 Good morning. It is true, sadly, that I can't be here much longer, but I'm around all of today and tomorrow. So please do feel free to ask questions both during the lectures, afterwards, lunch, dinner, whatever. And you can also email me later if you didn't remember something now and would like to know about it. So I've been asked to give three lectures to kind of set the frame to provide the background for the further lectures that will follow by various people, by Sujit Rajendra and Wilfried Buchmiller and others who will go into more advanced topics concerning the links between particle physics and cosmology, which are illustrated there in this picture of the Uroboros, which is a familiar symbol telling us in this context how the largest scales in the universe are supposed to have all resulted from the microphysics at the smallest scales. And I guess you are of mixed backgrounds, but I imagine most of you do have a grounding in particle physics and it has become a very profound realization in the last, I would say, 30 years that this part of the diagram, the part that we investigate in laboratories, in experimental facilities at the Large Hadron Collider and elsewhere, that these actually can tell us something about how the whole universe began and evolved to give us the complex sky that we see today and indeed where we occupy sort of the middle ground logarithmically speaking between the largest scales and the smallest scales. So obviously, this is a very large subject and I'm aware that many of you have not had any previous exposure to cosmology. At the same time, some of you may well be experts. So I'll try and keep the discussion at a very elementary level. That's what Giovanni urged me to do, but I will make the odd comment for those of you who are more expert to give you some food for thought. So let's go. So the first question one should ask is, what does the universe look like? If you look out at the universe, well, we are on this planet. We look out, we see stars. We see stars assembled in galaxies. We see groups and clusters of galaxies. If you make larger surveys, then we see large scale structure. For example, this famous stickman that was first spotted in the survey carried out by the Center for Astrophysics way back 30 years ago and a more modern child of the same kind of survey, the Sloan Digital Sky Survey, which is measuring the red shifts of several million galaxies and tracing out what has been called the cosmic web. This is extending over hundreds and hundreds of megaparsecs. I remind you that the distance to our nearest galaxy, Andromeda, is about three quarters of a megaparsec. So we are really talking about very, very large scales indeed. And our conceit is that we are going to be able to explain all this stuff by appealing to the fundamental laws of physics that hold down on scales down to 10 to the minus 18 centimeter, which is what you have been able to prove so far at the LHC. The question is, given this complex universe, you know, if you are, for example, a biologist or have a friend who does biology, they will tell you, you know, it's a very complicated subject. There's just so much variety. You know, a lot of it is just classification and etymology and so on. And it's very hard to get a fundamental understanding. How can we dare to do that here? Well, to see that, we'll take you through the usual formulation. But I can tell you already the bottom line in advance. We are going to be dealing with the early universe, which actually is a lot simpler than the late universe. And fortunately for us, the universe is mainly dominated by photons. There are about a billion photons per particle of matter in the universe. So to a very good approximation, we'll be able to treat it as a radiation gas and an ideal gas. And that really is going to help a lot in formulating a simple model of the universe. So the first point is that although the universe is indeed lumpy, as you have seen, it actually seems to get simpler as you average on larger and larger scales. It appears to evolve towards a homogeneous distribution with small fluctuations that have grown under gravity to give us all that structure, the cosmic web that we just saw. Now, this is best seen by looking at the variance. So if I plonk a box down of a given size anywhere in the universe and then look at the variance of mass fluctuations in it, density fluctuations, that gives me some measure of the lumpiness. And this nice plot, I should have credited it from Max Tegmark, it shows how the density contrast or the fluctuations change from small to large scales, sorry, this battery is dying, small to large scales. So here we have the scale of tens of millions of light years. So this is light years is of course something you know, but what we are really going to use is the parsec, which is the distance at which an object subtends a second, one second parallax, that is about 3.3 light years. And we are going to be talking about kiloparsecs, megaparsecs. But remember the scale in these terms. So as you see on small scales you have a pretty large contrast. These are nonlinear fluctuations. The density contrast is greater than 1. So in other words, delta rho by rho, square root of delta rho by rho squared is pretty large. This is where we have the Lyman alpha forest, which is gas that we see towards distant quasars through spectroscopy. And coming closer still, here is the scale of galaxies. And then when you come to larger and larger scales, we start seeing bigger and bigger objects. This is a cluster of galaxies whose potential well has been revealed by the fact that it acts as a gravitational lens for a distant quasar. And by simply doing great tracing, just geometric optics, we can determine what the gravitational potential is. I showed you this picture already. The so-called cosmic wave revealed through the measurement of redshifts of a million galaxies in the Sloan Digital Sky Survey. We can measure the abundances of rich clusters of galaxies and then we go to the largest scales of all, the cosmic microwave background, which through its anisotropies on scales of the sky, or the quadrupole, octopole, etc., is sampling the biggest scales in the universe, the size, if you like, of the universe, the present Hubble radius, which is of order about 10,000 MPa6. And you see that the fluctuations are continually decreasing as measured from nonlinear to linear to very, very small values, right? And this blue line, by the way, is not a fit to the data. It's a theoretical model. This is the standard cold-dark matter model transfer function. And you see it gives a pretty good description of the data when normalized to the fluctuations of the largest scales. It does match what we see on the smaller scales. Of course, this is not really a technical diagram in the sense that there is a lot of fudging been going on because many of these things we are interested in the fluctuations in the gravitational potential, but what we actually see are the galaxies of the visible matter. So something called Bayer centers that tells us to what extent the visible matter follows the true potential well set by the dark matter that you will see dominates the total mass budget. So leaving aside those issues, which only experts are concerned about, on the whole it appears to be a success story. Yeah? Well, he's asking what is the zero? This is not a scale in time. This is a scale. I'm just telling you what we observe when you look at the sky. I do not yet know if there is time, okay? So we'll come to that. What I'm inviting you to do is to join me on a journey with perfect telescopes and detectors without knowing anything about cosmology to look at the sky. This is what we observe. Let us then empirically deduce from that what we can and construct a world model. But to answer your questions, the largest scales that are right now entering our horizon are here. These are scales that have entered our horizon in the past up to seven, eight billion years ago. So these are actually older objects. This is younger. But that we do not know yet. We'll come to that. Okay? Right. Now, let us start back to when observations first began. I should tell you that this is a very young subject until the 1920s. We didn't even know that we lived in a galaxy. Okay? People thought that what we saw out in the sky is everything that there is our universe just consists of the stars and stuff and little gaseous nebulae that we see. We did not actually know that we live on one galaxy like zillions of other galaxies to be precise, 10 to 11 other galaxies. So it was only in the early part of the 19th century that we started actually realizing we live in a big universe and the mathematical tools to describe that had been just invented by Einstein as you know. And so cosmology was one of the first applications of general relativity. And we still have a lot to learn. We are still using the same mathematics that was used back in the 1930s to construct a model for the universe. Now, the first thing that we observe when you look at the sky is we see a picture. Now, this is a picture from the Sloan Digital Sky Survey. Each of those dots is a galaxy. And what they do is that they expose a picture of the sky. They take the plate and then at the position of each image of a galaxy, they drill a hole, put a fiber optic cable through the back so they can take the light out, put it in a spectrometer and measure the redshift. It's a very elaborate and expensive process which is why so far we only have a million or two redshifts. But when you do that, you get a plot like this and the brightness of each object actually is telling you something about how far away it is because as Hubble noticed, this is Hubble's law. Not the thing that you think is Hubble's law. This is the real Hubble's law. Hubble worked out that if I just look at a picture like that, it's two-dimensional. I don't know how far these objects are. But if there were objects distributed in a volume in three dimensions, in three spatial dimensions, then you know that the number of objects goes as the cube of the scale, right, or cube, but their flux is falling off as r squared. So the number that is brighter than some brightness s will go as s to the minus 3 halves. And this 3 over 2 is just reflecting the dimensions in which the objects are spread out and the