 Okay. Fasten your seat belt. And it's time to start a new course, the one on dark matter, and we have Tracy's later from MIT. Please. Thanks very much for the invitation. I'm very pleased to be here in beautiful Trieste. I've mostly gotten over my jet lag. I know many of you are still pretty jet lag having only come in yesterday, so I will try not to go too fast. Nonetheless, we've got a lot of material to get through. Okay, so first important question. People at the back, can you hear me okay? Okay, great. I'd also like to encourage you throughout this lecture and my subsequent lectures. If you have questions during the lecture, please raise your hand. I will try to pause at strategic points during the lecture to give you a chance to ask questions as well. It's just easier for me if we address points of confusion or things you want to hear more about when they arise rather than getting to the end of the lecture and then finding out that half the room has been very, very confused about something I said back on slide three. So, okay, so we have four lectures here. What I want to do in this first lecture is give you the historical justification for dark matter and in particular why we think that some new particle might be responsible for dark matter. I want to talk you through what our current observations are of the dark matter distribution in the cosmos and their implications. And then the other thing that I want to do this lecture is talk about what we already know about dark matter and how we might learn more before we start talking about models of dark matter and its interactions with the known particles just purely from looking at its gravitational effects. Then in subsequent lectures I'm going to go on to talk to you about some major classes of models of dark matter such as WIMP dark matter and axioms about their different cosmologies about the range of cosmological effects that you can get and then how we might search for them using both telescopes and terrestrial experiments. But for lecture one we're going to pretend for the moment that the standard model doesn't exist until later in the lecture and just think about dark matter. Alright so let me just begin by giving a bit of a historical review. So the search for dark matter and so for those of you who have seen this already you can nod along or go just take the moment to get some rest after your long trips yesterday. Okay so the search for dark matter goes back almost a century at this point. The first hints came in 1933 when an astronomer called Zwicky was studying the Kona galaxy cluster. He was trying to estimate the mass in this galaxy cluster and like a good scientist he came up with two methods to follow with the hope that these two methods would give a consistent answer and so they'd serve as a cross check on each other. So method one was essentially a counting argument was to count the galaxies in the cluster, add up the total luminosity in those galaxies based on a calibration to a local system where he had a better measurement, take what used the mass to light ratio from that local system and use that to estimate the total amount of mass in the galaxy. Method two was rather than a counting argument to look at the gravitational effects so he used measurements of the galaxy velocities obtained just looking at the Doppler shifts of those velocities to determine how fast they were going. Since it was an equilibrium system the virial theorem says that there's a relationship between the kinetic energy of those galaxies and the potential energy. In other words how fast they're going is determined by the enclosed mass. From this you can get a separate estimate of the total mass in the galaxy. So this is based on bright, so the first estimate is based on bright observable objects, the second estimate is just based on gravitation. And what you find unfortunately is that while you might have hoped these numbers would serve as a cross check from each other actually they're different by a couple of orders of magnitude with the second one being larger. So this is a little bit of a problem. Now and at the time Zwicky proposed that maybe what's going on is that there's a lot of gravitating non-luminous matter in this galaxy cluster. Are the possibilities that you might think of well maybe our theories of gravity are wrong on these scales. But at this point it seemed reasonable well maybe there's a lot of stuff there that we just can't see. Maybe it's burnout old stars, maybe it's gas that for some reason isn't radiating in the wavelengths that we can look at. It's just dark matter that isn't showing up in the in luminosity. So that was an interesting puzzle but the next real forward step in understanding this puzzle came in the 1970s and 1980s where people did a somewhat similar analysis but this time on galactic scales rather than cluster scales. So what Vera Rubin and her colleagues did was look at what are called galactic rotation curves so the velocity of stars and gas clouds as a function of distance from the center of the galaxy. Again you all know Newton's laws assume you know that in general the orbital velocity of object are determined by the mass enclosed within the orbital radius of those objects. So if we just do a little bit of undergrad gravitational mechanics we can write down the circular velocity v squared over r is given by this function where this is the enclosed mass. If we were to say that most of this mass is contained within some region towards the center the galaxy the bulge of a spiral galaxy the core of an elliptical galaxy then we'd expect the velocity to fall off as one over square root r as we move away from that center. In reality the galaxy has a disk so this calculation is a little bit more complicated but this plot on the right shows this circular velocity in kilometers per second as a function of radius where you would expect from the disk of the galaxy. So our galaxy is a spiral we can map out its disk in bright stars in hot gas and we'd expect the velocity distribution to fall off at large radii. Instead what the measurements look like are these data points and this is that's that's for one specific galaxy but this behavior is not isolated to one galaxy this plot on the left shows a large number of nuclei this is from paper by Ruben et al in 1980 and you see that this behavior this flat rotation to curve at large velocities is quite ubiquitous. So it turns out that to get that constant velocity we need the enclosed mass to be increasing with needs to be increasing with r which suggests that there's some additional component which extends out to much larger radii than the disk. So this curve here marked halo is is a as an example such thing where you have a large spherical halo of mass which you're not seeing in your ordinary observations and which is responsible for these flattened rotation curves. Now again this could be this could also mean well Newtonian gravity is not a good approximation on these scales. We know that Newtonian gravity is not the right answer that we need GR. Turns out when you include general relativity into this it doesn't really make much difference because the gravitational force at these scales is quite weak. So it's not just GR but maybe there's some additional modification to GR on galactic scales. So then you have this question or air is what we're looking at additional matter the gravitates normally or the same matter that we always had the different gravitational potential. So I think this was resolved at least from the perspective of much of the field in 2006 with the study of a system called the bullet cluster. So the difficulty in distinguishing these two explanations in galaxies like the Milky Way or in equilibrium clusters is that the ordinary matter and the putative dark matter are centered at the same location. So it's hard to tell the difference between gravity from the ordinary matter falls off more weakly and there's additional extended dark matter. But to get around this we can look at non-equilibrium systems where we might expect if there are two components of matter we might expect them to be actually specially separated from each other. So this study in 2006 paired data from the Chandra X-ray telescope with weak gravitational lensing. Now I'm not going to go into much depth about gravitational lensing techniques here but essentially the presence of mass bends light. Those of you who have done a TR course know all about this. The presence of mass acts like a lens thus by looking at those lensed images we can infer the distribution of mass. So this top plot here it shows the distribution of the stars, invisible light and then these green contours show the inferred mass distribution for these two colliding clusters which form the system called the bullet cluster. This