Okay, good morning. So today, in my last lecture, we're finally gonna get to alternatives. So I'm gonna discuss what I think is at the moment the best motivated alternative to inflation.  And there are many people who have worked on this, so I'll just list a few papers. There's a paper by Krabs, her dog, and Turok, 0712. And then there's a series of very nice papers by Rubakov, starting in 2009. Then there's a paper on Galilean Genesis, and the culprits are in the audience, a subset thereof. And our contribution to this was to simply point out that in fact, the Genesis scenario and Rubakov's scenario are in fact really part of the same, are just different realization of the same idea. And so in a paper with Kurt, if you want pointed out the similarities of the approaches. Okay, so in a nutshell, what is the idea? So let me just write some words to give you the basic outline of the idea. So of course, being an alternative to inflation, there is no inflation in this scenario. And contrary to inflation, where of course inflation relies on a rapid expansion of space-time, so the dynamics of space-time is very important in generating scale invariant perturbations. Here on the other hand, gravity is completely unimportant. And we will see that in fact, the dynamics, the space-time, instead of being approximately the sitter, which is the story for inflation, here it's approximately Minkowski. The main reason why I view this as probably the best alternative at the moment is that it relies on symmetries. Specifically, it relies on approximate conformal invariance in the early universe. And this is conformal invariance on Minkowski space-time. And so as a result in three plus one dimensions, the conformal, the group is the conformal group with algebra SO4 comma two. And the reason we obtain scale invariant perturbations is that this conformal group is spontaneously broken. As we will see, some fields pick up a time-dependent profile in such a way that the conformal symmetries are spontaneously broken. And they're spontaneously broken, as we will see as follows, that the SO4 two conformal group is spontaneously broken down to SO4 comma one. Okay. And based on our discussion of the first lecture, we already know that SO4 comma one are the symmetries of the sitter. They're the symmetries of spectator fields and inflation. So already our fluctuations will have the same properties as spectator fields and inflation, despite the fact that there's no the sitter. But moreover, they will be constrained by the symmetries that are spontaneously broken, the five symmetries that are spontaneously broken. Okay, so we'll see there are some very nice relations that we get at the end of the day. This scenario, I wanna emphasize, solves the flatness and horizon problem in a way akin to inflation, except with rather different dynamics. So once you turn on gravity, it's a mild evolution, but nevertheless it drives the universe to be flat and homogeneous. That's the good news. The bad news compared to inflation, let's say, is that you need a bounce in the scenario or and or, right? A violation of the null energy condition, okay? Which is, in many ways, the most robust of all the energy conditions in general relativity. Okay, so that's the price to pay. So that's the outline. So before we get to generalities, I wanna start with a specific example just to fix our ideas. So the simplest examples, apologies to Paulo and Alberto. The simplest example is not Genesis. It's a five to the fourth. So let's just consider a scalar field in Minkowski space. So again, for the moment, we're completely ignoring gravity. So a scalar field in Minkowski space with a negative quartic potential, okay? So of this type, but where lambda is negative. So the potential looks like this. Now, of course, the first pathology of this potential is that it's unbounded from below. But as we will see, we imagine the genesis, the phase that we're gonna be interested in doesn't rely on phi going all the way to minus infinity. At some point, the dynamics will describe, we'll, how to say, we'll break down. So it's natural to assume that this potential, we have to assume that it's regulated somehow. And we imagine that this is coming from corrections of order which are suppressed by powers of M-plunk. So for example, six order over M-plunk squared. And indeed, we will see that the dynamics we're interested in precisely breaks down at a point. It breaks down just in the sense that our approximations will break down, that nothing really singular is going on. But precisely at a place where one over M-plunk corrections are important. So it's just an assumption that things are well behaved past this point. Now, a nice curious feature is that a negative quartic potential is actually asymptotically free. Workout of the beta function is negative. So it's an asymptotically free theory. And moreover, at least classically, it is conformal. So it's classically conformal. Of course, quantum mechanically, there will be corrections that break conformal invariance, the famous common Weinberg corrections, those will be kept small at the end of the day. So let's just think of it as a classically conformal field theory. In other words, this theory is invariant under the following transformations. So this is the conformal group on full spacetime. So it has, of course, the four spacetime translations. These act on phi just simply with a gradient. As usual, you have the boosts, the six boosts and rotations. So these are the Poincaré transformation plus supplemented by dilation, spacetime dilation. Sorry, this is a weight one field. Special conformal transformations on spacetime, which act on phi as follows. So of course, you recognize these. They're the same as the transformations we wrote down for the sitter, but now it's a conformal group on spacetime. So the indices are Lorentz indices, in this case. And to see that this is really the conformal algebra, in disguise, we can do an automorphism of the algebra. We can take linear combinations of these generators as follows. So let's define delta J five mu to be the linear combination of delta P mu plus delta K mu. Let us take delta, let's define J six mu to be the difference of these two generators. And finally, I'll define J five six to be simply dilation. And you see, when you do this, that these 15 symmetries, they package themselves nicely into the following statement that the commutator of JAB JCD is eta AC, okay? Where this eta is a six dimensional metric, Minkowski metric with two times. So eta AB is by definition eta mu nu one minus one. So it's a metric with two times. And this, of course, is just a good old rotation group. So in this case, because of the signature, it describes SO four comma two, okay? All right. So now we have this theory, it has these symmetries, and we shall consider a particular solution. At first it would look, it's gonna look like a fine tune solutions, but then we were to argue that in fact, it's a dynamical attractor. So we're gonna consider on this negative quartic potential a homogeneous solution. So similar to what we do in inflation. So we're gonna assume a homogeneous background phi to be phi bar of T, okay? Now, since I don't have gravity, this theory is time translation invariant. So there's a conserved energy associated with it. So it's just the classical mechanics of a point particle rolling down this 1D potential. So there's a conserved energy, a half phi dot squared plus lambda over four phi to the fourth as the first integral motion that I get from the equation of motion. And so I can solve it easily for, I can solve this differential equation. We will be interested specifically in the solution that has zero energy. The zero energy solution, and I'm gonna argue in fact that this is an attractor. Okay, it looks like very specific solution, but it's an attractor at the end of the day. And so the solution in this case, if I set E equal to zero, I can solve for phi dot in terms of phi integrate, and the answer, so I should have put bars here, the answer for phi as a function of time is square root of two over minus lambda times one over minus T, where time here I'm taking to run from minus infinity to zero. So physically what this describes is a particle which in the infinite past started out at the top of the hill at T equals minus infinity and rolls down subsequently, okay? Again, it looks like a fine-tuned solution because it's precisely the initial conditions such that I started at the top asymptotically in the past, but we will see that this is in fact an attractor. For the moment, let's just take it as it is and study what this implies. Yeah, so somehow that's right. So for negative lambda, a bit of function is positive, so you're driven to zero lambda. It's not relevant for what I'm talking about, it's just the curiosity. No, I'm not using the asymptotic freedom in any way. It's a comment, a side comment. It's classically conformal because there is no mass scale in this problem. Yeah, ah, good, that's right. So this does not have a stable vacuum, obviously, but for our purposes, this will not be absolutely, so you're correct. For our purpose, this will not be so important. At the end of the day, we're only gonna