dimensions in which the light is spread out. So astronomers don't actually deal with this s. They deal with its logarithm because they look at a very, very wide range of objects and they define something called the magnitude which is the log of s respect to some fiducial value. And for some reason which nobody understands, they multiply it by 2.5 minus. So that means s becomes m and because of this minus, the larger the value of m, the fainter the object. Remember that. So s to the minus 3 halves, if I do this transformation, becomes 10 to the 0.6 of m. So in other words, the number of objects in the sky as a function of the magnitude should go as 10 to the 0.6 m if they are distributed uniformly in a three-dimensional space. And here is the test done on that Sloan Digital Sky Survey. You see this is the plot 10 to the 0.6 m and the data fitted pretty well which shows that these objects which we know are galaxies from the red ships are actually distributed more or less homogeneously in the distance that we have probed so far. This is not going very far. This is going about 800 mega-pass 6. So you see, even without knowing anything other than the two-dimensional plot, by doing this test which is a standard test in astronomy, it's sometimes called the Wim in Wimax test. But this is really a aspect of what you might have also have heard of called Albus Paradox. So Albus Paradox is that if this is the case, then if you add up infinite shells of galaxies, you will get an infinitely bright night sky. A paradox which was not Albus at all, but if you read a nice book by Edward Harrison called The Dark Night Sky or something, it was actually propounded much, much further back by Kepler and then by Deshesieu and so on. And its resolution was actually very important. Its resolution is that these galaxies do not extend forever. There is a boundary to the distribution. And the galaxies are so dilute today that the optical depth, well, the surface of the sky, the fraction of the sky that is covered by galaxies is very, very little. It's only 1% even in the Hubble D field, the deepest we have looked into the universe to where we see the edge of the galaxy distribution. We are just seeing 1% of the sky covered with galaxies. That is why there is no bright night sky. But that's another topic. You might wonder what that line is. You see, there are data points on that as well. That line is actually stars. Stars are not distributed in three dimensions. They're distributed in two dimensions. They're distributed in a disk. But the light is still falling off as 1 by r square. So in that exponent, it will be s to the minus 1 over 1. And therefore, stars will have a slope which is different from that of galaxies. And you see clearly that you can tell the difference between 10 to the 0.4 M and 10 to the 0.6 M. So I'm showing you this in order to impress on you that astronomers have a variety of pretty smart techniques to try to extract information from a flat sky. This is very interesting. So what we have established so far as Hubble did back in 1926 is that there seem to be so-called island universes which are distributed homogeneously. This other thing, of course, is the red ship. So you all know what the red ship is respect to some line, optic line in the laboratory when you look in the distant spectra of distant objects. Well, you don't know the distant yet. When you look at the spectra of object stars, galaxies, fainter galaxies, and so on, you see that the lines move gradually to the red. And if you plot that, assuming that that approximates to a velocity, so if you interpret this in special radivity, the red ship will look like a regular recession velocity. And then, therefore, if I just multiply the red ship by C and I get the velocity, and that velocity is plotted here. So for example, if I look at a galaxy at a red ship to 0.1, its velocity will be C, which is 310 to the 5 kilometers per second times 0.1. So that's 310 to the 4. And that red ship to 0.1 corresponds to a distance of about 500 megaparsecs. These distances were measured completely independently using, in fact, the fact that some stars, C-feed variable stars as they're called, fluctuate in intensity at a rate which is dependent on their absolute brightness. So they serve as what are called standard candles, depending on where you, how faint they are on the sky, you can deduce how far they are because they're like standard, you know, 100 watt bulbs. You know what the wattage is from the rate at which it is fluctuating. So that's a whole story. I will not have time to go into that. But basically, this is a recent Hubble diagram based on Type 1 supernovae, which are standardizable candles. They're not actually standard candles, but we will have found ways to use them as such. And using that, you can see that the expansion is linear, velocity is proportional to distance. And that allows us to read off from this that at a redshift of 0.1, the distance is 500 kilometers, 500 megaparsec. So let's remember that. So this is what we have established so far, and this was already known back in the 1930s, 40s, that there is what you can see on the sky are galaxies. They are island universes. They are going away from us, and the velocity is proportional to the distance. This was, by the way, not Hubble's discovery. This was a discovery of chap called Vesto Sliffer, who first measured the distances and redships. As an aside, Hubble was actually at Oxford, but he didn't study astronomy. He studied law. And afterwards, he was a bare-knuckle prize fighter. So he was a very interesting guy. Now, if you look at the sky, and we are following just what I told you earlier, we have some ideal detector, and you're looking at the sky. And we first, the thing that we see is the Milky Way. That's where we are. We are sitting somewhere here. We live in the outer suburbs. And if you go further to redshift of about 0.01, remember, keep this in mind, redshift of 0.1 is 500 megaparsec, redshift of 0.01 is 50 megaparsecs. So if I go out to 50 megaparsecs, which includes the local cluster of galaxies, then I see this tracery on the sky of many, many galaxies. If I go a bit further, this is to 100 megaparsecs, I'm still seeing the tracery, but it is becoming fainter, the contrast is getting less as you saw earlier, and so on. I keep tracing it out, and every time I do this, now that I've given you the formula, just multiply the redshift by the speed of light. These are all small numbers compared to 1, so it's a very good approximation. Multiply by the speed of light, you get the recession velocity, and then you can read off what distance it is. So redshift of 0.1, I remind you, is 500 megaparsec, 0.01 is 50 megaparsecs and so on. So up to here, you see, I've gone to about 250 megaparsecs, and you can see that the contrast is actually diminishing. Of course, by eye, and this is color coded, you can't really tell, but if I measure the power spectrum of galaxy clustering, then I can measure it explicitly. And then I go to the farthest point up to which we can do such service, which is about redshift of 0.06, which corresponds to about 300 megaparsecs. Beyond this point, we don't really have much data. So at this point, I should tell you that I've been talking about ideal detectors, but in real life, of course, you have to get time on a telescope and look for stuff, and it takes a lot of resources and a lot of energy to do that, so it's not easy. We don't have some magic detector that maps out the entire universe. We have a lot of data up to redshift of 0.1, which is most of astronomy and cosmology, 99% of it, and then we have now some high redshift data for which we have had to use objects like the Hubble Space Telescope and things flying up in space, because they are redshifted to the point where the radiation can only be seen in the infrared and far infrared, not from the ground. The atmosphere of the Earth absorbs all that light. But to cut a long story short, I'm now going to fast forward to redshift of 1,000. Now, you might say, what is the redshift of 1,000? How does it work? With the formula that you gave me earlier, and I did say that that only works for redshift less than less than 1. Redshift of 1,000 we'll see later what that is, but I'm just trying to make the point that the same contrast that you saw earlier becomes much, much smaller, and what we see at a redshift of 1,000 is just a smooth sky of radiation with my ideal detector, which can see in microwaves as well as in optical light, in that radiation there are just tiny patches of fluctuations which are something like one part in 10 to the 5. So, we see that if we go back, so when we look out at distance, we are looking back in time, we're looking back to the beginning of whatever, we're looking back in time, we still don't know where it all came from. What we see is a blackbody radiation with this precise temperature. This is the most precise to measure quantity in cosmology. It's the temperature of the microwave background spectrum and it seems to have fluctuations of no more than you can see. It's up to about 30, 40 micro kelvin. The temperature is 3 kelvin, so it's of order 1 part in 10 to the 5. So, this is the basis for my statement earlier that the early universe is actually going to be a lot simpler than the late universe. It's lumpy, inhomogeneous, modeling it is a nightmare. We do it in the way that I'm going to describe, but there could be lots of issues there that you have not yet sorted out. Whereas by contrast, the late universe is going to be very, very simple. Now, the point is that as I said, when you look out in distance, this is the Hubble Space Telescope looking out in the so-called Hubble D field and we see this universe of galaxies and then as you look further and further back, there are no more galaxies. We see that the sky is, we see all the galaxies we can, but they only cover about 1% of the sky, right? And beyond that, we see nothing until we get to a very high redshift when we start seeing the universe has got sufficiently hot and dense that it has become ionized and that radiation that we are seeing is the black body is surely evidence of a equilibrium thermal phase from some hot dense early state and that's exactly what we are seeing. So, this is the picture you must keep in mind. We just look out, we look back in