We will have a new lecture about inflation by Manich 살아j. Okay. Thanks. So... What we are going to do in the last two lectures. We are gonna do particle production during reheating, sorry heating and particle production during inflation, so that's the basic idea. We actually started reheating yesterday and we understood that essentially when we have a rate of gamma for any process, v srečenju h to je minus 1, je zelo gama dt, ki je gama over h. Zelo, v srečenju, zelo, kako jazem je vzelo, jazem se, da je proces vsečen, kako gama je vsečen h. Vsečen h je vsečen gama, zelo gama dekrize vsečen vsečen vsečen. in kaj drugi nespojoga je gamma danes na hoditelje, da je svepoje, ki se se vse pristaj začtenja. In, da ga so vse zazavilo, da boiledjamo to, za taj approachedi, se, da bi tukamo, da bi pa vsak padal si, tukaj tukaj s sre stabbedi , da je najbezdi, ki tukaj tukaj, ki je gamma vštot. Tukaj到jemo vse in da bomo delati te vsemčenje, tako kaj ga uživamo ni, of instantaneous decay of the inflaton. For example, this would be the moment where a gamma is equal to h. And if I, for example, do a row of radiation, the instantaneous inflaton decay would give me a plot like this. And the temperature would go as one over the scale factor, because then you have dominant thermal butt. So now let me, yesterday people asked me about an upper bound on the reading temperature, which is actually a very good question. So let me start to address this question. And let me just discuss an example of gravitino production. Now, what is amazing to me, at least, is that we can get very far without knowing very little about supergravity, right? You see, initially you read gravitino production, you say, OK, I have no idea what I'm gonna get, I don't know what supersymmetry, supergravity. But in fact, you don't need, to do this type of estimate, you just need to know the basic rules. And essentially, the basic rule is that each particle is on superpartner, and so the graviton has the partner, which is the gravitino. And then you ask yourself, OK, gravitino will be coupled like the graviton, gravitational, essentially. So in a sense, I can imagine a diagram like this. This will be some gluon. This will be some gluon. This is another gluon. And then here I would have gravitino and gluino. Gluino is the partner of the gluon. And you see, this is like a gravitational interaction of a particle with the graviton, it's just a supersymmetric version. Now, why do I take gluons? Well, here you could take photons, you could take anything from the thermal bath. But here we take gluons, because there is a strong interaction, so there's the strongest vertex we can imagine. Now, here, when you want to estimate a rate, sometimes you can do it very quickly, actually. This vertex is as a coupling constant, which is of order 1. We're not going to do the minutia of the numbers here, we just do estimates. And this vertex has a coupling constant, which is order 1 over n-plank, because this is a gravitational coupling. And then you understand that sigma, the cross-section, will be 1 over n-plank square, that's about it. You know cross-sections, you need to know the dimensions. Cross-sections are areas, so as dimension of 1 over mass square. In natural units, we have the unit of the mass, is equal to the unit of the energy, is equal to the unit of the temperature, is equal to 1 over the unit of time, is equal to 1 over the unit of length. Natural units means h bar is equal to c, is equal to bosman constant is equal to 1. They are lazy, for the lazy people, like theorists, they are beautiful, because you forget everything, you don't need to remember where c, h bar goes. And actually it allows you to do very simple estimates by just knowing this. Because cross-section is one of the area, but you see length is inverse of energy. So you need an energy, essentially to the minus 2. Cross-section is an area, so you need energy to the minus 2. And then you say rate. You see rate, let's look it for here. This is number of processes, dimension 0. This is dimension minus 1, this is dimension plus 1. So how do I get a gamma? Gamma will be sigma times something to make it the right dimension. What do I take? I take the temperature of the thermal butt. You see, I don't know anything about supergravity and I already know how to produce gravitinos. That's pretty remarkable, right? And now we can say, OK, when do I produce gravitinos? I produce gravitinos when gamma is equal h. Again, right? So let me just do it. Gamma is equal h. So t cubed divided by m-plank square is t-square over m-plank, right, the formula for h, forgetting all numerical factors. And then, of course, you see, actually, let me be more clear, because actually that's what I would really do. Let me compute gamma divided by h, and I will be clear in a moment why. So that's the calculation that I will do. Times m-plank divided by t-square, I get t divided by m-plank. So what you learn from this exercise is that this process would be equal to one, only when the temperature is equal to the plant mass, but plant mass is tend to the 18 gb, too much of a bigger temperature. So what we are saying, that this process here is essentially the number of gravitinos divided by the number of photons, because each, this is the probability for a process to occur. And from each process we are producing one gravitino. So essentially, this will tell me the number of gravitinos divided by the number of photons, and we understand that this process is dominated by the earliest possible times. So you see, this will be the moment where you produce most of the gravitinos, because then the temperature is going to decrease. And this gives me the abundance of gravitinos, so essentially n-3-2 is t-reating over m-plank times the number of photons. So this will tell me how many gravitinos that are around. Now, I can ask, of course you could worry about the inverse process, but since, right, this is such a rare thing, we disregard about the inverse process. Imagine this something of the order of 10 to the minus 10, or even smaller, you don't really care about the inverse process, because you have produced so few gravitinos that you really don't care. So this is a process that is always out of equilibrium. It takes place with the probability, the probability for this to take place is t-reating divided by m-plank. Yes? Do we compare to gamma? Because as I say, the question is, why do I compare with gamma? Because the number of processes is essentially gamma dt. And in one, sorry, let me imagine, dividing the time into many upper times. So each t is h to the minus 1. Ah, because this is what tells me the abundance, because after the production, the number of gravitino skazes one over volume, the number of photos skazes one over volume, so this ratio becomes constant. So that's the real quantity to look. So the question was, why do I look at the ratio? Because this ratio is the thing that eventually is constant after the production is done. This is called the abundance. OK? Good. One more question. Please. And what about phase transitions? I mean, in phase transitions, for example, or QCD phase transitions, are they also... OK, very... ...on the temperature drop, they have a higher temperature. Very good, very good question. So there is a question about, yesterday I said that the reading temperature needs to be greater than a few MEV. But then the question is, OK, but what about electro-whee phase transition, QCD phase transition that occur earlier? Well, the problem is that they don't need to occur. So the problem, of course, is how to generate barions. And then, OK, we will have to make something maybe difficult if you have only temperature to MEV. But in principle, it's not forbidden. I could be very smart and find a way to create barions, even if I don't go that high in temperature. So the question will be some maybe non-termal mechanism. And I don't know, right? I need to cook it up. Maybe some version of Aflek Dain. I can try imagine using it. But this is not mandatory, right? So if I really want to say, you know, what did the doctor order me? The doctor ordered me greater than two MEV. Then, OK, if I want to live better, maybe it's easier if I also eat wealthy. But if I just say, what is the minimum requiring from the doctor is two MEV. Now, let me ask what is about the upper bound. That's what we are really trying to go there. So this is a cloud. Why do I care about gravitino? I could have made millions of examples. Why exactly gravitino production? Because gravitino are very dangerous beasts. Because once they are produced, they live long. And maybe they live forever. And then there is the risk that gravitino overclose the universe. But let me discuss the situation where the gravitino is unstable. So the gravitino will decay, producing one other super particle. But the decay time of the gravitino is very long because it's gravitational decay. And so this can destroy the elements that you form during nucleosynthesis. Marko, sorry, I read a question from the chat. Does this ratio stay constant or it decreases with temperature? No, this ratio is constant. It says this, you need to compute it every apple time. You add it up. And then you see that this is dominated by the production here. So essentially, what you produce here, for example, is negligible with respect to what you produce here. Look here, I'm doing estimates. I'm not doing the full, exact, integral. I'm just telling you things in a quicker way. So once the production seizes, in a sense, because whenever the temperature drops to a factor of 10, the process becomes 10 times less probable. So this will become eventually completely negligible. And so this ratio essentially, at all effects, becomes constant. The ratio is constant because they both get diluted by the expansion of the universe. And gamma, I mean the amount of light particles in the plasma. No, no, no, I'm keeping details of factors of order of a few because then there would be a g star to the one-third proportionality. You see, I'm just giving orders of magnitude here. But you're totally right. Maybe I should define it divided by the entropy rather than by the number of photons. You're asking me more sophisticated things that they want to do here. I want to give you an order of magnitude estimates on t-eating. And that's all I really need for this. But you're right, I could be more sophisticated, but I'm not. I'm a simple-minded guy. So let me say now, I want to see the decay of the gravitinos. Again, I don't know anything about supergravity, but I know that the graviton couples to a particle like this. So I take the supersymmetric version of this. So here there will be the gravitino. Here there will be a particle. And here it will be the partner. Again, this coupling constant is 1 over m-plank. So now what is the decay rate? The decay rate will be sigma 1 over m-plank square. And then I need to put something that gives me dimension plus 1. So now here I will put m3f to the cube. And that's essentially an estimate on the decay rate of the gravitino. So when does the gravitino decay? It decays when this is t-square over m-plank. And so now I can have a relation between the gravitino mass and the temperature of the universe when the gravitino decays, which will be m3f is equal m-plank t-square to the power 1 third. I want now to see, for example, what happens under which conditions gravitinos decay before big bang-nucleosynthesis. So let me say that I want t, so gravitino decays before bbn. So here I will say m3f needs to be greater than essentially m-plank mev squared to the 1 third. So m-plank is 10 to the 18 gv. Mev is 10 to the minus 3, so 10 to the minus 6 gv squared. And this gives me 10, 1 third. And this gives me essentially 12 divided by 3, 4. 