 All right, if you are ready, I will introduce you. I'm going to start. OK, now. So we're happy to have Jonik again, who will deliver the second lecture on knowing their matter models. So please. OK, thanks so much. I hope everybody's been having a good day so far at the school and otherwise. So welcome to my second and final lecture for this school. And before we start with the material that I hope for us to work through together today, just a few sort of add-ons or sort of fill-ins from the discussion that we had yesterday. So first, as I mentioned, even though I spelled out an entire two-to-two zoo of processes, we're not going to go over the details of all of those. I chose a few of them and we walked through them yesterday together. But I did want to give you good references for reading about those. So if you want to read about co-scattering, semi-annihilations, or asymmetric dark matter, which also came up in the Q&A session yesterday. So I've given you here several different references to read. There's also a question that had come up yesterday in the Q&A session about a tool that we use very often, which is this red-shifting, where we're assuming that total entropy is conserved. And I just wanted to make the statement from yesterday more accurate. There's a shoot that doesn't always have to happen. But the reason that this is a good approximation of the universe is that you can show just using the first law of thermodynamics that, indeed, entropy is conserved when if either the chemical potentials are all small or there's just a small number of fields that have large chemical potentials that have a slowly varying number density. So just to be a little bit more focused on the statement that I made yesterday. And additionally, just to give you guys there are a few references. I'm sure you can find information online pretty easily. I just wanted to point to three references that I think might be very useful, in particular about things that we're not covering so much in this course. The first is one particular lecture by Ronnie Harnick from a school given in Mines a few years back. In the first lecture, he gives a lot of details about the evidence of how we know that dark matter exists. So that's certainly a topic that we didn't cover in our lecture series. If you want to hear a little bit more about what type of general constraints we have on dark matter, what type of constraints on dark matter masses, then you can take a look at a nice lecture by Josh Ruderman, who gave a series of lectures at the Israel Institute for Advanced Studies just last year, the winter school before COVID-19 took over the universe. And that's contained in, I think, in lecture four, actually. So I have a type over here. Whoops, this is lecture four of Josh's lectures. And there's also a really interesting set of notes from a cosmology course given by Daniel Bauman in Amsterdam, which if anybody's interested in reading in general about early universe cosmology, I highly recommend those notes. And after we finish our lectures, we're going to be posting on the school website these whiteboard notes that were the ones we've written together yesterday and the ones we're going to write together today. So you'll also have all of these references to these resources available to you as well. So with that, let's start our lesson for today. And as a reminder, we had finished yesterday talking about our two-to-two zoo. And what we're going to move on now are sort of slightly different types of interactions. And the one we're going to do right now are looking at three-to-two interactions. And these are going to be what I'll call SIMPS, strongly interacting massive particles. And we'll see exactly what this means. So until now, we looked at two-to-two interactions. And these two-to-two interactions were primarily interactions that took dark matter particles into other stuff. And let's imagine the following situation. So imagine that dark matter is the lightest state. In some, I'll call it, a nearly secluded dark sector. So imagine that I have a whole bunch of dark particles, and dark matter is going to be my lightest state. And the question I want to ask is what happens if what's important is not so much how dark matter speaks with other particles, but how it speaks with itself? And let me also give you a reference for this before we move forward. So this idea for the SIMPS is something that I put out a few years back together with Eric Kuflick, Tomer Wolanski, and Jay Wacker. So this is a PRL that you can find at the archive number 1402.5143. And let me add, we came up with this idea while we were hanging out on the beach. So good physics can always be done in happy, good situations. But getting back to what is the physics behind this. So as I said, we're going to think about what if what's important is how dark matter interacts with itself. So the first process you might think of doing is just doing two to two interactions of dark matter with itself. But that just gives us self interactions. We want self interactions to set the relic abundance. But if we just look at that first process we could think of, which is just chi, chi, going to chi, chi, this doesn't change the number density. It takes two dark matter particles into two dark matter particles. So this doesn't change the number density of dark matter. And the first process that does would be, for instance, three dark matter particles annihilating into two. So the first process that does change the number density is going to be, here we go, three dark matter particles colliding and two dark matter particles shooting up. So this is some three to two annihilation process of dark matter into a smaller number of dark matter particles. Just I know also yesterday in the Q&A session it came up, when is dark matter a fermion or a boson, et cetera. And so far I've been agnostic because I could tell every piece of the story, regardless of what its nature was. I do want to emphasize here that because I have a three to two interaction these are not fermions. So just note that dark matter over here is a scalar for these three to two interactions. So what we're going to do now is, yes, let's use this sort of our tool kit of estimating tools to try to understand what this means for dark matter if this process is what's responsible for our relic abundance. So let's use our estimating tools. And before we do, what we're going to do together is the back of the envelope estimate for when freeze out of this three to two process occurs. Just before that, let me write down the Boltzmann equation for you, even though we're not going to really solve it all. So it's dn dt plus 3 hn. That's our usual left-hand side. And then the right-hand side is the collision terms. And we always care about both the forward process and the back reaction. We take into account everything that's happening in the system. So I'm going to write it in the following way, minus some weird object. I'm going to call it sigma v squared of the three to two process. I'll explain everything in a moment. Times n chi cubed minus n chi squared and equilibrium of chi. So this isn't going to be very important for discussion right now. This is in case somebody wants to know what the actual Boltzmann equation looks like. But what we're going to do together is a workout. When does freeze out occur? Using the tools that we've developed yesterday. So freeze out occurs when the rate for the process that's changing the number density. So when the rate of this three to two process is of order, the expansion of the universe of order Hubble. So let's take a look at this left-hand side and right-hand side. So the rate for three to two has always we have these two factors. So first I have to ask myself, how many particles do I have to meet in order for the process to occur? And if I look at this three to two process, if I'm a dark matter particle over here, I have to meet two additional dark matter particles for the process to happen. And so my rate is going to be proportional to n chi squared. And this is, as we said, this is coming because if I am a dark matter particle, I need to meet another two. And the second thing that my rate is proportional to, so I have to wait until I meet other particles for the process to happen. And once I've met them, I care about the strength of my interaction. So this means that I'm proportional to the thermally averaged cross-section. And I'm going to write it in this funny way. Thermally averaged, I'm going to call it sigma v squared of the three to two process. And I write it here with this. Instead of sigma v sigma v squared, so this v squared, you can kind of think of it as if what I'm really proportional to is the flux of particles that are hitting me. And so what I really would think of is I have some flux that's, and flux would be n times v. And I have to meet two particles that'll be that squared. So this is just sort of a stand-in notation for the collision term inspired by what we expect to be happening. So that's this v squared factor. And of course, I have to now parametrize somehow what is this really weird object, this thermally averaged three to two cross-section. And I'm going to parametrize it in the following way. So this term over here, this whole thermally averaged cross-section, I'm going to parametrize it in the following way. So this is my sigma v squared of three to two. Let me call this now alpha effective, some coupling cubed over mass of dark matter to the fifth. And let me just point out that the dimensions of this, the dimensions that I'm putting in over here for this term, the dimensions of this sigma v squared three to two, you can either get from just looking at the fact that you look at how many dimensions are missing in order to get a rate, which would be dimension one, or using detailed balance. And we'll look through a slightly more complicated example in