Okay, so we are ready to give a brief overview of observational cosmology. Okay, so this is really going to be a brief overview and also it's kind of hot spot because it's a, I'm going to go through different things in observational cosmology in the video. Okay, so what I'm going to do is that show you kind of numbers and figures of blocks that people generally see in papers in cosmology and then try to see why we get the skeptics, how you do it. Okay, and other thing is, so most of you probably were there when I was doing the Wednesday colloquium where we had this one above. So actually one of the thing I would like to do is that I would go back to that talk and explain a few things in slightly more detail over there because over there it's very flashed to me. So things that you generally see in a cosmology talk is like this, which is basically one of the main of cosmology is to know at present with whatever data we have, is to know what are the things that make up the universe and what is the quantity of those things, the nature of these things. And we believe in certain models, of course those models believe that models do not have time for observations. So one of the models that we believe in is this standard hot big bang model and it's called the standard model of cosmology.  There can be competing models of course and at present other than models which are very different like let's say a steady state cosmology, we really don't have enough power to distinguish between very, very final space competing models. So what we do in cosmology is not model selection, in the sense like not like whether this model is right or just other models are right, but we mainly believe in this model and let's see whether we can know how good this model is. For example, we believe that inflation is one of the reasons that everything started, the structures and the work of it, the seeds were not there, but there will be other models also. Or even with inflation there will be different kinds of inflation and it's not so easy to differentiate between different models. What we can do is that we can say that inflation gives certain predictions, certain very general predictions that we can get. Similarly with dark energy for example, we know that the universe is going through an expanded expansion and there will be some new form of energy density and we can basically say whether it is whether we have it or what accuracy we have. So saying whether this is the model we are having, it still can happen. So things that we generally see is that for example, I have given here dark energy w, w is basically the equation of state which connects your pressure to the density. And over here it's omega matte, omega matte is whatever is your matte density. And what you see is that things like this will have plots which enclose our region in the parameter space meaning your data is saying that you must live in this region of your parameter space. That's the most important thing. Data is not saying that your omega matter is this value or omega w is this value. Can you please explain what the axes are? Yes, so as I said the axes are the x axes, these are omega matter and omega is basically let's say dark matter and normal matter here. So omega is basically any omega i, so let's say there are different kinds of species of matter, there's radiation, dark matter, dark matter, whatever. Any omega is the density divided by something called a critical density. And critical density is basically the amount of density that basically makes the universe absolutely flat. So if I add all of these two i's together it will become, so this is a fraction, so it will become equal to one. If I add this omega i's together. So if this is matter, if it's matter, matter meaning matter, matter means very often for example, or dark matter which are massive, then this is called omega matter. If it is dark energy then I would call it omega dark energy, if it's radiation I would call it omega radiation, neutrinos I would call it omega neutrinos. All of these added together will be summed. Just to be clear, so we studied in class this equation that said energy plus energy plus energy, in the case of a flat universe is E dot squared by E squared. So the normalization of this equation is E dot squared by E squared. And therefore, by matter you mean dust. By matter I mean dust or dark matter or anything which is zero pressure, which is zero pressure. By dark energy you mean this. So in general, whatever be the component you will have some pressure to be some time strong. And so this W is your equation of state and this will be different form. So if W can be zero, so that's like dust. It's polishing. So omega matter is P equals zero, it makes a radiation as P equals E by T. And omega times C is minus one or it may be something which is changing in time. This doesn't have to be a constant also. The minus one is because as we've seen for God's logic, it causes P plus C is equal to C. And what about the neutrinos? So neutrinos very much is like radiation. So we can just club it with it. And so if you look at your, so let me see. Sir, this order is row critical. So row critical is something that makes this thing. It's the right-hand side of the equation. It's energy plus energy plus energy was equal to something. That's something is wrong. So it's row critical. Then it was summed across the square. This is the equation. So this is basically just as we have different energy plus energy plus energy plus energy. It's an all-faction, of course. And these things behave differently with this time or with that shift. And the way it behaves, of course, depends on the W. But over here what we can see is that if it's omega matter, then it scales as that. So do you know what A is? So if it's omega matter, then it scales as 8 to the power minus 3. Meaning if the volume goes to the factor of 2, your density will go by a factor of 8. It's just like if I just count people over here, if I increase the size of the room by a factor of 2, your density will go to the factor of 8. For radiation, it has an extra value. That is because you think of the energy density of each variable as your expanding. So your lambda is sort of invisible. So your mu is the thickness. You generally don't use, I mean, because this is omega not much different. And that goes as 8 to the power minus 2. And then this x over here is basically dark energy, anything which is... And that has a thing like this. So if W is over here 8 to the power minus 1, you can see that this is just a constant. It doesn't have any dependence on that shift. So one of the things to notice immediately is that if I draw these densities over here, this is something which is actually very important, very simple but very important. If I draw these densities over here, the function of z, z is basically z equal to 0 when in present time, a is 1. So this is a normalized shift. In fact, if you could have normalized some more else also. z is the red shift. z is the red shift. So a is 1. So present time you say a is 1 and anything, any scale of the universe is scaling it present time. We could have scaled this and redshift off something different. Let's say I'm done then equal to 1 at the c and v redshift also. It doesn't matter. But this is our convention a is 1 and z is 1. So if I do it z, then you will see your omega radiation will fall faster. And this is something that I think she has already told. Then there will be a quantity where the radiation and matter will be equal. And later, because this third part is equal to minus 1. It's constant. So this will be omega r. This will be omega n. I still didn't get how to read the products in short in the previous video. Yes. So I'll come up to the basis of the rest of the term around here. So one of the things you see over here is the dynamics of the universe. And these are the wishes as big as the dynamics of the universe. Different times in redshift, it will be sort of influenced maximally by different. So in the early universe, it is the photon density that is sort of gliding there. Around redshift of around 10,000 degrees. And you can actually capture. I was just equating these two densities. You can capture what is in that shift. This meta density is there. And only very, very recently, around redshift of 1, when redshift of 0 is somewhere around here, it's infinity because it's infinity. You will see that this dark energy density is there. Now what can also show that in these different regimes, how the, if you look at the size of the universe, the half, how that goes