time. If we could look further, far enough back, we would see where it all came from, but we cannot see that because it is obscured from us by this opaque wall of plasma because the universe has become hot and ionized. Of course, if we found other probes than light to look back like neutrinos or gravitational waves, then we would be able to penetrate that, but that's not the story, right? Now, this is up to here what I've told you is the standard load. Now, I give you my first slightly more interesting issue for those of you who are actually going to be working in cosmology. I should tell you that there is a big open problem with all this. The open problem is the following. When we look at the microwave background, what we see is not what I showed you in the last slide. What we actually see is this. We see that there is a blackbody radiation in the sky, but that it is hotter towards one part of the sky by about 3 millik, right? The overall temperature is 3K. So, this is one part in a thousand. There is a hot spot and in the opposite direction there is a cold spot and the change of temperature from here to there is precisely a cosine function. It is exactly what you expect when you have a uniform isotropic bath of radiation and if you move through it with a certain velocity, if you do the Lorentz transformation, you will see that the temperature then varies with angle as cosine theta, okay? And that's exactly what we see. So, we seem to be moving through the microwave background at one part in a thousand, which means V over C is 10 to the minus 3 and therefore we are moving at something like a thousandth of the speed of light, 370 kilometers per second in some particular direction towards this direction, right? Why are we moving? If the universe was actually uniform, homogeneous and isotropic, then we should not be moving. We should be in our rest frame, the microwave background should look isotropic. It doesn't. It looks like that. Now, of course, you will see later that we can always have peculiar motions. We are all supposed to be expanding, all the galaxies are supposed to be expanding away from each other, but in fact Andromeda, our nearest neighbor, is falling towards us because the local gravity overcomes the expansion of the universe. So, you can have very local small velocities. So, if we are moving, then this velocity should not last forever. It should die out very soon. If I average over a bigger box, 100 megaparsecs, then this velocity should disappear. But in fact, if you look at this paper, you will see that if we use type 1 supernovae, which is what I showed you in the previous slide to trace the Hubble diagram, if we use them to do kind of tomography of the Hubble expansion, if we take shells of galaxies and we find the same dipole in the distribution of supernovae as we find in the microwave background, then we see that this dipole is continuing out to as far as we can make observations, which is this, and these are observations from the so-called nearby supernova factory. Up to 300 megaparsecs, we are still moving. We are still not converged to the frame in which the CAB is isotropic. And in my view, this is a paramount issue for the standard model. We have to understand why we are moving. Something is pulling us and we have to understand what that is, which is pulling us. Why is it there? It's a huge lump of matter. Such a lump of matter should not exist in a homogeneous isotropic universe. So this is what the universe actually looks like out to 300 megaparsecs. So we are, of course, in the center. We have mapped out the universe around us. We belong to the Virgo cluster and then around us there are superclusters, which are the biggest structures that we can see, clusters of clusters. However, they don't seem to be any clusters of clusters of clusters. There are no super-superclusters. The universe is not fractal. It stops after some point. It is fractal out to about 30-40 megaparsecs. The dimensionality of the galaxy distribution is actually 2, not 3. But after that, it is believed there is a transition to homogeneity. But if you look at this sky, which, remember, Einstein did not see any of this. When he constructed his world model, they had no observations those days. So you might be a little less confident of writing a simple mathematical model down if you had this data than when the cosmology first started. However, intuition is a great thing. Those guys had the right intuition and that model has actually, therefore, worked much, much better than could be expected. So let us see how this works. But this is the latest current picture of what the universe looks like from where we are. The statement that is usually made in the literature is that within 100 megaparsecs you should be about this distance, right? You see here that is 100 million light years is 30 megaparsecs. So 30-30. So this is the scale. On this scale, the universe is supposed to become homogeneous. Well, I leave you to judge for yourself if it does that or not. An important point which really is a limiting factor for cosmology and distinguishes it from other sciences is that we are stuck on this particular planet, this particular galaxy, okay? We are, of course, moving in time and we see everything that is there along our past light cone. We also see what is along our past world line, you know, fossils and radioactive rocks and all that stuff. But everything that we know about the universe is within this light cone, right? So we have a unique vantage point on the universe. We cannot move somewhere else in the universe and see what it looks like from over there. So in modern language, this is called cosmic variance. You might have differences in how the universe looks like depending on, well, if you are sitting at the back of the auditorium at the front or symmetrically in the middle or at one corner, the auditorium looks different to you, right? The overall dipole, quadrupole, whatever fluctuations will look different. So in the same way, the universe can and does really look different. But we think that to a large approximation, the universe is like a Gaussian density field. We will come to that later. So it doesn't matter too much. However, philosophically speaking, this is a fundamental limitation on cosmology that we can only observe from one point of view. And therefore, we have to assume something from a, of a philosophical nature and that is called the cosmological principle which was propounded by Mill, who was the civilian chair of geometry at Oxford. He said, well, you know, there's, you know, it's like an extension if you like of the Copernican idea, okay? Of course, historians of science will tell you that was a complex idea, not as simple as you might think. But nonetheless, this is what has stuck in the public imagination that we are not anything special. From being at the center of the universe, we have gone to the point of being nothing special at all. And our position is therefore typical. However, even if our position is typical, what we see from our vantage point may not be the same as what other people see. So we have to make an approximation or rather a assumption that things are more or less the same from any viewpoint of you. So now let's start working on the modeling of this universe because I have to cover quite a lot of material. So this is something that all of you are familiar with. This is the metric which allows you to determine the interval between two space-time events in special relativity, Minkowski space-time, okay? And you know that that metric is symmetrical. The distance between A and B is the same as between B and A. So there are 10 independent functions in this 4 by 4 matrix. The Minkowski metric in particular, I choose this particular signature of time and space and that tells me this is the interval, this is invariant for all inertial observers, okay? So Lorentz velocity transformations are actually the same as the inertial coordinates of Newtonian mechanics. Now, the question then arises how do I extend this metric to cover the situation when gravity comes in and curve space-time as Einstein taught us, right? So this immediately created a problem when Einstein tried to do this because in general relativity this Gij, the metric is related to the distribution of matter. In special relativity, the metric is independent of the test particles. That's why the test particles moving on the manifold. But Gij, if I equate it to the Minkowski metric, etaij, right? Then this is the solution in the absence of matter. If there is no matter, then you go back to the flat space-time of special relativity. However, you and I might see nothing wrong with that, but Einstein was very bothered by it because this is contrary to Marx's principle. Ernst Marx, a philosopher had said that inertial frames are determined relative to the motion of the distance stars in the universe. Something must define what an inertial frame is. And his point was the famous rotating bucket experiment of Newton, if some of you might know about that. If I take a bucket, fill it with water, hang it by a rope from the ceiling and twist the thing up and let it go, what's going to happen? The bucket will start spinning, okay? The water will gradually take a curved shape. And then, if I suddenly stop the bucket, the water will keep spinning still with the curved shape until friction or whatever, you know, stops it and then it becomes flat. So you could ask the question, when the thing is spinning and the water is a curved shape, you look at it and you say, well, it's in a non-inertial frame. That is why it is feeling all these pseudo forces and so on. And the question is, is it feeling it with respect to what? What is defining the inertial frame? Okay? This is actually a non-trivial question. It's something that still exercises people who worry about classical mechanics. But our answer is simply that, you know, something is defining the inertial frame that tells the water to get a curved shape. And Marx's principle said that this is the distant stars because in his view, mass or inertial mass was conferred by the distant star. Today we know that there is a Higgs field pervading the universe, that's what gives us mass or at least some part of the mass. Some part of the mass comes from breaking of carol symmetry in strong interactions. So I