Aspects of Eternal Inflation, part 1 - Leonard Susskind

first question how many people here consider themselves high energy physicist this is not a test how many astrophysicist okay um my lectures are basically very elementary I'm not going to do very very fancy things I see people in the audience that I know many of them are my students some of them know more than I do this is not for you guys oh but you can help me you'll be able to help me we're going to be talking about eternal inflation eternal inflation is a very subtle and slippery subject and in fact it's one that I think is yet to become a subject it's that it doesn't really exist yet in a proper form of a theoretical construction and so I don't find it conducive to the kind of formal presentation that if we were doing some aspect of quantum field theory instant times or spontaneous symmetry breaking or things like that which in the past I have given summer schools about where you start from a definite set of principles and you work your way through and in the end you have some kind of beautiful structure we're going to start with what looks like it might be a beautiful sub subject and watch it get uglier and uglier and uglier and so for that reason there's no real point in starting with a very very systematic starting point because there is none um let me nevertheless tell you what the Givens are for me you may have different Givens other people who will lecture at the school will very definitely have different Givens Givens you know our assumptions the first is that Einstein gravity is correct at least at relatively low energies there will be corrections and in fact those Corrections might be monstrously big things near black holes or near horizons but at least are in the beginning we're going to begin assuming that Einstein gravity is correct at low energies we're not going to be doing modified gravity we're not going to be doing Einstein we are going to be doing Einstein next we're going to assume there's a collection of scalar fields I'm sorry I I don't feel like writing this on the blackboard if you want to take notes be my guest but I'll just tell you anyway we're going to assume there's a collection of scalar fields how many scalar fields well that's somewhat in the eye of the beholder nobody knows how many scalar degrees of freedom there are some people think there are hundreds of them other people think that the Higgs boson is all there is but there is at least one and probably more than one because inflationary theory which is a successful theory requires another one so at least two and probably more whether it's 500 or whether it's two is not going to make a lot of difference for us in these course of lectures but they do exist scalar fields that's basically a postulate and those scalar fields have a potential V of Phi soon I'll start writing on the blackboard but everybody knows V of Phi anybody not know V if I everybody is a friend of V of Phi all right now the configuration space of these scalar fields together with the potential to find something called a landscape landscape is simply the space of all possible values of the scalar fields and you can think of points on the landscape as environments at every point in the landscape the scalar fields influence how other degrees of freedom interact with each other they provide masses for particles they may provide coupling constants and so forth so those scalar fields can also be just thought of as environments or specifying the environments the minima of the scalar field or scalar fields the minimum of the potential are stable or at least classically stable environments they're the points where the environment can settle down and stay there stabili but in quantum mechanics in general there will at best be metastable in other words possibly one can tunnel out of these minima from one environment to another next as I said we're assuming now I need to get up the right we have a potential many many minima or two minima it won't matter for us at least in the beginning we're not going to be studying incidentally questions about anthropic principle and things of that nature we're studying we're going to try to study the mathematics of eternal inflation and the phenomena of eternal inflation if it makes sense doesn't really depend on how many minima there are as long as there are a few minima some of those minima will have positive energy some of those minima will have negative energy and some of those minima will have exactly zero energy all right the solution of Einstein's equations at any one of these minima is de sitter space the sitter space is a space of uniform curvature a space of uniform positive curvature and it's characterized by a radius of curvature it's characterized by a radius of curvature or if you like a curvature which is one over the square of the radius of curvature and the connection is that the radius of curvature R which I'll call R the radius of curvature R which I'll also sometimes call 1 over H R is the radius of curvature H is the inverse radius of curvature it also happens to be the Hubble expansion coefficient of the Hubble expansion parameter for the expanding de sitter space that's why it's called H and it's given by 8 pi Newton's constant divided by 3 times the value of the potential energy at the minimum can everybody see this yeah ok sorry I got this wrong wait yes it's squared this should read H squared is equal to that H squared is equal to that so and that's 1 over R squared so R is large when V is small V is law R is large when V is small a very big universe corresponds to a small vacuum energy all right our first goal is going to be to understand the sitter space how many people here are expert at the sitter space I know a few of you are very expert at it all right good that means we can start with the sitter space and I won't feel guilty about wasting your time if I am wasting your time get up and leave ok the sitter space as I said is a space of constant positive curvature but it's a space of constant positive curvature with Minkowski signature with Euclidean signature a space of constant curvature would of course just be a sphere the world is four-dimensional and we will take the case of four-dimensional de sitter space but let's begin with the case of a four-dimensional sphere I'm just going to draw on the blackboard a sphere is a sphere and let's think of it as embedded in coordinates one of which I'll call T and the others I'll call X sub I X 1 through X 3 X 1 through X 4 this is a 5 dimensional space and embedded in the 5 dimensional space is a 4 dimensional sphere and it's given by T squared plus X I X I this by definition means the sums of the squares of the X is T squared plus X I X I equals R squared equals 1 over H squared now that's not the sitter space of course it's just a plain sphere and we can coordinate eyes the sphere or chop up the sphere in two layers these layers are going to correspond to space like surfaces are in several different ways the first way we might see of course is going to stand ultimately at the time we can chop them up into space like sections and think of time Euclidean time in this case is unfolding from the bottom to the top but of course in this case here P and X are completely symmetric so there's no sense in which P is time any more than X is time but to go to the sitter space we put a minus sign in here now we have a Minkowski space metric and a Minkowski space metric describes a well the space looks like a hyperboloid now so let's draw a hyperboloid in here now that is the sitter space but we can again divide it will not provide it but foliated I guess is the mathematical word foliated by a sequence