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 --> 00:00:24,875 at ocw.mit.edu 8 00:00:24,875 --> 00:00:27,270 PROFESSOR: OK, in that case I can 9 00:00:27,270 --> 00:00:31,270 begin by giving a quick review of last time. 10 00:00:31,270 --> 00:00:36,050 We began last time by talking about the data of measurements 11 00:00:36,050 --> 00:00:38,700 of the cosmic microwave background, 12 00:00:38,700 --> 00:00:43,185 starting with the data as it existed in 1975 which 13 00:00:43,185 --> 00:00:46,575 I advertised as being an incredible mess, which it was. 14 00:00:46,575 --> 00:00:52,640 You could easily believe that this data fit this solid line 15 00:00:52,640 --> 00:00:54,465 curve, which was what it's supposed to fit. 16 00:00:54,465 --> 00:00:58,120 But you could equally well believe that it didn't. 17 00:00:58,120 --> 00:01:00,720 Things got worse before they got better. 18 00:01:00,720 --> 00:01:02,980 There was the famous Berkeley-Nagoya Rocket Flight 19 00:01:02,980 --> 00:01:07,820 experiment of 1987 which had a data point which 20 00:01:07,820 --> 00:01:11,940 missed the theoretical curve by 16 standard deviations, which 21 00:01:11,940 --> 00:01:15,410 might seem fairly disappointing. 22 00:01:15,410 --> 00:01:17,504 It reminds one, by the way, of a famous quote 23 00:01:17,504 --> 00:01:19,420 of Arthur Eddington-- which you may or may not 24 00:01:19,420 --> 00:01:21,990 be familiar with-- but Eddington pointed out 25 00:01:21,990 --> 00:01:25,480 that while we always say that we should not believe theories 26 00:01:25,480 --> 00:01:28,340 until the confirmed by experiment, 27 00:01:28,340 --> 00:01:32,050 it's in fact equally true that we should not believe data 28 00:01:32,050 --> 00:01:34,980 that's put forward until it's confirmed by theory, 29 00:01:34,980 --> 00:01:37,280 and that certainly was the case here. 30 00:01:37,280 --> 00:01:39,110 This data was never confirmed by theory 31 00:01:39,110 --> 00:01:41,292 and turned out to be wrong. 32 00:01:41,292 --> 00:01:44,740 The beautiful data was achieved in 1990 33 00:01:44,740 --> 00:01:49,040 by this fabulous COBE satellite experiment, which 34 00:01:49,040 --> 00:01:53,370 showed-- unambiguously, for the first time-- 35 00:01:53,370 --> 00:01:56,180 that the cosmic background radiation really 36 00:01:56,180 --> 00:02:00,610 does obey an essentially perfect blackbody curve, which 37 00:02:00,610 --> 00:02:03,770 is just gorgeous. 38 00:02:03,770 --> 00:02:05,970 We then went on, in our last lecture, 39 00:02:05,970 --> 00:02:09,759 to begin to talk about the cosmological constant 40 00:02:09,759 --> 00:02:12,580 and its effect on the evolution of the universe-- 41 00:02:12,580 --> 00:02:17,680 completely changing gear here-- and the key issue 42 00:02:17,680 --> 00:02:22,580 is the cosmological effect of pressure. 43 00:02:22,580 --> 00:02:26,540 Earlier we had derived this equation. 44 00:02:26,540 --> 00:02:32,470 The equation shows the significant role of pressure 45 00:02:32,470 --> 00:02:35,650 during the radiation-dominated era, 46 00:02:35,650 --> 00:02:38,672 but it also shows that pressure-- 47 00:02:38,672 --> 00:02:40,130 if it were negative-- could perhaps 48 00:02:40,130 --> 00:02:42,610 have the opposite effect, causing 49 00:02:42,610 --> 00:02:46,020 an acceleration of the universe. 50 00:02:46,020 --> 00:02:48,820 Furthermore, we learned last time 51 00:02:48,820 --> 00:02:53,530 that vacuum energy-- first thing we learned, I guess, 52 00:02:53,530 --> 00:02:55,990 is just that's synonymous with Einstein's cosmological 53 00:02:55,990 --> 00:03:01,060 constant, related to Einstein's cosmological constant lambda 54 00:03:01,060 --> 00:03:02,602 by this equation. 55 00:03:02,602 --> 00:03:04,310 That is, the energy density of the vacuum 56 00:03:04,310 --> 00:03:06,309 is equal to the mass density of the vacuum times 57 00:03:06,309 --> 00:03:09,720 C squared and is equal to this expression in terms 58 00:03:09,720 --> 00:03:13,180 of Einstein's original cosmological constant. 59 00:03:13,180 --> 00:03:15,900 And most important, in terms of physics, 60 00:03:15,900 --> 00:03:22,010 we learned that if we have a non-zero vacuum energy-- vacuum 61 00:03:22,010 --> 00:03:24,440 energy by definition does not change with time 62 00:03:24,440 --> 00:03:26,280 because the vacuum is the vacuum, 63 00:03:26,280 --> 00:03:29,320 it's simply the lowest possible energy density allowed 64 00:03:29,320 --> 00:03:31,660 by the laws of physics, and the laws of physics, 65 00:03:31,660 --> 00:03:33,820 as far as we know, do not change with time, 66 00:03:33,820 --> 00:03:36,060 and therefore vacuum energy density does not 67 00:03:36,060 --> 00:03:39,460 change with time-- and that is enough to imply 68 00:03:39,460 --> 00:03:41,650 that the pressure of the vacuum has 69 00:03:41,650 --> 00:03:46,080 to be equal to minus the energy density in a vacuum, 70 00:03:46,080 --> 00:03:48,970 and therefore minus that expression in terms 71 00:03:48,970 --> 00:03:54,080 of the cosmological constant-- which is exactly what will give 72 00:03:54,080 --> 00:03:57,500 us a repulsive gravitational effect, 73 00:03:57,500 --> 00:03:59,725 where we put that into the Friedmann equation. 74 00:04:02,460 --> 00:04:05,470 Now I should emphasize here that the effect 75 00:04:05,470 --> 00:04:07,730 of the pressure that we are talking about 76 00:04:07,730 --> 00:04:12,060 is not the mechanical effect of the pressure. 77 00:04:12,060 --> 00:04:13,890 The mechanical effect of the pressure 78 00:04:13,890 --> 00:04:17,779 is literally zero because the pressure that we are discussing 79 00:04:17,779 --> 00:04:21,579 here is a uniform pressure, and pressures only 80 00:04:21,579 --> 00:04:23,810 cause mechanical forces when there's 81 00:04:23,810 --> 00:04:25,760 gradients in the pressure-- more pressure 82 00:04:25,760 --> 00:04:27,550 on one side than the other. 83 00:04:27,550 --> 00:04:29,500 And if all this pressure is always balanced, 84 00:04:29,500 --> 00:04:32,150 the mechanical force of the pressure is zero. 85 00:04:32,150 --> 00:04:36,260 But nonetheless, that pressure contributes-- according 86 00:04:36,260 --> 00:04:39,850 to Einstein's equations-- as a contribution 87 00:04:39,850 --> 00:04:44,870 to the gravitational field, and a positive pressure 88 00:04:44,870 --> 00:04:47,740 creates an attractive gravitational field 89 00:04:47,740 --> 00:04:51,790 but a negative pressure produces repulsive gravitational fields. 90 00:04:51,790 --> 00:04:54,350 And a positive vacuum energy corresponds 91 00:04:54,350 --> 00:04:57,140 to a negative pressure which, in fact, would dominate 92 00:04:57,140 --> 00:05:00,660 this equation, resulting in a gravitational repulsion. 93 00:05:03,530 --> 00:05:08,110 So it's useful to divide the total energy 94 00:05:08,110 --> 00:05:13,750 density into a normal component plus vacuum energy, 95 00:05:13,750 --> 00:05:15,450 and similarly we can divide the pressure 96 00:05:15,450 --> 00:05:19,880 into a normal component plus the vacuum contribution 97 00:05:19,880 --> 00:05:21,230 to the pressure. 98 00:05:21,230 --> 00:05:23,390 The vacuum contribution to the pressure 99 00:05:23,390 --> 00:05:25,766 will instantly disappear from all of our equations 100 00:05:25,766 --> 00:05:27,140 because we know how to express it 101 00:05:27,140 --> 00:05:29,720 in terms of the vacuum mass density. 102 00:05:29,720 --> 00:05:33,020 It's just minus Rho vac C squared. 103 00:05:33,020 --> 00:05:35,270 So we can then re-write the Friedmann equations 104 00:05:35,270 --> 00:05:39,610 making those substitutions, and we 105 00:05:39,610 --> 00:05:41,870 conceive-- in the second-order equation-- 106 00:05:41,870 --> 00:05:45,670 that the vacuum contribution, negative, negative 107 00:05:45,670 --> 00:05:49,510 is a positive, produces a positive acceleration-- 108 00:05:49,510 --> 00:05:53,850 as we've been saying-- and a positive vacuum energy also 109 00:05:53,850 --> 00:05:55,850 contributes to the right-hand side 110 00:05:55,850 --> 00:05:58,560 of the first-order Friedmann equation. 111 00:05:58,560 --> 00:06:03,370 And in many, many situations-- although not quite all-- 112 00:06:03,370 --> 00:06:07,100 this vacuum energy will dominate at late times. 113 00:06:07,100 --> 00:06:08,950 It definitely falls off more slowly 114 00:06:08,950 --> 00:06:10,780 than any of the other contributions. 115 00:06:10,780 --> 00:06:14,120 The vacuum energy is a constant, and every other contribution 116 00:06:14,120 --> 00:06:16,450 to that right-hand side falls off 117 00:06:16,450 --> 00:06:20,080 with A. The only way that Rho vac can 118 00:06:20,080 --> 00:06:22,629 fail to dominate if it's non-zero 119 00:06:22,629 --> 00:06:25,170 is if you have a closed universe that collapses before it has 120 00:06:25,170 --> 00:06:28,460 a chance to dominate, which is a possibility. 121 00:06:28,460 --> 00:06:32,610 But barring that, eventually the vacuum energy will dominate, 122 00:06:32,610 --> 00:06:35,870 and once the vacuum energy dominates, 123 00:06:35,870 --> 00:06:38,360 we just have H squared-- A dot over A 124 00:06:38,360 --> 00:06:41,990 is H. H squared equals a constant, 125 00:06:41,990 --> 00:06:46,280 and that just says that H approaches its vacuum 126 00:06:46,280 --> 00:06:51,230 value, which is the square root of 8 pi over 3 G Rho vac, 127 00:06:51,230 --> 00:06:53,720 and with H being a constant we can also 128 00:06:53,720 --> 00:06:56,650 solve for A. The scale factor itself 129 00:06:56,650 --> 00:07:00,591 is just proportional to an exponential of E to the H T, 130 00:07:00,591 --> 00:07:05,110 where H is the H associated with the vacuum. 