leave it to those of you who are budding philosophers to try to work out how that might or might not relate to Marx's principle. Anyway, the point is that this was the historical position that Einstein had and therefore he had to find some way to define an inertial frame in the absence of matter. So the first option that he took was to say that when away from all matter, the metric becomes singular, okay, so that Marx's principle is not violated. You never get a inertial frame in the absence of matter. However, Deseter who was in Leiden with whom he was corresponding said, that's nonsense. If I look at the light from a distant star it is coming to me through empty space where there is no matter. If you are going to mess around with the metric then something will happen to the light and I will not be able to see the star. So this clearly doesn't work. So this is ruled out, right? The other option that Einstein had was to change the manifold to take that metric curve the space in on itself. So this is the classic analog of this balloon, okay? And if you do that if you take a two-dimensional spherical surface embedded in three dimensions then you have a situation where the space is finite but unbounded, right? And it has because it has no boundaries it has a non-singular metric everywhere and this is the model that for curved space time that has come to be adopted and which we are still using today. So this thing works. So his world model is basically based on homogeneity which, by the way, he didn't know about at the time when he propounded this that was only established later by 1926 as I have already shown you by Hubble, right? And what he really relied on was what later came to be called Mellon's cosmological principle that everything is the same for everybody so you can define some, you know fictitious set of co-moving observers who are all in this space time they can all exchange the time with each other they're all carrying clocks and rulers, right? And they're all comparing notes with each other so therefore they'll be able to construct a close dense coordinate system over this entire manifold, right? And this is the standard model that we are using today to interpret all observations, right? So let me go very quickly through this because this will all be in the notes some of you would have done this before so the name of the game now is to construct an analog of that balloon which we normally use to picture the universe but we are going to construct a three-dimensional balloon embedded in a four-dimensional space the fourth dimension is the fourth spatial dimension entirely hypothetical it's just a fiducial dimension, it doesn't exist, right? So the idea is however sketched out here you know if you have a balloon a sphere then you can consider a little so this is the line element on the sphere and the set of points that define the sphere are x squared, y squared, z squared plus w squared this is the fourth dimension, right? But I can also define if you like a ring of latitude of course I can only show you a 3 and a 2 so imagine a 4 and a 3, okay? So the line of latitude here that little square will be x squared plus y squared plus z squared whereas what I am showing you here is just x squared plus y squared and r squared is x squared plus y squared plus z squared so just imagine I put an extra w there so in that case if you do the standard the maths so this is the line element I have got an extra dw square here but I can replace it by r squared dr squared by capital r squared minus little r squared where capital r is the radius of the big sphere and little r is the radius of the line of latitude, okay? So essentially I am just trying to generalize the line element from 3 to 4 dimensions, okay? So when I do that then I get a line element that looks like this, okay? So this is the angular part multiplied by r squared and this will largely play no role in what is going to come because the universe is isotropic so there is no angular dependence we are really going to be focusing on this part and I want you to recognize that this thing which will come later is actually just come from this fiducial fourth dimension of course I can always map to, you know from one set of coordinates to another so for example I can map into the polar angle chi and I can write it like this the polar angle being the angle subtended by a point here with respect to the z axis, right? So this is already beginning to give us quite interesting physics because basically now we have to ask the little r and capital r if capital r is very big compared to little r okay? Then of course we see space as more or less flat we don't see any effect of it just like we live on the earth it looks flat, right? You only start seeing the effects of the curvature when you sample a dimension which is comparable to the scale of the radius of the earth in other words when you fly in a plane or something you are 30 kilometers above the earth then you can begin to see the curvature of the horizon you don't see it from the ground, right? However when the two become comparable then you start seeing interesting effects for example if I take a standard bar and I move it along remember that light can only travel along this surface then you see it becomes a maximum at the equator and then it starts becoming smaller and smaller and smaller and when you reach the antipodal point it has shrunk to a point okay? So you get lots of interesting effects and those are things that people there is a nice little book by called George sorry Tomkins in Wonderland discussion of this by George Gamma, right? So in this picture remember always that the big bang is the antipodal point of the hypersphere that we sit on okay? That's one way to think about it right? Anyway these are the three possible geometries of maximally symmetric space that is what we are considering the maximal symmetry okay? And if it was flat space then angles of a triangle add up to 180 otherwise they are more than 180 if it is positively curved or less than 180 if it is curved like a saddle right? And now we have to start asking what is the role of time? So far you have not talked about time. Time is considered to be just as in Newtonian mechanics something that is background okay? Because you are going to be dealing with small gravitational fields so unlike in say black holes or something where time is very closely intertwined and in fact the direction can flip over here time is basically considered to be independent of the rest of it. So what we are going to be considering as was done by Friedman and Lemeth is the idea that we are in a dynamic space time that space time that we constructed is allowed to now evolve under gravity and what we will then do is to generalize that scale factor that radius of the big sphere that we talked about into something that is measured by some quantity called the scale factor A of t that varies with time and that is the only variation is time everything else is independent of time okay? Now the idea is that if I do that then the change any change dynamical is self-similar a triangle will remain identical to itself or become smaller or larger but it remains the same shape okay? In fact that is because the metric that we constructed is actually conformal to Minkowski metric as you will see later okay? If I want to describe a open expanding universe I can just change this chi to i of chi and then instead of signs I will get sign hyperbolic okay? It is trivial so trivial in the sense mathematically trivial we are describing the entire universe though don't forget that so when I put all this together I get the so called Robertson Walker line element named after Robertson and Walker of course and that is the one that you are familiar with which has got this part here which is largely unimportant because they don't play a role since when the galaxies evolved the theta and phi's don't change the angular part remains invariant we are only going to be concerned about the radial part the spatial part and the only time dependence is here right? Null geodesics as usual have interval equal to zero so we can trace the path of light rays we can trace the path of material particles using this metric the metric is everything that there is using the metric we have covered the universe with graph paper we can measure things okay? so you don't then ask questions like what is the universe expanding into there is nothing except the metric okay the universe is the metric visually this is what will look like these are some nice pictures I picked up from Martin Booker who shows if you filled an universe with grids that's what a negatively curved universe would look like as opposed to a flat universe as opposed to a positively curved universe this is the curvature of spatial sections okay you would still have curvature in space time even when space is flat and therefore now we are invited to contemplate the universe as just something which is positively curved becomes bigger and then might contract if it is flat it remains flat and just keeps expanding asymptotically forever similarly if it is negatively curved little saddle becomes a bigger saddle becomes a bigger saddle so these are all of course cartoons these are just depictions to try to give us a visual image do not follow these too far trust the maths these are just to make us feel that we understand what is going on but always the equation is the final arbiter otherwise you get into all kinds of paradoxes and complexities which are meaningless so now we can go back and ask if this is the metric then why are we seeing redshift we are seeing redshift because when you look at a null geodesic ds square equal to 0 that will be just dt over a of t because there are only two terms the rest of it has gone and then that is just this quantity equals constant so if I look at a particular galaxy which is emitting light and I look at the peaks or crests of two waves that has been emitted and I ask by the time it reaches me what has happened to the wave basically the wave has been stretched out because the scale factor has increased in between the light being emitted and the light being detected so the change in the wavelength of the light which is what we call the redshift is simply the change in the scale factor and that is the simplest way to understand the redshift