be interested in the dynamics on the spontaneously broken solution. Where conformal invariance is spontaneously broken. That's right. But what your comment is very relevant, in particular, because to get scale invariant perturbations, we will need a field which has weight zero, which would break unitarity explicitly. However, this weight zero field will turn out to be, if you want, in the simplest version, just an angular field. Okay, and so in fact, it doesn't break unitarity because this weight zero field ceases to exist at the would be conformal point. In fact, yeah, if you want, we never work around the point where conformal invariance is restored. We're always on the spontaneously broken solution. Okay, very good. So now let's indeed study the symmetries that this preserves. So first of all, it's clear that being just a time-dependent background, the symmetries that are preserved includes all the spatial stuff. So the symmetries that are preserved by this background are all the spatial symmetries, so clearly spatial translations, spatial rotations. The spatial part of the special conformal transformation, less obvious, although somewhat obvious, is that this also preserves dilation. Because phi is a weight one field and if I rescale time by a constant, then indeed phi shifts in the appropriate, rescales in the appropriate way. So in fact, another symmetry that's preserved via this, by this background is space-time dilation. So how many symmetries are preserved? There are 10. So three P's, three K's, three J's, one D. So there are 10 preserved symmetries and therefore five broken symmetries. So what is broken? What is broken is the rest. So clearly this is a time-dependent background. So what's broken are time translations, boosts and the time part of the special conformal transformations, or these are the five symmetries that are spontaneously broken. So now let's study a little bit more closely what these preserved symmetries are. So again, let us take linear combinations of these guys. So let us take the following linear combinations. So delta J, five I, which is a half, delta PI. So very similar to what we did over there. So delta PI plus delta K I, then I'll define delta J, six I. No, some of this is, I think this is wrong. You should be four and five, I think. Well, let's leave it as five and six for the moment. Okay. I don't have a four index, basically. Delta PI minus delta K I. And finally, delta J, five, six will be simply the dilation. Okay, so I'm just taking linear combinations of these unbroken guys, and you can check quite nicely that these, of course, obey a very similar algebra to what I wrote down earlier. So JAB, JCD is basically that same commutation relation as earlier plus dot, dot, dot, dot. So basically exactly what I wrote down over there, where now the metric eta that appears here where eta AB is delta IJ one minus one. So it's a five-dimensional Minkowski metric. And so this algebra, in fact, it's just the rotation group, but with signature minus one plus one plus one plus one. And so therefore, it is the SO four comma one algebra. Okay, so as advocated, this particular time-dependent background spontaneously break the original SO four comma two down to SO four comma one. So these are the same, the unbroken symmetries are the same as the isometries of the sitter space. And so in particular, just based on what we discussed the very first lecture, this is the conformal group on R three. So we expect fields whose weight dimension, whose mass dimension delta is much less than one. We expect those fields to have nearly scaling variance spectrum, scaling variance. So indeed, let us see that this expectation is borne out. Sorry, can you say again? The dilation is preserved. Sorry, where did I write nothing here? Sorry, guys, it's early in the morning. Oh no, it's sorry, thank you. Okay, thanks, yes, absolutely. Yeah, because in my notes, five was outside, yes. Thanks, yes, thank you. Yeah, it's just a weight dimension one, thank you. Very good. Yeah, because the theory clearly, the quartic linearly realized dilation. Okay, so indeed as an example of such field, this is related to the question that was asked a few minutes ago, let us consider as a particular example, let's consider a spectator field, which I'll call theta, whose weight dimension, whose mass dimension will be zero. Now, a general action at the quadratic level that I can write down for this guy, at least one term that will appear, let's just focus on that guy. We'll have the following operators. I wanna give this guy a kinetic term, d theta squared. But since theta has mass dimension zero, this has mass dimension two only, so I need two dimensions of mass out front. I cannot put an explicit mass scale because it would break in following variance. So the only thing I can do really is to multiply this by five squared. Let's put the minus a half here, okay. There are other things I can write down at the quadratic level, but for the kinetic term, this is what I can write down, okay. So in other words, by conformal invariance, and the fact that this guy has weight zero, manifestly, it must be multiplied by five squared. This also goes towards your point, obviously, because of course, in the conformal field theory, there are unitary bounds on the weight dimension, but you see that in the point where conformal invariance is restored, is five equals zero and this field ceases to make sense. So let us see, so now, let us think about what happens to this field. The background solution for this guy is zero, okay, so it doesn't do anything. So if I consider now quadratic fluctuations for this guy, you see that at the quadratic order, what I should do is simply put the five field on the background, theta will be fluctuations, and now you see that this field exactly behaves as if it were coupled to an effective metric, it's as if it's coupled to an effective metric, G effective mu nu, which is five bar squared of T, okay, it's identical, as far as it's concerned, it's as if it's coupling to a scale factor, which is five, it doesn't tell the difference, cannot tell the difference, and in particular, if you now plug in our solution for five, this is equal to two over minus lambda, which is a positive thing, T squared E mu nu, and this you recognize as the conformal slicing of the sitter, so this field behaves exactly as amassed this field on the sitter, this is the sitter metric, and so as a result, the two point function for theta will be scale invariant, okay? All right, now let's address the fact that the solution we considered was rather special, let's consider the stability of the solution, yeah, so you see, it's very nice, in this scenario, the background metric is doing nothing, it's remaining Minkowski, but scaling invariant perturbations are generated because there's a scalar field which evolves in time, and it acts effectively as a scale factor pumping energy into other fields in the theory. Okay, so for stability, let's consider perturbations around our time-dependent background, five bar of T plus pi, let's call pi the fluctuations, and it's a trivial exercise, you just perturb the action around our background, compute the equation of motion, and you find in the privacy of your room tonight that in fact this is the equation of motion for pi, and the fact that it has, the fact that you have this quartic potential, right, this is the quartic potential, pi to the fourth, and so at quadratic order, I get pi squared on the background times pi squared, and this actually gives a tachyonic master, not surprisingly, because the potential is points downwards with the following, with the following term, so minus six over t squared, and now in the limit, let's consider the limit of late time or equivalently when, so either when k goes to zero or equivalently when t goes to zero, so at late times we wanna figure out what is the growing mode solution for this perturbed equation, and it's easy, there are two solutions, one is pi going as one over t squared, and the other is pi going as t cubed, but remember t is going to zero, so in fact the growing mode is this one, it's one over t squared, so this is the growing mode solution, and at first sight this is alarming, because our background goes as one over t, the perturbation goes as one over t squared, so the perturbation seems to grow faster than the background, which seems to indicate an instability, but in fact that's not so, if you think about it for a moment, you realize that if I take the background solution phi, which we have over here, and imagine that I expanded it, I shift the time by a tiny amount epsilon, okay? You see that I can, let's Taylor expand for small epsilon, this will give me phi bar of t plus epsilon times phi bar dot of t, and phi bar dot will precisely go as one over t squared, okay? So if you want, this term goes as one over t, that's our background, this guy would go as one over t squared, that's the growing mode fluctuations, so said differently, all that this is saying is that the perturbation, the growing mode of our fluctuation resumms to simply a time shift of the background, so in this sense it's an attractor. If you perturb it, you wait long enough, and you find