 Alright, welcome back. I'm particularly happy to introduce Kazuya Koyama, because his flight was cancelled yesterday, so I was afraid that he couldn't make it. So we're glad to have him here, and he's going to talk about dark energy. Okay, can you hear me? Okay, so I'm very happy to be here. In fact, my flight was cancelled yesterday and I just arrived half an hour before my lecture. So I'll talk about the dark energy, and I think you had a very good introduction lectures of cosmology. So this is a continuation of lectures on cosmology. I guess it's better to wait. Okay, so I first give you some reading lists. By the way, so the slides are already available on the web page. So if you go to the web page, you can download your lecture note. So I uploaded all the lecture notes up to lecture 4. It doesn't mean that you don't need to come here. You have to come here. That means that you don't need to take notes. You can just download the lecture note. So now we have a textbook on dark energy. So this is a very good starting point if you are interested in dark energy. So this is a huge textbook. And then there is a very nice review on dark energy. So this is in 2006, so it's getting a little bit old, but this is still a very good review article you want to look at. And then I will also talk about modified gravity, and this is probably the most useful reference. So this is in 2011. And I also talk about some of the vaginal tests of these models, and this is a very good review, discussing not only theoretical aspects of dark energy, but also discussions about observational tests. And of course I include my review, so of course you can look at my review. And of course, probably you know that no one knows what is dark energy. Maybe I can finish my lecture here. So I think it's important to understand why we are so interested in dark energy. So why we need to spend four lectures learning about dark energy. So the first lecture is basically a try to critically examine the evidence of dark energy. So this is basically probably you already had in the cosmology lectures. But try to make sure that what is the assumption we make if we say there is dark energy. And understanding the assumption to get the evidence of dark energy is very important to construct models of dark energy. And I also talk about cosmological constant today. This is probably every one agrees that this is the most probably approachable, simplest answer to dark energy. But of course we have a lot of problems with cosmological constant. So this is the first lecture. And then the next lecture I will introduce models of dark energy and modification of gravity. And as you will see that there are millions of models. And I think there is no meaning to discuss all the models. So again the focus is why it's so difficult to construct dark energy modified gravity models. So that is the fundamental problem. So this is the second lecture. And in the final two lectures, I try to move on to a slightly different aspect to try to understand how we test these. These theoretical models using observations. And the most important thing is to understand how the structure in our universe is formed. And I think in the next week you will have lectures on CMV and large scale structures. So you will learn more about the details of observation and proofs. But again I will give you the basic idea how we test dark energy using observations. And I will show you some examples of current observational tests. And also discuss why the next 5 to 10 years are so excited. And in the final lecture, in the third lecture I will mainly focus on linear structure. So linear means that the inhomogeneity in the universe is very small on large scales. So you can treat everything deviations from isotropic homogeneous universe as a small perturbations. But you know that on very small scales these structures become very nonlinear. And we will probably already have very good lectures on numerical methods. So we will need a numerical method to understand the nonlinear structures. And again having dark energy modified gravity models, these nonlinear structure formations are very complicated. So I will try to explain all this. So feel free to ask questions during lectures if you have any questions. Okay, so I think I try to change to the first lecture. Okay, so let's look at the observational evidence of dark energy and what is the cosmological constant. So I will show you there are three basic assumptions you will make to show that there is dark energy. So let's remember the first assumption, so this is the first assumption. So the first assumption is that our universe on average on large scales is isotropic and homogeneous. Okay, so this is the first assumption. I will come back to this assumption, but isotropy is pretty much proved by cosmic microwave background radiations. So if you look at the temperature of CMB photons, the difference of temperature is 10 to minus 5. So this means that our universe looks very isotropic. But the second assumption is that our universe is homogeneous. And of course this is not the case on small scales because we have structures. So this is a distribution of galaxies in the less safe space. And if you look at this, you see some structures. However, the argument is that if you average over on very large scales, these distributions look very homogeneous. However, I come back to this. This assumption is very difficult to prove. You have to remember that this is indeed an assumption. So this one, we probably can say that we proved isotropy of the universe because of CMB, the homogeneity. And looking at the scales, the CMB, you can prove the isotropy up to 6000 megaprosec, but it's very difficult to prove the homogeneity. And in fact, the general relativity will play a role if you talk about homogeneity. So at the moment, this is the assumption. So once we assume these two, homogeneity and isotropy, you can write down your metric in this form. So the Friedman-Lawson-Werker metric. So the only degrees of freedom is the scale factor. And in the introduction of cosmology lecture, you learn that there are three possible carbatures for the three space. I'm sure you can read all these. But you can guess. So this is the closed universe. This is the flat universe. This is the open universe. And so depending on the carburetor, your three metric is, so this is the flat case. So this is the very simple coordinate. And if it is open, you have sine function here. If, sorry, if it is closed, you have sine. If it is open, you have sine hyperbolic function. So this is the open one. It doesn't look open, but it is