Aspects of Eternal Inflation

Transcription

1 Aspects of Eternal Inflation Lecturer: Prof. Leonard Susskind Transcriber: Alexander Chen July 21, 2011 Contents 1 Lecture Lecture Lecture Lecture

2 Aspects of Eternal Inflation Lecture 1 1 Lecture 1 We are going to discuss topics about eternal inflation. We have some assumptions. First is that Einstein gravity is correct at low energies. We are not going to do modified gravity. In addition to that we are going to assume some scalar fields. The question is how many. Some people would say there are many, but there is at least one, which is the Higgs field. The scalar field has a potential V (ϕ). The parameters and structure of the scalar field is called the landscape. The scalar field affects the surrounding things around it. There is a minimum of the potential V (ϕ) and it is metastable. It doesn t matter how many minima there are, and some have positive energy, and some have negative energy. The vacuum at the minimum of the popential is a de-sitter space, which is characterized by a radius of curvature R, which satisfies 1 R 2 = H2 = 8πG 3 V min (1) A de-sitter space is a space of constant positive curvature, like a sphere. We will consider a 4- dimensional de-sitter space, because our spacetime is 4-dimensional. We will consider a 4-sphere inside a 5-dimensional Euclidean space characterized by T 2 + X i X i = R 2 = 1 H 2 (2) This is just a sphere. Now if we change the metric to Minkowski, then this describes a de-sitter space The metric on the sphere looks like T 2 + X i X i = R 2 = 1 H 2 (3) ds 2 = R 2 [ dτ 2 + cos 2 τdω 2 ] 3 (4) But the de-sitter space, being in hyperbolic space has metric ds 2 = R 2 [ dτ 2 + cosh 2 τdω 2 ] 3 (5) This is the metric of de-sitter space in what is called global slicing where τ is sliced globally. This has the same structure as the FRW metric, which can be seen more conveniently as ds 2 = dt 2 + cosh2 Ht H 2 dω 2 3 (6) where we have redefined t = τ/h, and the scale factor is a(t) = cosh Ht/H. Let s look at other ways of looking at the de-sitter space. The above is a global slicing where the plane of constant τ is perpendicular to the center axis. We can also make a Lorentz transformation and tilt the planes for slicing. We can do a infinite Lorentz transformation and slice it using light-like planes to slice the hyperbloid. The light-like surface intersects the hyperbloid at a space-like surface. This is a space-like foliation of the same hyperbloid. The metric in this slicing is ds 2 = R 2 [ dτ 2 + e 2τ dx i dx i] (7) This only covers the half of the hyperbloid, but the space metric is the flat space metric. We can also write it as a FRW metric ds 2 = dt 2 + e2ht H 2 dx2 (8) 2

3 Aspects of Eternal Inflation Lecture 1 This, as opposed to the above FRW metric, is a flat universe, whereas the above one describes a closed universe. The next slicing again begins with a sphere. We cut it using a sequence of planes that intersects. One plane can be rotated to another plane, and the operation that go from one plane to another is a symmetry operation. If we use this method to slice a 3-dimensional flat space, the planes rotate around an axis, and in 4-dimensional space they rotate around a 2-surface. This is called the static slicing. The de-sitter space is sliced using hypersurfaces originating from the center of the asymptotic lightcone of the de-sitter space. Again this only covers half of the space. The metric is [ ] ds 2 = R 2 (1 r 2 )dτ 2 + dr2 1 r 2 + r2 dω 2 3 (9) This metric is static because there is no time dependence in the metric components. This looks like a blackhole metric, and it has a horizon, which is at r = 1. A spatial section is a 3-sphere, specifically the τ = 0 slice. And the r = 1 is the equator of the slice. There is no singularity there, so this is just a coordinate artifact. The next exercise is to prove that the spatial metric above describes a hemisphere. Let s draw the penrose diagram for the de-sitter space. The static slicing is just slicing using the t = t = Figure 1: Penrose Diagram of de-sitter space surfaces of constant proper time. The static slicing has a horizon, so the description is thermal. An observer sits on a static slice will not be able to receive signal from beyond the horizon, and he sees a thermal temperature of 1 T = 2π (10) 1 r 2 R Note that at r = 0 the temperature is T = 1/2πR and this is called the Hawking temperature. The entropy of the static patch of the de-sitter space is S = A G = 4πR2 G (11) 3