with time. And you will see that at this region, your A, because A tends to the size of the universe, the expansion factor. This goes as 2 to the power 2 hertz. Over here, A goes as 2 to the power half. And then over here, A goes as exponential of. So the universe was having some expansion. And then suddenly the expansion becomes exponentially faster. Can you give us a rough idea of the numbers? How are we going to do the universe with omega or omega in the process? So, what is the age today? So the age today is basically T0. That approximately is 14.2. And the error bars of this are what kind of position? Very, very small. It's actually plus minus 0.3 or something like that. So age of the universe is very, very multiple. One of the things is that, so the reason that we need to have this dark energy also comes from the age of the universe. So if you look at the age of the earth, or the age of some of the stellar systems like the sun, then one knows exactly how long a star lives. And some of the stars can live very long. It's not a massive star. A massive star lives for a short period. It's not a massive star. It can live for a short period. It can live for a long period. It's a small star. Then you can have an age which is quite high. Now it turns out this age of the universe is a function of all these things. So very simply, if I take this age, if I take one of our age, that, very 90 degrees of age of the universe. So the fact that we need an age which should be larger than the age of the star, or that age of earth, we would require a component of the universe. Because if you have only a matter, suppose you need many of these, many of these over here, but just only a matter, and put that 1 to 1, you will find that the age actually is smaller than the age of some of the stars. So that's an example. So this time is not calculated from this equation. The time is actually capital on this equation? But this equation, we don't know what the x side of the value of this. Yes, so the whole point of this quality is to find out this omega-matter, omega-matter omega, omega-head beta-x, and it's possible to find out the w's, that we are given at w, w, w, w, w, w, w. So how do we know that times were given? So, if you know this thing, you can integrate this, integrate the one of each, so that gives you the time. So, error parts of this how well we know any of these quantities, will tell us how well we know the inverse integral of this. Inverse can be estimated from the law of cost of integral of that part of the screen. Right. So, cost of integral of that part does not give you the edge of integral directly, gives you all these parameters. And then you put in this parameters and calculate the edge of the universe. So, for instance in a completely dust dominated universe, we can measure Hubble's constant now. Then there is a particular relationship between the inverse of that Hubble's constant and the edge of the universe. But in a radiation dominated universe, there will be a different relation. Exactly. Okay. So, in this, so depending on what the orbitals are by measuring Hubble's constant, you get different answers for the edge of the universe. All the same order. Yeah. Because television has to be one over each. But the numbers are different inside. So, Hubble constant there is an observational observed thing which we know what accuracy show. Hubble constant which is 70 plus minus 1.5. 1.5 in what units? You know which is plus second plus meter plus. So, it's almost a percent, couple of percent accuracy that we know. Right. So, we know Hubble constant is very accurate. Now, if we knew the model of the universe, we could get to the edge of the universe. But that depends on, if we are asked to get one answer, if we are asked to get another answer. This combination looks like another. So, if you are asked to get this, how much will it be? About 99 million. Yeah. Over 99 million. Yes. Which is problematic because some of the stars are older than 9 million. One of the thing that the dark energy automatically solves is the fact that we have an edge of the universe which is older than the stars. We don't want the, we don't want the Earth to be older than 9 million. So, that's that's all. It's long from this problem set that we really get today. Put it in this problem. Again, dust universe, give, dust universe, radiation universe and the real model find the relationship between Hubble stars and the H0. And remember, all of these are normalized to H0. So, it's very important that you know this H0. So, this is what local measurement of our consequence is. So, if you just take this H0 and get enough idea, but you need all these other things if you want to do the integral. That's my question. So, cosmic microwave, I'm not going to be caught up in the radiation from the universe that has been redshifted to the expansion. Yeah. So, what paintings was the radiation back then? I'll come to that. I'll come to the cosmic microwave. So, I'll come to all of these things. So, this part is still not clear. If we use that equation, we can do all the time. And we should have thought it could be omega and w. This one? The actual equation to determine the time. This is the master equation it tells us. What is the expansion? This H is the expansion. Because what is how the constant equation? It measures how fast something is expanded. So, this one will get the expansion of the universe. What is the last term? Let's show you the last plot. See. So, let's say the last plot has these two numbers. This is omega n and this is omega n. So, this is more of a data analysis. Suppose you, by some way, you know what is the value of H at omega 2. So, it will mean that I can have any choice of all these things that should be of nature, sir. What value of H? Suppose I draw these things very accurately. Let's sort of take a prior that we did the same. Get rid of this. We also add present H. Radiation is absolutely dynamic. So, we'll get rid of this as well. So, it will happen with this and this. So, the radiation will affect the H of the universe, but only very little. Yeah, exactly. So, and then because, as I said, I've taken a prior that this is, this is my future. Omega matter and omega x together is 1. So, I don't really need omega x. I can add it as 1 minus omega matter. So, there are two parameters, this omega matter and w. Now, I can understand if I get a value over here very accurately, I can always have two parameters around so that I can get the same value. So, what you're saying from this one observation, just one simple number, is that there will always be some, what I'm going to call as a degeneracy between these two parameters. I can always move this parameter over here, and move this parameter over here. So, when we look at plots like this, this is what is known as a degeneracy point. Of course, this plot doesn't come from one value of H. It comes from the same value, and I'm going to tell you how it is. There are two factors that you have to be in this region. So, your observation tells us that your value of w and your values of any combination of values over here is consistent with your observation. That's one important thing. The other important thing that I want to sort of point out over here is that if you look at this region, there are two kinds of, one is these straight lines over here. The other is this shaded region. What is the shaded region? The first one, this bigger region comes when you have observations from the same point, meaning one kind of observation. An observation, right, would add something extra, which comes from large scale structure. Over here, the same thing came from WMAP observation. The large scale structure comes from this, so our digital skies are the same. Then, of course, you have a smaller region because the more data you have, there is more degeneracy and you're broken into this region. Different data sets respond differently to your parameters. And so, you sort of make a small region. And the ideal for small region, of course, is to go for different data sets, think of ways to sort of shape this smaller and smaller and smaller. The more accurate you can get, the more accurate you can tell the system. What does it do? Similarly, over here, say, another example. So, over here, as you can see, there's, again, a W. And over here, what we have done is that instead of having a constant W, as I said, this W can also be a function of your time, meaning a function of region. So, here is the W equal to zero part and here is the part, which is telling you the function of