lower panel these colored regions show where the gas density is highest and where the gas is hottest. And you see that there's a pretty substantial offset between where all the gas is and where the mass is peaked. Now it's very hard to understand this in the context of a simple modification of gravity. I mean it's one thing to say gravity falls off more rapidly or more slowly as you move away from a large mass but it's something else entirely to say oh look there's this peak in the gravitational potential and there's very little mass there. So in general no matter how we modify gravity we would still expect mass to be the source. So the dark matter explanation here is that there are indeed two components of matter in these colliding halos. There's the component we know about formed of baryonic matter as Barbara as some was explained earlier and that matter well we know how gas works if you have two clusters of gas and they slam into each other they will they will experience collision pressure they will heat up they will emit a lot of x-rays. That's what's happening in these colored blobs. The fact that most of the mass in the galaxy however is separated like this suggests that the dark matter halos have just passed straight through each other and kept going. They haven't rammed into each other their shapes haven't been distorted they haven't heated up they've just flown straight through. So this suggests not just that there's a second component of matter but that it has very different properties and that it is at least to a first approximation almost collision-less. Okay so alright so then if you believe from this alright we really are looking at some new matter component we're not looking at modified gravity then you can ask another question you can say alright are we looking at some kind of new particle or are we just looking at ordinary matter that in a form that doesn't radiate much light and that is nearly collision-less. I mean stars are collision-less as an example they're not dark but if you have a large population of that if you have a bunch of stars and you run it through another bunch of stars because their filling factor is so low most of the time those stars will never collide with each other and will look like those two distributions just go straight through each other. Okay so stars are collision-less maybe burned out stars can be dark and collision-less. So such objects will be called massive compact halo objects or machos. I think this acronym is partly so the physicists could make jokes about wimps versus machos and I think most physicists are also rather satisfied that at this point the wimps appear to be winning the contest. Call back to our high school days. But alright so why do I say wimps are winning the contest? So there have been so if these machos do make up the whole halo of our galaxy again we should be able to use gravitational lensing to look for them because when one of these tiny objects passes between us and a distant star then it should lens the light from that star should briefly magnify it. We should see an amplification of that light coming from the presence of that gravitational lens. So Chisorand et al did a large study of this back in 2006 and they found that with the sensitivity of their search they would have expected to see about 40 such events if dark matter in the halo were entirely sub-solar mass objects. Yep see a question over there? Yeah so right so I said collision-less approximately collision-less so that means if I have a big population of them you know like a big population of gas particles if I have these two galaxy cluster-sized halos and I move them through each other they need to go through each other with very few collisions okay now we know that this doesn't happen for ordinary gas but it does happen for stars for example and the reason that it happens for stars is not because two stars can't collide I mean of course if you put two stars sufficiently close to each other they will collide and all sorts of interesting things will happen but the spacing between stars is enormous I mean their mass density may be comparable to the gas but it's concentrated in many high-mass objects that are separated by a huge distance so if we run them through each other they'll just pass through cleanly without collisions so they will look collision-less even though if we brought them close enough together they absolutely would collide similarly with black holes if we're talking about black holes as being dark matter or mattress we're talking about a large number of heavy objects separated by large distances right so if we run a collection of a thousand black holes in the collection of another thousand black holes through each other but I mean there are probably hundreds or thousands of light years between them and black holes in their gravitational pull is not hundreds or thousands of light years so unless they're event horizon so unless they're so dense that their regions of gravitational influence start significantly overlapping then they'll just slide past each other so in that sense black holes are a collision-less component that makes sense any other questions sorry just ask you yes that's yeah so right so you good okay so the question was in the bullet cluster are the stars centered around the maxima of the masses I mean I said the gas was offset from the maxima of the masses but where are the stars yeah so the stars behave like a collision-less component and as I'll talk about later you can actually treat the stars as a tracer of any collision-less component and the gas is a tracer of the collisional component so if you see something that's offset from both the stars and the gas then maybe that's a sign that the dark matter is not totally collision-less yeah so so the obvious follow-up question is well how are you sure that the dark matter isn't just all the stars then we can estimate the amount of mass in the stars and we can estimate the amount of mass in the gas and the amount of mass in the stars is much smaller so so this search this match was searched they claim to rule out these maturers of masses between about 10 to the minus 7 solar masses and 15 solar masses as the primary constituents of the halo these objects are still be there but in this mass range at least they can't make up all the halo now there are other ways to search for heavier for heavier such objects if you have really heavy much heavier than solar mass objects passing through the halo then they can disrupt loosely bound wide binary systems when they pass through so the existence of loosely bound binary systems can provide a counter argument to very heavy to very much heavy objects comprising 100% of the halo so this study from 2014 claims to rule out well the headline limit is they claim to rule out all maturers above about five solar masses comprising 100% of the halo over the over the last year or so there's been a lot of interest in maybe 30 solar mass black holes making up all the dark matter and so some people will say well you know this five number is pretty optimistic and the more conservative number would be to say that only things heavier than about 100 solar masses are ruled out in which case there's a little window between about 15 and 100 solar masses in which mattress could potentially make up the whole halo but there's a lot of debate about that constructs so in general if you want the halo to be made up of dark matter except possibly for this narrow range around a few tens of solar masses you need them to be pretty you need them to be pretty light objects you need them to be um there's actually a somewhat better constraint than this now but basically you need them to be the mass of the moon or smaller so if the galaxy is filled with death stars maybe they can be the dark matter but um but it's hard to do this with burnt out with burnt out stars or anything that's close to the sun and mass alright so that gives us some evidence against mattress but there's actually another piece of evidence which is important so you heard in the previous talk a bit about the cosmic microwave background and I'm not going to go into depth on it here either because you have all you have several lectures on it coming up later in the summer school but essentially just to recap the key points when the universe was about 400 thousand years old and about a thousand times hotter than it is now the hydrogen gas went from being almost entirely ionized to almost entirely neutral and the universe became transparent to microwave photons the CMB this cosmic microwave background radiation was last scattered at that time and so so just to be clear the CMB light most of it when it hits our telescopes that's the last time that that's the first time that it hit anything since the universe was