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 00:00:24,665 --> 00:00:27,040 PROFESSOR: OK, in that case, let's begin in our usual way 9 00:00:27,040 --> 00:00:32,450 by going through a review of last time's lecture. 10 00:00:32,450 --> 00:00:34,960 Last time, we talked really about two 11 00:00:34,960 --> 00:00:36,980 calculational problems. 12 00:00:36,980 --> 00:00:41,520 One was the calculation of the age of the universe, 13 00:00:41,520 --> 00:00:45,640 taking into account a universe model which 14 00:00:45,640 --> 00:00:51,550 has matter, radiation, vacuum energy, and curvature. 15 00:00:51,550 --> 00:00:52,870 And we got the general formula. 16 00:00:52,870 --> 00:00:56,230 And then for the same type of cosmological model, 17 00:00:56,230 --> 00:01:00,850 we also calculated how one finds the brightness 18 00:01:00,850 --> 00:01:04,860 of a distant source-- the energy flux in terms 19 00:01:04,860 --> 00:01:07,750 of the redshift of that source. 20 00:01:07,750 --> 00:01:11,750 So first, the age of the universe calculation-- 21 00:01:11,750 --> 00:01:15,310 that really just depends on the first-order Friedman equation, 22 00:01:15,310 --> 00:01:17,740 which I've rewritten here. 23 00:01:17,740 --> 00:01:20,580 We put three terms on the right hand 24 00:01:20,580 --> 00:01:24,070 side for the mass density-- a matter term, a radiation term, 25 00:01:24,070 --> 00:01:26,010 and a vacuum energy term. 26 00:01:26,010 --> 00:01:29,690 And we know-- and this is the important ingredient-- 27 00:01:29,690 --> 00:01:33,420 we know how each depend on the scale factor. 28 00:01:33,420 --> 00:01:35,420 Non-relativistic matter falls off like 1 29 00:01:35,420 --> 00:01:37,630 over the cube of the scale factor. 30 00:01:37,630 --> 00:01:40,110 Radiation falls off like 1 over the fourth power 31 00:01:40,110 --> 00:01:41,580 of the scale factor. 32 00:01:41,580 --> 00:01:45,830 And vacuum energy is just constant. 33 00:01:45,830 --> 00:01:51,360 Next step that we did was just to rewrite this equation, where 34 00:01:51,360 --> 00:01:53,260 we put in the explicit time dependence 35 00:01:53,260 --> 00:01:55,780 in the form of this x which is the ratio of a 36 00:01:55,780 --> 00:02:00,780 of t to the present value of the scale factor-- a of t 0. 37 00:02:00,780 --> 00:02:03,080 And furthermore, we expressed the matter density 38 00:02:03,080 --> 00:02:07,960 in terms of the present contribution to omega. 39 00:02:07,960 --> 00:02:13,720 And rewriting equation in that language, it takes that form. 40 00:02:13,720 --> 00:02:16,490 And then, I pulled a fast one. 41 00:02:16,490 --> 00:02:19,020 I said we could also write this last term 42 00:02:19,020 --> 00:02:20,560 to look pretty much like the others. 43 00:02:20,560 --> 00:02:24,640 It just is a constant that falls off like 1 over a squared. 44 00:02:24,640 --> 00:02:26,740 So if you define omega sub k 0, which 45 00:02:26,740 --> 00:02:29,940 is exactly what you need to make this look like that, 46 00:02:29,940 --> 00:02:34,140 and in terms of omega sub k 0, all four terms 47 00:02:34,140 --> 00:02:35,800 have the same characteristic. 48 00:02:35,800 --> 00:02:39,880 They're just a constant times a power of x. 49 00:02:39,880 --> 00:02:42,380 So this is, then, the rewriting of the Friedman equation one 50 00:02:42,380 --> 00:02:45,080 more time, just using this new definition of how we're 51 00:02:45,080 --> 00:02:47,930 going to treat the curvature of the universe. 52 00:02:47,930 --> 00:02:49,780 And simply by looking at this formula 53 00:02:49,780 --> 00:02:52,110 and applying it to x equals 1, you 54 00:02:52,110 --> 00:02:55,270 can see that that becomes then 1 is 55 00:02:55,270 --> 00:02:57,060 equal to the sum of these omegas. 56 00:02:57,060 --> 00:03:00,110 And that can be thought of as a clearer, perhaps, 57 00:03:00,110 --> 00:03:02,600 definition of what omega sub k 0 is. 58 00:03:02,600 --> 00:03:06,030 It's just 1 minus all of the other contributions to omega. 59 00:03:06,030 --> 00:03:09,974 So it's how much the actual mass density of the universe 60 00:03:09,974 --> 00:03:11,390 differs from the critical density. 61 00:03:13,980 --> 00:03:16,762 Then once we have this equation, which 62 00:03:16,762 --> 00:03:18,220 is the equation which tells us what 63 00:03:18,220 --> 00:03:23,630 x dot is as a function of x, we could just rewrite that 64 00:03:23,630 --> 00:03:25,940 by bringing dt to one side of the equation 65 00:03:25,940 --> 00:03:29,270 and dx to the other and integrating both sides. 66 00:03:29,270 --> 00:03:31,460 And that leads to our final result. 67 00:03:31,460 --> 00:03:34,890 The age of the universe is simply given by that integral. 68 00:03:34,890 --> 00:03:36,880 And this is a very neat expression 69 00:03:36,880 --> 00:03:38,690 for the age of the universe in terms 70 00:03:38,690 --> 00:03:41,930 of the present value of the Hubble expansion rate 71 00:03:41,930 --> 00:03:47,790 and each contribution to omega in terms of its present value. 72 00:03:47,790 --> 00:03:49,794 And you just plug those into this formula. 73 00:03:49,794 --> 00:03:51,960 In general, you have to do the integral numerically, 74 00:03:51,960 --> 00:03:53,960 because the integral's a little too complicated 75 00:03:53,960 --> 00:03:57,550 to have an analytic expression. 76 00:03:57,550 --> 00:03:59,740 And that will give you the age of universe 77 00:03:59,740 --> 00:04:04,180 for any model that meets this description. 78 00:04:04,180 --> 00:04:06,730 So any questions about that calculation before we go on? 79 00:04:11,470 --> 00:04:13,430 OK, very good. 80 00:04:13,430 --> 00:04:15,750 The next calculation we did last time 81 00:04:15,750 --> 00:04:20,800 was the calculation of radiation flux versus redshift. 82 00:04:20,800 --> 00:04:23,960 And this is exactly what the astronomers 83 00:04:23,960 --> 00:04:26,270 were measuring in 1998 when they concluded 84 00:04:26,270 --> 00:04:28,340 that the universe was accelerating. 85 00:04:28,340 --> 00:04:32,230 They were looking at distant supernova type 1a explosions. 86 00:04:32,230 --> 00:04:36,100 They made the assumption that all supernova type 87 00:04:36,100 --> 00:04:41,110 1a explosions have the same intrinsic power output. 88 00:04:41,110 --> 00:04:45,704 That's based roughly on observation and guesswork. 89 00:04:45,704 --> 00:04:47,370 There's not really a good theory for it, 90 00:04:47,370 --> 00:04:51,510 so it's mostly a matter of being consistent with observations. 91 00:04:51,510 --> 00:04:55,730 But then they could calculate for any given model 92 00:04:55,730 --> 00:04:59,450 in terms of these different omegas what you expect in terms 93 00:04:59,450 --> 00:05:04,190 of received radiation as a function of redshift. 94 00:05:04,190 --> 00:05:07,060 And they compared their data with the models-- 95 00:05:07,060 --> 00:05:09,010 and I'll show you that data shortly-- 96 00:05:09,010 --> 00:05:11,890 and found that the models only fit 97 00:05:11,890 --> 00:05:14,740 if one had a significant component of vacuum 98 00:05:14,740 --> 00:05:18,630 energy causing universe to accelerate. 99 00:05:18,630 --> 00:05:23,420 So to do the calculation, we need a metric for the universe. 100 00:05:23,420 --> 00:05:26,336 And I considered only the closed universe case. 101 00:05:26,336 --> 00:05:28,210 There's also the flat case and the open case, 102 00:05:28,210 --> 00:05:29,470 which are similar. 103 00:05:29,470 --> 00:05:32,560 And you'll actually be asked to do those on the homework set. 104 00:05:35,560 --> 00:05:37,390 So the metric for a closed universe 105 00:05:37,390 --> 00:05:41,320 can be written this way, where sine psi is 106 00:05:41,320 --> 00:05:44,932 the square root of k times r, to relate it 107 00:05:44,932 --> 00:05:47,140 to the other way-- the more standard way-- of writing 108 00:05:47,140 --> 00:05:49,750 the Robertson Walker metric. 109 00:05:49,750 --> 00:05:51,695 But for our purposes for this calculation, 110 00:05:51,695 --> 00:05:53,570 it's easiest to do it this way, because we're 111 00:05:53,570 --> 00:05:57,150 going to be interested in radio trajectories of photons. 112 00:05:57,150 --> 00:06:01,330 And this metric simplifies the radial direction 113 00:06:01,330 --> 00:06:02,360 as much as it can be. 114 00:06:02,360 --> 00:06:03,770 It's just d psi squared. 115 00:06:40,645 --> 00:06:42,020 Oh, it's the computer that froze. 116 00:06:53,960 --> 00:06:57,126 You never know with Windows. 117 00:06:57,126 --> 00:06:58,400 I think we're in business now. 118 00:07:00,824 --> 00:07:01,740 Back to where we were. 119 00:07:01,740 --> 00:07:03,440 We have the metric. 120 00:07:03,440 --> 00:07:05,960 Now what we want to do is imagine a light source 121 00:07:05,960 --> 00:07:07,340 being received by a detector. 122 00:07:07,340 --> 00:07:09,120 And we put the light source in the center 123 00:07:09,120 --> 00:07:10,161 of our coordinate system. 124 00:07:10,161 --> 00:07:14,020 We put the detector at some distance corresponding to psi 125 00:07:14,020 --> 00:07:17,900 equals psi sub D, where psi is our radial coordinate and psi 126 00:07:17,900 --> 00:07:21,594 sub D is the radial coordinate of the detector. 127 00:07:21,594 --> 00:07:23,510 We imagine a whole sphere with the same radius 128 00:07:23,510 --> 00:07:26,180 as the detector, because we expect 129 00:07:26,180 --> 00:07:27,980 the source to be spherically symmetric. 130 00:07:27,980 --> 00:07:30,530 And therefore, the light emitted by the source 131 00:07:30,530 --> 00:07:32,390 will be uniformly spread over that sphere. 132 00:07:32,390 --> 00:07:35,020 And that will allow us to calculate how much of it 133 00:07:35,020 --> 00:07:36,840 will hit the detector. 134 00:07:36,840 --> 00:07:38,386 The fraction hitting the detector 135 00:07:38,386 --> 00:07:40,260 will just be the area of the detector divided 136 00:07:40,260 --> 00:07:42,010 by the area of the sphere. 137 00:07:42,010 --> 00:07:44,030 The area of the detector is whatever it is. 138 00:07:44,030 --> 00:07:46,880 We call it capital A. The area of the sphere 139 00:07:46,880 --> 00:07:49,460 is 4 pi times the radius of the sphere. 140 00:07:49,460 --> 00:07:53,740 And the radius of the sphere in physical coordinates 141 00:07:53,740 --> 00:07:56,310 is the scale factor squared times the sine squared 142 00:07:56,310 --> 00:08:00,070 of psi sub D, coming from the metric. 143 00:08:00,070 --> 00:08:02,300 It's the radius that appears in the angular part that 144 00:08:02,300 --> 00:08:04,820 counts, because it's the angles that we're integrating over 145 00:08:04,820 --> 00:08:06,670 to get the area of the sphere. 146 00:08:06,670 --> 00:08:08,940 So the radius is just a tilde squared 147 00:08:08,940 --> 00:08:11,360 times sine squared is the radius squared. 148 00:08:14,360 --> 00:08:16,430 Then we also need to remember something 149 00:08:16,430 --> 00:08:19,680 we've said a number of times previously in this class, which 150 00:08:19,680 --> 00:08:22,430 is that when the photons travel from the source 151 00:08:22,430 --> 00:08:27,070 to the detector, their intensity is suppressed 152 00:08:27,070 --> 00:08:32,070 by two powers of 1 plus z, two powers of the redshift. 153 00:08:32,070 --> 00:08:34,419 And one of those factors in 1 plus z 154 00:08:34,419 --> 00:08:37,059 comes from redshifting each photon. 155 00:08:37,059 --> 00:08:38,850 The frequency of each photon is redshifted, 156 00:08:38,850 --> 00:08:42,340 and that means that the energy of each photon is redshifted-- 157 00:08:42,340 --> 00:08:44,860 goes down by a factor of 1 plus z. 158 00:08:44,860 --> 00:08:48,200 But in addition, the rate of arrival of the photons 159 00:08:48,200 --> 00:08:52,140 is essentially a clock which is also time dilated. 160 00:08:52,140 --> 00:08:57,750 So the rate of arrival of the photons as seen by the observer 161 00:08:57,750 --> 00:09:00,460 is suppressed by another factor of 1 plus z. 162 00:09:00,460 --> 00:09:05,480 So putting all that together, the received energy flux, 163 00:09:05,480 --> 00:09:08,820 which is the power received divided by the area, 164 00:09:08,820 --> 00:09:14,120 is just the power emitted by the source divided by 4 pi. 165 00:09:14,120 --> 00:09:16,990 We get this factor of 1 plus z squared, 166 00:09:16,990 --> 00:09:20,120 due to what we just discussed. 167 00:09:20,120 --> 00:09:25,270 And then the a squared of t sine squared psi sub D. 168 00:09:25,270 --> 00:09:27,290 So it's just the total power times that fraction 169 00:09:27,290 --> 00:09:32,160 that we receive times the two factors of 1 over 1 plus z. 170 00:09:32,160 --> 00:09:34,560 And this then is essentially the final answer, 171 00:09:34,560 --> 00:09:38,190 except we want to know how to evaluate a tilde squared of t 0 172 00:09:38,190 --> 00:09:40,850 and sine squared of psi sub D in terms of things 173 00:09:40,850 --> 00:09:44,030 that we more directly measure. 174 00:09:44,030 --> 00:09:46,969 So to do that, a tilde of t 0 turns out 175 00:09:46,969 --> 00:09:48,510 to be easy, because it really is just 176 00:09:48,510 --> 00:09:53,140 related by the definition of omega sub k 0 to omega sub k 0. 