region. So, we can write things like this. We can write a W like this. So, in this region, from the data, you can say that this is a region that we do. Extrane formation. Then you sort of, in this case, slightly move. So, we need to do extrane formation. One of the points is that we are kind of, measurements that are happening right now, we think it's going to change this kind of process much, much, much better. What are the actions? So, I was saying, these are the two parameters, but just, that is W prime. Now, this is just, the reason I am showing this process, just to show that this is how parameters whenever you open up this logic, you will see parameters being listed like this. So, if I look at the latest double map paper, then you will see something like this. You will see a list of parameters and you will see what is their value and then you will see, basically, what is their rewards. As you can see, at present, there are certain parameters which we call the private parameters. This is basically your model, you fit your data with these parameters and you get the error-passing parameters and there are certain very important parameters over here for double map, which we call derived secondary parameters, which are basically some combinations of this. So, you can think of something like this. You can think of a matrix, which tells us what is the error bar on each of the parameters and what is the correlation of the errors between these parameters. And this will be the jicomelain class for another set of different parameters. So, what are the private parameters? What are the main parameters of the standard responsible model? There are 1, 2, 3, 4, 5, 6, 10. One is the amount of variance, omega b. One of the reasons, another thing you want to see over here is that you see, over here, we are talking about omega i, omega matrix, omega variable. But when you go into observations, close to a time, you will see it has an h squared. This is because, what you are going to measure in the cosmology is scaled by h, pages of the Hubble's constant. So, your Hubble's constant that you generally use is h0. So, this is 100 times small h. Small h is the notion of this. Until you understand, you can look at the possibilities. So, the volume that you measure goes as this h. So, densities are always, because these are all densities, right? Because they are all scale. So, whenever we put something in cosmology, most of the time, these are in the possibilities. So, one of the basic parameters is the amount of variance. And we know, as you can see over here, if you look at the amount of variance, it is... It's always omega b today, right? Yeah, omega b today. And page is h today. H is h today. The other thing is the amount of whole dark matter. So, basically, I have the dark matter. So, we will not make the difference between whole dark matter and whole dark matter now. We will assume that dark matter is whole dark matter. One form of dark matter will be really big. These are really large, big sized dark matter. The other will be, say, a hot dark matter will be like nutrients. This is the number. That's what it is. Then there is this omega lambda in this dark energy. And there is something called the spectral index. What is the spectral index? Something which is called the tau, which is the realisation optical depth. Now, here comes what it is. And the last part is basically that amplitude of fluctuation. Now, first let's start with the spectral index and amplitude of fluctuation. So, we have learnt about fluctuation. And in fluctuation what you have is that fluctuation is generated. And you can write down the possible fluctuation. And if you write down the possible fluctuation as a function of some T a, possible fluctuation is basically the real transform of the fluctuation. One second. As far as the F1W model is concerned, only the first three parameters matter. And that do the sum of the first two and the third three. So, F1W doesn't see the last three parameters and doesn't see the first two parameters independently. So, this sees the sum of these three. So, sum of the first two. Lambda goes to what? There is omega lambda in this dark energy pattern. So, the sum of the first two has dust and the second one has dark energy. And then and the Hubble's his Hubble's it's mostly independently about it. How constant is and as you saw that equation, the eight squared have an eight zero squares in the front. So, you have to independently get this. Yes. From CME alone you can't cannot get up as fast. Unless you do something which is known as lensing of the CME. So, that's different. I'm not going to do lensing of the CME. Right. I'm going to do lensing of the CME. There is a different model. And then there's two parameters which are the coefficients. And then there's one parameter which is called the tau which is actually a pairs of logic parameter which is an antifaceted parameter but I will come to that tonight. So, talking about this s and this sigma squared r Then you have some more spectrum of protections, perturbations is in density and if you fully across from that you will get a possible and that can be done as some amplitude of times it will go, some spectral effects. Initially just from scaling arguments, because we can just say this in must be very close to 1. So, there is no preference scheme is basically this is not exactly this sigma square r, but this a is related to this amplitude. So, this value this thing tells us what was the amplitude of amplitude, what is the amplitude of mass fluctuations. So, it will from a link to everything, it will link to what is the amplitude of c, it will link to what is the amplitude of n axis that are forming structures all of these. So, more of this means more of signal division, more of amplitude of the axis. This n of s is the escape. So, what is tau? So, to know what is tau will come to. Now, the flash parameters are. Why is this fluctuation? Now, why is the fluctuation. Why is the fluctuation the density of the. Yeah. So, when you learn about inflation it tells you that inflation can actually generate fluctuations in the. And what we believe is that those are the seed fluctuations that grew due to gravitation and stability into whatever things that mean. Now, why there is a fluctuation that is as the story I would not want to go into that today. I mean I can talk about this later. It is a very small fluctuation, very small fluctuation. This is the fluctuation that plausibly is due to quantum fluctuations in the. Now, we discussed briefly that in the center space, even the most symmetric vacuum in the system, there is one fluctuations which get frozen in classical evolution and then enhanced by gravitation. So, this P of k is a measure of those. And these are, remember these are very, very early. So, these are before or during the time of inflation. So, this kind of fluctuations can be used in expansion. And the number of times it is exponentially inflated is actually one of the gravitation that inflation theory is trying to find out. Now, over here we talked about derived parameters. As you can see if you have this Frippel equation, then from this three you can grab one of it. This sigma 8 and this delta square r both are basically connected to this magnitude. The delta square r because what we are looking at is basically in the radiation area from the C of B. One can, from here one can actually get a value of sigma and I am showing how to do it.  