10 to the 4 gv, about 10 tv. So if the gravitino has a mass greater than about 10 tv, it decays before bbn, no problem. I can live my life happy, because the problem is that the gravitino destroys the elements that you form at bbn when they decay. But if they decay before bbn, they won't destroy anything, right? Yes? I'm assuming that the gravitino is not the lightest superparticle, so the gravitino will decay to another particle, which is lighter than the gravitino itself. Otherwise, I could break our parity, but this would, of course, give us a pressure on the rate. OK. Marko, in the chat somebody is asking, not sure I understand why decay occurs when gamma is equal to h of t. Well, so let me try to repeat the argument. Really, when you want to see how many processes take place, that's the definition of rate of a process, you do the calculation gamma dt. This is the number of processes, just how you would end up defining a rate. Now, if you look at an upball time, we just imagine slicing the time in upball, because upball is going down as 1 over t, then this in an upball time will be gamma h to the minus 1. So now I ask myself, when is this number equal to 1, because I'm looking when the decay takes place. This number is equal to 1, when gamma is equal to h. So essentially, when we do these estimates, we always compare gamma of a process with h. And when the two things are equal, then it means this is the moment in which the process occurs. Of course, I could do all the integral, in principle, I should do these, maybe sometimes gamma depends on temperature, I need to be careful about that, but in doing this estimate, you will see that what I'm doing here for this particular example, I can just do gamma divided by h. As I say, it's always very hard, because we all prefer to have exact calculation in front of us, but here I don't want to do exact, I just want to do estimates, and I'm doing things which are small, tiny mistakes, but at the end the results are the right orders of magnitude. Yeah, okay, that's a very good question, which I will not enter too much into details, but essentially, this is something, so this will be a super particle, and this will be a particle, for example, of the standard model. So this could be a photon, or this could be even a gluon, a quark, and this can generate either electromagnetic shower or hadronic shower. This thing can have the energy, because this could be, for example, 100 gV. This thing could have the energy to destroy the nuclei that you have formed at bbn. And so now we will see that all the calculation of bbn will be disrupted by this, and since we don't have an observational evidence for this, that's okay. We need to say that this process needs to be suppressed. Is it clear? So now, what is the point? Now you need to do very... So if you have this, no problem. Done deal, let's move on. But if instead you don't have this condition, then you need to do detailed calculations on how many gravitinos you have and how many nuclei you destroy. And these I cannot do quick estimate. These you need to do detailed calculation. But you see that you will always end up with a limit on how many gravitinos you have, right? Because essentially you will have a problem if you have a thousand gravitinos, or a million gravitinos, or a billion gravitinos. I don't know, but it's clear that the limit you get is proportional to the number of gravitinos. So the limit you have will become an upper limit on this quantity here that controls the number of gravitinos. So now let's see a figure. And here I will not be able to discuss the details of the figure, I just want to give you the flavor. So this is from a paper by these Japanese authors. And you can see that essentially here is a gravitino mass, here is a limit on reading temperature. These are four plus, because these are different versions of choice of supersymmetric particle masses. So let me not enter in detail, I'll just move one of them, but it's just to show you that it's very model dependent. But the concepts are not model dependent, so these lines here are the limit exclusion from BBN. So you see if the mass of the gravitino is of the order of 10 TV, then the limits disappear, for the reason I was just telling you before. If instead the mass of the gravitino is lighter, you need to include the limits. For example, here you don't want to destroy too much deuterium, for example. And notice that the limit is an upper limit on the reading temperature. And for example, notice that the limit could be very strong if you have in this limit here, for example. Here it could be as small as a T reading smaller than 10 to the 5GV, which of course is a very, very strong limit. So is it clear? So these limit, of course, are again not mandatory from the doctor, because maybe supergravity is not realized in nature, or maybe the gravitinos are very heavy and they decay early, or maybe the gravitino is the lightest particle, and then, of course, you need to consider different type of limits. But in a sense, this gives you a flavor of what traditionally people have in mind when they put limits on reading temperature. Okay? I'll do a stop sharing. Okay. Okay, so. Now I want to make, I want to drop one of the assumptions I've been doing. And I ask myself the question, is reading really instantaneous? Remember the plot that we were doing, right? This was time. This was, for example, a row of radiation. Here we say the inflaton decays. And so you have something like this. And now I want to question this picture here. Is this really what takes place? Now, why do I want to even question this? Because let me imagine that I want to remember that H is proportional to rot to the half divided by M-plank. And remember that H goes as 1 over T. So if I want to compute T-reeating, so let's say that this is the time that here I call T-reeating. And I want to divide this by T-end. Remember, this is the end of inflation. Now I can do estimate by looking at this scaling. This will be raw end of inflation divided by raw re-eating to the half. This will say, let's say that we disregard the change of energy during inflation in this estimate. Remember, for a thermal butt, this will be proportional to the re-eating temperature squared. And then the bottom, the top, with something that we discussed in one of the previous lectures, it will actually depend on the amount of gravitational weight that we observe, but that was essentially the result. Now let's take T-reating as an example, 10 to the 5GV. This was one of the limits that we saw from the previous figure. Then you put in this ratio here. And you'll find that T-reating is about 10 to the 23 R to the half T-end. I don't know what R is. R, remember, is the tensor to scalar ratio, that we don't know how much it is. But notice that the time of re-eating could be even a factor of 10 to the 20 greater than the time of end of inflation. Maybe you could say that inflation completes in 10 to the minus 30 seconds. Re-eating will complete in 10 to the minus 10 seconds, 20 orders of magnitude more. So you see that there is a huge separation of time between the end of inflation and re-eating. Re-eating could be happening on a timescale, which is 10 to the 20 times greater. Given this observation, we need to ask ourselves, can I really reasonably say that re-eating is instantaneous, because look how much time it takes to decay. And then, in particular, I can also think just based on very simple knowledge of what a decay does. If here I have the number of particles, I know that they exponentially decay. If you have a decay, n is equal to n0 e to the minus gamma t. Let's forget about the expansion of the unit. I just make this a simple discussion here. And if I ask myself, when do I produce the maximum amount of particles? If I do dn in dt, this is equal to n0 gamma e to the minus gamma t. This is the moment in which you produce a lot of particles, essentially. So it's not true that you produce all the particles at the end. You produce a lot of particles at the very beginning. So it's not true that here the temperature is zero. Certainly this needs to be modified. Am I clear? Let's discuss how this needs to be modified, and let's discuss whether this is a big deal or a small deal. At the end of the day, we will find that this is a small deal, but it's really non-trivial. No, it's not a good motivator to tell you this. But I want to be honest, but we will see that there is really a lot of non-trivial interplay in answering to this question. There is a question from Zoom. Since gravitinos themselves are massive, we will have a short period of matter domination epoch after eating and before bbn, right? How could that affect the usual hot big bang era? OK, this could be, again, it's model dependent. It depends on how many gravitinos you produce, but the amount of gravitinos I was considering was always very small, so they don't come to dominate. But you may imagine a different situation where you could produce many more of them, but the limits on reading temperature I was telling you here make sure that the gravitinos do not come to dominate. Now, I'm still making one assumption, in which you can find a discussion in the literature. I will post you the papers in the post-election notes about thermalization of the decay products, because this is a different issue, right? I'm telling you decay is not instantaneous, so this picture is wrong, but I will still assume that the decay products thermalize instantaneously. The inflaton keeps decaying in a continuous way, and the daughters keep thermalizing quickly. This is not trivial. This has been discussed in a few papers, and I will give you the literature about this. But for simplicity in this lecture, I'm going to the conventional idea that the decay is low, and so the thermalization is much faster on the time scale of the decay, which is a reasonable thing, but needs to be studied. Now, let