 Thank you. Okay. Okay. So welcome back to dark energy part two and the second episode. So I ran way out of time yesterday, so I don't want to go through much more calculations in motivating you that there is a cosmological constant problem. And then typically if we have a field of mass M, which I didn't write here of mass M, then typically it will contribute to radiative corrections to the cosmological constant in a way that scales like the mass of this particle to the power 4. And this is really the essence of the cosmological constant problem. If we had something, if we had our way that the radiative corrections of the cosmological constant were going like a cosmological constant itself, then they would be attuning, but attuning. So we could set the cosmological constant to be small, very small, it would be attuning, but that would be a technically natural tuning, because the quantum corrections to the cosmological constant wouldn't make it go way out of control, it would always be of the same order of magnitude. But yes, it does not depend on the form of the potential, no, no, no, no. This is just assuming you can include corrections from the interactions afterwards, but that does not depend on the form of the potential, no. The cosmological constant problem is really an issue due to the fact that the scales that enter in the radiative correction to the cosmological constant have nothing to do with the scale of the cosmological constant itself. It's coming from information about the field content of your universe, and not about the cosmological constant itself. I wanted to show you a much cleaner way to see this computing the one-loop effective action and seeing the cosmological constant problem. I'm not going to do it for you here because it's going to take too much time, and I want to focus on dark energy, not too much on the motivations, but if you could do follow something like that where you do a one-loop effective action, like Holman-Weinberg, so you can – you want to look here, it's in Euclidean, so that's why there's a minus there. You can look at the one-loop effective action by integrating out, for instance, a scalar field. It doesn't need to be a scalar field, and then you'll get something which will depend on the propagator of the scalar field. If it were which leaves in – which couples to gravity, which leaves in space time, and then when you integrate over that, the measure you get from this – from the fact that you have the metric coming in and some being exactly the square root of the determinant, and so you have a contribution to the cosmological constant. In whichever way you want to do it, you always end up having, for instance, at one loop an integral of a full momentum for the scalar field or whatever other field you want to consider, k squared plus the mass of the field, you have an m squared in front. This integral diverges, so a priori could be whatever you want, but if you do dimensional regularization, the only scale that comes in into this integral is the mass of the particle. So this, just by dimensional analysis, has to scale like the mass of the particle to the full. You can do a cut-off regularization if you wanted to, and then you would have a contribution from the cut-off as well, which would be even bigger than that. So no matter how you do it, you are always going to have a running of the cosmological constant that scale like the mass of the particle to the full. So typically we would expect contributions to the cosmological constant that scales like the masses of all the particles that we know of at the very least. So for instance, you will have a contribution from the mass of the Higgs and a contribution from the mass of the W bosons and everything that you want to think of. And so if you put the numbers in the game, that would correspond to H squared from the Friedman equation that scales roughly like the cosmological constant over the Planck scale. If I take just from physics that I know of, from physics in, yeah, yeah, yeah, fermions contribute as well. They may contribute negatively, but they contribute as well. So you have contributions from everything, yeah. Yeah, so let me take just the physics that I know from the standard model. You can go all the way to the mass of the top quark. And you'll have a contribution that would scale roughly like, let's say, 200 giga-electron volt to the full of the Planck scale here. I'm taking it to be, it's a reduced Planck scale. Turn to the 18 giga-electron volt, squared. So if I go back in numbers, the H0, that corresponds roughly to 10 to the 2 plus 9, so that's 11 squared. So that's 10 to the 22. Electron volt over 10 to the correct, so that's roughly 10 to the minus 14 electron volt, is that correct? Actually, I forgot my net, it's, yeah, that's wrong, yeah? Actually, is that correct? Paulo, I think I got confused in numbers, is that correct, actually? That's okay, something like that. Say it again? Do you understand? The units should be what? GEV. GEV, thank you. The units should be GEV, yes, that's right. Exactly, the units should be GEV. So that's giga-electron volt, and that is 10 to the minus 5 electron volt. Thank you, that's more, that's more like it. Okay, this in distance scale, you know what that is in distance scale, actually, the contribution from non-physics. This is actually two centimeters as a distance scale. So that would mean that if it was true that all the contributions from the physics that we know of the standard model, I'm not inventing new particles, I'm not inventing anything here. This is really something very clean. If they did indeed contribute to the vacuum energy with that amount, and if that vacuum energy did indeed contribute to an acceleration of the universe as predicted by GR, by standard gravitational laws, then the observable universe would be the size of, would be that big. We wouldn't be able to see beyond that. So I'm not very good with observation, but that doesn't seem to be exactly what we're observing today. This is rolled out by really many sigma. This is a huge discrepancy between what we said we had the other day, the observed value. Yesterday we said this was roughly 10 to the minus 33 electron volt. So this discrepancy between what we expect to see from the vacuum energy of particle physics to what we actually see is the cosmological constant problem. And this is a problem already within the physics that we know exists in the standard model. So some people already have asked me yesterday about the analogy or the difference between the cosmological constant problem and the hierarchy problem. So let me say this