three or four hundred thousand years old so it gives us a snapshot of what the universe looked like at those early times and what it tells us is that at that time the universe was indeed nearly homogeneous it was a hot soup of light and atoms but it wasn't perfectly homogeneous there were oscillations in the temperature and density from competing radiation pressure and gravity so overdensity is tended to fall in on themselves due to gravity but if those over densities contain baryons ordinary matter this electromagnetically interacting then radiation pressure tended to push the baryons away from each other again the photon temperature anisotropies which is in this show it in this famous plot from plank so this is showing the power in the oscillations at different scales that describes those temperature and density and in homogeneities a recombination now by studying these peaks we can understand the properties of these oscillations and in particular the degree to which these oscillations are driven and damped depends on how much dark matter you have relative to ordinary baryonic matter we're here by dark matter we mean matter that doesn't experience radiation pressure so the dark matter component only experiences the gravity which causes it to infall the baryonic component experiences gravity which causes it to infall and radiation pressure which pushes it back out the ratio of these two components determines how important gravity is relative to radiation pressure and that in turn affects the anisotropies so this is just an example of what so that's the physics this is an example of what changing the amount of dark matter would do to the CMB spectrum so this pink bar on the right is going to show us how much dark matter we have and this yellow curve is showing us the resulting CMB spectrum so you see is the amount of dark matter gets larger the peaks get suppressed in particular the ratio of the second and third peak changes quite a bit so that's a diagnostic for how much dark matter there is in the universe and what we find is that in order to fit the data I think we'll talk a bit more about the concordance model later but in order to fit the data we need a dark matter component which is about five times more abundant than the barionic matter now get this doesn't tell us what this dark matter is but it has to be something that doesn't interact electromagnetic okay whatever it is it can't experience radiation pressure so the last sort of important early probe of dark matter and what it tells us is we can look at structure formation so those density and homogeneities in the CMB give us the initial conditions for cosmic structure formation after the photons decouple from the ordinary matter from the barions so then there's not so much radiation pressure and these overdensities continue to grow under gravity eventually they collapse into structures this is a picture from the millennium simulation showing this cosmic web of structure now this cosmic structure formation is largely determined by the dark matter because it's most of the mass we just argued there has to be five times as much dark matter as visible matter so one question that we might ask one fairly simple question is all right should this dark matter be relativistic during this process or non-relativistic two regimes is it moving fast or not so if most of the dark matter is hot then so it's moving very quickly it's moving close to the speed of light at free streams through the universe this erases these small-scale structures you don't get small clumps small over density clumps to begin with because the dark matter particles are just moving so fast all those density and homogeneities get get suppressed what happens first instead is that you get very large scale structures scales such that the dark matter can't cross them and arise in time those large structure you do eventually get small structure small clumps of dark matter but the way that they're formed is the large structures are formed and then they break apart so this is a top-down scenario with structure formation now on the other hand if most of the dark matter is very very non-relativistic then what happens then we'll talk a little bit more about this later is the small clumps of dark matter form first and then they collect together to form larger structures so in this hot dark matter scenario you think the galaxy clusters should be much older than galaxies you should form clusters first and then they should break apart release the halos in which they live should follow that pattern in the cold dark matter scenario the smaller structures in the universe also the oldest and when we compare this to observation this tells us that the bulk of dark matter has to be cold if dark matter was hot we would not see galaxies in the way at the time at the red shifts that we do and with the properties that we do equivalently another way to say this hot dark matter predicts that we should see many fewer small halos than we do so most of the dark matter at least has to be cold now neutrinos hot dark matter does exist neutrinos are hot dark matter they at the time that they you know that they don't experience radiation pressure at the time of the CMB they're not luminous they're dark but they would count as hot dark matters during this process they can't be all the DM that gives you the wrong structure formation history alright so that's the historical review of how we know what we know about the basic properties of dark matter and this is already enough to tell us that there is nothing in the standard model that satisfies all these criteria we need a particle that's stable on cosmological time scales it doesn't experience radiation pressure and it's almost collisionless that means it had better be electrically neutral it can't be highly relativistic when structure formation begins it needs to be cold or maybe warm it's okay if it's just mildly non relativistic and in the standard model the only stable neutral particles in neutrinos and they would be hot dark matter so this means so the optimistic way of saying this is the dark matter gives us one of our most powerful pieces of evidence for physics beyond the standard model is one of the best clues we have as to what the fundamental physics of the universe might entail beyond our familiar standard model the downside of course is we still don't know what it is but everything that we've learned so far about dark matter comes from studying the gravitational effects of dark matter and the distribution of dark matter okay so I'm gonna pause and ask for questions again here but the so the remainder of today's lecture is going to be okay well let's continue in this vein before we start thinking about models of dark matter as a particle what more can we learn just from looking at observations of dark matter we know it's out there we can see it we can measure its distribution with gravitational lensing what can this tell us and then in subsequent lectures I'll go on to talk about other ways that we might search for dark matter and try to nail down its particle properties but any questions at this point okay so the question was what's the energy scale at which you differentiate hot and cold dark matter so we'll talk about this a little more in upcoming slides but and I'll show you some more exceptions to this in the next lecture later today but in general if you're talking about thermal dark matter temperature was the same as the CMB at some point and then a decoupled then if you're at the few kv scale you're in the warm regime the few kv is warm heavier than that is cold much heavier than that is cold above about 10 kv it's very hard to distinguish warm dark matter from cold dark matter with our current observations much lighter than that is hot so neutrinos are at the sub EV scale so they're very very definitely hot generally a few kv is the transition yep question over there sorry can you just say again yeah so the question is could we have some combination of cold dark matter and more more hot dark matter that isn't neutrinos yeah so there are I'll show you a little bit later some constraints on the fraction of hot dark matter but generally so long as the hot dark matter is like a percent or a couple of percent of the total abundance it's fine to have hot dark matter warm dark matter is even less constrained if warm dark matter I think warm dark matter I think I have the exact number later but I think it could be about 30 percent of the total dark matter without being ruled out you just need you just need most of the dark matter to be cold to generate structure formation it's it's hard to constrain a subdominant component yeah so the question is CDM has some inconsistencies with observation of the galaxies here for which I'll say wait for the second half