me look at the equations that we need to solve. So, let me first write the equations and then I will discuss them. Rho phi plus 3h plus gamma rho phi is equal to 0. Rho dot gamma plus 4h rho gamma is equal to gamma rho phi, and then rho phi plus rho gamma is equal to 3m-plank-square h-square. So, the first equation tells me that the inflaton deplets because of the expansion of the universe and because produces radiation. The second equation tells me that rho gamma, the energy of radiation, gets produced by the decay of the inflaton, and then it gets diluted by the expansion of the universe. This equation instead is the usual equation for a borate in a two-fluid system, inflaton and radiation. Gamma, I don't mean just photons, so, they mean all the particles in the thermal bath. Notice, for example, that if I have gamma is equal to 0, then the first equation will tell me rho phi proportional to a to the minus 3. And from the second equation, I will get h is equal to over 3t. This is just the usual assumption that we have done so far. If the inflaton doesn't decay, the inflaton is oscillating and is behaving like matter. This is exactly what we saw yesterday. But now there is the fact that the inflaton is decaying. Now, we can analytically solve these equations by doing some approximations. The approximation we are going to make, for example, is gamma is much smaller than h, because h is the one at the very start and the decay is very slow, as we saw. And rho gamma is much smaller than rho phi. So, the inflaton, we are looking at this situation here, where a lot of inflaton quanta are decaying, but the inflaton is still dominant. There is still a lot of inflaton left. So, the inflaton eventually will decay. So, re-eating will take place here. Here, re-eating completes, because here rho gamma will become greater than rho phi when almost all of the inflaton is decayed. Here we are in a situation where the inflaton starts to decay and the inflaton produces a lot of gamma, but there is still a lot of inflaton around. So, that's what we are trying to get out of this equation. And we can do some approximation, as I say. For example, we can neglect this term here. We can neglect this term here. So, what we say, we say that we have essentially inflaton dominating, behaving like matter. And now, since we know this, this is our consistent with the approximation that we... So, if you make this approximation, you get this equation here. And then you can put this equation here and you can just solve it, because this will become a differential equation in terms of something that you know very well. So, this is a couple system. The approximations are, we are very early. So, gamma is much smaller than h. OK, then I see the inflaton be as matter. Then, there is much less energy in gamma than in the inflaton. So, h is 2 over 3t. So, you see, I know this now. And now, I can put them in this equation and I can solve this equation. Is it clear? So, let's see the result. I'm not gonna solve this. This is our homework, actually. I will ask you to solve this differential equation. But let me just solve it. Let me write down the solution. You see, this becomes a differential equation where we know everything, essentially. We know h is 2 over 3t, and we know rho phi. So, let me just write the equation for simplicity. One second. d rho gamma in dt plus 4 2 over 3t rho gamma plus gamma, then rho phi is just a constant divided by volume cube. So, 1 over a cube is equal 1 over t squared. So, this will be t squared. Is equal gamma over t squared. You see, this is a simple differential equation that we can just solve, OK? No, no magic here. The result is rho gamma is equal 6 fifth m plank square h and gamma a to the 5 halfs minus 1 divided by a to the fourth, where a is equal 1 at the end of inflation. And this is also the upper rate at the end of inflation. You see that this is very different from what we had assumed before. Notice that when scale factor is equal 1 at the end of inflation, rho gamma is equal 0. That's correct. And now, this equation, it takes a second to do it. Just imagine doing the derivative, as you find the maxima of this function here. You find that there is a maximum at a nearly equal 1.5. So, let me just do the plot. You just can do this very quickly. So, essentially, you would have something like that. Here you have the scale factor. Here you start from a is equal 1. This is the end of inflation. This thing goes up and then it goes down like this. And it goes down as a to the minus 3 half. Yes. Why do I neglect? No, I don't neglect. This thing here is what here becomes this. No, no, no, look. OK, look, let me go slowly, because this goes maybe too quick. You agree with the three equations? Where are you? OK, do you agree with the three equations? So, here I neglect gamma with respect to h, because gamma is very small compared to above rate, because otherwise all the input would decay immediately. So, this is our regime. From here you immediately get rho phi is equal 1 over a cubed. Period. Then here you neglect this rho gamma. From here you get h is equal 2 over 3t. So, here I get essentially rho phi is equal c over a cubed, which is equal c divided by t square, another constant. And this is exactly what I'm writing here. So, once I do this approximation, this equation I solve it exactly under that approximation. You see that I don't neglect the right hand side, because this is what gives the production. OK? So, let's appreciate this result. There is a sudden production, and this is very, very quick, because this is a is equal 1.5. So, this is immediately right after inflation. I reach a peak immediately after inflation. And then the rho gamma decreases as a to the minus 3a. After the production, after inflation has decayed, we know that rho gamma goes as a to the minus 4, right? Once the inflaton has completely decayed, then I don't have the production anymore, rho gamma goes as 1 over a to the 4. Notice that here instead, this is decreasing much more slowly, because there is the combination of two effects. The universe is expanding, so rho gamma is decreasing, but the inflaton keeps decaying, so it keeps giving quanta of gamma. So, if the inflaton stops, at some point the heating stops, and this will go down, so this is inflaton totally decayed, and then there will be a change of slope a to the minus 4, which is much, much faster decay. Is it clear? Now, there is this phase here, right? If you instead did instantaneous inflaton decay, that's what you would assume for rho gamma. So, this is instantaneous phi decay approximation. Is this figure clear? So, what we did before, remember what we did before, before we said the inflaton decays all of a sudden when gamma is equal h, and then the radiation will dilute. So, that's what we were doing before, right? Remember the figure that we were doing before? The inflaton decays all of a sudden and then it dilutes radiation. But now we see that the things are very different. The inflaton starts decaying immediately, so there is an immediate growth of rho gamma, then there will be this intermediate period, and then rho gamma will go down as a to the minus 4. Notice that this is super conventional with imperturbative inflaton decay. I'm not inventing anything, I'm just dropping the assumption of instantaneous inflaton decay. So, this is the correct calculation to be done in the case in which the inflaton decays perturbatively. Any questions? It's interesting to compare, right? How big is this height here versus this height here? Let's try to compare. This was the mistake that we did up to 30 minutes ago, right? Up to 30 minutes ago, we were imagining this is how it goes, so we would say this is the maximum possible energy that radiation had. Now, instead, we see that this is the maximum possible energy that radiation had. So, let's discuss what is the ratio between this number and this number to see whether we were doing a big mistake or a small mistake. Is it clear? Let's do this. You see, we have already this. We know that A1.5 maximizes this. So, out of there, we get... Please. ...decay only in radiation. Is all the possible that the... ...decay in some non-relativist matter? Oh, this will change the picture, but what I'm assuming is that the inflaton decays in things which are relativistic and they thermalize. Now, imagine the inflaton has a huge energy. So, even if you produce something which would be 100 gV, this would be like massless from this point of view. You need to produce something that is super heavy to see the effect of its mass. For conventional particles, even if they are heavy for our standards, maybe they are bad. Oh, okay, now it's on again. Okay, good. Is it clear what I've done so far? Is it clear that we are trying to compare this height against this height to see what the mistake was? Okay, so let's just say A of order 1.5 in this calculation here and we find that this rho gamma max is 0.4 m-plank square h and times gamma. So, this is this value here. Let me now instead go back to the assumption of instantaneous inflaton decay. So, back instantaneous inflaton decay. And so, we go back to saying that the decay takes place when gamma is equal h. This means that rho gamma instantaneous is equal 3 m-plank square h square is equal 3 m-plank square gamma square. So, this quantity here instead is 3 gamma square m-plank squared. So, this is the mistake that we did 30 minutes ago and this is the result, the correct result that we now we know is true. Let's compute the ratio between these two numbers. So, I'm really computing the ratio between this height and this height here. Rho gamma max divided by rho gamma instantaneous is 0.4 m-plank square h and gamma divided by 3 gamma square m-plank squared. 