WEBVTT 00:00:00.090 --> 00:00:01.670 The following content is provided 00:00:01.670 --> 00:00:03.820 under a Creative Commons license. 00:00:03.820 --> 00:00:06.550 Your support will help MIT OpenCourseWare continue 00:00:06.550 --> 00:00:10.180 to offer high quality educational resources for free. 00:00:10.180 --> 00:00:12.700 To make a donation or to view additional materials 00:00:12.700 --> 00:00:16.620 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.620 --> 00:00:17.330 at ocw.mit.edu. 00:00:23.860 --> 00:00:26.550 PROFESSOR: OK, in that case, let's take off. 00:00:26.550 --> 00:00:28.190 There's a fair amount I'd like to do 00:00:28.190 --> 00:00:31.090 before the end of the term. 00:00:31.090 --> 00:00:35.190 First, let me quickly review what we talked about last time. 00:00:35.190 --> 00:00:39.290 We talked about the actual supernovae data, 00:00:39.290 --> 00:00:42.930 which gives us brightness as a function of redshift for very 00:00:42.930 --> 00:00:46.590 distant objects and produced the first discovery 00:00:46.590 --> 00:00:49.530 that the model is not fit very well by the standard called 00:00:49.530 --> 00:00:52.140 dark matter model, which would be this lower line. 00:00:52.140 --> 00:00:55.590 But it fits much better by this lambda CDM, 00:00:55.590 --> 00:01:01.090 a model which involves a significant fraction-- 0.76 was 00:01:01.090 --> 00:01:04.150 used here-- and significant fraction of vacuum energy 00:01:04.150 --> 00:01:07.370 along with cold, dark matter and baryons. 00:01:07.370 --> 00:01:10.420 And that was a model that fit the data well. 00:01:10.420 --> 00:01:12.380 And this, along with similar data 00:01:12.380 --> 00:01:15.370 from another group of astronomers, 00:01:15.370 --> 00:01:19.180 was a big bombshell of 1998, showing 00:01:19.180 --> 00:01:21.900 that the universe appears to not be slowing down 00:01:21.900 --> 00:01:25.550 under the influence of gravity but rather to be accelerating 00:01:25.550 --> 00:01:27.810 due to some kind of repulsive, presumably, 00:01:27.810 --> 00:01:30.140 gravitational force. 00:01:30.140 --> 00:01:32.940 We talked about the overall evidence 00:01:32.940 --> 00:01:34.760 for this idea of an accelerating universe. 00:01:34.760 --> 00:01:36.760 It certainly began with the supernovae data 00:01:36.760 --> 00:01:38.220 that we just talked about. 00:01:38.220 --> 00:01:40.530 The basic fact that characterizes 00:01:40.530 --> 00:01:43.050 that data is that the most distant supernovae are 00:01:43.050 --> 00:01:46.780 dimmer than you'd expect by 20% to 30%. 00:01:46.780 --> 00:01:50.310 And people did cook up other possible explanations 00:01:50.310 --> 00:01:53.380 for what might have made supernovae at certain distances 00:01:53.380 --> 00:01:54.460 look dimmer. 00:01:54.460 --> 00:01:56.890 None of those really held up very well. 00:01:56.890 --> 00:01:59.860 But, in addition, several other important pieces of evidence 00:01:59.860 --> 00:02:03.930 came in to support this idea of an accelerating universe. 00:02:03.930 --> 00:02:06.660 Most important, more precise measurements 00:02:06.660 --> 00:02:13.940 of the cosmic background radiation anisotropies came in. 00:02:13.940 --> 00:02:17.030 And this pattern of antisotropies 00:02:17.030 --> 00:02:18.980 can be fit to a theoretical model, which 00:02:18.980 --> 00:02:22.290 includes all of the parameters of cosmology, basically, 00:02:22.290 --> 00:02:25.850 and turns out to give a very precise setting for essentially 00:02:25.850 --> 00:02:28.150 all the parameters of cosmology. 00:02:28.150 --> 00:02:31.970 They now really all have their most precise values coming out 00:02:31.970 --> 00:02:33.680 of these CMB measurements. 00:02:33.680 --> 00:02:36.240 And the CMB measurements gave a value 00:02:36.240 --> 00:02:38.510 of omega [? vac ?], which is very close to what we'll 00:02:38.510 --> 00:02:40.301 get from the supernovae, which makes it all 00:02:40.301 --> 00:02:41.890 look very convincing. 00:02:41.890 --> 00:02:45.430 And furthermore, the cosmic background radiation 00:02:45.430 --> 00:02:49.000 shows that omega total is equal to 1 to about 1/2% 00:02:49.000 --> 00:02:52.900 accuracy, which is very hard to account for if one doesn't 00:02:52.900 --> 00:02:54.980 assume that there's a very significant amount 00:02:54.980 --> 00:02:55.790 of dark energy. 00:02:55.790 --> 00:02:57.290 Because there just does not appear 00:02:57.290 --> 00:03:02.310 to be nearly enough of anything else to make omega equal to 1. 00:03:02.310 --> 00:03:05.360 And finally, we pointed out that this vacuum energy also 00:03:05.360 --> 00:03:08.140 improves the age calculations. 00:03:08.140 --> 00:03:11.060 Without vacuum energy, we tend to define 00:03:11.060 --> 00:03:13.900 that the age of the universe as calculated from the Big Bang 00:03:13.900 --> 00:03:16.930 theory always ended up being a little younger 00:03:16.930 --> 00:03:20.660 than the ages of the oldest stars, which didn't make sense. 00:03:20.660 --> 00:03:22.950 But with the vacuum energy, that changes 00:03:22.950 --> 00:03:27.860 the cosmological calculation of the age producing older ages. 00:03:27.860 --> 00:03:32.010 So with vacuum energy of the sort that we think exists, 00:03:32.010 --> 00:03:36.570 we get ages like 13.7 or 13.8 billion years. 00:03:36.570 --> 00:03:38.940 And that's completely consistent with what 00:03:38.940 --> 00:03:40.820 we think about the ages of the oldest stars. 00:03:40.820 --> 00:03:43.280 So everything fits together. 00:03:43.280 --> 00:03:46.910 So by now, I would say that with these three arguments 00:03:46.910 --> 00:03:49.570 together, essentially, everybody is convinced 00:03:49.570 --> 00:03:51.520 that this acceleration is real. 00:03:51.520 --> 00:03:53.530 I do know a few people who aren't convinced, 00:03:53.530 --> 00:03:54.920 but they're oddballs. 00:03:54.920 --> 00:03:56.960 Most of us are convinced. 00:03:56.960 --> 00:03:59.460 And the simplest explanation for this dark energy 00:03:59.460 --> 00:04:00.735 is simply vacuum energy. 00:04:00.735 --> 00:04:02.610 And every measurement that's been made so far 00:04:02.610 --> 00:04:05.300 is consistent with the idea of vacuum energy. 00:04:05.300 --> 00:04:07.330 There is still an alternative possibility 00:04:07.330 --> 00:04:09.120 which is called quintessence, which 00:04:09.120 --> 00:04:12.642 would be a very slowly evolving scalar field. 00:04:12.642 --> 00:04:15.100 And it would show up, because you would see some evolution. 00:04:15.100 --> 00:04:17.490 And so far nobody has seen any evolution 00:04:17.490 --> 00:04:22.330 of the amount of dark energy in the universe. 00:04:22.330 --> 00:04:24.530 So that's basically where things stand as far 00:04:24.530 --> 00:04:28.090 as the observations of acceleration of dark energy. 00:04:28.090 --> 00:04:29.580 Any questions about that? 00:04:32.690 --> 00:04:36.230 OK, next, we went on to talk about the physics of vacuum 00:04:36.230 --> 00:04:39.680 energy or a cosmological constant. 00:04:39.680 --> 00:04:41.970 A cosmological constant and vacuum energy 00:04:41.970 --> 00:04:43.830 are really synonymous. 00:04:43.830 --> 00:04:45.900 And they're related to each other 00:04:45.900 --> 00:04:47.480 by the energy entering the vacuum 00:04:47.480 --> 00:04:50.960 being equal to this expression, where lambda is what Einstein 00:04:50.960 --> 00:04:52.775 originally called the cosmological constant 00:04:52.775 --> 00:04:56.264 and what we still call the cosmological constant. 00:04:56.264 --> 00:04:58.180 We discussed the fact that there are basically 00:04:58.180 --> 00:05:00.310 three contributions in a quantum field 00:05:00.310 --> 00:05:02.845 theory to the energy of a vacuum. 00:05:02.845 --> 00:05:04.720 We do not expect it to be zero, because there 00:05:04.720 --> 00:05:07.489 are these complicated contributions. 00:05:07.489 --> 00:05:09.530 There are, first of all, the quantum fluctuations 00:05:09.530 --> 00:05:12.750 of the photon and other Bosonic fields, 00:05:12.750 --> 00:05:14.950 Bosonic fields meaning particles that 00:05:14.950 --> 00:05:18.050 do not obey the Pauli exclusion principle. 00:05:18.050 --> 00:05:20.380 And that gives us a positive contribution 00:05:20.380 --> 00:05:23.790 to the energy, which is, in fact, divergent. 00:05:23.790 --> 00:05:27.870 It diverges because every standing wave contributes. 00:05:27.870 --> 00:05:31.740 And there's no lower bound to the wavelength of a standing 00:05:31.740 --> 00:05:32.610 wave. 00:05:32.610 --> 00:05:35.280 So by considering shorter and shorter wavelengths, 00:05:35.280 --> 00:05:37.880 one gets larger and larger contributions 00:05:37.880 --> 00:05:39.900 to this vacuum energy. 00:05:39.900 --> 00:05:44.030 And in the quantum field theory, it's just unbounded. 00:05:44.030 --> 00:05:46.020 Similarly, there are quantum fluctuations 00:05:46.020 --> 00:05:48.160 to other fields like the electron field 00:05:48.160 --> 00:05:50.290 which is a Fermionic field, a field that 00:05:50.290 --> 00:05:53.450 describes a particle that obeys the Pauli exclusion principle. 00:05:53.450 --> 00:05:56.710 And those fields behave somewhat differently. 00:05:56.710 --> 00:05:58.840 Like the photon, the electron is viewed 00:05:58.840 --> 00:06:02.610 as the quantum excitation of this field. 00:06:02.610 --> 00:06:06.550 And that turns out to be by far, basically, the only way we 00:06:06.550 --> 00:06:09.270 know to describe relativistic particles 00:06:09.270 --> 00:06:12.090 in a totally consistent way. 00:06:12.090 --> 00:06:13.960 In this case, again, the contribution 00:06:13.960 --> 00:06:15.420 to the vacuum energy is divergent. 00:06:15.420 --> 00:06:18.020 But in this case, it's negative and divergent, 00:06:18.020 --> 00:06:20.320 allowing possibilities of some kind of cancellation, 00:06:20.320 --> 00:06:22.131 but no reason that we know of why 00:06:22.131 --> 00:06:23.380 they should cancel each other. 00:06:23.380 --> 00:06:26.350 They seem to just be totally different objects. 00:06:26.350 --> 00:06:28.410 And, finally, there are some fields 00:06:28.410 --> 00:06:30.220 which have nonzero values in the vacuum. 00:06:30.220 --> 00:06:33.910 And, in particular, the Higgs field of the standard model 00:06:33.910 --> 00:06:37.170 is believed to have a nonzero value even in the vacuum. 00:06:40.170 --> 00:06:43.310 So this is the basic story. 00:06:43.310 --> 00:06:46.110 We commented that if we cut off these infinities 00:06:46.110 --> 00:06:48.550 by saying that we don't understand things at very, very 00:06:48.550 --> 00:06:53.050 short wavelengths, at least one plausible cut off 00:06:53.050 --> 00:06:56.300 would be the Planck scale, which is the scale associated 00:06:56.300 --> 00:06:59.460 with where we think quantum gravity becomes important. 00:06:59.460 --> 00:07:02.050 And if we cut off at this Planck scale, 00:07:02.050 --> 00:07:05.880 the energies become finite but still too large 00:07:05.880 --> 00:07:09.340 compared to what we observe by more than 120 orders 00:07:09.340 --> 00:07:10.470 of magnitude. 00:07:10.470 --> 00:07:12.650 And on the homework set that's due next Monday, 00:07:12.650 --> 00:07:15.010 you'll be calculating this number for yourself. 00:07:15.010 --> 00:07:18.720 It's a little bit more than 120 orders of magnitude. 00:07:18.720 --> 00:07:21.360 So it's a colossal failure indicating that we really 00:07:21.360 --> 00:07:24.790 don't understand what controls the value of this vacuum 00:07:24.790 --> 00:07:27.010 energy. 00:07:27.010 --> 00:07:28.707 And I think I mentioned last time, 00:07:28.707 --> 00:07:30.540 and I'll mention it a little more explicitly 00:07:30.540 --> 00:07:32.090 by writing it on the transparency 00:07:32.090 --> 00:07:36.490 this time, that the situation is so desperate in that we've 00:07:36.490 --> 00:07:40.150 had so much trouble trying to find any way of explaining why 00:07:40.150 --> 00:07:45.860 the vacuum energy should be so small that it has become quite 00:07:45.860 --> 00:07:50.080 popular to accept the possibility, at least, 00:07:50.080 --> 00:07:53.190 that the vacuum energy is determined by what is called 00:07:53.190 --> 00:07:55.720 the anthropic selection principal 00:07:55.720 --> 00:07:58.352 or anthropic selection effect. 