at least at large radii provide reasonably good fits. The dark matter density scales like this. So features of this profile, this is saying that there's some characteristic scale radius for the Milky Way, the scale radius is thought to be about 20 kiloparsecs. One way to think of this is it's the radius where the logarithmic slope of this profile changes from minus one to minus three. So with small distances, this r over rs term is small. The scaling, this is like a one over r profile in the density and at large radii, it's then this term dominates and we have a one over r cubed scaling in the density. Okay, so the intermediate step between these where the logarithmic radius changes from minus one to minus three where it goes through minus two, we call that point the scalar radius. So if we wanted to have this, if we wanted to have this mass, enclosed mass, scales as one over r scaling, so the enclosed mass is gonna be integral from r to zero of rho of r dr. So r squared, r squared rho of r dr. So this in a spherically symmetric distribution would correspond to a density that scales like one over r squared. So this kind of distribution scales like something that is rising a little bit more shallow. So basically, so flat density rotation curves corresponds to one over r squared density profile. The NFW profile goes from being flatter than that with r at large radii, to steeper than that with r at large radii and the flat rotation curve area will be sort of this intermediate region. Although you work this out in detail, the region of the rotation curve that is approximately flat can extend significantly out beyond the scalar radius. Okay, so that's what we think roughly what dark matter healers look like. Another way that it's often characterized is by what's called the Inastro profile. This is thought to be a somewhat better fit to simulation data than NFW. If you overlay them, you'll see that over the region where we can measure the rotation curves best, they give extremely similar results. So the Inastro profile looks like this. Alpha is a parameter that is just pulled from simulations. For milky waist-sized halos, we think that alpha should be about 0.17 and this r to the minus two is again like the scale radius where the logarithmic derivative of the density is minus two. So it's analogous to the scale radius in the NFW case. So, okay, so rotation curve studies tell us that galaxy, tell us that whatever it is that's increasing the overall mass scale of galaxies and galaxy clusters is also distributed in such a way that you get a flat rotation curve out to several tens of KPC where you would expect a steeper fall off from the visible matter. Now, today, we can take those observations out to larger radii and also into smaller radii and we can look at systems which have a much smaller number of stars associated with them. So it's in the Milky Way galaxy, it's pretty easy to find stars and gas clouds that act as tracer particles. It's much harder to do that in, for example, the dwarf satellite galaxies of the Milky Way that someone just asked about. These are systems which have much larger mass to light ratios than the galaxy as a whole. So we interpret that as they have much more dark matter relative to baryonic matter than the Milky Way as a whole. But so this on one hand makes them great laboratories to study the properties of this dark matter, but the downside is not much baryonic matter means very few stars. So it's hard to find tracers. But observations of these dwarf galaxies have in some cases been used to suggest that maybe these profiles in actual observations break down at sufficiently small radii, that this NFW profile rises quite steeply at small r, it's still rising like one over r. This dynastroprofile does eventually flatten out, but it can still have a pretty steep cusp at small radii. So this is, so as I said, these come from simulations, but there is some claimed disagreement between observations and these simulation based predictions. So even today, these kinds of rotation curve tests where you look at the velocities and orbits of tracer particles in systems and use that to infer properties about the dark matter are still extremely relevant. So there is, okay, so let me say, so the direction that these simulations go in is they typically suggest lower densities so by small scales, I mean subgalactic scales, I mean in the cause of dwarf galaxies. So the rotation curves that I was telling you about are measurements mostly on tens of KPC scales. These problems often show up when you try to do measurements at the order one KPC scales. Now, that said, the simulations from which people pull these profiles typically only include the dark matter. They do not include baryonic matter self-consistently. So there is an ongoing controversy in our field today as to whether these discrepancies can be entirely explained by the fact that the simulations aren't properly including the visible matter. Okay, so there's a review on this topic which you can look at up here. Two classic issues, two longstanding possible problems that have been raised in this context are referred to as the too big to fail and the cusp core problems. So what do we mean by this? So the cusp core problem is maybe simpler to explain was what I was just saying. These profiles drawn from dark matter only simulations suggest that the dark matter density should continue to rise pretty steeply into small distances from the galactic center. Sorry, DM only sims as the DM density continues to rise at small r. But there are claimed observations