 Next, then whenever you're ready, we can start. Okay, so I'll already start sharing my screen, just in case that takes a moment to respond. Ah, there we go. All right, then I guess we can start. Welcome everybody back to the second week of the school. And this week we will have completely changed gear. We have completely different lectures and lectures. So the first of today will be Yannit Hochberg from Hebrew University of Israel. And she's an expert in dark matter, WIMP and WIMP dark matter and also detection and models. And she's going to lecture us on no WIMP dark matter models, please. Okay, thanks so much. So thanks to the organizers for inviting me and thank you to all of you for joining this morning. So it's really nice to meet you, even if it's only in the virtual world. And hopefully we get a chance in the future to meet in person. And I hope you and your loved ones are all safe and healthy. So this is our plan. So the rough plan, we have two sets of lectures coming up. So each one that's today and tomorrow and each one is ballpark of an hour and a half. Of course, we're gonna take a short break in the middle. And what I really hope to give you in these lectures is sort of a nice taste of what I feel are exciting developments in the field of dark matter theory. And I really wanna give you sort of a sense of teach you different tricks and different methods to sort of give you a sense of the type of stuff that I think it's more difficult to sort of acquire and get a feel for if you're reading a paper or reading a book. Okay, so that's really my goal. I also wanna give you a taste of what I find to be exciting stuff. And in particular, sort of understand different methods and tricks so that maybe you go after the school and one day in your career, you write down the next amazing super cool dark matter theory, okay? And so the rough plan, we're gonna start off by setting the stage. So just to make sure that we're all sort of on the same page as we move forward. And then we're gonna be talking about new mechanisms for dark matter. And then we're also gonna be talking about some model realizations. And the lectures that we're gonna be going through, the lessons we're gonna be going through together are really complementary to the other. So complementary to the other lectures that you've already heard last week on different aspects of dark matter from Celine Bohm and lectures that you're gonna be hearing about this week on detection from Tong Yanlin, okay? So I really hope that all of these lectures together give you a really sort of a comprehensive picture of what's interesting in dark matter and what's exciting. So here's my more detailed outline for us. So the outline for our time together. So as I said, I'm gonna start off by setting the stage. And I know that you've had very detailed lectures last week by Celine, but I wanna sort of make sure that I'm like straining a line between all of us as we move forward. So this is gonna contain some very brief introduction. And then we're gonna be talking about a little bit about the early universe, just to make sure that we have a toolkit buildup. And then we're gonna move into talking about dark matter mechanisms. And we're gonna start off by thinking about two to two processes. And there's a whole zoo of processes over here, which we're gonna be discussing. This contains the WIMP, which you've started to learn about last week, but we're also gonna be going beyond. And after we finish talking about some examples of two to two, we're gonna be moving on to more intricate situations, like three to two. So we're here instead of WIMPs, we're gonna be talking a little bit about SIMPs and hopefully even get to go a little beyond that. And then towards the end, we're gonna move to talking about dark sectors. So this is my rough plan for us together. And most likely we'll sort of, today we'll probably be finishing, sort of going through the two to two, we'll see exactly how far we get. And then our plan for tomorrow will be to really pick up and talk about more intricate processes afterwards, okay? So that's our plan for today. So let's start at the beginning and setting the stage. So I don't have to give a full introduction to dark matter because of course, you've already learned about this in some amount of detail, but just to make sure we're all starting at the same point, our universe is dark, okay? This is a fact. We've measured, or we know the amount of dark matter in our universe. And this is, sorry, let me write this first as an equal. So this is the energy density in dark matter compared to everything that's out there. And this is measured to be 0.27. So we have 20% of the energy content in the universes in the form of dark matter. And set another way, I can relate the energy density in dark matter to that in baryons. And it's around five times the energy density sitting in baryons, okay? And so in our local environment, just to give you the numbers, we have 0.3 GV per centimeter cubed dark matter, okay? So this is what we do know about dark matter. We know that dark matter is massive, okay? It has a mass, but we have absolutely no idea what that mass is, okay? And if we want to think about scales, there are roughly, let me look for instance in GV units, there are, so dark matter could be as light as around 10 to the minus 30 GV or as heavy as 10 to the 50 GV, okay? Huge range of masses, of masses in which dark matter can reside. There's roughly 80 orders of magnitude over here. And we don't know where dark matter sits along this, along this axis, okay? And the reason that just to make sure we, so dark matter can't be smaller than 10 to the minus 30 because otherwise it's what we call too fluffy, okay? And that means that basically it's wavelength exceeds our galaxy sizes. And we also know that it can't be too heavy because then it can ruin, if it microlenses, it can ruin globular clusters, okay? So this roughly sets around 80 orders of magnitude for where dark matter can reside. A couple of other things that we know about dark matter, but we know and we don't know, we know that if at all it shouldn't interact too strongly with QCD and quantum electrodynamics, right? So shouldn't interact too strongly with let's say our known forces because otherwise we would have detected it already, which we haven't. And we also know that it shouldn't interact too strongly with itself because strong dark matter interactions distort dynamics in dark matter halos, okay? And also very importantly, and that's why you have so many lectures about dark matter, dark matter is a very, very important thing. We wouldn't be here without it, okay? Dark matter plays a crucial, a really major role in our cosmological history. And in the fact that we're even out here existing today asking the question of what it is. And so the big picture question that we're interested in understanding that we want to know is what is this dark matter, okay? And in particular, we're gonna be talking about together what are the mechanisms? What type of processes and interactions in the early universe that can set its abundance, okay? And then there was of course also model building. Once we have a mechanism, we also want to model build and understand what type of theories can realize these different mechanisms and complementary to this. And these are the parts that we're not gonna be talking about together today or how to detect this and also what are the constraints? Okay, so those will be contained in complimentary lectures by other people. So let's start with, let me give you an early universe cheat sheet, okay? Just to make sure we all have all the necessary ingredients to be moving forward together in this course. So here's my early universe cheat sheet, okay? So as you probably know, our universe is expanding, okay? So if I have some distance out, then universe is expanding and I get some different distance, which is A times L. So this little A is what we call the scale factor and volume