a few minutes together. But this is, in terms of dimensions, this is correct. It's this weird three to two object, and we're not used to thinking about three to two interactions. But this is how I can parametrize this cross-section. And now what we of course want to do is compare this, right, compare our rate for three to two that we have over here. We're going to want to compare it to Hubble, okay? And so now we're going to use our trick of shifting to T equality. So let's take the value of n chi, the dark matter number density at freeze out, just like we did yesterday by, from our trick where we redshift to T equality, okay? And so I remind you what we got yesterday was that the number density of dark matter at freeze out, I can write as T equality m dark matter squared over x freeze out cubed, where I'll just remind you that x freeze out is defined as the mass of chi over the temperature at freeze out. And so plugging this into our three to two rate, what we find is that, okay, so we have this number density squared, okay? So it's T equality squared, m chi to the fourth over x freeze out to the six, now times my thermally average three to two cross-section, which we called alpha effective cubed over m chi to the fifth. And we demand that this is a order T squared over m plank, which I can also write as m chi squared over x freeze out squared times m plank. And now we just, we shift things around and solve for m. And what we're gonna get is that m goes now like alpha effective times T equality squared m plank to the one over three power. And if we plug in numbers, this roughly comes out to be alpha effective times around 100 MeV. So this is the mass coupling relation that we find if a three to two process, this simp process is what's responsible for the dark matter abundance, okay? And again, what you see over here, similar to what we had, so this is a very different relationship than we had for the WIMP, it's different than what we had for the forbidden case, et cetera. What I wanna point out is that over here, if this coupling, okay? This effective coupling that we have over here is in order one coupling, a strong-scaled coupling, then the strong-scale emerges, okay? And this is why we call this the strongly interacting massive particle. Or in short, a simp, okay? And you see over here that unlike what we had for the case of the WIMP, so for the case of the WIMP, we had mass went like alpha times just an ordinary geometric mean between these two unrelated scales, T equality and m plank, what we now have is this generalized geometric mean, different powers of T equality and different powers of, over here, one over three, okay? And so importantly, what we find over here, just thinking about the mass scales is that this naturally gives us much lighter masses than the WIMP, okay? With very different interactions. Now, we've worked out the case here for three to two. What I want us now to do is walk together through, yeah, is there a question? Yep, from Badja. Please go ahead. Hi, so my question was that we estimate cross section as alpha effective divided to mass scale, but at the end it appears alpha effective is order one. So we have confinement scale in the problem, probably why it cannot appear in cross section? Yeah, so that's a great question. We're gonna also towards the end of the lecture today when we work through dark sectors, we'll see indeed, you know, we're led to sort of strongly couple of theories over here. What I did when I parameterized this cross section, and this is something that we did also yesterday, I'm parameterizing my ignorance, calling it alpha effective cubed over m dark matter to the fifth. In any particular model realization that you'll write down, you'll actually compute what is this three to two cross section in terms of the parameters of the theories, which might have decay constants, confinement scales, all sorts of other things and you'll translate what that means in terms of what this relationship between alpha effective and m means in terms of the parameters of your theory in order to match onto this, okay? So this is not meant to be, this is just a parameterization to use to understand what the mass scale that you're pointed to is. But absolutely there could be, once you have any time you have a model, you go ahead and you compute what is the relevant cross section in terms of the parameters of the theory and then you match it on. And we're gonna be doing something like that towards the end of the lecture today. Okay, thank you very much. Sure. All right, next question by Prisco. Yeah, I wanted to ask in terms of XF, do we still get the usual range of the two to two? So 28 to 30 for XF? Yeah, we do. So I can show you, yeah, let me actually, I can just show you that very briefly, okay? I mean, the reason is very simple. Yeah, do you remember that, what I showed you yesterday was that basically XF is just logarithmically sensitive to the parameters. So here, what has changed? I have now an N squared. Okay, so I have like instead of an E to the minus X times stuff, and then I solve and that's why I get the logarithm of E to the minus two X. So essentially within the log, there's like a factor of two, you know, roughly speaking, it's just an order one change. Because we're logarithmically sensitive, you're still getting dark matter that's freezing out at XFs that are a factor of 20 or 30. So you're always still freezing out at dark matter with temperatures that are around the factor of 20 beneath the mass. So that's why that's really the strength of, you know, if you understand what the YX freeze out is behaving as it is, it's just, once you're logarithmically sensitive to something, you're pretty much know that, noticapers, but it didn't a priori have to be that way, but that's indeed what happens here. And that's why that ballpark of a temperature for freeze out really holds across a broad range of mechanisms, okay? Any other questions? Not for the moment. Okay, great. So what I'd like us now to do together is more generally, instead of just looking at three to two, look at N to two, okay? So, you know, imagine that your dark sector, maybe dark matter is a fermion, and then you can't have three to two, or maybe you have some Z2 symmetry that's forbidding three to two, okay? So there are cases where maybe three to two isn't, before isn't, isn't actually a viable option. So then you might think of looking at N to two self-scatterings, okay? So either if we have Z2 symmetry, or if dark matter is a fermion, okay? So let's look at this sort of in general. Imagine that I have some dark matter, one, two through N, okay? All of these are, and they go to two other particles. I have some N to two process of self-interactions. So in this case, let me define the strength of this interaction of this N to two cross-section. I'll define it, I'll write it in the following way. So I'll call it sigma VN to the minus one. The V power is thinking again of myself as a dark matter particle, and then I need N minus one guys to come and interact with me. So that's just a notation really to fill in for the collision term. And I'm gonna parameterize this as some coupling to the Nth power times dark matter to, I'm gonna write down the mass dimensions and then we're gonna see together how to get them, okay? So it's two plus three N minus two, okay? Now let me show you how I know what the mass dimension is, okay? So let's work that out. So let's take a look at what happens. Let's use detailed balance for this process, okay? So detailed balance tells us that the forward process is equal to the backward process in equilibrium, okay? So that means that if I think of the forward process, that is N equilibrium, N times, times sigma VN minus one of this N to two process. And if I look at that compared to the back reaction where I only have two particles coming in, I have N equilibrium squared times an ordinary cross section sigma V, that's just a two to N cross section, okay? So not a weird object. And this has to equal zero in equilibrium. And so what this means is that the sigma VN minus one of the N to two process has to equal N equilibrium now to the two minus N power times sigma V of two to N. Now, let me write the N, sorry, a little less curly, okay? Now in terms of dimensions, mass dimensions, so a two to N, this guy over here, a two to N cross section, that's just an ordinary cross section, okay? So this has mass dimensions minus two, okay? This is my dimensions. And this I also can figure out what its dimensions are because each power of N is mass dimension three. So this gives me M to the three two minus N. Probably these are my mass dimensions. And so that tells me all together that the dimensions of the quantity that we've written over here are M to the minus, oops, let me write that down here, okay? So the mass dimensions of what we're interested in overall, this thing is just gonna be N to the minus two plus three N minus two, okay? So that's exactly the dimensions that we've written out before, okay? So it just shows you how one of couple ways you can figure out what the mass dimension for any weird object is, okay? So now let's take this weird object which hopefully you believe me now that this is a good parameterization for it. Yeah, Priska. Yeah, sorry, why do we waste the N equality raised to the power of N? Shouldn't it be N minus one and one for the two to N? Yeah, this is what I mean over here is that the right hand side of the Boltzmann equation, the right hand side for the Boltzmann equation has the full power of how many particles are meeting, okay? Think of it, the dN, dT, yeah, okay? So it's because it's the right-handed side, it's not compared to H times N, yeah, that's the extra factor, yeah, yeah, yeah, thanks. Okay, so now let's write out the rate for the N to two process. So the rate for the N to two process, I have to wait for N minus one, dark