4 Aspects of Eternal Inflation Lecture 1 This can be thought of as an entanglement entropy with the other side of the horizon. We are going to again write the metric in another slicing. This is called the conformal slicing and is good for causal structure. We start with the flat slicing and redefine dt = dτ/e τ. Then we can integrate this equation easily and get T = e τ. With this definition we have ds 2 = R 2 [ dt 2 + dx 2 T 2 This is just flat spacetime, but with an overall factor which is time dependent. Slicing the space in t is like slicing the space in a logarithmic manner in T. In a fixed amount of time t = 1/H the size of the universe double. Because this is conformal, lightcones are still 45 degrees, so it is very easy to see what observer can get signal from where. It is the overall conformal factor that gives the space curvature. We are going to look at the future boundary, which is the top horizontal line for de-sitter space. For a point A on the future boundary, the causal past of the point A is called the causal patch of A. For flat spacetime, the causal patch of the t = is everything of the spacetime. So for an observer at t = it can receive signal from anywhere in the spacetime. Now consider de-sitter spacetime, and consider the causal patch of a point A. If we foliate the spacetime using horizontal slices, then the slice of the causal patch is called an instantaneous causal patch. The important thing of causal patches is that the number of patches replicate exponentially, when you go at steps of time t 1/H. In 3 dimensions the number of causal patches the number increases by 8, which is the cube of 2. We can think of the inflating system with causal patches as points on a lattice. Each e-folding increases the number of points on the lattice of patches, and they are causally disconnected from each other. So now we come to eternal inflation. We will consider bubble nucleation. Consider a potential V (ϕ) which has two local minima and one global minimum. We will consider the tunneling from the false vacuum to the true vacuum. There are many phases and ways of eternal inflation, and we will consider them. When we tunnel through the potential barrier there will be phase transitions and there are many ways for this to happen. Let s call the metastable vacuum the ancestor vacuum and the true one the young vacuum. It is extremely unlikely that the whole space just pop into the new vacuum. The thing which happens is that a bubble is created at some time T, and it quickly grows at near the speed of light. Inside the domain wall, we have the new vacuum, and outside we have the ancestor vacuum. Many bubbles will nucleate, and there is a mean distance between bubbles. In flat spacetime, the growing bubbles will eventually collide and merge. However in the de-sitter ancestor vacuum, the spacetime will grow as well, and it can beat the speed that bubbles grow. This is seen from the fact that there is a line of future boundary for de-sitter space, and a bubble can never completely take over the whole spacetime. But if the nucleation is too frequent, then the bubbles can still collide and form a big bubble which fills all. We will investigate the condition when this happens. Let s consider the time of one time step t 1/H, and look at all the bubbles that nucleate in this time step. The eternal inflation will obviously be terminated if the mean coordinate distance between bubbles is smaller than the coordinate size of the bubbles. The size of the potential barrier controls the rate of bubble creation, P, which is the probability of creation per proper volume per unit time, Γ = γh 4, where γ is dimensionless. Consider the region of coordinate separation x, and the volume of the region at time T is ( ) x 3 V = (13) HT ] (12) 4

5 Aspects of Eternal Inflation Lecture 1 Multiply this by the nucleation rate and time step, we have the number of nucleation is N = V γh 4 t = 1 T 3 γ( x)3 (14) The mean distance between the bubbles will them be T/γ 1/3. The size of the bubbles are about T, so we essentially compare γ with 1. If γ is significantly smaller than 1, then the bubbles will have space to grow and will not take over the whole space. 5

