 All right, good morning. So the focus for today is gonna be on physics beyond the standard model. And in particular, I'm gonna take the angle of focusing on dark matter because arguably that's the most important piece of evidence we have that the standard model isn't complete. So what we're gonna do over the series of today's three lectures is to start off by kind of reviewing the evidence that we have for dark matter, then we'll move into discussing what types of guesses we have for the masses of dark matter and what types of particles it can be. And then in the third part, we're going to discuss how we actually search for dark matter and kind of what the current status is of these experiments. The lectures today are going to be hopefully somewhat interactive, so make sure that you're kind of sitting next to, I think everybody's kind of clumped together, but if you're finding yourself kind of far off on the side, you might wanna move closer towards the center, or not, depending on your preference. It's also gonna be a blackboard talk, so you might wanna make sure you have a good view of the blackboard. Okay, so first thing is how we know that dark matter exists. Most of the evidence that we have is coming from astrophysical observation, so one theme that you'll kind of end up seeing over the course of today's lectures is we're gonna veer back and forth between particle physics and astrophysics, because when it comes to dark matter and thinking about what it can be and how we can observe it, the two become really fundamentally linked together. So when we look out at the Milky Way at our own galaxy, we need to ask ourselves, what are the, where can we look to find evidence for dark matter? So to start off, let's just kind of make a map of what there is and what we know exists. So obviously none of this is gonna be drawn to scale, but all very schematic. So we have the galactic disk, which for scale is on the order of 10 kiloparsecs in radius. Can you see this or should I be drawing bigger? It's fine, okay, good. And height is roughly half a kiloparsec. So most of the disk is comprised of stars. And in particular, there is about 10 to the 11 stars in the disk with the total mass of roughly five times 10 to the 10 solar masses. So the unit M with a circle and a dot in it, I'm gonna denote one, well, one solar mass. So this is essentially five times 10 to the 10 suns. So this is about the number of stars we observe in the disk. At the center, there's a black hole. And that's roughly 10 to the six solar masses. And what else do we have? We have gas. The gas is spread out along the disk, consists primarily of molecular and gaseous hydrogen. And it's about, in terms of total mass, 10% of the mass that's in stars. This is everything that's observable in our galaxy. And if we look at all of these components and ask where, what can we use to search for dark matter? It turns out that stars provide the best clue. And the reason that stars provide the best clue is that stars are essentially, to first approximation, non-collisional, which means that their interactions are pretty much governed solely by gravity. And that's about what we, I mean, that's essentially what we expect dark matter to be as well with its interactions governed solely by gravity. So to kind of kick things off and to show this explicitly, I want you guys to do just a brief couple minute exercise to estimate essentially what the time is in years between collisions of stars in the galactic disk. So for this exercise, you can assume that we have everything I've written down here, but also that just, let's see, that every star has a radius that's equivalent to the sun's radius. And that every star has a random velocity of 50 kilometers per second. So take a couple minutes, feel free to chat amongst yourselves or not, as you wish. And using what I've written up on the board, just estimate the amount of time between collisions of stars in the Milky Way's disk. So maybe take another minute or so. What we're gonna want in order to be able to do this is essentially we just need the mean free path for the stars, which first approximation is just one over the number density of the stars times the cross section for their interaction. And then the collision time or the time between collisions is then roughly just going to be the mean free path divided by the velocity of the stars. So I'll just write this out, so we're very clear about this, but the number density is gonna be the 10 to the 11 stars in the disk times the volume of the disk. And this comes out to be roughly 0.6 inverse cubic parsecs. The cross section is roughly pi times the radius of the stars squared. Which is approximately 10 to the minus 15 parsec squared. And then the time between collisions is going to be lambda over the which if you substitute in the numbers, you get 10 to the 21 years. So the time between collisions of stars in the galactic disk is way longer than even just the age of the universe, so from this simple exercise, we can see that stars in the disk are pretty much governed solely by their gravitational interactions, not by the collisions between them. And as a result, they end up serving as very good tracers for dark matter, because when we talk about dark matter, we're talking