open. So it's like satellite. If you compute the 3D curvature, so this curvature, so this is a three dimensional Riemann curvature tensor. So this is zero for flat space. And this is positive for closed space. And this is negative for open space. So now here comes the second assumption. So we need to know the dynamics of this scale factor. So we have to use some gravitational theory, and we use general relativity. So general relativity is a theory to relate the geometry to matter. And the geometry is described by this Einstein tensor. So this is the combination of rich curvature and rich scalar. And Timu Nyu describes the matter. So this is the matter energy momentum tensor. So this is the second assumption. So we use GR to describe the dynamics of our universe. So using the assumption one, so homogeneity and isotropy, you have to have matter energy momentum tensor in this form. Because using the homogeneity, you say that your metric is given by Friedman-Robotzen-Walker metric. Then you compute a left-hand side. And then the right-hand side should have the same symmetry to have the solution. And then the matter, energy momentum tensor, should be written in this way. So this mu is for velocity. So this is time direction. So it's minus one. And there is no spatial direction because you assume spatial homogeneity. And you have two quantities, energy density and pressure. So the matter is described by these two quantities. And the important thing about GR, so this Einstein equation, is that there is an identity for the Einstein tensor. So the Einstein tensor satisfies this Bianchi identity. This has nothing to do with gravitational equation. This is the identity for this Einstein tensor. So if you take divergence, so this is the covariant derivative, it vanishes. So this means that if you take divergence, so this divergence of T mu nu should also vanish. And this is the conservation of energy momentum tensor. So now we apply this to the Friedman metric. So you get the Friedman equation. So this is giving the expansion rate of the universe in terms of the density and also 3D curvature. And combining the time-time component and space-space component, you get this acceleration equation. So A double dot measures the acceleration of the expansion. And this is determined by low energy density plus 3P. P is the pressure. So you get these two equations. And then from the conservation of energy momentum tensor, you get this equation. So this shows the conservation of energy density. So if there is no expansion of the universe, so this means that the energy density is constant. But due to the expansion of the universe, energy density basically becomes smaller and smaller. And this is determined by low plus P. So think about these equations. You have three unknown quantities. You want to calculate the scale factor and energy density and pressure. And you think that, well, there are three equations. So you may be able to solve for these three quantities. But this is not the case. So you have to remember there is a Bianchi identity. So this is the identity. So this means that combining these two equations, for example, you can derive this equation. So this means that you have only two equations for three unknown quantities. So this means that you need one additional assumption that is to specify the equation of state. So equation of state is given by pressure over density. So this is what you have to put by hand. So this is basically coming from microphysics of matter you have in the universe. So you cannot solve this equation of state from gravitational equations. So then this means that you need the final assumption. So final assumption is that we will introduce dark energy in addition to non-matter. So non-matter, what I mean is variance. So variance is a very common name for everything. So leptons is also, I include leptons in variance, but variance is basically the normal matter. I say non-matter. But of course we don't know much about called dark matter. That's the reason why you have lectures on dark matter. But to describe cosmology, what you have to know is just the equation of state. And the equation of state for dark matter is there is no pressure. So the equation of state for called dark matter is zero. So even if you don't know what is dark matter, you just need to specify the equation of state. So dark matter, we say it's zero. And for variance, I will say equation of state is zero. And for radiation, you have equation of state one-third. So this is for photons and neutrinos. For dark energy, we have no idea. So just put dark energy by hand. This is unknown energy. So that means that we don't know about the equation of state. So we want to know about this equation of state. As usual, we will introduce this density parameter. So you divide your density by half of the functions. And you can do the same for curvature. And the Freedman equation becomes a very simple sum. So the sum of each density parameter for the matter and the curvature is one. OK? OK, so now let's move on to the observations. So what we measure? So what we measure is the distance. But remember that in order to measure the distance, you only need the assumption one. So assumption one is the isotropy and the homogeneity. And we say that our universe is described by Freedman metric. That is enough. For the moment, we don't need to use any other assumptions. So using this, you want to calculate the distance. So distance, this is the commuting distance, chi. So in the flat case, this is just chi squared. So chi is just radius. And in order to calculate chi, so basically we use the null geodes, s6, so your line element. So forget about the angular part. Your line element is you have a time part and this spatial part. And so let's forget about this angular part. And then because the photons obeys the null geodes s6, so this means that the line element is zero, you can calculate this d chi in terms of dt. So you change chi to t and do the integration. And then I introduce this let's shift. So let's shift is the inverse of the scale factor. And then you can change integration over time to the integration over let's shift. And by doing this, you introduce this Hubble parameter. So this is just a change of variable. And then you notice that this distance is determined by this Hubble function. But remember that this is just a geometry. Given Friedman metric, you can calculate the distance and you notice that this is all determined by Hubble functions. And you already heard about luminosity distance and angular diameter distance, right? So you can calculate this from a disk-moving