 Okay, so welcome back to our last lecture of on dark energy. So last time we were looking at Galileo's As models that can potentially give you a kinetic screening. So we'll see how that works today We'll introduce the Galileo's last time. Let me just imagine now for concreteness to have a particular model Where we have a Lagrangian, which is the one of GR Let's imagine let me put the square minus G in front in it as well square minus G and then we have GR and then we have Let me say co-orientized version of the Galileo Lagrangian and I mentioned the other day. Let me not talk too much about how we co-orientize them But we'll have the second the cubic Galileo And in principle we could have Up to the quintic one of them. So we'll focus in the case where we have just L2 plus L3 Where this is going to be square root minus G because we're now going to want to work in a curve spacetime for a second to see how that can lead to dark energy models. The L2 is simply the standard kinetic term for our scalar field and then we had introduced a cubic Galileon Which we said was like so. In principle you could include All of them if you wanted to but let me just focus on that just for concreteness. That particular model has issues of its own but That can be resolved if you introduce the other ones I just want to be as simple as possible At this stage. So we have introduced a scale lambda and we had some motivation Yesterday that this special type of Galileon Operators they satisfy at the level of the action. So if I were to put an integral in here Performing some integration by parts that satisfy The Galileon and shift symmetry and so we said that any quantum corrections Generated within this low-energy effective field theory, but at least loops of these Owned scalar field should only involve operators that directly generate the same Symmetry and since these Galileon operators they only do so after integration by parts The coefficients of these Galileon operators cannot be renormalized cannot be Modified by quantum corrections This statement will no longer be true if you take some generalization of the Galileon Well, you put an arbitrary function of phi in front or an arbitrary function of phi and Derivatives of phi in front and you would lose the non-normalization theorem You would lose the Galileon symmetry. So this satisfies as we mentioned the non Renormalization which is a motivation for thinking that the scale lambda could potentially be quite small and not needing to worry Just yet about quantum corrections So now we're gonna look at how We can have some dark energy solutions And this is gonna be very symbolic you can do it exactly Particularly I would suggest to include the other Galileon terms if you want to do it exactly because you will strongly help with the the well-behavedness of The perturbations, but let me not do perturbations on top of that right now about the solution So we're gonna want to look at ffw solutions or even just to see the solutions For the I'll start with the the city ffw. Sorry mini super space approximation Just like we did in the very first lecture the mini super space approximation where I put myself in ffw I'm gonna keep the lap so I can derive the Freeman equation with it and as soon as I derive the freedom equation normal I can set it back up to one if I want to and then I have my three coordinates in there so I have the labs and and the scale factor and my Scala field I will say is only a function of time. This is what it means to be in the ffw Symmetry and this is the mini super space approximation We're only gonna consider variation of the action with respect to the labs The scale factor if I wanted to and the scala field if I wanted to So we already saw in the first lecture what this term Coming from the standard Einstein term is within that mini super space approximation And now we can put this thing in we know of course what that is. We'll have the d phi squared In this case will be Phi dot Squared over and squared the laps that comes in and a dot is The ratio is back to time the time here And we can work out what the box is a priori I would need to involve some derivatives of the lab so there is a scale factor Second rate of the of the scale of field when I derive the box But I cannot perform integration by parts and it is very much the virtue of these Galilean operators that Even though it looks like it's gonna involve lots of derivatives in particular derivatives of the laps after integration by parts All of those get removed and we're gonna be left with the laps generating the constraint as it is the case in GR it's very much the virtue of these operators is very much the virtue of the fact that they don't have Any Ostrowski ghost and the constraint remains there So I'm just gonna write you for you symbolically what we get in the after integration by parts We're gonna have not the whole Yeah, let me do the whole action We know from this part here. What was it? We had something like six and blank squared Over two and it should have been what a a dot squared Of a n I think you can check that From the notes of the first lecture There should be roughly something like that and then we're gonna have from this term We will have plus Five dot squared will have an a cube in front and then we'll have an n maybe a factor of two And then from this thing here just from scaling argument We know that it will have to scale like one of a lambda cube It will need to have five dot cube after integration by parts. There's no five double dots and then one of the derivative must be taken by an a dot And so we'll need to have an a square in front and then there's four derivatives here Every derivative carries an inverse power of the laps and then there's one power of the laps coming from the measure The square root minus g. So it should be an n-cheap coming in here So that's roughly the scaling exactly what coefficient we have here is Is that too it happens to be it That doesn't matter too much Okay, and so now we can derive our Friedman equation