I come around Wafa. Only the ones who are attending online need to register. Those who are present do not need to register for it. Registration is only because of using, because those who are connecting via Zoom, they need to be able to connect to the Zoom room. OK, so now I guess we can start. Please, Mark. Good morning. So let me continue from where I stopped yesterday. One small thing I left to say was about the statistics of perturbations. And we know they are very close to Gaussian. It is also a prediction of slow roll inflation in the sense that imagine you have a free field, and which means Ray doesn't interact at all. And then imagine you want to compute the three-point function, which is not being recorded. It was all false, what I said. Giving lectures used to be much easier. Now, even when I teach at my university, I'm always afraid that I forget to click to register. No, no, no, no. Anyway, we're recording with three devices. Shall we start again? Thank you. So we were saying that imagine you want to compute the three-point function. And you know that if you have a free field, this will vanish. Just imagine making quantum operator, right? Annihilation creation operators, and this clearly gives zero. So if these are free fields. And then you can imagine, OK, let me assume that the inflaton is not free because it's interacting with itself. And so how do you encode the interaction of the inflaton? Well, simply you put them in the potential, right? So you can tailor expand the potential. So this would be something like this. Let me tailor expand it around the origin, assuming that I have a minimum in the origin where it vanishes. And then however you see that the interactions are proportional to the derivative of the potentials. And the lower all parameters being small, remember, constrain the derivative of the potential to be small. And so essentially, also, non-gaussianity is going to be small just because these are small self-interactions. Of course, we could consider interactions with other fields. This could enhance the non-gaussianity. And that's something that we will discuss on last lecture. OK, so yesterday I made some claim about perturbations. I wanted to make a quick calculation to support or to justify some of these claims. But what I'm going to do is really a toy model. And it is a test massless field in the sitter. So this is really a toy model for the perturbations. So I'm not going to perturb the metric. And by mean test, I mean that I neglect the energy of this field from the point of view of the expansion of the universe. So you know that when you give the line element for the sitter, we can use the physical time. And in this case, A is equal E to the HT, essentially. So this is how you do it. But you can also use conformal time. You can do many things, but this will be two coordinates that we will discuss. In this case, the scale factor, we write it minus 1 over h tau. And then inflation, the sitter, sorry, is done in this portion here. So tau goes to 0 minus during inflation. And the scale factor grows like this. So why we are using conformal time? Because it's easier to do calculations for fields. So because we will write s, the action of the field, d4x minus g, g mu nu, d mu phi, d nu phi. Massless field, so I didn't add any potential. And you see that's what I'm writing for the action of the field. And now you see that from the square root of the determinant, you get the scale factor to the fourth power. And then from here, you get scale factor to the minus 2.  If I go to momentum space, this will be a half dT d3k. Then I have a square phi dot square plus k square phi square. Then what I can do, this would be the Fourier modes of the field. And there is an a square also here, of course. Now we like to have canonically normalized field, because you see this is not the canonical normalization. So we are going to define chi a phi. This is canonically normalized. And the equation that we get for chi, that's essentially what I want to discuss, is this a question here? Chi and here, sorry, let me use a consistent notation. Derivative with respect to conformal time. People usually call it prime, so let me just put a prime. So the question of motion for the field chi is going to be the following one. And you can see, right, that this term in reality is 2a squared h squared. And now we can see that there are two regimes. You see from the equation. Remember, yesterday I was telling you that the field is initially inside the horizon, then it goes outside the horizon. And you can see from the equation, because inside the horizon means that the scale factor initially is very small. So this term initially is subdominant. And what we get, we get the same equation as a simple harmonic oscillator in Minkowski. This justifies what I was telling you yesterday, right? And we know what it is the solution here. We take the solution, which is the normalized one in Minkowski, e to the minus i kt divided by square root of 2k. And then when you go out of the horizon, you get this different term. So you see that there is really a different behavior. And you see that this is a minus sign, which means that you are going to amplify the mode. That's the amplification due to the expansion of the universe. And the solution, which is the one that gets the right initial condition, is chi is equal 1 minus i k tau divided by square root of 2k e to the minus i k tau. Remember that the time as I was doing before. So this is tau. This is a. This is the scale factor. So early times means tau goes to minus infinite. So if tau goes to minus infinite, the numerator here becomes 1. And you see the usual Minkowski-like normalization. Instead, late time. So if I now do look at this late time, it means that I take tau goes to 0. And so you see now that it's different. You get minus i square root of 2k to the 3 half times tau. So that's what the mode does when tau goes to 0. The phase vanishes. And this term becomes the dominant one. Now, remember that the original field was phi. So phi is chi over a, which is minus h tau chi. Because that's what the scale factor is. And now if I put everything together, I get i square root of 2k to the 3 half h. And particularly what is, we see that indeed the field becomes constant. This is the frozen field that I was telling you yesterday. You see there's no more dependence on time. So there is the oscillation during the inside the horizon. And then the field freezes to a constant value. Now we can ask what is the power of this field. And to get the power, I'm going to do it shorter. But essentially, we can just take the average of the square of the field. Because this is a perturbation. So the average of the field itself will be 0. But here, if you go to Fourier space, you do some algebra, you're going to find what I also wrote yesterday. This is k cubed divided by 2 pi square phi squared. And this is what we call the power spectrum of phi as a function of k. So now let's put these together. So you see that when you take phi squared, you get 1 over k cubed. That exactly counts as with this k cubed. And what we end up with is h over 2 pi squared. So you see that this is a scaling variant and constant. So these are the properties that I was telling you yesterday. Yesterday we were arguing, right? But now we see it from the calculation. So now in reality, we know that inflation is not the sitter. So for example, h decreases during inflation. So this will tell me that the power decreases. So when I do evaluate this quantity, if h decreases, I should evaluate it at horizon crossing. So most that cross the horizon at a different time, they will have different power. But in reality, as I told you, this is not the full calculation. When you do the full calculation, you need to add also the metal perturbation. And you can find very nice reviews, which I will not have time here to cover. But when you add also the metal perturbation, the calculation that I did becomes a bit more complicated. And what one ends up ends up with the things that I was telling you yesterday. So now we would add delta phi, delta