1 00:00:00,080 --> 00:00:01,670 The following content is provided 2 00:00:01,670 --> 00:00:03,820 under a Creative Commons license. 3 00:00:03,820 --> 00:00:06,550 Your support will help MIT OpenCourseWare continue 4 00:00:06,550 --> 00:00:10,150 to offer high quality educational resources for free. 5 00:00:10,150 --> 00:00:12,700 To make a donation or to view additional materials 6 00:00:12,700 --> 00:00:16,620 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,620 --> 00:00:17,327 at ocw.mit.edu. 8 00:00:24,790 --> 00:00:28,054 PROFESSOR: OK, in that case, let's get going. 9 00:00:28,054 --> 00:00:30,095 Today's lecture will be mainly on the blackboard. 10 00:00:30,095 --> 00:00:32,880 I have a few slides I want to show. 11 00:00:32,880 --> 00:00:36,670 And what we want to talk about is the inflationary universe 12 00:00:36,670 --> 00:00:37,170 model. 13 00:00:41,770 --> 00:00:45,390 So I'll start by describing the mechanism of inflation, 14 00:00:45,390 --> 00:00:46,025 how it happens. 15 00:00:50,450 --> 00:00:52,210 Inflation is based on the physics 16 00:00:52,210 --> 00:00:56,380 of scalar fields and gravity. 17 00:00:56,380 --> 00:01:03,470 As I think we've said, in present day particle theory, 18 00:01:03,470 --> 00:01:07,930 and by present day I mean not yet string theory, 19 00:01:07,930 --> 00:01:10,230 all particles are described as fields, 20 00:01:10,230 --> 00:01:12,530 quantum excitations of the field. 21 00:01:12,530 --> 00:01:16,390 The analogy that most people are at least qualitatively 22 00:01:16,390 --> 00:01:18,930 familiar with, is the photon, which 23 00:01:18,930 --> 00:01:22,290 is a quantum excitation of the electromagnetic field. 24 00:01:22,290 --> 00:01:25,190 But in fact, to describe a relativistic theory 25 00:01:25,190 --> 00:01:27,440 of interacting particles, the only way 26 00:01:27,440 --> 00:01:29,520 we really know for any kind of particle 27 00:01:29,520 --> 00:01:32,240 is to introduce a field and describe the particle 28 00:01:32,240 --> 00:01:35,450 as a quantized excitation of the field. 29 00:01:35,450 --> 00:01:37,950 So when we talk about a scalar field, 30 00:01:37,950 --> 00:01:40,850 that's the quantum representation 31 00:01:40,850 --> 00:01:42,990 of some kind of a scale or particle. 32 00:01:42,990 --> 00:01:45,530 And scalar in this case, means spinless, 33 00:01:45,530 --> 00:01:48,270 same in all directions. 34 00:01:48,270 --> 00:01:50,210 A scalar field then, is just a number 35 00:01:50,210 --> 00:01:53,010 defined at each point in space. 36 00:01:53,010 --> 00:01:57,580 The only scalar field that we've actually seen in nature so far, 37 00:01:57,580 --> 00:01:58,990 is the Higgs field. 38 00:01:58,990 --> 00:02:00,910 And indeed, inflation is very much 39 00:02:00,910 --> 00:02:02,860 modeled on the Higgs field. 40 00:02:02,860 --> 00:02:05,200 Although the field that drives inflation, 41 00:02:05,200 --> 00:02:08,669 which is by definition, called the inflation, 42 00:02:08,669 --> 00:02:12,000 is probably not the Higgs field of the standard model. 43 00:02:12,000 --> 00:02:14,210 Although recently, actually, in the past few years, 44 00:02:14,210 --> 00:02:16,900 people have written a number of papers proposing 45 00:02:16,900 --> 00:02:19,640 that maybe the Higgs field of the standard model 46 00:02:19,640 --> 00:02:22,640 could, in fact, be the field that drives inflation. 47 00:02:22,640 --> 00:02:23,380 So we don't know. 48 00:02:23,380 --> 00:02:24,770 It's an open question. 49 00:02:24,770 --> 00:02:26,660 But in any case, the field drives inflation 50 00:02:26,660 --> 00:02:29,010 is some kind of cousin, at least, 51 00:02:29,010 --> 00:02:31,040 of the Higgs field of the standard model 52 00:02:31,040 --> 00:02:34,130 and that has many of the same properties. 53 00:02:34,130 --> 00:02:36,300 In particular, the properties of a scalar field 54 00:02:36,300 --> 00:02:40,280 are pretty much summarized by its potential energy function. 55 00:02:40,280 --> 00:02:44,230 Energy density is a function of the value of the field. 56 00:02:44,230 --> 00:02:46,680 And there are two kinds of potential energy functions 57 00:02:46,680 --> 00:02:50,690 that I like to talk about. 58 00:02:50,690 --> 00:02:55,420 One is the kind of that is used in New inflationary models. 59 00:02:59,970 --> 00:03:08,070 And potential energy versus field value. 60 00:03:08,070 --> 00:03:13,360 And it has a plateau with a peak at someplace, which is usually 61 00:03:13,360 --> 00:03:17,060 assumed to be phi equals zero and a potential energy function 62 00:03:17,060 --> 00:03:19,860 which may or may not be symmetric about phi equals zero 63 00:03:19,860 --> 00:03:21,855 but I'll assume it is, just for simplicity. 64 00:03:25,280 --> 00:03:29,510 And the second type, which I'd like to talk about, 65 00:03:29,510 --> 00:03:31,280 mainly for comparison-- This is really 66 00:03:31,280 --> 00:03:33,690 the one that will be interesting. 67 00:03:33,690 --> 00:03:39,560 But one could also imagine a potential energy function 68 00:03:39,560 --> 00:03:42,920 which really has a local minimum someplace, which 69 00:03:42,920 --> 00:03:44,160 is not the global minimum. 70 00:03:44,160 --> 00:03:46,850 And I'll draw it with the local minimum 71 00:03:46,850 --> 00:03:51,270 at the origin and the global minima, two of them 72 00:03:51,270 --> 00:03:54,280 degenerate, elsewhere. 73 00:03:54,280 --> 00:03:56,980 And this, again, is a graph of mi versus phi. 74 00:03:56,980 --> 00:03:59,080 And the reason I'm drawing this is partly 75 00:03:59,080 --> 00:04:01,296 for historical interest. 76 00:04:04,760 --> 00:04:07,415 This was what was used in the original inflationary model. 77 00:04:07,415 --> 00:04:08,730 It is my original paper. 78 00:04:15,960 --> 00:04:17,200 It does not work. 79 00:04:17,200 --> 00:04:20,000 But we'll want to talk about why it does not work. 80 00:04:22,560 --> 00:04:26,070 In both cases we're interested in a state, which 81 00:04:26,070 --> 00:04:34,250 can be called a false vacuum, which is a state where 82 00:04:34,250 --> 00:04:37,670 the scalar field is just sitting at phi equals zero. 83 00:04:49,260 --> 00:04:52,870 In the case of the second of these potentials, 84 00:04:52,870 --> 00:04:55,260 phi equals zero being a local minimum, 85 00:04:55,260 --> 00:04:58,140 is classically completely stable. 86 00:04:58,140 --> 00:05:00,540 If one had a scalar field in some region of space 87 00:05:00,540 --> 00:05:02,750 just sitting in that minimum, there'd 88 00:05:02,750 --> 00:05:05,760 be no place where energy could come from that would drive it 89 00:05:05,760 --> 00:05:09,610 out of that minimum over the barrier. 90 00:05:09,610 --> 00:05:13,040 In the second case, the field is classically stable. 91 00:05:13,040 --> 00:05:16,260 But it's still possible for it to quantum mechanically tunnel 92 00:05:16,260 --> 00:05:17,750 through the barrier. 93 00:05:17,750 --> 00:05:22,100 And that process has been calculated and understood. 94 00:05:22,100 --> 00:05:25,600 Originally by Sidney Coleman and collaborators. 95 00:05:25,600 --> 00:05:29,600 And an important feature of that tunneling 96 00:05:29,600 --> 00:05:31,550 is that does not happen globally. 97 00:05:31,550 --> 00:05:34,250 You might think that there would be some probability that 98 00:05:34,250 --> 00:05:36,014 suddenly everywhere in the universe 99 00:05:36,014 --> 00:05:37,930 the scalar field would tunnel over the barrier 100 00:05:37,930 --> 00:05:40,020 and go down to the other side. 101 00:05:40,020 --> 00:05:42,050 The probability of that is zero, as you 102 00:05:42,050 --> 00:05:44,152 might realize if you thought about a bit more. 103 00:05:44,152 --> 00:05:45,610 There's just no way that the scalar 104 00:05:45,610 --> 00:05:47,499 field that far over there is going 105 00:05:47,499 --> 00:05:49,790 to know the tunnel at the same time as the scalar field 106 00:05:49,790 --> 00:05:50,920 far over there. 107 00:05:50,920 --> 00:05:54,010 So the timely happens locally and it really 108 00:05:54,010 --> 00:05:57,250 happens in a very small region, which 109 00:05:57,250 --> 00:06:00,040 then tunnels over the barrier and the scalar field 110 00:06:00,040 --> 00:06:02,870 start rolling down on the other side in this small region. 111 00:06:02,870 --> 00:06:04,010 And then that region grows. 112 00:06:04,010 --> 00:06:06,650 The scalar field, as it rolls over the barrier, 113 00:06:06,650 --> 00:06:09,780 pulls the scalar field nearby. 114 00:06:09,780 --> 00:06:12,070 And the region grows with a speed 115 00:06:12,070 --> 00:06:15,420 that rapidly approaches the speed of light. 116 00:06:15,420 --> 00:06:17,640 In the New inflationary potential, 117 00:06:17,640 --> 00:06:21,070 where we have actually a local maximum here, 118 00:06:21,070 --> 00:06:26,410 the situation is classically meta stable, 119 00:06:26,410 --> 00:06:29,330 in the sense that the smallest possible fluctuation can start 120 00:06:29,330 --> 00:06:32,700 the field rolling down the hill. 121 00:06:32,700 --> 00:06:34,510 And in particular, quantum fluctuations 122 00:06:34,510 --> 00:06:36,550 will, if nothing else, start the field 123 00:06:36,550 --> 00:06:41,780 rolling down the hill in some finite amount of time. 124 00:06:41,780 --> 00:06:43,850 We're interested in the case where 125 00:06:43,850 --> 00:06:47,430 the amount of time it takes for quantum fluctuations 126 00:06:47,430 --> 00:06:51,370 to push the field off the hill is relatively long compared 127 00:06:51,370 --> 00:06:54,610 to time scales involved in the early universe. 128 00:06:54,610 --> 00:06:57,220 And the key time scale involved in the early universe 129 00:06:57,220 --> 00:06:59,896 is the Hubble time. 130 00:06:59,896 --> 00:07:02,270 And the Hubble time is just driven by the energy density. 131 00:07:02,270 --> 00:07:04,800 So one can calculate the Hubble time. 132 00:07:04,800 --> 00:07:07,350 And one is interested in the case 133 00:07:07,350 --> 00:07:09,480 for building inflationary models, 134 00:07:09,480 --> 00:07:12,480 where the top of the hill is smooth enough, gentle enough, 135 00:07:12,480 --> 00:07:15,950 has a small enough second derivative 136 00:07:15,950 --> 00:07:19,220 so that the amount of time it will take for the scalar field 137 00:07:19,220 --> 00:07:21,480 to roll off the hill is long compared 138 00:07:21,480 --> 00:07:26,160 to the Hubble time of the universe. 139 00:07:26,160 --> 00:07:29,150 So in both cases, for short times, 140 00:07:29,150 --> 00:07:32,300 the scalar field is just stuck at the origin. 141 00:07:32,300 --> 00:07:35,050 And that's what's important, as far as 142 00:07:35,050 --> 00:07:37,590 what we want to talk about next. 143 00:07:37,590 --> 00:07:39,150 So the characteristic of this state 144 00:07:39,150 --> 00:07:47,960 called the false vacuum is that the scalar field is 145 00:07:47,960 --> 00:07:52,590 pinned at a high energy state. 146 00:08:17,220 --> 00:08:21,106 Scalar field is pinned at a high energy density. 147 00:08:21,106 --> 00:08:26,790 And by pinned, I mean in advance you just can't change quickly. 148 00:08:26,790 --> 00:08:30,590 In general, particle physicists use the word vacuum 149 00:08:30,590 --> 00:08:34,549 to mean the state of lowest possible energy density. 150 00:08:34,549 --> 00:08:36,760 When we call this a false vacuum, 151 00:08:36,760 --> 00:08:41,530 we're really using the word false in the sense of the word 152 00:08:41,530 --> 00:08:43,780 temporary. 153 00:08:43,780 --> 00:08:45,880 These states are temporary vacuums, 154 00:08:45,880 --> 00:08:49,020 in that for some period of time, which is long by early universe 155 00:08:49,020 --> 00:08:52,602 standards, the energy density can't get any lower. 156 00:08:52,602 --> 00:08:53,810 So it's acting like a vacuum. 157 00:08:57,110 --> 00:08:59,530 Now what are the consequences of that? 158 00:08:59,530 --> 00:09:01,870 The important consequence of that 159 00:09:01,870 --> 00:09:06,640 is that the pressure has to be large in negative 160 00:09:06,640 --> 00:09:10,280 and in fact, equal to the negative of the energy density. 