Which is again the constraint generated by the laps which is not dynamical as we see here The variation with respect to the laps Gives us From here we get the standard Turn from the Friedman left-hand side, which is familiar. It's gonna have a three in here or h At this stage is a dot of an a But I'm gonna set n is one once after having derived the equation of motion From here, we're gonna have Is equal to a half five dot squared and This is the standard thing. So if we didn't have the cubic Galilean interaction This would be a standard kinetic term a standard a scalar field and this is indeed the energy density that we have Usually if it doesn't have a potential, right? It's just the contribution from that But now there's a new contribution to the energy density coming from the cubic Galilean term and this will have to scale like five dot Cube of a lambda cube and this should be an H Just to take care of the number of derivatives Okay, so this is roughly the Equation of motion now we can see that there is a solution And we're just gonna look at the scaling and I'm gonna go through the exact number An exact an exact solution where we have that five dot Well, let me see just fine is equal to a constant and this constant should be oh well Let me just say there should be a solution. We're roughly speaking All of these terms are of the same order of magnitude so I can compare any two of them For instance, I can compare this term this term and that time for instance, and that tells me that five dot H is of order lambda cube and So that means that five dot there's a solution where five is equal to lambda cube of Let me say a constant h zero times t and then we'll by plugging that back into here we'll get that h squared is Equal to We'll have the M plank squared in front here and I put that back in here That's gonna give us lambda to the six over these h zero so h squared We say h is a constant equal to h zero If this is a constant then this will be a constant this will be a constant and h will be a constant So we'll be on the setter and we'll have the h zero Squared in there right and so that gives us a Solution where we have h zero squared is Equal to Have h to the four I'll take the square root so it's h cube of a M plank roughly There's a numerical factor in here, okay So depending if we want to put that exactly be that then it'll be a numerical factor, so we need have at the setter Solution and we have the right order of magnitude for the amount of acceleration today If the scale lambda cube is of the order of h zero squared today times the plan scale So this in energy scale is Roughly h zero is 10 to the minus 33 electron volt Where the plan scale is 10 to the 18 plus 9 electron volt to the one third and if I Able to count if we able to count they should give us roughly 10 to the minus 13 electron volts So this is tiny once again, it requires a very small scale for the scale of This interaction Operator here, so that means that it's operator here becomes important at an extremely low scale So we really strongly relying on this operator to play the role of for dark energy and not in Just to give you an idea In this can scale correspond to around a thousand kilometers. So for gr We would we have that the Strong coming scale if it were of gr is the plan scale and so the scale which the non-unit is important if we Ignore the curvature, but we can trust it all the way down to the inverse plan scale Which is a tiny amount 10 to the minus 34 centimeter if I remember correctly was for these models Modified well of a dark energy model at least we'll see how that connects to modify gravity The scale I wish non-air it is becoming important a huge. I mean in distance. This is huge This is astronomical most is the size of Things on us. So so it's a very a very different Scale the only reason why we are lying ourselves to do so without being too worried Although we should still be worried is because there may be some essence in the normalization theorem that doesn't correct this thing in here and once again the level of tuning that we have here, so there's a Unfortunate notation accident here that lambda is the scale is not the cosmetical constant. So let me call lambda cc the cosmetical constant So lambda cc is different than lambda This is the cosmetical constant and this is the scale of the interactions in this model We don't have a cosmetical constant. We have that dark energy fluid instead We have set the cosmetical constant But if we want to see the level of tuning that we have on this scale lambda as compared To the plank scale. Let me put it in cube in here just to simplify my life We have something that goes like h zero squared of a m plank squared And that's precisely the level of tuning. We were worried originally. This is precisely level of tuning We are we are worried for the original cosmetical constant problem. This is the same level of tuning We are worried about for the mass of dark energy fluid quintessence Fluid here it doesn't have a mass in this formulation, but we see that we still involve our scale In the game so that if we if we want dark energy At the right scale to come out at the end of the day the scale of the interaction has to be tuned to The same amount. There's no there's no winning this game without putting a small scale somewhere And at this level we haven't solved the cosmetical constant problem in the first place So we have the original problem of setting this value to zero and then we involve a new scale Which has the same tuning as well. We had originally The only hope you may have in playing this game is saying that maybe it's more stable to consider a scenario where this tuning could happen and not be as unstable under quantum corrections as Compared to simply setting the cosmetical constant to zero as we've seen when you look at radiative corrections to the Cosmetical constant They are they are of order of the mass of other particles was you expect here to be protected with some form of normalization theory Okay, so that's how