 The last two lectures to deviate from sort of the canonical series of lectures one might give on inflation and Instead of going further into details of how to calculate perturbations Or of specific inflationary models to talk about challenges and controversies and possible research topics Which I imagine a lot of people are interested in So before I get into that there's a few things Which I do need to discuss That I didn't manage to get to in the in the second lecture so in particular what are the sort of primary observables That are connected to inflation so the primary things you could observe in cosmology Which are related to inflation the most directly And so we talked about how inflation is supposed to solve the flatness problem and the way it does that is it produces an omega k Which can be either positive or negative because k can be either positive or negative, so let's put absolute value of science which scales as e to the 2 so if you remember by the way the definition of omega k it's k over a h squared that's the definition and so what we saw last time is that is That omega k scales like e to the 2 times a number I'll call it n minimal Minus n total the total number of e-folds of inflation And we gave a kind of upper bound on n minimal Which was 62 so What this equation says is that if inflation lasted a few more e-folds than this number Which is somewhat uncertain, but it's of order 62 So if inflation lasts a few more e-folds than that then the curvature is exponentially small Right, so if inflation lasted even 10 more e-folds and then minimal then you have an e to the minus 20 Which is a pretty small number 10 to the minus 7 or something. It's way below anything will observe We'll ever be able to measure So this was sort of the upper bound on n minimal and It could be 50 in a somewhat more typical scenario it's somewhere in this range though generally Anyway, so that's so one of the observables in cosmology that's directly related to inflation is the spatial curvature the spatial curvature were large that would be Very informative or even if it's detected it so it would have to be below the current observational limit Which by the way the current observational limit is Something like a little less than 0.01 That's the observational limit So if we end up discovering that omega k is not zero, let's say 0.005 or something then that teaches us something very important It tells us that if in fact inflation happened it only lasted a few more e-folds than this n minimal value It didn't last very long Okay, so that's one thing another thing is The amplitude of the power spectrum of scalar perturbations, so we talked about how when There's a fluctuation in the inflect on it gives rise to a fluctuation in the time at which inflation ends inflation ends when the inflect on reaches a certain value and So if there's a fluctuation somewhere in the universe during inflation which say pushes the inflect on towards its end point that inflation ends early in that part of space and If it ends early it means the universe is effectively older it reheated longer ago in that region and therefore it's colder than the average So we quickly worked out an equation For for delta n that's the change in the number of e-folds First of all, this is h delta t And we can rewrite this as h delta phi over phi dot And if we use and this is also the reason it's interesting is it's the curvature perturbation It's delta a over a so it's the perturbation in the scale factor divided by the scale factor and we can If we know that delta phi is of order h and here I'm going to put the 2 pi n Delta phi is h over phi is h over 2 pi Then plugging this in we have h squared over 2 pi phi dot, okay, and normalize this way This is the amplitude. It's usually called delta s. It's proportional to the power spectrum of perturbations But it's the typical size in a in a fluctuation in in curvature Now how do you observe this So the cleanest way to observe these fluctuations is to look at the cosmic microwave background If this is t now And this is us at earth now When we look at the at the CMB sky, we're looking at light which reached us Along a light cone so we're seeing a Back along this cone and when we look at the CMB what we're seeing is the State of the universe at this time here T last scattering This is T not T now So CMB photons for the most part last interacted with anything at T last scattering Which is a few hundred thousand years after the Big Bang. So it's in the early universe At the time of T last scattering the universe was full of plasma of ionized ionized plasma protons and electrons After T last scattering Electrons were captured by protons to form neutral atoms and that's why CMB light stops scattering Okay, so you can think of the universe before this time as a kind of opaque solid I'm not really a solid but an opaque substance through which light can't propagate So what you see when you when you look at the at the CMB is is simply you're just looking at This slice of the universe at that time And then what do you measure? Well, you measure the temperature as a function of angle So this is your sky you look at for instance in this direction You measure the temperature of the CMB the CMB is almost a perfect black body So when you look in a particular direction, you see a black body spectrum. There's a temperature associated associated to that So if we call this angle here n hat then you measure T of n hat And you do that for every point on the sky and that's your CMB data set Okay, so what does that have to do with this? Well the temperature at this point? is directly related to delta n at that Pro-moving point at the end of inflation for the reason that I already described that if inflation ends later Then the universe reheats later at that point would be hotter Okay, so there's a direct relation between the temperature at any point on the CMB sky and the fluctuation in the number of e-folds At that point in that part of the universe All right So one thing we can measure Given this T event hat So given T event hat one thing we can measure is delta T over T so this means T event hat minus the average value of the temperature Divided by the average value of the temperature. This is a quantity whose average is zero But which has a non-zero variance and a non-zero standard deviation and It's that standard deviation that's directly related to to these qualities over here Again remember delta n is another quantity with average zero It's the fluctuation in the number of e-folds with respect to the background. It's average is zero and So the variance in that quantity, which is what I mean when I write this I Mean that I mean this the standard deviation when I write this equation That's precisely very closely related to this temperature fluctuation over there Okay, so we can do this and We have more I mean so the standard deviation is Is a function of scale All