131 00:07:05,110 --> 00:07:13,252 So this will be a very easy to obtain asymptotic solution 132 00:07:13,252 --> 00:07:15,210 to the equations of the universe, and, in fact, 133 00:07:15,210 --> 00:07:20,360 we think that our real universe is approaching an exponentially 134 00:07:20,360 --> 00:07:22,790 expanding phase of exactly this sort today. 135 00:07:22,790 --> 00:07:25,420 We're not there yet, but we are approaching it. 136 00:07:25,420 --> 00:07:25,920 Yes? 137 00:07:25,920 --> 00:07:27,816 AUDIENCE: So does an exponential A of T 138 00:07:27,816 --> 00:07:30,090 mean that the universe just keeps expanding forever 139 00:07:30,090 --> 00:07:33,911 and just spins out into nothingness? 140 00:07:33,911 --> 00:07:35,660 PROFESSOR: Yes, an exponentially expanding 141 00:07:35,660 --> 00:07:39,360 A means the universe will continue expanding forever 142 00:07:39,360 --> 00:07:41,910 and ordinary matter will thin out to nothing, 143 00:07:41,910 --> 00:07:43,530 but this vacuum energy density will 144 00:07:43,530 --> 00:07:47,560 remain as a constant contribution. 145 00:07:47,560 --> 00:07:50,250 So the universe would go on expanding exponentially 146 00:07:50,250 --> 00:07:51,830 forever. 147 00:07:51,830 --> 00:07:56,000 Now, there is the possibility that this vacuum that we're 148 00:07:56,000 --> 00:08:00,630 living in is actually what might be called a false vacuum-- that 149 00:08:00,630 --> 00:08:03,310 is, an unstable vacuum-- a vacuum which 150 00:08:03,310 --> 00:08:05,220 behaves like a vacuum for a long time 151 00:08:05,220 --> 00:08:08,730 but is subjected to the possibility of decay. 152 00:08:08,730 --> 00:08:10,600 If that's the case, it's still true 153 00:08:10,600 --> 00:08:16,390 that most of our future space time 154 00:08:16,390 --> 00:08:19,080 will continue to exponentially expand exactly 155 00:08:19,080 --> 00:08:23,400 as this equation shows, but a kind of a Swiss cheese 156 00:08:23,400 --> 00:08:26,120 situation will develop where decays in the vacuum 157 00:08:26,120 --> 00:08:30,860 would occur in places, producing spherical holes 158 00:08:30,860 --> 00:08:34,643 in this otherwise exponentially expanding background. 159 00:08:34,643 --> 00:08:36,059 We'll be talking a little bit more 160 00:08:36,059 --> 00:08:37,830 about that later in the course. 161 00:08:40,557 --> 00:08:41,390 Any other questions? 162 00:08:46,900 --> 00:08:50,130 OK, well what I want to do now is 163 00:08:50,130 --> 00:08:53,920 to work on a few calculations which 164 00:08:53,920 --> 00:08:55,547 I'd like to present today. 165 00:08:55,547 --> 00:08:58,130 If all goes well, we might have as many as three calculations, 166 00:08:58,130 --> 00:09:00,320 or at least two calculations that we'll do 167 00:09:00,320 --> 00:09:03,070 and one that we'll talk about a little bit. 168 00:09:03,070 --> 00:09:04,990 The first thing I want to do-- and I 169 00:09:04,990 --> 00:09:08,100 guess we just talked about starting it last time-- 170 00:09:08,100 --> 00:09:12,140 is calculate the age of the universe in this context. 171 00:09:12,140 --> 00:09:14,490 How do we express the age of the universe 172 00:09:14,490 --> 00:09:17,400 in terms of measurable, cosmological parameters, 173 00:09:17,400 --> 00:09:20,770 taking into account the fact that vacuum energy is 174 00:09:20,770 --> 00:09:23,080 one of the ingredients of our universe 175 00:09:23,080 --> 00:09:25,752 along with radiation and non-relativistic matter, which 176 00:09:25,752 --> 00:09:26,835 we have already discussed. 177 00:09:34,410 --> 00:09:50,227 So to start that calculation we can write down 178 00:09:50,227 --> 00:09:51,685 the first-order Friedmann equation. 179 00:09:55,680 --> 00:10:06,050 A dot over A squared is equal to 8 pi over 3 G times Rho 180 00:10:06,050 --> 00:10:08,960 and now I'm going to divide Rho into all contributions 181 00:10:08,960 --> 00:10:10,480 we know about. 182 00:10:10,480 --> 00:10:17,430 Rho sub M, which represents non-relativistic matter, 183 00:10:17,430 --> 00:10:24,070 plus Rho sub radiation, which represents radiation, 184 00:10:24,070 --> 00:10:28,930 plus vacuum energy, which is our new contribution, which 185 00:10:28,930 --> 00:10:32,280 will not depend on time at all. 186 00:10:32,280 --> 00:10:34,310 And then to complete the equation 187 00:10:34,310 --> 00:10:38,835 there is minus KC squared over A squared. 188 00:10:42,640 --> 00:10:44,890 And the strategy here is really simply 189 00:10:44,890 --> 00:10:47,570 that because we know the time evolution of each 190 00:10:47,570 --> 00:10:49,850 of the terms on the right-hand side, 191 00:10:49,850 --> 00:10:52,270 we will be able to start from wherever we are today 192 00:10:52,270 --> 00:10:55,650 in the universe-- which will just take from data, 193 00:10:55,650 --> 00:10:59,180 the values of these mass densities-- 194 00:10:59,180 --> 00:11:02,050 and we will be able to integrate backwards and ask how far back 195 00:11:02,050 --> 00:11:04,080 do we have to go before we find the time 196 00:11:04,080 --> 00:11:07,180 when the scale factor vanished, which is the instant of the Big 197 00:11:07,180 --> 00:11:07,680 Bang. 198 00:11:15,160 --> 00:11:19,660 So what we want to do is to put into this equation 199 00:11:19,660 --> 00:11:22,660 the time dependents that we know. 200 00:11:22,660 --> 00:11:25,020 So Rho sub M of T, for example, can 201 00:11:25,020 --> 00:11:39,370 be written as A of T naught divided by A of T cubed, 202 00:11:39,370 --> 00:11:43,020 times Rho sub M zero. 203 00:11:43,020 --> 00:11:46,720 And all these zeroes, of course, mean the present time. 204 00:11:46,720 --> 00:11:48,690 So this formula says, first of all, 205 00:11:48,690 --> 00:11:51,440 that the mass density falls off as 1 206 00:11:51,440 --> 00:11:53,000 over the cube of the scale factor. 207 00:11:53,000 --> 00:11:56,070 A of T is the only factor on the right-hand here 208 00:11:56,070 --> 00:12:00,830 that depends on T. The numerator depends on T sub zero, 209 00:12:00,830 --> 00:12:04,260 but not T. The other constant, T is 210 00:12:04,260 --> 00:12:07,080 zero in the numerator and Rho sub M zero. 211 00:12:07,080 --> 00:12:08,990 Rho sub M zero denotes the present value 212 00:12:08,990 --> 00:12:11,241 of the mass density. 213 00:12:11,241 --> 00:12:12,990 And the constants here are just rearranged 214 00:12:12,990 --> 00:12:15,280 so that when T equals T naught, you just 215 00:12:15,280 --> 00:12:17,220 get Rho is equal to its present value. 216 00:12:20,370 --> 00:12:26,040 And we can do the same thing for radiation, 217 00:12:26,040 --> 00:12:28,340 and here I won't write everything out 218 00:12:28,340 --> 00:12:31,240 because most things are the same. 219 00:12:31,240 --> 00:12:33,000 The quantity in brackets will be the same 220 00:12:33,000 --> 00:12:35,910 but this time it will occur to the fourth power 221 00:12:35,910 --> 00:12:38,080 because radiation falls off like the fourth power 222 00:12:38,080 --> 00:12:41,110 of the matter-- fourth power of the scale factor-- 223 00:12:41,110 --> 00:12:45,550 and then we have Rho radiation zero. 224 00:12:45,550 --> 00:12:54,151 And then finally, for the vacuum energy, 225 00:12:54,151 --> 00:12:56,900 we will just write on the blackboard what we already 226 00:12:56,900 --> 00:12:59,700 know, which is that's independent of time. 227 00:13:02,270 --> 00:13:04,640 So this gives us the time dependents of all three terms 228 00:13:04,640 --> 00:13:05,140 here. 229 00:13:12,640 --> 00:13:15,940 Now we could just go from there, but cosmologists 230 00:13:15,940 --> 00:13:17,820 like to talk about mass densities in terms 231 00:13:17,820 --> 00:13:20,970 of the fraction of the critical density, omega. 232 00:13:20,970 --> 00:13:22,720 So we're going to change the notation just 233 00:13:22,720 --> 00:13:25,020 to correspond to the way people usually 234 00:13:25,020 --> 00:13:27,520 talk about these things. 235 00:13:27,520 --> 00:13:31,190 So we will recall that the critical density-- which 236 00:13:31,190 --> 00:13:33,960 is just that total mass density that 237 00:13:33,960 --> 00:13:36,830 makes little K equal to zero and hence 238 00:13:36,830 --> 00:13:41,430 the universe geometrically flat-- so Rho sub C, 239 00:13:41,430 --> 00:13:45,450 we learned, is 3 H squared over 8 pi G 240 00:13:45,450 --> 00:13:51,280 and then we will introduce different components of omega. 241 00:13:51,280 --> 00:13:54,450 So I'm going to write omega sub X here where X really is just 242 00:13:54,450 --> 00:13:59,460 a stand-in for matter or radiation or vacuum energy. 243 00:13:59,460 --> 00:14:01,560 And whichever one of those we're talking about, 244 00:14:01,560 --> 00:14:07,310 omega sub X is just a shorthand for the corresponding mass 245 00:14:07,310 --> 00:14:10,050 density, but normalized by dividing 246 00:14:10,050 --> 00:14:11,235 by this critical density. 