177 00:09:53,140 --> 00:09:54,940 So this formula is just a rewriting 178 00:09:54,940 --> 00:09:58,440 of the definition of omega sub k 0. 179 00:09:58,440 --> 00:10:01,200 To figure out what psi is, we want 180 00:10:01,200 --> 00:10:05,870 to integrate along the line of sight 181 00:10:05,870 --> 00:10:08,510 to be able to figure out the time of emission 182 00:10:08,510 --> 00:10:09,880 in terms of psi. 183 00:10:09,880 --> 00:10:11,380 And that time of emission could then 184 00:10:11,380 --> 00:10:15,200 be related to the redshift, because the redshift is just 185 00:10:15,200 --> 00:10:21,980 the ratio of the scale factors between reception and emission. 186 00:10:21,980 --> 00:10:25,890 So we look first at the metric. 187 00:10:25,890 --> 00:10:27,540 And say we're going to be looking 188 00:10:27,540 --> 00:10:31,430 at null geodesics in the radial direction. 189 00:10:31,430 --> 00:10:34,310 And null means ds squared equals 0, 190 00:10:34,310 --> 00:10:36,230 and that's minus c squared dt squared 191 00:10:36,230 --> 00:10:39,170 plus a tilde squared of t times d psi squared. 192 00:10:39,170 --> 00:10:41,520 And that implies immediately that the psi dt 193 00:10:41,520 --> 00:10:45,450 is just equal to the speed of light divided by a tilde. 194 00:10:45,450 --> 00:10:50,140 And then we can get the total increment 195 00:10:50,140 --> 00:10:54,030 in psi between the source and us by integrating 196 00:10:54,030 --> 00:10:56,150 between the time of emission-- the time 197 00:10:56,150 --> 00:10:59,420 of the source-- to the present time-- t sub 0. 198 00:10:59,420 --> 00:11:02,340 And then it's just a matter of changing variables 199 00:11:02,340 --> 00:11:04,790 to express the variable of integration. 200 00:11:04,790 --> 00:11:10,140 Instead of st, we could express it as z-- the redshift itself. 201 00:11:10,140 --> 00:11:14,230 And that brings in a factor of h, because h is a dot over a. 202 00:11:14,230 --> 00:11:17,119 And I showed the manipulations last time, 203 00:11:17,119 --> 00:11:18,410 but it brings in a factor of h. 204 00:11:18,410 --> 00:11:22,010 But we know what h is as a function of z. 205 00:11:22,010 --> 00:11:24,680 It comes from the Friedman equation. 206 00:11:24,680 --> 00:11:28,150 And that then gives us an expression for psi of z sub 207 00:11:28,150 --> 00:11:31,290 s as an integral over z. 208 00:11:31,290 --> 00:11:34,610 And writing in what h of z is and what a tilde of z 209 00:11:34,610 --> 00:11:40,180 is from that expression, the expression for psi of z 210 00:11:40,180 --> 00:11:44,290 becomes the equation that's boxed. 211 00:11:44,290 --> 00:11:46,820 Just a matter of algebraic substitutions involving 212 00:11:46,820 --> 00:11:51,970 the Friedman equation, which determines what h of z is. 213 00:11:51,970 --> 00:11:53,780 And then putting everything together, 214 00:11:53,780 --> 00:11:58,560 J is just given by this expression, where all I've done 215 00:11:58,560 --> 00:12:04,610 is to substitute the expression for a tilde of c 0. 216 00:12:04,610 --> 00:12:06,540 And sine squared psi is still here, 217 00:12:06,540 --> 00:12:09,560 but it gets evaluated according to that formula. 218 00:12:09,560 --> 00:12:14,100 And putting these together, we have a complete calculation 219 00:12:14,100 --> 00:12:18,790 of the received radiation flux as a function 220 00:12:18,790 --> 00:12:23,240 of cosmological parameters-- the omegas and the h 0-- 221 00:12:23,240 --> 00:12:28,180 and the redshift of the source. 222 00:12:28,180 --> 00:12:30,030 And that's the end of the calculation. 223 00:12:30,030 --> 00:12:32,420 And that's where we finished last time. 224 00:12:32,420 --> 00:12:34,195 So any questions about that calculation? 225 00:12:39,070 --> 00:12:40,770 OK, fine. 226 00:12:40,770 --> 00:12:45,650 In that case, moving on, the next thing I wanted to show you 227 00:12:45,650 --> 00:12:48,920 was some real data. 228 00:12:48,920 --> 00:12:52,830 So here are some real data from one of those two teams 229 00:12:52,830 --> 00:12:55,890 that made the original announcements in 1998. 230 00:12:55,890 --> 00:13:00,470 This is from the High-Z Supernova Search Team. 231 00:13:00,470 --> 00:13:03,665 And I should write some definitions on the blackboard. 232 00:13:16,820 --> 00:13:19,484 The vertical axis there is essentially brightness. 233 00:13:19,484 --> 00:13:21,650 But you wouldn't expect the astronomers to just call 234 00:13:21,650 --> 00:13:25,230 it brightness, because they like to use fancier words. 235 00:13:25,230 --> 00:13:27,680 So they write it as little m minus capital 236 00:13:27,680 --> 00:13:31,130 M-- measured in magnitudes, they put in parentheses. 237 00:13:31,130 --> 00:13:34,960 And little m minus capital M has the name, 238 00:13:34,960 --> 00:13:43,770 it's called the distance modulus, 239 00:13:43,770 --> 00:13:45,777 meaning it's a way of measuring distance. 240 00:13:45,777 --> 00:13:48,110 They think of brightness as a way of measuring distance, 241 00:13:48,110 --> 00:13:51,400 which indeed is what it's being used for. 242 00:13:51,400 --> 00:14:00,820 And it's defined as 5 times the logarithm 243 00:14:00,820 --> 00:14:10,530 base 10 of d sub L over 1 megaparsec, 244 00:14:10,530 --> 00:14:14,551 which means the luminosity distance-- I'll define this 245 00:14:14,551 --> 00:14:16,050 in more detail in a second-- d sub L 246 00:14:16,050 --> 00:14:19,010 is the luminosity distance-- distance as inferred 247 00:14:19,010 --> 00:14:21,310 from the luminosity. 248 00:14:21,310 --> 00:14:23,250 And they're measuring it in megaparsecs 249 00:14:23,250 --> 00:14:25,820 and taking the logarithm base 10. 250 00:14:25,820 --> 00:14:29,110 And then, by convention, there's an offset here of 25. 251 00:14:29,110 --> 00:14:30,630 Why not? 252 00:14:30,630 --> 00:14:33,530 So this is the definition of the distance modulus. 253 00:14:33,530 --> 00:14:39,480 And d sub L is defined by the relationship of what 254 00:14:39,480 --> 00:14:42,350 J would be in a flat Euclidean universe 255 00:14:42,350 --> 00:14:44,630 if you were receiving that luminosity. 256 00:14:44,630 --> 00:14:48,460 So J is equal to the actual power 257 00:14:48,460 --> 00:14:55,650 output of the source divided by 4 pi d sub L squared. 258 00:14:55,650 --> 00:14:58,760 This defines d sub L. So d sub L is the distance 259 00:14:58,760 --> 00:15:02,420 that that source would have to be at in a static Euclidean 260 00:15:02,420 --> 00:15:05,320 universe for you to see it with the brightness 261 00:15:05,320 --> 00:15:06,430 that you actually see. 262 00:15:12,972 --> 00:15:14,680 This, I guess, completes the definitions, 263 00:15:14,680 --> 00:15:15,929 but we can put these together. 264 00:15:18,230 --> 00:15:30,940 And m minus M is then equal to minus 5/2 265 00:15:30,940 --> 00:15:44,270 times the logarithm base 10 of 4 pi J times 266 00:15:44,270 --> 00:15:51,980 1 megaparsec squared, divided by the actual power 267 00:15:51,980 --> 00:15:59,595 output of the source, and then, of course, plus 25. 268 00:16:02,620 --> 00:16:05,433 So this relates this distance modulus 269 00:16:05,433 --> 00:16:10,100 to the energy flux and the power output of the original source. 270 00:16:21,860 --> 00:16:29,220 There's also on this slide the acronym MLCS. 271 00:16:29,220 --> 00:16:33,790 MLCS stand for multi-color light curve shape. 272 00:16:33,790 --> 00:16:38,460 And what that refers to is the High-Z Supernova Search Team 273 00:16:38,460 --> 00:16:41,170 invented a method of compensating, to some extent, 274 00:16:41,170 --> 00:16:44,920 for small variations in the actual power 275 00:16:44,920 --> 00:16:47,285 output of the supernovae type 1a. 276 00:16:47,285 --> 00:16:48,660 Instead of assuming that they all 277 00:16:48,660 --> 00:16:50,670 have exactly the same brightness, 278 00:16:50,670 --> 00:16:54,270 they discovered by looking at nearby supernovae of this type 279 00:16:54,270 --> 00:16:59,480 that there's a correlation between the absolute brightness 280 00:16:59,480 --> 00:17:03,090 of the supernovae and the shape of the light curve-- 281 00:17:03,090 --> 00:17:05,582 that is, light versus time. 282 00:17:05,582 --> 00:17:07,790 So they were careful to measure the light versus time 283 00:17:07,790 --> 00:17:09,849 for the supernovae that they used in this study. 284 00:17:09,849 --> 00:17:13,400 And they used that as a way of applying a small correction 285 00:17:13,400 --> 00:17:16,040 to what they interpreted as the intrinsic brightness 286 00:17:16,040 --> 00:17:18,869 of each supernova. 287 00:17:18,869 --> 00:17:22,599 And the results are these points. 288 00:17:22,599 --> 00:17:24,450 [INAUDIBLE] the top are the raw points, 289 00:17:24,450 --> 00:17:28,280 and three different curves for three different models. 290 00:17:28,280 --> 00:17:30,410 And they characterize the models in the same way 291 00:17:30,410 --> 00:17:33,480 we would-- in terms of different contributions to omega. 292 00:17:33,480 --> 00:17:40,680 So the top model is the cosmological constant dominated 293 00:17:40,680 --> 00:17:43,690 model, where omega sub lambda, which 294 00:17:43,690 --> 00:17:48,450 is what we've been calling omega sub vac is 0.76. 295 00:17:48,450 --> 00:17:52,620 And it's a flat model, so 0.24 for omega matter. 296 00:17:52,620 --> 00:17:54,340 And radiation is ignorable. 297 00:17:57,180 --> 00:18:04,580 They compared that with the middle model of these three, 298 00:18:04,580 --> 00:18:08,250 which was a model that had no vacuum energy, 299 00:18:08,250 --> 00:18:10,720 and omega matter of 0.2. 300 00:18:10,720 --> 00:18:13,060 That was essentially the dominant model 301 00:18:13,060 --> 00:18:15,890 at the time, the belief that the universe was open then 302 00:18:15,890 --> 00:18:21,060 had about a critical density of 1/5 or 1/4. 303 00:18:21,060 --> 00:18:22,980 And then they also compared it with a model 304 00:18:22,980 --> 00:18:27,480 where omega was 1-- entirely of matter with no vacuum energy. 305 00:18:27,480 --> 00:18:29,730 And that was this dashed curve, which 306 00:18:29,730 --> 00:18:32,380 is the lower of these three curves. 307 00:18:32,380 --> 00:18:34,050 And when the data is just plotted, 308 00:18:34,050 --> 00:18:36,710 it's a little hard to see how much difference there 309 00:18:36,710 --> 00:18:39,180 is between the three curves. 310 00:18:39,180 --> 00:18:43,940 So they re-plotted the data, plotting the middle curve 311 00:18:43,940 --> 00:18:46,150 as a straight line by construction. 312 00:18:46,150 --> 00:18:48,420 And then they plotted deviations from that line. 313 00:18:48,420 --> 00:18:51,280 And they did that for both the theoretical curves 314 00:18:51,280 --> 00:18:52,530 and the data. 315 00:18:52,530 --> 00:18:54,220 And in this magnified picture, you 316 00:18:54,220 --> 00:18:57,410 can see a little bit better that this top curve fits 317 00:18:57,410 --> 00:18:58,760 things the best. 318 00:18:58,760 --> 00:19:01,570 And that's what they call the lambda CDM model. 319 00:19:01,570 --> 00:19:05,090 It corresponds to omega m equals 0.24. 320 00:19:05,090 --> 00:19:07,740 Omega lambda equals 0.76. 321 00:19:07,740 --> 00:19:10,000 So it's the model with a cosmological constant, 322 00:19:10,000 --> 00:19:12,120 with a vacuum energy. 323 00:19:12,120 --> 00:19:17,760 And lambda CDM stands for lambda and cold, dark matter. 324 00:19:17,760 --> 00:19:19,349 And cold, dark matter is just what 325 00:19:19,349 --> 00:19:21,140 we've been calling non-relativistic matter. 326 00:19:24,060 --> 00:19:27,470 So the claim is that these data points, even though there's 327 00:19:27,470 --> 00:19:30,430 a fair amount of scatter, fit the top curve much, 328 00:19:30,430 --> 00:19:32,290 much better than they fit the middle curve 329 00:19:32,290 --> 00:19:34,650 or the bottom curve. 330 00:19:34,650 --> 00:19:37,600 And statistically, that's true. 331 00:19:37,600 --> 00:19:41,300 It really is a much better fit, even though by eye, it's 332 00:19:41,300 --> 00:19:43,290 not that clear what's going on. 333 00:19:43,290 --> 00:19:46,020 I think by eye it looks clear that the top one fits it better 334 00:19:46,020 --> 00:19:48,310 than others, but it's not that clear 335 00:19:48,310 --> 00:19:49,914 how important the difference is. 336 00:19:49,914 --> 00:19:52,330 But nonetheless, the astronomers were thoroughly convinced 337 00:19:52,330 --> 00:19:54,650 that this was a real effect. 338 00:19:54,650 --> 00:19:56,960 There was considerable discussion 339 00:19:56,960 --> 00:20:00,660 about possible systematic errors. 340 00:20:00,660 --> 00:20:05,310 And I guess next, I'll say a few words about that. 341 00:20:05,310 --> 00:20:06,720 First of all, I should maybe just 342 00:20:06,720 --> 00:20:11,690 clarify a little bit better what's being seen. 343 00:20:11,690 --> 00:20:16,600 What's being seen is that for a given redshift, this curve, 344 00:20:16,600 --> 00:20:20,270 which basically shows brightness in a funny, funny way, where 345 00:20:20,270 --> 00:20:26,780 dimmer is upward, larger values of little m minus M-- 346 00:20:26,780 --> 00:20:29,220 there's a minus sign in this formula-- 347 00:20:29,220 --> 00:20:32,095 means a dimmer galaxy, one that looks further away. 348 00:20:32,095 --> 00:20:36,660 And basically, when astronomers see this, they think distance. 