g mu nu. We would include the delta phi in the Einstein equation. So this will not be any more a test field. And at the end of the day, you find what I was telling you yesterday. The spectral tilt is ns 1 minus 6 epsilon plus 2 eta. Remember, epsilon is m planck square half v prime over v square eta is m planck square v prime prime over v. And remember that this quantity are much smaller than 1. And r, so the tensor to scalar ratio, are 16 epsilon. Remember, we defined essentially p zeta goes as lambda 1 minus ns and r p gravitational wave divided by p zeta. So just to summarize, this calculation was done for a test field in the center. And the purpose was to show you the transition between the sub-horizon regime to the super-horizon regime. And also to show you concretely that indeed the field freezes. And the power, you see, becomes scaling variant. Now, this is the calculation that we did. And now I'm making some statements without fully proving them that when you add also the metric perturbation and when you also include the relation between them in the Einstein equations, then you'll find some more general properties. You will find that the spectral tilt, the scale dependence of the power, is given by the Zohl-Roll parameters and also the tensor to scalar ratio is given by the Zohl-Roll parameters. Now, during inflation, we use the number of e-folds as a measure of time as well. And we define them n to be the logarithm of the scale factor at the end of inflation, divided by the scale factor with some function of time. So during inflation, a is, of course, smaller than the final result. And so n is greater than 0, because you do the logarithm of something greater than 1. At the end of inflation, a is equal to a n. Therefore, n is equal to 0. So n decreases during inflation. So n is a parameter that is decreasing. And when n is equal to 0, it means the end of inflation. So now, imagine that I give you all these. And I ask you, OK, now tell me the phenomenology for a given potential of inflation. For example, I'm asking you, OK, let's take this model here. Let me copy the relations upstairs. So ns 1 minus 6 epsilon plus 2 eta r 16 epsilon. And then I tell you, OK, here is the model of inflation. Let me take the simplest. This model is now ruled out. But let me just discuss this, because it's the simpler one to do calculations. I just tell you, I have a massive inflaton field with quadratic potential. Tell me what is the value of ns for this model and what is the value of r for this model.  And now you see, naively, it seems that you say, oh, but the potential becomes steeper when I go higher up. It doesn't look flat. But remember that these are the flatness parameters that you need to evaluate. So for example, if you compute epsilon, this is equal m-plank squared divided by 2. Then here you have the derivative of the potential, so m squared phi divided by the potential itself. So you see this gets 2 m-plank squared divided by phi squared. So you see that for this potential, the greater phi is, the more flat indeed the potential is. It doesn't because flatness, remember, you need also to take this into account. If you take eta, now you do the second derivative. This is m squared phi squared. No, m squared, that's it. A half m squared phi squared, you get the same result. So you see here that the inflation ends. If here is phi, here is v, inflation essentially ends when phi is the square root of 2 m-plank. So phi comes from here, goes down. And when phi reaches square root of 2 m-plank, that's the moment that inflation ends, because the slow-roll parameters are no longer smaller than 1. Now, but what you need to do, essentially, you need to evaluate these relations at some given moment during inflation. Because this is the end of inflation. So this is the value of phi at the end of inflation. But the value of phi when the C and B modes were produced was higher. So this is the value phi C and B. And the question is, what should we put here for this value phi C and B? The first thing that we can do, we can solve the equations of motion. And this will be left to you as homework. So the first thing that you do, you need to find phi as a function of n. And this will be a homework that I will post later. I will guide you. You just look at the equation of motion for the scalar field, and you will be able to find phi as a function of n. For this potential, what you find, phi nearly equal 4n plus 2n-plank. And in the homework, I will guide you to solve this. It's not very hard. When you see n is equal to 0, indeed, you find that phi is square root of 2n-plank. But when n is greater than 0, phi is bigger. So this tells me, the farther away phi is, the bigger is the number of e-foil. It means the earlier I am during inflation. And now, if you just now take this number and you put it here in the formula for epsilon and for eta, and then you put this formula here, you will get a phenomenological prediction for this model. You have phi as a function of n. So you can compute the slow-roll parameters as a function of number of e-foils. And so you can compute these two quantities as a function of e-foils. And what we get, we get the prediction ns is 1 minus 2 over n. And r is 8 over n. So now, you see that essentially the question still remains in the sense, which value of n should I use? These are the predictions, but the question is, what? So is it clear? We use n as a measure of time. So we give the value of the inflaton as a function of number of e-foils. And so we can do the phenomenological prediction as a function of the number of e-foils. If you change the potential, you can just use the same machinery. You will just get some different functions here. But once you solve the homework, you will see that you have a tool to do it for any potential that you want. But now the question remains, and maybe you have heard that n is equal to 60, but why, right? So that's what we have to discuss now. Do you have any questions so far? Yes, please. I see that you just have the same scale as one scale. Should we be concerned about correction of the theory? Yeah, very much so. And in fact, there's a lot of problems in these models. Because, for example, if you try to embed them in string theory, it is really not simple. And even if you just try to have a consistent theory of quantum gravity, you don't know whether you can trust your potential for these field values. There's a huge literature about it. So indeed, it's a concern that people address. What do you mean by acceptable? No, this model has been ruled out, but the latest CMB data. But it's not by much, but this model by itself is ruled out. Very important. When I told you, I was making delta phi. These for tau went to minus infinity. This was e to the minus ik tau divided by square root of 2k. The literature, this is called the Bunch Davis vacuum. It essentially is the same vacuum that you would write in Minkowski. This is the least energy vacuum. And so that's what people typically assume. But for example, you could have a model where you would say, no, my vacuum initially is populated. So my vacuum is different from that. And then the results will be different. And people have also studied this as well. But let me just say that this is the simplest choice, and it fits the data. So that's why this is probably the most popular one. Arbitrary in the extent that is the simplest choice, and it fits the data. So always in science, if you have something, you can make it more complicated. The question is whether you should or you should not. But indeed, there is a huge literature on this. Yes, please. Very good. Think about where we're at now. Two points. There is a phase that I didn't discuss here about quantum diffusion. And if phi is very, very large, just the addition of quantum perturbations will become more important than the classical motion. But this happens far, far away. So it doesn't concern us. So our question is phenomenology. And that's what I'm going to address now. What is the value of n? Because you see, when it starts, it's a badly posed question in the phenomenological