importantly then expands like A cubed. Volume expands like A cubed. And if we look at, let's look at a length element or our metric. So we have dS squared is dT squared minus A of T squared. That's this, the scale factor times dX squared. And we define something called Hubble, which I write as capital H as A dot over A. So A dot is just the time derivative of A, okay? So this is our Hubble constant. And what we know from the first Friedman equation is that this Hubble constant squared should equal the energy density over three M plank squared. And since the energy density row goes like T to the fourth, this is just black body radiation, what this tells us, and this is gonna be the piece that we're gonna be using is that this Hubble goes like temperature squared over M plank, okay? So this is gonna be appearing all over the place in our estimates together, Hubble goes like temperature squared over M plank. Now, our early universe is a thermal environment, okay? And what this means is that all the species that we have have some, we can write down some phase-based distribution for a species that's in thermal equilibrium. And from that we can get number densities and energy densities, okay? So we have phase-based distributions for different species that are in thermal equilibrium. And from this we can get number densities and energy densities, okay? And so here's the type of estimates that we're gonna need. So the number density, if the particles are relativistic, which I'll denote by R, number density goes like T cubed and the energy density goes like T to the fourth, okay? So this is what happens for relativistic species and we're gonna be needing this. And when something is non-relativistic, if a species is non-relativistic, then the number density goes like M times T to the three halves, E to the minus M minus chemical potential mu over T and R just goes like M times N, okay? So this is what happens for non-relativistic species. And very importantly, in particular, if I have some rapid number changing processes, okay? So when the chemical potential equals to zero, if I have some number changing processes that are fast, then what happens is that the number density is exponentially suppressed, okay? Number density goes like E to the minus M over T, exponential suppression, okay? So this is the type of behavior that we're gonna be using throughout. And the last little bit that we need here is of course also entropy density. So entropy density, which is denoted by S goes like temperature cubed, okay? And the entropy density in the universe is governed by the relativistic species that contribute to it, okay? So these are the types of scalings that we're gonna need for number densities, energy density, and for entropy density. And now we're set to write down our Boltzmann equations. So here we go, so here's my Boltzmann equation, okay? So let's start by considering a consider a system in which I have no collisions, okay? So just free particles. So consider a system with no collisions, just free particles, okay? In this case, that means that the change in time of the total number of particles dn dt just vanishes, right? Constant amount of particles over here. But now let's look at what happens to the number density of particles, okay? So I can write this as dn times the volume, number density times the volume dt, and this will just be v times dn dt plus n times dv dt, and this should equal zero, okay? Let me write this another way. So I'm just gonna divide by the volume to get the dn dt plus n over v dv dt equals zero. And now I remind you that our volume, our universe is expanding. So our volume scales with the scale factor cubed, and what this means is that if I look at one over v dv dt, this is just three over a dA dt, okay? And so if we just take this and plug it in to our above equation, what we find is that we now have that dn dt plus three nA dot over a equals zero. Now remind you that A dot over a was exactly what we defined as Hubble, okay? So this would be the Boltzmann equation for a completely free particle with no collisions. If the particle is not free and it does have collisions, then all we have to do is augment this equation on the right-hand side by including these collision terms. And that's what we're gonna be doing throughout. Okay, so if our particle is not free, we have collisions and then the right-hand side is contains the collision terms, okay? So we just write this as dn dt plus three nH equals in general minus the collision term, which I'll just denote here by C. Okay, and different processes we're gonna be talking about have different collision terms and that's a slightly different Boltzmann equation. So this is our cheat sheet for the early universe, which I think is gonna contain everything that we're gonna need moving forward today and tomorrow. And having set that up, we can now start talking about different mechanisms. So the idea is that the early universe in principle can have many different processes that are occurring depending on particle content and what the Lagrangians are. So when I say mechanisms, I'm just talking about types of processes in the early universe that set the relic abundance of dark matter, okay? And what we're gonna be starting off with are a various set of two-to-two processes, so I'll call this the two-to-two zoo, okay? And moving forward, I'm gonna be denoting, dark matter will usually be denoting as chi, okay? And in all the pictures and diagrams that I'm drawing, time always goes in my writing from left to right, okay? So let's see what type of processes people have worked out already for the two-to-two case. So the first one is, here's my two-to-two process, okay? Time again, going this way. So I could have chi, chi, annihilating into two other particles, I'll call them phi, phi, and this is just the WIMP process, this ordinary two-to-two annihilations. Next type of process that I could have is actually, looks very similar, look at this, it's still two chi's annihilating into two phi's, but let's now consider the kinematic regime where M chi is smaller than M phi, okay? A process of this sort, if it's what governs the relic abundance, is what we call forbidden dark matter. So these are forbidden processes. We could have something of the following, we could have two chi's annihilating to one chi and one other particle. These are processes that are called semi-annihilations. Next one we could have is, again, still two-to-two, but maybe instead of having just one type of dark matter particle, we actually have some whole tower of particles. So maybe I have some chi-i and chi-j that are annihilating into two phi's, and these processes are called co-annihilations, and we'll be talking about them as well. One more, for more recent times, we could have chi that scatters off of phi's and changes its identity to some other particle sign. So these are called co-scattering, and a very recent one just from the last few months, I'll draw here, I could have chi that meets a phi, and the phi changes it into a phi, so this is called a zombie process, okay? And the last one that I'm gonna draw here, and I'm gonna put it sort of in these dotted parentheses, we could also have a case where actually the elastic scattering of chi's off of phi's is what sets the abundance, so these are called elastically decoupling relics or ensured elder processes, okay? So this is a whole bunch of different, each one of these represents a different process in the early universe, and if it governs the relic abundance, then we say the dark matter, that the mechanism is this process, and what I'd like to do is sort of pick a few of these and understand what happens over here and what type of dark matter we're pointed to, and so actually in order to really set up a toolkit, the one that we're actually gonna start with is, we're gonna phrase it in a particular way, is the one that you've already seen last week, we're gonna talk about the WIMP, okay? And then from the WIMP, we're