is the scale, this is 10 to the minus 33 electron volt. This is the scale of H0. We know physics very well up to roughly TV scale. And then there may be physics that we don't know. So this is beyond the standard model physics all the way up to the plan scale. In the Higgs hierarchy problem, it really is a problem if you consider that there should be physics beyond the standard model that would contribute to the Higgs mass and therefore give radiative corrections to the Higgs mass which are going to be much larger than the Higgs mass. But we actually don't know exactly what happens beyond the standard model. And if all the particles contributed in a symmetric way, for instance, with supersymmetry, then the contribution to the Higgs mass would cancel out and there wouldn't be such an issue for the Higgs hierarchy problem. That resolution could work out for the cosmological constant problem if supersymmetry was restored all the way up down to at least the Hubble parameter today. But clearly that's not the case. We're not living in a supersymmetric world. Contributions to the cosmological constant don't cancel out within the particles that we know of already. And so it's really an issue within the physics that we know already. Okay, so this led to the old cosmological constant problem which is how can we set lambda to zero? And when it was first formulated, the old cosmological constant problem, it wasn't called old at the time, people had the conviction that the universe was accelerating at the time. And so the real question at the time was really, how can we set? And it could be that we don't understand the way the vacuum energy gravitates as well as we thought we did. It could be that we don't really understand these loop calculations although we have some very strong indication that, I mean these are loops of matter and we understand extremely well how these loops of matter works out in the in LHC. There's no confusion that it's probed within many loop accuracy. So, but there could be possibly something that misses us. And at the time we could say, well, maybe there is something that just sets lambda to zero, exactly equal to zero. We don't know exactly what it is. So that's the essence of the old cosmological constant problem. But maybe there's something that does that first. But since then it's been observed that actually the universe is accelerating after all. And the new cosmological constant problem is really a vaccine problem, is why lambda is actually not quite zero. We might prefer having it zero rather than a really tiny amount. If it was really zero there, I wouldn't say, well, there's not really any discrepancies actually. This calculation is wrong and it should actually be zero. But if it's just a tiny amount, then we see that discrepancy between the two. So the new cosmological constant is actually why lambda is so small. Or if lambda is actually zero, then what can source this acceleration? In a way which is technically natural. So the key thing is about technical naturalness here. At this level we would, most models of dark energy are happy in, well not happy, but they are okay as a first step to set the cosmological constant to zero. And then just postulate the existence of another fluid, another type of field, that leads to the acceleration of the universe. But even when you do that, that scale, that tiny scale, which is much smaller than anything we know of within so-called physics, has to come in somewhere. And so we really need to make sure that this scale is technically natural, that it doesn't receive radiative corrections that make it go out of control. And that's really the essence of the new cosmological constant problem. So a lot of the models of dark energy that I'll be talking about today and tomorrow, they postulate indeed the existence of new fields, and they really address more than new cosmological constant problem. They have in mind setting lambda being exactly equal to zero, and then seeing what is the dynamics that could lead to an acceleration of the universe. And most of them, when you look at them after the fact, you see that they don't even solve the new cosmological constant problem, or they don't even solve the technically naturalist issue. But you have to investigate them in the first place to see whether or not they would be able to solve them. And going through the different machineries help you possibly finding a way to go through this technically naturalist way. So we'll look at that hopefully tomorrow. Okay, so that's it for my motivations. Now we're going to go through models of dark energy. And we'll have a look at Quintessence's quintessence to start with, because that's one of the first models of dark energy. Quintessence was really proposed to address the coincidence problem, but since then it's really understood that you could have a scalar field. I could play the role of dark energy, which would be dynamical. A lot of the language that we'll be using to describe quintessence is very much reminiscent to what happens during inflation, because it's a phase of accelerated expansion as well. It's just the scales involved are very different. So it'd be very unfair to say that whatever happened during inflation is the same physics as what happens today, because we have a huge level of discrepancy in scales between the two. So there's no reason to believe that they wouldn't be changing physics between the two this. Very likely, many layers of physics between what happened in inflation and what happens today at the late time period of the universe. So the idea behind inflation is rather than having a cosmetical constant, which will have an equation of state parameter being equal to precisely minus one. And we have the coincidence problem, which stated that as a function of time, or as a function of, let me put the scale factor, the energy density of radiation scale like 1A to the 4, that of matter like A to the cube, and then a cosmetical constant would just be a constant and so only dominates N in B or 4. The same order of magnitude as matter today, and not throughout the age of the universe. So the idea is extremely simple. If you have anything which is dynamical, where the equation of state parameter is not exactly equal to minus one, then it won't be a constant anymore. And so it will just probably be a little bit bigger in the past and therefore it wouldn't be as subdominant in the past as it is for the cosmetical constant. That's really the whole essence behind quintessence. So we can do that, we can make dark energy dynamical by saying, for instance, there's a scalar field. So let us take