context. Because essentially, most probably, you cannot see experimentally when it starts. So maybe the question we need to ask is the scale that we see in the CMB. When were they produced? Were it produced when the inflator was here, here, here, here? And in a sense, that's exactly what this question is about. What value of n should I put when I compare with CMB experiment? That's what we're going to discuss now. Marco? Yes? Some questions in Zoom about m, little m. Can you explain again what is little m? Little m? Yeah. Little m is just a parameter. It's the mass of the inflatum field, right? So imagine I have a generic potential. And if I have a generic potential, I can ask, what is the mass of the field from the potential? By definition, m squared is a half the second derivative of the potential. Evaluate it in 0 if I have a minimum in 0. So if this is the definition of a mass, if I have a potential which is only the quadratic term, that's why I'm going to write it. And then there may be other terms, but this is the mass term. Now, if you look for this model, if you just look at the normalization of the perturbations, m is of the order 10 to the 13 gv. But for our discussion, m is just a constant parameter. So OK, we understood that it's very important to know the value of n to do any phenomenology. And now the question that comes about is which value of n should I choose when I compare with the CMB data? In order to answer this question, I must do a little regression. And I need to speak about reheating after inflation because we are going to see these two are very much connected. So during inflation, volume goes as essentially a cube. This is how much the volume grows. So this will be e to the number of e-folds to the cube. So e to the 3n. And we are going to see that typically we expect n, we will see, n could be order 50 to 60. So during inflation, you see that you devolve the space grows by a lot. So if there are particles at the onset of inflation, these particles just get diluted away and you don't see them anymore. So everything that is left after inflation, essentially when this simplest version is the inflaton field, so everything you see and also everything that you don't see was created by the decay of the inflaton. Maybe not directly by the decay of the inflaton, but maybe the inflaton decays in matter, let's say, where matter has just been standard model fields, or maybe decays in something as any decays to matter. We don't know, but essentially everything that we see in this scenario needs to start from the decay of the inflaton field. So this is an extremely important process, right? And we have essentially reheating. I was mentioning yesterday, but let me say it again. For me, it's a very broad word, and it is all processes from decay of the inflaton to establishment of the hot thermal bath of Big Bang theory. So right naively, if you think about the Big Bang, that if you ask everyone on the street, either they say what? Or they will say that the temperature was infinite at early time. But this is in the context of inflation. This is an extrapolation. We will say, no, at some point the temperature was created because the inflaton decays, right? That's the basic idea. But from here on, this would be the standard Big Bang theory. And here instead is reheating. So that's what we need to discuss. There are some requirements, not very stringent, but nonetheless, one is the temperature. So it should be greater than about 2 MeV because you need a Big Bang nucleosynthesis. And also then you need to have a cold dark matter. Let's say dark matter. And then you need a barionosymmetry. So these are three things that need to be taken into account during your reheating process. Now, why do we speak about the reheating? Well, reheating is the most unknown cosmological period. As I told you, essentially if I do an hour of time, right? This is one second. This is when BBN, Big Bang nucleosynthesis, starts. And then we know very well the history, right? We have radiation domination. Then we have matter domination. We know what the equality time is. Then today we have lambda domination. And then before, right, we have inflation. And we can at least test some part of the inflation through the CMB. But then, reheating, we really have very little, if known, direct observational reach to it. And so this is certainly the most unknown part. And what there is uncertainty associated to that. And as we're going to see, this uncertainty reflect on the value of N that is the number of efforts. So let me precisely discuss this. So what I want to emphasize now is that there are uncertainties in reheating. I want to show that this reflect into uncertainty in inflationary predictions. Even if I know the model of inflation, suppose really I knew that this is the right model of inflation. Still, I would have some problems in making predictions due to the fact that there is this uncertainty in reheating. That's what we're going to discuss now. So you know that Planck gives the most accurate CMB measurement on its own. So Planck pivot scale is chosen to be 0.05 megaparsec to the minus 1. So Planck gives a result. You open the abstract of Planck, and Planck tells you millions of numbers. For example, what is the spectral tilt? What are the limits on tensor to scalar ratio? But then these numbers depend on the scale, as we discussed. And Planck typically gives these numbers at these pivot scales. And so the question that we ask is at which value of n, at which if n was this scale produced? That's the question. So let me do a cartoon to visualize what we are saying. So here we have the scales. Here we have time. And now I need to put a few lines. This is what I call the time tk. Here is what I call the time tn. Here is what I will call the time reheating. And here is what I call today. So this is today. This is an end of inflation. This is end of reheating. And here is when the model left the horizon. So let me just do inflation. During inflation, this would be h to the minus 1. Remember, h grows very slightly, but it's essentially very close to constant. And then here there will be some power law. And here instead is what physical scales do. And this is the Planck pivot scale. So this is the wavelength. That's what it did. So the scale has re-entered in our sky. I'm neglecting, here I should take a small amount of energy density, our dark energy domination here. But I am neglecting this part. And that's essentially the calculation that I need to do. And the number of e falls this year. So what I need to say, I need to find, essentially, given the value of lambda, I need to find. And given that today I observe the scale here, I need to trace it back. And I need to find what is the value of n. So here is the heating stage. And here is radiation domination, matter domination. And there will be also lambda domination for a little bit. So what are the unknown parameters that I would like to know in order to really do this plot? So there are some unknown inflationary parameters. One is rho k, which is equal nearly the inflaton potential. This is the energy density at horizon crossing. Would tell me the energy of the universe at this moment here. Since the inflaton is going very slowly, I can neglect the kinetic energy of the inflaton. Then another parameter that I don't know is the energy density at the end of inflation. This is no longer equal to the potential of the inflaton, because at the end of the inflation the inflaton is moving faster. And these are instead the unknown reading parameters. We is essentially a rho reading. So is the energy density at the end of reading. And also I don't know the equation of state. So let me say this is an effective equation of state during reading. Remember, we discussed yesterday the equation of state. What we say, for example, if we have radiation, this is equal 1 third. If you have matter, the equation of state is equal to 0. If you have a cosmological constant, the equation of state is minus 1. But what was the equation of state during reading? And why should it be constant? It probably was not