of small dwarf galaxies both around the Milky Way and in the field so between galaxies that suggest that the systems actually have flattened cores so the density profile flattens out and no longer changes with r at scales of about 0.1 to one kiloparsec in size. So on this plot, this is, you know, on this rotation curve, this is impossible to see, this is way down here in the central region but with better observations of dwarf galaxies and the stars that compose them we can now actually do these measurements, sorry. So this has been observed in what's called the things and little things surveys. There are papers on this by Oedau from 2012 and 2015. I can give you more references if you're interested in them. The too big to fail problem is a related issue that says that when we compare to simulations and look at the number of dwarf galaxies in our Milky Way and the masses of those dwarf galaxies then we don't see as many big dense dwarf galaxies as we would have expected. And so this term was first coined in a paper by Boil and Culture in 2012 and then it was showing that you can also see relationships like that outside the Milky Way in the field by Garrison Kimmel in 2014. Okay, so that's the, so when I say the reason it's called too big to fail is because one answer for why you might not see dark matter halos that you expect from simulations is well maybe they're just not forming stars. Like just in my simulations with dark matter only I can track clumps of dark matter moving through the universe but to actually see those in our telescopes we need there to be some baryonic matter attached to them. So we need them to have some stars but so this is a longstanding answer to what's sometimes also called the missing satellite problem which is just does the Milky Way have as many satellites as we would expect from simulations although that problem seems to have been getting less of a problem as we find more satellites. But the, so the reason it's called too big to fail is because these massive dense objects, I mean they've got a lot of mass, they should attract a lot of gas, they should attract quite a lot of baryonic matter. They should be too big to fail at forming stars and thus we should see them. Okay, so I think it's pretty well understood at the moment. I think it's pretty well accepted that simulations that only include dark matter that only include a component that is, you can hypothesize that dark matter is something that is completely collisionless, the only interactions it has are through gravity. This is nice and predictive and it's relatively easy to simulate and it turns out that this naturally gives you the density profiles that I've written down here, those density profiles naturally give you a region of flat rotation curves which can explain these observations. The heuristic picture here is that because your dark matter does not experience electromagnetic forces, it finds it difficult to lose energy, it finds it difficult to shed angular momentum and consequently it forms large puffy halos around galaxies rather than spinning down into a visible disk, like the visible matter. But those simulations, they work fine at these large, they work more or less fine at these larger radii, they fail when trying to describe the behavior of small satellites and a small radii. So the controversy is whether that can be fully explained by simply properly including the effects of the visible matter because once we get into this part of the rotation curve, you can see that you are not outside the range where the visible matter is negligible. That's exactly where all the gas is piling up, where all the stars are piling up and you might expect the effects of a visible matter on the dark matter halo to be large. There's also some claim, I should say, I've talked mostly about dwarfs here, but there's also some claimed evidence of this cost-core problem in other systems in low-surface brightness galaxies, in high-surface brightness spiral galaxies and in some of these systems, there's still a fair bit of debate about whether it could just be a systematic in the calculation that makes it look like there's a core, like whether you're measuring the wrong stars, you're doing the analysis wrong. But I think in these dwarf galaxies, there's reasonably robust evidence for some kind of coring. Okay, so once we include ordinary matter in the simulations, then this picture changes. We do expect these density profiles to get modified in the regions where there's a lot of, where there's considerable baryonic matter, which means that small radii. So ordinary matter concentrated, so I just said ordinary matter systematically more concentrated at small radii than any collisionless component is, just because it interacts through electromagnetism can radiate energy in angular momentum. So if we expect to see the effects of baryonic matter showing up, they're likely to be most pronounced at small scales. Okay, so how can ordinary matter affect the dark matter distribution on small scales? I mean, obviously, there's the zero-thorner, well, if there's more baryonic matter, you might think that will make my potential deeper at small radii compared to a dark matter only simulation. I'll have more gravitational pull. So shouldn't I just end up with more dark matter at the centers of systems? Once I add baryonic matter in, just from the gravitational pull increasing. Like that is an effect that happens. It's called adiabatic contraction. In baryonic systems, the dark matter density can go up just because the collapse of the baryons makes the