of space like surfaces that provide a notion of space and the notion of time ok let's write down a metric if we were writing the sphere let's write the sphere first is the sphere d s squared is equal there's an R squared on the outside and then what multiplies that can be thought of as the metric of a unit sphere and that can be written B tau squared I'll tell you eight hours in a moment plus cosine squared tau times D Omega 3 squared the Omega 3 squared stands for a unit sphere what is tau tau is angle measured from the equator over here tau is angle measured from the equator d tau squared is just that term cosine squared tau times the metric of a unit sphere of one less dimension in this picture here the unit sphere of one less dimension of the sphere one less dimension would be a circle alright so this is the metric of a sphere when you go from the sphere to the hyperboloid all you do is change the sign of time like variables and this will become D s squared is equal to R squared times minus D tau squared but the cosine squared tau becomes not cosine squared cosine of course our tower for an angle runs over a finite range on the hyperboloid the corresponding quantity runs over an infinite range and it becomes hyperbolic cosine the omega-3 squared the hyperbolic cosine of or not the hyperbolic cosine but tau itself instead of being an ordinary angle becomes a hyperbolic angle along the hyperboloid a hyperbolic angle whose trigonometric functions are hyperbolic sines and cosines and so forth this is the metric of the sitter space it's the metric of the sitter space in what is called global slicing where you slice where the space-like surfaces are slices which slice right through the whole thing it has the form of a freedman robertson-walker metric I will make a little bit of a change of variables let's make a little bit of change of variable let's call R times tau let's call it T so this will become minus DP squared plus hyperbolic cosine squared of H times T divided by HS H squared 1 over H squared is the same as R squared upstairs so R squared becomes 1 over H squared and cow is equal sorry do I have this right yeah I do tau is equal to H times P times the omega-3 squared this has the form of a freedman robertson-walker metric with the time conventional proper time and a scale factor the scale factor let's write it down the scale factor a of time a of T is equal to 1 over H hyperbolic cost of H T now thought of as a freedman robertson-walker metric it has some unusual properties the scale factor increases exponentially at late time but it also has a contracting phase as a contracting phase and a bounce and expanding phase and that's one presentation of the sitter space it's one way of writing the metric of the sitter space let's consider some more ways of writing these are of the sitter space let's see I guess we can put this up is this large number of blackboards here makes me very uncomfortable Matias is this whole room makes me uncomfortable the paneled walls Stanford we would never have a room like this but there is something good about the Institute for events that it could particularly somebody of my age it's a wonderful place to come to lose weight okay no more insults yeah okay notice that the space like surfaces have been defined by introducing planes and slicing the space by the sequence foliated over the sequence of planes incidentally because this geometry here has Lorentz invariance a five dimensional Lorentz invariants in this case T and X defined five dimensional variables you can Lorentz transform your initial surface here and tilt it tilted by ordinary lengths transformation that won't change the metric so there are many presentations of the same de sitter space all of which are global slicing which are sort of related by tilting the surface with slices right through it okay so that's that's global slicing of the sitter space what other kinds of presentations are there basically there's one for every possible not every possible way but for a large number of ways of choosing planes to slice it up with two foliate it with the next way ah is instead of choosing space like planes in fact what we're actually doing is taking the limit in which the boost here the boost or the change of the angle of these slices goes all the ways up to light like so this would be slicing the decision space would like like planes would look something like this that would be a light light plane that would be one which is asymptotic and divides the hyperboloid right in half the light like slicing only takes into account half of the space and here's what you do you take light like planes like this you take one generator I guess it's not called a generator but one hyperboloid along the hyperbola here one hyperbola along the hyperboloid and you mark off equal proper time in other words what was called equal P or equal towel over here you mark off equal time and then you draw a light like surface to each one of these now the like like surface intersects the hyperboloid on a space like surface how do you see that think of the intersection of the light like surface with the hyperboloid the light like surface intersects the hyperboloid with some motion so to speak in the plane perpendicular in the line perpendicular to the blackboard so if we learn well let's just draw it here's the hyperboloid and one of these planes intersect something like that I'm not a very good drawer ax but that's I think you can get the idea and you notice right over here that the intersection is sticking right out of the blackboard and it's purely space like anywhere is along here this intersection is spaced like so these are a space like foliation again of the same hyperboloid and you can work out what the metric is in terms of a time variable which proceeds from minus infinity over here up to plus infinity over here all right so let me tell let me give you the metric but this is your first problem prove that in this light like slicing of the sitter space the metric is given as follows Oh each one of these surfaces is characterized by the time at which it intersects one of the one of the hyperbolas here all right so here's the metric the metric is again the S squared is equal to R squared and now it's really simple - D tau squared sin tau hyperbolic angle along here plus e to the 2t e to the 2 tau DXi DXi now this DX IV X I this is three-dimensional space not four-dimensional space the whole space is four-dimensional this is flat space flat space space not spacetime flat space and this is sometimes called the flat slicing of the sitter space it's the same geometry it only covers half of the full geometry that was that was characterized by the full hyperboloid here and it covers the half from an asymptotic plane right through the middle here out to infinity this is for you to prove that this is the same geometry except sliced up in this other way here we can also rewrite it as minus DT squared plus 1 over H squared e to the 2 H T DX squared the x squared again being flat space alright so this is the flat slicing and in this slicing it just looks like a frw universe looks like let's come over here this was a closed frw universe in cosmological language at Coruscant corresponds to K equals plus 1 this is the same geometry but written so that it looks like a K equals 0 in other words a flat frw geometry with a scale factor which is e to the H tau HT divided by H so in this geometry it's a K equals 0 frw with a scale factor which again increases exponentially now these different presentations are actually closely related to each other in particular if you let t get very late over here then the cash increases like a simple exponential and so