gonna easily be able to move beyond it, okay? So that's gonna be our plan for now, we're gonna start off by working through the WIMP and phrasing it in a very particular way to make sure we understand what's going on, okay? And I think it's no secret that the WIMP has been the star of the show for, I don't know, 40 plus years at this point, okay? So this is just some process where two dark matter particles annihilate into two other particles, two other bath particles, okay? And you saw this, I said, you saw this already with Celine and I wanna phrase it in a very particular way, it's gonna be useful for us moving forward. So here's my dark matter process, okay? I have two kais annihilating into two fies, okay? Now if this process is fast, what it does is it sets the, oops, let me take a different color here, if this process is fast, then it sets the chemical potential of chi to be equal to the chemical potential of phi, okay? And we can write down the Boltzmann equation for this, for this, so we have dn of chi, dt plus three h and chi is equal to minus some collision term that we're gonna write out in a moment. Now I just wanna point out already here as a side note that if I have a diagram like this, okay, I'm actually always also elastically scattering, okay? So here's a side note. We're always elastically scattering, okay? So what I mean by that is that I flip this diagram on its side and I have chi-phi scattering to chi-phi, okay? Chi-phi, chi-phi, okay? And we're assuming that phi is some bath particle that's either the standard model itself or some particle that's in thermal contact with the standard model, and so that's gonna set the chemical potential of phi to be zero and also give it a temperature, okay? So the bottom line from this is that this chemical potential of chi is equal to the chemical potential of phi, which is equal to zero, okay? And that means that in our estimates, the number density for chi, if it's non-relativistic, will be exponentially suppressed. And in fact, also something that's gonna be relevant for us later, almost always, what you'll find is that this two-to-two annihilation process that's happening over here shuts off before the scattering process, okay? Well, let's write down the collision term now, okay? And roughly speaking, what you find is that the rate for some process is basically like the strength of the interaction, so some thermally-average cross-section times the number density of the particle that you have to meet. So this is gonna be the thermally-average cross-section times the number density. And so what this means for us is that I have dn dt plus three hn equals minus thermally-average cross-section for this chi-chi goes to phi-phi process times n chi squared, okay? So this is taking care of this forward process where I had chi's annihilating into phi's. And of course, we also have to take into account the back-reaction process, the inverse process, where two phi's are annihilating and producing two chi's. And the way that I can write down this term and I'll explain in a moment why is that this just looks like minus n equilibrium of chi squared times that cross-section, okay? So the second term over here is really taking care of the inverse process where phi-phi is annihilating into chi-chi. And so just to explain why, how do I get this factor over here? Okay, so why is this the right thing to write down? What we do is we use a trick that's called detailed balance, okay? So let's write that. Use a trick that we're gonna be using later today as well. It's called detail balance, okay? And with detail balance, it's just a statement that if I'm in equilibrium, then forward reactions and backward reactions, they're both very rapid and they should just cancel out, okay? So in equilibrium, forward and backward reactions are both rapid, okay? And thus, they should cancel out, okay? And that's why instead of writing a cross-section for the forward process times the number densities minus the cross-section for the backwards process times the number densities, I know that we can write it this way because in equilibrium, I know that my right-hand side has to vanish, okay? So this is why we can write it in this way, okay? Now, so this is the equation that describes the WIMP two-to-two annihilations and let's just make sure we understand what happens in the system, okay? And the way that we sort of like to think of this is instead of thinking about how particle densities are evolving as a function of time or temperature in a box, let's think about how, think of these particle densities in a box that's expanding with the universe, okay? So think of this, well, let me say this differently. So instead of a particle density in a box, we're thinking of particle densities in a box that's expanding with the universe, okay? So this is, sometimes this is called a yield and it's denoted often by Y, it's just the number density divided by the entropy density, which is just the same as number density times a cubed, okay? So number density in a box that's expanding with the universe and I'm gonna draw a plot here that I know you've seen already with Celine, I wanna make sure though that we all understand what's happening here because an identical plot is what's gonna be accompanying us throughout all of the different mechanisms that we're gonna be talking about, okay? So here it is, what's happening to this, let's understand what's happening, what this equation is telling us. So here I'm writing this as a function of a variable called X, which is mass of dark matter over temperature, okay? And let's look at n over s divided by the measured value, okay? And here's what the solution, okay? What the solution to this equation up here gives us. It looks roughly like this, okay? So let me just draw that out and we'll explain the different parts. So what you see over here is that at very early times, so let me remind you very early times are sitting, this is early times or high temperature, okay? At very early times forward backward reactions are both happening rapidly, okay? So that's this region over here, both sides of this story are rapidly occurring. Once the temperature reaches the mass of the dark matter, okay, that's when the particle starts to become non-relativistic and then it becomes much more difficult for energetically to be able to produce the dark matter particles. And so we're in the non-relativistic regime, that's what's happening over here, non-relativistic regime where the production of dark matter is suppressed. But then as our universe is cooling, it's also expanding and at some point it expands so much that the particles can no longer meet each other. They can't find each other and so they can't annihilate and that is the point at which dark matter stops rapidly annihilating away and we're left with a constant dark matter density, okay? So this region over here is where the particles are so dilute that our annihilations shut off and this is when we're left with a constant relic abundance, okay? And of course how much we have left, how much dark matter is left is a function of the strength of the interaction, a function of the thermally average cross section, okay? So this picture that we've drawn over here, this is the standard picture for dark matter freeze out, okay? And as I said, we've drawn this now for the WIMP but this is actually a very general picture that holds for almost all of the mechanisms that we're gonna be talking about, okay? And so what I'd like us to do now together is sort of write out a back of the envelope estimate to understand at what point is this freeze out problem in the process occurring and what does it tell us about the mass scales and the strength of the interaction, okay? So that's gonna be the next thing that we're gonna do. We're gonna do a back of the envelope estimate. So and what we're basically doing is we're kind of trying to focus on this region over here of when freeze out is occurring. So when is freeze out