you get the same background solution just time shifted by some amount. So the solution is an attractor in this sense. Are there questions so far? Yeah, sorry, can you speak louder? I can't hear. Yeah, yeah, that's right, so at the end of the, that's right, so let's compute that in a little while. We'll see that the pi pi correlation function is actually very red, but I wrote you the fact that it has this tachionic mass term. So we'll come back to that, that's an important issue in our story. Sorry, I can't hear, can you scream? Yes, so I just wanna argue that, so why introduce this delta, this theta field, yeah? So this theta field is just a field which has, I'm presuming, it's just a field in a theory, I'm presuming it's existence, which has weight dimension zero, that's it. And then I argue that it has, in particular, it can have a kinetic term of this form, it must couple to pi, and as a result must have a scaling variance spectrum at the end of the day. So the point is that I cannot write a simple kinetic term for this guy, it must, by conforming variance, this kinetic term must be multiplied by pi squared. And so I guess I should have said this, but obviously in this interpretation, this is just a U1, it's to think of this what inverse quartic potential with a U1 symmetry and theta is just the angular variable for this potential. And that was Rubik's model, you just have a theory with U1 global invariance, and that's it. That was another question, I think, there. Shift symmetry, yes, exactly, that's right. So I could write down, I could also allow operators that would break the shift symmetry, so I could have also pi squared, M where M is a dimensionless, hold on a second, how five to the fourth? Right, where this guy, let's call it kappa, he's at a dimensionless number, so all that we have to assume in order for this to be a scaling variant is that this kappa is sufficiently small, this will induce some tilt to the spectrum. But indeed, one model you can consider is just when it's a U1 invariant potential, in which case this operator is absent. Okay, where was I? I'm here. Okay, so now we've seen this nice explicit example, and we can take a step back and describe the scenario in more general terms. This is what we did in the paper with Kurt that I mentioned earlier, so let's abstract ourselves from this particular example. So the general scenario, and I would say this is why I think this scenario is kind of interesting, is that it's quite generic in some sense, so our postulate is that let's suppose that in the early universe you have an approximate conformal field theory, and such that the dynamics of the CFD, such that there will be primary operators in the theory with weight delta I, so there could be more than one, such that these guys acquire some time-dependent VEV, so some time-dependent profiles, and the particular profile we're interested in is one in which these operators O sub I of T will be precisely proportional to one over T to the weight dimension of the corresponding operator. So our scalar field was just a particular example of this where delta was one, but in general you can assume that first of all, there are many such operators, and as long as all of these guys have time-dependent profiles which go as T to the particular weight, then if at least one of these fields, if at least one of the deltas is non-zero, okay so that really there's time-dependence in the story, then you can convince yourself rather easily that you get the same symmetry-breaking pattern, SO4 comma 2 goes to SO4 comma 1, and now you're done, you can just argue, first of all, based on just SO4 comma 1 invariance, like we did in the very first lecture, you can argue that the correlation functions must take on particular forms, so just based on SO4 comma 1 alone, so just based on SO4 comma 1, you can now deduce that the correlation functions of let's say perturbations of these O's, X1, X2, let's just take the two-point function, just like we did in the first lecture, this object must be, must scale as some coefficient divided by the distance X1 minus X2 to the delta I plus delta J, if the deltas are the same, these are just the rules of conformal field theory on R3, so this is if the deltas are the same, and it's zero otherwise. Similarly, the three-point function is also completely fixed up to a normalization, so the three-point function, three of these fields, Cijk, X1 minus X2, so this is just a usual combination, X1 minus X3 delta I plus delta K, minus delta J, X2 minus X3 delta J plus delta K minus delta I, and so on and so forth. Yes, I don't think so. No, so somehow at equal time, well, no, sorry, at unequal time, let me think about it for a moment. Yeah, I'm tempted to say at unequal times the correlation function, so this SO4 1 must be the same as inflation, so you could probably say it must be functions of the desitter invariant distance between the points. Yes, because as far as these fields are concerned, it's like they're living on a desitter background. So this, if you want, is the same story as what we discussed in the first lecture with respect to spectator fields and inflation, but now here we have more symmetries, we have the full SO4 comma two, five of those symmetries are spontaneously broken, and so associated with the five broken symmetries, time translations, the boosts, and the time component of the special conformal transformation, there will be a soft pi on theorems, and we'll come back to that. Now, one thing that's nice about the scenario is that there are many different realizations, so there are many explicit realizations that people have thought about, so one of them is the spot of the fourth example we talked about by Rubakov, then there's Galilean genesis of Paolo and Alberto, there are DBI generalizations of both these guys, okay? So, and presumably so on and so forth. In fact, you can possibly even contemplate generalizations of the scenario. So here we were interested in SO4 comma two down to SO4 comma one, but really from a group theoretic point of view, all we've used is that the fact that we have SO4 comma one as a subgroup as well as Minkowski space as another subgroup, okay, because that's the symmetry of the background if you want, and one can argue that SO4 comma two is the smallest group that contains a subgroup to bore SO4 comma one and SO3 one, but in principle you can consider larger groups, okay? You could imagine some larger group on the left-hand side giving rise to SO4 comma one and Minkowski as subgroups. This is not by the way of violation of common mandula because in fact the symmetries we're considering are nonlinearly realized. So in principle there's a lot more freedom. Okay, let me say a word about cosmology. So far we've just been working in a static Minkowski background. So what happens when you turn on gravity? So there are many ways you can couple to gravity. The minimal way is to simply couple minimally the CFT to gravity, which will of course break explicitly the conformal invariance of the theory. So we're gonna assume minimal coupling. So in other words we envision the action simply being given by Einstein-Hilbert gravity, couple to our CFT in this way. And like I said, the Einstein-Hilbert term because it introduces M plot, this explicitly breaks conformal invariance, but only very mildly. So it breaks conformal invariance mildly at the one over M Planck level. And indeed we'll see that our results will be defined as a perturbative expansion in one over M Planck with a zero-thorter result being what we describe up till now. So it's very nice that the cosmology, even the cosmology can be described in a model-independent way, just based on the symmetries of the problem. So for any model that we would choose to consider. So first of all, at zero-thorter in M Planck, so zero-thorter in one over M Planck, so this guy to the zero, at this order we can write down immediately what the density of the CFT, the energy density and the pressure will be given by, okay? Now, by symmetry both these guys must be purely time dependent. There cannot be explicit space dependence by homogeneity and isotropy. So there can only be functions of time. Moreover, by dilation invariance, they can only scale as their mass dimension, which is four. So they must go as one over T to the fourth. So both of these quantities, both rho and P must scale as one over T to the fourth, just based on dilation invariance. But moreover, at zero-thorter in M Planck, the energy density must be strictly conserved at this order in one over M Planck. So in fact, alpha must be zero. Does that make sense? Because energy density must be conserved in the absence of gravity. So this guy is zero, okay? At zero-thorter in one over M Planck. So we fully fixed the form of rho and P up to an arbitrary coefficient and all the model dependence of the cosmology will sit in this particular coefficient. In particular, for the fight of the fourth example, beta is related to lambda in a simple way, in simple way. Very nice. So now we can go ahead and solve Einstein's equations in the presence of these sources. In particular, we can integrate the H dot equation to find out the expansion rate. So now we're moving away from the