the redshift you can try and interpret it as a Doppler shift on small scales and you can even interpret the redshift at larger distances as the sum of a lot of local Doppler shifts but that is not a very good way of thinking about it because that gives rise to the usual paradoxes about what can something be going faster than light away from me and so on the expansion of space has no limit it is not carrying information it does not have to preserve the central tenet of special relativity that nothing can move faster than light it is actually moving in fact as you will see the expansion is just illusory it is just a coordinate transformation nothing is actually expanding depending on which coordinate system you are in but basically this is the picture the wavelength of light has been extended and the change in the scale factor is just the redshift think of it as a different kind of redshift from Doppler shift or the gravitational thing that you would see if you put a gamma ratio here at the ceiling that is the gravitational redshift this is another kind of gravitational redshift one that operates only on the very largest scales in the universe right so in this picture therefore if this guy goes to zero z of infinity that is the big bang and the big bang I try to give you an idea of how to think about it think of it as the antipodal point of the hypersphere on which we sit and that kind of makes sense because when I look out at the universe I see things getting bigger and bigger and bigger right so how am I ever going to reconcile that with the idea that at one point the whole universe was inside smaller than a nucleus or whatever it is I tell you right because things can get bigger and then smaller thanks to the geometry of Kurtz space time okay which is not something that we are part of our intuition at all we live in flat space time this is a point that frequently puzzles people is everything expanding of course everything is not expanding otherwise you would never know how are you going to measure it right in fact the precise statement is that bound structures anything which feels any other interaction electromagnetism strong interactions or whatever does not expand because those are far far far stronger than gravity they're bound objects even our solar system is a bound object our solar system is not expanding the galaxy is not expanding it is bound by gravity so over density of 100,000 compared to the universe as a whole it's only when you get to the space between galaxies with the density contrast is much much much smaller than one where the potential is more or less unperturbed that you start seeing this overall stretching of the metric that is what I'm talking about and as I mentioned there is no restriction on the rate at which that metric can be stretched because it's not carrying any information there is nothing physical about it and in fact as I already hinted this expansion is illusory because I can always go into a coordinate system where I just take the expansion out it's expanding at the constant rate anyway I just move to a frame where the expansion is taken out and then I can see in the so called co-moving frame all the galaxies are stationary with respect to each other so nothing is expanding now that is basically the essential principles of the standard cosmology and this is what we are basing our entire framework on so ok so now I come to the dynamics and then I'll take some questions so far I have constructed a metric I have shown you that you can use that to understand why the light from distant galaxies should be redshifted how that is a proxy for the distance for small red ships and how this allows us to conceive of a evolution back towards when the scale factor was much much smaller when the galaxies are closer together the universe was denser and since we know some physics we know that would mean that it gets also hotter so I have to start talking about matter I have to start telling you what is in this universe so the classic way that the cosmology started was by assuming that the universe is full of ideal matter matter whose world lines are like this they do not intersect each other they just move without colliding with each other it is a collision less gas of particles ok it might have pressure if it is relativistic but typically it is non-radivistic they called it dust you know galaxies are like dust particles they do not hit each other they just move up in straight lines like that ok so you can think of it as a fluid the equation of that is diagonal there are no there is no vorticity no angular momentum no messiness of any kind ok no dissipation nothing and the only thing that is bringing here from general radivity is that Poisson's equations which you normally would write with 4 pi g rho on the right hand side now has 3 p added to it why 3 p because pressure is kinetic energy the gas in this room has got kinetic energy it's got pressure and by according to general radivity all energy gravitates including kinetic energy so you have to add the pressure to the total budget of stuff that gravitates this is of course the mass energy density for dust however apart from that little change nothing else has been introduced here from general radivity so I am giving you a sort of a nice simple version of how to derive dynamical cosmology the work of theorem still holds on Newton's iron sphere theorem if I make a little hole then whatever happens outside it I don't care about it all cancels out inverse square law ok so the field equations of Einstein which are in general very very complicated I have a metric from the metric I can construct this the Riemann tensor I can construct the curvature I can construct the Ritchie scalar so I can write down Einstein's equations in terms of projects that have been constructed from the metric on the left hand side and Einstein tells us that the behavior of that all that is governed by what is in this right hand side which is the energy momentum tensor and the connecting factor is the Newton's constant ok now this is a very very complicated equation in general you can see how many indices there are on these tensors right and obviously you need this full machinery if you are going to be dealing when there is strong gravity say near a black hole and so on fortunately as I have already explained to you cosmology is much much simpler we have chosen the most symmetric possible space time that there is ok maximally symmetric space time the Robertson-Walker metric you can't get any more symmetries than that and we are populated with an ideal gas with a diagonal T mu nu ok you have made life as simple as possible for ourselves therefore we can actually solve this equation and this equation basically only the 00 and the 11 components are relevant and they give us something that looks like things that we can understand this is why this model by the way is so popular because everybody understands it or everyone can do maths with it you can construct observables based on it you can go out and look for them and of course it leads us to very startling conclusions namely that two thirds of our universe is supposedly made of dark energy so at that point maybe you should go back and look at these equations and ask if you really formulated it properly but fortunately that is not the subject of my lecture today ok so this is the equation the Friedman equation as it is called which tells us that the rate of change of the scale factor is governed by the balance between the energy density of matter which is gravitating and the possible curvature of space section and the acceleration which is sometimes called Rai Choudhury's equation has got this 3p from their added to it and there is a minus sign there if this quantity is positive then the universe expansion rate is always slowing down as is natural because gravity is attractive force ok but sometimes if this guy is depends on what the sign of this pressure is you can get a reversal of this sign and you can get an expansion so that depends on the equation of state ok now just to step back a little bit because general relativity even at this level even with this maximally symmetric solution is pretty complicated let us look at the Neutronian example because that is simply to understand so that equation the Friedman equation that we just saw just looks like the equation for the acceleration of a test particle which is at the surface of a sphere of radius R which contains you know the R cube times rho amount of matter and you each shell you know how to do the exercise each shell is attracting the particle and you sum them all up and then you will get this equation which tells us using that second equation that the acceleration goes with proportion to rho plus 3p times the scale of this sphere ok so I am just doing this simple Neutronian exercise which I am allowed to do because the Berkhoff's theorem still works in general relativity and I supplement that with the first law of thermodynamics that tells me that if I have got pressure of the gas that is contained in a volume that is expanding or contracting the kinetic energy the particles are bouncing off the wall they are doing work on it and so they will cool down or heat up according to whether I am expanding or contracting ok that is all straight forward when I take that equation and I solve it then I find that for an ideal fluid I should expect this relationship integrating that one so therefore this v dot by v is just 3 times l dot by l because v goes as l cube and when I plug that in into that top equation I get that the acceleration is therefore the sum of these two terms and that allows me to integrate it once and I get that the velocity square is going as this ok this is an integration constant sorry integration constant so at this point Einstein who did the same exercise decided because it was the 1920 something nobody had told him that there were actually island universes and we are this all red shifted and all that he did not know any of that he thought we lived in a static universe and he tried to therefore get a static solution you can get a static solution by choosing that integration constant correspondingly and you can see that if I choose this guy this thing equal to 0 then I can get a static universe l double dot is 0 right and that will give me a universe of this