of this lecture because we're going to talk about that in depth okay more questions okay great then then let me then let me proceed onwards okay so what are our gravitational probes of dark matter from which we have already learned a fair bit and can maybe say more so one number is well I told you there's about five times as much dark matter as ordinary matter we can be a lot more precise than this the fact the experiments that measure the cosmic microwave background anisotropies have told us that this is the so omega c here is the fraction of the critical density that is in dark matter h here describes the Hubble parameter so it's just h at the present day rescaled to 100 kilometers per second per megaparsec so you get this number there's nothing special about 100 kilometers per second per megaparsec is just so that we don't have to cut around units all the time for this age so so this is one thing any dark matter model has to explain why dark matter has the abundance that it has however that's mostly governed by the particle physics of dark matter so I'm going to talk about in the next lecture we have the power spectrum of matter fluctuations how much how matter clusters on different scales we can measure this at some level from the cmb we can also measure it in the present day by looking at galaxies and looking at clusters we can look at the detailed distribution of dark matter day provided that we're looking at objects close enough that by stellar motions or gravitational lensing we can prove the dark matter content directly something like the bullet cluster tells us a lot about how the dark matter is distributed we can look at how stars are orbiting around dwarf galaxies of the Milky Way and use that to probe their internal structure so that can give us information as well we are fortunate enough to live and we don't just live in an isolated solar system our cosmic neighborhood provides us with many examples of dark matter structures at a range of mass scales so this can actually tell us quite a lot about the micro physics of dark matter before we ever start talking about looking at interactions between dark matter and the known particles okay so I want to say a little bit about about cold dark matter structure formation here I know you have talks later in this workshop on numerical methods in cosmology and on large-scale structure so I'm not going to go into a ton of depth here but I want to give you a sense of how it works so there's a formalism called the preschechter formalism which is used for simple estimates of dark matter halo formation now it's it's worth noting that the preschechter formalism is on pretty shaky theoretical ground but it but it seems to work out pretty well pretty well as an empirical statement so if the theory seems shaky to you that's okay okay so what preschechter did in their original paper on this topic was okay let's model dark matter halos as spherically symmetric isolated systems now it turns out that the solution for that system is very much like the Robertson Walker or Friedman Robertson Walker or Friedman LaMaitre Robertson Walker metric that you saw in the last lecture from that estimate you can work out that what should happen is that these over densities grow and grow as they accrete more mass and then at some point they start to collapse onto themselves they shrink down now in a perfectly spherical collapse or in a universe that had a very high density of matter you would expect this collapse to eventually crunch back to a point I've heard about the big crunch in cosmology but of course now that we're talking about dark matter particles not an idealized cosmos the real collapse isn't perfectly spherical you don't collapse down to a point what you end up with are these very realized equilibrium halos the important thing that you get out of this formalism is that we can estimate that this collapse should begin when the what I'm going to call the over density delta so this is the average density inside the halo minus the average density everywhere else normalize the average density everywhere else this is the relative over density is about 1.7 so that comes out of this simple spherical collapse model so one thing that you can do to estimate how many halos there are is essentially to say well okay let's assume that we have some Gaussian random field of perturbations with some variance and this is sourced by the crossing microwave background anisotropies you can read off the pattern of anisotropies from the CMB will be interested only in fluctuations above a certain above a certain scale say at some mass scale M we're going to smooth these fluctuations by a top hat function which basically corresponding to a radius such that the mass includes that radio within that radius is the mass scale we're interested in so then now if we want to work out how much mass is there in halos greater than that mass scale we're basically just going to say okay how many fluctuations are there above this collapse threshold above this delta of 1.7 so we're just going to say that the fraction this is a little bit of math but we're just going to say the fluctuations above some collapse threshold delta C give you collapsed regions so the fraction of the mass in collapsed regions with a mass scale greater than M this is just a normalization factor this is just the integral from this threshold upwards of a distribution of a Gaussian distribution of fluctuations okay so it just us just saying okay we know the distribution of fluctuations we just want to pick out all the ones what it's above the collapse density over a mass scale larger than M that gives you this expression simple expression now this is where it gets really dodgy if you take the limit you say okay well let's suppose the variance becomes really really large so we have lots you know we have lots of large fluctuations we're looking at very small scales eventually we'd expect everything to be contained in a collapse structure of some kind some definition but by the formalism that we've used here we've argued that only over densities participate in collapse to get any collapse you need to have higher density than the regions around you what press check that I'll said was basically okay we want this number to asymptote to one eventually so we're just going to multiply this now and it asymptotes to a half because only the over dense regions collapse so we're just going to multiply it by two there is a way to justify this better I want you to take the press check to formalism basically as a as a rough motivation for scaling not as a detailed theoretical prediction so if we take this estimate and we then ask alright this tells us the total amount of mass in helos greater than a certain scale if we can differentiate it with respect to M to get the total amount of power in helos at a given scale and that gives us this expression so this is what's called the mass function this case the press check to mass function this is saying how many dark matter helos do we have as a function of mass so just two features that I want you to notice here one is that when the is that there's a cutoff here once the variance in the mass scale becomes small relative to this critical threshold for collapse then there's an exponential cutoff in the number of halos above that mass that makes sense if fluctuations above that scale are really really rare then you're not going to make many halos that heavy and the other thing is that the number of small halos dnd log M so if in the in the limit where it has this one over M scaling so at low so this tell that so in this formalism you predict many many small halos for each large halo and that's just a function of the fact that there are a lot if you have a Gaussian distribution of fluctuations there are more small scale fluctuations than big ones now as I said you shouldn't treat this as a detailed prediction there are other mass functions that you can use calibrated to simulations that are inspired by press sector so chef torman has a mass function that you'll see has the same sort of similar overall features that has a one over M scaling has a one over M overall scaling it has this exponential suppression again you is this ratio of the critical density to the variance of the fluctuations that has this exponential suppression at large masses this is another similar and this is another empirical mass function from a 2001 paper but to do this properly what you do is this calculation properly you can look at simulations at structure formation okay so so our naive expectation for this mass function so you have a roughly you have a roughly power law multiplied by an exponential pattern of halo masses you have this overall one over M scaling and then this exponential cutoff at high masses there's also in principle