Aspects of Eternal Inflation, part 2 - Leonard Susskind

good morning all right I want to start with something that has on the face of it very little to do with eternal inflation but it will come up and just think of it as a side comment before we even begin it has to do with quantum mechanics and in particular founder the foundations of quantum mechanics which I may or may not come back to for the end of these lectures but there was one particular point that I wanted to make because it will come up later today and I think an interesting way um it has to do with the phenomena of decoherence decoherence is a phenomena which is very important to the interpretation of quantum mechanics it has to do with when the virtual realities in the wavefunction become turned into real realities objective realities by the phenomena of sometimes we call it measurement or sometimes we just call it entanglement and tangle Mint's between the system under consideration and possibly some environmental degrees of freedom and then when those environment the environmental degrees of freedom might just be a big big detector or it might just be a photon which interacts with Schrodinger's cat and carries off the record of whether the cat was alive or dead it's when that happens that we say the virtual realities that are encoded in a wave function get turned into objective realities in older language it's when the wavefunction collapses or the wave packet collapses so all I really wanted to say about it was the following imagine a laboratory there's a laboratory moving upward in time and experiments take place phenomena happen the entire laboratory if it's a sealed laboratory is in some quantum state and it remains in a quantum state and at no point does the information as to what occurred inside that laboratory get out unless something escapes from the laboratory a photon for example it's that photon which is carrying off a record for example it could be a record of whether Schrodinger's cat is dead or alive it could be a record of whether a photon went through an upper slit or a lower slit and a two-slit experiment that there happen to be a detector nearby that detector might detect the photon and then as I forget photon the detector might detect an electron going through the two slit experiment get excited and then radiate a photon and it's in the process of radiating that photon that the irreversible phenomena happens that we call wavefunction collapse if the photon doesn't get radiated then the wave function of the entire system doesn't collapse and we would say as far as the entire system goes it remains in a pure quantum state and there's no record of what happened it's important to the interpretation of quantum mechanics that decoherence happens or entanglement between the system and the environment is entanglement or decoherence and irreversible phenomena in practice when you have a big detector of some kind - a big apparatus with so many degrees of freedom when a record is left on it by some experiment in practice it is irreversible it's just too hard to reassemble all the degrees of freedom and have them go backward and Rico here throwing goes cat so to speak in a linear superposition States but in principle one can imagine mirrors that radiate the or that reflect the outgoing photons the carrier signal the carrier record of a system bring them back into the system and reco hear it in other words the wave function which collapsed uncollapse --is now it is important to the interpretation of quantum mechanics that at least some decoherence does happen so that you can say that things really happened really did decohere and the question is what is the mechanism for that well the mechanism in ordinary here our thinking is photons which just come off and never come back but you can never be sure that there really aren't mirrors way out there which could bring them back together in inflation or into eternal inflation or Indesit are like spaces there's another mechanism which can absolutely ensure that the phenomena of decoherence is permanent and cannot be undone and it's the existence of horizons if here's the sitter space and here's our experiment that entire experiment takes place within a causal patch and when a photon passes out through the event horizon it is gone it cannot come back and that is a mechanism for absolute irreversible entanglement or absolute irreversible B coherence to take place and I just wanted to mention it because it is going to come up again it's going to come up in our discussions about eternal inflation alright so that was - first step of my lecture today what's that say it again exactly so abs is a cosmology where nothing ever permanently dqo here's just because its walls are reflecting right okay um yeah so let's come back I think I mentioned the problem that we're going to attack today and discuss today it's the problem of determining what kinds of different things there are that we would call eternal inflation there's not just one phenomena there's a range of phenomena that are controlled by a control parameter many made the in general live with many control parameters the control parameter in this case would be the decay rate for bubbles to nucleate in a background the sitter space so here's the setup we imagine a potential we imagine a scalar field potential of some kind for simplicity I'm going to take one of the vacuum to have absolutely zero cosmological constant and the other one to to have a positive cosmological constant tunnelling transitions can take place and those tunneling trends the transitions that take place produce bubbles a bubble without worrying too much about the fine detail that happens right at the bubble nucleation simply looks like an outgoing light cone and let's let's examine a little more closely the bubble the interior of the bubble and let's give these places names let's call this the live place and let's call this the dead place why is this called the dead place because it stops inflating here inflation takes place here no inflation takes place here and I'll imagine that inflation is live meaning it can reproduce it reproduces interesting things happen expansion this is dead inflation has stopped happening of course real life can't exist here and real life possibly could exist here so we're not talking about life as we know it we're talking about inflation inflation takes place here we'll call it live okay now here's what we're interested in yes as I said if the potential barrier in between is high then basic phenomena which happens is bubble nucleation and bubble nucleation let's take a look at what the diagram looks like fairly precisely let's so let's do a Penrose diagram all right so we start with the Penrose diagram of the center space a bubble nucleate at some point let's put that point over here at R equals 0 that's arbitrary any point could be any point in space could be rotated around until it's R equals 0 that's the nucleation point and after the bubble nucleates it starts to expand with this which close to the speed of light and so it moves out about like that well that's not the end of the diagram something happens afterward and to understand what happens afterwards you go down here to the dead region and you say ok that's going to be basically flat space 0 cosmological constant so we draw a Penrose diagram for flat space that's what flat space looks like this portion of the diagram was misrepresented as the sitter space but let's patch into it the corresponding piece of flat space here's the domain wall going out here's the domain wall going out notice that R is equal to zero over here R stands for the size of the two sphere that's attached at every point to this diagram and it goes to infinity up here monotonically likewise R is equal to 0 over here R is equal to infinity over here likewise monotonically and so this can always be matched by a little bit of squashing and squeezing this can always be matched on to here and now we have a picture of a bubble of dead vacuum nucleating in a live vacuum what we're interested in is the global behavior of this and the statistical structure of the pattern of nucleation of bubbles as a function of a control parameter which we're going to take to be let's say the height of this ball this bump here when the height is very high the bubble nucleation rate is very slow and bubbles are widely dispersed and far from each other eternal inflation takes place as we discussed last time as