our action to be that of GR, coupled to a scalar field. So let me take a standard scalar field like so with a given potential. So this is really a standard scalar field like you've probably seen during inflation. And we wrote last time that the stress energy term, so we wrote it on flat spacetime, but now if I derived it even on cos spacetime, it would be given by d mu phi, d mu phi minus g mu nu a half d phi squared plus a potential. And so from there, on FRW, let me set the lapse to be one now, because I can't because we already derived the Freemann equations, so I can just do that, so I am in FRW. And we have the scalar field just depending on time. That's my symmetry for the system. Then we have that the energy density rho is equal to minus T00. And in this case, we're going to have phi dot squared here, minus, minus, minus phi dot squared, so it's going to be half phi dot squared. And then, yes, and then minus, minus plus v as the potential. And then the pressure is equal to Tij or T11, T22, et cetera. And that's equal to, there's nothing from here. And then from here, we have an a squared, but I'm taking the one up down, so it gives us a half phi dot squared, minus v. We see that equation of state parameter, which is the pressure over the energy density, is a half phi dot squared, minus v, over a half phi dot squared, plus v. And so we have something which looks very close to minus 1. If the potential dominates over the kinetic energy, and that's something that should be very familiar as compared to inflation, we want kind of the same type of phase. What is different is the scale, but otherwise, at this level, we want something which is dominated by the potential. So if we have that the kinetic term is smaller than the potential, then the equation of state parameter is approximately equal to minus 1, and a small correction, which will always be positive in this context. So this is always positive. So we always have an equation of state parameter for which it's close to minus 1, but with a small, slightly positive contribution. And so that means that instead of having a cosmological constant throughout the age of the universe, the small correction here will make it slightly dynamical. And depending on how large that contribution is, you could have something that decays a little bit like that, or that decays much more. So there's a whole region of study of models of quitescence, where you can see that depending on the form of the potential, you can allow for yourself to have some tracker solutions or scaling solutions where the energy density present in the dark energy fluid, so this quantity here, is made out of the potential, but it is also made out of the kinetic energy density. The potential, you may have been slightly higher up your potential in the past, and you may have had a little bit more kinetic energy in the past. So the contribution to the dark energy fluid may have been higher in the past, and you may have tracked the behavior of radiation during the radiation era, and of matter during the matter era, without yet being entirely, without yet being dominant. So you could have regimes, scaling regimes, where they're just slightly sub-dominant, and then only at late time, when a slight instability happens and you come out of the tracking solution, is that fluid allowed to dominate, or its potential energy allowed to dominate, over dark matter, and the matter component, and you start having the dark energy-dominated era, that would correspond to a tracker solution. So we'll leave that as an exercise, but you can check that if you have a potential of the form, an exponential potential, minus alpha phi of n-plank, with the scale here associated with the appropriate scale for dark energy. If you have a potential like that, you can check that you have something that looks like w roughly equal to minus one, when you account for the dynamics at late time. And then if you do a more careful analysis, and I would probably involve doing a little bit of numerics, something you can do easily on Mathematica, but you would see that you could go through a tracker solution that you follow that up. I'm not gonna go through the details of that. This is something well-documented, it's just to give you a taste of what can be done in this type of setup. So this is the idea behind quintessence, and the quintessence is there because you, the terminology is there because of course you would have a fifth mode, fifth force, so then you flew it in your universe. Yeah, so you want, of course, you build this model so that you have what you want, right? Yeah, so you want to have, you want by the end of the day to be dominated by something positive. We're in a period of the sitter with positive acceleration, we're not in a period of ADS. So yes, you could put something which is negative in there with the potential being slightly negative, but then it would definitely not do what you want at late time. Yeah, so then it would be in a period of ADS rather than the sitter. Yeah, so this we know is not what we're observing, so you wouldn't want to do that. Of course, it's a very important question of why is the cosmological constant or seems to be positive rather than a negative? That's a whole set of study in itself, and I was troubling and still troubling many people. What I want to address in this, oh yes, observations in the past and even a little bit today, with that much significance, so it's maybe one or two sigma significance, seem to always see equation of state parameter for dark energy with it slightly smaller than minus one. It's not of a high significance, so it's not something I would say we really need to all drop and doing what we're doing and focus on that. It's not clear that that's not gonna go our way. W is minus one is still entirely consistent with it and W is slightly larger than minus one is slightly still completely consistent with it. But still, it got people thinking on whether we could have possibly something, a fluid, which is less than minus one. And the answer in that language is, well, it's simple. I don't want to have V being negative because it would lead me to a period of ADS instead of the center. But what we can do at this level is say, hey, let me put that being the wrong sign. Let me put a minus in here and that track down. So it means a minus here, minus there. So it means a minus here, minus there. So it means, oh, let me put a plus here. Okay, another story. Let me do that and then I have something which is called phantom, dark energy. And it will give us W is smaller than minus one. So that's great, no? We can do whatever we