order one over M Planck. So to be precise, we're now moving to the one over M Planck squared order to discover what the cosmology is doing. So let's integrate the H dot equation, the H dot equation which reads one over two M Planck squared rho plus P. The right-hand side being down by one over M Planck squared, I can set the rho and P to be their zero-thorter value. So this guy's zero and P goes as one over T to the fourth. So this is minus beta over two M Planck squared T to the fourth and so I can integrate and I find that the Hubble parameter as a function of time is equal to beta over six and Planck squared T cubed. So the Hubble parameter in magnitude grows in time since T is going to zero. And now you'll notice that the sine of H is directly related to the sine of beta. So T is negative, okay, so T is negative. So this is the denominator is negative and now depending on the sine of beta, you either get an expanding universe or a contracting universe. So specifically for beta positive, the universe is contracting and for negative beta it's expanding. And of course this is related to the non-energy condition. The five to the fourth example has a negative beta. You can show it's related to lambda directly, so it's negative and so therefore you're contracting. Sorry, I said this completely the other way around. So beta is positive for the negative quartic potential, so you're contracting. In the Genesis scenario you can consider coefficients in such a way that you're in the expanding branch. The key point is that the evolution is very slow. If I integrate once more to get the scale factor as a function of time, you find the following. You just integrate this order by order in one over M Planck, you find that the scale factor, the constant bit can be set to one without loss of generality and then there's a time dependent correction which is of the form beta over 12 M Planck squared T squared. Plus high order corrections, dot, dot, dot. And so indeed for T large and negative, this is a small correction to Minkowski space. So we have a controlled expansion away from Minkowski, sorry, a controlled perturbative expansion away from Minkowski space suppressed by one over M Planck squared.  So defining the equation of state as being proportional to minus H dot over H squared. I didn't bother to work out the coefficients, but anyways, minus H dot over H squared, the point is that this scales as T squared and Planck squared over beta, okay? So this is the key point. If beta is positive, namely you're contracting, so this is a contracting branch, then W, the equation of state for the CFT is much bigger than one at early times. It diverges in the infinite past. And if beta is negative, corresponding to the expanding solution, then instead W is much less than minus one. Does that make sense so far? Okay, so this is the cosmology. You're either W is much bigger than one or much less than minus one. These are the two branches. Yeah, so okay, perfect. So that equation turns out is obeyed. So now, yes, exactly. But you see that at this order, very good. It better be obeyed because we know that the Friedman equation, the H dot equation, and the energy conservation equation, they're redundant. There's one that's redundant. So if you want Friedman equation as the first integral of motion of the other two, but you can see that explicitly. So if I tried to, at this step, write down immediately the Friedman equation, which would be rho over M Planck squared, I find nothing at this order. Because rho was zero at zero third and M Planck. So the right hand side is zero to order one over M Planck squared. Similarly, the left hand side, based on what we found, is also zero because it goes as one over M Planck to the fourth. So to really fish out the appropriate correction, what you have to do is you have to correctly find the leading correction to rho, okay? And you can do that by solving the conservation equation. Plugging what H is, solve for rho, and so on and so forth. You find the correction to rho and you can check order by order the Friedman equation is satisfied. Now I want to argue that this, such a background, so remember, let's think about the logic of what we said. So I gave a specific example at the beginning of this fight of the fourth. I showed, we started with a particular time-dependent solution. We showed that this was stable under perturbation, though the scalar field itself, but now you wonder, now that we've introduced gravity, does gravity mess everything up? Is the universe gonna become inhomogeneous due to gravity, okay? And I want to argue that that's not the case. In other words, the solution is stable, even taking into account gravity. And the argument, for this purpose, all I'm gonna do is a heuristic argument, which is the following. So let's consider in general, this is a general argument, if you want, that let's consider a very general Friedman equation of the following form. So I'm gonna allow all kinds of form of stress energy on the right-hand side. So in particular, let's allow for a non-relativistic dust component, a relativistic radiation component, a spatial curvature term. I also can allow for the kinetic energy of a scalar field, let's call that C kinetic, which would go as eight of the six, plus dot, dot, dot, dot, dot. And we also allow ourselves a component, which is an additional component. In this case, will be the CFD component, let's just call it C sub-phi to be generic. And this C sub-phi will have some generic equation of state, which I'll take to be constant to be W sub-phi, okay? So W sub-phi is the equation of state of this postulated component, and I'm taking it to be constant, so that the energy density component goes as precisely this particular power. So this is an argument that's separate from what we've said so far. It's more general, if you want, than our particular considerations. And now we wanna ask, under what circumstances do we solve the flatness problem? So what I mean by the flatness problem, people even now disagree on exactly what is meant by it, what I mean by the flatness problem is quite simple. I wanna consider an initial time of my evolution in which these components on the right-hand side, they all make comparable contributions to the Freeman equation, okay? So I wanna think in particular of the curvature being roughly comparable to the Hubble radius at this initial time. It cannot be exactly of the same order, it may be a little bit sub-dominant to start the evolution, but in any case, I'm not fine-tuning it. And now I wanna ask, what happens subsequently? So if the universe is expanding, the scale factor is growing, then clearly under the standard set of components, since A is expanding, the term on the right-hand side, which dominates ultimately will be the spatial curvature. And that's a flatness problem, and expanding FRW universe tends to become dominated by curvature at late times. So if you wanna avoid domination by curvature, what you must ensure is you have to throw in a new component whose power of A must be less than two, so that it dominates over the curvature component as the universe expands. So in other words, what we need is three, one plus W sub-phi to be less than two, and this implies that W must be less than minus a third, okay? So that immediately leads you to inflation, or accelerated expansion, because W is minus a third is exactly the threshold between deceleration and acceleration. So this is inflation, right? So in fact, there are two branches if you want. One of them is that W lies between minus one and minus a third, and that's the usual, what's usually assumed for inflation. And our solution over there, this Genesis-type solution, in fact corresponds to W being less or much less than minus one, okay? So this is if you want the Genesis branch. In particular, in the limit that W becomes much less than one, it corresponds to very slow expansion, okay? But nevertheless, what it means is very slow expansion, W must be much larger than much less than minus one, and so this guy actually grows tremendously fast as A grows, while the other components are remaining pretty much constant. Does that make sense? In a contracting universe, on the other hand, if the universe is contracting, then instead the scale factor is shrinking to zero. So if you're contracting, then instead the scale factor is going to zero, and so now the dangerous contribution is the one that blue-shifts the fastest, namely this contribution, kinetic energy of a scalar field, or anisotropy of the collapse. So you have to worry about this term, and so in this case, if you wanna win over the one over eight of the sixth contribution, W must be bigger than one, okay? So the other possibility here is that W sub phi be bigger than one, and that is our other branch. Of course, I've conveniently erased it, but here I had W somewhere being bigger than one in our case, so this would be the contracting branch of the conformal scenario. Now in either case, you see that, again, assuming these contributions are roughly comparable at the initial time, as time goes on, you're completely dominated by the CFT contribution, or the inflationary contribution in the standard story, and so you see that