349 00:20:36,660 --> 00:20:39,670 So larger values means further away. 350 00:20:39,670 --> 00:20:41,920 So what's being seen is that these distant supernovae 351 00:20:41,920 --> 00:20:44,560 are a little bit dimmer than what 352 00:20:44,560 --> 00:20:49,029 you would expect in either of the other two models, 353 00:20:49,029 --> 00:20:51,195 either of the models that do not have vacuum energy. 354 00:20:55,280 --> 00:20:57,050 And the amount by which they're dimmer 355 00:20:57,050 --> 00:20:59,580 is a few tenths of a magnitude. 356 00:20:59,580 --> 00:21:03,030 And each tenth of a magnitude corresponds to about 10% 357 00:21:03,030 --> 00:21:04,180 in brightness. 358 00:21:04,180 --> 00:21:07,530 So what they're saying is that these distant if supernovae, 359 00:21:07,530 --> 00:21:10,000 if we assume they really fit this curve, 360 00:21:10,000 --> 00:21:13,750 are 20% to 30% dimmer than you would 361 00:21:13,750 --> 00:21:17,340 have expected in other models. 362 00:21:17,340 --> 00:21:19,590 It might be worth saying a little bit about why dimmer 363 00:21:19,590 --> 00:21:22,354 is the right sign to correspond to acceleration, which 364 00:21:22,354 --> 00:21:24,270 is by no means totally obvious, I don't think. 365 00:21:33,890 --> 00:21:35,075 So we're plotting-- 366 00:21:45,646 --> 00:21:47,120 AUDIENCE: What year is this? 367 00:21:47,120 --> 00:21:48,330 PROFESSOR: What year? 368 00:21:48,330 --> 00:21:55,540 This was old data that was published in 1998. 369 00:21:55,540 --> 00:21:58,470 It has gotten better. 370 00:21:58,470 --> 00:22:02,020 Now it's much more unambiguous that this works. 371 00:22:08,120 --> 00:22:10,295 So this is distance as inferred by brightness. 372 00:22:20,970 --> 00:22:24,160 So this is basically what's being plotted. 373 00:22:24,160 --> 00:22:26,920 If one thinks about a fixed z in which way 374 00:22:26,920 --> 00:22:29,390 that you go-- up or down-- I find that totally cryptic. 375 00:22:29,390 --> 00:22:32,130 I don't really parse that very well in my own head. 376 00:22:32,130 --> 00:22:34,590 But it's much clearer if you think about the other way. 377 00:22:34,590 --> 00:22:37,330 You could think about a galaxy-- or a supernova 378 00:22:37,330 --> 00:22:40,006 in this case-- at a fixed distance, and ask, 379 00:22:40,006 --> 00:22:42,380 suppose I compare different models-- ones that accelerate 380 00:22:42,380 --> 00:22:44,660 and models that don't accelerate. 381 00:22:44,660 --> 00:22:48,070 So if we fix the distance and say, what would we 382 00:22:48,070 --> 00:22:49,710 expect for the redshift of a given 383 00:22:49,710 --> 00:22:54,690 galaxy, in an accelerating model versus a non-accelerating 384 00:22:54,690 --> 00:22:57,289 model-- remember, the redshift is basically 385 00:22:57,289 --> 00:22:59,080 a measure of the velocity, or at least it's 386 00:22:59,080 --> 00:23:02,520 strongly influenced by the velocity of the object. 387 00:23:02,520 --> 00:23:05,380 So if the universe is accelerating, 388 00:23:05,380 --> 00:23:08,870 it means that the universe was expanding slower in the past 389 00:23:08,870 --> 00:23:10,550 than you would have thought otherwise. 390 00:23:10,550 --> 00:23:14,350 It's speeded up to reach its present expansion rate. 391 00:23:14,350 --> 00:23:16,420 So an accelerating universe is a universe 392 00:23:16,420 --> 00:23:19,710 that was expanding slower in the past. 393 00:23:19,710 --> 00:23:22,360 And slower in the past means that a galaxy at a given 394 00:23:22,360 --> 00:23:24,330 distance would have been moving slower, 395 00:23:24,330 --> 00:23:27,090 and hence would have had a lower value of z. 396 00:23:27,090 --> 00:23:31,320 So the effect of acceleration for a given distance-- we'll 397 00:23:31,320 --> 00:23:35,830 fix the distance-- should be to move the line that way, 398 00:23:35,830 --> 00:23:36,835 towards lower z. 399 00:23:50,370 --> 00:23:52,960 And by moving the dot that way, it puts it above the curve. 400 00:23:52,960 --> 00:23:54,730 So it's the same as shifting things up, 401 00:23:54,730 --> 00:23:59,420 which is the more natural way of describing 402 00:23:59,420 --> 00:24:02,620 what's seen in the graph. 403 00:24:02,620 --> 00:24:04,494 The points are higher than the curve. 404 00:24:07,709 --> 00:24:10,000 So the bottom line, though, is that what they're saying 405 00:24:10,000 --> 00:24:15,160 is distant supernovae are 20%, 30% dimmer than you 406 00:24:15,160 --> 00:24:16,130 might have thought. 407 00:24:16,130 --> 00:24:18,379 And from that, they want to infer that the universe is 408 00:24:18,379 --> 00:24:21,634 accelerating, which is a rather dramatic conclusion. 409 00:24:21,634 --> 00:24:23,300 So naturally, you want to ask, are there 410 00:24:23,300 --> 00:24:26,279 other things that can cause supernovae to look dimmer? 411 00:24:26,279 --> 00:24:27,820 And of course, there are other things 412 00:24:27,820 --> 00:24:29,486 that can cause supernovae to look dimmer 413 00:24:29,486 --> 00:24:31,220 than you might have thought. 414 00:24:31,220 --> 00:24:32,620 And there are two main ideas that 415 00:24:32,620 --> 00:24:33,840 were discussed at the time. 416 00:24:57,680 --> 00:25:00,594 One of them is just plain dust. 417 00:25:00,594 --> 00:25:03,010 If you're looking at something through a dusty atmosphere, 418 00:25:03,010 --> 00:25:05,740 it looks dimmer than it would otherwise. 419 00:25:05,740 --> 00:25:08,280 And that is a genuine possibility 420 00:25:08,280 --> 00:25:11,250 that was strongly considered. 421 00:25:11,250 --> 00:25:25,470 The arguments against dust were mainly twofold. 422 00:25:28,650 --> 00:25:30,860 The first is that dust very rarely 423 00:25:30,860 --> 00:25:33,450 absorbs uniformly across the spectrum. 424 00:25:33,450 --> 00:25:37,390 Dust usually-- depending on the size of the dust grains-- 425 00:25:37,390 --> 00:25:41,210 absorbs more blue light than red light, 426 00:25:41,210 --> 00:25:43,320 leaving more red light coming through. 427 00:25:43,320 --> 00:25:45,460 So the effect of seeing something through dust 428 00:25:45,460 --> 00:25:47,690 is normally to cause it to look more red. 429 00:26:12,930 --> 00:26:14,710 And this reddening was not seen. 430 00:26:14,710 --> 00:26:17,445 The spectrum of the light from the existing supernovae 431 00:26:17,445 --> 00:26:19,150 was analyzed very carefully. 432 00:26:19,150 --> 00:26:20,790 And the spectrum of the distant ones 433 00:26:20,790 --> 00:26:23,850 looked just like the spectrum of the nearby ones-- appropriately 434 00:26:23,850 --> 00:26:27,160 redshifted, of course, but otherwise not distorted 435 00:26:27,160 --> 00:26:27,790 in any way. 436 00:26:27,790 --> 00:26:30,270 There was no sign of this reddening. 437 00:26:30,270 --> 00:26:32,970 Now, it's possible to have what the astronomers refer 438 00:26:32,970 --> 00:26:37,200 to as gray dust, which is by definition dust that absorbs 439 00:26:37,200 --> 00:26:40,780 uniformly across the spectrum of what you're looking at. 440 00:26:40,780 --> 00:26:43,080 But the grains have to be unusually large. 441 00:26:43,080 --> 00:26:45,200 And nobody was ever able to figure out 442 00:26:45,200 --> 00:26:47,520 a source for dust grains of that sort. 443 00:26:47,520 --> 00:26:50,040 So based partly on theoretical grounds 444 00:26:50,040 --> 00:26:52,790 and partly on what nobody has ever found, 445 00:26:52,790 --> 00:26:55,000 there's no evidence for dust grains 446 00:26:55,000 --> 00:26:57,510 that would possibly cause dimming 447 00:26:57,510 --> 00:26:59,715 that would look this way, that would be dimming 448 00:26:59,715 --> 00:27:01,791 that was uniform across the spectrum. 449 00:27:01,791 --> 00:27:02,290 Yes? 450 00:27:02,290 --> 00:27:03,956 AUDIENCE: How do you tell the difference 451 00:27:03,956 --> 00:27:07,406 between reddened light from dust and redshifted light? 452 00:27:07,406 --> 00:27:09,280 PROFESSOR: OK, how do you tell the difference 453 00:27:09,280 --> 00:27:13,549 between reddened light from dust and plain old redshift? 454 00:27:13,549 --> 00:27:15,840 The difference is that the plain old redshift uniformly 455 00:27:15,840 --> 00:27:17,820 shifts everything by the same factor. 456 00:27:17,820 --> 00:27:20,420 So the whole spectrum is just moved down uniformly 457 00:27:20,420 --> 00:27:21,670 towards the red. 458 00:27:21,670 --> 00:27:23,090 This reddening effect really means 459 00:27:23,090 --> 00:27:24,506 that the blue part of the spectrum 460 00:27:24,506 --> 00:27:26,310 is depressed relative to the red part. 461 00:27:26,310 --> 00:27:27,976 So the shape of the spectrum is changed. 462 00:27:36,202 --> 00:27:38,160 So one argument is that we don't see reddening, 463 00:27:38,160 --> 00:27:43,640 and we don't know any way to make dust that would be gray. 464 00:27:43,640 --> 00:27:48,280 The second argument is that if dust was a major factor, 465 00:27:48,280 --> 00:27:50,390 presumably most of the dust that would be relevant 466 00:27:50,390 --> 00:27:53,080 would be dust in the same galaxy as the supernova explosion 467 00:27:53,080 --> 00:27:55,075 itself, because there's not that much dust 468 00:27:55,075 --> 00:27:58,250 in intergalactic space. 469 00:27:58,250 --> 00:28:04,350 And if dust in the galaxy of the supernova itself were relevant, 470 00:28:04,350 --> 00:28:11,150 then-- let me draw a little picture here. 471 00:28:20,390 --> 00:28:22,950 So if dust in what's called the host 472 00:28:22,950 --> 00:28:38,730 galaxy-- the galaxy which has the supernova in it-- 473 00:28:38,730 --> 00:28:40,850 then you would have a picture where 474 00:28:40,850 --> 00:28:47,620 there would be a ball of dust filling the galaxy. 475 00:28:47,620 --> 00:28:54,090 And the supernova that you're looking at might be there, 476 00:28:54,090 --> 00:28:55,300 or it might be there. 477 00:28:55,300 --> 00:28:58,790 And let's say we're looking from over here. 478 00:28:58,790 --> 00:29:02,380 So depending on where the supernova was in the galaxy, 479 00:29:02,380 --> 00:29:06,740 we would see very different amounts of intervening dust. 480 00:29:06,740 --> 00:29:09,040 And if dust were causing this dimming, 481 00:29:09,040 --> 00:29:11,290 it would mean we would be seeing a significant scatter 482 00:29:11,290 --> 00:29:14,330 in the amount of dimming depending on where 483 00:29:14,330 --> 00:29:17,670 the supernova happened to be in its host galaxy. 484 00:29:17,670 --> 00:29:20,350 And that spread was not seen. 485 00:29:20,350 --> 00:29:23,070 The spread that one sees in that curve 486 00:29:23,070 --> 00:29:28,050 could be measured and calibrated against known uncertainties 487 00:29:28,050 --> 00:29:30,205 in the brightness of supernovae and then 488 00:29:30,205 --> 00:29:31,820 the detection apparatus. 489 00:29:31,820 --> 00:29:33,570 And the spread that was seen was just what 490 00:29:33,570 --> 00:29:36,730 you expect without any additional spread associated 491 00:29:36,730 --> 00:29:40,730 with a dusty galaxy acting as the host. 492 00:29:45,450 --> 00:30:15,760 So no evidence for the spread of brightnesses 493 00:30:15,760 --> 00:30:17,880 that would be expected from a dusty host. 494 00:30:38,240 --> 00:30:40,400 Another item that was considered-- 495 00:30:40,400 --> 00:30:43,280 these are the main arguments against dust-- 496 00:30:43,280 --> 00:30:45,300 another argument that was considered, 497 00:30:45,300 --> 00:30:48,930 another possible source of dimming, is galactic evolution. 498 00:31:02,260 --> 00:31:05,090 And there, the main effect that people worried about 499 00:31:05,090 --> 00:31:07,570 was the production of heavy chemical elements 500 00:31:07,570 --> 00:31:09,810 during the life of a galaxy. 501 00:31:09,810 --> 00:31:13,339 As you've certainly learned about from your reading-- 502 00:31:13,339 --> 00:31:15,630 I don't know if we've talked about it in class or not-- 503 00:31:15,630 --> 00:31:19,130 the early universe was essential all hydrogen and helium. 504 00:31:19,130 --> 00:31:22,170 Heavier elements were made later in stars 505 00:31:22,170 --> 00:31:24,610 that produce supernovae explosions. 506 00:31:24,610 --> 00:31:29,730 And these supernovae explosions gradually 507 00:31:29,730 --> 00:31:32,450 cause galaxies to become more and more enriched 508 00:31:32,450 --> 00:31:34,050 with heavy elements. 509 00:31:34,050 --> 00:31:37,830 And by heavy, I mean anything heavier than helium. 510 00:31:37,830 --> 00:31:40,950 And that could affect, in principle, 511 00:31:40,950 --> 00:31:44,570 the behavior of supernovae explosions. 512 00:31:44,570 --> 00:31:46,600 So the evidence against that was simply 513 00:31:46,600 --> 00:31:51,210 that every other characteristic that astronomers 514 00:31:51,210 --> 00:31:53,120 could measure of these supernovae 515 00:31:53,120 --> 00:31:57,150 in the distant galaxies looked exactly 516 00:31:57,150 --> 00:32:01,020 like what was seen for nearby galaxies. 