00:07:58.352 --> 00:07:59.810 And Steve Weinberg was actually one 00:07:59.810 --> 00:08:03.062 of the first people who advocated this point of view. 00:08:03.062 --> 00:08:06.080 I'm sort of a recent convert to taking this point of view 00:08:06.080 --> 00:08:07.790 seriously. 00:08:07.790 --> 00:08:11.380 But the idea is that there might be 00:08:11.380 --> 00:08:13.620 more than one possible type of vacuum. 00:08:13.620 --> 00:08:15.720 And, in fact, string theory comes in here 00:08:15.720 --> 00:08:16.949 in an important way. 00:08:16.949 --> 00:08:18.490 String theory seems to really predict 00:08:18.490 --> 00:08:21.830 that there's a colossal number of different types of vacuum, 00:08:21.830 --> 00:08:25.680 perhaps 10 to the 500 different types of vacuum or more. 00:08:25.680 --> 00:08:27.840 And each one would have its own vacuum energy. 00:08:27.840 --> 00:08:31.560 So with that many, some of them would have a, by coincidence, 00:08:31.560 --> 00:08:33.220 near perfect cancellation between 00:08:33.220 --> 00:08:35.730 the positive and negative contributions producing 00:08:35.730 --> 00:08:38.770 a net vacuum energy that could be very, very small. 00:08:38.770 --> 00:08:42.840 But it would be a tiny fraction of all of the possible vauua, 00:08:42.840 --> 00:08:45.970 a fraction like 10 to the minus 120, 00:08:45.970 --> 00:08:48.030 since we have 120 orders of magnitude 00:08:48.030 --> 00:08:50.730 mismatch of these ranges. 00:08:50.730 --> 00:08:54.170 So you would still have to press yourself to figure out 00:08:54.170 --> 00:08:56.550 what would be the explanation why we should be living 00:08:56.550 --> 00:08:59.260 in such an atypical vacuum. 00:08:59.260 --> 00:09:03.840 And the proposed answer is that it's anthropically selected, 00:09:03.840 --> 00:09:07.280 where anthropic means having to do with life. 00:09:07.280 --> 00:09:11.110 Whereas, the claim is made that life only 00:09:11.110 --> 00:09:16.840 evolves in vacuua which have incredibly small vacuum 00:09:16.840 --> 00:09:18.090 energies. 00:09:18.090 --> 00:09:20.910 Because if the vacuum energy is much larger, if it's positive, 00:09:20.910 --> 00:09:24.520 it blows the universe apart before structures can form. 00:09:24.520 --> 00:09:27.590 And it it's negative, it implodes the universe 00:09:27.590 --> 00:09:30.760 before there's time for structures to form. 00:09:30.760 --> 00:09:34.750 So a long-lived universe requires a very small vacuum 00:09:34.750 --> 00:09:35.664 energy density. 00:09:35.664 --> 00:09:37.580 And the claim is that those are the only kinds 00:09:37.580 --> 00:09:39.810 of universes that support life. 00:09:39.810 --> 00:09:42.100 So we're here because it's the only kind of universe 00:09:42.100 --> 00:09:44.961 in which life can exist is the claim. 00:09:44.961 --> 00:09:45.460 Yes? 00:09:45.460 --> 00:09:46.655 AUDIENCE: So, different types of energies, 00:09:46.655 --> 00:09:48.806 obviously, affect the acceleration rate and stuff 00:09:48.806 --> 00:09:49.472 of the universe. 00:09:49.472 --> 00:09:52.106 But do they also affect, in any way, the fundamental forces, 00:09:52.106 --> 00:09:54.064 or would those be the same in all of the cases? 00:09:54.064 --> 00:09:56.510 PROFESSOR: OK, the question is, would the different kinds 00:09:56.510 --> 00:09:59.630 of vacuum affect the kinds of fundamental forces 00:09:59.630 --> 00:10:03.760 that exist besides the force of the cosmological constant 00:10:03.760 --> 00:10:05.839 on the acceleration of the universe itself? 00:10:05.839 --> 00:10:08.130 The answer is, yeah, it would affect really everything. 00:10:08.130 --> 00:10:10.350 These different vacuua would be very different from each other. 00:10:10.350 --> 00:10:12.000 They would each have their own version 00:10:12.000 --> 00:10:14.585 of what we call the standard model of particle physics. 00:10:14.585 --> 00:10:16.960 And that's because the standard model of particle physics 00:10:16.960 --> 00:10:18.890 would be viewed as what happens when 00:10:18.890 --> 00:10:22.380 you have small perturbations about our particular type 00:10:22.380 --> 00:10:23.434 of vacuum. 00:10:23.434 --> 00:10:24.850 And with different types of vacuum 00:10:24.850 --> 00:10:27.670 you get different time types of small perturbations 00:10:27.670 --> 00:10:29.530 about those vaccua. 00:10:29.530 --> 00:10:32.020 So the physics really could be completely different in all 00:10:32.020 --> 00:10:36.330 the different vacuua that string theory suggests exist. 00:10:40.840 --> 00:10:43.030 So the story here, basically, is a big mystery. 00:10:43.030 --> 00:10:45.880 Not everybody accepts these anthropic ideas. 00:10:45.880 --> 00:10:48.160 They are talked about. 00:10:48.160 --> 00:10:49.780 At almost any cosmology conference, 00:10:49.780 --> 00:10:52.446 there will be some session where people talk about these things. 00:10:52.446 --> 00:10:55.920 They are widely discussed but by no means 00:10:55.920 --> 00:10:58.080 completely agreed upon. 00:10:58.080 --> 00:11:00.230 And it's very much an open question, 00:11:00.230 --> 00:11:03.840 what it is that explains the very small vacuum energy 00:11:03.840 --> 00:11:07.300 density that we observe. 00:11:07.300 --> 00:11:10.370 OK, moving on, in the last lecture 00:11:10.370 --> 00:11:14.779 I also gave a quick historical overview of the interactions 00:11:14.779 --> 00:11:16.320 between Einstein and Friedmann, which 00:11:16.320 --> 00:11:18.680 I found rather interesting. 00:11:18.680 --> 00:11:24.460 And just a quick summary here, in 1922 June 29, to be precise, 00:11:24.460 --> 00:11:27.530 Alexander Friedmann's first paper about the Friedmann 00:11:27.530 --> 00:11:31.450 equations and the dynamical model of the universe 00:11:31.450 --> 00:11:35.150 were received at [INAUDIBLE]. 00:11:35.150 --> 00:11:37.820 Einstein learned about it and immediately decided 00:11:37.820 --> 00:11:42.840 that it had to be wrong and fired off a refutation claiming 00:11:42.840 --> 00:11:44.820 that Friedmann had gotten his equations wrong. 00:11:44.820 --> 00:11:46.140 And if he had gotten them right, he 00:11:46.140 --> 00:11:48.389 would have discovered that rho dot, the rate of change 00:11:48.389 --> 00:11:50.500 of the energy density, had to be zero 00:11:50.500 --> 00:11:53.170 and that there was nothing but the static solution allowed. 00:11:55.840 --> 00:12:00.030 Einstein then met a friend of Friedmann's Yuri Krutkov 00:12:00.030 --> 00:12:05.110 at a retirement lecture by Lawrence in Leiden 00:12:05.110 --> 00:12:07.210 the following spring. 00:12:07.210 --> 00:12:09.400 And Krutkov convinced Einstein that he 00:12:09.400 --> 00:12:11.570 was wrong about this calculation. 00:12:11.570 --> 00:12:13.390 Einstein had also received a letter 00:12:13.390 --> 00:12:14.960 from Friedmann, which he probably 00:12:14.960 --> 00:12:17.005 didn't read until this time, but the letter 00:12:17.005 --> 00:12:19.290 was apparently also convincing. 00:12:19.290 --> 00:12:21.450 So Einstein did finally retract. 00:12:21.450 --> 00:12:26.760 And at the end of May 1923, his refraction 00:12:26.760 --> 00:12:28.425 was received at Zeitschrift fur Physik. 00:12:31.240 --> 00:12:33.700 And another interesting fact about that 00:12:33.700 --> 00:12:37.400 is that the original handwritten draft of that retraction 00:12:37.400 --> 00:12:38.620 still exists. 00:12:38.620 --> 00:12:42.010 And it had the curious comment, which was crossed out, 00:12:42.010 --> 00:12:46.270 where Einstein suggested that the Friedmann solutions could 00:12:46.270 --> 00:12:48.670 be modified by the phrase, "a physical significance 00:12:48.670 --> 00:12:51.290 can hardly be ascribed to them." 00:12:51.290 --> 00:12:53.330 But at the last minute, apparently, Einstein 00:12:53.330 --> 00:12:55.800 decided he didn't really have a very good foundation 00:12:55.800 --> 00:13:00.414 for that statement and crossed it out. 00:13:00.414 --> 00:13:01.830 So I like the story, first of all, 00:13:01.830 --> 00:13:03.884 because it illustrates that we're not 00:13:03.884 --> 00:13:05.300 the only people who make mistakes. 00:13:05.300 --> 00:13:07.140 Even great physicists like Einstein 00:13:07.140 --> 00:13:09.065 make really silly mistakes. 00:13:09.065 --> 00:13:12.114 It really was just a dumb calculational error. 00:13:12.114 --> 00:13:14.030 And it also, I think, points out how important 00:13:14.030 --> 00:13:16.620 it is not to allow yourself to be caught 00:13:16.620 --> 00:13:20.066 in the grip of some firm idea that you cease to question, 00:13:20.066 --> 00:13:21.440 which apparently is what happened 00:13:21.440 --> 00:13:24.830 to Einstein with his belief that the universe was static. 00:13:24.830 --> 00:13:26.330 He was so sure that the universe was 00:13:26.330 --> 00:13:29.150 static that he very quickly looked at Friedmann's paper 00:13:29.150 --> 00:13:32.100 and reached the incorrect conclusion that Friedmann 00:13:32.100 --> 00:13:34.310 had gotten his calculus wrong. 00:13:34.310 --> 00:13:37.960 In fact, it was Einstein who got it wrong. 00:13:37.960 --> 00:13:39.630 So that summarizes the last lecture. 00:13:39.630 --> 00:13:40.900 Any further questions? 00:13:44.280 --> 00:13:47.380 OK, in that case, I think I am done with that. 00:13:47.380 --> 00:13:49.750 Yeah, that comes later. 00:13:49.750 --> 00:13:52.890 OK, what I want to do next is to talk 00:13:52.890 --> 00:13:58.800 about two problems associated with the conventional cosmology 00:13:58.800 --> 00:14:01.280 that we've been learning about and, in particular, 00:14:01.280 --> 00:14:02.941 I mean cosmology without inflation, 00:14:02.941 --> 00:14:04.440 which we have not learned about yet. 00:14:04.440 --> 00:14:06.530 So I am talking about the cosmology 00:14:06.530 --> 00:14:09.140 that we've learned about so far. 00:14:09.140 --> 00:14:12.340 So there are a total of three that I 00:14:12.340 --> 00:14:16.080 want to discuss problems associated 00:14:16.080 --> 00:14:19.400 with conventional cosmology which serve as motivations 00:14:19.400 --> 00:14:23.570 for the inflationary modification that, I think, 00:14:23.570 --> 00:14:27.710 you'll be learning about next time from Scott Hughes. 00:14:27.710 --> 00:14:30.570 But today I want to talk about the problems. 00:14:30.570 --> 00:14:37.030 So the first of 3 is sometimes called the homogeneity, 00:14:37.030 --> 00:14:38.220 or the horizon problem. 00:14:46.280 --> 00:14:49.870 And this is the problem that the universe 00:14:49.870 --> 00:14:53.130 is observed to be incredibly uniform. 00:14:53.130 --> 00:14:56.370 And this uniformity shows up most clearly 00:14:56.370 --> 00:14:59.390 in the cosmic microwave background radiation, 00:14:59.390 --> 00:15:01.960 where astronomers have now made very sensitive measurements 00:15:01.960 --> 00:15:04.390 of the temperature as a function of angle in the sky. 00:15:04.390 --> 00:15:07.770 And it's found that that radiation in uniform to one 00:15:07.770 --> 00:15:10.970 part in 100,000, part in 10 to the 5. 00:15:26.770 --> 00:15:29.260 Now, the CMB is essentially a snapshot 00:15:29.260 --> 00:15:33.610 of what the universe looked like at about 370,000 years 00:15:33.610 --> 00:15:36.160 after the Big Bang at the time that we call t sub 00:15:36.160 --> 00:15:37.590 d, the time of decoupling. 00:16:00.088 --> 00:16:00.588 Yes? 00:16:00.588 --> 00:16:02.560 AUDIENCE: This measurement, the 10 to the 5, 00:16:02.560 --> 00:16:05.026 it's not a limit that we've reached 00:16:05.026 --> 00:16:06.150 measurement technique-wise? 00:16:06.150 --> 00:16:09.186 That's what it actually is, [INAUDIBLE]? 00:16:09.186 --> 00:16:11.630 PROFESSOR: Yes, I was going to mention that. 00:16:11.630 --> 00:16:15.710 We actually do see fluctuations at the level of one part in 10 00:16:15.710 --> 00:16:17.540 to the five. 00:16:17.540 --> 00:16:18.730 So it's not just a limit. 00:16:18.730 --> 00:16:22.230 It is an actual observation. 00:16:22.230 --> 00:16:25.840 And what we interpret is that the photons that we're 00:16:25.840 --> 00:16:30.480 seeing in the CMB have been flying along on straight lines 00:16:30.480 --> 00:16:31.906 since the time of decoupling. 