the universe is driven to be flat, homogeneous, and empty. All these other sources are driven to zero. So in fact, the solution we've been discussing is a dynamical attractor, even in the presence of gravity. Now of course, the key difference with inflation, as some people would say, is that of course here, by virtue of the fact that you're slowly evolving in time, the patch of the universe you start with has to be already pretty large, okay? So it's not gut scale. It's something much larger, like millimeter or so. So when I say these contributions have to be comparable, I'm assuming that the curvature scale of the universe is millimeter size to start with, not gut scale. So some people would say, well, inflation is better on that count, whatever, okay? So the version of the flatness problem that I'm considering is simply the statement that all these components are comparable. Okay, very nice. Yes, Meddad? That's right. So what we're saying is, so now in the presence of gravity, the universe will have some, it will be in general an FRW metric, which can have different amount of curvature. Physically, as usual, this would correspond to the local patch you're considering. Physically it would correspond to the local patch being either slightly underdense or overdense compared to the rest of the matter. But the point is that as time goes on, that curvature becomes irrelevant. The question in the back, yeah? Yeah, so that's a matter of taste at some level. Yeah, in fact, this maybe what I'll discuss later today. These issues are fine-tuning, they're very difficult to quantify them. Of course, if you start with the universe that starts very hot at the plank scale, okay? Or gut scale. Now, if the universe is very hot, then clearly you shouldn't assume that it's already very homogeneous to start with. Because being at hot temperature means there are many states that are nearly degenerate, so why choose a particular state? If your philosophy is instead that the universe starts out very cold, as is the case here, you imagine that you start with a slow evolution, then who knows? Maybe it's much more, at least philosophically, if the universe starts outside cold, then it's reasonable to assume there is a preferred state to start with. That was always the philosophy in these alternatives. Yeah, yeah, the matter density. Omega, yeah, the point is that Omega, by definition, is always one if I include curvature, okay? Now, if you throw in curvature, it can be less than one, la-la-la. So the assumption is like inflation, right?  Omega can be bigger than one or less than one. But you cannot have arbitrarily large amount of curvature, otherwise inflation doesn't start, okay? So it's the same story here. You can tolerate some amount of curvature, like I said, order one, but usually a bit below one. But the point is that as the universe evolves, Omega is driven to zero, sorry, to one. Excuse me, curvature is driven to zero, so Omega goes to one. Yeah, very nice question. So there is a scale now, which is M-plank. So the symmetry is being broken by M-plank, okay? Other than that, yeah, so there is a, yeah, if you want other than that, the symmetries are broken by initial conditions. So if you take our five to the fourth example, we precisely chose, there's another solution in that example, which was you stay at the top of the hill, perched, and then the conformal invariance is forever, forever there, okay, classically. But we chose a solution in which you roll. So that's it. That's how we break the symmetries. And that introduces a scale, in a sense it introduces one over T. Okay, let's see. Yeah, very nice question. That's right, that's right, so very nice question. So in fact, as you know, in a collapsing universe, you can get into a regime where you enter a chaotic billiard regime, in which some dimensions will shrink, other will grow in a way that's chaotic. That's precisely what's avoided here, if you want. So this onset of chaotic behavior is triggered by the fact that this term, which also comes from, so this is the average anisotropy, the contribution to the average scale factor is a one over eight of the six contribution, and indeed, if I don't have this component, it's driven to dominate everything else. And once it's dominating, it's the onset of this mixed master behavior. But that's precisely the point, it's somewhat counterintuitive, but if I had this additional component with W bigger than one, in fact, the mixed master behavior is squenched, okay, because this guy comes to dominate. If you want one way to think about it is the scale factor doesn't change very much. The expansion is very, the contraction is very slow, but this is boosted by a huge power. So the other guys stay roughly constant. Okay, let's, let me move on. Yeah, I wanna come back to a question that was asked earlier about the fluctuations of pi, because this will play a role in what we're about the observational consequences of the story. Okay, so let's move on to observational consequences, and in fact, so these, so many of these, in fact, all of the observational consequences I'm gonna discuss were basically derived by Rubikov and collaborators in a series of papers, and then our contribution with Paolo and Marco, and my former student, Austin Joyce, was in fact to show that these observational consequences are purely the result of symmetries, whereas Rubikov was deriving them in his fight of the fourth model, we show that they're really a consequence of symmetries. So first of all, indeed, coming back to the question that was asked, let's consider the spectrum for pi. Okay, so we've already understood the growing mode. So just to recap, right, this was the mode function equation for pi with the famous tachyonic term. By the way, pi, of course, here is really the goldstone for the spontaneous breaking pattern. Okay, it's really the goldstone for SO4 comma 2 going to SO4 comma 1, and at first sight it's puzzling the fact that pi has a mass, okay, a tachyonic mass moreover, given that it's a goldstone should be massless, but again, this is an artifact of being space times, the symmetries are space-time symmetries, the ones that are spontaneously broken. In fact, the generalization of a goldstone mode, in this case, is just a statement that in the long wavelength limit, this mode just becomes a redefinition of the background, which indeed this is what it is, it just becomes a time shift. So we remember that the growing mode solution at long wavelength was one over t squared, and so without doing calculation, we could just solve this guy, yadda yadda yadda, but it's simple just based on dimensional analysis, this is a massless, it's a dimension, it's a massless field, and so as a result, just based on dimensionality, the power spectrum for pi must precisely scale as one over k squared, t squared. Now, as a result, the spectrum is very red, which is just a statement that pi grows as t goes to zero, so correspondingly in the infrared, it's very large. Nevertheless, this is not a disaster for observations, because if you actually compute a final observable, in particular the curvature perturbation, you find that the curvature spectrum, the curvature perturbation spectrum is actually very blue, so it scales as k cubed, or k squared, I think. So it's a very blue spectrum for the curvature perturbation, and what it means is that simply the spectrum, the curvature perturbation that's due to pi, so this is due to pi, is very small at long wavelength. So in fact, from that point of view, this guy is unobservable. It also means therefore that pi is not the source of the density perturbation we observe in the late universe. The source of it must be the spectator field theta, which must convert to zeta somehow later on in the process, okay, through some conversion mechanism like kerbaton or modulated reheating, something or other that will generate, that will transfer the scaling spectrum of theta onto zeta. Nonetheless, the fact that pi has a red spectrum has observational consequences. So this is what I wanna discuss next in the last part of the lecture. So pi is the goldstone for the symmetries, this point, the broken symmetries. And so the broken symmetries, remember, are time translations, boosts, and the time part of the special conformal transformations. You can easily work out, based on the field transformations, that pi under these transformations shifts as follows. So delta pi is one over t minus dt pi under boosts, it's xi over t. And finally, under the time part of the special conformal transformation, it goes as x squared with a minus, okay? So coming to Diana's question earlier, so now it's not a typo, okay? So these three transformations you see are manifestly nonlinear on pi. So pi really is the goldstone for these transformations. And based on our discussion in the last lecture, we therefore expect that pi will be associated with soft pi on theorems, namely, insertions of soft pies will be related to symmetry transformations on the lower order point function, exactly like we saw last time. So I'm gonna move on to slides at this point. Okay, so this is the story. These are the soft pions for pi. So just like we did yesterday, we imagine taking the soft