a cutoff at low masses there are a couple of ways to to get this cutoffs this is sort of the the most generic if dark matter does have some non negligible interaction with the standard model then dark matter will stay will keep will talk to the CMB to some degree I said it doesn't experience radiation pressure but early in the universe when the universe is very hot the dark matter will still scatter on the photons and this will keep this will keep its temperature up will keep its temperature comparable to the CMB temperature now if this is the case then dark matter density fluctuations at small scales will be suppressed if you have fluctuations that are smaller in the Hubble horizon when the dark matter is strong is coupled to the photons and is still being heated up by the photons then you can transform momentum between dark matter particles by scattering off a photon which then freestreams and then hits another dark matter particle so when you do this calculation you find that the typical temperatures of this kinetic decoupling in typical dark matter in typical dark matter models this cuts off the power spectrum at mass scales of about three times ten to the minus six solar masses so this prevents halos from forming or suppresses them when they do form at you know earth mass or smaller okay these small scale halos just get wiped out there's some now in the limit that the decoupling temperature is very large and dark matter just doesn't talk to the standard model at all these limits go to zero but over a pretty broad range of dark matter models you get masses for this cut off between about one solar mass in ten to the minus 12 solar masses as I'm about to show you on the next page that is way way way away from the range that we can possibly constrain at this point but it's but it's a cut off so so this is what we generically expect cut off at high masses from there aren't many there just aren't many fluctuations at those high scales at those are large scales cut off at low masses from the coupling between the dark matter and the standard model so we generically expect we call dark matter more small halos than large halos in between all right so once we have this these density fluctuations we can work out the power spectrum of density fluctuations if you haven't seen this kind of math before don't worry too much about it those of you it's basically just describing how much power we have in fluctuations and different scales this now we can probe this as I said by looking at present-day galaxies and also by looking at the cosmic microwave background anisotropies so in terms of the sea so we can write this power spectrum at z equals zero there's a primordial spectrum of fluctuations there's a transfer function between that and the matter power spectrum which just depends on k on the scale of these fluctuations and then there's a redshift dependent term which describes how these fluctuations grow with redshift so this is from a paper by Jose Cadel in 2012 where so this is plotting out the matter power spectrum as a function of k and you can see that we can measure over several orders of magnitude so k here is inverse length scale this is measured in one over an inverse megaparsecs here scribing the radial extent these we can convert this into a mass by looking at the ambient density at this time so these so these measurements at small k that means very that means very large scales we can predict what this matter power spectrum should be from the CMB these were the new results in this paper this is from that act the act CMB experiment this solid line is a model from lambda CDM cold dark matter you see this cutoff so this small k means large length scales means large mass scales this is the high mass cutoff in the spectrum that we're talking about then as we go down to as we go down to one larger k and smaller length scales we can measure galaxies and clusters in the present day these pink dots come from measurements of the limon alpha forest which I'll talk more about in in a moment but in general you see we have a pretty wide range of measurements here that agree pretty well with the expectations from simulations this is another way of plotting the same thing so now we're putting the mass scale on the on the x-axis the relationship between k and mass so the mass and close in a volume scales like our cubed constant density right so scales like one over k cubed so that's why we're covering many more orders of magnitude on this mass scale plot than we were on the k scale plot here we see masses from about so these constraints show up for masses between about 10 of the 12 solar masses which is comparable to the mass of the Milky Way these show the results for galaxy clusters and out here we have eventually the the Hubble scale so I just said this is in pretty good agreement with CDM expectations with cold dark matter expectations so deviations from this expectation from these expectations are constrained so I said earlier that hot dark matter suppresses the growth of these perturbations earlier times and so it damps the matter power spectrum on small scales so here hot dark matter will induce a cutoff a low mass cutoff in this spectrum there's some characteristic mass scale in hot dark matter models at which this at which this quantity is largest so and here are so I'll refer you to these papers for a detailed analysis of these constraints and the detailed limits depend somewhat on on how many species you're talking about so if neutrinos there are three fermions axions which I'll talk about this afternoon just talking about a single scalar but in general we find that so the mass of the neutrinos has to be less than about 0.3 EV and their contribution to the dark matter density has to be less than has to be less than about 2% axions the mass scale again has to be less than about an EV the contribution that we expect to the dark matter for masses that small there's about a percent olala there's a big caveat for axions but this is hot this is for hot dark matter contribution so we can constrain the constituent that is hot dark matter pretty strongly now for limits on warm dark matter and somewhat heavier dark matter they also produce a cutoff in the spectrum in the matter in the matter power spectrum at low masses but it's but it's somewhat less pronounced so the strongest constraints on this that I'm currently aware of say that the mass of the warm dark matter if it's all the dark matter has to be heavier than about three KV that cutoff in the present day would correspond to suppressing all structure between about three times ten to the eight solar masses this is the mass sale of small dwarf galaxies in the Milky Way there's a claim from 2012 that they actually saw a two times ten to the eight solar mass dark satellite with gravitational lensing so if that's true then you also don't want to push this cutoff much lower than this scale okay that's it as the question was asked earlier a subdominant component of warm dark matter is hard to constrain the estimate in this paper was that it was fine you could have any mass of warm dark matter provided less than about 35 percent of the dark matter was warm so this is just a visual depiction of what I've been seeing of how you set these constraints this point on the left is showing the cosmic web and on the left hand side called dark matter simulation and on the right hand side a warm dark matter simulation at three different red shifts like this is in the present day I think this is regif one and this is regif five and if you look particularly if you look at these left and right images I don't know how well you can see it but there's a much less small scale structure in the warm dark matter case so this is our direct observable warm dark matter suppresses small scale structure this plot on the right shows in terms of that matter power spectrum that we were just looking at how would the power spectrum be suppressed as a function of scale this is for 4 kv 2 kv or 1 kv dark matter the green line is a theory predict is a theory model from another paper don't worry about it and whether they're that short or to show you the impact impact of different red shifts but so you see generally few kv dark matter starts to really suppress the power in the matter spectrum at um k of about 10 inverse megaparsecs which is now into the region that we are observationally probing with with studies of the with studies of the mass distribution at higher edges okay so first any questions about this yeah this one all right can you just can you speak okay so this is so on the y-axis here is the mass variance so it's looking at the right so so we're looking at the so yeah so this is the so this is the mass variance so the this is the so this is a smoothing scale appropriate to the mass