247 00:14:19,210 --> 00:14:23,080 And then I'm just going to rewrite these three equations 248 00:14:23,080 --> 00:14:27,150 in terms of omega instead of Rho. 249 00:14:27,150 --> 00:14:42,150 So Rho sub M of T becomes then 3 H naught squared over 8 pi G 250 00:14:42,150 --> 00:14:51,026 times the same A of T zero over A of T cubed, 251 00:14:51,026 --> 00:14:57,140 but not I'm multiplying, omega sub M zero. 252 00:14:57,140 --> 00:14:59,210 And from the definitions we've just written, 253 00:14:59,210 --> 00:15:03,890 this equation is just a rewriting of that equation. 254 00:15:03,890 --> 00:15:06,925 And we can do the same thing, of course, for radiation. 255 00:15:10,250 --> 00:15:13,610 Rho radiation of T is equal to the same factor 256 00:15:13,610 --> 00:15:17,770 out front, the same quantity in brackets 257 00:15:17,770 --> 00:15:21,000 but this time to the fourth power, 258 00:15:21,000 --> 00:15:26,130 and then times omega radiation at the present time. 259 00:15:29,650 --> 00:15:37,020 And finally, Rho vac doesn't really depend on time-- 260 00:15:37,020 --> 00:15:41,220 but we'll write it as if it was a function of time-- 261 00:15:41,220 --> 00:15:44,450 it consists of the same factor of 3 H naught squared over 8 pi 262 00:15:44,450 --> 00:15:50,670 G, and no powers of the quantity in brackets, 263 00:15:50,670 --> 00:15:55,960 but then just multiplying omega sub vac zero. 264 00:16:02,304 --> 00:16:03,280 [ELECTRONIC RINGING] 265 00:16:07,669 --> 00:16:09,960 Everybody should turn off their cell phone, by the way. 266 00:16:09,960 --> 00:16:11,852 [LAUGHING] 267 00:16:18,010 --> 00:16:19,615 OK, sorry about that. 268 00:16:32,010 --> 00:16:40,160 OK, now to make the equation look prettier, 269 00:16:40,160 --> 00:16:43,470 I'm going to rewrite even this last term 270 00:16:43,470 --> 00:16:45,480 as if it has something to do with an omega. 271 00:16:48,110 --> 00:16:59,970 And we can do that by defining omega sub 272 00:16:59,970 --> 00:17:10,319 K zero to be equal to minus KC squared over A squared of T 273 00:17:10,319 --> 00:17:21,450 naught times H naught squared, which is just 274 00:17:21,450 --> 00:17:23,880 the last term that appears there [INAUDIBLE] of factor 275 00:17:23,880 --> 00:17:28,260 of H squared, which we'll be able to factor out. 276 00:17:28,260 --> 00:17:32,580 And doing all that, the original Friedmann equation can now 277 00:17:32,580 --> 00:17:47,670 be rewritten as H squared-- also known as A dot over A squared-- 278 00:17:47,670 --> 00:17:50,500 can be written as H naught squared-- oh, I'm sorry, 279 00:17:50,500 --> 00:17:54,230 one other definition I want to introduce here. 280 00:17:54,230 --> 00:17:58,150 This ratio-- A of T naught over A-- keeps recurring, 281 00:17:58,150 --> 00:18:04,400 so it's nice to give it a name, and I'm 282 00:18:04,400 --> 00:18:06,230 going to give 1 over that a name. 283 00:18:06,230 --> 00:18:12,690 I'm going to let X equal the scale factor normalized 284 00:18:12,690 --> 00:18:13,960 by the scale factor today. 285 00:18:17,627 --> 00:18:19,960 And I might point out that in Barbara Ryden's book, what 286 00:18:19,960 --> 00:18:22,330 I'm calling X is just what she calls the scale factor, 287 00:18:22,330 --> 00:18:25,460 because she chooses to normalize the scale factor so that it's 288 00:18:25,460 --> 00:18:27,030 equal to 1 today. 289 00:18:27,030 --> 00:18:29,630 So we haven't done that yet but we are effectively doing it 290 00:18:29,630 --> 00:18:34,640 here by redefining a new thing X. Having done 291 00:18:34,640 --> 00:18:39,290 that, the right-hand side of the Friedmann equation 292 00:18:39,290 --> 00:18:41,470 can now be written in a simple way. 293 00:18:41,470 --> 00:18:45,430 It's H naught squared over X squared 294 00:18:45,430 --> 00:18:49,360 times a function F of X-- which is just an abbreviation 295 00:18:49,360 --> 00:18:53,720 to not have to write something many times-- this is not, 296 00:18:53,720 --> 00:18:55,330 by any means, a standard definition. 297 00:18:55,330 --> 00:18:57,020 It really is just for today. 298 00:18:57,020 --> 00:19:01,500 It allows us to save some writing on the blackboard. 299 00:19:01,500 --> 00:19:04,920 So I'm going to, for today, be using the abbreviation 300 00:19:04,920 --> 00:19:19,140 F of X is equal to omega sub M zero times X plus omega sub 301 00:19:19,140 --> 00:19:25,780 radiation zero times no powers of X plus omega 302 00:19:25,780 --> 00:19:33,310 sub vac zero times X to the fourth, and finally 303 00:19:33,310 --> 00:19:39,260 plus omega sub K zero times X squared. 304 00:19:42,450 --> 00:19:44,237 And this just lists all the things 305 00:19:44,237 --> 00:19:46,820 that would occur in parentheses here if we factored everything 306 00:19:46,820 --> 00:19:49,220 out. 307 00:19:49,220 --> 00:19:51,300 Notice I factored out some powers of X squared, 308 00:19:51,300 --> 00:19:55,610 so the powers of X that appear here do not look familiar, 309 00:19:55,610 --> 00:19:57,700 but the relative powers do. 310 00:19:57,700 --> 00:20:00,320 That is, omega should fall like-- omega matter should 311 00:20:00,320 --> 00:20:03,670 fall like 1 over X cubed. 312 00:20:03,670 --> 00:20:07,830 Omega radiation should fall off like one power faster 313 00:20:07,830 --> 00:20:09,660 than that, and it does. 314 00:20:09,660 --> 00:20:12,400 This is one less power of X there than there, 315 00:20:12,400 --> 00:20:15,970 and omega vacuum should fall off like four powers of X 316 00:20:15,970 --> 00:20:18,970 different from radiation, and it does, et cetera. 317 00:20:18,970 --> 00:20:22,590 But there's no real offset here that makes 318 00:20:22,590 --> 00:20:24,410 the factors there not look familiar. 319 00:20:33,990 --> 00:20:38,270 OK, all of this was just to put things in a simple form, 320 00:20:38,270 --> 00:20:42,790 but there's one other very useful fact to look at. 321 00:20:42,790 --> 00:20:46,850 Suppose we now apply this for T equals 322 00:20:46,850 --> 00:20:51,132 T zero, which means for X equals 1. 323 00:20:51,132 --> 00:20:54,535 OK, it's true at any time, but in particular 324 00:20:54,535 --> 00:20:56,285 we can look at what it says for X equals 1 325 00:20:56,285 --> 00:20:58,243 and it tells us something about our definitions 326 00:20:58,243 --> 00:21:01,526 that we could have noticed in other ways-- 327 00:21:01,526 --> 00:21:06,510 but didn't notice yet-- which is that we set X equal to 1 here. 328 00:21:06,510 --> 00:21:08,090 These just becomes the sum of omegas. 329 00:21:08,090 --> 00:21:11,870 The powers of X's all become just ones. 330 00:21:11,870 --> 00:21:15,690 And the left-hand side is just H zero squared, 331 00:21:15,690 --> 00:21:18,300 which matches the H zero squared here 332 00:21:18,300 --> 00:21:21,760 because at T equals T naught H squared is H zero squared, 333 00:21:21,760 --> 00:21:23,800 so these H zeros squareds cancel. 334 00:21:23,800 --> 00:21:27,820 So applying it to T equals T naught X 335 00:21:27,820 --> 00:21:38,800 equal to 1, what you get is simply 336 00:21:38,800 --> 00:21:47,520 1 is equal to omega sum M zero plus omega sub radiation zero 337 00:21:47,520 --> 00:21:54,020 plus omega sub vac zero plus omega sub K 338 00:21:54,020 --> 00:21:58,080 zero, which gives us the simplest 339 00:21:58,080 --> 00:22:01,630 way of thinking about what this omega sub K zero really means. 340 00:22:01,630 --> 00:22:05,680 We defined it initially up there in terms of little K, 341 00:22:05,680 --> 00:22:09,250 but for this equation, we can see that omega sub K 342 00:22:09,250 --> 00:22:12,260 zero really is just another way of writing 343 00:22:12,260 --> 00:22:15,020 1 minus all the other omegas. 344 00:22:26,914 --> 00:22:29,330 So you could think of this as being the definition of what 345 00:22:29,330 --> 00:22:32,220 I'm calling omega sub k zero. 346 00:22:32,220 --> 00:22:34,260 So it's a language in which you essentially 347 00:22:34,260 --> 00:22:37,130 think that the total omega has to equal 1, 348 00:22:37,130 --> 00:22:41,410 and whatever is not contained in real matter becomes 349 00:22:41,410 --> 00:22:44,290 a piece of omega sub k, the curvature or contribution 350 00:22:44,290 --> 00:22:44,850 to omega. 351 00:22:52,940 --> 00:22:54,390 OK, now it's really just a matter 352 00:22:54,390 --> 00:22:58,322 of simple manipulations and I-- the main purpose 353 00:22:58,322 --> 00:23:00,030 of defining F of x is to be able to write 354 00:23:00,030 --> 00:23:02,790 these simple manipulations simpler than they would be 355 00:23:02,790 --> 00:23:05,484 if you had to write out what F of x was every time. 356 00:23:05,484 --> 00:23:07,400 We're first just going to take the square root 357 00:23:07,400 --> 00:23:11,390 of the key equation up there-- the Friedmann equation-- and we 358 00:23:11,390 --> 00:23:18,290 get a dot over a is also equal to x dot over x. 359 00:23:18,290 --> 00:23:21,430 Note that the constant of proportionality there, a 360 00:23:21,430 --> 00:23:23,160 of t naught-- which is a constant-- 361 00:23:23,160 --> 00:23:25,260 cancels when you take a dot over a. 362 00:23:25,260 --> 00:23:27,960 So a dot over a is the same as x dot over x. 363 00:23:30,950 --> 00:23:34,770 And that-- just taking the square root of that equation-- 364 00:23:34,770 --> 00:23:37,720 is H naught over x. 365 00:23:40,240 --> 00:23:41,248 Hold on a second. 