want. Everybody happy with that? W would be smaller than minus one, yeah? So there were some, yes, there's a complaint there. Very much so, there's the wrong sign here so it does affect the kinetic term. Yeah, that's entirely unstable. If you look at the Hamiltonian associated with that, its kinetic energy will be unbounded from below but it's not just like you have a small instability and with the potential for instance and then you can scale that instability so that it's tamed down, so that instability is as slow as you want. It's an instability in the kinetic term. So if you give the field as much momentum as you want, the Hamiltonian will go as low as you want. So that's a dramatic instability. This is a ghost-like instability and that model has no stable solution, it doesn't make sense to consider it just like that. So this is, it has a ghost instability. If it's what? Yeah, is it better to put the minus sign in here than in there? So phenomenology, well, sorry, theoretically, it would be much better to have the potential being the wrong sign for some region and then maybe later on it would go back the right way up. That's much better rather than having the kinetic term being negative because that's really an instability related to the, with a scale related to how fast or the momentum of the field itself and there's no way you can switch that off because if you have a potential, it may just be an instability for region in the potential and maybe later on it will be, okay. So that's a very bad instability. You don't want to do that, okay? Keep the minus sign here. And the question I'd like to address now is that of naturalness because in principle, if I have the right sign here, that gives us some models of dark energy, of dynamical dark energy. And if I put the right scale here, if this potential behaves appropriately, it can mimic the late time evolution of the universe perfectly. So then there's a question about how the fluctuations behave and what matters is how things go at the perturbative level but at the level of the background, a priori we could do that. So if we want to address at the very least the new cosmological constant problem, you can ask the question about skills and how technically natural things would be. So let's think about the mass of the scalar field, the effective mass of the scalar field. For the question of naturalness, let me put in the scales to outside. So let's say V of phi is equal to a scale mu to the four. This has dimension energy to the four and then I'll have a dimensionless potential which is phi over a scale F. So now U zero is dimensionless and I can say that U zero, all the coefficients that come in in U zero are such that this is all of order one. I put in the scales in there. So there's a small assumption that I come in here that there's not a big hierarchy between the coefficients that come in building it U zero but let me assume that I can do that. And so we'll have that V prime of phi is, so let me, sorry, no scale in terms of scale. V roughly scales like mu to the four. V prime roughly scales like mu to the four over F and V double prime roughly like mu to the four over F squared. And so roughly speaking, this looks like V prime squared over V. But now we'll borrow what I know because I've heard it yesterday, you've seen in inflation. During inflation, you define the slow roll parameters. I don't know exactly how you define it but I suspect it was something like the second derivative of the potential over the potential squared one over M plank squared. Was it that? Yeah? Okay, there may have been a factor or two in there but roughly the square of whom? Oh, sorry. Yes, yeah, like this and there's a half here. Yes, that's one, I'll have that. Okay, so during inflation, if you want a slow rolling potential so that you really in a period of accelerated expansion to inflation, you want epsilon to be smaller than one, maybe even much smaller than one. Observation tells us that maybe it's roughly of the order of 0.96 or something like that. It's smaller than one. Sorry, one minus epsilon is that. During dark energy, we want roughly the same in the most natural model. Natural in the most simple model of dark energy, you want roughly the same. You don't want to be fastly rolling on your potential because otherwise you'll be dominated by that. You want to be mainly dominated by your potential. The kinetic energy is small, so you want this epsilon to be smaller than one or much smaller than one, which means in our language here that you want v prime squared over v to be much smaller than vm plank squared, yeah? But now from our Friedman equation, we roughly have that the Hubble parameter is related to the potential because that's what dominates the energy density. So that roughly goes like one over m plank squared times the potential. So that potential here is roughly equal to h and then we have m plank squared. So that's m plank to the full. That's the Hubble parameter today. Okay, so now the effective mass for our scalar field for the dark energy fluid. Effectively, this is equal to v double prime, which from over there we wrote down like so. So we wrote down this as v prime squared over v and there's some dimension problems here. What did I do? There's an h square, yes, h squared here. Yeah, but there's still dimensions are not right, right? Should be of m plank squared here, what did I do? Okay, is that better? Yeah, I should have divided by m plank squared, I'm not applied. So this thing here, the v that come in here, it's equal to h squared m plank squared. So the m plank squared cancels out and so I have an h zero squared here. And so that gives me something which goes like h zero squared here. And that we said roughly of the order of magnitude of 10 to the minus 33 electron volt squared. So what does that mean? Is it m plank squared over two? Yes, is that right? Yes, that is right. Yes, that's dimensionally correct, yes. Yes, is that right? Is that what you've seen? Yes, okay, thank you. We end up, we want to tackle the cosmological constant problem with the very least the new one. And we say well if we had a cosmological constant, we don't like it because the scale of the cosmological constant should be tiny of the order of h zero today, that's really not natural. So let's postulate this whole dynamics instead. Let's say we have a scalar field and it's an dynamics, it's such that at late time we have dark energy and the scale associated with that field, the mass of the scalar field today should be of the order of the herbal parameter today. And at this level there's nothing that will prevent