scale that we are looking at so I mean so what this is saying is like if you have um right I mean this is this is delta m over this is delta m over m so it's a so when some so this is um describing I mean this this is in some ways like this is a measure of the variances met is large you'll have a lot of halos so the mass scale is on the x-axis here right so as we move up it as we move up in mass scale this is um as we move up in mass scale this is how the the mass variance changes yeah so that I mean that plot is mostly just to I mean it's it's expressed so the mass variance is very close is very closely related to the is very closely related to this power spectrum so if it's easier to think of I mean it's easier to just think about in terms of the power spectrum the main point of showing both plots is to show you how the k scale relates to the is how to show you how the k scale relates to the mass scale so the so the black line here is the non-linear matter power spectrum okay the dashed line is the um the dashed line is the linear matter power spectrum these purple point these pink points from the lineman alpha are data inferences for the linear part the matter power spectrum sorry so yeah so so you should you should be comparing the you should be comparing the purple points here to the to the to the linear case not the not not the not the non-linear case that's how this comparison was done yeah so right so this is yeah this is not just this lion does not just come from let's take the print let's take the pressure to model and put it and put it on a plant this is a more this is a more detailed CDM prediction I I need to double check this paper to remember exactly where they got this lion from but um it's it's it's based on a CDM simulation code okay the reason I wrote out the pressure yet so the pressure to formalism if you ever want to do a detailed comparison between data about structure formation and a model of structure formation don't just use the pressure to formalism okay it's um it's good for getting a sense for how things should scale they're saying all right at some scale we expect at some large enough scale we expect the matter power spectrum to cut off because there are just no fluctuations of that size and it's good for getting a sense of all right at at small scales we expect there to be a lot more halos you would have a mid to large scales but it's um I put it I put it in there just to give you a sense of how you might estimate spherical collapse it's not it it's not on good enough theoretical or empirical foundation to do detailed comparisons between that model and the data yeah um so right so this is saying that at small yeah so well so okay so okay I mean this is partly that choice of maximum is partly just the way that this is plotted right I mean if I would have plot this is k squared p of k whatever then it would look it would look different I mean you can see that but but so if I go to so if I go to sufficiently small k we're talking about long length scales right so we're talking about large math so we're talking about large mass scales there's sufficiently large mass scales you should start to get a suppression in the power spectrum just because there are many fluctuations at that scale so this gets so if we want to compare so this is you can see what we can use as calibration this red point it's about at the maximum here so that point here in this way of expressing the part is is over here so this is mass scales of like 10 to the 19 or 10 to the 20 solar masses today this is a regime where we're starting to see a cutoff because they just aren't many hallows that big yes yeah so that so the G squared C so right so what this is saying is just if you're translating from the CMB I can infer what the primordial I can infer what the primordial power spectrum was okay what the primordial spectrum of fluctuations was if I want to translate that into a matter power spectrum at any point I first need to take into account that all right what I'm what I'm measuring is what I'm measuring in the CMB is not exactly the matter power spectrum so I have a transfer function for that but I also need to take into account that there's a red shift dependence that because these matter fluctuations are going to grow over time so this is just so we need to model this transfer function using something like the press check to formalism only in reality we would use some simulation if we want to say okay I'm gonna take my CMB data and then I'm going to compare it to data at a much later time one structure has had a chance to grow that's what I mean that that G of Z here is a transfer function okay so there was a question earlier about where the cold dark matter has problems on small scales so I'll give you the spoiler version now my take on this is that it probably does not have significant pro is that there are probably no insurmountable problems with CDM on small scales I'll tell you how I come to that conclusion and what the status is so one of the oldest suspected problems with CDM is what's called the missing satellite problem and so when you apply press check to formalism or or you or you do something more sophisticated you run an embody simulation that includes only dark matter so called perfectly collisionless dark matter that interacts only by gravity you evolve it using the cosmology including the primordial power spectrum determined by the CMB then you predict many many low mass halos and I say high mass subhalos of the Milky Way still mean pretty small halos so smaller than the Milky Way in the 10 or 10 to the 11 solar masses then you find it you have a lot of them there's that one over M in the press check to formula and the predicted number of the subhalos the Milky Way exceeds the observed number of satellites at the same mass scale by about an order of magnitude so this is the original this is the original missing satellite problem pointed out in 1999 so okay it's not 1999 anymore it's been seven so has anything changed in the next 17 years well one thing that's changed is that people are now doing simulations of dark matter that include baryonic matter so because you know of sixth of the matter in the universe is baryons they interact strongly they can have a significant effect so when we say observable halos in the Milky Way we say we mean halos that have formed stars if they haven't formed stars they would just be dark we're not going to see them unless we get very lucky so in these small halos well maybe there are obstacles that stop them from forming stars maybe they're still there they're just not making stars or maybe they're not still there and the reason that they're not still there is that they formed but then they were disrupted so these effects were so there were attempts made to take these effects into account in this paper for example by Brooks et al in 2012 so this plot is showing as a function of the mass of the subhalo this is mass on the x-axis and this is the circular velocity in the subhalos of one kiloparsec on the y-axis so this is again a measure of the measure of the mass enclosed within that radius this top panel shows how many halos you predict just from a dark matter only simulation but then once you include baryons this plot this lower plot shows how many halos you would actually expect to form stars and be observable and it's not a very large fraction because many of these halos don't actually don't actually form stars they don't have enough gas for significant star formation and others if they're inhabiting a galaxy which has a large baryonic disk as these small dark clumps of dark matter pass through the disk they can be tidally disrupted with the gravity and essentially ripped apart so and these black dot ones are cases where that processes partially happen to the point that we probably wouldn't see the halo even if it is still there so that's one answer to the missing satellite problem maybe we did form these dark matter halos but some of them didn't form stars and others have been destroyed so in any case it seems at least not a clear cut problem anymore there's another variation on the missing satellite problem which says well okay maybe the little tiny halos could be disrupted maybe the little tiny halos wouldn't form stars but there are some large halos in this distribution they're fairly massive they're fairly dense and this is known as the too big to fail problem as of its as of 2012 because the idea being that rather than being too big to be allowed to fail like a bank they're too big to fail at forming stars if these halos are there the argument goes we should see them so this plot shows again the so this is a plot of the radio so this is a plot of the radius of the Milky Way sub halos and their maximum velocity which measures their mass so