gravitational potential well deeper. There are also other effects, and this is what makes this a hard problem, that there are nontrivial effects that go in opposite directions from the effect of baryons on dark matter. The biggest effect that people talk about in the other direction is supernova feedback or stellar feedback. So when a supernova goes off or several supernovae go off in quick succession in a region with a lot of stars and gas in it, that can trigger a large outflow of gas from that region. As somebody brought up earlier, most of the mass in our galaxy is in gas, not in stars. So as that gas outflows from this region, that can actually be a big transport of mass out of the region. That perturbs the gravitational potential. If there is a steep cusp of dark matter, when that potential is perturbed, it will slosh around this cusp and break it up. So, okay. These are effects which are pretty hard to study analytically, can be studied in the context of large-scale simulations that try to include both dark matter and a prescription for the baryonic physics and the star formation rate. But this kind of behavior also depends pretty strongly on how bursty your supernova rate has been. If it's just a steady stove flow, a little bit of gas is continually flowing out of the galaxy, that doesn't have much of an effect on the potential. If you have highly non-adiabatic behavior where many supernovae go off in quick succession and an order one fraction of the gas is moved out of some region, that can have a very large effect on the gravitational potential. And unfortunately, the level of burstiness of the star formation history and supernova history are not super well-known from observation. So, our situation at the moment is that the most sophisticated simulations, including baryonic matter, seem like they can often produce cause of a similar size to what we see in observations. They can help resolve issues like this too big to fail problem by reducing the density of dark matter on small scales. But that is contingent on turning some knobs in the simulation, which you cannot predict a priori. Sorry, this remains an open question. And there's a somewhat older review about the effects that can matter in these simulations. A few years old by now, but still has a lot of relevant physics in it. So that's an open question, but in the presence of that open question, a large number of people in my area have asked, okay, can we use these kinds of measurements, these inheritors of self-interaction, of a rotation curve measurements from the 1970s to try to understand what if it's not baryonic matter that's making the cause? Are there other aspects of dark matter physics that we can constrain or even find evidence for in this small scale behavior? So, and one thing that people talk about a lot is, well, okay, I sort of jumped from there's something out there that isn't distributed like ordinary matter to let's suppose that there's some new component that is effectively collisionless, that doesn't really interact. And the rationale for that at this point in the story is more or less, well, if I make that assumption, then I naturally find that there are density profiles that approximately match the rotation curves. But what if we didn't make that assumption? What if we allowed this extra component, whatever it is, to have some level of self-interactions, some level of collisional behavior? So, this was suggested back in 2000, Meisberg and Steinhardt as a possible way to get cause in these small scale systems. Before I go to this, I guess I'll just ask, did people have any questions that I missed? Cause I've been talking for a while. Yep, question up there? Yes, there will be, yes, there will be, yes, I'll be, sorry, yes, I will make them available online. I was just still doing some editing this morning, but yes, I will make sure it's available online. So yeah, don't worry about like copying down every word that I say. Yeah, sorry, can you say? Okay, the question is what could the effects of dark matter at the center of galaxies be to the formation of supermassive black holes? This is actually a very interesting question, but so this was actually one of the first questions that my PhD advisor asked me to look at when I was a grad student. And I know of like several other people who were asked the same question by their PhD advisors when they were grad students. So as a warning, if your PhD advisor asked you to look into this question, it's a really interesting question, but the, so what happened in my, so the issue, okay. So how can dark matter affect supermassive black holes? So there's an ongoing problem in the formation of supermassive black holes that says basically that we see very large black holes very early in the universe's history. And it's not obvious how they got that big. Now, and in fact, if you say, all right, we know when the first stars formed, let's assume that these supermassive black holes formed by the way that we know black holes formed today, a star went supernova, it collapsed into a black hole. And then we assume that from that point onwards, the black hole grew by a creating of what we would usually think is the maximum possible rate, which is what's called the Eddington limited rate. So then you find that it's very hard to get the kind of supermassive black holes that we see today, or even that we observe as quasars at Redshift seven and eight in time. It's, so okay, so you might say, okay, well this Eddington limit seems inconvenient. It's stopping my black holes from growing fast enough. Where