Cosmology (Lecture - 01) by Nima Arkani Hamed

okay so we'll start the afternoon session so we're very glad to have NEMA with us here from Princeton he's going to tell us about cosmology back to the future all right it's fantastic to be here at the school and at the ICTs as always so when I think it was right Josh emailed me and and suggested that I talk about cosmology at the school I was really delighted because actually it's a it's a subject that I've I've thought about now and again over the years and it's something I've been meaning to think more about there's a good excuse to start thinking about to start thinking more about it again as well and in a in a very in a very real sense very much of the work that that I've been pursuing over the past decade or so especially in the context of trying to understand and new kinds of structures that might underlie scattering amplitudes engage theories and gravity have really been a warm-up to thinking about questions about cosmology and the wave function of the universe and that's one of the things that I will try to explain in these set of lectures so this is not going to be a canonical introduction to cosmology there won't be any pictures of the CMB I won't review slow-roll inflation how many of you know about slow roll inflation something okay many of you know very good okay so those of you don't know you'll learn what you need to know from the lectures they'll be relatively self-contained except for this lecture that'll be a little slightly more impressionistic in in in parts but they'll be pretty self self-contained so you can learn all those things in in many many places by now down very very beautifully I want to develop cosmology from very particular point of view which is the point of view of trying to get rid of times in our description of cosmology and so let me let me just start off making a few a few general remarks about about the whole subject I mean cosmology is the ultimate historical science there are many historical Sciences cosmologies you know the most glorious of all the historical Sciences its main driving question um well theoretically and experimentally is what happened in the early universe and I think either stated or unstated most people's motivations for pursuing this question is that it might help us get closer towards the still still vague still not well-defined but of course extremely fascinating question of what might what might have been the origin of the universe so it's a historical science with very grand questions but it's a historical science unlike every historical science it has an interesting relationship to the concept of time okay because all the interesting stuff happened a long time ago when no one was around none of us were around back then all we see are some are some patterns today that we infer the existence of a past in order to make sense of and we're much more familiar with this you know even as kids in the less glorious still fascinating historical sciences like paleontology right a paleontologist says there used to be dinosaurs walking around the earth it's tens of millions of years ago why because no one was around back then we weren't around back then but because today in the ground we see these big bones and then there are little bones inside what looks like the stomach of the big bones I'm making this up ok but but it's probably never quite so quite so so so direct but but but you sort of infer yeah there are these big monsters walking around they ate little small things right because that makes sense of the pattern you see a little closer to home the actives are historical scientists right a detective said says that person a murdered person B yesterday why he wasn't around yesterday it's because the bullet from the gun owned by person a is found the body of person B today okay and so this pattern in space today is explained by imagining the existence of a past which gave rise to it by some rational rules okay so so that's different than the notion of time that we talked about in other things it's it's a distinction between the fact that cosmologies and observational science versus the fact that for example particle physics is an experimental one because we can get to set up initial conditions and wait in time and watch it evolve into the future whereas here something happened a long time ago we had nothing to do with it and we have to infer the existence of a time in order to make sense of it now set more formally what we can get to talk about in cosmology are measurements of spatial correlations in the very very late universe and especially from the point of view of sort of holographic observables when you have quantum gravity to get very fancy for a moment the only kind of observables that we can talk about when we have quantum mechanics and gravity forced us to go to some boundaries of space and time because we have to be able to do observations with infinitely large measuring apparatus and we have to be able to do the observations infinitely many times in order to get precise quantum mechanical observations okay and in cosmology so in in in an TV studio space we all know what that means in flat space it's a little little bit more it's a little bit less familiar but there are some null boundaries at the boundaries of Minkowski space in cosmology had we been in the situation where we had an expanding universe and then the universe opened up and eventually became infinitely big in and it just continued the salary became eventually infinitely big then the only kind of observable we could talk about in such a situation you could still wait you would see the universe become you'd see an infinite amount of sky and you could lie on your back and you could ask the question what fraction of stars are red what fraction of stars are blue ok and so now that's an averaged question so because the world is quantum mechanical we have to we can't say anything deterministically the only thing we can do is do an experiment over and over again to get a totally precise answer now here we don't get to do the experiment over and over again there's only one experiment that was done but the proxy for doing the experiment over and over again is averaging over space and that uses the fact and assuming translational invariance it uses the fact that the same experiment happened over and over and over again in all the different parts of the universe so that if you average things over everything you see in space and if what you have access to becomes infinitely large you get to do an infinite amount of averaging then those are the precise observables that you're allowed to talk about ok so we go from anti-de sitter space where we have boundary correlation functions very rich set of objects to flat space where we have much lots to talk about but still a fair amount we have the scattering matrix which is the what we're of course used to talking about as in practical physics all the time and when one step further we go to cosmology or these at least these cosmologies that open up into infinitely large universes and we don't even get we don't get the control the initial state but we get to ask these average questions about spatial correlations so if we ask more if we ask more practical questions what our cosmologist friends have been doing for it for 20 years and what they will be doing for another 20 or 30 is measuring density perturbations ok so so here's an example of something so you can imagine and and hopefully we'll eventually really be able to to measure these things three dimensionally but the first thing you might imagine doing is looking at that that's the two-point function and what does that average mean that average means that literally you measure what Delta Rover is here you measure what it is there and then I mean if you literally measure it here or there it might be you know negative to ten to the minus five plus three ten to the minus five at literally these two points but you don't just do that you take that product and then you average it over all the places x and y can be with the same fixed separation right so that gives you a function of X minus y and that's the two point functions all right so because of our assumption of translational invariance we can nicely do this in position space we can actually it's really effective to go to momentum space to do this and if we go to momentum space and lanterns face so via Delta K 1 plus plus K 2 and now there's something here that that that depends on the overall K and you know in the real world when we say we have density perturbations at are around 10 to the minus 5 and that there are approximately scale invariance then that's what that's what we're saying okay we're saying that this two-point function in momentum space is around 10 to the minus 10 1 over K cubed the 1 over K cubed is there just in order to make up the unit's this thing is dimensionless and when we transform to Fourier space those are the mass dimensions it has to have if we say this back in position space just directly in polluted space this is a this is around 10 to the minus 10 and it has a logarithmic dependence on X minus y over some IR cut off okay but of course we could talk about all kinds of other things in principle we they have not been measured yet partially because this is these perturbations are so tiny these things are all going to be very very small but in principle we could talk about endpoint functions and once again we could go to Fourier space and so there is some function of a bunch of momenta ki and there's a momentum conservation and so for every sort of polygon you can draw on the sky so I have a bunch of K 1 K 2 K 3 and so on they all add up to 0 so for every polygon you can draw in k space there's a number associated with it right and those are all the sort of spatial correlations we have in the pattern of density perturbation so there's an ocean of information in these objects so there's lots and lots of information and a big part of the next 30 years of the experimental program in cosmology as I said is to try to see if it's possible to measure and much better bound already the three point function which is just a function of a triangle and a triangle is really fully specified by just giving it's three side lengths so just K 1 K 2 K 3 so so this is a function of three lengths K 1 K 2 and K 3 and so that's something that if it's there we'd like to we'd like to see it we'd like to measure it and it would have an enormous enormous amount of information much much more information than just this one essentially one number there's a little bit more because the