the barrier comes down bubbles nucleate faster and faster at some point the bubbles will nucleate so fast that they will simply fill up the whole volume and the lot the live vacuum will be extinguished it will become extinct no inflation after a while in between there are several phases several different statistical phases of eternal inflation all of which are interesting and in fact given that we are in a vacuum which descended from eternal inflation which may be wrong but given that we are we don't actually know which of these phases we live in so it's worth exploring what the various phases are ok the control parameter is gamma remember what gamma was the decay rate per unit space per unit time was gamma times H to the fourth where H is determined by the height of this potential over here it's a dimensionless number and basically we can think of it as going from zero to one it's the probability that within a Hubble volume within a Hubble time a bubble is nucleated in one Hubble time in one Hubble volume it's the probable probability for nucleation to take place and the question is what happens as we vary gamma okay so let me begin drawing another diagram here let me divide up space into causal patches let's have an initial condition over here the initial condition will be that the vacuum is in the live sector and now let's just these are guidelines they're not real phenomena the guidelines and they divide the initial surface into causal patch sized regions or horizon sized regions regions of size H inverse as time goes on let's draw more of them in such a way that we fill the space time over here at this point we have more causal patches or more of these little wedge shaped objects and as time goes up the number of them exponentially increases in any one of these a bubble could nucleate now just for simplicity really just for simplicity let's assume that the bubbles can can look nucleate at these points or at this point or at this point here you don't have to assume this of course where I'm going is I'm going to make a simple lattice model of what happens all right so just for simplicity let's place the nucleation but they could take place in between that's not important and when a nucleation happens it creates one of these dead regions there's a dead region there's another dead region back here a bigger dead region bigger in the sense that it looks bigger on the coordinate patch up here a very late one will look small and if you wait long enough this system will fill up with bubbles in such a way that it forms a fractal in the future here we're interested as I said in classifying the different statistical behaviors that you would see if you looked down from above at this bubble nucleation the trick is to cut it off we can study the entire infinite structure all simultaneously so what we do is we cut it off at a given time we look at the bubble population that would be formed by all bubbles before that time that gives us a pattern of bubbles and then we let the cutoff go to infinity thus creating smaller and smaller coordinate patch bubbles right it looks like a statistical system not that unfamiliar from other areas of physics where there are big things small things smaller things smaller things and that's that's the structure that we want to understand our to do that I'm going to make a simplified model the main reason for studying this model is because in totally different contexts it has been completely solved by mathematicians it's called Mandelbrot percolation theory and the two regions the dead regions and the live regions are called the curds and the way somehow this has to do with the making of cheese but I don't know what it has to do with the making of cheese cheese when it's made is full of little fractal bubbles I guess I don't know um in any case here's the model instead of these causal instead of these diamond-shaped regions we're going to slice through at any given time and we're going to divide space here's here's a region of space a regional coordinate space and we're going to begin by dividing it up into cells each cell about Horizon size right and just to make the model very concrete we're going to do it on a square grid actually a cubic grid excuse me a cubic grid all right so we start with a cubic grid representing this initial state here broken up into into cells all of the cells start live that's a boundary condition we could start some of them dead but let's so let's start with an all live and so they're all colored now at Stanford we had great the problems deciding whether we should call the dead regions white or black we started calling them black but then I started drawing them on the blackboard and coloring them in with white chalk and we got very confused so we will use the terminology live and dead okay so everything starts out live which means white which on the blackboard means black all right what is the problem that now all right now we now we decide in each box whether a bubble nucleates there or not if a bubble nucleates and it nucleates with probability gamma each one of these boxes is imagined to be horizon size we're going to also discretize time and the time step is also going to be inverse h that means that the probability to nucleate a bubble is just gamma so with probability gamma we either color this dead or we don't color it dead probability gamma is the the control parameter for the system here all right after going through the first round this corresponds to the first that all of the bubbles that are nucleated in the first round meaning the first a folding or the first two folding we then break up the boxes the remaining boxes into things of half the size which means each box is broken up into eight new boxes you can break up the dead ones also but the dead ones are dead and if they they don't care whether they're broken up or not they're finished they're out of the picture and then we go through the remaining live part of the lattice again with the same probability gamma colouring-in killing it or not killing it and so forth whatever is left over we divide up again now this dividing up is of course inflation if we instead draw the picture at each stage so that each box is Hubble size of course the entire picture has to grow exponentially since I can't do are pictures that grow exponentially what I'll do instead is break things up into smaller and smaller boxes okay this is the inflating region this is the dead region and the first question is what is the probability that the entire thing gets extinguished it could happen that in the first step everybody is colored dead there's a probability for that all right if it happens it's extinguished but it might live for a while it might live through several steps is there a probability that it will live forever and continue to inflate and the answer depends on gamma so let's see if we can work it out here's the equation I'm going to write down let's let's follow a particular trajectory through here concentrating for forum for the sake of just expediency I'm also going to divide up the dead region use the dead wages are dead they don't reproduce anymore but I'm going to divide them up just for counting purposes first question is how many boxes are there after the nth step and the answer is easy it's 8 to the N 8 to the N each doubling takes each box and multiplies it by 8 and so we get 8 to the end boxes all together after n steps now imagine following a trajectory that are wanders at each step wanders into a box and ask what's the probability that that box is dead the probability or what's the probability that that what's the probability that a given box is either live or dead in particular let's take the probability that it's live after n steps Oh incidentally the probability that it's live is equal by definition to the number of live boxes after n steps divided by the total number which is 8 to the N ok so the probability events live satisfies an equation and the equation is P live in the n plus first step right for a given box is first of all the probability that the at the nth step its parent was live let's see - gamma x p live that's it why why - gamma this is just basically the probability that it gets killed if it gets killed it's no longer live so it's 1 minus gamma to make it simple it's 1 minus gamma times P live we can also write an equation for P dead I'm not going to use it but let's write it anyway he did we don't need it he did of n plus 1 is equal to P dead of n