potential Late-time acceleration solutions could emerge from this type of solutions now. Let's him. Let's as we mentioned If you have a scalar field leaving now space-time Driving the acceleration of the universe we would expect this color field to also couple to matter so also couple to the stress energy tensor of The Stunning model fields that lives in in our universe and therefore not to be rolled out by 5th 4th 5th forces experiments where we need we need a screening mechanism to occur and In this type of models the screening mechanism involved is the Weinstein Mechanism or kinetic screening mechanism, and then we'll see how that appears automatically in this type of models So let's put aside the FRW solution or example sit a solution that we looked at Just now let's simply look now for the sake of sake of simplicity at the simple Scala field model with its coupling to To matter and we know what the GR component is doing. We know the force mediated by GR So the tensor mode so we don't need to worry too much about that now We just want to look at the additional component that comes in from this color field when coupling to matter And so we can do that in flat space some we don't really need to look into the The fact that the metric may be Schwarzschild or something like that So the configuration we're going to put for ourselves is just to be able to understand a little bit What's going on is that of a static and spherically symmetric configuration because that's where the function mechanism is the best understood So we're going to consider to have a source t Which is a mass at the center of the center of a space time So let me imagine we have a mass here Localize at the center. So we this goes like m of m plank and then delta 3 Of all we're gonna be working in the spherical coordinates We have a spherical symmetry and then we want to see how that scale of field behaves as a Function of all so we have fine as a function of r to start with and then we'll do perturbations on top of that But let's have a look at that if we had none of the Galileo interactions. So we've just had L is minus a half d phi squared Plus directly the coupling between phi and t over the Planck scale And we consider a source which is Localized mass For instance the Sun the mass of the Sun in here localized at the origin of our spherical coordinates What would be? The form of the force mediated by that's killer field That would be what we have usually in Newton and gravity And so we will recover something that goes like the Newton square. Hello We go like one of the r squared where I would have the mass and we would have the Planck scale in here We'll need to behave roughly like that Okay, up to a factor in here that I don't know exactly what it is you could see that by Putting it explicitly we're gonna see that in a more general case So I'm not gonna do it just there. We're just gonna look at the most general case. Okay, so if we had that We would see that this additional scalar field would mediate an additional force on top of what? Gravity GR usually do and that would be clearly ruled out by observations. Okay So now what we're gonna consider is a case where we potentially may have screening. I'm gonna see how we have screening so that would be Newton's quello, but it's an additional one So if we're working in the Newtonian approximation, you would have an order one correction to what we have in in the Newtonian force So instead the whole point of this interaction is that they can lead to some screening So instead we're gonna consider the case where we have For instance just the cubic Galilean It's much easier if I just focus on the cubic Galilean and then as an exercise you can do it for the other ones if you wanted to and Okay, so we can do that Assuming in a spherical symmetry So you have that and then you have the coupling to matter. So phi of M flag T and T is Minus M the mass of the center of us space time and then delta R of the R squared So I'll let you derive explicitly what the equation of motions Are when you do so don't forget that you're working in spherical coordinates So you're gonna have to need to put the measure Back in it. Okay, this R square here and this R square there and then here looks like you would have double derivative with respect to R for phi But as we said before we can integrate those by parts Okay, this roughly speaking So it looks like phi prime squared and then phi double prime and you have other terms for this you can rewrite it as Phi prime cube, okay So you can derive your equation of motion with respect to phi in here And what you get is something that looks like Phi prime of R Plus one of a lambda cube Phi prime of R Where is equal to M of M plank In deriving this equation There's a derivative with respect to R and then there's a delta function. Let me let me write it We'll have something like that R squared in here from the measure And there is Delta Delta R I think there's an R cube even Okay, so you can perform this is some type of Birkhoff theorem that you have for this particular Solution It's not true in general. And so if you integrate by parts, sorry if you integrate some both sides You integrate here and you get rid of the delta function in there. What you get is that your equation of motion for phi is one of a lambda cube by prime of R squared is equal to M The mass of the blank scale and then one of the R cube And they may be factors of 4 pi from the measure of the delta function, which are gonna ignore for now, right? We're just looking at things symbolically Hopefully this is right. And so we can see from this Equation two type of behaviors depending on which term on the left-hand side dominates Which term is the most important? So if we are at very large distances, you're not you find what I mean by the very large distances in a second But if we are all going to infinity Then clearly this thing is very small. And so we don't expect phi prime to be very large and Then therefore since phi prime is quite small. This thing is