dependence is not exactly scale invariant but there's obviously a huge amount of extra information in measuring the 3 point and higher point function okay so these are and precisely because these numbers are small there is perturbation theory in whatever's going on the underlying physics if you imagine it's inflation giving rise to these things it's very weakly coupled so perturbation theory is is going to be great and that's why all the calculations of course people doing these computations explicitly are all all carried out perturbatively so I'm just mentioning this right now is just the concrete example of the kind of observable that we're talking about again we assume in order for them to be precise quantity observables we have to assume the universe becomes infinitely large blah blah blah our own universe is not like that because of our de sitter expansion or accelerate expansion now and that's one of the that's the largest I think conceptual challenge probably in all of physics it's so hard it's very unlikely I think anyone's gonna make any progress on it in any time period I could imagine right now it's a very hard problem to figure out how to deal with the fact that the accelerating universe gives us a finite amount of stuff we have access to and that means that it simply makes it impossible to divide the world into an infinite measuring apparatus and the finite systems that are being looked at that's absolutely necessary for quantum mechanics to make precise predictions to have precise observations in quantum mechanics that might be the precise prediction of a quantum mechanical theory so we don't have access to that of course in any practical sense universe is infinitely is very very large and so it's perfectly it's perfectly good enough although there are of the important instances where we're making measurements involving modes that are almost as big as Hubble where this issue of cosmic variance comes in the fact that there's a very small number of modes means the experiment was done a very small number of times and so we're really in the soup there that there are things that we won't be able to do better okay but so I'm so I'm definitely going to ignore that very deep fact about the absence of proper quantum mechanical observables in our physical universe because of all the complications and mysteries associated with de sitter space okay now what is the object that's supposed to calculate all of these things okay the object that's supposed to calculate all these things is the pompously named wave function of the universe okay and that's because while all we're trying to do is make predictions for all these spatial correlators so if there's let's say a scalar field associated something gives rise to Delta R over o like an inflow town or something like that well then there's going to be a wave function that just tells you sigh mod squared will tell you what the likelihood is that that field configuration you'll see in you use that wave function to compute expectation values and those expectation values will be exactly these things that we're talking about okay now so we could just directly jump to talking about the correlation functions or we could talk about the wave function first and then still leave as a step to go to do the averaging to do the to compute the expectation values using this wave function and most just technically in most of the inflationary literature people like to use a formalism where they calculate the correlation functions directly if you do that use the stringer Keldysh horrible ism etc are perfectly fine the purpose of these lectures I want to focus more directly on the wave function it has slightly nicer properties and I'll be able to talk about a few more things if I do that okay but now but now comes the sort of main the the the main question and the sort of attitude of these lectures that I want to explain the question is what are the rules what are the rules for that govern what this object is what I mean is if someone comes along we know what the rules are if someone hands me some some theory some inflationary theory I have an action I can calculate we're gonna review that we're gonna read that and talk about it in some detail as we become comfortable with doing perturbative calculations of the wave function in the universe okay no problem we're gonna do that I said this Lex will be slightly more impression but we'll do that in the remaining lectures that'll be a big big big part of it so then you shout okay then III know what the rules are but I'm not asking that you see the this thing doesn't depend on time he knows nothing about time this thing is just a property of the late universe okay so all the measurements were making our measurements in the late universe and I'm asking if someone came along and said here here's a wavefunction the universe I did the calculation hey I'm just um don't worry I did it right okay you're like gosh I don't want to do all that work they probably did it right but what do you check to see if they were right or wrong okay what are the invariant properties that the that the wavefunction is supposed to have in order to be consistent with having arisen from reasonable unitary causal local evolutions in the past history of the universe okay we don't know the answer to this question actually I think we're not particularly close to knowing the answer to this question and I think it's a very important question to try to do some to do some work on and try to understand more more sharply okay let me contrast the situation with I'll just go go down from here so one step down from cosmology is the question about the S matrix okay now you can say do we know the rules for the S matrix so I do the same thing all those finding diagrams are very complicated someone computed them they hand you the final answer how do you know if they're right or wrong okay now here you say okay well the answers to be Lorentz invariants okay yeah fine this Lorentz invariant it should be unitary that that seems pretty tough but needs to satisfy s dagger s equals 1 and so you might think just Lorentz invariant and unitarity are very very powerful constraints and and if you satisfy them more or less you you have to have a reasonable theory completely false okay it is a triviality to write down a totally non perturbative exact exactly unitary s matrices that are Lorentz invariant and complete garbage ok and the reason is a simple one it's not unrelated to what we're talking about here because again you know you're an experimentalist at CERN you wake up in the morning you collide your protons you have your coffee you spent a long time doing it I'm not saying they're lazy you go back in the afternoon and you look at what came out right there's this a bunch of histograms a lot of stuff came out so you weren't around you weren't in the proton when the gluons hit each other then another gluon hit it another quark hit it and so on just you threw some things in you close your eyes you came back they all came out and now I'm asking the question what in that data it carries the imprint that I came from consistent unitary local evolution in the interior of the space-time okay ah we don't know the answer to that question we don't know the answer to that question even in perturbation theory today and the people in the 1960s this was the heart of the s-matrix program you see this is what they this is well we roughly know the answer to this question we roughly know the answer is that the S matrix has got to be the scattering amplitudes that need to be suitably analytic functions of the kinematic variables okay they can't have like random sharp very terrible singularities they have to be politican ufff but they're not they're definitely not that they're definitely not free of singularities they're as branch cuts they're sort of a complicated singularity structure but the causality is somehow imprinted in the analytic properties of the S matrix but you can ask is there a god-given can we derive do we know what those analytic properties are a priori ahead of time no we do not and if you open up if you have the misfortune of opening up any of those old books on the analytic s matrix they strongly discourage you from doing then you will find exactly this that they have all these great wonderful words and then they're like gosh what should these I don't like properties being I don't know let's look at some Fineman diagrams okay let's look at some time two and then and then the rest of the books are about looking at Fineman diagrams and seeing the analytic properties and then pretending that they didn't know and going back and so on oh we still need to know look at some more Fineman diagrams so they didn't know okay we still don't know today even though we know much much more about the property of scattering happily we still don't know today the situation is slightly better the site the situation is a little better in perturbation theory we know exactly what locality and unitarity mean a tree level precisely that a tree level the amplitudes are just rational functions of momenta and and and maybe huh Lissa polarization vectors they have poles their poles have got to be in very particular locations only when a sum of the subset of the momenta goes on shell that's an imprint of locality and the amplitudes have to factorize on those poles that's the imprint of unitarity and so you can check if someone hands you the answer and they say you know you know I did the Fineman diagrams for 10 glue on scattering you can check if they're lying to you by seeing if the poles are on the right place and whether we factorize properly we also know the answer at one loop because at one loop the structure of the transcendental functions that show up now now they're not they don't just have poles they have branch cuts but there are well enough understood it at one loop there are in four dimensions there die logarithms and D dimensions they're they're like in six dimensions there are four logarithms and so on and we understand their structure well enough to be able to say exactly what kind of discontinuities they need to have in order to be compatible with what the causal evolution description but beyond one loop that's a it's a topic of active active research to try to figure out what that means things are a little better still if we back up from asking for the final amplitude at the level of yet the final functions but we stop at the level of what you can call the integrand of the amplitude what you get before you integrate over the internal loop momentum and at that level things are almost