517 00:32:01,020 --> 00:32:05,635 So no evidence for any kind of evolution was seen. 518 00:32:05,635 --> 00:32:07,010 And there are many properties you 519 00:32:07,010 --> 00:32:08,670 could measure that are independent of distance, 520 00:32:08,670 --> 00:32:11,050 like the shape of the spectrum and things like that, 521 00:32:11,050 --> 00:32:15,160 and the pattern of the light curve versus time. 522 00:32:15,160 --> 00:32:18,250 So all those characteristics that astronomers can measure 523 00:32:18,250 --> 00:32:22,030 seem to be exactly the same for the very distant supernovae 524 00:32:22,030 --> 00:32:24,260 which happened billions of years ago, 525 00:32:24,260 --> 00:32:28,530 and the more nearby ones that happened recently. 526 00:32:28,530 --> 00:32:32,000 And furthermore, among the nearby ones, 527 00:32:32,000 --> 00:32:37,722 there's a big spread of abundances of heavy chemical 528 00:32:37,722 --> 00:32:39,430 elements, just because different galaxies 529 00:32:39,430 --> 00:32:41,300 have had different histories. 530 00:32:41,300 --> 00:32:43,160 So among the nearby ones, you could 531 00:32:43,160 --> 00:32:45,290 look for is there an effect caused 532 00:32:45,290 --> 00:32:48,880 by the relative abundance of heavy elements, 533 00:32:48,880 --> 00:32:52,600 and astronomers didn't find any. 534 00:32:52,600 --> 00:32:55,270 So there was no sign that galactic evolution 535 00:32:55,270 --> 00:32:57,334 could be playing a role here, even though one 536 00:32:57,334 --> 00:32:58,500 does need to worry about it. 537 00:33:02,430 --> 00:33:13,120 So the point is that distant supernovae 1a 538 00:33:13,120 --> 00:33:14,380 look like nearby ones. 539 00:33:23,260 --> 00:33:25,130 I'll call that a in my outline. 540 00:33:25,130 --> 00:33:58,830 And b is that among the nearby 1a's, heavy element abundance 541 00:33:58,830 --> 00:34:00,195 had no perceptible effect. 542 00:34:12,469 --> 00:34:14,639 So the dominant opinion gradually shifted, 543 00:34:14,639 --> 00:34:19,889 and now I think it's almost 100% that this acceleration is real. 544 00:34:19,889 --> 00:34:22,250 The acceleration, by the way, is further 545 00:34:22,250 --> 00:34:26,110 confirmed by measurements of fluctuations 546 00:34:26,110 --> 00:34:29,300 in the cosmic background radiation measurements that 547 00:34:29,300 --> 00:34:31,300 have been done by some ground-based experiments, 548 00:34:31,300 --> 00:34:33,920 and also the satellite experiments of WMAP 549 00:34:33,920 --> 00:34:37,510 and now Planck, which measure the anisotropies-- 550 00:34:37,510 --> 00:34:40,822 the ripples-- in the cosmic background radiation. 551 00:34:40,822 --> 00:34:42,530 It's hard to see what those ripples would 552 00:34:42,530 --> 00:34:46,372 have to do with the amount of vacuum energy. 553 00:34:46,372 --> 00:34:48,580 But it does turn out-- and we'll talk more about this 554 00:34:48,580 --> 00:34:51,010 a little bit later-- that we really 555 00:34:51,010 --> 00:34:53,260 do have a detailed theory of what makes these ripples. 556 00:34:53,260 --> 00:34:55,560 We can calculate what the spectrum of those ripples 557 00:34:55,560 --> 00:34:56,889 should look like. 558 00:34:56,889 --> 00:34:58,960 And the calculations depend on parameters 559 00:34:58,960 --> 00:35:01,790 which include the amount of vacuum energy. 560 00:35:01,790 --> 00:35:03,890 And in order to make things work, 561 00:35:03,890 --> 00:35:06,520 one does have to put in essentially exactly 562 00:35:06,520 --> 00:35:10,390 the same amount of vacuum energy as has been detected 563 00:35:10,390 --> 00:35:13,200 in these supernova 1a observations. 564 00:35:13,200 --> 00:35:15,810 So everything fits together very tightly. 565 00:35:15,810 --> 00:35:19,750 And I think now, just about everybody is convinced 566 00:35:19,750 --> 00:35:22,630 that the universe really is accelerating. 567 00:35:22,630 --> 00:35:24,450 The acceleration could, in principle, 568 00:35:24,450 --> 00:35:28,010 have at least two different causes that we can talk about. 569 00:35:28,010 --> 00:35:30,590 One is vacuum energy, which is the one that I'm focusing on, 570 00:35:30,590 --> 00:35:32,940 which is the simplest explanation. 571 00:35:32,940 --> 00:35:35,680 The other possibility that is discussed in the literature 572 00:35:35,680 --> 00:35:38,950 is something called quintessence, 573 00:35:38,950 --> 00:35:40,740 which is a made-up word. 574 00:35:40,740 --> 00:35:43,830 And what it refers to is the possibility 575 00:35:43,830 --> 00:35:46,310 that the acceleration of the universe today 576 00:35:46,310 --> 00:35:48,950 could be caused by a mechanism which is really in principle 577 00:35:48,950 --> 00:35:51,640 exactly the same as what we talk about for inflation 578 00:35:51,640 --> 00:35:55,450 in the early universe and will be talking about later. 579 00:35:55,450 --> 00:35:59,540 Specifically, there could be a slowly evolving scalar field 580 00:35:59,540 --> 00:36:03,080 which is essentially uniform throughout the universe, 581 00:36:03,080 --> 00:36:06,525 and changing slowly with time so it looks like it's a constant. 582 00:36:06,525 --> 00:36:09,040 And it could be the energy density of that scalar field 583 00:36:09,040 --> 00:36:11,840 that is looking to us as if it were vacuum energy. 584 00:36:14,920 --> 00:36:17,050 But that's the minority point of view. 585 00:36:17,050 --> 00:36:19,140 And that introduces extra parameters 586 00:36:19,140 --> 00:36:24,154 that don't seem to be necessary. 587 00:36:24,154 --> 00:36:25,070 But it's up for grabs. 588 00:36:25,070 --> 00:36:25,903 Nobody really knows. 589 00:36:28,420 --> 00:36:31,090 OK, any questions about what we just talked about? 590 00:36:36,140 --> 00:36:38,970 In that case, let me go on to my next topic, which 591 00:36:38,970 --> 00:36:40,500 is I want to talk a little bit more 592 00:36:40,500 --> 00:36:44,000 about the physics of vacuum energy. 593 00:36:44,000 --> 00:36:45,810 What is it that we understand about it, 594 00:36:45,810 --> 00:36:48,830 and why is it that most physicists say 595 00:36:48,830 --> 00:36:52,520 it's the least understood issue in physics? 596 00:36:52,520 --> 00:36:54,380 We really don't understand vacuum energy, 597 00:36:54,380 --> 00:36:57,470 even though we do understand why it might be nonzero. 598 00:36:57,470 --> 00:37:01,980 Where we're totally at a loss is trying to make any sense out 599 00:37:01,980 --> 00:37:06,040 of the value of the energy density that is actually 600 00:37:06,040 --> 00:37:08,140 observed. 601 00:37:08,140 --> 00:37:14,590 So where does vacuum energy come from in a quantum field theory? 602 00:37:44,740 --> 00:37:48,390 There are basically, I would say, three contributions. 603 00:37:50,799 --> 00:37:52,590 Maybe I should say in quantum field theory. 604 00:38:05,757 --> 00:38:07,840 The other context in which this might be discussed 605 00:38:07,840 --> 00:38:10,885 would be string theory. 606 00:38:10,885 --> 00:38:13,010 I may or may not say something about string theory, 607 00:38:13,010 --> 00:38:14,960 but I won't say much. 608 00:38:14,960 --> 00:38:17,090 But in quantum field theory, there are basically, 609 00:38:17,090 --> 00:38:18,960 I think, three contributions. 610 00:38:18,960 --> 00:38:21,230 The first is the easiest to understand, 611 00:38:21,230 --> 00:38:40,100 which is quantum fluctuations in bosonic fields, 612 00:38:40,100 --> 00:38:43,050 where the best example is the photon, 613 00:38:43,050 --> 00:38:44,530 or the electromagnetic field. 614 00:38:49,530 --> 00:38:52,934 Now, in a classical vacuum, e and b-- the 615 00:38:52,934 --> 00:38:54,350 the electric and magnetic fields-- 616 00:38:54,350 --> 00:38:56,766 would just be 0, because that's the lowest possible energy 617 00:38:56,766 --> 00:38:58,510 density. 618 00:38:58,510 --> 00:39:03,032 But just as you are probably aware that 619 00:39:03,032 --> 00:39:05,490 there's an uncertainty principle in quantum mechanics which 620 00:39:05,490 --> 00:39:09,840 tells you that the momentum and position of a particle cannot 621 00:39:09,840 --> 00:39:12,630 be well-defined at the same time, 622 00:39:12,630 --> 00:39:14,970 it is also true that e and b cannot be well-defined 623 00:39:14,970 --> 00:39:16,190 at the same time. 624 00:39:16,190 --> 00:39:19,850 So the uncertainty principles applied to the field theory 625 00:39:19,850 --> 00:39:24,370 imply that e and b cannot just be 0 and stay 0. 626 00:39:24,370 --> 00:39:28,150 E and b are constantly fluctuating. 627 00:39:28,150 --> 00:39:30,480 And that means that there's energy 628 00:39:30,480 --> 00:39:33,540 associated with those fluctuations. 629 00:39:33,540 --> 00:39:38,560 And the mathematics of it is actually incredibly simple. 630 00:39:38,560 --> 00:39:41,030 If one imagines the fields inside a box, 631 00:39:41,030 --> 00:39:45,210 to be able to at least avoid the infinity of space, 632 00:39:45,210 --> 00:39:47,130 the fields inside a box could be described 633 00:39:47,130 --> 00:39:50,510 in terms of standing waves, where each standing wave is 634 00:39:50,510 --> 00:39:54,470 either a half wavelength or a full wavelength across. 635 00:39:54,470 --> 00:39:58,830 And by the way, you'll be doing a homework problem on this. 636 00:39:58,830 --> 00:40:02,070 And each standing wave has the physics 637 00:40:02,070 --> 00:40:04,260 of a harmonic oscillator. 638 00:40:04,260 --> 00:40:06,800 It oscillates sinusoidally with time, the wave. 639 00:40:06,800 --> 00:40:09,660 And when one works out the mathematics, and even 640 00:40:09,660 --> 00:40:11,570 the quantum mechanics, it's exactly the same 641 00:40:11,570 --> 00:40:13,610 as a harmonic oscillator. 642 00:40:13,610 --> 00:40:18,320 So each standing wave has a zero-point energy. 643 00:40:18,320 --> 00:40:21,490 You may know that the zero-point energy of a harmonic oscillator 644 00:40:21,490 --> 00:40:26,170 is not 0, but it's 1/2 h bar omega, or 1/2 h nu, 645 00:40:26,170 --> 00:40:29,040 depending on whether you're using nu or omega to describe 646 00:40:29,040 --> 00:40:32,250 the frequency of the oscillator. 647 00:40:32,250 --> 00:40:35,660 So each standing wave contributes 1/2 h bar omega. 648 00:40:59,270 --> 00:41:03,830 And then the problem is how many standing waves are there? 649 00:41:03,830 --> 00:41:06,270 And the answer is, there's an infinite number of them, 650 00:41:06,270 --> 00:41:10,810 because there's no limit to how short the wavelength can be. 651 00:41:10,810 --> 00:41:13,590 So there's no limit to how many ups and downs you 652 00:41:13,590 --> 00:41:15,980 can have in your standing wave from one 653 00:41:15,980 --> 00:41:18,299 end of the box to the next. 654 00:41:18,299 --> 00:41:19,715 So the answer you get is infinite. 655 00:41:19,715 --> 00:41:20,410 It diverges. 656 00:41:41,660 --> 00:41:45,610 Now, the fact that it diverges at short distances 657 00:41:45,610 --> 00:41:49,294 can be used as an excuse for getting the problem wrong. 658 00:41:49,294 --> 00:41:50,210 Obviously, it's wrong. 659 00:41:50,210 --> 00:41:52,190 The answer's not infinite. 660 00:41:52,190 --> 00:41:55,291 But we have an excuse, because we certainly 661 00:41:55,291 --> 00:41:57,290 know there are wavelengths that are short enough 662 00:41:57,290 --> 00:41:59,748 that we don't understand the physics at those length scales 663 00:41:59,748 --> 00:42:00,760 anymore. 664 00:42:00,760 --> 00:42:03,270 We're basing everything on extrapolating from wavelengths 665 00:42:03,270 --> 00:42:06,450 that we can actually measure in the laboratory. 666 00:42:06,450 --> 00:42:08,445 So one could imagine that there's 667 00:42:08,445 --> 00:42:10,820 some wavelength beyond which everything we're saying here 668 00:42:10,820 --> 00:42:13,236 is nonsense, and we don't have to keep adding up 1/2 h bar 669 00:42:13,236 --> 00:42:15,050 omega anymore, because the arguments 670 00:42:15,050 --> 00:42:18,940 that justify the 1/2 h bar omega no longer apply. 671 00:42:18,940 --> 00:42:22,500 So we can use that as a cutoff for the calculation. 672 00:42:22,500 --> 00:42:24,990 And a typical cutoff-- by typical, I 673 00:42:24,990 --> 00:42:28,820 mean typical in arguments that physicists talk about, so 674 00:42:28,820 --> 00:42:30,670 typical in physics speak. 675 00:42:38,810 --> 00:42:40,440 So a cutoff that's often invoked here 676 00:42:40,440 --> 00:42:53,960 is the Planck scale, which is the square root 677 00:42:53,960 --> 00:43:00,280 of h bar times G divided by c cubed. 678 00:43:00,280 --> 00:43:05,180 And that has units of length, and it's 679 00:43:05,180 --> 00:43:13,412 equal to about 1.6 times 10 to the minus 33 centimeters. 680 00:43:17,770 --> 00:43:19,270 And what makes the scale significant 681 00:43:19,270 --> 00:43:22,370 is it's the scale at which we expect the effects of quantum 682 00:43:22,370 --> 00:43:24,992 gravity to start to be important. 683 00:43:24,992 --> 00:43:26,450 And we know that this quantum field 684 00:43:26,450 --> 00:43:27,720 theory that we're talking about does not 685 00:43:27,720 --> 00:43:29,230 include the effects of gravity. 