matter particles to come and interact with me if I'm a dark matter particle, and then I have to, I'm proportional to the size of my cross-section, which is this sigma vN minus one N to two, and of course I imply that this has to, I demand that this has to be equal to Hubble, and so what you end up getting over here, I'm going to do that each power of N chi at freeza carried one power of T equality, okay? And in the way that we used our trick to relate it to measured quantities. And so what you find is that M of dark matter goes like this coupling alpha times T equality to the N minus one power times M plank, all this to the one over Nth power, okay? So this is sort of the general result for an N to two process of self-interactions. And so for instance, if you know the next one after three to two, it would be four to two. So for instance, if we take N equals four, what this would give you is that Mx should be of order alpha times 100 KV, okay? And this has some issues with kinetic, with bounds coming from BBN, but there are ways to evade it, but I just wanted to show you sort of how how you take a general process and understand what would be a mass scale that would be affiliated with it, okay? And even though I didn't, we didn't really go through toy models for many of the two to two processes because two to two processes are so familiar. I know that N to two processes are less familiar than two to two. So let me just give you here a very easy toy model, a very simple toy model. And later we're gonna see this, we're gonna see something much more complicated but in a way much more natural through dark sectors. But here, let me just provide you with a toy model. So indeed we can write down these theories. So here it is, it's a three to two toy model that has a Z three symmetry, okay? So imagine that I just have a single scalar, okay? A single scalar with a Z three symmetry, it can write down interactions like in melagrangian that have three point interactions and I can also write down four point interactions like this for my field chi. And so this means if I think about just gluing together these vertices in order to get three to two interactions that I actually have about two types. So I can for instance take three, three point interactions. So here's the first one and here's the second one and here, here I go, okay? So here's my C, let me just, whoops, elongate that line over here, okay? So this is one way that I could do it and over here I have three vertices, okay? And this is the original inspiration for choosing to parameterize our three to two cross, sorry for scrolling, parameterizing our three to two cross section is some coupling cubed, okay? Because I can take three, three point interactions and construct this type of, so this is the inspiration for that alpha effective cube parameterization that we did for the three to two. In this theory, I can also actually just have using one three point and one four point. So I can also have, you know, let's say here's my four point, alpha going to two, okay? So this is another way that I could get this, okay? So you see it's actually very easy to get these types of interactions even though we're not so used to thinking of them, okay? So this first one use three, three point interactions and this one uses a three point and a four point. Okay, so hopefully I've convinced you that the story is very nice, that there's actually a caveat and that's that I've been cheating you. So let me tell you exactly how I've been cheating, okay? Okay, so I've been cheating you because I've implicitly assumed that I can write down one temperature that describes my entire system, okay? So implicitly assumed there was one temperature for the entire system. But if all I had was really just three to two interactions then that's a process that's pumping heat into the system. Okay, so here's the three to two pumps heat into the system. Now, if I think of these three particles that are coming in and these are, you know, some cold particles and the ones that are shooting out are each more energetic than the incoming ones, okay? And so this is in principle, if this is the only thing that's happening and there are bad ramifications for structure formation, it's really a problem. So to make this story that we've told consistent, there has to be a way to cool the dark sector, okay? We need to be able to cool. There has to be some way to shed this heat or set another way to dump the entropy, okay? And you can do this by interacting and sort of dumping your entropy into some other light state or directly into the standard model, okay? So it can be done or let me say that this can be to some other light state or directly to the standard model, okay? So basically we wanna be able to exchange heat with the standard model. And so the process that we wanna be active, we wanna have something where we're cooling. So here's Kai, for instance, scattering off of some standard model particle, okay? So this is a cooling process and I wanna be able to cool and I want this cooling process to be active while three to two freeze out is occurring. But at the same time, if this process is happening, then we have to worry, sorry, this is a dark matter particle over there. If this process is happening, then you might just think, well, let me take this diagram, flip it on its side and then I'm just back to the ordinary two to two annihilation process. I don't want that to govern the dark matter abundance because if it is, we're back to the case of the WIMP and I'm not teaching you anything new right now, okay? So this would be an annihilation process. And the big question is, can I cool without annihilating, okay? And a priori it seems like this would be problematic because the cross sections for these two processes would roughly be the same, okay? So if I wanna write down what's the rate for my kinetic process, then it's gonna be proportional to this two to two cross section and the rate for the annihilation process is also proportional to the two to two cross section. But as we've been learning, rates are also proportional to number densities, okay? So what's important is who I have to meet. It's who we need to meet because for the kinetic process, and again, I remind you, I always look at things from the perspective of I'm a dark matter particle. Who do I need to meet? So for the first case, I need to meet a standard model particle for the kinetic process. Whereas for the annihilation process, I have to meet, oops, sorry about that. For the annihilation process, I have to meet a dark matter particle. And so I'm proportional in one case to a standard model number density, and in another case to a dark matter number density, okay? And so if I write down the ratio of these rates, so here it is, the ratio between, let me write it actually reverse. Let me write down the annihilation rate over the kinetic rate, okay? So the cross section is roughly gonna cancel out and what I have is ratio of number density of dark matter to number density of standard model. And so the number density of dark matter, as we said at freeze out is non-relativistic, okay? So it's exponentially suppressed. And so if what we're doing is scattering off of light standard model species, okay? Alpha photons or electrons and retrinos particles that are that are abundant in an early universe, then this is going to be just exponentially suppressed. Okay, so I'm gonna get some exponent of minus M over T. You know, this is of order 10 to the minus eight or so. So it's much, much smaller than one. So I can easily cool without annihilating. So I'm just right here, smiley face, okay? And the way that I get this again is, and this was important, that we wanna scatter off of light abundant standard model species, okay? So retrinos, photons, electrons, can all do the job, okay? And by the way, if you remember when we first described the WIMP yesterday, I mentioned to you that also for the case of the WIMP, the annihilations shut off long before the kinetics, the elastic scattering shuts off. And that's exactly summing from this relationship, the fact that annihilations are proportional to number of densities of a non-relativistic particle compared to the elastic scattering, which cares about abundant particles, okay? So this is exactly the same story that happens for the WIMP. Yeah, Gothen, does have a question? So given the newer results we have in terms of other dark energy, would you also have heat pumping into dark energy and stuff standard model or could you have in traffic which dark matter would go into dark energy particles as well to make sure you don't have this extra heat? Well, nothing I've said here has to do with dark energy. You don't have to necessarily cool into standard model particles. It could be that you're cooling into some other dark, light dark states. There are some constraints on what you can and can't do there. And later today, maybe in the Q and A session, we're probably not gonna manage to do it in the lecture itself. We can also talk about what happens in a different variation of this where actually you cannibalize, so dark matter sort of does pump this heat into itself and then there's some interesting results from that. Okay, great. Okay, okay. So we've now essentially arrived at a set of conditions such that everything that we're discussing in the context of these three to two interactions of the SIMP mechanism is working. So here they are, our conditions, okay? So on the one hand, we want, if I look at the ratio between the kinetic rate, the kinetic, the elastic scattering compared to the three to two rate at the time of freeze out, we want this to be larger than one because we want the kinetic process to be active. But at the same time, we want the annihilation rate compared to the three to two rate at the time of freeze out to be suppressed, okay? And so if I just parameterize my SIMP