perfect still not quite but their own thorough they're almost fully understood because they're we know that unitarity and causality have to do with where now we have rational functions again you see what we're not they still just have sort of normal poles and now we know they have to have poles in the right kind of spot and again if you put internal particles on shell now there should still be a kind of factorization so the sort of textbook picture of the cutting rules and unitarity and so on is something that makes a lot of sense at the level of the of the integrand but but I want to stress that even for something much cleaner a much more familiar people have done decades more work on after all this time we still don't know the rules even for the s-matrix now yeah I think that's that's the the the the problem is that you're thinking about the obvious branch cuts and there are lots of unobvious ones associated with these funny anomalous thresholds and and other things like that so if what you're saying was true there would be a there would be a well-developed program of just of using this kind of direct unitarity methods to calculate things even at one loop it's not even that easy at one loop although you can do it but you could keep going if you've done it you keep going even numerically you could do it a tool no one has done it ok so anyway let's let's go one step further the things that that are and by the way notice that in this in this passage we're we're going sort of further and further from the real world right we started from the center we can't say anything about at all or accelerating you can't say anything ignore that cosmology ok so our world isn't Minkowski space this cosmological we know almost nothing Minkowski space much closer to the real world we know fair bit definitely don't know the answer even in perturbation theory not not particularly 80s we know everything ok we go to the 80s box now everything is wonderful ok so when things aren't the real world things are wonderful when things are close to the real world we are confused I think there's an important clue there ok and the clue has everything to do with with time and the questions of the night of the interesting kinds of functions and and singularities that we get when time evolution matters because in 80s we know the rules perfectly might be it that we know if someone hands us a bunch of boundary correlators they have to they they have to they have to have the correct short distance behavior you have to have the operator product expansion we're done so the rules are perfectly well-defined ok mmm now going back to cosmology I want to make another qualitative comment and actually Surat is one of the people whose us responsible for bringing this to many of our attentions that in fact so here's a one surprising consistency condition on the wave function of the University there's a very basic question the stuff that gives rise to these spatial correlators is some normal kind of particle physics elementary particle physics something right so for instance the physics of the of the scattering of electrons and photons or all kinds of other stuff ordinary ordinary physics of even weakly coupled scattering goes in to giving rise to the late time wave function of the universe so so one consider today should somehow be that that that that wave function is compatible with coming from a unit being associated with something that in flat space will give you a unitary S matrix but it but if you're just thinking sort of naively you would think how is that possible you're asking a completely static question at one late time how is it possible this would know anything about scattering of elementary particles and so on but in fact it's very possible um because if we just stare at our data here let me go to imagine we're talking about a four point cosmological correlation function so this is way a cosmologists would draw it with a k1 k2 k3 k4 the way particle physicists might draw it would be K 1 K 2 minus K 3 minus K 4 it looks exactly like a scattering process hey draw both the correlator and the scattering involved for spacial momento that add up to zero because of momentum conservation ok so the amount of data is looks exactly the same in fact it's these things depend on precisely one extra variable relative to these ones and that's because here we have something else here we have in scattering processes we have the sum of momenta of the particles but we also have time translational invariance so we have a sum over energies a delta function of the sum of reneges and then we imagine some particles are incoming some are outgoing in order to be able to make that delta function equal to zero some energies are set to be positive or something set to be negative here there is no analog of energy conservation there's no analog of time translational invariance so all we have is the Delta function for the sum of the momenta so a cosmological correlations or the data that goes into the wave function of the universe depend on exactly the same things as a scattering amplitude except for one extra variable and the depend on precisely one extra variable which is the O is the sum of all the energies and where if by energy there's no actual particle here I just mean literally like square root of K square plus M square or or something like that okay so so this actually has despite the fact that it's purely spatial blah blah blah it has actually more a little more information in it and in fact in a precise sense and I'll just give you a an impressionistic example now in a precise sense the cosmological wavefunction actually contains in some approximation the scattering amplitude this is a really remarkable thing it's it's it's it's mathematically extremely simple but I think it's physically very important and let me just give you a rough feeling for it so let's say we have a 2/2 to process with a fight of the four coupling then the amplitude that we'd get for that the amplitude we get for that is you know some coupling lambda and then there is that delta function for momentum conservation and a delta function for energy conservation but and again we'll review this at greater length in the rest of these lectures when we do this when we instead ask for a four-point correlator in cosmology then instead of just things coming in and going out to infinity we're doing calculations over some conformal time a de that ends somewhere okay so just to get a rough idea of how these calculations could be related to each other if we go back and think about where all of this came from for the ordinary amplitude of course everything is Lorentz invariant so we treated them on the same footing but but this Delta function it's coming it's coming from the fact that we had time translation variance and it arose from an integral over time which I'll call ADA here e to the I II 1 a 2 e 1 plus e 4 ADA this integral is from minus infinity to infinity ok so so that's the that's the that's where these Delta functions in energy space come from in their origin in position space ok I was just positioned spaced for time now again we'll see it in quite a bit of detail later but but I think you'll find it pretty reasonable that when we do this calculation it's something very similar except the integral it's not going to go from minus infinity no go for - and being 0 and it's gonna get get cut off and so instead I get what I'll get for the what I'll get for the correlator here is just still the spatial momentum but I get the integral minus infinity to 0 e to the I II 1 beta and so this gives me instead lambda over ignoring all factors of I and so on lambda over u1 up to e4 that's some right so whereas before hand we had sores before hand we had a delta function of e 1 plus e for now it's replaced by 1 over e 1 plus e 4 now in fact when we do cosmological calculations this is literally what we'd get if we were in flat space and if we're actually calculating the vacuum wave function with no time dependence here at all that's exactly what we get you know more generally we can get powers here and we can get other things upstairs but that the basic fact remains is that where before we had a delta function like singularity when the sum of the energies goes to 0 now now there's nothing happening there when all these energies are like they're normal there are positive numbers because here they're literally just square root of K squared plus M Squared ok however if you if I just go back to this simple example where it's literally that if you analytic ly continue these energies so that some become positive some become negative then it's possible to reach this thing as a pole and the residue on the pole of this cosmological correlator is the s-matrix okay in this case it's pretty trivial because it's lambda in in both cases but there's a simple reason for this the simple reason for it is that is that where where is this some of the energies coming from it's because the only possible place I could get a divergence when I'm integrating over time is when I do these calculations everything is chopped off in in the future the only possible place I can get a divergence is when of all the times go off into the past and so what I can do is is and whatever whatever complicated thing I might be I might have going on I might have loops I might have whatever I have I can take a big circle and surround all of them together and if I take all of these times to minus infinity together so there's some sort of center of mass of all those times a to center of mass if I take all those center of mass off to minus infinity together then there is in front of the whole thing again just by but by the approximate time translation invariance in front of the whole thing is there's a 1 over the sum of all the energies of the external particles ADA ADA center of mass and then I have lots of other integrals left left to do but minimally I have this one okay and this integral could give me a divergence as I go off into the past it's being chopped off by this oscillation here okay so if you continue so that this sum goes to zero then you're gonna get a singularity right you're gonna get it you're going to get a singularity as as the overall center of mass time goes to minus infinity but precisely in that limit you see precisely when all these things are going off to minus and pain together the answer doesn't really know about this boundary far far far off in the future and therefore the result of your calculation is essentially the same as if the boundary wasn't there which is the flat space scattering amplitude alright so in this