161 00:09:10,280 --> 00:09:18,130 And there are two ways we can convince ourselves of that. 162 00:09:18,130 --> 00:09:20,920 The first is that if we remember the cosmological equation 163 00:09:20,920 --> 00:09:25,426 that we derived somewhere in the middle of course for row dot. 164 00:09:25,426 --> 00:09:33,050 We learned that row dot is equal to minus 3 a dot over a 165 00:09:33,050 --> 00:09:36,770 where a is the scale factor times row 166 00:09:36,770 --> 00:09:39,610 plus the pressure over c squared. 167 00:09:43,400 --> 00:09:48,420 Now what we're saying here, is that as the universe expands, 168 00:09:48,420 --> 00:09:52,080 the scalar field is just stuck at this false vacuum value. 169 00:09:52,080 --> 00:09:55,410 The energy density is stuck at the energy density associated 170 00:09:55,410 --> 00:09:57,800 with that value of the field, the potential energy 171 00:09:57,800 --> 00:10:00,460 density of the field itself. 172 00:10:00,460 --> 00:10:04,330 And therefore, row dot will be zero as the universe expands. 173 00:10:04,330 --> 00:10:06,120 And if row dot is zero, we can just 174 00:10:06,120 --> 00:10:10,050 read off from this equation what the pressure has to be. 175 00:10:10,050 --> 00:10:13,800 Row dot equals zero implies that the pressure is just 176 00:10:13,800 --> 00:10:18,080 equal to minus the energy density times c squared, 177 00:10:18,080 --> 00:10:20,710 which is another way of saying it's minus-- Excuse me, 178 00:10:20,710 --> 00:10:22,730 the mass density times c squared, 179 00:10:22,730 --> 00:10:25,590 which is another way of saying it's minus the energy density. 180 00:10:25,590 --> 00:10:29,330 I'm using u for energy density and row for mass density. 181 00:10:29,330 --> 00:10:32,540 And they just differ by a factor of c squared. 182 00:10:32,540 --> 00:10:36,820 So this is the straightforward equation method 183 00:10:36,820 --> 00:10:39,120 of seeing the answer here. 184 00:10:39,120 --> 00:10:42,690 But if you want to explain this to your roommates or somebody 185 00:10:42,690 --> 00:10:45,280 who is not taking this class, there's 186 00:10:45,280 --> 00:10:48,916 also a simple argument based on a thought experiment, which 187 00:10:48,916 --> 00:10:50,290 I think is worth keeping in mind. 188 00:10:50,290 --> 00:10:52,206 And we've used this argument before, actually, 189 00:10:52,206 --> 00:10:54,690 in similar contexts. 190 00:10:54,690 --> 00:11:02,095 If we imagine a piston chamber in our thought experiment. 191 00:11:07,017 --> 00:11:08,600 And in our piston chamber, we're going 192 00:11:08,600 --> 00:11:10,215 to put false vacuum on the inside. 193 00:11:17,570 --> 00:11:27,160 And the false vacuum will have an energy density, 194 00:11:27,160 --> 00:11:28,440 I'll call it u sub f. 195 00:11:31,250 --> 00:11:32,200 And on the outside. 196 00:11:36,970 --> 00:11:39,270 we're going to have a zero energy vacuum. 197 00:11:45,770 --> 00:11:50,440 Now we've learned that since 1998, we've 198 00:11:50,440 --> 00:11:54,510 known that our vacuum is not a zero energy vacuum. 199 00:11:54,510 --> 00:11:58,410 We seem to be seeing a non-zero vacuum energy in our universe. 200 00:11:58,410 --> 00:12:01,390 However, even if that's true, the vacuum energy 201 00:12:01,390 --> 00:12:03,050 of our universe is incredibly small 202 00:12:03,050 --> 00:12:05,660 compared to the false vacuum energy density 203 00:12:05,660 --> 00:12:08,020 that we're talking about in terms of the early universe. 204 00:12:08,020 --> 00:12:10,260 So you could still very well approximate it as zero 205 00:12:10,260 --> 00:12:11,950 and not worry about it. 206 00:12:11,950 --> 00:12:13,740 So that's what we'll be doing here. 207 00:12:13,740 --> 00:12:16,020 So we'll think of the outside as being 208 00:12:16,020 --> 00:12:19,110 either a fictitious vacuum, which by definition, has zero 209 00:12:19,110 --> 00:12:21,250 energy density and we can talk about it 210 00:12:21,250 --> 00:12:23,250 even if it doesn't exist. 211 00:12:23,250 --> 00:12:25,410 Or we could think of it as being the real vacuum 212 00:12:25,410 --> 00:12:27,790 in our universe, which has an energy density which 213 00:12:27,790 --> 00:12:29,580 is approximately zero on this scale. 214 00:12:32,290 --> 00:12:34,830 Now what we want to do is just imagine 215 00:12:34,830 --> 00:12:36,030 pulling out that piston. 216 00:12:48,200 --> 00:12:54,600 So we have now created an extra region 217 00:12:54,600 --> 00:12:57,630 on the interior of the piston chamber. 218 00:12:57,630 --> 00:12:59,920 And we're going to be assuming that we've somehow 219 00:12:59,920 --> 00:13:04,730 rigged the walls of the chamber so that the false vacuum will 220 00:13:04,730 --> 00:13:07,370 be stretched as we pull out the piston. 221 00:13:07,370 --> 00:13:12,010 The piston is attached to the false vacuum in some way. 222 00:13:12,010 --> 00:13:15,440 So this entire area inside region is now false vacuum. 223 00:13:23,820 --> 00:13:26,260 And therefore, the volume of the false vacuum region 224 00:13:26,260 --> 00:13:26,801 has enlarged. 225 00:13:29,050 --> 00:13:33,370 And if we call the extra region here 226 00:13:33,370 --> 00:13:37,150 delta v, the volume of that region, 227 00:13:37,150 --> 00:13:48,200 we now have a situation where the energy has increased 228 00:13:48,200 --> 00:13:52,510 by the energy density of the false vacuum times delta v 229 00:13:52,510 --> 00:13:54,250 by changing the volume of the chamber. 230 00:13:57,990 --> 00:14:01,770 Now, energy has to be conserved. 231 00:14:01,770 --> 00:14:04,440 So this energy has to be equal to the work that 232 00:14:04,440 --> 00:14:08,290 was done by whatever force pulled out on this piston. 233 00:14:08,290 --> 00:14:12,760 We won't need to specify who was pulling on the piston, 234 00:14:12,760 --> 00:14:22,440 but the work done when one pulls out on a piston 235 00:14:22,440 --> 00:14:28,691 is just equal to minus p times delta v, 236 00:14:28,691 --> 00:14:31,790 the work done by the person pulling on the piston. 237 00:14:31,790 --> 00:14:35,550 So the normal case, the pressure would be positive, 238 00:14:35,550 --> 00:14:37,780 the piston would be pushing out on the person 239 00:14:37,780 --> 00:14:42,040 holding the piston and the interior would be doing work 240 00:14:42,040 --> 00:14:43,530 on the person pulling out. 241 00:14:43,530 --> 00:14:47,440 And it would be positive if the pressure were positive. 242 00:14:47,440 --> 00:14:49,590 The work done on the person, but this 243 00:14:49,590 --> 00:14:54,340 is supposed to be the work done on the gas. 244 00:14:54,340 --> 00:15:17,920 And that's minus p times delta v. 245 00:15:17,920 --> 00:15:20,420 So if energy is conserved, the work done on the gas 246 00:15:20,420 --> 00:15:23,070 has to be equal to the change in energy of the gas. 247 00:15:23,070 --> 00:15:26,420 And the change in energy of the gas is that. 248 00:15:26,420 --> 00:15:33,970 So conservation of energy implies 249 00:15:33,970 --> 00:15:44,100 that delta e equals delta w or use of f times delta v 250 00:15:44,100 --> 00:15:49,840 equals minus p times delta v, which of course implies 251 00:15:49,840 --> 00:15:57,220 that p is equal to minus use of f, as we said before. 252 00:15:57,220 --> 00:16:01,050 So the point is that if the energy inside the piston 253 00:16:01,050 --> 00:16:04,250 is going to increase, the person pulling out on the piston 254 00:16:04,250 --> 00:16:08,690 had better be doing work, had better be doing positive work. 255 00:16:08,690 --> 00:16:10,711 And if the pressure inside were positive, 256 00:16:10,711 --> 00:16:12,210 the person pulling out on the piston 257 00:16:12,210 --> 00:16:13,418 would be doing negative work. 258 00:16:13,418 --> 00:16:16,140 The piston would be pushing on him. 259 00:16:16,140 --> 00:16:18,740 So for it to make sense here, with the energy 260 00:16:18,740 --> 00:16:22,810 in the piston increasing, the person pulling out 261 00:16:22,810 --> 00:16:25,310 has to really be pulling against a suction. 262 00:16:25,310 --> 00:16:28,170 He has to do work to pull out. 263 00:16:28,170 --> 00:16:30,900 And a suction means a negative pressure, 264 00:16:30,900 --> 00:16:32,670 if we have zero pressure outside. 265 00:16:32,670 --> 00:16:35,164 Pressure inside has to be negative. 266 00:16:35,164 --> 00:16:37,330 So we could reach this conclusion either of two ways 267 00:16:37,330 --> 00:16:39,010 and we get the same conclusion. 268 00:16:39,010 --> 00:16:42,260 The pressure is just equal to minus the energy density. 269 00:16:42,260 --> 00:16:43,035 Yes? 270 00:16:43,035 --> 00:16:44,935 AUDIENCE: I'm a little confused about why 271 00:16:44,935 --> 00:16:46,880 the energy's increasing inside. 272 00:16:46,880 --> 00:16:48,255 Because why couldn't you just say 273 00:16:48,255 --> 00:16:50,664 the energy density decreases with the increased volume? 274 00:16:50,664 --> 00:16:52,580 PROFESSOR: OK the question is why couldn't you 275 00:16:52,580 --> 00:16:53,990 just say that the energy density would 276 00:16:53,990 --> 00:16:55,540 decrease with the increased volume. 277 00:16:55,540 --> 00:16:56,956 That certainly is what will happen 278 00:16:56,956 --> 00:16:59,060 if you have normal gas inside. 279 00:16:59,060 --> 00:17:03,110 What makes this particular false vacuum odd 280 00:17:03,110 --> 00:17:05,190 is the origin of this energy density, which 281 00:17:05,190 --> 00:17:08,030 is the potential energy density of the field. 282 00:17:08,030 --> 00:17:10,290 So if we were talking about the situation, 283 00:17:10,290 --> 00:17:12,619 for example, which is the clearest cut, 284 00:17:12,619 --> 00:17:15,260 the only way the energy density here could go down 285 00:17:15,260 --> 00:17:17,349 is if the scalar field goes up over the barrier 286 00:17:17,349 --> 00:17:19,410 and then comes down over here. 287 00:17:19,410 --> 00:17:21,790 And there's no way to drive it there, except to wait 288 00:17:21,790 --> 00:17:25,520 for a quantum fluctuation, which is a very slow process. 289 00:17:25,520 --> 00:17:28,680 And similarly, here there's no barrier. 290 00:17:28,680 --> 00:17:30,230 So it can't just roll down. 291 00:17:30,230 --> 00:17:32,410 But that takes a certain amount of time. 292 00:17:32,410 --> 00:17:35,880 And we're assuming that all the things we're talking about here 293 00:17:35,880 --> 00:17:37,400 are happening on a time that's fast 294 00:17:37,400 --> 00:17:39,900 compared to the amount of time it takes for the scalar field 295 00:17:39,900 --> 00:17:41,410 to roll. 296 00:17:41,410 --> 00:17:45,190 So what makes this peculiar false vacuum special is that it 297 00:17:45,190 --> 00:17:48,620 cannot lower its energy density quickly. 298 00:17:48,620 --> 00:17:51,340 And that's what the word false vacuum implies. 299 00:17:51,340 --> 00:17:53,676 And there are states like that. 300 00:17:53,676 --> 00:17:55,050 And then those states necessarily 301 00:17:55,050 --> 00:17:56,591 have a negative pressure, or pressure 302 00:17:56,591 --> 00:18:02,730 that's equal to good accuracy to minus the energy density. 303 00:18:02,730 --> 00:18:05,500 Or I say to good accuracy only because the energy density 304 00:18:05,500 --> 00:18:06,950 could change a little bit slowly, 305 00:18:06,950 --> 00:18:09,010 at least for the top case. 306 00:18:09,010 --> 00:18:11,780 But it's limited how much it can change. 307 00:18:15,466 --> 00:18:16,840 OK, now what are the consequences 308 00:18:16,840 --> 00:18:18,920 of this cosmologically? 309 00:18:18,920 --> 00:18:23,400 Well we've also learned that we could write the second order 310 00:18:23,400 --> 00:18:26,590 Friedman equation, which is the equation really tells us what 311 00:18:26,590 --> 00:18:29,440 the force of gravity is doing. 