WEBVTT 00:00:00.500 --> 00:00:02.395 [SQUEAKING] 00:00:02.395 --> 00:00:04.311 [RUSTLING] 00:00:04.311 --> 00:00:06.227 [CLICKING] 00:00:11.080 --> 00:00:12.080 SCOTT HUGHES: All right. 00:00:12.080 --> 00:00:14.840 So at this point we're going to switch gears. 00:00:14.840 --> 00:00:20.030 Everything that we have done over the past several lectures 00:00:20.030 --> 00:00:22.970 has been in service of the approach 00:00:22.970 --> 00:00:25.100 to solving the Einstein field equations in which we 00:00:25.100 --> 00:00:29.510 assume a small perturbation around an exact background. 00:00:29.510 --> 00:00:32.570 Most of it was spent looking at perturbations around flat space 00:00:32.570 --> 00:00:34.100 time. 00:00:34.100 --> 00:00:36.380 A little bit in the last lecture we touched 00:00:36.380 --> 00:00:38.510 on some of the mathematics and some of the analysis 00:00:38.510 --> 00:00:39.530 when you curve-- 00:00:39.530 --> 00:00:42.440 you expand around some non-specified curved 00:00:42.440 --> 00:00:43.640 background. 00:00:43.640 --> 00:00:46.280 I didn't tell you where that curve background comes from. 00:00:46.280 --> 00:00:48.710 Today we'll be-- this lecture will 00:00:48.710 --> 00:00:50.480 be the first one in which we begin 00:00:50.480 --> 00:00:55.820 thinking about different forms of different kinds of solutions 00:00:55.820 --> 00:00:58.340 that arise from different principles. 00:00:58.340 --> 00:01:00.560 We're going to begin this by studying cosmology. 00:01:03.464 --> 00:01:06.027 It's the large-scale structure of the universe. 00:01:21.140 --> 00:01:24.810 So from the standpoint of-- 00:01:24.810 --> 00:01:29.270 from the standpoint of the calculational toolkit 00:01:29.270 --> 00:01:30.950 that we will be using, this is going 00:01:30.950 --> 00:01:33.060 to be the first example of a spacetime 00:01:33.060 --> 00:01:35.310 that we construct using a symmetry argument. 00:01:46.380 --> 00:01:48.350 We are not going to make any assumptions 00:01:48.350 --> 00:01:52.970 that anything is weak or small or any kind of approximation 00:01:52.970 --> 00:01:54.770 can be-- 00:01:54.770 --> 00:01:56.900 any kind of an approximation can be applied. 00:01:56.900 --> 00:02:01.610 What we're going to do is ask ourselves, suppose spacetime-- 00:02:01.610 --> 00:02:06.020 at least spacetime on some particular very large scales-- 00:02:06.020 --> 00:02:08.720 is restricted by various symmetries. 00:02:13.260 --> 00:02:21.040 So we will apply various restrictions to the equations 00:02:21.040 --> 00:02:30.150 and to the spacetime by the assumption 00:02:30.150 --> 00:02:33.356 that certain symmetric symmetries must hold. 00:02:33.356 --> 00:02:34.720 Let me reword this. 00:02:34.720 --> 00:02:36.470 By demanding that certain symmetries hold. 00:02:47.370 --> 00:02:50.640 Doing so will significantly reduce 00:02:50.640 --> 00:02:54.630 the complicated non-linear dynamics of the field equations 00:02:54.630 --> 00:02:55.770 of general relativity. 00:02:55.770 --> 00:02:58.170 This will allow us to reduce those complicated 00:02:58.170 --> 00:03:01.362 generic equations into something that is tractable. 00:03:25.490 --> 00:03:27.300 So let me describe-- whoops-- let's 00:03:27.300 --> 00:03:29.963 me give a little bit of background to this discussion. 00:03:29.963 --> 00:03:31.380 Let me get some better chalk, too. 00:03:34.780 --> 00:03:38.335 So as background, I'm going to give 00:03:38.335 --> 00:03:40.210 a little bit of a synopsis of some stuff that 00:03:40.210 --> 00:03:42.580 is described very nicely in the textbook by Carroll. 00:03:47.468 --> 00:03:49.010 So for background, part of what we're 00:03:49.010 --> 00:03:50.690 going to consider as we move into this 00:03:50.690 --> 00:03:55.190 is a notion of what are called maximally symmetric spaces. 00:04:10.640 --> 00:04:13.580 So I urge you to read Section 3.9 of Carroll 00:04:13.580 --> 00:04:16.610 for extensive discussion of this. 00:04:16.610 --> 00:04:20.029 But the key concept of this is that a maximally symmetric 00:04:20.029 --> 00:04:23.990 space is a space that has the largest number-- 00:04:23.990 --> 00:04:28.780 so let's say MSS, maximally symmetric space, 00:04:28.780 --> 00:04:38.120 has the largest number of allowed Killing vectors. 00:04:46.450 --> 00:04:50.170 If your space has n dimensions, it 00:04:50.170 --> 00:04:54.030 has n times n plus 1 over 2 such Killing vectors. 00:04:54.030 --> 00:04:56.320 And recall, if you do a lead derivative 00:04:56.320 --> 00:04:59.140 of the metric along the Killing vector, you get 0, OK? 00:04:59.140 --> 00:05:02.560 So it's that these n times m plus 1 Killing 00:05:02.560 --> 00:05:05.590 vectors all define ways in which, as you 00:05:05.590 --> 00:05:07.240 sort of flow along these vectors, 00:05:07.240 --> 00:05:09.960 spacetime is left unchanged. 00:05:09.960 --> 00:05:20.000 Intuitively, what these do is define a spacetime 00:05:20.000 --> 00:05:23.215 that is maximally homogeneous-- 00:05:23.215 --> 00:05:24.590 I shouldn't say spacetime yet, we 00:05:24.590 --> 00:05:26.870 haven't specified the nature of this manifold. 00:05:26.870 --> 00:05:29.900 So this defines a space that is maximally homogeneous. 00:05:43.010 --> 00:05:44.480 And homogeneous means that just-- 00:05:44.480 --> 00:05:46.760 it has uniform properties in all locations. 00:05:59.880 --> 00:06:02.460 And it is maximally isotopic. 00:06:11.020 --> 00:06:12.520 Which is a way of saying essentially 00:06:12.520 --> 00:06:15.400 that it looks the same in all directions. 