00:16:31.906 --> 00:16:33.530 And therefore, what they show us really 00:16:33.530 --> 00:16:35.071 is an image of what the universe look 00:16:35.071 --> 00:16:37.080 like at the time of decoupling. 00:16:37.080 --> 00:16:39.200 And that image is an image of the universe which 00:16:39.200 --> 00:16:42.767 is almost a perfectly smooth mass density and a perfectly 00:16:42.767 --> 00:16:44.350 smooth temperature-- it really is just 00:16:44.350 --> 00:16:47.620 radiation-- but tiny ripples superimposed 00:16:47.620 --> 00:16:50.210 on top of that uniformity where the ripples have 00:16:50.210 --> 00:16:53.240 an amplitude of order of 10 to the minus 5. 00:16:56.049 --> 00:16:57.840 And those ripples are important, because we 00:16:57.840 --> 00:17:00.110 think that those are the origin of all structure 00:17:00.110 --> 00:17:01.760 in the universe. 00:17:01.760 --> 00:17:05.260 The universe is gravitationally unstable 00:17:05.260 --> 00:17:08.069 where there's a positive ripple making the mass density 00:17:08.069 --> 00:17:10.160 slightly higher than average. 00:17:10.160 --> 00:17:12.170 That creates a slightly stronger than average 00:17:12.170 --> 00:17:15.609 gravitational field pulling in extra matter, 00:17:15.609 --> 00:17:18.010 creating a still stronger gravitational field. 00:17:18.010 --> 00:17:20.339 And the process cascades until you ultimately 00:17:20.339 --> 00:17:22.660 have galaxies and clusters of galaxies 00:17:22.660 --> 00:17:24.990 and all the complexity in the universe. 00:17:24.990 --> 00:17:28.960 But it starts from these seeds, these minor fluctuations 00:17:28.960 --> 00:17:32.260 at the level of one part in 10 to the five. 00:17:32.260 --> 00:17:33.970 But for now we want to discuss is simply 00:17:33.970 --> 00:17:36.960 the question of how did we get so uniform. 00:17:36.960 --> 00:17:40.210 We'll talk about how the non-uniformities arise later 00:17:40.210 --> 00:17:42.640 in the context of inflation. 00:17:42.640 --> 00:17:47.030 The basic picture is that we are someplace. 00:17:47.030 --> 00:17:49.540 I'll put us here in a little diagram. 00:17:49.540 --> 00:17:53.690 We are receiving photons, say, from opposite directions 00:17:53.690 --> 00:17:54.190 in the sky. 00:17:57.100 --> 00:17:59.930 Those little arrows represent the incoming patterns 00:17:59.930 --> 00:18:03.310 of two different CMB photons coming 00:18:03.310 --> 00:18:05.020 from opposite directions. 00:18:05.020 --> 00:18:09.480 And what I'm interested in doing to understand 00:18:09.480 --> 00:18:12.570 the situation with regard to this uniformity is I'm 00:18:12.570 --> 00:18:15.240 interested in tracing these photons back 00:18:15.240 --> 00:18:19.570 to where they originated at time t sub d. 00:18:19.570 --> 00:18:21.169 And I want to do that on both sides. 00:18:21.169 --> 00:18:22.460 But, of course, it's symmetric. 00:18:22.460 --> 00:18:25.052 So I only need to do one calculation. 00:18:25.052 --> 00:18:26.520 And what I want to know is how far 00:18:26.520 --> 00:18:28.260 apart were these two points. 00:18:28.260 --> 00:18:30.690 Because I want to explore the question of whether 00:18:30.690 --> 00:18:32.360 or not this uniformity in temperature 00:18:32.360 --> 00:18:34.290 could just be mundane. 00:18:34.290 --> 00:18:36.590 If you let any object sit for a long time, 00:18:36.590 --> 00:18:38.580 it will approach a uniform temperature. 00:18:38.580 --> 00:18:42.230 That's why pizza gets cold when you take it out of the oven. 00:18:42.230 --> 00:18:44.730 So could that be responsible for this uniformity? 00:18:44.730 --> 00:18:46.860 And what we'll see is that it cannot. 00:18:46.860 --> 00:18:49.040 Because these two points are just too far apart 00:18:49.040 --> 00:18:51.600 for them to come in to thermal equilibrium 00:18:51.600 --> 00:18:54.830 by ordinary thermal equilibrium processes 00:18:54.830 --> 00:18:59.210 in the context of the conventional big bang theory. 00:18:59.210 --> 00:19:01.560 So we want to calculate how far apart these points were 00:19:01.560 --> 00:19:03.252 at the time of emission. 00:19:07.220 --> 00:19:09.180 So what do we know? 00:19:09.180 --> 00:19:13.900 We know that the temperature at the time of decoupling 00:19:13.900 --> 00:19:15.920 was about 3,000 Kelvin, which is really 00:19:15.920 --> 00:19:18.850 where we started with our discussion of decoupling. 00:19:18.850 --> 00:19:20.540 We did not do the statistical mechanics 00:19:20.540 --> 00:19:22.370 associated with this statement. 00:19:22.370 --> 00:19:24.485 But for a given density, you can calculate 00:19:24.485 --> 00:19:27.880 at what temperature hydrogen ionizes. 00:19:27.880 --> 00:19:30.740 And for the density that we expect for the early universe, 00:19:30.740 --> 00:19:34.254 that's the temperature at which the ionization occurs. 00:19:34.254 --> 00:19:35.670 So that's where decoupling occurs. 00:19:35.670 --> 00:19:39.900 That's where it becomes neutral as the universe expands. 00:19:39.900 --> 00:19:44.977 We also know that during this period, aT, 00:19:44.977 --> 00:19:46.560 the scale factor times the temperature 00:19:46.560 --> 00:19:48.950 is about equal to a constant, which 00:19:48.950 --> 00:19:52.370 follows as a consequence of conservation of entropy, 00:19:52.370 --> 00:19:54.879 the idea that the universe is near thermal equilibrium. 00:19:54.879 --> 00:19:56.170 So the entropy does not change. 00:19:59.690 --> 00:20:04.260 Then we can calculate the z for decoupling, 00:20:04.260 --> 00:20:09.070 because it would just be the ratio of the temperatures. 00:20:09.070 --> 00:20:11.630 It's defined by the ratio of the scale factors. 00:20:11.630 --> 00:20:15.250 This defines what you mean by 1 plus z decoupling. 00:20:15.250 --> 00:20:17.690 But if aT is about equal to a constant, 00:20:17.690 --> 00:20:20.990 we can relate this to the temperatures inversely. 00:20:20.990 --> 00:20:24.510 So the temperature of decoupling goes in the numerator. 00:20:24.510 --> 00:20:28.810 And the temperature today goes into the denominator. 00:20:28.810 --> 00:20:34.680 And, numerically, that's about 1,100. 00:20:34.680 --> 00:20:37.780 So the z of the cosmic background radiation 00:20:37.780 --> 00:20:42.020 is about 1,100, vastly larger than the red shifts associated 00:20:42.020 --> 00:20:43.930 with observations of galaxies or supernovae. 00:20:49.350 --> 00:20:53.160 From the z, we can calculate the physical distance today 00:20:53.160 --> 00:20:59.870 of these two locations, because this calculation we already 00:20:59.870 --> 00:21:00.370 did. 00:21:06.900 --> 00:21:11.770 So I'm going to call l sub p the physical distance between us 00:21:11.770 --> 00:21:14.520 and the source of this radiation. 00:21:14.520 --> 00:21:18.951 And its value today-- I'm starting with this formula 00:21:18.951 --> 00:21:20.450 simply because we already derived it 00:21:20.450 --> 00:21:30.430 on a homework set-- it's 2c h naught inverse times 1 minus 1 00:21:30.430 --> 00:21:35.250 over the square root of 1 plus z. 00:21:35.250 --> 00:21:38.480 And this calculation was done for a flat matter dominated 00:21:38.480 --> 00:21:55.064 universe, flat matter dominated. 00:21:55.064 --> 00:21:56.730 Of course, that's only an approximation, 00:21:56.730 --> 00:21:59.930 because we know our real universe was matter 00:21:59.930 --> 00:22:02.380 dominated at the start of this period. 00:22:02.380 --> 00:22:04.830 But it did not remain matter dominated 00:22:04.830 --> 00:22:07.980 through to the present at about 5,000 or 6,000 years ago. 00:22:07.980 --> 00:22:10.990 We switched to a situation where the dark energy is actually 00:22:10.990 --> 00:22:13.330 larger than the non-relativistic matter. 00:22:13.330 --> 00:22:14.830 So we're ignoring that effect, which 00:22:14.830 --> 00:22:16.996 means we're only going to get an approximation here. 00:22:16.996 --> 00:22:19.770 But it will still be easily good enough to make the point. 00:22:22.830 --> 00:22:26.400 For a z this large, this factor is a small correction. 00:22:26.400 --> 00:22:28.390 I think this ends up being 0.97, or something 00:22:28.390 --> 00:22:31.160 like that, very close to 1, which means what we're getting 00:22:31.160 --> 00:22:33.500 is very close to 2c h naught inverse, which 00:22:33.500 --> 00:22:35.610 is the actual horizon. 00:22:35.610 --> 00:22:37.894 The horizon corresponds to z equals infinity. 00:22:37.894 --> 00:22:39.310 If you think about it, that's what 00:22:39.310 --> 00:22:41.055 you expect the horizon to be. 00:22:41.055 --> 00:22:43.026 It corresponds to infinite red shift. 00:22:43.026 --> 00:22:44.650 And you don't see anything beyond that. 00:22:50.100 --> 00:22:56.000 So if we take the best value we have for h naught, which I'm 00:22:56.000 --> 00:23:00.970 taking from the Planck satellite, 57.3 00:23:00.970 --> 00:23:06.130 kilometers per second per megaparsec, 00:23:06.130 --> 00:23:14.980 and put that and the value for z into this formula, 00:23:14.980 --> 00:23:24.080 we get l sub p of t naught of 28.2 billion light years 00:23:24.080 --> 00:23:27.700 times 10 to the 9 light year. 00:23:32.440 --> 00:23:35.020 So it's of course larger than ct as we expect. 00:23:35.020 --> 00:23:38.450 It's basically 3ct for a matter dominated universe. 00:23:38.450 --> 00:23:42.520 And 3ct is the same as 2ch 0 inverse. 00:23:47.930 --> 00:23:50.010 Now, what we want to know, though, 00:23:50.010 --> 00:23:53.510 is how far away was this when the emission occurred, 00:23:53.510 --> 00:23:55.200 not the present distance. 00:23:55.200 --> 00:23:56.840 We looked at the present distance 00:23:56.840 --> 00:24:00.030 simply because we had a formula for it from our homework set. 00:24:00.030 --> 00:24:03.290 But we know how to extrapolate that backwards, l sub 00:24:03.290 --> 00:24:06.070 t at time t sub d. 00:24:06.070 --> 00:24:08.550 Distances that are fixed in co-moving space, 00:24:08.550 --> 00:24:10.050 which these are, are just stretched 00:24:10.050 --> 00:24:12.420 with the scale factor. 00:24:12.420 --> 00:24:16.280 So this will just be the scale factor 00:24:16.280 --> 00:24:19.180 at the time of decoupling divided by the scale factor 00:24:19.180 --> 00:24:21.450 today times the present distance. 00:24:26.290 --> 00:24:30.790 And this is, again, given by this ratio of temperatures. 00:24:30.790 --> 00:24:33.400 So it's 1 over 1,100, the inverse 00:24:33.400 --> 00:24:35.170 of what we had over there. 00:24:35.170 --> 00:24:37.160 So the separation at this early time 00:24:37.160 --> 00:24:41.070 is just 1,100 times smaller than the separation today. 00:24:41.070 --> 00:24:46.350 And that can be evaluated numerically. 00:24:46.350 --> 00:24:54.900 And it gives us 2.56 times 10 to the seven light years, 00:24:54.900 --> 00:24:59.747 so 25 million light years. 00:24:59.747 --> 00:25:01.830 Now, the point is that that's significantly larger 00:25:01.830 --> 00:25:05.724 than the horizon distance at that time. 00:25:05.724 --> 00:25:07.140 And remember, the horizon distance 00:25:07.140 --> 00:25:09.770 is the maximum possible distance that anything 00:25:09.770 --> 00:25:12.280 can travel limited by the speed of light 00:25:12.280 --> 00:25:14.710 from the time of the big bang up to any given 00:25:14.710 --> 00:25:17.790 point in cosmic history. 00:25:17.790 --> 00:25:23.060 So the horizon at time t sub d is just 00:25:23.060 --> 00:25:26.480 given by the simple formula that the physical value 00:25:26.480 --> 00:25:30.180 of the horizon distance, l sub h phys, 00:25:30.180 --> 00:25:36.570 l sub horizon physical, at time t sub d is just equal to, 00:25:36.570 --> 00:25:41.720 for a matter dominated universe, 3c times t sub d. 00:25:41.720 --> 00:25:44.670 And that we can evaluate, given what we have. 00:25:44.670 --> 00:25:52.050 And it's about 1.1 times 10 to the sixth light years, which 00:25:52.050 --> 00:25:54.840 is significantly less than 2.56 times 10 to the seven 00:25:54.840 --> 00:25:56.790 light years. 