6 Aspects of Eternal Inflation Lecture 2 2 Lecture 2 Before we begin, let s discuss one aspect about quantum mechanics. It has to do with the phenomenon of decoherence. This has to go with measurement, or entanglement, which is the interaction between virtual reality, which is quantum mechanical state, and actual reality which is the macroscopic world. Decoherence happens when a wave function collapses, which is an irreversible process. Now in quantum mechanics in principle we can recohere the event. For example an atom decaying and emitting a photon, we might have a mirror which reflects the photon and the system reabsorbs it. In de-sitter space there is another mechanism which prevents this from happening. This is because in de-sitter space when the photon passes through the bountary of a causal patch, it can never enter it again. Let s come back to eternal inflation. Consider a potential with a minimum and an absolute minimum. We call the previous one live vacuum and the latter dead vacuum. Let s say a bubble nucleates in de-sitter space at r = 0. We can consider it in a Penrose diagram r = r = 0 Figure 2: Penrose diagram of bubble nucleation In the process of eternal inflation, there are a lot of phases of bubble nucleation, and we want to investigate what can happen. Remember the nucleation rate is defined as Γ = γh 4. We discussed last time that the number of causal patches increases exponentially. For simplicity we assume bubbles can only nucleate at intersections of the coordinate patches. Because at t = there will be infinitely many bubbles, we want to cutoff this process by looking at the bubbles at a specific time. Let s put spacetime in a box which is discretized in both space and time. If a bubble nucleates in one box, then the box is labeled dead. This probability is controled by Γ. For the next time step, we subdivide each box into 8, and label boxes as dead according to Γ. And we repeat it indefinitely. The question is whether the whole box will be labeled dead. This obviously depends on Γ. After n steps, there will be 8 n boxes. The probability of a box to be live after n steps, by definition, is P L (n) = N L(n) N T (15) where N L (n) is the number of live boxes at step n. We have a recursion formula P L (n + 1) = P L (n) γp L (n) (16) whose meaning is self-evident. The boundary condition is that P (0) = 1. So we know that P L (n) = (1 γ) n. Now the population of live boxes is the probability times the total number of boxes N L (n) = (1 γ) n 8 n (17) 6

7 Aspects of Eternal Inflation Lecture 2 and as long as γ < 7/8 then the number of live boxes will increase exponentially. Only when γ > 7/8 will the box be extinguished. At γ = 7/8 there is a phase transition from slow roll inflation to eternal inflation. Now there are other phase transitions. Let s consider an infinite lattice and bubbles nucleate in this lattice. The bubbles will form perculation clusters. As long as they don t merge into one cluster, we can find a surface which separate the clusters but lies totally in the live region. Now as the bubble clusters perculate, they will form a infinite network, which interrupts live crossing surfaces, but not live crossing lines. This is another phase transition, which is at γ = 1/3. We call the region γ < 1/3 the deadisland phase, which means that the bubble clusters form connected pocket universes. And because this construction is symmetric between γ and 1 γ, so at γ = 1/2 the perculation will form a tube structure which is symmetric in live and dead region. We call this the tubular phase. Similarly at γ = 2/3 we have live island phase. The above discussion is only for one iteration. For two iterations the γ = 1/3 point will get pushed to the left, and when the number of iteration goes to infinity the point converges to a finite position. The transitions between the dead island phase, tubular phase, and the live island phase are first order, whereas the transition to the extinct phase is second order. Let s consider the live island phase in more detail. In this phase the live islands will be unstable under the time steps, and they will crack due to the new dead regions created inside them. Due to the relatively large magnitude of γ we can have something which seems very unprobable, which is a surface of bubbles forming inside a live island, separating the island into two. The most striking thing is that when a crack forms a singularity is formed and thus a black hole, which in this case is an eternal black hole. When many cracks happen, more and more singularities will be formed and the entire space will be unstable. This makes this phase very difficult to study. Now let s consider the tubular phase. In the island phase an observer inside an island will see a surface that spread out and eventually sees the whole space. However inside a tubular universe the constant time surface is distorted and the observer will eventually see double imaging, etc. due to the topology. The bubble creation picture, or tunneling mechanism, will not create a universe with many things in it. When the field tunnels out, in order to make the universe as we know today, it needs to hit a slow-roll region to produce our universe. So eternal inflation is not to substitute slow-roll inflation. Let s study a world where the scalar potential has many minima, but all of which have positive cosmological constant, and all of them are alive. To describe this we generalize our description to many colors for the box and sub-boxes. We allow for the possibilities of transitions between colors, and the transition rates are governed by rate equations. Let s define the probability of a box having color a to be P a (n) = N a (n)/n T. We have the recursion formula P a (n + 1) = P a (n) + b γ ba P a (n) + b γ ab P b (n) (18) where γ ab is the transition rate from a to b. These are linear equations and there is a matrix structure connected to this. In addition, we have the principle of detailed balance, which says that γ ba = M ba e S b (19) where S b is the entropy associated with the b-th vacuum. With this in mind, if we redefine P a = e Sa/2 φ a and rewrite the matrix equation in terms of φ a, then the matrix φ a = M ab φ b will be symmetric. Try to prove that there is a zero eigenvalue, and try to find that eigenvector for this eigenvalue. The eigenvector turns out to be just P a e Sa (20) 7