distance. So this is what you can measure using supernovae or using CMV. So I will not repeat the derivation, but this is one of the very important equations you have to remember. But everything is determined by disk-moving distance, which is the integration of this Hubble function. You can also calculate the stage of the universe. Again, you find this e. So maybe it's better to remember e is the Hubble function normalized by the Hubble parameter today. Everything is determined by this function and this is the only dimension of the full quantity. And this number will become very important for dark energy. So let's remember this number. So inverse of this H0 has the unit of years. So this is roughly 10 to 10 years. And if you multiply the speed of light, so this becomes the length. So this is 3,000 megaprosec. So these two numbers you have to remember. And we often use this unit, natural unit, so I will explain later. So in terms of energy scale, this is 10 to minus 42 giga electron volt. So remember this number. So this number will come up quite often later. So now you want to measure first this number. So this number can be measured from luminosity, distance, or angular diameter distance. But for small z, this is basically determined just by a less shift and this H0 inverse. So you can measure directly H0. Or you can use CMB or supernovae to measure this H0. So this is a measurement of this H0, so in a bit funny unit. So this is kilometer per second per megaprosec. So it's around 70. But you see that the measurement from CMB, and so this is a measurement from this small less shift. There is some discrepancies. And this is a hot topic at the moment. So if you are interested in, maybe you can look at this recent paper. And at the moment this tension is not explained. So this may have some interesting implications to dark energy. So remember that this H0, we know roughly it's 70. But in fact the measurements are not so great at the moment. But what we want to know is not just H0. So what we want to measure is the equation of state of dark energy. So now we combine assumption two and three and try to parameterize this E function. So this E function is the helpful function. You can do that because if you know your freedom of equation, you can write down this H in terms of matter. And if you assume what kind of matter you have in the universe, so you assume you have dark matter and the variance, you have radiation, you have curvature, and you have dark energy. So then you can compute this function, and then you can compute all these distances. So lambda CDM model, so this is the affiducial model of our universe, is the assumption that the dark energy equation of state is minus one. So I will come back to this lambda. So basically this means that if this is minus one, this is just a constant. So this is cosmological constant. So this is just a number. So it doesn't depend on time. And then you try to see if this model can fit the observations. So you can measure these distances using supernovae, CMB, and baleonacosic oscillations in the distribution of galaxies. So then this is what we find. So in lambda CDM, there is only one parameter. One parameter is constant this omega lambda, and you have the density parameter for omega matter. And all the observations intersect and indicate that this omega lambda is not zero. So this means that there is something like cosmological constant. So cosmological constant just means that the equation of state is minus one. You can also try to measure W. So you can try to use this and try to also change this equation of state. And you can see that the constraint on W is not very precise, but minus one is consistent with all the observations. Okay, so what is the cosmological constant? So I introduced the cosmological constant. So it's very easy to do in general relativity. So you just put constant in your action. So remember that GR is described by this action. So this is the rich curvature. And you can introduce this constant in your action. And you can derive all the equations including this lambda. So you get this contribution lambda, which is proportional to g mu nu. And in the Friedman equation, you have this constant. So you say that there is a density which is related to this lambda divided by 8 by g. So this is the density of the lambda, but this is just a constant. And the interesting thing about lambda is that the pressure is minus this number. Okay, so equation of state is W is P over low. So this means that the W is minus one. And then you can calculate the acceleration. So the acceleration is proportional to lambda. So this means that the A double dot, the second derivative is positive. So having lambda, you can accelerate the universe. So what's the problem? So we have the answer, right? So we have observations. Everything is consistent. If you have cosmological constant, you can explain everything. So why we are talking about our energy? That's the question. So what's the problem? So lambda CDM works very well. It can expand all the observations. You can include cosmological constant in Einstein GR without changing any theory. So why we bother? So I think that's the question we want to understand. Right? So to understand this, it's very important to understand the number, so energy scales. So let's remember, so as I said, I will use a natural unit. So this is a very good exercise. I recommend you to do this exercise to change everything into a natural unit. So the natural unit is to set a Planck constant and the speed of light and Boltzmann constant to one. So then everything can be expressed in terms of the energy. For example, Newton constant in this unit is half the dimension of energy. And so this we call Planck scale. And this is given by 10 to 19 GB. So this is the number you have to remember. At the same time, you can also express this Hubble constant just in terms of the energy. So I said that this is 10 to minus 42 giga-electron Boltzmann. So now you already see a huge hierarchy between the scales. So now you compute the energy density associated with this lambda. So the low lambda was lambda divided by G. So G is M Planck to the minus 2. So you get M Planck squared times lambda. And remember that our universe, if there is a lambda, the Hubble parameter is just given by H squared over lambda over 3. And it seems that our universe is mostly determined by lambda. So this means that the lambda is related to today's horizon scale, H0. So this means that because lambda is constant and the lambda dominates now, so this is H0. So you can use this number. Then you compute the energy density associated