00:25:56.790 --> 00:26:01.920 And, in fact, the ratio of the two, given these numbers, 00:26:01.920 --> 00:26:10.870 is that l sub p of t sub d over l sub h is also of t sub 00:26:10.870 --> 00:26:19.720 d is about equal to 23, just doing the arithmetic. 00:26:19.720 --> 00:26:23.170 And that means if we go back to this picture, 00:26:23.170 --> 00:26:27.820 these two points of emission were separated from each other 00:26:27.820 --> 00:26:29.995 by about 46 horizon distances. 00:26:40.000 --> 00:26:42.900 And that's enough to imply that there's no way 00:26:42.900 --> 00:26:45.880 that this point could have known anything whatever about what 00:26:45.880 --> 00:26:48.100 was going on at this point. 00:26:48.100 --> 00:26:50.690 Yet somehow they knew to emit these two photons 00:26:50.690 --> 00:26:54.100 at the same time at the same temperature. 00:26:54.100 --> 00:26:56.300 And that's the mystery. 00:26:56.300 --> 00:26:58.630 One can get around this mystery if one simply 00:26:58.630 --> 00:27:03.460 assumes that the singularity that created all of this 00:27:03.460 --> 00:27:07.359 produced a perfectly homogeneous universe from the very start. 00:27:07.359 --> 00:27:09.150 Since we don't understand that singularity, 00:27:09.150 --> 00:27:11.630 we're allowed to attribute anything we want to it. 00:27:11.630 --> 00:27:14.330 So in particular, you can attribute perfect homogeneity 00:27:14.330 --> 00:27:15.729 to the singularity. 00:27:15.729 --> 00:27:17.270 But that's not really an explanation. 00:27:17.270 --> 00:27:19.260 That's an assumption. 00:27:19.260 --> 00:27:23.060 So if one wants to be able to explain this uniformity, 00:27:23.060 --> 00:27:25.760 then one simply cannot do it in the context of conventional 00:27:25.760 --> 00:27:27.010 cosmology. 00:27:27.010 --> 00:27:28.880 There's just no way that causality, 00:27:28.880 --> 00:27:31.240 the limit of the speed of light, allows this point 00:27:31.240 --> 00:27:34.150 to know anything about what's going on at that point. 00:27:34.150 --> 00:27:34.650 Yes? 00:27:34.650 --> 00:27:36.910 AUDIENCE: How could a singularity not be uniform? 00:27:36.910 --> 00:27:39.520 Because If it had non-uniform [INAUDIBLE], 00:27:39.520 --> 00:27:41.537 then not be singular? 00:27:41.537 --> 00:27:42.953 PROFESSOR: OK, the question is how 00:27:42.953 --> 00:27:46.020 can a singularity not be uniform? 00:27:46.020 --> 00:27:47.480 The answer is, yes, singularities 00:27:47.480 --> 00:27:49.560 can not be uniform. 00:27:49.560 --> 00:27:53.090 And I think the way one can show that is a little hard. 00:27:53.090 --> 00:27:55.810 But you have to imagine a non-uniform thing collapsing. 00:27:55.810 --> 00:27:58.060 And then it would just be the time reverse, everything 00:27:58.060 --> 00:27:59.750 popping out of the singularity. 00:27:59.750 --> 00:28:01.850 So you can ask, does a non-uniform thing 00:28:01.850 --> 00:28:03.534 collapse to a singularity? 00:28:03.534 --> 00:28:05.450 And the answer to that question is not obvious 00:28:05.450 --> 00:28:08.510 and really was debated for a long time. 00:28:08.510 --> 00:28:11.830 But there were theorems proven by Hawking and Penrose 00:28:11.830 --> 00:28:16.170 that indeed not only do the homogeneous solutions that we 00:28:16.170 --> 00:28:18.722 look at collapse but in homogeneous solutions also 00:28:18.722 --> 00:28:19.805 collapse to singularities. 00:28:23.505 --> 00:28:25.380 So a singularity does not have to be uniform. 00:28:30.190 --> 00:28:34.150 OK, so this is the story of the horizon problem. 00:28:34.150 --> 00:28:35.950 And as I said, you can get around it 00:28:35.950 --> 00:28:38.325 if you're willing to just assume that the singularity was 00:28:38.325 --> 00:28:39.040 homogeneous. 00:28:39.040 --> 00:28:41.550 But if you want to have a dynamical explanation for how 00:28:41.550 --> 00:28:44.100 the uniformity of the universe was established, 00:28:44.100 --> 00:28:46.730 then you need some model other than this conventional 00:28:46.730 --> 00:28:49.670 cosmological model that we've been discussing. 00:28:49.670 --> 00:28:51.660 And inflation will be such a twist 00:28:51.660 --> 00:28:55.120 which will allow a solution to this problem. 00:28:55.120 --> 00:28:57.830 OK, so if there are no questions, 00:28:57.830 --> 00:28:59.400 no further questions, we'll go on 00:28:59.400 --> 00:29:01.270 to the next problem I want to discuss, 00:29:01.270 --> 00:29:05.520 which is of a similar nature in that you can get around it 00:29:05.520 --> 00:29:10.670 by making strong assumptions about the initial singularity. 00:29:10.670 --> 00:29:13.020 But if one wants, again, something 00:29:13.020 --> 00:29:15.530 you can put your hands on, rather than just an assumption 00:29:15.530 --> 00:29:19.550 about a singularity, then inflation will do the job. 00:29:19.550 --> 00:29:22.590 But you cannot solve the problem in the context 00:29:22.590 --> 00:29:24.055 of a conventional big bang theory, 00:29:24.055 --> 00:29:26.430 because the mechanics of the conventional big bang theory 00:29:26.430 --> 00:29:28.720 are simply well-defined. 00:29:28.720 --> 00:29:31.510 So what I want to talk here is what 00:29:31.510 --> 00:29:36.980 is called the flatness problem, where flatness 00:29:36.980 --> 00:29:40.840 is in the sense of Omega very near 1. 00:29:40.840 --> 00:29:42.470 And this is basically the problem 00:29:42.470 --> 00:29:48.930 of why is Omega today somewhere near 1? 00:29:48.930 --> 00:29:52.360 So Omega naught is the present value of Omega, 00:29:52.360 --> 00:29:55.795 why is it about equal to 1? 00:29:55.795 --> 00:29:57.170 Now, what do we know first of all 00:29:57.170 --> 00:29:59.750 about it being about equal to 1? 00:29:59.750 --> 00:30:03.900 The best value from the Planck group, 00:30:03.900 --> 00:30:05.890 this famous Planck satellite that I've 00:30:05.890 --> 00:30:08.820 been quoting a lot of numbers from-- 00:30:08.820 --> 00:30:10.970 and I think in all cases, I've been quoting numbers 00:30:10.970 --> 00:30:14.310 that they've established combining their own data 00:30:14.310 --> 00:30:16.897 with some other pieces of data. 00:30:16.897 --> 00:30:18.480 So it's not quite the satellite alone. 00:30:18.480 --> 00:30:20.730 Although, they do give numbers for the satellite alone 00:30:20.730 --> 00:30:22.800 which are just a little bit less precise. 00:30:22.800 --> 00:30:26.410 But the best number they give for Omega naught is minus 00:30:26.410 --> 00:30:34.560 0.0010 plus or minus 0.0065. 00:30:34.560 --> 00:30:36.460 Oops, I didn't put enough zeroes there. 00:30:44.030 --> 00:30:47.657 So it's 0.0065 is the error. 00:30:47.657 --> 00:30:49.740 So the error is just a little bit more than a half 00:30:49.740 --> 00:30:52.320 of a percent. 00:30:52.320 --> 00:30:56.330 And as you see, consistent with-- I'm sorry, 00:30:56.330 --> 00:30:57.650 I meant this to be 1. 00:31:00.910 --> 00:31:01.680 Hold on. 00:31:04.480 --> 00:31:09.750 This is Omega naught minus 1 that I'm writing a formula for. 00:31:09.750 --> 00:31:12.060 So Omega naught is very near 1 up to that accuracy. 00:31:28.200 --> 00:31:30.620 What I want to emphasize in terms of this flatness problem 00:31:30.620 --> 00:31:33.920 is that you don't need to know that Omega naught is very, very 00:31:33.920 --> 00:31:36.180 close to 1 today, which we now do know. 00:31:36.180 --> 00:31:43.090 But even back when inflation was first invented around 1980, 00:31:43.090 --> 00:31:44.630 in circa 1980 we certainly didn't 00:31:44.630 --> 00:31:47.740 know that Omega was so incredibly close to 1. 00:31:47.740 --> 00:31:50.950 But we did know that Omega was somewhere 00:31:50.950 --> 00:31:57.790 in the range of about 0.1 and 0.2, which is not 00:31:57.790 --> 00:32:02.140 nearly as close to 1 as what we know now, but still close to 1. 00:32:02.140 --> 00:32:04.420 I'll argue that the flatness problem exists 00:32:04.420 --> 00:32:06.640 for these numbers almost as strongly 00:32:06.640 --> 00:32:09.000 as it exists for those numbers. 00:32:09.000 --> 00:32:11.990 Differ, but this is still a very, very strong argument 00:32:11.990 --> 00:32:15.240 that even a number like this is amazingly close to 1 00:32:15.240 --> 00:32:18.350 considering what you should expect. 00:32:18.350 --> 00:32:22.255 Now, what underlies this is the expectations, how close should 00:32:22.255 --> 00:32:24.070 we expect Omega to be to 1? 00:32:24.070 --> 00:32:26.170 And the important underlying piece 00:32:26.170 --> 00:32:31.560 of dynamics that controls this is the fact that Omega equals 1 00:32:31.560 --> 00:32:34.370 is an unstable equilibrium point. 00:32:34.370 --> 00:32:37.290 That means it's like a pencil balancing on its tip. 00:32:37.290 --> 00:32:39.920 If Omega is exactly equal to 1, that 00:32:39.920 --> 00:32:41.650 means you have a flat universe. 00:32:41.650 --> 00:32:43.270 And an exactly flat universe will 00:32:43.270 --> 00:32:46.390 remain an exactly flat universe forever. 00:32:46.390 --> 00:32:48.055 So if Omega is exactly equal to 1, 00:32:48.055 --> 00:32:51.580 it will remain exactly equal to 1 forever. 00:32:51.580 --> 00:32:54.420 But if Omega in the early universe 00:32:54.420 --> 00:32:56.390 were just a tiny bit bigger than 1-- 00:32:56.390 --> 00:32:58.220 and we're about to calculate this, 00:32:58.220 --> 00:33:05.170 but I'll first qualitatively describe the result-- 00:33:05.170 --> 00:33:08.522 it would rise and would rapidly reach infinity, 00:33:08.522 --> 00:33:10.980 which is what it reaches if you have a closed universe when 00:33:10.980 --> 00:33:13.010 a closed universe reaches its maximum size. 00:33:13.010 --> 00:33:17.830 So Omega becomes infinity and then the universe recollapses. 00:33:17.830 --> 00:33:19.395 So if Omega were bigger than 1, it 00:33:19.395 --> 00:33:20.870 would rapidly approach infinity. 00:33:20.870 --> 00:33:23.300 If Omega in the early universe were just a little bit less 00:33:23.300 --> 00:33:26.250 than 1, it would rapidly trail off towards 0 00:33:26.250 --> 00:33:29.990 and not stay 1 for any length of time. 00:33:29.990 --> 00:33:34.000 So the only way to get Omega near 1 today 00:33:34.000 --> 00:33:36.310 is like having a pencil that's almost straight 00:33:36.310 --> 00:33:39.470 up after standing there for 1 billion years. 00:33:39.470 --> 00:33:41.195 It'd have to have started out incredibly 00:33:41.195 --> 00:33:42.320 close to being straight up. 00:33:42.320 --> 00:33:44.250 It has to have started out incredibly 00:33:44.250 --> 00:33:46.240 close to Omega equals 1. 00:33:46.240 --> 00:33:49.080 And we're going to calculate how close. 00:33:49.080 --> 00:33:53.160 So that's the set-up. 00:33:53.160 --> 00:33:56.700 So the question we want to ask is how close did Omega 00:33:56.700 --> 00:33:58.940 have to be to 1 in the early universe 00:33:58.940 --> 00:34:02.110 to be in either one of these allowed ranges today. 00:34:02.110 --> 00:34:04.100 And for the early universe, I'm going 00:34:04.100 --> 00:34:07.220 to take t equals one second as my time at which I'll 00:34:07.220 --> 00:34:08.389 do these calculations. 00:34:08.389 --> 00:34:10.639 And, historically, that's where this problem was first 00:34:10.639 --> 00:34:17.610 discussed by Dicke and Peebles back in 1979. 00:34:17.610 --> 00:34:20.040 And the reason why one second was chosen by them, 00:34:20.040 --> 00:34:23.150 and why it's a sensible time for us to talk about as well, 00:34:23.150 --> 00:34:25.729 is that one second is the beginning 00:34:25.729 --> 00:34:27.770 of the processes of nucleosynthesis, which you've 00:34:27.770 --> 00:34:30.370 read about in Weinberg and in Ryden, 00:34:30.370 --> 00:34:34.610 and provides a real test of our understanding of cosmology 00:34:34.610 --> 00:34:35.969 at those times. 00:34:35.969 --> 00:34:38.721 So we could say that we have real empirical evidence 00:34:38.721 --> 00:34:40.929 in the statement that the predictions of the chemical 00:34:40.929 --> 00:34:42.678 elements work. 00:34:42.678 --> 00:34:44.719 We could say that we have real empirical evidence 00:34:44.719 --> 00:34:47.900 that our cosmological model works back to one second 00:34:47.900 --> 00:34:49.900 after the Big Bang. 00:34:49.900 --> 00:34:51.745 So we're going to choose one second 00:34:51.745 --> 00:34:54.120 for the time at which we're going to calculate what Omega 00:34:54.120 --> 00:34:56.840 must've been then for it to be an allowed range today. 