about matter that does not collide very much with itself, that its dynamics are governed primarily by gravity. So this means that we can look to the motion of stars in galaxies and from that try to infer what's going on with the dark matter that's there. And from exercises such as this, we've gotten the strongest pieces evidence for dark matter. So in particular, rotation curves of stars. So this is what I wanna discuss next. The evidence for rotation curves, sort of the best measurements for rotation curves came in the 1970s and then have continued to improve since, but it was really the 1970s that the measurements became robust enough that people were able to trust the inferences that you can make from them. And so what we're doing with rotation curves is looking at the motion of stars about the center of a galaxy. So if I put a star here, I'm looking at its rotational velocity around the center of the galaxy. And then I do this as I move further and further away so I can determine what the circular velocity is as a function of radius from the center of the galaxy. Now, once I get out beyond the disk, the radius of the disk, then we know just from Gauss's law that the circular velocity should be dominated by all the mass that's inside, which if we're just looking at the visible matter should just be the total mass that we can observe stars in the disk. And so from Newton, we just know that the circular velocity is the gravitational constant times the mass that's enclosed divided by the radius. So once I'm outside, once I'm sort of sampling the motion of stars beyond the edge of the disk, then I would expect something like where I'm just being dominated by the mass that's enclosed, which is just the mass of the disk if that was all the mass that existed. So this means that if I make a plot of the circular velocity as a function of radius, if Newtonian, if the mass in the disk with what was dominating, then Newtonian gravity tells me that the circular velocity should be falling off as one over our square root of r. But that's not actually what's observed. What's observed is flattening of this velocity. So the first measurements were done with M31 were the Andromeda galaxy. It's our nearest neighbor. And the measurements by Rubin and Ford, which came in the 1970s and then were followed up by measurements by Whitehurst were essentially sampling this rotation curve in M31 out in this region here where they could see with pretty good accuracy that the data showed that the velocity were flattening out and were not actually falling as one over square root of r. So this flattening of the rotation curve tells us a few things. We can interpret it one way is that we have some additional mass that's there so that even as we move beyond the visible matter outside of the galactic disk, where we're actually, we're being dominated by invisible matter that's there that we can't observe that's actually just governing the dynamics of the stars out at these really large radii. So one interpretation is that there's this new matter that's there, it's invisible to us so we'll call it dark matter. Other interpretation is that maybe we shouldn't be using Newtonian gravity. And maybe this is what needs to be modified. Now to date, so these types of models are called modified Newtonian gravity or Mond. To date, these types of models haven't really been mostly phenomenological and haven't been sort of absorbed into a full cosmological picture. And so the cognitive dominant hypothesis is that what's there is a new matter particle and that this matter particle has to be neutral otherwise we would have observed it because it would have emitted photons. And the only possible neutral particle in the standard model is neutrinos. But neutrinos actually don't give us enough of the density. So from that we infer that there needs to be some new neutral stable particle beyond what's predicted in the standard model. Oh good, yeah. So I should note that there's lecture notes that are all online. So everything that I drop on the board that's super schematic you can find detailed version of in these notes. But yeah, so let's see. This is roughly 220 kilometers a second. This is for the measurements in M31 and the data is going out beyond roughly 10 kiloparsec. But they have the data only, well in these first measurements the data was going out to roughly 30 kiloparsec. So we observe the flattening in a particular range of radius. But not beyond, which is important because actually we expect that something different should be happening beyond that point. All right, so we're gonna take the first interpretation and run with it in the course of these lectures. And what I wanna do now for the first part is see just how much we can actually infer about dark matter from this simple picture of the rotation curve. And it turns out we could actually do quite a bit at the level of like till the level approximation. So that's what we're gonna do now. We're gonna get a sense of just how much dark matter there is and how it's distributed relative to the galactic disc. Okay, so the fact that we observe that the rotation curve flattens means that the mass that's enclosed in that energy range where the velocity is flatt, this has to scale as r, right? So that I get a