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 --> 00:00:10,160 to offer high quality educational resources for free. 5 00:00:10,160 --> 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 00:00:21,827 --> 00:00:22,410 PROFESSOR: OK. 9 00:00:22,410 --> 00:00:24,440 In That case, let's get started. 10 00:00:24,440 --> 00:00:28,680 First with our review of what we talked about last time. 11 00:00:28,680 --> 00:00:31,800 We began by talking about the dynamics 12 00:00:31,800 --> 00:00:35,080 of a flat radiation-dominated universe, which 13 00:00:35,080 --> 00:00:36,900 is very straightforward. 14 00:00:36,900 --> 00:00:40,440 We start with Friedmann's equation for a flat universe. 15 00:00:40,440 --> 00:00:42,910 For radiation, rho is proportional to one over a 16 00:00:42,910 --> 00:00:44,610 to the fourth. 17 00:00:44,610 --> 00:00:47,320 So that gets us a dot over a squared 18 00:00:47,320 --> 00:00:51,140 is equal to a constant over a to the fourth. 19 00:00:51,140 --> 00:00:54,850 Rearranging that we can write it as a times da 20 00:00:54,850 --> 00:00:59,420 is equal to the square root of that constant times dt. 21 00:00:59,420 --> 00:01:02,450 And then we can just integrate both sides. 22 00:01:02,450 --> 00:01:05,190 So you get one half a squared is equal to the square root 23 00:01:05,190 --> 00:01:10,270 of the constant times t, plus a new constant, constant prime. 24 00:01:10,270 --> 00:01:13,060 OK, then we say that we can adjust this constant prime 25 00:01:13,060 --> 00:01:14,570 by resetting our clocks. 26 00:01:14,570 --> 00:01:16,570 And we haven't said anything yet that determines 27 00:01:16,570 --> 00:01:18,130 how our clocks are set. 28 00:01:18,130 --> 00:01:20,540 So the standard convention is to set your clocks 29 00:01:20,540 --> 00:01:23,960 so that t equals 0 is the time when the scale factor is zero. 30 00:01:23,960 --> 00:01:26,950 And that corresponds to constant prime equals zero, 31 00:01:26,950 --> 00:01:30,090 as one can see by just looking at that equation. 32 00:01:30,090 --> 00:01:33,060 So that's what we do when the constant prime term disappears. 33 00:01:33,060 --> 00:01:35,760 And then we don't care about the constant of proportionality 34 00:01:35,760 --> 00:01:36,850 anyway. 35 00:01:36,850 --> 00:01:39,530 For a flat universe the constant of proportionality 36 00:01:39,530 --> 00:01:40,660 is completely meaningless. 37 00:01:40,660 --> 00:01:43,820 It just tells you how many meters per notch 38 00:01:43,820 --> 00:01:46,590 you're dealing with, so it defines your notch, 39 00:01:46,590 --> 00:01:49,490 but otherwise has no physical meaning whatever. 40 00:01:49,490 --> 00:01:52,400 So the important bottom line is that a of t 41 00:01:52,400 --> 00:01:55,310 is proportional to the square root of t 42 00:01:55,310 --> 00:01:57,990 for a flat [INAUDIBLE] dominated universe. 43 00:01:57,990 --> 00:01:59,920 And once one knows that, one can easily 44 00:01:59,920 --> 00:02:03,890 get quite a few other things. 45 00:02:03,890 --> 00:02:04,390 OK. 46 00:02:18,730 --> 00:02:21,030 As I was just saying, once you now how a of t behaves 47 00:02:21,030 --> 00:02:23,780 you can immediately calculate lots of other things. 48 00:02:23,780 --> 00:02:27,360 And in particular h is a dot over a and that is just 1 49 00:02:27,360 --> 00:02:28,590 over 2t. 50 00:02:28,590 --> 00:02:30,740 So we know what h is, a function of time, 51 00:02:30,740 --> 00:02:32,370 even without putting any more details 52 00:02:32,370 --> 00:02:34,450 about what kind of radiation we have. 53 00:02:34,450 --> 00:02:35,620 It doesn't matter. 54 00:02:35,620 --> 00:02:38,960 You still get h is equal to 1 over 2t. 55 00:02:38,960 --> 00:02:42,940 The horizon distance is given by the formula a of t times 56 00:02:42,940 --> 00:02:45,830 the integral from 0 to t of c over a of t 57 00:02:45,830 --> 00:02:48,466 prime, dt prime, the interval represents 58 00:02:48,466 --> 00:02:50,090 the total co-moving distance that light 59 00:02:50,090 --> 00:02:53,140 could travel between time 0 at time t. 60 00:02:53,140 --> 00:02:55,200 And then one multiplies that by the scale 61 00:02:55,200 --> 00:02:57,930 factor time t to turn it into a physical distance, 62 00:02:57,930 --> 00:03:00,380 and that then becomes the horizon distance of time t. 63 00:03:00,380 --> 00:03:01,939 And that is two times ct. 64 00:03:01,939 --> 00:03:03,480 You might remember it was three times 65 00:03:03,480 --> 00:03:06,550 ct for the matter-dominated case. 66 00:03:06,550 --> 00:03:10,640 And then finally you even know exactly what 67 00:03:10,640 --> 00:03:13,240 the mass density is as a function of time. 68 00:03:13,240 --> 00:03:19,140 Because 8 squared is 8 pi over 3 rho a pi g over 3 times rho. 69 00:03:19,140 --> 00:03:20,340 And we know what h is. 70 00:03:20,340 --> 00:03:22,340 So that tells me what row is also. 71 00:03:22,340 --> 00:03:25,940 We actually know the mass density as a function of time, 72 00:03:25,940 --> 00:03:27,280 independent of anything else. 73 00:03:30,730 --> 00:03:31,230 OK. 74 00:03:31,230 --> 00:03:35,920 Then we began talking about black-body radiation, which 75 00:03:35,920 --> 00:03:41,160 is basically a gas of massless particles at a temperature t 76 00:03:41,160 --> 00:03:42,810 that is a gas of massless particles 77 00:03:42,810 --> 00:03:44,166 in thermal equilibrium. 78 00:03:44,166 --> 00:03:45,790 And it turns out that the temperature t 79 00:03:45,790 --> 00:03:48,620 determines almost everything. 80 00:03:48,620 --> 00:03:52,580 So the energy density of black-body radiation, u, 81 00:03:52,580 --> 00:03:56,160 which is the same as rho mass density times c squared, 82 00:03:56,160 --> 00:03:59,820 is equal to a kind of a fudge factor g times pi 83 00:03:59,820 --> 00:04:06,120 squared over 30 times kt to the fourth, over h bar c cubed. 84 00:04:06,120 --> 00:04:09,914 And this fudge factor g is equal to 2 for photons. 85 00:04:09,914 --> 00:04:12,330 And the reason we gave it a letter instead of just writing 86 00:04:12,330 --> 00:04:14,070 a two in the formula in the first place 87 00:04:14,070 --> 00:04:16,403 is that we'll soon be talking about black-body radiation 88 00:04:16,403 --> 00:04:18,329 of other kinds of particles, and g 89 00:04:18,329 --> 00:04:21,339 will be different for different kinds of particles. 90 00:04:21,339 --> 00:04:25,090 The two for photons is given the number 91 00:04:25,090 --> 00:04:28,920 two because there are two spin states of photons. 92 00:04:28,920 --> 00:04:32,670 Photons that are massless spin one particle and massless 93 00:04:32,670 --> 00:04:35,935 particles, all we have maximum spins, they don't have spins 94 00:04:35,935 --> 00:04:39,250 in the middle, so they spin along one axis for a photon 95 00:04:39,250 --> 00:04:42,650 is either plus or minus 1, which is two spins. 96 00:04:42,650 --> 00:04:44,760 That's the quantum mechanical description. 97 00:04:44,760 --> 00:04:48,330 It corresponds completely to the classical description 98 00:04:48,330 --> 00:04:50,552 that you can separate any light wave 99 00:04:50,552 --> 00:04:52,760 into left-circularly polarized and a right-circularly 100 00:04:52,760 --> 00:04:55,450 polarized part. 101 00:04:55,450 --> 00:04:57,880 Or equivalently, you could separate it into an x-polarized 102 00:04:57,880 --> 00:05:01,290 and and y-polarized part. 103 00:05:01,290 --> 00:05:04,260 And you can get x-polarization by superimposing 104 00:05:04,260 --> 00:05:07,960 left and right, so these are not alternative polarizations. 105 00:05:07,960 --> 00:05:10,480 They're just two different ways of describing 106 00:05:10,480 --> 00:05:12,200 the general polarization. 107 00:05:12,200 --> 00:05:14,390 The general causation is a linear combination 108 00:05:14,390 --> 00:05:20,150 of either left and right or linear combination of x and y. 109 00:05:20,150 --> 00:05:22,900 In any case, the basis has two basis elements, 110 00:05:22,900 --> 00:05:25,620 and that's where the two comes from. 111 00:05:25,620 --> 00:05:27,490 Next item on our list is the pressure 112 00:05:27,490 --> 00:05:30,100 which could be calculated from the statistical mechanics, 113 00:05:30,100 --> 00:05:32,634 and we've already calculated by other means. 114 00:05:32,634 --> 00:05:34,467 But the answer, not matter how you calculate 115 00:05:34,467 --> 00:05:36,820 is is p equals 1/3 u. 116 00:05:39,860 --> 00:05:41,707 Next we talked about the number density. 117 00:05:41,707 --> 00:05:44,040 And these are all calculations that we didn't really do. 118 00:05:44,040 --> 00:05:46,510 I'm just quoting standard calculations 119 00:05:46,510 --> 00:05:48,450 from statistical mechanics. 120 00:05:48,450 --> 00:05:50,800 And you can learn how to do them presumably 121 00:05:50,800 --> 00:05:53,630 by some [INAUDIBLE] class. 122 00:05:53,630 --> 00:05:56,370 The number density is a different constant g 123 00:05:56,370 --> 00:05:58,450 star, in general it's different. 124 00:05:58,450 --> 00:06:00,400 For photons it ends up being the same. 125 00:06:00,400 --> 00:06:02,260 But in general a different constant, 126 00:06:02,260 --> 00:06:06,130 g star times zeta of 3 where zeta of 3 127 00:06:06,130 --> 00:06:10,430 is the Riemann zeta function evaluated at argument three. 128 00:06:10,430 --> 00:06:13,460 And that's equal to 1 over 1 cubed plus 1 over 2 cubed, 129 00:06:13,460 --> 00:06:15,512 plus 1 over 3 cubed dot dot dot. 130 00:06:15,512 --> 00:06:20,570 And it adds up to 1.202 to three decimal places. 131 00:06:20,570 --> 00:06:24,150 And then the rest divided by pi squared then times kt cubed, 132 00:06:24,150 --> 00:06:25,880 over h bar c cubed. 