standard model interaction in the following way, let me call this thermally average cross-section for the kinetic process, which is roughly the same one for the annihilation process. I'll just denote this as some epsilon squared over m chi squared, okay? And again, in any particular model, you can go ahead and write down what is the actual description in terms of the parameters of the theory, but this is just a way to write things in general. Then you kind of think of this, here's my dark matter, okay? And here's my standard model. And I have some coupling between them that's of order epsilon. Then these conditions tell us a range for these interactions with standard models such that everything works, okay? So what we have is a range of epsilon where things work. And this is what it is. So there's some minimal value of epsilon min. Roughly goes like alpha effective to the power half t equality over m plank to the third power. And we have some maximum value of epsilon, which is roughly alpha effective times t equality over m plank to this one over six power. Okay, so this is some epsilon min and epsilon max value. And if epsilon is in between them, then things work. So this roughly speaking, here's let's say, let me look here as a function of dark matter mass. And here's my epsilon, then I have these two curves epsilon min and epsilon max that define the range of validity. Okay, so if I'm in between these two, this is where I have simp dark matter. If I look beneath epsilon min, I do not have kinetic equilibrium and I'm in trouble. And if I'm along this curve, this is where the two to two annihilations take over. So this is sort of the Wimplake curve that we discussed yesterday. Okay, and to give you a sense of how broad this range is. So, let's say I start off here at masses of MEV and I go up to, I don't know, 100 MEV. Then we have values like 10 to the minus 11 all the way up here to 10 to the minus five. This is probably around 10 to the minus nine. So as you see, there's really a broad range, a few orders of magnitude for where this coupling is. And once we're in that regime, the size of that coupling doesn't matter and our relic abundance is dictated by the three to two annihilations that we described earlier. Okay, so as long as epsilon is in between this value, these two values between epsilon min and max and epsilon max, it's precise value doesn't matter. So this is, this is the Simp mechanism. And so, and this relates actually to another mechanism that, let me write it over here. So this relates to another mechanism which is called an elastically decoupling relic or in short, it's called an elder. And let me give you the reference over here. So this is an idea that was put forward by Cufflick, Perlstein and Ray Lee Warrior and also with Tsai in a PRL paper and the archive number is 1512.04545, okay? And let me just briefly sketch out what happens here and if we wanna go more in detail I'm happy to do that in the Q and A session. So let's just take a look at these two processes that we've come to understand, they're crucial in SIMPs. Okay, so in SIMPs we have these two processes. I had three to two self-interactions of Kais and I also had elastic scattering of Kais of light standard model species, okay? And in the Simp mechanism, what we said happens is that the three to two process decouples first and it determines the relic abundance and the elastic scattering is active during freeze out so it's gonna decouple later. So with decouples, seconds, okay? And it's active during freeze out. You can ask yourself what would happen if the order of these two processes one day decouple is reversed? And that's the case in which you arrive at these elastically decoupling relics or elders. So what happens here is that this one, now the couple seconds and this one, the elastic scattering is the one that actually decouples first, okay? And when you work out the math for what happens over here and again I'm happy to talk about this to describe this is related to something called cannibalization. So we can have cannibal dark matter in our universe. When you work through the implications for this, what you find is that the abundance of dark matter in our universe kind of surprisingly is exponentially sensitive to the size of the elastic scattering cross-section, okay? So it's roughly goes like e to the minus sigma v, this elastic scattering cross-section, okay? So dark matter number density can actually be determined through a process that completely preserves the number density of dark matter, okay? And the way that this, so as I said, I'm happy to give details, oops, details in Q&A session and this relates to something that are called cannibalization. So if anybody's interested to hear more, we can talk about that later today, okay? But sort of just to sketch how this happens, let's just write that out here. What happens is that when you work through what happens here is that your density of dark matter, the relic abundance of dark matter depends on the ratio of entropies at the time that these two sectors decoupled from each other. So it depends on the entropy ratio at the point where this process is decoupling. And then you can just, and then because the entropy is just getting exponentially suppressed, you end up with sort of an e to the minus temperature and the temperature is a polynomial in the elastic scattering cross-section. So let me just write this out here. So the dark matter density abundance, excuse me, depends on the entropy ratio at the time that it's decoupled. And the dark matter entropy is getting exponentially suppressed, okay? So you end up with something like e to the minus x and you can work out that x or t, their equivalent is some power law that depends on the elastic. Scattering cross-section, okay? So that's sort of the gist of it and we can work through the math together later today. And so just to sort of finish this line of thought, what actually happens is that if I think now just generally about, you know, I have these two knobs and I have some, these two different types of processes, three to two self interactions and something that's related to the couplings to the standard model. And if I just look at the relative size between them, then I actually, we actually find an interesting phase diagram. So let me just draw that out here. So, you know, let me think in general, let's say on the y-axis I have here, the three to two self interactions versus the dark matter standard model interactions. Okay, so these are these two to two scatterings that we're talking about. And what you find is that depending on the relative importance of the relative size of these couplings, you flow in your theory from a regime where dark matter is WIMP-like to a regime where it's SIMP-like to a regime where it's an elder. Okay, so in the WIMP regime, what I mean is that my dark matter abundance is being determined by this chi-chi to standard model, standard model. Okay, that's the case that 40 years ago was already discovered. When I'm in a SIMP regime, what's really important in dominating everything is when three chi's meet and annihilate into two chi's and in the elder regime, what's important is how dark matter scatters off of standard model particles. Okay, so this is the phase diagram of the theory. Okay, and this is something that that's also very generic in general. Anytime you write down a particular model realization as well, you're gonna have a whole bunch of processes. And depending on the relative importance of the processes, different regions of the parameter space might realize different mechanisms for dark matter and then have very different predictions. So I think this is a good spot to take a break for five minutes. Okay, perfect, let's take the break. Okay, thanks, everyone. Hi, so maybe we can resume. Sure. There is a question in the chat. Okay. Do you want me to read it for you? Ah, okay, do you have a particular model that matters? Okay, excellent question because perfect leeway into our next, our last lesson, okay? Our last part of the lesson today. So so far, everything that we've been discussing together are what I called mechanisms, okay? So these are mechanisms in the early universe that could set the relative abundance. And we've understood what they generally would imply about masses and couplings and types of interactions that are important. And what I want us to discuss for the rest of our time together now is different types of model realizations for these ideas. So actually let me change back my color, sorry about that. What we're gonna talk about now are what I'll call dark sectors. So as I said, just to spell this out, we've developed several mechanisms and we've understood the behavior, okay? Understood behavior. So far, we've talked about M and mass and coupling and what type of process it is. And what we're gonna move on now is to talking about models, which are theories that realize these mechanisms. And in terms of models, as I mentioned though, so yes, you know, in a way, supersymmetry is the poster child for WIMPs, possibly also for co-initializations, of course. And we've already seen one example in the context of SIMPs. We've seen the toy Z3 model for they gave us three to two interactions of SIMPs. So these are just types of examples of what I mean by models that will realize these theories. But what I really wanna show you by talking now about dark sectors is that all of these different ideas that we've been talking about these mechanisms are really generic in theory land, okay? So let's, I hope to give you a little bit of a sense of what I mean by that. So let's start talking about a dark matter, a dark sector zoo, okay? So if you think about our beautiful standard model of particle physics, which I'm sure you're all fairly familiar with, you know, here's our standard model, okay? And as we know, it's a whole zoo of particles, okay? Zoo