00:06:31.820 --> 00:06:35.780 In the kind of spacetimes that we are familiar with, 00:06:35.780 --> 00:06:38.690 something-- a spacetime that is highly isotropic 00:06:38.690 --> 00:06:42.020 is one that is invariant with respect 00:06:42.020 --> 00:06:46.040 to rotations and boosts, and one that is homogeneous 00:06:46.040 --> 00:06:48.968 is something that is invariant with respect to translations. 00:06:48.968 --> 00:06:50.135 So let me give two examples. 00:06:56.700 --> 00:07:05.930 In Euclidean space, that is a maximally symmetric 00:07:05.930 --> 00:07:17.310 three-dimensional space, n times n plus 1 over 2 equals 6. 00:07:17.310 --> 00:07:19.950 And those 6 Killing vectors in Euclidean space 00:07:19.950 --> 00:07:26.879 correspond to 3 rotations and 3 translations. 00:07:32.150 --> 00:07:42.370 Minkowski flat spacetime: n equals 4. 00:07:42.370 --> 00:07:48.730 n times n plus 1 over 2 is equal to 10. 00:07:48.730 --> 00:08:01.407 I have 3 rotations, 3 translations, and 4 boosts, OK? 00:08:04.360 --> 00:08:09.550 The requirement that your space satisfy these properties, 00:08:09.550 --> 00:08:12.190 it leads to a condition that the Riemann tensor 00:08:12.190 --> 00:08:16.510 must be Lorentz-invariant within the local Lorentz frame. 00:08:33.830 --> 00:08:36.630 So for these two examples that I talked about, 00:08:36.630 --> 00:08:38.309 the Riemann tensor actually vanishes, 00:08:38.309 --> 00:08:41.640 and 0 is certainly Lorentz-invariant, 00:08:41.640 --> 00:08:42.820 so there's no problem there. 00:08:42.820 --> 00:08:45.150 But as I start thinking about more general classes 00:08:45.150 --> 00:08:48.000 of spacetimes, which I'm going to consider to be examples 00:08:48.000 --> 00:08:50.910 of massively symmetric spaces, they 00:08:50.910 --> 00:08:53.730 might not have vanishing Riemann tensors. 00:08:53.730 --> 00:08:55.980 But the Riemann tensors, in order to be massively 00:08:55.980 --> 00:08:59.430 symmetric, if I go into a freely-falling frame, 00:08:59.430 --> 00:09:00.720 that has to look-- 00:09:00.720 --> 00:09:06.630 everything has to look Lorentz-invariant. 00:09:06.630 --> 00:09:14.220 This leads to a condition that my Riemann tensor must take-- 00:09:14.220 --> 00:09:19.140 it is constrained to take one particular simple form. 00:09:19.140 --> 00:09:37.753 It must be R over n times n minus 1 times metric like so. 00:09:41.537 --> 00:09:44.290 This is-- so Carroll goes through this in some detail. 00:09:44.290 --> 00:09:46.480 Essentially what's going on here is 00:09:46.480 --> 00:09:48.730 this is the only way in which I am guaranteed 00:09:48.730 --> 00:09:51.400 to create a tensor that is Lorentz-invariant 00:09:51.400 --> 00:09:53.110 in a local Lorentz frame. 00:09:53.110 --> 00:09:54.490 So I go to my local Lorentz frame 00:09:54.490 --> 00:09:56.930 and I must have a form it looks like this, 00:09:56.930 --> 00:10:01.270 and this is a way of putting all my various quantities 00:10:01.270 --> 00:10:03.490 on my metric tensors together in such a way 00:10:03.490 --> 00:10:07.210 that I recover the symmetries of the Riemann tensor, 00:10:07.210 --> 00:10:09.467 and is my number of dimensions. 00:10:19.710 --> 00:10:21.150 Because it will prove useful, let 00:10:21.150 --> 00:10:23.940 me generate the Ricci tensor and the Ricci scalar from this. 00:10:26.670 --> 00:10:34.470 So R mu nu is going to be R over n times m minus 1. 00:10:34.470 --> 00:10:38.240 I'm taking the trace on indices alpha and beta. 00:10:38.240 --> 00:10:41.398 If I trace on this guy, I get n. 00:10:41.398 --> 00:10:43.440 The trace in the metric always just gives me back 00:10:43.440 --> 00:10:46.860 the number of dimensions. 00:10:46.860 --> 00:10:50.820 And when I trace on alpha and beta here, 00:10:50.820 --> 00:10:54.190 I basically just contract these two indices, 00:10:54.190 --> 00:11:08.760 and so I get the metric back over n times g mu nu. 00:11:08.760 --> 00:11:10.350 Take a further trace and you can see 00:11:10.350 --> 00:11:12.210 that that R that went into this thing 00:11:12.210 --> 00:11:14.730 is indeed nothing more than the Ricci curvature. 00:11:14.730 --> 00:11:16.577 Excuse me, the scalar Ricci curvature. 00:11:30.990 --> 00:11:31.500 OK? 00:11:31.500 --> 00:11:33.830 You can construct the Einstein tensor out of this, 00:11:33.830 --> 00:11:37.220 and what you see is that the Einstein tensor must 00:11:37.220 --> 00:11:40.110 be proportional to the metric. 00:11:40.110 --> 00:11:42.140 And in fact, there are-- 00:11:42.140 --> 00:11:45.580 the only solutions for the Einstein tensor that-- 00:11:45.580 --> 00:11:47.630 the only solution is the Einstein field equations 00:11:47.630 --> 00:11:51.200 in which the Einstein tensor is proportional to the metric are 00:11:51.200 --> 00:11:55.860 either flat spacetime or a cosmological constant. 00:11:55.860 --> 00:11:58.850 So empty space, empty flat spacetime 00:11:58.850 --> 00:12:01.700 and cosmological constant are the only maximally 00:12:01.700 --> 00:12:03.740 symmetric four-dimensional spacetimes. 00:12:06.890 --> 00:12:10.130 That does not necessarily describe our universe. 00:12:10.130 --> 00:12:18.510 So our universe, how am I going to tie all this together? 00:12:18.510 --> 00:12:21.510 We begin with the observation that our universe 00:12:21.510 --> 00:12:24.210 is, in fact, homogeneous and isotopic 00:12:24.210 --> 00:12:26.400 on large spatial scales. 00:12:52.700 --> 00:13:00.680 I emphasize spatial because the spacetime of our universe 00:13:00.680 --> 00:13:02.480 is not homogeneous, OK? 00:13:02.480 --> 00:13:04.847 In fact, the past of our universe 00:13:04.847 --> 00:13:06.305 is very different from the present. 