133 00:06:25,880 --> 00:06:28,436 So the number [INAUDIBLE] like the cube of the temperature, 134 00:06:28,436 --> 00:06:30,560 while the energy density went like the fourth power 135 00:06:30,560 --> 00:06:31,351 of the temperature. 136 00:06:34,260 --> 00:06:37,810 And the g star that appears here as I mentioned already I think 137 00:06:37,810 --> 00:06:39,070 is 2 for photons. 138 00:06:39,070 --> 00:06:42,100 And we'll learn more later about how 139 00:06:42,100 --> 00:06:45,420 to apply this to other kinds of particles. 140 00:06:45,420 --> 00:06:51,490 And finally the entropy density is the same g 141 00:06:51,490 --> 00:06:56,670 as we had for energy densities times 2 pi squared over 45 k 142 00:06:56,670 --> 00:06:58,440 to the fourth t cubed, again it goes 143 00:06:58,440 --> 00:07:02,570 like t cubed, divided by h bar c cubed. 144 00:07:02,570 --> 00:07:06,780 And entropy is a somewhat subtle concept. 145 00:07:06,780 --> 00:07:09,730 Fortunately for our purposes we will not 146 00:07:09,730 --> 00:07:13,935 need to know much at all about what entropy actually is. 147 00:07:13,935 --> 00:07:19,450 I might just say for some sense of completeness that entropy 148 00:07:19,450 --> 00:07:22,590 is a measure of quote the disorder of a state. 149 00:07:22,590 --> 00:07:25,170 And this "disorder" means some measure 150 00:07:25,170 --> 00:07:29,040 of the number of different microscopic quantum states 151 00:07:29,040 --> 00:07:32,165 that contribute to a given macroscopic classical 152 00:07:32,165 --> 00:07:32,665 description. 153 00:07:35,870 --> 00:07:39,470 The important thing for us is, well first of all, 154 00:07:39,470 --> 00:07:41,200 that the second law of thermodynamics 155 00:07:41,200 --> 00:07:43,677 tells us that entropy never decreases, 156 00:07:43,677 --> 00:07:46,260 but we're going to make use of a much stronger statement which 157 00:07:46,260 --> 00:07:49,040 holds very well for the early universe, which 158 00:07:49,040 --> 00:07:54,680 is that if the system stays close to family equilibrium, 159 00:07:54,680 --> 00:07:57,230 then the entropy doesn't change. 160 00:07:57,230 --> 00:08:00,070 And in the early universe I think for every process 161 00:08:00,070 --> 00:08:04,150 that we're going to discuss that condition holds. 162 00:08:04,150 --> 00:08:06,840 The system stays close to thermal equilibrium and entropy 163 00:08:06,840 --> 00:08:08,210 is conserved. 164 00:08:08,210 --> 00:08:09,780 The one exception will be inflation, 165 00:08:09,780 --> 00:08:11,905 which we'll learn about near the end of the course. 166 00:08:11,905 --> 00:08:15,840 At the end of inflation there's a humongously entropy producing 167 00:08:15,840 --> 00:08:17,276 phase transition. 168 00:08:17,276 --> 00:08:19,794 And if inflation is right essentially all of the entropy 169 00:08:19,794 --> 00:08:21,210 that we have int he universe today 170 00:08:21,210 --> 00:08:23,610 was produced during that phase transition. 171 00:08:23,610 --> 00:08:26,170 Before that there was only negligible entropy. 172 00:08:32,620 --> 00:08:33,120 OK. 173 00:08:33,120 --> 00:08:35,549 That's it for my summary. 174 00:08:35,549 --> 00:08:36,345 Any questions? 175 00:08:40,850 --> 00:08:41,503 Yes? 176 00:08:41,503 --> 00:08:43,918 AUDIENCE: I know it's a law that entropy only increases. 177 00:08:43,918 --> 00:08:46,450 Is there anything behind that or is it 178 00:08:46,450 --> 00:08:50,180 just it's a law of physics? 179 00:08:50,180 --> 00:08:52,440 PROFESSOR: Yeah, that's a subtle question, 180 00:08:52,440 --> 00:08:56,952 which I think if you ask 10 different physicists you'll 181 00:08:56,952 --> 00:08:58,285 likely get 10 different answers. 182 00:09:00,800 --> 00:09:04,830 But one thing's for sure is that it does not 183 00:09:04,830 --> 00:09:07,120 follow as a consequence from the other laws of physics 184 00:09:07,120 --> 00:09:08,167 that we know. 185 00:09:08,167 --> 00:09:09,750 The other laws of physics that we know 186 00:09:09,750 --> 00:09:12,290 are essentially time, reversal, and variant. 187 00:09:12,290 --> 00:09:16,690 So why entropy always increases is something of a mystery. 188 00:09:16,690 --> 00:09:21,310 The usual story is something like the early universe, 189 00:09:21,310 --> 00:09:24,390 for reasons unknown, started in the state of peculiarly 190 00:09:24,390 --> 00:09:24,905 low entropy. 191 00:09:27,690 --> 00:09:30,430 And because it started low it's approaching the equilibrium 192 00:09:30,430 --> 00:09:32,670 value, which is much larger. 193 00:09:32,670 --> 00:09:34,735 So that's a possible explanation. 194 00:09:34,735 --> 00:09:36,360 I actually am in the process of writing 195 00:09:36,360 --> 00:09:39,296 a paper about the growth of entropy in the hour of time, 196 00:09:39,296 --> 00:09:40,920 and maybe I'll get a chance to tell you 197 00:09:40,920 --> 00:09:42,961 about that sometime before the end of the course, 198 00:09:42,961 --> 00:09:44,404 but I won't try to explain it now. 199 00:09:44,404 --> 00:09:45,820 But it's a slightly different idea 200 00:09:45,820 --> 00:09:47,927 about what could explain it. 201 00:09:47,927 --> 00:09:49,260 But it's something of a mystery. 202 00:09:49,260 --> 00:09:52,680 Nobody really knows what determines the arrow of time, 203 00:09:52,680 --> 00:09:54,300 why entropy only goes one way. 204 00:09:58,432 --> 00:09:59,265 Any other questions? 205 00:10:02,860 --> 00:10:03,801 OK. 206 00:10:03,801 --> 00:10:05,300 What I want to do now is to continue 207 00:10:05,300 --> 00:10:06,960 talking about black-body radiation. 208 00:10:09,977 --> 00:10:11,810 I think we've said everything that we wanted 209 00:10:11,810 --> 00:10:15,100 to say about photons, but now we want 210 00:10:15,100 --> 00:10:18,350 to apply to other kinds of particles. 211 00:10:18,350 --> 00:10:22,160 And we're going to begin with neutrinos, because neutrinos 212 00:10:22,160 --> 00:10:25,680 certainly do account for a significant fraction 213 00:10:25,680 --> 00:10:27,783 of the entropy in the universe today even. 214 00:10:31,490 --> 00:10:37,090 Neutrinos were, until around 1990 or so, 215 00:10:37,090 --> 00:10:39,940 thought to be massless. 216 00:10:39,940 --> 00:10:43,860 Now we know that they in fact have a small mass. 217 00:10:43,860 --> 00:10:47,190 I think I mentioned last time that we have never 218 00:10:47,190 --> 00:10:51,520 measured the mass of a neutrino and largely for that reason, 219 00:10:51,520 --> 00:10:53,837 we don't actually know what the masses are. 220 00:10:53,837 --> 00:10:55,420 But what we have measured is something 221 00:10:55,420 --> 00:10:58,620 that's quantum mechanical and rather peculiar, which 222 00:10:58,620 --> 00:11:02,220 is that one type of neutrino, and neutrinos 223 00:11:02,220 --> 00:11:04,220 come in three types, or flavors. 224 00:11:13,990 --> 00:11:22,770 And that is nu sub e, nu sub mu, and nu sub tau. 225 00:11:22,770 --> 00:11:25,980 And what we've seen is that neutrinos of one flavor 226 00:11:25,980 --> 00:11:28,800 can turn into, just by traveling through space, 227 00:11:28,800 --> 00:11:31,360 a neutrino of the other flavor. 228 00:11:31,360 --> 00:11:35,810 And that can only happen if there's a non-zero mass. 229 00:11:35,810 --> 00:11:40,540 And what it particular measures is 230 00:11:40,540 --> 00:11:44,720 delta m squared between the neutrinos. 231 00:11:44,720 --> 00:11:47,240 And I think I even wrote on the formulas 232 00:11:47,240 --> 00:11:52,180 on the blackboard for the known limits of that. 233 00:11:52,180 --> 00:11:54,100 So I won't write it again. 234 00:11:54,100 --> 00:12:00,464 But neutrino oscillation, which is 235 00:12:00,464 --> 00:12:02,130 what we call this, the conversion of one 236 00:12:02,130 --> 00:12:06,480 kind of neutrino to another, implies 237 00:12:06,480 --> 00:12:12,649 that delta m squared is not equal to zero. 238 00:12:12,649 --> 00:12:14,190 So it's still, in principle, possible 239 00:12:14,190 --> 00:12:16,106 that one of these neutrinos could be massless, 240 00:12:16,106 --> 00:12:17,552 but they can't all be massless. 241 00:12:17,552 --> 00:12:19,385 If one is massless, the other two have mass. 242 00:12:28,580 --> 00:12:30,150 And of delta m squared is very small. 243 00:12:30,150 --> 00:12:32,525 So these would be smaller masses than any other particles 244 00:12:32,525 --> 00:12:35,740 that we know of in the universe. 245 00:12:35,740 --> 00:12:40,800 Now for purposes of cosmology it turns out 246 00:12:40,800 --> 00:12:44,120 that you can get by by thinking of the neutrinos 247 00:12:44,120 --> 00:12:46,220 as being massless. 248 00:12:46,220 --> 00:12:48,970 And then we'll start by discussing it that way. 249 00:12:48,970 --> 00:12:50,800 It's not quite as trivial as just 250 00:12:50,800 --> 00:12:52,490 saying that the mass is a small number. 251 00:12:52,490 --> 00:12:54,580 So if you have a formula that involves the mass, 252 00:12:54,580 --> 00:12:55,952 you can usually neglect it. 253 00:12:55,952 --> 00:12:58,410 It's a little bit more subtle because whether the neutrinos 254 00:12:58,410 --> 00:13:01,489 are massless or massive affects the number of spin states. 255 00:13:01,489 --> 00:13:03,780 And we're going to even treat the number of spin states 256 00:13:03,780 --> 00:13:06,740 as if the neutrino is massless, which obviously 257 00:13:06,740 --> 00:13:09,010 is a bit of a cheat and needs further explanation. 