This is a realization is basically integral over the realization of the derived. So, this is tau is basically how many times it is going to be scattered at that sheet which is closer to us. So, remember that what the C B is doing is that C B has some photon fluctuations at the sheet of 1000 and this photons are coming to us because at that sheet of 1000 temperature has just become such that the electrons and the photons can become atoms and so the photons are no more scattered. So, it comes to us after that. Now, that very close to us what happens is that the stars and galaxies begin to realize the difference and so this photons starts getting re-scattered in the game. And so we would not really get a snapshot of exactly the C B field at 0, we get a snapshot of what you know the fact that some of it has been re-scattered at low reactions. So, whatever we get has been slightly modified and to do that we have to put in a parameter which has sort of in some sense linked with other parameters. So, if you do not know this tau very well, you will not know the other things because that is the imprint and then come to the imprint is basically modified by this thing. So, this tells us what are the main parameters, what are the branch parameters and of course, this is the basic set. You can go on adding parameters to this for example, you can say I will look around the neutral, simply neutral omega and neutral over here. So, you can talk about the number of species of neutral that can be a parameter. It does not have w over here, you can put in a w over here, you can put in a w prime over here. So, it is a long list of parameters that we need, but basically this is a parameter and remember something, if your number of observational points are not in it, it is some basic set of observational points. So, the number of parameters that you put in dilutes the accuracy with which you can actually determine one of these parameters. So, it does not make sense always to have more and more parameters. For example, I can say well are you putting a parameter which tells us how Gaussian or non-Gaussian is this, is this primary operation by suitability parameters. What will mean is that you may have to have accuracy to get it. So, it is like flow or observation is still the same observations. So, when we put in a parameter, we add a parameter, it has to be a judicial choice. How much we should, so that that is what it is a parameter that you observe directly. We do not observe any parameters, parameters are fitted to observations. Observations are made of these parameters. So, observations are CMD or Laskan structure and this is what I am going to talk about. So, let me put very briefly you observations there. Then it just means that derived are functions of the other. So, there is some set of independent parameters and then some other set which can be required as functions. So, all of these parameters are derived from observations. Do not observe these parameters, observe some quantity which depends on these parameters. Any of these parameters, suppose I give you different functions of these parameters, that would be absolutely. So, is there any particular reason to choose these particular? What is the difference between the parameters and the derived parameters? In the first place, the parameters are decided. So, for example, your a better parameter choice is let us say omega baron 8 square times omega baron because what you are getting from your observations, the way omega billion goes in is basically as a density and your densities are scaled by 8 square times omega baron. So, that is good. Through this derived parameter, you have to put in certain conditions. What is that? One of the conditions is that of the azur flatness. If you azur flatness, then then you can relate from these and get an interest. But, when you are doing this primary parameters you do not have to issue an interest number from the regret. For example, when you deny this parameter sigma 8, it is right you have some assumption in the sense of this thing goes to sigma 8. So, your primary parameters in some sense are more robustly linked with your data than your derived parameters in the sense that these derived parameters takes in more assumptions. So, if the fluctuation at this scale is this, if I want to know the fluctuation at some smaller scale, how do I link this fluctuation to this fluctuation? That's an assumption, that's an assumption of how the fluctuation is. For example, how, what we can get is the total amount of scattering along your line of sight. That is sort of what you get from this. Whereas, same realization it tells us at what red shift the scattering starts. Now, I can have different kind of you know, the scattering may be small and larger red shift. So, you can have different history of scattering. The integral will be the stop. The integral is what you get, but what is the form of the integral that can be very more. So, by here for example, the realization is, as you can see over here, the red shift of realization. If the universe was realized in statings, since the very, very above model, it says that the universe was, you know, it says the universe was like this. You know, the realization was completely un-realized and then that's some red shift itself. You know, be like this, maybe this can be your realization function of red shift. What you are getting is basically tau is the integral of this. Now, when you have more and more data, of course, you will be able to look into the history of realization instead of what? That's the next one. So, these things are much more common. That's why it's often the realization. How does it, I'll come to that. I would have to say like the omega, like the critical quantity will be divided into normal and that also. The critical density is something we define, right? Critical density is something we define which basically gives closure. It's basically this, as you know, from some observations, then you know what is critical density. Critical density is a function of the small angle concept. So, critical density goes up and down with something. So, remember because the critical density has a small h squared. This is why all your omega b e are multiplied with this h squared, omega even multiplied with this h squared because omegas are respect to critical density. So, it's k s up and down by h squared. So, you always have this h squared over here. You should cannot get directly from this observation. So, somehow you will have to get h squared from some other observations. So, when you look at all these observations, what are the observations that you look at? Supernova is something that you already know. CMB is what we are going to talk about. Then there are objects which are formed due to the expansion between these small small perturbations going. And one of our observations is this really large structure for the process of the axis. There is something very interesting called baryonic acoustical situations, which is basically just like the CMB, but not in photons, but in matter. It's actually very simple. So, you know that g r can take dense light. So, that means it's some density. And we have also been talking about gravitational waves approach at this point. So, let's go. We want to look at. So, I'm giving you an idea that if you add different components, then you sort of reduce the region in parameter space, which is the reason you want to look at different assumptions. There's another way to work in that. So, over here, this is like shift. And this is a main density. And remember, I told you that I did it in terms of omega matter. This is again the same thing. There's no way going for matter. It is going as what does it do for possibility of constant. And what you can do is that from these different observations, we can prove what was the density of the universe, the density as a function of your reaction. So, for example, when you are very, very far away in the CMB region, so reaction to 1000, in this region, the densities are very small. This is a case for meter space. And then, by looking at different, different probes, you are probing different epochs of the expansion. So, if you really want to know about the history of the universe, you need to know different epochs of the expansion. And those different epochs of the expansion, those different density scales. And those are basically done through different epochs. So, one of the interesting thing that you can see directly is that, suppose I fix my data at this height of after shift of 1000, this is again, but I mean you can go straight line. Then, if I extrapolate it, you should have got data points over here. This is completely dominated by omega matter. The data points that you find is over here. So, by fixing your scale, if you couldn't fix the scale, then you could have done it like this. Because the CMB fixes your scale at this kind of densities, it means that if you have to have densities that enhanced over here mean densities, then you have to have some extra output. So, by probing something over here, CMB, and let's say, lensing of clusters, you are having a better angle on your two different densities. If you didn't have this part, then you could have said that it is like this. So, you could have fixed it with something, the normalization would have changed. Similarly, if you just had the CMB, you didn't have much clue over what was happening over here. So, look at that. So, different probes, different probes, most different size scales, different densities, different perturbations. Only when you sort of combine these different probes, you actually break it, because you have that problem. This is something that you have already done in some sense. Forget about this dark energy, any cosmological parameter, it sort of orders the whole expansion, which is this. So, this then modifies distances to an object, because if the expansion changes, then distances to some objects will change. The other thing is that different cosmological parameters also modifies that evolution of structures. Why is this so? What will happen is that, if there was no expansion, structures go due to merging of small, small structures due to gravitational instability. So, if there is no expansion, things will go faster, because things will go closer and closer.  