0.4 divided by 3 is 0.1, as we all know, 0.1 h and divided by gamma. And we see, so the ratio, again, 0.1 h and divided by gamma and this could be much, much greater than 1 by many orders of magnitude. Because remember, h is the rate at the beginning of inflation but remember that we were considering a situation where gamma was, the decay was taking place on a timescale which was 20 orders of magnitude greater. So, you could imagine that this could be some 10, 20 orders of magnitude. So, here the board doesn't give justice to this. We would need to have a board, which is enormous to get a feeling, but you understand that this could be many orders of magnitude greater than this. Is the answer clear? Remember, I'm not doing anything fancy. Standard perturbative inflatum decay. I prove to you that the assumption that most books do makes a mistake of some 10, 20 orders of magnitude on the maximum amount of energy in the radiation. This is the assumption that you make and this is instead what you should have done. Now, given this, the question is, why do we always do instantaneous inflatum decay? Is it all wrong? Because now we need to understand what is the effect of this on some real thing. For example, on the gravity in abundance. Because before, we computed the abundance under this assumption. And under this assumption, here, we found N3f divided by Ngam is equal to t, rejting divided by Nplank. But now, this was done following the wrong thermal history. If I take the right thermal history, what do I get as a ratio? Maybe I get something which is enormously greater, because this could be many orders of magnitude greater than this. And the answer, in fact, is no. You essentially get the same result. Nearly the same result. Mako, I interrupt you with a question from Zuma. The question is, which is referring, I think, to these three equations, which is the underlying theory action in parentheses, from which these differential equations are derived? Yes, very good question. In a sense, what we are really doing here, we are just doing essentially a conservation equation on the energy momentum tensor of these objects here. So what we are really applying, we are applying d mu t mu nu, in the case in which gamma is equal to zero, this will be equal to zero. And then you will say that separately, radiation goes down as 1 over a to the 4, and matter goes down as 1 over a to the cube. If you instead consider the interaction between the two fluids, that's what is account by this factor of gamma. So you see that essentially, the decay of the inflaton grows the energy of gamma, and the decay of the inflaton decreases the energy of gamma. So this will be interaction of these two fluids, and this instead is the Freeman equation, the zero-zero Einstein equation. Other questions? OK. Maybe I can ask you the question, because the answer is not really trivial. I'm telling you, even if this is many orders of magnitude greater than these, when you compute the amount of gravitinos from the wrong history, from the approximated history in the right history, you essentially get the same result. Why do you think this can be the case? It's not obvious, right? This will be one of the homeworks that I will guide you to solve. But anyone wants to take a guess? Yes, we should look at the ratio between what and what. Very good, very good, very good, very good. You see, this is the quantity that we really are after. So it's true that here you produce a lot of gravitinos, because remember that the production of gravitinos is proportional to the temperature. If this is a lot of many orders of magnitude greater than these, you will produce a huge amount of gravitinos, really a huge amount of gravitinos. But the inflaton has yet to decay. And so the inflaton is still in its bag, a lot of will be photons. And so the inflaton, during this period, this is 8 to the minus 3, instead of 8 to the minus 4, because the inflaton keeps producing a lot of photons. And so eventually, as she said, this thing needs to be computed at the end of re-eating, not here, because what you produce here will be diluted by the following production of photons. And you know, when we studied the BBN limit, in this time scale, BBN is here. So we don't really care how many gravitinos we have at this moment. We care how many gravitinos we have at this moment. And the amount of gravitinos that we have at this moment is related to the amount of gravitinos that we have here, because we need to study N3 half divided by N gamma at the end of re-eating, because this will be N3 half divided by N gamma at the time of BBN. Now, I will show you a plot now to simplify this. It doesn't mean that this is completely never useful. In fact, among the first papers that studied this, they discussed the production of super heavy dark matter. I don't know if you remember the name Wimpzillas. This is a rocky cove that made this fantastic name, super heavy dark matter. Imagine that you have a particle, this is raw, but imagine that you have a particle, let's say that this would be raw to the 1,4, the temperature, that has a mass here. If you have a super heavy particle that has a mass like here, you will never be able to produce, if you do the instantaneous decay approximation, but if you do the correct thermal history, you will produce this super heavy particle. In a sense, if you have a particle that has a mass between T max and T re-eating, then if you do instantaneous decay approximation, you will not see the production of this particle, but in reality you have production of this particle. So let me make it clear. The energy density of radiation here is much, much greater than what it will be here. But here there is still the inflaton around. So the thermal energy here is much greater than the final re-eating energy, but there is still the inflaton around and this needs to be taken into account. So imagine that you have, before I show you the plot, let me tell you another possible way in which this could be relevant. So what we are saying, right? We are saying that here there is the inflaton field and the inflaton decays into radiation and then there is a particle x that gets produced out of the thermal bath. This is the decay of the inflaton and here you could imagine that you have a particle that is produced in this way, for example. Gamma, gamma goes in x, x and you could imagine that the cross-section for producing this particle is temperature to the power of n. In the case of gravitinos, the cross-section was 1 over n-plank square. So for gravitino n is equal to zero. But imagine, for example, that you have a different particle for which n is greater than zero. The greater n is, the more you are sensitive to the temperature. And if you have a theory which has n greater or equal to 6, which you need to have special models, but you could imagine that, then in this case, most of the production takes place here. So if n is smaller than 6, the dilution wins and most of the production takes place here. If n instead is greater than 6, you will have production here. Let me... This will be the homework. I will give you some hint to solve the homework. But let me just show you the plot of this result. And if I forgot how to do it, just a second, ajuta, zoom, OK? So this is one model, one calculation in one given model. I will not enter into the details of the model. This is essentially time and this is the abundance of gravitinos. Why is the number of gravitinos divided by number of photos normalized to the value that you have under the instantaneous decay approximation? And you can see that this is the right thing that you will need to do. The black one is really what takes place. And however, the orange one is what you get in the instantaneous decay approximation. And you see that eventually the result becomes the same. This is another maybe more complicated way to show that what you produce here essentially gets diluted by eventual production of radiation from the inflaton. Here instead is a different example in which there is a model where I have a cross section for the production of the particle x. There goes a temperature to the 6. It's a model where I exchange a mediator of spin 2, a heavy mediator. Then you will see that the black curve is the production in the real case and the orange curve is the approximation. And you see that there is indeed dilution. You produce a lot of this particle here, then this gets diluted. But eventually the final result is about an order of magnitude greater than the one in the instantaneous decay approximation. So in a sense, the message I want to give you from all this discussion is that this is what really goes on, at least provided that this is the right picture of the world. But eventually, depending on what processes you are really interested in, this may or may not be relevant. Is it clear? Questions? Is something being created that is indicated to gravitinos? Or do we have some other gravitinos that help? No, in this case, you produce a lot of gravitinos. That's what I've discussed so far. You are complicating it more than what I've done so far. Here you produce gravitinos from the thermal butt. So these are the gravitinos that are produced here. And you see that in the instantaneous decay approximation, instead you are producing zero gravitinos. So at these early times, there is a lot more gravitinos than in the instantaneous decay approximation. But eventually, this gets diluted, because eventually the instantaneous decay calculation catches up. So the second case is a different model, in which there is a particle, x, that gets produced in a cross-section that depends to the sixth power of the temperature. If you're interested after this lecture, I can tell you what the model is, but the model is just one example. I just want to focus here on this temperature to the sixth dependence. For this model, I produce a lot, and eventually the final result is greater than what I would have in instantaneous decay approximation. The y-axis is the same quantity. It is the ratio between the abundance divided by the naive abundance. Yes? Well, I see the dilution, because you see the problem is that here we decided to plot this ratio, which now I realize maybe is not the most convenient thing to plot. But the crucial point is to understand that eventually this thing become together. Oh, the second one, again, here, again, you see the dilution. Here you see, if there was no dilution, you wouldn't say, okay, you see I produce ten order of magnitude more than the instantaneous decay. But there is dilution, but the dilution is not erasing everything, and so there is some leftover, because there is a bigger dependence on the temperature. So if there is bigger dependence on the temperature, you are more sensitive to what takes place here. It's true that we still, there will be some dilution, but it's more strong this effect here. It's just mathematical calculation, and once you solve the differential equation, you will see this. Because the inflaton keeps producing photons. You see, you have the thermal bath here, which is huge, but there is the inflaton, which is even huger, and the inflaton keeps producing photons. So what you produce here gets diluted. Does it get diluted a lot or a little? Depends on what is the coefficient here. And I'm telling you that if you have n equals 6 or greater, then this effect becomes important. Marco, there is a, just somebody is asking, if you can put the reference to reproduce this calculation. I will put the reference, these calculations were done in a paper of mine, and I will add the reference, and then I will post it. Is the cut-off scale of this model? As I say, I am willing to discuss the model in private. After all, right now, this will be time wasted for this lecture, because it's just an example, not really the main focus of the lecture. If you care about the model, I'll be super happy to talk later. OK, how much time do