one way of thinking about this presentation over here is it just corresponds to the late time limit of this one but it also stands on its own right as a different slicing covering half the geometry alright so remember now your job is problem number one prove that that's the same metric is that one over there next slicing let's come over to here next slicing that my students think they know what the next slicing is and they'll probably do the next slicing begin again with a sphere here's the sphere and let's cut it up in a totally different way let's cut it up with a sequence of planes like so let's just rotate now what's the advantage of this the advantage of this is that going from one time to another we're going to think of this angle here is time going from one time to another is a symmetry operation before we do let's just point out that these geometries are not static they do not have a time translation invariance they're not good geometries for discussing Hamiltonian physics for the constant Hamiltonian you would need at best a time-dependent Hamiltonian and it's probably much worse than that for various reasons but this slicing of the sphere going from one slice to the next is a symmetry operation so when you write the metric you will discover that the coefficients of the metric have no time dependence now let's just be a little careful about what this in two dimensions in two dimensions like I think I better draw can you picture a three-dimensional sphere cut by can you can you picture a four dimensional sphere cut by planes this way not so easy a little bit hard to picture okay but when you do this point over here on the sphere where all these sections intersect is not a point just to show you what it is if I were dividing up flat space this way just ordinary flat four dimensional space or whatever let's do three dimensional space we would be breaking it up in the following way let's see if I can draw this is this picture clear it's about as clear as I'm ever going to get it so you better say yes it's a sequence of planes a sequence of planes which rotate about an axis in three dimensions in two dimensions it will be a sequence of lines rotating about a point a three-dimensional space it would be a sequence of planes rotating about a line what would be in four dimensions and four dimensions it would be a sequence of hyper planes three dimensional hyper planes rotating about a two-dimensional axis or about a two-dimensional surface so this point here really corresponds not to a point but to a two-dimensional surface to a two-dimensional surface one more than that I can picture here where I can picture a line alright this type of slicing is called the static slicing of the sort of space and I'll write down what the metric is and the thing that though of course is to study this metric and get familiar with it study the geometry is studied the particular set of coordinates they are very important coordinates what they look like in the sitter space is something like this they do not cover the whole space again there's our de sitter space here's the analog of that point and now we draw a light cones light cones these are these are not light cones these are light like planes and then you foliate with a sequence of light like planes and these light like planes are generated by Lorentz transformation one from the other by Lorentz transformation here they're generated by rotation the portion of the de sitter space covered by the static slicing this is called the static slicing is definitely not the full de sitter space it's the region in here now of course you can extend it to the other side also but we're going to focus on this region in the pink region here and I'm going to write down again what the metric is the metric in this kind of slicing again this is your second problem prove that the metric of the static patch of the sitter space sliced in this way is given by R squared V s squared equals R squared is always that R squared out there times minus one minus R squared where R is a coordinate which basically measures spatial distance from a point over here 1 minus R squared that's part of your job to figure out what R is the Tao squared same Tao there's just proper time along this trajectory deepal squared 1 minus R squared plus the R squared over 1 minus R squared plus R squared D Omega 2 squared Omega 2 squared D Omega 2 squared stands for the metric of a unit to sphere now this is a metric in which the sitter space is presented in a way which is time independent there are no times in the coefficient functions here in the metric functions G naught naught G 1 1 G Omega Omega R all time independent so this is a static description of the sitter space it has a radial coordinate which can be thought of go to the goal go right to the center here the center here is a three sphere ordinary three sphere of space the center there is a three sphere on that three sphere we pick a pole that pole can be thought of as the position of an observer R is the distance from the observer and the observer could be thought of as moving along a trajectory like that so our measures this or as a coordinate which measures distance from the observer is the metric and it will not have escaped you that this metric has a look which looks sort of like a black hole metric it has a coefficient here an inverse coefficient here and it has a horizon the horizon is the place where G naught naught vanishes and for the place where G 1 1 diverges so R equals 1 little R equals 1 little R equals 1 is the place where the horizon of the sitter space is and um if I were to try to draw let's try to draw a spacial section through here here's a spacial section what is it it's a three sphere unfortunately I can't draw a three sphere but I can draw two sphere so there's space at an instant of time this point over here is one of the poles of a sphere the west pole the other one over here is the east pole and that point over here is the equator of the sphere I could have made it the North Pole in the South Pole but then I would have had to turn it on its side and I would have lost the correspondence with directions here so these are the two poles of the sphere we're going to be interested these this metric here incidentally diverges at R equals one where is our equal first of all where is R equals zero R equals zero is over here and over here that's where the circle shrinks to zero R equals one little R equals one is on the equator okay there's nothing singular happening on the equator this is a singularity that's associated with the coordinates nothing singular happening on the equator one observer here we can imagine another observer over here but we're going to study the left-hand side of it and that's the geometry at any in time the spatial geometry right in other words it's this geometry over here the spatial part of the geometry space how many problems have I given you already - okay problem number three prove that this is a hemisphere this should be a hemisphere that's where I got it from prove it prove that this is a hemisphere a hemisphere means a three-dimensional hemisphere now prove that that's a three-dimensional hemisphere and as I said this line here plays the role of horizon we're going to study this in a little more detail a few minutes R equals one is a horizon but it's very very convenient and helpful to redraw these pictures as Penrose diagrams now I do not have either the time or the inclination to go through in detail what Penrose diagrams are if you don't know find out fast because we're going to be drawing a lot of Penrose diagrams in this class that easy I'll tell you quickly what they are they're useful for rotationally