negligible. I can as compared to that one So if R is large enough and I define what I mean by large enough the larger than some given value or star Which we will define in a second then this should be small so if Then phi prime of R one of a lambda cube should be Small and then I'm just the left-hand side is dominated by the first linear term and we recover something Like the Newton square law where phi prime goes like M over M plank one over R squared There's factors of two pies and cetera that which I haven't put it in but this is just to give you an impression So we recover in the standard note Newton square law behavior at large distances And this is what happens in the weak field approximation Okay, so then we have order one corrections to GR gravity because this is in addition to the standard Newton to the standard Newton force Okay So if that was the end of the story and if this behavior was true all the way from infinity down to very close to the mass What's at the origin of your spacetime? Then we will be clearly will died by Test of gravity. However, we see that at smaller distances And now we can define what we mean by smaller distances a smaller distances is when this is not Much smaller than one but instead much larger than one and now we can take this behavior as a first approximation to define What we mean by smaller distances? So there's any limit. Okay. Let me just say there's an intermediate scale an intermediate scale Where both this term and that term of the same order of magnitude that happens when this is of order one So when five prime of R is of order lambda cube and So to see when that scale happens we can plug in here We can plug in here the expression for 5q for 5 prime and that tells us that this is at a radius Which goes like M of a M plank Us to the cube is of order lambda cube and let me call it our star so our star goes one like one of a lambda and then M of a M plank To the one side the way and we go from infinity a very large R We have something that looks like a Newton square low mediated by the new degree of freedom all the way up till we hit a radius Our star which is in virtually proportional to this strong coupling scale So we see that the smallest is strong complex scale is The quickest coming from infinity are we gonna hit that new regime? Okay, and the biggest the mass is the quickest coming from infinity as well We're gonna feed the new regime So we're gonna have a more efficient Vanston mechanism as we will see if we have a smaller scale lambda or if we have a bigger mass so if we have Just the Sun as opposed to if you account for all the mass present in the galaxy, then it will give you slightly different Or slightly more efficient to a Vanston behavior Okay, so this is the intermediate scale and now we can define what we mean by short distances as Compared to that as you enter what we call the strong coupling radius or the Vanston radius You will see this being called as well the Vanston radius that this distances This equation is satisfied with the left-hand side being dominated by the interacting term by The second term and we have an equation rather that goes like five prime over R squared Goes like lambda cube M of an plank and then one over R squared, right? No, one of our cube. So we see that five prime now No longer scales like one of our squared But with a scale like one of R to the one-half and then the rest is You have to you can complete the rest with the units You will need to go like M of M plank Half and then it will need to go like lambda to the three half This is roughly speaking. And so if you look at the force mediated By this color field I'm gonna multiply it by R squared so that I can compare with what I would have had if we are in the Newton square Low, so let me say it's R squared. I'm gonna divide by the whole thing. They would it be and then M plank It's M plank of M So what happens a large distances is something? which is Afford a one for this quantity So it's our order one correction as compared to what we have in GR at large distances but as We reach a strong coupling radius our star which is determined by the scale lambda and by the mass of the object we considering so it's it's depending on the environment again is dependent on Explicit configuration. We're looking at then we'll see that Phi prime R squared Phi prime in this case goes like R to the three-half Yeah So this as compared to what we would have had for Newton's well, oh goes to zero R squared Phi prime Here it goes like R to the three-half Whereas here R squared Phi prime Goes like one. So this means there's an order one Corrections to gravity on distance scales larger than the strong coupling radius whereas there's a suppressed Correction to gravity on shorter distances And this behavior this transition between that Order one effect to a suppressed effect is precisely what is called the vanstein rate the vanstein screening mechanism So if you wanted to then you could see More precisely what the effect of such a force would be in the solar system And if you look for just this cubic allele on case Different constraints that you have from purpose of gravity in the solar system There's actually the strongest one for this type of consideration is the one coming from lunar laser ranging experiment so apparently Americans said they've been on the moon And they put a mirror on the moon from the Apollo mission from this mirror on the moon We can shoot a laser from the earth to the moon and have a very good Accuracy of the motion of the moon around the earth and determine the advance of the Parallelion of the moon Along its different orbits and and we can compare with why it should be injured what it should be in this case and the fact that The angle of the position of the moon around the earth Is now measured with a precision of 10 to the minus 11 and Is in complete agreement with what we would have had just from the force of gravity without an additional fifth force That can help us put a constraint on the parameters of such a theory