beautiful way this some singularities of the the cosmological wave function of the cosmology correlators actually contained in them information about the flat space scattering amplitude so have we not gone through this argument known only these things I think this is not this is not something super obvious ahead of time but this is one consistence you can do someone comes up to me hands me a perfectly healthy looking wave function universe I checked that it's normalizable everything looks fine then I might then go investigate say there let's see words where where are its singularities and I would look to see if there is a singularity when the sum of all the energies goes to zero and then if that thing does not look like a healthy s matrix and including it needs to be unitary it needs to have all the properties all the unknown properties of the estimation X for its consistency then there's something wrong right so that tells us two things it tells us that that these these cosmological questions are a beautiful one parameter generalization of scattering amplitudes to really single variable the sum of all the energies since we know there's been so much magic found in the structure of scattering amplitudes B it seems very unlikely that that magic is restricted to this contention one surface where the of all the energies goes to zero it's sort of very likely that there's something that extends beyond there and actually controls the structure of the cosmological of the cosmological wave function itself okay any questions about this okay so the first thing that we're gonna so we're gonna try to do two things in these lectures we'll see how we do but the first thing that we're gonna do is just try in a few examples or a few classes of examples to learn how to calculate the wave function of the universe and we'll begin by you know we'll begin by doing things completely totally standardly with time integrals and the way everyone does them but then we're going to sort of take a we're gonna approach it from a slightly funny philosophy of of trying to get rid of any reference to time in the calculations okay because what we want to do is understand more directly what is the what's the purely boundary property of these objects that determines them and when we begin with these things where they have a perfectly standard representation in perturbation theory we have a good start we can play around and see whether we can think about it in that way this is gonna have as we'll see it'll have some concrete applications because in in even many of the very very simplest and some of the very simplest questions involving you know something as elementary as if you imagine you have something like this you know a four point function where you are exchanging a massive particle okay so something where if it was an amplitude you could do it when you're a child that object is is not calculated in the literature okay and well we'll see why I mean in total generality even you know if this particle here is a a general mass it's just a function of four momenta when we do our inflationary when we care about inflation in fact there's even a slightly simpler version of this where essentially one of these legs is put to a background and it's really it's really a for particle calculation but really with one of them a method going close to zero so it only depends on three momenta okay so this is something that's relevant for the measurement of the three-point function in non non Gaussian ities if you imagine there are particles with the mass close to Hubble during inflation and they may have been produced or close to being produced with some small probability during inflation and you want to know you know exactly what you should be looking for what what what this this tree-level thing what is it you can write down some integral we can compute it numerically but there is no nice sort of analytic expression understanding for what it is what it's understood in various limits but not in but not in a not in an accessible way analytically this attitude that we're gonna take to try to get rid of time to do cosmology without x which in these cases are turn to very elementary things nothing very very deep but they're gonna allow us to do these calculations so we can get analytic answers to these things and yes just because in general master that's right something as simple as that right and but in particular and and I'll talk about this a little more detail in just a second there are aspects of these of these calculations that really seem that I have to do with the physics of time in a crucial way for example something we'll just talk about in a moment is that during the inflationary phase there might be some small probability that due to the time depends on the background you actually produce this physical particle M hey might have physically produced that the particle m and you can ask how does that show up in the how does that show up in the correlators and we'll give the answer that it shows up as a particular asil Ettore pattern in the three point function and the four point function and that oscillatory pattern in the observable the spatial observable is the fingerprints of time evolutions in the interior you really had a time-dependent background you produced a particle it's wave function oscillated e to the I MT time time time right so of course so when I say that we're going to try to understand everything without time what we're going to do is we'll begin by writing these things down as normal time integrals but we all quickly see that they satisfy differential equations with an interesting property purely in terms of the boundary variables okay and then we'll start staring at those differential equations start trying to understand them and we'll see those differential equations force on us these oscillatory pieces and all the rest of them without any reference the time obviously okay so there's a purely boundary understanding of something that we normally ascribe to something very time-dependent so that's something I'd like to get through today all right but anyway so one thing so there are third there are two things that I want to do in these lectures so one thing is to is to compute it's to play and compute play with and compute the wave function of the universe but with this with the time without time attitude so much as we can and but more more generally however you do it one one could hope that if we if we just get more data and see what these objects look like that we could start to learn what the rules are the analog of what we know for for amplitudes yeah no longer the cutting rules or what we know a tree level in one loop all of those things we should we should learn what the answers to those properties are just in perturbation theory for the wave function of the universe okay and we can generate more data in that way and these things could also be potentially useful for experimentalists at least for providing templates for things for them to look for examples of things things like this is that many limits of these things I'm you can also give it to them numerically many limits are known analytically but it would be nice to have a you know a really good a solid answer to this to this question that can even be useful to experimentalist but there's a second thing that we could do again and these are these are all in analogy with things that have been done in the context of of amplitudes so you can say those people in the 60s they thought that they could derive somehow what the imprint of causality was and the analytic properties of the s-matrix and they failed so why should we do any better now and well I mean it's never a bad idea to try again when you have more more clues more more data but I think there's also a there's also a growing new attitude about that about that the question which is to not try to derive these properties to not try to say we take a locality in unitarity and those are our god-given principles and then we derive what the consequences are on the properties the s-matrix but to do something that's kind of the opposite of that and to guess guess new patterns new principles new structures look for new kinds of questions they'll enough much more abstract there'll be an alien they'll be unusual just by the by the nature of the beast we're trying to we're trying to we're trying to remove the reliance in our language and our the way we think about things on on on on on on local causal unitary evolution so whatever if we're not going to make those things primary yeah anything else that might be primary is definitely going to look strange to a begin with and it's not obvious that it's that it's even possible in general but you can try you can try to see are there especially since even in some cases if we know how to check whether the answer is compatible with the rules after all even in even in cases perturbation Theory tree-level just at the level of the integrand whatever if we know how to get if we know how to check if we know what the properties are then we can wonder whether there is some underlying different kind of mathematical structure and a different kind of questions that you ask of that structure that produces these answers which have the properties of being compatible with the things that we know to check to give us locality and unitarity in other words we can look for structures that don't have the sings as inputs would have them as outputs okay now we've seen things like that in the context of scattering amplitudes the past five years in the context of N equals 4 super yang-mills we've seen it in the context of this strange geometric object that lives in grassman Ian's the Amitha he'd Rijn in the past few months we've seen it in much more down-to-earth context for a much wider variety of theories including you know Oh sigh cube theories theories of gluons and a number of dimensions and nonlinear signal model and here but it's essentially exactly the same basic structure there is a geometric object that lives in the space of the kinematical data of the of the scattering problem this turns out to be much older mathematical object it's known as the associate Hren and various generalizations of it generalized per meter Hedra and so on but it's the same story there is a geometric object you ask a certain kind of question of that geometric object the answer to that question are local and unitary scattering amplitudes and so you then see that where the locality in the unitarity how they come out they're nowhere in sight in the definition of the object you see them come out as properties of the question