312 00:18:29,440 --> 00:18:37,960 A double dot is equal minus 4 pi over 3 g times row 313 00:18:37,960 --> 00:18:42,200 plus 3 p over c squared. 314 00:18:46,910 --> 00:18:49,430 Now for the false vacuum, p is equal to minus row 315 00:18:49,430 --> 00:18:52,220 c squared minus the energy density. 316 00:18:52,220 --> 00:18:55,060 And that means that this term is negative and three times as big 317 00:18:55,060 --> 00:18:57,650 as that term. 318 00:18:57,650 --> 00:19:00,300 So for the false vacuum, this quantity, 319 00:19:00,300 --> 00:19:03,160 which we normally think of as being positive, 320 00:19:03,160 --> 00:19:05,500 becomes negative. 321 00:19:05,500 --> 00:19:07,760 I should write factor of a here. 322 00:19:07,760 --> 00:19:10,230 And that means that instead of gravity slowing 323 00:19:10,230 --> 00:19:14,940 down the expansion the universe for a false vacuum, 324 00:19:14,940 --> 00:19:16,980 the expansion is accelerated. 325 00:19:16,980 --> 00:19:18,790 And that's also what we're seeing today 326 00:19:18,790 --> 00:19:21,050 with the vacuum energy, which behaves 327 00:19:21,050 --> 00:19:23,590 the same way as this false vacuum 328 00:19:23,590 --> 00:19:26,340 and produces gravitational repulsion in exactly 329 00:19:26,340 --> 00:19:27,080 the same way. 330 00:19:30,220 --> 00:19:41,490 So false vacuum implies gravitational repulsion. 331 00:20:06,870 --> 00:20:10,680 OK, this basically is the mechanism of inflation. 332 00:20:10,680 --> 00:20:12,900 So we're sort of finished with this chapter. 333 00:20:12,900 --> 00:20:15,390 Are any questions before I go on about how 334 00:20:15,390 --> 00:20:17,496 this gravitational repulsion arises? 335 00:20:17,496 --> 00:20:17,995 Michael. 336 00:20:17,995 --> 00:20:20,994 AUDIENCE: So, for the top vacuum that you've drawn up there, 337 00:20:20,994 --> 00:20:23,210 where there's no barrier, you just roll slowly, 338 00:20:23,210 --> 00:20:25,614 are we assuming that it takes a long time for it 339 00:20:25,614 --> 00:20:27,598 to begin to roll or after it's started 340 00:20:27,598 --> 00:20:30,078 rolling that it also takes a very long time to reach 341 00:20:30,078 --> 00:20:31,080 the bottom. 342 00:20:31,080 --> 00:20:34,039 PROFESSOR: I guess I'd say both, Begin to roll 343 00:20:34,039 --> 00:20:35,080 is not that well defined. 344 00:20:35,080 --> 00:20:37,750 Because it may have it an infinitesimal velocity 345 00:20:37,750 --> 00:20:39,980 from the time you start discussing it. 346 00:20:39,980 --> 00:20:41,480 And then that infinitesimal velocity 347 00:20:41,480 --> 00:20:43,004 gets bigger and bigger. 348 00:20:43,004 --> 00:20:45,670 But I think what we're saying is that the whole process, however 349 00:20:45,670 --> 00:20:49,060 you divide it up, it's going to take a long time compared 350 00:20:49,060 --> 00:20:51,540 to the time it takes for the exponential expansion 351 00:20:51,540 --> 00:20:52,140 to set in. 352 00:20:59,440 --> 00:21:00,420 OK. 353 00:21:00,420 --> 00:21:09,040 So now, I'd like to take this physics 354 00:21:09,040 --> 00:21:11,410 and just put a scenario around it. 355 00:21:11,410 --> 00:21:13,760 And we'll call the new inflationary scenario 356 00:21:13,760 --> 00:21:17,660 because that's what it is. 357 00:21:17,660 --> 00:21:19,790 Maybe now I should mention a little bit more 358 00:21:19,790 --> 00:21:22,950 about the history here. 359 00:21:22,950 --> 00:21:24,660 When I wrote my original paper, I 360 00:21:24,660 --> 00:21:26,270 was assuming a potential of something 361 00:21:26,270 --> 00:21:30,790 like this, because it seemed generic and created inflation 362 00:21:30,790 --> 00:21:33,860 and I was able to understand that inflation would solve 363 00:21:33,860 --> 00:21:37,692 a number of cosmological problems, which 364 00:21:37,692 --> 00:21:39,400 are the problems that we've talked about. 365 00:21:39,400 --> 00:21:41,330 And we'll come back to talk about how 366 00:21:41,330 --> 00:21:43,510 inflation solves them. 367 00:21:43,510 --> 00:21:44,010 But 368 00:21:44,010 --> 00:21:48,670 But, one still has to end inflation. 369 00:21:48,670 --> 00:21:52,530 In this model, inflation would end only 370 00:21:52,530 --> 00:21:55,590 by the tunneling of the scalar field 371 00:21:55,590 --> 00:21:57,400 through the barrier, which as I said, 372 00:21:57,400 --> 00:22:01,510 happens in small regions which then grow. 373 00:22:01,510 --> 00:22:05,970 Those regions are spherical so they're called bubbles. 374 00:22:05,970 --> 00:22:10,190 And the whole process really is very much the way water boils. 375 00:22:10,190 --> 00:22:12,970 When you boil water, it forms very small bundles initially 376 00:22:12,970 --> 00:22:15,790 and the bundles grow and then start colliding with each other 377 00:22:15,790 --> 00:22:17,890 and making a big frothy mess. 378 00:22:17,890 --> 00:22:21,520 And it turns out that that's exactly what would happen 379 00:22:21,520 --> 00:22:24,790 in the early universe if you had this model. 380 00:22:24,790 --> 00:22:27,010 When I first started thinking about it, 381 00:22:27,010 --> 00:22:31,500 I hoped that these bundles could collide with each other 382 00:22:31,500 --> 00:22:33,500 while they're still small and merge 383 00:22:33,500 --> 00:22:38,970 into a uniform, hot region of the new phase, a phase where 384 00:22:38,970 --> 00:22:43,430 the scalar field is not there, but there. 385 00:22:43,430 --> 00:22:46,010 But that turned out to be the case. 386 00:22:46,010 --> 00:22:50,680 It turned out that the bubble formation process produced 387 00:22:50,680 --> 00:22:53,280 horrible inhomogeneities that there did not 388 00:22:53,280 --> 00:22:55,270 seem to be anyway to cure. 389 00:22:55,270 --> 00:22:57,810 And that then was the downfall of 390 00:22:57,810 --> 00:23:00,920 the original inflationary model. 391 00:23:00,920 --> 00:23:05,590 But a few years later, Andrei Linde in the Soviet Union, 392 00:23:05,590 --> 00:23:09,560 and independently, Albrecht and Steinhardt in the US, 393 00:23:09,560 --> 00:23:12,880 proposed what came to be called the new inflationary model, 394 00:23:12,880 --> 00:23:16,060 which started with a different assumption 395 00:23:16,060 --> 00:23:17,810 about what the underlying potential energy 396 00:23:17,810 --> 00:23:19,890 function for the scalar field was. 397 00:23:19,890 --> 00:23:22,080 Instead of assuming something like this, 398 00:23:22,080 --> 00:23:24,559 which might be called generic in some sense, 399 00:23:24,559 --> 00:23:26,100 they instead assumed something that's 400 00:23:26,100 --> 00:23:28,970 a little bit more special, a potential energy 401 00:23:28,970 --> 00:23:31,530 function with a very flat plateau 402 00:23:31,530 --> 00:23:35,820 somewhere, which, well, we normally put in the middle. 403 00:23:35,820 --> 00:23:41,690 And this has the advantage, that the inflation ends, not 404 00:23:41,690 --> 00:23:46,150 by bubbled nucleation by tunneling, but instead, 405 00:23:46,150 --> 00:23:49,130 by just small fluctuations building up and pushing 406 00:23:49,130 --> 00:23:51,940 the scalar field down the hill. 407 00:23:51,940 --> 00:23:54,550 And what makes it work, basically 408 00:23:54,550 --> 00:23:57,900 is that those small fluctuations have some spatial correlations 409 00:23:57,900 --> 00:23:58,740 built into them. 410 00:23:58,740 --> 00:24:03,700 So over some small region, which I will calla coherence region, 411 00:24:03,700 --> 00:24:07,210 the fluctuations are essentially uniform. 412 00:24:07,210 --> 00:24:09,430 And the other important feature is 413 00:24:09,430 --> 00:24:12,540 that once the scalar field starts to roll, 414 00:24:12,540 --> 00:24:16,930 it still has some nearly flat hill to roll on. 415 00:24:16,930 --> 00:24:19,900 So a significant amount of inflation 416 00:24:19,900 --> 00:24:25,560 happens after this homogeneous coherence region forms. 417 00:24:25,560 --> 00:24:29,890 So the initial coherence region can be microscopic, 418 00:24:29,890 --> 00:24:32,110 but it is then stretched by the inflation that 419 00:24:32,110 --> 00:24:35,470 continues as the scalar field rolls down 420 00:24:35,470 --> 00:24:37,990 the hill towards the bottom. 421 00:24:37,990 --> 00:24:41,990 So that process of stretching the coherence region 422 00:24:41,990 --> 00:24:47,150 after it has already formed is what makes this model workable, 423 00:24:47,150 --> 00:24:48,260 while this model was not. 424 00:24:51,910 --> 00:24:56,000 So that's the basic story of how new inflation succeeded 425 00:24:56,000 --> 00:24:58,990 in allowing inflation to end gracefully, 426 00:24:58,990 --> 00:25:00,910 is the phrase that was used. 427 00:25:00,910 --> 00:25:02,830 The problems associated with this model 428 00:25:02,830 --> 00:25:05,310 came to be called the graceful exit problem. 429 00:25:05,310 --> 00:25:10,170 And this is the first solution to the graceful exit problem. 430 00:25:10,170 --> 00:25:11,380 They're now other solutions. 431 00:25:11,380 --> 00:25:12,796 But they're very similar actually. 432 00:25:23,052 --> 00:25:24,510 So I'll just write here that's it's 433 00:25:24,510 --> 00:25:54,010 a modification of the original inflationary model 434 00:25:54,010 --> 00:25:56,335 to solve this graceful exit problem problem. 435 00:26:00,640 --> 00:26:04,095 Now I should say a little bit about how inflation starts. 436 00:26:06,654 --> 00:26:08,320 But I can only say a little bit about it 437 00:26:08,320 --> 00:26:11,795 because the bottom line really is we don't know. 438 00:26:11,795 --> 00:26:15,320 We still don't have any real theory of initial conditions 439 00:26:15,320 --> 00:26:17,950 for cosmology, whether it's inflationary cosmology 440 00:26:17,950 --> 00:26:19,850 or any kind of cosmology. 441 00:26:19,850 --> 00:26:22,600 The nice feature of inflation is that it 442 00:26:22,600 --> 00:26:27,912 allows a significantly broader set of initial conditions 443 00:26:27,912 --> 00:26:30,370 than is required, for example, in the standard cosmological 444 00:26:30,370 --> 00:26:34,220 model, where, as we discussed, the needed initial conditions 445 00:26:34,220 --> 00:26:37,530 are very precisely specified. 446 00:26:37,530 --> 00:26:40,480 I might say a few things though, about ideas people have had. 447 00:26:45,480 --> 00:26:49,280 One idea, which I think sounds very reasonable, 448 00:26:49,280 --> 00:26:51,810 is due to Andrei Linde. 449 00:26:51,810 --> 00:26:55,492 And it's a vague idea, so it really 450 00:26:55,492 --> 00:26:57,450 needs to be more precise before it could really 451 00:26:57,450 --> 00:26:58,408 be considered a theory. 452 00:27:08,700 --> 00:27:11,630 But this is just the idea that the universe started out 453 00:27:11,630 --> 00:27:15,150 with some kind of chaotic random initial conditions. 454 00:27:15,150 --> 00:27:19,580 And then the hope is simply that inflation will start somewhere. 455 00:27:19,580 --> 00:27:23,620 That somewhere in the initial chaotic distribution there'll 456 00:27:23,620 --> 00:27:25,430 be a place where the scalar field will 457 00:27:25,430 --> 00:27:29,130 have the right properties, the right configuration 458 00:27:29,130 --> 00:27:31,680 to initiate inflation. 459 00:27:31,680 --> 00:27:37,130 There are also models by Vilenkin, 460 00:27:37,130 --> 00:27:41,849 Alex Vilenkin of Tufts, and independently, Andrei Linde, 461 00:27:41,849 --> 00:27:43,140 who by the way, is at Stanford. 462 00:27:48,080 --> 00:27:50,920 They both worked on models where the universe 463 00:27:50,920 --> 00:27:53,760 could begin by a quantum tunneling process, 464 00:27:53,760 --> 00:27:55,330 starting from absolutely nothing. 465 00:28:18,850 --> 00:28:21,690 I wrote here absolutely nothing and that's 466 00:28:21,690 --> 00:28:24,590 more nothing than nothing. 467 00:28:24,590 --> 00:28:27,599 You might think of nothing as just empty space. 468 00:28:27,599 --> 00:28:29,640 But from the point of view of general relativity, 469 00:28:29,640 --> 00:28:32,080 as you already know enough to understand, 470 00:28:32,080 --> 00:28:34,080 empty space is not really nothing. 