cancellation between the r and the numerator and the radius in the denominator. So this cancellation would then be able to explain the flattening. If I take that the mass scales as radius, then I can write down how the mass density should scale. So the mass density is just going to be the mass of the dark matter divided by volume. The mass, I'm gonna say scales as r and the volume is going to scale as r cubed. There's an important assumption that's being made right in that step there. So I wanna just make sure to delineate it, which is that colored chalk, there we go, which is right here. What I've assumed there is that the dark matter is distributed in a big sphere and not in a plane. That is making the assumption that the dark matter is surrounding the disc and hasn't actually collapsed to form a disc. The reason why I can make that assumption is the following. So I've made, the fact that we can't observe dark matter means that it's not interacting very strongly with the matter that we do see. As a result, we don't think it should dissipate energy very much. That's in contrast to the baryons that end up forming the disc. Baryons interact strongly with each other so they can dissipate energy and in that process they can collapse to form a disc. But because the dark matter is non-dissipative, it's not going to collapse to form a disc and so it's just gonna sort of live in this big, what's called a halo around our galaxy. And the reason we can say it's in this halo is because it's not dissipating energy in the same way that the baryons are. So if I make that assumption here, then that tells me that the mass density scales as one over r squared for the dark matter. So just how big is this halo then? Relative to our disc. So let's get a sense of the size scales here. The total mass of the halo, and again this is super schematic, but it'll give us a sense of order of magnitude, is going to be the integral of the mass density over the volume and where I'm integrating from zero to roughly the edge of this dark matter halo. We know some of these numbers from experiments so we know roughly from looking at measuring the dynamical motions of stars. Again, all of this information is coming from stars. That the mass of the halo is approximately 10 to the 12 solar masses. This is the mass of r halo, so the Milky Way's halo. We also know that the local density of dark matter, so by local I mean the dark matter that's very close to where the earth is. The local density of dark matter is roughly 0.3 dB per centimeter, cute. So given these two inputs, we can solve this and get an estimate for what the radius of the halo is. And if we do that, we find that the dark matter halo in our galaxy has a radius that's roughly 100 kiloparsec. So compare that for a sense of scale to the disk. So what I've drawn here is the halo is way not in proportion, right? This is 10 kiloparsec. From our very simple estimate we see that the dark matter halo extends way out much further than that, right? Almost 10 times the radius of what's in the disk. So what we're seeing is really only a really tiny fraction, tiny, tiny fraction of what's there. Oh, I forgot to draw on here where we are. We have about eight kiloparsecs from the center. So we're close to the outer edge of the disk. Yeah, that's neat. So the way we know that is by measuring the motions of the stars near us. So if this is the sun here, there's measurements that are done looking at the relative velocities of stars moving in and out of the galactic disk near us. And based on that, we can infer how much mass density there is near us. So that's how we get estimates on what the local dark matter density is. I should mention that there is a big uncertainty on this number. It's roughly a factor of two uncertainty on this. As the measurements continue to improve, that number is going to get better. But yeah, it can go up as high as, so there's some estimates that pin it as high as 0.6. This gives us an estimate of the halo mass. Let's get an estimate for how fast the dark matter is moving in the halo. So to estimate the average velocity, we're going to use just the varial theorem. So that tells us that the average velocity is G times the mass of the halo divided by the radius of the halo. And we've got these numbers now. So mass of the halo, roughly 10 to the 12 solar masses. And then approximate order of magnitude of 100 kiloparsec radius. If you plug in these numbers, you find that the average velocity of dark matter in the Milky Way is 200 kilometers a second. These particles are moving really slowly in the Milky Way. We're talking non-relativistic speeds. And the fact that the velocities are non-relativistic is actually going to play a really important part later on when we come back to how we can actually look for dark matter. It sort of changes the phenomenology. And in some cases, make certain calculations manageable versus not being manageable if the dark matter was moving at relativistic speeds, as opposed to non-relativistic speeds. Say that one more time. Oh, why this? Oh, good. OK, yeah. So I should be clear about what's coming from data and what's coming from theorist approximation. So this is a number that's inferred from data. And this is also a number that's inferred from