with lambda, you get 10 to minus 48 dB to the fourth. So this is 10 to minus 3 electron Boltzmann to the fourth part. So this is the number you will see. That's right, yes. That's an interesting question. There is an interesting theory relating dark energy to the mass of neutrinos. However, at the moment we don't know there is any connection. But it's very interesting if there is a connection. It's a very good point. In fact, this number, if you are a particle of physicists, you probably are familiar with this. So this is related to the mass of the neutrino. So remember that this is what we observed. So it seems we have cosmological constant for the energy density is 10 to minus 3. And in fact, we have this cosmological constant everywhere. So this is a prediction of quantum field theory, so known as vacuum energy. If you remember your quantum physics, if you have a harmonic oscillator, there is a zero point energy. So zero point energy is just given by Planck constant times frequency divided by 2. So I said that I would set h bar to 1, but I restored it just to say that this is the quantum physics. So this is the prediction of quantum mechanics. But if you apply this to quantum field theory, like massive field and bosons and helmions, you will get the same kind of zero point energy. Later I will use this fact that for bosons, so bosons is like photons and scalar field. So spin 0, spin 1 field, zero point energy is positive. But for helmions, the zero point energy is negative. So now this P is the momentum, and you want to sum up all these momentum. So you do the integration over this zero point energy. So now you want to compute this, and so P is the momentum, and this can start from zero to high energies, so infinite number. So you can expand this square root and calculate the integration by expanding this, assuming that P is larger than m. And you immediately notice that this integration diverges. So the first term you get is P max to the fourth power. So P max is that if you set infinity, this becomes infinity, so I set some maximum number for the wave number. The next one is also diverging, and then you get this m to the fourth power contribution. So this means that you really need to know the physics at very high energies. Remember P is the wave number associated with the energy. So if you want to calculate this up to some max wave number, this means that you have to know, we say ultraviolet physics, so the high energy physics to calculate these quantities. And the problem is that these contributions diverge, and this depends on how you regularize, so how you introduce cutoff. But this term depends on P max only logarithmically, and this part is quite robust. So independent of UV physics, you always get this kind of contribution. So the vacuum energy, conservative estimation is that it is mass to the fourth power. So what is the problem? So you say that you now observe vacuum energy of 10 to the minus 3 electron volt. So this means that the contribution to the dark energy, vacuum energy, the cosmological constant vacuum energy, the mass must be smaller than 10 to the minus 3 electron volt. So this is a neutrino mass. So if you have electron in our universe, the vacuum energy is 0.5 M E B to the fourth power. And if you have pranks scale mass, which is a natural scale, for gravity the vacuum energy is pranked to the fourth power, and this is the famous 10 to 120 times the observed lambda. So now you see the problem. So now you can calculate the vacuum energy from particle physics. Compared with the observed vacuum energy, there is a huge difference. So this is not only the problems in the early universe. We know that there is a phase transition. So we have an electro-weak phase transition and QCD phase transition, and this phase transition will generate again vacuum energy. For the electro-weak phase transition, you get contribution like 200 giga electron volt to the fourth power. For QCD, you get 0.3 giga electron volt to the fourth power. So any number you get, it's much larger than the observed vacuum energy. So then you wonder whether this vacuum energy is real. It's very strange. You calculate its diverges. Is that just a mathematical quantities? In fact, it is not, and this vacuum energy was already observed in some sense. So this is known as the Kashmir energy or Kashmir force. So let's consider some scalar field, and you basically have waves of scalar fields, but then you have two plates, and you put the boundary conditions that the scalar field punishes on this boundary, and then you compute this scalar field, the momentum. And due to this boundary conditions, this momentum for this direction, so this x direction, the momentum becomes discrete. So n is 1, 2, 3. So you have a discrete momentum inside this two plate. So now you want to compute the same thing, 0-point energy. So 0-point energy was one-half times frequency. So you basically want to sum up this frequency. The only difference is that this direction, the momentum is quantized. So you have a discrete momentum. So instead of integration, you have this term of n for this direction. And you find that this energy will diverge. So you have to introduce some regularization. So this is a very simple way to regularize. So you put some function by hand, so that for large p, this becomes 0, so the energy stays finite. But of course, if you take a to 0, this becomes 1, and everything diverges. Yeah, this is a very simple toy example. And then you compute the total energy. So energy between these two plates and energy away from this plate. And this energy diverges if you set this regularization scale a to 0. However, you can compute the force between these plates by taking derivative with respect to d. So this is just coming from vacuum energy. And the vacuum energy is different because of these two plates. And you find the force between these plates is proportional to 1 over d to the force power and given by this precise number. So this is a very simple example using scalar field. But you can do the same calculation for electromagnetic field, for example. And this is a tiny force, but this has been measured. So this says that vacuum energy, if you naively calculate vacuum energy, it diverges, but if you calculate the physical force between two plates, it exists. And it's finite. So this means that the 0-point energy really exists. So this is known as the old cosmological constant problem. So remember that the 0-point energy is not very important in the usual quantum field theory. Why? Because usually in the quantum field theory, you don't care about the 0-point of the