471 00:28:34,080 --> 00:28:36,470 Empty space is really a dynamical system. 472 00:28:36,470 --> 00:28:38,640 Empty space can bend and twist and stretch and do 473 00:28:38,640 --> 00:28:40,420 all kinds of complicated things. 474 00:28:40,420 --> 00:28:44,700 It's really no different, in some basic sense, 475 00:28:44,700 --> 00:28:47,420 from a big piece of rubber. 476 00:28:47,420 --> 00:28:50,100 So nothingness really is intended 477 00:28:50,100 --> 00:28:54,920 to mean a state where there's not only no matter present, 478 00:28:54,920 --> 00:28:59,150 but also no space and no time, really nothing. 479 00:28:59,150 --> 00:29:01,550 One way to think of it, perhaps, is 480 00:29:01,550 --> 00:29:04,770 as a closed universe, the limit as the size of the closed 481 00:29:04,770 --> 00:29:07,785 universe goes to zero so that there's nothing left. 482 00:29:11,670 --> 00:29:14,690 None of these theories are precise. 483 00:29:14,690 --> 00:29:18,700 We don't really know how to precisely formulate them. 484 00:29:18,700 --> 00:29:20,560 In this tunneling from nothing, one 485 00:29:20,560 --> 00:29:22,644 is talking about tunneling in the context where 486 00:29:22,644 --> 00:29:25,060 the structure of space itself changes during the tunneling 487 00:29:25,060 --> 00:29:26,010 process. 488 00:29:26,010 --> 00:29:28,710 So it's tunneling in the context of general relativity. 489 00:29:28,710 --> 00:29:30,750 And we don't really have a successful quantum 490 00:29:30,750 --> 00:29:33,040 theory of general relativity. 491 00:29:33,040 --> 00:29:35,780 So these ideas are very speculative and quite vague. 492 00:29:35,780 --> 00:29:39,540 But they do indicate some possibilities 493 00:29:39,540 --> 00:29:42,600 for how the universe might have started. 494 00:29:42,600 --> 00:29:46,610 An idea closely related to this tunneling from nothing 495 00:29:46,610 --> 00:30:08,270 is the Hartle and Hawking-- This is Jim Hartle, of UC Santa 496 00:30:08,270 --> 00:30:11,370 Barbara, who's also the author of a general relativity 497 00:30:11,370 --> 00:30:12,790 textbook now. 498 00:30:12,790 --> 00:30:15,870 And Stephen Hawking, who you must 499 00:30:15,870 --> 00:30:18,326 know from Cambridge University. 500 00:30:18,326 --> 00:30:19,950 They proposed something called the wave 501 00:30:19,950 --> 00:30:20,991 function of the universe. 502 00:30:32,950 --> 00:30:36,890 From their point of view, it's self contradictory 503 00:30:36,890 --> 00:30:39,610 to talk about the universe having an origin 504 00:30:39,610 --> 00:30:42,860 because before the origin of the universe, space and time 505 00:30:42,860 --> 00:30:44,137 we're not even defined. 506 00:30:44,137 --> 00:30:46,470 And therefore, you could not think of there being a time 507 00:30:46,470 --> 00:30:48,560 before the universe was created. 508 00:30:48,560 --> 00:30:51,270 And therefore universe didn't actually get created. 509 00:30:51,270 --> 00:30:55,460 It just is and has some earliest possible time. 510 00:30:55,460 --> 00:30:58,400 And that's what this wave function of the universe 511 00:30:58,400 --> 00:30:59,979 formalism reflects. 512 00:30:59,979 --> 00:31:01,770 But otherwise, it's pretty similar, really, 513 00:31:01,770 --> 00:31:03,740 to the idea of tunneling from nothing. 514 00:31:03,740 --> 00:31:06,170 The idea is that the universe had some kind of a quantum 515 00:31:06,170 --> 00:31:11,030 origin, which determined the initial state of the universe. 516 00:31:11,030 --> 00:31:18,570 In any case, for the purpose of inflation, what we really 517 00:31:18,570 --> 00:31:23,120 need to assume, and this could be an assumption which follows 518 00:31:23,120 --> 00:31:30,330 from any of these theories, we need 519 00:31:30,330 --> 00:31:40,260 to assume that the early universe contained 520 00:31:40,260 --> 00:31:47,440 at least a patch, and we don't know exactly 521 00:31:47,440 --> 00:31:49,800 know how big the patch has to be, but greater 522 00:31:49,800 --> 00:31:55,930 than or about equal to the inverse Hubble constant times 523 00:31:55,930 --> 00:31:57,820 the speed of light, the Hubble length. 524 00:32:11,530 --> 00:32:14,460 And this initial patch also has to be expanding, 525 00:32:14,460 --> 00:32:16,870 or else it would just collapse. 526 00:32:16,870 --> 00:32:18,439 It really has to be expanding faster 527 00:32:18,439 --> 00:32:19,480 than a certain threshold. 528 00:32:19,480 --> 00:32:22,280 But I won't try to put that all into the one sentence. 529 00:32:28,342 --> 00:32:29,800 Oh, I didn't say patch of what yet. 530 00:32:36,381 --> 00:32:36,880 Whoops. 531 00:32:42,630 --> 00:32:46,072 That's where the average value of phi is about zero. 532 00:32:46,072 --> 00:32:48,460 And by average, I mean averaging over rapid fluctuations, 533 00:32:48,460 --> 00:32:49,560 if there are any. 534 00:32:57,270 --> 00:32:59,820 And if one has this, no matter where one got it from, 535 00:32:59,820 --> 00:33:01,830 inflation will begin. 536 00:33:01,830 --> 00:33:03,810 And once inflation begins, it doesn't 537 00:33:03,810 --> 00:33:05,280 matter much how it begins. 538 00:33:31,080 --> 00:33:35,870 To see what happens next, it's easiest to at least pretend, 539 00:33:35,870 --> 00:33:37,760 that to a good approximation, you 540 00:33:37,760 --> 00:33:41,740 can treat a small region of this patch 541 00:33:41,740 --> 00:33:43,990 as if it were homogeneous and behaving 542 00:33:43,990 --> 00:33:48,630 like a Robertson-Walker universe of the type we know about. 543 00:33:48,630 --> 00:33:52,577 Then we can write the first order Friedman equation, 544 00:33:52,577 --> 00:33:54,160 which is a little bit more informative 545 00:33:54,160 --> 00:33:55,370 than the second order one. 546 00:34:01,521 --> 00:34:03,270 I'm going to leave out the curvature term. 547 00:34:03,270 --> 00:34:05,650 We'll argue later that the curvature term becomes small 548 00:34:05,650 --> 00:34:07,160 quickly. 549 00:34:07,160 --> 00:34:08,977 But for our first pass, we'll just 550 00:34:08,977 --> 00:34:11,060 assume that the universe is described by something 551 00:34:11,060 --> 00:34:14,860 as simple as the Friedman Robertson-Walker 552 00:34:14,860 --> 00:34:19,090 equation for a flat universe. 553 00:34:19,090 --> 00:34:22,542 For row, we're just going to put row sub f. 554 00:34:22,542 --> 00:34:24,000 We'll assume that our space is just 555 00:34:24,000 --> 00:34:27,920 dominated by this false vacuum energy density. 556 00:34:27,920 --> 00:34:31,520 And this can easily be solved. 557 00:34:31,520 --> 00:34:35,219 It just says that the first derivative divided 558 00:34:35,219 --> 00:34:37,830 by the function itself is a constant. 559 00:34:37,830 --> 00:34:39,840 Just take the square root of this equation. 560 00:34:39,840 --> 00:34:42,179 And that is an equation which just immediately get 561 00:34:42,179 --> 00:34:45,210 solved and gives you an exponential. 562 00:34:45,210 --> 00:34:53,190 So you find that for late times, you just get a of t 563 00:34:53,190 --> 00:34:59,490 behaving as a constant times an exponential of time where 564 00:34:59,490 --> 00:35:02,500 the exponential time constant, which I'm calling chi. 565 00:35:02,500 --> 00:35:06,980 Chi is just the square root of this coefficient. 566 00:35:06,980 --> 00:35:14,290 The square root of 8 pi g over 3 times row sub f. 567 00:35:20,640 --> 00:35:23,710 So clearly this is the solution to that equation. 568 00:35:23,710 --> 00:35:26,500 And for late times, it is the solution that will dominate. 569 00:35:29,615 --> 00:35:30,990 Actually it is the only solution, 570 00:35:30,990 --> 00:35:32,130 the way I've already simplified this. 571 00:35:32,130 --> 00:35:34,670 But if we started with the full system of equations, 572 00:35:34,670 --> 00:35:38,455 there'd be other solutions with different initial conditions. 573 00:35:38,455 --> 00:35:39,830 But this is always what you would 574 00:35:39,830 --> 00:35:41,540 be led to, for late times. 575 00:35:41,540 --> 00:35:46,366 The exponential expansion would dominate. 576 00:35:46,366 --> 00:35:47,990 So that's what innovation basically is. 577 00:35:47,990 --> 00:35:50,630 It's a period of exponential expansion. 578 00:35:53,740 --> 00:36:01,370 There are a few features of inflation, 579 00:36:01,370 --> 00:36:08,630 which helps to understand why it is so robust. 580 00:36:08,630 --> 00:36:10,860 That is, why no matter how it starts, 581 00:36:10,860 --> 00:36:13,060 it leads to the same result. 582 00:36:16,880 --> 00:36:22,190 So one feature of inflation I'd like to mention 583 00:36:22,190 --> 00:36:28,135 is the cosmological no-hair. 584 00:36:28,135 --> 00:36:30,010 Some people call it a theorum and some people 585 00:36:30,010 --> 00:36:30,884 call it a conjecture. 586 00:36:33,990 --> 00:36:36,390 I think the more precise statement about this theorem 587 00:36:36,390 --> 00:36:39,530 is that you can prove it as a theorem perturbatively. 588 00:36:39,530 --> 00:36:44,790 That is, if all initial deviations are small, 589 00:36:44,790 --> 00:36:47,070 you can really prove it, but people 590 00:36:47,070 --> 00:36:49,450 think it's true, even beyond perturbation theory. 591 00:36:49,450 --> 00:36:52,280 And in that case it's a conjecture. 592 00:36:52,280 --> 00:37:04,330 But it's basically the statement that if one has a system with p 593 00:37:04,330 --> 00:37:13,680 equals minus row c squared and row is greater than zero, 594 00:37:13,680 --> 00:37:17,430 if that describes the matter, then essentially 595 00:37:17,430 --> 00:37:20,160 any metric that you start with will evolve 596 00:37:20,160 --> 00:37:22,405 into this exponentially expanding flat metric. 597 00:38:17,800 --> 00:38:19,850 Any system will evolve to locally resemble 598 00:38:19,850 --> 00:38:22,690 a flat exponentially expanding space time. 599 00:38:22,690 --> 00:38:26,980 And the word locally there is needed to make it true. 600 00:38:26,980 --> 00:38:29,520 If, for example, you start with a closed universe, as just 601 00:38:29,520 --> 00:38:33,770 a simple example, which has this kind of matter filling it. 602 00:38:33,770 --> 00:38:35,547 It will start to grow exponentially. 603 00:38:35,547 --> 00:38:37,130 It will always stay a closed universe. 604 00:38:37,130 --> 00:38:38,960 It will never become literally flat. 605 00:38:38,960 --> 00:38:41,120 But as it gets bigger and bigger, any piece of it 606 00:38:41,120 --> 00:38:42,610 will look flatter and flatter. 607 00:38:42,610 --> 00:38:45,151 And it will keep getting bigger and bigger exponentially fast 608 00:38:45,151 --> 00:38:45,920 forever. 609 00:38:45,920 --> 00:38:47,570 So it will rapidly approach a space 610 00:38:47,570 --> 00:38:50,470 which looks like an exponentially expanding 611 00:38:50,470 --> 00:38:52,970 flat space. 612 00:38:52,970 --> 00:38:54,970 Now this exponentially expanding flat space time 613 00:38:54,970 --> 00:39:06,810 has a name, which is de Sitter space, named 614 00:39:06,810 --> 00:39:09,736 after a Dutch astronomer. 615 00:39:09,736 --> 00:39:11,814 It was discovered early on in the history 616 00:39:11,814 --> 00:39:12,730 of general relativity. 617 00:39:12,730 --> 00:39:16,750 1917, I think, was the date that de Sitter wrote his paper 618 00:39:16,750 --> 00:39:18,764 about de Sitter space. 619 00:39:18,764 --> 00:39:20,430 It has some very interesting properties, 620 00:39:20,430 --> 00:39:24,040 which De Sitter do not notice all of them. 621 00:39:24,040 --> 00:39:29,970 In spite of the fact that I'm describing it 622 00:39:29,970 --> 00:39:32,380 as a flat exponentially expanding space time, 623 00:39:32,380 --> 00:39:34,500 that's not the only possible description. 624 00:39:34,500 --> 00:39:36,840 It turns out that the same space time, 625 00:39:36,840 --> 00:39:40,400 by changing what you call equal time services, 626 00:39:40,400 --> 00:39:43,100 can be described as either an open or closed 627 00:39:43,100 --> 00:39:45,740 Robertson-Walker universe, completely homogeneous 628 00:39:45,740 --> 00:39:46,970 in both cases. 