So then we will take the mass really becoming from the mass of the earth in this system And we say that even if you have a tiny difference in here Coming from this force is still sufficient to put constraint on this type of models and having a Correction from this force which should be less than one in 10 to the 11 because this is a precision of the angle Tell us that the scale lambda Has to be smaller roughly and 10 to the minus 13 electron volt That's interesting. No, I found that very interesting. This is actually precisely the scale we needed to for these models to Explain dark energy This is a pair coincidence. It's roughly of that. Maybe it's it's not quite 13 Maybe it's 12 or it's really very much on the edge for this cubic Galileo I'm precisely cubic and I got a little interaction. It'll be slightly different if you added the quartet Galileo and the cetera But not dramatically different. Maybe a few one or two orders of magnitude, but not dramatically different So this is really one This tells you that having this type of model could in principle just be on the edge of Giving you dark energy solution with a scale that would still be consistent with solar system constructs Now it does happen for that for this cubic Galileo and actually for more the observation cosmological constraint This is actually ruled out just having this cubic Galileo but People are now trying with different all the type of models and I'll discuss a little bit more about that But I want to go a little bit more detail about Weinstein screening Because it's a little bit more than just just that Now if we consider this solution as the background solution, so we have a large mass at the center of the system that we want to consider and then we want to look at A probe system on top of that. So let's imagine we have a large mass Which is the Sun for instance and then we are here in the middle of space and we have two test bodies and we want to Check what the force between these two test bodies are In the vicinity of a large mass that means that the background is Transcreening how we have a large background configuration for pine. I'm sorry for fine here for the skeletal fine And then we like to look at how fluctuations how we're going to have a small force between two test Bodies on top of this configuration. So the situation we considering is the skeletal fine has the background value Which is the solution we have just found For instance, so you may want to consider more complicated configurations where it may be a little bit more complicated and then you have Fluxorations on top of that. So this is sourced by a large mass a large Stress and a trace of a stress energy density and then once we solve for that We know exactly the solution of that We want to look at corrections or small fluctuations on top of that small perturbations on top of that that are sourced by a small test contribution a small test source That's the situation we want to explore And so we can do perturbations On top of our solutions and this when you do perturbations It's always best to do it at the level of Lagrangian or the action because he really first of all it allows you to determine stability Things like ghosts. It's very it's much harder to find this criteria if you just focus on the equation of motion because You can multiply the equation version by minus. You wouldn't see the difference was at the level of the action It's much more clear But also the level of skills is much clearer to understand what's going on if you're working at the level of the action so we have D4x Let me just rewrite for a second the configuration. We're looking at So we're looking at a cubic Galileo theory Again, just for simplicity. This is ruled out by observations, but let me just For simplicity give you this essence of the manstein mechanism. And so T. Let me say this is T bar T will have a contribution from t bar and delta t and then phi has a contribution from phi bar and delta phi So I do fluctuations on top of that and I'm using the fact that the background Satisfies the equation of motion for the source t bars and I'm not gonna follow The terms that are just linear in delta phi because this would be zero from my equations of motion And so what we have by performing fluctuations is something that goes like that In color And I tell you in a second what this effective inverse metric is and we have the interaction Then we wrote down and then we have the coupling to our perturbed source So I ain't all you what Z was it looks like any effective inverse metric on which the fluctuations are living in But clearly this Z Should depend on the background So if we didn't have this interaction here This Z would just be time you knew right at this level. We have ignored gravity We just focus on this part of gravity if you want to think of it like that So if we didn't have that term this Z menu would just be a time you knew right this is a time you knew D me phi D me phi Delta Phi Delta Phi Well, actually, this is D me fighting if I but now from this part in here We clearly gonna have something that goes like box five background It time you knew for the fluctuations and then there's also a case where this is part of the fluctuation one of the five He is part of the fluctuations and then the other one is part of the background and to express it like that I need to do an integration by parts and so The derivative of the one that came for the fluctuation will need to act back on to the one for the background So we'll have two degrees acting on the background so putting all of that into account what we end up having is Something like that Which will be the end of the story if we didn't have the cubic Galileo interactions But then we have correction that have to come from lambda cube and I said that they have to be secondary it is acting on the background and They happen to be box five background minus, sorry it's a new minus D mu D mu phi background It may not be entomatically obvious that this is what it is but It has to be the case that