00:13:24.310 --> 00:13:27.040 Because light travels at a finite time, when 00:13:27.040 --> 00:13:28.900 we observe two large distances, we 00:13:28.900 --> 00:13:31.270 are looking back into the distant past. 00:13:31.270 --> 00:13:34.540 And we see that the universe is a lot denser in the past 00:13:34.540 --> 00:13:36.280 than it is today. 00:13:36.280 --> 00:13:38.410 It remains the case, though, that it is still 00:13:38.410 --> 00:13:40.240 homogeneous and isotropic spatially, 00:13:40.240 --> 00:13:42.700 at least on large scales. 00:13:42.700 --> 00:13:44.530 So we are going to take advantage 00:13:44.530 --> 00:13:47.110 of these notions of maximally symmetric spaces 00:13:47.110 --> 00:13:53.890 to define a spacetime that is maximally symmetric spatially, 00:13:53.890 --> 00:13:56.950 but is not so symmetric with respect to time, OK? 00:13:56.950 --> 00:13:59.910 So we'll get to that in just a few moments. 00:13:59.910 --> 00:14:01.390 There is a wiggle word in here. 00:14:01.390 --> 00:14:03.490 I said that our universe, by observation, 00:14:03.490 --> 00:14:07.840 is homogeneous and isotropic on large spatial scales. 00:14:07.840 --> 00:14:08.740 What does large mean? 00:14:14.320 --> 00:14:17.320 Well the very largest scales that we can observe of all-- 00:14:17.320 --> 00:14:21.280 so that when we go back and we probe sort of the largest 00:14:21.280 --> 00:14:24.310 coherent structure that can be observed in our universe, 00:14:24.310 --> 00:14:28.270 we have to go all the way back to a time which is 00:14:28.270 --> 00:14:31.090 approximately 13-point-something or another-- 00:14:31.090 --> 00:14:33.790 I forget the exact number, but let's say 00:14:33.790 --> 00:14:38.200 about 13.7 billion years ago, and we see the cosmic microwave 00:14:38.200 --> 00:14:39.170 background. 00:14:58.783 --> 00:15:00.200 So the cosmic microwave background 00:15:00.200 --> 00:15:03.950 describes what our universe looked like 13.7 billion years 00:15:03.950 --> 00:15:08.390 ago or so, and what you see is that this guy is 00:15:08.390 --> 00:15:22.235 homogeneous and isotropic to about a part in 100,000, OK? 00:15:22.235 --> 00:15:23.610 With a lot of interesting physics 00:15:23.610 --> 00:15:26.510 in that deviation from-- 00:15:26.510 --> 00:15:28.290 that sort of part in 100,000 deviation, 00:15:28.290 --> 00:15:31.600 but that's a topic for a different class. 00:15:31.600 --> 00:15:34.410 We then sort of imagine you move forward in time, 00:15:34.410 --> 00:15:36.520 you look on-- so that tells you about the largest 00:15:36.520 --> 00:15:39.630 scales in the earliest times. 00:15:39.630 --> 00:15:41.920 Look at the universe on smaller scales. 00:15:41.920 --> 00:15:43.923 I mean, clearly you look in this room, OK? 00:15:43.923 --> 00:15:45.840 I'm standing here, there's a table over there, 00:15:45.840 --> 00:15:47.700 this is not homogeneous and not isotropic, 00:15:47.700 --> 00:15:49.620 things look quite different. 00:15:49.620 --> 00:15:58.360 What we start to see is things deviate from homogeneity 00:15:58.360 --> 00:16:21.650 and isotropy on scales that are on the order of several tens 00:16:21.650 --> 00:16:24.300 of megaparsecs in size, OK? 00:16:24.300 --> 00:16:26.990 Parsec, for those of you who are not astrophysicists, 00:16:26.990 --> 00:16:29.330 is a unit of measure. 00:16:29.330 --> 00:16:33.930 It's approximately 3.2 light years. 00:16:33.930 --> 00:16:36.200 So once you get down to boxes that 00:16:36.200 --> 00:16:40.190 are on the orders of 50 million light years or so on a side, 00:16:40.190 --> 00:16:44.110 you start to see deviations from homogeneity and isotropy. 00:16:44.110 --> 00:16:48.507 And this is caused by gravitational clumping. 00:16:48.507 --> 00:16:50.090 These are things like galaxy clusters. 00:17:01.700 --> 00:17:03.470 So when I talk about cosmology and I 00:17:03.470 --> 00:17:06.990 want to describe the universe as large-scale structure, 00:17:06.990 --> 00:17:10.490 I am going to be working on defining 00:17:10.490 --> 00:17:14.660 a description of spacetime that averages out 00:17:14.660 --> 00:17:19.369 over small things like clusters of galaxies, OK? 00:17:19.369 --> 00:17:23.420 So this is sort of a fun lecture in that sense, in that anything 00:17:23.420 --> 00:17:25.579 larger than an agglomeration of a couple 00:17:25.579 --> 00:17:27.170 dozen or a couple hundred galaxies, 00:17:27.170 --> 00:17:28.712 I'm going to treat that like a point. 00:17:37.110 --> 00:17:41.490 So here's what I am going to choose for my spacetime metric. 00:17:44.070 --> 00:17:46.710 This is where you start to see the power of assuming a given 00:17:46.710 --> 00:17:47.210 symmetry. 00:18:00.660 --> 00:18:02.770 So the line element, I'm going to write it 00:18:02.770 --> 00:18:08.410 as minus dt squared plus some function R 00:18:08.410 --> 00:18:18.310 squared of t gamma ij dx i dx j. 00:18:18.310 --> 00:18:24.960 The function R of t I've written down here. 00:18:24.960 --> 00:18:27.168 It's one variant of-- there's a couple functions that 00:18:27.168 --> 00:18:28.293 are going to get this name. 00:18:28.293 --> 00:18:29.590 We call this the scale factor. 00:18:32.300 --> 00:18:36.440 Caution, it's the same capital R we used for the Ricci scalar, 00:18:36.440 --> 00:18:38.140 it's not the Ricci scalar. 00:18:38.140 --> 00:18:39.890 Just a little bit of unfortunate notation, 00:18:39.890 --> 00:18:43.130 but it should be clear from context which is which 00:18:43.130 --> 00:18:45.410 when they come up. 00:18:45.410 --> 00:18:55.160 I have chosen gtt equals minus 1 and gti equals 0. 