So, now there is a, there is a play between the things moving away due to expansion and coming closer due to your gravitational attraction. So, your cosmological parameters will also change how the structures are going. And this is a denominator. Well, the interesting thing is that one can sort of combine this and this, fix your cosmology and then this will be only due to your gravity. Meaning, if you have a different kind of gravity, then you will be probing this, you can combine this too, because this is just due to cosmology, this is due to cosmology and gravity. If you combine this thing, then you can prove gravity. So, the large structures can be used to prove different forms of gravity. So, suppose there is, it is not g r, it is something else, there is some added term in the action. And those will change in this g r section, in this space. So, this is something which is becoming very popular these days, probing gravity or what is known as modified gravity in this kind of cosmological. So, this is something that we, so the first thing is this expansion. And as I said, due to the expansion, you have this different distance coordinates. So, Shrestha, did you do this luminosity distance? We talked about how luminosity was, I don't need to know luminosity distances, we discussed how we could plot a luminosity dimension. So, when you talk about distance in cosmology, it is slightly different than talking about distance in everyday life. So, in everyday life, distance between me and you is the same distance that we will put in if I want to think of how much flux we are getting given a luminosity. Or if you have certain scale over there, then the angular size that we will give to that scale, to the angle over here, will be the same distance. In cosmology, because the expansion of the universe, it turns out that all these distances are different. So, the simplest distance that you can think of is what is known as the coordinate distance. If you know what is the red sheet of the object, I want to know what is the coordinate distance to the object. A very simple way of thinking is that if we leave it in a two-dimensional world, so let's say we leave it in a sphere. So, the distance between Bombay and London, if I go like this, it is a coordinate distance. That is the same distance. It is not the same distance. And if, of course, if it is really large, if it is flat, then it can just, it is a flat x squared y squared z squared. This is a coordinate distance. And this coordinate distance is unchanging. This coordinate distance is unchanging. Just like the coordinates on this flat metaphor, the homogenous isotropic metaphor, galaxy statement. Now, if you want to make it slightly more realistic, remember that universe is expanding. So, if you are outside this expansion, so basically your scale is not expanding in the universe, then the distances that you will see will be actually the coordinate distance multiplied by this expansion rate, which is basically multiplied by this scale factor e. So, if we are on the universe, if we are expanding the universe, then we have a coordinate distance, which now is called differentiate between these two. It is not even homo-moving distance. And if you are outside the expansion, it is on the proper distance. The only difference between proper distance and homo-moving distance, homo-moving meaning moving with the expansion. Proper meaning is opposite to the expansion. So, it is multiplied by t. The thing to notice over here is this coordinate distance is an integral of this h of z. So, if you know what is that h of z, you can take a coordinate distance or inversely actually integral of this. So, you can find out what are the components that was in the h of z. It is an inverse problem. I do not understand this. What is the distance between two points? Just a little bit more. Yes, a little bit more. How do you relate that? What is that component thing? So, it is basically just a… This is the way that the component goes, right? It is a zero to r inverse of t, two to z, something like that. So, you are going along the component. Yes. And you integrate one over the other. Yes. Along the path to the trajectory of the center. That is why I said it is a good center. Tell us all that. And these are the electrons. That is what you call this coordinate. Just a coordinate. This is a coordinate. Distance to distance to the source. This is the unchanging component. Yes. Depending on what is your geometry of your universe, whether it is flat or closed or open, you can write it down. You can write it down. Now, this is not very useful because at the end of it what we measure in the universe is signals. Those are basically coming due to light coming to us. The first thing in the case of flat universe is just D r prime. Which is the statement that a photon moves as much in the eta coordinate. Eta is r. They are flat universe. That thing in the sphere of the universe is what we put i in class. That angle. And eta is that angle. Now, what happens is that because the distances are different, so if you are looking at a standard candle and we talked about standard candle, suppose there is a standard candle which has some distance to the universe. And we know what is the luminosity now. Suppose we know both the actual luminosity and the classical luminosity. Then, in Earth, we will just divide it by d square, which is just the same distance as the coordinate distance. But over here, just now, there is an extra 1 plus z term coming. And why is it coming to this? Think of it very simply. Let's say this to come about whatever. It is giving out n photons of some frequency in a frequency range delta nu. And within that time, because luminosity is downward per time in a delta t. And it is spread over a distance which is some d k. And we need the flux. Flux will be again taken area. How many photons are passing through the area? In another delta t prime. Now, the d that we get will not only be the d that we are getting, we also have to connect this delta nu, the frequency delta nu with the frequency delta nu prime that we are getting. And also the delta t in which it is emitted and in which we are getting. And if you do that, you will see that there is this 1 plus z factor. Just to say what we discussed, you know, it comes by 1 over d square. So, this 1 plus z is appearing square. Now, 1 plus z is simply a, a then by a now. So, this is the factor of a then by a now square which we discussed. You remember one factor of a then by a now where the times are different. So, you measure fewer photons per unit time. The second factor of a then by a now where the wavelengths are different. So, e photon is which. So, you remember we had that a then by a now square. That is the 1 over 1 plus z over d square. And so, if you do that, you will have this kind of. I have described this 1 plus z square which is 2 1 plus z now. So, actually a then by a now is 1 plus z because as we saw lambda by lambda observed by lambda emitted is a observed this should be bigger. So, is a now by a then. Remember we discussed this just because the conformal distance between emission and absorption is constant. Therefore, the proper distance is So, this is the first different distance. The second different distance is basically what is known as standard ruler distance or the angular tangent distance. Meaning if you have a ruler which I know the length actually and the angle that we are getting over here I will just do basically r by r is going to d theta which over r is d will be the same as this but over there because we have to get the photons from here to know this distance we need to have the photons from here and here come to us at the same time that gives our 1 plus z. And here I am related. Let us do this things possible. And so, we have another different is the angular tangent distance. So, if you are looking at fluxes then we will use luminosity distance and