of particles. And it's not just any zoo. We have, it's governed by a symmetry structure, right? So we have an SU3 color cross SU2 week cross U1 hypercharge. And this is what we see in our visible sector. And, you know, a question you could ask yourself is why couldn't the dark sector likewise be intricate, be complicated, have new particles, have a whole zoo as well? So why not in the dark sector too, okay? And perhaps even there are some new gauge symmetries that are governing the interactions of these whole new zoo of particles, okay? And so just taking inspiration from the standard model, so inspired by the standard model, you know, imagine that we had some SU3, let's say we had an SU3 instead of having our ordinary QCD, we had some dark version of it, okay? So maybe there's some dark SU3. And actually it doesn't even have to be so, so similar to the standard model. You can think of, in general, maybe there's some, you know, maybe what we have, you know, maybe we can think of some general SUN with some number of colors, or SON, gauge theory, or some SPNC gauge theory, okay? We could have symmetries of this type. So these types of theories are just, you know, sort of generally speaking, maybe what we have are some, the dark sector could have some strongly coupled gauge theories. You know, these are just QCD-like theories so I'll sort of call these dark QCD. I don't mean that it's an identical copy, I just mean these types of strongly coupled gauge theories. And you know, maybe just like the standard model has UN electromagnetism, maybe there's some dark version of UN electromagnetism. So maybe I have some UN dark, okay? And that would mean that I could have a dark photon and it could, in principle, kinetically mix with the standard model, which would give me some way for these two sectors to talk to each other, okay? So if I have something like this, I have a dark U1, this would mean that I could have now a kinetically mixed dark photon. Sometimes it'll be denoted by V, sometimes in the literature it's denoted by gamma D, it's just the same thing, okay? And you know, maybe the simplest version of the dark sector would be just to take this dark U1 and forget about anything related to QCD-like pieces of the theory. That's certainly a possibility. So it could be that what we have is just the dark U1. So basically imagine some dark version of quantum electrodynamics with some dark particles that are charged under it, okay? So in my Lagrangian, so my Lagrangian can have some mixing, let's say epsilon, some epsilon gauge field strength of the photon that could couple to the field strength, I'll call it F prime of my dark photon. And there could be some dark version of the fine structure constants or equivalently of E, okay? And all this does for us, I mean, what this does, it's a lot of stuff, but it gives us an ability for standard model stuff to talk through photon to now kinetically mix into this dark photon, which now talks to all sorts of dark particles, okay? So the kinetic mixing gives us some way for the dark sector and the standard model to speak with each other, okay? So kinetic mixing gives dark matter standard model a way to communicate, okay? And this is probably, this is a very, very simple dark sector, but things don't have to be quite as simple as this, right? So, and it could be more complicated, so it could be more complicated. So we could have also these QCD-like dark sectors, which is what I'm gonna focus on now, okay? And these theories are very rich, okay? So they're rich theories. And what's nice is that that means that it's a rich playground for many of these dark matter mechanisms that we've been talking about to many of these processes to occur in these theories. So rich theories, a rich playground for many dark matter mechanisms and processes to occur, okay? And as I said, this is true in general. When you have a particular theory, given this model or this theory, you often have multiple processes that can happen. And depending on the relative strength and size and the different cases, it could be a different mechanism that's controlling the abundance. And you always have to figure out in your theory, take into account everything and figure out which region and parameter space you're interested in, which mechanism is realized over there and what it implies for those parameters, okay? So we always get these sort of different phases of theories just like we saw for the WIMP and for the SIMP and for the elder, okay? And so in these QCD-like theories, let's look what happens. So in these QCD-like theories, we have lots of mesons, okay? So we have the equivalent dark versions of the pions, dark versions of the rows, dark versions of the kions, et cetera, et cetera, okay? So these are dark mesons. And in these theories, in these theories, the pions, which are the pseudo-Nambu-Goldstone bosons of the theory, they can play the role of dark matter. So let me give you a few examples of two-to-two processes that could happen over here. And then we're gonna delve in a little bit more detail to actually looking at the three-to-two, which again is sort of the type of process that's maybe a little less intuitive. So here are examples of two-to-two processes that we mentioned as mechanisms and let's see what type of processes we could have in these QCD-like theories. So here's a whole bunch of them. So the first, maybe what I could have are dark pions. So these are all dark particles. I can have dark pions that are annihilating into two dark rows. In these theories, the mass of the row particle is typically above the mass of the pion. And so this gives us forbidden channels, okay? So this is a process that if it's controlling the abundance would give us forbidden dark matter. We could also just have pions annihilating into these dark photons, okay? So this would be sort of ordinary two-to-two annihilations, which are just the wimp case or what we call the wimp. I could also have two pions annihilating through this hidden photon, mixing into the ordinary photon and going to standard model particles. So this again would be a type of contribution to two-to-two annihilations, which we related to wimp dark matter. Another type of process that we could have are pions that are scattering through exchange of a hidden photon, okay? So through exchange of this vector, scattering off of standard model particles. So this is the elastic scattering process, okay? So we could have, this is what would give us, for instance, kinetic equilibrium between the standard model and the pions, and so this can certainly give us elder dark matter, elastically decoupling relics. And we can also have another type of process that could happen here is what we call semi-annihilation, which is a process where two dark matter particles are calling the pions over here. My dark pions, two dark pions annihilate to one dark pion, and they can add to the mixed hidden photon, okay? So this is a semi-annihilation process that could happen. It happens through the gauged West of Mina wind term, which I'll be discussing in a moment, but in case somebody's interested, this is another type of process that could happen in this theory, okay? And so in general, given this whole host of processes that could happen, what you go ahead and do is in each case, you're gonna go ahead and compute from your Lagrangian. So let's write that down in each case. You compute from your Lagrangian. And your Lagrangian is gonna have some, your dark coupling and the mass of the pions and the decay constant of the pions and the mass of the vector, et cetera, et cetera. So you're gonna have particular parameters that describe your Lagrangian, your theory, and where you're gonna go ahead and compute from this Lagrangian, the cross-section that you need for the dark matter abundance, okay? So that means that, you know, any sigma v or sigma v squared or whatever it was for whatever process you're interested in, you're gonna write this now as a function of the parameters of the theory, okay? And what this now gives you is the ability to translate between the cross-sections that we developed previously before we had a model when we just used some general parameterization into what are the ramifications in terms of the couplings and masses, et cetera, in your particular theory, okay? So you use this to translate the cross-section we developed previously for a particular mechanism into this model, okay? To understand what types of parameters, what parameters are needed here, okay? So this is that translation between a mechanism level to a particular model. So these are examples for the two-to-two case, but we can also have three-to-two interactions in this theory. So let's look at these three-to-two interactions, which we said we needed for simps, okay? And in a way, you know, as I said, in a way, these three-to-two interactions seem more exotic, okay? Just because they're less familiar. But they actually occur even in the standard model itself. So they're really not that exotic, okay? We actually have them in the standard model, okay? So let me show you how that happens in the standard model. Okay, so let me show you how that happens in the standard model. So let me just remind you what happens in QCD, okay? So in QCD, QCD is an SU3 gauge theory, right? It's an SU3 gauge theory, and I have three light flavors, right? The up, down, and the strange. This theory has an SU3 left cross SU3 right, global symmetry. And then what happens is that the theory confines, and chiral-symmetry-breaking occurs, okay? So the theory confines, so chiral-symmetry-breaking occurs. And what happens is that this SU3 left cross SU3 right breaks down to a diagonal component of SU3 that's preserved, okay? Now, in the standard model itself in QCD, these global symmetries are only approximate because the masses for the quarks are not actually degenerate, but it's a