 Lorentz will sit down, sorry. Okay, welcome back. We're the second lectures by Kazuya about the models of dark energy and modified gravity. Okay, so in this lecture, I will look at models of dark energy and modified gravity. And as I said, there are so many models. I think there's no meaning to review all the models. But I will pick up some interesting examples and explain why it is so difficult to construct models of dark energy. So let's start from the simplest alternative to cosmological constant. This comes from the scalar field. So the scalar field, good thing is that we now know our scalar field exists. So we know Higgs exists. So the scalar field is described by this action. So phi is a scalar field. So this is the usual Einstein-Hebert action. And you have a kinetic term. So this is a derivative term. And you can have a potential. And you see that if there is no kinetic term and if potential term is constant, this is just a cosmological constant. So this is a very simple extension of the cosmological constant. So this is a very good exercise if you do not study GR. So calculate the energy momentum tensor for the scalar field using this action. So the definition is given by this. And then you can also derive the equation motion for the scalar field. So you take variation of scalar field and then you get this equation for the scalar field. So this is the covariant derivative. So this v prime is the derivative with respect to phi. So my notation is v prime is the v prime. And the same story applies. So if you calculate the conservation of this energy momentum tensor, you reproduce the scalar field equation motion. So this is the equation for the scalar. And this is consistent with the conservation of energy momentum tensor. Okay? So now let's consider homogeneous solutions. Homogeneous solution means that phi is just a function of time. Then you compute energy density and the pressure. So energy density is minus t upper zero, lower zero. And you find that density is given by kinetic energy. So phi dot is the time derivative. So this is the definition of phi dot. I defined as d phi dt. But this first term is the kinetic energy and you have potential. And pressure, you have kinetic energy minus potential. And then you can derive the equation motion for the scalar field. And this looks like just a harmonic oscillator if potential is m square phi square. So this becomes m square times phi. And then this becomes a harmonic oscillator. However, there is an interesting term here which is proportional to half blue expansion. So this comes from the expansion of the universe. So this gives the friction. So the motion of the scalar field, for example, for given potential. So there is no expansion of the universe. This scalar field just oscillates around this minimum. But thanks to this friction, the scalar field can be rolling down this potential very slowly. And in fact, if this happens, the kinetic energy of the scalar field is very small compared with the potential energy. If phi depends on t very weakly, so then you can ignore phi dot squared. And then you notice that, of course, this is the cosmological constant. So the equation of state is close to minus one. So this means that this is a very simple extension of cosmological constant, but the equation of state is not exactly minus one. So it can be different from minus one. And this is determined by the potential. So what we are interested in the equation of state, so you can compute the equation of states, the equation of state, if you remember, the equation of state by definition, equation of state is given by pressure over energy density. And we are interested in the one plus equation of state. So this is given by pressure plus density divided by density. And if you remember, the energy density is given by kinetic energy plus potential. And then pressure is the kinetic energy minus potential. Okay, so if you add energy density and pressure, the potential disappears. And so one plus w is determined by just phi dot squared. So again, if phi is constant, it's minus one. And defining this psi parameter, this is the second derivative of scalar field divided by 3h phi dot. This can be written in this phi. So this is just writing down phi dot squared in a complicated way. So phi, we want to do that. This psi phi basically measures how slowly the scalar field is moving. So if phi is moving very fast, the second derivative is very large. So psi, psi becomes large. So in order to get w close to minus one, this psi must be smaller than one. And using this condition, this equation can be rewritten in this way. So here I make the assumption that the expansion of the universe h squared is 8 pi e over 3 rho phi. And once the kinetic energy becomes small, this can be approximated by just the potential. So this is the approximation. So you can know the kinetic term. And using these approximations, everything can be written in terms of the potential. And here I define 8 pi g as kappa squared. And this is related to the Planck scale. So before I define g equals one over small Planck scale squared, but this is the reduced Planck scale. So it's 8 pi is included. So this kappa squared is just 8 pi g. And then this epsilon phi is known as the slow roll parameter. So this is just written by potential. And the equation of state is written by this slow roll parameter and this density parameter for the scalar field. So this is the ratio of the scalar field energy density to the total energy density. And if you are familiar with inflation, this is very similar parameter you use. So this is a slow roll parameter. But there is one important difference between inflation and dark energy because during inflation, scalar field is a dominant energy. But late times you cannot ignore matter density. So this omega phi is not one. So you have to take into account this. So this is the function describing how scalar field dominates the energy. So this epsilon describes how slowly the scalar field is moving the potential. And this epsilon is related to the potential. So you can parameterize this epsilon, so I will come back to this, okay? So basically if epsilon is small, w becomes close to minus one. And the good thing of having models is that equations of state is fixed for given potential and initial conditions. And of course there can be many potentials. But looking at different potentials, you notice that in fact equations of state the parameter space is limited. So it is not arbitrary. So that's a good thing. If you say there is dark energy with some equation of state, equation of state can be anything. But you say it's coming from a scalar field and the given potential you can predict what is the equation of state. And most of the