366 00:23:54,720 --> 00:23:57,030 Yeah, we're taking the square root of the equation, 367 00:23:57,030 --> 00:23:59,130 so yeah, we had H naught squared over x squared. 368 00:23:59,130 --> 00:24:01,130 Here we have H naught over x and then just times 369 00:24:01,130 --> 00:24:02,170 the square root of f. 370 00:24:09,630 --> 00:24:12,280 And these x's cancel each other. 371 00:24:15,620 --> 00:24:16,440 Wait a minute. 372 00:24:28,390 --> 00:24:29,884 Oh, I'm sorry. 373 00:24:29,884 --> 00:24:31,550 They're not supposed to cancel because I 374 00:24:31,550 --> 00:24:33,357 didn't write this quite correctly. 375 00:24:33,357 --> 00:24:34,940 That should have been x to the fourth. 376 00:24:34,940 --> 00:24:35,440 Apologies. 377 00:24:42,020 --> 00:24:43,665 And now here we have x squared. 378 00:24:51,990 --> 00:24:54,310 And this can just be-- by manipulating 379 00:24:54,310 --> 00:25:02,289 where the x's go-- rewritten as x times dx dt. 380 00:25:02,289 --> 00:25:03,830 So I multiply the whole equation by x 381 00:25:03,830 --> 00:25:06,570 squared to get rid of that factor on the right, 382 00:25:06,570 --> 00:25:08,770 and now on the right-hand side we just 383 00:25:08,770 --> 00:25:23,510 have H zero times square root of F. 384 00:25:23,510 --> 00:25:25,940 And now I just want to do the usual trick of separating 385 00:25:25,940 --> 00:25:30,270 the dx pieces from dt pieces in this equation. 386 00:25:30,270 --> 00:25:34,260 And we can rewrite that equation as dt 387 00:25:34,260 --> 00:25:40,540 is equal to 1 over H naught times 388 00:25:40,540 --> 00:25:46,380 x dx over the square root of F. And maybe I'll rewrite it 389 00:25:46,380 --> 00:25:49,310 as the square root of F of x to make it explicit 390 00:25:49,310 --> 00:25:50,990 that F depends on x. 391 00:25:50,990 --> 00:25:54,190 So this is just the rewriting of this equation, 392 00:25:54,190 --> 00:25:57,550 moving factors around, and in this form 393 00:25:57,550 --> 00:25:59,080 we can integrate it and determine 394 00:25:59,080 --> 00:26:02,510 the age of the universe. 395 00:26:02,510 --> 00:26:06,280 The present age of the universe can be obtained just 396 00:26:06,280 --> 00:26:08,820 by integrating this expression from the big bang 397 00:26:08,820 --> 00:26:10,720 up to the present. 398 00:26:10,720 --> 00:26:14,150 And that will be the integral dt from the big bang up 399 00:26:14,150 --> 00:26:15,670 to present, the sum of all the time 400 00:26:15,670 --> 00:26:19,150 intervals from the big bang until now. 401 00:26:19,150 --> 00:26:23,620 And it's just equal to 1 over H zero times 402 00:26:23,620 --> 00:26:32,320 the integral of x dx over the square root of F of x. 403 00:26:32,320 --> 00:26:36,400 And now-- just to think about the limits of integration-- 404 00:26:36,400 --> 00:26:39,810 what should limits of integration be? 405 00:26:39,810 --> 00:26:41,630 AUDIENCE: Zero to one. 406 00:26:41,630 --> 00:26:44,400 PROFESSOR: I hear zero to one, and that's correct. 407 00:26:44,400 --> 00:26:48,190 We're integrating from the big bang up to the present. 408 00:26:48,190 --> 00:26:50,860 At the big bang, a is equal to zero 409 00:26:50,860 --> 00:26:53,790 and therefore x is equal to zero. 410 00:26:53,790 --> 00:26:56,665 And, at the present time, t is equal to t naught 411 00:26:56,665 --> 00:26:58,750 and therefore x is equal to 1. 412 00:26:58,750 --> 00:27:00,480 So we integrate up to one if we want 413 00:27:00,480 --> 00:27:02,480 the present age of the universe. 414 00:27:02,480 --> 00:27:04,760 We could also integrate it up to any other value 415 00:27:04,760 --> 00:27:08,150 of x that we want, and it will tell us the age of universe 416 00:27:08,150 --> 00:27:09,750 when the scale factor had that value. 417 00:27:16,850 --> 00:27:19,250 So this is the final, state of the art 418 00:27:19,250 --> 00:27:22,370 formula for the age of the universe, 419 00:27:22,370 --> 00:27:25,810 expressed in terms of the matter contribution 420 00:27:25,810 --> 00:27:28,390 to omega, the radiation contribution to omega, 421 00:27:28,390 --> 00:27:30,970 and the vacuum contribution to omega, 422 00:27:30,970 --> 00:27:32,501 and the value of H naught. 423 00:27:32,501 --> 00:27:35,000 Those are the only ingredients on the right-hand side there. 424 00:27:35,000 --> 00:27:36,740 And then you can calculate the age. 425 00:27:36,740 --> 00:27:39,140 And it's the completely state of the art formula. 426 00:27:39,140 --> 00:27:42,320 It's exactly what the Planck people did when they told you 427 00:27:42,320 --> 00:27:47,780 that the age of the universe was 13.9 billion years, using 428 00:27:47,780 --> 00:27:50,050 that formula. 429 00:27:50,050 --> 00:27:53,470 Now as far as actually doing the integral, in the general case, 430 00:27:53,470 --> 00:27:55,940 the only way to do it is numerically. 431 00:27:55,940 --> 00:27:58,060 That's how it's usually done. 432 00:27:58,060 --> 00:28:02,020 Special cases can be done analytically. 433 00:28:02,020 --> 00:28:05,110 We've already talked about the case where there's 434 00:28:05,110 --> 00:28:17,900 no cosmological constants, no vacuum energy, but just matter 435 00:28:17,900 --> 00:28:20,765 and curvature-- omega, in this language. 436 00:28:23,730 --> 00:28:27,060 There's another special case which can be done, 437 00:28:27,060 --> 00:28:34,130 which is the case that involves vacuum 438 00:28:34,130 --> 00:28:36,845 energy and nonrelativistic matter. 439 00:28:45,430 --> 00:28:49,990 And this is the flat case, only, that can be done analytically. 440 00:28:49,990 --> 00:28:54,190 So it corresponds to omega radiation 441 00:28:54,190 --> 00:29:00,400 equals omega k equals zero, and that 442 00:29:00,400 --> 00:29:05,959 means that omega matter plus omega vac is equal to 1, 443 00:29:05,959 --> 00:29:08,250 because the sum of all the omegas is always equal to 1. 444 00:29:08,250 --> 00:29:12,730 So in this case I can write an answer for you. 445 00:29:12,730 --> 00:29:21,147 And I don't intend to try to derive this answer, 446 00:29:21,147 --> 00:29:23,480 but it's worth knowing that can be written analytically. 447 00:29:23,480 --> 00:29:26,300 That's the main point, I guess. 448 00:29:26,300 --> 00:29:28,930 So it does get divided into three cases 449 00:29:28,930 --> 00:29:32,730 depending on whether omega matter is larger than, smaller 450 00:29:32,730 --> 00:29:35,120 than, or equal to 1. 451 00:29:35,120 --> 00:29:39,830 So the first case will be if omega matter 452 00:29:39,830 --> 00:29:43,780 zero is greater than 1. 453 00:29:43,780 --> 00:29:45,480 And if omega matter is greater than 1, 454 00:29:45,480 --> 00:29:49,030 that corresponds to omega vac less than 1 455 00:29:49,030 --> 00:29:51,720 because the sum of the two is equal to 1 in all cases, here. 456 00:29:55,052 --> 00:29:58,064 So omega vac has to be less than 0. 457 00:29:58,064 --> 00:30:00,355 So this is not our real universe but it's a calculation 458 00:30:00,355 --> 00:30:06,640 that you can do, and it's 2 over 3 H naught 459 00:30:06,640 --> 00:30:14,590 times the inverse tangent of the square root of omega sub 460 00:30:14,590 --> 00:30:25,150 m zero 1 over the square root of omega sub m zero minus 1. 461 00:30:28,590 --> 00:30:30,945 So if you plug this integral into mathematica, 462 00:30:30,945 --> 00:30:33,320 you should get that answer or something equivalent to it. 463 00:30:36,340 --> 00:30:40,770 For the case-- the borderline case, here-- 464 00:30:40,770 --> 00:30:45,140 where omega matter zero equals 1, 465 00:30:45,140 --> 00:30:47,950 that's the special case where omega vac is equal to 0 466 00:30:47,950 --> 00:30:50,410 because the sum of these is always one. 467 00:30:50,410 --> 00:30:52,006 So this special case in the middle 468 00:30:52,006 --> 00:30:53,380 is the case we already knew, it's 469 00:30:53,380 --> 00:30:55,660 just the matter-dominated flat universe. 470 00:30:55,660 --> 00:31:00,230 So that's two thirds H inverse. 471 00:31:00,230 --> 00:31:05,600 So it's 2 over 3 H naught. 472 00:31:05,600 --> 00:31:17,310 And then, finally, if omega matter zero is less than 1 473 00:31:17,310 --> 00:31:19,690 and then omega vac is positive. 474 00:31:22,334 --> 00:31:23,750 And this [INAUDIBLE] approximation 475 00:31:23,750 --> 00:31:27,780 is our universe, that is, that our universe has possibly zero 476 00:31:27,780 --> 00:31:31,080 curvature-- in any case, unmeasureably small curvature-- 477 00:31:31,080 --> 00:31:35,310 and very, very small radiation for most of its evolution. 478 00:31:35,310 --> 00:31:37,360 So this last case is our universe 479 00:31:37,360 --> 00:31:43,840 except for cases that are near the radiation-dominated era, 480 00:31:43,840 --> 00:31:53,610 and the formula here is 2 over 3 H naught times 481 00:31:53,610 --> 00:31:57,700 the inverse hyperbolic tangent of 1 482 00:31:57,700 --> 00:32:07,390 minus-- square root, excuse me-- of 1 minus omega sub m 483 00:32:07,390 --> 00:32:17,470 zero over the square root of 1 minus omega sub m zero. 484 00:32:20,592 --> 00:32:23,110 OK, so this is just a result obtained 485 00:32:23,110 --> 00:32:27,430 by doing that integral for the special case 486 00:32:27,430 --> 00:32:30,570 that we're talking about. 487 00:32:30,570 --> 00:32:33,430 Now, I don't know any simpler way 488 00:32:33,430 --> 00:32:36,764 to write it except as these three cases. 