occurring? Well the condition for when, roughly for when freeze out occurs, freeze out occurs when the rate of these two to two annihilations is roughly of order the expansion, okay? That's what's at this point at which the particles can't meet each other before the universe expanded and then the process stops occurring. So freeze out happens when the rate of two to two is of order Hubble, okay? And now let's try to write out what is the rate for this two to two annihilation to happen and the thing that I kind of want you to keep in mind throughout all of the exercises like this, what we're gonna do, think of yourself as if you're the dark matter particle, okay? And then what's my chance of annihilating with somebody else for this process, okay? So think of it as if you are the dark matter particle, okay? So using that here is my two to two rate. So the rate for two to two process, it has two factors in it, okay? First, I'm a dark matter particle and so in order for this two to two annihilation to happen, I have to meet another dark matter particle. So the rate is proportional to number density of the other guy I need to meet, okay? And of course it also depends on the strength of my interaction, so it's gonna be proportional to this thermally average cross section, okay? And again the freeze out condition requires that this is of order Hubble and I remind you that Hubble we said goes like temperature squared over I'm think, okay? And so just to spell this out, this factor over here happens because we have to meet one other dark matter particle, okay? And let me give a name to this equation just so we'll be able to easily refer to all denoted with this red star, okay? And now the trick that I wanna do is to take this number density of dark matter at the time of freeze out and relate it to measured quantities, okay? So let's relate this to measured quantities. And this is a trick that's really very often very often useful to do and the way that we're gonna do this is by playing with red shifts, okay? So play with red shift. So let me explain what I mean by this. So in our system our total entropy is conserved, okay? So let me write total entropy is capital S, okay? So this is entropy density times a cubed and the fact that this is completely conserved we're gonna use this to red shift between different epochs, okay? So use to red shift because this is a constant, okay? So what that means is that entropy density goes like one over a cubed which is the same as saying it goes as t cubed like we just saw, okay? And this is because temperatures are proportional to one over the scale factor. And so the way that we're gonna do this is basically after freeze out occurs the co-moving number density, okay? The number density that doesn't care about the fact that the universe is expanding the co-moving number density is also just constant, okay? And so we're gonna red shift from freeze out to a particular time. The time we're gonna use is t equality, okay? Matter radiation equality. So after freeze out, after freeze out occurs the co-moving number density, okay? So that's a number density whenever moved the aspect of expansion the co-moving number density of dark matter is constant and so we're gonna red shift back using this. And the trick that we're gonna use which I find to be often there are many, many different tricks one can play. This one I think is particularly useful. We're gonna red shift to t equality which is the time or temperature of matter radiation equality. Radiation equality which happens of order 0.8 EV, okay? So that's the trick that we're gonna use. And also let me now parameterize this general two to two cross section, okay? It's a two to two cross section that we're looking at. So I'm gonna parameterize it in the following gray parameterize in the following way. Let me call this thermally average cross section. It's just a name. I could parameterize it any way I wanted to. It's pretty natural for a two to two cross section to be parameterized by some coupling squared. I'll call it alpha effective squared divided by m chi squared by the mass of the dark matter squared, okay? This is just a parameterization so we can understand what type of relationship between mass and coupling this mechanism predicts. And now let's do some redshifting, okay? So at matter radiation equality, at matter radiation equality, what matter radiation equality really means is that at this time the energy density and matter at equality, which is the sum of the dark matter energy density plus the baryonic one, this exactly equals the energy density sitting in photons. Okay, that is the definition of this temperature. And I'll mind you that we've measured in our universe today that the energy density of dark matter is roughly five times that that sits in baryons, okay? And so the ratio between these, this ratio, which is an order one number, also holds back at the time of equality. And so what we do is use this relationship to write down that dark matter energy density at the time of equality is roughly that of sitting in photons, okay? So I've neglected here the order ones that come from the five and the five minus one, but you can see also how in principle to hold them and they're not gonna be very important for us right now which is why everything now is with squiggly is I'm not gonna care about order one factors, okay? This is how I use a measure quantity, the amount of dark matter that we observe in our universe and I look at what happens at t equality, okay? So at t equality, energy density in dark matter, roughly that that sits in photons, okay? So I'm neglecting over here some order ones, but you can see from this, if you wanted to do it more accurately how to keep them into the story, okay? And so now let's do some redshifting. Actually, looking at the clock, maybe this is a good point for us to take a five minute break with that work. Yes, perfect. Yeah? Okay, so let's take a five minute break and we'll come back at a quarter to the other. Yeah, 45 years. Okay, thanks everyone. I on it, could you try to lower a little bit the volume of your input because the voice is a little bit distorted, maybe. The voice is distorted, okay? Just a little bit. You know how I do that? I'm not sure. Actually, let me try. I have an idea. Let me see if putting on headphones improves the situation. Maybe then... When you talk normally, it's fine. Maybe when you get louder, I think you just get distorted. Okay, let's see. Yeah, now it's lower, yes. Okay, okay. Okay, and there was a request if you could write a little bit bigger. Okay. Okay, oh, and there is also one raised hand if you want to take a question. Okay, sure. By Max, please. Okay, next, please. And I can ask you, so how do we know that it's totally entropy of the universe is conserved? Well, this is my assumption walking in that total entropy is, I don't have something that's like a sink outside of the universe, right? Everything in the universe is, as a whole, I have a constant amount of entropy. But for me, it is not completely clear why this assumption is so natural because if I think of a closed system, then I would think that normally the entropy would increase there, no? My total entropy is conserved. But is this really natural as an assumption for a closed system to think that we would be conserved? And I'm just not sure about this. Maybe you can comment a little bit on that. Yeah? Yeah, I think so, I think so, yeah. For my closed system, when I'm encompass everything, everything, then I just, I don't have my change in entropy. Okay. Okay, thank you. Any other questions? I don't see any more. Okay, no, I assume. Okay, so thanks everyone and welcome back. So as I said, we're gonna be using this trick of redshifting. So we use this trick all the time, we're gonna redshift between all sorts of different temperatures. And right now we're gonna choose to redshift via T equality. And so let's do that. So my number density of dark matter at freeze out, let me