00:35:00.910 --> 00:35:20.480 How close must Omega have been to 1 at t equals 1 second? 00:35:20.480 --> 00:35:23.530 Question mark. 00:35:23.530 --> 00:35:25.030 OK, now, to do this calculation, you 00:35:25.030 --> 00:35:26.930 don't need to know anything that you don't already know. 00:35:26.930 --> 00:35:29.070 It really follows as a consequence of the Friedmann 00:35:29.070 --> 00:35:33.710 equation and how matter and temperature and so on behave 00:35:33.710 --> 00:35:37.902 with time during radiation in matter dominated eras. 00:35:37.902 --> 00:35:40.360 So we're going to start with just the plain old first order 00:35:40.360 --> 00:35:46.680 Fiedmann equation, h squared is equal to 8 pi over 3 g Rho 00:35:46.680 --> 00:35:51.370 minus kc squared over a squared, which 00:35:51.370 --> 00:35:56.640 you have seen many, many times already in this course. 00:35:56.640 --> 00:35:58.320 We can combine that with other equations 00:35:58.320 --> 00:36:00.140 that you've also seen many times. 00:36:00.140 --> 00:36:03.920 The critical density is just the value of the density 00:36:03.920 --> 00:36:05.540 when k equals 0. 00:36:05.540 --> 00:36:07.900 So you just solve this equation for Rho. 00:36:07.900 --> 00:36:14.484 And you get 3h squared over h pi g. 00:36:14.484 --> 00:36:15.900 This defines the critical density. 00:36:15.900 --> 00:36:20.650 It's that density which makes the universe flat, k equals 0. 00:36:20.650 --> 00:36:22.130 And then our standard definition is 00:36:22.130 --> 00:36:26.412 that Omega is just defined to be the actual mass density divided 00:36:26.412 --> 00:36:27.620 by the critical mass density. 00:36:27.620 --> 00:36:31.990 And Omega will be the quantity we're trying to trace. 00:36:31.990 --> 00:36:34.430 And we're also going to make use of the fact 00:36:34.430 --> 00:36:37.250 that during the era that we're talking about, 00:36:37.250 --> 00:36:40.990 at is essentially equal to a constant. 00:36:40.990 --> 00:36:43.870 It does change a little bit when electron and positron 00:36:43.870 --> 00:36:44.910 pairs freeze out. 00:36:44.910 --> 00:36:46.600 It changes by a factor of something 00:36:46.600 --> 00:36:49.470 like 4/11 to the 1/3 power or something like that. 00:36:49.470 --> 00:36:51.760 But that power will be of order one for our purposes. 00:36:51.760 --> 00:36:53.180 But I guess this is a good reason why 00:36:53.180 --> 00:36:54.679 I should put a squiggle here instead 00:36:54.679 --> 00:36:56.910 of an equal sign as an approximate equality, 00:36:56.910 --> 00:36:59.240 but easily good enough for our purposes, 00:36:59.240 --> 00:37:02.565 meaning the corrections of order one. 00:37:02.565 --> 00:37:03.940 We're going to see the problem is 00:37:03.940 --> 00:37:05.170 much, much bigger than order one. 00:37:05.170 --> 00:37:07.010 So a correction of order one doesn't matter. 00:37:10.510 --> 00:37:14.210 Now, I'm going to start by using the Planck satellite limits. 00:37:14.210 --> 00:37:18.230 And at the end, I'll just make a comment about the circa 1980 00:37:18.230 --> 00:37:19.160 situation. 00:37:19.160 --> 00:37:25.922 But if we look at the Planck limits-- I'm sorry. 00:37:25.922 --> 00:37:28.380 Since I'm going to write an equation for a peculiar looking 00:37:28.380 --> 00:37:30.560 quantity, I should motivate the peculiar 00:37:30.560 --> 00:37:31.667 looking quantity first. 00:37:34.650 --> 00:37:36.700 It turns out to be useful for these purposes. 00:37:39.482 --> 00:37:40.940 And this purpose means we're trying 00:37:40.940 --> 00:37:44.220 to track how Omega changes with time. 00:37:44.220 --> 00:37:46.580 It turns out to be useful to reshuffle the Friedmann. 00:37:46.580 --> 00:37:49.220 And it is just an algebraic reshuffling 00:37:49.220 --> 00:37:52.680 of the Friedmann equation and the definitions 00:37:52.680 --> 00:37:54.200 that we have here. 00:37:54.200 --> 00:37:56.360 We can rewrite the Friedmann equation 00:37:56.360 --> 00:38:02.560 as Omega minus 1 over Omega is equal to a quantity called 00:38:02.560 --> 00:38:07.174 a times the temperature squared over Rho. 00:38:07.174 --> 00:38:09.840 Now, the temperature didn't even occur in the original equation. 00:38:09.840 --> 00:38:12.460 So things might look a little bit suspicious. 00:38:12.460 --> 00:38:14.440 I haven't told you what a is yet. 00:38:14.440 --> 00:38:26.925 a is 3k c squared over 8 pi g a squared t squared. 00:38:26.925 --> 00:38:29.550 So when you put the a into this equation, the t squares cancel. 00:38:29.550 --> 00:38:32.410 So the equation doesn't really have any temperature 00:38:32.410 --> 00:38:33.660 dependence. 00:38:33.660 --> 00:38:36.360 But I factored things this way, because we 00:38:36.360 --> 00:38:38.789 know that at is approximately a constant. 00:38:38.789 --> 00:38:40.830 And that means that this capital a, which is just 00:38:40.830 --> 00:38:42.205 other things which are definitely 00:38:42.205 --> 00:38:45.260 constant times, a square t square in the denominator, 00:38:45.260 --> 00:38:47.420 this a is approximately a constant. 00:38:51.770 --> 00:38:53.330 And you'll have to check me at home 00:38:53.330 --> 00:38:55.705 that this is exactly equivalent to the original Friedmann 00:38:55.705 --> 00:38:58.870 equation, no approximations whatever, just substitutions 00:38:58.870 --> 00:39:03.620 of Omega and the definition of Rho sub c. 00:39:03.620 --> 00:39:06.040 So the nice thing about this is that we can read off 00:39:06.040 --> 00:39:07.790 the time dependence of the right-hand side 00:39:07.790 --> 00:39:09.290 as long as we know the time dependence in the temperature 00:39:09.290 --> 00:39:11.280 and the time dependence of the energy density. 00:39:11.280 --> 00:39:15.350 And we do for matter dominated and radiation dominated eras. 00:39:15.350 --> 00:39:17.870 So this, essentially, solves the problem for us. 00:39:17.870 --> 00:39:20.120 And now it's really just a question 00:39:20.120 --> 00:39:22.500 of looking at the numerics that follow 00:39:22.500 --> 00:39:25.920 as a consequence of that equation. 00:39:25.920 --> 00:39:28.840 And this quantity, we're really interested 00:39:28.840 --> 00:39:30.572 in just Omega minus 1. 00:39:30.572 --> 00:39:32.780 The Friedmann equation gave us the extra complication 00:39:32.780 --> 00:39:34.372 of an Omega in the denominator. 00:39:34.372 --> 00:39:36.580 But in the end, we're going to be interested in cases 00:39:36.580 --> 00:39:38.121 where Omega is very, very close to 1. 00:39:38.121 --> 00:39:41.200 So the Omega in the denominator we could just set equal to one. 00:39:41.200 --> 00:39:43.200 And it's the Omega minus 1 in the numerator that 00:39:43.200 --> 00:39:47.390 controls the value of the left-hand side. 00:39:47.390 --> 00:39:57.080 So if we look at these Planck limits, we could ask how big 00:39:57.080 --> 00:39:59.220 can that be? 00:39:59.220 --> 00:40:02.030 And it's biggest if the error occurs on the negative side. 00:40:02.030 --> 00:40:04.120 So it contributes to this small mean value 00:40:04.120 --> 00:40:06.260 which is slightly negative. 00:40:06.260 --> 00:40:10.970 And it gives you 0.0075 for Omega minus 1. 00:40:10.970 --> 00:40:15.140 And then if you put that in the numerator and the same thing 00:40:15.140 --> 00:40:20.560 in the denominator, you get something like 0.0076. 00:40:20.560 --> 00:40:23.760 But I'm just going to use the bound 00:40:23.760 --> 00:40:32.330 that Omega naught minus 1 over Omega is less than 0.01. 00:40:32.330 --> 00:40:34.950 But the more accurate thing would be 0.076. 00:40:34.950 --> 00:40:37.430 But, again, we're not really interested in small factors 00:40:37.430 --> 00:40:37.930 here. 00:40:41.170 --> 00:40:42.830 And this is a one signa error. 00:40:42.830 --> 00:40:45.010 So the actual error could be larger than this, 00:40:45.010 --> 00:40:46.730 but not too much larger than this. 00:41:12.980 --> 00:41:16.010 So I'm going to divide the time interval between one second 00:41:16.010 --> 00:41:18.940 and now up into two integrals. 00:41:18.940 --> 00:41:20.920 From one second to about 50,000 years, 00:41:20.920 --> 00:41:22.940 the universe was radiation dominated. 00:41:22.940 --> 00:41:25.680 We figured out earlier that the matter to radiation equality 00:41:25.680 --> 00:41:27.470 happens at about 50,000 years. 00:41:27.470 --> 00:41:29.830 I think we may have gotten 47,000 years or something 00:41:29.830 --> 00:41:32.530 like that when we calculated it. 00:41:32.530 --> 00:41:41.962 So for t equals 1 second to-- I'm sorry, 00:41:41.962 --> 00:41:43.420 I'm going to do it the other order. 00:41:43.420 --> 00:41:45.711 I'm going to start with the present and work backwards. 00:41:50.130 --> 00:42:01.966 So for t equals 50,000 years to the present, 00:42:01.966 --> 00:42:03.340 the universe is matter dominated. 00:42:08.514 --> 00:42:09.930 And the next thing is that we know 00:42:09.930 --> 00:42:11.710 how mattered dominated universe's behave. 00:42:11.710 --> 00:42:14.030 We don't need to recalculate it. 00:42:14.030 --> 00:42:17.360 We know that the scale factor for a matter 00:42:17.360 --> 00:42:22.340 dominated flat universe goes like t to the 2/3 power, 00:42:22.340 --> 00:42:25.290 I should have a portionality sign here. 00:42:25.290 --> 00:42:29.390 a of t is proportional to t to 2/3. 00:42:29.390 --> 00:42:31.602 And it's fair to assume flat, because we're always 00:42:31.602 --> 00:42:33.935 going to be talking about universes that are nearly flat 00:42:33.935 --> 00:42:36.059 and becoming more and more flat as we go backwards, 00:42:36.059 --> 00:42:37.866 as we'll see. 00:42:37.866 --> 00:42:39.990 And again, this isn't an approximate calculation. 00:42:39.990 --> 00:42:41.890 One could do it more accurately if one wanted to. 00:42:41.890 --> 00:42:43.960 But there's really no need to, because the result 00:42:43.960 --> 00:42:46.610 will be so extreme. 00:42:46.610 --> 00:42:51.101 The temperature behaves like one over the scale factor. 00:42:51.101 --> 00:42:53.225 And that will be true for both the matter dominated 00:42:53.225 --> 00:42:57.070 and a radiation dominated universe. 00:42:57.070 --> 00:43:00.680 And the energy density will be proportional to one 00:43:00.680 --> 00:43:01.910 over the scale factor cubed. 00:43:08.180 --> 00:43:11.040 And then if we put those together and use the formula 00:43:11.040 --> 00:43:14.200 on the other blackboard and ask how Omega minus 1 over Omega 00:43:14.200 --> 00:43:19.600 behaves, it's proportional to the temperature squared 00:43:19.600 --> 00:43:22.160 divided by the energy density. 00:43:22.160 --> 00:43:23.820 The temperature goes like 1 over a. 00:43:23.820 --> 00:43:26.990 So temperature squared goes like 1 over a squared. 00:43:26.990 --> 00:43:30.160 But Rho in the denominator goes like 1 over a cubed. 00:43:30.160 --> 00:43:33.310 So you have 1 over a squared divided by 1 over a cubed. 00:43:33.310 --> 00:43:36.990 And that means it just goes like a, the scale factor itself. 00:43:36.990 --> 00:43:41.740 So Omega minus 1 over Omega is proportional to a. 00:43:41.740 --> 00:43:44.310 And that means it's proportional to t to the 2/3. 00:43:47.990 --> 00:43:55.240 So that allows us to write down an equation, 00:43:55.240 --> 00:43:57.810 since we want to relate everything 00:43:57.810 --> 00:44:00.630 to the value of Omega minus 1 over Omega today, 00:44:00.630 --> 00:44:08.410 we can write Omega minus 1 over Omega 00:44:08.410 --> 00:44:21.660 at 50,000 years is about equal to the ratio of the 2 times 00:44:21.660 --> 00:44:37.210 the 50,000 years and today, which is 13.8 billion years, 00:44:37.210 --> 00:44:41.640 to the 2/3 power since Omega minus 1 00:44:41.640 --> 00:44:42.989 grows like t to the 2/3. 00:44:42.989 --> 00:44:44.530 I should maybe have pointed out here, 00:44:44.530 --> 00:44:46.800 this telling us that Omega minus 1 grows with time. 00:44:46.800 --> 00:44:48.762 That's the important feature. 00:44:48.762 --> 00:44:51.670 It grows t to the 2/3. 