data, both coming from measurements of stellar kinematics. So again, this is why we started off with that approximation about the collisional time between stars, because so much is done looking at the motions of stars in galaxies and based on the motions of stars inferring things like what the total mass is of the dark matter and also what the local dark matter density is. So the star-serviced tracers and based on measurements of their motion, we can make these kinds of inferences here. It's everything. So most of the stars are in the disk, but we also do end up with stars far away. They're just they don't, there's not as many as there are in the disk, but you do have stars that are also in the halo. So measurements of their velocities, but then in addition, you're right. Also if we look at orbits of satellite galaxies, then that also affects the determination of that number. So a satellite galaxy is galaxies that are in orbit around our Milky Way. So our Milky Way is some large 10 to the 12 solar mass halo, but we can also have small little galaxies that are orbiting around us where these small galaxies can be 10 to the 10, 10 to the 9 solar masses. So that's what I mean when I say satellite galaxies. A very naive way of doing the estimate. And it sort of makes this inference that there's kind of a hard cut off in the dark matter distribution, and that's not the case. So it's a much more subtle point to actually to discuss how you actually define what the radius of a halo is, because technically it's just, it's all the dark matter particles that are bound to the, let's say, the center here of the galaxy. So, and that can extend out right far out. And so there's different measures that people use for what the actual halo mass is. So sometimes it's the virial mass, so it's all the mass within the virial radius. There's different, yeah, so there's different ways of defining the mass and the radius, and that was something I kind of wanted to avoid, because it sort of depends on the application. So this is like the simplest possible thing, but yeah, you're right. You wouldn't want to think of it as sort of a hard cut off. That's, because it's just sort of the density falls off as a function of radius, and there's no reason for it to be cut off. Oh, that the density falls off. So the only thing that we can say is that the density is falling off roughly as one over r squared in this region. We don't know what's going on outside of what's been measured with rotation curves, but we know that it can't be flat forever, because if you were to integrate this to get the total mass, it would blow up if it were to extend out like this forever. So yeah, so this really only applies in this region where we've actually been able to do the measurements. How far do you have to go for what? So for those kinds of measurements, we'd need to go, we'd look at large scale. Yeah, yeah, so definitely not on these scales. So this is very local in comparison. Good, so let's see. What to cover next? So I just want to make a few points before moving on to estimates of the actual masses of the dark matter particles that could be constituting this dark matter. So like I've said, these estimates that I've done up here are just very tilde level approximations and just based off of the rotation curve measurements. If you want to do something that's more realistic, you need to turn to numerical simulation. And the reason is that you need to be able to account for the gravitational interactions between all of the different galaxies and all of the dark matter halos between those galaxies. So what the simulations do is they start off with a certain set of initial conditions and then they watch as the dark matter halos kind of grow with time and merge together. So it's an end body gravitational problem and as a result, we can't actually do that. Analytically, I need to rely on simulations to get the answers. When we do these simulations, we can look at what comes out for the dark matter density distribution. And what we find is that our estimates here are actually not too far off, but they're also not the exact answers. So for example, it's from end body simulations that only include dark matter. And this is important for reasons that I will say in a second, but the results from these end body simulations suggest that the dark matter density is given by what's called an NFW distribution. So that stands for Navarro-Frank-White, which is actually a double falling power law. RS is the scale radius, but for our purposes, you can think of it just as some numbers. So simulation suggests that this dark matter density on where this is log, on a log log plot looks like this. So essentially, it has some slope here at small radiuses and then at large radius described by another slope. So an inner and outer slope. If you were to go through the literature, there's a lot of different ways that other different kinds of profiles that people get from these simulations. The NFW one is the one that's most commonly used in a literature, which is why I've written it up on the board. But the important thing to note from here is that our one over R squared estimate for what this mass density is only applies, I guess, for NFW around, well, I don't wanna quote the number right