00:18:55.160 --> 00:18:59.180 Remember from our discussion of linearized theory 00:18:59.180 --> 00:19:05.110 around a flat background, that the spacetime-- the 10 00:19:05.110 --> 00:19:10.350 independent functions of my spacetime metric, of those 10, 00:19:10.350 --> 00:19:12.540 four of them were things that I could 00:19:12.540 --> 00:19:14.850 specify by choosing a gauge. 00:19:14.850 --> 00:19:18.870 Well here, I have specified four functions pretty much by fiat. 00:19:18.870 --> 00:19:21.800 Think of this as defining the gauge that I am working in, OK? 00:19:21.800 --> 00:19:24.900 In a very similar way, I have chosen a coordinate system 00:19:24.900 --> 00:19:29.340 by specifying gtt to be minus 1 and gti to be 0. 00:19:29.340 --> 00:19:31.530 This means that I am working in what are 00:19:31.530 --> 00:19:34.260 called co-moving coordinates. 00:19:39.500 --> 00:19:55.810 So if I am an observer who is at rest in the spacetime 00:19:55.810 --> 00:20:02.110 so that I would define my four velocity like so, 00:20:02.110 --> 00:20:04.750 I will be essentially co moving with the spacetime. 00:20:04.750 --> 00:20:06.367 Whatever the spacetime is doing, I'm 00:20:06.367 --> 00:20:08.200 just going to sort of homologously track it. 00:20:31.890 --> 00:20:36.420 It's worth noting-- so those of you who think about astronomy, 00:20:36.420 --> 00:20:39.060 astrophysics, and observational cosmology, 00:20:39.060 --> 00:20:42.420 the earth is not co-moving, OK? 00:20:42.420 --> 00:20:44.250 We build our telescopes on the surface 00:20:44.250 --> 00:20:46.110 of the Earth which rotates. 00:20:46.110 --> 00:20:49.320 The Earth itself orbits around the Sun. 00:20:49.320 --> 00:20:52.387 The Sun is in a solar system. 00:20:52.387 --> 00:20:54.720 Or excuse me-- the Sun at the center of our solar system 00:20:54.720 --> 00:20:57.300 is itself orbiting our galaxy. 00:20:57.300 --> 00:21:00.420 And our galaxy is actually falling 00:21:00.420 --> 00:21:05.460 into a large cluster of galaxies called the Virgo Cluster. 00:21:05.460 --> 00:21:09.360 This basically means that when we are making cosmologically 00:21:09.360 --> 00:21:12.390 interesting measurements, we have to correct for the fact 00:21:12.390 --> 00:21:16.140 that we make measurements using a four velocity that is not 00:21:16.140 --> 00:21:18.150 a co-moving four velocity. 00:21:18.150 --> 00:21:19.650 This actually shows up in the fact 00:21:19.650 --> 00:21:21.120 that when one makes measurements, 00:21:21.120 --> 00:21:23.078 one of the most impressive places that shows up 00:21:23.078 --> 00:21:25.650 is that when you measure the cosmic microwave background, 00:21:25.650 --> 00:21:28.380 it has what we call a dipole isotropy. 00:21:28.380 --> 00:21:32.730 And that dipole is just essentially a Doppler shift 00:21:32.730 --> 00:21:36.300 that is due to the fact that when we make our measurements, 00:21:36.300 --> 00:21:41.980 we are moving with respect to the co-moving reference frame. 00:21:41.980 --> 00:21:42.670 All right. 00:21:42.670 --> 00:21:45.140 So that's the metric that we're going to use here. 00:21:45.140 --> 00:21:47.410 Setting gtt and gti like so means 00:21:47.410 --> 00:21:50.860 I have chosen these co-moving coordinate systems. 00:21:50.860 --> 00:21:58.820 I'm going to take gamma ij, I'm going 00:21:58.820 --> 00:22:12.410 to take this to be maximally symmetric, OK? 00:22:12.410 --> 00:22:16.950 So this is my statement that at any moment of time, 00:22:16.950 --> 00:22:20.720 space is maximally symmetric. 00:22:20.720 --> 00:22:23.540 So a few words on the units, a few things 00:22:23.540 --> 00:22:24.810 that I'm going to set up here. 00:22:24.810 --> 00:22:34.940 So my coordinate, I'm going to take my xi to be dimensionless, 00:22:34.940 --> 00:22:36.800 and all notions of length-- 00:22:36.800 --> 00:22:45.950 all length scales and the problem 00:22:45.950 --> 00:22:53.860 are going to be absorbed into this factor R of t. 00:22:56.570 --> 00:23:00.200 We're going to see that the overall scale of the universe 00:23:00.200 --> 00:23:02.330 is going to depend on the dynamics of that function 00:23:02.330 --> 00:23:05.310 R of t. 00:23:05.310 --> 00:23:08.660 So let's imagine that on a given-- 00:23:08.660 --> 00:23:10.790 at some given moment of time, you 00:23:10.790 --> 00:23:14.030 want to understand the curvature associated 00:23:14.030 --> 00:23:16.070 with that constant time slice. 00:23:20.600 --> 00:23:34.530 So the Riemann tensor that we build from our spatial metric, 00:23:34.530 --> 00:23:38.930 I'm going to write this as 3 R-- 00:23:38.930 --> 00:23:40.600 and I'm doing purely spatial things, 00:23:40.600 --> 00:23:45.120 so I'm going to use Latin letters for my indices. 00:23:45.120 --> 00:23:47.360 It's going to equal to some number k-- 00:23:50.860 --> 00:23:52.670 not to be confused with the index k. 00:23:52.670 --> 00:23:55.477 It's unfortunate, but there's only so 00:23:55.477 --> 00:23:56.560 many letters to work with. 00:24:01.510 --> 00:24:02.500 That looks like so. 00:24:02.500 --> 00:24:19.650 And if I take a trace to make my Ricci curvature, I get this, 00:24:19.650 --> 00:24:24.510 and your Ricci scalar will turn out to be equal to 6k. 00:24:24.510 --> 00:24:26.670 We won't actually need that, but just so you 00:24:26.670 --> 00:24:28.440 can establish what that k actually means. 00:24:28.440 --> 00:24:30.385 It's simply related to the Ricci curvature. 00:24:30.385 --> 00:24:33.170 Oops, that should have a 3 on it. 00:24:33.170 --> 00:24:37.110 Ricci curvature of that particular instant in time. 00:24:41.470 --> 00:24:48.980 Now I'm going to require my coordinate system 00:24:48.980 --> 00:24:53.790 to reflect the fact that space is isotopic. 