right, so why is it a function of scale? What does that mean? Well when you when you do this When you met when you measure this difference on the sky You can do it as a function of angle, so If you'd like you can measure the difference in temperature between two points Which are separated by some angular scale? So for example, you can measure the temperature here and you can measure the temperature here And you can see by how much they deviate and then divide by the average and you can Take the average of that quantity across the sky All right, so you can have a fixed angular scale here if you want in other words You can do a Fourier transform. You can do a spherical Fourier transform of this T of n hat and decompose it into modes and What you'll find is that the differences in temperature are almost the same they almost it almost doesn't matter what the angular separation is there's almost a scale invariance in In the spectrum. It's almost the same amplitude regardless of angle, but not quite So why is that? well If you look at this equation over here, I Told you that during inflation H is approximately constant so H dot over H squared is Small so H is not changing very much during inflation and the same thing is true of Phi dot Phi dot Phi satisfies an equation like this And in slow roll we can neglect the Phi double dot term So we can neglect it if this is small and if Ada the other slow roll parameter is also small and Then we find that Phi dot is approximately V prime over 3h and again this approximation is valid in slow roll and Neither of these quantities is changing very much during inflation. So Phi dot is also close to constant during inflation Okay, so this ratio here Changes a little bit during inflation, but not very rapidly So what does that mean? It means that these perturbations They're generated as inflation proceeds and They have almost the same amplitude As time passes, but not exactly there is a slight change in their amplitude And the way that's usually characterized is by what's called the tilt the tilt of the spectrum Defined like this There's some sort of historical reason And like this Where K is the momentum So imagine To make it even easier Forget about what we can see for a moment and suppose we could measure the temperature of the universe everywhere on some sort of Slice at this time T last scattering. So suppose we know the temperature everywhere. We know T of X At T last scattering then we can just do a linear Fourier transform of this And we'll have T of K We'll have a power spectrum for T of K. That's what this delta S here is So this derivative means with respect to that K So this is the way the tilt is defined and it can be inferred from this data. We have on the CMB sky And we can compute this derivative Using slow roll equations That's the result where Should write it somewhere over here epsilon One half Prime over V with an m-plank sitting here and eta is M-plank squared V double prime These are these same two slow roll parameters we defined last time remember that's 1 over 8 pi G M-plank squared Okay, so these are these two slow roll parameters we defined last time And so what we find is this is this logarithmic derivative of the of the power Is a small number in slow roll? And again the reason for that physically is just that nothing much is changing during inflation inflation is almost static the physical quantities H and Phi dot V and so forth don't change rapidly during inflation So think of inflation as a kind of a stationary phase almost with just a gradual decrease Doesn't have to be a decrease, but a gradual change in In the physical quantities that parameterize it. So there's a gradual shift in the power spectrum Over time You might wonder why it's D log K that we're taking here Yes well It's possible. You're right. I didn't derive this Directly in my notes, but I did check that it works for so you think it should be six epsilon minus two eta That would not give the right answer for the for at least one case Six you think it's six epsilon minus two eta. Yeah, that would give the wrong answer for at least for this model So I don't think so Well, let me come back to that I have not derived it in my notes I this I took from a review So perhaps there's a typo or I made a mistake, but I think well Well, we'll compute this in just a second for one model and we'll see if the answer we get is reasonable Okay, so in any case it's proportional to these slow roll parameters, so it's small right so now why is it Why is it D by D log K? It's because of the exponential growth in the scale factor remember that a during inflation is Some a initial times e to the ht h inflation times t and what this means is that If you generate a perturbation on some scale that scale grows exponentially For the rest of inflation, so K the corresponding Momentum decreases exponentially And so when you take D by D log K, you're basically taking D by DT Okay, so K goes as K initial times e to the minus ht And so when you take D by D log K, it's like taking D by DT and that's how you can compute this I'm not going to do it on the board because it'll take some time, but that's how you would derive this formula or whatever the correct version of it is That will give the right answer. Yes, that will give the right answer. Okay. Thank you Okay Good. Yeah for five squared these are these are equal so you can't distinguish that formula from the other one Okay, very good Yeah, so let's actually compute them let's compute epsilon and eta for the model that we that we discussed so the model we discussed was We have Phi is one half five squared so epsilon is One half V prime over V squared and so And so this is going to give us two Like squared over Phi squared Eta is the double prime over V squared and So this is just going to give us. Oh, I forgot that's correct This is going to give us two I'm like squared Or Phi squared as well. So for this particular example, Eta and epsilon are equal which indeed is why The other formula gave the right answer So if we compute minus 6 epsilon plus 2 eta yet minus 8 Like squared Okay, and you'll recall probably that last time We derived a formula for the number of e-folds as a function of Phi The number of e-folds was proportional to Phi squared And if we write this in terms of the number of e-folds, it's minus 2 over n This is a maybe a nicer way to express it and This n remember is roughly well This n changes during inflation. That's that's That's the fact that inflation is not exactly Is not exactly constant in time the quantities are changing slowly But the part of inflation that we observe when we look at the CMB sky is The part that's close to this number n minimal. It's it's it's the it's the part that that was produced Sort of as far back as as as we can see in inflation anything was produced earlier than that is outside of horizon today And anything that was produced much later than that Has