489 00:32:36,764 --> 00:32:38,555 It is, however, a single analytic function, 490 00:32:38,555 --> 00:32:40,680 and when you graph it-- and I'll show you a graph-- 491 00:32:40,680 --> 00:32:44,080 it is one smooth function right across the range of these three 492 00:32:44,080 --> 00:32:47,160 cases, which is similar to what we found 493 00:32:47,160 --> 00:32:50,855 the earlier for the flat, matter-dominated case. 494 00:32:57,819 --> 00:32:58,360 So let's see. 495 00:32:58,360 --> 00:33:00,400 This is not that yet. 496 00:33:00,400 --> 00:33:03,260 This is the case that we did a long time ago, 497 00:33:03,260 --> 00:33:09,610 actually, the case of a matter-dominated universe 498 00:33:09,610 --> 00:33:11,590 with nothing but nonrelativistic matter 499 00:33:11,590 --> 00:33:13,930 and possibly with curvature. 500 00:33:13,930 --> 00:33:19,830 And I can remind you, here, that what 501 00:33:19,830 --> 00:33:22,210 we found for that model is that we tended to get ages 502 00:33:22,210 --> 00:33:24,540 there were too young. 503 00:33:24,540 --> 00:33:28,090 So if we take a reasonable value for H 504 00:33:28,090 --> 00:33:33,150 of 67 to 70 kilometers per second per megaparsec-- 505 00:33:33,150 --> 00:33:36,140 which is in this range-- and take a reasonable value 506 00:33:36,140 --> 00:33:40,010 for omega-- which is somewhere between, say, 0.2 and 1 507 00:33:40,010 --> 00:33:41,940 depending on what you consider reasonable, 508 00:33:41,940 --> 00:33:44,530 this model doesn't work anyway-- but if you 509 00:33:44,530 --> 00:33:48,820 take omega anywhere between 0.1 and 1, 510 00:33:48,820 --> 00:33:52,780 you get numbers for the age which 511 00:33:52,780 --> 00:33:56,164 are in the order of 10 billion years, which is not old enough 512 00:33:56,164 --> 00:33:57,580 to be consistent with what we know 513 00:33:57,580 --> 00:33:59,700 about the ages of the older stars. 514 00:33:59,700 --> 00:34:02,000 And especially if you think that omega should be one, 515 00:34:02,000 --> 00:34:05,530 you get a very young age of more like 9 billion years, 516 00:34:05,530 --> 00:34:06,970 which is what we found earlier. 517 00:34:06,970 --> 00:34:10,989 This is just a graph of those same equations. 518 00:34:10,989 --> 00:34:13,600 But, if we include vacuum energy, 519 00:34:13,600 --> 00:34:15,330 it makes all the difference. 520 00:34:15,330 --> 00:34:20,090 So this now is a graph of those equations. 521 00:34:20,090 --> 00:34:25,530 What's shown is the age, T naught, as a function of H 522 00:34:25,530 --> 00:34:30,360 and for various values of omega sub m, the same omega sub 523 00:34:30,360 --> 00:34:35,570 m that's called omega sub m zero on the blackboard. 524 00:34:35,570 --> 00:34:44,110 And shown here are the Barbara Ryden a benchmark point, 525 00:34:44,110 --> 00:34:48,429 which is the left-most of these two almost overlapping points. 526 00:34:48,429 --> 00:34:52,409 And also shown here is the favored point 527 00:34:52,409 --> 00:34:56,420 from the WMAP satellite seven-year data. 528 00:34:56,420 --> 00:34:58,530 They lie almost on top of each other. 529 00:34:58,530 --> 00:35:01,190 I didn't get a chance to plot the Planck point, which 530 00:35:01,190 --> 00:35:03,020 is the one that we would consider the most 531 00:35:03,020 --> 00:35:04,940 authoritative these days, but I'll 532 00:35:04,940 --> 00:35:08,370 add that before I post the lectures. 533 00:35:08,370 --> 00:35:11,020 It lies almost on top of these, and it corresponds 534 00:35:11,020 --> 00:35:16,600 to a Hubble expansion rate of a little under 70, 535 00:35:16,600 --> 00:35:24,070 and a vacuum energy contribution of about 0.7, 536 00:35:24,070 --> 00:35:30,700 and therefore a matter contribution of about 0.3. 537 00:35:30,700 --> 00:35:33,010 This curve. 538 00:35:33,010 --> 00:35:38,530 And it gives an age of 13.7, 13.8 billion years-- perfectly 539 00:35:38,530 --> 00:35:42,390 consistent with estimates of the age of the oldest stars. 540 00:35:42,390 --> 00:35:45,480 So this age problem which had been, 541 00:35:45,480 --> 00:35:48,630 until the discovery of the dark energy, a serious problem 542 00:35:48,630 --> 00:35:51,210 in cosmology for many, many years 543 00:35:51,210 --> 00:35:55,800 goes away completely once one adds in the dark energy. 544 00:35:55,800 --> 00:35:57,960 So that's it for the age calculation. 545 00:35:57,960 --> 00:36:01,471 Are there any questions about the age of the universe? 546 00:36:01,471 --> 00:36:01,970 Yes? 547 00:36:01,970 --> 00:36:03,329 AUDIENCE: So when you say dark energy, 548 00:36:03,329 --> 00:36:05,524 are you using that synonymously with vacuum energy? 549 00:36:05,524 --> 00:36:06,440 PROFESSOR: Sorry, yes. 550 00:36:06,440 --> 00:36:07,840 I used the word dark energy there 551 00:36:07,840 --> 00:36:09,904 and I've been talking about vacuum energy, 552 00:36:09,904 --> 00:36:11,070 and what's the relationship? 553 00:36:13,910 --> 00:36:17,240 When I said dark energy I really meant vacuum energy. 554 00:36:17,240 --> 00:36:20,100 In general, the way these words are used 555 00:36:20,100 --> 00:36:24,230 is that vacuum energy has a very specific meaning. 556 00:36:24,230 --> 00:36:27,230 It really does mean the energy of the vacuum, 557 00:36:27,230 --> 00:36:29,210 and by definition, therefore, it does not 558 00:36:29,210 --> 00:36:32,180 change with time, period. 559 00:36:32,180 --> 00:36:34,090 We don't know for sure what this stuff 560 00:36:34,090 --> 00:36:36,690 is that's driving the acceleration of the universe, 561 00:36:36,690 --> 00:36:41,650 and hence the name dark energy, which is more ambiguous. 562 00:36:41,650 --> 00:36:44,120 I think the technical definition of dark energy 563 00:36:44,120 --> 00:36:47,510 is it's whatever the stuff is that's driving 564 00:36:47,510 --> 00:36:49,420 the acceleration of the universe. 565 00:36:49,420 --> 00:36:54,480 And the other conceivable possibility-- and observers 566 00:36:54,480 --> 00:36:57,800 are hard at work trying to distinguish, experimentally, 567 00:36:57,800 --> 00:37:00,890 between these two options-- the other possibility is that it 568 00:37:00,890 --> 00:37:03,910 could be a very slowly evolving scalar 569 00:37:03,910 --> 00:37:08,080 field of the same type that drives inflation 570 00:37:08,080 --> 00:37:10,100 that we'll be talking about later. 571 00:37:10,100 --> 00:37:12,000 But this would be a much lower energy scale 572 00:37:12,000 --> 00:37:14,550 than the inflation of the early universe, 573 00:37:14,550 --> 00:37:17,560 and much more slowly evolving. 574 00:37:17,560 --> 00:37:22,270 So far, we have not yet found any time variation 575 00:37:22,270 --> 00:37:24,829 in the dark energy. 576 00:37:24,829 --> 00:37:27,370 So, so far, everything we' have learned about the dark energy 577 00:37:27,370 --> 00:37:29,210 is consistent with the possibility 578 00:37:29,210 --> 00:37:32,910 that it is simply vacuum energy. 579 00:37:32,910 --> 00:37:33,410 Question. 580 00:37:33,410 --> 00:37:35,326 AUDIENCE: Is the amount of dark energy related 581 00:37:35,326 --> 00:37:36,626 to the amount of dark matter? 582 00:37:36,626 --> 00:37:38,750 PROFESSOR: Is the amount of dark energy related to. 583 00:37:38,750 --> 00:37:40,370 the amount of dark matter? 584 00:37:40,370 --> 00:37:41,210 No. 585 00:37:41,210 --> 00:37:43,300 They're both numbers and they differ 586 00:37:43,300 --> 00:37:45,990 by a factor of 2 and 1/2 or so, but there's 587 00:37:45,990 --> 00:37:49,200 no particular relationship between them that we know of. 588 00:37:49,200 --> 00:37:50,655 AUDIENCE: But doesn't dark matter 589 00:37:50,655 --> 00:37:53,192 imply that they have a certain attraction to bodies 590 00:37:53,192 --> 00:37:56,370 around it, which is a form of energy? 591 00:37:56,370 --> 00:37:59,706 PROFESSOR: Yeah, well let's talk about this later. 592 00:37:59,706 --> 00:38:02,081 AUDIENCE: Do we have any idea what dark energy is at all? 593 00:38:02,081 --> 00:38:03,622 PROFESSOR: OK, the question is, do we 594 00:38:03,622 --> 00:38:05,760 have any idea what dark energy is at all? 595 00:38:05,760 --> 00:38:07,090 And the answer is probably yes. 596 00:38:07,090 --> 00:38:09,780 That is, I think there's a good chance it is vacuum energy. 597 00:38:09,780 --> 00:38:11,950 Now if you ask what is vacuum energy, what is it 598 00:38:11,950 --> 00:38:14,236 about the vacuum that gives it this nonzero energy, 599 00:38:14,236 --> 00:38:15,610 there we're pretty much clueless. 600 00:38:15,610 --> 00:38:17,443 I was going to talk about that a little more 601 00:38:17,443 --> 00:38:19,260 at the end of today, if we get there. 602 00:38:19,260 --> 00:38:23,080 But whatever property of the vacuum 603 00:38:23,080 --> 00:38:26,310 it is that gives it its energy-- we know of many, 604 00:38:26,310 --> 00:38:27,810 it's just a matter of what dominates 605 00:38:27,810 --> 00:38:31,180 and how they add up-- the end result is pretty 606 00:38:31,180 --> 00:38:34,680 much the same as far as the phenomenology of vacuum energy. 607 00:38:34,680 --> 00:38:37,040 So we understand the phenomenology of vacuum energy, 608 00:38:37,040 --> 00:38:39,580 I would say, completely. 