plus gamma times P live of n notice the some of these stays equal to 1 if P live plus P dead is 1 plus 1 then it's 1 over here these cancel we don't need the second equation the first equation is all we need it's easy to solve at each step you get a factor of 1 minus gamma you lose the probability of 1 minus gamma so after the nth step the probability is 1 minus gamma to the in that you're still live ok that says that the probability that you're live goes to 0 sounds like your extinguished but no the number of the probability is the ratio if I want to know the number the absolute number of live cells I multiply by 8 to the N this is the number of live cells at time n it's if n live goes to 0 that it becomes extinguished not if the probability for a given cell ok what's the what's the condition that the system remains alive forever the answer is that 1 minus gamma times 8 should be bigger than 1 if 1 minus KR gamma times 8 is bigger than 1 then the population will exponentially increase this will be exponentially increasing if it's less than 1 then the average population goes to zero let's say I think if I remember that you get extinguished if gamma is bigger than 7 8 so that's the first fact and the point 7/8 in this example corresponds to the transition between no eternal inflation and eternal inflation that's where it occurs R we can calculate something else in this model I'm not going to do it it's a little bit intricate but it can be done it can be worked out it's just a basically algebraic equations you can calculate the probability that eternal inflation takes place now eternal inflation will definitely not take place if gamma is bigger than 7/8 and it has a finite probability of taking place if gamma is less than 7/8 we're starting with a single box as I said it's always possible that the single box can get killed in the first step or the first two steps so there's always a possibility of extinction but the question is what's the possibility of probability of no extinction and that's the situation for gamma less than 7/8 and when it's calculated it's of course 1 when gamma is equal to 0 and it smoothly goes to zero smoothly goes to zero the fact that it smoothly goes to zero the people who study these kind of statistical problems mandelbrot and other mathematicians call this a second order phase transition because it's smooth in the language of eternal inflation this transition is the transition from no eternal inflation the basically slow-roll eternal inflation it's the situation when the decay rate is large in other words when you pull this the height of this barrier down low enough that you're just at the edge of internal inflation takes place and the eternally inflating side of it is what's called slow roll eternal inflation right now there are two more phase transitions as a function of gamma let's plot gamma over here again this is gamma going from zero let's say to one we found one phase transition but we're going to find more phase transitions in here characterizing different kinds of behaviors which is quite different than what happens in the slow roll eternal inflation alright to explain it I need first to explain a concept called percolation percolation Theory percolation theory is a standard statistical mechanical problem and it goes something like this you start you don't we're not going to iterate for the first round just ordinary product percolation theory does not subdivide the cells after they've been subdivided you just start with a big let's say infinite lattice an infinite cubic or an infinite square lattice from now on I will not draw a cubic lattices I will do a square lattices you start with a big infinite square lattice it is extends in both directions and also in all directions and you randomly with probability gamma either paint it with chalk or leave it blank in other words in the current terminology you kill it or you leave it live but we're not going to follow it yet as it breaks up into smaller pieces so we just want the pattern of things that are created if the probability is small if gamma is small then on the average there will be a large separation between these clusters of of dead regions in particular if the probability is very small the typical cluster will be no bigger than one cell and they'll be very well separated now as you start to jack up the the probability for four or transition the there's a notion of cluster size is the notion of percolation clusters here's the idea if you haven't percolated percolated means these things get connected together in a big infinite network if you haven't percolated then it is possible to run a surface a two-dimensional surface but I'll draw it as one-dimensional through the whole lattice from one side to the other even though it's infinite you'll always be able to find the place to put that surface if these two clusters have grown together never mind go around it here there'll always be a way to run a surface and so we could say we characterized the phase here by saying there exists crossing surfaces across the whole geometry which stay completely in the live region let's call them live crossing surfaces live crossing surfaces LX s they're also of course exist live crossing curves that's obvious if you can send surfaces from one end to the other you can put lines in them and make curves so the characterization of the percolation for small gamma and it's the order parameter which characterizes it is the existence of lie of live crossing surfaces now we increase gamma the percolation klephts clusters on the average begin to grow the typical connected cluster will grow and if we follow the average size of a percolation cluster it will smoothly go to infinity the percolation clusters will run into each other they will get bigger and the average percolation cluster will grow in fact its size will grow as you go to some particular value of gamma I think it's a gamma some ways about equal to a third it's not exactly equal to a third ah and at that point the clusters will grow together and percolate but when they percolate they don't completely fill up the entire geometry what happens is the percolation clusters interrupt the crossing surfaces but they don't interrupt the crossing curves you can still find curves through it and I'll tell you roughly what the percolated region looks like the percolation region is kind of an infinite network of tubes these things connect together to form tubes and it looks something like this if you start with a collection of lines like that and then fatten them now take away the lines and now extend this out to infinity and create an infinite network of tubular structures like this in here is the dead region out here is the live region you can also have disconnected perkele are dead regions over here and you can have buried in the solid mass of the dead weight of the dead region you can also have some live regions but the characteristic is the existence of this infinite network and when that happens the network interrupts the live there live crossing surfaces but it does not interrupt the live crossing lines or live crossing curves you can still have curves go through this from one end to the other but not line but not surfaces so that's a new phase that's the second phase it begins at about a gamma of about a third and the dead regions which might be the regions where life is possible incidentally form these big tubular networks anything else to say about them are and that persists for a range of gamma now this model happens to be symmetric between black and white if you want to change gamma and one minus gamma remember all we're doing is painting a cell white or dead with probability gamma we could do exactly the same and start with a completely white thing and paint the thing black with probability 1 minus gamma it would be the same model so there's a symmetry at gamma equals 1/2 at exactly gamma equals 1/2 it is completely symmetric between live and dead and that's an interesting fact and in fact gamma equals 1/2 is in this phase where you have these tubular structures it's called the tubular yes it would no no we're not doing the iteration yet we're not we're not doing the Mandelbrot the sequence of iterations this is just a one-shot deal so far conventional percolation is that the answer yeah okay right so we're going to come back to that but once you die well first of all once you die you don't come back to life once you die you're in the terminal vacuum it's