invariant geometries when is it when there's an axis of a rotation or a point of rotation they're useful they represent only the radial and timelike coordinate so that at each point of the Penrose diagram you should think of a sphere of directions a two sphere of directions that's sort of implicitly there but it's a two-dimensional figure which shows the radial and timelike directions number one number two it has taken the geometry and mapped it by appropriate coordinate transformation so that all of it I see my students falling asleep that's okay good get a nap you're gettin that you'll need it okay you take the whole geometry and you map it in such a way as to squeeze it all down onto the finite blackboard by some appropriate mapping that means a mapping which will map infinity to finite places but one special rule when you do so radial light waves should always move at 45 degrees you map it in such a way that you preserve the fact that light rays move at 45 directly their degrees the this is the Penrose diagram of flat space what does it correspond to here's the center of coordinates R equals zero here is P equals infinity up here you've taken all of P equals infinity and pushed it here here's P equals minus infinity is R equals infinity and here are the space-like surfaces all of which move out toward R equals infinity these are lines of constant radius like that that's the Penrose diagram of flat space-time okay what I want is the Penrose diagram for the sitter space so let me draw the Penrose diagram for the sitter space it's extremely simple it's a square okay so let's see what's on the square there's the West Pole in the East Pole those are the positions of two observers they correspond to these lines over here they're over here R equals 0 and R equals 0 going up to the Future infinity of the hyperbola that's up here K equals infinity K equals minus infinity okay how about this point over here the thing that I call the horizon the horizon is right at the center and the light cones that spread away from it they're the light like surfaces the away from it just divide the square like so whoops like so the global coordinates foliated in the obvious way as horizontal slices the flat slicing I'll draw the flat slicing for you it actually looks like this but more interesting for right now is the static slicing so let's draw the static slicing the static slicing looks like this and incidentally these space-like surfaces bunch up near the horizons over here they bunch up because there's got to be an infinite amount of time along this axis alright so this is the static slicing and those are the various presentations of the sitter space that we'll be working with the static slicing here it is has a horizon and because it has a horizon the description within a static slice is thermal there are another there's a way to see that their thermal a mathematical way to see that that the properties of this chunk over here are described by thermal equilibrium that's something for you to explore you can explore it in various places I will simply tell you the rules the rules are that an observer in this static patch here this is called the static patch and notice we should make clear that for an observer moving along here this really is an event horizon that observer cannot get signals from behind this line here it really is an event horizon and it has many of the properties of the event horizon of a black hole all right the little going what we're going to take from the black hole the fact that the description of an observer in here was restricted to live on that line watching things from outside sees a thermal world you see the thermal world with temperature let's see what the temperature is the temperature is one over two pi times the square root of G naught naught let me be clear now temperature in general relativity there are two kinds of temperature this coordinate temperature which is typically taken to be dimensionless and then it's proper temperature it's temperature that would be read by an honest thermometer all thermometers made in the same standard thermometer factory and you move them around from place to place if you take such a standard thermometer and you take it upward in the gravitational field that will show a lower temperature than downward and that's a redshift phenomenon all right so as you move around in here in particular as you move in are the temperature changes the proper temperature changes the proper temperature changes and what is G naught not G naught naught is 1 minus R squared so this is a square root of 1 minus R squared but there's some dimensional factor G naught North had an R in it so there's an R downstairs and that has the proper units of temperature one over length incidentally I am setting the speed of light equal to 1 and everything we're doing so inverse distance and also H bar inverse distance as units of energy and temperature has units of energy this is the temperature notice that it R equals 0 the temperature the proper temperature at R equals 0 is just 1 over 2 pi our that's usually called the Hawking temperature of the disorder space the bigger the de sitter space the smaller cosmological constant the beginner this is a space the cooler is the observer sitting at the center but the temperature always diverges as you go out to our equals one incidentally R equals 1 is a hyperboloid or the limit of a hyperboloid which gets closer and closer to this point over here and it really means these light cones so if you get real close to the horizon it's hot if you're near the center of this sort of space is it's as cool as it can be the center meaning R equals zero it's as cool as it can be anywhere and it has a temperature in addition it has an entropy the entropy of the de sitter space if I wrote it down well it's one-quarter the area of the horizon in plunk units and plunk units it's one-quarter right so if we if we want to get it out of plunk units we put a G downstairs and the area of the horizon is 4 pi 4 PI R squared so it's PI R squared over G which is the same as PI over G H squared ok so this is the Hawking bekenstein and some other people entropy of the sitter space of the static patch of the city space incidentally the reason that it has an entropy is because there's entanglement entropy between the side and the side whenever you're looking at only a part of a space there's entanglement entropy with the rest of the the rest of the space and one way of thinking about this entropy is that it's entanglement entropy ok let's see how much time is it now let me take a quick break for questions with you already yes okay one one the one thing I I'm a little bit deaf okay I think the right thing to say is one way of thinking about it is it as entanglement entropy there are other ways and I think it would be a good idea for me to spend a little bit of time talking about black holes so let me defer it for now let me defer it for now but that's an extremely subtle question of whether it is or isn't entanglement entropy and what is its entanglement with in one view its entanglement with the other side of the horizon in another view its entanglement with the heart where the horizon degrees of freedom themselves the ones which make up the the entropy so this is a subtle business that we would like to come back to in fact I'm sure we'll come back to it any other questions yes sir yeah I suppose it probably is but then when the temperature is defined in a frame where the gas or whatever it is is instantaneously at rest so the whole notion of temperature means that you're that but yeah I think there is a I think there is a covariant definition of it but in this case well