limit of a pi insertion times an operator O that involves a bunch of k's, one through N. And now you immediately intuit that in momentum space, you see, you can already guess the answer, because first of all, it all involves one over t, okay? So time dependence is explicitly involved. You expect that the transformation on the soft, on the hard modes will involve time derivatives of the hard modes. That's number one. And number two, you see that you have different powers of x's. So constant, linear, and quadratic. So in momentum space, you expect, just like we discussed yesterday, a constant shift, a piece linear in q, and a piece quadratic in q. And that's what's found, okay? There's a minor subtlety, which I'm gonna gloss over, which is the fact that the q squared bit is actually not the full q squared, but it's a q squared that's averaged over all the angles of q, okay? But aside from that, this is the answer, okay, which we worked out. Now, it's very nice, because you would say, well, on the one hand, there are more relationships than in spectator field inflation. There are all these new soft limits, which we can test. Unfortunately, pi is not itself directly observable. As we saw, pi doesn't contribute to zeta at the end of the day, so how can you test this? What it means is that there's still some tests you can do, but the tests will be more model dependent, so to speak, a bit less direct. So let's discuss a few of them. Yeah, so, and these crucially rely on the power spectrum for pi being very red, as we saw. Okay, so here's one particular example. So if you take, so pi does participate through correlation functions of spectators through exchange diagrams. So it can be an eternal leg, okay? Contributing a four point function, for example, for the spectator field. Now, in the limit that this mode in between becomes very soft, namely when the hard guys are basically co-linear, so that they add up to almost zero momentum. In that limit, you basically have the same phenomenon as in field theory, where you have Tupkowski rules. So the diagram factorizes, so in this particular limit as q goes to zero, the diagram factorizes into the product of two three point function with a soft pi inserted in each. So at the end of the day, given that pi acts non-trivial in the correlation functions, you find a particular shape for this four point function. You see, number one, the angular dependence is peculiar. Okay, it's a particular angular dependence, which in fact vanishes when integrated over. And secondly, I keep clicking because I'm used to, I don't know. And moreover, you see that the q dependence is one over q, so this four point function becomes very large, in fact, in this co-linear limit. Okay, so that's one signature. The shape is special, and also the amplitude is quite large as q goes to zero. There's also some contributions from pi coming from loop diagrams, such as this one. And you can show this gives a usual tau NL shape for the experts, a usual shape for four point function, which has a particular q dependence. And finally, even if pi is not itself observable, it does affect, so if you imagine the global profile for pi, locally the local value of pi will affect the local observables, in particular the theta theta correlator. Here I'm calling it chi, but really chi is the same as theta. So the local, in any given realization, the local value of this of pi will affect the local statistics of our correlation functions. And so in particular, you find the following. So the two point functions of theta, or chi, in the presence of a long wavelength pi acquires, by virtue of the coupling to pi, it acquires some particular anisotropic contributions. So the power spectrum will have, at some level, anisotropies, which are quadrupolar, in fact. So cosine theta is the angle between the hard mode k and q. And moreover, so you see there's some scale dependence. This piece is scale invariant, and the other is actually dying off on small scales. So these are things that in principle, one can look for in the data. And again, our contribution with Paolo and Marco was simply to point out that these features are all governed by symmetries. So at the end of the day, if you were to see these signals, it would be a strong indicator of this conformal mechanism. Now you could ask, could I do this in inflation? Can I just have these features in inflation? The answer is yes, you can do anything you want in inflation. You could cook up a pi, which couples in precisely the right way, so it has this particular spectrum and the couples in another field precisely in the right way, and you can do anything you want in inflation. But it would be very unnatural in some sense. Nothing would force you to do that, whereas here really the very existence of pi is forced upon you based on the symmetry breaking pattern, the fact it is a goldstone. Okay, so that's all I had to say about the conformal mechanism, I think. So there's some, ah, and finally, of course, the very last thing I wanna say are gravity waves, which you've heard about yesterday's colloquium. Gravity waves are really very hard to generate in anything other than inflation. And the simple fact, the reason is that, so here, the reason we were generating scale invariant perturbations for the scalar fields was because in the CFD fields had some time dependent value, they were coupling, yadda, yadda, yadda. The point, though, is that gravity waves only care about gravity directly, the gravitational background. In this case, the gravitational background is very slowly evolving. It's very close to Minkowski's pace, and so naturally gravity waves are not appreciably excited. So had the bicep, brouhaha, been really confirmed, this would have ruled out these kinds of alternatives. But it was more brouhaha than anything else, and so therefore, jury is still out. Okay, so that's all I had to say about the conformal story. Let me pause for questions. Yes, again. Very good, very good question. So that's right, that's a great question. So first of all, so R is zero. NS, NS is anything you want, in the sense that NS, at the end of the day, so let's go back to our theta field. So when I wrote down the action for theta, I had that it was phi squared d theta squared. That gave rise to an exactly scaling variance spectrum. But now you can have corrections as discussed. I could have phi to the fourth theta squared, so this will give rise to a effective mass term for theta once phi picks up at expectation value. And so depending on the magnitude of kappa, this is still conformally invariant, classically. Now depending on the magnitude of kappa, this will induce a small tilt red or blue, depending. So there's no preference for blue or red from this point of view. Of course, this is a multi-field scenario, so naturally you expect significant non-Gaussianities. So the fact that theta could have non-vanishing three point or four point function that when you convert to zeta, this will give rise to some local FNL for zeta. And so again, this is very model dependent. However, the fact that you haven't observed FNL personally doesn't necessarily rule this out because the three point function, there could be a symmetry that protects this three point function from being non-zero. And indeed in the model of Rubikov, this three point function vanishes. Now there will be FNL generated from the conversion. It's a non-linear conversion process, but these can be small, okay? These can be order one. So, but like any multi-field scenario, if you keep on measuring higher point function, constraining tau NL and so forth, at some point, yeah, at some point it becomes unnatural not to have significant non-Gaussianities in these models. Yes, Hassan, very good, yeah, very good. So people have studied this. In fact, Franz Pretorius and Princeton, together with Paul Steinhardt and a couple of other collaborators, they actually did the numerical simulations to see what happens when you consider more general initial conditions for the scalar field. So I think they considered simply, they only considered the scalar field coupled to gravity, so no other field in the story, but now with generic initial conditions. And you see the attractor mechanism taking place, namely regions where the scalar field was sufficiently homogeneous. Those regions evolve very slowly. Regions where the scalar field was more anisotropic or things were wilder, they collapse into a big crunch singularity. So you see, the only difference with inflation is inflation generates a lot of space in the process, a lot of space time. In this case, all that happens is you start with a patch and that patch keeps on its initial size, so to speak. But relative to the rest, it wins in volume as you wait long enough. So what I said, which is a perturbative argument, is borne out by these non-perturbative numerical calculations. You wanna put the measure, yeah, that's right. So at the end of the day, absolutely it's a valid point. At the end of the day, it becomes a similar constraint as an inflation. Yeah, no, it depends how much you start with. So, no, I agree with you, but no, I'm saying, I think the statement I'm making is simply that in order at the end, for the end of