258 00:13:09,010 --> 00:13:12,290 But I'm going to do it first for massless neutrinos 259 00:13:12,290 --> 00:13:15,020 and we'll come back later to discuss 260 00:13:15,020 --> 00:13:18,610 why you can get the same answers if the neutrino has a mass. 261 00:13:29,230 --> 00:13:35,695 So we're going to start out imagining 262 00:13:35,695 --> 00:13:36,820 that the nu's are massless. 263 00:13:47,230 --> 00:13:49,890 And in that approximation there is only 264 00:13:49,890 --> 00:13:54,410 one spin state for a neutrino and it's left-handed. 265 00:14:21,450 --> 00:14:27,520 And that means that the spin points the opposite direction 266 00:14:27,520 --> 00:14:29,930 from the momentum. 267 00:14:29,930 --> 00:14:32,280 So if the neutrino is spinning like this, 268 00:14:32,280 --> 00:14:33,770 which my right hand will correspond 269 00:14:33,770 --> 00:14:36,180 to a spin in that direction, the momentum 270 00:14:36,180 --> 00:14:39,380 will always be in that direction. 271 00:14:39,380 --> 00:14:41,350 And I can do the same thing this way. 272 00:14:41,350 --> 00:14:43,519 If it's spinning like this, the spin is that way, 273 00:14:43,519 --> 00:14:44,810 the momentum would be that way. 274 00:14:44,810 --> 00:14:49,520 The momentum's always the opposite direction of the spin. 275 00:14:49,520 --> 00:14:51,490 Now you might realize and that that 276 00:14:51,490 --> 00:14:55,200 leads to a question about Lorentz invariance. 277 00:14:55,200 --> 00:14:58,620 If the spin and the momentum point in opposite directions 278 00:14:58,620 --> 00:15:01,640 in some frame, it's not obvious if they would also 279 00:15:01,640 --> 00:15:04,590 point in opposite directions in some other frame. 280 00:15:04,590 --> 00:15:06,420 But it turns out to be true. 281 00:15:06,420 --> 00:15:10,070 One can verify that, which we're not going to do, 282 00:15:10,070 --> 00:15:13,400 but it is a Lorentz invariance statement, 283 00:15:13,400 --> 00:15:15,110 as long as the particle's massless. 284 00:15:15,110 --> 00:15:16,680 If the particle's not massless it 285 00:15:16,680 --> 00:15:20,580 is clearly not a Lorentz invariance statement. 286 00:15:20,580 --> 00:15:22,400 And that's easy to say. 287 00:15:22,400 --> 00:15:25,240 If the particle were not massless, and say 288 00:15:25,240 --> 00:15:28,450 it was spinning this way, so the spin is that way, 289 00:15:28,450 --> 00:15:31,080 and the momentum is that way, that way 290 00:15:31,080 --> 00:15:33,850 the situation and the particle had a mass, 291 00:15:33,850 --> 00:15:36,710 then I could change Lorentz frames by jumping into a rocket 292 00:15:36,710 --> 00:15:39,510 ship, and shooting off in this direction, 293 00:15:39,510 --> 00:15:41,700 the same direction the particle is moving, 294 00:15:41,700 --> 00:15:44,710 and since the particle is moving with some finite velocity, 295 00:15:44,710 --> 00:15:47,230 if it has a mass, there has to be a finite velocity, 296 00:15:47,230 --> 00:15:49,930 in principle the spaceship can overtake it. 297 00:15:49,930 --> 00:15:52,724 And when the spaceship overtakes it, 298 00:15:52,724 --> 00:15:54,390 from the point of view of the spaceship, 299 00:15:54,390 --> 00:15:56,670 the momentum will now be pointing that way, 300 00:15:56,670 --> 00:16:00,220 but the spin doesn't change when the spaceship overtakes it. 301 00:16:00,220 --> 00:16:02,180 So this relationship between spin and momentum 302 00:16:02,180 --> 00:16:04,830 is manifestly not Lorentz invariant 303 00:16:04,830 --> 00:16:06,530 when the particle has a mass, which 304 00:16:06,530 --> 00:16:08,450 is why we are cheating in a somewhat big way 305 00:16:08,450 --> 00:16:10,580 here, by saying that we're going to treat 306 00:16:10,580 --> 00:16:11,950 the neutrinos is massless. 307 00:16:11,950 --> 00:16:14,680 We're going to even be ignoring this fact 308 00:16:14,680 --> 00:16:16,790 that what we're saying about the spin and momentum 309 00:16:16,790 --> 00:16:19,590 is not even Lorentz invariant if the particle has a mass. 310 00:16:19,590 --> 00:16:21,640 And we'll make better excuses for that later, 311 00:16:21,640 --> 00:16:25,954 but for now we'll see simple picture first, and then 312 00:16:25,954 --> 00:16:27,870 talk about the more complicated picture later. 313 00:16:38,580 --> 00:16:41,800 Anti neutrinos, which I'll write with a bar over it, 314 00:16:41,800 --> 00:16:43,075 also have one spin state. 315 00:16:51,760 --> 00:16:53,830 And as you might guess since anti particles are 316 00:16:53,830 --> 00:16:55,705 kind of the opposite of particles, 317 00:16:55,705 --> 00:16:56,580 that is right-handed. 318 00:17:09,550 --> 00:17:17,020 Now in addition neutrinos differ from photons in that neutrinos 319 00:17:17,020 --> 00:17:20,640 belong to a class of particles called fermions. 320 00:17:20,640 --> 00:17:23,140 Photons are bosons. 321 00:17:23,140 --> 00:17:25,530 How many of you already know about [INAUDIBLE] 322 00:17:25,530 --> 00:17:27,511 fermion or boson? 323 00:17:27,511 --> 00:17:28,010 OK. 324 00:17:28,010 --> 00:17:29,380 Most, but not quite all. 325 00:17:29,380 --> 00:17:31,240 Fair enough. 326 00:17:31,240 --> 00:17:35,620 So fermions are particles that obey the Fermi exclusion 327 00:17:35,620 --> 00:17:37,640 principle that you're very likely 328 00:17:37,640 --> 00:17:39,600 learned about in a chemistry class someplace. 329 00:17:39,600 --> 00:17:41,100 It says that no two particles can 330 00:17:41,100 --> 00:17:44,360 be in exactly the same quantum state. 331 00:17:44,360 --> 00:17:46,240 Bosons do not obey such a principle. 332 00:17:46,240 --> 00:17:47,990 In fact, bosons are even more likely to be 333 00:17:47,990 --> 00:17:50,460 found in the same quantum state. 334 00:17:50,460 --> 00:17:52,460 And that's the important difference. 335 00:17:52,460 --> 00:17:56,690 In quantum field theory, and only in quantum field theory, 336 00:17:56,690 --> 00:17:59,474 you can't do this in just non relativistic quantum mechanics, 337 00:17:59,474 --> 00:18:01,890 but in quantum field theory you can prove something called 338 00:18:01,890 --> 00:18:04,730 the spin statistics theorem, which 339 00:18:04,730 --> 00:18:08,640 says that particles with half integer spins, that is 1/2, 340 00:18:08,640 --> 00:18:12,920 3/2, et cetera, are necessarily fermions. 341 00:18:12,920 --> 00:18:17,500 And particles with integer spins are necessarily bosons. 342 00:18:17,500 --> 00:18:19,840 So we know whether a particle is a fermion or a Boson 343 00:18:19,840 --> 00:18:21,500 as soon as we know its spin. 344 00:18:21,500 --> 00:18:23,320 And in the case of neutrino, it's spin 1/2. 345 00:18:32,160 --> 00:18:42,872 So nu's have spin one half. 346 00:18:42,872 --> 00:18:44,080 And that makes them fermions. 347 00:18:59,450 --> 00:19:03,440 Now the statistical mechanics formulas that we just went over 348 00:19:03,440 --> 00:19:05,980 come about from counting states. 349 00:19:05,980 --> 00:19:07,820 Basically the underlying principle 350 00:19:07,820 --> 00:19:12,269 is that the system is likely to be in any state 351 00:19:12,269 --> 00:19:14,810 that you could imagine with the probability of e to the minus 352 00:19:14,810 --> 00:19:17,690 the energy of that state, divided by kt 353 00:19:17,690 --> 00:19:19,570 all in the exponent. 354 00:19:19,570 --> 00:19:21,900 So it's state counting that determines these formulas. 355 00:19:21,900 --> 00:19:24,149 And that means it's going to be different for fermions 356 00:19:24,149 --> 00:19:26,032 and bosons because for bosons you're 357 00:19:26,032 --> 00:19:27,740 going to count situations where there are 358 00:19:27,740 --> 00:19:30,140 many particles in the same state. 359 00:19:30,140 --> 00:19:32,567 And for fermions you're not. 360 00:19:32,567 --> 00:19:34,025 And in particular that means you're 361 00:19:34,025 --> 00:19:38,090 going to be counting more states for the bosons 362 00:19:38,090 --> 00:19:41,230 and to be counting fewer states for the fermions. 363 00:19:41,230 --> 00:19:44,440 So you would expect these constants g and g star 364 00:19:44,440 --> 00:19:49,780 to be smaller for fermions than they are for those bosons. 365 00:19:49,780 --> 00:19:51,530 And that indeed is true. 366 00:20:06,540 --> 00:20:23,418 So for fermions g gets an extra factor 367 00:20:23,418 --> 00:20:30,610 that is multiplied by 7/8s. 368 00:20:35,290 --> 00:20:46,760 And g star is multiplied by 3/4. 369 00:20:57,430 --> 00:20:59,790 So I think we could have predicted that these g's would 370 00:20:59,790 --> 00:21:01,672 get factors that are less than 1, 371 00:21:01,672 --> 00:21:03,670 simply because we're counting fewer states. 372 00:21:03,670 --> 00:21:06,890 I think we can also predict without doing any calculations 373 00:21:06,890 --> 00:21:10,870 that g should have a bigger factor than g star. 374 00:21:10,870 --> 00:21:12,740 Remember g star is the factor that 375 00:21:12,740 --> 00:21:15,140 appears in the number density. 376 00:21:15,140 --> 00:21:18,600 So if we want to calculate the average energy per particle 377 00:21:18,600 --> 00:21:21,170 at a given temperature, we would take 378 00:21:21,170 --> 00:21:25,460 a formula which has a g in it for the energy density, 379 00:21:25,460 --> 00:21:26,930 and divide it by a formula that has 380 00:21:26,930 --> 00:21:29,060 a g star in it for the number density. 381 00:21:29,060 --> 00:21:31,170 The energy density divided by the number density 382 00:21:31,170 --> 00:21:32,610 is just the energy per particle. 383 00:21:32,610 --> 00:21:34,610 And this tells us that we'll get a number that's 384 00:21:34,610 --> 00:21:38,150 bigger than 1 for the fermions. 385 00:21:38,150 --> 00:21:39,690 So for fermions, there's slightly 386 00:21:39,690 --> 00:21:44,380 more energy per particle than there is for bosons. 