halos which are out to the right hand side of this plot have a high density they have a large mass for their radius and that's what these so this gray band shows what we believe the dwarfs were idols of the Milky Way to follow these dots show the predictions from a simulation you can see generally the simulated halos are denser and heavier than anything really that we see in the Milky Way there are other this is not just restricted to the Milky Way there are other studies of dwarf galaxies in the local groups but away from the Milky Way and Andromeda galaxies so they're not in a big halo with a large baryonic disk again so here again we're plotting radius of the halo here circular velocity of the halo which is measuring mass on the y-axis so if you're higher up on this plot then you're heavier and more dense and these are these lions correspond to predictions from a host of simulations these data points correspond to the observed satellites of the Milky Way and of these field galaxies and again you see there seems to be this distinct preference for the simulations to overestimate how dense and heavy these satellites should be so this isn't just isolated to the Milky Way now you might ask why me okay could this just be a fluke as you will note by counting the data points in here this is not really a high statistics problem there was a paper last year that looked at the that looked at the probabilities of of this happening by chance not using a simulation but just using a semi-analytic prescription for the sub halos and they found that there was about a 1% probability that this was a fluke just looking at only the large known Milky Way satellites and that the probability dropped further when lower mass satellites were considered so I mean it could be a fluke but it doesn't in that case it doesn't seem super likely again though that these are based on dark matter only simulations and possibly this can change when Barry answer included there's actually another way of looking at this because we're saying okay if the dark matter density is too is too high in simulations versus observation how is that density distributed what is the how does the dark matter behave inside these halos so dark matter and body simulations typically predictor roughly university universal density profile for halos common parameterizations include what's called the Navarro Frank white profile and the I nasta profile which is shown by the red and the blue dash lines on this plot so now this is density as function of radius from the center of the galaxy there are now these other two lines here this green line and this pink line showcases where the where the density profile flattens out to a constant value inside some radius okay for the NFW and I asked her profiles the density rises towards the center of the galaxy so there's a separate but not wholly independent problem which is called the cusp core problem with CDM that there is some observational evidence for flatter profiles that are more like the green or pink lines in this plot where the profile is so cool it has a core at the center these observations go back to 1994 least there's a review by de bloc in 2009 which is fairly good so you we see this in dwarf sorority galaxies the Milky Way we see it in galaxy clusters we see it in low-surface brightness spiral galaxies see in high surface brightness spiral galaxies there is a long-standing debate over whether some of these results at least could be due to systematic problems if you for example many studies assume that the hellos are spherical they're probably not really spherical if you do this wrong then you can then you can bias your results but nonetheless there's a reasonably broad hint of suggestions that the profile is that the profile is shallower than we might have thought towards the center this plot shows another way of looking at the same thing so this is this is a dwarf galaxy sample from the things and little things surveys of the local group so this dashed line so this y-axis here is now the inner density slope so for an nfw like profile you might expect this to be around minus one scales roughly as one over r towards the center of the galaxy and the x-axis here is the mass in stars of these of these small galaxies so this dashed line shows the CDM prediction see the slopes typically around minus one of the minus one this blue lion shows the prediction from a simulation that included barrions which I'll talk about a bit more in a second and these and these are the data points these stars are against simulations only things with error bars are data points here so you do see that there appears to be some significant evidence for a much shallower slope towards the center of these halos then you would have naively expected from CDM likewise there's a 2004 paper that claims evidence for shallow profiles in the cause of several massive galaxy clusters these are their best fit slopes this is the expectation from the simple nfw profile again though you worry a little about systematics so you can sum up these small scale discrepancies by essentially saying that the predictions from cold dark matter only simulation seem to systematically over predict the density of dark matter on small scales you can frame this as all being the same thing essentially is a general mass deficit the dwarf galaxies with small stellar masses they appear less concentrated than predicted they don't their density doesn't rise as steeply towards the center they're less dense overall may have flattened cause and in general there are just fewer massive and dense dark matter subhalos than expected both in satellites and in the field as I've said in several of the previous plots maybe this just means that simulations that don't include the ordinary matter are bad simulations of our universe there's a pretty good review in this 2014 paper if you want to look into this in more depth but essentially the idea is that baryonic matter can be pushed out of the halo by supernovae or other energetic events that outflow of baryonic matter can pull dark matter after it by gravity by gravitational effects doesn't need to be any non gravitational interaction between the dark matter and the baryons for this to happen so this can disrupt those steep cusps of dark matter at the centers of halos if all the dark matter halos as a result of these processes are less massive or rise less steeper or the density rises less steeply towards the center that can in turn reduce the predicted abundance of big subhalos for those galaxies of the Milky Way is significantly lighter than we think it is then it will not form as many heavy subhalos in the first place now difficulty with this kind of explanation well the the caveat to this kind of explanation is that we don't really know if baryons can do this this effect depends pretty strongly on whether the star formation of the history of the galaxy is smooth and continuous over time or whether there were several large bursts of star formation the latter are much more effective at causing the gravitational potential to change and disrupting these cusps of dark matter but baryons could in principle solve all these problems at least at present now you might ask well okay can we will we ever be able to test this will we ever be able to tell the difference but tell whether baryons are really responsible for this or whether it's a problem with the cold dark matter paradigm so this this plot shows an estimate of the core slope of the slope in the center of the halos as a function of the mass of the as a function of the halo mass and essentially so here red is red and orange of minus one like nfw profiles zero a cause and you see essentially that it's just in this intermediate that is mostly in this intermediate mass range that that we predict that we predict cause to form basically because at low masses the halos don't have any baryons in them and at high masses the potential is deep enough that dark matter still forms that is very hard to disrupt the dark matter cast so if those plots that I showed you before about large cause in clusters hold up then that might actually be somewhat difficult to explain with baryons if on the other hand on the other hand if we could measure much lower mass dwarfs look at there and we could see cause in those dark matter halos that would again be quite difficult to explain with baryons it's just that most of the observations that we have and most of the statistics that we have are in this galaxy scale region in which baryons are pretty are potentially pretty effective but support so suppose it's not baryons suppose that at some point in the future we we were to find out that baryons just can't do this maybe out we find evidence that our star formation history is really smooth so what if instead some novel