you from having quantum corrections to the scalar field that will be of the same order of magnitude as the mass of the particles that you will couple to. So we end up with a natural question here which is we introduce a new whole dynamics but then the mass of the scalar field has to be tuned to be extremely light. That would be the lightest particle that we've seen that would exist. But there's no symmetry that would protect this mass of dark energy to be as small as what we'd expect it to be. So that's a naturalness. Okay, so what can we do? Well we can live with that maybe. You may say that's not a problem but people have explored different possibilities. So as a second example, we can have a look at F of our gravity. The idea there is that we started with GR, we had a scalar field and then we had the matter lagrangian which would couple to the metric G and may couple to phi as well possibly and we'll couple to all the other species that exist in the world. That is the standard model. So what we have done so far is thinking, well maybe we could have this part here generating dark energy possibly. Maybe we could have that part there instead possibly generating dark energy. We don't want to switch the sign here but maybe we can have some generalizations of that possibly and we'll have a look at that maybe later today or tomorrow. People have explored as well the way the scalar field could enter in the coupling to matter. In most of these models where you're trying to have dark energy being mediating here with a scalar field or even mediated there to a less extent but at least very much in here or in there we run out, we run off with the similar type of tuning issues and naturalness issues where you always have a scale in the end of the day which is tuned to be extremely small but in a way which is not technically stable so in a way that would receive kind of corrections that make it run away from its very small value. The one possibility that people have explored is to modify this part instead and the first thing that people have tried or I don't know if it's the first thing but one thing that people have tried is to promote this curvature term to a function of the curvature. People could have done many other things but if you are in four dimensions, if you are, for instance, Riemann curvature like so this can be written in terms of gasmanate terms. Yeah, in terms of the Riemann. This combination here can be rewritten in terms of the gasmanate term and the scalar curvature squared if it doesn't have higher derivatives and therefore there's nothing new compared to that. Otherwise, if you have any arbitrary combination of this and not set that to zero, for instance, you will typically end up having with equation of motion for your metric which will be higher derivative and so this changes quite dramatically the way you want to solve for that. It's okay if you want to do that perturbatively in an effective theory sense and say that these higher derivatives are gonna be small and they just correspond to new physics that would come in so you would expect them to be suppressed by the plan scale or maybe a slightly lower scale but suppressed in such a way that you're working in a regime where those are not allowed to dominate because when they start dominating it really means that you should account for the new physics that generated them but if you want to do dark energy and you want to find exact solutions you want to rely on operators which possibly could be as large as the ones you're considering here already and therefore you want something which is simply second order in derivatives and F of R is the only thing that you can do at that level as a modification of gravity if you didn't have a scalar field. There was a question over there. The topological term? Yeah, I guess the gas beneath term in four dimension is a topological term. That's right, that's right, that's right. Exactly, exactly. So you could make it appear by adding a scalar field in front of it but the idea here is to give up the scalar field and just look at corrections to gravity itself. Okay, so people have explored F of R gravity so there's no, let me just write it, the action for F of R, so there's no ambiguity. It will be D for X, growth minus G and let me just write it as F of R and then you have coupling to matter. Yes, you will see. Yes, yeah. It's a very good question, that's an important question because we know that Einstein gravity works very well so it better be that we can recover it in some regime. That's an important question, yeah. Okay, so this is my action for F of R. Let's see if that's the way I wanted to write it. Let me put the blank square here. So then the equation of motion you could do, you could derive them. So you vary your action with respect to the metric, one over square root minus G and you end up with your modified Einstein's equations which are, instead of having just your Einstein tensor, you have something that looks a little bit like that. So you have your function F prime of R times R mu nu and then you have minus a half F of R mu nu. So this would be the Einstein term if F of R was just R. But when F of R is not R, it's slightly different. And then you have other terms, minus mu nu F prime of R plus mu nu box F prime. And all of that is equal to the standard right-hand side term, one over two and blank squared T mu nu. So very naively what you see appearing here is something that would look like your Freedman equation, sorry, something that would look like your Einstein equation from this part, at the very least if F was exactly R. And then you have a new combination that comes in from the derivatives of this function and this could possibly play the role of dark energy, play the role of the new source that leads to an acceleration of the universe that sources this part without necessarily needing to invoke a cosmological constant on this side. So that is the idea. Then you would have that universe just likes to accelerate by itself, not because it's filled with some dark energy type of fluid. It would appear, but rather because of the way gravity behaves. But clearly if you're working in a regime where F of R is as close as you want to R, you recover standard GR. But then people are interested in solution where you would self-grabitate by yourself. And so a popular example, popular choice a few years ago is the function F of R is equal to R. That's fine. But then minus mu squared over R. So this is something quite dramatic. This is not a small departure from GR because you see that you can no longer set R being equal to zero. So the Stan and