particular radius which is just one 8 piracy 0 inverse in fact he wrote it down on a black board which we have in the museum of the history of science in Oxford it is even got a typo in it he got the dimensions wrong but that was a static universe actually this is one example I know one a very rare example where Einstein's intuition went wrong he should have realized that a static universe is unstable if I put up the metric a little bit you can do this exercise if you put up the metric you will see that the perturbation goes exponentially in an universe which is maintained static by doing that balancing act it is not a viable universe and in fact when this was pointed out to him Einstein wanted to do away with that term but actually he is not allowed to do that it is not his choice because once you write down an equation subject to certain symmetries you have to accept whatever it is that the symmetries allow you have to write down so basically the statement is that the symmetry that underlies this equation which is general coordinate invariance allows you to add any term which is multiplying the metric okay this is the term that we must allow and this is called the cosmological constant and this in fact is something which is not a matter of choice but a necessary and inevitable consequence of the symmetry underlying general relativity okay you can only do away with it by going to something other than general relativity in fact what was recognized later by Zeldovich and Pauli and others was that this lambda term there is also in field theory this Tij we are not talking about an ideal gas but later we might talk about field theory Tij has also got the freedom to be scaled by so called super normalizable operator in effective field theory terms which also is proportional which is like a cosmological constant and therefore I can interpret this I can take it to this side as interpreted as some kind of a fluid which has got pressure which is the negative of lambda over 8 pi gm right but in principle there is actually a cosmological constant associated to the geometry and another cosmological constant associated with the field theory and the net cosmological constant is the sum of these two which are normally completely dependent and yet we will see later that it interpreted in the simple model that we are using we are being told that the sum of this guy and this guy which are two completely different things on some background which does not know about this guy at all is of order the present Hubble parameter square that is what we are being told okay so now let us go to the dynamics of the full Friedman-Remeth-Robertson-Walker metric as universe as we now call it right so now we have that a dot by a we have seen this equation before but now we have written the curvature in terms of plus minus we have chosen a particular signature which to reflect whether the space is positively or negatively curved or flat right and then we can split it into a part that depends on the background which is ordinary matter and radiation and then we can add this lambda term as we are required to do because of the coordinate invariance of generativity right and we also have conservation of energy momentum which is just what we you know when we expand the volume work will be done on it right and that then finally gives us the standard Friedman equation which is what is the basic workhorse of current cosmology and the plus minus now refers to the open or closed universe I have specifically put R here to make you realize that that is a reflection of the radius of curvature of that internal space the fiducial fourth dimension that we constructed in order to embed our three dimensional spacing so there are two solutions describing an expanding universe which were first discovered one by Einstein decider where you put say that the pressure is negligible you have dust particles and lambda is 0 and the curvature is 0 in this case you can simply integrate that a dot by a squared goes as 8 pi 0 by 3 and then a goes as t to the two thirds because the energy density of matter is being diluted just as a cube as the space becomes bigger and therefore the time is just two thirds of inverse of H and you can relate it to the energy density of the dust which is filling the universe and then decider found another solution the opposite extreme where you have no matter at all that is 0 but you have a cosmological constant lambda right you just have that so a dot by a square is equal to lambda by 3 so therefore a goes as e to the square root of lambda by 3 times t is exponentially expanding universe without any matter in it a direct violation of Marx principle if you remember that it's an anti-marquian solution so the joke at that time was that this was motion without matter and that Einstein static universe was matter without motion and of course you can now construct many things in between but these are two standard models that have come to to analyze and of course in more general terms you can have both matter and lambda and you have to ask for the consequences of having a lambda term if it is comparable to H squared then it will be of order similar to the matter density so you can have limits universe which is like expanding and then goes into a coasting phase bit like Einstein static universe but then lambda picks up again and goes exponential and so on so this is all I have to say about the construction of the standard cosmology we are now going to use this standard cosmology as it was done in the 1930s nearly 100 years ago to now start exploring the early universe and I see that I have another 25 minutes left so I will take another 15 minutes to do the radiation 5-10 minutes maybe and then allow time for questions so are there any questions at this stage before I go on or would you rather wait yes yes I have not discussed topology at all actually I had a couple of slides on that which I left out since I meant to be talking about the early universe in general the universe's topology is an open question none of what I have said decides the topology however if once you are stuck in one particular geometry negative or positive it is very hard to imagine how you can have a transition to the other geometry because that would be a so called folding transition in order to fold you need one more dimension to fold in well so you could imagine doing it if you are a particle phenomenologist you can do anything you can imagine an extra dimension and fold in that but not in the classical space time that you are considering I was wondering that result the decider's result can also be found just by taking pure vacuum energy this one yes that's exactly what the decider universe was and the so called steady state theory was based on that metric and so we have had decider space time right from the beginning of cosmology it was ejected because in order to have that because the universe is getting diluted to have anything in it you have to constantly create matter at not a very great rate you have to create one atom per century per galaxy some small number like that right nonetheless it was the steady state theory was ruled out because it will not give you a hot big bang which is what we are going to come to right but actually all the mathematics of the steady state is now still present in the so called inflationary universe things just go round and round right and people rediscover them you said that the universe is the metric and I've heard several times that the big bang was something like a huge amount of energy concentrated in a point what is the meaning of that you better ask the guy who wrote that in the new scientist or something it doesn't mean anything so the big bang is really something which happens in the whole universe we don't know the size and everything else right I'm only concerned with what I can see and construct as I've done for you okay now the next question is having done this and as I said I'm still stuck back in 1935 I better fast forward to the modern day so I'm going to construct for you the early universe you're going to go back 14 billion years to the very first second and all that the first question you might ask is how do I know that the laws of physics that I measure in the laboratory today are the same 14 billion years ago you might say why not but okay that's a good answer but it could also be that they're different I mean you might ask a string theorist and they'll tell you all couplings of nature or expectation values of moduli fields or something and they're evolving so maybe we're different in the past well is that true we can do experiments here is an experiment done by optical astronomers where battal in Australia who are looking at the fine structure constant and what you see there is the variation in the fine structure constant which of course as its name says determines the hyperfine splitting of spectral lines and what they can measure is that as you look back in redshift up to redshift of 4 which is basically most of the age of the universe the fine structure constant stays within a few parts of 10 to the 5 of its present value okay we'll say 1 part into the 4 actually at one time they were claiming they could see a trend but what I take away from this is that there is no evidence for a change in the fine structure constant over whatever year over 12 billion years I think that's a very interesting result so I'm going to use that as my justification for extrapolating back physical laws back whenever I like because you would have expected to see so if those moduli fields are changing and varying the fine structure constant they stop doing that in the first second of the universe that's a very powerful constraint by the way on those ideas in fact we'll see that everything should have frozen by 0.1 second otherwise you would have seen it in big magnetosynthesis so now we can start constructing a dynamical evolution of the early universe because for matter we know that the number of particles are conserved in a co-moving volume so this is the number of particles in the volume it's not changing nothing is being created or destroyed that is 0 so rho therefore is going as 1 by a cube 1 by a cube is a shorthand for 1 plus z cube and therefore if I can now simply solve the Hubble equation and I get that A goes as t to the 2 third you have already