potentials can be classified into two categories of models. So one is known as freezing models. So these freezing models, the potential looks like this. So the scalar field potential looks like this. And the scalar fields start from here and then moves down the potential. And initially the scalar field is moving very fast. But the late times are moving very slowly. So around here it stops. So this means that the equation of state becomes close to minus one at late times. So that's the reason why this is called freezing model. So the scalar field dynamics freezes at late times. And another example of the potential is the opposite. So let's say you have a potential like this and you start from here. The scalar field is nearly constant here. But then at late times the scalar field is moving down the potential at late times. In this case equation of state start from minus one. But then it becomes different from minus one at late times. So this figure shows our equation of state and this is a derivative of equation of state. So this describes time dependence of equation of state. So in this freezing model you start from W very different from minus one and then moves to this minus one. So this is lambda CDM because equation of state is minus one and there is no derivative. On the other hand, thawing model moves from here and then moves away. And then in this paper it was noticed that by choosing many different potentials most of the models are within this yellow band. This is very good because if we say equation of state is anything you can have any numbers for W and derivative of W but saying that it comes from scalar field with potential you can basically predict what the equation of state looks like in this model. So this is the good thing of having a theoretical model you can predict the equation of state. And in fact the scalar field model gives one interesting example where cosmological constant is not a constant at early times. So let's consider this exponential potential. So you have exponential potential so the potential is exponential minus kappa so kappa is again just eight by G square root and lambda is the parameter and phi is a scalar field. And this exponential potential has a very interesting solution if lambda satisfies this condition. So if lambda squared is larger than three times one plus Wm so Wm is the equation of state for matter. If this condition is satisfied if you compute the scalar field equation of state it is the same as the equation of state of matter. And if you compute the density parameter of the scalar field it is determined by this lambda and the equation of state of matter. So what does it mean? So let's say you have a matter so this is energy and this is less shift so this is scale factor. So matter decays like eight to minus three and if you have this exponential potential the scalar field energy basically scales like matter so it's not constant it decays like eight to minus three and this density parameter the ratio between the two densities are determined by this lambda. And the good thing about this solution is that this is an attractor meaning that no matter how you choose the initial conditions you can start from energy density of the scalar field here eventually you will get this solution. So this kind of solves the problem of cosmological constant because cosmological constant is constant you have to tune this constant so that this constant dominate at late times. So it's a tuning of the initial condition but in this case independent of initial conditions you always go to this solution and the energy density is always very similar to matter density. So this may solve the coincidence problem. However there is a problem you also want to get acceleration from this potential and the condition to get acceleration is lambda squared is smaller than two. Compared with this we know matter equation of state is zero so this means that the lambda squared must be larger than three to have this nice solution. So this means that you never get acceleration. Okay so the scalar field energy tracks matter but then you never get acceleration. So it doesn't really solve the problem. And what people did is to add another exponential potential with small coefficient here mu which is smaller than mu squared is smaller than two. So this potential eventually accelerates the universe. So this still solves the sensitivity to the initial conditions because independent of initial energy of the scalar field you get this nice solution. So it's tracks matter energy and then accelerates the universe. But think about it it doesn't solve the coincidence problem. So what determines when this acceleration happens is basically the parameters. So you have to tune these parameters so that's the acceleration that happens today. So this is not really solving the coincidence problem. However by adding the dynamics to the cosmological constant or dark energy you can say something about this initial condition problem. So sensitivity to the initial density. Okay. So these are the examples of the quintessence model and you can come up with millions of models depending on the potential but the fact all these potentials have the same issue. So this can be understood using this very simple potential. So let's consider m square phi square so quadratic potential. As I said that in order to have acceleration so w close to minus one you need to satisfy that the slow roll parameter is smaller than one. So this is the condition so that the scalar field example in this case you have m square phi square potential and you want to make sure that scalar field is moving very slowly. So you have to satisfy this condition and you notice that for this potential m square disappears because we have v prime over v so m square disappears. So this is the condition for phi. So phi must be larger than Planck scale. So again if you are familiar with inflation this is indeed the slow roll condition. The scalar field must be larger than Planck scale. So now then you want to have this potential energy of the scalar field is the dark energy density. We know that the dark energy density is 10 to minus 48 GB to the fourth power. Phi is larger than Planck scale. It's 10 to 18 GB. So you can calculate the mass and mass is energy density potential divided by phi squared and you notice that mass is 10 to minus 42 GB. Now you should be familiar with this number. This is nothing but the horizon scale today. So this means that the scalar field is very light. The mass is very, very small. It's 10 to minus 42 GB. So we call this Compton wavelength. So look at this solution. So if we consider a length scale less than this Compton wavelength the solution is given by this. So this looks like usual