even constant. But we could imagine an average equation of state. So if I want to consider an average equation of state, what was the nature of the dominant source during reading? And I will discuss different possibilities. So now I need to have horizon crossing in this plot here. Horizon crossing is defined to be when lambda divided by 2 pi, which is a over k. Remember, this would be the usual relation that we have for Minkowski between the wavelength and the wave number. But now lambda grows as the scale factor. And when this is equal h to the minus 1, this is the moment. You see there is one horizon crossing here. And then, of course, there will be the horizon crossing today when it re-enters. And then remember the number of e-folds that we just defined was this one here, a and e to the minus n. So the scale factor was smaller, and this was, therefore, positive. Now, you can show. And this will be another homework. n is about 55.6 plus 2 logarithm vk to the half divided by 10 to the 16 gv plus logarithm 10 to the 16 gv rho n to the 1 fourth plus 1 minus 3w 12 1 plus w logarithm rho reading rho n. So you see, if you know the evolution of these, here you know the evolution. It depends on the inflationary model. Here you know the evolution if you know the equation of state. And here you certainly know the evolution. So if you know the equation of state here, you will know the slopes. Then, if you know these times, in order to know these times, you will need to know what was the energy at that given amount of time. Then you can redo this plot. And so you can take a given scale on your sky. This will be the length that you measure in your sky. And you can trace it back to this moment, and you can compute the number of e-folds. That's what this relation is telling us, that if you know the scale, this is true for the pivot scale. And this seems a bit mysterious, but if you do the homework, I'll try to guide you. It is not particularly hard to see this. Essentially, it really means that we just need to take this number here, and we need to trace it back through this history until we find this value here. And you can see here, there is this magical number that is almost 60. That is essentially what justifies our choice. You need to just do the algebra, and it's not particularly difficult in a sense.  There is these terms that are uncertainty in the inflaton model, because we know the scale of the inflaton at horizon crossing and the scale of the inflaton at the end. Instead, the last line is a term that encodes the physics of reheating. This is the equation of state-during reheating. And this is the energy density at the end of reheating. And so this is uncertainty due to the fact that we don't know exactly what went on through reheating. And so I want to now evaluate this term, because imagine that this term is always 0.5. I don't know what it is, but I don't care. If I see that this term would be 0.5, who cares? I don't know what it is, but essentially, this doesn't really play any role. But imagine that instead this term is 10 or 15. Then it plays a big role, because out of 60, out of 55, you have an uncertainty, which is a big number.  And so effective, I mean that imagine that the reheating, there was maybe a species that dominates and is massive. So at that moment, W is equal to 0. Then the species decay, then W becomes 1 third. Then another massive species take over. W is equal to 0. And then you would compute an average. So essentially, it's an effective number. Just tell me what is the average behavior during the reading. This is, you see, since I don't really know anything, this is the simplest thing I can try to do. At least I will learn what was the average thing going on at that moment. And I want to discuss with you two extreme cases and what is this value here for the quickest possible reading and for this, OK, maybe I should keep this. I should evaluate this term for the quickest possible reading and for the slowest possible reading. Let me call this term delta n is the number of e-folds that is associated to the physics of reading. Now, what is the quickest possible thing that you can imagine? The inflaton decays immediately. And therefore, essentially, rho n is equal to rho reading. And so this term is equal to 0. So if the immediate decay, this gives me delta n is equal to 0. Essentially, it's obvious, right? The immediate reading means that this time coincides with this time. Inflaton decays right away. Then these times will be the same time. And therefore, there is delta n is equal to 0. So delta n is how much you accumulate during this period. OK? Now let's instead discuss what is the slowest possible decay. And let me assume, right, that there was a cartoon that I already drew about the potential of the inflaton. So this is phi v. And the inflaton at reading is oscillating about the minimum of the potential. So heuristically, we can imagine that the inflaton does some oscillations if I add these t and phi. And maybe the amplitude of the oscillation is decreasing due to the expansion of the universe. This is the end of inflation, right? The inflaton arrives here. If the inflaton decays right away, OK, that's done. Delta n is equal to 0. But if the inflaton stays around for a long time, the inflaton will keep oscillating. It could do billions, or I don't know. A lot of it, we will see how much. It can do a lot of oscillations. And it does not decay. It's just there that it keeps oscillating. And its amplitude will, the oscillations will decrease because of the expansion of the universe. But this essentially will just go on for a long amount of time. And the equations that govern these are these equations here. And it's essentially, let's take just a massive inflaton. So let's just take the potential for a massive inflaton field. This would be m squared phi is equal to 0. This will be 1 half m squared phi squared. And so just if you look at the first equation, you recognize almost the equation of the simple harmonic oscillator, right? If there was not this term, this would be the equation of the simple harmonic oscillator. But then you see that you have a friction term. Remember, in the simple harmonic oscillator, a friction term would be something that is acceleration is equal minus gamma velocity, right? This would be a friction term. I teach mechanics at a steer, so I remember about this. But what is the point here that this gamma depends on the inflaton itself? So it's a bit more complicated system because, you see, the friction is proportional to the above rate. And the above rate depends on the value of the inflaton. So it's a system which is a bit more complicated. It's not just constant friction. However, you know, since we all have done physics one, that we expect damped oscillations, right? Oscillations that are decreasing with time. And now we can ask ourselves, OK, how does it really decrease? And in order to do how it really decreases, we can just do the virial theorem. We can apply the virial theorem. Neglecting the decrease of the amplitude during a few oscillations. Because the amplitude is decreasing with the time scale h to the minus 1. And h at this point becomes much smaller than m because phi is smaller than Planck mass. So the above rate becomes much smaller than the oscillation rate. So the inflaton oscillates with a time scale m to the minus 1. And the amplitude decreases with a time scale h to the minus 1. So the amplitude decreases adiabatically. It decreases very, very slowly. So during a few oscillations, we can neglect the decrease of the amplitude. And we can use the virial theorem. And remember that the virial theorem tells you that the average of the kinetic energy is equal to the average of the potential energy. This will be for the simple harmonic oscillator, for example. So if you have saw, remember that the pressure, what is the average of the pressure, the pressure is exactly 1 half phi dot square minus v, which is a kinetic minus potential. And this is equal to 0. So in this model, during oscillations, w effective is equal to 0. Now the question of state keeps