no longer inflating and the assumption is that when you're in a terminal vacuum you don't come back to life okay so here we are live dead tubular structure and that tubular theory is correct right at gamma equals 1/2 that's an interesting fact this is not entirely apparent that this tubular infinite structure throughout space is symmetric between the live and dead regions but in fact there is it is there are for example cycles in here there are cycles around here and this tubular topology the tubular topology like this is symmetric between live and dead its tubular with respect to life and its tubular with respect to dead so there's a phase in here which is tubular that's just a name that we made up at Stanford I don't know whether I don't know what the the professionals call it okay now what happens beyond gamma equals two-thirds well white and black oh oh let's give this one a name over here let's give this one a name let's call it the Dead Island phase Dead Island before we get to the tubular phase we have islands of dead region islands of dead region let's call this the live no it's the dead the dead island phase the eye dead islands between here and here and it means that the regions which have stopped inflating form connected small clusters sometimes we call them pocket universes in the eternally inflating world or bubbles there could be colliding bubbles they could be bubbles which collided but they form a connected cluster okay so that's the Dead Island phase but now guess what happens when you move over to here by the symmetry there's a wider others a alive Island phase over here that phase would have the property that it has islands of live stuff now live just me live and dead just mean the two colors in this diagram so far we're not iterating it okay so that's the that's the phase structure symmetric about 1/2 tubular in the middle Dead Island on one side live Island on the other side okay now let's consider the next step where we start breaking this up and undergoing the iteration which allows us to mock up eternal inflation so we go to the next step and then the next step in the next step in the next step I'm going to tell you what mathematicians know about this at each step at each step after each step the pattern is the same the only difference is that these points move somewhat to the left the fact that they move to the left is obvious when you go to the next round you're going to put more dead stuff in you're going to make it easier to percolate if you start over here with no dead region and you go under go one round you find out that the transition is it about 1/3 if you add more dead stuff in the next round it will clearly push this point to the left and it will also push push this point to the left it's no longer symmetric everything gets pushed to the left here that's what would happen if you went through two iterations if you go through three iterations you prove push it a little more if you allow the system to run to infinity you push these points to some finite come Virgen places so that in here you still have Dead Island phase in here you have tubular phase and in here you have live online live Island phase but there's another phase and the other phase is the extinction phase if gamma is bigger than 7/8 then extinction this is the phase diagram then cannot be read if it can't be read I'll read it dead island out to some out to some finite gamma tubular in between and next is live island phrase now the nature of these transitions this transition over here is second-order that's what I showed you before that's this curve over here these transitions are now first-order let's go back to standard the percolation for a moment where we don't iterate and make things chop them up into smaller and smaller regions let's keep with the original percolation model the original percolation model with all of the transitions of second-order what that means what is the second order transition the second order transition is one in which the correlation length becomes infinite at the transition point that's the definition of a second order transition the correlation length is just the average size of the percolation clusters what happens is you increase gammas the percolation clusters grow they begin to get connected together but before they actually form you tubular phase the typical percolation cluster grows and becomes infinite that's the second order transition something else happens which is different in this Mandelbrot sequence of pure percolation theories so let me show you let me give you a rough idea of what it looks like let's start in the dead island phase so we have some dead islands with some value of gamma so that you have dead islands now that's the first round the next round when you subdivide the lattice and add in more dead stuff may make these dead they make these dead islands somewhat bigger but will also create small dead regions here in between go a little more go to the next step the next step in other words the next iteration will make a little more structure on here they may collide if they collide will make some structure and there are lots more smaller islands at some point before the typical large percolation cluster gets infinite they become connected together by the small structure in between that's what happens the transition happens suddenly all of a sudden this cluster which is a finite size suddenly finds himself connected to this one by the kind of micro structure of small bubbles in-between small bubbles mean late formed bubbles so it will generally happen that the percolation happens suddenly in in the Mandelbrot construction and I would say in eternal inflation when you decrease the rate of bubble nucleation you'll go through a transition which will take you to the tubular phase in fact it does inherit the tubular kind of phase but the tubular phase is now constructed out of tubes which have all kinds of microstructure it's hard to draw tiny tiny filaments connecting things it's hard to draw I'm not going to try to draw it but it maintains the tubular structure in the next between here and here and then it maintains the white island starting at the white island that's the old term the live island structure in here okay but with first order transitions in between now the easy thing to understand is the Dead Island phase you just form bubbles bubbles of decayed vacuum alright that's that's this famous picture which people are always drawing the bubbles can collide that's okay but they form distinct separated things this is what usually analyzed sometimes people analyze it from this side they study they study the potential when it's close to slow roll inflation and then they find the transition across here that's the transition from extinct to live Island this transition would be hard to find by any methods that weren't the discrete like this perturbation theory or solving stochastic equations perhaps I don't think it's would be easy to find this transition or this one alright so any questions about this ah no you should have questions about this nobody has questions about this all right I have a question about this what does the the live island phase actually look like this is the phase in which the inflating region forms islands what happens in those islands all right the live regions are susceptible to a phenomena called cracking cracking is a term that the people who do this kind of mathematics use and what cracking is is basically that it's unstable in each round of the sequence of the Mandelbrot sequence see this Mandelbrot sequence being the subdividing of the lattice into smaller and smaller lattices it's unstable with respect this is dead this is live you create more dead stuff and it gets unstable with respect to these regions cracking by creating dead region in between them the dead regions sorry the live regions get cracked into non inflating regions and it keeps happening at any given stage there's always live stuff left over but the live stuff keeps getting cracked and cracked and cracked so the question is what does this cracking look like if you think you live in this island phase or this phase of Island eternal inflation or slow-roll eternal inflation it would be important to know what the the dead region looks like and how this phenomena affects it so I'm going to show you the simplest example of cracking and show you what it does it does something I think spectacular extremely interesting it creates something that