686 00:43:29,230 --> 00:43:30,604 And we don't really even know how 687 00:43:30,604 --> 00:43:33,735 to modify it so that it would include the effects of gravity. 688 00:43:33,735 --> 00:43:35,110 So the quantum effects of gravity 689 00:43:35,110 --> 00:43:37,580 are still something of a mystery. 690 00:43:37,580 --> 00:43:40,750 So it makes sense to cut the theory off, 691 00:43:40,750 --> 00:43:43,870 if not earlier, at least at the Planck scale. 692 00:43:43,870 --> 00:43:44,749 Yes? 693 00:43:44,749 --> 00:43:47,040 AUDIENCE: So I would imagine what we're doing, in order 694 00:43:47,040 --> 00:43:49,498 to say that we have a standing wave, we have to have a box. 695 00:43:49,498 --> 00:43:52,952 And then in order to realize the fact that the universe may 696 00:43:52,952 --> 00:43:56,680 be large, you just take the limit as the box gets large. 697 00:43:56,680 --> 00:43:59,350 But is it really OK to do that? 698 00:43:59,350 --> 00:44:01,402 I mean, to treat an infinite system 699 00:44:01,402 --> 00:44:05,700 as the limit of a finite system? 700 00:44:05,700 --> 00:44:08,305 PROFESSOR: OK, the question is-- what 701 00:44:08,305 --> 00:44:09,680 we're going to be doing here is I 702 00:44:09,680 --> 00:44:11,610 talked about putting the standing waves in a box. 703 00:44:11,610 --> 00:44:13,110 And then at the end, we're going to take the limit 704 00:44:13,110 --> 00:44:14,530 as the box gets bigger and bigger. 705 00:44:14,530 --> 00:44:16,810 And the question is, is that really a valid way 706 00:44:16,810 --> 00:44:20,920 of treating the infinite space? 707 00:44:20,920 --> 00:44:24,170 And the answer is, in this case, it is. 708 00:44:24,170 --> 00:44:30,497 I'm not sure how solid an argument I can make. 709 00:44:30,497 --> 00:44:32,580 Certainly what one does find is what you'd expect, 710 00:44:32,580 --> 00:44:34,460 that as you make the box bigger and bigger, 711 00:44:34,460 --> 00:44:36,610 the energy that you get is proportional to the size 712 00:44:36,610 --> 00:44:37,630 of the box. 713 00:44:37,630 --> 00:44:40,830 So you're calculating an energy density. 714 00:44:40,830 --> 00:44:48,690 And probably the most precise thing I can say at the moment 715 00:44:48,690 --> 00:44:52,160 is that if it were not true, if the answer you got really 716 00:44:52,160 --> 00:44:56,070 depended on the way in which the space was infinite, 717 00:44:56,070 --> 00:44:59,610 then you'd be learning something about the infinite universe 718 00:44:59,610 --> 00:45:01,990 by doing an experiment in the lab, which 719 00:45:01,990 --> 00:45:03,980 is a little far-fetched. 720 00:45:03,980 --> 00:45:05,760 That is, if you do an experiment in a lab, 721 00:45:05,760 --> 00:45:06,780 it really doesn't tell you anything 722 00:45:06,780 --> 00:45:08,946 about whether the universe is infinite or turns back 723 00:45:08,946 --> 00:45:10,940 on itself and is closed. 724 00:45:10,940 --> 00:45:12,679 And calculations certainly do show 725 00:45:12,679 --> 00:45:14,220 that you get the same-- you could do, 726 00:45:14,220 --> 00:45:16,820 for example, a closed universe without a box. 727 00:45:16,820 --> 00:45:18,470 And you get the same energy density, 728 00:45:18,470 --> 00:45:22,420 as long as the universe was big, as we're getting this way. 729 00:45:22,420 --> 00:45:26,630 So I think there's a pretty solid calculational evidence 730 00:45:26,630 --> 00:45:28,940 that what you get does not depend on the box. 731 00:45:32,164 --> 00:45:33,193 Yes? 732 00:45:33,193 --> 00:45:34,609 AUDIENCE: Going off that question, 733 00:45:34,609 --> 00:45:37,054 do we use the maximum size of our box 734 00:45:37,054 --> 00:45:40,390 as the size of our observable universe, then? 735 00:45:40,390 --> 00:45:42,140 PROFESSOR: OK, the question is, what do we 736 00:45:42,140 --> 00:45:43,795 use as the maximum size of the box? 737 00:45:43,795 --> 00:45:46,045 Is it the size of the observable universe? 738 00:45:46,045 --> 00:45:47,670 The answer really is that what you find 739 00:45:47,670 --> 00:45:49,836 is that you get an energy density that's independent 740 00:45:49,836 --> 00:45:52,620 of the size of the box, as long as the box is big. 741 00:45:52,620 --> 00:45:55,680 And it's that energy density that we're looking for. 742 00:45:55,680 --> 00:45:58,240 We don't claim to know anything about the total energy. 743 00:46:01,102 --> 00:46:02,810 And we don't really need to know anything 744 00:46:02,810 --> 00:46:03,680 about the total energy. 745 00:46:03,680 --> 00:46:05,720 Everything that we formulated here in terms of energy 746 00:46:05,720 --> 00:46:06,219 densities. 747 00:46:13,260 --> 00:46:17,645 Now, the catch is, that if one puts in this cutoff 748 00:46:17,645 --> 00:46:20,920 and takes into account only the energies of 1/2 h bar 749 00:46:20,920 --> 00:46:25,160 omega going up to this cutoff and stopping there-- or down 750 00:46:25,160 --> 00:46:29,392 to the cutoff if one's thinking of length as the measure-- 751 00:46:29,392 --> 00:46:31,350 you could then ask, do we get an energy density 752 00:46:31,350 --> 00:46:34,710 that's in any way close to what the astronomers tell us 753 00:46:34,710 --> 00:46:37,350 the vacuum energy actually is? 754 00:46:37,350 --> 00:46:41,130 And the answer is emphatically no. 755 00:46:41,130 --> 00:46:42,570 We don't get anything close. 756 00:46:42,570 --> 00:46:44,990 We're in fact off by about 120 orders of magnitude, 757 00:46:44,990 --> 00:46:49,620 which even in cosmology is a significant embarrassment, 758 00:46:49,620 --> 00:46:54,090 which is why physicists consider this question of the vacuum 759 00:46:54,090 --> 00:46:56,460 energy density to be such an incredible mystery. 760 00:46:56,460 --> 00:46:59,200 We really have no idea how to get 761 00:46:59,200 --> 00:47:01,875 a number as small as what we observe. 762 00:47:04,760 --> 00:47:07,107 Let us go on to talk about other contributions, 763 00:47:07,107 --> 00:47:09,190 because they are certainly important in the way we 764 00:47:09,190 --> 00:47:10,005 think about things. 765 00:47:37,360 --> 00:47:39,130 So far I have number one, right? 766 00:47:39,130 --> 00:47:41,220 So next comes two. 767 00:47:41,220 --> 00:48:02,150 And that is the quantum fluctuations of Fermi fields, 768 00:48:02,150 --> 00:48:05,431 where the best-known example here is the electron. 769 00:48:05,431 --> 00:48:07,430 Now, in quantum field theory, I should point out 770 00:48:07,430 --> 00:48:10,855 that all particles are described by fields, not just the photon. 771 00:48:10,855 --> 00:48:13,090 The electron is described by a field also. 772 00:48:13,090 --> 00:48:16,180 It's called the electron field. 773 00:48:16,180 --> 00:48:19,830 And because the electron is a fermion and not a boson, 774 00:48:19,830 --> 00:48:22,130 the electron field has somewhat different properties 775 00:48:22,130 --> 00:48:25,410 than bosonic fields, reflecting the fact 776 00:48:25,410 --> 00:48:29,490 that the fermions themselves obey the exclusion principle. 777 00:48:29,490 --> 00:48:31,160 It turns out that for fermions, there 778 00:48:31,160 --> 00:48:34,170 are also quantum fluctuations. 779 00:48:34,170 --> 00:48:35,791 They're also of order 1/2 h bar omega. 780 00:48:35,791 --> 00:48:37,540 But actually, it's a little bit different. 781 00:48:37,540 --> 00:48:41,130 They're in some sense h bar omega and not 1/2. 782 00:48:41,130 --> 00:48:43,810 But what's peculiar is that for electrons, the contribution 783 00:48:43,810 --> 00:48:44,435 is negative. 784 00:48:58,010 --> 00:49:03,250 And the origin of this negativity I think 785 00:49:03,250 --> 00:49:05,170 has a fairly simple explanations, 786 00:49:05,170 --> 00:49:09,710 although the explanation is not ever given, actually. 787 00:49:09,710 --> 00:49:13,240 The exploration that's used in quantum field theory books 788 00:49:13,240 --> 00:49:15,659 involves looking at equation 47 and seeing 789 00:49:15,659 --> 00:49:17,200 that there's an anticommutator there. 790 00:49:17,200 --> 00:49:18,510 And because the fields anticommute, 791 00:49:18,510 --> 00:49:19,384 there's a minus sign. 792 00:49:19,384 --> 00:49:21,440 And that means the energy is negative. 793 00:49:21,440 --> 00:49:25,640 And that is basically the way it's described in textbooks. 794 00:49:25,640 --> 00:49:28,080 That certainly says where the minus sign 795 00:49:28,080 --> 00:49:29,280 appears in which equation. 796 00:49:29,280 --> 00:49:31,446 But I don't think it's really an explanation of what 797 00:49:31,446 --> 00:49:33,120 the minus sign is talking about. 798 00:49:33,120 --> 00:49:35,328 But I think there is an explanation of what the minus 799 00:49:35,328 --> 00:49:37,990 sign is talking about, which goes back to the old picture 800 00:49:37,990 --> 00:49:40,480 that Dirac himself introduced when he first 801 00:49:40,480 --> 00:49:42,780 invented the Dirac equation. 802 00:49:42,780 --> 00:49:45,180 When Dirac first invented the Dirac equation, 803 00:49:45,180 --> 00:49:46,940 he was trying to interpret it more or less 804 00:49:46,940 --> 00:49:50,360 in the same language as the Schrodinger equation. 805 00:49:50,360 --> 00:49:53,240 We don't quite do that anymore. 806 00:49:53,240 --> 00:49:56,340 But in doing that, Dirac discovered 807 00:49:56,340 --> 00:49:58,760 that his Dirac equation, which was 808 00:49:58,760 --> 00:50:01,020 the natural relativistic generalization 809 00:50:01,020 --> 00:50:04,280 of the Schrodinger equation to a particle which 810 00:50:04,280 --> 00:50:10,004 has spin 1/2, which I'm not sure how Dirac knew it had spin 1/2, 811 00:50:10,004 --> 00:50:11,420 but in any case, it's the equation 812 00:50:11,420 --> 00:50:13,690 for a particle of spin 1/2. 813 00:50:13,690 --> 00:50:16,940 And what he found was that if you just look at the energy 814 00:50:16,940 --> 00:50:19,720 the spectrum that the equation itself gives you, 815 00:50:19,720 --> 00:50:21,620 it's symmetric about 0. 816 00:50:21,620 --> 00:50:26,560 So if we plot energy going this way, if there's a state here, 817 00:50:26,560 --> 00:50:29,299 there's also a state there at negative energy. 818 00:50:29,299 --> 00:50:30,840 And if there's a state there, there's 819 00:50:30,840 --> 00:50:33,460 another state there, exactly opposite it. 820 00:50:33,460 --> 00:50:37,620 It's completely symmetric up and down. 821 00:50:37,620 --> 00:50:40,040 Now, the interpretation that Dirac gave to that 822 00:50:40,040 --> 00:50:44,030 was not that there are a lot of ways of making negative energy. 823 00:50:44,030 --> 00:50:46,570 He realized that the vacuum is by definition 824 00:50:46,570 --> 00:50:49,500 the state of lowest possible energy. 825 00:50:49,500 --> 00:50:52,950 And if you can lower the energy by adding a particle 826 00:50:52,950 --> 00:50:55,487 to these negative energy states, that 827 00:50:55,487 --> 00:50:57,820 would mean that there'd be a way of lowering the energy, 828 00:50:57,820 --> 00:51:00,350 and the state would not be the vacuum. 829 00:51:00,350 --> 00:51:04,860 So the vacuum, Dirac proposed, is the state 830 00:51:04,860 --> 00:51:08,105 in which all of these negative energy levels are filled. 831 00:51:19,630 --> 00:51:23,570 And the action of putting all these x's on the picture 832 00:51:23,570 --> 00:51:25,980 is often called filling the Dirac sea. 833 00:51:38,790 --> 00:51:43,340 S-E-A-- sea, where sea refers to this ocean of negative energy 834 00:51:43,340 --> 00:51:45,710 states, which is infinite. 835 00:51:45,710 --> 00:51:47,810 It just keeps going down. 836 00:51:47,810 --> 00:51:51,140 You can imagine filling all of them to describe the vacuum. 837 00:51:51,140 --> 00:51:53,140 Then if you ask what is the physics after you've 838 00:51:53,140 --> 00:51:54,930 done that-- what are the possible excitations 839 00:51:54,930 --> 00:51:56,721 of the vacuum, what states does this theory 840 00:51:56,721 --> 00:51:58,240 contain other than the vacuum? 841 00:51:58,240 --> 00:52:01,930 And the answer is that there could be occupations 842 00:52:01,930 --> 00:52:06,020 of these positive energy states, and those are called electrons. 843 00:52:06,020 --> 00:52:08,652 It's also possible to remove-- if you 844 00:52:08,652 --> 00:52:10,110 put in the right amount of energy-- 845 00:52:10,110 --> 00:52:12,420 one of the negative energy states, which is filled, 846 00:52:12,420 --> 00:52:16,040 but we could take away the particle that's there. 847 00:52:16,040 --> 00:52:18,144 And the absence of a particle there-- a hole 848 00:52:18,144 --> 00:52:19,935 in the negative energy sea-- is a positron. 849 00:52:30,247 --> 00:52:31,205 So electrons are there. 850 00:52:34,630 --> 00:52:42,250 The e plus is a hole in the Dirac sea. 851 00:52:46,517 --> 00:52:48,100 Now, the difficulty with this picture, 852 00:52:48,100 --> 00:52:51,184 and the reason why it's not often use these days, 853 00:52:51,184 --> 00:52:52,725 is that it makes it look like there's 854 00:52:52,725 --> 00:52:56,450 an intrinsic difference between electrons and positrons. 855 00:52:56,450 --> 00:52:58,080 Nonetheless, Dirac was perfectly aware 856 00:52:58,080 --> 00:52:59,580 that when you went through the math, 857 00:52:59,580 --> 00:53:00,895 they were completely symmetric. 