629 00:39:46,970 --> 00:39:48,600 So it's very weird. 630 00:39:48,600 --> 00:39:50,770 But the easy way to think of it for most purposes, 631 00:39:50,770 --> 00:39:57,060 is as this flat exponentially expanding picture. 632 00:39:57,060 --> 00:40:05,339 OK Next thing I want to point out about de Sitter spaces is 633 00:40:05,339 --> 00:40:07,255 that they have what are called event horizons. 634 00:40:25,740 --> 00:40:27,980 Now early in the course we talked about horizons 635 00:40:27,980 --> 00:40:31,040 and didn't really try to quantify the name. 636 00:40:31,040 --> 00:40:33,150 The horizons that we used to talk about 637 00:40:33,150 --> 00:40:35,884 are technically called particle horizons. 638 00:40:35,884 --> 00:40:37,300 Those are horizons that have to do 639 00:40:37,300 --> 00:40:39,350 with the past history of the universe 640 00:40:39,350 --> 00:40:41,930 and are related to the fact that since the universe has 641 00:40:41,930 --> 00:40:46,380 only a finite past history, or as a cosmological model 642 00:40:46,380 --> 00:40:49,425 at least, there's a finite distance that light could 643 00:40:49,425 --> 00:40:51,350 have traveled up until this time. 644 00:40:51,350 --> 00:40:54,980 And we cannot have any way of seeing anything that's further 645 00:40:54,980 --> 00:40:58,650 away than that maximum distance that light could have traveled. 646 00:40:58,650 --> 00:41:00,600 That's the particle horizon. 647 00:41:00,600 --> 00:41:02,430 These event horizons are different. 648 00:41:02,430 --> 00:41:04,680 They're related really, to the future of the universe, 649 00:41:04,680 --> 00:41:06,200 rather than the past. 650 00:41:06,200 --> 00:41:08,450 It's a statement that, because of the fact 651 00:41:08,450 --> 00:41:11,250 that these universes are exponentially expanding, 652 00:41:11,250 --> 00:41:16,620 if two events that happen at a particular time 653 00:41:16,620 --> 00:41:21,080 are separated from each other by more than a certain distance, 654 00:41:21,080 --> 00:41:22,650 then the light from one will never 655 00:41:22,650 --> 00:41:26,270 reach the future evolution of the other. 656 00:41:26,270 --> 00:41:33,720 And one can see that by looking at the total coordinate 657 00:41:33,720 --> 00:41:37,780 distance that light could travel between any two times. 658 00:41:37,780 --> 00:41:42,650 So I going to let delta r of t1 t2 659 00:41:42,650 --> 00:41:52,890 be equal to the coordinate distance 660 00:41:52,890 --> 00:42:04,990 that light travels from t1 to t2. 661 00:42:04,990 --> 00:42:06,850 And I'm going to assume that a of t 662 00:42:06,850 --> 00:42:08,700 is given by exactly this formula. 663 00:42:08,700 --> 00:42:10,385 And I'll write out const because I 664 00:42:10,385 --> 00:42:12,700 don't want to write it too many times. 665 00:42:12,700 --> 00:42:17,860 I could give it a one variable symbol if I wanted to. 666 00:42:17,860 --> 00:42:23,646 This delta r of t1 t2 is just the integration 667 00:42:23,646 --> 00:42:27,090 of the coordinate velocity of light 668 00:42:27,090 --> 00:42:34,650 from t1 to t2 of c divided by a of t dt. 669 00:42:34,650 --> 00:42:37,610 The coordinate velocity of light is just c divided by a of t. 670 00:42:37,610 --> 00:42:41,430 We've seen that formula before. 671 00:42:41,430 --> 00:42:46,780 And this can easily be done by putting in what a of t is. 672 00:42:46,780 --> 00:42:50,050 And we get c over the constant that 673 00:42:50,050 --> 00:42:51,510 appeared in that formula, whatever 674 00:42:51,510 --> 00:42:56,010 it is, times chi, the exponential expansion 675 00:42:56,010 --> 00:43:05,990 rate, times e to the minus chi t 1 minus e to the minus chi t 2. 676 00:43:14,197 --> 00:43:16,280 And now, the question we want to ask ourselves is, 677 00:43:16,280 --> 00:43:19,180 suppose we let this light ray travel 678 00:43:19,180 --> 00:43:21,590 for an arbitrarily long amount of time, which 679 00:43:21,590 --> 00:43:25,700 means taking t2 to infinity. 680 00:43:25,700 --> 00:43:29,890 And the important feature of this expression 681 00:43:29,890 --> 00:43:33,610 is that as t2 goes to infinity, the expression 682 00:43:33,610 --> 00:43:35,510 approaches a finite value. 683 00:43:35,510 --> 00:43:38,890 The second term just disappears. 684 00:43:38,890 --> 00:43:40,700 And you're left with the first term. 685 00:43:40,700 --> 00:43:44,380 So no matter how long you wait, anything 686 00:43:44,380 --> 00:43:46,740 that started out with a coordinate separation larger 687 00:43:46,740 --> 00:43:50,216 than that value, that asymptotic value, 688 00:43:50,216 --> 00:43:52,090 will just never be reached by the light pulse 689 00:43:52,090 --> 00:43:54,590 that you've sent. 690 00:43:54,590 --> 00:43:57,240 And that's what this event horizon is. 691 00:43:57,240 --> 00:43:59,450 And it's easy to see what it actually 692 00:43:59,450 --> 00:44:01,550 amounts to numerically. 693 00:44:01,550 --> 00:44:04,220 If you want to know how far away the object has to be now, 694 00:44:04,220 --> 00:44:07,890 in physical terms, so that its coordinate distance is larger 695 00:44:07,890 --> 00:44:12,375 than the maximum we get here, we know how to do that. 696 00:44:15,010 --> 00:44:19,740 The maximum value can be written as just the limit as t2 697 00:44:19,740 --> 00:44:25,945 goes to infinity of delta of r1 r2. 698 00:44:25,945 --> 00:44:27,820 And we have the expression for it right here. 699 00:44:27,820 --> 00:44:32,120 It's just the first piece of this answer. 700 00:44:32,120 --> 00:44:34,050 And this is the coordinate distance. 701 00:44:34,050 --> 00:44:37,160 If we want to know the present physical distance of something 702 00:44:37,160 --> 00:44:39,220 which is at that coordinate distance, 703 00:44:39,220 --> 00:44:41,800 we would just multiply it by the present scale factor. 704 00:44:41,800 --> 00:44:48,610 And present here means, t1 and t2 are the arguments here, 705 00:44:48,610 --> 00:44:53,160 and we just want to multiply by a of t1 to get the physical 706 00:44:53,160 --> 00:44:56,576 distance of an object which is at this boundary, 707 00:44:56,576 --> 00:44:58,950 the boundary of what we'll be able to receive a light ray 708 00:44:58,950 --> 00:45:00,000 from and what we won't. 709 00:45:02,780 --> 00:45:13,530 So this is the event horizon distance, physical distance, 710 00:45:13,530 --> 00:45:21,410 and it's just equal to c times chi inverse. 711 00:45:21,410 --> 00:45:23,280 When you multiply by a of t1, you 712 00:45:23,280 --> 00:45:25,285 cancel the constant of the denominator 713 00:45:25,285 --> 00:45:26,800 and you cancel the e minus chi t1. 714 00:45:26,800 --> 00:45:29,550 And you're just left with c times chi inverse. 715 00:45:29,550 --> 00:45:33,717 Which is the Hubble length. 716 00:45:33,717 --> 00:45:35,300 It's the inverse Hubble constant times 717 00:45:35,300 --> 00:45:38,110 the speed of light, which is the Hubble length. 718 00:45:38,110 --> 00:45:42,990 So anything that is further away one Hubble length, from us now, 719 00:45:42,990 --> 00:45:48,530 if that object emits a light ray, we will never receive it. 720 00:45:48,530 --> 00:45:50,800 And that's called the event horizon. 721 00:45:50,800 --> 00:45:55,010 Now the reason this is important is nothing travels 722 00:45:55,010 --> 00:45:56,640 faster than light. 723 00:45:56,640 --> 00:45:59,410 And that means that in a de Sitter space, 724 00:45:59,410 --> 00:46:03,610 everything is limited in how far it can ever get. 725 00:46:03,610 --> 00:46:05,780 And an important implication of that 726 00:46:05,780 --> 00:46:10,220 is that if, in our full space, which may not be entirely 727 00:46:10,220 --> 00:46:14,530 de Sitter space, if we have a de Sitter region, 728 00:46:14,530 --> 00:46:17,640 but junk outside that, which we don't understand, 729 00:46:17,640 --> 00:46:20,800 don't know how to predict, could be anything, 730 00:46:20,800 --> 00:46:23,860 we would still know, even without knowing what's outside, 731 00:46:23,860 --> 00:46:25,760 that whatever's outside can never 732 00:46:25,760 --> 00:46:27,950 penetrate into the de Sitter region 733 00:46:27,950 --> 00:46:32,340 by more than one event horizon, by more than one Hubble length. 734 00:46:32,340 --> 00:46:34,090 So the interior of the de Sitter region 735 00:46:34,090 --> 00:46:37,200 is protected from anything on the outside. 736 00:46:37,200 --> 00:46:39,970 And that is a rigorous theorem of general relativity, 737 00:46:39,970 --> 00:46:42,077 this protection. 738 00:46:42,077 --> 00:46:44,410 And that means that once you have a sizable region of de 739 00:46:44,410 --> 00:46:46,800 Sitter space, no matter what's going on outside, 740 00:46:46,800 --> 00:46:48,890 it's never going to disappear. 741 00:46:48,890 --> 00:46:52,170 It will always be protected by this event horizon. 742 00:47:42,864 --> 00:47:44,280 I should give you now a few sample 743 00:47:44,280 --> 00:47:45,993 numbers associated with this scenario. 744 00:47:53,279 --> 00:47:55,070 And here I have to say that we don't really 745 00:47:55,070 --> 00:47:58,990 know very accurately what are the right numbers to give here. 746 00:47:58,990 --> 00:48:04,090 So I think the word sample numbers was well chosen there. 747 00:48:04,090 --> 00:48:06,510 What we don't know is what energy scale inflation actually 748 00:48:06,510 --> 00:48:08,640 happened at in the history of universe. 749 00:48:08,640 --> 00:48:10,890 Turns out that the consequences are pretty much 750 00:48:10,890 --> 00:48:14,380 identical for most questions, or all the questions 751 00:48:14,380 --> 00:48:19,450 that they have been able so far to investigate observationally, 752 00:48:19,450 --> 00:48:23,050 regardless of what energy scale inflation happened. 753 00:48:23,050 --> 00:48:25,160 Inflation was originally invented 754 00:48:25,160 --> 00:48:27,450 in the context of Grand Unified Theories. 755 00:48:27,450 --> 00:48:30,320 And I think that's still a very plausible context 756 00:48:30,320 --> 00:48:31,950 in which inflation might have happened. 757 00:48:31,950 --> 00:48:33,449 And the sample numbers I'll give you 758 00:48:33,449 --> 00:48:36,085 will be numbers associated with Grand Unified Theories. 759 00:48:44,890 --> 00:48:48,950 And what starts the whole story is the energy scale 760 00:48:48,950 --> 00:48:52,510 of Grand Unified Theories, which is about 10 761 00:48:52,510 --> 00:48:59,680 to the 16 GeV billion electron volts. 762 00:48:59,680 --> 00:49:04,950 And this number is arrived at by measuring, at accessible energy 763 00:49:04,950 --> 00:49:08,590 with accelerators, the interaction 764 00:49:08,590 --> 00:49:10,890 strengths of the three fundamental interactions 765 00:49:10,890 --> 00:49:13,410 of the standard model of particle physics. 766 00:49:13,410 --> 00:49:15,030 The standard model of particle physics 767 00:49:15,030 --> 00:49:19,390 is based on three different gauge groups, su 3, su 2 768 00:49:19,390 --> 00:49:21,174 and u1. 769 00:49:21,174 --> 00:49:22,590 Each one of those gauge groups has 770 00:49:22,590 --> 00:49:26,119 associated with it an interaction strength. 771 00:49:26,119 --> 00:49:27,160 And they can be measured. 772 00:49:27,160 --> 00:49:30,240 And that's where we start this discussion. 773 00:49:30,240 --> 00:49:33,010 Then once you measure them at accessible energies, which 774 00:49:33,010 --> 00:49:35,490 is like 100 GeV, or something like that, 775 00:49:35,490 --> 00:49:39,050 then you can theoretically extrapolate 776 00:49:39,050 --> 00:49:41,780 to much higher energies. 777 00:49:41,780 --> 00:49:47,860 And what is found is that to good accuracy, 778 00:49:47,860 --> 00:49:52,112 the three actually meet at one point. 779 00:49:52,112 --> 00:49:53,820 And that is the underlying basis, really, 780 00:49:53,820 --> 00:49:55,660 of Grand Unified Theories. 