off the top of my head, but only applies at essentially one point where you have a slope that's roughly one over R squared. But yeah, so for NFW, the inner slope is roughly R to the minus one and the outer one is roughly R to the minus three. Lot of debate as to whether or not this is the correct profile. And the reason is that the simulations that have been done, that have obtained these kinds of profiles only include dark matter particles. They do not include any gas physics. And the gas physics is expected to actually change the result because what you end up getting are things like explosions by the black hole or supernova explosions along the desk. And these explosions essentially can redistribute the dark matter. And by this sort of redistribution, what it can potentially do is flatten the density of the dark matter close to the center of the galaxy where you expect these explosions to be most relevant. And so at the moment, there is a lot of effort to improve the simulations to include this gas physics. It's a really hard problem because it's just computationally more intensive to include that. And so those simulations right now are not at the same level of resolution as the dark matter only simulations. But the improvements have been happening steadily and very quickly. So probably in the next few years, this is something that'll be resolved in more detail coming from the simulation and the spectrum. It's a very interesting problem. So as I'm coming from the particle physics end because if the gas physics can explain this core, that's one thing. But if it turns out that observations tell us a core that a core should be there and the simulations are not finding the gas physics can explain it, then that can actually tell you that something interesting is happening with the dark matter particle itself so that it must be self-interacting or something like that. So it can actually potentially end up telling you something about the particle physics properties of the dark matter. Yeah, so this flattening is, the evidence for it is a little bit tenuous, but people have started trying to measure these density distributions in some satellite galaxies that we know are dominated by dark matter. It's called dwarf galaxies. And some groups find that in those satellite galaxies, there is evidence for these cores. And if that's the case, the cores can either be coming from these supernova explosions, AGM feedback, things like that, the baryonic physics, or if it can't be explained by that and we'll know as these simulations improve, it might be coming from the fact that the dark matter interacts with itself. And if it interacts with itself, it's gonna mostly end up affecting what's happening near the center of the galaxies where the density of the dark matter is the largest. Yeah, right now we don't know. It's a big debate that's kind of going on. I think that'll become much clearer, both as the observational evidence improves with the measurements from the satellite galaxies and also as the n-body simulations, the resolutions improve, the simulations that include the baryons. And then, but right now, this is kind of a big question mark. We don't know if there's cores in these. Oh yeah, so the simulations that have been done so far that include baryons seem to indicate that you can easily form cores if you have this kind of these baryonic feedback mechanisms. So the first results coming out from those suggest that you can actually explain these cores almost entirely with these feedback mechanisms. But it might be as that, again, the resolution isn't quite at the level where it is for the dark matter only simulation, so it might be that as the resolutions improve, the story might change, we don't know yet. So yeah, so with the simulation students, so there's several kind of interactions that could potentially be happening here. One is just gravitationally. So essentially, if I have a supernova that explodes here, what it does is it blows out gas at extremely high velocities and redistributes the gas locally in that region. And by redistributing the gas locally in that region, it changes the local gravitational potential, which then ends up affecting the dark matter distribution. So the kinds of processes that are being simulated are those kinds where you have these explosions or the feedback from the black holes that redistribute the gas locally and then from that change the gravitational potential which changes the dark matter density. So that's one way that you can get these cores. A separate way is if you just have the dark matter interacting amongst itself. So if you have that kind of interaction, then you get scattering processes just between the two dark matter particles, and then that can change the slope here. Does that answer your question? Okay. Yep. Oh, here? Yeah, yeah. So when you do the simplest possible thing here, you don't expect that there should be the core. The only place, so the reason, because that's sort of the zero-th order estimate of what can be done. The presence of the core is something that we're starting to see with these simulations which are taking into account much more, I mean, if you wanna think about it as a perturbative expansion in what you understand about the dark matter