now redshift it to T equality, okay? So I'll write this as number density at T equality. Actually, let me, let's do this. A little bit more clear, okay? So here at T equality, the way that I redshift is again, just by the ratios of temperatures between the two epochs and I'm redshifting between to the third power. So I have T freeze out over T equality cubed, okay? And now number density of dark matter, if it's a T equality, it is non-relativistic. So I can write this as the energy density of dark matter at T equality over the mass of dark matter. And then I just keep these temperature ratios T freeze out cubed over T equality cubed. And now we're gonna use this fact that we've developed over here, which is the fact that at T equality, up to order ones, the energy density in the photons is the same as that sitting in dark matter. So we're gonna use that now and plugging that in, let me squiggle over here. So what this means is now I'm gonna swap out energy density of dark matter for energy density of photons. This at T equality and I keep over mass of dark matter and I still have the ratio here, T F cubed over T equality cubed. And now the fact that I have energy density of photons makes my life very simple because we know that the energy density of photon just goes like temperature to the fourth, okay? And so what this gives us is temperature T equality to the fourth. I have temperature T equality to the third downstairs. So I just have one more power of T equality left then times T freeze out cubed over mass of dark matter. And one more thing that we're gonna do over here is move to, instead of talking about temperatures, it's often convenient. I'm gonna move the temperature of freeze out to be that variable X that I mentioned this quantity, which is mass of dark matter over the temperature at freeze out. And this will all together give me T equality M chi squared over X freeze out cubed. So this is the number density of dark matter at freeze out written in terms of T equality, okay? And this is what we're gonna be using. And now let's plug this in to our freeze out condition. So let's compare it to our freeze out condition what we define to be our equation that I denoted with a star. And so what we find is that here's my two to two rate. So I reminded that this was the number density of dark matter at freeze out times my thermally average cross section, which we're parameterizing as alpha effective squared over mass of dark matter squared. And so plugging in and dark matter at freeze out to be what we just found, M dark matter squared is gonna cancel out from up here and over here, okay? And so what we're left with is alpha effective squared T equality over X freeze out cubed. Now remind you that we're gonna require that this is of order Hubble at the time of freeze out. Hubble at the time of freeze out is temperature squared over M plank. And again, temperature I'm gonna move to write in terms of this variable X. So this is gonna be M X squared over X F squared M plank. And now all I have to do is move parts around from left hand side and right hand side. And what we're gonna find when we do this and solve for M chi for mass of dark matter. And what we basically arrived at is that the mass of dark matter should go like the coupling alpha effective times square root of T equality M plank. And when we plug in properly of so all the fudge factors that have been neglected over here, what this means is that alpha effective, the mass is alpha effective times a number that turns out to be around 30 TV, okay? So this is the mass coupling relationship that we find for this two to two annihilation mechanism, often called the wind. And so you see over here that if this coupling that parameterized the cross section happens to be of order, the electro wick coupling happens to be of order 10 to the minus two, then the scale that's gonna emerge for dark matter is around 300 GV of order, the weak scale, okay? So this is of course a big if, so if alpha effective is of order 10 to the minus two of order wick coupling, then we find that the weak scale emerges, okay? Well, what I'd really like to stress is this is a coincidence of scales, okay? A coincidence of scales between M plank and T equality that can thrive together to give this geometric mean, which if we plug in a weak coupling gives us the weak scale. But if our coupling is substantially smaller, then of course we get something different, right? So if alpha effective is much smaller than a weak coupling, then you see here that we get dark matter masses that are much smaller than the weak scale, okay? So what this shows you is that we naturally get in the setup, we can also have here light dark matter, okay, dark matter that is much smaller than what we typically think of for the win, okay? But this is what's often called the win miracle, okay? We've derived it now in a particular way, which I hope makes very transparent that really there's no, first of all, there's no miracle. There's just a coincidence of scales and if we plug in a particular size coupling, we arrive at a particular size mass, but that definitely need not be the case. Now, before continuing here, a natural question to ask is what is the value of X freeze out? What is the temperature at which the dark matter is freezing out? And let's work through that over here for the WIMP and something very similar holds for all of the other mechanisms we're gonna be talking about. So what value is X freeze out? So let's assume instantaneous, instantaneous freeze out, okay? So that means that we're gonna ignore sort of subtleties of what's happening exactly at freeze out, but let's just plug in at our freeze out condition equilibrium distributions, okay? So plug in densities, well, let me say it differently, let's plug, we're gonna plug in equilibrium densities for the chi's at the time of freeze out. And so our freeze out condition is then N chi equilibrium at the time X freeze out times sigma V should be of order Hubble at the time of freeze out. And so what this gives us, let me assume that the dark matter is non-relativistic as we've been doing and we'll see that this is a self-consistent assumption. So the number density in equilibrium is exponentially suppressed. And so what we get over here is something that looks like M chi squared over X freeze out and I'm dropping factors of order one just to show you how things are behaving to the three halves power. And here's my exponent of minus X freeze out minus mass over temperature times sigma V and this should be of order Hubble which is T freeze out over M plank which we write as M chi squared over X freeze out squared M plank. And now you see that if I wanna solve for what is the value of XF, XF just goes like logarithm of all these parameters. Okay, so and the reason that this is really nice is because it means that the value of XF, the value of the temperature at which dark matter is freezing out is really very insensitive to all sorts of changes only logarithmically dependent on any of these parameters. So it's really broadly over a broad range of masses and parameters, it roughly gives us the same answer. So this is roughly the same over a broad range. And when you plug in numbers, what this gives you is that X freeze out, which is the mass of dark matter over the temperature is really of order between 20 to 30 for dark matter masses, typically let's say in the MEV to TV mass range. Okay, so when you think of dark matter freeze out and throughout all of the mechanisms that we discuss really basically dark matter is freezing out non-relativistically at temperatures that are roughly a factor of 20 beneath its mass. This is something that's gonna hold throughout. Now another way to write this relationship that we found over here between the mass and the coupling that gave us the right amount of relic abundance is if I just think back of what was my, how we parameterized our cross-section which was alpha squared over M squared. So another way