00:44:51.670 --> 00:44:55.420 So the value at 50,000 years is this ratio 00:44:55.420 --> 00:45:00.870 to the 2/3 power times Omega minus 1 over Omega 00:45:00.870 --> 00:45:04.060 today, which I can indicate just by putting 00:45:04.060 --> 00:45:06.280 subscript zeros on my Omegas. 00:45:06.280 --> 00:45:07.310 And that makes it today. 00:45:10.200 --> 00:45:13.430 And I've written this as a fraction less than one. 00:45:13.430 --> 00:45:15.810 This says that Omega minus 1 over Omega 00:45:15.810 --> 00:45:18.840 was smaller than it is now by this ratio 00:45:18.840 --> 00:45:23.720 to the 2/3 power, which follows from the fact 00:45:23.720 --> 00:45:28.380 that Omega minus 1 over Omega grows like t to the 2/3. 00:45:28.380 --> 00:45:32.230 OK, we're now halfway there. 00:45:32.230 --> 00:45:38.010 And the other half is similar, so it will go quickly. 00:45:38.010 --> 00:45:41.020 We now want to go from 50,000 years 00:45:41.020 --> 00:45:46.054 to one second using the fact that during that era 00:45:46.054 --> 00:45:47.595 the universe was radiation dominated. 00:45:53.790 --> 00:46:06.660 So for t equals 1 second to 50,000 years, 00:46:06.660 --> 00:46:08.350 the universe is radiation dominated. 00:46:13.490 --> 00:46:18.390 And that implies that the scale factor 00:46:18.390 --> 00:46:20.040 is proportional to t to the 1/2. 00:46:23.960 --> 00:46:27.000 The temperature is, again, proportional to 1 00:46:27.000 --> 00:46:28.880 over the scale factor. 00:46:28.880 --> 00:46:32.070 That's just conservation of entropy. 00:46:32.070 --> 00:46:36.212 And the energy density goes one over the scale factor 00:46:36.212 --> 00:46:36.795 to the fourth. 00:46:40.080 --> 00:46:42.630 So, again, we go back to this formula 00:46:42.630 --> 00:46:45.500 and do the corresponding arithmetic. 00:46:45.500 --> 00:46:46.960 Temperature goes like 1 over a. 00:46:46.960 --> 00:46:49.140 Temperature squared goes like 1 over a squared. 00:46:49.140 --> 00:46:50.616 That's our numerator. 00:46:50.616 --> 00:46:51.990 This time, in the denominator, we 00:46:51.990 --> 00:46:55.130 have Rho, which goes like one over a to the fourth. 00:46:55.130 --> 00:46:57.650 So we have 1 over a squared divided by 1 00:46:57.650 --> 00:46:58.647 over a to the fourth. 00:46:58.647 --> 00:47:00.230 And that means it goes like a squared. 00:47:06.930 --> 00:47:10.130 So we get Omega minus 1 over Omega 00:47:10.130 --> 00:47:13.080 is proportional to a squared. 00:47:13.080 --> 00:47:16.630 And since goes like the square root of t, 00:47:16.630 --> 00:47:19.230 a squared goes like t. 00:47:19.230 --> 00:47:20.962 So during the radiation dominated era 00:47:20.962 --> 00:47:22.420 this diverges even a little faster. 00:47:22.420 --> 00:47:24.294 PROFESSOR: It goes like t, rather than like t 00:47:24.294 --> 00:47:28.060 to the 2/3, which is a slightly slower growth. 00:47:28.060 --> 00:47:32.170 And using this fact, we can put it all together now 00:47:32.170 --> 00:47:41.970 and say that Omega minus 1 over Omega at 1 second 00:47:41.970 --> 00:47:47.520 is about equal to 1 second over 50,000 00:47:47.520 --> 00:47:55.020 years to the first power-- this is going 00:47:55.020 --> 00:47:57.950 like the first power of t-- times 00:47:57.950 --> 00:48:12.420 the value of Omega minus 1 over Omega at 50,000 years. 00:48:12.420 --> 00:48:16.380 And Omega at 50,000 years, we can put in that equality 00:48:16.380 --> 00:48:18.560 and relate everything to the present value. 00:48:42.640 --> 00:48:46.760 And when you do that, putting it all together, 00:48:46.760 --> 00:48:53.220 you ultimately find that Omega minus 1 in magnitude at t 00:48:53.220 --> 00:48:59.440 equals 1 second is less than about 10 to the minus 18. 00:48:59.440 --> 00:49:01.780 This is just putting together these inequalities 00:49:01.780 --> 00:49:05.121 and using the Planck value for the present value, the Planck 00:49:05.121 --> 00:49:05.620 inequality. 00:49:08.597 --> 00:49:10.930 So then 10 to the minus 18 is a powerfully small number. 00:49:10.930 --> 00:49:13.810 What we're saying is that to be in the allowed range today, 00:49:13.810 --> 00:49:16.010 at one second after the Big Bang, 00:49:16.010 --> 00:49:17.870 Omega have to have been equal to 1 00:49:17.870 --> 00:49:19.870 in the context of this conventional cosmology 00:49:19.870 --> 00:49:23.680 to an accuracy of 18 decimal places. 00:49:23.680 --> 00:49:26.900 And the reason that's a problem is 00:49:26.900 --> 00:49:28.630 that we don't know any physics whatever 00:49:28.630 --> 00:49:31.390 that forces Omega to be equal to 1. 00:49:31.390 --> 00:49:33.680 Yet, somehow Omega apparently has 00:49:33.680 --> 00:49:37.930 chosen to be 1 to an accuracy of 18 decimal places. 00:49:37.930 --> 00:49:41.760 And I mention that the argument wasn't that different in 1980. 00:49:41.760 --> 00:49:45.520 In 1980, we only knew this instead of that. 00:49:45.520 --> 00:49:48.590 And that meant that instead of having 10 to the minus 2 00:49:48.590 --> 00:49:51.840 on the right-hand side here, we would have had 10 00:49:51.840 --> 00:49:54.050 differing by three orders of magnitude. 00:49:54.050 --> 00:49:56.270 So instead of getting 10 to the minus 18 here, 00:49:56.270 --> 00:49:58.255 we would have gotten 10 to minus 15. 00:49:58.255 --> 00:50:00.880 And 10 to minus 15 is, I guess, a lot bigger than 10 00:50:00.880 --> 00:50:03.620 to minus 18 by a factor of 1,000. 00:50:03.620 --> 00:50:05.530 But still, it's an incredibly small number. 00:50:05.530 --> 00:50:07.975 And the argument really sounds exactly the same. 00:50:07.975 --> 00:50:10.660 The question is, how did Omega minus 1 00:50:10.660 --> 00:50:12.610 get to be so incredibly small? 00:50:12.610 --> 00:50:15.100 What mechanism was there? 00:50:15.100 --> 00:50:16.830 Now, like the horizon problem, you 00:50:16.830 --> 00:50:20.250 can get around it by attributing your ignorance 00:50:20.250 --> 00:50:21.340 to the singularity. 00:50:21.340 --> 00:50:22.840 You can say the universe started out 00:50:22.840 --> 00:50:25.470 with Omega exactly equal to 1 or Omega equal to 1 00:50:25.470 --> 00:50:28.459 to some extraordinary accuracy. 00:50:28.459 --> 00:50:30.000 But that's not really an explanation. 00:50:30.000 --> 00:50:32.740 It really is just a hope for an explanation. 00:50:32.740 --> 00:50:34.330 And the point is that inflation, which 00:50:34.330 --> 00:50:35.970 you'll be learning about in the next few lectures, 00:50:35.970 --> 00:50:37.540 provides an actual explanation. 00:50:37.540 --> 00:50:39.710 It provides a mechanism that drives 00:50:39.710 --> 00:50:44.280 the early universe towards Omega equals 1, thereby explaining 00:50:44.280 --> 00:50:47.810 why the early universe had a value of Omega 00:50:47.810 --> 00:50:49.640 so incredibly close to 1. 00:50:49.640 --> 00:50:51.800 So that's what we're going to be learning shortly. 00:50:51.800 --> 00:50:53.675 But at the present time, the takeaway message 00:50:53.675 --> 00:50:56.540 is simply that for Omega to be in the allowed range today 00:50:56.540 --> 00:51:00.130 it had to start out unbelievably close to 1 at, for example, t 00:51:00.130 --> 00:51:01.800 equals 1 second. 00:51:01.800 --> 00:51:04.170 And within conventional cosmology, 00:51:04.170 --> 00:51:07.920 there's no explanation for why Omega so close to 1 00:51:07.920 --> 00:51:10.610 was in any way preferred. 00:51:10.610 --> 00:51:13.630 Any questions about that? 00:51:13.630 --> 00:51:14.424 Yes? 00:51:14.424 --> 00:51:16.844 AUDIENCE: Is there any heuristic argument 00:51:16.844 --> 00:51:21.684 that omega [INAUDIBLE] universe has total energy zero? 00:51:21.684 --> 00:51:25.556 So isn't that, at least, appealing? 00:51:25.556 --> 00:51:28.440 PROFESSOR: OK the question is, isn't it maybe appealing 00:51:28.440 --> 00:51:31.510 that Omega should equal 1 because Omega equals 00:51:31.510 --> 00:51:34.525 1 is a flat universe, which has 0 total energy? 00:51:34.525 --> 00:51:36.650 I guess, the point is that any closed universe also 00:51:36.650 --> 00:51:38.430 has zero total energy. 00:51:38.430 --> 00:51:42.200 So I don't think Omega equals 1 is so special. 00:51:42.200 --> 00:51:46.590 And furthermore, if you look at the spectrum of possible values 00:51:46.590 --> 00:51:52.270 of Omega, it can be positive-- I'm sorry, not with Omega. 00:51:52.270 --> 00:51:55.564 Let me look at the curvature itself, little k. 00:51:55.564 --> 00:51:57.230 Little k can be positive, in which case, 00:51:57.230 --> 00:51:58.035 you have a closed universe. 00:51:58.035 --> 00:51:59.451 It can be negative, in which case, 00:51:59.451 --> 00:52:00.800 you have an open universe. 00:52:00.800 --> 00:52:02.510 And only for the one special case of k 00:52:02.510 --> 00:52:07.460 equals 0, which really is one number in the whole real line 00:52:07.460 --> 00:52:10.576 of possible numbers, do you get exactly flat. 00:52:10.576 --> 00:52:12.450 So I think from that point of view flat looks 00:52:12.450 --> 00:52:17.420 highly special and not at all plausible as what you'd 00:52:17.420 --> 00:52:20.620 get if you just grabbed something out of a grab bag. 00:52:20.620 --> 00:52:24.372 But, ultimately, I think there's no way of knowing for sure. 00:52:24.372 --> 00:52:26.830 Whether or not Omega equals 1 coming out of the singularity 00:52:26.830 --> 00:52:29.080 is plausible really depends on knowing something about 00:52:29.080 --> 00:52:30.740 the singularity, which we don't. 00:52:30.740 --> 00:52:32.550 So you're free to speculate. 00:52:32.550 --> 00:52:34.430 But the nice thing about inflation 00:52:34.430 --> 00:52:36.220 is that you don't need to speculate. 00:52:36.220 --> 00:52:38.500 Inflation really does provide a physical mechanism 00:52:38.500 --> 00:52:41.070 that we can understand that drives Omega 00:52:41.070 --> 00:52:44.830 to be 1 exactly like what we see. 00:52:44.830 --> 00:52:47.500 Any other questions? 00:52:47.500 --> 00:52:49.540 OK, in that case, what I'd like to do 00:52:49.540 --> 00:52:53.120 is to move on to problem number three, 00:52:53.120 --> 00:52:56.010 which is the magnetic monopole problem, which unfortunately 00:52:56.010 --> 00:52:59.410 requires some background to understand. 00:52:59.410 --> 00:53:01.770 And we don't have too much time. 00:53:01.770 --> 00:53:04.240 So I'm going to go through things rather quickly. 00:53:06.920 --> 00:53:09.860 This magnetic monopole problem is 00:53:09.860 --> 00:53:14.100 different from the other two in the first two problems 00:53:14.100 --> 00:53:18.270 I discussed are just problems of basic classical cosmology. 00:53:18.270 --> 00:53:19.750 The magnetic monopole problem only 00:53:19.750 --> 00:53:24.200 arises if we believe that physics at very high energies 00:53:24.200 --> 00:53:27.080 is described what are called grand unified theories, which 00:53:27.080 --> 00:53:30.400 then imply that these magnetic monopoles exist and allow us 00:53:30.400 --> 00:53:32.410 a root for estimating how many of them 00:53:32.410 --> 00:53:33.950 would have been produced. 00:53:33.950 --> 00:53:35.520 And the point is that if we assume 00:53:35.520 --> 00:53:38.290 that grand unified theories are the right description 00:53:38.290 --> 00:53:41.060 of physics at very high energies, then 00:53:41.060 --> 00:53:43.750 we conclude that far too many magnetic monopoles would 00:53:43.750 --> 00:53:46.970 be produced if we had just the standard cosmology that we've 00:53:46.970 --> 00:53:48.534 been talking about without inflation. 00:53:48.534 --> 00:53:50.575 So that's going to be the thrust of the argument. 00:53:50.575 --> 00:53:52.220 And it will all go away if you decide 00:53:52.220 --> 00:53:54.595 you don't believe in grand unified theories, which you're 00:53:54.595 --> 00:53:55.120 allowed to. 00:53:55.120 --> 00:53:57.370 But there is some evidence for grand unified theories. 00:53:57.370 --> 00:53:59.900 And I'll talk about that a little bit. 00:53:59.900 --> 00:54:03.230 Now, I'm not going to have time to completely describe 00:54:03.230 --> 00:54:04.320 grand unified theories. 00:54:04.320 --> 00:54:06.892 But I will try to tell you enough odd facts 00:54:06.892 --> 00:54:08.100 about grand unified theories. 