858 00:53:00,895 --> 00:53:02,520 The fact that you described it this way 859 00:53:02,520 --> 00:53:05,070 is just really a feature of your description, 860 00:53:05,070 --> 00:53:07,710 but it doesn't make any measurable difference. 861 00:53:07,710 --> 00:53:11,220 So a positron really is just a perfect image of an electron, 862 00:53:11,220 --> 00:53:13,340 but with the opposite charge, with otherwise 863 00:53:13,340 --> 00:53:15,656 all the same physical properties. 864 00:53:15,656 --> 00:53:17,030 And there's ways of describing it 865 00:53:17,030 --> 00:53:19,570 where you don't make this distinction between particles 866 00:53:19,570 --> 00:53:20,550 and holes. 867 00:53:20,550 --> 00:53:22,550 But the particle hole way is I think the easiest 868 00:53:22,550 --> 00:53:25,190 way of understanding where the negative energy is coming from. 869 00:53:25,190 --> 00:53:27,820 The negative energy came by saying that the energy was 0 870 00:53:27,820 --> 00:53:30,190 before we filled any of these levels. 871 00:53:30,190 --> 00:53:32,710 And as you fill the negative energy sea, 872 00:53:32,710 --> 00:53:34,580 you're lowering energy all the time. 873 00:53:34,580 --> 00:53:37,460 And it's that contribution which makes up 874 00:53:37,460 --> 00:53:41,100 the infinite negative contribution coming 875 00:53:41,100 --> 00:53:43,190 from the Fermi fields. 876 00:53:43,190 --> 00:53:45,230 And the algebra is certainly exactly right. 877 00:53:45,230 --> 00:53:47,100 The energy that people write down 878 00:53:47,100 --> 00:53:49,170 for the negative energy of the Fermi fields-- 879 00:53:49,170 --> 00:53:53,870 what they get by anticommuting two operators in equation 37-- 880 00:53:53,870 --> 00:53:55,550 is exactly the expression you get 881 00:53:55,550 --> 00:53:59,300 for what it takes to fill the Dirac sea. 882 00:53:59,300 --> 00:53:59,800 Yes? 883 00:53:59,800 --> 00:54:01,288 AUDIENCE: Are we pretty confident 884 00:54:01,288 --> 00:54:03,215 that the smallness of the vacuum energy 885 00:54:03,215 --> 00:54:05,256 can't come from cancellations between the bosonic 886 00:54:05,256 --> 00:54:06,512 and the Fermi fields? 887 00:54:06,512 --> 00:54:08,720 PROFESSOR: OK, the question is, are we confident that 888 00:54:08,720 --> 00:54:11,450 the cancellation cannot come from the cancellations between 889 00:54:11,450 --> 00:54:13,810 the Fermi fields and the bosonic fields. 890 00:54:13,810 --> 00:54:16,850 No, we're all confident that it cannot come from that. 891 00:54:16,850 --> 00:54:18,900 it very likely does come from that. 892 00:54:18,900 --> 00:54:22,110 But we are confident that we have no idea why that happens. 893 00:54:22,110 --> 00:54:24,150 And therefore, it's a big mystery. 894 00:54:24,150 --> 00:54:26,090 Certainly our ignorance allows for any answer. 895 00:54:26,090 --> 00:54:28,131 Because we have a positive infinite contribution, 896 00:54:28,131 --> 00:54:30,340 we're just going to cut off and make it large. 897 00:54:30,340 --> 00:54:32,872 And we're going to have a negative contribution, which 898 00:54:32,872 --> 00:54:35,080 we're going to cut off and make it large in magnitude 899 00:54:35,080 --> 00:54:35,540 but negative. 900 00:54:35,540 --> 00:54:36,520 And then we're going to add them, 901 00:54:36,520 --> 00:54:38,354 and we have no idea what we're going to get. 902 00:54:38,354 --> 00:54:39,770 But the fact that we get something 903 00:54:39,770 --> 00:54:42,910 that gets incredibly close to 0, and not something that's 904 00:54:42,910 --> 00:54:46,330 at all the same magnitude as the pieces you're adding together-- 905 00:54:46,330 --> 00:54:48,520 the positive piece or the negative piece-- 906 00:54:48,520 --> 00:54:51,550 means there's something going on that we don't understand. 907 00:54:51,550 --> 00:54:53,800 There's a cancellation that's happening that we cannot 908 00:54:53,800 --> 00:54:55,170 explain. 909 00:54:55,170 --> 00:54:56,670 Now, I should maybe add that there's 910 00:54:56,670 --> 00:54:59,540 one context where we would expect a cancellation. 911 00:54:59,540 --> 00:55:01,210 And that is, there are theories that 912 00:55:01,210 --> 00:55:03,150 are what are called supersymmetric, 913 00:55:03,150 --> 00:55:06,740 which have a perfect symmetry between bosons and fermions, 914 00:55:06,740 --> 00:55:11,780 which would relate the positive energy from the photon 915 00:55:11,780 --> 00:55:15,280 to the infinite negative energy you would get from particles 916 00:55:15,280 --> 00:55:18,350 called photinos, which would be the supersymmetric partner 917 00:55:18,350 --> 00:55:20,280 of the photons-- a spin 1/2 particle 918 00:55:20,280 --> 00:55:23,210 that's a mirror image of a photon 919 00:55:23,210 --> 00:55:26,390 but has a fermionic character. 920 00:55:26,390 --> 00:55:28,340 So in an exactly supersymmetric theory, 921 00:55:28,340 --> 00:55:30,370 you would get an exact cancellation 922 00:55:30,370 --> 00:55:33,840 between the positive and the negative contributions. 923 00:55:33,840 --> 00:55:37,570 And the answer has to be 0 in an exactly supersymmetric theory. 924 00:55:37,570 --> 00:55:43,010 However, the world is clearly not exactly supersymmetric. 925 00:55:43,010 --> 00:55:46,522 This photino has never been seen. 926 00:55:46,522 --> 00:55:48,730 And there'd be a particle called the selectron, which 927 00:55:48,730 --> 00:55:51,840 would be the scalar partner of the electron, which 928 00:55:51,840 --> 00:55:52,900 also has not been seen. 929 00:55:52,900 --> 00:55:55,240 And every known particle would have 930 00:55:55,240 --> 00:55:59,200 a partner, which has not been seen. 931 00:55:59,200 --> 00:56:02,716 There are no supersymmetric pairs which are known. 932 00:56:02,716 --> 00:56:04,340 So supersymmetry is still a possibility 933 00:56:04,340 --> 00:56:06,240 as a broken symmetry of nature. 934 00:56:06,240 --> 00:56:09,000 And a lot of people think-- for pretty good reasons, I thin-- 935 00:56:09,000 --> 00:56:12,030 that it's very likely that the world does 936 00:56:12,030 --> 00:56:14,710 have an underlying supersymmetry. 937 00:56:14,710 --> 00:56:16,530 But as long as the supersymmetry is broken, 938 00:56:16,530 --> 00:56:18,500 it no longer guarantees this cancellation. 939 00:56:18,500 --> 00:56:20,890 And you could estimate what the mismatch is. 940 00:56:20,890 --> 00:56:24,770 And it does make things a little bit better here. 941 00:56:24,770 --> 00:56:27,300 If we just take this Planck scale cutoff, 942 00:56:27,300 --> 00:56:32,100 we miss an energy density by a factor of about 10 to the 120. 943 00:56:32,100 --> 00:56:34,250 If we apply supersymmetry and make 944 00:56:34,250 --> 00:56:37,420 an estimate of what the supersymmetry breaking scale is 945 00:56:37,420 --> 00:56:40,920 and what effect that has on the mismatch of these calculations, 946 00:56:40,920 --> 00:56:41,870 then it gets reduced. 947 00:56:41,870 --> 00:56:44,080 Instead of being 120 order of magnitude problems, 948 00:56:44,080 --> 00:56:46,490 it's got a 50 order of magnitude problem, which 949 00:56:46,490 --> 00:56:51,300 is a lot better, but not good enough. 950 00:56:51,300 --> 00:56:55,330 Now, I do want to mention a third contribution here 951 00:56:55,330 --> 00:56:57,410 for completeness. 952 00:56:57,410 --> 00:56:59,180 The third one is likely be finite, 953 00:56:59,180 --> 00:57:04,812 so it's not as problematic as the other two. 954 00:57:04,812 --> 00:57:06,884 AUDIENCE: [INAUDIBLE] 955 00:57:06,884 --> 00:57:07,800 PROFESSOR: Same thing. 956 00:57:07,800 --> 00:57:08,440 Planck scale. 957 00:57:08,440 --> 00:57:09,436 AUDIENCE: Oh, OK. 958 00:57:16,910 --> 00:57:18,970 PROFESSOR: The third contribution 959 00:57:18,970 --> 00:57:27,630 is that some fields are believed to have 960 00:57:27,630 --> 00:57:29,080 nonzero values in the vacuum. 961 00:57:39,223 --> 00:57:47,311 And the famous example of that is the Higgs field, 962 00:57:47,311 --> 00:57:49,310 for which the particle associated with the Higgs 963 00:57:49,310 --> 00:57:54,370 was discovered a year ago at CERN, 964 00:57:54,370 --> 00:57:56,967 after over 50 years of looking for it. 965 00:57:59,650 --> 00:58:01,597 And the Higgs field is maybe the only field 966 00:58:01,597 --> 00:58:03,180 that's part of the standard model that 967 00:58:03,180 --> 00:58:07,460 has a nonzero expectation value, a nonzero value in the vacuum. 968 00:58:07,460 --> 00:58:08,894 But in more sophisticated theories 969 00:58:08,894 --> 00:58:10,310 like grand unified theories, there 970 00:58:10,310 --> 00:58:13,680 are many more fields that have nonzero values in the vacuum. 971 00:58:13,680 --> 00:58:17,370 So that's a likely extension of our standard model 972 00:58:17,370 --> 00:58:19,990 of particle physics. 973 00:58:19,990 --> 00:58:25,980 So the bottom line is that it's easy for particle physicists 974 00:58:25,980 --> 00:58:30,430 to understand why the vacuum energy should be nonzero, 975 00:58:30,430 --> 00:58:35,900 but damned hard to have any idea of why it has the value 976 00:58:35,900 --> 00:58:36,420 that it has. 977 00:58:39,492 --> 00:58:41,200 We'll talk maybe at the end of the course 978 00:58:41,200 --> 00:58:44,960 about the possibility that the value of the vacuum energy 979 00:58:44,960 --> 00:58:49,100 density is, quote, "anthropically selected." 980 00:58:49,100 --> 00:58:50,971 That is one possible explanation, 981 00:58:50,971 --> 00:58:52,720 which maybe shows how desperate physicists 982 00:58:52,720 --> 00:58:55,120 are to look for an explanation here. 983 00:58:55,120 --> 00:58:58,230 One possible explanation begins with ideas 984 00:58:58,230 --> 00:59:00,920 from string theory, where string theory tells us 985 00:59:00,920 --> 00:59:03,000 that there isn't just one kind of vacuum, 986 00:59:03,000 --> 00:59:05,760 but in fact, a huge number of different types of vacuum, 987 00:59:05,760 --> 00:59:09,360 perhaps 10 to the 500. 988 00:59:09,360 --> 00:59:11,310 And that would mean that if there 989 00:59:11,310 --> 00:59:16,680 were sort of random values for these infinite numbers that 990 00:59:16,680 --> 00:59:19,239 get cut off, that get cut off with different values-- 991 00:59:19,239 --> 00:59:21,280 and there are other ways of looking at the vacuum 992 00:59:21,280 --> 00:59:23,820 energy in string theories-- you'd 993 00:59:23,820 --> 00:59:25,530 expect coming out of string theory 994 00:59:25,530 --> 00:59:28,670 that the typical vacuum energy would be about the same as what 995 00:59:28,670 --> 00:59:30,772 you get when you cut off the quantum 996 00:59:30,772 --> 00:59:32,480 fluctuations of the electromagnetic field 997 00:59:32,480 --> 00:59:33,700 at the Planck scale. 998 00:59:33,700 --> 00:59:36,680 That is, the typical vacuum energy coming out 999 00:59:36,680 --> 00:59:39,160 of a string theory would be at the Planck scale, which 1000 00:59:39,160 --> 00:59:41,679 is this huge number compared to what we observe. 1001 00:59:41,679 --> 00:59:43,095 But string theory would be predict 1002 00:59:43,095 --> 00:59:46,700 that there would be a spread of numbers going essentially 1003 00:59:46,700 --> 00:59:49,150 from plus the Planck scale to minus the Planck scale, 1004 00:59:49,150 --> 00:59:52,150 with everything in between. 1005 00:59:52,150 --> 00:59:54,800 There'd be a tiny fraction of those vacua that 1006 00:59:54,800 --> 00:59:58,695 would have a very small vacuum energy like what we observe. 1007 00:59:58,695 --> 01:00:00,570 That's what you'd expect from string theory-- 1008 01:00:00,570 --> 01:00:03,560 a large number, but a tiny fraction of vacua that 1009 01:00:03,560 --> 01:00:05,760 would be in that integral. 1010 01:00:05,760 --> 01:00:07,690 And then the only problem would be 1011 01:00:07,690 --> 01:00:13,010 to explain why we might likely be living in such an unusually 1012 01:00:13,010 --> 01:00:17,290 small fraction of the set of all possible vacuums. 1013 01:00:17,290 --> 01:00:20,920 And the answer to that that's discussed 1014 01:00:20,920 --> 01:00:23,330 is that it may be anthropically selected. 1015 01:00:23,330 --> 01:00:27,010 That is, life may only form when the vacuum 1016 01:00:27,010 --> 01:00:29,650 energy is incredibly small. 1017 01:00:29,650 --> 01:00:32,560 And that is not built entirely from whole cloth. 1018 01:00:32,560 --> 01:00:34,220 There is some physics behind that. 1019 01:00:34,220 --> 01:00:38,440 We know that this vacuum energy affects the Friedman equation, 1020 01:00:38,440 --> 01:00:41,800 which means it affects the expansion rate of the universe. 1021 01:00:41,800 --> 01:00:44,820 So if we had a Planck scale vacuum energy, 1022 01:00:44,820 --> 01:00:47,330 that would cause the universe to essentially blow 1023 01:00:47,330 --> 01:00:51,760 apart at the time scale of the Planck scale, which is about 10 1024 01:00:51,760 --> 01:00:55,142 to the minus 40 something seconds, 1025 01:00:55,142 --> 01:00:57,100 due to the huge repulsion that would be created 1026 01:00:57,100 --> 01:00:59,410 by that positive vacuum energy. 