inflation, by the end of inflation to have a universe that's sufficiently flat corresponding to observations today, you need to have a certain number of refold as you well know. It's a similar story here. So your initial conditions must be such that you have a sufficient number of e-folds of contraction, but now the e-fold, since the universe is not evolving very rapidly, then the time scale is set by the scaler. So five goes as one over T. So you should ensure that this evolution is preserved for a range of time, which is very similar. You get a number of e-folds just like in inflation. The only difference, like I said, and I'm not selling you a used car, I'm being very honest, that at the end of the day, the difference with inflation is you start with a large patch as opposed to a microscopic patch. Yes, yeah, there are no ghosts, so that's right. So those theorems, it's very interesting. So in general, it's a very deep question as to whether that's right. So what is it that you violate? What sacred principle do you violate if you violate the non-energy condition? So first, indeed, people thought, well, you have a ghost, it's none. But then people found example, ghost condensate, the Galilean in which you can violate the neck without having ghosts. But then you have other problems associated with it. So for ghost condensate, there is no Lorentz invariant. Vacuum associated with it, you're always working in the spontaneously broken phase. In the Galilean case, you have superluminal perturbation. So it's a long story. But it's an interesting question. And at the end of the day, let me see. So at the end of the day, I think, so one example that we cooked up, so it's a whole history of refinements, if you want. So pushing, trying to push the limit as to what of these principles, so what are these pathologies are really fundamental and which can be avoided. And in one of the papers, so I think it was a paper by Rubikov that there was one by us, in fact by, what's his name? By David. Yeah, exactly. So David and also, and Rico, in fact, in Keene. So you can find, now you can find solutions that, theories in which you violate the neck, in which you start from a theory that has a well-defined Lorentz invariant vacuum. You move on continuously to a solution that violates the null energy condition without developing ghost or gradients in the process. The theory around the neck violating solution is subluminal. The only leftover pathology is that the theory around Minkowski is superluminal, if I remember correctly. Okay, so at the end of the day there's, but that's the furthest that we've taken it or that people have taken it as far as I know. But it's a very interesting question, right? What is it that fundamentally do you violate, if you want? Yes? Yeah, but the patch is not, how to say, the initial patch is not 10 to the 28 centimeters, it's a millimeter, okay? So you solve it just by saying, in the same way that inflation solves it, all that inflation says is that you start inflation in a Hubble radius. Within that Hubble radius, by causality, things have had time to communicate and homogenize themselves to some extent, and then inflation takes on. Here it's the same except that the initial patch is a millimeter, that's the difference. But otherwise, you just say, I start homogeneous within that patch, I let things evolve. The point is that the contraction is completely asymmetric with respect to the expansion subsequent to the hot big bang. So a millimeter turns into a large universe. Yes? Yeah, that's right, so we were never probably very precise about reheating, but I guess it's quite model dependent.  But it's quite model dependent. I don't know, in Genesis, I guess you hit strong coupling, so you can say maybe this is where you reheat. Ah, that's a good question, right? That's a good question, so the answer is yes. So if you have to convert to zeta, if you convert to zeta before, you convert to zeta, and now, so in the models that we've studied in which you have a non-singular bounce, for example with a ghost condensate, or as a toy model to get a smooth evolution, you find that zeta on scales much larger than the Hubble scale at the bounce is constant, is preserved. So the spectrum of zeta is preserved through the bounce. The what, for granted? Oh yeah, absolutely, but that's a much better assumption than in inflation, if you think about it. So Bunch Davey's vacuum here, we're starting cold, okay? Now I don't believe in transplanting story, but all the modes that we're talking about that end up forming galaxies in the late universe, they're all macroscopic, okay? So assuming, the universe starts out cold, assuming that these modes are in their ground state, it's perfectly reasonable. In inflation, I mean, of course, the story starts hot, so I mean there, I also think it's reasonable, but for different reasons, I mean, it's okay. Yes, there is no what? There's no gravitational waves, no what? Non-Gaussianities. Yeah, but Planck, I mean Planck doesn't go down. I mean, it's what, F and L of two plus minus something, right? What is it, five plus minus two? So for example, in Matthias's modulated reheating, yeah, it's too bad that Planck really didn't go further down, because if you had gone down to F and L of one, even, that would have been great. So for example, in Matthias's modulated reheating, you get F and L of three, you can get F and L of three, so no, I mean, it's a multi-field scenario, so you do get non-Gaussianities, but no, the real motivation is to try to have an alternative to inflation, right? Now speaking of which, I can tell you in two minutes my alternative, is it okay? So my favorite alternative is Abercadabra and I'm absolutely serious, okay? It isn't being recorded. I'll find I'm fired from Penn tomorrow and then get a bit posted on Twitter or something, but I'm absolutely serious, okay? So I'll tell you for what it's worth, I really think now this is the best alternative. For the following reason, so in a nutshell, right? What inflation does is it does the following. So first of all, you don't know the initial state at the onset of inflation, that's also Abercadabra, and it's Abercadabra in such a way that some scalar field in our past history started out perched on top of its potential, okay? And then you have some evolution and beautifully enough this generates scaling variant perturbation, and this is the beginning of the hot big green phase and then everything follows. So my alternative is Abercadabra, okay? So you just start from a question mark and you go straight to a scaling variant spectrum on super horizon scale of flat universe and all that. Now, this you'll say has many undesirable feature. Number one, it's not falsifiable, okay? So good, but this is like Matthias' story. If you find an observation tomorrow, I'll just learn something about the initial state. So you learn that there's gravity waves, perfect. I'll put in gravity waves in my initial state. You find there's non-Gaussianity, I'll put in non-Gaussianities in my initial state. Perfect, it's not falsifiable. You say, ah, but inflation is falsifiable, is it? So if inflation is the same story, you find a new observable that contradicted your favorite in floton potential, you change it, you find a new in floton potential. In fact, I'm hard pressed if anybody could tell me one observable that would falsify inflation. I've never heard one, okay? Because every time there's something that comes up, people change their story, okay? There's tension between Planck and Bicep. No problem, we'll put some running and fit the data, okay? So it's just equally unfalsifiable as inflation is, okay? It's just I'm putting everything here. But now you're gonna say, okay, that's nice, but I like inflation better because from arbitrary, or for some arbitrary initial conditions, I can get a universe like I see, whereas here you're putting it by hand. There's no quantified way that I know of that makes inflation favored over what I just said. And this is an old argument, goes back to Penrose, and in fact, there's a very nice paper by Holland and Wald, Holland and Wald from 2002. And I remember reading it as a grad student and I thought these guys are cookie, okay? It didn't make any sense. And now I'm older, I read it again, and I think these guys were geniuses, okay? This makes absolute sense what they're talking about. So the point is the following. Of course, it's the famous arrow of time problem that if you say, you wanna say that there's some, you start from some chaotic arbitrary initial conditions and inflation will take over because it's an attractor and you'll be led dynamically very often to this state. If you assume ignoring weak interactions that the physics is time reversal invariant and the measure that the, yeah, so the measure you put is also on the evolution is also time reversal invariant, then by Lueville's theorem, the measure on final states is the same as the measure on the initial states. Again, it's nothing new what I'm saying, but what this means is the following. So you have a beautiful story for inflation, which is you start from a patch, a small patch, which was somehow fluctuating, chaotic a little bit. Inflation takes over and out