properties, that's the zero-th order term, and then this is much higher, this is including all the higher-order corrections that are accounted for by the simulation. So, and also from observation, because there's some observations that suggest that this might be the case. So if it is core, then our simple approximation, they're just didn't quite capture all of the physics. And the reason for that is because these cores are most important very close to the center part of the galaxy, so like here, and the rotation curve measurements, if you notice the scale, right? The flattening is on 10 kiloparsec, 30 kiloparsec, so rotation curve measurements are kind of probing what's happening out here, and the core would be in here. So we can't actually really rely on the rotation curve measurement to be telling us what's going on here. What I wanna do next is given what we've done here in terms of just setting up the scales of the problem and kind of mapping out where the dark matter is relative to where the baryonic matter is in the galaxy, what I wanna do now is use this as a way of making our first set of guesses about what the masses of the dark matter particles can be. So what we're gonna do now is gonna essentially be the most generic statement that you can possibly make about dark matter in the sense that it makes essentially zero assumptions about the particle other than its distribution in the galaxy and whether or not it's a fermion or a boson. Okay, so what we're gonna need for these estimates are one, the fact that dark matter forms halos, which I hope I've convinced you of given our previous discussion and the second thing is whether or not the dark matter is a boson or a fermion, that's it. So let's start off with the bosonic case first. If the dark matter is a boson, then imagine, so we've got our dark matter halo and what we wanna do now is essentially pack it as much as we can full of dark matter particles. And when we do that, we're essentially putting in a particle in a given cell of phase space. And if it's a boson, then just the, Fermi boson statistics tells us that we can continue packing as many bosons as we want in any volume of phase space here and there's nothing preventing us from doing that, right? The spin statistics doesn't care. If it's a boson, it doesn't care. I can have as many particles as I want in each volume, in each phase space volume. So because I can do this, essentially that I can treat the dark matter field as being coherent. I've now packed so many of these particles in a single volume of phase space that it essentially acts like a coherent field. And if it's acting as a coherent field, then there's actually, our arguments for the stability of the halo are gonna come down to, well these arguments are actually gonna look really similar to the kinds of arguments you make for the stability of the hydrogen atom. So for example, the, so the stability of the halo is gonna be set by the uncertainty principle. In the same way that the stability of the hydrogen atom is set by the uncertainty principle for the electron that's orbiting around. So if this is my halo here, and I say that it's radius is r halo, then delta x is roughly twice, the radius of the halo. And delta p is roughly the dark matter mass times its velocity. If I substitute this in here, and so I get twice dark matter mass v r halo about one, and I solve for the dark matter mass that, if I make the assumption that v is on the order of 200 kilometers a second, non-relativistic, right? This is, matches what we said before, using the Varyl theorem. And I put in the radii of the smallest dark matter halos that we can observe, which is coming from dwarf galaxies. If I put in these numbers, what I find is that the dark matter mass for, if it's a boson, cannot be any smaller than roughly 10 to the minus 22 EV. Matter that's at the bottom of this scale, so roughly 10 to the minus 22 EV, is sometimes called fuzzy dark matter. So the super ultralight, these are ultralight scalars getting it somewhere in the sense that now we know that, there is a lower bound on the mass of bosonic dark matter. It's very small, but we can't actually make things any smaller than that, because if we did, then the halos would not be stable, and we would not be able to form any, we just wouldn't be able to form any halos. We can repeat this whole set of argument now, but doing the case for fermions, and the result's gonna be a lot more constraining. The reason is because of poly exclusion. So when we have fermions, we can't just pack them continuously into a single cell of phase space. Fermi statistics tells us that fermions are picky, they don't wanna be living very close to their neighbors, and so you can only pack a maximum of one fermion in each unit of a phase space, and that restriction is enough to tighten this bound considerably. So for fermionic dark matter, poly exclusion becomes important, yes. So okay, so our estimate here is coming from, this is the average velocity, right? But you can have higher velocities than this, but you're still, you can't have arbitrarily high velocities because what'll end up happening is after a certain point, your dark matter particle can just escape the halo. There's an escape velocity, and so you can look at, you actually