this quantity here has to be Yeah Has to be there has to be that so you can derive that explicitly There is is probably is to put that back into an epsilon structure and you will see that it is exactly exactly that And I'm gonna spend too much time doing that. So this is the effective metric that the fluctuations are living in but now what we've seen is that if you had very large distances then You are dominated the you're in the weak field approximation and then you're for you'll be dominated by the standard Ita menu and we'll see Who's turned it cubic Galileo? Very far away and that makes sense because if you're very far away from your background source Then you shouldn't be sensitive to what what was going on at the origin, right? If you are then doing something wrong very far away You should be quite insensitive of that But as you go closer and closer to your original background source in here Then you start becoming more and more sensitive to that and these terms are the one that dominates And as I go closer and closer we said we we've seen that it means I Raised it but he meant that five prime of a r was much larger than lambda cube So then this thing here says becoming much larger than one as R is much larger than r star You have Z menu is simply Ita menu But as are you start entering the strong coupling radius of the vanishing radius then Z menu is Dominated by the interactions and something that goes like two derivatives acting on five background of a lambda cube the two derivatives Acting on five background of a lambda cube This is roughly speaking one of the r five prime of a lambda cube Roughly speaking and So this is roughly You can put back your solution from here You'll have something like there's like one of our to the three half and then the rest is to the three half and of an plank to the half so Z The eigenvalues of Z roughly speaking are much larger than one if Lambda is very small some scale Blank but don't really mean in plank So the smallest the strong coupling scale is the more efficient the national radio or national mechanism will be Or if the mass is very large And so if this starting to become quite different from all the one Then what we really need to do is kind of can only clean normalize our field so to do that explicitly when This Z menu is not diagonal is a It's entirely doable, but it's not something I want to spend an hour on so let's just you can bargain on eyes You can change your coordinate. I didn't analyze your effective metric and then just make it Absorb the eigenvalues in the scalar field fine So but I just gonna do it symbolically You know define a new canonically normalized color field with roughly speaking these girls like The square root of Z where what we mean here The eigenvalues the typical eigenvalues that come in into this effective metric Z delta fine Work. Let's let's just imagine that we in a scenario where Z menu Is roughly speaking going like see it. I mean, that's not quite the case, but let's imagine just for To understand what's going on Let's imagine we could do that Really what we should put there is the eigenvalues, but doesn't matter too too much when we do that And we plug that back into our action what we get a for the normalized Scala field is something that is now normalized just like what we wanted And then this is the essence of the van Schnoen mechanism that I mentioned yesterday What we get for the interactions is that What they see on this On this background is no longer a scale Z along a scale lambda, but I redress scale Lambda square root Z and then the coupling to matter is no longer at the plank scale But actually at a much larger scale Carried by and plank times square root Z. Well, we end up with a new Strong coupling scale. This is the redressed scale lambda star, which is of the order of lambda square root Z and a new Coupling to scale coupling the matter scale Or the coupling to matter. Let me just say the coupling to matter is suppressed by one of a square root Z as compared to draw in Situation I considered before if you just consider the mass of the The earth and then if you consider the scale lambda as I mentioned before to be a thousand kilometers minus one so 10 to the minus 13 Electron volt You put these numbers back in What you get is Z of the order of 10 to the 16. So it's huge In that regime When when you close to the When you're on the surface of the earth, that's on the surface of the earth. That's huge and the associated lambda star is Of roughly of the order of a centimeter. So you manage to Increase your strong coupling scale, which is a good thing And when you start doing perturbations on top of that background You don't need to worry about these interactions Until you get too much shorter distances and that's better because you want to be able to do things at the linear level For as long as possible once you have accounted for this range time screening And so you see that the coupling to matter of these perturbations on top of the diagram is very suppressed now as compared to what you have For the standard Newton's qualia. So that's how you can avoid Forces experiments 15 minutes. Is that I can't hear Yes, these are perturbations, yeah now this is This is I can say this is the this is the leading term This is what would be the linear order at the level of the equation motion And it's at the quadratic order at the level of the equation of motion. So this corresponds to linear perturbations if you want, yeah So it's okay to be working with linear perturbations Okay, so this is the essence of the van Steen mechanism We looked at different type of screening mechanism between the cameleon one which leads to the mass Being environment-dependent and being so the field becoming massive and effectively Suppressing the force it can mediate in dense environments to this function mechanism to also weak coupling Someone asked a very interesting question yesterday on whether we can put all of this well The question was much better formulated than that But let me just say what