sizes we are going to use angular tangent distance. And a lot of cosmology can be done just like this. Remember the whole of supernova cosmology uses this basically I believe what is the flux of the supernova or the luminosity of the supernova. I know what is the flux of the supernova from measuring. So, then the ratio is just like this. And here but then what this deal has this deal has is over here the r has the stage of z. So, if you change your cosmological parameters you will get different angular tangent distance of the same luminosity and the same force of the flux. So, you can say what is the the force of the process. Similarly, how do you measure the angular diameters? I will measure. You have to have some standard. So, I will come to standard. What interesting thing that can be said is that if we have these two distances for one object which is actually not very easy to do it is very very difficult. Suppose there is some red shape where I know for that red shape of course it does not have the same object it has the same red shape. I know what is this angular diagonal distance to that red shape and what is the luminosity distance to that red shape. So, suppose there is red shape. Yeah. So, suppose there is a galaxy where supernova goes in the galaxy from the supernova I know what is the dL the luminosity distance. Then somehow I know what is the mass size of this galaxy. Somehow. How I know somehow. So, from there I know what is the dL. If I take the ratio of this dL by dL it should just go as what does this do? Well, when you have done this you have cancelled all the effects of all the metatons how different constituents are you do not care anymore.  So, basic Einstein's equation then filter's equation. So, if you can find that the vision of this and this is known by as something known as ethering and relation. Very interesting. So, there is a dL so, basically whenever you look at the distances dL by dL then it is of course always you do dL by dL then it is always you do dL by dL. So, there is a 1 plus z 1 is divided by 1 plus z 1 is divided by 1 plus z dL by dL by dL it is 1 it is not 1 upon 1 plus z So, dL is r z by 1 plus z so, generally since we always do it in squares filter j in terms of so, one very interesting thing just by looking at angular denoted distance and the and the university distance you have in principle the test whether it is fundamental assumption of this generally given into the statement equation as well and people have done this and at present it is ok it is actually there is no deviation function another thing can do it can do something different. Suppose there is a deviation so what people generally do is they will have an epsilon over here and try to see how good we can do this epsilon over here. That can also tell us a lot about non-standard physics suppose there is some kind of extra energy some action So, people have talked about this so that is something very interesting that can be done and with data this is going to be done in more and more precision at present the precision is around 0. So, let us now go into cp we are going very slow the same of course we will be using this thing very much so what are the same things same of course if you look at the 0th order it has this that one is really good that one is good and this is what Koges applied all of it that you do and if you look at the rest thing this is the this is really good in fact this is the cmp and, well some of this from right here I think this is not one sigma this is not one sigma so one of the classic pictures of Koges' legends we call this cmp and we call the airbus and the cmp and because the airbus are not one sigma, two sigma because they are practicing the airbus the width of the airbus is almost the width of the line so this is because of the no no no this is really because of the beam so whenever a satellite looks at something whenever a telescope looks at something it has a resolution that is called the beam of a telescope so the way I had a beam which was 7 degrees inside so it was really large beam so with that beam there is so the fluctuation is much bigger than the beam then you can specify the fluctuation the fluctuation is smaller than the beam then what happens is that this fluctuation gets erased because of the conclusion and so the airbus are due to be another thing that is the best thing that you see that is very good so this is what we have the 0th order if we look at if we zoom into fluctuation which is one part in 1000 to 10 to the power of minus 3 then what you see is that there is a diaper pattern why is this diaper pattern because the earth is moving around the sun and the sun is moving around the galaxy and that is like 1000 kilometers per second so v by c is about 10 to the power of minus 3 so the center of the galaxy is more or less at rest with respect to the no you have all of the states but the dominant thing is this and take in the middle of course is a galaxy but if you zoom in even more this is what W maps so you have a center which is dominated by the galaxy and these are the spores this blue and the red which are basically differences in temperature at the power of minus 1 so delta t by t is more or less at the power of minus 1 it is like 100 times this is of course possible so we really don't see like this it just tells you that some spores are higher than the mean and some spores are smaller than the mean way how do you do it, observation very very good what we do is that the C and V we know what is the spectral dependence but what we get in the observation is not only the C we get the C and V then there is dust in our galaxy which radiate and radiate then there is electrons which are spinning there is synchronicity so what you get is basically your C and V plus dust plus synchronicity so everything has all of the four parameters now we believe that we know in some sense what is the spectral dependence of all the systems so what we do is that we look at your C and V map for different frequencies so these are at different frequencies so this is basically more of a map and as we go to different frequencies for example this is a frequency where as you can see in the center there is a large contusion in the center for what was galactic so it's very contaminated and this is basically dust galactic form which is very dust but this is a high frequency and then no frequency you don't see the galaxy much but then we see some other form so what we do is that we combine all these different frequency maps we combine the 50 or the subtract there are different ways to do that and then what we are left with is so if you subtract this galaxy then we will take something right and then we also do frequency subtraction then we will do this final and what we can do is that we can just cut out this region over here and then say this is very much like a primordial signal fluctuation we do the fluctuations then you will see that the characteristic size of a hotspot or a post velocity is approximately a degree in size that's very important that's a characteristic size it's not completely random and of course there are other sizes but the most prevalent size is like a D so what we do is that we statistically analyze the size we have this same thing and from there we calculate what is known as the same in the hospital basically if it was a ordinary 3D what you have done is that you take a Fourier transform this is a sky ok so what we do is basically we do the yl's instead of u and then this is basically the anm squared so yl will have an amplitude anm take the model of circles so basically this is how so we have a delta t around any direction theta phi on this side and that's basically that thing minus this dLp aLm is basically you take this to the spherical representation and then what is the result see basically it tells us what is the power at different scales this is just a equivalent of a k squared what is the power at different scales very nice picture for every yl there will be 3m's so this is the 4 volt around dipole term and then we call it the 4 volt and we have to of course have to so there are three regimes of this thing so this is what we do if you do that you have something like this so the y axis is what is this aLm squared aLm squared basically L into L plus 1 aLm squared because that gives you dipole power in logarithmic limit L squared basically aLm squared haven't you made L squared or sum over sum over sum over but divided by L divided by L no no what is that L into L plus L squared this is the average and this is basically C L