probably in all cases I what is it I'm not sure what it is I never thought about it very much but um in a frame of reference where things are static that's what temperature is defined so it's a good question but yeah well it is the temperature that a thermometer would record but now let's think about it is the temperature that a thermometer located at that position would record the only problem is that in a typical situation let's say our universe our universe has a Hubble constant which is or an R inverse Hubble constant which is what 20 billion light-years something of that order of magnitude 20 billion light-years it's an extremely big distance H is an extremely small number the temperature is impossibly low but not impossibly low it's extremely low so what kind of thermometer can register temperatures which are well I'll tell you what units I know and I know in plunk units its Candela - sixtieth and Clarke units does that help you now convert that to it's 10 to the - a large number in any units you know so would take a rather spectacular thermometer to be able to measure it but it is what a thermometer would measure at the center here if you'll have a thermometer now keep in mind if your trajectory along here is accelerated this trajectory is not accelerated a trajectory which moves on a curved path like this is accelerated and keep it accelerated you would have to provide some force here's what you could do you have a fishing line okay the universe because of its The Sitter character is accelerating outward that means there's a kind of gravitational anti-gravity pulling everything away from the earth and that force is in fact proportional to distance that's what the cosmological constant does all right now you take your fishing line and on the end of your fishing line you you put a thermometer and you cast it out there and you hold on to it you don't let it pass through the horizon you don't let it pass through the horizon but you'll hold on to it and you keep it steady at a distance which you've already calculated is almost the distance to the her you also are connected by a by a cable through that thermometer so that you can read off its temperature the temperature will be hot and the temperature will increase as that thermometer gets closer and closer to the horizon the horizon in proper distance from you is about a distance R okay about a distance R now not a proper distance is about a proper distance R not not exactly and so yeah the temperature would depend on how far you cast out the other line okay any more yeah that's why I say you get the temperature is so low basically you have one photon in a distance of order 10 billion light years of wavelength 10 billion years or light years so it's not something however keep in mind that we are going to be interested in de sitter spaces with much larger Hubble constants in our past was inflation perhaps before inflation there was even a bigger Hubble constant it is going to be important to take into account that that the sitter spaces with large Hubble constant have temperatures so in our world it's of no interest whatever from the point of view of measuring things in a world with a very very much larger Hubble constant maybe a thousand of the be ten to the seventh times smaller than the me plunk mass that temperature is important it creates fluctuations and those fluctuations are important okay if there are no more questions I'm going to go on to another set of coordinates now I I know coordinates are pretty dull things but it's necessary to master them it is really necessary to master them another set of coordinates which I'm going to write I'm going to use them in the flat space slicing they're called conformal coordinates and they're very very good coordinates for exploring the causal structure what's in causal contact with what who can send messages to him all right so let's start again with the flat slicing of the sitter space here it is I've got it right here let me write it again d s squared is equal to minus D tau squared plus e to the to tell the Exide DXi all right this much is flat space over here wouldn't it be nice if we have an e to the 2 tau over here so that we could take the e to the 2 calamy outside well we don't have an e to the 2 tau over here but we can define a new coordinate DT such that DT is equal to D tau divided by e to the Tao do I have that right in other words that DT times e to the Tao is equal to D tau if I do that then I can place it I can replace this with DT squared and pull an e to the Tao on the outside now this is an easy Quay ssin to integrate and what do we get we get T is equal to minus e to the minus tau that's a little bit funny T is negative but that's ok T is negative and it runs from minus infinity to zero when tau is deep in the negative past this is large and negative and then when tau goes to the remote future T goes to infinity with this Oh incidentally there is an r-squared here I forgot the r-squared that should always be an r-squared on the outside and with this definition we have the d s squared is equal to R squared times minus VT squared plus the x squared divided by T squared so where do I get the T squared from t squared T squared is equal to e to the minus 2 tau but it's e to the plus 2 tau which appears in the metric so there's a T squared downstairs here this is the metric in conformal coordinates we can write it this way I usually write it this way most of the time H squared downstairs very simple flat space-time but with an overall factor of 1 over H squared T squared T runs as I said from minus infinity to infinity so this geometry only has it only has a past of course it doesn't only have a past it has a future too but the remote future little T or little tau equals infinity or little T equals infinity is all on this horizontal slice here so here's the equals infinity and slicing things up into tau sorry slicing things up into T it's dividing up the time interval of the sitter space and a kind of logarithmic way each one of these intervals is the same in proper time and it corresponds to an interval of order of magnitude T equals 1 over H during each one of these intervals the size of the universe doubles or the size of the scale factor doubles during each one of these intervals here's the scale factor 1 over H T during each interval the size of the scale factor doubles the same amount of proper time elapses and that's not surprising because the because the scale factor is exponential okay so this is the sitter space in conformal coordinates and the special value of it is light rays look at well light ray is a light ray is just DT equals DX in other words the motion of light rays doesn't give a damn about this factor in the denominator here the motion of light rays is exactly the same as it would be in ordinary flat space-time in other words light rays just move along 45 degrees oops that one's not very 45 in every direction light rays move along 45 degrees and from this point of view it's very easy to tell who can send a message to home this person over here can send messages to everywhere is in here this person up here can receive messages from anybody in the backward light column all right the only thing R about it from this point of view is it comes to an end so it's as if you had flat space-time that just came to an end but not quite you do have to keep track of this factor here it's what gives the space-time curvature okay on another bit of terminology another bit of terminology this terminology is more general than just the sitter space it's an important concept so I'll introduce it typically space times have future boundaries I'll give you some examples of some future boundaries