can we try to combine these different mechanism with one another so that they all come in and Join forces, I mean that sense that's that sounds good And it's interesting because this van Steen mechanism is actually working in the other way as The other two the weak coupling to gravity I'm sorry the weak coupling to matter or the the cameleon mechanism in the sense that you actually need This contribution from the mass to be very large To redress the background to have the fluctuations on top of them being very small so if you start making this very small for the whole Perviant formulation, then you're not gonna have a van Steen mechanism and This van Steen mechanism we see that any of the interactions It makes it as suppressed as compared to Normal kinetic term in particular if we had had a mass If I had added a mass here for the scalar field if I squared And then we see that when I canonically normalize it I'll have a Z coming in here And so it suppresses the effective mass of the scalar field as well So it's working in the absolutely opposite direction as a cameleon mechanism So it's actually not something that you can do very easily of having both a cameleon mechanism and a van Steen mechanism at the same time Okay, okay, so I have 15 minutes to tell you all about modified gravity Which are clearly Not gonna do so Let me just give you a few motivations, okay So first of all when we're talking about modified gravity here. This is not None it's modified gravity for dark energy and It's a modification of gravity at very large cosmological distances to help us with the cosmological constant problem Possibly or with dark energy as a source for dark energy and this all this realm of possibilities where you could have self Acceleration where the acceleration of the universe is not coming from an external that fluid But the degrees of freedom leading to an acceleration of the universe are intrinsic to the gravity on itself That's the idea behind Self-acceleration gravity just wants to accelerate by itself for a reason or another possibly From the fact that it has so many city zero modes that want to condensate itself and lead to the acceleration of the universe if we considering modifications Modifications of gravity At large distances so in infrared So we're not yet to consider the inclusions of operators which are infrared operators as compared to the standard one The Sun and Einstein Hilbert one so it this Direction has absolutely nothing to do with trying to have renormalized theory of gravity of UV completing gravity Including operators that come in at the plank scale. It's the opposite sign of the spectrum. You really trying to see what's going on Cosmological scale so it's infrared corrections So you want to include very naively you want to include new operators in your action for gravity Which are gonna be the ones relevant at very low energy So you would want to include something which has fewer derivatives than the Einstein Hilbert term So of course you can include a cosmological constant But that's not really what we mean by modified gravity and then that's it Covariantly, this this is it. You can't really include any Anything else if you really do want to modify gravity you could say well I'm gonna include functions of that the scalar curvature and we talked about that and that's That's really gravity in a scalar field and it's not really modifying gravity in it right in this sense So it's been shown in some very nice papers that Any effective description of gravity in infrared Will ultimately need to effectively look like the graviton has a mass at large distances It can be an effective mass. It can be a resonance. It can be a local or not local mass But effectively needs to be an operator that looks like a mass and this is I mean Let me just think of a Scala field That's trivial if you think of a scalar field right if you want to have something local for a scalar field Let me see how this kind of field Being sourced like that if I want to modify the behavior of this kind of field in infrared The only thing I can do is add a mass here okay So if we want to add a modify the behavior of gravity in infrared effectively We will need to include something that looks like a master Yeah, this is not local So if I want to have something local I need to consider a mass, but indeed and what you're talking about you can put them Into an effective mass and then it's a non-local effective mass Yeah, and you can consider them as being part of a resonance So you could have indeed something that will look like fewer number of derivatives by taming the number of derivatives as a log Like you said precisely and effectively that would look like a mass At no local mass, but a master nonetheless So this can be function of Box in such a way that this is the one that dominates when you send the derivatives to zero at very low momentum indeed, so It doesn't this doesn't need to be necessary. I can't stand with some type of of master now, you know let me say Two statements and may seem disconnected. So if you have a GR which is this interacting theory of a massless spin to fill fundamental object gravitation waves which have two Polarizations we have two degrees of freedom which are going to call them plus or minus two in 40 working 40 if you want an effective Local or not resonance or not mass Massive spin to field then in 4d. It will have five excitations I spin s in 4d If it's massive to us plus one Excitation so degrees of freedom So in 4d for spin to that corresponds to five degrees of freedom and those corresponds to the same two as we had for GR for the massless case and then you have plus or minus one excitations and then you have Helicity zero excitation These are helicity. I can see think of them as helicity modes, of course and in the massive case But let me just think of them as helicity modes their helicity modes in Special limit of the theory when you kind of decouple the behavior of the non-unit is of gravity you recover some notion of Lorentz in vines and