I think every relative this thought could not be created so let me show you what it does let's start with de sitter space let's take a space like surface through it it's a sphere and now let's imagine something which would be extremely improbable if the decay rate was small but if the decay rate is pretty large it's not that improbable namely a whole bunch of bubbles this is of course a three sphere not a two sphere a whole bunch of bubbles nucleate connected together separating the left side from the right side separating this region from this region over here now I've drawn it as a kind of chain of beads but of course in higher dimensions it would be a surface that would be a surface of nucleation that would separate one three-dimensional region from another three-dimensional region that would be a crack we start with everything live and a crack forms that separates the geometry into two inflating regions with a region in between abdominus ladings so let me show you what that looks like on the Penrose diagram are this two sphere this is on a two sphere this chain of pearls is not a chain it's a surface of pearls on a two sphere here is a two sphere let's put the two sphere over here on the Penrose diagram every point on the Penrose diagram is a two sphere and once it forms it starts to grow it starts to spread it starts to thicken the wall or the the region in here starts for thickened and so it thickens out like that it eventually hits time it equals infinity now what's in here since it corresponds to no cosmological constant what's in here has to be a solution of vacuum Einstein equations it is a solution of the vacuum Einstein equations you might think it's just some sort of flat space but it's not it's a black hole it's a Schwarzschild black hole an eternal Schwarz a black hole this is the picture that you draw or the top half of the picture that you draw to study the eternal black hole which is of course an unphysical thing usually you can't form one of these in the usual way with what you form is basically half of it normally well in other picture where is it whatever you form this eternal black hole which has a singularity in the middle two flat regions one over here and one over here separated from each other they cannot communicate with you live each other they're separated by this region in between a singularity forms and everything in here is a solution of the flat Einstein equations although not empty space Schwartzel black hole the most important thing is that the crack is accompanied by a singularity so that when the crack occurs is the crack a singularity forms in between now the constant cracking here the continuous cracking is much more complicated I don't know how to follow it I don't think anybody knows how to follow it but the expectation is that the cracking sends out signals that eventually collapse the entire the entire dead region in here instead of the dead region persisting the dead region appears there always cracks between between different dead regions lead to an instability or it's expected that they lead to an instability and so while alive while alive regions continue they get cracked smaller and smaller but they remain they continue to inflate they continue to spit off dead regions but any given dead region probably undergoes collapse that's the expectation you can't live forever even though the zero cosmological constant out here in here you probably can't live forever in this geometry the dead regions eventually collapse it's an extremely complicated phase nobody knows how to analyze it and therefore that's the main reason why we don't study that phase why eternal inflaters like to study this phase let's talk for it let's talk a little bit about the tubular phase what would life be like if you happened to be living in the dead vacuum yeah work say it again well you have to patch something in here that's a solution of the vacuum Einstein equations okay you'll have to patch in something which is a solution of the vacuum Einstein equations there aren't very many solutions of the vacuum Einstein equations which have spherical symmetry all Penrose diagrams are spherically symmetric and it's just a matter of it's just a matter of a little bit of mathematics sewing in here a solution of the Einstein field equations the thing that fits in there the only thing that is the only thing around is what yield black holes one possibility is a short yield black hole of zero mass that's not what happens shorter black hole of zero mass would not have a singularity what does happen is the Penrose diagram of a Schwarzschild black hole I didn't intend to go through this you can find this in the literature it's a if I remember I'll write down the references to it I think it was first discovered by Sasaki and his and his co-workers in Japan was rediscovered by by our group at Stanford and Berkeley and you can read about it the main point though is that I'll give you a reason why yeah let me go let me try to explain yeah okay let's suppose the singularity didn't form I don't know something like that let's go from here right through here back to here the size of the 2-sphere over here is infinite all right we're up near the boundary of the pseudo space the time size of a two sphere is infinite so let's draw the two sphere right about over here it's very big the geometry is very big over here it's very big over here and in here the metric is finite and so that means a float through there a finite geometry basically this geometry looks like two big geometries connected together by a narrow throat this is an unstable situation this is called a wormhole it's a thing a feature of ordinary general relativity that wormholes are not stable they collapse and that's what's going on here this geometry just slicing right through here would be a wormhole between two very large geometries and it would collapse okay so this geometry I don't know how are these dis cracking there phase I don't know how to analyze the tubular phase is a little bit easier to think about it does not it does not appear that it's unstable there's no reason to think it's unstable but it's kind of interesting to think about what an observer who was in this tubular phase would see now he'll only see this if the cosmological constant is zero and only see it if he has much better telescopes than I think we're ever going to have but still here's what the well let's first go to the island phase let's start with the island phase here's the island phase and an observer at the center looks out it looks out looks back on his backward light cone and from any given time let's say he's looking at some surface of constant time he sees a surface thick - this could be for example looking at the surface of last scattering he sees a sphere around him as time goes on that sphere he sees more and more of that sphere more and more of that sphere if the cosmological constant is really zero that observer can live forever and look back and eventually see the whole thing so with time he sees a surface that spreads out and eventually fills the entire geometry what about some person in here what would they see they would look out after a small amount of time and see a surface of last scattering look like that go a little bit further in time see a surface which looks like this as time goes on start to see a surface which looks like this a little more time and the surface will spread around here eventually these things will cross and I'll be able to see right around the universe this is a this is a phenomena which is common in topologically complicated universes and so with time it eventually be able to see this whole tubular structure and not only tubulin structure but it's topology by seeing multiple images on this guy now don't don't go out and try to do this we don't live in such a world or at least we don't live in a world with a vanishing cosmological constant the implication of not having is a vanishing Kozmo of having a non vanishing cosmological constant is you only get to see so much growth before you encounter horizon problems and so forth so this is what you would see if you lived in this tubular phase in the island phase what you just see is a sky a spherical sky around you and you see more and more of it with time okay so those are the phases of eternal inflation in a very simple model if the model is a lot more complicated for example many minima then the structure is very much more complicated as Alan Guth says