387 00:21:44,380 --> 00:21:47,330 And I think that's easily believable because fermions 388 00:21:47,330 --> 00:21:48,580 obey this exclusion principle. 389 00:21:48,580 --> 00:21:50,871 It means once you put one particle in the lowest energy 390 00:21:50,871 --> 00:21:52,400 state, you can't put any more there. 391 00:21:52,400 --> 00:21:55,210 You have to put them in higher energy states. 392 00:21:55,210 --> 00:21:57,042 So the expectation would be that you 393 00:21:57,042 --> 00:21:59,000 have more energy per particle with the fermions 394 00:21:59,000 --> 00:22:03,170 and that is indeed what the [INAUDIBLE] calculation 395 00:22:03,170 --> 00:22:05,980 indicates. 396 00:22:05,980 --> 00:22:16,130 So now we're ready to write down a formula for g, 397 00:22:16,130 --> 00:22:18,560 for the neutrinos. 398 00:22:18,560 --> 00:22:20,150 And this will be the overall g that 399 00:22:20,150 --> 00:22:22,390 occurs in the formula for the energy 400 00:22:22,390 --> 00:22:28,250 density and the entropy density, and the pressure for neutrinos. 401 00:22:28,250 --> 00:22:33,690 And it will, we'll write it out as a string of factors. 402 00:22:33,690 --> 00:22:37,284 We'll first of all have this factor 7/8 coming from the fact 403 00:22:37,284 --> 00:22:38,200 that they're fermions. 404 00:22:45,910 --> 00:22:49,320 Then we'll have a factor of three 405 00:22:49,320 --> 00:22:51,492 because there are three flavors. 406 00:22:51,492 --> 00:22:53,450 And we're just going to add them together here. 407 00:22:53,450 --> 00:22:54,830 So we get a factor of three. 408 00:22:59,430 --> 00:23:03,050 Then we get a factor of two because there's 409 00:23:03,050 --> 00:23:04,525 particle anti particle pairs. 410 00:23:12,100 --> 00:23:14,255 That is, we have to count both neutrinos and anti 411 00:23:14,255 --> 00:23:17,040 neutrinos, and the total energy density. 412 00:23:17,040 --> 00:23:19,240 For photons, a photon is the same as an anti photon, 413 00:23:19,240 --> 00:23:21,530 so you don't get an extra particle connected 414 00:23:21,530 --> 00:23:22,600 with the anti particle. 415 00:23:22,600 --> 00:23:24,650 But for neutrinos you do, so that gives you 416 00:23:24,650 --> 00:23:26,920 a factor of two here. 417 00:23:26,920 --> 00:23:31,347 And then you always have the number of spin states, 418 00:23:31,347 --> 00:23:33,430 but in this case, the of number spin state is one, 419 00:23:33,430 --> 00:23:35,300 but I'll write it just so we can keep 420 00:23:35,300 --> 00:23:39,140 track of the general pattern. 421 00:23:39,140 --> 00:23:44,340 And when you do that multiplication, you get 21/4. 422 00:23:44,340 --> 00:23:48,630 So g sub nu for all three neutrinos put together is 21/4. 423 00:23:51,852 --> 00:23:53,810 And you could put this into all of our formulas 424 00:23:53,810 --> 00:23:55,410 and get the energy density pressure, 425 00:23:55,410 --> 00:23:57,465 and entropy density for neutrinos. 426 00:24:00,550 --> 00:24:03,557 Then we're also interested in g star, which 427 00:24:03,557 --> 00:24:04,890 will give us the number density. 428 00:24:08,940 --> 00:24:11,060 So g star sub nu. 429 00:24:11,060 --> 00:24:15,150 And it differs only by the first factor. 430 00:24:15,150 --> 00:24:19,770 3/4 for the fermion nature of the particle. 431 00:24:30,530 --> 00:24:34,620 And the rest is just dittos. 432 00:24:34,620 --> 00:24:37,665 And what you get in the end is 9/2. 433 00:24:50,425 --> 00:24:53,200 OK, now this does not end our discussion 434 00:24:53,200 --> 00:24:57,200 of black-body radiation in the early universe 435 00:25:00,570 --> 00:25:03,750 Because as we go back in time to earlier times 436 00:25:03,750 --> 00:25:07,440 we can come to a times when KT, the main thermal energy, 437 00:25:07,440 --> 00:25:11,360 is large compared to the rest energy of an electron, 438 00:25:11,360 --> 00:25:12,540 m sub e c squared. 439 00:25:40,490 --> 00:25:42,115 So at these very high temperatures even 440 00:25:42,115 --> 00:25:47,510 e plus, e minus pairs contribute to the black-body radiation. 441 00:26:04,930 --> 00:26:09,985 So we'd like to write down a g four e plus, e minus pairs. 442 00:26:15,885 --> 00:26:18,510 Actually, maybe I shouldn't call them pairs because they really 443 00:26:18,510 --> 00:26:22,700 are contributing individually, but g for e plus, e minus. 444 00:26:22,700 --> 00:26:23,286 Yes? 445 00:26:23,286 --> 00:26:24,744 AUDIENCE: So I know said that we're 446 00:26:24,744 --> 00:26:27,740 assuming, even though it's a spin 1/2 particle that's 447 00:26:27,740 --> 00:26:29,608 just one spin state. 448 00:26:29,608 --> 00:26:32,048 I was wondering is it possible for a spin 1/2 particle 449 00:26:32,048 --> 00:26:33,024 to be massless? 450 00:26:33,024 --> 00:26:37,032 I can't think of any offhand, because don't they 451 00:26:37,032 --> 00:26:41,662 have to have m equals negative 1/2 state? [INAUDIBLE] 452 00:26:41,662 --> 00:26:42,427 in general. 453 00:26:42,427 --> 00:26:43,135 PROFESSOR: Right. 454 00:26:43,135 --> 00:26:45,140 No, it is perfectly consistent, would 455 00:26:45,140 --> 00:26:48,020 be perfectly consistent for neutrinos to be massless. 456 00:26:48,020 --> 00:26:50,520 And then they would only have and m equals 1/2 state 457 00:26:50,520 --> 00:26:52,900 in spite of the fact that their spin 1/2. 458 00:26:52,900 --> 00:26:57,170 And the m equals minus 1/2 state would just be missing, 459 00:26:57,170 --> 00:26:58,580 but that's OK. 460 00:26:58,580 --> 00:27:01,030 It's similar to what happens with photons. 461 00:27:01,030 --> 00:27:03,100 Photons are spin one particles. 462 00:27:03,100 --> 00:27:06,150 And the m equals 0 state is missing. 463 00:27:06,150 --> 00:27:08,700 And that can only happen with massless particles. 464 00:27:08,700 --> 00:27:12,010 For massive particles, by basically the argument 465 00:27:12,010 --> 00:27:14,250 that I gave about catching up to the neutrino, 466 00:27:14,250 --> 00:27:16,520 it's not possible to have one helicity 467 00:27:16,520 --> 00:27:19,570 and not have the other. 468 00:27:19,570 --> 00:27:21,960 But for massless particles it is possible. 469 00:27:21,960 --> 00:27:23,260 The photon does it. 470 00:27:23,260 --> 00:27:25,260 We used to think the neutrino did it. 471 00:27:25,260 --> 00:27:26,815 Turned up the neutrino doesn't do it, 472 00:27:26,815 --> 00:27:29,740 but we can still describe the old theory, which 473 00:27:29,740 --> 00:27:31,300 is simpler than the new theory. 474 00:27:31,300 --> 00:27:32,383 And that's what I'm doing. 475 00:27:35,070 --> 00:27:38,227 AUDIENCE: Is there any reason they're missing those? 476 00:27:38,227 --> 00:27:38,810 PROFESSOR: OK. 477 00:27:38,810 --> 00:27:40,360 The question is is there any reason 478 00:27:40,360 --> 00:27:43,150 why they're missing those spin states. 479 00:27:43,150 --> 00:27:47,967 The answer basically is that there's no solid reason, which 480 00:27:47,967 --> 00:27:50,550 is why we didn't know if it was going to be missing the states 481 00:27:50,550 --> 00:27:52,350 or not. 482 00:27:52,350 --> 00:27:55,580 But for the case of massless particles 483 00:27:55,580 --> 00:28:00,330 only the m states are all completely 484 00:28:00,330 --> 00:28:04,780 disconnected from each other under Lorentz transformations. 485 00:28:04,780 --> 00:28:08,230 For any non-zero mass, all the spin states mix. 486 00:28:08,230 --> 00:28:10,680 And because they all mix under Lorentz transformations 487 00:28:10,680 --> 00:28:13,710 you can't have any one without having all of them. 488 00:28:13,710 --> 00:28:15,460 But that statement disappears when 489 00:28:15,460 --> 00:28:17,730 the particle becomes massless. 490 00:28:17,730 --> 00:28:19,350 So when the particle becomes massless, 491 00:28:19,350 --> 00:28:22,470 essentially, each spin state is its own kind of particle. 492 00:28:22,470 --> 00:28:25,360 And it might be part of a spectrum of real particles, 493 00:28:25,360 --> 00:28:26,770 or might not. 494 00:28:26,770 --> 00:28:29,230 And we don't know any fundamental principle 495 00:28:29,230 --> 00:28:31,700 that answers that question, other than experiment. 496 00:28:48,380 --> 00:28:48,890 OK. 497 00:28:48,890 --> 00:28:50,264 So I was just going to write down 498 00:28:50,264 --> 00:28:52,560 a g for e plus, e minus pairs. 499 00:28:52,560 --> 00:28:55,775 Remember this g determines the energy density, the pressure, 500 00:28:55,775 --> 00:28:56,775 and the entropy density. 501 00:28:59,380 --> 00:29:04,860 And it again has a factor of 7/8 because these are fermions. 502 00:29:11,960 --> 00:29:15,710 It has a factor of one to sort of follow the general pattern 503 00:29:15,710 --> 00:29:18,960 because there's only one species of electrons. 504 00:29:18,960 --> 00:29:23,800 For neutrinos we had three, but for electrons we just have one. 505 00:29:23,800 --> 00:29:27,140 We do get a factor of two for particles, 506 00:29:27,140 --> 00:29:30,850 anti particles because an e plus is different from an e minus. 507 00:29:41,240 --> 00:29:44,335 And there are two spin states for the electron. 508 00:29:44,335 --> 00:29:47,430 If could be spin up, or spin down, or spin equals 1/2, 509 00:29:47,430 --> 00:29:49,050 and there's spin equals minus 1/2. 510 00:29:49,050 --> 00:29:50,750 So there are two spin states. 511 00:29:55,570 --> 00:29:59,625 And that gives us a factor of 7/2 for g. 512 00:30:03,670 --> 00:30:12,960 And for g star e plus, e minus, the only difference again, 513 00:30:12,960 --> 00:30:20,825 is in the first factor, which is 3/4 for fermions for g star. 514 00:30:23,550 --> 00:30:27,970 And then times dot, dot, dot. 515 00:30:27,970 --> 00:30:32,060 And that then is equal to just a nice round factor of three. 