Minkowski solution, sorry, the Stan and Minkowski geometry is no longer a solution of your modified Einstein equations. This would be pathological. Instead, what you have, if you have your function F of R being like so, you have, you can see that you have the Sitter or ADS solution of the form. Let me just say the scalar curvature is a constant. So just a constant like that. That means that the Einstein tensor would be one of a four C gemini. You want that because you want to see if you can have something that would look like a cosmological constant, so that you want to see if you could have something that looks like precisely the Sitter in this type of solution, in this type of equation, a theory without the need of a stress energy tensor on the right-hand side. So if you plug that ansatz back into your Einstein equation in here, what you get, I don't want to go too much through the details, you get one plus mu to the four of a C over four gemini from here and there. And then your F prime, yeah, that was your F prime, and then your F is minus a half. Your F is C minus mu squared over C, and then you have gemini. And you have the derivatives here that don't come in in this case. This is equal to zero. And if I haven't made any mistake, this should lead you to C is equal to plus or minus three mu squared. And so by choosing appropriate scale mu here, you can end up with that choice of F of R function gives you the Sitter or ADS solution in the absence of matter, in the absence of cosmological constant, where the Hubble parameter H squared is associated with the scale here, on that scale there, related to C, and there's some factors that come in and I'm gonna worry too much about, but it's exactly equal to H squared, oh sorry, mu squared, that's scale there. And you can have ADS as well, but we're interested in the Sitter solutions. And so of course you need to tune that scale to be once again the scale of dark energy today, to the minus 33 electron volt, to have that working out and give you the right order of magnitude. Principal, this is something one can do. So they may or may not be naturalist issues related to that, but there is one way which is easier to answer these questions and it's simply to realize, and this is something I know some of you know very well because some of you have posters on that, is to realize that F of R is nothing other than what? Couple quintessence, yes, exactly, exactly. It's nothing other than couple quintessence. So where's the scalar field? You're saying it's couple quintessence, where's the quintessence scalar field? Yes, exactly. We can see that we can, in these guys in here, these guys in here, it's actually gravity and a scalar field and then we will write it down as gravity and a scalar field, we will see that these scalar field couples non-triviality to matter. So this scalar field can also be seen as being a quintessence scalar field or dark energy scalar field. So F of R is nothing other than GR plus scalar field and to see that, we can take a priori the unrelated action which is given by plus square root of two F of phi, F prime of phi R minus F prime of phi plus F of phi. Okay, so up here this is an action. It seems that this is a completely unrelated theory because this is a theory of a function of the scalar curvature whereas this is a theory where you, as a functional you have, as a variables you have the metric and you have the scalar field and it's not a function of the scalar field of the scalar curvature. However, you can see at this level that if you were to integrate, so at the level of the path integral, we're dealing with something which at this level is the integration of your scalar field phi and your metric E to the, let me work in Euclidean, it doesn't matter too much, this function, this action of phi and G. Probably you'll have that. So let me perform at tree level the integration of the scalar field. We see that the scalar field here has no dynamics. I don't have any derivatives acting on the scalar field. So what it means to perform the integration at tree level is simply means that I'm gonna look at my standard equations, solve for them and plug them back in at this level. So the equations for the scalar field are m-plank squared over 2 f double prime of phi r minus f prime, minus f prime, minus phi f double prime plus f prime is equal to zero. This part here has been conveniently chosen so that these two things cancels. And now we see that everything gets multiplied by f double prime. And so long as the second derivative of this, of phi is not different than zero. And that's the case, if the second derivative of f is different than zero, that's the case if we have at least a quadratic contribution or something different than just a linear contribution to the function f. And that would be the case if we have something different than g r. So if f was just a linear function of r, it would just be g r. And it departed from g r corresponds to f having a non vanishing second derivative. So f double prime is different than zero if we're not in g r. That's again how we recover g r. If f double prime is equal to zero, then this equation has no content. And then we see that then this equation tells us that the scalar field phi is equal to m-plank squared over two, the scalar curvature. So now when we thought we had a function of the scalar field in here, we can plug it back into a function of the scalar field, sorry, of the scalar curvature instead. And what we have here after integration of a phi of the classical level is back to our f of r theory. So we have the m-plank squared over two. And then the f prime of phi is now gonna be an f prime of our function of r. And we have r minus r f prime of r plus f. And I forgot about the m-plank squared. I should come in here over two. And the m-plank squared I should come in here over two. When we're gonna do that, you see that this part precisely cancels that one. And we end up with our f of r action. So we see that f of r is precisely equivalent to g r with a scalar field, so long as I'm not coupling to matter yet. Now we can see how the coupling to matter is affected by that. Any questions? Yeah, yeah. Yes, yes, gravity is dynamical as well. I just haven't integrated over it. If I knew what the whole thing, if I could integrate over gravity, our life would have been much easier. Yes, exactly. We'll consider them at the part in a second. But for now, let me just focus on the facts, on this equivalence here. Okay? Yes, we're gonna have a look at that right now. Oh, it's afternoon. You're asking questions. Yes. It doesn't look like Einstein