a much larger value of k according to this formula and so it's not going to be in the CMB It's going to be in large-scale structure or even on a smaller scale. Okay, so we should plug in n of order 50 or so 50 or 60 into this formula And so this gives us something like minus 0.03 or so In other words ns is approximately 0.97 ish Somewhere around there. Okay, so this is another observable of inflation. So there's the amplitude of the power spectrum And I should have told you what it is So this is roughly a few Times ten to the minus five That's the standard deviation power spectrum is the square of that. That's the standard deviation in fluctuations in temperature And then there's the tilt. So that's the degree to which this changes Delog Delta squared delog K as you change the scale and This number in fact is roughly correct In other words, it's it's roughly what data seems to indicate Is the correct value So those are the two of the Several of the primary observables in inflation and that let's let's talk about just one more and then we can Go ahead to discuss the objections that people race So the last one the last observable I want to mention is The amplitude of tensor modes So you may remember bicep 2 Announced that it had detected primordial gravity waves It was very exciting And then it turned out in fact it detected polarization due to dust So it was not primordial. It was a Foreground a sort of a systematic error or a foreground that needed to be subtracted If you wanted to to measure what was behind it And the reason this was so exciting is that inflation doesn't only generate fluctuations in the infleton Phi It would generate perturbations in any massless field Or any field whose mass is less than than the Hubble roughly and so in particular it generates perturbations in gravity in the metric and fluctuations in the metric Have a tensor structure. So you've heard now a lot about gravity waves, you know that gravity waves have two polarizations So inflation excites those gravity waves it it perturbs the metric in such a way as to produce Gravitons with those Polarizations and you can distinguish the imprint of those fluctuations on the CMB from the imprint of these scalar perturbations that come from fluctuations in the in the infleton So I'm not giving you the details. I'm just telling you the bottom line that there is a power spectrum of perturbations in the metric H is a metric perturbation and it has a simpler form Than the scalar perturbations the scalar perturbation power spectrum, which is written over here. It depended on Phi dot Right, so it was h over Phi dot times Delta Phi Delta Phi is universal. It's just a border H But Phi dot can be very different in different inflation models tensor perturbations on the other hand Only depend the amplitude only depends on On h and I'm like right it like that. So this is sort of like the fluctuation and this is the thing that that normalizes it Okay, so so there's power in these tensor perturbations you can distinguish them from the scalar perturbations and people usually express this in terms of what's called are which is the ratio of the power spectra and that's 16 epsilon now data tells us that R is less than about point oh eight So there's only at most eight percent as much power in tensor perturbations as there is in scalar perturbations and This number is not quite consistent With the epsilon that we would have had over here so this is minus four epsilon So Epsilon is a bit less than than than point oh one And that means that sixteen epsilon is something like point one four. Okay, so for that model for M squared Phi squared Epsilon is roughly point oh one and therefore are should be about it turns out a little less than point one six It's something like point one five or point one four and that is Incompatible with this with this bound. Okay, so so this is actually a model which is ruled out by observation That might give rise to various reactions. I mean you might wonder why I'm telling you about it if it's ruled out Well, the answer is it's it's a very simple model It illustrates all of the relevant physics and it was only quite recently that was it was ruled out So I think it's perfectly fine to sort of learn from it but another reason I did this is because one of the criticisms of inflation is that it's not falsifiable and In fact Here you can see very easily that this sort of simplest model of inflation has in fact been falsified has been ruled out by The lack of observation of tensor power, so We'll come back to that in a moment Okay Are there any questions about this? We'll come back to that. Yeah, that probably is what they mean. Yeah, how exactly is Yeah, sorry, you don't exactly you don't know how which form say it again which formula or an understanding Index Sorry, sorry. Yes. Yes. Good. Okay. What is it and hat? Good, right. So so let me let me make this more clear. So this is supposed to represent time and Because I'm not very good at drawing. I'm drawing something which is like a two spatial dimensional universe Okay, so if you live in a If you live in a two-dimensional universe then you can look around you Right, you live in a plane. So you look around you at some at some angle That's what this n hat is supposed to represent. So there's some theta here and how does a unit vector? That's just at angle theta with respect to some to some x-axis and And so different n hats just represent different directions that you're looking Okay, and in in in our actual universe, which is three dimensional There's a theta and a phi and how does a unit vector on a sphere? It's just pointing towards a particular location on the sky And so it corresponds a particular value event had corresponds to looking in a particular direction When you look in a particular direction, you're seeing light which came from that direction And if you didn't scatter all the way back to last scattering then it originated at this point down here At the on the last scattering surface Does that make sense so think of the last scattering surface? It's not a surface because we're in three dimensions. It's a volume and The CMB sky is a spherical slice of that volume and what you're measuring when you look at the CMB What you're seeing is the temperature on that spherical slice of the universe at the time of last scattering Okay, so n hat is just a vector that points to a particular point on the sky. Okay Other questions Okay Yeah Yes. Yes. Yes. This this relation here is not general the fact that this ended up being minus 2 over n It's not exactly specific to m squared phi squared, but it would be different for for some other models. Yeah So when I calculate I don't really calculate n total I can't calculate that unless I measure a mega total So if I if I could measure a mega not a mega well a mega k if I could measure spatial curvature Then modular the uncertainty and precisely what this number is I would