781 00:49:55,660 --> 00:49:58,030 That's what allows the possibility that all three 782 00:49:58,030 --> 00:50:02,050 interactions are really just a manifestation of one underlying 783 00:50:02,050 --> 00:50:07,160 interaction, where the one underlying interaction is made 784 00:50:07,160 --> 00:50:10,490 to look like three interactions at lower energies 785 00:50:10,490 --> 00:50:12,550 through this process called spontaneous symmetry 786 00:50:12,550 --> 00:50:15,700 breaking, which was talked about a little bit in a lecture I 787 00:50:15,700 --> 00:50:17,880 gave the time before last, I think, in, 788 00:50:17,880 --> 00:50:21,160 probably in Scott's lecture. 789 00:50:21,160 --> 00:50:23,930 Now this meeting of the three lines 790 00:50:23,930 --> 00:50:28,010 is decent in the context of what is literally 791 00:50:28,010 --> 00:50:30,800 the standard model of particle physics. 792 00:50:30,800 --> 00:50:34,240 But if one modifies the standard model of particle physics 793 00:50:34,240 --> 00:50:38,450 by incorporating supersymmetry, a symmetry between fermions 794 00:50:38,450 --> 00:50:41,510 and bosons, and that involves adding a lot of extra particles 795 00:50:41,510 --> 00:50:43,860 because none of the particles that we know of 796 00:50:43,860 --> 00:50:45,930 make up a fermion boson pair. 797 00:50:45,930 --> 00:50:48,890 So in a supersymmetric model for every known particle, 798 00:50:48,890 --> 00:50:51,030 you introduce a new unknown particle. 799 00:50:51,030 --> 00:50:54,930 Which would be it's supersymmetric partner. 800 00:50:54,930 --> 00:50:59,440 In that minimal supersymmetric extension 801 00:50:59,440 --> 00:51:02,000 of the standard model, the meeting of the lines 802 00:51:02,000 --> 00:51:03,290 works much better. 803 00:51:03,290 --> 00:51:07,200 So it's a piece of evidence in favor of supersymmetry. 804 00:51:07,200 --> 00:51:10,460 In any case, where the lines meet 805 00:51:10,460 --> 00:51:13,370 to good approximation in either one of these two discussions, 806 00:51:13,370 --> 00:51:18,360 whether it's supersymmetric or not, is it about 10 to 16 GeV. 807 00:51:18,360 --> 00:51:22,380 So that becomes the fundamental math scale 808 00:51:22,380 --> 00:51:25,515 of the end unified theories. 809 00:51:32,240 --> 00:51:34,020 Hold on a second. 810 00:51:34,020 --> 00:51:37,156 That' what I'm looking for. 811 00:51:37,156 --> 00:51:38,530 Now once one has this mass scale, 812 00:51:38,530 --> 00:51:42,680 one can figure out an appropriate mass density. 813 00:51:42,680 --> 00:51:44,596 And that's what we're really interested in, 814 00:51:44,596 --> 00:51:45,970 what would be an appropriate mass 815 00:51:45,970 --> 00:51:49,520 density for a false vacuum in a grand unified theory. 816 00:51:49,520 --> 00:51:52,120 And one can develop that, and we really 817 00:51:52,120 --> 00:51:53,560 don't know how to do any better. 818 00:51:53,560 --> 00:51:55,476 Because as I've told you, we don't know really 819 00:51:55,476 --> 00:51:57,740 know how to calculate vacuum energies anyway. 820 00:51:57,740 --> 00:52:01,197 But as a dimensional analysis answer, 821 00:52:01,197 --> 00:52:03,780 we can get the answer because it is really uniquely determined 822 00:52:03,780 --> 00:52:05,995 by dimensional analysis up to factors. 823 00:52:08,670 --> 00:52:12,520 If one wants to make an energy density out of E gut 824 00:52:12,520 --> 00:52:17,330 plus constants of physics, the only way 825 00:52:17,330 --> 00:52:20,510 to do that is to take E gut to the fourth power 826 00:52:20,510 --> 00:52:25,280 and divide it by h bar cubed c to the fifth. 827 00:52:25,280 --> 00:52:27,840 And you can convince yourself at home 828 00:52:27,840 --> 00:52:29,610 that that gives you an energy density. 829 00:52:29,610 --> 00:52:32,680 And you could even evaluate it numerically, by mass density, 830 00:52:32,680 --> 00:52:34,870 excuse me. 831 00:52:34,870 --> 00:52:41,280 And this is about equal to 2.3 times 10 832 00:52:41,280 --> 00:52:46,050 to the 81 grams per centimeter cubed. 833 00:52:46,050 --> 00:52:48,350 So it's a fantastically high mass density, 834 00:52:48,350 --> 00:52:50,690 10 to the 81 grams per centimeter cubed. 835 00:53:05,020 --> 00:53:10,060 And if one puts this into the formula for chi, 836 00:53:10,060 --> 00:53:26,640 the exponential expansion rate, chi 837 00:53:26,640 --> 00:53:33,890 turns out to be about 2.8 times 10 to the minus 38 seconds. 838 00:53:38,750 --> 00:53:43,200 And c times chi inverse, the Hubble length, 839 00:53:43,200 --> 00:53:49,520 it turns out to be about 8 times 10 to the minus 28 centimeters. 840 00:53:53,030 --> 00:53:57,069 So all these numbers, off scale by human standards. 841 00:53:57,069 --> 00:53:58,610 And that's just a feature of the fact 842 00:53:58,610 --> 00:54:02,530 the Grand Unified Theories are off scale by human standards. 843 00:54:13,107 --> 00:54:16,516 AUDIENCE: [INAUDIBLE] 844 00:54:16,516 --> 00:54:18,790 PROFESSOR: Do I have this backwards? 845 00:54:18,790 --> 00:54:20,270 No, this is incredibly small. 846 00:54:20,270 --> 00:54:21,888 This is 10 to the minus 28. 847 00:54:21,888 --> 00:54:23,855 AUDIENCE: So then it's chi [INAUDIBLE] 848 00:54:23,855 --> 00:54:24,730 PROFESSOR: I'm sorry. 849 00:54:24,730 --> 00:54:26,740 Hold on. 850 00:54:26,740 --> 00:54:28,192 Yeah, no this-- 851 00:54:28,192 --> 00:54:29,650 AUDIENCE: Chi should be [INAUDIBLE] 852 00:54:29,650 --> 00:54:31,024 PROFESSOR: Chi inverse is a time. 853 00:54:31,024 --> 00:54:32,900 C times the time is a distance. 854 00:54:32,900 --> 00:54:34,574 So I think that's right. 855 00:54:34,574 --> 00:54:36,615 AUDIENCE: So is chi inverse 10 to the [INAUDIBLE] 856 00:54:42,500 --> 00:54:45,580 PROFESSOR: Yeah, if we're in cgs units, Chi inverse by itself 857 00:54:45,580 --> 00:54:48,300 would differ by a factor of 10 to the 10. 858 00:54:48,300 --> 00:54:57,601 So it would be 10 to the minus 38, Hm. 859 00:54:57,601 --> 00:54:58,100 Hold on. 860 00:54:58,100 --> 00:54:59,736 This must be chi inverse. 861 00:54:59,736 --> 00:55:01,085 AUDIENCE: Oh, OK. 862 00:55:01,085 --> 00:55:02,835 PROFESSOR: There is an inconsistency here. 863 00:55:02,835 --> 00:55:03,840 You are right. 864 00:55:06,440 --> 00:55:07,520 Yes, that's chi inverse. 865 00:55:07,520 --> 00:55:08,061 This is time. 866 00:55:10,910 --> 00:55:15,220 And then this just multiplies by c. 867 00:55:15,220 --> 00:55:17,920 OK so the way this scenario would work is, 868 00:55:17,920 --> 00:55:20,480 we would start with the early universe with some patch 869 00:55:20,480 --> 00:55:24,610 or order of magnitude this size. 870 00:55:24,610 --> 00:55:26,760 Which I might point out is 14 orders of magnitude 871 00:55:26,760 --> 00:55:28,970 smaller than the size of a single proton, which 872 00:55:28,970 --> 00:55:31,970 would be about 10 to the minus 13 centimeters. 873 00:55:31,970 --> 00:55:35,060 So 15 orders of magnitude, maybe. 874 00:55:35,060 --> 00:55:38,110 And then we would need enough inflation, 875 00:55:38,110 --> 00:55:49,460 so that at the end of inflation, the patch should 876 00:55:49,460 --> 00:55:56,386 be on the order of maybe one to 10 centimaters or more 877 00:55:56,386 --> 00:56:01,450 It has to be at least about this big, but could be much bigger. 878 00:56:01,450 --> 00:56:04,177 There's no problem with the being much bigger. 879 00:56:04,177 --> 00:56:06,260 Much bigger would just mean there's more inflation 880 00:56:06,260 --> 00:56:08,730 than you minimally needed. 881 00:56:08,730 --> 00:56:12,630 There's no problem with having too much inflation. 882 00:56:12,630 --> 00:56:16,020 And then it's a matter of checking 883 00:56:16,020 --> 00:56:22,200 and a calculation, which I'll tell you the answer of. 884 00:56:22,200 --> 00:56:26,340 If we want to go from some size of the end of inflation 885 00:56:26,340 --> 00:56:28,760 to the present universe-- And that's really what 886 00:56:28,760 --> 00:56:32,510 we're interested in, ultimately, getting the present universe. 887 00:56:32,510 --> 00:56:36,710 --there'd be a further coasting expansion 888 00:56:36,710 --> 00:56:39,140 from the end of inflation until now, 889 00:56:39,140 --> 00:56:43,740 which can just be calculated by using the idea that a times 890 00:56:43,740 --> 00:56:47,130 temperature, scale factor times temperature, is a constant. 891 00:56:47,130 --> 00:56:49,130 So the increase in the scale factor 892 00:56:49,130 --> 00:56:54,420 is proportional to decrease in the temperature. 893 00:56:54,420 --> 00:56:57,890 And the reheat temperature of this model-- 894 00:56:57,890 --> 00:56:59,640 Maybe I didn't describe reheating exactly, 895 00:56:59,640 --> 00:57:01,910 I'll describe it quickly in words. 896 00:57:01,910 --> 00:57:05,510 At the end of inflation, the scalar field 897 00:57:05,510 --> 00:57:07,980 is destabilized by these fluctuations 898 00:57:07,980 --> 00:57:12,060 and rolls down the hill, then oscillates about the bottom. 899 00:57:12,060 --> 00:57:14,450 And when it oscillates about the bottom, 900 00:57:14,450 --> 00:57:17,630 we need to take into account the fact that this field interacts 901 00:57:17,630 --> 00:57:19,710 with other fields. 902 00:57:19,710 --> 00:57:22,590 And it then gives its energy to the other fields, basically 903 00:57:22,590 --> 00:57:25,170 the standard model fields ultimately, 904 00:57:25,170 --> 00:57:28,790 heating them up, producing the hot soup of particles 905 00:57:28,790 --> 00:57:31,610 that we think of as the starting point for the conventional Big 906 00:57:31,610 --> 00:57:33,000 Bang Theory. 907 00:57:33,000 --> 00:57:34,580 So this reheating process at the end 908 00:57:34,580 --> 00:57:38,750 of inflation as the inflaton field 909 00:57:38,750 --> 00:57:43,770 oscillates about its minimum, reproduces the starting point 910 00:57:43,770 --> 00:57:46,587 of the conventional Big Bang Theory. 911 00:57:46,587 --> 00:57:48,045 And it produces a temperature which 912 00:57:48,045 --> 00:57:50,253 is comparable to the temperature that you started at, 913 00:57:50,253 --> 00:57:53,040 which is the temperature scale of the theory. 914 00:57:53,040 --> 00:57:55,410 So if it's Grand Unified Theory scales, 915 00:57:55,410 --> 00:58:00,120 we would reheat to a temperature of order 10 to the 16 GeV. 916 00:58:00,120 --> 00:58:02,570 And then, to ask what will be the expansion factor 917 00:58:02,570 --> 00:58:08,370 between then and now, it would be 10 to the 16 GeV times 918 00:58:08,370 --> 00:58:12,922 the Boltzmann constant times 2.7 Kelvin. 919 00:58:12,922 --> 00:58:14,630 This is the ratio of the temperature then 920 00:58:14,630 --> 00:58:18,716 to the temperature now, both expressed as energies. 921 00:58:18,716 --> 00:58:20,340 And then we might want to multiply this 922 00:58:20,340 --> 00:58:24,262 by 10 centimeters, if we say that at the end of inflation, 923 00:58:24,262 --> 00:58:26,160 the universe was 10 centimeters across. 924 00:58:32,610 --> 00:58:34,003 Size at end of inflation. 925 00:58:43,330 --> 00:58:48,810 This, I worked out at home, is about 450 times 10 926 00:58:48,810 --> 00:58:50,280 to the 9th light years. 927 00:58:55,700 --> 00:58:57,950 And we would want something like 40 times 10 928 00:58:57,950 --> 00:59:01,406 to the 9th light years to explain the present universe. 929 00:59:01,406 --> 00:59:03,820 So this is about 10 times too big. 930 00:59:03,820 --> 00:59:06,320 And that's OK. 931 00:59:06,320 --> 00:59:09,689 It means that we could get by with one centimeter 932 00:59:09,689 --> 00:59:11,480 and 10 centimeters is being a bit generous. 933 00:59:16,000 --> 00:59:18,240 So inflations would start with this tiny patch. 934 00:59:18,240 --> 00:59:20,160 At the end of inflation the patch 935 00:59:20,160 --> 00:59:23,150 would have grown to one or maybe 10 936 00:59:23,150 --> 00:59:26,300 or perhaps more, centimeters in length. 