00:54:08.100 --> 00:54:10.760 So there will be kind of a consistent picture that 00:54:10.760 --> 00:54:14.314 will hang together, even though there's no claim that I can 00:54:14.314 --> 00:54:16.730 completely teach you grand unified theories in the next 10 00:54:16.730 --> 00:54:18.510 minutes and then talk about the production 00:54:18.510 --> 00:54:20.884 of magnetic monopoles and those theories in the next five 00:54:20.884 --> 00:54:21.390 minutes. 00:54:21.390 --> 00:54:23.780 But that will be sort of the goal. 00:54:23.780 --> 00:54:26.342 So to start with, I mentioned that there's 00:54:26.342 --> 00:54:28.300 something called the standard model of particle 00:54:28.300 --> 00:54:30.790 physics, which is enormously successful. 00:54:30.790 --> 00:54:33.720 It's been developed really since the 1970s 00:54:33.720 --> 00:54:37.710 and has not changed too much since maybe 1975 or so. 00:54:37.710 --> 00:54:40.950 We have, since 1975, learned that neutrinos have masses. 00:54:40.950 --> 00:54:43.200 And those can be incorporated into the standard model. 00:54:43.200 --> 00:54:46.460 And that's a recent addition. 00:54:46.460 --> 00:54:48.380 And, I guess, in 1975 I'm not sure 00:54:48.380 --> 00:54:53.040 if we knew all three generations that we now know. 00:54:53.040 --> 00:54:54.840 But the matter, the fundamental particles 00:54:54.840 --> 00:54:57.277 fall into three generations, these particles 00:54:57.277 --> 00:54:58.110 of a different type. 00:54:58.110 --> 00:54:59.464 And we'll talk about them later. 00:54:59.464 --> 00:55:00.505 But these are the quarks. 00:55:00.505 --> 00:55:02.890 These are the spin-1/2 particles, 00:55:02.890 --> 00:55:05.620 these three columns on the left. 00:55:05.620 --> 00:55:09.680 On the top, we have the quarks, up, down, charm, strange, top, 00:55:09.680 --> 00:55:10.180 and bottom. 00:55:10.180 --> 00:55:14.460 There are six different flavors of quarks. 00:55:14.460 --> 00:55:18.502 Each quark, by the way, comes in three different colors. 00:55:18.502 --> 00:55:19.960 The different colors are absolutely 00:55:19.960 --> 00:55:21.510 identical to each other. 00:55:21.510 --> 00:55:23.870 There's a perfect symmetry among colors. 00:55:23.870 --> 00:55:25.730 There's no perfect symmetry here. 00:55:25.730 --> 00:55:28.354 Each of these quarks is a little bit different from the others. 00:55:28.354 --> 00:55:30.630 Although, there are approximate symmetries. 00:55:30.630 --> 00:55:32.860 And related to each family of quarks 00:55:32.860 --> 00:55:35.320 is a family of leptons, particles 00:55:35.320 --> 00:55:37.660 that do not undergo strong interactions 00:55:37.660 --> 00:55:41.310 in the electron-like particles and neutrinos. 00:55:41.310 --> 00:55:42.750 This row is the neutrinos. 00:55:42.750 --> 00:55:45.480 There's an electron neutrino, a muon neutrino, and a tau 00:55:45.480 --> 00:55:47.590 neutrino, like we've already said. 00:55:47.590 --> 00:55:50.180 And there's an electron, a muon, and a tao, 00:55:50.180 --> 00:55:52.820 which I guess we've also already said. 00:55:52.820 --> 00:55:56.690 So the particles on the left are all of the spin-1/2 particles 00:55:56.690 --> 00:56:00.060 that exist in the standard model of particle physics. 00:56:00.060 --> 00:56:03.240 And then on the right, we have the Boson particles, 00:56:03.240 --> 00:56:05.560 the particles of integer span, starting 00:56:05.560 --> 00:56:07.910 with the photon on the top. 00:56:07.910 --> 00:56:09.870 Under that in this list-- there's 00:56:09.870 --> 00:56:12.080 no particular order in here really-- is 00:56:12.080 --> 00:56:14.840 the gluon which is the particle that's 00:56:14.840 --> 00:56:17.340 like the photon but the particle which describes 00:56:17.340 --> 00:56:20.530 the strong interactions, which are somewhat more complicated 00:56:20.530 --> 00:56:24.890 and electromagnetism, but still described by spin-1 particles 00:56:24.890 --> 00:56:26.940 just like the photon. 00:56:26.940 --> 00:56:30.190 And then two other spin-1 particles, the z0 and the w 00:56:30.190 --> 00:56:33.130 plus and minus, which are a neutral particle 00:56:33.130 --> 00:56:35.000 and a charged particle, which are 00:56:35.000 --> 00:56:37.355 the carriers of the so-called weak interactions. 00:56:37.355 --> 00:56:40.490 The weak interactions being the only 00:56:40.490 --> 00:56:43.380 non-gravitational interactions that neutrinos undergo. 00:56:43.380 --> 00:56:45.660 And the weak interactions are responsible 00:56:45.660 --> 00:56:47.340 for certain particle decays. 00:56:47.340 --> 00:56:50.870 For example, a neutron can decay into a proton giving off 00:56:50.870 --> 00:56:53.940 also an electron-- producing a proton, 00:56:53.940 --> 00:56:57.810 yeah-- charge has to be conserved, proton is positive. 00:56:57.810 --> 00:57:00.250 So it's produced with an electron and then 00:57:00.250 --> 00:57:04.750 an anti-electron neutrino to balance the so-called electron 00:57:04.750 --> 00:57:06.480 lepton number. 00:57:06.480 --> 00:57:08.045 And that's a weak direction. 00:57:08.045 --> 00:57:09.920 Essentially, anything that involves neutrinos 00:57:09.920 --> 00:57:12.474 is going to be weak interaction. 00:57:12.474 --> 00:57:13.640 So these are the characters. 00:57:13.640 --> 00:57:15.070 And there's a set of interactions 00:57:15.070 --> 00:57:17.720 that go with this set of characters. 00:57:17.720 --> 00:57:21.510 So we have here a complete model of how 00:57:21.510 --> 00:57:24.020 elementary particles interact. 00:57:24.020 --> 00:57:26.580 And the model has been totally successful. 00:57:26.580 --> 00:57:29.130 It actually gives predictions that 00:57:29.130 --> 00:57:32.580 are consistent with every reliable experiment that 00:57:32.580 --> 00:57:36.930 has been done since the mid-1970s up to the present. 00:57:36.930 --> 00:57:38.920 So it's made particle physics a bit dull 00:57:38.920 --> 00:57:42.140 since we have a theory that seems to predict everything. 00:57:42.140 --> 00:57:44.340 But it's also a magnificent achievement 00:57:44.340 --> 00:57:46.510 that we have such a theory. 00:57:46.510 --> 00:57:49.570 Now, in spite of the fact that this theory is so unbelievably 00:57:49.570 --> 00:57:54.660 successful I don't think I know anybody who really regards this 00:57:54.660 --> 00:57:59.360 as a candidate even or the ultimate theory of nature. 00:57:59.360 --> 00:58:02.559 And the reason for that is maybe twofold, first 00:58:02.559 --> 00:58:04.225 is that it does not incorporate gravity, 00:58:04.225 --> 00:58:07.060 it only incorporates particle interactions. 00:58:07.060 --> 00:58:10.057 And we know that gravity exists and has to be put in somehow. 00:58:10.057 --> 00:58:12.640 And there doesn't seem to be any simple way of putting gravity 00:58:12.640 --> 00:58:15.300 into this theory. 00:58:15.300 --> 00:58:20.152 And, secondly-- maybe there's three reasons-- second, 00:58:20.152 --> 00:58:22.610 it does not include any good candidates for the dark matter 00:58:22.610 --> 00:58:25.390 that we know exists in cosmology. 00:58:25.390 --> 00:58:29.860 And third, excuse me-- and this is given a lot of importance, 00:58:29.860 --> 00:58:33.600 even though it's an aesthetic argument-- this model has 00:58:33.600 --> 00:58:37.690 something like 28 free parameters, quantities that you 00:58:37.690 --> 00:58:40.190 just have to go out and measure before you can use the model 00:58:40.190 --> 00:58:41.540 to make predictions. 00:58:41.540 --> 00:58:44.110 And a large number of free parameters 00:58:44.110 --> 00:58:49.500 is associated, by theoretical physicists, with ugliness. 00:58:49.500 --> 00:58:52.460 So this is considered a very ugly model. 00:58:52.460 --> 00:58:55.530 And we have no real way of knowing, 00:58:55.530 --> 00:58:57.650 but almost all theoretical physicists 00:58:57.650 --> 00:59:00.130 believe that the correct theory of nature 00:59:00.130 --> 00:59:03.850 is going to be simpler and involve many fewer, maybe 00:59:03.850 --> 00:59:06.147 none at all, free knobs that can be turned 00:59:06.147 --> 00:59:07.730 to produce different kinds of physics. 00:59:11.470 --> 00:59:14.500 OK, what I want to talk about next, 00:59:14.500 --> 00:59:16.100 leading up to grand unified theories, 00:59:16.100 --> 00:59:19.730 is the notion of gauge theories. 00:59:19.730 --> 00:59:22.270 And, yes? 00:59:22.270 --> 00:59:26.206 AUDIENCE: I'm sorry, question real quick from the chart. 00:59:26.206 --> 00:59:28.666 I basically heard the explanation 00:59:28.666 --> 00:59:31.454 that the reason for the short range of the weak force 00:59:31.454 --> 00:59:34.324 was the massive mediator that is the cause of exponential field 00:59:34.324 --> 00:59:35.062 decay. 00:59:35.062 --> 00:59:37.030 But if the [INAUDIBLE] is massless, 00:59:37.030 --> 00:59:39.490 how do we explain that to [INAUDIBLE]? 00:59:39.490 --> 00:59:41.724 PROFESSOR: Right, OK, the question 00:59:41.724 --> 00:59:43.265 is for those of you who couldn't hear 00:59:43.265 --> 00:59:45.915 it is that the short range of the weak interactions, 00:59:45.915 --> 00:59:48.500 although I didn't talk about it, is usually explained 00:59:48.500 --> 00:59:51.970 and is explained by the fact that the z and w naught 00:59:51.970 --> 00:59:53.170 Bosons are very heavy. 00:59:53.170 --> 00:59:55.270 And heavy particles have a short range. 00:59:55.270 --> 00:59:57.770 But the strong interactions seem to also have a short range. 00:59:57.770 --> 01:00:01.290 And yet, the gluon is effectively massless. 01:00:01.290 --> 01:00:04.210 That's related to a complicated issue which 01:00:04.210 --> 01:00:05.990 goes by the name of confinement. 01:00:05.990 --> 01:00:08.940 Although the gluon is massless, it's confined. 01:00:08.940 --> 01:00:11.820 And confined means that it cannot exist as a free 01:00:11.820 --> 01:00:13.260 particle. 01:00:13.260 --> 01:00:14.960 In some sense, the strong interactions 01:00:14.960 --> 01:00:18.670 do have a long range in that if you took a meson, which 01:00:18.670 --> 01:00:20.620 is made out of a quark and an anti-quark, 01:00:20.620 --> 01:00:23.520 in principle, if you pulled it apart, 01:00:23.520 --> 01:00:25.720 there'd be a string of gluon force 01:00:25.720 --> 01:00:27.865 between the quark and the anti-quark. 01:00:27.865 --> 01:00:29.490 And that would produce a constant force 01:00:29.490 --> 01:00:32.970 no matter how far apart you pulled them. 01:00:32.970 --> 01:00:34.680 And the only thing that intervenes, 01:00:34.680 --> 01:00:36.870 and it is important that it does intervene, 01:00:36.870 --> 01:00:39.480 is that if you pulled them too far apart 01:00:39.480 --> 01:00:41.380 it would become energetically favorable 01:00:41.380 --> 01:00:45.776 for a quark anti-quark pair to materialize in the middle. 01:00:45.776 --> 01:00:48.150 And then instead of having a quark here and an anti-quark 01:00:48.150 --> 01:00:50.112 here and a string between them, you 01:00:50.112 --> 01:00:52.070 would have a quark here and an anti-quark there 01:00:52.070 --> 01:00:55.140 and a string between them, and an anti-quark here 01:00:55.140 --> 01:00:57.046 and-- I'm sorry, I guess it's a quark here 01:00:57.046 --> 01:00:59.170 and an anti-quark there and a string between those. 01:00:59.170 --> 01:01:01.160 And then they would just fly apart. 01:01:01.160 --> 01:01:02.890 So the string can break by producing 01:01:02.890 --> 01:01:04.210 quark anti-quark pairs. 01:01:04.210 --> 01:01:07.110 But the string can never just end in the middle of nowhere. 01:01:07.110 --> 01:01:08.967 And that's the essence of confinement. 01:01:08.967 --> 01:01:10.675 And it's due to the peculiar interactions 01:01:10.675 --> 01:01:14.772 that these gluons are believed to obey. 01:01:14.772 --> 01:01:16.230 So the gluons behave in a way which 01:01:16.230 --> 01:01:18.490 is somewhat uncharacteristic of particles. 01:01:18.490 --> 01:01:20.990 Except at very short distances, they behave very much 01:01:20.990 --> 01:01:21.990 like ordinary particles. 01:01:21.990 --> 01:01:24.970 But at larger distances, these effects of confinement 01:01:24.970 --> 01:01:28.570 play a very significant role. 01:01:28.570 --> 01:01:30.570 Any other questions? 