does it come from and can I get around it? So the Eddington limit is a radiation pressure limit. It basically says that if you try to cram too much matter onto a black hole in a certain period of time, and the matter is the ordinary matter that we know about and interacts with the radiation field, then the radiation pressure will push the matter apart and stop you from accreting it all into the black hole sufficiently quickly. So if you can get dark matter to efficiently fall onto the black hole, that can evade this Eddington limit because dark matter, as we usually think about it, does not experience radiation pressure. It does not have interactions. So you can cram as much of it onto a black hole as you want and it won't cause problems with the dark matter. Although if you allow the dark matter to have self interactions or to interact with its own radiation field, then of course that can change that picture. Okay, so accreting dark matter onto black holes seems to be a promising way to make dark matter, to make black holes grow faster. The problem is that again, if you assume that dark matter doesn't have any self interactions, it doesn't interact with the radiation field, then basically the black hole will scoop up whatever dark matter was present there when the black hole first collapsed. But it's very hard for dark matter particles to lose energy and lose angular momentum. And so it's difficult to get dark matter to fall onto the black hole for the same reason that dark matter doesn't collapse down into a disk in our galaxy. So you can try to get around that by turning on dark matter self interactions. And if dark matter is sufficiently self interacting, that it behaves like a fluid in regions of high density, such as the center of the galaxy in regions around black holes, then it's easier to get it to fall onto the black hole. But it turns out that it's still pretty hard because the kinds of self interactions people usually talk about for dark matter don't include like, they still don't allow for radiation. Like you're not coupling to some massless particle like the photon that you can radiate off. That in turn means it's very hard to shed angular momentum. Which again means that it's hard to get your dark matter to be close to the black hole and going slowly enough that it will fall in. So it seems like in order to get dark matter to really feed black holes effectively, you probably needed to have some kind of dissipative interaction so that it can actually lose energy and angular momentum. And in that case, then yeah, maybe you can feed black holes with dark matter. There's also the sort of inverse problem which is just what does the black hole do to the dark matter distribution. Now in a galaxy like the Milky Way, the region in which the black hole's gravity dominates the gravity of everything else is extremely small and is much, much smaller than the scales that we're talking about here. It's not 0.1 to one KPC. It's a very sub-parsec scale. But people sometimes consider situations where maybe there could, where you could form a very sharp cusp of dark matter right around the black holes. There would be a dark matter spike there which isn't falling in again because it's hard for the dark matter to lose angular momentum. And in that case, you know, maybe you could get some kind of signature from dark matter particles colliding with each other and annihilating. So that's a really good question. But yeah, if your advisor is like, oh, I've had this great idea about feeding black holes with dark matter, just keep in mind, I know of like five of us that did that project and came to the conclusion that it's really hard to feed black holes with dark matter. So yeah, but it's a cool idea. Okay, so what if dark matter had self-interactions? I just told you that it's not necessarily a great way to feed black holes and make them large. But suppose we just wanted to do it to try to change the distribution of dark matter on these sort of 0.1 to one KPC scales. Okay, well, we can just ask, what kind of cross-section would you expect to need to change the overall dynamics of dark matter at all? So the criterion for this is basically that you want the average dark matter particle to scatter on order once per dynamical time. If only 10 to the minus five of the dark matter particles are ever scattering with another dark matter particle, it's not gonna change the overall energy distribution. Okay, so let's work out for something like a dwarf galaxy of the Milky Way. What should that number be? What kind of interaction cross-sections would we need to significantly change this collisionless dark matter picture? So what this means is that we want the number density. So the scattering rate is the number density of dark matter, which I'll call n, times the cross-section, times sigma, times the collision velocity of the two dark matter particles. So we want this rate to be comparable to, so this is gonna give me a rate. We want this to be comparable to one over some timescale, which I'll call tau. Okay, so how do we get this number density? So what we can actually measure is the mass density of dark matter. So we'll take the mass density divided by the dark matter mass. We'll give us the number density. So what we're gonna require is that the cross-section divided by the dark matter mass. These are the particle physics quantities that we don't know and that are pretty hard to measure. This should be comparable to one over the mass density of the dark matter, which we can hope to measure, the velocity of the dark matter, which we