know what that is right in particular I would know the difference between these two and there's not that much uncertainty here It's only logarithmically sensitive to various things like the reheating temperature and so forth So if I if I ever measure spatial curvature to be non-zero Then I will be able to say what this is if all I can say is that the spatial curvature is less than something like this Then it gives me some bounds that this difference must be larger than something But it doesn't tell me what n total is right, so inflation could have gone on for a million e-folds and All I can see are the last 60 basically it's so the amount of it I can see are the last n minimum whatever that is 60 ish or so that makes sense Okay Good, okay, so so so let's get to the juicy part. So let's talk about the Controversies and problems Well, first I should say probably what does everyone agree with? I think everyone agrees that if you're given Suitable V so this means in particular that epsilon and eta are small and So and that n is greater than 60 or so so the number of e-folds Calculated from that potential is sufficiently large Then everyone would agree that inflation can occur and if it does occur it'll produce a Quote-unquote good universe, which means Flat homogeneous It has a good spectrum of perturbations. So perturbations as described over there. Okay, so I'm So I think there's no there's no debate over sort of the mechanics of Of anything that I've told you so far That's one thing and the other Is that Given suitable initial conditions Inflation will occur Okay, so that I think is is uncontroversial for instance an example of suitable initial conditions would be that the universe is homogeneous and isotropic before inflation and The value of the infleton is Somewhere on the inflating part of the potential and the time derivative of the infleton is not too large Okay, so in our m squared phi squared toy model. There's v of five versus five You'll recall that there was a Region in here where there's no inflation where epsilon is greater than one This was something like root two times m plank So suitable one example of suitable initial conditions would be Phi Takes a value somewhere over there and Phi dot equals zero or it equals the slow roll value Okay, then no one would dispute that this universal inflate and all this stuff will follow So if we start here and Phi dot initial is equal to zero let's say okay, so that's Right that's that's that's hard to argue with Okay, but it's not very satisfying because at least if those were the only initial conditions that would give rise to inflation then it would not be very satisfying because You don't want to have to assume that the universe is homogeneous and isotropic before inflation The primary the thing we started with them sort of main point of inflation as a theory is to explain why the universe is so homogeneous And also so flat. I should probably have said that Let's say it's a flat universe that that's part of these initial conditions So you want to explain why the universe is flat why it's homogeneous and isotropic You don't want to have to assume that in order for inflation to begin that would be a circular argument So What are the criticisms? So one of them I'll come to this question of initial conditions in a moment, but one of them is I'll just shorthand tuning of the potential So I'll explain I'm gonna explain all these in in some depth in a moment But let me just make a list of the four that I'm going to discuss Another one is That it's unpredictable Because there exists Backwards e means there exists many inflation models Many V if you want the models different models The third one is this issue of the initial conditions are they tuned and The fourth one Which to my mind is the most interesting is The problem of eternal inflation and the measure all right, so let's see how far we can get with these Yeah, so let's start with this this issue of tuning of V. So what does this mean? so We've said that that that you have to have a suitable V meaning Epsilon and eta have to be small okay, so both epsilon and eta Take the form of m-plank to some power Yes, these four I'm gonna I'm gonna discuss them one after another why why these are these are Let's say Challenges to inflation. Yeah Right so both epsilon and eta Take the form m-plank to the end and derivative of Phi times V over V and You want this to be you want this let's say to be much less than V now If your inflation model has the characteristic that Phi is much greater than m-plank Which we saw was the case for m squared Phi squared if you'll remember in m squared Phi squared Phi needed to be about 15 times m-plank or so We had some discussion over that So if that's the case then this is really not very problematic In other words you could just have So then it's okay say dV by d Phi To be approximately simply V over Phi And you don't have an issue So for example for any monomial potential, that's that's the case. Yeah. Why do I require it for all n? Yeah, it's I mean it's really It's enough That v prime and v double prime are small I mean if you had some higher derivative that was enormous I think that would probably cause a problem actually because it would mean that eventually something gets large but You normally assume some kind of smoothness on v so yeah, I mean That there's yeah, if you want let's just restrict to n equals one or two So if Phi is bigger than m-plank Then yeah, then there's no You don't you don't require any sort of feature in the potential you can have for example m squared Phi squared or any single power What we're just fine. There's no problem there You might still wonder how you got that potential in the first place, but but certainly there's no Sort of extra feature that needs to be put in however, if if Phi is much less than m-plank then It's not okay. So then dv by d Phi Must be much smaller than v over Phi and so you need something like you know that You need some sort of feature Where v prime is small a plateau so So one form of tuning of the potential that you can complain about applies in this case where Where Phi is much less than in plank I think I said before models where Phi is bigger than I'm planning to call it large field and and conversely Models where Phi is much less than in plank are called small field And it's also the case that large field models have large R relatively large R and Small field models have relatively smaller and since there's now some observational pressure on The simplest of these large field models as we as we said I'm squared Phi squared seems to be ruled out Well It doesn't mean that we're well into this small field regime We can certainly be somewhere on the border, but but the data is starting to disfavor Some of these large field models So that's one thing you can say about it There's another complaint that sometimes made So sometimes when you have and I