fairly good description. And this theory, given that we've had this symmetry-breaking pattern, has eight pseudo-numbu-goldstone bosons. So we have eight pseudo-numbu-goldstone bosons. These are the cans and the pions and the eta. And this theory, QCD itself has five point interactions. So we do have interactions where two k-ons annihilate into three pions. This is a process that happens in the standard model itself. This happens in the standard model and QCD through topological terms. So this is something that happens through a topological term in our Lagrangian. That's called the West's-Omena-Witton term. Okay, and if you wanna read more about that, there are a few beautiful papers. The first is by West's-Omena and it dates back to 1971. And then there's a set of two papers by Witton, both of them in 1983, I believe. Really beautiful papers that explain the origin of this term. But the standard model itself has five point interactions. And it turns out that even if you were just take this term that exists in QCD itself and calculate the rate for the three to two annihilations, you'd find that it's just right to be a simp if the mass was of order 100 MeV. So let me just write that out. If you calculate the rate, you find that it's just right to be a simp if the mass of the participating particles is of order 100 MeV. Okay, and so inspired by this, we can just write down QCD-like theories in the following way. So inspired by this, let's look at QCD-like theories. Okay, so I'll just write this out. For instance, for the SU case, you can do it for any of these different gauge groups. So imagine that you have some SU gauge theory. Okay, and I have some general number NF flavors. And I'm gonna write them with a degenerate mass. Okay, this is gonna make sure that all of my symmetries, my global symmetries are not gonna be approximate anymore. They're actually gonna be able to be exact. And so that's gonna give me complete stabilization of my dark matter particles. This theory has an SUM left across SUM right global symmetry. And once chiral symmetry breaking occurs, this breaks down to a diagonal component. Again, just identical to what happened in QCD. So there's some diagonal component that's now an exact symmetry of the theory. And we have some, depending on the gauge group and on the number of flavors, we have some number, call it N-pi of Pseudo-Nambu-Golstern-Bozans. Okay, so I'll collectively call these guys the pions. And these pions can play the role of dark matter. Okay. These theories have five point interactions. In these theories, the West Amino wind term exists, the topological term is met, the topological condition, excuse me, is met. And here's what the Lagrangian looks like. Okay, so it'll be very explicit here. So it's two N, N is the number of colors, over 15 pi squared f pi to the fifth. f pi is the pi on decay constant. Let me just write that down. This is the pi on decay constant. So that's my coefficient. I have an epsilon tensor, epsilon mu rho sigma times trace of the following. Pi d mu pi d nu pi d rho pi d sigma pi. Okay, so this is the West Amino wind term. And you see over here it has a very particular structure. Okay, my coefficient, I've written now in terms of the parameter of the theory, which is the decay constant of the pion. And you see it has interactions between five pions, and they're contracted in a very particular way, anti-symmetrically with four derivatives, et cetera. Okay, so this is what I have over here, our five pions that are talking to each other, that are contracted in a particular way. Okay, and so given this Lagrangian, now I can go ahead and compute what is the size, what is the relationship between the parameters of a theory, in this case, the mass and the decay constant of the pions, such that we produce the relic abundance. So, and I've written this out here for the SU case, but just to emphasize that you can do this for any SU gauge theory, or SON gauge theory, or SPN gauge theory, you just have to have enough flavors. Over here, you have to have three, oops, for the SP, for the SU and SO cases, you need to have three flavors or more. For the SP, you have two flavors or more. But provided that you have enough flavors, the West Amino wind term exists, and we have a realization of our three pions, our three dark matter particles, annihilating into two, okay, where these are not just all pions, dark pions, okay? And very importantly, we've gotten this interaction completely generically, okay? There's nothing special about a particular thing, this is just completely generic and QCD-like dark sectors. So this is generic in QCD-like dark sectors, okay? And as I said, now that we have a Lagrangian, we go ahead and compute what is our thermally average three to two cross-section in terms of the parameters of the theory, and then we can match that onto our previous result to get the relationship that we need between the different couplings in our, between the different parameters in our theory. So let me just show you what that looks like, okay? So when you do this, what you find is that, let me write it down here. So you find that this thermally average three to two cross-section, you know, there's some numbers, there's five square root five over two pi to the fifth. Then there's n squared, the number of colors over x freeze out squared, m pi to the fifth over f pi to the 10th. And then there's some additional numerical factor, combinatorical factor, I'll call it some t squared over n pi cubed, okay? The details here are less important. What I want to show you, so this piece is some combinatorics, combinatorical factor that depends on the group, on the group and on an f. Now over here, I have some piece that again is depending on the number of colors, but what's really important is you see over here now, we have this mass to the fifth over the k constant to the 10th. And so, if I was translating this to, what is the, like what is the alpha effective that we used before, then we have to translate this and you get something like alpha effective, goes like some numbers, which I don't care about right now, m pi over f pi to the 10 over three powers. Okay, but there's a clear relationship that we can just draw a translation between our Lagrangian, the parameters in our theory to our previously obtained results. And what this means is that now if I want to think about, what is the relationship between my parameters m pi over f pi as a function of dark matter mass and some units, let's say GV, then there's some curve that corresponds to obtaining the right relic abundance, okay? So this would be the curve that's the solution to the Boltzmann equation or equivalently very similar, just the equivalent of our back of the envelope estimate. And maybe just worth saying over here, especially given a question that was asked earlier is that you see over here that of course, there's gonna be some cap over here on how large I can take my coupling to be, such that the theory is still perturbative and what I'm doing makes sense. And so really, once I've written down a particular realization, I can now translate everything really in terms of what are the parameters and what are the bounds on my parameters such that I really can understand what is the mass and how do these particles interact. And another thing that's really nice about being able to write down a particular model realization is that these model realizations are also very predictive, okay? So let me just write down units here. So what you typically find is that in these theories, if you look at the bounds on m pi over f pi, shouldn't be larger than two pi for sort of the story to make sense. So you're typically bounded to be roughly beneath a GV and let's say up to 0.1 GV, okay? So there's around two orders of magnitude of mass in which this sort of makes sense. Okay, but going back to talking about the predictivity. So, oh, sorry. And one more thing, super sorry about this. I just wanna give you guys another two references in case you wanna read more about playing around with playing around with pions for three to two interactions. So let me just, sorry about that, note this. So here are just two, a couple of references. So what we always find over here is that the mass is the ballpark of 100 MEB and you are in the sort of strongly coupled regime of the theory, okay? So that's sort of also where the name originally was inspired. And as I said, two references. So the first is where we wrote out these pion realizations and you can see all the details of how you play around with this type of Lagrangian. That's a paper that we put out. That was with Eric Kuffler, Kitoshi Murayama, Tomor Volanski, and Jay Wacker in a PRL. That was at 1411.3727. And then there's another paper where we implemented the kinetic mixing between the dark photon and the ordinary photon. So that's a paper with Kuffler and Murayama. That's 1512.07917. Okay, so just in case you want additional reading on this. And these theories, I mean, I've been focused on the pion story, but you can actually do something very similar also by looking at blue balls. So if you wanted to look at information about three to two blue balls, then that was something that was actually originally proposed in an ancient paper back in the 90s from Carlson Hall and Machakek in 1992. But there are also some papers by Sony and Zeng from 2016 and from Forestel et al in 2017. So there are many different ways in which you could think of realizing these three to two interactions, even though originally you would have thought that they're very exotic, they're actually not. Okay, so in our last 10 minutes together, I wanna tell you about how these theories are also not just a rich playground in which we can realize these mechanisms, but also give you a sense of the fact that they're also very predictive. So also predictive. Okay, and let me show you how this