to write this would be that what we found is that the size of this cross-section that enter our Boltzmann equation which was alpha effective squared over M chi squared. This should be of order one over T equality and M prank. Okay, so this is this cross-section. This cross-section should be of order T equality one over T equality and M prank. Now, if you just think about, think about ordinary two to two cross-sections then a priori you'd think that there is, I can't take the coupling that's sitting over here to be as large as I want. So usually we have what's called the uniterity bound, okay? Which is just the statement that in general you'd expect the coupling to be at most four pi. And so given how we've parameterized this cross-section that tells us that you'd expect an upper bound on the mass of dark matter that comes out to be of order around 300 TV, okay? So this is what's often called the uniterity bound on dark matter. This was derived originally in a paper by Greist and Kamiokowski in 1989. But actually in recent times, they've shown that there are several ways in which you can evade this bound. And so there are several ways to evade this bound. So there are ways to get thermal relics, relics that are in thermal contact in the early universe just like the WIMPs, but that have math that exceeds this bound. But I think the ones that have been developed so far go up to 10 to the 12, 12 orders of magnitude. I'm gonna write that differently, 12 orders of magnitude higher than this, okay? And so just to give you the references in case we're not gonna be able to cover these in class, but the first of them is called super heavy thermal relics or super heavy thermal dark matter, I think is it's official title. And this is a paper by Kuflik and Kim. It's a PRL, I'll give you the archive number 1906.00981. And a process that we've written down together but we're not going to be solving is this zombie process. And this is by Kramer, Kuflik and many others. This is a PRL also just recently 2003.04900. Okay, so these are two different ways to that exist in the literature today by no means the only ones. This one uses zombie processes and this one uses a chain of nearest neighbor interactions to be able to evade this uniterity bound that otherwise at least naively one would expect to be placed on the mass of dark matter that annihilates the two-to-two interactions, okay? And so to sort of summarize where we're at now what we've basically arrived at the WIMP process which I really prefer to call just sort of ordinary two-to-two annihilations because as we saw it by no means has to be a weekly interacting particle. This roughly gives us dark matter that's in the KV and above mass range with possibly some higher order, higher uniterity bounds on its mass but we have caveats of how one can evade that. And so this concludes our discussion of the WIMP where what I really hope we've done is developed a good toolkit for now moving forward to going to other two-to-two processes in the two-to-two zoo and understanding very easily how they differ. So having finished talking about the WIMP we're now gonna move to talking about the next process which are forbidden processes, okay? So forbidden dark matter. And this is an idea that was actually well it originated in a paper by Greist and Sackl in 1991. This paper that I know Celine Hindo so mentioned about the three exceptions to the relic abundance. But really was still phrased there in the context of a week scale and has been revived in more modern days by Ruderman and Diangelo, Diagnolo and 1505.07107 and this was a nice PRL paper. Okay, so these are the references for that. And so here is the process for forbidden dark matter here is some stuff is happening and we are agnostic right now what it is but it's some process that takes kai-kai to 5-phi which is just the annihilation that we've been discussing so far. But now I wanna look at a different kinematic regime. Let's look at the case when m-kai is smaller than m-phi. So the sum of the incoming masses is smaller than the sum of the outgoing masses. Okay, so if I was looking at zero temperature this process would be forbidden, right? This process simply can't occur. But as we know, our early universe is a thermal environment, okay? And so at the early universe, this process can occur by sort of living off of the Boltzmann tail of the distribution, okay? So, but the early universe is a thermal environment and that means that this process can occur by living off of the Boltzmann tail of the distribution, okay? So let's see and let's see exactly how this happens, okay? So what we're gonna do is write down our Boltzmann equation for the system. And so here it is. So it's the NDT plus 3NH. This is how everything always starts off. And now we have to write down both the forward process where two chi's are annihilating. That's the forbidden direction. And also the back reaction where two chi's are being produced which is actually like the allowed region, okay? So what we have here is minus N chi squared. Sorry, let me move this all down. Oops. So let's make sure we have enough room for this. So here's my Boltzmann equation, the NDT plus 3NH equals. So first I have my forward process which is two chi's meeting each other. So it's minus N chi squared times sigma V for this process chi chi goes to phi phi. Okay, it has a minus sign in front of it because it's removing two chi's from my story. And then I have the inverse process which is N phi squared. And I'm assuming these particles are bath particles and they're in equilibrium times sigma V of this forward process phi phi to chi chi, okay? And now in order to make progress, we're gonna use the same trick that we saw for the one but detailed balance, okay? So here's a trick, detailed balance, okay? And as we said that in equilibrium, both of the forward process and the backward process, everything should be rapid and the right hand side should vanish, okay? So in equilibrium, the right hand side should vanish. And what that tells us is that I can write the thermally average cross section for the forbidden case, chi chi goes to phi phi, the forbidden side. I can write this in the following way as N phi equilibrium squared divided by N chi equilibrium squared times sigma V of the not forbidden direction phi phi goes to chi chi, okay? This is what detailed balance tells me I can do in equilibrium. And the reason that this is useful is that this ordinary not forbidden process, this is just an ordinary two to two cross section. So this is just ordinary. And so we can just write it again as some alpha effective squared over M chi squared. So let's see what we got over here. So here's my squiggly down here. So you see we have the ordinary cross section times the ratio of N phi in equilibrium squared over N chi in equilibrium. Now, if we just think of the scalings for equilibrium distributions, what we have is an exponential suppression for phi and for chi, but importantly they're different because they depend on the mass, okay? So this is gonna give us an E to the minus two M phi minus M chi over temperature times sigma V of this phi phi to chi chi the non forbidden direction, okay? And so you see what we got over here is that our forbidden cross section, normally average cross section goes like exponential suppression depending on the mass difference between the particles times an ordinary cross section, okay? So let's plug this into our Boltzmann equations. So we have dN dt plus three NH. So this equals minus sigma V. I'm gonna write the non forbidden direction phi phi to chi chi. And so what I have now, and let's, sorry, let's move back over here. So you see this term over here, okay? This term is what I'm writing down now, okay? So I have N chi squared times the forbidden cross section which we just derived. So that means that what I really have is N chi squared times the non forbidden cross section which we just wrote and this exponential. So I'll call that E to the