energy. What is important is the difference of energies. For example, for Cassemere force, I said that the energy is infinite if you do not regularize, but if you compute the derivative, it's finite. So this is what matters in the flat universe with no gravity. The problem of GR is that GR matter curves the spacetime. Even vacuum energy curves the spacetime. So you cannot say that you don't care the 0-point energy. If you have energy, this changes your spacetime. So this means that if you have electron in our universe, this creates the vacuum energy of 0.5 megaeb to the fourth power. If you trust GR, this gravitate, you create the expanding universe. And the fact you find, so this is Planck scale, sorry, the Hubble scale becomes 10 to 6 kilometer. So having the electron in our universe, the horizon scale becomes 10 to 6 kilometer. But we know that our universe is much larger. The electron is not that heavy particle, but just having electrons in our universe, you have a huge energy and this curves the spacetime and your universe becomes so small. So that's the problem. But you say that while you have freedom to add lambda in Einstein gravity, so just fine tune it. So you introduce some lambda, you know vacuum energy is huge. You can tune your lambda in the Einstein field action so that you reproduce this small lambda. The problem is that as I said, the vacuum energy is very sensitive to the ultraviolet physics. The cutoff of high energy momenta was a mass. So if you go to high energy and more and more high energy, you don't know how many particles you have. And then all these new particles contribute to vacuum energy. So basically you have to know all the knowledge of UV physics to do the tuning to get this observed lambda. So this is the fundamental problem. So in technical terms, this means that this tuning you can do at low energies is not stable, meaning that if we introduce new particles, you have to tune it again. So this is not stable. So we say this is unstable under radiative corrections. So you may think that how we solve this problem. And in fact, this problem is nothing to do with the accelerator expansion of the universe, right? So vacuum energy exists. If it exists, it curves the spacetime too much. And you have to solve this problem before looking at the acceleration problem. So one possibility is to have some symmetry. So the supersymmetry is one example. So as I said, the vacuum energy depends on you have bosons or helmiens. And this sign changes whether you have bosons and helmiens. And you notice that if you have the same number of bosons and helmiens, you can cancel the vacuum energy. And if this is the case, this works very well. The only problem is that we never see super particles, right? So if supersymmetry is exact, you should see the same number of bosons and helmiens, but this is not what we observe. And we believe that the supersymmetry is broken at some high energy, like RTEB. So this means that if supersymmetry is broken, you again get vacuum energy of this order. This is, again, too large compared with the observed vacuum energy. So this does not solve the problem. So people argue that maybe there is some argument to say that if lambda is zero, there is some special symmetry. We don't know what it is, but maybe there is a symmetry with this cosmological constant zero. And assume that this is a very precise symmetry which exists at quantum level. Then you can do the tuning because the small cosmological constant breaks this symmetry and creates all these collections. But these collections are created because you break the symmetry. So this means that if you do the tuning and you get a very small number, this is a very, very fine-tuned value. But assuming that there is an exact symmetry with zero cosmological constant, this tuning is natural because quantum collections only create the same orders of cosmological constant. So this is known as the naturalness. So people try to come up with the theory of having this symmetry. So this sounds all very good, but what is the symmetry? So still no one knows. So another interesting attempt is known as the self-tuning. So let's imagine that you have a huge vacuum energy. So somehow we do not see this huge vacuum energy. But there may be some extra field and this absorbs this huge vacuum energy so that we do not see the vacuum energy. However, there is a very famous Nogo theorem by Weinberg and saying that basically this is the same as fine-tuning. This proof is very simple but very difficult to understand. I recommend you to look at original Weinberg's Nogo theorem paper. I forgot to put the reference but you can usually maybe Google it. You can find the original paper by Weinberg. The idea is quite simple. So let's say you have a huge vacuum energy but somehow there is an additional field so that this field absorbs this large vacuum energy so that the spacetime remains flat. And then because the spacetime is flat, this additional field phi is also stays constant. Assuming these conditions, what he showed is that the action you get is this form. So you have the vacuum energy from usual matter. So this is the standard model particles. So this creates a huge vacuum energy. But then you have an additional field so that you want to compensate this huge vacuum energy so that you want to tune this. However, in the end what you are doing is fine-tuning between this potential energy coming from this field and the vacuum energy. There is no dynamical tuning. So this is just fine-tuning. So this is the argument. This sounds trivial but surprisingly many attempts try to do this dynamical tuning to illuminate vacuum energy but basically fails because of this Nogo theorem. So this is a very strong theorem. And of course you may try to change this assumption. This is what people are doing. For example, even having vacuum energy and you tune it to make it small, you don't need to have a flat space. You can have a dynamical space. So this is one way. But then I have to say that still we do not know the very good model for it. So there are many attempts. So I'm showing this because we don't know the answer. So you need to find the answer. So you have to choose which one you are interested in. So another idea is to change gravity. So the problem of vacuum energy is that if you have a huge vacuum energy in general relativity, this vacuum energy gravitate and this is not what our universe looks like. So you want to change your Einstein's gravity so that if you have a cosmological constant, there is some function so that this cosmological constant does not change your space time. So this