516 00:30:36,380 --> 00:30:41,800 And then if we go back to this earlier time, 517 00:30:41,800 --> 00:30:44,150 we have not only the e plus, e minus pairs, 518 00:30:44,150 --> 00:30:47,430 which started to exist when we crossed this threshold, 519 00:30:47,430 --> 00:30:49,530 we also have photons and neutrinos. 520 00:30:57,440 --> 00:31:02,840 So for kt, large compared to m sub ec 521 00:31:02,840 --> 00:31:15,500 squared, and I should say small compared to the next threshold, 522 00:31:15,500 --> 00:31:21,280 m sub nu c squared, mass of a muon. 523 00:31:24,291 --> 00:31:27,790 The mass of the muon as 106 MeV Mass of electron 524 00:31:27,790 --> 00:31:28,670 is a half of an MeV. 525 00:31:28,670 --> 00:31:31,670 So there's are good long range here where 526 00:31:31,670 --> 00:31:32,740 this is the right number. 527 00:31:37,030 --> 00:31:49,090 So g total is equal to 2 plus 21/4 plus 7/2 528 00:31:49,090 --> 00:31:55,440 equals 10 and 3/4. 529 00:31:55,440 --> 00:31:57,950 So this number 10 and 3/4 plays an important role 530 00:31:57,950 --> 00:32:01,030 in a long segment of the history of our universe. 531 00:32:57,684 --> 00:32:59,600 Another number that you might be interested in 532 00:32:59,600 --> 00:33:03,090 is the energy density of radiation in the universe 533 00:33:03,090 --> 00:33:05,840 today. 534 00:33:05,840 --> 00:33:08,860 And the e plus, e minus pairs are of course 535 00:33:08,860 --> 00:33:14,450 long since frozen out because kt is a lot less than half 536 00:33:14,450 --> 00:33:17,160 of an MeV. 537 00:33:17,160 --> 00:33:21,334 So we just have neutrinos and photons. 538 00:33:21,334 --> 00:33:22,750 You might think that we could just 539 00:33:22,750 --> 00:33:27,570 add the formulas for photons to the formula for neutrinos, 540 00:33:27,570 --> 00:33:29,920 but that turns out not to be right. 541 00:33:29,920 --> 00:33:32,860 And the reason it's not right now 542 00:33:32,860 --> 00:33:37,700 is that we believe that the temperature of the neutrinos 543 00:33:37,700 --> 00:33:41,820 is not the same as the temperature of the photons. 544 00:33:41,820 --> 00:33:43,400 And this is a problem that you'll 545 00:33:43,400 --> 00:33:45,290 be working out on the next homework set, 546 00:33:45,290 --> 00:33:47,640 so I won't give all the details here 547 00:33:47,640 --> 00:33:51,260 because I want you to have the fun of working out 548 00:33:51,260 --> 00:33:52,420 those details. 549 00:33:52,420 --> 00:33:54,650 But I'll tell you the outline of what it is. 550 00:33:54,650 --> 00:34:00,040 The transition occurs when kt crosses 551 00:34:00,040 --> 00:34:02,080 this magic threshold of m sub e c 552 00:34:02,080 --> 00:34:06,110 squared, which means that the electron positron pairs are 553 00:34:06,110 --> 00:34:11,460 going to start to disappear from the thermal equilibrium mix. 554 00:34:11,460 --> 00:34:12,940 Now those electron positron pairs 555 00:34:12,940 --> 00:34:16,139 have both energy and entropy. 556 00:34:16,139 --> 00:34:20,639 It turns out that the energy is very hard to keep track of. 557 00:34:20,639 --> 00:34:24,800 And the reason why that is true is 558 00:34:24,800 --> 00:34:38,949 that we know that du dt is equal to minus 3h, times rho plus p, 559 00:34:38,949 --> 00:34:41,620 I think i have this right. 560 00:34:41,620 --> 00:34:42,120 times u. 561 00:34:44,639 --> 00:34:46,780 No, not times u. 562 00:34:46,780 --> 00:34:47,909 And this might be a u. 563 00:34:51,650 --> 00:34:53,770 Yeah, I think this is right. 564 00:34:53,770 --> 00:34:57,160 But what is definitely true is that it involves 565 00:34:57,160 --> 00:34:59,344 the pressure as well as the energy density. 566 00:34:59,344 --> 00:35:01,010 And keeping track of what the pressure's 567 00:35:01,010 --> 00:35:02,620 doing as a function of time is complicated 568 00:35:02,620 --> 00:35:04,870 because depending on exactly what the mix of particles 569 00:35:04,870 --> 00:35:07,590 are and the pressure is even given 570 00:35:07,590 --> 00:35:09,820 by a more complicated formula when you're 571 00:35:09,820 --> 00:35:11,560 very near the threshold, that is when 572 00:35:11,560 --> 00:35:13,970 kt is near the mass of a particle, 573 00:35:13,970 --> 00:35:18,040 there's a more complicated formula for the pressure. 574 00:35:18,040 --> 00:35:20,890 So the pressure is hard to keep track of. 575 00:35:20,890 --> 00:35:24,800 But the entropy turns out to be easy to keep track 576 00:35:24,800 --> 00:35:29,620 of because entropy is simply conserved during this process. 577 00:35:29,620 --> 00:35:33,510 So on the problem that you'll be doing for the next problem set 578 00:35:33,510 --> 00:35:38,280 you'll be looking at the entropy contained in these e plus 579 00:35:38,280 --> 00:35:40,340 e minus pairs. 580 00:35:40,340 --> 00:35:42,720 And then there's an important assumption which is valid. 581 00:35:42,720 --> 00:35:45,610 We will not really try to justify it, 582 00:35:45,610 --> 00:35:49,450 but at the time when the e plus, e minus pairs go out 583 00:35:49,450 --> 00:35:51,480 of equilibrium, when they disappear 584 00:35:51,480 --> 00:35:55,120 from the thermal equilibrium mix as kt falls below m sub 585 00:35:55,120 --> 00:36:00,930 ec squared, at that point the photons are still interacting 586 00:36:00,930 --> 00:36:04,710 strongly with everything else, and with each other. 587 00:36:04,710 --> 00:36:06,770 But the neutrinos have pretty much decoupled. 588 00:36:06,770 --> 00:36:09,880 They're not really interacting with anything anymore. 589 00:36:09,880 --> 00:36:11,570 So when the e plus, e minus pairs 590 00:36:11,570 --> 00:36:13,880 disappear they give essentially all 591 00:36:13,880 --> 00:36:16,810 of their entropy to the photons, and essentially not 592 00:36:16,810 --> 00:36:23,069 of their entropy to the neutrinos. 593 00:36:23,069 --> 00:36:24,610 And that means that you can calculate 594 00:36:24,610 --> 00:36:26,690 the entropy density of the photons, 595 00:36:26,690 --> 00:36:30,676 and the entropy density of the neutrinos 596 00:36:30,676 --> 00:36:33,050 and you can calculate from that the relative temperatures 597 00:36:33,050 --> 00:36:34,520 between the two. 598 00:36:34,520 --> 00:36:36,560 And the net effect is that the photons end up 599 00:36:36,560 --> 00:36:38,435 with a higher temperature than the neutrinos. 600 00:36:41,010 --> 00:36:44,720 So cosmology makes a clear prediction here. 601 00:36:44,720 --> 00:36:46,330 The universe today should be bathed 602 00:36:46,330 --> 00:36:49,590 in a thermal distribution of neutrinos. 603 00:36:49,590 --> 00:36:53,090 The temperature I think ends up being about 2.4 degrees Kelvin, 604 00:36:53,090 --> 00:36:57,490 a little colder than the photons. 605 00:36:57,490 --> 00:36:59,530 And it's really the great challenge 606 00:36:59,530 --> 00:37:01,090 of observational cosmology to try 607 00:37:01,090 --> 00:37:03,640 to measure those thermal neutrinos. 608 00:37:03,640 --> 00:37:06,270 Because neutrinos interact so weakly, 609 00:37:06,270 --> 00:37:08,690 nobody has come close to measuring 610 00:37:08,690 --> 00:37:10,780 the existence of those neutrinos. 611 00:37:10,780 --> 00:37:12,580 Everybody thinks they must be there, 612 00:37:12,580 --> 00:37:14,360 if anybody's discovers they're not there, 613 00:37:14,360 --> 00:37:18,230 it'll be a major shift in our understanding of cosmology. 614 00:37:18,230 --> 00:37:20,730 But nobody really knows how to measure them. 615 00:37:20,730 --> 00:37:24,071 So you guys could try to do that sometime 616 00:37:24,071 --> 00:37:25,445 during your career as physicists. 617 00:37:31,400 --> 00:37:43,460 So putting these things together what you'll show 618 00:37:43,460 --> 00:37:46,540 is that the temperature of neutrinos 619 00:37:46,540 --> 00:37:51,700 should be equal to 4/11 to the 1/3 620 00:37:51,700 --> 00:37:54,740 power times the temperature of the photons. 621 00:38:01,050 --> 00:38:02,540 And given that, we can write down 622 00:38:02,540 --> 00:38:05,705 a formula for the energy density in radiation today. 623 00:38:12,870 --> 00:38:23,120 So u [? rad ?] 0 for today is equal to g for photons, 624 00:38:23,120 --> 00:38:29,804 plus 21/4, which is g for the neutrinos, 625 00:38:29,804 --> 00:38:31,470 but then we have to correct for the fact 626 00:38:31,470 --> 00:38:33,470 that the neutrinos are at the lower temperature, 627 00:38:33,470 --> 00:38:35,890 and remember energy density's go like the fourth power 628 00:38:35,890 --> 00:38:36,692 of the temperature. 629 00:38:36,692 --> 00:38:38,150 So there's a correction here, which 630 00:38:38,150 --> 00:38:41,730 is the fourth power of this 4/11. 631 00:38:41,730 --> 00:38:49,580 So this multiplies 4/11 to the 4/3 power, 632 00:38:49,580 --> 00:38:51,080 the fourth power of the temperature. 633 00:38:54,860 --> 00:38:57,810 And then the rest of the formula for energy density. 634 00:38:57,810 --> 00:39:04,560 Pi squared over 30 times k, times the temperature 635 00:39:04,560 --> 00:39:10,880 of the photons to the fourth power, divided by h bar 636 00:39:10,880 --> 00:39:16,820 c cubed. 637 00:39:16,820 --> 00:39:20,360 And if you plug numbers into this, 638 00:39:20,360 --> 00:39:22,730 you get a number which I wrote on the blackboard 639 00:39:22,730 --> 00:39:26,470 when we started talking about radiation for the radiation 640 00:39:26,470 --> 00:39:28,190 density of today's universe. 641 00:39:28,190 --> 00:39:39,235 It's 7.01 times 10 to the minus 14 jewels per meter cubed. 