Newtonian potential so M over R and the strength is determined by alpha. So this is a usual solution for the person equation. But then if we consider length scale larger than this Compton wavelength because of this exponential factor scalar field disappears. So physically this is because if scalar field has a mass it does not propagate beyond this Compton wavelength and it decays. So then solution goes to zero. So the problem of having this very small mass is that the inverse of this mass Compton wavelength is horizon scale. So this means that this solution applies to all the universe. Inside the horizon scales the scalar field behaves like gravity. So this is mediating a long range force so there is no suppression. So it looks like gravity. So why this is a problem? So this means that if this scalar field couples to some matter and this coupling create the additional force so if we consider the acceleration of some object this acceleration of object is determined by usual Newton potential. So this is the usual GR. But if this matter couples to the scalar field you get additional force from this scalar field. And this is known as the HIFS force because this is the force beyond what we know. So this HIFS force is created by this additional scalar field. And if you look at the length scale less than Compton wavelength the solution for the scalar field looks like gravity. So this HIFS force looks like gravity. So it has exactly the same dependence from radius so it forces one over R squared proportional to mass. And this strength is determined by this coupling constant alpha squared. So why this is the problem? Because we do not see any HIFS force in the solar system. I say that there is a very strong constraint on the deviation from GR. This is indeed the constraint on this alpha parameter. Alpha squared, so this determines the strength of HIFS force. In the solar system this must be less than 10 to minus five. So that's the problem in the solar system we do not see this new force. Imposing this condition, branch stick parameter. So this alpha is indeed related to this branch stick parameter and this must be larger than 40,000. And I say that they for our gravity has branch stick parameter zero. So now I believe this is already excluded. Okay, so this is the problem. So if you have a very light scalar field and if this couples to matter and you have a very strong constraint from solar system. Imposing this constraint your theory looks like GR. Okay, so that's the problem. Why this is not the problem for quintessence because we assumed that there is no coupling between scalar field and the matter. So we assumed that this is zero. So there is no constraint. But this is another hidden small number. So this must be less than 10 to minus five and we tuned this parameter to be zero in quintessence model. But there is no symmetry reason to say that this scalar field does not couple to matter. Okay, so this is a hidden assumption of quintessence. So we have to explain why we do not see this scalar field. One possibility is to violate the equivalence principle. This constraint comes from the observation in the solar system meaning that this basically saying that coupling between scalar field and the variance are suppressed by 10 to minus five. But this does not mean that there is a coupling between scalar field and the cold dark matter because there is no observational constraint on that. You could have all the one coupling between cold dark matter and the scalar field and this coupling can change your gravity for dark matter. And that is enough to change gravity for our universe because dark matter is a dominant matter in our universe. But you do not see this because variance do not see this. But this means that you have to have a different coupling between CDM and the variance. So this is violating so-called strong equivalence principle. So equivalence principle means that this scalar field couples to CDM and the variance in the same way but you have to violate this. But this is one option. And the other option is known as the screening mechanism. So you want to realize this situation. So you have a very small mass and a very large coupling in cosmology. This is what dark energy is doing. So you have a small mass and if you want to modify gravity for example, you want to have a large coupling. The problem is that in the solar system you do not see this modification. So you want to have this mass becomes very large and the coupling becomes very small. So you want to achieve this by this time mechanism. So this is known as a screening mechanism. And I'm not sure I have enough time to go through both of the mechanisms. So I will focus on this chameleon mechanism because this is related to the, for gravity I just discussed. So how we achieve this? So let's consider this example. So you have a scalar field and you have a coupling of this scalar field to matter through this coupling function A. So the matter field couples to scalar field because the scalar, matter field is described by this combined metric are determined by the scalar field. So this gives the coupling between scalar field and matter and this can be seen very easily if you compute the equation motion for the scalar field. The scalar field equation contains the contribution from matter. So matter energy density gives the contribution to the scalar field. Okay. So this is the example where the scalar field is coupled to matter. So this alpha is the strength of the coupling. And we want to change the coupling between scalar field and the matter depending on the density. And in fact looking at this potential and in fact this theory has this mechanism in a very natural way because looking at these equations the potential depends on density. So this means that the scalar field equation changes according to the density. So remember what we want to achieve is that for very low density you have a very small mass and for very high density you have a very large mass. So you want to achieve this. And in fact you can do this because for small density basically you can ignore this coupling to matter. So let's say you have a potential like this. And if you have a small density this contribution is very small. So you get this effective potential but the mass around this minimum is very small. However if you go to very high density region you get a very large contribution from matter. So that the potential looks like this and then the mass becomes very large. So this means that depending on the density the scalar field mass becomes very light to very heavy. And this is what we want to achieve. So in cosmology we want to have a very light scalar field so behave like dark energy but in the solar system it has a very large