wouldn't say this is very general, but sometimes when you have these plateau models So there's some region of the potential which is a small field model But when you write one down you sometimes find that there's another region of the potential which is a large field model Okay, so in fact the same potential may have two regions One of which is a small field model of inflation and one of which is a large field model and which one happens depends on the initial conditions Okay, so for example in this picture. Let's make this very steep here So here you don't inflate here you do here you don't but maybe over here. This is just m squared Phi squared Okay, so you might have a large field model which would describe inflation if the initial condition of the field is over there And a small field model would describe it if it's over there and then another criticism that sometimes made is that these large field models Tend to have a large number of evil a large number of e-folds And if you have a large number of e-folds you produce a lot of volume, and so maybe that's more probable Okay, so this is another criticism That you'll sometimes find that nature seems to prefer in that sense these large field models, but the data Is starting to put pressure on them? so Should we do this should we go through? We go through the criticisms, and then I'll tell you my view or should I tell you my view as we go that's Kind of hard to resist yes So how do you compare infinity to infinity? Yeah, that's an excellent question We're gonna come we're gonna come to that when we come to item four here Yeah, it's a it's not obvious how to do it at all. I completely agree with you But the universe could be finite Okay, well, I think I Think I'll tell you So one at a time because I'm not sure we're gonna get through all four of these today So let me let me tell you as I go what I think about this So for this one so first of all for the tuning of the potential itself It's really unclear how much of a problem it is for two reasons One is that the data has not actually put very much pressure on these large field models in the sense that M squared phi squared was only ruled out by about a factor of two right so it predicted are to be roughly twice the observational bounds and So it doesn't require a very a very big change to evade that there are a number of ways You could modify M squared phi squared to evade it or you could simply consider a different potential Which would look roughly the same and would still be large field It would still have phi larger than M plank, but predict a value of R That's consistent with this so it's not like there's some you know really really strong bound that tells you that you cannot possibly be in this regime So that's one reason why this is not a big concern or at least not not yet The other is that how much you should worry about Having features in the potential like that after all the standard model of particle physics is full of Small dimensionless parameters Right it has some tiny yukawa couplings in it And has lots of stuff in it that isn't very minimal or very simple so It's not very clear. How much you should really worry about that I think you should think about it, but whether it should cause you to throw the theory away. Maybe not yet Did you have a question? Yeah, they are yeah. Yeah Yeah, yeah, no, so so I didn't write there. There's a Yeah, it's something called the lift bound which directly relates the range of phi to R But this This value of R doesn't yet tell you that the range of phi is sub plank in you also have to be careful Whether you're talking about the reduced plank mass or the plank mass that sometimes get but it doesn't tell you that the range of Phi is below the reduced plank mass for sure. It's not even close Anyway, it's a question of order one factors. So it's not like there's some sort of really sharp bound there Okay, so that's one one response the other response which I can't resist telling you is just sort of a specific model Which I think serves as something of a counter example to this worry so Just consider the following V So this is a function of many fields It was one up to N. So there are N. This is a multi-field model of inflation and the potential is A function of all of these fields like this so What is this crazy thing? it's a sum of Periodic functions of these fies these cues You could take to be plus one minus one or zero and these lambdas You could take to be some very high scale like maybe the gut scale So a few orders of magnitude below the plank scale now. Why do I mention this? Well, it turns out that? The that this this kind of a potential has a very complicated structure It has a huge number of local minima. It has of order if P is bigger than but not much bigger than N Then it has roughly the square root of n factorial distinct minima That's a very large number And it has even more saddle points of all types Right, so when you're living in n dimensions a minimum is a point where the first derivative where the gradient vanishes in all directions And all the second derivatives are positive right that the Eigen values of the Chikobi and are all positive and a saddle point is a point Where some of the eigen values of the Chikobi and are negative and some are positive and a maximum is where they're all negative So there's lots more saddle points than there are minima Because there's a lot more ways to make say one of the eigen values negative than to have all of them be positive So there are there are this many minima. There's something like n choose Little p saddles with p negative modes or something like this Okay, so why is that important? Well? This is something like a saddle. I drew it in one dimension, but it's a little bit like a saddle and so in a model like this you have an enormous number of flat regions Which are Which most of which might not be suitable for inflation, but some of them will be and if you just randomly scatter Your starting point in this landscape, so you just take this field space here, and you just pick random points uniformly and You only keep the points which give rise to enough inflation 60 e-folds and you make a plot of our Versus NS You find points everywhere. So this is a logarithmic scale So this is something like one Zero or one minus one minus two minus three and this is point eight Nine one you find points all over the place in this plot Actually more than just in this range In other words, there are all different sorts of inflationary trajectories Some of which have tiny are so they're small field some of which have large are so they're large field Some of which have really big tilt. I mean tilts that are totally rolled out observationally Some of which are in the nice range, which is in here thoughts. Anyway, there's there's lots and lots of points filling the space So this is not the way people usually think about inflation miles historically people have