642 00:39:48,282 --> 00:39:50,740 So now we know how to derive this formula that I pulled out 643 00:39:50,740 --> 00:39:54,160 out of hat when we first started talking about energy density 644 00:39:54,160 --> 00:39:55,947 radiation. 645 00:39:55,947 --> 00:39:57,780 You might remember we used this to calculate 646 00:39:57,780 --> 00:39:59,840 when radiation matter equality took place. 647 00:40:15,920 --> 00:40:16,420 OK. 648 00:40:16,420 --> 00:40:17,530 Any questions about this? 649 00:40:21,001 --> 00:40:21,500 OK. 650 00:40:21,500 --> 00:40:23,430 Well in that case now I'm ready to come back and talk 651 00:40:23,430 --> 00:40:25,138 about neutrino masses more realistically. 652 00:40:50,064 --> 00:40:52,830 OK. [INAUDIBLE] the mass we now know. 653 00:40:52,830 --> 00:40:55,770 Or at least two out of the three do. 654 00:40:55,770 --> 00:41:00,900 And still I argued, or told you, that the mass 655 00:41:00,900 --> 00:41:03,169 didn't matter for the calculations we just did. 656 00:41:03,169 --> 00:41:05,460 And that seems a little strange because the calculation 657 00:41:05,460 --> 00:41:08,250 we just did depended on counting the numbers of spin states, 658 00:41:08,250 --> 00:41:09,958 and the number of spin states, of course, 659 00:41:09,958 --> 00:41:12,690 changes if the particles have a mass. 660 00:41:12,690 --> 00:41:16,880 Neutrinos could not have just one spin state 661 00:41:16,880 --> 00:41:19,050 if they had a mass. 662 00:41:19,050 --> 00:41:22,370 So the question then is what happens 663 00:41:22,370 --> 00:41:25,017 to these right-handed neutrinos. 664 00:41:25,017 --> 00:41:27,350 Remember the neutrinos we know and love are left-handed. 665 00:42:04,712 --> 00:42:05,230 OK. 666 00:42:05,230 --> 00:42:10,170 Nobody has ever seen a right-handed neutrino. 667 00:42:10,170 --> 00:42:12,792 But if the neutrino has a mass, they must exist. 668 00:42:12,792 --> 00:42:14,250 And one way to see is that argument 669 00:42:14,250 --> 00:42:16,083 I gave you about catching up to the neutrino 670 00:42:16,083 --> 00:42:17,242 and going faster than it. 671 00:42:17,242 --> 00:42:18,700 And then a left-hand neutrino would 672 00:42:18,700 --> 00:42:20,770 turn into a right-handed neutrino, which 673 00:42:20,770 --> 00:42:24,080 is, this is all you did, was change frames, 674 00:42:24,080 --> 00:42:26,020 and the world is Lorentz invariant. 675 00:42:26,020 --> 00:42:29,450 It means that right-handed neutrinos must in principle 676 00:42:29,450 --> 00:42:30,870 exist in any frame. 677 00:42:38,630 --> 00:42:39,130 OK. 678 00:42:39,130 --> 00:42:41,400 It turns out that there are two theories 679 00:42:41,400 --> 00:42:43,360 of how the neutrino mass works. 680 00:42:43,360 --> 00:42:46,175 And we don't know yet which of these theories is correct. 681 00:42:57,110 --> 00:43:05,310 And they are called the possibility of a Dirac mass 682 00:43:05,310 --> 00:43:07,815 or the possibility of a Majorana mass. 683 00:43:16,810 --> 00:43:19,280 And these are both named after people, 684 00:43:19,280 --> 00:43:22,280 so don't try to parse anything about the physical meaning 685 00:43:22,280 --> 00:43:24,300 of those words from the name. 686 00:43:27,150 --> 00:43:29,950 The Dirac mass is the easier one for us 687 00:43:29,950 --> 00:43:32,550 to understand because it's the same kind of mass 688 00:43:32,550 --> 00:43:35,900 that an electron has. 689 00:43:35,900 --> 00:43:39,940 So if the neutrinos have a Dirac mass 690 00:43:39,940 --> 00:43:43,880 it really would mean that there are right-handed neutrinos, 691 00:43:43,880 --> 00:43:45,692 which are a new species a particle 692 00:43:45,692 --> 00:43:47,150 that we haven't seen yet, but which 693 00:43:47,150 --> 00:43:51,410 would be implied by the existence of this mass. 694 00:43:51,410 --> 00:43:55,080 The catch is that because of the very peculiar way 695 00:43:55,080 --> 00:43:57,790 that the neutrinos interact it is 696 00:43:57,790 --> 00:44:00,830 possible for the right-handed neutrinos 697 00:44:00,830 --> 00:44:04,556 to interact vastly more weakly than the left handed neutrinos. 698 00:44:04,556 --> 00:44:05,930 And that would be the explanation 699 00:44:05,930 --> 00:44:08,500 for why we've never seen them because they interact 700 00:44:08,500 --> 00:44:10,550 so extraordinarily weakly. 701 00:44:10,550 --> 00:44:13,570 And even in the early universe where everything almost reaches 702 00:44:13,570 --> 00:44:16,210 thermal equilibrium, the cross-section 703 00:44:16,210 --> 00:44:18,350 for producing right-handed neutrinos 704 00:44:18,350 --> 00:44:21,405 would be so small that they would essentially never 705 00:44:21,405 --> 00:44:21,905 be produced. 706 00:44:45,750 --> 00:45:06,900 So right-handed nu's interact so weakly 707 00:45:06,900 --> 00:45:22,640 that they're not seen in lab, or in early universe. 708 00:45:38,395 --> 00:45:38,895 OK. 709 00:45:38,895 --> 00:45:40,228 The other type is more peculiar. 710 00:45:46,260 --> 00:45:48,330 And there are no other particles in nature 711 00:45:48,330 --> 00:45:50,290 that are known to have this type of mass, 712 00:45:50,290 --> 00:45:54,190 but it's a theoretical possibility. 713 00:45:54,190 --> 00:45:56,480 Majorana masses can only be possessed 714 00:45:56,480 --> 00:46:00,940 by particles which are neutral, but neutrinos are neutral. 715 00:46:00,940 --> 00:46:05,330 And if neutrinos have a Majorana mass 716 00:46:05,330 --> 00:46:08,400 it would mean that the right-handed partner 717 00:46:08,400 --> 00:46:11,620 of the left handed neutrinos that we see 718 00:46:11,620 --> 00:46:14,090 would in fact be a particle that we already know about, 719 00:46:14,090 --> 00:46:16,214 but it will be the particle that we have previously 720 00:46:16,214 --> 00:46:19,820 identified as the right-handed anti neutrino. 721 00:46:19,820 --> 00:46:22,860 So if the Majorana hypothesis is true, 722 00:46:22,860 --> 00:46:24,450 the anti neutrino and the neutrino 723 00:46:24,450 --> 00:46:26,072 would be the same particle. 724 00:46:26,072 --> 00:46:27,780 And one would be the right-handed version 725 00:46:27,780 --> 00:46:29,738 and the other would be the left-handed version. 726 00:46:51,950 --> 00:46:54,280 And that is also a possibility, we just know know. 727 00:46:57,370 --> 00:46:58,860 And in this case, clearly what we 728 00:46:58,860 --> 00:47:00,120 said about the early universe still 729 00:47:00,120 --> 00:47:01,786 works because we did the counting right. 730 00:47:04,170 --> 00:47:05,880 If it's the direct possibility we 731 00:47:05,880 --> 00:47:07,390 would have to do a further argument, 732 00:47:07,390 --> 00:47:11,129 we justify the fact that these right-handed partners interact 733 00:47:11,129 --> 00:47:12,920 so weakly that we would never see them even 734 00:47:12,920 --> 00:47:16,240 in the early universe, but that is the case. 735 00:47:16,240 --> 00:47:18,420 And for the Majorana case there clearly 736 00:47:18,420 --> 00:47:21,890 is no real difference from the calculations that we did. 737 00:47:21,890 --> 00:47:25,220 Any questions about that? 738 00:47:25,220 --> 00:47:25,900 Yes? 739 00:47:25,900 --> 00:47:27,316 AUDIENCE: What exactly do you mean 740 00:47:27,316 --> 00:47:31,552 when you say the direct mass is like the electrons? 741 00:47:31,552 --> 00:47:32,260 PROFESSOR: Right. 742 00:47:34,830 --> 00:47:38,500 What I mean is that it appears in the equations of motion 743 00:47:38,500 --> 00:47:42,700 in the same way that the that the electron mass appears, 744 00:47:42,700 --> 00:47:47,555 and therefore like the electron there's a right-handed electron 745 00:47:47,555 --> 00:47:51,279 and there's also a left-handed handed electron, which are just 746 00:47:51,279 --> 00:47:53,320 related to each other by a parity transformation. 747 00:48:00,670 --> 00:48:02,002 Yes? 748 00:48:02,002 --> 00:48:10,349 AUDIENCE: Is it not a problem in the Majorana case [INAUDIBLE] 749 00:48:10,349 --> 00:48:12,804 particle? 750 00:48:12,804 --> 00:48:15,937 Doesn't there have to be an [INAUDIBLE]? 751 00:48:15,937 --> 00:48:16,520 PROFESSOR: OK. 752 00:48:16,520 --> 00:48:17,520 The question is doesn't there have 753 00:48:17,520 --> 00:48:19,353 to be an anti particle, how can the neutrino 754 00:48:19,353 --> 00:48:22,730 and the anti neutrino be the same particle. 755 00:48:22,730 --> 00:48:24,190 The answer is no. 756 00:48:24,190 --> 00:48:26,780 The photon does not have an anti particle. 757 00:48:26,780 --> 00:48:28,920 So particles that have a charge of any kind 758 00:48:28,920 --> 00:48:31,700 must have an anti particle, but if the particle is really 759 00:48:31,700 --> 00:48:35,147 neutral then it can just be it's own anti particle, 760 00:48:35,147 --> 00:48:36,480 and that would be the case here. 761 00:48:39,030 --> 00:48:39,777 Yes? 762 00:48:39,777 --> 00:48:41,360 AUDIENCE: Sorry, I have two questions. 763 00:48:41,360 --> 00:48:45,330 One is what is the difference between a neutrino 764 00:48:45,330 --> 00:48:47,420 and a anti neutrino besides? 765 00:48:47,420 --> 00:48:49,870 I mean, how do we know they're the same, that they 766 00:48:49,870 --> 00:48:51,392 are the same particle [INAUDIBLE]. 767 00:48:51,392 --> 00:48:52,891 It's not just a matter of semantics. 768 00:48:52,891 --> 00:48:58,111 If the only difference that we can see is [INAUDIBLE]. 769 00:48:58,111 --> 00:48:59,610 Whether we call it the anti neutrino 770 00:48:59,610 --> 00:49:03,870 or say that it's the same particle, just [INAUDIBLE]. 771 00:49:03,870 --> 00:49:04,630 PROFESSOR: Right. 772 00:49:04,630 --> 00:49:05,130 OK. 773 00:49:05,130 --> 00:49:08,340 You're saying how do we know whether the anti neutrino was 774 00:49:08,340 --> 00:49:10,656 the same particle as the neutrino, 775 00:49:10,656 --> 00:49:12,530 and what do we mean by that statement anyway. 