01:01:30.570 --> 01:01:33.810 OK, onward, I want talk about gauge theories, 01:01:33.810 --> 01:01:35.400 because gauge theories have a lot 01:01:35.400 --> 01:01:38.530 to do with how one gets into grand unified theories 01:01:38.530 --> 01:01:41.380 from the standard model. 01:01:41.380 --> 01:01:44.520 And, basically, a gauge theory is a theory 01:01:44.520 --> 01:01:49.500 in which the fundamental fields are not themselves reality. 01:01:49.500 --> 01:01:51.680 But rather there's a set of transformations 01:01:51.680 --> 01:01:54.020 that the fields can undergo which take you 01:01:54.020 --> 01:01:56.320 from one description to an equivalent 01:01:56.320 --> 01:01:58.370 description of the same reality. 01:01:58.370 --> 01:01:59.940 So there's not a one to one mapping 01:01:59.940 --> 01:02:01.490 between the fields and reality. 01:02:01.490 --> 01:02:03.364 There are many different field configurations 01:02:03.364 --> 01:02:05.500 that correspond to the same reality. 01:02:05.500 --> 01:02:07.560 And that's basically what characterizes 01:02:07.560 --> 01:02:08.824 what we call a gauge theory. 01:02:08.824 --> 01:02:10.740 And you do know one example of a gauge theory. 01:02:10.740 --> 01:02:12.896 And that's e and m. 01:02:12.896 --> 01:02:16.990 If e and m is expressed in terms of the potentials Phi and A, 01:02:16.990 --> 01:02:19.120 you can write e in terms of the potential that way 01:02:19.120 --> 01:02:23.140 and b as the curl of A, you could put Phi and A together 01:02:23.140 --> 01:02:25.000 and make a four-vector if you want 01:02:25.000 --> 01:02:28.052 to do things in a Lorentz covariant way. 01:02:28.052 --> 01:02:30.260 And the important point, though, whether you put them 01:02:30.260 --> 01:02:32.440 together or not, is that you can always 01:02:32.440 --> 01:02:35.380 define a gauge transformation depending on some arbitrary 01:02:35.380 --> 01:02:38.595 function Lambda, which is a function of position and time. 01:02:38.595 --> 01:02:40.720 I didn't write in the arguments, but Lambda is just 01:02:40.720 --> 01:02:43.190 an arbitrary function of position and time. 01:02:43.190 --> 01:02:46.450 And for any Lambda you can replace 5 by 5 prime, 01:02:46.450 --> 01:02:50.365 given by this line, and a by a prime, given by that line. 01:02:50.365 --> 01:02:51.990 And if you go back and compute e and b, 01:02:51.990 --> 01:02:53.860 you'll find that they'll be unchanged. 01:02:53.860 --> 01:02:55.610 And therefore, the physics is not changed, 01:02:55.610 --> 01:02:59.140 because the physics really is all contained in e and b. 01:02:59.140 --> 01:03:01.590 So this gauge transformation is a transformation 01:03:01.590 --> 01:03:04.570 on the fields of the theory-- it can be written covariantly 01:03:04.570 --> 01:03:08.065 this way-- which leaves the physics invariant. 01:03:08.065 --> 01:03:09.940 And it turns out that all the important field 01:03:09.940 --> 01:03:13.770 theories that we know of are gauge theories. 01:03:13.770 --> 01:03:15.970 And that's why it's worth mentioning here. 01:03:15.970 --> 01:03:20.290 Now, for e and m, the gauge parameter is just this function 01:03:20.290 --> 01:03:23.087 Lambda, which is a function of position and time. 01:03:23.087 --> 01:03:24.670 And an important issue is what happens 01:03:24.670 --> 01:03:26.480 when you combine gauge transformations, 01:03:26.480 --> 01:03:29.850 because the succession of two transformations 01:03:29.850 --> 01:03:32.150 had better also be a symmetry transformation. 01:03:32.150 --> 01:03:35.620 So it's worth understanding that group structure. 01:03:35.620 --> 01:03:37.430 And for the case of e and m, these Lambdas 01:03:37.430 --> 01:03:40.930 just add if we make successive transformations. 01:03:40.930 --> 01:03:42.430 And that means the group is Abelian. 01:03:42.430 --> 01:03:44.910 It's commutative. 01:03:44.910 --> 01:03:47.060 But that's not always the case. 01:03:47.060 --> 01:03:48.530 Let's see, where am I going? 01:03:48.530 --> 01:03:50.646 OK, next slide actually comes somewhat later. 01:03:50.646 --> 01:03:52.020 Let me go back to the blackboard. 01:03:57.100 --> 01:04:00.340 It turns out that the important generalization of gauge 01:04:00.340 --> 01:04:03.710 theories is the generalization from Abelian gauge 01:04:03.710 --> 01:04:06.680 theories to non-Abelian ones, which was done originally 01:04:06.680 --> 01:04:16.130 by Yang and Mills in 1954, I think. 01:04:16.130 --> 01:04:19.140 And when it was first proposed, nobody knew what to do with it. 01:04:19.140 --> 01:04:21.310 But, ultimately, these non-Abelian gauge theories 01:04:21.310 --> 01:04:26.170 became the standard model of particle physics. 01:04:26.170 --> 01:04:28.570 And in non-Abelian gauge theories 01:04:28.570 --> 01:04:31.720 the parameter that describes the gauge transformation is a group 01:04:31.720 --> 01:04:35.850 element, not just something that gets added. 01:04:35.850 --> 01:04:39.030 And group elements multiply, according 01:04:39.030 --> 01:04:41.040 to the procedures of some group. 01:04:41.040 --> 01:04:43.772 And in particular, the standard model 01:04:43.772 --> 01:04:44.980 is built out of three groups. 01:04:47.431 --> 01:04:49.430 And the fact that there are three groups not one 01:04:49.430 --> 01:04:52.880 is just an example of this ugliness 01:04:52.880 --> 01:04:54.980 that I mentioned and is responsible for the fact 01:04:54.980 --> 01:04:58.470 that there's some significant number of parameters 01:04:58.470 --> 01:05:01.120 even if there were no other complications. 01:05:01.120 --> 01:05:07.460 So the standard model is based on three gauge groups, SU3, 01:05:07.460 --> 01:05:10.370 SU2, and U1. 01:05:14.459 --> 01:05:16.250 And it won't really be too important for us 01:05:16.250 --> 01:05:17.920 what exactly these groups are. 01:05:21.460 --> 01:05:25.280 Let me just mention quickly, SU3 is a group of 3 01:05:25.280 --> 01:05:28.530 by 3 matrices which are unitary in the sense 01:05:28.530 --> 01:05:33.130 that u adjoint times u is equal to 1 and special in the sense 01:05:33.130 --> 01:05:35.390 that they have determinant 1. 01:05:35.390 --> 01:05:36.980 And the set of all 3 by 3 matrices 01:05:36.980 --> 01:05:38.438 have those properties form a group. 01:05:38.438 --> 01:05:39.820 And that group is SU3. 01:05:39.820 --> 01:05:41.430 SU2 is the same thing but replace 01:05:41.430 --> 01:05:44.400 the 3 in all those sentences by 2s. 01:05:44.400 --> 01:05:47.306 U1 is just the group of phases. 01:05:47.306 --> 01:05:48.680 That is the group of real numbers 01:05:48.680 --> 01:05:51.080 that could be written as either the i phi. 01:05:51.080 --> 01:05:55.380 So it's just a complex number of magnitude 1 with just a phase. 01:05:55.380 --> 01:05:58.510 And you can multiply those and they form a group. 01:05:58.510 --> 01:06:00.740 And the standard model contains these three groups. 01:06:00.740 --> 01:06:03.370 And the three groups all act independently, 01:06:03.370 --> 01:06:05.952 which means that if you know about group products, 01:06:05.952 --> 01:06:09.120 one can say that the full group is the product group. 01:06:12.640 --> 01:06:16.670 And that just means that a full description of a group element 01:06:16.670 --> 01:06:20.980 is really just a set of an element of SU3, 01:06:20.980 --> 01:06:24.490 and an element of SU2, and an element of U1. 01:06:24.490 --> 01:06:28.094 And if you put together three group elements in each group 01:06:28.094 --> 01:06:29.510 and put them together with commas, 01:06:29.510 --> 01:06:31.840 that becomes a group element of the group SU3 cross, 01:06:31.840 --> 01:06:33.660 SU2 cross U1. 01:06:33.660 --> 01:06:37.322 And that is the gauge group of the standard model 01:06:37.322 --> 01:06:38.155 of particle physics. 01:06:42.010 --> 01:07:08.050 OK, now grand unified theories, a grand unified theory 01:07:08.050 --> 01:07:10.800 is based on the idea that this set of three groups 01:07:10.800 --> 01:07:13.220 can all be embedded in a single simple group. 01:07:37.760 --> 01:07:40.160 Now, simple actually has a mathematical group theory 01:07:40.160 --> 01:07:40.660 meaning. 01:07:40.660 --> 01:07:42.660 But it also, for our purposes, just 01:07:42.660 --> 01:07:46.580 means simple, which is good enough for our brush 01:07:46.580 --> 01:07:49.220 through of these arguments. 01:07:49.220 --> 01:07:52.052 And, for example-- and the example 01:07:52.052 --> 01:07:54.010 is shown a little bit in the lecture notes that 01:07:54.010 --> 01:07:57.900 will be posted shortly-- an example 01:07:57.900 --> 01:07:59.570 of a grand unified theory, and indeed 01:07:59.570 --> 01:08:02.070 the first grand unified theory that was invented, 01:08:02.070 --> 01:08:06.260 is based on the full gauge group SU5, which is just a group of 5 01:08:06.260 --> 01:08:10.750 by 5 matrices which are unitary and have determinant 1. 01:08:10.750 --> 01:08:15.380 And there's an easy way to embed SU3 and SU2 and U1 into SU5. 01:08:15.380 --> 01:08:17.490 And that's the way that was used to construct 01:08:17.490 --> 01:08:19.930 this grand unified theory. 01:08:19.930 --> 01:08:29.200 One can take a 5 by 5 matrix-- so this is a 5 by 5 matrix-- 01:08:29.200 --> 01:08:36.029 and one can simply take the upper 3 by 3 block 01:08:36.029 --> 01:08:40.850 and put an SU3 matrix there. 01:08:40.850 --> 01:08:46.080 And one can take the lower 2 by 2 block 01:08:46.080 --> 01:08:47.830 and put an SU2 matrix there. 01:08:51.840 --> 01:08:53.939 And then the U1 piece-- there's supposed 01:08:53.939 --> 01:08:57.359 to be a U1 left over-- the U1 piece 01:08:57.359 --> 01:09:01.120 can be obtained by giving an overall phase to this 01:09:01.120 --> 01:09:03.250 and an overall phase to that in such a way 01:09:03.250 --> 01:09:05.590 that the product of the five phases is 0. 01:09:05.590 --> 01:09:08.620 So the determinant has not changed. 01:09:08.620 --> 01:09:17.660 So one can put an e to the i2 Phi there and a factor of e 01:09:17.660 --> 01:09:22.020 to the minus i3 Phi there for any Phi. 01:09:24.810 --> 01:09:28.130 And then this Phi becomes the description 01:09:28.130 --> 01:09:31.950 of the U1 piece of this construction. 01:09:31.950 --> 01:09:35.020 So we can take an arbitrary SU3 matrix, and arbitrary SU2 01:09:35.020 --> 01:09:39.240 matrix, and an arbitrary U1 value expressed by Phi 01:09:39.240 --> 01:09:42.810 and put them together to make an SU5 matrix. 01:09:42.810 --> 01:09:44.810 And if you think about it, the SU3 piece 01:09:44.810 --> 01:09:47.634 will commute with the SU2 piece and with the U1 piece. 01:09:47.634 --> 01:09:49.800 These three pieces will all commute with each other, 01:09:49.800 --> 01:09:51.341 if you think about how multiplication 01:09:51.341 --> 01:09:52.620 works with this construction. 01:09:52.620 --> 01:09:53.953 So it does exactly what we want. 01:09:53.953 --> 01:09:57.310 It decomposes SU5. 01:09:57.310 --> 01:10:01.710 So it has a subgroup of SU3 cross SU2 across U1. 01:10:01.710 --> 01:10:06.560 And that's how the simplest grand unified theory works. 01:10:06.560 --> 01:10:11.060 OK, now, there are important things that need to be said, 01:10:11.060 --> 01:10:14.540 but we're out of time. 01:10:14.540 --> 01:10:18.590 So I guess what we need to do is to withhold 01:10:18.590 --> 01:10:21.690 from the next problem set, the magnetic monopole problem. 01:10:21.690 --> 01:10:25.110 Maybe I was a bit over-ambitious to put it on the problem set. 01:10:25.110 --> 01:10:27.320 So I'll send an email announcing that. 01:10:27.320 --> 01:10:29.474 But the one problem on the problem 01:10:29.474 --> 01:10:31.390 set for next week about grand unified theories 01:10:31.390 --> 01:10:32.290 will be withheld. 01:10:32.290 --> 01:10:37.250 And Scott Hughes will pick up this discussion next Tuesday. 01:10:37.250 --> 01:10:40.319 So I will see all of you-- gee willikers, 01:10:40.319 --> 01:10:42.360 if you come to my office hour, I'll see you then. 01:10:42.360 --> 01:10:45.110 But otherwise, I may not see you until the quiz. 01:10:45.110 --> 01:10:49.990 So have a good Thanksgiving and good luck on the quiz.