varying during the oscillations. And whenever the inflaton is at a minimum, the equation of state is like the one of a vacuum. And then it's the one which is dominated by kinetic energy. So the equation of state here is w is equal to plus 1. And here it is like if it was a vacuum. So w is equal to minus 1. The equation of state of the inflaton keeps oscillating between minus 1 and plus 1. On an average, it's equal to 0 due to the virial theorem. So essentially, the inflaton behaves as matter. So now you see the two different scenarios. First of all, the first immediate inflaton decay. So you immediately produce the thermal bath. Delta n is equal to 0. And then there is this other decay that is extremely slow. And for a lot of time, the equation of state of the universe is w is equal to 0. So now let's evaluate this term, assuming that now we understand that w is equal to 0. So we have delta n is essentially 1 over 12 logarithm rho reading divided by rho n. And now let's take the slowest possible decay. This thing could be of the order of MeV. I told you that you just need to have a thermal temperature. It's sufficiently to have the BBN thermal bath. And this could be as high as 10 to the 16 GV. These are energy density, so you need to put to the full power. You see, this is really the slowest possible ever decay of the inflaton field. And this ratio, if you compute it, it will give the number minus 15. So essentially, it's delta n. We don't know what it is. But it's something between minus 15 and 0. So now you see that having a term that is delta n, which is something between 0 and minus 15, it's huge compared to 55. And if we now go to even the plank paper, so let me see if I can show it. I'm sure we are all familiar with this. But nonetheless, this was the plank paper, the latest, I guess, plank paper. And you can see that these are the same predictions that I was telling you for the primordial tilt and the tensor to scalar ratio. And the ellipses or the half ellipses will be from the CMB data. And now you see that they studied a lot of different models. But you see that one inflaton model is not a point. And here they evaluate the phenomenological predictions for n that goes between 50 and 60. So they take something. They say, OK, you see 55 is the central value, in a sense. And then we think reasonable to have an uncertainty of order 10. And so you see that's what we get. So imagine this was the model that we discussed a moment ago. You see now it's ruled out, m squared phi squared. Yes, yes, yes, yes. OK, so no, this is the point. OK. And you can see that every given model is not given as a point, but it is given as a line because of these uncertainties during reading. And you see that it's sizeable. So I think I justified why we take delta n is equal to 60. And then, however, I think I try to make the point that there is uncertainty associated to this. How much time do I have, 10 minutes, still? I think I started a little later. So OK, I'm going to finish one more thing that I want to discuss. Sorry, maybe I should stop share. I want to now start a little more to discuss about reheating, because I think I proved the relevance. It's a question about VK, maybe not. Before you erase, they can repeat what was VK. VK is the potential of the inflaton when the model left the horizon. So we are going to discuss reheating, making initially some very simple assumptions. And then we are going to simply make it more realistic by dropping the assumptions. So the first assumption I want to make is instantaneous inflaton decay and instantaneous thermalization. By instantaneous, I don't mean exactly at the end of inflation. I mean there will be some decay rate. So there will be a moment when the inflaton decays. So let me just compute what in this context will be the reheating temperature. Now, when you have a process in the time t, which is of the order inverse h, the number of processes taking place integral gamma dt, which is of the order of gamma over h. Because delta t is of the order of 1 over h. So gamma is the rate for a process. Notice that gamma has mass dimension plus 1. Because you see this is number of processes. So this is dimensionless quantity. And this is mass dimension minus 1 in natural units. So gamma is the dimension of an energy. This is the decay rate of the inflaton field. And the number of processes that take place is the integral of gamma in one upper time. This is the amount of processes that take place in that time, which is parametrically gamma divided by h. So you see, by looking at this relation here, we can really say the decay happens when gamma is equal h. Typically, h is a function of time. h initially is greater than gamma. At some point, h decreases. At the moment when gamma is equal h, we say this is the decay of the inflaton. And here, we are going to say that the inflaton decay is instantaneously, immediately at that time. So now, we can also find the temperature. Because if the inflaton decays and you have a thermal bat, because we are saying that there is instantaneous thermalization of the decay process of the inflaton, we are just going to write gamma. Let's do gamma squared. h squared is equal to rho divided by 3m Planck squared. And this will be the rho of the thermal bat. So the idea is that, at some point, when gamma is equal h, all the inflaton decays. This is just a very sudden approximation. And the decay products thermalize immediately. If you do so, then this will be 1 over 3m Planck squared, a pi squared 30, if I remember right. G star t re-eating to the 4. This is the number of relativistic degrees of freedom, which is of order 100, at least in the standard model, is of order of 100. And essentially, from this relation, we'll find that, parametrically, the re-eating temperature is of the order of gamma squared, gamma and Planck squared root. So you know the decay of the inflaton. From the decay rate of the inflaton, you can immediately find what is the re-eating temperature. And the figure, the plot I was telling you, is the one that I was telling you before. So here is time. Here is when there is the value gamma is equal h. So here is the temperature. Initially, there is no temperature. And then, all of a sudden, you form the temperature. And then here, the temperature goes as 1 over a. And this value here is the re-eating temperature, which is parametrically gamma and Planck. So this will be that you see the inflaton. Here is oscillating. So there are the inflaton oscillations. And then, all of a sudden, the inflaton decays. And it's going to produce the thermal. Probably, I will stop here. Do you have any questions? Yes, please. Now, this will be a different scenario. And for example, this will be what you want. Sorry, Martin, can you repeat? Yes. So the question was, I was doing a special case in which the inflaton was oscillating about the minimum of the potential. But then the question is, what about if I have a potential, for example, like this, so where the inflaton rolls, and then it keeps rolling? And then in this case, you would have a different scenario where here you have kination. And here, the question of state is steady plus 1 because you just have kinetic energy. And this is a plausible scenario, but I was just more simply minded, imagine that there is a minimum in the potential. Maybe here is OK, but in principle, one could imagine something like that. Or could imagine that the inflaton decays into a field that is massless, and so it gives all kinetic energy. Yeah, it's possible. You see, it's part of the uncertainties related to reading. Yes? You said that we want reheating to happen before we bend local synthesis, but there is also, is there also an upper bound energy? We'll discuss this tomorrow. OK. There is depending within models. So if you assume some models, for example, there's an upper bound for not overproducing gravitinos, for example. Depends on what the model you have in mind. But the lower bound is really robust. The upper bound depends on the model. Thank you. OK? OK, so I'll end up here, and I'll see you tomorrow. Thank you. Thank you.