00:24:53.790 --> 00:24:56.150 So if it's isotropic, it must look the same 00:24:56.150 --> 00:24:59.210 in all directions, and in a three-dimensional space, 00:24:59.210 --> 00:25:01.160 anything that is the same in all directions 00:25:01.160 --> 00:25:03.089 must be spherically symmetric. 00:25:19.226 --> 00:25:31.965 And so what this means is that when I compute gamma ij dx i dx 00:25:31.965 --> 00:25:41.050 j, it must be equal to some function of radius. 00:25:41.050 --> 00:25:43.540 The bar on that radius just reminds you this 00:25:43.540 --> 00:25:45.580 is meant to be a dimensionless notion of radius. 00:25:45.580 --> 00:25:46.750 Remember, all length scales are going 00:25:46.750 --> 00:25:48.417 to be absorbed into the function capital 00:25:48.417 --> 00:25:56.226 R. It's R squared d omega. 00:26:06.450 --> 00:26:08.580 And my angular sector is just related 00:26:08.580 --> 00:26:14.660 to circle coordinate angles the usual way. 00:26:14.660 --> 00:26:28.600 So it's convenient for us to put this f of R-- 00:26:28.600 --> 00:26:35.423 I can rewrite this as an exponential function, OK? 00:26:35.423 --> 00:26:37.590 Just think of this as the definition of the function 00:26:37.590 --> 00:26:39.350 beta. 00:26:39.350 --> 00:26:42.920 The reason why this is handy is that suppose 00:26:42.920 --> 00:26:48.650 I now take gamma ij, I compute the three-dimensional 00:26:48.650 --> 00:26:52.160 Christoffel symbols, I compute my three-dimensional Riemann, 00:26:52.160 --> 00:26:54.770 ignoring for a moment that my Riemann is meant 00:26:54.770 --> 00:26:56.480 to be maximally symmetric, OK? 00:26:56.480 --> 00:26:58.700 I'm just going to say, I know the recipe 00:26:58.700 --> 00:27:01.340 for how to make Riemann from a metric. 00:27:01.340 --> 00:27:02.407 I will do that. 00:27:02.407 --> 00:27:04.115 I will then make Ricci from that Riemann. 00:27:14.540 --> 00:27:20.725 When you do this, what you find is 00:27:20.725 --> 00:27:23.100 that-- let's just look at the Rr component of this thing. 00:27:28.400 --> 00:27:31.220 This turns out to be 2 over R times a regular derivative 00:27:31.220 --> 00:27:32.420 of beta, OK? 00:27:32.420 --> 00:27:34.910 It's just a little bit of tensor manipulation 00:27:34.910 --> 00:27:39.400 to do that using some of the tools, the mathematical tools 00:27:39.400 --> 00:27:41.660 I'm going to post in the 8.962 website, 00:27:41.660 --> 00:27:43.210 you can verify this yourself. 00:27:48.070 --> 00:28:00.730 If I compute Ricci from the maximally symmetric assumption, 00:28:00.730 --> 00:28:13.180 what I find is that this is equal to 2k times gamma, which 00:28:13.180 --> 00:28:15.460 is itself exponent of 2 beta. 00:28:23.460 --> 00:28:25.440 Let's equate these and solve for beta. 00:28:49.160 --> 00:28:58.162 So one side I've got 2 over r bar dr bar of eta. 00:28:58.162 --> 00:29:03.200 On the near side I have 2k and gamma r bar 00:29:03.200 --> 00:29:09.320 r bar is itself e to the 2 beta. 00:29:09.320 --> 00:29:11.075 Cancel, cancel. 00:29:11.075 --> 00:29:13.670 A little bit of algebra. 00:29:13.670 --> 00:29:15.240 First let's write it this way. 00:29:15.240 --> 00:29:17.570 Let's move this to the other side... 00:29:17.570 --> 00:29:26.240 e to the minus 2 beta is equal to this. 00:29:26.240 --> 00:29:30.590 Let us make the assumption-- so we have a choice of a boundary 00:29:30.590 --> 00:29:31.970 condition. 00:29:31.970 --> 00:29:37.250 Let's put beta equals 0 at r bar equals 0. 00:29:37.250 --> 00:29:40.190 This is basically saying that on my-- 00:29:40.190 --> 00:29:41.780 so r bar is sort of the origin. 00:29:41.780 --> 00:29:43.322 We're just sort of saying that things 00:29:43.322 --> 00:29:47.720 look like a flat spacetime in the vicinity of the origin 00:29:47.720 --> 00:29:49.220 of the coordinates we're using here, 00:29:49.220 --> 00:29:51.470 that's a fine assumption to make. 00:29:51.470 --> 00:30:02.250 Doing so, we can easily integrate this guy up, 00:30:02.250 --> 00:30:04.500 and here's what we get. 00:30:04.500 --> 00:30:08.540 So with this, we now have a full line element. 00:30:32.690 --> 00:30:36.380 So there's a few unknown quantities in here. 00:30:36.380 --> 00:30:38.240 What is k? 00:30:38.240 --> 00:30:41.180 What is R? 00:30:41.180 --> 00:30:46.640 So far I have only talked about the geometry of this spacetime. 00:30:46.640 --> 00:30:50.000 We haven't yet connected this-- any of the dynamics 00:30:50.000 --> 00:30:53.120 of the spacetime to a source. 00:30:53.120 --> 00:30:54.930 It is when we hook this up to a source 00:30:54.930 --> 00:30:57.097 that we're going to learn something about these two. 00:30:57.097 --> 00:30:58.400 So hold that thought for now. 00:30:58.400 --> 00:31:00.380 This essentially has just said that here 00:31:00.380 --> 00:31:05.060 is what my maximally spatially symmetric spacetime looks like, 00:31:05.060 --> 00:31:07.730 allowing for there to be a difference between the past 00:31:07.730 --> 00:31:09.770 and the present. 00:31:09.770 --> 00:31:17.640 Before I move on, so I can't tell you what k is yet, 00:31:17.640 --> 00:31:19.920 but I can make the following observation which 00:31:19.920 --> 00:31:24.963 allows me to restrict what values of k 00:31:24.963 --> 00:31:25.880 I need to worry about. 00:31:44.220 --> 00:31:58.870 Suppose I take k and I replace it with k prime equal alpha k. 00:32:01.960 --> 00:32:07.870 But in doing so, I define R tilde 00:32:07.870 --> 00:32:10.060 to be square root of alpha-- 00:32:15.230 --> 00:32:15.730 yeah. 00:32:15.730 --> 00:32:21.030 Square root of alpha times R bar. 00:32:21.030 --> 00:32:36.670 And I also require that my overall scale factor 00:32:36.670 --> 00:32:38.650 look like the original scale factor divided 00:32:38.650 --> 00:32:41.170 by square root of alpha. 