of this produces a beautiful big universe, which at late times has these beautiful galaxies and all kinds of complicated physics from a rather simple initial condition. But if physics is time reversal invariant, it means that there is a perfectly fine solution, fine solution to the equation of motion, which is that you start from our messy universe with galaxies and varions and all the stuff that Matthias was talking about, you start collapsing this universe and you see what happens. Now if you play that movie in your mind, you'll see that more often than not, of course, you collapse to a crunch. Things become very, very chaotic, very messy as you collapse this universe. Instead, the movie I'm proposing to you is that all this messy galaxies and so forth, they collapse together, they form a pristine cosmic microwave background, which is homogeneous at one part and 10 to the five and moreover all this radiation, this entropy that sits there, further collapses and coalesces to a microscopic patch in which a single degree of freedom comes back and rolls up the potential, okay? So this movie played forward in time is absolutely nonsensical and you would say it must have tiny probability of occurring. Now if the laws of physics and the measure you put on initial and final states are the same by time reversal invariance, then it tells you that equally well, this story of inflation being a natural, that starting from inflation is a natural initial state must be very unlikely, okay? Again, it's nothing new. People have pointed this out for a long time. But somehow as far as I know, people never pushed it to its logical conclusion, which is if that's true, then it means that we might as well just postulate the initial state to be what we observe it to be. And the point is more than philosophical in cosmology, unlike anything else in science, we don't get to choose initial conditions. Usually in physics you wanna learn something about dynamics. You play with initial conditions, you change them and then you see what happens as a function of initial conditions and you determine the dynamics. Here all we have is things at the present time and we're trying to deduce what happened dynamically based on some postulate for the initial state, for the initial conditions. You cannot disentangle the two. So at some level without a theory of initial conditions, how can I believe that inflation was more favorable over what I just postulated as my alternative? So that's, I believe in dinosaurs, very good. So of course we believe that things happen in the past. We believe in the past, that's of course the story. And I know you know this, but for the purpose, for the benefit of the students, because this is not something that was obvious to me, is that of course this is all tied to the second law of thermodynamics. And so you have to be careful how you use it because of the following reason. You walk into your favorite bar, and at the bar sits your drink, your favorite drink, and in it sits a half melted ice cube because the bartender knows you. He puts it there already before you step in. And now you say, let me based on my statistical physics book predict what will happen to the ice cube, to my drink. And of course you say, well, entropy is gonna increase. So here's the entropy of my system as a function of time and the ice cube will melt and entropy will grow and so forth. And now I ask you, predict for me based on the same principle what happened five minutes before you stepped into the bar. And you'll say, oh, of course, well, the bartender put it in my glass with a full ice cube and so the entropy of the system was even lower. But that's wrong. That's not what your physics textbook would tell you because if physics is time reversal invariant, what led you to think that entropy would rise in the future would also tell you that the entropy will rise in the past. So in fact that the, I have melted ice cube should have been a full ice cube so if it should have been water that spontaneously formed a half melted ice cube. That's what you'll deduce. So indeed, right? Of course that's, of course, of course, of course. That's right. So ultimately we are forced to postulate a past hypothesis. We postulate that there was a low entropy initial state. Yeah. Yeah. Your core's grain. Yes. But, but. No, of course I agree.  I mean, this is clear. So I believe the CMB was generated because the, that's right. But we, that's the point, right? We observed that we're not in a maximum entropy state. That's the fact. So then we asked, what's our previous history? Our previous history is that it was an even lower entropy state or the same. Okay, but my point is the following. So I agree that of course I'm being provocative, obviously, right? Ultimately, but I do have a point at the end. I don't believe that it's an initial state that's just specify out of the box, but I do think this is more preferred over inflation. That's what I believe. So there's an alternative here to be found, which is what you call abracadabra. No, I don't have to postulate a creator. I'm just postulating initial conditions. Abracadabra, yeah. Okay, okay, okay, right, right, right. But that's important, that's very important. No, no, that's a very important distinction. Very good. Because that kind of logic would say it's much more probable that we just spontaneously created a second ago and all that, okay. But I think there's a key difference. So I'm willing to tolerate the past hypothesis. I believe that the CMB, of course, didn't, you know, it's not just an illusion. We actually went through, I'm not crazy. No, because we have a theory of physics, right? It's the stuff we understand, photons and so forth. It was the same stuff back then. Nuclear synthesis, it was the same stuff back then. The part where I start to become skeptical is when we have to postulate the scalar field, something that we've never seen, some previous complicated history. But you agree, but then I, you know, but you completely agree that you need a theory of initial conditions to explain how inflation started. The statement that people make that, oh, inflation, you can start with arbitrary initial conditions and inflation will start, that's crap. That's not true. Yeah. The origin of life, how did life, less than computing how probable it is. No, no, I'm not disputing this. No, of course. Of course. Of course, of course. And people have been debating this, and Boltzmann could, yeah. Okay, so absolutely I agree with you. Just the history of time, what happened, I believe, right, in the past, obviously. But the point that I don't accept is to say, oh, we've proven inflation, based on what? Based on what? Based on a few observations, based on AE, there needs to be, I think we all agree, there needs to be a theory of initial conditions that makes somehow, that tells me quantitatively why the inflaton started out where it started in our past history. There has to be something. There's equally an acrocadabra here that tells me how out of quantum gravity I end up in this history. Otherwise, I mean I'm just, I mean dinosaurs, I observe them. I dig, I see bones. I see bones, right? I see bones, I see the cosmic microwave background, I measure it, I observe it. I see something. What do you see about inflation? What is it that you see? You see scale and varying perturbations. I'm not defending that. Okay, of course, of course, but that's what, of course that's my message, right? That something else, I think, a more compelling, if there is something else, would be some theory here, a dynamical theory, that would actually make this evolution entropically favorable. That would be nice, you know? That's all I'm saying. For example, I have no problem. Yeah, of course, because, of course, that's the fun of being provocative, but that's the point, I think. I'm not at all convinced that inflation occurred. That's it, yeah, yeah. You have also made some state reports. Yeah. Now, the state is a thermal state, it looks very nice, and you really don't see this occur. Now, here, if not so different, also there, you have to specify initial conditions to make a new change. That's right. And to specify reasonable initial conditions and the inflation is also different in a sense. Well, I mean, I don't know. It's the same, right? When the number of percolation makes it very small, then maybe in the case of nucleosynthesis, you measure a few numbers with these regions, okay? So then it looks like more credible. But I think to me, the difference is that in the case of nucleosynthesis, I agree we can make the same story, but there it's all physics that we know.  I think it's just a matter, I'm just saying, I'm not, all I'm saying is that, it's not that I don't really believe in abracadabra initial state, okay? I'm being provocative. All I'm saying is that there's certainly room for a better alternative, which will make this, which will generate the hour of time. I mean, it is, it is, it is a logical possibility. Well, that will explain more naturally, let's say, how this, how we started out in this particular state at the end of inflation. Anyways, that's it. I meant it to be provocative, okay? For fun, I knew, yeah, so, yeah, we stop. Thank you.