just estimate, you can calculate the escape velocity for a halo of that mass with that radius, and you would find that it's consistent with the escape velocity, like essentially the fastest stars that we can measure in our galaxy, and that's roughly on the order of 550 kilometers per second. So anything faster wouldn't actually be bound, it would just be passing through. And that happens, I mean we have, not everything is bound to our halo, you can have stuff that's passing through, but it's not gonna be the dominant. It won't be the dominant thing that's in the halo. Well, yeah? Yeah, that's what I'm gonna do next after I do this. Like I said, this is the most generic estimate that you can make, in the sense it's the most model independent one. The only thing I'm requiring is that I form a dark matter halo. What we're gonna do next when we finish with this is consider what happens when we make the additional assumption that the dark matter's thermal in the early universe. Then the mass range is gonna be more constrained. But it's coming with an additional assumption. So yeah, so it's always important to remember that what the most generic possibility is. Is there another question? What I'm using here is for the sort of the smallest ones we've seen, so these dwarf galaxies. It's just because that gives you the tightest bound. You could put in the radius of the Milky Way and you'd get a bound, it would be weaker than this. But we know we've seen smaller halos, so we can use that number instead and that would be the halo associated with these dwarf galaxies. Any other questions? Good, so let's finish up with the fermionic example and then I think we'll be ready for the break. And then when we come back, we'll consider a thermal dark matter in more detail. So for the fermionic case, I can take the mass, say that the mass of the halo is roughly the mass of each individual fermionic dark matter particle and multiply that by the velocity of the velocity. The volume of the halo times the phase space density, f of p, d cubed p. For fermions, we know that this can't be, the maximum value that's gonna take is one. So that gives me that constraint here, times d cubed p. This step here, I was making the assumption that fp max is one. And I can estimate this as being four thirds pi r cubed and then d cubed p as being fermion mass velocity altered cubed. Again, if we put in estimates for the mass of the halo and the radius of the halo and then infer from the, oh, and the velocity, which we also know, then we can solve for what the limit is on the mass of the fermionic dark matter particle. And what we find is that this is larger than order nev. And what I wrote up on the board is sort of the simplest possible way that you can do this, but you can refine these phase space arguments. And if you do that, you can make this even tighter. The current limit that is in the literature is 0.7 keV. Fermionic dark matter particle has to have a mass greater than 0.7 keV in order to be able to form the halos that we observed today. Notice that this constraint is much larger than what we got for the bosonic case. And that's coming entirely from the fact of just poly exclusion and how you can pack these particles into phase space to form the halos. So these are the lower limits. What's the upper limit? The upper limit on the dark matter mouse is really high. It's 10 to the 59 eV. And where that's coming from is measurements of what's called machos. So massive compact halo objects. So you can look for these things using gravitational lensing because if they happen to be coming in front of something that's emitting light, you can look for the lens signal. And based on that, we know that you can't have dark matter that's larger than 10 to the 59 eV in mass. So this is what we have to work with. So we now know that we've, well, over the last hour, we've used information gained from the motions of stars. So looking at these rotation curves to make inferences about the distribution of dark matter in the Milky Way. We've gotten estimates for the total mass of the halo, the size scale. We've seen that it's much, extends out much further than the galactic disc. We've seen that the velocities of the dark matter in these halos are non-relativistic. And we've used the fact that these halos can form to then tell us something about what the masses of the individual dark matter particles have to be. And what we get is something that falls in a humblingly large mass range. So if it's bosonic, it could be anywhere from 10 to the minus 22 eV all the way up to 10 to the 59. If it's fermionic anywhere from 0.7 keV up to 10 to the 59 eV. So this large range that we know is possible really means that we need a very diverse set of experiments to try to probe this. We can also then begin to make some additional assumptions about the properties of the dark matter to kind of limit this mass range a little bit further. So when we come back after the break, that's what we're gonna do. We're gonna talk about dark matter that's thermal in the early universe and discuss what that does to the allowed mass range and how we can actually probe for it. So I guess let's break now and then come back. How much for one? 15 minutes, so 1130. 1130, okay, so we're back at 1130.