Hilbert term yet. That's right, that's right. So at the moment, it doesn't look quite like, sorry, at this level, let me just say, okay, it's gravity and a scalar field. At the moment, let me just say it's gravity and a scalar field, and this doesn't look quite like g r quite yet. But we'll see in a second, we'll see in the afternoon, as we're running out of time, how this can actually be rewritten in back into the Einstein Hilbert form, and the difference comes in from the coupling to matter. Yeah, yeah. So yeah, let me just say it's gravity so far. I said it was g r because it will happen to be g r. Yeah, so let me stop here for this morning. Is there any questions? Yeah. Oh, okay, perfect, okay. Oh, okay, no, no, no, no, good. Ah, perfect, yeah. No, it's not supposed to be at all. No, no, no, no, no, no, no, no, no, no, no. We're dealing here with modifications, so we're dealing here with trying to tackle the cosmological constant problem. Things that happen at very low energy. We, in these modifications of gravity, we really try to see what's going on at low energy. If anything, it would make the UV completion worse. It's definitely not trying to find a UV completion for gravity. No argument, just the fact that it works out. There's no. The one over r? No, no, r squared. Ah, yes, yeah, exactly, exactly. This, we would expect them to be there. They may be suppressed by the Planck scale as compared to if I just have r here. They probably will be suppressed by the Planck scale at the very least, maybe the strength scale, maybe something slightly lower. We need to be quite lucky for it to be too, too low, but certainly, I mean, GR is working very well on many scales. We don't expect these corrections to come in at too low a scale, but certainly not at the scale of dark energy. These corrections, they expect them to come in at the scale of dark energy. So you will expect to have an infinite of corrections here that I haven't written down of that form, and then also any number of derivatives acting on that, all of that. Yeah, so none of these corrections need to be and won't be second order in derivatives and it doesn't matter. Yes, but if you want to rely on these operators to derive some of your phenomenology and not need to worry about everybody else, then you better have that what you rely on is second order in derivative, or that it is small. But if it's a significant contribution, it would need to be second order. If it's a... Yes, yes, yes, yeah, yeah, exactly, exactly. Whatever is gonna UV-complete GR or this theory will come in at some scale like that and will bring in all of these corrections here, and this have to be that, yeah, yeah. But this won't help you with dark energy. Okay, so yeah, so we've seen that F of R is equivalent to this theory, okay? Now, at this level I could have kept the coupling. Let me put the coupling to matter here. As I had it over there, so I here, and when I perform this integration, it's an integration of a phi. This thing remain completely unaffected, so we still have the same coupling as we had before, okay? But now, it'll be easier to understand or to answer many of the questions we may have about this F of R theory. If we're dealing with its formulation in gravity in a scalar field plus coupling to matter rather than in its F of R formulation, because the mass is coming into the scalar field, rather than the gravity part itself, and that's, it's always, if you can have complicated things, you better have it in a scalar field. Now, as many of you have pointed out, this is just a conformal transformation away from GR actually. So in here we're dealing with a metric G, but I can always, this is what we call the Jordan frame. In this Jordan frame, we see that the kinetic term for gravity is not standard. We have a function of the scalar field in front of it, which is not standard, but the coupling to matter is standard in the sense that all the species that you may want to consider living in this manifold couple to the metric G and have at this level seem to have no knowledge directly about phi. They would have it through G, but not directly, not directly. So this is what we call the Jordan frame. From this Jordan frame, we can always go to our Einstein frame. And by Einstein frame, we mean that the kinetic term for gravity is the standard one. So we'll have just the scalar curvature in here. We won't have this contribution in here. And this at this level just looks like a conformal factor. So we can deal with that by just performing a conformal transformation. There will perform a conformal transformation here we're dealing with a metric G menu. We go to an Einstein frame metric G menu, which is equal to, I always have it. The wrong one, yes. So this is gonna happen to be a prime of phi G menu. And when we do that, you will see that if you had G menu in here, now in the Einstein frame, you will end up having a coupling to a new G bar menu, which will depend on phi with a dependence, a conformal dependence on the scalar field as well. So we can look at that explicitly. We perform this conformal transformation. So first off, the simplest part, under this transformation, what would be squared minus G in terms of square root minus G bar? So we have G menu equal to one over F prime G bar menu. This will be one over F prime squared in four dimensions. Yeah, this particular coordinate transformation dispends on the number of dimensions. And then something which you can look it up or you can derive it exactly yourself. I didn't write it down. Okay, this thing here, you will see, we'll end up going like F prime R bar plus after integration by parts actually, it will be minus three half F double prime over F squared, D bar phi squared. Yeah, performs an integration by parts. So I'm lying a little bit here. It's only true once I take the appropriate measure in here. You cannot see? Okay. The second term. Let me just say, because the rest depends on integration by parts, which I'm not really allowed to perform just yet. Let me just say that these goes like F prime phi bar and then terms that go like box of phi and then terms that go like do it is a phi squared. So when you put that back into this Lagrangian in Jordan frame, I call that the Jordan frame metric. We go to Einstein frame or the Jordan frame actions. All right. We'll have a D for X. This thing becomes square root minus G bar over F prime squared. And then we have M plank squared over two. We have the F prime from here and we have an additional F prime from there. So it's an F prime squared. And this is precisely why we chose that particular conformal factor