is the idea. Again, the idea is always good. But how to get this equation? So this must be a very non-local equation because normal matter, it looks like GR. But for a cosmological constant, it looks very different. So you have to change the nature of gravity depending on the scale you have in your system. Again, we do not have a very good theory to realize this. So there is one interesting idea is using these extra dimensions. So if you consider some two-dimensional object, so let's say in four dimensions you have two-dimensional string. So this is two-dimensional object and so this is four-dimensional universe. So you have two-dimensional string. So you put some cosmological constant on this string. So in this case, this is just a tension. And what happens is that this tension or a cosmological constant does not gravitate. And in fact, this tension only changes these two-dimensional extra dimensions. And you want to apply this to six dimensions. So it's very difficult to imagine. But your universe is 4D. And you have two extra dimensions. So you put a cosmological constant on this brain. But this cosmological constant does not change the geometry of these four-dimensional universe. Instead, it only changes the geometry of two-dimensional extra dimensions. So this means that even if you have a huge vacuum energy, you do not feel this cosmological constant. And this is just changing six extra dimensions. Yeah. So five dimensions, you have one extra dimension. And this nice property happens only when you have two dimensions. Just two. Yes. Do you have questions? That's a good question. So the question is, is there any action? No. That's a problem. So it's a very non-local equation. I don't know how we get this. And in fact, the problem is Bianchi identity. So you have to have Bianchi identity for g mu nu. And you want to have matter conservation so how you satisfy Bianchi identity. That's also very difficult. So all of what I said is that we don't know the answer. So this is a fundamental problem. And the fact that the acceleration did is to make the problem worse. So now you get the additional problem. So let's imagine that we have this old cosmological constant problem. Five vacuum energy does not gravitate. But let's assume that we solve this somehow. Then you need to explain why the universe is accelerating. We need a cosmological constant. So this means that the accelerated expansion of the universe makes things worse. And we seem to have more problems. So this is known as a coincidence problem. If you look at the energy density of radiation and the matter, so radiation scales like A2 minus 4 and the matter scales like A2 minus 3 and the cosmological constant is constant. And it's surprising that somehow today all these three densities have similar densities. Especially the matter density and the lambda density is very similar today. But looking at this picture, they have a very different density in the early universe. They will have a very different density in the future. So far today we have this agreement between density of matter and density of lambda. Is it just a coincidence? So this is a coincidence problem. Or there is a deep meaning to it. So of course there is some very simple answers, the anthropic principle. I have no comment on this. So that's the problem. So I hope I convinced you that this is a huge problem. We had the old cosmological constant problem. Even if we didn't find accelerated expansion of the universe, this was the problem. So the problem was that vacuum energy exists, but it does not gravitate in the way it should be. And in fact this accelerated expansion of the universe may help us because understanding why the universe accelerates now, whether this is caused by some fine-tuned cosmological constant, why it's entirely different from a cosmological constant. This may have something to say about the old cosmological constant. So remember that when we talk about the energy, people talk about usually accelerated expansion but you have to remember that we are always having this old cosmological constant problem and we are also hoping to find solutions to this very old theoretical problem. Okay, so I think from now on I focus on the accelerated expansion of the universe to dark energy problem. So now we want to see which assumption we want to change because assuming these three assumptions we have the evidence that we need dark energy. So this means that we have to change one of them. Okay, so the first one was homogeneity and isotropy. Second one was GR and third one was the matter content of the universe. So which one you want to choose? So shall we do the both? If you want to doubt this assumption of homogeneity and isotropy please raise your hand. Okay, it's not very popular. Maybe GR is wrong. If you think GR is the problem. Okay, that's... Okay, so maybe we don't understand what is matter in our universe. Okay? Okay, so let's look at this in turn. So we come back to this freedom of metric. So let's look into that more precisely. So there are two principles we use in cosmology. One is the Copernican principle. So this says that we are not at a special location in the universe. So this is the principle. And the cosmological principle says that on large scales the universe is homogeneous and isotropic. So we used this second principle to get the freedom of metric. So the mathematical proof is this. If all observers measure isotropic distance relative to relation, then the space term is freedom. So this is the mathematical statement. In fact you do not need homogeneity here. So you just say that if you have an observer and if this observer measure isotropic expansion and then you can show that this is the case for all observers, then the metric is freedom. Then you can immediately see the problem, right? So how you show this is the case for all observers in the universe. That's impossible. Because we only see from the Earth. You never know if you go to the other side of the universe. You never know you see the isotropic universe. So in order to show that your space term is described by freedom of metric, you need to have this assumption that the universe we see is the universe people see in the other side of the universe. That's the reason why we need Copernican principle. Assuming Copernican principle, we are not at a special location in the universe meaning that if we measure the isotropic expansion of the universe from this assumption, this means that all observers will measure isotropic expansion. Then it's enough to use this mathematical proof to show that our universe is described by freedom of