sort of Considered one model at a time they took m squared phi squared because well That's the simplest thing you could think of and then lambda phi to the fourth. That was the next one and so forth Okay, so this is a different way of doing it take a model where we randomly choose the parameters It's got lots of fields. So it has a big configuration space And you immediately find this huge variety of different inflation models Some are small field have small or some are large field have larger you can get almost anything unfortunately So, you know, I think this just shows that it's not it's not at all clear that there's any tuning that needs to be done To find small field models. I should have said that You can choose these randomly and you can choose these randomly You can also choose the kinetic matrix for these fields randomly if you want it doesn't change anything very much So there's no there's no tuning or anything like that. I Promise to provide some research opportunities. So here's one. Maybe I should write them nobody cares, but everybody agrees on so let me erase that part and Can write some research opportunities up there or even when everybody agrees So I think one interesting thing to do is to study models like that multi-field random models of inflation and in particular one One thing that has not been done is to study non-gassianity, which we did not discuss but It's interesting because these trajectories Especially the ones with small r Tend to take turns. They're not in other words. They go around curves. They're not in straight lines And this will lead to to interesting signal signals Okay, anyway, so that's so that's a I Think that's one way of looking at tuning if you if you choose everything randomly and the average of everything is You know the gut scale you can't really be accused of tuning anything But but but you still managed to find a big variety of inflationary trajectories Okay, which brings us nicely to the next one that it's unpredictive Because there are many possible potentials and those different potentials give different results M squared phi squared gave us gave us our Which is now ruled out and a particular value of the tilt which is about right If we studied other models, we would have would have gotten different values for for RNNs And here you see a model that gives lots of different values for them depending on where you start So how big of a problem is that? So for a while it seemed like neutrinos might be massless and then Through various means to the Sun Presence of neutrino oscillations and so on it became more and more clear that neutrinos have mass We still don't know what the masses are exactly. We have various constraints so The standard model of particle physics with all neutrino masses zero Was falsified and it's now been replaced by a new model in which neutrino masses are not zero But we don't know exactly what they are No one's worried That that means that Quantum field theory is unpredictable. No one's worried that that means that the standard model is not a good theory or something That's total nonsense Really, you have some class of quantum field theories You collect data and you zero in on the one that correctly describes your data And then you try to test it and see if you can zero in even more So maybe it would be fair to say that we didn't know what the neutrino masses where there were some constraints We knew they were light, but and with data we zeroed in more On those but really what people did actually is add the new parameters you need to add To tank into account with you know neutrino masses when it was shown they were non-zero so You know in that in that analogy if someone asked you is quantum field theory falsifiable Okay, maybe You can maybe think of something but there's a huge class of quantum field theories and they all make different predictions So it's really hard It's really hard to find something which would falsify all of them And that's not what you mean when you say a theory is falsifiable What you mean is that you have a specific theory and it makes specific predictions Which can be tested and you can throw it away if they don't match experiment So by that standard inflation is certainly falsifiable because any specific model makes very specific predictions for NS for R for the tensor amplitude of tensors, etc and We can see that in action. We've already ruled out The one I used as an example M squared phi squared so I Think you know, this is a bit philosophical But but I think it's important to say in this context that that you know It's it's it's a big ask to try to rule out an entire class or a whole paradigm It's not very easy because these are big kind of ill-defined big things with lots of theories and yeah Yeah, it no it's a good question and I'm gonna come back to it next time if that's okay I'm gonna talk about trying to build inflation models from the top down Yeah, no, I mean the vast majority of the literature totally ignores that it just deals with Classical plus small perturbations around the background doesn't worry about we normalizability doesn't worry about anything You know people deal with with models with very high powers of phi and it's just sort of Or or I mean what's a little bit sort of more? Well grounded is to look at just effective field theories of inflation where you explicitly Don't worry about the UV and you just write down the most general effective field theory you can You know, but yeah, I'll come back to that sort of question in the next lecture Okay, so so What was I saying right anyway, so so yeah, so I think falsifiability. It's really It's really something you should think about in the context of a specific individual model not some big class Right and It's also worth pointing out that if the universe was very different than it is Okay, good. Yes. You're worried about this. Yes So so these these kind of potentials come straight from a UV they come for example from string theory compactifications So those field are acts fields are axioms and there's a there's a shift symmetry Which is why it's a periodic function. It doesn't have to be cosine exactly, but there's a some kind of periodic function there So this is still an effective description. It's it's the it's the theory of the moduli It's a theory of the light degrees of freedom. It's not the whole theory, but But it comes from a from a UV completion. So it's reasonably solid Yeah, so so Right, so if we lived in a very different universe like one that had large curvature Let's say we knew the universe was spherical because we could send light rays and they would come back Or something or we look at Andromeda and we see it that way and that way because we live on a torus, right? So something like that then Or if we looked around and we saw a spectrum of perturbations, which was totally not scale invariant Maybe we saw galaxies at the vertices of a cubic lattice or