in an eternally inflating universe anything that can happen will happen and will happen an infinite number of times if there are regions on the landscape which allow the tubular phase and we started nearby we might see a region in what's called the multiverse which looks tubular in other places we might be controlled by some other kinds of decays which look more Island like and so there would be patches of space with all these various different kinds of phases taking place and the whole thing with is a horrible mess it's not easy to classify I have no idea how to classify it main reason I'm telling you this is to tell you there's more than one thing called eternal inflation it is not one thing it is many things most of them highly intractable ok any questions about the topological phases of eternal inflation yes uh I think so yeah ah yes I think you would just form a deceit or black hole yeah I I think the answer is yes um so let's say you start in the sitter space and you tunnel not by one tunneling event to the sitter space but you create this surface or shell of tunneling events that looks like a cou sphere and then I would guess that would you yeah I'm pretty sure that you've created here will be some sort of considered black hole so yes I think the answer is yes if you tunnel to the city space by this phenomena of surface cracking yeah you will still have singularities any other question say it again oh all right let it depends on I I don't remember the answer in detail but I do remember what the parameters are the parameters are how high up it nucleates and the size of the critical bubble wall and I don't the life of me remember the the connection so it's a good question I don't remember the connection it I think the bigger the domain wall the bigger the critical bubble here the larger the mass of the of the black hole but I can't remember the details so I won't try to reconstruct them we'll come to that I think we'll come to that eternal inflation is not a substitute for slow-roll ordinary inflation let's let's get that straight let's talk about that for a minute bubble nucleation like this typically makes an empty universe a universe with nothing in it or very little in it this kind of tunneling event is not an interesting candidate for making an observable universe an observed universe what you need which is rather baroque you need something very complicated in order to make a viable picture out of this you need that after you've tunneled out and we're going to discuss the structure of bubbles but for now just let me say it quickly what you need is after you've tunneled out then you have to settle down on a on a plateau so it would look something like this the tunnel the tunnel out but after you've tunneled out here you're in the tunneled out to about over here and then in here there's a region of inflation slow roll inflation before you roll to the final cosmological constant over here so eternal inflation which takes place in these regions over here is not a substitute for ordinary slow roll inflation they are two different things in fact you could characterize them this way to make slow roll inflation you need a degree of fine-tuning to make this potential flat enough that's not what you need to make eternal inflation what you need to make eternal inflation is just minima The Sharper the minimum of the better the more likely you are to stay in that region so they're quite different phenomena and they are not substitutes for each other you might ask why do we need eternal inflation um I don't think that's the issue I think the issue is to say start with a given a theory do we or don't we get eternal inflation that's if there are lots of minima out here even just one more minimum out here you will get eternal inflation even more than that I would say there's a reason to believe that eternal inflation has to be in our in our description of things and the reason is the existence of the current cosmological constant the current cosmological constant will lead if it really is a cosmological constant in other words if it really is a stable minimum it will itself lead to eternal inflation it's enough to make eternal inflation ah just being here just having one minimum makes you turn inflation incidentally a world with just one minimum like this is not a viable world it's a world which will come to thermal equilibrium at the bottom of this well and we'll just sit there and sit there and sit there literally let's sign up I'm getting I'm getting too many things thrown in altogether now what I want to do is study a generalization of this model a little bit now you this is an interesting generalization which represents a complicated landscape but I'm going to study an example in which there's a complicated landscape which does not have anything but positive energy vacuums again that's not a viable world but let's just study it and see what kind of things come out of it it has lots of minima they all inflate some inflate faster than the others and transitions are possible between them a network of transitions I want to equip that model with enough complexity that it can study a multiverse that would be produced by such a complicated landscape all right this is not so hard to do you just say every box can have any number of colors all boxes are alive because they all inflate all boxes are alive none of them terminate the eternal inflation and you just give them many colors the colors represent the different vacuums here and you can play exactly the same game you start with the whole checkerboard three-dimensional checkerboard painted one color the initial condition whatever it happens to be we can start here and then let it run tunneling events recolor some of the boxes with various probabilities let it run a little more running means subdividing the boxes into sub boxes and red boxes can make transitions to blue boxes blue boxes can make transitions to green boxes green boxes can make transitions back to red boxes you allow for the possibility that not only are their transitions down but you allow the possibility that a freak accident can happen and allow a transition up and you write a bunch of rate equations rate equations determining the probabilities for each color of box as a function of time okay let me show you what those rate equations look like they're in two Woodleigh well more or less obvious let's define PA to be the probability that a box has the eighth color let's define it to be the number of boxes after a given number of iterations in other words at the nth stage it's the number of boxes of type a divided by the total number of boxes all right let's consider the rate equations for the evolution of PA and I'm going to write them down R yeah here they are PA at the n plus first step alright this is the probability that this box has color a after the n plus first box is first of all the probability that it's inside a bigger box of the same color previously in the previous step that would be what you would write down if there were no transitions now if there are transitions you have to add to that I'll write it down summation over B and I'll play what they mean - gamma a B PA not only play with that is first the minus sign means depletion you start with the probability that it was in the eighth kind of box and that it made a transition that the smaller box that you're interested in made a transition from A to B so you read gamma a B this way this is the transition probability for a could be transition from A to B is gamma a B that's a probability pure probability times the probability that the previous box that it was that it was embedded in as color a all right so that's depletion and then there is replenishment replenishment would be also the sum over P of gamma a B P B so this is the probability that the previous box was of a different color B and that you made a transition from B to a that's it those are the rate equations the input of a set of gammas and you solve them and you determine what the population and what the probability of the given box looks like as a function of time okay let's work on that a little bit first of all the linear equations for the peas the linear equations with a matrix the reason I put those summation signs in there is because these equations are not amendable to Einstein summation convention z' here we some of them I would see where is it yeah here we we are not this is not summed over a it's the probability that were in the