you can you can think of these modes as Scala vector tensor in some particular limit of the theory and so this additional mode for cosmological Purposes will look like a scalar mode Yes Okay, okay, okay Okay, but not that's good. Okay. So this will behave as a Galilean and I'm sorry In some limit this behaves as a Galilean and these behaves as vector fields in some limits And they don't really couple to matter. Let me just say in half a second, and then I'll show you a slide the intuition of why these behaves as a Galilean and so why this whole story about the Galilean That I talked about may be relevant for models of modified gravity So if you want to add a master here, it's not gonna be a cosmological constant it's gonna Effectively break for the different variants. So if I if I the most if I have a scalar field What I mean by us and the amass for the scalar field is adding a phi squared In at the level of my Lagrangian for the for gravity for sorry for a spin to field I would like to add something which goes like the matrix square, but this Anything which is covered is what this would just give me a number, right? Or a cosmological constant. So what we need to do is really consider Just been to feel them more Back to the roots. We're gonna consider Flux iterations around flat space time and what we mean by adding a master is really a master for these fluctuations around flat space time So the master is Always going roughly like H squared doesn't matter too much what the form of this H squared is so you have each square You have each menu squared or something like that But what does matter is that this thing here is actually? The square of the difference between your dynamical metric and your Minkowski metric So it's key menu minus it a menu Squat and by that you want to put at the level of your action and this is Coordinate transformation this transforms as a tensor and the coordinate transformations trivial as so but this Does not transform properly these are not transformed at all under coordinate transformations And so if you just wrote it like that this would not be coordinate transformation environment So in writing the master kind of in this form, you're really working in a preferred free And so you can restore your coordinate transformation environment by promoting this object to a tensor You're gonna promote it to a tensor By writing it as No flat Minkowski metric and then introducing for what is what I call Stuckeberg Skeller fields so fine a's For Stuckeberg Gallifields they transforms a Skeller field and a coordinate transformation in and So that this whole thing is eta tilde menu transform as a tensor And so if I put a tilde in here, hey, this transforms at the tensor and I have restored my coordinate transformation environments now I Can't go back to the preferred frame that we had before By setting simply this color field to be x a this is what is called unitary gauge but for a second it may be useful To split it to keep the excitations that live into phi a Write them like that as something that would look like a vector field in some limit and something that would look like a Skeller field in some limits, okay, let me just write it like that and So the way they come in into it a menu. Let me set that to zero I Set it By hand for now just to focus on the point. It's not that it's error. It's just it's too complicated for me So then it a tilde menu in that case is it a menu plus to B mu D mu phi plus B mu D mu phi squared So now we see that built in into this master into this Like range and for your modified theory of gravity. No matter precisely what it is. No matter whether this is Local or not local a real hard mass or a soft mass a resonance. It doesn't matter too much This whole story will go through and we see that for that Field that behaves as a scalar field in some limit the Galileo and shift symmetry is built in So this color field phi the way it enters phi satisfy or the the horse theory satisfies shift and Galileo symmetry and so If you want your theory to be ghost-free not to have been playing by the higher derivative terms It has to be the case that the Galileo type of interactions that are wrote down yesterday and we use today Have to emerge from your theories of modified gravity in infrared no matter what you do. So this is the connection between Galileo's and Modified gravity. Yes Yeah, yeah, yeah, okay, okay, okay, can I Let me just show you So I'll post that online It's just to give you some motivation on having a look at it I wrote down. Can you see? It's a bit Not so very bright. So you probably can't see and I Showed for you a diagram of how these different things Matching together between modifications of gravity to Galileo's to Hadeski theories to beyond Hadeski theories to beyond beyond Hadeski theories to generalize poker theories to generalize non-abillions to Multigravity etc. How all of these things fitting together and how the mansion mechanism works for most of these different things So even if you're working for a generalized poker theory, which is really a vector field It will have and dark energy would really be somehow linked to the helly city's error mode that comes into this theory And you'll have a function mechanism that protects you hopefully within some Some regions please some dance regions and then there's been some recent Detection of gravitational waves that allows you to Determine the speed of gravitational waves as compared to that of light and that has ruled out many of this model ask Palo For more details for that. So it's it's quite kind of nice So in this whole spectrum math from correct observations You have many of these models here that are ruled out. They're no longer viable candidates for dark energy So it's not only that we're populating our whole spectrum with new ideas without any ways to probe them There's it's quite nice how we've been able some people have been able to Discriminate between them and then there's a whole way to observe them, which I don't have time to Okay, thank you