metric on large scales. But then how we show the Copernican principle is correct. That's a huge problem. There's a very nice review by Chris Clarkson and there's a very nice textbook discussing about this principle. So I recommend you to look at this review paper and textbook. This is in fact deep problem. So how we show the spatial homogeneity in GR? That's very difficult. So this is a diagram in GR. So this is the world line of observer. So this is time. So this is the constant time hyper surface. So this is the spatial space. And we want to show the homogeneity on this space. But what we observe is this past light cone. And in fact this surface of light cone is the observations. So you only observe the intersection between light cone and this constant hyper surface. So then how we show this spatial hyper surface is homogeneous. So that's a very difficult problem in GR. And in fact it's very difficult to show. So I think it's natural to doubt this assumption and this was done probably 10 years, 15 years ago a lot. And people come up with this void model. So let's forget that we are living in a... We are not living in a special place. Let's assume we are indeed living in a very special place. Let's assume that we are living inside this huge void. Huge under dense region. Let's say this is 10 gigaphertz exercise. And you have a huge difference between density around us and the density away from us. And in fact there is some very famous metric describing this kind of inhomogeneous universe. This looks like Friedman universe but the scale factor now depends on time and this radial direction. And then you have different expansions for radial part and angular part. And what you can do is to basically find this density profile so that you can calculate the distance from us and compare it with supernovae and CMB. And what people find that indeed it's very easy to explain supernovae in this model without cosmological consent. Having this kind of profile you can easily explain supernovae observations. You don't need dark energy. So this is one indication that... But basically you are saying that you have to live at a very special point in the universe. However, after that people studied this kind of model a lot and people found a lot of problems with this model. So this simple model. And if you calculate the H0, the horizon constant today we measure like 70. But in this model H0 must be like 50. So this is already excluded. And also if we observe clusters in this model because there is a radial profile clusters have a large radial velocity. And if you have a cluster, clusters of galaxies and so this is moving because you have this spatial profile in the universe. And then CMB photon comes and then this will be scattered by electrons and this creates additional CMB anisropies. And this is known as the kinetic scenario as a little bit of effect. And this effect is huge in this model because clusters fill this profile. And in fact even current data this simple model is already excluded. The idea is very interesting but again to explain accelerations without using lambda is very difficult. And this is very important. You have to combine many observations to show that but at the moment I think the consensus is that it's very difficult to explain accelerations without violating this assumption. And there is a related argument which is related to the coincidence problem which is about reaction. So we know that the universe becomes very inhomogeneous at late times. So in some sense the homogeneity assumption is broken on small scales. But we say that on large scales on the average it's homogeneous and that's how we prove that. But again this is a very difficult problem in GR and there are a lot of arguments about the effect of this small scale inhomogeneity on the expansion of the universe on large scales. And one nice idea of this back reaction is that due to this small scale inhomogeneity these inhomogeneitys back react and then the expansion accelerates. So if this is the case you can solve the coincidence problem. So if you look at the energy density of dark energy and this is the energy density of matter when the matter energy density becomes the same as dark energy energy density this is basically when nonlinear structure forms. So if dark energy is not really a dark energy this is due to this back reaction from nonlinear structure formation you can explain why you see this dark energy today because this is because you create nonlinear structures today and we exist because of nonlinear structure formation. Again the idea is good but of course the question is this can you get this kind of very large back reaction from small scale inhomogeneity? I think the answer is no because that's the consensus. But this does not mean that there is no effect so people are discussing about the magnitude of this effect this is 1% or 10 to minus 5 that is not decided yet. However probably it's very difficult to explain the acceleration along this line. Okay so let's look at the second assumption general relativity we use GR from the beginning. One reason is that of course observationally we know that GR is a very good theory of gravity it can explain solar system test and other things so in the next lecture I will talk about more theoretical point of view why GR is special but let's look at this observational evidence of GR so in the solar system what you calculate is again the metric and you compute the time-time component of the metric for example sourced by the sun so you have a usual potential M over R and the special part you can have the same potential and this gamma parameter so this will appear later is known as the post parameterized Newton parameter. GR predicts gamma is 1 and a Newton theory you don't care about spatial perturbations so gamma is 0 is a Newtonian theory and these two potentials determine the bending of light so in the solar system because of the sun's gravity the position of the stars look different because the gravitational field bend the trajectory of the light and you compute this bending and in GR it is given by this number and this is the measurement and this is very close to 1 and in fact in terms of this gamma parameter the deviation of this parameter from GR is 10 to minus 4 also you can compute the time delay so you send signal from Saturn there is a time delay due to the gravitational field of the sun again you can compute and Cassini satellite went to Saturn and we did this experiment and again you get GR and the accuracy is again 10 to minus 5 so remember that GR is a very precise theory of gravity so if you change gravity you