in the set of space-time the future boundary would simply be T equal capital T equals 0 more generally the future boundary could be more complicated it could be wavy or it could be light like example of a light like future boundary would be flat space-time flat space-time has a Penrose diagram which looks like this here is the future boundary this light cone this asymptotic light cone together with P equals infinity define the future boundary of flat space-time all kinds of space times they typically have future boundaries and I now want to define what's called a causal patch a causal patch is a very important concept maybe more pigging more important than a lot of people recognize who work in these things pick a point on the future boundary if it's the sitter space and it's just a horizontal line if it's flat space-time it either corresponds to T equals infinity up here or a point on the future light color look and simply draw the backward light cone of it the backward light cone is called the causal past let's call this point a it's called the causal past of AE or this region here in the interior of it the interior is called the causal past or just the causal patch associated with a every point in the future boundary has its own causal patch sometimes they overlap other in some regions they overlap in some regions they don't overlap and if you think about it any observer who lives on a time like trajectory which ultimately arrives at AE can see things only within his causal patch all of his physics all the things that he can detect is within the causal patch that's the meaning of the causal patch everything that a particular observer can see now of course different observers will have the same causal patch if they go to the same point they'll have different causal patches if they go to different points how about over here in flat space-time the causal patch associated with vehicles infinity up here is everything the backward light cone and everything inside it is simply everything so flat space-time are from the point of view of a very distant future observer distant instant in time that observer can look back and see everything in you can also ask what about the causal patch of somebody lives over here causal patch of somebody not somebody lives over here but I observer who goes to that point now that observer has to accelerate and in order to get out to that point has to have constant acceleration in constant eternal acceleration basically to get out to that point but there is also a causal path service if I fill in this diagram over here this is for the specialists now if I fill in this diagram here this becomes a what's called a causal diamond and that causal diamond this Rindler space it's the space-time is seen by an Excel or uniformly accelerated observer but never mind it's not important if you don't know what I what I mean so causal pasts are associated with points in space-time another example would be in a black hole there's a black hole geometry the causal past of anybody outside the black hole at P equals infinity is everything outside the horizon of a black hole but there are people who for own have the misfortune of falling into the black hole and their causal past looks like this okay so that's the idea of a causal patch one other thing about causal patches in particular a causal patch of the sitter space yeah here's the sitter space everything below that line here's the causal patch of a point now I'm going that the causal patch is a region of space-time but if I foliate the space-time by space like sections like that then I can speak of a causal patch at an instant of time this region over here which is of course a sphere I will call the causal past at an instant of time that's too big a phrase so let's just call it the instantaneous causal causal patch of point eight now here's an interesting thing about the sitter space and an important thing a really important thing there's a sense in which as time goes forward the number of causal patches replicate their causal patches replicate and the causal patches the number of them grow exponentially with time so let me show you how you think about that take again the sitter space and take a slice of it at a particular time draw a bunch of causal patches I can't quite fit them together to fill up the space and a non-overlapping way except in one dimension but take this volume and that volume defines by definition the volume associated with a causal patch at a given time all right so at a given time tau or T we could say that a given region of space occupies a certain number of causal patches incidentally the size of a causal patch is about R but the definition of causal patch doesn't depend on the definition of are you just draw these light cones backward and you can define in that way volumes associated with causal patches and roughly speaking I can say a given coordinate volume here's a given chord that volume not a given proper volume but a given coordinate volume is X number of causal patch sizes big now what happens as time goes on the causal patches replicate time goes on a little more we can say the causal patches reproduce and each one produces a certain number of new causal patches depends on the dimensionality how many new ones it produces roughly speaking in three dimensions every causal patch when you double when you when you go one step when you go one step what does it mean that means a time T of order inverse H one step in in capital T in three dimensions the number of causal patches in a given coordinate area gets multiplied by eight why eight because it's 2 to the third okay in two dimensions the number of causal patches would get multiplied by four in three dimensions the number of causal patches sorry and one dimension the number of causal patches would double and so forth this replication of causal patches is in some sense the the most important thing about eternal inflation the construction that I did here did not really depend very much on the exact definition of the sitter space let's just said take a slice continue to slice it in different ways and the number of causal patches will grow and typically it will grow exponentially okay the causal patch the the meaning of the causal patch should be you should think of it as defining a lattice spacing as defining a cell size on a surface in various mathematical theories of the sitter space which relate to the sitter space the quantum field theories on space like sections here each causal patch plays the role of a point in the lower dimensional space it plays the part of the role of a point of course there's lots of degrees of freedom in here many many people can exist inside a causal patch but all of those people are counted as degrees of freedom associated with one cell that's theme that theme of causal patches in the sitter space being like cells of a quantum field theory that has become somewhat pervasive by now among some class of theorists and eternally in other inflating kinds of theorists have become used to that idea so that suggests that we think of these inflating spaces as lattice systems where the number of lattice points replicates each efj Olding or each yeah each a folding or perhaps two folding each two folding of the geometry where length scales get doubled or whatever Ichi folding replicates the number of causal patches and therefore replicates the number of lattice sites on some kind of construction which we're going to study in some detail in fact I think we're going to come to it right now how much time do we have 20 minutes okay all right I'm interested in the following problem it's a problem that has been studied very beautifully by Matthias