happens. So let's again stick to this example we've been working with so far, which are QCD-like theories times a dark U1. Okay, so remind you what this tells me is that here's my photon and I kinetically mix with some strength let me call it epsilon to some dark photon. And so if I look at sort of what's the size of this coupling compared epsilon compared to the mass of that dark photon. Okay, so let me fix, I'm gonna fix the gauge group and also fix the dark matter mass. Okay, they're not gonna play an important role here. And this very simple construction, different regimes in this parameter space realize different dark matter mechanisms. So let me just draw that here for you. So I have some dip going like this and I have some curve going like this. Okay, so just to show you this over here is, so the regime over here in between the two lines is where we have simp dark matter. This curve over here is where we have ordinary two to two annihilations, we have wimps. And this bottom curve is the thermalization curve. And so this is the curve along which elder dark matter lives, okay. And if we're below, we are non-thermalized with the standard model. And in fact, if we're in this regime over here, this is where semi annihilations contribute, okay. And sort of the border where this is dipping down is where the mass of the vector is of order twice the dark matter mass just because we hit a resonance in the ordinary annihilation. What I want you to take away from this picture is not so much my very not good drawing skills, but just the fact that different regions of this parameter space realize different mechanisms for dark matter, okay. So different regions of the parameter space. And this is parameter space of a very concrete theory. These give different mechanisms for dark matter. And moreover, we have existing constraints on this parameter space, as well as hopefully future projections for what experimentally we can hope to detect or exclude. So just to sort of show you what that looks like. So we have some constraints on the parameter space. And these come from a whole bunch of different machines. So it comes from high energy and low energy machines, low energy colliders. And we also have a whole bunch of future probes that we could hope to probe regions of this parameter space and be sensitive to these different dark matter mechanisms. And that's again from either high energy and low energy machines, such as LHC and Bell II and the ILC, et cetera, et cetera. Also from beam dumps and fixed target experiment. So things like Apex and ship. And also from direct detection, which hopefully you've started to learn about with Tongyuan primarily from the electron, the interesting prospects for electron recoils. Currently relevant are super CDMS and also the new Sensei experiment. And I'm sure others will be joining. And just to sort of give you a sense of what I mean by we have constraints and we have probes, I'm gonna draw you a very funny picture over here just to sort of give you a little bit of a sense. So let me now just overlay, take this picture that we drew before. Okay, so here was my curves in different regions or different dark matter mechanisms. Let me just roughly draw for you, here's blue. Let me show you sort of the size of existing constraints is something like, something like this, okay? Where this is of order 10 to the minus three or 10 to the minus two. Okay, so these are existing constraints of where we can't live. So you can roughly think that epsilon should be beneath 10 to the minus two to 10 to the minus three across a broad dark photon mass. And we have all sorts of future probes into this parameter space, I'll draw those in pink. And no, no, no, no, here's one from ship. Bell two is gonna be doing something crazy like this. And then there's LHC and all sorts of different ones that are sort of poking in different directions into deep into uncharted parameter space. So this is what I mean when I say that we have, that these ideas are very predictive because they definitely give us many ways in which we can hopefully be pushing and testing these theories and hopefully learning more and more. Okay, so there are many future probes. So that's what I wanted to share about dark sectors in general and maybe just to summarize my whole set of lectures. Hopefully the most important thing for me is hopefully I've given you some sense that dark matter is exciting. I think dark matter lectures that were given in schools even five years ago, certainly 10 years ago or beforehand would have sounded very different and contained very different, very different types of prospects. And I think that's one of the things that I find most exciting that there are so many new mechanisms and so many models to be thinking about. So we've talked about mechanisms and models and other aspects that pertain to dark matter that have been covered in other lecture series. And I hope I've given you a little bit of an idea of how to think about this type of stuff creatively. I think thinking outside of the standard box, think outside of standard box because maybe you guys will be the ones in the not even so far future to really write down the next amazing dark matter mechanism and I look forward to reading those papers on the archive and thank you so much for your attention. That's it for me. Thanks a lot, Jonit. So next we can transition to the Q and A. So we're gonna stop the recording. Sure, wait, I just want to add my crocodile. Okay. It's finished without a crocodile. Hold on guys. Okay, there we go. Perfect. So I stopped the recording. Okay. So don't be afraid to ask any kind of question because it's not going to be recorded. Next question is by Niklas. Hi, Jonit. I have two questions. Actually the first one I think could be a bit quick one. In regards to self-interacting dark matter when you get the essentially like non-trivial velocity dependence is that essentially what provides you with solving in regards to the bullcaster that you can't get something that's just instead of not let's say V to the power of one or something that can actually get some kind of power in terms of the velocity to regards to the bull cluster. Yeah, so great question. In general, I didn't, you know, because I wasn't I wasn't really focused on constraints over here but of course one of the types of constraints that we have on dark matter is the fact that it shouldn't interact too strongly with itself. And at the same time, you know that there are some indications, although, you know, with appropriately stated caveats that maybe if there's some self-interactions of dark matter that that could perhaps be relevant to solving some puzzles. I won't call them problems, I'll call them puzzles related to all sorts of observations in different systems. And certainly for the case of simps which are strongly self-interacting dark matter with three to two, you would expect that any theory that has strong three to two interactions is likely to have unsuppressed strong two to two interactions. Okay, so even just first at the mechanism level if I just take, you know, I take the general sort of I'll let's say parametrize, it makes sense to parametrize the two to two self-scattering of dark matter as the same, you know, as ballpark the same coupling squared over mass squared. And what this tells you is that with the couplings of order one we're ending up with is exactly a cross-section that's, you know, it's basically of order, a barn per hundred maybe. I mean, it's of order or strong scale like a strong interaction. So it's really in the ballpark to be relevant to address those puzzles. And once you're in the ballpark to address those puzzles then it's exactly, you know, are you in the ballpark and it's good or are you, is it a constraint and it's bad, you know, bad, quote unquote. And so the way to really be able to resolve this in general is once you write down a particular theory then in that theory you compute, okay I compute what's my three to two cross-section and this gives me a translation to the parameters of the theory and likewise I should be computing the two to two cross-section to understand what's going on. Now one thing that you might have noticed in this, like in this pion realization that I wrote so you see in the pion realization for the three to two cross-section I have over here, whoops, you see I have this X-freeze out squared sitting down there, okay. So things aren't just proportional to mass, you know, mass over F. There's some additional like temperature dependence over here, X-freeze out is still, it's roughly of order 20 but it's already showing you that something slightly different can happen compared to the original parameterization and certainly when you'll go ahead and compute the two to two cross-section there's some other dependence and it turns out in this theory that actually you keep being pushed to this sort of strongly coupled regime of the theory where you're exactly on that limit of like perfect I'm exactly expecting to get a cross-section that's good but like on the border of being not good and in fact I can tell you from a variety of different projects that we're working on now and with different ideas of how we can hope to evade this and using different aspects of of course of so velocity dependence, et cetera it seems to be sort of really a generic feature for any simp dark matter model that I've successfully written down so far that you really are always pointed to the strongly coupled regime of the theory where your two to two cross-sections are large. So it seems difficult, I don't think it's impossible but our original naming of the theory was almost like too good it keeps being really spot on that it's strongly coupled. All right, nice, interesting. My second part regards to dark gloomballs since if you were to let's say consider some kind of like dark purion mill theory you have some