seen that that is the Einstein DC dissolution however if I have radiation then the thing that is conserved in a co-moving volume is not the number of particles but the total energy density and that involves the energy of the photon itself because as the box gets bigger the energy the energy of the photon is of course redshifted right so now it is A times 4 and therefore rho is now going as 1 plus r to the 4 and the radiation therefore the rate of change will go as t to the half rather than t to the 2 thirds right and because this is climbing faster with redshift than matter radiation will come to dominate in the early universe at high redshift right at redshift is over about 10,000 radiation is the dominant component radiation is also the dominant component even today in the sense that the number of photons in the universe as already mentioned is a billion times larger than the number of particles of any kind so that makes life very simple when you come to studying the microwave background as you will see later so what is the time at which these two things become equal I just said one to the other so I get a 1 plus z factor by equating the 2 and that then this the equality value of the scale factor is just about 10 to the 4 as you will see right but notice that rho goes as 1 by t square at all times whether radiation or matter so this is log rho versus log of A and so the power law of 1 by A cube is this 1 by A4 is that and this is the point of equality right as I mentioned astronomers tell us that today they see some evidence for a lambda which has come to dominate at the present epoch it is about 2 times more than the matter density and that of course is a serious issue because the early universe was radiation dominated this is going less slowly therefore it is natural for the universe to become matter dominated at some point doesn't matter how much matter is there this line can go up and down but I can draw the line here or here or here that will determine the equality redshift but ultimately the universe will become matter dominated but for the universe to become dominated by lambda ok just today is very very weird because it was clearly completely negligible at early times I don't have to worry about it you can see how many orders of magnitude smaller than or it is but the value of lambda is meant to be the Hubble parameter square this is the actual value deduced from the data and this is a very unusual number because H naught is the present rate of expansion so lambda cannot be a fundamental parameter because how can a theory, fundamental theory know at what rate the universe is expanding today but be there as it may this is something that we fortunately don't need to worry about in these lectures because we are going to be up there we are going to be in the radiation dominated area so I am going to describe to you the construction of the thermal history of the early universe which is normally shown in pictures such as this that you might have seen we are starting today where we see this 3 degree K background we look back through all these galaxies, quasars etc and we are looking back to a red shift of 1000 where the universe becomes opaque because it becomes the radiation the universe becomes a plasma it becomes ionized and you can't look through it and matter domination occurs just a little bit before that and that is significant because once matter comes to dominate fluctuations can start growing under gravity so the large scale structure of the universe is forming then before that the universe was totally smooth with small density fluctuations as we see in the microwave background we can trace the expansion back as I will show you back to reliably to about a second after the big bang when the temperature was high enough for nuclear reactions to happen before that neutrons and protons were a free ideal gas and at that point they combined to make nuclear and that is one of the big successes of the big bang going back to 1953 when alpha, fallen and herman gave a complete and still correct theory of big bang nucleosynthesis and because we now know that we have a fundamental theory that goes up to 100 GeV we can actually construct the history of the universe back to about 10 to the minus 11 seconds however this part is only based on our understanding of the theory and on lattice conditions of non prohibitive dynamics we do not have any direct evidence of any relic from the early universe which is coming from either the quark hydrogen or the electric phase transition so we know everything reliably up to there more or less reliably up to here beyond that is total speculation beyond that is particle cosmology and although it is speculation it is essential to speculate because none of this part explains anything about the universe as it is today. We are here our antiparticles are not here that is a very big fundamental problem that we have to understand because all the laws of physics that we know are symmetric between particles and antiparticles okay there is a tiny tiny violation of Cp in k on decays now seen in b meson decays at some you know one part in 10 to the 5, 10 to the 8 okay but what we are seeing is the order of one asymmetry which is a very big problem and there has been a lot of excitement about it so this will be discussed in later lectures I suppose we do not have any candidate for the dark matter that seems to be much more than the baryons that you have made of 6 times more what is that that could be a new particle possibly in new physics beyond the standard model we do not have any explanation for why there are density fluctuations in the universe we need those fluctuations to grow under gravity to give us these galaxies but also fluctuations random fluctuations of a gas are not sufficiently large to give us what we need so these are at least three relics of the early universe we have no idea when they were created when they come from so we speculate away about you know possibility of the genesis of baryons perhaps through genesis of a neutrino asymmetry first leptogenesis the fluctuations might be created by a period of primordial inflation the dark matter could be a new candidate a new particle new stable particle in some physics beyond the standard model so that is all speculation so now let me give you a couple of more interesting issues about the early universe which in my view are open problems in case any of you are looking for PhD problems or whatever to do so the interesting question that is very rarely asked and I think should be asked more often is does the universe have any net quantum numbers because some people talk about the universe coming as a fluctuation from the vacuum a vacuum has no quantum numbers what do you actually know the chemical potential which is obviously derived from chemistry that is how it is derived is the quantity that characterizes a conserved quantum number and it is additively conserved in all reactions because number of particles is conserved so it is zero for photons because you can create of course photons in arbitrary numbers and so it is equal and opposite for a particle and its antiparticle if you can annihilate into things which have no chemical potential remember it is additively conserved in all reactions so if you have a finite chemical potential then that is a particle antiparticle asymmetry so in other words some non-zero value for a conserved quantum number now what are the non-zero charges that we expect obviously have to be associated with gauge forces because global symmetries can be violated if on maybe by Planck level effect global symmetries we know are not sacrosanct but gauge symmetries are so we know electromagnetism is a very good gauge symmetry remains invariant all the time so the net electric charge of the universe is conserved but what is it is it positive negative zero we do not know it is consistent with being zero because right back from 2016 when Bondi and Gold wrote a paper about it and then there was a long gap until 2005 when these guys wrote a paper about it if you consider a possible difference between the electron and the proton charge which would then give you a net charge for the universe you can easily work out as Bondi and Gold did that the moon's attraction towards us by gravity would be overcome by the electric repulsion if it exceeded something like 10 to the minus 24 e and these people have improved it by another factor of 100 by looking at fluctuations in the microwave background the barrier on acoustic oscillations would be affected if there was electric charge so we just have a upper bound we do not actually have a number the net barrier on number is very very small compared to the number of photons I have mentioned this several times it is less than one in a billion but that is not surprising in the sense that barrier on number is a is a global quantum number there is no long range force connected to the baryons we know that from all the Etowage experiments there are no fifth forces so you could imagine creating baryons out of nothing and indeed that is those are the theories of parallelism and similarly left on number right there could be in principle a large left on asymmetry in neutrinos because B minus L is an exact symmetry of the standard model but it could be violated by beyond standard model interactions you know so in principle you could have a large left on asymmetry but because neutrinos are now known to mix pretty strongly with each other you cannot just keep it in one particular species one particular flavor it will mix with the others and a large left on asymmetry in electron if you know is strongly constrained by nucleosynthesis it will change the neutron to proton chemical abundance so that is really very very small as well really of the same order I have not mentioned the question of a net color charge for the universe which is very very interesting I have not written that down because there is some interesting issues to even formulate the question because you know there is something called confinement having a net charge would of course violate confinement confinement of course is a empirical question to what extent we know that confinement is true but I would not go into that here but I just mentioned it in case some of you want to think about it the dark matter may have a large chemical potential