can hope to measure. And well, just by assuming that it's bound in the same gravitational potential as all the visible matter we can see and the dynamical timescale of the system. So if we put in numbers like this for something like a dwarf galaxy or something in the neighborhood of our galaxy, so the local density of dark matter at the radius of the Milky Way of the Earth from the center of the Milky Way around 10 KPC, this is ballpark one GB per cubic centimeter. So I'm gonna use this just for an auto-magnitude estimate. Like at the Earth is 0.4 GB per cubic centimeter in a dark matter clump, it could be somewhat higher. The V that corresponds to the Milky Way's rotation, well, okay, the V that corresponds to the Milky Way's rotation curve velocity is a couple of hundred kilometers per second. So it's about 10 to the minus three times C. For dark matter particles within a dwarf galaxy, that's a smaller region with a lower velocity dispersion. So we'll take come V of 10 to the minus four times C. So a few tens of kilometers per second, which is pretty typical for dwarf galaxies. And for the dynamical timescale of a dwarf galaxy, it's gonna be somewhere in order of a billion years. So if we put these three numbers in together, then what we find is that this corresponds to a cross-section, and you can check this yourself, of about between 0.1 and one, and you can see we've made a bunch of order of magnitude estimates here. But you end up with a cross-section around one square centimeter per gram. Now these are astrophysicist's favorite units, but since this is a particle physics school, we might wanna convert this into more particle physicist-friendly units. So length has units of inverse mass, this has units of one over mass cubed. If we do the conversion between centimeters and grams and GV, we find that this corresponds to a scale of about one over 100 MeV cubed. So if the dark matter mass was about 100 MeV and the cross-section corresponded to, this is like a QCD scale cross-section, right? The QCD scale is 100 MeV. So if I had a sort of QCD scale cross-section and QCD scale mass, then I would have a large enough self-interaction rate to mean that the average dark matter particle was scattering once in a billion years in the core of a dwarf galaxy. If I have a cross-section appreciably smaller than this, then nothing will happen. So this could be telling us about self-interactions, but if that's so, like, I mean, that's super interesting. It means that the dark matter is with itself actually reasonably strongly interacting by standard model standards. Like this is much larger than the lecture week cross-section, for example. So these self-interacting dark matter models tend to pick out relatively light dark matter compared to most of the particles in the standard model except the electrons and the neutrinos with pretty large scattering cross-sections. But you can ask, okay, if I turn on a scattering cross-section like this, what happens? And indeed, the effect is to reduce the height of these dark matter casps. It's to lower the dark matter density at small scales. And I mean, it does go in the right direction to explain these observations. But as I said previously, it's also possible that all these observations are telling us is that star formation and supernova feedback has been an important effect in the evolution of these dwarf galaxies. Okay, so that was all sort of like going from rotation curve studies in the 1970s, following that along, what could that potentially tell us about dark matter? And I see that I'm getting pretty close to the end of my time. So that's fine, I'll just say. So this leads us to another piece of evidence, which was one of the key pieces of evidences for why what we're looking at is some kind of new matter and not simply a modification to gravity. Just from the rotation curves, you might say, well, okay, fine, you just build up this whole structure with this collisionless particle called dark matter that gets the rotation curves right. And maybe it's non-collisionless, maybe it is collisional on sufficiently small scales and that explains these other observations. But can I just say that Newtonian gravity is different on galaxy scales? That we have measured a deviation from our standard theories of gravity. So the thing that I think really changed the community's mind about this was observations of the bullet cluster in 2006. So all the systems that I've told you about so far were systems that were approximately in equilibrium, where whatever the potential was, you could assume that both the dark matter and the visible matter had relaxed into the same potential. But if you wanna be able to tell the difference between how dark matter behaves and how modified gravity from visible matter behaves, you might wanna be able to separate, you would like to be able to physically separate the dark matter and the ordinary matter. A way that you can do this is by looking at non-equilibrium systems and in particular collisions of galaxy clusters. So that was the observation done by the Chanda Telescope back in 2006. So the picture here is supposed, the cartoon here is supposed that we have two colliding galaxy clusters and in the dark matter framework, each of these clusters, as I've said to you, has these large, puffy dark matter halos around them and then the visible matter confined to a smaller region within the halos. Now, when these clusters run into each other, then what's going to happen? Well, suppose the dark matter halos are collisionless, so these dark matter particles will just