have to always worry about this that's the reason why we trust GR but this is not only the reason why we trust GR we have these binary pulses so we have two binary pulses and they basically rotate together but then they emit gravitational waves and this orbit decays because they lose energy because of gravitational waves so this decay of the orbit this is the prediction of GR and they agree very well and in fact this is not the only the story so you can calculate all these collections due to GR and of course the problem of testing gravity is that usually you do not know the mass of these two binaries so you don't know the mass so what you can do is to assume GR try to combine many observations in this case they are using a lot of different post capillary motion parameters try to find the mass and assuming GR so this is the allowed region so this line is basically what you observe and then you can pinpoint the mass of these pulses the idea is that if you use long theory of gravity there is no consistent solution and GR remarkably gives a very consistent mass so this is the way to test gravity even if you do not know about the matter, the mass and this is very important for cosmology too so I will come back to this in tomorrow's lectures so how we test gravity even if you do not know about matter but this also means that GR is very accurate so just summarize where we test gravity so let's imagine you have some spherical object so you have a spherical object with radius r and mass m so there is a very nice paper summarizing all the tests of gravity using two parameters so this object gravity is described by first usual GM overall so this is the gravitational potential so this is the gravitational potential so if gravitational potential becomes zero this means that this is a very strong gravity and another quantity you can compute is the curvature of this object so this measures the spatial curvature created by this object so this scales like GM over R cubed so this is the curvature so if you go up you have a very high curvature means that your spacetime is very different from flat space and all the tests we do is basically around here so this is the solar system test so this is the binary pulses so this is all very high curvature so if you think about cosmology you are looking at a very low curvature so maybe it's probably easier to see the scale so you fix mass and change your radius of this object so this is one solar radius this is one AU and if you increase radius you go in this way so this is one megaparsec so the horizon scale is 3,000 megaparsec around here and you notice that there is no test of gravity here you tested GR very well around here then you see a lot of problems in cosmology for example you need dark matter and you need dark energy if you go to this very low curvature regime and if we trust GR because we have a lot of tests of GR around here but we never tested GR around here so this is the reason why there may be a reason to doubt GR given the existence of dark energy and maybe you heard about this discovery of gravitational waves so this is probably not the proof of GR yet but this is testing gravity at very strong gravity so very large potential and very high curvature its scale is very different but around here so now we get a very good handle of gravity using gravitational waves but you can do similar things using cosmology down here but remember that we are testing gravity in a very different regime and only the precise test of GR has been done around here ok so this is the final question so what is matter so this is very simple because we don't know so we just add dark energy right so that's very easy the problem is that if you're just looking at the expansion of the universe you only have one parameter so it's the equation of state you don't need to know what is dark energy you just need to know what is equation of state so how you distinguish between different dark energy if you can observe only background expansion which is only determined by equation of state so that's the question I want to ask in the tomorrow's lecture and especially I distinguish between changing gravity and adding different matter but this is not very difficult this is not easy to do because if you change gravity you change basically left hand side of Einstein's equations you change gravity you add dark energy you change right hand side so how you distinguish between the two whether you are changing gravity or you are just having weird matter so that's the reason why in fact distinguishing between dark energy and modified gravity is not well defined and you have to be more careful how you distinguish between different models so changing matter and changing gravity in fact probably there is no reason to separate but we want to understand how we can separate different models from observations okay so to finish my first lecture just to mention that there are a lot of astronomical surveys so let's try to understand dark energy so this is the year so this is the planned astronomical surveys so now dark energy survey is ongoing we have extended both surveys ongoing but then in Europe we have this U-Grid so the last week we had a meeting in Lisbon so this survey is more than 1,000 people involved to understand the nature of dark energy so this is the mission statement of U-Grid so this is a satellite mission so it's try to understand the nature of acceleration of the universe try to test gravity on cosmological scales from the measurement of cosmic expansion history and growth of structure so all these statements you want to understand why more than 1,000 people are interested in doing this I think I hopefully convince you that there is a strong theoretical motivations but this is not enough why people try to do astronomical observations to understand the nature of dark energy so this is what I want to explain in my lectures so basically to understand why this is the mission status for U-Grid why this is important but in the next lecture I will try to explain the examples of the models of dark energy and modified gravity this will give you the taste of current theoretical models as I say there is no answer so I cannot give you the answer but you can see how difficult it is then tomorrow based on these theoretical models I will discuss how we can distinguish between these theoretical ideas using structure formation and I will discuss about current observational tests and then discuss about nonlinear structures I think I'm a bit over time so you can ask any questions during a coffee break and I try to have more question time in the next lecture