that you asked now I think that these things are happening and and the strangeness of these objects they're out their unfamiliarity they involved much more basically combinatorial ideas than we're then we're used to seeing I think these things are all related to the fact that they were butting up this against this question about doing away with with time but if anything like that has has something going for it we should also see it in the context of cosmology ok so that's the second thing that I want to tell you about is is the just the very start at the beginning of seeing something analogous to the Association and the Appleton hedron's it's really closer even to the story of the of the Association you don't need to know any of those things that'll be itself self-contained but uh there's something that we talked about in the fall known as cosmological polytopes but the again the general structure will end up being essentially identical so I'm going to define an interesting class of shapes polyhedra really polytopes in in in general high numbers of dimensions and you asked essentially a question about the volume of this object or a closely related notion of a certain kind of canonical differential form and the answer to that question will calculate the wave function of the universe okay and then we'll be able to see how the how the properties that that that that are supposed to go along with the perturbative wave function univers actually encoded in these objects yes yes that's right no no that's right I'm really talking about the wayfinder universe and not and not the correlators there is actually a correlator polytope we could talk about that too but uh but it's um but this is this is the primitive object underneath it yeah yeah yes absolutely yes yes and for insights and for for for literally the toy model that I'm talking about for literally the toy model we'll be talking about here you could absolutely apply it for ABS boundary correlators as well however you'll if I if I get there at the rate I'm going if I get there you'll see why the structure of the thing that I'm talking about is much more naturally associated with time so despite the fact that I could interpret as the ABS boundary correlators - there are many things you can ask about it which would make no sense in ABS what make perfect sense in cosmology for example there there are the various singularities associated with facets and boundaries of this thing are really associated with particle productions and and are not seen in in 80s and the basic definition of the objects once we understand it as as yeah I hope it'll become apparent that that that time matters okay alright so that's it for the totally introductory remarks that's that's the plan what I want to do in the rest of this lecture is is go through one example and I think for the purposes of coherence of the lectures on we'll start next lecture with with a very systematic introduction data to do all the calculations and first in the free theory and then with interactions and all all the rest of it because that will very naturally lead in to the story of the apologises what I want to end with but I want I want to talk in the rest of this lecture about [Music] some of the things that we also also mentioned slightly more impressionistic Li but but we'll will compute some things too so so the sort of general the general topic is inflation as the cosmological collider at a paper with one a few years ago on this basic topic and so this is something we all we always tell a journalist is that if we have inflation at very very high energy scales and it's the highest energy that that there's a highest energy event the universe have ever seen you know Hubble can be 10 to the 14 GeV unbelievably high energy scales so to give us access it should be like a big Collider but how how can that be true precisely well let me make and let me make a little analogy so when we have normal particle physics colliders again this is an analogy between amplitudes and correlators the first thing you have to do is see what are the stable particles in the world before you collide them they have to be stable enough for you to uh to have them around and collide them okay so the first thing you have to do is see what are the stable particles what does it mean to have a stable particle it means that it's two-point function is large right its two-point function doesn't die off so you want to see what are the things that have macroscopic two-point functions okay so now because of Lorentz invariants the behavior of the two-point function there's nothing to it you just need to know that it's there right you need to know you have a stable particle it lives a long time but after that what the two-point function is is completely fixed by Lorentz invariant so that's the symmetries totally fix it you just need to know that it's there and so this is fixed by Lorentz so where's all the dynamical information all the dynamical information is in scattering processes not nonlinear questions okay so you won't discover what's what's what's going on what the new particles are what the interactions are and so on so there are in these nonlinear things like scattering amplitudes okay and so now let's ask the let's go back and ask the question about that experimentalist at the LHC they do their they do their collisions in the morning they come back in the afternoon how do they know that they made a themed boson in the middle of the day somewhere or that they made a Higgs or something else well let's think about it in the very simplest in the very simplest case of something which we sort of normally ascribed to tree exchange of some particle if we're if we're exchanging a particle of something of some of some mass m and some spin s then that shows up as the presence of a pole in the scattering amplitude and therefore so for instance if there's something in the s channel here there is there's a pole one over s minus M Squared and then there are some residue upstairs that can depend on on key for example okay so first of all the way this experimental scan no something happened that they produced a particle and then it decayed first then you know the whole picture is that they see a particular kind of of singularity in the scattering amplitude by the way sorry let's take one step before that how do I know anything is happening at all right there are things aren't just missing each other well even if you haven't made the Z particle yet or you're just barely making it but let's say you're at low energies the the first thing you see is just the amplitude isn't zero right um so it's not zero but it has a simplest possible analytic structure it's just a polynomial okay so so even before this I might find that the amplitude is a constant and it might have something looks like s or something looks like s squared with different coefficients okay and T and so on okay so it could be polynomials here contact interactions so that's the first simple that's the simplest possible analytic structure the next simplest possible analytic structure could have a pole alright and if you have a pole you know that more and more is going on than just something is happening you've made a particle okay and you know what the mass of the particle is now you also know what the spin of the particle is and you can determine what the spin of the particle is by decom fight doing a partial wave expansion of that numerator okay and so what we know is that is that this is that the this residue of one over s minus M squared has to be written as a sum with some coupling constants squared that you see there so that might depend on the spin of the particle being exchanged but the exchange particle will give you a dependence on the scattering angle that's the S LeSean polynomial of cosine theta okay that's how experimentalist the turbine spins okay so you sit on a resonance and and you see that you have to be able to think of the numerator as a sum over over over species of different spin of some probability to have produced the particle which is the square of some coupling constants really even a mod square of some coupling constants times the s LeSean polynomial cosine theta now that's just group Theory uh how can we how can we see this how can we see it in a more down-to-earth way well uh let's say I asked you please couple this scalar particle to a particle of spin 10 and just calculate the sort of Fryman diagram the tree exchanges that spin spin ten particle how many of you would be afraid of doing that calculation okay and why would you be afraid of doing it I would be afraid of doing it because because well even writing down the Lagrangian for the spin ten particle there's an awful lot of indices there lots of different ways of contracting the indices how do you know you did it right fact there's many choices how what even fixes what the choices are right so you don't tend to run into this for low spin particles for instance although you should you know we we write lagrangians down all the time lively so so if I if I asked you to write down the Lagrangian for a massive spin one particle all of you would write something like this down immediate right that's what we're supposed to write down do you know why you write that down it's not obvious right because why is this piece without masses after me a new squared why would you say it should be happy new squared anybody yeah but it's not gauge invariant what's your problem right so why isn't it deem you a new squared or in fact any combination ad mu nu squared plus B D mu mu squared plus you know M squared mu mu what's wrong with that and then that kind of problem gets worse and worse and worse as you go to higher and higher points okay so that's why you might be scared of doing the calculation now of course they're doing it well how is it the way people standardly do it is they find the equation of motion and then they make sure you're not propagating anything other than the spin the particle to spin that you're talking about you're not party but you're not propagating any ghosts you know propagating any particles of lower spin so you have to try to make sure it's an e-wrap but there's a lot of gymnastics and taking out traces and stuff stuff like that okay and this is gymnastics that's reinventing the wheel that was solved by mr. Legendre in the 1800s okay so what I want to describe just just in a moment well what I want to describe now is how we can actually get this answer without ever talking about the lagrangian just directly by looking at this this on shell picture i'm stressing this in this very simple