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So a couple of days ago us Bond fans finally got the big live review for Bond 25 straight from Ian Fleming's iconic house GoldenEye on Jamaica. Now normally these things are incredibly exciting and you would think that would especially be the case this time after such a long wait all set in such an iconic location. But somehow it was all pretty underwhelming. All that really happened was that Fugunaga listed the cast in a pretty unexcited way and we got to look at some montages of previous Bond films and some presenter that nobody ever heard of showed us the beach that Ursula emerged from nearly 60 years ago. We were all really hoping for the big title review but it seems they haven't even got one yet. I was going to ask, dare I say, do we have a title bit completely nothing at the moment? Bond 25. I can't remember the last time we had a title when we announced the beginning of the film. I really don't know what Micky G is on about here. He doesn't remember the last time a title was announced at an event like this? Well how about the actual last time? Or the time before dead? Or the one before dead? Heck I'm having more trouble finding a time in which they didn't announce the title at such an event. So again, debates on what a title for Bond 25 is going to be continues. I was in that group of people rooting for Shadowhand. We all know that Babs confirmed that that is off the table, which to me does make it seem like they do actually have a title but they just don't want to reveal it yet for whatever reason. There were also rumors for Eclipse, which I really hope won't be the title. Sounds way too twilight to me. Barbara also went on to talk about the importance of the Me Too era nowadays and wanting to incorporate that in Bond 25, which honestly really gave me the chills. Same holds for giving Bond an electric Aston Martin now, I mean what is that about? I might just be the guy that misses the good old days talking here but meh. To me, the less political correctness in Bond the better. We have enough of that in the real world. So other than that we got to see the new cast answer some random questions from social media and before you knew it the event was over again. Now, at this day I was filming my casting video that I recently uploaded because I was trying to get on a Dutch TV quiz about Bond, so I was fully dressed up. It just happened to be on the day that this event was taking place as well, so just so you know I wasn't dressed up for this event, it's just pure coincidence. In between shooting the video we watched the event and my best bro left the camera on purpose without my knowledge to film my genuine reaction. And you can tell I'm not as excited about this whole thing as I should be. I'm just looking at messages from other Bond fans on my phone, casually sipping my drink. I'm pretty bored throughout. Oh, except for the part where Anna de Armas was announced. Some of you may remember I talked about her months ago during a live chat when somebody asked me which actress I would love to see as a future Bond girl. I mentioned her. I would have never dreamed she would actually be casted too. I mainly knew her from War Dogs, a fantastic film by the way, and her eyes and that accent that really appealed to me. You know and apart from that she's really freaking hot as well, so hooray for my dream casting come true. I was also really happy to see Jeffrey Wright returning for his third outing as Felix Leiter. I'm all for continuity and it's great to see him return. Some pictures of him and Craig working together showed up today, that's cool. Romy Malak being the villain was no surprise to any Bond fan, but it's cool that he's now confirmed. He had this little pre-recorded cameo in the most creepy way possible. I really think this dude is going to be a great villain. It's pretty cool to have the third Oscar winning Bond villain in a Roa board as well. Other than that there really isn't a whole lot to talk about with this event. The cast seem pretty bored and clueless about what is going to happen. I'm getting similar vibes to the one from Quantum when people really had no idea about what is going on. But yes, let's talk a little bit about the cast. As a matter of fact, let's make it a game. Let's play... Guess the actor's part. Ok, so let's start with Dali Bensala. Clearly he's going to be a henchman type of some sort. He gives me a bit of a gobinda vibe. He's French-Algerian though. Never heard of him until now. Billy Magnuson, another guy I've never really heard of. He has that look to be a slimy bureaucrat like Denby Inspector, something along those lines. I really think he'll begin as a good guy and then turns out to be a villain. Ah yes, Anada Armas. You know, let's just take a moment to appreciate her. I already have a feeling she's going to be one of my new favorite Bond girls. Already a serious contender for the top 10 hottest Bond girls ranking 2 if you ask me. Anyway, you would guess he would be the main Bond girl, but I gave it some thought. Drake's Bond never really had a true femme fatale, a chick truly working for the bad guy out to kill Bond. Maybe Anna could fulfill that part, and LaShana Lynch would end up being the main Bond girl instead. I know Leia Seydoux returns too, but there's no way she makes it true to the end credits. Heck, she probably doesn't even make the opening credits. The story is probably going to be centered around Leia Seydoux and around her dying and all. I hope Anna becomes the main Bond girl. I would love to see Craig and his tenure with her in his arms, if that even happens in this new Me Too era nonsense they're on about, but I wouldn't mind seeing her play a Xenia on a top type of chick either. David Densig. I'm clueless about this one. My guess is that he's going to be a secondary villain. Maybe he'll play some corrupt guy working for the CIA that gives Felix some trouble and is later revealed to be in cahoots with the main villain, Rami Malek. Who knows? Well, those are just some of my guesses. Share yours in the comments too. Let's hope things get going and get going well. Craig talked about wanting to go out on a high note, and that's certainly what I'm rooting for too. Here's to the start of production for Bond 25. Finally. Are you a fan of my work, and you just can't wait for other episodes or content to come up to the channel? Well, you can now join the DbF community on my Patreon page. Supporting my channel grants you access to all my latest videos two weeks before the regular viewers get to see it. Your name will also be included on the supporter Hall of Fame, and you get the vote for the new content. And as a bonus for signing up, you will receive a personal thank you video from me as well as these custom Blu-ray covers that I designed myself. These will all be received by email immediately, so you can pop them out of your printer and put them in empty Blu-ray covers to have your own DbF Blu-ray collection on your shelf. All support for giving back to my channel is appreciated immensely. You can find the link in the description and make sure you subscribe to the channel. Thanks a lot, guys.


Hi, here we are. Listen folks, I know it's been a long haul for you as the viewer of this vlog and for everybody as far as this injury is concerned. You know, we've been talking about it a lot and even yesterday trying to jog once again after another little break just because I want to make sure we don't go backward with this injury. But I got to make a quick shout out to, so because of that, in the comments yesterday I read Jason's comment. Thank you. Shout out to Jason. Remember Jason? We did the 5k time trial. No, yeah, 5k time trial, any PR'd. It was amazing. Well, Jason commented yesterday about this phantom feeling, pain, however you want it, whatever category you want to put it into after bone injuries, specifically stress fractures. So we're gonna break it down today. Talk about this because who knows?  