dark matter physics was responsible well what could that tell you so there are a few possibilities for this we could have warm dark matter we could have dark matter that isn't a hundred percent collisionless could be somewhat collision all we could have dark matter that undergoes dissipative interactions some form and in all cases you might ask well this could it could this be all the dark matter or could it be some small fraction of the dark matter so warm dark matter there was a question about this earlier about what this could do we know the warm dark matter suppressor structure at small masses so you might say well okay so can warm dark matter produce the press these small scale structure these cusp in the centers of dwarf galaxies can the free streaming in the early universe move out those halos so unfortunately if you try to do the simplest thing here then to create the kinds of core sizes that we see in these dwarfs which are about a kiloparsec or three thousand light years across you estimate that you need to the dark man needs to be less than a key in mass and that's in conflict with the bounds that we showed earlier in the matter power spectrum yep yep so another way to say it is we know that we can't suppress our best suppression our maximum suppression scale is about 10 to the a solar masses we can't possibly affect we can't affect anything heavier than that because we've seen our evidence for halos it's going to be heavier than that size so that's a bit too low to really significantly affect our little missing satellite problem so you can alleviate the too big to fail problem because small structures get wiped out that means the structure formation starts later halos that form at later times are forming in a universe that is less dense overall because it's been expanding so that means that they're less dense and less concentrated which alleviates the too big to fail problem fully solving the too big to fail problem seems to require a warm dark matter mass at about 2k via lighter which is kind of on the borderline for those bounds that we showed earlier so maybe warm dark matter can do it but it's difficult to do it without time without running into other constraints the sort of much less studied idea that's been proposed is but it's it's just kind of neat so I'll mention it here is if there was enough there were multiple states of dark matter there was the ground state and also a slightly heavier excited state then if there was a significant amount of dark matter in this excited state then as it decayed down to the ground state it could get a velocity kick that could cause these dark matter particles to stream out of the centers of halos could flatten out the centers of halos and deplete their density this doesn't require you to have a pretty small mass splitting though and the dark matter has to couple to something else that allows it to decay to conserve energy and momentum more generally you could have some dark matter self interaction what if dark matter isn't fully collisionless what if from what it what if it's collisional to some degree we have some limits of this from the bullet cluster but they don't rule out all forms of collisionality and if we did have such self interactions those interactions will exchange energy and momentum and angular momentum between dark matter particles that too can deplete dense casps this is a plot of constraints on self interacting dark matter as a function of the velocity normalize the typical velocity of dark matter particles in waffles essentially this red region this picture is the cross sections that you need to get cause forming in dwarf galaxies these over here we have mostly limits upper limits on the cross section from making sure well we would like there to be some dwarf galaxies in the milky way if your self interaction causes all the dwarf galaxies to evaporate that's a problem likewise if you're dwarf if your self interaction makes all your hellos extremely spherical that can be a problem that's what these halo shape constraints come from and then there are limits from the bullet cluster and from yeah and from the electricity of x-ray clusters so putting all these constraints together we find that the kind of cross sections you need in these particular units so this is the cross section divided by the dark matter mass normalized to a particular cross section so I'll just tell you what it is you need cross sections of around one cross sections divided by mass of the dark matter about one square centimeter per gram or about two times ten to the minus twenty four centimeters square centimeters per GV we'll talk more about dark matter models in the next paper but many people are interested in thinking about dark matter models that are somewhere around the weak scale so somewhere around the GV to TV scale that corresponds to cross sections that are about ten to the minus twenty four square centimeters to significantly affect the dwarf galaxies now this is the unit that's called a barn in particle physics because it's as big as a barn it's a really large unit in terms of cross sections just for comparisons this is for dark matter scattering with itself dark matter collisionality just for comparison if we wanted to set bounds on dark matter scattering with ordinary particles there's a plot we'll see again later in the talk this is the current limit from the Lux experiment on the scattering cross section between dark matter and nucleons protons and neutrons and it's about ten to the minus forty five square centimeters so it's twenty orders of magnitude smaller than this cross section so if dark matter is really self-interacting at this level it's some it's a very large cross section compared to its interactions with the standard model the other particles are quite a large cross section by particle physics terms so that that means in turn for those particle particles in the audience you need a light particle to do the scattering okay so we're coming to the end of my time so I'll just I'll just show you a couple more plots so if we ask what is the effect of self-interacting dark matter on the halo mass function we find that the so on that dn by dm thing that we were computing earlier we find that the impact is pretty small for models that are ruled out for other reasons it can be large but for models that are allowed by other regions the the the impact is not large so self-interacting dark matter doesn't really help with the missing satellite problem it can help with cause this this plot is showing this plot is showing various self-interacting dark matter models this is the profile of density as you move into larger it's just smaller radii the black lines are the case with no dark matter self interactions blue lines here are the case with the largest of interactions so self-interacting dark matter can make shallow of the density profile it's more radii however if you're talking about the centers of galaxies like the Milky Way you really can't ignore the existence of barion's they greatly dominate the dark matter density and once barion's are involved it gets more complicated turns out that including barion's reduces the actually reduces the core size relative to pure self-interacting dark matter that might mean that to generate cause and small dwarfs you might need a very you might need actually quite a large cross-section and this self-interacting dark matter also alleviates the too big to fail problem with similar cross-sections again this is just something for the particle physicists in the audience for though this is this there are simple models of self-interacting dark matter that have been studied in some detail for example most of them require the dark matter to be coupled to a new force carrier that's somewhere around the MEV to GV scale this is a plot of various constraints on self-interacting dark matter and it finds find the cause you know the best fit is sort of hundred hundred GV dark matter coupled to a 10 MEV force carrier could do it that's a little non-trivial to achieve dark matter models but it can be done okay now the last thing the last topic that I was considering covering today but we'll leave it for considering covering this morning but we'll leave it for this afternoon is talking about self-interacting dark matter and mergers like the bullet cluster what constraints that we can place and the reason why I'm gonna leave it for this afternoon is that there's actually a possible hint of a detection of dark matter self-interaction in this channel although it's very very tentative so we'll call it there for now and I'll come back to that first thing this afternoon I'll talk about the I'll talk about these last constraints and tests of self-interaction and then we'll talk about dark matter models thanks very much for listening