in the transformation so that it cancels that F prime at the top and bottom and you simply have end up with a scalar curvature term. And then you'll have some terms that can look like corrections to the kinetic term. So let me put that here. And you end up having minus three half phi double prime, so F double prime over F prime squared. A kinetic term for phi squared. I put a bar on the derivative table here just to mention that the indices are gonna be raised with respect to this new metric G bar menu. And then we have some sort of potential term for the scalar field, which is just what it is. And then we'll have minus phi F prime over F prime squared. F, sorry it was phi F prime over F prime squared, yeah. So over F prime. And then this thing is plus F over. And now the coupling to matter occurs G which is equal to P bar over F prime. We see that if we had for instance, let me stop there for a second. So we see that we end up with this is just equal to one. This is a kinetic term for the scalar field. So the scalar field is dynamical. And we see that this is a potential for the scalar field. It'll depend on the function F. And then we have coupling to matter which couples to the standard metric but then a different conformal factor in front. So in that language, since it is GR, it is manifestly GR, we see that the whole acceleration of the universe at late time is really coming from the scalar field itself from its given potential, from its given potential. And so as you mentioned, this is nothing other than quittances. The only difference is that it couples non-trivially to matter. But the fact that we had dark energy solution is not really related to the, to high couples to matter. Once again, emphasize that all of that derivation relies on having this function when you want, you want a real modification of gravity. So you want the second derivative not to be zero. And of course you want the first derivative not to be zero as well. It can be a constant if it is a constant then it's just, I can, you didn't need to go through any of these things. Maybe I'll stop, yeah, maybe I'll stop there. Any questions about that? I can't hear you, sorry. Yes? Yes, from, yes, yes, thank you, yes. Yes, yeah, yeah, yeah, yeah, yeah, yeah, yeah. If you have it, yeah, absolutely. If you have this square root here, yes, it's like that. Thank you, yeah, yeah, yeah, yeah, yeah, yeah. Yeah, so what I have in mind is indeed that you have actually, yes, yeah, yeah. So if you have your action for your matter field, one of the F prime G bar sine I. So that means in particular, let me just give you an example. But if you had just Maxwell, that's a bad example. Maxwell is a bad example. If you had a massive scalar field, let's imagine you thought you had a massive scalar field minus a half d chi squared minus a half m squared chi squared. That was your action for your scalar field. So you have d4x, you have this. Now it will start coupling, this field chi will start coupling through the field phi as well. So you'll have d4x square root minus g bar, and then you'll have an F prime, sorry, squared here. And here you'll have a minus a half F prime d squared and then minus a half m squared chi squared. So you have a non-trivial coupling between your field chi and your scalar field phi, which is your dark energy fluid. And that affects even more the way the scalar field chi the effective mass of your quintessor scalar field of your dark energy scalar field. Because you have a non-trivial coupling between the two. But of course that's not limited to having the coupling to another scalar field. So what we're gonna see later this afternoon is see that if you have a coupling between not only between standard gravity, your Einstein metric and your standard model field, but also your dark energy fluid, dark energy field and your standard model field, that will change the force of gravity as we know it in the solar system. So we need to deal with that. Okay, any questions? There's a mistake here. The last time here. Ah, there's a square here, okay, thank you. Yes, yes, yes. These are the same theories. These are the same, these are the same theory. This theory is exactly equivalent to that one. There's no difference. It's just a change of frame. It's a change of frame. We introduce auxiliary variables, but that's no difference. We integrate, we can integrate over them or not. And then it's just a change of frame. So between f of r and that theory here, we're matter couples to gravity and then that field phi like so. There's no difference. It's exactly the same. They are the same theories. Yes, so it's a change of frame. There's no physical content in that. Yes, so in that language, it's clear where the physics lies. It's all in here. So the acceleration of the universe is just coming from the scalar field in there. There's no, so there's no mystery. You can rewrite it in a way that may appear to be as a modification of gravity, but where the modification of gravity really is is not into what the tensor mode of gravity are doing, but more in how gravity couples to external matter. That's where the modification of gravity lies, not into the kinetic term for gravity. Yes? Ah, yeah, absolutely. So I don't know if you'll see that. There's a starbin scheme model of inflation. You can rewrite it as a, well, the way he introduced it was indeed as f of r is r plus r squared. And that's a model of inflation. So you can see it like so, or you can go back and rewrite it as a theory with a scalar field with a particular potential. It's gonna be a particular exponential potential. And so if you write it like that, I may say, oh, well, that's quite natural. And then from that perspective, you say, well, I would never have come up with that type of potential for my scalar field. But it doesn't matter too much actually. It's the naturalness issue in any ways that much better understood in that language. I should say, I didn't say that. I should say that don't worry about this factor in here. This can always be absorbed into the field where definition. So you can always make a transformation of your field so that this overall factor is one and then you have corrections in setting here. So you can always canonically normalize your scalar field. Yeah. So when you have it for what? For r or for? Yeah, for r. Yeah, don't do it for f prime is equal to zero. I mean, you don't, yeah. Yes, yes, yeah. This is the case which is done for inflation. There's no singularity whatsoever. Yeah, yeah. Sarabinsky inflation is just entirely equivalent to inflation rated in a standard way with a particular potential.