something like this Okay, so you can imagine all sorts of crazy scenarios where inflation would certainly not be the right description any theory of inflation Be careful, maybe someone can come up with one that produces perturbations that vertices of a cubic lattice, but I doubt it so so In other words the reason we came to it the reason people are interested in it is it does a good job of describing the universe We live in and you shouldn't hold that against it that would be absurd, right? Okay, it's hard to falsify now because it does a good job of describing reality. That's not a reason to throw it away anyway, so so I think Again a little bit philosophical, but enough said about that Okay What about this initial conditions? so So what's the issue with these initial conditions? Well? One argument is that Let's say you don't want to assume that the whole universe is Homogeneous and isotropic before inflation because that's that seems really much too much. That's basically assuming the answer But you do want to say that at the moment that inflation begins it begins in a patch So at TI for initial or inflation That there's a patch a homogeneous patch and it's of size But over hi, okay, so you have a Hubble patch at the beginning of inflation Which is which is homogeneous, so you need that to use Any of the approximations we use in deriving this at least at the beginning Okay, so let's say you just want to assume that and then you ask what does that mean? About the past we know what it means about the future the universe is going to start to inflate But what does it mean about the past? So let's go back down towards T plank Well, we talked about this On Monday You have to assume something about what? Dominated the universe prior to inflation before inflation. Well, it certainly wasn't accelerating because otherwise it would have been inflating Let's say it was radiation dominated just to pick something simple And then what we saw is that if you extrapolate back I Should start with one Hubble patch. Sorry, so here's my initial Hubble patch So this is this one over hi and size and now let me extrapolate back What we learned was that this Contains many disconnected causal regions and if you plug in the constraint on how big that can be and You ask how many of these regions are there you get something like 10 to the 9? so a billion Hubble regions Hubble volumes at T plank So if you needed this to be homogeneous on the Hubble scale at the beginning of inflation You would have had to assume that in the early universe you had this vastly super horizon region, which was homogeneous and That certainly doesn't sound good because again the whole point of inflation was to explain Why the universe today doesn't have this problem, right? This is the horizon problem. So So yeah, if you really needed to assume this if you really needed to assume that there was this homogeneous patch And that you could just smoothly evolve it back You would run into this problem There wasn't a very good counter argument to this Because if you assume anything other than that Then it's hard. So if you don't have any homogeneous patches in the universe Then the universe is very inhomogeneous and since gravity is non-linear it acts on those inhomogene in homogeneities in complicated ways It forms black holes forms horizons. There's gonna be singularities that develop and you can't use perturbation theory So the standard tool of cosmology which works well when you're close to FRW is to treat all the perturbations Well, it's to is to have small perturbations and then expand in them right to do perturbation theory If you don't have that if you have large fluctuations big in homogeneities, you can't use perturbation theory You can't do anything analytically basically because Einstein's equations are too hard to solve. So you've got to do it numerically but until relatively recently like 10 or 15 years ago Even solving Einstein's equations numerically was impossible when black holes form Now but thanks to all the work that people like Frans Petorius and the Compact object community has put in we now have numerical codes which can handle the formation of horizons and even things like colliding black holes So it can handle gr in quote-unquote the strong field regime the regime where there are horizons So and some of those codes so some of these black hole merger codes use periodic boundary conditions, but some of them use Periodic boundary conditions if you have periodic boundary conditions in three dimensions There's a name for that manifold It's t3 topologically Okay, so What does it mean? It means that what those codes are simulating is a universe which is t3 Times time this is a cosmology according to the definition. I gave at the very beginning So we can use those same codes to simulate early universe cosmology In a situation where we don't assume anything about any homogeneity, so we can take initial conditions which are dominated by Gradients by in homogeneities. So if we have our scalar fields, we can make del phi squared Facial grad phi squared the largest term in the energy Okay, so that means that there are big in homogeneities in Phi and then we can stick that into the computer We can do that and run it And see what comes up So here's another research opportunity Take a numerical gr code and study cosmology with it people are starting to do this But just starting really Mostly those codes are used for black hole mergers for good reason those black hole mergers are very interesting and it's a Obviously a very timely topic, but again those same codes can very easily be adapted to study cosmology and problems in cosmology All right, so what do you find if you do this? Well, you find that essentially all the time Although you started with the very unhomogeneous universe so you did not have a large Look like you were gonna have a large Super horizon homogeneous patch. That's what this argument seemed to imply But if you start from here without that With everything dominated been by your homogeneities and you evolve forward You find that you always or almost always produce patches like that and inflation begins So inflation starts Anyway, and this by the way is work with Leonardo and will East and under Linda Okay, so So why is it well basically it's because when you have in homogeneities they tend to clump gravity is attractive in homogeneities collapse into black holes in these simulations Because there's nothing to stop them from forming black holes, but they collapse into gravitationally bound dense regions and meanwhile the rest of the universe goes on expanding and sort of leaves them behind and Eventually it consists of a bunch of voids which are empty and homogeneous With a kind of dust scattered about which are these black holes and so when those voids get large enough It's this kind of patch and inflation begins. So it's very much not frw It's not homogeneous But