609 00:38:39,580 --> 00:38:42,020 The big issue, which I'll talk about either 610 00:38:42,020 --> 00:38:43,920 at the end of today or next time, 611 00:38:43,920 --> 00:38:47,164 is trying to estimate the magnitude of the vacuum energy, 612 00:38:47,164 --> 00:38:48,830 and there we're really totally clueless, 613 00:38:48,830 --> 00:38:50,040 as I will try to describe. 614 00:38:53,600 --> 00:38:56,700 OK, that's it for my slides. 615 00:38:56,700 --> 00:38:58,920 OK, I wanted to now talk about another very important 616 00:38:58,920 --> 00:39:02,300 calculation, which is basically the calculation which 617 00:39:02,300 --> 00:39:08,270 led to the original evidence that the universe is 618 00:39:08,270 --> 00:39:10,630 accelerating to begin with. 619 00:39:14,190 --> 00:39:18,020 OK, discovery that the universe was accelerating 620 00:39:18,020 --> 00:39:20,810 was made, as I said earlier, by two groups of astronomers 621 00:39:20,810 --> 00:39:28,355 in 1998, and the key observation was using a type 1a supernovae 622 00:39:28,355 --> 00:39:31,860 as standard candles to measure the expansion 623 00:39:31,860 --> 00:39:33,810 rate of the universe versus time, 624 00:39:33,810 --> 00:39:35,580 looking back into the past. 625 00:39:35,580 --> 00:39:38,530 And basically what they found is that when they look back 626 00:39:38,530 --> 00:39:42,180 about 5, 6 billion years, the expansion rate then 627 00:39:42,180 --> 00:39:45,530 was actually slower than expansion rate now, 628 00:39:45,530 --> 00:39:47,480 meaning that the universe has accelerated. 629 00:39:47,480 --> 00:39:49,270 And that was the key observation. 630 00:39:49,270 --> 00:39:51,700 So the question for us to calculate 631 00:39:51,700 --> 00:39:56,040 is, what do we expect, as a function of these parameters, 632 00:39:56,040 --> 00:40:00,940 for redshift versus luminosity? 633 00:40:00,940 --> 00:40:03,890 These astronomers, by using type 1a supernovae 634 00:40:03,890 --> 00:40:06,050 as standard candles, are basically 635 00:40:06,050 --> 00:40:09,200 using the luminosity measurements of these type 636 00:40:09,200 --> 00:40:13,810 1a supernovae as estimates of their distance. 637 00:40:13,810 --> 00:40:16,390 So what they actually measured was simply luminosity verses 638 00:40:16,390 --> 00:40:18,960 redshift, and that's what we will learn how to calculate, 639 00:40:18,960 --> 00:40:20,700 and the formula that will get will 640 00:40:20,700 --> 00:40:22,570 be, again, exactly the formula that they 641 00:40:22,570 --> 00:40:27,420 used when they were trying to fit their data-- to understand 642 00:40:27,420 --> 00:40:28,920 what their data was telling them-- 643 00:40:28,920 --> 00:40:31,610 about possible acceleration of the universe. 644 00:40:40,550 --> 00:40:42,240 So the calculation we're about to do 645 00:40:42,240 --> 00:40:43,990 is really nothing new to you folks 646 00:40:43,990 --> 00:40:45,830 because we have calculated luminosities 647 00:40:45,830 --> 00:40:47,220 in another contexts. 648 00:40:47,220 --> 00:40:50,630 Now we will just write down the equations in their full glory, 649 00:40:50,630 --> 00:40:55,210 including the contribution due to vacuum energy. 650 00:41:13,609 --> 00:41:15,150 So we'd like to do these calculations 651 00:41:15,150 --> 00:41:16,925 in a way that allows for curvature, even 652 00:41:16,925 --> 00:41:18,800 though-- in the end-- we're going to discover 653 00:41:18,800 --> 00:41:20,550 that the curvature of our universe 654 00:41:20,550 --> 00:41:23,055 is-- as far as anybody can tell-- negligible. 655 00:41:23,055 --> 00:41:25,180 But people still look for it and it still very well 656 00:41:25,180 --> 00:41:28,077 could be there at the level of one part in 1,000 657 00:41:28,077 --> 00:41:29,035 or something like that. 658 00:41:29,035 --> 00:41:32,770 But at the level of 1 part in 100, it's not there. 659 00:41:32,770 --> 00:41:39,380 So we begin by writing down the Robertson-Walker metric, 660 00:41:39,380 --> 00:41:43,375 ds squared is equal to minus c squared 661 00:41:43,375 --> 00:41:51,850 dt squared plus a squared of t times dr 662 00:41:51,850 --> 00:42:02,290 squared over 1 minus little k times r squared plus r squared 663 00:42:02,290 --> 00:42:11,320 d theta squared plus sine squared theta d phi squared, 664 00:42:11,320 --> 00:42:14,660 end curly brackets. 665 00:42:14,660 --> 00:42:17,620 OK, so this is the metric that we're familiar with. 666 00:42:17,620 --> 00:42:20,280 We're going to be interested, mainly, in radial motion, 667 00:42:20,280 --> 00:42:22,630 and if you're interested mainly in radial motion, 668 00:42:22,630 --> 00:42:25,540 it helps to simplify the radial part of this metric 669 00:42:25,540 --> 00:42:28,030 by using a different radial variable. 670 00:42:28,030 --> 00:42:31,200 And we've done this before, also. 671 00:42:31,200 --> 00:42:33,310 At this point, we really need to pick 672 00:42:33,310 --> 00:42:35,102 whether we're talking about open or closed. 673 00:42:35,102 --> 00:42:36,643 If we're talking about flat, we don't 674 00:42:36,643 --> 00:42:37,890 need to do anything, really. 675 00:42:37,890 --> 00:42:39,810 If you eliminate k, here, the radial part 676 00:42:39,810 --> 00:42:41,362 is as simple as it gets. 677 00:42:41,362 --> 00:42:43,070 But if we want talk about open or closed, 678 00:42:43,070 --> 00:42:46,070 it pays to use different variables, 679 00:42:46,070 --> 00:42:47,650 and the variable that we'd use would 680 00:42:47,650 --> 00:42:49,507 be different in two cases. 681 00:42:49,507 --> 00:42:51,590 So I'm going to consider the closed-universe case. 682 00:43:00,010 --> 00:43:07,870 And I'm going to introduce an angle, sine of psi 683 00:43:07,870 --> 00:43:10,580 being equal to the square root of k-- which 684 00:43:10,580 --> 00:43:14,870 is positive in this case-- times little r. 685 00:43:14,870 --> 00:43:18,920 And this psi is, in fact, if you trace everything back, 686 00:43:18,920 --> 00:43:22,462 the angle from the w-axis that we originally 687 00:43:22,462 --> 00:43:23,920 used when we constructed the closed 688 00:43:23,920 --> 00:43:26,320 Robertson-Walker universe in the first place. 689 00:43:26,320 --> 00:43:29,160 But now we're essentially working backwards. 690 00:43:29,160 --> 00:43:32,426 We've learned to know and love this expression, 691 00:43:32,426 --> 00:43:33,800 so we're going to just rewrite it 692 00:43:33,800 --> 00:43:36,440 in terms of the new variable, sin of psi 693 00:43:36,440 --> 00:43:41,690 equals the square root of k times r. 694 00:43:41,690 --> 00:43:46,330 And from this, by just differentiation, 695 00:43:46,330 --> 00:43:49,730 you discover that deep psi is equal to the square root 696 00:43:49,730 --> 00:43:57,840 of k times dr over cosine psi. 697 00:43:57,840 --> 00:44:00,100 And that is equal to the square root 698 00:44:00,100 --> 00:44:07,430 of k times dr over the square root of 1 minus kr squared. 699 00:44:07,430 --> 00:44:11,270 So this, then, fits in very nicely with the metric itself. 700 00:44:11,270 --> 00:44:13,489 The metric is just the square of this factor, 701 00:44:13,489 --> 00:44:15,530 and therefore it is just proportional to deep psi 702 00:44:15,530 --> 00:44:18,910 squared all by itself. 703 00:44:18,910 --> 00:44:23,670 And rewriting the whole metric, we can write it 704 00:44:23,670 --> 00:44:29,640 as ds squared is equal to minus c squared dt squared 705 00:44:29,640 --> 00:44:33,560 plus a new scale factor-- which I'll define 706 00:44:33,560 --> 00:44:42,990 in a second in terms of the old one-- times deep psi squared 707 00:44:42,990 --> 00:44:46,560 plus-- now, the angular term becomes 708 00:44:46,560 --> 00:44:49,090 nonstandard instead of just having an r squared here, 709 00:44:49,090 --> 00:44:52,052 we have sine squared of psi. 710 00:44:52,052 --> 00:44:54,010 Which is, of course, proportional to r squared. 711 00:44:59,780 --> 00:45:05,030 And that multiplies d theta squared plus sine squared 712 00:45:05,030 --> 00:45:12,770 theta d phi squared, end curly brackets. 713 00:45:12,770 --> 00:45:25,372 And a tilde is just equal to our original 714 00:45:25,372 --> 00:45:30,822 a divided by the square root of k. 715 00:45:30,822 --> 00:45:34,020 So we scaled it. 716 00:45:34,020 --> 00:45:37,480 And I should mention that I'm putting a tilde here 717 00:45:37,480 --> 00:45:40,250 because we've already written an a without a tilde there, 718 00:45:40,250 --> 00:45:41,730 and they're not equal to other. 719 00:45:41,730 --> 00:45:44,200 If you want to just start here, you can, 720 00:45:44,200 --> 00:45:46,290 and then there's no need for the tilde. 721 00:45:46,290 --> 00:45:48,245 You could just call this the scale factor 722 00:45:48,245 --> 00:45:49,680 and it doesn't need a tilde. 723 00:45:49,680 --> 00:45:52,600 The tilde is only to distinguish the two cases from each other. 724 00:46:17,998 --> 00:46:21,202 AUDIENCE: [INAUDIBLE] 725 00:46:21,202 --> 00:46:21,910 PROFESSOR: Sorry? 726 00:46:21,910 --> 00:46:24,200 AUDIENCE: Do a and a tilde have different units? 727 00:46:24,200 --> 00:46:25,348 PROFESSOR: Didn't hear you? 728 00:46:25,348 --> 00:46:27,440 AUDIENCE: Do a and a tilde have different units? 729 00:46:27,440 --> 00:46:29,690 PROFESSOR: They do have different-- yes, a and a tilde 730 00:46:29,690 --> 00:46:31,660 do have different units. 731 00:46:31,660 --> 00:46:37,710 That's right, and that's because in what 732 00:46:37,710 --> 00:46:40,130 one might call conventional units here, 733 00:46:40,130 --> 00:46:43,070 r is some kind of a coordinate distance. 