8 Aspects of Eternal Inflation Lecture 2 This is the biggest eigenvalue, and the other eigenvalues are negative. This means that anything we start with will end up being in this state, where the states are populated according to equal probability distribution, and the universe is in thermal equilibrium. The way out of this is to have a dead vacuum, which does not recycle into the other vacuums. Now the principle of detailed balance breaks down. But we can still solve the equations, and the final state is a totally dead vacuum. However we are interested in the transient state, as our universe has a little cosmological constant. Now let s come back to the beginning about our discussion on decoherence. Eternal inflation in general has a tree-like structure. When we break the boxes as we did in the above procedure, it is like a tree growing branches, where the production of a dead vacuum means a death of the branch. A single causal patch correspond to a single path from the root of the tree to the end of the branch. Now consider a cat which lives in vacuum a but will be dead in vacuum b. The cat forms a coherence state with the vacuum. However when the cat travels outside a causal patch then the system decoheres. This is the mechanism for decoherence in de-sitter space. 8

9 Aspects of Eternal Inflation Lecture 3 3 Lecture 3 Consider dimensional quantum electrodynamics. It is very simple and we can do it in a short time. There is only electric field, and there is no electromagnetic waves, and the electric field is totally determined by the Gauss s law. Let s consider a unit electric field on the 1 dimensional space, produced by a plus charge and a minus charge at L and L. This configuration is metastable because if there is a discharge occurring in the vacuum, then the pair produced will be pulled to the ends and neutralize the electric field. The problem is to calculate the rate of this process. We will do this in Euclidean spacetime and use the method of instantons. Let s consider a configuration where a pair is created and they trace out a circle in the Euclidean space and eventually annihilate. Because there is no electric field inside the circle, the action of this instanton will be S = πr 2 E 2 + 2πrm (21) The first term is the change in energy due to the instanton, and the second term is associated to the masses of the pair. The integration is over the size of the instanton r. We will use the saddle point method. The maximum of the action is at r = m/e 2, and maximum is S max = πm2 E 2 (22) So the decay rate is just exp ( πm 2 /E 2). This path integral is the probability of finding a pair of virtual electron and positron at time t = 0. The wave function is the square root of this probability. Now after the pair is created they will be attracted to the ends. The trajectory is governed by the equation of motion t 2 + x 2 = m2 E 2 (23) and there will be hyperbolic trajectory for both particles. What does this have to do with bubble nucleation? In 1+1 dimension the domain wall is just a particle, and the bubble is just the region between the pair where there is no electric field. The only difference in eternal inflation is just that the pair production will not discharge the whole capacitor. Now let s carry out the same procedure for bubble nucleation, where the vacuum tunnel from a place of positive cosmological constant to a zero cosmological constant. The Euclidean metric for this is ds 2 = dt 2 + a(t) 2 dω 2 3 (24) We will write down the Friedmann equations, but will leave out the 8πG/3 coefficients. We first write down the coventional equations (ȧ ) 2 = ϕ2 a 2 + V (ϕ) 1, ϕ + 3 ϕȧ a2 a = V (25) ϕ Now we change the signature to Euclidean, and we get (ȧ ) 2 = ϕ2 a 2 V (ϕ) + 1, ϕ + 3 ϕȧ a2 a = + V ϕ The effect is like reversing the sign of the potential, and we will be looking at the bounce solution for the instanton. If we start at the live vacuum A, which corresponds to a hill in the Euclidean potential, then we 9 (26)