776 00:49:15,850 --> 00:49:21,340 I guess the answer is really something that comes out 777 00:49:21,340 --> 00:49:24,230 of the fundamental equations in the context of the quantum 778 00:49:24,230 --> 00:49:26,220 field theory. 779 00:49:26,220 --> 00:49:35,680 But maybe I can say something more concrete, however. 780 00:49:35,680 --> 00:49:40,925 There's a quantum number called Lepton number, 781 00:49:40,925 --> 00:49:43,300 and actually divides into three kinds of quantum numbers, 782 00:49:43,300 --> 00:49:47,910 so there's an electron number, muon number, and a tau number. 783 00:49:47,910 --> 00:49:49,940 And for all the interactions that we've 784 00:49:49,940 --> 00:49:53,480 seen that number is conserved, that is 785 00:49:53,480 --> 00:49:55,340 the sum of the number of electrons 786 00:49:55,340 --> 00:49:57,340 minus the number of anti electrons, 787 00:49:57,340 --> 00:49:59,500 plus the number of electron neutrinos, 788 00:49:59,500 --> 00:50:02,330 minus the number of anti electron neutrinos 789 00:50:02,330 --> 00:50:03,740 is conserved. 790 00:50:03,740 --> 00:50:06,670 And if that were a rigorous conservation law, then 791 00:50:06,670 --> 00:50:09,250 it would really mean that the neutrino was not neutral. 792 00:50:09,250 --> 00:50:11,440 It would have this nonzero charge 793 00:50:11,440 --> 00:50:15,190 called electron charge for the electron neutrino. 794 00:50:15,190 --> 00:50:20,160 So the Majorana possibility requires 795 00:50:20,160 --> 00:50:25,580 that that quantity is not really conserved. 796 00:50:25,580 --> 00:50:28,440 So that's an important fact about nature 797 00:50:28,440 --> 00:50:30,690 that we haven't really learned yet, whether or not 798 00:50:30,690 --> 00:50:33,880 it's exactly conserved or only highly approximately conserved. 799 00:50:33,880 --> 00:50:36,830 The belief is that it's only highly approximately conserved, 800 00:50:36,830 --> 00:50:40,080 but we don't know. 801 00:50:40,080 --> 00:50:40,580 Yes? 802 00:50:40,580 --> 00:50:42,163 AUDIENCE: Does a neutrino [INAUDIBLE]? 803 00:50:47,610 --> 00:50:48,470 PROFESSOR: Yeah. 804 00:50:48,470 --> 00:50:49,290 You are right. 805 00:50:49,290 --> 00:50:50,510 You're very quick. 806 00:50:50,510 --> 00:50:53,551 You are right that neutrino oscillations are certainly 807 00:50:53,551 --> 00:50:55,800 enough to prove that the individual Lepton numbers are 808 00:50:55,800 --> 00:50:58,640 not conserved, but they still have the possibility 809 00:50:58,640 --> 00:51:00,932 that total Lepton number can be conserved, 810 00:51:00,932 --> 00:51:02,390 which we don't think is true, but I 811 00:51:02,390 --> 00:51:04,140 don't think we've ruled it out yet either. 812 00:51:07,342 --> 00:51:08,175 Any other questions? 813 00:51:12,070 --> 00:51:12,804 Yes? 814 00:51:12,804 --> 00:51:17,040 AUDIENCE: In the Majorana case, if they are the same particle, 815 00:51:17,040 --> 00:51:20,930 do they still come in left and right-handed forms 816 00:51:20,930 --> 00:51:25,196 so as to keep the [INAUDIBLE] the same value, 817 00:51:25,196 --> 00:51:28,633 or do they only come in the left-handed phase, 818 00:51:28,633 --> 00:51:32,259 in which case the [INAUDIBLE] would be decreased [INAUDIBLE]? 819 00:51:32,259 --> 00:51:33,800 PROFESSOR: Well, the counting that we 820 00:51:33,800 --> 00:51:36,310 had before would be correct. 821 00:51:36,310 --> 00:51:42,120 And the only question is whether the factor of two 822 00:51:42,120 --> 00:51:44,810 that we put here for particle, anti particle 823 00:51:44,810 --> 00:51:47,170 is here where maybe the particle's really neutral 824 00:51:47,170 --> 00:51:49,400 and the factor of two is the number of spin states, 825 00:51:49,400 --> 00:51:51,272 but it's the same result either way. 826 00:51:51,272 --> 00:51:52,730 It's all in the question of whether 827 00:51:52,730 --> 00:51:55,740 the left-handed neutrino is the anti particle 828 00:51:55,740 --> 00:51:57,800 or the right-handed neutrino, or another spin 829 00:51:57,800 --> 00:51:59,221 state of the right-handed. 830 00:51:59,221 --> 00:51:59,720 I'm sorry. 831 00:51:59,720 --> 00:52:01,350 I'm saying this wrong. 832 00:52:01,350 --> 00:52:03,440 The question is whether the right-handed neutrino 833 00:52:03,440 --> 00:52:06,210 is the anti particle of the left-handed neutrino 834 00:52:06,210 --> 00:52:09,230 or whether it's another spin states 835 00:52:09,230 --> 00:52:10,837 of the left-handed neutrino. 836 00:52:10,837 --> 00:52:13,170 But either way the number particles that we think exists 837 00:52:13,170 --> 00:52:15,086 would be the same in the mayorana description. 838 00:52:20,570 --> 00:52:21,070 OK? 839 00:52:40,860 --> 00:52:41,360 OK. 840 00:52:41,360 --> 00:52:43,340 Now we're ready to actually write down 841 00:52:43,340 --> 00:52:46,685 for example a formula for kt. 842 00:52:55,300 --> 00:53:03,870 So sticking to this time range of kt is much, much bigger 843 00:53:03,870 --> 00:53:10,090 than m sub eb squared, but kt is much, much less 844 00:53:10,090 --> 00:53:14,660 than m sub nu c squared. 845 00:53:14,660 --> 00:53:17,310 We can write down a formula-- I'm sorry. 846 00:53:17,310 --> 00:53:18,810 Actually the formula's more general. 847 00:53:18,810 --> 00:53:21,840 I didn't really look carefully what I wrote. 848 00:53:21,840 --> 00:53:22,389 Forget this. 849 00:53:22,389 --> 00:53:24,930 As long as we know what little g is, and this formula's going 850 00:53:24,930 --> 00:53:26,346 to have a little g in it, and then 851 00:53:26,346 --> 00:53:29,210 you can fill in the right value for any time period you want. 852 00:53:29,210 --> 00:53:32,670 We can write down a formula for kt as a function of time. 853 00:53:32,670 --> 00:53:34,350 And that's because we've learned how 854 00:53:34,350 --> 00:53:37,287 to write energy density as a function of time, 855 00:53:37,287 --> 00:53:39,620 and we've learned how to express energy density in terms 856 00:53:39,620 --> 00:53:40,810 of the temperature. 857 00:53:40,810 --> 00:53:42,630 And putting those two together gives us 858 00:53:42,630 --> 00:53:45,040 a formula for the temperature as a function of time. 859 00:53:45,040 --> 00:53:57,300 And that formula has the form of 45 h bar cubed c to the fifth, 860 00:53:57,300 --> 00:54:05,230 divided by 16 pi cubed little g times capital 861 00:54:05,230 --> 00:54:11,550 G, Newton's constant, whole thing to the 1/4 power, times 1 862 00:54:11,550 --> 00:54:13,030 over the square root of t. 863 00:54:19,770 --> 00:54:21,500 So if we had nothing but radiation, 864 00:54:21,500 --> 00:54:24,010 and if little g were constant, this 865 00:54:24,010 --> 00:54:25,980 would be the formula for the temperature 866 00:54:25,980 --> 00:54:28,600 as a function of time. 867 00:54:28,600 --> 00:54:31,210 Now what actually happens is that little g 868 00:54:31,210 --> 00:54:34,720 changes as we go through these different thresholds for which 869 00:54:34,720 --> 00:54:37,639 particles are contributing to the black-body radiation. 870 00:54:37,639 --> 00:54:39,180 And that means that the exact formula 871 00:54:39,180 --> 00:54:41,980 is a little bit more complicated than this. 872 00:54:41,980 --> 00:54:44,910 But as long as you're well into any one of these periods, 873 00:54:44,910 --> 00:54:47,790 as long as you're not near any of these border lines, 874 00:54:47,790 --> 00:54:51,900 this formula is in fact a very accurate approximation 875 00:54:51,900 --> 00:54:55,940 to the temperature as a function of time. 876 00:54:55,940 --> 00:55:00,380 And an interesting time period is about one second 877 00:55:00,380 --> 00:55:01,400 after the Big Bang. 878 00:55:22,148 --> 00:55:29,440 And that corresponds to this g equals 10 and 3/4 879 00:55:29,440 --> 00:55:32,510 that we talked about earlier, where we have neutrinos 880 00:55:32,510 --> 00:55:35,040 e plus, e minus pairs, and photons. 881 00:55:35,040 --> 00:55:37,340 And since this is before the e plus, e minus pairs have 882 00:55:37,340 --> 00:55:39,870 disappeared, the temperature of the neutrinos at this stage 883 00:55:39,870 --> 00:55:42,450 is still the same as the temperature of the photon. 884 00:55:42,450 --> 00:55:45,040 So everything is at the same temperature here. 885 00:55:45,040 --> 00:55:47,700 We just add up the g's. 886 00:55:47,700 --> 00:56:08,210 And the end result for that is that kt is equal to 0.860 MeV 887 00:56:08,210 --> 00:56:16,240 million electron volts, divided by the square root of t 888 00:56:16,240 --> 00:56:16,750 in seconds. 889 00:56:22,756 --> 00:56:24,630 There are the units that you could express it 890 00:56:24,630 --> 00:56:25,860 in in the notes. 891 00:56:25,860 --> 00:56:28,300 Other sets of units are given. 892 00:56:28,300 --> 00:56:32,670 So if 0.86 is about 1, which it is for many purposes, 893 00:56:32,670 --> 00:56:34,730 then roughly speaking we're saying at 1 second 894 00:56:34,730 --> 00:56:38,739 after the Big Bang, kt was about one MeV, 895 00:56:38,739 --> 00:56:40,280 which means it's higher than the rest 896 00:56:40,280 --> 00:56:43,136 mass of the electrons, which is 1/2 MeV. 897 00:56:43,136 --> 00:56:45,750 So the electron positron pairs were still pretty much present 898 00:56:45,750 --> 00:56:47,700 at one second after the Big Bang, 899 00:56:47,700 --> 00:56:49,930 but they start to disappear pretty much immediately 900 00:56:49,930 --> 00:56:50,630 after that. 901 00:56:56,970 --> 00:57:00,270 One can also write down what the temperature itself is. 902 00:57:00,270 --> 00:57:13,390 The temperature is 9.98 times 10 to the ninth k, divided 903 00:57:13,390 --> 00:57:15,210 by the square root of t in seconds. 