mass it doesn't show up. So that's the screening mechanism. And you can find the solution for the scalar field in this theory in this way. So let's consider the very simple situation you have a very low density environment like cosmological density. And you have a very high density like solar system. And you want to find the solution for the scalar field and I assume that the scalar field in the minimum so for cosmology you are here but in the solar system you have very deep potential and the your scalar field is here. So basically scalar field transit from cosmological scalar field to the solar system scalar field and the mass inside this dense object is very heavy. So this means that the scalar field does not propagate inside the solar system but the mass becomes very small in the cosmological background. So it behaves like dark energy. So in this way you can avoid the very strong constraint from solar system. And by solving this equation you can get this solution. So this is a solution for the scalar field which looks exactly the same except for this factor. So remember you had m over r so this is the usual solution for the gravity and you have this exponential damping factor. But in addition you get this suppression factor and this suppression factor is determined by this formula. So this is the difference between scalar field at infinity and the scalar field inside divided by the Newtonian potential of this object. So this is a bit complicated equation but this is the criteria. So let's consider some spherical object. You compute your Newtonian potential and compare this with the difference between the scalar field inside this body and outside. And if this condition is satisfied your coupling is effectively suppressed. And then the scalar field disappears. So this is known as the chameleon mechanism. And this means that whether you see the screening or not depends on the gravitational potential of the object. Yeah, so this depends on, so due to this formula in fact it's not the length scale it is determined by the potential of the object. Okay so if you consider for example the sun you can compute the gravitational potential of the sun. You compute the scalar field so this depends on your potential and compared with this difference with the sun's potential if this condition is satisfied you can explain solar system constraint. This gives the conditions for your potential in the end this gives the conditions for your theory or model. You mean quantum correction? Yes, quantum correction is another question, yes. In fact this potential from the beginning must be very flat and this is assumption we have to assume that this potential is preserved by quantum corrections. It helps that this matter gives deep potential but still this matter is not huge enough. So I think I probably need to think about time so let's see. Yeah, so just mentioned about the screening mechanism so this is basically an interesting idea and you need to have this screening mechanism if you create a modified gravity model because usually the solar system constraints are so strong and you cannot have the energy model which satisfies the local constraint. And you may think that the screening mechanism is a bit contrived so you try to recover GR on small scales and you want to modify gravity on large scales but think about that in fact in the photo gravity this indeed has so called the chameleon mechanism I explained. If you go to the scalar tensor theory you notice that this theory exactly have that chameleon mechanism. And if you look at the action in fact this is very natural because if you look at this action you have lambda CDM and the collection becomes small if the curvature becomes large and naively you expect that you go back to GR if you go to the high curvature regime and this is achieved by this chameleon mechanism. So screening mechanism is not that contrived so some of models for example DGP model I explained also has this screening mechanism. So again look at this action you have exactly the four-dimensional Einstein theory here and just dimension analysis because you have one dimension of length here if you consider very small so this is the opposite if you consider very small lengths you can ignore five-dimensional gravity and you expect that you go back to four-dimensional GR and in fact there is a screening mechanism known as Einstein mechanism to achieve this. So some of modified gravity models can have this screening mechanism in a natural way but the point is that you have to have these screening mechanism otherwise you cannot explain the local constraint. So just to sum so I just show some example of the dark energy model and you can look at this review and you're kind of a bit depressed by looking at this review with so many models. Okay so this is 2006 and after that people are interested in the modified gravity and then this is the 2011 review. So this is not dark energy this is just modified gravity so now you get more and more models. Okay. But we have to remember that we are not solving fundamental problems yet all these models have the program that they have small numbers. For modified gravity models you have to make sure you recover solar system constraint on small scales. Okay. You can modify gravity on cosmological scales but if you modify gravity in the solar system then your model is already excluded. So you have to check your local constraint. So I think situation is pretty much similar to early universe or cosmology before double map satellite came. Probably all of you do not know this situation so in 1998 so this is a CNB spectrum. You could have any models of cosmology at that time because we only have COVID data. A lot of balloon data they are not consistent. You draw a lot of lines they are all different theoretical models. Okay. And of course you know that what happened after prank is that this is the only model you can have. It's a lambda sedan. Okay. But in order to achieve this we needed to have this observation of CNB. Okay so the situation is exactly the same. We have so many models, theoretical models. But it's very difficult to distinguish between theoretical models without having this kind of observations. So we need observations. So that's a reason why it's exciting now because we will have these observations. Okay. So in the next two lectures I will more focus on the observational test of these models. So try to find out the way to distinguish between these models and what we expect to achieve from future observations. And in order to understand that we need to understand the formation of structure with dark energy and modified gravity. So in the next lecture I will look at this and then I will discuss about observation of test.