context where probably most of you know what i'm talking about because because everything that we're talking about here is gonna have a parallel in the side of cosmology where all the complications that we run into here get infinitely harder if you actually try to do the calculations in in a straight way if I ask you to exchange a spin part 10 particle in de sitter space you would really you would probably kill me it's it's very hard to do it even for spin too it's hard it's not even easy to do for spin once but but but but thinking about it like this what will make life a lot easier so where does this formula come from it's just it's just group theory but where does it come from in a more nuts and bolts way well you see if all I care about is if all I care about is the is the numerator of one abreast minus M Squared that is given that's what unitarity tells me is that that's the sum over all the polarizations that's the sum of all the polar Asians of the product of these two three particle amplitudes in other words I'm producing a non shell particle here on the pole and so I just take the product of these two three particle amplitudes now what are the amplitudes do I need to write down Oleg Ranjan to write down the amplitudes I do not because if I say that this is some spin s particle so it has a polarization vector epsilon mu up to epsilon s and there is momenta one and two these polarization vectors have to satisfy that their transverse to the momentum obviously and that their traceless so that I'm not propagating anything other than the spin s particle okay but even without this condition what this three particle amplitude is is completely unique because I have to contract this mu index with something what can I contracted with there is the P 1 P 2 P 3 is negative P 1 negative P 2 so that's not new and so all I can do every index can be contracted in some combination of alpha P 1 mu plus beta P 2 mu but the combination P 1 plus P 2 is negative P 3 and that gives me 0 because of this so the only thing that I'm left with this P 1 minus P 2 and therefore the only thing the amplitude can be up to an overall constant I'll call G is P 1 minus P 2 mu 1 up to P 1 minus P 2 us epsilon mu 1 mu s alright so that's what the three particle amplitude is okay and so now when I exchanged the particles what I get on this side is uh P 1 minus P 2 u 1 P 1 minus P 2 us on the other side I get P 3 minus P 4 nu 1 through P 3 minus P for us and in the middle I get the sum over all the polarization right up to sum over all the polarizations epsilon lambda u 1 through mus epsilon negative lambda a new one through newest okay but what is this sum this sum is just gonna give me a whole bunch of Delta deltas or a 808s right just by its invariance and in fact we can make our life easier by going to the center-of-mass frame if we go to the center-of-mass frame all the zero components of the if we go to the center-of-mass frame the massive particle is at rest that we've produced because we're again we're right on thrust with the mass particles at rest so the polarization vectors are zero in the time direction they're only nonzero in the spatial directions first of all so in those contractions everything is replaced by just the spatial components and also p1 minus p2 is just some vector and then let's say in the initial Direction p3 minus before some vector in the final direction so all I have this has the structure let me let me write it like this it looks like X I 1 through X is I'm just stressing these are spatial YJ 1 through YJ s and now I have some big tensor here TI 1 up to is j1 up to Jay ass which is just made out of Delta deltas but it needs to have an important property that it's traceless its traceless in if I take the trace in each one of these in in the eyes or on the J's separately okay okay well you can do it by hand at low orders it's it's it's pretty easy but that's exactly what the Legendre polynomials are okay that's their precise definition is that thing exactly a tensor that's made out of Delta Delta's exactly a temperature that's made out of Delta Delta's with the property of being traceless in each component separately it's just a definition of the legendre Lee gangin Bower polynomials in general and if I just normalize these things to be unit vectors this is this is literally the s Legendre polynomial of x dot y and if they're not unit I would take that out and put an X to the S why do they yes so that's just the definition okay so you see yes yes that's right because when I when I sum over all the when I sum over all the polarizations the only thing I'm left with or are back to the immune the new indices and and so it's um in very intensive ok yes yes sorry I should have said I'm considering now the sort of simple case where these things are just scalars when they start having more actually if these are massless and massive they're still continues to be precisely one structure that you can have but if they if it gets more complicated and and if more things are massive here you can have you can have more of Moreh structures but anyway this is not what I wanted to spend much time talking about but but but but what I wanted to point out is that the choice you make for the Lagrangian is in fact exactly dictated by the requirement that it gives you I can say it literal analog of this is that you know if you have a Lagrangian you'd have some numerator n mu 1 mu s nu 1 nu s over P squared minus M Squared that would be the propagator right all your work in choosing a Lagrangian is to choose a numerator and it needs to have the property of this it needs to be a tensor which is which which is made out of a 2 a 2 s and is traceless in taking out each one of them separately ok and that's that's that's what fixes that's what fixes the Lagrangian so so it's it's it's exactly this property of propagating physical particles it fixes the Lagrangian not the other way around and in fact so it's sort of particularly silly to write down the Lagrangian derive the propagator go back shove through the final rules and bring it back because it's inverting the logic so we can actually directly directly write it down ok so all right so so now we've learned the next most complicated thing that we could have okay we could have things that are polynomials the next most interesting kind of analytic structure we could have we could have poles and we learned what the rules are the rules are that the residue of the poles have got to be expandable as a sum over LeSean polynomials with positive coefficients okay then yes sure sure there's certainly not certainly not and and that's as I told you you can't even go beyond no sir certainly not but that's related to not knowing what the rules are we see here I know exactly what unitarity means unitarity means that it has to factorize like this that's what I was telling you a tree level we know exactly what it means that one loop we pretty much know it means the two of you don't know what it means okay but but at this at this basic level we we do okay then if we wanted to go beyond that then then the the next most complicated kind of struck a singular you could have branch cuts that's corresponds to loop amplitudes and so on okay and that's that's where what we just said we know what one loop means two loop is more complicated and so on all right so what's the analog of that all of that in cosmology so so this is what we do in in particle physics and and I'm stressing it this way just so you see that the physics that we've seen so far is almost completely dictated by symmetries okay the only thing here was not dictated by symmetries was the actual particle content and the strength of the interactions now what's the analog of all of the story in cosmology step one is to identify the object that has a large to point function okay well that's what inflation has wonderfully given us when we look at the spectrum of nearly scale invariant perturbations on the sky that's the big two-point function that we're seeing okay so so we have the in photon and the two-point function for the two-point function for the in photon so if I even think about fluctuations of the in photon this is just what we talked about in position space hmm ah and this this shows how little we actually have to know about the physics that gives rise to it if we just know that it's scale invariant then then it could either be a constant or or could have a logarithmic dependence on X minus y but just by dimensional analysis the thing that in front needs to have units of mass squared and if it's coming from inflation will be hubble squared during inflation okay so that's the big two-point function and again the analog of the first interesting kind of interactions that we could have between the photons are just contact interactions okay so just contact interactions might be like let's say the the in photon has some nonlinear self coupling a cubic coupling or a quartic coupling let's say it has a as a defy to the four coupling something like that then again since I'm looking at perturbations around the perturbations around the inflationary trajectory I can even sort of put one of these things on the background if I want so really even though the things that are cosmologists care about are three-point functions they really arise in the they really will arise fundamentally from four point functions in the underlying theory but one way or another all of these things have the property that the interaction is local in time here and all of these things they have a they have an the analogue of the just the contact interactions are the sort of simplest possible singularity structures that we already talked about these things go like 1 over K 1 plus K 2 plus dot dot maybe to some power and they have some polynomial dependence on the rest of the case upstairs so that's that's what contact interactions look like okay now there's a very important interaction that's totally Universal we should always talk about it's not so relevant for particle physics is very important in inflation its gravity and fact back in the early 2000s one mel de santé did this a beautiful calculation of the inflationary three point function that in this language we can think of as coming from graviton exchange now this is something you might think is now this is more exciting you're you're exchanging a particle it's a graviton two massless particle it looks like this should this could give you something more interesting but in fact the the effect that you get here is really coming from what you can think of as the sort of instantaneous Newtonian attraction at this