eighth box times the transition from A to B so Einstein summation convention doesn't work very well for these equations right here in term is okay we sum over B the left hand side we do not sum over a we sum over B in other words we sum of all the things you could get to from a by making a transition from A to B nevertheless these are linear equations and there's a matrix structure connected within the matrix let's see what the matrices look like the first term is diagonal it's the probability a being proportional to the probability a PA all right so what we have in here is on the diagonal we have let's see if I can written down some words we have gamma 1 2 plus gamma 1 3 plus gamma 1 4 that's all on the diagonal here the diagonals are big they take a lot of room to write transition from 1 to 2 glad that right this is no yes no yes the what a minus sign this is depletion of the first vacuum by nucleating the second depletion of the first vacuum by nucleating the third depletion of the first vacuum by nucleation the fourth then in the next diagonal here we have minus gamma let's say - 1 minus gamma 2 3 I think and so forth and so on and that's what the diagonals are built like the off diagonals are simpler than just gamma a B and I think this one is gamma 2 1 and this one is gamma 1 2 now these matrices are not symmetric incidentally what we're interested in is the eigenvalues and the eigenvectors of this matrix if we know the eigenvalues we know the rate at which the eigenvalues control the rates at which the probabilities change alright the eigenvectors those are the stable sort of equilibrium our configurations in particular what we're going to look for is an eigenvector with zero eigenvalue and eigenvector with zero eigenvalue is a stationary state that doesn't change with time i ghen values of these of this matrix over here you can shift the first term to the left hand side it's just the first time to the left hand side and call it AB or I can't reach it okay in your notes erase the PA of n and call it Delta PA on the left hand side the change in PA from one step to the next that's equal to the matrix on the right hand side if you know the eigenvalues of the matrix on the right hand side you know the rate or you know the evolution of the of the peas so the problem is to compute the eigenvalues of this matrix here now how do you even know it has eigenvalues it's not a symmetric matrix how many eigenvalues does it have does it have n eigenvalues does it have no I gained value x' okay there's a trick to make it photos this act on this acts on a column vector P 1 P 2 P 3 P 4 if you use the principle of detailed balance then you can convert this to a symmetric matrix what is detailed balanced detailed balance is a relationship between the decay rate of 2 to 1 sorry this is I want this is to 1 it's it's a relationship between the decay rate from 1 vacuum to another versus the decay rate in the other direction or the transition rate in the other direction these transition rates are not equal but they're related by detailed balance so let me tell you what detailed balances detailed balance says that gamma a B is equal to a symmetric matrix M VA this could be the quantum mechanical transition amplitude times a factor and the factor is the density of states let's write it as e to the entropy of the final vacuum B each of these vacuums has an entropy the entropy represents the exponential of the entropy represents the degeneracy or represents the level density represents the number of states associated with that de sitter space and this is what detailed balance says that the ganas are not symmetric but there's a symmetric matrix which you multiply by the density of final states to find the transition matrix if you do this this is now an exercise I want you to do this exercise if you do this and you write let's see if I have it written here correctly and you redefine you write the key a is equal to e to the si over two times Phi a in other words you rescale the pas by a factor of e to the si over to rewrite this not for the peas but for the Phi's you will find that the matrix in here magically becomes symmetric so that's an exercise to show that these rate equations that the rate equations become symmetrized the symmetric matrix appearing in them is NAB and all the factors non symmetric factors here are disappear four five that ensures that there's a complete set of eigenvectors alright so that's the first that's the first part of the problem the second part of the problem is to find one of the eigenvectors to prove that the matrix after redefinition old-people it doesn't matter whether it's after redefinition the matrix here has a zero eigenvalue prove that it has a zero eigenvalue and find the eigen vector zero eigenvalue means a state and all other eigenvalues incidentally are negative that means that the other eigenvectors d Griese with time a negative eigenvector in these kind of rate equations indicate a decrease in time a positive eigen vector would correspond to an exponential increase of probabilities that's impossible probabilities can't be bigger than one and A zero eigenvalue is excuse me would be a stationary probability distribution there is yeah it's clearly what there is a zero eigenvalue yeah the promising yesterday's action obvious but but the problem is to find the eigenvector all right and i will tell you what the eigenvector is because the eigenvector is interesting the eigenvector is that p a is equal to e to the si you could plug it into this equation here you just make the substitution you don't need to symmetrize the equation you do need to symmetrize the equation if you want to discover the complete set of orthonormal eigenvectors but just to just to discover this fact all you have to do is plug into here and you'll discover that this eigenvector here in other words e to the S 1 e to the S 2 dot dot that that's an eigenvector of course it's not normalized these each of the SS are huge you have to normalize it if you want the total probabilities to add up to 1 and what this statement is is that if you choose the probabilities to be equal to the density of states up to the number of states in each de sitter space that's a stationary solution it persists forever it doesn't change with time in fact because it's the biggest eigenvalue all the others being negative whatever you start with you will tend asymptotically to this eigenvector all the others exponentially go to 0 with time so whatever initial state you start with if you wait long enough this will be the population of the different vacuums what is this population this population is thermal equilibrium it's thermal equilibrium where every state is populated with equal probability basically every state on an energy surface is proper is with equal probability it's what you might expect if you had a landscape like this and the landscape point the the configuration point was hopping around in here and got ugly it was hopping around in here air gotta CLE in other words equal time and equal volumes of phase space this is what you would get this is thermal equilibrium for each of these de sitter patches in other words if you follow this Mandelbrot construction moving along from one box to the next box in time to another box to another box to another box and it doesn't matter what route you take just take a route through the time evolution of it from a box to a box inside that box to a box inside that box to another box inside that box what you will find is that after a while it settles down to an equilibrium configuration which is just thermal equilibrium this is not a good thing this is not a good thing because it's not consistent with our world our world is not thermal equilibrium there are transients and so forth but worse than that it says that every state is populated according to equal probability a world like this would be a Boltzmann disaster there will be a world in which the only things that happen are fluctuations away from equilibrium and our world is not a fluctuation away from equilibrium it's not a viable a viable description of universe it's a universe that's full of Boltzmann brains um what's wrong how do we fix it we fix it presumably hypothetically this is uh what eternal inflators think by having terminal vacuums dead vacuums dead vacuums of either 0 or negative cosmological constant which can't recycle in that case this procedure breaks down there is no detailed balance anymore there is no