1027 01:00:59,410 --> 01:01:03,360 And conversely, if there was a huge negative vacuum 1028 01:01:03,360 --> 01:01:05,140 energy on the order of the Planck scale, 1029 01:01:05,140 --> 01:01:08,420 the universe would just implode on a time scale 1030 01:01:08,420 --> 01:01:09,830 of order of the Planck scale-- 10 1031 01:01:09,830 --> 01:01:12,610 to the minus 40 something seconds. 1032 01:01:12,610 --> 01:01:15,870 So assuming that life takes billions of years to evolve 1033 01:01:15,870 --> 01:01:18,470 and assuming nothing else about life, 1034 01:01:18,470 --> 01:01:22,070 one can conclude that life can only exist 1035 01:01:22,070 --> 01:01:24,980 in the very narrow band of possible vacuum energy 1036 01:01:24,980 --> 01:01:27,610 densities which are incredibly small, like the one 1037 01:01:27,610 --> 01:01:29,380 that we're living in. 1038 01:01:29,380 --> 01:01:31,660 So it could be that we're here only because there 1039 01:01:31,660 --> 01:01:32,990 isn't any life anyplace else. 1040 01:01:32,990 --> 01:01:36,620 So all living things see a very, very small value of this vacuum 1041 01:01:36,620 --> 01:01:40,110 energy density, even though if you plunk yourself down 1042 01:01:40,110 --> 01:01:43,830 at a random place in this multiverse, 1043 01:01:43,830 --> 01:01:45,810 you'd be likely to see a vacuum energy that's 1044 01:01:45,810 --> 01:01:49,180 near the Planck scale. 1045 01:01:49,180 --> 01:01:52,360 OK, I'm done talking about this for now. 1046 01:01:52,360 --> 01:01:55,020 Any further questions about it before we leave the topic? 1047 01:02:04,690 --> 01:02:07,435 I had suggested that we go on to talk about problems 1048 01:02:07,435 --> 01:02:09,650 with the conventional big bang model, 1049 01:02:09,650 --> 01:02:13,210 but, there is actually something else I wanted to do. 1050 01:02:13,210 --> 01:02:15,130 I don't know how long it will take exactly, 1051 01:02:15,130 --> 01:02:17,800 but I have a little historical interlude to talk about here. 1052 01:02:21,780 --> 01:02:24,140 We've been talking about the Friedman equations 1053 01:02:24,140 --> 01:02:26,790 and how they're modified by the cosmological constant, which 1054 01:02:26,790 --> 01:02:33,190 of course is an item that was very dear to Einstein's heart. 1055 01:02:33,190 --> 01:02:35,260 So I'd like to tell you a little history 1056 01:02:35,260 --> 01:02:39,530 story about Albert Einstein and Alexander Friedman, 1057 01:02:39,530 --> 01:02:41,250 which I think is very interesting. 1058 01:02:41,250 --> 01:02:44,310 The punchline of the story is that Einstein made pretty much 1059 01:02:44,310 --> 01:02:46,080 of a fool out of himself on this. 1060 01:02:46,080 --> 01:02:49,020 And the reason why I like the story is maybe twofold. 1061 01:02:49,020 --> 01:02:51,490 One is, I find it very comforting 1062 01:02:51,490 --> 01:02:55,750 to know that even perhaps the greatest physicist of all time 1063 01:02:55,750 --> 01:02:57,890 can make dumb mistakes just like the rest of us 1064 01:02:57,890 --> 01:02:59,760 make dumb mistakes. 1065 01:02:59,760 --> 01:03:02,310 I think that's a very comforting thing to keep in mind. 1066 01:03:02,310 --> 01:03:06,100 And the other moral of the story is, 1067 01:03:06,100 --> 01:03:07,640 I think, the importance of trying 1068 01:03:07,640 --> 01:03:11,150 to be open-minded about issues in physics. 1069 01:03:11,150 --> 01:03:15,220 Einstein was very much convinced that the universe was static, 1070 01:03:15,220 --> 01:03:17,160 and so convinced that, in fact, he really made 1071 01:03:17,160 --> 01:03:21,020 stupid mistakes trying to defend his static universe. 1072 01:03:21,020 --> 01:03:24,690 So this will be a story of such a mistake. 1073 01:03:24,690 --> 01:03:27,720 So those are the two people. 1074 01:03:27,720 --> 01:03:31,287 Friedman was a Russian natural-- he was really a meteorologist. 1075 01:03:31,287 --> 01:03:33,620 They didn't really have that many theoretical physicists 1076 01:03:33,620 --> 01:03:35,060 back in those days. 1077 01:03:35,060 --> 01:03:37,450 But as a meteorologist, he was an expert 1078 01:03:37,450 --> 01:03:40,629 in solving partial differential equations, 1079 01:03:40,629 --> 01:03:42,920 and got himself interested in general relativity, which 1080 01:03:42,920 --> 01:03:44,950 was a new theory at this point. 1081 01:03:44,950 --> 01:03:49,547 And in 1922, he published an actual physics paper, 1082 01:03:49,547 --> 01:03:51,630 I think the first physics paper he ever published, 1083 01:03:51,630 --> 01:03:52,440 and one of two. 1084 01:03:52,440 --> 01:03:55,250 He wrote basically two papers about the Friedman equations-- 1085 01:03:55,250 --> 01:03:59,050 one for closed universes, and one for open universes. 1086 01:03:59,050 --> 01:04:00,800 So the first of those papers was published 1087 01:04:00,800 --> 01:04:05,440 in June 29, 1922 in the premier physics journal of the day-- 1088 01:04:05,440 --> 01:04:08,805 the Zeitschrift fur Physik, a German journal. 1089 01:04:11,410 --> 01:04:15,530 And almost immediately-- or a few months later, 1090 01:04:15,530 --> 01:04:18,610 when Einstein noticed this article-- 1091 01:04:18,610 --> 01:04:22,620 Einstein submitted a comment about the article claiming 1092 01:04:22,620 --> 01:04:24,810 that the article was entirely wrong, just 1093 01:04:24,810 --> 01:04:26,620 mathematically wrong. 1094 01:04:26,620 --> 01:04:30,790 And the article was titled "Remark on the Work of A. 1095 01:04:30,790 --> 01:04:34,420 Friedmann 'On the Curvature of Space' " by A. Einstein, 1096 01:04:34,420 --> 01:04:38,510 Berlin, received September 18, 1922. 1097 01:04:38,510 --> 01:04:40,990 Looking at these dates-- the original article 1098 01:04:40,990 --> 01:04:45,500 was received in June 1922, and Einstein 1099 01:04:45,500 --> 01:04:50,030 was responding by September 18, a few months later. 1100 01:04:50,030 --> 01:04:53,720 And this is a translation, which comes from a book called 1101 01:04:53,720 --> 01:04:55,920 Cosmological Constants, which is basically 1102 01:04:55,920 --> 01:04:58,330 a book of famous articles in cosmology, 1103 01:04:58,330 --> 01:05:00,380 like Friedman's, and all these original articles. 1104 01:05:00,380 --> 01:05:02,546 It's a great book if you can still get a copy of it. 1105 01:05:02,546 --> 01:05:03,930 It's no doubt out of print. 1106 01:05:03,930 --> 01:05:08,120 It was written by Jeremy Bernstein and Gary Feinberg. 1107 01:05:08,120 --> 01:05:10,125 And I'm taking the translation from there, 1108 01:05:10,125 --> 01:05:12,930 because this was written in German. 1109 01:05:12,930 --> 01:05:14,650 I don't know German. 1110 01:05:14,650 --> 01:05:17,010 "The works cited contains a result concerning 1111 01:05:17,010 --> 01:05:21,250 a non-stationary world which seems suspect to me. 1112 01:05:21,250 --> 01:05:23,380 Indeed, those solutions do not appear 1113 01:05:23,380 --> 01:05:25,400 compatible with the field equations." 1114 01:05:25,400 --> 01:05:27,670 And I guess A is the label of the field equations 1115 01:05:27,670 --> 01:05:30,360 as they appeared in Friedman's paper. 1116 01:05:30,360 --> 01:05:32,560 "From the field equation, it follows necessarily 1117 01:05:32,560 --> 01:05:36,435 that the divergence of the matter tensor Tik vanishes." 1118 01:05:36,435 --> 01:05:39,350 That is, energy momentum is conserved 1119 01:05:39,350 --> 01:05:43,200 as a four-vector quantity. 1120 01:05:43,200 --> 01:05:47,120 "This along with ansatzes C and D"-- equations from the paper-- 1121 01:05:47,120 --> 01:05:49,820 leads, according to Einstein, to an equation which we can all 1122 01:05:49,820 --> 01:05:53,050 recognize the meaning of-- the partial of rho with respect 1123 01:05:53,050 --> 01:05:56,000 to x sub 4-- time-- is 0. 1124 01:05:56,000 --> 01:05:59,560 Einstein convinced himself that the equations of general 1125 01:05:59,560 --> 01:06:03,140 relativity led to the conclusion that rho cannot change with 1126 01:06:03,140 --> 01:06:05,280 time. 1127 01:06:05,280 --> 01:06:07,430 And he then goes on to say "which together with 8 1128 01:06:07,430 --> 01:06:10,690 implies that the world radius R"-- that's the scale factor. 1129 01:06:10,690 --> 01:06:14,594 That's what we call a of t-- "is constant in time. 1130 01:06:14,594 --> 01:06:16,260 The significance of the work, therefore, 1131 01:06:16,260 --> 01:06:18,830 is to demonstrate this constancy." 1132 01:06:18,830 --> 01:06:20,750 All Friedman does once you correct 1133 01:06:20,750 --> 01:06:22,560 his equations, according to Einstein, 1134 01:06:22,560 --> 01:06:25,620 was prove that the only cosmological solution is rho 1135 01:06:25,620 --> 01:06:29,600 equals a constant, which was Einstein's static solution. 1136 01:06:29,600 --> 01:06:33,852 This was entirely wrong-- no basis whatever in mathematics. 1137 01:06:33,852 --> 01:06:35,310 But it took a while before Einstein 1138 01:06:35,310 --> 01:06:36,715 got himself straightened out. 1139 01:06:36,715 --> 01:06:39,660 And he did actually publish this. 1140 01:06:39,660 --> 01:06:44,370 The sequence of events was, June 29, Friedman submits his paper. 1141 01:06:44,370 --> 01:06:49,880 September 18, Einstein submits his rebuttal to the paper. 1142 01:06:49,880 --> 01:06:53,410 Friedman didn't learn about this until the following December. 1143 01:06:53,410 --> 01:06:57,960 Friedman had a friend who played a key role in this story-- Yrui 1144 01:06:57,960 --> 01:07:02,830 Krutkov, who was visiting in Berlin during this time. 1145 01:07:02,830 --> 01:07:08,020 And Friedman actually learned from Krutkov 1146 01:07:08,020 --> 01:07:11,860 that Einstein had submitted a rebuttal. 1147 01:07:11,860 --> 01:07:15,280 So Friedman apparently was able to track it down and read it. 1148 01:07:15,280 --> 01:07:17,730 And he wrote a detailed letter to Einstein 1149 01:07:17,730 --> 01:07:20,031 explaining to Einstein what he got wrong, 1150 01:07:20,031 --> 01:07:21,780 which is a gutsy thing to do, but Friedman 1151 01:07:21,780 --> 01:07:23,827 was right in this case. 1152 01:07:23,827 --> 01:07:26,410 But Einstein was traveling and actually never read the letter, 1153 01:07:26,410 --> 01:07:28,980 at least not until much later. 1154 01:07:28,980 --> 01:07:33,100 Then the following May, Krutkov and Einstein 1155 01:07:33,100 --> 01:07:36,920 are both at a conference in Leiden, a conference that they 1156 01:07:36,920 --> 01:07:40,735 were both attending, which was a farewell lecture by Lorentz, 1157 01:07:40,735 --> 01:07:44,660 who was retiring at that time. 1158 01:07:44,660 --> 01:07:48,560 So they met and started talking, and continued talking. 1159 01:07:48,560 --> 01:07:51,320 And we know most about it from a series of letters 1160 01:07:51,320 --> 01:07:57,790 that Krutkov wrote to his sister back in Saint Petersburg. 1161 01:07:57,790 --> 01:07:59,670 And according to those letters-- and I'm now 1162 01:07:59,670 --> 01:08:03,500 quoting from a rather lovely book called Alexander A. 1163 01:08:03,500 --> 01:08:06,070 Friedmann-- The Man who Made the Universe Expand, 1164 01:08:06,070 --> 01:08:07,760 by Tropp, Frenkel, and Chernin. 1165 01:08:10,830 --> 01:08:13,780 Krutkov wrote to his sister that on Monday, May 7, 1166 01:08:13,780 --> 01:08:16,790 1923, "I was reading, together with Einstein, 1167 01:08:16,790 --> 01:08:20,069 Friedman's article in the Zeitschrift fur Physik. 1168 01:08:20,069 --> 01:08:23,250 And then on May 18, he wrote, "I defeated Einstein 1169 01:08:23,250 --> 01:08:25,310 in the argument about Friedmann. 1170 01:08:25,310 --> 01:08:27,870 Petrograd's honor is saved!" 1171 01:08:27,870 --> 01:08:30,840 Petrograd is what we now call Saint Petersburg, 1172 01:08:30,840 --> 01:08:36,830 and where they were all from-- that is, Friedman and Krutkov. 1173 01:08:36,830 --> 01:08:40,240 And then shortly after that, on May 31, 1174 01:08:40,240 --> 01:08:44,279 Einstein submitted a retraction of his refutation 1175 01:08:44,279 --> 01:08:47,840 of Friedman's paper. 1176 01:08:47,840 --> 01:08:50,550 And the retraction is-- again, I'm 1177 01:08:50,550 --> 01:08:52,830 quoting from Cosmological Constant, which 1178 01:08:52,830 --> 01:08:56,399 translates all these nice papers into English. 1179 01:08:56,399 --> 01:09:00,790 Einstein wrote, very briefly, "I have in an earlier note 1180 01:09:00,790 --> 01:09:04,890 criticized the cited work-- Friedmann 1922. 1181 01:09:04,890 --> 01:09:07,609 My objection rested however, as Mr. Krutkov 1182 01:09:07,609 --> 01:09:10,420 off in person and a letter from Mr. Friedmann 1183 01:09:10,420 --> 01:09:14,000 convinced me, on a calculational error. 1184 01:09:14,000 --> 01:09:17,000 I am convinced that Mr. Friedmann's results 1185 01:09:17,000 --> 01:09:19,540 are both correct and clarifying. 1186 01:09:19,540 --> 01:09:21,720 They show that in addition to the static solution 1187 01:09:21,720 --> 01:09:23,689 to the field equations, there are 1188 01:09:23,689 --> 01:09:27,529 time varying solutions with a spatially symmetric structure." 1189 01:09:27,529 --> 01:09:31,870 Anyway, the expanding universe that we now talk about. 