and his collaborators of senatorial saga there are too many of them criminally a little but he has helped the problem the problem of phase transitions are from one kind of behavior to another we now come to the problem of eternal inflation eternal that internal eternal inflation internal inflation also happens but when it does you come to Princeton and that fixes you up real good okay now we have to come to the issue of bubble nucleation bubble nucleation occurs because this landscape what we're going to take a very simple model to begin with the very simple model is a potential of simple landscape which has two minima one of them happens to be right at zero cosmological constant our own vacuum is very close to zero cosmological constant not quite and another inflating vacuum which is up here this is not the kind of figure that you would draw to describe slow conventional slow roll inflation it's the kind you would use to study eternal inflation and to get from here to here you have to tunnel tunneling incidentally is not something that happens simultaneously everywhere we will come to what a tunneling transition looks like but this is the picture tunneling from here to here and we're going to explore the question of in the space of parameters but in particular the height of the potential here in the space of parameters when does eternal inflation occur when doesn't eternal inflation occur and how many different phases how many different types of eternal inflation are there it turns out eternal inflation is not one phenomenon it's many phenomena many different kinds of eternal inflation are very quickly if you have a potential like this then basically there's no inflation at all you just roll down and you go down to the bottom the frw universe just grows and does whatever it does eventually just comes to rest or whatever ah if you have a potential like that then you get trapped in here and essentially you get trapped forever in there although here and there locally in space there may be tunneling phenomena as you lower this potential there are phase transitions there's a sequence of phase transitions and a sequence of different behaviors and it's interesting to to try to classify those behaviors just to know what kind of thing we're talking about what kind of different phenomena are we grouping together under the rubric of eternal inflation our first thing to know is that when a tunneling event happens here we are we're sitting up here in the vacuum a metastable vacuum up here let's give it a name let's call this the ancestor vacuum and let's call this one here are the descendant vacuum you go from here to here after giving birth to a to a younger generation of vacuums starting up here so we start with the ancestor vacuum here and what does not happen we have this infinite slicing of the sort of space let's think in terms of the infinite slicing of this inner space it's extremely unlikely that the whole thing is simultaneously going to go plop from one vacuum to the other that's not the way phase transitions happen in quantum field theory it's not the way phase transitions happen in condensed matter physics what happens is bubbles nucleate someplace a bubble nucleate let's say that that bubble nucleates at time minus capital T in conformal coordinates then the bubble starts to grow if it's big enough if it's big enough it will start to take over and will start to grow we're going to study bubble nucleation a little bit but basically it grows very quickly it's the domain wall that separates it from the original vacuum spreads out quickly and very quickly reaches the speed of light or get asymptotically close to the speed of light a simple enough approximation in one which is very close to the truth is it just a domain wall spreads out with the speed of light inside the domain wall we have the defendant vacuum outside the domain wall we have the ancestor vacuum now the phenomena of bubble nucleation is very similar to bubble nucleation in condensed matter physics you have some how many people have read cat's cradle okay what happens in cat's cradle cat's cradle is a metastable phase of our water called ice 9 no no it's the water which is meta meta meta stable and it can decay to ice 9 it's the tunneling rate is very small so it doesn't readily happen but some bad guy kicks off a transition and makes ice 9 a little bit of ice 9 and then the ice 9 bubble expands well no big tragedy it only explains a certain amount until it's finished but cat's cradle was not written in de sitter space it was written in ordinary flat space-time what will inevitably happen in flat space-time his flat space-time is a bubble will nucleate but other bubbles will nucleate there will be on the average a mean separation between the bubbles but an infinite number of bubbles will nucleate and they will come crashing together and eventually the whole sample will make a transition to the to the lower energy state that's what happened to the world in cat's cradle all the water turned into ice 9 all the bubbles come crashing together but the new phenomenon in in eternal inflation is at the same time that these bubbles are being nucleated the ancestor vacuum is inflating inflating means exponentially growing so that means the distance between these ball a distance between points here is getting larger and larger and that allows the possibility that the bubbles the separation of the bubbles due to the inflation allows them to outrun the the tendency for them to grow together now you can see this right on this diagram here it's immediately evident here that here's a situation where infinitely many bubbles nucleate but the course of this termination at late time they simply won't come crashing together you could have a uniform distribution of bubbles nucleate at some given instant of time in flat space they would come crashing together and cause the whole space to make a transition in the sitter space no such thing this is the phenomenon of inflation of the background allowing the bubbles are allowing the inflation to beat out the tendency of the bubbles to grow so you see it right on this diagram here here's what we want to investigate we want to investigate under what circumstances well of course if the bubble nucleation is too frequent if the barrier here is too low so that the bubble nucleation rate is too large then you can get into a situation where the bubbles nucleate too frequently and too with two smaller distance between them and the whole thing does go kablooey into the into the ice nine phaser into the into the lower phase there in that case you would say that inflation was aborted or exterminated or whatever by bubble nucleation which eventually put everything down here on the other hand if the bubble nucleation is low enough then the mean separation between these bubble nucleation czar such that they can outrun our they can their own tendency to expand so we would like some sort of qualitative semiquantitative way to investigate whether whether you eternally inflate which means these regions in between never quite get filled up they always continue to inflate the bubbles never completely take over the whole thing or whether or whether the eternal inflation is killed let me give you a way to think about it a simple way to think about it okay let's take into account let's begin here's here I think is the best way to think about this these kind of questions let's take all bubbles which nucleate within a time go back step let's begin the world all up in here now we take one a folding of inflation one a folding or one two folding one two folding means at time T of order one over H it is a step of capital P one step of capital