937 00:59:26,300 --> 00:59:28,820 And then by coasting up to today it 938 00:59:28,820 --> 00:59:32,560 becomes something that's larger than the region 939 00:59:32,560 --> 00:59:35,240 that we now observe. 940 00:59:35,240 --> 00:59:37,420 And that's basically how the scenario works. 941 00:59:40,140 --> 00:59:43,525 Any questions about those numbers or the general pattern 942 00:59:43,525 --> 00:59:44,900 of what we're talking about here? 943 00:59:48,121 --> 00:59:48,620 OK. 944 00:59:48,620 --> 00:59:52,270 What I want to talk about next, and this will pretty much 945 00:59:52,270 --> 00:59:54,950 be where we'll stop, although a few other things 946 00:59:54,950 --> 00:59:57,260 we might mention if we have time, 947 00:59:57,260 --> 00:59:59,180 I want to talk about how it solves the three 948 00:59:59,180 --> 01:00:01,805 cosmological problems that we have discussed 949 01:00:01,805 --> 01:00:07,120 of the conventional Big Bang model. 950 01:00:07,120 --> 01:00:10,750 And the explanations are actually quite simple. 951 01:00:10,750 --> 01:00:13,890 So we can go through them pretty quickly. 952 01:00:13,890 --> 01:00:20,410 First we had the horizons slash homogeneity problem. 953 01:00:33,450 --> 01:00:38,300 Remember that was caused, or could be stated as, the problem 954 01:00:38,300 --> 01:00:42,195 that the early universe expanded so fast 955 01:00:42,195 --> 01:00:43,820 that the different pieces of it did not 956 01:00:43,820 --> 01:00:45,710 have time to talk to each other. 957 01:00:45,710 --> 01:00:49,620 And, in particular, when the cosmic microwave background was 958 01:00:49,620 --> 01:00:52,850 released, points at opposite sides of the universe 959 01:00:52,850 --> 01:00:55,870 were separated from each other by about maybe 50 horizon 960 01:00:55,870 --> 01:00:57,594 distances, we calculated. 961 01:00:57,594 --> 01:01:00,010 And that means there's no way they could have communicated 962 01:01:00,010 --> 01:01:02,540 with each other, and therefore no way we 963 01:01:02,540 --> 01:01:04,510 could explain how they turned out 964 01:01:04,510 --> 01:01:07,930 to have the same temperature at the same time. 965 01:01:07,930 --> 01:01:09,970 In this case, what we've done is, 966 01:01:09,970 --> 01:01:12,350 we've inserted into the history of the universe 967 01:01:12,350 --> 01:01:16,910 an extra phase of evolution, the inflationary phase. 968 01:01:16,910 --> 01:01:18,510 And if we go back to the beginning 969 01:01:18,510 --> 01:01:20,390 of the inflationary phase, we see 970 01:01:20,390 --> 01:01:22,035 that that problem is just not there. 971 01:01:22,035 --> 01:01:24,700 And if it's not there, it doesn't develop later. 972 01:01:24,700 --> 01:01:26,890 At the beginning of the inflationary phase, 973 01:01:26,890 --> 01:01:30,000 by assumption, the region that we're starting to talk about 974 01:01:30,000 --> 01:01:32,570 was about horizon length in size. 975 01:01:32,570 --> 01:01:35,990 And if we had enough inflation to produce 10 centimeters out 976 01:01:35,990 --> 01:01:38,550 of that, that was 10 times more than we needed, 977 01:01:38,550 --> 01:01:40,990 it would mean that the entire observed universe would 978 01:01:40,990 --> 01:01:42,910 be coming from a region that would be only 979 01:01:42,910 --> 01:01:46,480 about a tenth of the size of this Hubble length. 980 01:01:46,480 --> 01:01:50,290 So that would therefore be well inside the horizon 981 01:01:50,290 --> 01:01:52,230 at that time. 982 01:01:52,230 --> 01:01:55,810 And that means that if you allow a little bit of leeway 983 01:01:55,810 --> 01:01:59,130 with these numbers by having a little bit of extra inflation, 984 01:01:59,130 --> 01:02:02,240 there can be plenty of time for the entire region that's 985 01:02:02,240 --> 01:02:05,900 going to become our personally observed region, 986 01:02:05,900 --> 01:02:09,250 to come to a uniform temperature by the ordinary processes 987 01:02:09,250 --> 01:02:10,990 of thermal equilibrium. 988 01:02:10,990 --> 01:02:14,940 Because they're much less than the horizon distance apart. 989 01:02:14,940 --> 01:02:18,410 And then once the uniformity is established, before inflation, 990 01:02:18,410 --> 01:02:20,440 when the region that we're talking about 991 01:02:20,440 --> 01:02:25,170 is incredibly tiny, inflation takes over and stretches 992 01:02:25,170 --> 01:02:27,400 that tiny region so that today, it's 993 01:02:27,400 --> 01:02:30,290 large enough to encompass everything that we see. 994 01:02:30,290 --> 01:02:31,790 And therefore everything that we see 995 01:02:31,790 --> 01:02:34,710 had a causally connected past and had 996 01:02:34,710 --> 01:02:38,186 time at the early stages to come to uniform temperature, which 997 01:02:38,186 --> 01:02:42,080 is then preserved as the whole thing expands. 998 01:02:42,080 --> 01:02:46,240 So that gives a very simple explanation 999 01:02:46,240 --> 01:02:49,830 for the homogeneity problem. 1000 01:02:49,830 --> 01:03:04,190 Basically before inflation the region was tiny. 1001 01:03:09,430 --> 01:03:13,770 Second on our list was the flatness problem. 1002 01:03:20,970 --> 01:03:24,390 And the basis of that problem was the calculation 1003 01:03:24,390 --> 01:03:28,440 that we did about how omega minus 1 evolves in time. 1004 01:03:28,440 --> 01:03:29,950 And we discovered that omega minus 1 1005 01:03:29,950 --> 01:03:33,680 always grows in magnitude during conventional evolution 1006 01:03:33,680 --> 01:03:35,190 of the universe. 1007 01:03:35,190 --> 01:03:41,560 And that therefore, for omega minus 1 to be small today, 1008 01:03:41,560 --> 01:03:44,885 it would have to be amazingly small in the early universe, 1009 01:03:44,885 --> 01:03:51,520 as small as 10 to the minus 18 at one second after the Big 1010 01:03:51,520 --> 01:03:54,570 Bang to be consistent with present measurements of omega 1011 01:03:54,570 --> 01:03:55,330 minus 1. 1012 01:03:58,570 --> 01:04:00,975 The key element there was this unstable equilibrium 1013 01:04:00,975 --> 01:04:04,230 and the fact omega L minus 1 always grew with time. 1014 01:04:04,230 --> 01:04:06,440 And that depended on the Friedman equations. 1015 01:04:06,440 --> 01:04:09,350 During inflation, the Friedman equations 1016 01:04:09,350 --> 01:04:11,350 in some schematic sense were the same equations, 1017 01:04:11,350 --> 01:04:13,840 but the rows that go into it are different. 1018 01:04:13,840 --> 01:04:15,840 So the equations basically are different. 1019 01:04:15,840 --> 01:04:18,040 And if we look at the key equation, the first order 1020 01:04:18,040 --> 01:04:21,600 Friedman equation, H squared equals 1021 01:04:21,600 --> 01:04:29,764 8 pi over 3 G row minus Kc squared over a squared. 1022 01:04:29,764 --> 01:04:31,180 This was the equation that we used 1023 01:04:31,180 --> 01:04:34,932 to derive this flatness problem. 1024 01:04:34,932 --> 01:04:36,390 We could see immediately, if we now 1025 01:04:36,390 --> 01:04:39,580 think about it, during the inflationary process, 1026 01:04:39,580 --> 01:04:41,750 things are completely reversed. 1027 01:04:41,750 --> 01:04:45,670 Omega is driven towards 1, and exponentially fast. 1028 01:04:45,670 --> 01:04:48,440 And the way I see that is to just ask what is this equation 1029 01:04:48,440 --> 01:04:50,260 do during inflation. 1030 01:04:50,260 --> 01:04:52,120 And during inflation, we just replace row 1031 01:04:52,120 --> 01:04:54,640 by this constant value row sub f, 1032 01:04:54,640 --> 01:04:58,230 the energy density of the false vacuum is fixed. 1033 01:04:58,230 --> 01:05:01,510 And that means that during inflation, this term is fixed. 1034 01:05:01,510 --> 01:05:04,120 This term is falling off like 1 over a squared. 1035 01:05:04,120 --> 01:05:08,120 And a is growing exponentially with time. 1036 01:05:08,120 --> 01:05:12,020 So that means that this term is decreasing 1037 01:05:12,020 --> 01:05:16,720 relative to that term by a huge factor, 1038 01:05:16,720 --> 01:05:20,010 by the square of the expansion factor. 1039 01:05:20,010 --> 01:05:23,850 So in our sample numbers over there, 1040 01:05:23,850 --> 01:05:27,660 we were talking about an overall expansion from 10 1041 01:05:27,660 --> 01:05:31,110 to the minus 27 centimeters to 10. 1042 01:05:31,110 --> 01:05:33,163 That's expansion by a factor of 10 to the 28. 1043 01:05:52,870 --> 01:05:54,910 In that case, during inflation, this term 1044 01:05:54,910 --> 01:05:58,350 decreases by a factor of 10 to the 56 1045 01:05:58,350 --> 01:06:01,190 while this term remains constant. 1046 01:06:01,190 --> 01:06:03,130 And that means that by the end inflation, 1047 01:06:03,130 --> 01:06:05,890 this is completely negligible and this equation 1048 01:06:05,890 --> 01:06:09,240 without this extra term means you have a flat universe. 1049 01:06:09,240 --> 01:06:11,140 So during inflation, the universe 1050 01:06:11,140 --> 01:06:15,450 is driven towards flatness, like one over a squared, which 1051 01:06:15,450 --> 01:06:18,360 is 1 over the square of this exponential expansion factor, 1052 01:06:18,360 --> 01:06:19,330 so very, very rapidly. 1053 01:06:46,250 --> 01:06:49,410 And finally, the third of the problems that we talked about 1054 01:06:49,410 --> 01:06:50,555 was the monopole problem. 1055 01:06:57,070 --> 01:07:01,230 We argued, originally Kibble argued, 1056 01:07:01,230 --> 01:07:03,890 that you'd expect approximately one of these monopoles 1057 01:07:03,890 --> 01:07:07,530 to form per horizon volume, just because the monopoles are 1058 01:07:07,530 --> 01:07:12,760 basically twists in the scalar fields. 1059 01:07:12,760 --> 01:07:15,760 And there's no way the scalar fields can organize themselves 1060 01:07:15,760 --> 01:07:18,360 on distances larger than the horizon distance. 1061 01:07:18,360 --> 01:07:20,146 So you'd expect something on order 1062 01:07:20,146 --> 01:07:21,270 of-- It's a crude argument. 1063 01:07:21,270 --> 01:07:23,410 --but something on the order of 1 not 1064 01:07:23,410 --> 01:07:27,710 in the scalar field per horizon volume. 1065 01:07:27,710 --> 01:07:30,030 And that led to far too many magnetic monopoles, 1066 01:07:30,030 --> 01:07:32,880 fantastically too many. 1067 01:07:32,880 --> 01:07:36,960 And the formation of one monopole per horizon volume 1068 01:07:36,960 --> 01:07:38,840 is hard to avoid. 1069 01:07:38,840 --> 01:07:40,540 I don't know of any way of avoiding it. 1070 01:07:40,540 --> 01:07:42,980 But what gets us out of the problem here, 1071 01:07:42,980 --> 01:07:47,380 is that we can easily arrange in our inflationary model 1072 01:07:47,380 --> 01:07:49,640 for the bulk of the inflation to happen 1073 01:07:49,640 --> 01:07:51,726 after the monopoles form. 1074 01:07:51,726 --> 01:07:53,100 And that means the monopoles will 1075 01:07:53,100 --> 01:07:57,230 be diluted by the exponential expansion that 1076 01:07:57,230 --> 01:08:00,260 will occur after the monopoles form. 1077 01:08:00,260 --> 01:08:01,850 The rest of the matter is not diluted, 1078 01:08:01,850 --> 01:08:03,400 because when inflation takes place 1079 01:08:03,400 --> 01:08:06,200 it's at a constant energy density 1080 01:08:06,200 --> 01:08:09,420 so the amount of other stuff that will be produced 1081 01:08:09,420 --> 01:08:12,840 is not diminished by this extra expansion. 1082 01:08:12,840 --> 01:08:14,940 But the monopolies, which will produce first, 1083 01:08:14,940 --> 01:08:18,000 will be thinned out by the expansion. 1084 01:08:18,000 --> 01:08:26,790 So the basic idea here is that the volume goes 1085 01:08:26,790 --> 01:08:34,760 by a factor of the order, using our sample numbers, 1086 01:08:34,760 --> 01:08:38,729 it's linearly growth by a factor of 10 to the 28. 1087 01:08:38,729 --> 01:08:41,680 Volumes go like cubes of linear distances. 1088 01:08:41,680 --> 01:08:45,160 So 10 to the 28 cubed is 10 to the 84, I think. 1089 01:08:49,050 --> 01:08:49,905 Probably right. 