734 00:46:43,070 --> 00:46:45,960 So in my language I'd measure it in notches, 735 00:46:45,960 --> 00:46:49,810 and then a has units of meters per notch. 736 00:46:49,810 --> 00:46:52,250 On the other hand, here psi is an angle. 737 00:46:52,250 --> 00:46:54,900 It is naturally dimensionless. 738 00:46:54,900 --> 00:46:57,360 So one doesn't introduce notches in this case, 739 00:46:57,360 --> 00:46:59,470 and therefore a just has units of length-- 740 00:46:59,470 --> 00:47:01,532 a tilde, rather-- just has units of length. 741 00:47:06,730 --> 00:47:11,670 OK, now we want to imagine that some distant galaxy is 742 00:47:11,670 --> 00:47:15,410 radiating-- or a distant supernova, perhaps-- 743 00:47:15,410 --> 00:47:17,815 and we want to ask, what is the intensity 744 00:47:17,815 --> 00:47:20,780 of the radiation that we receive on earth? 745 00:47:20,780 --> 00:47:22,530 And we'll draw the same picture that we've 746 00:47:22,530 --> 00:47:25,390 drawn at least twice before, if not more. 747 00:47:25,390 --> 00:47:27,600 We'll put the source in the middle. 748 00:47:32,520 --> 00:47:35,740 We'll imagine a sphere surrounding the source, 749 00:47:35,740 --> 00:47:38,570 with the source of the center, and we'll 750 00:47:38,570 --> 00:47:42,270 imagine that the sphere has been drawn so that our detector is 751 00:47:42,270 --> 00:47:44,357 on the surface of the sphere. 752 00:47:44,357 --> 00:47:45,440 This will be the detector. 753 00:47:54,650 --> 00:47:59,760 And we'll give a symbol for the area of the detector. 754 00:47:59,760 --> 00:48:00,560 It will be a. 755 00:48:05,240 --> 00:48:07,860 And we'll imagine drawing this in our co-moving coordinate 756 00:48:07,860 --> 00:48:11,590 system where psi is our radial variable. 757 00:48:11,590 --> 00:48:17,070 So the sphere here will be at some value, 758 00:48:17,070 --> 00:48:22,327 psi equals psi sub d-- where d stands for detector-- and psi 759 00:48:22,327 --> 00:48:23,410 equals zero at the center. 760 00:49:07,780 --> 00:49:09,780 OK, I'm going to make the same kind of arguments 761 00:49:09,780 --> 00:49:11,260 we've made in the past. 762 00:49:11,260 --> 00:49:18,690 We say that the fraction of light 763 00:49:18,690 --> 00:49:24,480 that hits the sphere-- which hits the detector-- 764 00:49:24,480 --> 00:49:36,359 is just equal to the area of the detector 765 00:49:36,359 --> 00:49:37,525 over the area of the sphere. 766 00:49:40,612 --> 00:49:44,682 Now, the area of detector is, by definition, a. 767 00:49:44,682 --> 00:49:46,182 The area of the sphere we have to be 768 00:49:46,182 --> 00:49:47,840 a little bit careful about because we 769 00:49:47,840 --> 00:49:51,540 have to calculate the area of the sphere using the metric. 770 00:49:51,540 --> 00:49:54,000 Now, the metric is slightly nontrivial, 771 00:49:54,000 --> 00:49:58,070 but the sphere is just described by varying theta and phi. 772 00:49:58,070 --> 00:50:00,580 And if we just vary theta and phi, this piece of the metric 773 00:50:00,580 --> 00:50:04,180 is what we're used to-- it's the standard Euclidean spherical 774 00:50:04,180 --> 00:50:07,730 element-- and the coefficient that multiplies is just 775 00:50:07,730 --> 00:50:10,100 the square of the radius of that sphere. 776 00:50:10,100 --> 00:50:16,850 So the radius of our sphere is a tilde times sine psi. 777 00:50:16,850 --> 00:50:20,640 That's the important thing that we get from the metric. 778 00:50:20,640 --> 00:50:22,930 The thing that multiplies d theta squared and d phi 779 00:50:22,930 --> 00:50:24,513 squared, et cetera, is just the square 780 00:50:24,513 --> 00:50:26,750 of the radius of the sphere that determines distances 781 00:50:26,750 --> 00:50:28,460 on the surface of the sphere. 782 00:50:28,460 --> 00:50:32,690 So what goes here is 4 pi times the radius squared. 783 00:50:32,690 --> 00:50:41,570 So it's 4 pi times a tilde squared of t naught times 784 00:50:41,570 --> 00:50:45,860 sine squared of psi sub d. 785 00:50:45,860 --> 00:50:49,770 It's t naught because we're interested in what 786 00:50:49,770 --> 00:50:53,410 happens when we detect this radiation today. 787 00:50:53,410 --> 00:50:56,710 Our detector is detecting it today and has area a today, 788 00:50:56,710 --> 00:51:01,480 and we want to compare it with entire sphere that surrounds 789 00:51:01,480 --> 00:51:04,990 this distant source as that sphere appears today, 790 00:51:04,990 --> 00:51:07,960 so that all of the distances are measured today, 791 00:51:07,960 --> 00:51:09,900 and therefore can be properly compared. 792 00:51:14,580 --> 00:51:18,090 The other thing we have to remember 793 00:51:18,090 --> 00:51:19,340 is the effect of the redshift. 794 00:51:24,040 --> 00:51:25,555 The redshift, we've said earlier, 795 00:51:25,555 --> 00:51:29,810 and it's just a repetition, it reduces the energy 796 00:51:29,810 --> 00:51:34,790 of each photon by a factor of 1 plus z, the redshift, 797 00:51:34,790 --> 00:51:38,480 and similarly it reduces the rate at which photons 798 00:51:38,480 --> 00:51:43,570 are arriving at the sphere by that same factor-- 1 plus z. 799 00:51:43,570 --> 00:51:46,420 It basically says that any clock slows down 800 00:51:46,420 --> 00:51:49,320 by a factor of 1 plus z, and that clock 801 00:51:49,320 --> 00:51:51,230 could be the frequency of the photon-- which 802 00:51:51,230 --> 00:51:53,870 affects its energy proportionally-- 803 00:51:53,870 --> 00:51:55,490 or the arrival rate of the photons. 804 00:51:55,490 --> 00:52:02,110 That's also a clock that get time dilated in the same way. 805 00:52:02,110 --> 00:52:15,440 So we get two factors of 1 plus z sub s, I'll call it. 806 00:52:15,440 --> 00:52:20,310 s for z of the source, the z between the time 807 00:52:20,310 --> 00:52:22,780 of emission at the source to a time 808 00:52:22,780 --> 00:52:26,350 where it arrives at us today. 809 00:52:26,350 --> 00:52:32,580 So 1 plus z is equal to a of t naught divided 810 00:52:32,580 --> 00:52:34,885 by a of t emission. 811 00:52:42,705 --> 00:52:44,900 I'll just put it here to remind us. 812 00:52:48,280 --> 00:53:00,240 One from redshift of photons and one from arrival rate. 813 00:53:15,310 --> 00:53:17,960 OK, putting that together we can now 814 00:53:17,960 --> 00:53:26,200 say that the total power received is 815 00:53:26,200 --> 00:53:30,830 equal to the power originally emitted by the source-- p 816 00:53:30,830 --> 00:53:37,060 will just be the power emitted by the source-- divided 817 00:53:37,060 --> 00:53:42,810 by 1 plus the redshift z of the source squared, 818 00:53:42,810 --> 00:53:44,330 and then just times the fraction. 819 00:53:44,330 --> 00:53:57,130 A over 4 pi a twiddle squared sine squared psi d. 820 00:54:28,630 --> 00:54:30,720 And then, finally, what we're really interested in 821 00:54:30,720 --> 00:54:35,370 is J-- the intensity of the source 822 00:54:35,370 --> 00:54:39,300 as we measure it-- which is just the power received 823 00:54:39,300 --> 00:54:47,430 by our detector divided by its area. 824 00:54:47,430 --> 00:54:51,370 So from this formula we just get rid of the A there. 825 00:54:51,370 --> 00:54:55,080 We can write it as the power emitted by the source, capital 826 00:54:55,080 --> 00:55:02,480 P, divided by 4 pi 1 plus z sub s 827 00:55:02,480 --> 00:55:10,660 squared a twiddle squared of t naught times sine 828 00:55:10,660 --> 00:55:14,350 squared psi sub d. 829 00:55:25,220 --> 00:55:27,040 Now that effectively is the answer 830 00:55:27,040 --> 00:55:31,950 to this question except that we prefer to rewrite it 831 00:55:31,950 --> 00:55:35,342 in terms of things that are more directly 832 00:55:35,342 --> 00:55:36,425 meaningful to astronomers. 833 00:55:39,160 --> 00:55:42,050 a twiddle is not particularly meaningful to the astronomer. 834 00:55:42,050 --> 00:55:43,970 The redshift is, that's OK. 835 00:55:43,970 --> 00:55:45,550 But a twiddle is not particularly 836 00:55:45,550 --> 00:55:49,899 meaningful to an astronomer, nor is sine squared of psi sub d. 837 00:55:49,899 --> 00:55:51,940 Now, many astronomers who know general relativity 838 00:55:51,940 --> 00:55:54,030 can figure this out, of course, but it's 839 00:55:54,030 --> 00:55:55,197 our job to figure it out. 840 00:55:55,197 --> 00:55:56,780 We would like to express this in terms 841 00:55:56,780 --> 00:56:01,490 of things that are directly measured by astronomers. 842 00:56:01,490 --> 00:56:08,460 So to do that, first of all, a tilde-- 843 00:56:08,460 --> 00:56:10,551 to get a tilde related to other things, 844 00:56:10,551 --> 00:56:12,300 it really just goes back to the definition 845 00:56:12,300 --> 00:56:15,180 that we gave for omega sub k sub 0. 846 00:56:15,180 --> 00:56:19,930 And if you look back at that definition, 847 00:56:19,930 --> 00:56:24,730 you'll find that a of t naught tilde 848 00:56:24,730 --> 00:56:30,130 is just equal to c times the inverse of the present Hubble 849 00:56:30,130 --> 00:56:36,800 expansion rate times the square root of minus omega 850 00:56:36,800 --> 00:56:38,280 sub k comma zero. 851 00:56:41,840 --> 00:56:45,000 And this is for the close-universe case. 852 00:56:45,000 --> 00:56:48,260 The closed-universe case, little k is positive. 