10 Aspects of Eternal Inflation Lecture 3 will sit there indefinitely. If we start with A which is a little away from A, then we will roll down the hill, and it will roll to the other side till point B and come back. This is the bounce solution. The boundary conditions for this solution are ϕ(a, B ) = 0, ȧ(t = 0) = 1, ȧ(b ) = 1 (27) This instanton allows the tunneling from near A vacuum to near B vacuum, and spend a lot of time around both, but it spends little time in between, which is the domain wall. The de-sitter space at vacuum A has O(5) symmetry, but the instanton has reduced O(4) symmetry. The calculation is similar to the electrodynamics case, and we calculate the path integral of field configurations up to the time t with appropriate boundary conditions. In the end of calculation we need to analytically continue it to Minkowski space. The Euclidean origin in polar coordinates will turn into a lightcone in Minkowski space. The Penrose diagram for this instanton is The curved line is the domain wall. In region I the metric looks I III B II A IV V like Figure 3: Penrose diagram of the instanton ds 2 = dt 2 + a(t) 2 dh 2 3, dh 2 3 = dr 2 + sinh 2 RdΩ 2 2 (28) This is the FRW metric that is produced inside the bubbles, and it is not exactly Minkowski space. Now we have assumed that the bubble nucleates at the center. We can actually move it around, which corresponds to shifting the point B up and down, but the features of the result remain the same. Now we will come closer to phenomenology. Because in the bubble we have negative curvature from the above metric, and curvature dominates, so everything will move away faster than escape velocity and there is no way of structure formation. To dilute this, we need to have a slow roll inflation inside the bubble enough to dilute the curvature such that we have galaxy formation. But with a flat potential there is the overshoot problem where the field will overshoot the final minimum. In this sense the negative curvature is good because it provides friction that slows down the rolling of the field, and prevents the overshooting. 10

11 Aspects of Eternal Inflation Lecture 4 4 Lecture 4 Up till now we have done some rigorous math, but today we will do some very deeply speculative work. This is the measure problem. The problem is like waking up in the morning and you don t know your nationality, and your best guess is Chinese because there is more Chinese than anybody else. Suppose we live in a universe consists of only big and small planets. Big planets can contain 10 9 observers, and little planets can only contain 10 6 observers. Suppose a city on a planet called IAS figured that in their cosmological model that there are only two planets, and there are as many big planets as small planets. If the people at IAS want to know whether they are on a big planet or a little planet. Because the number of these two planets are equal, there should be 50% possibility of each. But there are a lot more people that live on big planets, so it is actually much safer bet for living on big planets. But because the total number of planets is infinite, the ratio is not well defined. The usual process of regulating this kind of infinite ratio is to use a cut off. Like we use a big sphere of radius N and only count the numbers in the sphere, and the ratio is N B lim (29) N N L But there might be the problem of surface effect. This is avoided usually because the surface area grows slower than the volume. But if we are in a hyperbolic space, then the surface area grows at the same rate as the volume. Now this makes it hard to count, and it s very unclear whether there are well defined rules. Let s try to look at this problem. As Alan Guth said, in eternal inflation, everything that can happen will happen, and will happen infinitely many times. If we draw the multiverse in a hyperbolic plane. Let s draw a regulator using a fixed time line, do the counting on the slice, and take the regulator to infinity. The bubbles at late times will have fixed coordinate size, as measured in the de-sitter space. We assume that observations can be made only at a fixed time interval t 0 after the creation of the bubble. In order to ask the number of observers in vacuum type one or type two, we draw a time regulator that intersects some of the bubbles, and eventually take the regulator to infinity. We label first cutoff surface as t = 0. Say a bubble nucleated at t n < 0. There is a problem here because the bubble changes the geometry, and we have to choose a way to extend the regulator into the bubble. The rest of the setup is just t 0 which is the fix time interval after which the observers occur. We count the observers that occur before the cutoff, but not those after the cutoff. This is expressed as an integral dt N dx N e 3Ht N (30) The first and easiest way of extending the cutoff into the bubble is to just pass it into the bubble unchanged. Now if the cutoff time is at t = 0 then the above integral can easily be evaluated as t0 dt N e 3Ht N = 1 3H e 3Ht 0 (31) This can be thought as a probability measure for the time for evolution, or the time t 0. The ancestor vacuum is assumed to have a very large Hubble parameter, so this measure highly suppresses larger t 0, and strongly favors worlds where evolution happens quickly. Let s try other types of measure. These fall into the same logic, which is called takeout-putback logic. This says that there is an extensive quantity on the cutoff surface we are using, which we call Q. When we put a bubble into the cut-off surface, we take the part of Q out, and put it back inside the bubble, at 11