one time there isn't any there isn't any communication between between different times in an interesting way in this calculations that that should become more apparent when we talk about how to do these calculations properly later but for now I want to say is that even this thing that comes from graviton exchange despite the fact that it looks like it's exchanging something it's in fact all dominated by occurring at the same time once again and so once again the kind of seeing the kind of three point function and higher point function you get from this is because of exactly the same sort okay so they look like contact interactions okay all right so now let's go to the analogue of this okay so here you see here when you when you expand let's say you're at low energy so we haven't seen this particle of course we know I can integrate it out I get an infinite Tower of higher and higher dimension operators and so the low-energy amplitude will be a polynomial like this okay so to no order in this low energy effective theory perturbation theory to no order do you see that there is a singularity coming your way right and you just have a so but it has it has a finite radius of convergence okay so now let's ask in cosmology the analogous question suppose that we have a massive particle that we're integrating out so let's say we have the in photon and a couple system massive particle and I'm integrating it out well at first at first you might think again if it's heavy you know ten times heavier than the then then hubble's during inflation something I integrated out I just get a whole tower of higher and higher dimension operators right just a whole tower of contact interactions and so I'll just get a bunch of things that look like that okay I'll get something with like one over the sum of the k's four and then something with one of the some of the case to a higher power cubed with some pollen or only upstairs P D prime P double Prime and so on okay and this will be suppressed by some by higher and higher powers of M so to every order in effective field theory in to every order those in fact to feel theory expansion uh you just get what that's all we get we get a bunch of contact interactions and there's nothing different in the analytic structure of this correlation function but this is clearly missing some bit of physics what is the bit of physics that it's missing it's the same bit of physics that we're missing in the accelerator but reflected in a different and an interesting in somewhat different wedding we're missing the fact that the time-dependent background could actually physically create on shell this massive particle in fact physically create a pair of them physically create a pair of massive particles and once they're created they're sitting there and though they'll you know one will go this way one will go that way their wavefunction will oscillate and then eventually one will decay to a pair of influence on this side one will decay to a pair on the other side or maybe if one of them is on the background you would say that it oscillates into it on the other side okay but anyway and there should be some fingerprint just like we have been beam as on oscillations or came as on oscillations there should be some imprint of these oscillations in in the in the correlation function so just roughly what does that look like well let's first get a rough idea of what the what the sort of two point function of this massive particle might look like so let's say we have a massive particle here M I'm calling the particles Sigma let's let's say so what does it's two point function look like in space and just just just on the on the on the boundary well what does it look like without any fancy-schmancy cosmological stuff okay you have a very massive particle it's two point function let's say I'm talking about it on you know not on scales or a short compared to its constant wavelength or I can't ignore its mass but I'm talking about on scales long compared to a constant wavelength what do you think it is well this has units of mass squared but since it's a massive particle the right-hand side can just be purely contact okay so it's just a delta function some delta cubed X minus y and so the leading order it's just 1 over m Delta cubed X minus y right so very contact okay and of course it can have the corrections but this is what what we're talking about however that the effect that we're talking about here with the with the particle actually physically being created by the time-dependent background it's going to be exponentially suppressed by the mass of the particle divided by the by the scale of Hubble you can think of that as a Boltzmann flexure as a Boltzmann factor if you think of the decision space is being hubble or even more simply just as as a as a as the as the obvious depression that comes from the the fact that this scale that this is happening on a much faster time scale than the than the natural expansion but there's there should be a correction here that goes like e to the minus some constant turns out to be pi but it doesn't matter m over H there's valid when M is quite a bit bigger than H but now there should be something that actually oscillates it was like cosine X minus y hey / some IR scale and what makes up the units here it's not one over em what makes up the units instead of a delta cubed X minus y is a one over X minus y cubed okay so what is the what is the interpretation of this the interpretation of this is that you created a particle yes so we have to do we created it we we created a particle at some at some time there is the oscillation of its wave function e to the I mass times time let me be a little schematic here there's the oscillation of its wave function that goes like e to the I M T okay now we have to remember that the scale factor an inflation is going like e to the Hubble times T and so I can think of this equivalently as a to the power of M over h of I am over H or if I think of it in terms of spatial momenta this is in in momentum space there's something that's going like k ^ i m / h because right because the scale factor is stretching the scale factor is stretching the wavelengths but if you produce it at some time the number density of these particles is being diluted by the rest of the expansion of the universe right so so whatever you get should be down by a factor of of the amount of it should be down by an overall factor of one over a cubed from the rest of the expansion that's exactly this one over X minus y cube factor okay so this says that you prepare producing particles that are being diluted by the inflationary expansion of the universe with an amplitude that is Boltzmann suppressed okay all right so yes ah well you see here there was no choice for it okay here here we knew that it had to be a contact and had to be Delta cubed x minus y and it's just that now so you know that the this is what we'll see in a lot more detail again I'm doing pretty well do all of this we'll see it a little more precisely but you know that the wave function of a harmonic oscillator is e to the minus omega x squared right and if you ignore if it's Omega is just M then then it's then then the two-point function is just 1 over m right okay and in fact it's really a similar argument here because we know ahead of time there has to be this one of X minus y cube there is this effect from the dilution with the expansion Universal then again unit just tells you I think that sits in front there's got to be one aground alright but if I'm again being a little bit sloppy I would say that there's a sort of a typical of fluctuation in k-space of just one Sigma now this is in free space that goes like ke to the three-halves plus i plus I am and this is in order for for Sigma Sigma to have something that goes like K cubed or one over X minus y cubed and the with with the correct oscillatory dependence okay so so what does that tell us for how we can so so that's that's that's the physics of that's the physics of a particle production but now let's go see how that effects the how that shows up in the three point function so all we're doing here all we're doing is looking for the analog of sitting on the pole and seeing the resonance at a Collider we're just trying to see the analog of that you're cosmologically okay well if we if we think about what what we're what we're used to seeing when we have an oscillation phenomenon you produce something it and it has lots and lots of oscillations and then you make two measurements at one time and a different time and then you have oscillations and the time difference between the two of them so the analog of that here will be to imagine that at some point here we pair produced these these particles okay so one is going this way one is going that way but this one goes a long long way and it and at the case in photons late this goes let's say some some shorter time and it the case was pair of in photons and as I said this one is set to the inflationary background so it's sort of oscillating into this guy's turning into that guy in the inflationary background this is happening early so let's say this is some k1 k2 associated with those momenta and this associated with some momentum k3 ok so again something that you know is that these things are going to be associated with short scales because they're produced later they had less time to be stretched by inflation before they came back into the horizon these things were produced earlier so there are longer distances and so if I so therefore this picture corresponds to something with a short k3 a small k3 and too much bigger k1 and k2 so in order to access this regime where we can see this large time separation between these these two events we have to probe the three-point function and an interesting regime where I keep k3 small and I make k1 and k2 bigger and bigger and bigger right and now roughly what do we expect for what the 3-point function should look like well this is the sort of amplitude for this happening so it has this it's going to have uh it's going to have a factor if I want e to the I M you know T so let's say this is event to and this is event 1 e to the I M T 1 minus T 2 is also K 3 over K 1 or K 2 which are about the same in length to the power of i m over H ok so again by the same logic that connects that says that the scale factor is e to the HT and so we expect that there is a there's a feature in the there's a feature in the 3 particle amplitude there's a feature than in the through particle core later that's proportional to on the one hand there is something that says that the whole