1190 01:09:31,870 --> 01:09:34,090 Einstein did have to admit, ultimately, 1191 01:09:34,090 --> 01:09:39,470 that algebra is algebra, and you can't really futz with algebra. 1192 01:09:39,470 --> 01:09:42,850 And the Einstein equations do not imply that rho cannot 1193 01:09:42,850 --> 01:09:47,160 change with time, and that Friedman was right. 1194 01:09:47,160 --> 01:09:51,000 There's an interesting twist on this retraction letter. 1195 01:09:51,000 --> 01:09:54,149 This is just a photo of Einstein at this time period, 1196 01:09:54,149 --> 01:09:57,470 and Krutkov. 1197 01:09:57,470 --> 01:10:00,270 There's an interesting twist on the retraction letter, which 1198 01:10:00,270 --> 01:10:02,810 is that the original draft still exists. 1199 01:10:02,810 --> 01:10:04,512 I forget what museum it's in. 1200 01:10:04,512 --> 01:10:06,220 But it's quoted in another marvelous book 1201 01:10:06,220 --> 01:10:07,803 about this history called The Invented 1202 01:10:07,803 --> 01:10:11,542 Universe, by Pierre Kerzberg. 1203 01:10:11,542 --> 01:10:15,510 And I Xeroxed this from the book. 1204 01:10:15,510 --> 01:10:17,030 And this is the original draft. 1205 01:10:17,030 --> 01:10:21,270 And notice there are some cross-outs. 1206 01:10:21,270 --> 01:10:26,690 And the last cross-out, which followed this explanation 1207 01:10:26,690 --> 01:10:31,270 that there is this expanding solution-- in Einstein's 1208 01:10:31,270 --> 01:10:34,350 original draft, he wrote but then crossed out 1209 01:10:34,350 --> 01:10:39,120 "a physical significance can hardly be ascribed to them." 1210 01:10:39,120 --> 01:10:41,272 So his initial instinct, even after having 1211 01:10:41,272 --> 01:10:43,230 been convinced that these were a valid solution 1212 01:10:43,230 --> 01:10:46,210 to the equations, was to say that they couldn't possibly 1213 01:10:46,210 --> 01:10:48,900 be physical, because they're not physical. 1214 01:10:48,900 --> 01:10:51,480 The universe is static. 1215 01:10:51,480 --> 01:10:53,230 But somehow, before he submitted it, 1216 01:10:53,230 --> 01:10:55,690 he did realize that there wasn't actually 1217 01:10:55,690 --> 01:10:59,042 any solid logic behind that reasoning. 1218 01:10:59,042 --> 01:11:03,180 So logic did prevail, and he decided that he really 1219 01:11:03,180 --> 01:11:05,670 had no right to say that the solution has 1220 01:11:05,670 --> 01:11:08,440 no physical significance, which is a good thing, because now, 1221 01:11:08,440 --> 01:11:10,356 of course, it is the solution that we consider 1222 01:11:10,356 --> 01:11:16,490 physically significant-- the expanding solution of Friedman. 1223 01:11:16,490 --> 01:11:20,040 So [INAUDIBLE] is a mystery, I think. 1224 01:11:20,040 --> 01:11:22,697 OK, we have just a couple minutes left in the class. 1225 01:11:22,697 --> 01:11:24,530 So I think that is nearly enough time for me 1226 01:11:24,530 --> 01:11:26,805 to at least introduce what I want to talk about next. 1227 01:11:35,260 --> 01:11:38,650 What we'll be talking about next time-- and I'll just introduce 1228 01:11:38,650 --> 01:11:43,380 it now-- are a set of two problems associated 1229 01:11:43,380 --> 01:11:45,370 with the conventional big bang theory. 1230 01:11:45,370 --> 01:11:47,300 And by the conventional bang big bang theory, 1231 01:11:47,300 --> 01:11:49,677 I mean basically the theory we've been talking about, 1232 01:11:49,677 --> 01:11:51,260 but in particular, the big bang theory 1233 01:11:51,260 --> 01:11:53,880 without inflation, which we will be talking about later. 1234 01:11:53,880 --> 01:11:56,220 But so far, we've been talking about the big bang theory 1235 01:11:56,220 --> 01:11:57,990 without inflation. 1236 01:11:57,990 --> 01:11:59,740 And the two problems that we'll talk about 1237 01:11:59,740 --> 01:12:02,350 are called the horizon or horizon homogeneity 1238 01:12:02,350 --> 01:12:06,150 problem, and the flatness problem. 1239 01:12:06,150 --> 01:12:09,970 Both of these are problems connected 1240 01:12:09,970 --> 01:12:13,500 with the initial conditions necessary to make the model 1241 01:12:13,500 --> 01:12:15,710 work. 1242 01:12:15,710 --> 01:12:21,240 So this horizon homogeneity problem 1243 01:12:21,240 --> 01:12:23,840 is a problem about trying to understand 1244 01:12:23,840 --> 01:12:27,800 the uniformity of the observed universe, which we've just 1245 01:12:27,800 --> 01:12:30,000 put in as part of our initial conditions. 1246 01:12:30,000 --> 01:12:32,790 The model that we've constructed was just completely 1247 01:12:32,790 --> 01:12:37,640 homogeneous and isotropic from start to present. 1248 01:12:37,640 --> 01:12:42,380 The evidence for the uniformity of the universe 1249 01:12:42,380 --> 01:12:45,410 shows up most strongly, as I think we said before, 1250 01:12:45,410 --> 01:12:47,400 in the cosmic background radiation, which 1251 01:12:47,400 --> 01:12:50,850 can be measured to fantastic precision. 1252 01:12:50,850 --> 01:12:54,390 And this radiation is known to be uniform in all directions 1253 01:12:54,390 --> 01:12:57,260 to an accuracy of one part in 100,000, 1254 01:12:57,260 --> 01:13:01,010 which is really a phenomenal level of accuracy. 1255 01:13:01,010 --> 01:13:03,390 Now, what makes this hard to understand 1256 01:13:03,390 --> 01:13:05,820 in the conventional big bang theory is 1257 01:13:05,820 --> 01:13:07,440 that if instead of just putting it 1258 01:13:07,440 --> 01:13:10,340 in as an assumption about the initial conditions, 1259 01:13:10,340 --> 01:13:13,490 you try to get it out of any kind of dynamics, 1260 01:13:13,490 --> 01:13:15,630 that turns out to be impossible. 1261 01:13:15,630 --> 01:13:18,610 And in particular, a calculation that we'll do next time 1262 01:13:18,610 --> 01:13:21,940 is we'll imagine tracing back photons 1263 01:13:21,940 --> 01:13:25,260 from the cosmic background radiation arriving at the Earth 1264 01:13:25,260 --> 01:13:28,719 today from two opposite directions in the sky. 1265 01:13:28,719 --> 01:13:30,760 Now, the phenomenology is that those photons come 1266 01:13:30,760 --> 01:13:32,950 with exactly the same temperature to an accuracy 1267 01:13:32,950 --> 01:13:34,790 of one part in 100,000, and that's 1268 01:13:34,790 --> 01:13:36,920 what we're trying to explain. 1269 01:13:36,920 --> 01:13:39,830 Now, we all do know that systems do 1270 01:13:39,830 --> 01:13:42,470 come to a uniform temperature. 1271 01:13:42,470 --> 01:13:45,009 If you heated the air in this room in a corner 1272 01:13:45,009 --> 01:13:46,550 and then let the room stand, the heat 1273 01:13:46,550 --> 01:13:48,570 would scatter throughout the room, 1274 01:13:48,570 --> 01:13:51,459 and the room would come to a uniform temperature. 1275 01:13:51,459 --> 01:13:53,500 If you take a hot slice of pizza out of the oven, 1276 01:13:53,500 --> 01:13:56,810 it gets cool, as everybody knows. 1277 01:13:56,810 --> 01:13:58,820 So there is this so-called zeroth law 1278 01:13:58,820 --> 01:14:01,650 of thermodynamics which says that everything tends 1279 01:14:01,650 --> 01:14:03,840 to come to a uniform temperature. 1280 01:14:03,840 --> 01:14:07,340 And it's a fair question to ask, can we perhaps explain 1281 01:14:07,340 --> 01:14:10,460 the uniformity of the universe by invoking 1282 01:14:10,460 --> 01:14:12,700 this zeroth law of thermodynamcs? 1283 01:14:12,700 --> 01:14:14,770 Maybe the universe just had time to come 1284 01:14:14,770 --> 01:14:17,330 to a uniform temperature. 1285 01:14:17,330 --> 01:14:21,500 But one can see immediately when one looks at details 1286 01:14:21,500 --> 01:14:23,480 that that's not the case. 1287 01:14:23,480 --> 01:14:25,560 Within the context of our conventional model 1288 01:14:25,560 --> 01:14:27,760 of cosmology, the universe definitely 1289 01:14:27,760 --> 01:14:31,360 did not have time to come to a uniform temperature. 1290 01:14:31,360 --> 01:14:34,220 And the easiest way to drive that home 1291 01:14:34,220 --> 01:14:38,250 will be a calculation that we will do first thing next time, 1292 01:14:38,250 --> 01:14:40,500 which is that we will trace back photons coming 1293 01:14:40,500 --> 01:14:42,440 from opposite directions in the sky 1294 01:14:42,440 --> 01:14:44,730 and ask, what would it take for them 1295 01:14:44,730 --> 01:14:47,270 to have been set equal to the same temperature 1296 01:14:47,270 --> 01:14:49,220 when they were first emitted? 1297 01:14:49,220 --> 01:14:51,990 And what we'll find is that when we trace them 1298 01:14:51,990 --> 01:14:59,140 back to their emission sources, that those emissions took place 1299 01:14:59,140 --> 01:15:02,130 at two points which were separated from each other 1300 01:15:02,130 --> 01:15:05,370 by about 50 horizon distances. 1301 01:15:05,370 --> 01:15:08,130 So assuming that physical influences are limited 1302 01:15:08,130 --> 01:15:10,010 by the speed of light-- and according 1303 01:15:10,010 --> 01:15:12,600 to everything that we know about the laws of physics, that's 1304 01:15:12,600 --> 01:15:18,120 true-- there is no way that the emission of that photon coming 1305 01:15:18,120 --> 01:15:21,990 from that direction could have had any causal connection 1306 01:15:21,990 --> 01:15:23,590 with the emission of the photon coming 1307 01:15:23,590 --> 01:15:25,510 from the other direction. 1308 01:15:25,510 --> 01:15:28,560 So if the uniformity had to be set up 1309 01:15:28,560 --> 01:15:30,210 by physical processes that happened 1310 01:15:30,210 --> 01:15:33,297 after the initial singularity, there's just no way 1311 01:15:33,297 --> 01:15:35,130 that that emission could have known anything 1312 01:15:35,130 --> 01:15:36,824 about what was going on over there, 1313 01:15:36,824 --> 01:15:39,240 and no way they could have arranged to be emitting photons 1314 01:15:39,240 --> 01:15:42,592 at the same energy at the same time. 1315 01:15:42,592 --> 01:15:44,050 Now, everything does work if you're 1316 01:15:44,050 --> 01:15:47,030 willing to just assume that everything started at uniform. 1317 01:15:47,030 --> 01:15:48,750 But if you're not willing to assume that, 1318 01:15:48,750 --> 01:15:52,550 and want to try to derive the uniformity of the universe 1319 01:15:52,550 --> 01:15:54,890 as a dynamical consequence of processes 1320 01:15:54,890 --> 01:15:56,760 in the early universe, there's just 1321 01:15:56,760 --> 01:15:58,960 no way to do it in the conventional big bang 1322 01:15:58,960 --> 01:16:01,570 theory because of this causality argument. 1323 01:16:01,570 --> 01:16:04,531 And later, we'll see that inflation gets around that. 1324 01:16:04,531 --> 01:16:05,030 Yes? 1325 01:16:05,030 --> 01:16:08,000 AUDIENCE: How do we know that the homogeneity wasn't 1326 01:16:08,000 --> 01:16:10,240 just created when the universe was smaller, 1327 01:16:10,240 --> 01:16:12,802 in such a way that the speed of light limit 1328 01:16:12,802 --> 01:16:15,010 wouldn't be violated, and that it would just maintain 1329 01:16:15,010 --> 01:16:16,822 [INAUDIBLE]? 1330 01:16:16,822 --> 01:16:18,530 PROFESSOR: OK, the question is, how do we 1331 01:16:18,530 --> 01:16:20,530 know that the uniformity wasn't established when 1332 01:16:20,530 --> 01:16:23,490 the universe was very small, and then the speed of light 1333 01:16:23,490 --> 01:16:25,145 might not have to be violated? 1334 01:16:25,145 --> 01:16:27,651 Well, the point is that if the dynamics is 1335 01:16:27,651 --> 01:16:29,650 the conventional big bang model, what we'll show 1336 01:16:29,650 --> 01:16:31,660 is that there's not really enough. 1337 01:16:31,660 --> 01:16:34,300 No matter how early you imagine it happening, 1338 01:16:34,300 --> 01:16:36,882 it still is 50 horizon distances apart. 1339 01:16:36,882 --> 01:16:39,340 And there's no way that those points could've communicated, 1340 01:16:39,340 --> 01:16:41,895 no matter how close you come to t equals 0. 1341 01:16:41,895 --> 01:16:43,895 Now, you are of course free to assuming anything 1342 01:16:43,895 --> 01:16:47,060 you want about the singularity at t equals 0. 1343 01:16:47,060 --> 01:16:50,120 So if you want to just assume that somehow the singularity 1344 01:16:50,120 --> 01:16:53,010 homogenized everything, that's OK. 1345 01:16:53,010 --> 01:16:54,440 But there's no theory behind it. 1346 01:16:54,440 --> 01:16:56,970 That's just speculation. 1347 01:16:56,970 --> 01:16:58,880 But it is satisfactory speculation. 1348 01:16:58,880 --> 01:17:00,882 There's nothing it contradicts. 1349 01:17:00,882 --> 01:17:03,090 But the beauty of inflation is that it does, in fact, 1350 01:17:03,090 --> 01:17:06,350 provide a dynamical explanation for how this uniformity could 1351 01:17:06,350 --> 01:17:09,820 have been created, which, at least to many people, 1352 01:17:09,820 --> 01:17:12,000 is better than just speculating that somehow it 1353 01:17:12,000 --> 01:17:13,166 happened in the singularity. 1354 01:17:16,720 --> 01:17:18,170 OK, I think that's it for now. 1355 01:17:18,170 --> 01:17:19,795 I will tell you about the other problem 1356 01:17:19,795 --> 01:17:21,650 we'll talk about next time next time. 1357 01:17:21,650 --> 01:17:25,360 And I will see you all next Tuesday.