kind of phase transition then you would essentially assume that you have a phase transition probably could give you some gravitational waves. Meanwhile, you essentially would use the phase transition then go to lower temperature create your gloomballs to have them as essentially the stable dark matter. Would you essentially be constrained in terms of gravitational wave wise to use that as let's say an external probe? That's a good question. I haven't looked into that what I would recommend in terms of reading material I don't want to answer yes or no when I haven't worked that out myself but you can have a look. I mean I gave three references for these three to two interactions. One that's really easy to read is the Caltech one, the Sony and Jean Guen. I mean they're all easy to read but that's one in particular that yeah I think if you take a look over there you might be able to start sort of playing around with checking that out. I don't, the three to two gloomballs have not had as much there hasn't been as much like research activity on it I think it's really interesting and maybe there's something interesting that you can check. Maybe that hasn't been checked. I really can't say. All right, that was more out of curiosity but I know, thanks. Thanks. All right, next question is by Gautam. Yeah, I was just gonna ask you about like other possible ways in which you can dump out the heat. Like you mentioned like candlestick dark matter so if you just like, let's see what we'll learn how that works out. Yeah, sure, so is it a, so do you want me to, I can literally like, is it okay if I take like five and like I do a proper answer to cannibals? Is that okay? Sure, sure. Yeah, okay, so I'm gonna, and I'm gonna write out for this, okay? Cause I think. Love time. Are tricky enough without trying to explain them completely just with words, okay? So first I hope that it's clear. I just need some way to dump. Okay, if I don't want to cannibalize and I just need a way to shed the heat for the story that we told so far to be true. Okay, and I can do this by elastically scattering off of some light abundant species. I can do this by, I don't know, decays an inverse to get any process that can allow me to dump entropy is good and is viable. Okay, and it can be, it doesn't have to be into the standard model itself. It just has to be into something else, okay? But cannibalism is what happens when I don't have a light abundant species in which I can dump my entropy. Okay, so let me, let me, let's write this out. So here are cannibals. Okay, and this way, this idea was actually originally proposed in this paper, as I said, in the 90s by Paul Carlson and Machakek to paper by 1982. It was the first paper to introduce this three to two, this three to two process, but it exactly didn't have the ability to, to shed this heat, okay? So, so we have this three to two process and the question is what if can't shed the heat? Okay, what if we can't shed heat? That means that we don't have these light abundant species to dump entropy into, okay? Okay, so this is what happens when we have the three to two without that additional process, okay? Now, first thing that should be very clear is that there's absolutely no reason for the temperature of the dark sector now to be tracking that of to be identical to the temperature of the standard model, okay? And what's happening is that, you know, dark matter is becoming, when dark matter becomes non-rotivistic, basically this three to two process, but what it's doing is it's converting the mass of the incoming dark matter particle into kinetic energy of the other particles, okay? And then, and then, you know, because we have these scatterings of the particles amongst themselves, that's sort of redistributing, distributing that kinetic energy and that's what heats up the bath, okay? And that raises the temperature, so that's why we have what we call cannibals, okay? It means, I think this is literally a sentence that appears in this paper, that the kais eat themselves to stay warm, okay? And now let's just work through how much do they heat up, okay? So what's happening, so my kais, my dark matter particles that are going through these three to two interactions and self-scattering with themselves, they don't talk to any other particles, right? That's the setup over here. And so given that fact, the co-moving entropy density is conserved, okay? So the entropy density times a cubed is constant, okay? Now entropy density, I can just write, these are non-relativistic, so this is like the energy density over the temperature, okay? Energy density non-relativistic is just mass times number density over their temperature. So you see, I'm calling it a different, I'm calling it T with an index that it's the dark matter temperature because this is not the photon temperature, this is not a standard model temperature. And if it's non-relativistic, sorry, this is all, of course, squiggly, I'm not carrying any factors over here, but this is the mass over the temperature and mass temperature over two pi to the power of three halves, e to the minus m over T, okay, okay? And so given the fact that conservation of entropy is happening, that means that s is going like, s goes like inverse a cubed and a goes like scales inversely with the temperature of the photons and so what conservation of entropy is implying over here is that the temperature of chi is going like one over a logarithm of the scale factor, okay, which is the same as one over a logarithm of one over a temperature of the photons, okay? And so the chi temperature, the dark matter temperature, okay, the chi temperature is growing exponentially compared to the standard model bath, okay? So this is that cannibalization, so there's an exponential growth of the temperature of the dark matter that's cannibalizing compared to the standard model bath temperature, okay? And you can also estimate from this very easily what's the abundance of these particles, okay? Particles that are undergoing this process. And so let me just write, I'll write this out in case anybody wants to follow it. So I have, you know, let's look at m chi times n chi. Let's look at this abundance today, okay? So I can write this as m chi number density at freeze out over s at freeze out. I just redshift the two in the same way. And again, for a non relativistic particle, I can just write this as the temperature, the dark temperature at freeze out times s chi of freeze out over s of freeze out, okay? And now if I look at what's my relic abundance of dark matter, okay? It's the mass of dark matter times number density today. That's just rho divided by rho critical. So this looks like m chi n chi at freeze out over rho critical times s. Sorry, let me write this differently. And pull out a factor over here times s today over s at freeze out, okay? And I can now redshift this to freeze out. And I just continue this way. And what you find is that using, I'm using what we had over here, what you find is that I get to 0.6 m chi over ev, okay? So in ev units over x chi freeze out times x chi freeze out over sf, okay? And now once these two sectors, assuming at some point the two sectors were coupled, each sector is now decoupled and they're redshifting separately, but I can redshift back to the time at which they maybe were touching each other thermally, okay? So I can write this factor over here the same way. I can just redshift to the time of, I'll call it decoupling, okay? Time of decoupling. And now you ask yourself, okay, well what happens to this ratio of entropies at the time of decoupling? And what happens is that at the time of decoupling, so s chi at decoupling over s at decoupling, if these particles are non-relativistic, then this is just gonna have the exponential, you know, an exponential suppression in it, okay? So I'm gonna end up with m chi over e to the minus m chi over t dark, okay? If t dark is much smaller than m chi, t, sorry, t decoupling, okay? And now you wanna ask yourself, okay, so now what we found is that the relic abundance of dark matter is exponentially sensitive to e to the minus m over t, okay? And now let's take a look at what is this temperature at which we're decoupling, okay? So let's say that we had some, when do we decouple? Imagine that we had some elastic scattering where chi was scattering off of photons, but then that stopped at some points, okay? So whenever this stops, that's the point where these two sectors decoupled from each other in the early universe. And when does decoupling stop? It stops when the rate is of order Hubble. So the rate is n gamma times sigma v elastic scattering. And this has to be of order t decoupling squared over m plank. And the number density of photons is just the temperature cubed, okay? And so what you see over here is that this sigma v elastic has a scaling, okay? It's proportional that way. Basically it scales with mx over t decoupling. And so if I just plug this in to here, which was plugged in to here, what we're all together is what I said earlier that I now have a situation where my relic abundance is proportional to e to the minus this elastic scattering cross-section, okay? So this is how you get elastically decoupling relics. So basically in the early universe, you have some phase, okay? Here's my mass, and here's my yield. And so there's some phase in the early universe where I'm thermalized and then with the standard model. Now is decoupling time, I'm no longer thermalized and I'm a cannibal over here. I'm cannibalizing until at some point I freeze out. And I'm left with a constant relic abundance. Okay, so this is sort of the picture to keep in mind for that. So hopefully this helped a little in the context of discussion. Yeah, I just have one more question. It could just be me not understanding it correctly. So at some point these animals that are marked after are decoupled, right?