way to go from the dead vacuum back to their live vacuum you can still write down the rate equations but the dead vacuum just doesn't doesn't recycle you can still write down the equations running out of time so I won't write them down now or maybe write them down next time and you can still solve them but instead of finding there is still a stationary state could anybody guess what the stationary state is now everything dead just everything dead solid mass of dead vacuum that's all it would be in other words a solid a solid mass of this no transients just everything has has right so that's not what we're interested in our world our world incidentally is not a dead vacuum our world has a little bit of cosmological constant it's still live so a solid mass of dead vacuum is not what we want what we want is the transient to that so in other words you look at the next leading eigenvalue the next eigenvalue of slightly negative eigenvalue and that corresponds to a transient words exponentially decreases in probability the probability exponentially decreases with time but it's a transient transients can have interesting things happen during them ah you're going from one vacuum for the next vacuum to the next vac into the next vacuum and then eventually getting into the into the dead vacuum but during the course of pumping from one to the next and the next to the next interesting things can happen so and it's as I said it's not as it's in it it's not it I guess you would call it a stationary behavior you wouldn't call it a static behavior user misusing the terminology thermal equilibrium I recall static a flow of a system as a whole system that flows from one thing to the next thing to the next thing here's an example you put a ball in here it jumps down to here jumps down here jumps down to here you keep putting balls into here and they keep jumping down one after another after another that's stationary if you keep putting the balls in at a certain rate that's stationary not static things flow from here to here if you have a terminal vacuum like this then the rate equations tell you that that the behavior as you follow in particular as you follow a trajectory through this Mandelbrot in structure you see a flow a flow down eventually always ending here but there are always new live regions which create more balls so what does it look like it looks like at every stage there is still a source of balls because there are still live regions in particular live regions up here anywhere along here these live regions are eternally inflating and they're a constant source of supply of balls which roll down the hill when they get down here they finished the dead so that is what what things would look like with or without terminal vacuums without terminal vacuums a disaster you just sit in one of them or you sit at the bottom from now and then hopping up and and making transitions but everything being static static in the sense that the probability just doesn't change and with terminal vacuums you have a flow which is a better situation interesting things can happen during flows okay I let's see do I have five minutes there's only one more thing I wanted to say and it has to do with this issue that I brought up in the very beginning about decoherence I showed you how in when there's a horizon experiment can give rise to radiation which passes out through the through the horizon that way there's a sense in which this happens an eternal inflation that I just wanted to point out this Mandelbrot model or in general eternal inflation has a tree like structure to it there's a tree like structure to it ah the tree like structure you can see very clearly in this Mandelbrot model let's say you start with one box just one think of that as the root of a tree and then it breaks up let's say into four boxes or eight boxes the tree gives rise to eight branches who is enough for the for the purpose of the discussion those branches never talk to each other anymore once this box has done what it's going to do it never talks to this box anymore the reason is I lost a picture but it's out of causal contact there's separate this different horizon patterns they're out of causal contact so this one never talks to this one anymore what happens next each one breaks up again and again each time spitting out eight branches every now and then a branch gets killed it can no longer it can no longer branch that's dead that one's dead that one's dead but the tree keeps growing okay the tree keeps growing you'll have two ways that you can think about this tree I'll come back to this decoherence in a minute but you have two ways that you can think about this tree you can think about it globally and count the number total number of branches that's just 8 to the annum and it counts the total number of branches of a given type you multiply PAE by 8 to the end to find the total number of branches of a given kind okay or you can say look I am a fellow who likes to travel along branches I don't like to split up at these branches here I'm a real observer I travel along a trajectory through through my own causal patch a single causal patch corresponds to following the sequence of branches along here like that and in that case what you see is the probabilities what is the probability after n branching that you're in a given vacuum that's P of a so there's this dual description global and local global studying the whole tree local following it one case you're interested in the total number of things in the other case you're interested in the probability that you're in a given situation okay now let's say let's come back to this business of of decoherence let's suppose you're one of these observers who follows himself along a sequence of branches then what is world looks like starts with the original vacuum all right then a nucleation occurs that could be this branch over here so it follows it into this branch and then another nucleation happens and then another nucleation happen and that observer is following his world line through this branching structure which means following it from one vacuum to another until he finally achieves extinction well not extinction but just that until he finally gets to a terminal vacuum and then he stops okay all right let me let's think about a following crazy situation there's a cat over here this cat can survive a trends transition to vacuum a but it cannot survive a transition to vacuum B all right so it's a cat who can live in vacuum a but can't live in vacuum B what happens after the transition happens and the transition could either happen to a or B what does the quantum state look like all right so what the quantum state looks like is live cat happy cat x state AE plus dead cat this is a state dead cat x state b this is an entangled state this is a state in which the cat is entangled with his environment and in this case the environment just corresponds to whether he's in vacuum a or vacuum B now the question is does this state decohere is there any mechanism to have it decohere decohere means that a record is form of which vacuum you're in and that that record is irreversible well the answer is yes let's follow this cat or this entire experiment to the end and draw the causal patch this record of whether you're a or B goes out through the horizon here once it's out through the horizon it can no longer communicate what's with what's inside so this horizon has a permanent record that cannot go away of whether the vacuum was a transition to a or b that mechanism alone would be enough to decohere the state and decohere it means collapse the wave function from dead cat from this from the superposition to either one of these in other words remove the coherent phase between these two states so it's interesting there is a mechanism in eternal inflation to cause decoherence it's permanently coherence and here's an example of it how the sequence of bubble nucleation is really in a certain sense I lost my diagram I don't have the diagram anymore but represents the same phenomena in an experiment as the radiation of a photon okay I think we'll finish for now next time I'm going to come back and discuss the measure problem try to make the measure problem it's simple for you as I can and to discuss the various proposals for measures on the landscape you

Channel: Institute for Advanced Study

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Length: 95min 45sec (5745 seconds)

Published: Fri Jul 07 2017