I have? 15? OK, fantastic. One more question. Ah, OK, very good. The question is that the production of X particle reduces n gamma. Yes, but we are discussing cases in which the production is always very limited. For example, remember that the gravitinos here was 10 to the minus 10. So essentially I extract a little energy from the thermal butt, but I can neglect it in these considerations. I want to show now a different type. Now I will move to these final 15 minutes, and tomorrow lectures will be from the computer, so I can show you a bit more things. I'll try to go slowly, but I think I'm done with more introductory stuff, and I just want to discuss some example where it's easier for me to use the computer. So what I want to discuss, I want to discuss a completely different way of having a reading, which is called non-perturbative reading. So far, when we computed this decay rate, this gamma that was entering in this calculation, this was the perturbative decay rate. This is the rate that I could imagine if I have a particle in Minkowski and just diluted that just decays. However, the inflaton is not really a collection of independent quanta. The inflaton is more a condensate that keeps oscillating. And the fact that the inflaton is a condensate that is oscillating should have some effects that go beyond the perturbative decay that I was telling you. When you do Feynman diagram to do the decay of a particle, you treat the particle as an isolated quantum that just decays. But now the inflaton has equivalent oscillations, and in this paper you see now very old papers, people realize that this can actually be important. And I'm gonna show you an example taken from Leti's simulation, which is probably the simplest example. You see there is a quadratic inflaton potential. Maybe this is true at the minimum. I don't know what the inflaton potential is during inflation, but here I care about the potential during reading, so there will be massive inflaton potential. And then there is another scalar particle, chi, that represents matter. Of course, this is a toy example, I just do calculation. But this represents a particle, chi, that is produced by the inflaton. And what I will show you, I will show you some Leti simulation. There are some cause available on the market. This was the first code, which is the simplest one, it's the only one that I can use. So that's why I put this one there. And in the movie, we will see a slicing. So what we will see? We will see x, y plane. So this is z suppressed. And on the vertical axis, we will see either the value of the inflaton or of the field chi. So we will see that the inflaton is oscillating, initially. And then during the very quickly, you will see that the particle chi gets excited. And then you will see that also the inflaton will get excited. So now this is the movie. So on the left, you see the inflaton. This is the first half oscillation, one complete oscillation. And you see, just after already the first oscillation, you already start producing the field chi. And you see that the field chi, this is the fragmentation of chi, this is just quanta. So this is particle production of the field chi. And then you see that the field chi produces quanta of the inflaton field. So what we see here is an extremely quick production of particles that takes place not after 10 to the 30 times. It takes place right after at the end of inflation. Due to this coherent effect, that was discussed, that I tried to argue a second ago. Maybe let me finish 10 minutes and then I can collect the final questions. So the lattice simulation that I showed you have essentially, let me show it again, the first stage, this production of quanta of chi can be done from linearized calculation. And this can be done analytically. It goes under the name of parametric resonance. And I can discuss in private how to set up the equation to do it. But then you see you end up in a regime which is completely nonlinear. What you see here is highly nonlinear, it's not small perturbations. And these can be understood only from the lattice. And we can try to make a sense from what we see from the lattice simulation. The first period can be done analytically. And essentially the basic idea is that you have a mass for the field chi that depends on the inflaton field. You see the mass of the inflaton is m, is a constant parameter, so the inflaton is oscillating with frequency m. And instead the mass for chi is the value of the inflaton field. And the inflaton field, remember, is a huge vev. In this model here is essentially comparable to Planck mass at the end of inflation. So the inflaton is having a spanning a huge amount of value. So the mass of the field chi changes by a huge amount. And its effective frequency changes, is evolving in that way. There is, yesterday there was a nice talk and it was mentioning also this question here, one of the students talk. Essentially you can show analytically that when this quantity here essentially is greater than one, then you have a lot of particle production. And what is beautiful, you can see it here, here is the analytical calculation of the growth, the occupation number of the field chi. You can see that the production happens at step. Why the production happens at step? Because remember that phi is the mass of the inflaton. So when there is the mass of chi, when phi is very away from the origin, then chi is very massive and is not produced. But whenever phi swings to the origin, remember phi is essentially oscillating about this potential, so phi keeps moving, whenever phi swings through the origin, here m chi is equal to zero at this moment here. And this is the moment in which chi is effectively massless and you produce a lot of it. And so you can see that the production takes place in steps, which is exactly the moment in which phi crosses the origin. There are two effects here. The first effect is that whenever phi is equal to zero, you can indeed have a great omega. The second effect is that this is not just once, this is a periodic production, and so you can get some resonance effect. And what we see here is indeed the effect of the resonance. This is the first stage that can be done analytically. After that, we can only follow the lattice simulations. And this would be the spectra for the inflaton field and for the field chi as a function of time. This is like rainbow colors. So you see that the first field that gets produced is the field chi. But then you saw that the field chi backreacts and rescutters and destroys the coherency of the inflaton field. You saw from the movie chi goes produced first, but then you see chi is completely homogeneous, so it backreacts on the inflaton that becomes homogeneous, and then the process stops, but you end up with these distributions. These distributions are very far from thermal, and nobody has been able to compute the thermalization in this type of models. People argue what it could be, but there is no full calculation of thermalization, because the lattice after some point breaks and it leaves you a state which is very far from thermal. These, I'm sorry, the colors are not very good, but this shows the equation of state of this plasma, which is neither matter nor radiation. Remember, so this is evolution of time, this is the equation of state of the totality of this plasma. Remember yesterday we talked about effective equation of state during reading? Remember, matter is zero, radiation is one-third, and you see that you get this plasma that at this moment has an equation of state that it is in between. So you see the complication of all this business. If you want, I have not prepared the questions here, but at least the first stage can be understood analytically, and if you want, after the lecture, I'd be happy to show you if anyone is interested about the analytical calculation. OK, I will stop here and I take questions if there is time, or we go to lunch and we do the questions after lunch. Yeah, there is one question from the room. So since the radiation perfect fluid can be modeled, I think coming back to these equations on the board, since the radiation perfect fluid can be modeled by the Lagrangian of the standard electromagnetic field, can we model re-eating with some specific coupling of the inflaton field with the electromagnetic one? We can, certainly if we know what the inflaton is, we can put it in a Lagrangian and then we can work out the equations out of that Lagrangian. Here, I try to be, you see, as model independent as possible, so here I'm ignoring the microscopic description of the model, but if we have a concrete model, even better, for the point of view of doing calculations. Yes. OK, let's thank Marco for the lecture. OK, thank you.


FIRED AGAIN Unauthayankaii allegedly dropped as host of GQ Awards. If the unconfirmed reports on Twitter are anything to go by, the saying when it rains it powers might ring true for former KIA 959 radio DJ Unauthayankaii, who has allegedly been dropped as host of the upcoming GQ's Men of the Year Awards. This is just days before the ceremony is set to take place this Saturday. Entertainment commentator Filmfella said GQ Magazine has decided to replace Unauthay as host of their upcoming awards. It is unclear what led to their decision to drop Unauthay as host. However, social media users have already linked the decision to her sacking from KIA FM. In a now-deleted post, Unauthayi shared her excitement on Monday about hosting the GQ Awards, dubbing the gig a dream come true and an honor. A dream come true and an absolute honor to host this year's Men of the Year Awards for at, South Africa on the back of the Brits celebration hosted by MR and Mrs Alba. Kamigo, read the caption of the IG post. After her controversial sacking from the radio station, she has remained mum on the issue and has posted about her future endeavors and therapy sessions. The singer has been focusing on the things going well in her life at the moment, like her music career. Unauthayi posted a video where she is seen working with renowned producer Prince KB on a musical collaboration yet to be released. Took Prince KB on a tour of Iblowa E Pisa during our breaks and he has one question. Why do we call it New Bright or New Brighten and not New Brighten? she wrote, sharing snippets of her new song. F5 News reached out for confirmation and comment from Unauthayi and GQSA. GQSA has since replied to F5 News with an official statement stating that Unauthayi will not be part of the ceremony due to unforeseen circumstances. It is with regret that singer-songwriter, businesswoman, entertainer and reality show judge Unauthayi will no longer be joining us for our event due to unforeseen circumstances. However, we continue our long-standing relationship with Unauthayi and look forward to upcoming opportunities in the future.