that average is average over N so in the definition of the average you divide by 2 by 2 and so this L over here is approximately 180 by theta so if you look at two directions so what is L that's what it's because we are taking the delta T into the aLm remember delta T had one aLm what we are bringing aLm squared so what we are doing is taking the average of delta T times delta T delta T in this direction and delta T in the other direction and that has a separation of theta so in aLm what we have is this theta is sort of inversely proportional so when theta is 180 degree it is 21 so what we have is this this kind of very very very interesting kind of plot we have peaks and bounds and there are three regions the first region of course which is known as the acoustic oscillation it goes out and the jump is up it's an acoustic phase of the scene there is a large scale battle over here and then the third in this as you can see over here it's not that it's not there but it means more and more now different physics it lies to this different kind and if we can identify this physics then we can basically use this to get out of it so let's go very first to what is the gap so as you are doing the CAP photons when they come to us they are scattered again by this low red shift they are raised regions because we have the same we have a red shift of 1000 photons decoupled at the shift of 1000 and below red shift of around say 15 10 the universe is again realized so these photons are getting re-scattered because it is getting re-scattered all the features that they had at that red shift beginning features those features are getting sort of killed at some scale why was the universe re-indexed at so what happens is that we have so photons decoupled as I told you the temperature is around delta T by T power minus 5 same it also means that the density perturbation is the kind of perturbation now these perturbations will be growing in the gravitational instability and so by a red shift of 100 so in the factor of 10 this will go to 10 to the power minus 4 to 10 to the power minus 3 depending on so below red shift of 110 the first object starts forming so at a red shift of 50 the first early galaxy or the first early star the galaxy forms the early galaxy which is very small not like a galaxy our typical galaxy is at 11 solar masses the first galaxy is around 5 solar masses the ratio around 50 stars can form inside these galaxies and as soon as the stars form inside the galaxies then there is the clear reaction photons are emitted photons are emitted from the stars starts reacting to the region around it so blocking of electrons blocking of electrons and so what happens is this regions around let's say these galaxies, small galaxies will have shells of realised region more and more structures are growing so they will be more and more taken of spherical realised shells in the universe these shells will also increase because remember there are more photons coming out pushing away this realisation France slowly larger and larger at some scale these shells will also increase because the universe are increasing so everything is sort of bigger these shells at low red shift ratio around 5 or 6 will overlap when it overlaps we can essentially say that the whole universe is realised now till the universe is completely realised a photon statistical may pass through one of this realised region may not pass through now remember these shells are small this is not the size of realisation the shells are not small so handle substituted by the shells are small it basically means that L values will affect as much so any perturbations at these L values are getting damped these are small but also the shells act at low redness at low red shift closer to us because it is not at this scale it also means the angle that the shells are subtending to us are much so what it does is that it sort of get heats of perturbations in this scale generates a new scattering and increases stuff at this scale because there is some scattering scattering will give some correlation these are at low angles at this scale there is another reason why this region gets damped that is known as safety much of the matter of the universe is realised not a significant fraction when it is ionised it is basically ionised what is the stuff that is ionically dust no it is hydrogen but in the sun on the earth it is not ionised so you can think of the main stuff that is the valium when we talk about ionised we are talking about valium so valium is hydrogen 99% is hydrogen and where is it lying it is lying in different phases we are talking about interstellar gas actually ionised is intergalactic intergalactic gas that is what takes maximum of the space but also that is maximum of the amount because think of galaxies are like big clouds but small spaces over here so we are talking about ionisation this is intergalactic gas most of the valium of the universe are intergalactic gas not a gas I understand because the gas was too thick gas was too thick early in the universe but it did get too thick gas was too thick so it had to absorb the water and still it has quite a bit of ionisation happening intergalactic intergalactic interstellar gamma so think of okay think of initially so think don't think of the structure that you see around us now in the universe early in the universe ionisation happened because the density of the gas around it was too too to allow photons to pass so there is a big debate so there is not a debate whether it ionisation propagates into the thin region first or does it ionise the thicker in some region first there is not a debate about that but what we know is that there are some galaxies small galaxies so there will be stars inside the galaxy the stars will start in the region at some point of time the radiation pressure will be such that it will escape out of the galaxy so we have a ten years intergalactic media around it intergalactic media density we don't know as what does it do so we know what is the density of intergalactic medium now so what will happen is that around this galaxy it will start forming the spheres that grow at the speed of light that grow at the speed of light no it will not grow no it must grow at the speed of light why? because of the interaction with the ions and it will also start to combine and then it will generate a balance between realisation and accommodation so what happens is that ultimately this funds will merge now remember this funds will also merge when two galaxies are closer so some regions it will merge some regions it will not it is a very interesting topology of this France so the interstellar hydrogen intergalactic hydrogen today is largely ionised it is absolutely ionised completely 100% almost 100% except there are small small clumps which are now ionised very small very small very small very small density yes but it is ionised very small 3 can be the temperature of the of the force now the intergalactic gas is of the temperature of the form gas is the temperature of the form below this cloud because then it becomes you have to think of the density of the photons and the density of the of the gas so below this it can set up by the same degree so it cannot go below the degree of force but the gas is at the degree of force below the degree of force now what happens the same is starting it up same is 3 degrees but it is very diffused same is even more diffused so the different things happen so there is some there is a combination so the intergalactic gas and also you know as the gas density changes as the gas density becomes smaller and smaller this temperature also becomes smaller and smaller so the gas is more or less ionised only in clumps of matter or in galaxies the gas is not used so why has that affected you very much only at small states so basically it is actually experiential why do all these photons keep scattering off because it is very dangerous very dangerous photon has to travel the long distance before it scatters only at this regions so only at only at regions which is of this k it can have a chance of getting multiple scattering if the photon is slightly larger then it will go through so number of sites it goes through is much smaller it is not that so it does not scatter that only at this regions you know what we are doing is that we are looking at a correlation so this absolute thing is not we are not looking it is very important this is a correlation so the correlation will be destroyed if two kind of sites goes like this but if the two kind of sites goes like this