inflation begins anyway There are some caveats You need large field inflation to make this very generic and you need So it depends on your potential if your potential has I don't know Some region that's suitable for inflation and some region that isn't like down here Then if you start with a field that's homogeneous and sitting in the minimum you're not going to inflate But if you start with a field which has large fluctuations in it, but its average value is somewhere up there Then you will Okay, so it's not like inflation always happens What does always happen is you always form homogeneous regions? But it's not guaranteed that those will begin to inflate it depends a bit on the inflationary potential there has to be one and on the Initial conditions for five if you have a small field inflation model, it's harder it turns out in a situation like this that even if the average value in space of the Inflaton is sitting on the inflationary plateau and so if the universe was homogeneous it would inflate if The fluctuations are big enough so that in some part of the universe the field is over there Then this large gradient over here pulls the average down into the minimum rather efficiently And so that can prevent inflation from beginning So it's certainly not the case that all inflation models inflate all the time with any initial condition That's certainly not true. What is true though is that? Initial conditions, which without inflation would have produced a universe that looks nothing like the one we live in it would have been very inhomogeneous highly curved etc So initial conditions like that In many cases will still Begin to inflate if there's an inflationary potential and then produce the universe like the one we live in Okay, so then in other words inflation does solve the horizon and flatness problem problems for at least some set of initial conditions All right, so that's What I want to say about that I Have five more minutes. Okay questions. Okay, so yeah, no these don't include any quantum effects. That's another That's hard So I'm not gonna write it, but if you're brave you could try to try to include quantum effects Yeah, I mean so I'm I'm starting after the plank time When it's at least plausible that quantum effects are small But still it might be interesting to try to include them. Okay And I just repeat you mean yeah, so so Well, let me let me run it starting from from the initial time So let me start with the universe that's very inhomogeneous. So it doesn't have any smooth patches on any scale certainly not on super-horizon scales run so what happens is that the The large inhomogeneities do something complicated. There's lots of dynamics But they tend to clump together after a while So in other words regions with high density tend to clump together because they attract each other So they may orbit they may fly around for a while, but eventually they attract each other and They attract each other a lot they form a black hole Okay, at least in this simulation which only contains a scaler So there's not much pressure to prevent them from forming a black hole. So they form a black hole sometimes more than one Meanwhile so those regions stop expanding because they're inside a black hole or near it But the regions around them which have been sort of evacuated by this black hole pulling the over densities into them those regions expand and they grow and So after a while this three tourists it has some complicated metric on it. It's not at all homogeneous It's it's got black holes in it, but it contains these big voids Which are empty of much of anything and in those voids after a while After they get sufficiently empty you'll be dominated by via Phi Even though via Phi was a very small component of the energy density originally well It actually is going on today. Yeah because of dark energy, but Yeah, I mean the point is a few black holes don't prevent inflation from starting actually We know that now there's plenty of black holes in the universe and yet the universe is expanding exponentially Right, so a few small black holes are just some kind of very minor pollution Inflation doesn't care. It's something that takes place on much larger scales than that Of course, this is more interested than that because there are large perturbations on horizon scales from the beginning So it's less obvious what will happen, but that's what happens. Yeah The tuning for the initial conditions you mean for inflation to happen. Is that what you mean? Yeah, it's Yeah Well, it's I mean, okay It's it's hard to put a number to it because to put a number to it you have to assign a probability to a given initial condition you need to measure on the space of initial conditions and Once you've done that there's a definite answer for the tuning, but people can come and argue with your choice of measure Right, so there's there's no con on it. There's more than one canonical way to do that. Let me say it like that so so in this simulation what was done was to take the initial perturbations the initial value of Phi to be a Sum of sine waves with random coefficients and random phases And so when I say random that was with some uniform distribution over some range Okay, so so within that set it almost always underwent inflation But there are obviously points in that set where it would not for instance if all the coefficients are zero and the field is sitting at its minimum Then it won't inflate because there's nothing to drive it. So Yeah, so so how probable is that well not very probable if you're choosing these things uniformly But if you think that the point where it's sitting in the minimum is much more probable So you have to have a measure on your on your set of initial conditions So it's it's hard to answer that question all I can really say is that there's an open set of initial conditions which without an inflationary potential would have given rise to a very inhomogeneous universe That with this inflationary potential give rise to a very homogeneous universe So it certainly solves the horrendous flatness and horizon problems for those initial conditions That's the thing you can say for sure so Well, okay, I guess this will remain this is good because this fits nicely into the bigger picture So what we'll do next time? We'll talk about eternal inflation The cosmological constant problem, so I understand that that Claudia did a great job explaining dark energy Telling you about models of dark energy, but she didn't tell you how to solve the cosmological constant problem so I'll tell you how to solve it and And I'll solve it with a landscape. So I'll tell you how that works And and part and parcel of that is eternal inflation, so we'll talk about that So this is both a problem and an opportunity in some sense And then I'll give you some more ideas for research opportunities and try to give you sort of an overview of Where we are right now, so thank you