00:32:41.170 --> 00:32:45.850 Rewriting my spacetime, my line element in terms of k prime 00:32:45.850 --> 00:33:20.090 and the tilde R into-- the two tilde R's, I get this. 00:33:20.090 --> 00:33:24.200 Basically, that transformation leaves the line element 00:33:24.200 --> 00:33:28.390 completely irrelevant to me. 00:33:28.390 --> 00:33:30.030 That was completely the wrong word. 00:33:30.030 --> 00:33:35.610 That re-prioritization of k and R and the two different R's 00:33:35.610 --> 00:33:39.060 here, that leaves the line element completely invariant. 00:33:39.060 --> 00:33:40.920 It is unchanged when I do this. 00:33:46.210 --> 00:33:48.413 So what this tells me is-- by the way, 00:33:48.413 --> 00:33:50.830 alpha has to be a positive number so that the square roots 00:33:50.830 --> 00:33:51.910 make sense there. 00:33:51.910 --> 00:33:58.060 It tells me that the normalization associated with k 00:33:58.060 --> 00:34:05.460 can be absorbed into my scale factor. 00:34:10.830 --> 00:34:15.870 And so what it suggests we ought to do is just-- 00:34:15.870 --> 00:34:18.210 you don't need to worry about whether k 00:34:18.210 --> 00:34:23.250 is equal to 15 or pi or negative the 38th root of e 00:34:23.250 --> 00:34:24.540 or anything silly like that. 00:34:29.620 --> 00:34:34.659 The only three values of k that matter for us 00:34:34.659 --> 00:34:39.190 are whether it is negative 1, 0, or 1. 00:34:39.190 --> 00:34:42.750 This stands for all negative values of k, 00:34:42.750 --> 00:34:49.320 0 as a set onto itself, and all positive values of k, OK? 00:34:49.320 --> 00:34:53.328 So we will use this to say, great, the thing which 00:34:53.328 --> 00:34:54.870 I'm going to care about, once I start 00:34:54.870 --> 00:34:57.360 looking at the physics associated with this, 00:34:57.360 --> 00:34:59.590 is whether-- 00:34:59.590 --> 00:35:01.800 let's go back over to this version of it-- 00:35:01.800 --> 00:35:06.630 I'm going to care about whether k is negative 1, 0, 1, 00:35:06.630 --> 00:35:10.170 and I want to understand how my scale factor behaves. 00:35:19.320 --> 00:35:21.900 So before I start hooking this up to my source 00:35:21.900 --> 00:35:25.650 and doing a little bit of physics, many of you 00:35:25.650 --> 00:35:27.660 are going to do something involving cosmology 00:35:27.660 --> 00:35:30.480 at some point in your lives, and so it's 00:35:30.480 --> 00:35:34.950 useful to introduce a few other bits of notation that 00:35:34.950 --> 00:35:42.928 are commonly used here, as well as to describe 00:35:42.928 --> 00:35:45.220 some important terminology that comes up at this point. 00:36:00.240 --> 00:36:03.650 So here's some common notation and terminology. 00:36:03.650 --> 00:36:21.770 Let us define a radial coordinate chi 00:36:21.770 --> 00:36:25.910 via the following definition-- 00:36:25.910 --> 00:36:32.330 d chi will be equal to d R bar over square root of 1 00:36:32.330 --> 00:36:37.350 minus kR bar squared. 00:36:37.350 --> 00:36:39.500 Now remember, we just decided that k can only 00:36:39.500 --> 00:36:41.510 take on one of three interesting values. 00:36:44.210 --> 00:36:46.130 I can immediately integrate this up, 00:36:46.130 --> 00:36:50.840 and I will find that my R bar is equal to sine 00:36:50.840 --> 00:37:00.250 of chi of k equals plus 1 is equal to chi of k equal 0. 00:37:03.560 --> 00:37:08.160 And it's the sinh of chi of k equals minus 1. 00:37:15.115 --> 00:37:17.800 So let's take a look at what this 00:37:17.800 --> 00:37:23.005 means with these sort of three possible choices. 00:37:23.005 --> 00:37:26.320 The three possible values that k can take. 00:37:26.320 --> 00:37:27.820 What is our line element looks like? 00:37:32.000 --> 00:37:34.050 So let's look at k equals plus 1 first. 00:37:38.920 --> 00:37:47.360 I get minus dt squared R square root of t d chi squared plus-- 00:37:47.360 --> 00:37:57.650 I'm going to use the fact that R bar, 00:37:57.650 --> 00:38:04.160 this describes a spacetime in which every spacial slice is 00:38:04.160 --> 00:38:06.020 what is called a 3-sphere, OK? 00:38:06.020 --> 00:38:09.020 You're all nicely familiar with the 2-sphere. 00:38:09.020 --> 00:38:12.110 So a 2-sphere is the three-dimensional surface 00:38:12.110 --> 00:38:16.090 in which you pick a point and every point that is, 00:38:16.090 --> 00:38:19.340 let's say, a unit radius away from that point that, 00:38:19.340 --> 00:38:23.550 defines a 2-sphere in three-dimensional space. 00:38:23.550 --> 00:38:26.560 So this defines the space-- 00:38:26.560 --> 00:38:28.590 the spatial characteristics of pick 00:38:28.590 --> 00:38:31.670 a point in four-dimensional space 00:38:31.670 --> 00:38:35.360 and ask for all of the points that are a unit distance away 00:38:35.360 --> 00:38:37.670 from it in three dimensions, that is a 3-sphere. 00:38:53.430 --> 00:38:56.910 Notice that my 3-sphere has a maximum-- 00:38:56.910 --> 00:39:00.330 there's a maximum distance associated with it, OK? 00:39:00.330 --> 00:39:02.970 So there's no bounds on chi, OK? 00:39:02.970 --> 00:39:04.950 Chi can go from 0 to infinity. 00:39:04.950 --> 00:39:06.840 But this one's periodic, isn't it? 00:39:06.840 --> 00:39:12.690 So as chi reaches pi over 2, the separation between any two 00:39:12.690 --> 00:39:14.310 points on that single slice, they've 00:39:14.310 --> 00:39:17.550 reached their maximum value, and as chi continues to increase, 00:39:17.550 --> 00:39:19.680 the distance gets smaller again, OK? 00:39:19.680 --> 00:39:22.230 And eventually, when chi gets up to pi, 00:39:22.230 --> 00:39:24.120 you come back to where you started. 