I don't even know a better word to use for describing what is going on in the foot right now. So anyway, shout out to Jason. Let's get ready, rock and roll. And bottom line though, we're back to some grass fields, new grass fields to run at today. So that's exciting. Something a little different and just see how it does. You know what I'm saying? Just got keep going, keep going, keep going. Thanks for being here. You too. Okay, just tying my Terakiger fives and I got to make a quick shout out to the medical professionals. If you are watching, I know there's a few doctors and physical therapists that watch this vlog. Thank you for watching. But I need your insight. Do you have experience with phantom pain after stress fractures, after stress reactions? I think a lot of folks who are struggling with bone injuries right now or who have in the past or hopefully who don't in the future, but you never know, these things can pop up as runners. Could you share your opinions, your ideas, your insights into this phantom pain that some of us do struggle with? So we'd appreciate it because listen, I'm a runner. I've got a lot of experience, too much experience with stress fractures. But at the end of the day, I didn't study the science behind the bones and how they heal up. I'm learning a lot more right now, which I'll share more about back in the studio and like the whole calcification of the bones. And yeah, anyway, I won't even get into it right now. So if you could let us know your thoughts and opinions. We'd appreciate it down below. So it's a little breezy out. Hopefully you can hear me. Basically, I just want you to know for start here, I have zero pain, zero pain while I'm running. Okay. I don't feel a thing in my left foot, but I will say, you know, last night there was a sensation in my left foot when I was sitting editing the vlog last night. And what is this phantom pain? It means the injury is actually healed, but the nerves, the ligaments, the tendons around the stress fracture area, the stress reaction are reacting to this calcium deposit that forms right on your foot, right at the stress fracture area. It's all part of the healing process. And it might feel like a stress fracture. It might feel like it's coming back, but and sometimes it does big asterisk, big warning. Sometimes it does. You have to be careful. That's why I stopped last week. And in addition to this calcium deposit aggravating or yeah, basically aggravating what's around it, the nerves, there can also be a mental side to phantom pain that we're going to talk more about back in the studio where it's literally all in your head and this can create a phantom pain as well. It's just which just blows my mind that the brain is that kind of connected to obviously the rest of our body. So anyway, phantom pain right now, I'm telling you, it's a zero folks. It's a zero. I don't feel a thing and want to keep it that way with the zero pain scale. And so this phantom pain, I'll continue to monitor and shout out to all the new subscribers just so you know. I am about eight to nine weeks out from the initial pain in my foot. So I have been as patient as possible, but frankly, now it's just time to see like, I worry, am I dealing with this phantom pain? And again, like I remember this phantom pain from 10 years ago, but a shout out to Jason for just kind planting the seed again in my mind.   Oh, what am I doing in all these colors? Excuse the crazy hair just getting out of the pool and cross training. It's begun. It's a little strange. I must say I've never been much of a cross trainer, except when I'm injured. So now that I'm approaching health, full health emphasis on approaching and not there yet, I'm trying to figure out, okay, what is the balance with running cross training and making sure that, frankly, this doesn't happen again. So anyway, it's good to be back in the pool. And I took that time off maybe start, you know, about 10 days ago, I said, okay, we're going to knock this out. And that means I'm just going to do nothing. I'm just going to sit and be quote unquote lazy. And so far, so good. Alright, let's dive into phantom pain with respect to stress fractures. That's probably why you're here. If you found this video, welcome. If you're searching, maybe you're struggling with stress fractures. I'm just going to read a couple things that I found. Yes, online, some just from personal blogs from, you know, runners who have struggled with stress fractures, but then also from some cited sources, medical journals. Alright, so I'm just going to read here. Here we go. When this calcium deposit presses outward on other tissue, including nerve tissue, one can feel different sensations. Remember, I was talking about tightness the last like 10 days or last two weeks, like it just felt tight in that area. I don't know. Maybe that's what I was feeling. One can feel different sensations, including, including numbness or tingling while running for several months afterwards. Now, I don't really feel it when I'm running. It's really just when I'm kind of sitting around or maybe walking around in the boot. It just felt it feels a little tight. So who knows? Okay, then it goes on. There are there are a few ways to tell if the twinges of pain you are experiencing are the real thing, a stress fraction or stress reaction or not. Does the pain remain steady or progressively get worse throughout the run? If so, check with your doctor. I always say that. Don't listen to me. Check with your doctor. If not, and the pains are sporadic and very, very in intensity and location, chances are that the pains are phantom pains and are not a cause for worry. This is beyond exciting. Like I can't even tell you how exciting it is to just read those words. And that was from a medical journal. And I all have to find the citation and market and link it down below. So basically like it is moving around for me. It's it's not always in the same spot and it comes and goes. Like it's and that's why I've told maybe you, I've definitely told true love like it's confusing. I think I said that last week to you guys. It's just I'm confused. Like I don't know what's going on. So the fact that it's not like one spot, it's not, it definitely doesn't get worse throughout a run. This is good. And as I said earlier, like it's basically a zero on the pain scale, especially today. Like it feels really, really good. Okay. What else do I want to share? One little else. Oh yeah. This is just a um okay. This is a quote from an actual from a personal blog from a runner. Phantom pains can be caused by aforementioned calcium buildup that can also be mentally induced by fear of re-injury. Guys, ladies, this, this, I mean what I felt over the last two weeks. It might, I told true love. I haven't always told you guys and I'm telling you now like I've been afraid. Like I just told, I don't want to go backward and there's a fear factor there and I'm not saying that's a good way to approach living, to approach running, just getting back into running. It's not a good place to live in fear. So they go on to say, think of it as post-traumatic stress fracture disorder where instead of mental flashbacks to the time of trauma, there are phantom pains. So, oh man. Remember yesterday I talked about, I purchased a book, Running Is My Therapy. Like all of this connection to how our brain is reacting to different stimulus stimuli around our bodies connected to pain, connected to just the sensation of getting out there and running. All right. So there you have it on phantom pain. Just kind of reading from different sources whether it's personal experiences or medical journals. I had remembered phantom pain and I had told true love and we're going to talk to true love in a minute. I had told true love about phantom pain like two or three weeks ago but it's just nice to be reminded about it from various people, various sources and right now I'm excited. All right. We'll get you the question of the day in a minute. Let's go outside, get a little grub in our tummies. Does it get better than fries, burger and a shake? I submit that it does not. Right true love? I agree. Basically, today we're talking all about phantom keyword. Key word, not phantom of the opera. Phantom pain. So shout out to Jason again. This commentary today, yesterday and he's the guy that did the 5k PR that I filmed. He just reminded, and remember we talked about it two or three weeks ago, the phantom pain and basically it's you know I did more research today and we might never know for sure but I think that might be what's going on and it's very exciting because the pain scale is so low and it's moving around. Yes good pain tolerance. So anyway question of the day, who has suffered or struggled with phantom pain for anything? I don't even know. Does that doctors out there remember I asked earlier, can you suffer from phantom pain in like a soft tissue injury like a hamstring or a calf strain or I don't know any sort of soft tissue? Bone, it obviously I think it really can happen and anyways doing a bunch of research today. Hun four miles nice and easy on grass and no pain during the run really no pain after the run and remember last night I was like just a little little something something and so today nothing it's very exciting phantom boom phantom come on seek beauty work hard and love each other see you tomorrow on it this is the guy I keep thinking of the phantom drone Is it there drone?  Oh yes the phantom four yes