then the correlation is not this so suppose this is the size of the block correlation destroyed if two lines of sites goes like this however if one line of sites is here and the other line of sites is here why is it not destroyed if it is not scattered ultimately you need it is elaborate so you have to take this thing and take all sorts of combinations this is destroyed but it is not exponentially which is lower another is not lower but at the end of it you are what you are doing this is in some sense arbitrary so what you what you get what you can this amplitude so this amplitude that you get over here you can always get this amplitude in a theory by adjusting your amplitude of possible you have to that is completely arbitrary this is what we get you can do with your theory because you can always increase your A up and down what we measure actually is this the real degeneracy is this real degeneracy is this is what we measure this amplitude and this tau can you can always change the amplitude and tau to you make each other so it is you can see it is being destroyed by it by minus tau but here it is completely okay I suggest that we since it is one o'clock we give the ten more minutes with almost no questions except very pressing ones let us do sorry for harassing you like this is good so this is what we let's go so these are the components whatever those things and so this is very interesting you can even go to where those website so we measure this how we measure the parameters from this so over here what we have done is we have changed this omega barrier okay and you have seen how the safety parts which we do as you can see when you change the omega barrier there are two things that happen first of all related to this so very important this is fixed okay so this is fixed first thing is jumping up okay also the ratio of the peak say p to 30 no and I know I said no questions but then you why does it happen what is the connection between omega barrier so this is first thing okay answer this question frequency of the state without frequency between the question the height no it just keeps on changing so the second p to 30 keeps on changing all the p keeps on changing everything is fixed the distance is the same actually there is a little movement in the distance but in general the L positions are more just ask the same thing happens if you change or if you keep your omega barrier as you can see if you change your total omega you will see that things are changing so this first peak is moving shifting right let me see that so changing your omega barrier your omega total also changes so what are the things that change changes is first the height change the absolute height changes absolute height is respect to L equal to 0 the relative heights of the peaks change and the peaks move and of course the gap it happens more or less so all these things are there so why is that so this is because what happens is that if you think of the same so we have some density perturbations this density perturbations is maybe due to dark matter because dark matter is the maximum density so you have a density very dark matter the variants fall into the well the variants fall into the well the variants are strongly coupled with the photons so it drags the photon now the photon as the photon gets more or less compressed it starts applying a opposite pressure because the photon we have an impression of that so what will happen is that you will find the standing wave in form which will be governed by how much you can compress and what is the perturbation how much you will compress the photon it will depend on two things it will depend on the sources it will depend on the amount of the initial potential the potential it can drag so it will make a matter it will also depend on how much variants are there so if the variants are more the photons will be it's like a loaded spring it's a spring which is loaded so what we are getting compressed more means that the wavelength the maximum compression will be more, so peak height will decrease so this is what must happen peak height will increase because of the photon similarly but what will happen here is the peak height will increase the order of peak height will decrease one is the compression, one is the rare infection one is the compression and one is the rare infection whereas if the omega delta increases So, you can think of this as very simple analogy, you think of when you drop balls from different heights and instead of like after a minute, one minute you start dropping, the ball is dropped from smaller height you go and bounce, the ball is dropped from very high, it still not come after a moment and there will be exactly one scale in which the ball will just come. So, that one scale that comes gives you your in some sense your first big ball because that is an expansion of the universe as how much it would expand. So, the ball bouncing let us open very much hitting fluid and if you look at the age of the universe it tells us how much it can go and the original height of the first peak basically tells us how much this thing would expand in that time in the universe. So, when it first started it went to decoupling habit and decoupling what happens atoms come. So, electrons and variance protons combine. So, then photons are free, but till then it is compressed on the wave properties. So, that is the first thing and so all the other peaks are basically just harmonics. So, if we look at in terms of all these peaks are basically harmonics of the first peak. So, these are in I think I have not very many as you will see that it is like 100, 200, 400 and then 600 peaks. There is a small shift of course and that small shift has to be kept because think to see over here is that if we have different kinds of cosmology we have very very different. So, that cosmology that the elements that which to the effort of the universe this dense parameters those are given by the peak heights plus what is the next thing we want to know not only this different parameters you also know whether the universe is flat and open or closed. Then comes from the this L position of the first peak how does it come. If we know what is the cosmology from this peak heights, then we know what is the omega matter, omega density. So, we know how much the wave would have propagated from basically just of the big band to that big sheet. So, you know what is the scale is this part. So, you know what is exactly you know what is the scale.  It gives us the angle. So, physical scale we can compute having issue from whatever this different things are. Physical scale of what? Of the propagation of. So, if you think of a of a perturbation propagating when the perturbations of freezes or when the decoupling happens just propagated. So, this circle has been up to some scale that is the maximum scale that is the physical scale. And that we know if we know what is the omega matter. So, that gives us how sound speed is there basically sound speed times the nature of the universe. Sound speed of nature of the universe gives us the physical scale from this what I can tell what is the angular scale. This is the angular scale. So, you have 1 by this L value gives us theta. Now, we know physical scale by angular scale is the angular diameter distance. So, we know whether we are living in a flat universe or a curved universe because the flat universe we know exactly what should be the L value in which this picture. So, if it is a closed universe then what will happen is that the heat will come at a higher level because same size it will give us. So, if you think of it like this what is the physical scale right.  So, it is larger theta it is a closed universe it sort of gives another this angle is not angular subject. So, same size is sometimes 3 different angles. So, you are looking at the L value meaning looking at the angle we can tell whether it is a flat or a closed or a closed universe. And this is actually the best term in parameter in the proxy.  You know that this is 1.01 plus 1 is 0.02. In fact, it is bigger than that. So, that is flat to 1 percent. That is not the same. And then basically one person accuracy you know that the universe is flat. So, that is that is that is it.  So, it is a just in case we want to do this calculation of these peaks just to understand. So, we would need fluid dynamics of photons, dust. Just solve the Navier-Stokes equations linearized. Just the including gravitational interactions. Actually, it is much more I think even easier than that. Basically, I have basically.