00:39:24.120 --> 00:39:28.733 We call this a closed universe. 00:39:28.733 --> 00:39:30.150 This is something where if it were 00:39:30.150 --> 00:39:33.300 possible to step out of time and just run around 00:39:33.300 --> 00:39:38.050 on a spacial slice, you would find that it is a finite size. 00:39:38.050 --> 00:39:40.650 The best you could do is run around 00:39:40.650 --> 00:39:44.660 on that three-dimensional sphere in four-dimensional space. 00:39:57.530 --> 00:39:59.290 Let's do k equals 0 next. 00:40:17.850 --> 00:40:34.580 If I do k equals 0, there's my line element. 00:40:34.580 --> 00:40:38.170 Each spacial slice is simply Euclidean space. 00:40:47.790 --> 00:40:52.660 So this is often described as flat space. 00:40:58.310 --> 00:41:01.670 A significant word of caution. 00:41:01.670 --> 00:41:04.730 When you talk to a cosmologist, they will often 00:41:04.730 --> 00:41:06.778 talk about how the best-- 00:41:06.778 --> 00:41:09.320 we're going to talk about sort of the observational situation 00:41:09.320 --> 00:41:12.410 in the next lecture that I record a little bit. 00:41:12.410 --> 00:41:15.460 Our evidence actually suggests that this 00:41:15.460 --> 00:41:17.210 is what our universe looks like right now. 00:41:17.210 --> 00:41:21.530 We're in a k equal 0 universe in which space-- 00:41:21.530 --> 00:41:24.120 each spatial slice is flat. 00:41:24.120 --> 00:41:27.920 That does not mean spacetime is flat, OK? 00:41:27.920 --> 00:41:29.540 So when they say that it's flat, that 00:41:29.540 --> 00:41:31.280 is referring to the geometry only 00:41:31.280 --> 00:41:34.020 of the spatial slices in this co-moving coordinate system. 00:41:38.940 --> 00:41:59.580 k equals minus 1, you get a form that looks like this. 00:41:59.580 --> 00:42:02.830 This describes the geometry of a hyperbole. 00:42:17.540 --> 00:42:20.120 We call this an open spatial slice. 00:42:20.120 --> 00:42:23.018 So notice for both choices 2 and 3, 00:42:23.018 --> 00:42:24.560 if you could sort of step out of time 00:42:24.560 --> 00:42:28.280 and explore the full geometry of that spatial slice, 00:42:28.280 --> 00:42:29.480 it goes on forever, OK? 00:42:29.480 --> 00:42:31.950 Again, there's really no boundary on that chi 00:42:31.950 --> 00:42:34.037 as near as we can tell, and so that spatial slide 00:42:34.037 --> 00:42:35.120 can just kind of go, whee! 00:42:35.120 --> 00:42:37.190 And take off forever. 00:42:37.190 --> 00:42:41.330 This one sort of goes to large distances a little slower 00:42:41.330 --> 00:42:42.170 than this one does. 00:42:42.170 --> 00:42:44.270 This hyperbolic function means that this 00:42:44.270 --> 00:42:48.500 is really bloody large, OK? 00:42:48.500 --> 00:42:54.800 So both of these tend to imply a universe that 00:42:54.800 --> 00:42:57.040 is sort of spatially unbounded. 00:42:57.040 --> 00:43:04.142 The closed universe, because each slice is a 3-sphere, 00:43:04.142 --> 00:43:05.100 it's a different story. 00:43:09.640 --> 00:43:14.070 So another bit of notation which you should be aware of-- 00:43:17.040 --> 00:43:19.320 and I unfortunately am going to want 00:43:19.320 --> 00:43:22.230 to sort of flip back and forth between the notation I've 00:43:22.230 --> 00:43:25.157 been using so far and this one I'm about to introduce. 00:43:28.887 --> 00:43:31.470 It can be a little bit annoying when you're first learning it, 00:43:31.470 --> 00:43:34.710 but just keep track of context, it's not that hard. 00:43:34.710 --> 00:43:38.520 So what we're going to do is let's 00:43:38.520 --> 00:43:41.440 choose a particular value of the scale factor, 00:43:41.440 --> 00:43:43.500 and we will normalize things to that. 00:43:58.310 --> 00:44:04.540 So what I'm going to do is define some particular value 00:44:04.540 --> 00:44:10.570 of k such that the scale factor there I will call it R sub 0. 00:44:10.570 --> 00:44:17.800 And as we'll see, a particularly useful choice for this 00:44:17.800 --> 00:44:20.323 is to choose the value right now, OK? 00:44:20.323 --> 00:44:22.740 What we're doing, then, is we're kind of norm-- what we're 00:44:22.740 --> 00:44:25.073 going to see in a moment is this means we're normalizing 00:44:25.073 --> 00:44:27.880 all the scales associated with our universe 00:44:27.880 --> 00:44:30.180 to where they are right now. 00:44:30.180 --> 00:44:30.680 OK. 00:44:30.680 --> 00:44:37.240 Having done this, I'm going to define a of t 00:44:37.240 --> 00:44:43.923 to be R of t divided by this special value. 00:44:43.923 --> 00:44:45.340 For dimensional reasons, I'm going 00:44:45.340 --> 00:44:49.750 to need to put this into my radial coordinate. 00:44:49.750 --> 00:44:51.180 So notice, what's going on here is 00:44:51.180 --> 00:44:54.300 that my R will now have dimensions associated with it, 00:44:54.300 --> 00:45:00.655 and so essentially everything is just being scaled by that R0. 00:45:00.655 --> 00:45:03.030 And this is the bit where it gets a tiny bit unfortunate, 00:45:03.030 --> 00:45:09.995 you sort of lose the beauty of k only having three values. 00:45:09.995 --> 00:45:11.745 So I'm going to replace that with a kappa. 00:45:14.580 --> 00:45:16.920 This is unfortunately a little bit hard to read, 00:45:16.920 --> 00:45:19.310 so whenever I make it sort of with my messy cursive, 00:45:19.310 --> 00:45:21.870 it will be k; whenever it looks a little bit more 00:45:21.870 --> 00:45:23.600 like a printed thing, it will be kappa. 