T and let's take all of the all of the bubbles which nucleate in that step during that time and just for simplicity let's imagine them nucleating all at the same time at a conformal time minus T in the next round we will add in all those bubbles which nucleate in the next two folding and then in the next two folding but let's stop let's not do that yet let's just take the bubbles which nucleate within the first two folding here let them expand and then look at the system from the top look at it from above look at it from the perspective of T equals infinity and what we will see is a collection of bubbles all of the same size they're all of the same size is because they nucleated at the same time when I speak of size I'm talking about coordinate size there are coordinates laid down on here not proper size all proper sizes go to infinity up here we'll see a statistical collection of them like this the inflation will clearly be aborted if the mean separation between these is smaller than the size of them then they're going to be crowded on top of each other and if there are too many of them they will completely abort the eternal inflation so the right question then to try to answer whether eternal inflation takes place or whether at least you escape one round of e folding here is whether the distance the coordinate distance the coordinate distance the mean coordinate distance between bubbles is larger than or equal or less than the coordinate size of the bubbles what's the I think it's getting all right let's let's see if we can do the calculation do we have 10 minutes literally to take a look dude it won't take very long okay now the control parameter is the height of the barrier here what does it control it controls the nucleation rate let's call the nucleation rate gamma and the meaning of gamma is it's the probability per unit for volume per unit proper for valid volume the probability per proper units per you the probability that in a unit for volume of bubble nucleates it's not a probability per unit time it's a probability per unit time per unit space so it's probability per unit proper volume 4 volume so it has units of 1 over length to the fourth we can write it in terms of a pure number gamma times H to the fourth H has units of inverse length if we measure everything in Hubble units then gamma is the is the probability per unit for Hubble volume that a nucleation takes place so let's define the dimensionless variable small gamma in this way it's small gamma which is going to be the figure of Merit of whether we do or we don't deter kn'l inflate okay good so now let's see if we can calculate the mean separation as a function of gamma we're doing steps take a coordinate volume big coordinate volume Delta X the coordinate volume is Delta X cubed now when I say coordinate volume its dimensionless it's just a box of size Delta X of coordinate size Delta X by Delta X by Delta X and let's say as how many bubbles nucleate within the first be folding here I'll use the word a folding and two folding synonymously during during the first doubling time here how many bubbles nucleate okay so the first so you first calculate the volume in this region you first calculate the volume of this region the volume of this region is Delta X terms the scale factor cubed and the scale factor where is the scale factor the scale factor is 1 over H capital T that's the scale factor a alright so this looks like it is Delta X / h t cubed that's the volume of this region here at time T now multiply it by the rate the number of the rate is the number of nucleation in a unit for volume so that's gamma H which it was H to the fourth and now multiply it by delta T we still have yet to multiply it by delta T this was the rate per unit space per unit time so we have to multiply it by by delta T but delta T is simply 1 over H so that's another one of H downstairs upstairs downstairs downstairs that's the number of bubbles that are nucleated in here in one evil-lyn okay let's cancel out the HS there's three of them here four of them four of them upstairs well boy H cancels out altogether now let's consider the density of bubbles this is the number in a four volume Delta X and three volume Delta X cubed the density of them the density of the of nucleation means the density per coordinate volume per coordinate volume the density of them is just one over well multiplied by Delta X cubed this is just the inverse density here as being the inverse density it's also the cube of the mean spatial separation between the inverse of the density is the mean separation between things now it's the cube of the mean separation so this is the cube of the mean separation between bubbles one over T cubed so Heather's right and again no gantlos and I have it upside down yeah this is the density they multiply Delta X cubed we have to turn it over to find the mean distance so the mean distance cubed is T cubed divided by gamma notice when gamma gets small the mean distance gets large now what is the size of one of these bubbles the size of one of these bubbles when it's projected onto future infinity is just P so the size of the bubble is PE the volume of the bubble is P cubed and it's being compared with P cubed over gamma so have it right I think I have a right yeah okay P cubed don't the T is not the temperature incidentally whenever I see T's always hits me temperature it's not it's the conformal time here that's the size of a bubble so we're comparing this with T cubed itself let me take the cube root of it to define the length between them we're comparing T divided by gamma to the one-third with T it's clear that when gamma is significantly less than one the distance because we could have guessed this of course we could have guessed this but when gamma is significantly less than one the distance between these bubbles when projected onto future infinity is larger than their size significantly larger than their size when gamma is much less than one so when gamma is much less than one and that's the only parameter that this here is gamma you escape the first round of the foldings here and you are left with some inflating volume well it can now take all the region which has not been killed by bubble nucleation killed means stop from inflating drop down to here you can take it out and ask again let's take this region over here do exactly the same thing for the next be folding here and what you will find is again exactly the same thing if gamma is significantly less than one the next a folding will not kill you for the same reason the bubbles will now be more frequent this is more volume in here because of the exponential increase but they'll also be geometrically smaller when projected on to infinity because the geometrically smaller everything scales and you'll find exactly the same formula you escape the next round of our eternal inflation escapes for the next round if gamma is significantly less than one and so forth and so on so if gamma is significantly less than one bubbles do not take over the entire geometry there's always regions in between which are eternally inflated there are transitions and the transitions are as a function of gamma I guess it's too late to go into those transitions today we'll go into them tomorrow a little bit and I'll describe the various the various phases that exist and the various types of behavior statistical behaviors which which is which simple models seem to indicate exists okay tomorrow

Channel: Institute for Advanced Study

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Id: 18VfbG8ty18

Length: 87min 19sec (5239 seconds)

Published: Fri Jul 07 2017

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