pass through each other. The only effect will be that the clusters will drag on each other gravitationally as they go through. But the gas is definitely not collisionless and so if this collision is head-on, when the gas clumps run into each other, they will exert pressure on each other and they will hit each other up. So then, okay, so then what happens? So these gas clouds will then slow down and the next step in the evolution, the dark matter halos will just keep going their own way. Incidentally, so will the stars inside the two galaxy clusters because stars are also collisionless in the sense that they're sufficiently compact that they don't run into each other very often. So what will happen in this scenario is that the dark matter halos will eventually come out the other side of each other along with the stars, but the gas clouds will still be stuck having slowed each other down, having heated each other up. So what you'll have is a clump of gas in the center, dark matter halos heading off in both directions and because this gas is hot, it will emit x-rays. And so this is exactly what was observed in the bullet cluster system in 2006. They looked for x-rays coming from the gas and saw this bright region of x-ray emission in the center of the cluster and they used gravitational lensing techniques, as suggested by Zwicky in 1930, to map out where most of the gravity was where most of the mass in the cluster was actually concentrated. And what they observed was that there was this central region where the x-rays were bright, but the mass peaks were actually well separated from this region. And this is very hard to explain by just a simple modification of gravity. If I change how fast gravity falls off as a function of the distance from the gas, you know, okay, that's fine, but it should still be true that the center of the potential is where most of the matter is. What we saw in the bullet cluster is that that's not true if you assume that the only matter is the visible matter. This also, by this argument that I just gave you, sets a limit on how well, on how self-interacting dark matter can be because if these dark matter halos had interactions with each other, then they would slow down just like the gas. Yep, question. Okay, good, so the question is how probable is something like the bullet cluster given dark matter simulations? Like if you, you know, other velocity dispersions are rather high. So, and I guess that also relates to just the question of do you see other systems like the bullet cluster? There have been some claims of systems which are sort of anti-bullet clusters where you appear to see the dark matter not separating out like this. Now, yeah, on the anti-bullet cluster systems, the opinion has sort of gone back and forward a lot over the years. There have been suggestions that maybe what we're seeing there is that the collision is sort of happening along our line of sight. So the reason why the dark matter doesn't look like it's separated is because they're just on opposite sides of the central collision along our line of sight. The question about the probability of the velocities in the bullet cluster in CDM. Yeah, so I can, I'd want to look through the literature for that. I actually don't know what the latter status of that argument is. I mean, my understanding is it's meant to be, like the velocity dispersion is meant to be in the order of a couple of thousand kilometers per second, which I didn't think was that unlikely for galaxy clusters, but I'm not up to date on this argument. So that's a great question. Okay, so this sets an upper bound on the self-interaction cross-section. Now, early on, the early bounds set on this were right around this 0.1 to 1 centimeters squared per gram range. However, there's been more recent studies which say you have to be pretty careful about setting these limits. You have to make sure that the way in which you're doing the analysis is actually respects what was done in the analysis of the actual data. So originally, so there's a paper from a couple of years ago which says that rather higher cross-sections around a couple of centimeters squared per gram are actually okay. So I don't actually know of a good analysis that sets what is the really solid limit on the self-interaction cross-section from the baller cluster, but it's probably in the ballpark of a few centimeters squared per gram. Now, that said, of course, the velocities in these systems are very different. The typical velocity of the dark matter in these colliding systems is about 1,000 kilometers per second. The typical velocity in these dwarf galaxies is more like 10 or 20 kilometers per second. So depending on your particle physics model, your self-interaction rate may be velocity dependent and that can give you a situation where you expect to see signals in one of these systems but not in the other. So it's 1230 and I imagine that people are hungry. So that's where I want to leave you for today. The one piece of evidence for dark matter that I haven't told you yet is actually in some ways the most important piece of evidence, but so we'll get to that the start of next lecture where I will also talk more about cosmology and the behavior of dark matter in the early universe and that last piece of evidence is looking at the cosmic microwave background. So I hope from what we've talked about today, you understand from galaxy dynamics what we think we can say about the behavior of dark matter, what limits we can set on how collisional it is and yeah, next time, cosmology. Thank you very much.