904 00:58:24,770 --> 00:58:25,300 OK. 905 00:58:25,300 --> 00:58:27,490 The next item we want to talk about following 906 00:58:27,490 --> 00:58:29,460 the history of the early universe 907 00:58:29,460 --> 00:58:34,250 is recombination and decoupling. 908 00:58:43,850 --> 00:58:46,880 Until the temperature fell to about 4,000 909 00:58:46,880 --> 00:58:50,610 Kelvin, the hydrogen in the universe, and the universe 910 00:58:50,610 --> 00:58:55,510 was mostly hydrogen, about 20% helium and the rest hydrogen, 911 00:58:55,510 --> 00:58:59,390 and until the temperature fell to 4,000 Kelvin, 912 00:58:59,390 --> 00:59:02,800 the hydrogen would be ionized. 913 00:59:02,800 --> 00:59:04,800 This is another [? stat net ?] calculation 914 00:59:04,800 --> 00:59:07,280 that we're not going to be doing. 915 00:59:07,280 --> 00:59:09,610 4,000 degrees is not some magic temperature 916 00:59:09,610 --> 00:59:11,510 associated with hydrogen. 917 00:59:11,510 --> 00:59:13,310 The point at which the hydrogen ionizers 918 00:59:13,310 --> 00:59:15,350 depends on its density. 919 00:59:15,350 --> 00:59:18,090 But for the densities in the early universe, 920 00:59:18,090 --> 00:59:22,420 the ionization point is about 4,000 Kelvin. 921 00:59:22,420 --> 00:59:43,750 So at t equals 4,000 Kelvin the hydrogen recombines. 922 00:59:49,745 --> 00:59:52,270 Now the word recombine has somehow 923 00:59:52,270 --> 00:59:55,340 historically taken hold. 924 00:59:55,340 --> 00:59:57,239 So everybody calls it recombination 925 00:59:57,239 --> 00:59:59,280 in spite of the fact that according to our theory 926 00:59:59,280 --> 01:00:02,320 it was never combined previously at any time. 927 01:00:02,320 --> 01:00:05,919 So the prefix re there has absolutely no meaning whatever, 928 01:00:05,919 --> 01:00:07,835 but nonetheless it is completely conventional. 929 01:00:51,110 --> 01:00:51,610 OK. 930 01:00:51,610 --> 01:00:55,280 To estimate when this happens we can 931 01:00:55,280 --> 01:00:59,040 use an important fact, which we haven't said yet. 932 01:00:59,040 --> 01:01:12,440 Because entropy is conserved and entropy density 933 01:01:12,440 --> 01:01:20,630 is equal to some constant times t cubed, 934 01:01:20,630 --> 01:01:28,570 if entropy is conserved then like mass density 935 01:01:28,570 --> 01:01:31,790 in a matter-dominated universe, as the universe expands, 936 01:01:31,790 --> 01:01:33,460 the entropy thins out. 937 01:01:33,460 --> 01:01:35,430 And therefore just like the mass density 938 01:01:35,430 --> 01:01:38,260 in a matter-dominated universe, the density 939 01:01:38,260 --> 01:01:40,680 should go down like 1 over a cubed, 1 over the volume. 940 01:02:00,840 --> 01:02:03,145 Now strictly speaking this is only true if g is fixed. 941 01:02:09,267 --> 01:02:10,600 Well, actually that's not right. 942 01:02:10,600 --> 01:02:13,220 It's always true. 943 01:02:13,220 --> 01:02:17,230 What I'm about to write next is only true if g is fixed. 944 01:02:17,230 --> 01:02:32,770 If g is fixed then entropy density 945 01:02:32,770 --> 01:02:34,620 is proportional to the temperature cubed. 946 01:02:41,270 --> 01:02:46,210 And if you put these together you 947 01:02:46,210 --> 01:02:51,290 could see that t is just proportional to 1 948 01:02:51,290 --> 01:02:52,340 over the scale factor. 949 01:02:56,610 --> 01:02:59,420 So as long as little g, as long as the number of degrees 950 01:02:59,420 --> 01:03:02,707 of freedom contributing to this black-body radiation is fixed, 951 01:03:02,707 --> 01:03:05,040 the temperature just falls like 1 over the scale factor. 952 01:03:15,920 --> 01:03:16,420 OK. 953 01:03:26,400 --> 01:03:26,900 OK. 954 01:03:26,900 --> 01:03:30,417 So we said that recombination occurs at 4,000 degrees. 955 01:03:30,417 --> 01:03:32,000 There's actually another number that's 956 01:03:32,000 --> 01:03:35,974 more interesting, which is decoupling, 957 01:03:35,974 --> 01:03:37,015 which is slightly colder. 958 01:04:03,960 --> 01:04:09,410 Now the definitions are that recombination is usually 959 01:04:09,410 --> 01:04:13,040 defined as the temperature at which half of the hydrogen 960 01:04:13,040 --> 01:04:15,945 has recombined, but some significant fraction of it 961 01:04:15,945 --> 01:04:17,450 is still not yet recombined. 962 01:04:17,450 --> 01:04:18,575 It doesn't happen suddenly. 963 01:04:18,575 --> 01:04:19,740 It happens gradually. 964 01:04:19,740 --> 01:04:21,600 So you have to pick some point along the way 965 01:04:21,600 --> 01:04:25,520 to say this is the temperature that defines the recombination. 966 01:04:25,520 --> 01:04:27,270 Usually take us to the halfway point, 967 01:04:27,270 --> 01:04:30,000 which I think is the number used to calculate that. 968 01:04:30,000 --> 01:04:34,040 But perhaps of more interest is the temperature 969 01:04:34,040 --> 01:04:37,230 at which the photons decouple in the sense 970 01:04:37,230 --> 01:04:39,890 that as the photons scatter between then 971 01:04:39,890 --> 01:04:44,940 and now, a typical photon has not scattered at all. 972 01:04:44,940 --> 01:04:48,530 And that's colder because when you cross 4,000 K 973 01:04:48,530 --> 01:04:50,895 you still have half of the hydrogen ionized, which 974 01:04:50,895 --> 01:04:52,520 means there's still plenty of electrons 975 01:04:52,520 --> 01:04:55,220 around for these photons to scatter off of. 976 01:04:55,220 --> 01:04:57,230 So you have to cool to a colder temperature 977 01:04:57,230 --> 01:05:00,360 until the photons cease to interact with the electron 978 01:05:00,360 --> 01:05:02,950 positrons in any significant way. 979 01:05:02,950 --> 01:05:05,670 And that's why decoupling temperature 980 01:05:05,670 --> 01:05:10,040 is somewhat lower than recombination temperature. 981 01:05:10,040 --> 01:05:13,085 We can estimate at what time decoupling happened. 982 01:06:01,656 --> 01:06:03,390 Now this is only a crude estimate, 983 01:06:03,390 --> 01:06:06,260 but will in fact be pretty accurate. 984 01:06:06,260 --> 01:06:09,020 Since we don't yet even know about dark energy, 985 01:06:09,020 --> 01:06:12,060 we're going to estimate the evolution of the universe 986 01:06:12,060 --> 01:06:15,990 between the time of decoupling and now as being entirely 987 01:06:15,990 --> 01:06:17,550 matter-dominated. 988 01:06:17,550 --> 01:06:19,930 The time of decoupling is long after the time 989 01:06:19,930 --> 01:06:22,097 of matter radiation equality. 990 01:06:22,097 --> 01:06:23,555 So the universe is matter dominated 991 01:06:23,555 --> 01:06:25,292 at the time of decoupling and we're 992 01:06:25,292 --> 01:06:26,958 going to assume it's matter dominated up 993 01:06:26,958 --> 01:06:27,930 to the present day. 994 01:06:27,930 --> 01:06:32,960 And that means that we know that the scale factor will evolve 995 01:06:32,960 --> 01:06:36,050 like t to the 2/3, and therefore the temperature 996 01:06:36,050 --> 01:06:38,860 will evolve, like 1 over t to the 2/3. 997 01:06:38,860 --> 01:06:40,500 And that will be enough to tell us 998 01:06:40,500 --> 01:06:44,160 how much time is needed to go from 3,000 Kelvin 999 01:06:44,160 --> 01:06:48,450 to the present temperature 2.7 Kelvin. 1000 01:06:48,450 --> 01:06:53,070 So the time of decoupling making this approximation 1001 01:06:53,070 --> 01:06:56,350 is just the ratio of the temperatures 1002 01:06:56,350 --> 01:07:05,310 t zero over t decoupling, to the 3/2 power, 1003 01:07:05,310 --> 01:07:09,660 times the time today. 1004 01:07:09,660 --> 01:07:13,270 [INAUDIBLE] this fraction in the past. 1005 01:07:13,270 --> 01:07:24,766 And plugging in numbers that's 2.7 over 3,000 1006 01:07:24,766 --> 01:07:39,450 to the 3/2 times 13.8 trillion years. 1007 01:07:39,450 --> 01:07:47,461 And that's about 380,000 years. 1008 01:07:47,461 --> 01:07:48,960 And this in fact this is pretty much 1009 01:07:48,960 --> 01:07:51,501 exactly the number that people quote when they calculate this 1010 01:07:51,501 --> 01:07:52,280 more accurately. 1011 01:07:52,280 --> 01:07:56,050 So we really hit it just about on the nose. 1012 01:07:56,050 --> 01:07:58,950 So the time of decoupling was about 400,000 years 1013 01:07:58,950 --> 01:08:01,610 after the Big Bang. 1014 01:08:01,610 --> 01:08:04,140 And I should add, and then we'll stop, 1015 01:08:04,140 --> 01:08:06,850 that when we look at the cosmic background radiation, 1016 01:08:06,850 --> 01:08:10,590 what we are really seeing is an image of the universe at this 1017 01:08:10,590 --> 01:08:15,360 time, at the time of decoupling, because from this time onward 1018 01:08:15,360 --> 01:08:18,636 photons have pretty much just travelled in straight lines. 1019 01:08:18,636 --> 01:08:21,010 And that means that when we look at the cosmic background 1020 01:08:21,010 --> 01:08:24,850 radiation we're really seeing an image of the early universe 1021 01:08:24,850 --> 01:08:27,229 in exactly the same way as you're seeing an image of me 1022 01:08:27,229 --> 01:08:29,979 when you observe the photons there traveling from my face 1023 01:08:29,979 --> 01:08:32,120 to your eyes along straight lines. 1024 01:08:32,120 --> 01:08:34,497 It's the same principle. 1025 01:08:34,497 --> 01:08:36,080 So this determines what we're actually 1026 01:08:36,080 --> 01:08:38,279 seeing in the cosmic background radiation. 1027 01:08:38,279 --> 01:08:40,810 And therefore it's a very, very important number. 1028 01:08:40,810 --> 01:08:45,070 And we'll stop there now and continue next time.