Oh, hi beautiful. How are you? You look stunning. Thank you for being here. Today we're watching people do a balayage, not just your average old balayage hair painting methods. I see everywhere all the time somebody somewhere, somehow coming up with a new balayage technique every single day. And today I wanna highlight some of those techniques I've been seeing. And they're really strange. I'm 100% open to learning something today. I don't know how useful these things are gonna be, but we shall see. Today we have a sponsor. It is the RealReal.  If you love luxury clothing, well this is a luxury consignment website and app. And I know what you're thinking. Brad, that sounds horrifying because I'm gonna buy way too much stuff. Well, I say go for it. It's a lot of fun. They have a location in LA, New York City, Brooklyn, Chicago, and San Francisco. So you can actually shop in RealLife as well. 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And if you're ever thinking of selling your things on consignment, there are multiple ways you can sell your items on the RealReal even while sheltering in place. One of their newest offerings is booking a virtual consignment appointment where you can book a remote video call with one of their luxury managers. You can also ship your items directly in or safely drop off your items at the curbside drop off at one of the RealReal's locations. And you're welcome for this one. You can click the link right down below in the description box to get $25 off of your first purchase, as well as 20% off select items with the code Real. Now that I have this cute outfit on, I'm excited to see what new things we have going on in the world of hair color. Cause I gotta stay up to date, baby. I gotta stay in the know. And I'm gonna help you do that as well. Let's do it. Up first, we have a video by the channel, You're Gorgeous. And this is called How To Ombre Dark Hair.  I have a lot of questions and it just started. We're really going faster. Okay. I've never seen a frosting cap like that before. It looks very thick and very heavy duty. I don't know what that is or where you got it from. But I would like to know. You just plucked out so much hair out of that cap. I'm already taken aback by this and we just started. Oh my God. Yo, this girl's never gonna get this teasing out of her hair. I'm not gonna knock anybody today. I don't care.  Personally, I don't like teasing and everybody else out there can tease all they want. I just don't have the patience to take out teasing. Okay. So now we are using a board and just painting on some highlights. What really confuses me is like this section going on. Do we know like where this hair is coming from or do we just kind of pick up whatever we think? Because you can't see the sections. It's all around very intriguing to me. And I just don't know how this is going to come out well by any means. We're clearly going for a very bright bold look. These are chunky, chunky balayage pieces and they are incubating with foil and adding a lot of lightener, which I love to see. That way the hair lifts an appropriate amount instead of putting the tiniest amount on the hair and not having it lift at all. I've seen a lot of people do this like back combing with the brush before. And I always found that weird too because I always go down on the hair but it seems to work well for people. So maybe I should try it. We've made it to the cap and it's gotten weirder. Dude, what are we doing? Okay, first up, this is a whole lot of work. I could never and would never. We first did the frosting cap, pulled out all the pieces and now we're taking hair bands and putting different sections in our hair to kind of over direct the hair. That way when you paint the highlights on you're getting some nice blending going on. It's just a lot of a lot of a lot of things. So now we're going in and we're painting. I wonder if she like weaved any sections or she's just painting I think the entire section of hair. Those are thick. I would have probably sectioned it out more than that. Oh, that's a big piece of hair we're coloring. I hope we don't get spotting on that, baby. I mean, this girl's going for a different look clearly. She also has great technique though the way she's painting that hair, the way she's getting really in there, being precise with this. So I don't think there's gonna be any spotting luckily and beautiful folding of the foils. It's very nice, okay? There's definitely things that I wouldn't do, but it's nice. Okay, we have made it to the pulling out of the foils and her hair is really light to the point where it looks possibly over processed. What I'm just realizing right now is that she has to take all of this stuff out of this girl's hair with the lightener on it. That is going to hurt so bad. Pulling that hair through with all that lightener it is going to get caught and she's take the elastics off, I'm nervous. They rinse it beforehand and probably condition and then they take the cap off. That makes a lot more sense to me and that's a lot easier than I was thinking in this little head of mine. Okay, well I mean it looks light. Oh, okay. Where'd the highlights go Miss Thang? Okay, okay. Okay, interesting. Oh, all right. Am I confused? Yes, yes, yes, yes, yes. Reason being we ended up somehow with a foil highlight. Girl, you could have just popped some foils in there did a little root shadow and called it a day. It would have taken you half the time, a third of the time even and it would have been so much easier for you to do but maybe she has fun doing that. So I wouldn't personally be using this knowledge anytime soon on any real living people that I do hair for. Seemed like a waste of time to me. I don't know how to say that in a nice way but I guess that's it. I'm gonna say this one is a no for me. Big fat, no, no, no, no. Let's see what else we have. Next video up is by Freesalon Education and this is called Air Touch Balayage Technique full step-by-step tutorial. He's picking up the section. Everything looks really normal right now. What are we doing? This seems like a pretty good idea.  That way he's just painting I guess the healthier, longer hair on top and it kind of divides the section and gives you a little bit of like separation between the pieces I think. He also hair sprays it which I find interesting beforehand and he's really trying to do very minimal painting going on here. Most of it is being pushed out from the back. I think this is cool. I fully understand and I'm still intrigued by it and I still wanna know how this is going to come out. However, I feel like we're adding an extra step that doesn't need to be there.  I understand that. It just seems like a lot of effort. He has obviously a great technique with painting. Taking all those little baby hairs out before he does those heavy strokes is going to make it so that they aren't so stripey and there's a little bit of variation in color between the dark and the light once he puts everything together and all those little baby hairs go back into place. So the only other thing with this I can think about is that some people have really layered haircuts and some people's haircuts aren't even. So you get rid of that shortness underneath on some parts. It's gonna be more than others. You're gonna end up with heavier painting on one side and less than the other. I don't know, I'm just playing devil's advocate here and I'm trying to come up with the reasons why this may not work, but it probably will be fine. Okay, so we're doing very, very, very thick chunky pieces on the top. I've never seen anybody section hair like that. Unless she's doing a really far side part I've honestly have never seen that. So pretty interesting and this is the final results. Let's see. Oh, okay. It's definitely not ugly, okay? It's not bad, it's not ugly. You know, the person clearly has experience. What I have liked is to be more impactful and a little bit brighter towards the ends instead of like kind of the midsection being brighter. To me it looked like there was kind of like a lot of focus and lightness in the mids and less in the ends, which you know, usually you like to have a lot of the ends bright. That's kind of how hair grows naturally. It gets lighter towards the ends and darker at the top where the new growth is. I wouldn't be using this for my own practice and maybe you guys would, but I'm gonna say this is a no for me again. Just feel like it's a lot more work than needs to be. The results weren't as amazing as I maybe thought they were going to be. I mean, it was cute. I think there's just easier ways of doing it. Up next we have a video by The Life of Hair and this is called, Is This The Fastest Balayage Technique? Okay, you got me with that title. I wanna know what the most fastest technique is because I am all about speed mixed with technique and precision and I wanna get done quick but cute and if this is it, I'm gonna do it. We're going to do a six section balayage. Put ponytails in those six sections and this is gonna create elevation. This is an express balayage technique. So if you wanna get that little bit of lightness through the ends of your client's hair and you haven't got a massive amount of time, then this is the one for you. So we're doing some ponytails, six sections of ponytails. Ponytails are scary. We saw it in the first video. We're seeing it again. I'm feeling good about this one a little bit better than the first one. It's just a lot of hair at once. That's my only hang up about the ponytail thing. It's a lot of hair to color. I don't even know how it's gonna be easier for him or faster. So okay, so we're doing this sectioning and he's basically going in like a pinwheel type style thing, I think. Kinda taking out pie slices all around her head and putting into really high ponytails. That way he gets a lot of over direction within his sections and when he paints it, he'll get some really diffused highlights going on. That's the idea. It's worth noting that this girl has the perfect hair texture to do this because her hair is fine. She really doesn't have a lot of it. So it's gonna be easy to color it very quickly. And we're using some foils. We're gonna get really nice and light and he's just really going in and foiling