minus two delta M where delta M is this mass difference over temperature. And then the next term is very simple. It's just minus N chi and phi equilibrium squared. That was the direction that was very simple from the gecko. Okay, so this is the Boltzmann equation that describes this forbidden system. So now let's write down my freeze out condition. So freeze out happens when the rate for chi chi to phi phi, the rate for that process that's annihilating stuff away is of order Hubble. Okay? So what is the rate for the forbidden direction? Okay, what is the rate for chi chi to phi phi? Well, it's just gonna be that forbidden cross section times if I'm one dark matter particle, I have to meet another dark matter particle. So I have to be proportional to its number density. So I have one number density of chi times the forbidden cross section which as we saw is just E to the minus two delta X. I'll define delta X in a moment times alpha effective squared over M chi squared. Okay, where I've just defined over here, just as a short-hand notation, the delta X is M phi minus M chi normalized to M chi. Okay, so this first piece over here is the fact that I need to meet a dark matter particle in order for the process to happen. And this second part over here is just the forbidden cross section where I've written it as the exponential suppression times an ordinary cross section. And at freeze out, this thing should be of order Hubble which is T squared over M plank, which again, I can write as X freeze out squared over M chi squared times M plank downstairs. Okay, so this is the freeze out condition for this forbidden process. And we can now shift things around and solve for M as we did for the case of the WIMP. And what we find when we do this is that we start off looking very similar to the WIMP. So we have M chi is of order alpha effective times square root of T equality M plank. But now we have an extra exponential factor E to the minus delta X. Okay, so this is the mass coupling relationship for forbidden dark matter. And importantly, what you see over here because of this exponential is that we can easily get dark matter that is exponentially smaller than the WIMP. Okay, so we have here exponentially smaller masses than the WIMP. Okay, so this is naturally much smaller than what we got in the WIMP case. And so dark matter that we massively expect in this case is usually above, again, of order KV and above just like for the WIMP. But over here for the same size couplings we would get exponentially lighter dark matter. And I just wanna note over here, we're not gonna go through it, but the same exact freeze out picture that we saw for the case of the WIMP still remains the same over here. Okay, so freeze out conceptually, the freeze out picture looks exactly like the WIMP. Okay, it looks exactly the same. If we were look at what is N over S as a function of X it still looks exactly like this. Okay, nothing is changing. And also just to note over here that likewise, temperature at which freeze out is occurring X freeze out is still of order 20. So temperatures are roughly factor of 20 beneath its mass. Okay, so all of those things that we've developed for the case of the WIMP hold here as well. So this is forbidden dark matter. And what I hope to do for the remainder of our time together today is to now walk us through another one of the two to two zoo, which is called co annihilations. Okay, and this originally started from the same was proposed in the same paper in the 90s by Grist and Seco in this three exceptions paper. And again, originally proposed in the context of the week scale, but as I hope to show you it has nothing to do with the week scale. And it's a concept that just has to do with any two to two, any type of two to two interactions. So here's the process for co annihilations. So now let's imagine that instead of having let's say one dark matter particle, what we really have is a tower. Okay, we have some spectrum of spectrum of let's say N dark states or dark particles. And we consider annihilations of the following form. So maybe we have Chi I that meets Chi J, both of them from this tower of particles and annihilate into some other particle, I'll call it Phi. And so the idea is that in principle, this whole tower of particles, the whole tower can participate at freeze out and participate at freeze out, okay? And so here's my, you know, here's my tower. Whoops, here's my tower of particles. Chi I, the lightest one is gonna be the dark matter and this tower goes all the way up to some particle Chi N. So the lightest one in this tower is gonna be our dark matter. And in general, to describe this whole system, what we have to do is describe N coupled Boltzmann equations, you know, an equation for each one of these particles and since they're speaking to each other, we have N coupled equations. So in general, this would be described by N coupled Boltzmann equations. But we can use a trick, which is what we're gonna do is assume that there's rapid exchange between these Chi I and Chi J. So assume rapid exchange between Chi I's and Chi J's in general throughout this tower. And what that does is it sets the chemical potentials of all of the particles in the towers to be the same. This rapid change could maybe be happening through, you know, maybe there's some decays where Chi J is going to some Chi, whoops, some Chi I and another bath particle, or maybe it's happening through scatterings where Chi I is changing itself into a Chi J by scattering off of fives. Okay, so what this basically means is that this tower of particles is in chemical contact with itself. Okay, so the tower is in chemical equilibrium with itself. And what this enables us to do as the bottom line, which we need here, is that we can write the ratios of the number densities of these different particles in the tower, roughly as just the equilibrium ratios. And what this does is it removes N minus one equations from the story. Okay, and so at the end of the day, there's gonna be just one equation that we can write down and it's gonna be, we're gonna consider the equation for N, which is gonna be the sum of all of the number densities of these different particles in the tower. So this is the total number density, total number density, okay? The idea is that during freeze out, everything is participating, but eventually, all of these particles in the tower are gonna sort of decay down to be in the dark matter. So whatever is happening, whatever number densities are sitting, the total number density of all these particles that freeze out eventually will reside today in the dark matter. And so we're gonna write down the equation for this total density. Let's just write that out that eventually, everything sits in the dark matter, okay? And so here is the equation for this total number density. It's dN dt plus three N h equals minus sigma v effective, which I'm gonna define in a moment. There's some effective cross-section that comes in times N squared minus N equilibrium squared. And this, and you see this looks, this equation is identical to the equation that we saw for the WIMP, except now I have some effective cross-section which has the following form where this sigma v effective is just the sum over ij summing over all of the particles over i and j of Ni equilibrium times Nj equilibrium normalized by the total and equilibrium squared times the individual ij cross-sections of these particles, okay? So what we've arrived at is an equation that is identical to the WIMP. And so given that it's identical to the WIMP, we know what the size of this effective cross-section has to be in order to explain the observed relic abundance. It's exactly what we already showed for the WIMP. So let's write that out. So this is identical to the WIMP equation other than this effective cross-section. So we already know that what we need is for sigma v now effective to be of order one over T equality M plank, which is what we saw for, already for