not a large about the same value as the baryons if it is asymmetric it is quite possible that the dark matter has an asymmetry it is just like baryons but it is 5-6 times heavier and I think Zurich will be lecturing next week she might talk about that okay so now I have well now I have almost finished my time so I think I will stop at this point and I will continue with this in the next lecture constructing the thermodynamics of the early universe constructing the equation of state constructing how we actually know all these numbers that are this picture that you see here and other similar pictures how we actually construct that I will wait for the next lecture thank you okay now we have time for questions yeah so in one of your slides you displayed that the evolution of A dot over A depends on whether the universe is closed or open so I was wondering since A dot by A determines the value of the red shape so by measurement of red shape can we see whether our universe is open or closed indeed that's how you do it and the answer is that it is flat that's exactly how we actually did it we looked at the Hubble parameter as a function of redshift and look to see if there is a change in the curvature of that in practice actually the determination was done by looking at an angular size rather than at an expansion rate but it's a similar geometric measurement yeah but I mean can you say it is absolutely flat but then you are measuring absolutely zero I mean quite right you can't ever measure something to be exactly zero obviously we think it is close to zero to within one person that is the current state of that I mean in the open side or in the close side that we do not know we can only say it is plus minus one percent the theoretical prejudice would be it would be towards the open side but that's the theoretical prejudice you have said that we cannot exclude lambda by pure coordinate invariance what did you mean by that it means that the symmetry so the theory that we wrote down generally adbd has underlying symmetry which allows you so when you want to write down the equation of motion that symmetry allows you to write down a term which is proportional to the metric the multiplier of that term that is called the cosmological constant is completely arbitrary it can have any value but it also allows to write down an infinite number of different terms which we don't include I mean you can write any power of curvature right anything that is allowed by the symmetries you can write down and lambda non-equal zero is not more mentioned by it than r squared or r cube right no it is more fundamentally more of course you can write down higher powers of r you can add to the equation and people do that now but lambda is more fundamental in the sense that reflects the actual the basic coordinate invariance that is what it's directly related to and that is of more paramount issue the higher order effects in r are only going to be important when the curvature is strong okay lambda is going to be important even if you have no curvature at all as it is now shout out shout out with regards to the first I think second slide with the promoter fluctuations you had a theoretical curve that had the cold arc meta line the first few points in the small scales it clearly does not fit now this is a known problem with lambda or cold arc meta people have been talking about self-interacting in dark meta do you know how that improves that to fit no that is so let me go back to that slide because that's an interesting question in fact one that I'm working on presently but not in a position to answer it in more detail here so you're referring to the fact this is the old plot so obviously this is not a fair reflection of the data today but as I said I'm inviting you simply to observe that the overall shape is captured by the theory right not in detail because in any case a lot of fudging has gone on to make those data points sit on the curve right so what he's referring to is the fact that this is the theory based on collision less non-radivistic particles clustering under gravity and having no other interactions and in detail on small scales that may be mismatches with what is observed now those mismatches may well be due to the fact that our understanding of the formation of small structures based on numerical simulations does not yet include an understanding of baryons which are after all there it's an important component of the galaxy and baryons indulge in all kinds of behavior like you know they lose momentum they radiate they lump together the simulations cannot capture all that physics so there is a belief that if we could include the behavior of baryons properly then that might well account for some of these discrepancies which are there at present however an alternative and more exciting to a particle physicist idea is that the dark matter itself is not entirely non-interacting but has some interactions with itself or maybe it has some interactions with the relic neutrinos who knows maybe so the only thing we can say is that the dark matter does not interact with photons that is why it's dark matter but the other aspects of its interactions are pretty unconstrained in particular self interactions are unconstrained to the level that they could be of a hadronic size and very recently this is within the past month there has been some preliminary suggestion that dark matter does interact with itself with the hadronic cross section and if that turned out to be true then obviously it would have a significant impact on the formation of small structures but much of that is not you are not going to see from a power spectrum a power spectrum by definition is just looking at the two point correlation that would be reflected in more complex issues like for example the slope of the density profile of the inner parts of galaxies or dorsferoidals and stuff like that it would be reflected in the shape of the halo whether it's elliptical or spherical and things like that excuse me you say the chemical potential is conserved in all interactions yes okay but some processes like electron, positron annihilation of some process in which a large amount of entropy produced the chemical potential is not conserved in some processes like these is that a statement or a question I think how can it not be look at the right hand side what is E plus E minus annihilating into into photons for example photons I told you don't have a chemical potential because they are bosons they have no conserved quantum number these photons are being created in arbitrary numbers and you're absorbing them in arbitrary numbers they clearly don't have a chemical potential for some process that a large number of entropy produce you doesn't matter how much you produce if they do not carry a conserved quantum number you cannot keep track of them right you got to have some way to keep track so if you cannot keep track of a particles number density right then that means it doesn't have a chemical potential and if I've got some species that can annihilate into it particles are anti-products then that can't have a chemical potential either so you're right in a sense that if electrons and positrons annihilate into photons the parts that do annihilate have no chemical potential but I might have one extra electron or one extra positron for every billion electron-positron pairs that is a chemical potential and that is the guy that we actually have our chemical potential today is almost zero almost negligible normalized to photons is 10 to the minus 9 we are a little sprinkling of froth on top of a huge sea of photons so to a very good approximation the universe is just a gas of photons which in fact dynamically dominated in the early universe as I'll describe but even today the number of particles massive particles is completely negligible compared to the number of photons thank you so you mentioned this problem with the velocity dipole on the CMB and I wanted to ask you if this actually is just in the dipole still so over to other lower L multiples like quadrupole and so on well if you do the Lorentz transformation you'll find that actually you should see an effect not just in the dipole but also in the subsequent multiples right so the dipole is affected as beta then there is a beta square, beta cube we can do the expansion actually the plant people only realized this last year they were using only the first order thing now they have corrected it but the important point is not sorry the important point I was trying to make here is that effect is seen, Planck actually sees the effect on the quadrupole etc etc there is no question that what we are seeing is a kinematical effect we are moving through the microwave background there was a possibility earlier people had suggested the universe was tilted or something right I think it is completely clear we are moving if we are moving what that means if I can go to the next picture what that means is that we are here and we are moving towards shapely we are moving in this direction what we have done is we have looked at shells of galaxies going up to shapely right and you see that we are still moving even beyond shapely there are there is some data now from the something called the nearby supernova factory and the motion is continuing beyond shapely because earlier people had thought that somewhere here there was something called the great attractor that is what is pulling us if that was so then when I look beyond it I should see things falling in from the other side we do not see that so the motion is continuing in that direction and it has already got to about this point here 300 megapastics which means that there is still some as yet unidentified huge lump of matter okay of something like 10 to the 15 16 solar masses right that is about million you know 100,000 galaxies we do not see it nobody knows what it is right it could be of course a lump of dark matter but then you should not expect such a huge lump on that scale in a homogeneous universe so that is why I am highlighting this issue that it is a very important problem I think to understand why it is that we are moving okay remember this is I should have called this eppur see move away yet it moves we do not know why any other questions okay so it is time for a coffee break let us thank the speaker again