every piece of hair within those ponytail sections, which is going to give her a lot of lightness. So I split the ponytail section into two. I've got the bottom of the section here. I put my foil underneath this section. I mean, so I've done this before when I was younger on some people and I always ended up with spotting. And maybe that was just an hour of my technique when I was younger, but I always ended up with little spots of places I didn't bleach because the sections were so thick. And I make sure I've got lots and lots of product and I heavily apply the bleach to the ends of the hair. Key word there, heavy amounts of bleach, fully saturating the hair. And then I apply a lot of bleach to that section as well. And just make sure that I apply tons and tons of products because the more we apply, the better it will lift in the long run. That's right. The more we apply, the better it will lift in the long run. Guys, put enough freaking bleach on your hair. And then I put another foil on top and then I flip over the foil and I remove the bottom foil I just painted onto. That's genius. That's a really good idea. The whole flipping you do that, you just did. See, I never did that. And maybe that's where my error was. So we just make sure we work that also diagonally, side to side to get the product onto the hair thick enough that we can be guaranteed we get maximum lift. I like this guy a lot. I feel like this is gonna look bomb. So I'll just give you a close up of the sixth section or elastic band balayage. Gives the hair that very, very soft finish. Listen, it looks pretty good for doing that so quickly with just some ponytails. So we've left some little orange pieces through her top layers, which are quite short, just to veil that very, very black fringe in there. And you can just see from the side angle there, a great blend going through. And we've got the dark running right through the hair. So she's got that kind of very shadowy, rooty vibe going on. And it creates lovely, soft blended balayage-esque color. Okay, that black bang was gorgeous on her. She looks amazing. Also, see that? He just made his life so much simpler. And I'm definitely gonna be using this on people with hair that's on the finer side. I think this is a great idea. And it makes your life easier and makes everything go a lot faster. So I very much love this concept. And I'm excited to try it on somebody. He did an amazing job. All right, let's watch one last and final technique. And let's see how it goes. Up next with a video by Ryan Superno Hairdresser Top Indonesia Haircolor 3D Balayage Comb Technique. He is going in with some back combing. What is that? What are we using on that? What is it? I've seen these combs before.  And you can just put it in there and it sections everything out for you. That is cool. I very much enjoy this and would like to have one in my hands and use it. So he's basically getting like flawless weaving with just the comb. I don't know why we all had to hand weave out all these foil highlights. I feel like it would be so much easier if we just took a freaking boiling comb that has those little teethies in it and just sectioned everything out with like this guy's doing it. I'm ordering one of these combs today. Okay, so now he's going with a single process or a toner of sorts in between to add more of a red tone to her hair, I believe. This is totally normal. And I wanna know what these highlights are gonna look like. Hopefully it's gonna be like this beautiful blend between like some lighter blonde pieces and some more berry color in between. I like this a lot so far. Where did the highlights go? Okay, see, here's the issue. He is rinsing everything together. And maybe this is what he wanted. But you know, obviously when you rinse everything together you're gonna get a lot of that darker color on the lighter color and it's going to probably get rid of every single light piece you put in there. I'm not seeing a whole lot of lightness going on but it looks like he's toning the highlights. So maybe he wanted it like this. Ooh, I do love a good little gloss moment though. This looks shiny and pretty and like, ooh. I lied, it's really freaking gorgeous. Oh wow, that's really pretty. It had like the softest melt of highlights going on. I loved it. I think that is a really great technique to use.  Where do I buy it? That was cool watching other hairstylists show off their techniques. I love learning from other stylists and I always have room to grow and I never react to professional hairdressers. This was cool to watch and I'll definitely be using some of these techniques on my own clients and friends and family and myself probably. If you guys aren't already subscribed make sure you do right down below. Hit the like button and the bell icon to be notified every time I post a new video. If you would like to text me, this is my number. Yes, it's really connected to my phone. Yes, I really do see your messages and yes, we were applying to a few of you guys who text me right now. This will also keep you notified every time I post a new video. Make sure you follow me everywhere else here are all my social media handles. Go for it, I'm literally everywhere. You can find my videos everywhere.  It is linked right down below at xmodohair.com. If you guys like to shop my hair care products and my color, you can do so now at xmodohair.com. It is linked below for you. And if you would like some inspiration on what you should do next to your hair, maybe a different hairstyle, maybe a different hair color, you can head to our Instagrams at xmodohair and xmodocolor.  Thank you, you too. I have really long blonde hair and I wanna change but don't know what to do. Would you give me advice? Yes, of course that's what I'm here to do. I say chop it. I say bring it up to the shoulders. Get a little root shadow done. It'll give that new, cute feeling while still having your hair blonde because I know, I know girl, it is hard to get rid of that blonde hair. I like it on you a lot. So I think just a root shadow would change up a lot, having those money pieces in the front and then bringing it a lot shorter would look really pretty. If you end up doing it, I would love to see a photo and good luck. And thanks again to the RealReal for sponsoring today's video. Make sure to check out the link in the description box right below. That's all for today guys. Thank you so much for watching.  I'll see you next time. Bye guys.


Well, first of all, it was a very different job. I was an academy director at Newcastle and then an academy director at Everton. There was two very different roles because academies were so different by that time. Academies had just started when I started at Newcastle. Yes, I was in charge of the academy but it was quite a small department and I coached. I worked with the new intake every year and looked to develop them into then passing them on for the next academy team. By the time I got to Everton, the department was absolutely enormous and I found myself stuck behind the desk which was frustrating but necessary because my role was academy director. What's important in terms of developing players? Well, first of all, technique because they don't have the technique. We talked about it before the session that if they don't have the technique then they can't actually put things into practice. They can understand tactics, they can understand what it is that you want them to do but if they can't actually perform the technique then they're not going to be successful. I get coached at tennis and it has to teach me technique. We could play games all day long but that won't make me any good as far as a topspin backhand is concerned so it needs to work on the technical part of it. Many years ago I heard Ruth Hull saying there are no tactics without technique and I agree with him you have to have those techniques embedded first of all. It's great nowadays for academies because you have the players so long that actually the technical work should be done at that youngest age and you should be continuing to improve the technique making sure that you don't lose techniques that you've learnt as a younger player but it means that you can move them on to the more tactical side of the game quicker. When I first started in academies we didn't have the players out of school and things like that so you were starting them at a technical level and the first couple of months of me working with players who had left school was to really polish up the techniques. Once the sooner you can move on from them because you're satisfied with the level of technique of the players then the better because then you can start to really teach them the game and it then becomes about their understanding it comes about even down to game management so we wrote a full programme for the academy as academies do I feel coaching curriculum and it was very much at the youngest age groups, technique, technique, technique and that was a way up here in very little tactics and it gradually swung until it was keeping them top top making sure that the techniques are still as they need to be but let's you know the oldest groups need to know about game management it can take them 12 years to get into the first team in 10 minutes to get out if they don't know how to manage the game properly


Thank you so much for serving our country. What makes the game of baseball so great? Well, it's just unexpected actions, you know, how players react to different situations and get your blood running up, yeah? We understand you watched Babe Ruth play. Can you share any memories attending games here at Yankee Stadium? Well, it wasn't that I didn't see him play here. I'm not old enough. He was with the Boston Braves. They were playing the Dodgers, Boston Braves. That's where I seen Babe Ruth play it. Way back, I always know what it cost to go to a game now. I once sat in the bleachers Yankee game, fifth sense. What does it mean to be recognized as veteran of the game at the iconic Yankee Stadium? Oh, boy. You don't know how much of a thrill it is. They've been very good. Yeah, they're just marvelous as they do that for veterans. I really appreciate it. I really do wholeheartedly. It's just a nice feeling. Thank you again, Henry.