1090 01:08:53,609 --> 01:08:56,600 And that means that we can dilute these monopoles 1091 01:08:56,600 --> 01:08:59,840 by a fantastic factor and make everything work, if we just 1092 01:08:59,840 --> 01:09:01,569 arrange for the monopoles to be produced 1093 01:09:01,569 --> 01:09:03,661 before the exponential expansion sets in. 1094 01:09:07,439 --> 01:09:08,939 OK. 1095 01:09:08,939 --> 01:09:11,420 Finally, and I think this is probably 1096 01:09:11,420 --> 01:09:16,762 the last thing we will talk about, 1097 01:09:16,762 --> 01:09:18,720 another problem that we could have talked about 1098 01:09:18,720 --> 01:09:20,029 and we'll talk about the solution of it 1099 01:09:20,029 --> 01:09:22,750 now, even though we never really talked about as a problem, 1100 01:09:22,750 --> 01:09:27,234 is the small scale nonuniformities 1101 01:09:27,234 --> 01:09:27,900 of the universe. 1102 01:09:36,177 --> 01:09:37,760 And if we look out around the universe 1103 01:09:37,760 --> 01:09:39,609 we don't see a uniform mass distribution, 1104 01:09:39,609 --> 01:09:43,529 we see stars and stars collected in galaxies and galaxies 1105 01:09:43,529 --> 01:09:45,640 collected in clusters and clusters collected 1106 01:09:45,640 --> 01:09:49,819 in super clusters, a very complicated array of structure 1107 01:09:49,819 --> 01:09:51,410 in matter. 1108 01:09:51,410 --> 01:09:53,290 Those are all nonuniformities. 1109 01:09:53,290 --> 01:09:56,890 And we think we understand how they evolve from early times 1110 01:09:56,890 --> 01:09:59,880 because we also see in the cosmic microwave background 1111 01:09:59,880 --> 01:10:03,930 radiation, small fluctuations, which we can now actually 1112 01:10:03,930 --> 01:10:07,050 measure to very high degree of accuracy. 1113 01:10:07,050 --> 01:10:09,230 Those small fluctuations provide the seeds 1114 01:10:09,230 --> 01:10:12,350 for the structure in the universe that happens later 1115 01:10:12,350 --> 01:10:16,380 because of the fact that the universe is gravitationally 1116 01:10:16,380 --> 01:10:18,370 unstable. 1117 01:10:18,370 --> 01:10:21,060 So at very early times what we're seeing directly 1118 01:10:21,060 --> 01:10:23,340 in the CMB, these nonuniformities 1119 01:10:23,340 --> 01:10:26,660 were only at the level of one part in 100,000. 1120 01:10:26,660 --> 01:10:28,950 But nonetheless, in regions where 1121 01:10:28,950 --> 01:10:30,510 there was slightly more mass density, 1122 01:10:30,510 --> 01:10:32,680 that pulls in slightly more matter, 1123 01:10:32,680 --> 01:10:34,940 producing still stronger gravitational field 1124 01:10:34,940 --> 01:10:38,590 pulling in more matter and that amplifies the fluctuations. 1125 01:10:38,590 --> 01:10:41,180 And that affect, we believe, is enough to account 1126 01:10:41,180 --> 01:10:44,000 for all the structure that we see in the universe 1127 01:10:44,000 --> 01:10:47,920 as originating from these tiny ripples on the cosmic microwave 1128 01:10:47,920 --> 01:10:49,004 background. 1129 01:10:49,004 --> 01:10:50,420 But that still leaves the question 1130 01:10:50,420 --> 01:10:53,100 of where do these tiny ripples come from. 1131 01:10:53,100 --> 01:10:57,740 And in conventional cosmology, one really 1132 01:10:57,740 --> 01:10:59,810 had no idea where they come from. 1133 01:10:59,810 --> 01:11:03,220 One knew they had to be there even before they were seen 1134 01:11:03,220 --> 01:11:05,850 because we had to account for the structure in the universe 1135 01:11:05,850 --> 01:11:07,250 and how it evolved. 1136 01:11:07,250 --> 01:11:10,021 When they finally were seen, it was seen just right. 1137 01:11:10,021 --> 01:11:11,020 Everything fit together. 1138 01:11:13,950 --> 01:11:17,780 In inflationary cosmology, the exponential expansion 1139 01:11:17,780 --> 01:11:20,600 tends to smooth everything out. 1140 01:11:20,600 --> 01:11:23,220 And for a while, those of us working on it 1141 01:11:23,220 --> 01:11:26,470 were very worried that inflation would produce a perfectly 1142 01:11:26,470 --> 01:11:28,980 smooth universe and we'd have no way of accounting 1143 01:11:28,980 --> 01:11:32,320 for the small fluctuations that were needed to explain 1144 01:11:32,320 --> 01:11:35,230 the existence of stars and galaxies. 1145 01:11:35,230 --> 01:11:38,340 But then it was realized that quantum theory 1146 01:11:38,340 --> 01:11:41,210 can come to our rescue. 1147 01:11:41,210 --> 01:11:43,530 Classically, inflation would smooth everything out 1148 01:11:43,530 --> 01:11:46,560 and produce a uniform mass density everywhere. 1149 01:11:46,560 --> 01:11:49,310 But quantum mechanically, because 1150 01:11:49,310 --> 01:11:54,400 quantum mechanical theories are fundamentally probabilistic, 1151 01:11:54,400 --> 01:11:56,510 the classical prediction of a uniform density 1152 01:11:56,510 --> 01:11:59,920 turns into a quantum mechanical prediction of an almost uniform 1153 01:11:59,920 --> 01:12:03,240 density, but with some places being slightly higher 1154 01:12:03,240 --> 01:12:06,330 than that uniform density, other places being slightly lower. 1155 01:12:06,330 --> 01:12:07,960 And qualitatively, that's exactly what 1156 01:12:07,960 --> 01:12:11,396 we see in the cosmic microwave background radiation. 1157 01:12:11,396 --> 01:12:13,270 And furthermore, we can do it quantitatively. 1158 01:12:13,270 --> 01:12:15,790 One can actually calculate the effects 1159 01:12:15,790 --> 01:12:18,070 of these quantum fluctuations. 1160 01:12:18,070 --> 01:12:19,990 And that's what I want to show you now. 1161 01:12:19,990 --> 01:12:22,690 The actual data on that, which is just gorgeous. 1162 01:12:25,330 --> 01:12:28,095 Shown here is the Planck seven year data. 1163 01:12:34,510 --> 01:12:37,410 Shown here is the Planck seven year data, 1164 01:12:37,410 --> 01:12:39,590 where what's being plotted is the amplitude 1165 01:12:39,590 --> 01:12:43,600 of the fluctuations versus the angular wave length. 1166 01:12:43,600 --> 01:12:46,070 One is seeing these as a pattern on the sky. 1167 01:12:46,070 --> 01:12:49,800 So the wavelength you see as an angle, not as a distance. 1168 01:12:49,800 --> 01:12:53,060 And long wave lengths are at the left. 1169 01:12:53,060 --> 01:12:54,940 Short waive lengths are at the right. 1170 01:12:54,940 --> 01:12:56,830 It's really done as a multiple expansion, 1171 01:12:56,830 --> 01:12:57,996 if you know what that means. 1172 01:12:57,996 --> 01:13:00,700 And those numbers are showing on the top. 1173 01:13:00,700 --> 01:13:05,600 And the data points are shown as these black bars 1174 01:13:05,600 --> 01:13:07,980 with their appropriate errors. 1175 01:13:07,980 --> 01:13:10,540 And the red line is the theoretical prediction 1176 01:13:10,540 --> 01:13:13,920 due to inflation, putting in the amount of dark matter 1177 01:13:13,920 --> 01:13:16,710 that we need to fit the data that we also 1178 01:13:16,710 --> 01:13:19,140 measure from the supernovae. 1179 01:13:19,140 --> 01:13:20,712 And it's absolutely gorgeous. 1180 01:13:20,712 --> 01:13:22,420 So I have a little Eureka guy to show you 1181 01:13:22,420 --> 01:13:25,240 how happy I was when I saw this graph. 1182 01:13:25,240 --> 01:13:27,960 And with the help of Max Tegmark, 1183 01:13:27,960 --> 01:13:31,040 we've also put on this graph what 1184 01:13:31,040 --> 01:13:34,830 other theories of cosmology would give. 1185 01:13:34,830 --> 01:13:36,950 So if we had an open universe, where 1186 01:13:36,950 --> 01:13:40,390 omega was just 0.2 or 0.3, as many people believed 1187 01:13:40,390 --> 01:13:45,080 before 1998, we would have gotten this yellow line. 1188 01:13:45,080 --> 01:13:47,720 If we had inflation without dark energy, 1189 01:13:47,720 --> 01:13:51,650 making omega equal what out of matter, out of ordinary matter, 1190 01:13:51,650 --> 01:13:53,650 we would get this greenish line, which 1191 01:13:53,650 --> 01:13:55,760 also doesn't fit the data at all. 1192 01:13:55,760 --> 01:13:57,760 And there's also something called cosmic strings 1193 01:13:57,760 --> 01:13:59,165 that we haven't talked about. 1194 01:13:59,165 --> 01:14:01,890 It was for a time, thought to be a possible source 1195 01:14:01,890 --> 01:14:03,800 of the fluctuations in the universe. 1196 01:14:03,800 --> 01:14:08,447 But once this data came in, that became completely ruled out. 1197 01:14:08,447 --> 01:14:10,030 Now this is not quite the latest data. 1198 01:14:10,030 --> 01:14:12,820 The latest data come from the Planck satellite. 1199 01:14:12,820 --> 01:14:14,850 And it was released last March. 1200 01:14:14,850 --> 01:14:17,890 And I don't have that plotted on the same scale, 1201 01:14:17,890 --> 01:14:20,010 but this is the latest data which, as you see, 1202 01:14:20,010 --> 01:14:24,209 fits even more gorgeously than the data from WMAP. 1203 01:14:24,209 --> 01:14:25,750 The more accurately it gets measured, 1204 01:14:25,750 --> 01:14:29,700 the better it fits the theoretical expectations. 1205 01:14:29,700 --> 01:14:32,120 Now I should mention for truth in advertising, 1206 01:14:32,120 --> 01:14:36,400 that this data is to some extent fit to the model. 1207 01:14:36,400 --> 01:14:40,590 It's actually a six parameter fit. 1208 01:14:40,590 --> 01:14:42,820 But of those six parameters, I don't have time 1209 01:14:42,820 --> 01:14:44,930 to talk about them in detail, but four of them 1210 01:14:44,930 --> 01:14:49,420 are pretty much determined by other features. 1211 01:14:49,420 --> 01:14:50,976 Two of them are just fit to the data. 1212 01:14:50,976 --> 01:14:52,350 And one of them is something that 1213 01:14:52,350 --> 01:14:55,030 changes the shape a little bit. 1214 01:14:55,030 --> 01:14:57,900 It's the opacity of the space between us 1215 01:14:57,900 --> 01:15:00,440 and the surface of last scattering. 1216 01:15:00,440 --> 01:15:03,280 An important parameter that's fit that you should know about, 1217 01:15:03,280 --> 01:15:04,860 is the height of the curve. 1218 01:15:04,860 --> 01:15:07,110 The height of the curve can, in principle, 1219 01:15:07,110 --> 01:15:09,240 be predicted by inflation if you knew 1220 01:15:09,240 --> 01:15:11,340 the full details of this potential energy 1221 01:15:11,340 --> 01:15:13,700 function that I've erased for the scalar field. 1222 01:15:13,700 --> 01:15:14,330 But we don't. 1223 01:15:14,330 --> 01:15:15,930 We just have some qualitative idea 1224 01:15:15,930 --> 01:15:17,440 about what it might look like. 1225 01:15:17,440 --> 01:15:19,990 So the height of the curve is fit to the data. 1226 01:15:19,990 --> 01:15:23,010 But nonetheless, the location of all these peaks and everything 1227 01:15:23,010 --> 01:15:25,640 really just come out of the theory and it's just gorgeous. 1228 01:15:25,640 --> 01:15:28,070 And it works wonderfully. 1229 01:15:28,070 --> 01:15:30,410 So the bottom line is I think inflation 1230 01:15:30,410 --> 01:15:33,680 does look like a very good explanation for the very 1231 01:15:33,680 --> 01:15:34,830 early universe. 1232 01:15:34,830 --> 01:15:37,750 It's kind of bizarre since it talks about times like 10 1233 01:15:37,750 --> 01:15:41,140 to the minus 35 seconds after the Big Bang which 1234 01:15:41,140 --> 01:15:45,310 seemed like a totally incredible extrapolation from physics 1235 01:15:45,310 --> 01:15:46,410 that we know. 1236 01:15:46,410 --> 01:15:48,750 But nonetheless, marvelously, it produces 1237 01:15:48,750 --> 01:15:53,760 data that agrees fantastically with what astronomers are now 1238 01:15:53,760 --> 01:15:55,157 measuring. 1239 01:15:55,157 --> 01:15:55,990 So we'll stop there. 1240 01:15:55,990 --> 01:15:58,387 I want to thank you all for being in the class. 1241 01:15:58,387 --> 01:15:59,970 It's really been a fun class to teach. 1242 01:15:59,970 --> 01:16:02,990 I have very much enjoyed all of your questions 1243 01:16:02,990 --> 01:16:04,710 and enjoyed getting to know you and hope 1244 01:16:04,710 --> 01:16:06,630 to continue to see you around. 1245 01:16:06,630 --> 01:16:08,330 Thank you.