12 Aspects of Eternal Inflation Lecture 4 later time if necessary. First we can assume Q = V which is the proper volume. The amount of Q we take out because of a bubble is V T O = e 3Htn (32) We allow the volume inside the bubble to grow until it is equal to the above take-out volume, and put our cut-off surface back at time t, and we solve the equation V P B = e 3h(t t n) = e 3Htn, t = h H t n (33) h where h is the Hubble constant inside the bubble. Now using this as a cutoff inside the bubble, we should have t n + t 0 < t. Using this to do the integral we have ht0 /H dt N e 3Ht N = 1 3H e 3ht 0 (34) Now H is basically scaled out. We are now comparing different vacuums of different h. In literatue there is a factor of h 2 in front, and the probability concentrates at ht 0 1. This seems to explain the coincidence problem, that observations take place at time about t 0 1/h. This is a great triumph, until we look at it closer. We now look at other Qs. Using Q = V is called scale-factor cutoff. We can use the number of Hubble patches in a given volume as Q Q = V H 3 (35) Note the volume here will be different when taking out and putting back. When taking out it is Q = V T O H 3, and when putting back it is Q = V P B h 3. So we are not conserving the volume now. Working through all the algebra we will get M = 1 3h e 3ht 0 (36) This makes the measure singular at h 0. This suggests that there is a concentration of vacuum of very small Hubble parameter. Now we can look at the so-called information measure, which means Q is the amount of information inside the volume. This gives you a measure like M = h 3 e 3ht 0 (37) The important thing is not the logic of choosing Q, because nobody knows which one is correct. Up to now we assumed that when bubble nucleates, it has a constant volume and keep to it. Now we want to add a period of slow-roll inflation inside the bubbles, and a number N e of e-foldings. It turns out that this gives no change the all the formulae we got above. To get this result we need a bit of work. Could there be a fictitious observer who can observe all the multiverse? Or when does a specific theory setup is allow to be infinitely precise. In flat space, we seem to have an exact prescription, even if there is a black hole. In AdS, we know that at least the boundary theory it is very well modeled, and it can be precise. How about de-sitter space? It would seem that this space is inheritly imprecise, because the vacuum can always tunnel to another of a different cosmological constant, unless it is the terminal vacuum. So there is an inherit imprecision of e S where S is the entropy of the vacuum. Let s see what makes such difference. If we consider a causal patch, there is a bound for the total entropy of this patch S max. Now if 12

13 Aspects of Eternal Inflation Lecture 4 we move the final point at future infinity, and find the max of S max. If the maximum of S max then in this patch there will be observers which has the ability to make infinitely precise measurements. This is the case of flat space. An observer close to the future infinity t = will see the whole space, and inside of which the entropy is infinite, or the information content is infinite. This is true even with a black hole, where an observer close to the horizon will observe a space of infinite information. Now we come to eternal inflation. Eternal inflation ends in bubbles in different ways, and one of them is the hats, where the penrose diagram is a diamond, and observers here in principle can see the whole bubble. So the information in the diamond is infinite, but the de-sitter space causal patch of this bubble has finite entropy. So where do the entropy in the bubble come from? What does all that information describe? The only thing it could be is the information from the rest of the multiverse! 13