853 00:56:48,260 --> 00:56:51,247 But if you remember the definition of omega sub k 854 00:56:51,247 --> 00:56:53,580 naught-- maybe I should write it back on the blackboard, 855 00:56:53,580 --> 00:56:54,570 or is it findable? 856 00:56:54,570 --> 00:56:55,540 It's not findable. 857 00:57:06,630 --> 00:57:09,840 The original definition big A for this omega sub k naught 858 00:57:09,840 --> 00:57:20,120 was just minus Kc squared over a squared of t naught H naught 859 00:57:20,120 --> 00:57:21,620 squared. 860 00:57:21,620 --> 00:57:23,340 So this is just rewriting of that, 861 00:57:23,340 --> 00:57:26,230 and for our closed-universe case, k is positive, 862 00:57:26,230 --> 00:57:28,395 omega sub k naught is negative, this is then 863 00:57:28,395 --> 00:57:30,260 the square root of a positive number with that extra minus 864 00:57:30,260 --> 00:57:31,680 sign, so everything fits together. 865 00:57:36,600 --> 00:57:38,580 So that takes care of expressing a tilde 866 00:57:38,580 --> 00:57:41,060 in terms of measurable things. 867 00:57:41,060 --> 00:57:42,290 We use this formula. 868 00:57:42,290 --> 00:57:44,875 Expansion rate is measurable, omega sub k comma zero 869 00:57:44,875 --> 00:57:47,500 is measurable. 870 00:57:47,500 --> 00:58:06,290 And then, in terms of sine squared psi sub d, 871 00:58:06,290 --> 00:58:09,750 we obtain that by reminding ourselves 872 00:58:09,750 --> 00:58:15,800 that we know how to trace light rays through this universe. 873 00:58:15,800 --> 00:58:18,250 Light waves just travel locally at the speed c, 874 00:58:18,250 --> 00:58:21,240 they travel locally on null geodesics. 875 00:58:21,240 --> 00:58:23,650 So if we're looking at a radial light ray, 876 00:58:23,650 --> 00:58:31,550 this metric tells us-- if we apply it to a radial light ray 877 00:58:31,550 --> 00:58:36,600 where ds squared equals minus dc squared-- where ds squared has 878 00:58:36,600 --> 00:58:44,080 to be zero-- that says that minus c squared dt 879 00:58:44,080 --> 00:58:54,502 squared plus a twiddle squared of t d psi squared equals zero. 880 00:58:54,502 --> 00:58:55,960 This is just the equation that says 881 00:58:55,960 --> 00:59:00,060 we have a null line, a null radial line. 882 00:59:00,060 --> 00:59:07,180 That implies that deep psi dt has 883 00:59:07,180 --> 00:59:17,354 to equals c over a twiddle of t, which 884 00:59:17,354 --> 00:59:18,770 is a formula that, in other cases, 885 00:59:18,770 --> 00:59:21,650 we try to motivate just by using intuition. 886 00:59:21,650 --> 00:59:23,410 But, in that case, we were probably not 887 00:59:23,410 --> 00:59:24,868 talking about curved universe where 888 00:59:24,868 --> 00:59:26,670 the intuition is a little bit less strong. 889 00:59:26,670 --> 00:59:28,480 But you see it does follow immediately 890 00:59:28,480 --> 00:59:31,140 from assuming that we're talking about a null geodesic 891 00:59:31,140 --> 00:59:32,610 in the Robertson-Walker metric. 892 00:59:51,260 --> 00:59:54,130 Now, the point is that the Friedmann equation, which we've 893 00:59:54,130 --> 00:59:58,835 been writing and rewriting, tells us what to do with that. 894 01:00:07,076 --> 01:00:08,700 The Friedmann equation basically allows 895 01:00:08,700 --> 01:00:12,310 us to integrate that because it allows 896 01:00:12,310 --> 01:00:19,750 us to express a in terms of x, and we 897 01:00:19,750 --> 01:00:21,580 know some things about x. 898 01:00:21,580 --> 01:00:24,010 So let me try to get that on the blackboard, here. 899 01:00:27,910 --> 01:00:35,120 We know that H squared-- which is x dot over x squared-- 900 01:00:35,120 --> 01:00:40,050 can be written as H zero squared over x to the fourth times 901 01:00:40,050 --> 01:00:42,700 this famous function F of x. 902 01:00:49,440 --> 01:01:00,840 And psi of a given redshift, according to this equation, 903 01:01:00,840 --> 01:01:07,380 could just be obtained by integration of the time 904 01:01:07,380 --> 01:01:09,960 that the source emits the radiation up 905 01:01:09,960 --> 01:01:16,990 to the present time of c over a twiddle of t times dt. 906 01:01:26,400 --> 01:01:30,130 And now to rewrite this in terms of redshift, 907 01:01:30,130 --> 01:01:35,370 we can use the fact that 1 plus z is equal to 1 908 01:01:35,370 --> 01:01:38,680 over x because we know how to relate 1 plus z 909 01:01:38,680 --> 01:01:40,179 to the scale factor. 910 01:01:40,179 --> 01:01:41,970 1 plus z is just the ratio of scale factors 911 01:01:41,970 --> 01:01:46,200 and it's precisely the ratio that we called 1 over x. 912 01:01:46,200 --> 01:01:52,770 And we can then differentiate this equation 913 01:01:52,770 --> 01:02:02,560 and find that dz is equal to minus a twiddle of t naught 914 01:02:02,560 --> 01:02:11,040 over a twiddle of t squared times a twiddle dot times 915 01:02:11,040 --> 01:02:17,460 dt, rewriting x in terms of a of t naught over a of t. 916 01:02:20,780 --> 01:02:27,095 And this, then, is equal to minus a twiddle of t naught 917 01:02:27,095 --> 01:02:36,420 times H of t times, oops, times dt over a tilde of t. 918 01:02:42,640 --> 01:02:52,450 And this allows us to replace the dt that appears there 919 01:02:52,450 --> 01:03:00,080 and the final relationship is that psi of s 920 01:03:00,080 --> 01:03:06,020 is equal to 1 over a tilde of t naught times the integral 921 01:03:06,020 --> 01:03:17,910 from zero up to z sub s of c over H of z dz. 922 01:03:25,130 --> 01:03:26,970 Yeah, I think that looks like it works. 923 01:03:35,770 --> 01:03:38,260 So it really is just a matter of changing variables 924 01:03:38,260 --> 01:03:42,570 to express things in terms of H and integrating over z 925 01:03:42,570 --> 01:03:44,340 instead of integrating over t. 926 01:03:46,781 --> 01:03:48,280 And the usefulness of that is simply 927 01:03:48,280 --> 01:03:50,280 that z is the variable that astronomers 928 01:03:50,280 --> 01:03:51,300 use to measure time. 929 01:03:55,600 --> 01:03:58,120 And this then can be written in more detail, 930 01:03:58,120 --> 01:04:02,820 and it really finishes the answer more or less. 931 01:04:02,820 --> 01:04:07,840 Psi of z sub s can be written just-- writing 932 01:04:07,840 --> 01:04:14,060 in what a tilde is according to our definition here-- 933 01:04:14,060 --> 01:04:18,030 square root of the magnitude of omega comma k 934 01:04:18,030 --> 01:04:22,660 zero-- this could also have been written as minus omega of k 935 01:04:22,660 --> 01:04:26,380 comma 0 because we know it's a negative quantity-- 936 01:04:26,380 --> 01:04:39,940 and then times the integral from 0 to z sub s and integral dz. 937 01:04:39,940 --> 01:04:43,680 Now I'm just writing H as a function of z. 938 01:04:43,680 --> 01:04:48,910 Earlier we had written H-- it's no longer on the screen, 939 01:04:48,910 --> 01:04:52,980 I guess-- earlier we had written H in terms of F of z, 940 01:04:52,980 --> 01:04:56,040 oh, excuse me, F of x. 941 01:04:56,040 --> 01:05:00,890 x is related to z simply by this formula. 942 01:05:00,890 --> 01:05:02,340 So since the integral was written 943 01:05:02,340 --> 01:05:04,430 with z as the variable of integration, 944 01:05:04,430 --> 01:05:06,600 I'm going to rewrite the integrand in terms of z, 945 01:05:06,600 --> 01:05:08,835 but it really is just our old friend F of x. 946 01:05:13,720 --> 01:05:18,970 So it would be the square root of omega sub m zero 1 947 01:05:18,970 --> 01:05:31,420 plus z cubed plus omega sub radiation zero times 1 plus z 948 01:05:31,420 --> 01:05:36,090 to the fourth, all inside the square root, 949 01:05:36,090 --> 01:05:43,950 here, plus omega sub vac zero plus omega sub k 950 01:05:43,950 --> 01:05:47,025 zero times 1 plus z squared. 951 01:05:54,120 --> 01:05:58,920 And this, then, is the answer for psi of z. 952 01:05:58,920 --> 01:06:04,470 And then we put that into here and replace a twiddle by this, 953 01:06:04,470 --> 01:06:07,950 and we get a formula for what we're looking for, 954 01:06:07,950 --> 01:06:11,360 an expression for the actual measured intensity 955 01:06:11,360 --> 01:06:16,190 of the source at the Earth in terms of the parameters chosen 956 01:06:16,190 --> 01:06:19,530 here-- the current values of omega 957 01:06:19,530 --> 01:06:21,270 and the redshift of the source. 958 01:06:21,270 --> 01:06:24,010 And that's all that goes into this final formula. 959 01:06:24,010 --> 01:06:25,960 So if you know the current values of omega 960 01:06:25,960 --> 01:06:27,640 and the redshift of the source, you 961 01:06:27,640 --> 01:06:32,450 can calculate what you expect the measure intensity to be 962 01:06:32,450 --> 01:06:34,685 in terms of the intrinsic intensity. 963 01:06:34,685 --> 01:06:36,600 And that's exactly what the supernova 964 01:06:36,600 --> 01:06:39,970 people did in 1998 using exactly this formula-- 965 01:06:39,970 --> 01:06:43,350 nothing different-- and discovered that, 966 01:06:43,350 --> 01:06:46,640 in order to fit their data, they needed a very significant 967 01:06:46,640 --> 01:06:49,585 contribution from this vacuum energy, namely a contribution 968 01:06:49,585 --> 01:06:50,970 in the order of 60 or 70%. 969 01:06:53,800 --> 01:06:56,460 So we will stop there for today. 970 01:06:56,460 --> 01:06:59,470 We will continue on Thursday to talk a little bit more 971 01:06:59,470 --> 01:07:03,630 about the physics of vacuum energy.