thing is being the whole thing is being uh diluted by the expansion of the universe so that was that three-halves factor that we saw before there is an overall sorry let me write it here there's something that's e to the minus PI I'm over H that's the fact that it's an exponentially depressing there is the there is this K 3 over K 1 2 to the 3 halves and then what I can have is well I can have the I can have the K 3 over K 1 or 2 to the I mu and I can have it to the minus I mu and there can be some coefficient here and there could be some coefficient there well but there can be some phase here so let me just write it like this so this is the sort of thing that I should expect to see in this in this limit so in order to see whether a particle has actually been produced not just see that there was some kind of interaction there the analog of seeing the low-energy scattering that comes from from sitting at energies much lower than the Z or something right but in order to see that a particle is actually produced I see it by going to this limit I go to this limit where I keep K 3 small and make K 1 and K 2 larger and larger and in that limit I should see this fascinating oscillatory behavior right you know the I M / H I apologize see there's fascinating oscillatory behavior it's suppressed by two things it's suppressed by e to the minus it's suppressed by the sort of Boltzmann factor and it's suppressed by the overall dilution between when this guy was produced and that guy between when when one decayed and and the other one okay but so it's a small effect but that's how you know in principle sitting there in the structure of three point correlations between the density of galaxies in the sky can be clues to the presence of particles with mass close to Hubble during inflation and if the mass of these heavy particles is not you know much heavier than Hubble the exponential doesn't necessarily have to be a very big factor these are small effects but we're talking about potentially seeing things up at 10 to the 14 GB and so it's it's sort of interesting that that we can look for them in this way but ah there's one more factor I should have told you about the very fact that I'm that to do these calculations I was putting one guy to its inflationary background turns out to cost another factor of something that knows about slow rolling of the in photon but I was more talking about this very interesting non analyticity this oscillatory behavior which is the fingerprint of producing the particle alright I'll come back a little bit to saying how we can determine this more more precisely but I just want to say that this structure is the exact analog in cosmology of sitting on the pole sitting on the residence at a Collider however at a Collider you have the luxury of dialing the energy if you have enough money right so if you're a very low energies you can't see this if you have you can change the initial state and you can sit on the residence and then you do see it in cosmology we don't get to do the experiment again experiment was done for us once we had a wine initial state that we had nothing to do with but fortunately it's a somewhat generous initial state because of a time dependent everything is actually produced and all the particles out there in the world are actually produced maybe at an exponentially small rate but they're all they're all actually produced and so instead of dialing the energy of the machine in order to see the non analyticity we go to this regime in the three point function the squeeze limit in order to see what it looks like now I said oops can I just take three more minutes yes yes no no I won't it won't be more than three it won't be more than three at this level of Impressionism it will not be more than three yeah yeah unsurprisingly what somewhat surprisingly I'm not just a factor too slow I'm a factor of five slow so so I'll have to do something about that but okay so I just want to I just want to quickly end by pointing out how this is entirely determined by symmetries as well okay so we just said that the two-point function is determined by scale invariance and of course it's not exactly scale invariant but it's approximately determined by by scale invariance you don't even need to full the sitter invariance just scale invariance will do to determine the structure of the two-point function what about the this three-point coupling that we had between our two and photons in this and the signal particle okay we know the when when we have when we have the sitter on the inside the now now we really do use the sitter but when it's visitor on the inside just like an ABS on the boundary that's the actor that of the desiderata matriz is that of euclidean conformal symmetry on the boundary and therefore the Euclidean conformal symmetry fixes the structure of these three particle couplings okay so this is again fixed by symmetry after the strength of the coupling constant exactly like in the amplitude and finally what about in this limit so what's special about this limit in the amplitude when we took the limit where we sat on the pole we had factorization that told us that we got just a product of the two three particle amplitudes what is the analogue here the analogue is that when I take this limit where these things are very far apart let's say I'm just talking about a I'm talking about the four point function again always one of them is sort of put put on on the background here in this limit this propagator is almost the same as if this guy went to the boundary and then came back from the boundary and so the propagators become closer and closer to the product of those two things in the limit as the distance between them gets very large that's why you have to make the distance large in order for that aprox and factorization that approximate representation of the of the propagator to be correct but that means that in this limit again the four-point function or the inflationary three-point function once again becomes the product of these two three-point couplings that are fully fixed by symmetries it's the identical logic as for the case of scattering amplitudes and finally what about the spin of the particle and this is this is the thing ah one one one last little thing what about these phases these phases are a little interesting they're not fixed by conformal symmetry okay so just conformal symmetry by itself and nothing else what allow you to have any phases here however the what we get in normal cosmological calculations that's very very fixed special phases there and they're fixed by an interesting consistency condition on the full three point three point correlator you see the three point corollary we see is an interesting this oscillatory behavior as you make two of the sides long but in principle it could have singularities elsewhere it could have a singularity for example as you collapse the triangle to look like more and more sort of flattened out like this okay this is a garbage singularity you can see it seems totally random where should it happen should it happen when it comes here it should happen there should happen there there's nothing canonical about where the singularity should happen and one of the general sort of features of every all the singularities that we get from interesting observables and physics is that they're there if they have to be and for no other reason okay these things are completely spurious there's no reason for them to exist and in fact the absence of these singularities is really in one-to-one correspondence with putting in the correct a punch Davies hartle-hawking vacuum in the deep past year so if you demand the no bad singularities and these collapse triangle configurations that on the full three particle amplitude that actually fully fixes even these phases but they're not just fixed by symmetry they're fixed by one more assumption of the absence of spurious singularities and finally how's the spin of the particle encoded so there's an angle in this triangle as these sides get long longer there's still that that angle in the triangle and well the only thing the answer could possibly be is it gets multiplied by the s LeSean polynomial of cosine theta exactly as for scattering amplitudes okay and so you don't have to do a calculation for a spin seven particle and de sitter space and kill yourself to do it this entire effect the analog of sitting on the pole for the scattering amplitude is essentially fully fixed by symmetries and this structure is how the presence of massive particles can be encoded in cosmological perturbations what we'll do next time is I'll just quickly I'll quickly motivate what we're trying to do beyond this so this tells us what things look like in these squeezed limits we know what things look like just with contact interactions but what if we want to know what the whole four point function looks like when we have three level exchange everywhere for our scattering amplitudes we know the answer it's just 1 over s minus M squared times that numerator plus contact pieces right that's the most general thing that we get with tree level exchange and we'd like to know the analog of that in cosmology okay if that's that's something we're already here if we did the calculation flat out it would be very complicated even to see this simple behavior if you want to get the whole thing you're just left with a pretty messy difficult looking integral over Hankel functions and you can look up all the special functions in all the books it doesn't have a name so so but that's thing that we all think about from this again from this perspective of trying to do these calculations purely from the boundary point of view see already we've figured out what this looks like purely from the boundary point of view right just purely thinking about what the consistency conditions on the boundary are together with some extra input about the absence of stupid singularities but will go significantly beyond this in in in talking about the structure of the full four point function with tree level exchange for scalars and then once we have gotten some practice with calculating wave functions the universe in the final two lectures I will talk about in a slightly simpler set of examples these cosmological polytope objects [Applause] so we don't have much time because the general shaker lectures will start soon but maybe we can have one quick question [Music] you

Channel: International Centre for Theoretical Sciences

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Id: CU7m8m-9Hw4

Length: 98min 45sec (5925 seconds)

Published: Wed Jan 24 2018