Hey, what's good y'all should boy down from the down showing friends and we back for another k-pop reaction and today I am going to do a tease in session dance practice so I Don't know what to expect But I can already tell you from what I'm seeing on the screen It's gonna be some army move. That's gonna be crazy Let's see what happens. Oh, by the way, if you are enjoying this page, right? And you want to continue supporting by hitting the like button subscribe and sharing check this out Please do me a favor if you look at the heart There's a thank you on there if you want to support this page hit that Thank you and we appreciate your support for real and I could also really prioritize these comments because some y'all leave a lot Of good information and it really helps my journey in learning the whole k-pop culture and more So I really want to make sure that y'all know that I'm actually really paying attention me and my bros are definitely All right. Well, what I'm for to do. Let's go and get started on this video. All right Very theatrical Very very theatrical very theatrical very dramatic with character with attitude and Nothing like too over-aggressive, but a connection to the song and I'm not gonna cap I have only heard 12 seconds of the song and I hope the song is like this the whole time Let's go Yeah Okay Oh I see why y'all requested me to do this this has a lot of Different approach. So what I mean by that is this is so you have very like very theatrical Transitions nothing that's hard hitting like at the beginning if we watch the beginning if we go back It was like it remind me of waving and it reminds me of like robotics and animation of how they was moving a body coming up and Moving like that, right? And then it just changed. Let's watch that again Because we really about to watch this video. Okay Like very like yeah, you see how sharp it is right boom and then right here just Whoa and then right here It was hitting a tack tack tack So very just simple But I feel like when the music started to change and kind of go upbeat That everything just started to like change the way they was hitting the music like right now.    I know I know what we talking about Boom look at that Now they go sharp with it. Yeah Hey, I like that I like that combination.  Okay, listen all you guys out there You can't be scared to hit that part right there. You saw it. Let's go back You know the time it is we put it on mute and we look at this choreography check this out y'all eating this shit, right? Boom slide in bow. You got it. You got to show them. Oh, boom You got to hiss that if you ain't never seen Prince and Michael Jackson hit that before Go do your homework. You got to hit that move right there You better watch some James Brown Michael Jackson and some Janet while you at it. Mine's well Make sure you put that Prince in there.  I just want to pause that because I was dope I Love how they came up.  I just love that part. I just want to go back and see that part again I love this. I won't put on mute real quick.  That shit was hard See what I'm saying that keep changing I love that and and it's clean it's not like change up the textures change other check there and it's choppy. Oh don't worry.  I Love this Okay, somebody's been taking house class Slow motion. Way to play with the music. Uh huh. Did y'all hear the ba ba ba ba? And they hit it. Watch this, watch this.  Come on, what? You better know the trick that you made. What we doing right here? Is this gonna hurt the choreography or make the choreography? Let's see. Bring it back. Okay. Oh, wait a different way to... Yeah, yeah. Listen. Listen, listen, listen, listen. I want to say this because I think the video about to do a commercial because of YouTube ads. But let me say this right now. The choreographer and the director behind this choreography, whoever you are, and I know in the comments, I already know what y'all are going to do. Y'all are going to tell me. That's why I love y'all. Y'all let me know. Like, hey, this was the choreographer. I love y'all for that. I really do. Genius. The reason why I say that is because they're genius because of the fact that when it comes to the chorus, you know that these bands, these groups are going to show a part that they already did at the beginning. But the way that they're making the choreography where they be like, yeah, they are, but we're gonna set them up different. We're gonna have them come in different, but it's able to keep the audience together because when you see the chorus of the choreography, it's the part that everyone wants to do. That's when you see the men and women out here doing the choreography at the concerts. They're like, oh my God, this is my favorite part. But they did was change the transition how they came in. Exhibit A. Let's go back. Y'all remember these parts, right? Let's go back a little bit right here. Check this out. Y'all remember this, right? Y'all remember this, right? Just watch. It's coming back. It's your spot. Pull it. See what I'm saying? Da, da, da, da, ba, ba. And come in, man. These boys is bad. Let's go. Yeah, see? And now I'm coming back and I gotta hit it again. I gotta hit it again. Yeah, see what I'm saying? Smart. Smart choreography. And now you bring it up. Yeah. Playing with all their textures and how their bodies hit it. If you don't know what I mean by texture, okay, let's talk dance real quick. In Crump, we talk about texture and character. My teachers and my elders taught me about texture and character when I dance. You can take a move and do this, right? You see how that is real simple, right? I can change the texture of it. Boom. I can make it aggressive. You see what I'm saying? There's so many ways you can hit the boom, boom, boom, boom, boom, boom, boom, with the ta ta ta ta ta. Boom. Okay. They're playing with the character and the texture. Hey. Ugh. Very clean. That was clean. That was very clean. Ooh. Ah. Ah, this was great. This was great. I like this. I like this. This was really good. You know what? I'll be honest. You know what I was thinking they was going to do?  But you know when it went down, you know, I love how they went down and turned. I always thought they was going to make the body a ripple. So imagine me going down, boom, boom, boom, and it was like a ground move. And then they'll pop right back up in like a slow motion. But I saw that they, they like, you know, did like the ripping part. That's just something that I saw. I'm like, that's a great move. I think I would have had something crazy, something from the B-Boy element. See that? I would have, yeah, you see? But this still works. Yeah, this still works. This works perfect. Yeah. Yeah. Yeah. It still works. It still works. Yeah. It still works.  It was great. I know I said a lot in this video when I was talking about like how they was changing it up and texture. All I'm saying is it's like, it's so great to see a group. A group that can change up with the music that they already created in the studio. So now you got to learn this choreography that either you as a group or band make or a choreographer makes. And what's tight about it is the fact that they was able to follow that same attitude in their choreography, in the dance practice. So that, you know, I already saw that they have a music video. So I am definitely going to check it out. Either I'm going to watch it by myself or I'm going to see the bros want to watch it with me. But yeah, I just, um, it was really, really good. Like, you know, I love to see that, you know, you was able to change up and follow with the music, how your music is changing up the feeling throughout the whole song, right? And then, you know, you connect the choreography. So it's like hand in hand. So any creativity about the transition was really like, I didn't say too much about transitions and stuff like that in the formations, but I really did bang with that. I really, really did because it looked so good. Um, and the way they did it was it was like, it was, it wasn't so small transition. It was like they was, they were like, do something where the movements will happen and things was just like different changes and then different speeds was played. That stuff was tight. That was really good. Um, these are one of those videos that I would have did a part two and went back and found something different because with dancing, it's just sometimes, it's the little things that you'll miss or be the little things that you do catch. It just depends on the moment of what, where your eyes are at when you're watching, um, a movie, choreography, your favorite cartoon, your favorite anime, depending on what you're, what you're watching on that TV or your monitor, you can find something different every single time. So that's my little tangent and I appreciate y'all. And once again, if you'd like to support this page, hit the like and subscribe button. Um, make sure you subscribe to my second channel as new videos are already getting ready to be uploaded. I'm just trying to make sure they don't get blocked or copyrighted. So, yeah. But other than that, y'all, I appreciate y'all. I'll see y'all soon with more videos. We out of here. Peace.