Matt Unger is about to take us on a walk through the Govee booth where you do smart lighting and smart home stuff for the average person, right? That's our goal. We started as a very small company and now three years later we make all this type of lighting, but we'd love to take you through, show you the different areas and everything we do. Alright, this is an audio and video podcast, audio by itself too, so I may describe a little extra as we walk through, but this is visually the coolest booth we've been to so far. Well of course, you can do a lot of lighting in a smart home and we love very practical lighting. Some people just very practical lighting, but we also do a lot of creative stuff. The first area we're in here is our home theater area and what's really cool is everyone talks about AI, that's a big buzzword, but we want to use intelligent in really targeted, really cool ways. So for example, a lot of people know us for our home theater lights. As you can see, all the lights in the room are reacting to what's on TV, it's reacting to the colors, but we're doing really cool stuff too. For example, when you watch a TV or movie, if there's an explosion, we actually know there's an explosion and do a big lighting effect, or if there's fireworks, we know that and we do firework effects. And the last thing I'll say is we have our dream view sync system and so dream view, it means you can take any of our lights that we have, there's no bridge, there's no hub and you can literally tell those lights, do what the TV light does and really quickly and affordably you can have a whole home theater that's reacting. Nice, so how is it knowing that there's fireworks, is it by the sound? No, this is our algorithms, our intelligence we built in. I'll show you later, we started with our gaming so that we could actually tell what events were going on in the game and we're using that same intelligence now and putting it towards our home theater. So for example, if there's fireworks, there might be fireworks on screen, but maybe the outside of the screen is very dark, so normally you wouldn't get a lot of lighting from that. But because intelligently with our algorithms, we can tell there's fireworks in the scene. But I'm asking, how are you telling there's fireworks by the visuals or by the audio? By visuals, so this is all based on the visuals. There's some sensor on the front side of your TV looking at the video. So two ways, so you can see on here we actually have a camera above the screen. Okay, there it is, there it is. So we have two types of systems, we have one that's a camera system, you literally just put the camera on top, the back lights behind, it sees everything on the screen, it sees the colors and can react to those. That's great, so you don't have to worry about the programming telling you information, you're doing it from the visuals. That's great. We want things that are super simple. So we have packs that are literally a single pack that includes the back lights and the camera, you open it up, you put the camera on top, the back lights behind, and you have a completely reactive system. No other things to buy, no other hubs, it's super simple. Very cool, where are we going next? Well, let's go to the bedroom, right? Wow, Matt, that's that's a little forward, my husband's standing right behind me. We love this type of lighting, right? Everything from in there, we have our star projectors, till we have lamps, and we really believe that lighting can also be calming and nice. So we build actually some speakers into our lighting. So you have white noise and lighting that's reacting to the white noise. As you can see, we do very practical lighting up there on the ceiling, strip lighting, neon skirt lighting. So this is very much for, again, even homeowners that just want some elevated, really cool lighting that fits into their place, but it's very affordable and very easy to do. It's fabulous. I'm intrigued by the LED lights around the edge of the ceiling. How are those mounted? They're like ones every six or eight inches. Yeah, well, we want stuff to be very simple. So it's literally just stickers on the back, 3M stickers.  You don't have to hire anyone. You don't need to contract anyone. You literally just take this strand and put it up, plug it in, and it connects right away and you can have it very colorful or just sort of warm, soft colors. That's really a nice effect. Are we going into the starlight right now? Yeah, it might be hard to see in there, but no, we also, yeah, we also make star projectors. And it's really cool. Galaxy projectors, star projectors. We have white noise built in. It's really cool, again. So what for the audio audience? We're actually looking at constellations on the sky, on the ceiling and on the wall. We're really proud of this one. Not only does it have sort of the digital stars, but it actually has swappable discs. So you can put in planets or constellations. We have music that goes along with it. You've got a beautiful planet with a ring on it. Not Saturn, evidently, but some sort of planet. Yeah, we love, again, stuff that's really creative, but also really relaxing. People use lighting and home products in a lot of different ways. A lot of people for relaxation. And so that's why we love stuff like this. Very cool. All right, we're leaving the bedroom. Now we're going back in here. Let's go to the gaming room next. We're off to gaming next. Let's take a look in here. Ooh, I like this. This is more of our creative stuff, right? We have so many gamers and streamers. And so, you know, we have a lot of different types of wall panels are very creative. The same thing I showed in the home theater. Here's our boxes. So these don't use a camera. This uses a sync box. This is our AI sync box. And what's really cool, yeah, you see it below the screen there, is not only is it reacting to the colors on the screen, but we actually train it to games.  And again, that same dream view I talked about, you can literally tell any light in this room, from the light around the table to the wall panels, do what this light's doing to have it all react together. We've got cubic light panels that are changing all kinds of color. We've got a giant light panel behind a display. We've got, that's another one of those. We've got another kind of artsy looking hexagon. Yeah, and so I'm on social media marketing. The really cool thing is not only do we have streamers who are setting this up for gamers, but we even have parents who are getting this stuff, putting it in their kids' bedrooms. They're never going to leave those rooms. But the parents are really proud. They're like, I actually did this myself. The kids love it. Very cool. Very cool. We've got more rooms to see. Sure. Just two more things. So Govee Life is our home product brand. And this past year we've been going really hard into our home products. Things like ice makers and humidifiers and smart heaters. Even my mother, I can tell you, I even bought her our kettle. A lot of people know us just for our kettles or our temperature sensors. Tell us about the kettle. It's very pretty. What do we get? What's cool about it? Well, I mean the cool thing, it's a normal kettle, right? To boil water everything. But it fits into our whole ecosystem. So for example, if you're in your bedroom, you want to start the kettle or put on a schedule, you can easily do that. The very cool thing is we have all sorts of sensors, like motion sensors, present sensors, door and window sensors. So you can literally set it up when you walk into your kitchen. It knows you're in there and it starts your kettle. And it turns off your lights and it turns on your heater. And it kills somebody in the game in the other room, you know. I get, well, you might have to do that yourself. But no, we do a lot of people know us, even our leak sensors. Some people only know us for that. And so it's really cool. The leak sensor people. Yeah, you don't think about it much. But a lot of people are like, thank you so much. I had a leak in my home. You saved me. Very cool. The last thing, I can show you the outdoor lights. That's the last thing we can do. All right, we're walking out of the kitchen. So again, we do a ton of outdoor lighting now, especially for the holidays. People want to put up outdoor lighting that looks beautiful, that they only put up one time. So you can see along the top here, those are permanent outdoor lights. It used to be, you know, you'd have to pay professional installation $2,000, $5,000, $10,000 to do lighting around your home like this. Now we designed it. This is our Permanent Light Pro. This is our second version. You can buy a single pack, nothing else. You can put it on your home yourself. It stays up permanently year around. No, those aren't little stickers. Yes. Really? Still stickers? Yeah, they're still outdoors. Yeah, and actually it works really well because we want this to be super easy for people to put up. Works year-round, completely weatherproof. And again, the cool thing is when it's a holiday, you can have, you know, your Christmas lights showing, your Halloween lights.  The last fun thing I'll show you for people watching, our curtain lights have also blown up. We do these curtain lights, both indoors and outdoors. You can set any type of things on them. You can make, you know, you can put text across them. So what we're looking at is a series of lines that are going straight down with LEDs at maybe every three inches. The thing that reminds me of, you remember, like the 70s or 80s, you had those bead doorways, right? Yeah. It's sort of like, and some people have used them for things like that, but that's also really creative and really cool product. But no, we're at GoVee. We have our own factory, we make all this ourselves, and it's just the breadth of stuff we've been able to make the past few years. It's growing so much. I'm really glad we stepped in because I had heard of GoVee, but I didn't have any idea of the depth of products that you do. So Matt, what are, where would people find GoVee? Well, we're always on Amazon, anywhere in the world. GoVee.com. You can see everything we have. And that's GoVee.com. And we're also in Walmart, a lot of stores now. So thank you so much. We hope we have some fans out there. And it's great to meet you guys. This was great. Thank you, Matt.


Welcome to another ML Talks. Anybody who is watching this, you can tweet to ML Talks. We will be making this a dialogue. First I'll be asking questions of our guests, but we also expect questions from you in the audience as well. Right now we're in the midst of one of the greatest civil rights and human rights crises in our country. Storytelling has to be on some level a response to life, a processing of life. What were some other things that you came across? I like to talk to neurons. All of us have this little secret thing that we don't tell anybody. Education, technology, sports, music, all of those different things are separate. And a lot of times we forget to build those bridges and connections in between them. Be excited about what you're making and try making new things.