the case of the WIMP, okay? And so what's interesting over here is at freeze out, this whole tower of particles is populated and eventually they decay down. Well, what you see over here is that even though this effective cross-section has to be of a particular size, just like an individual cross-section in the case of the WIMP, here it's an effective cross-section. And the importance of this is that, what this means is that my dark matter density at the end of the day can be determined by interactions of other particles in the tower, okay? So it's not, it doesn't necessarily even have to be the dark matter itself is participating in the number changing process that's setting the abundance. So just to write that out. So first, just to make sure this is clear that at freeze out, the whole tower is populated. You only, please, a question. Yeah, let me just finish up writing out this one sentence, okay? So at freeze out, the whole tower is populated and then at late times, they decay down. Please, Gautam. Gautam? Yes, hi. I have a question regarding the particle tower. So this is just a spectrum of particles that all react along around the same time? Or is it like a Kalooza flying tower where you get a particle? No, no, it's, yeah, okay, great question. So the word tower is not meant to imply anything about, you know, what's the origin of the tower? What I just mean is that I have some spectrum of states, okay? So my dark sector, we're gonna be talking more about this in our lesson tomorrow. But just, I don't know, my dark sector certainly doesn't have to be comprised of just one dark matter particle. I could have a whole bunch of that and they each have their own mass and they each have their own interactions. In order for co annihilations to be what's controlling the relic abundance, what I need is that there are rapid, basically, that this pool of, I have some pool of particles that are in chemical equilibrium with each other. So basically what that means is that if some particle that I have is undergoing number changing processes, if it's in chemical equilibrium with the dark matter, then that's gonna, the dark matter density is gonna track, is gonna be reducing just by the fact that a different particle is reducing its number density, okay? So I don't have to have a particular, there's nothing over here cared about or was dependent on any mass splitting, any particular mass splitting, any particular structure within this theory just on the fact that I have rapid interactions between the particles. Okay, so it introduced a spectrum of single particles that get composite state. Yeah, yeah. And we'll be, yeah, and tomorrow we'll talk, this is true in general for what I call just generically dark sectors, which are just sectors of particles that could be governed by certain symmetries or not governed by certain symmetries, but nothing related to that was important over here. Hopefully I've spelled out to you what all we needed was that this tower is undergoing these rapid number changing processes, which tell me or give me a way to describe how the number density of dark matter is affected by interactions of other particles with which it's in chemical contact. Okay, and I just have one more question. In terms of the various particles, how different are they from each other? Like what would be things that he'll have to follow to really because it's like a lot of particles are rapidly exchanging and so on. It could create some phase transitions to not to like to. Yeah, so you see in terms of the only again, the only thing that I needed that like that I needed is the wrong way. The only thing that's necessary for this to be the process, this to be the type of process that's governing the relic abundance of dark matter is that I have a whole bunch of particles that are interacted with each other. These interactions are rapid. They're setting the chemical potential of each particle in the tower to be the same. And you see, I don't even care which process it is that's doing this. And what this means at the end of the day is that there's some effective cross-section which cares about the equilibrium just densities of the individual particles but also just cares about what you see over here. Let me, sorry, let me use a color. So it cares about the cross-section, the individual cross-sections over here. And so there are many different, depending on what your theory is that has a bunch of particles, you'll be, you'll find, you can find different things but it's all described by this one easy equation. And what's nice, what this shows you over here. So for instance, this happens in supersymmetry. Okay, supersymmetry is maybe like the poster child for co-annihilation. Also the poster child for wimps in general. But for instance, if you have, not just winos, but any of your neutralinos, they can, you can have co-annihilation between them. I think what's, the point that I really wanna stress over here, which is something that I feel when I learned about co-annihilation in the context of wimps, like when I was a graduate student, is that the number density, your dark matter abundance can be determined by processes that have nothing to do with the dark matter. It's not that dark matter itself will co-annihilate with another particle coming in. That could also happen. You could have dark matter with another particle going to something else, but you could even have two other particles in your system be the ones that are annihilating away. And as long as dark matter is in chemical contact with them, you're gonna be setting its abundance. Okay, so that's the point that I really wanna stress over here is that dark matter abundance through co-annihilations can be set by processes that have nothing to do with it. It doesn't even have to participate in those cross-sections. Okay. Great, Ian, thank you. Okay, so I'm just gonna actually write out that last point, and that'll be a good point for us to finish for today. So let's just write out that last point is that the sum that we've arrived at over here for this effective cross-section, it needed to be dominated by the whitest particle. And in particular, the abundance of the whitest particle could be determined by processes that have nothing to do with it, okay? In particular, the abundance could be set by interactions entirely of other particles without even involving the whitest, okay? And so what this really tells you about my abundance at the end of the day, my abundance can be independent of the interactions themselves of dark matter, okay? And this is something that happens and happens also in supersymmetry, but this is obviously much more general than that. And it's true for any types of these two-to-two processes and different mass scales. And this will of course have, this fact has important implications for phenomenology, right? So when we wanna think about what is the mass of dark matter and what type of interactions, if I have co-inilations, then in a way, the size of the interaction of dark matter, how strongly it's gonna interact with my detector in a direct detection experiment or anything like that is in a way somewhat detached from the cross-section that's setting its abundance, okay? So this will have implications for phenomenology. Okay, so this is all I wanted to, these are the example that I was hoping we'd walk through today for the two-to-two zoo. And then our plan for tomorrow will be to start talking about three-to-two processes. We'll talk about strongly interacting mass of particles or simps, maybe learn a little bit about cannibals, elders, and talk about dark sectors. So some combination of those. So that's it. Thank you very much, Yonit, for a very nice lecture. And I guess we can move now to the Q&A session. Any questions you wanna ask, please remind you to raise your hand. All right, so let's start with Manuel, Manuel, sorry. So thank you very much for the lecture. I have a question regarding the last mechanism. And we have this tower of,