00:45:27.970 --> 00:45:32.730 And so kappa is k divided by R0 squared. 00:45:32.730 --> 00:45:53.150 And when you do that, your line element becomes this. 00:46:00.800 --> 00:46:01.490 OK? 00:46:01.490 --> 00:46:03.750 So that's a form that we're going to use a little bit. 00:46:03.750 --> 00:46:08.177 What's a little bit annoying about it is just that my-- 00:46:08.177 --> 00:46:10.010 the kappa that appears in there doesn't just 00:46:10.010 --> 00:46:12.190 come as a set of one of these parts of three, 00:46:12.190 --> 00:46:15.660 but basically if kappa is a negative number, 00:46:15.660 --> 00:46:19.010 then you know k must be minus 1; kappa equals 0 corresponds to k 00:46:19.010 --> 00:46:22.360 equals 0; if kappa is a positive number, then k equals plus 1. 00:46:22.360 --> 00:46:24.860 This form where we're using this sort of dimensionless scale 00:46:24.860 --> 00:46:28.970 factor a is particularly useful. 00:46:28.970 --> 00:46:30.680 If you look at this, this is telling you 00:46:30.680 --> 00:46:43.150 that with the choice that R0 defines a scale factor now, 00:46:43.150 --> 00:46:47.860 this means a now equals 1. 00:46:47.860 --> 00:46:50.470 And so this gives us a nice dimensionless factor 00:46:50.470 --> 00:46:54.760 by which we can compare all of our spatial scales 00:46:54.760 --> 00:46:56.800 at different moments in the universe to the size 00:46:56.800 --> 00:46:57.550 that they are now. 00:47:00.800 --> 00:47:01.950 OK. 00:47:01.950 --> 00:47:04.890 Everything I have said so far has really 00:47:04.890 --> 00:47:06.990 been just discussing the geometry 00:47:06.990 --> 00:47:08.970 that I'm going to use to describe 00:47:08.970 --> 00:47:13.060 the large-scale structure of spacetime. 00:47:13.060 --> 00:47:16.150 I haven't said anything about what 00:47:16.150 --> 00:47:19.570 happens when I solve the Einstein field equations 00:47:19.570 --> 00:47:22.210 and connect this geometry to physics. 00:47:25.730 --> 00:47:27.680 So what we need to do is choose a source. 00:47:27.680 --> 00:47:29.560 And so what we're going to do is we 00:47:29.560 --> 00:47:33.010 will do what is sort of the default choice in many analyses 00:47:33.010 --> 00:47:34.190 in general relativity. 00:47:34.190 --> 00:47:38.170 We will choose our source to be a perfect fluid. 00:47:48.330 --> 00:47:51.180 What's nice about this is that it automatically 00:47:51.180 --> 00:47:57.390 satisfies the requirements of isotropy and homogeneity. 00:47:57.390 --> 00:48:03.930 At least it does so if the fluid is at rest 00:48:03.930 --> 00:48:05.627 in co-moving coordinates. 00:48:30.000 --> 00:48:34.560 So let's fill this in: t mu nu with everything 00:48:34.560 --> 00:48:42.850 in the downstairs position looks like rho plus P mu nu mu nu 00:48:42.850 --> 00:48:46.000 plus Pg mu nu. 00:48:49.122 --> 00:48:53.430 And this becomes in my co-moving coordinate system. 00:48:58.510 --> 00:49:01.820 So then it looks like this, OK? 00:49:01.820 --> 00:49:06.080 A handy fact to have, this is going 00:49:06.080 --> 00:49:09.418 to be quite useful for a calculation or two 00:49:09.418 --> 00:49:10.710 that we do a little bit later-- 00:49:13.670 --> 00:49:16.180 actually, not just a minute later, almost right away. 00:49:37.610 --> 00:49:43.175 This looks like a diagonal of all this stuff, OK? 00:49:46.640 --> 00:49:47.490 All right. 00:49:47.490 --> 00:49:53.110 So what we want to do is use this stress energy tensor 00:49:53.110 --> 00:49:54.260 as the right-hand side. 00:49:54.260 --> 00:49:57.670 So we've worked out our Ricci tensor. 00:49:57.670 --> 00:50:01.180 With a little bit of work, we can make the Einstein tensor, 00:50:01.180 --> 00:50:03.490 couple it to this guy, we can set up our differential 00:50:03.490 --> 00:50:07.090 equations, and we can solve for the free functions that 00:50:07.090 --> 00:50:09.190 specify the spacetime. 00:50:09.190 --> 00:50:13.420 Before doing this, always a good sanity check, remind yourself, 00:50:13.420 --> 00:50:16.718 your fluid has to satisfy local energy conservation. 00:50:21.020 --> 00:50:22.360 Actually, let's just do the 0. 00:50:22.360 --> 00:50:25.120 So this is energy and momentum conservation, 00:50:25.120 --> 00:50:30.339 we set that equal to 0, this is local energy conservation. 00:50:39.930 --> 00:50:51.210 Expanding out these derivatives, what you 00:50:51.210 --> 00:50:53.190 find is that this turns into-- 00:51:09.050 --> 00:51:12.280 so it looks like this. 00:51:12.280 --> 00:51:16.960 And plugging in-- so using the spacetime-- 00:51:21.040 --> 00:51:24.660 by the way, I made a small mistake earlier. 00:51:24.660 --> 00:51:34.870 I should have told you that this spacetime, 00:51:34.870 --> 00:51:40.080 this is now called the Robertson-Walker spacetime. 00:51:50.522 --> 00:51:52.730 So this was actually first written down in the 1920s, 00:51:52.730 --> 00:51:54.620 and Robertson and Walker developed this basically 00:51:54.620 --> 00:51:56.328 just as I have done it here, just arguing 00:51:56.328 --> 00:51:58.160 on the basis of looking for something that 00:51:58.160 --> 00:52:03.800 is as symmetric as possible with respect to space if not time, 00:52:03.800 --> 00:52:05.702 and they came out with that line element. 00:52:05.702 --> 00:52:07.660 My apologies, I didn't mention that beforehand. 00:52:07.660 --> 00:52:09.077 This is my third lecture in a row, 00:52:09.077 --> 00:52:10.370 I'm getting a little bit tired. 00:52:10.370 --> 00:52:13.130 So if I take that Robertson-Walker metric, 00:52:13.130 --> 00:52:17.610 plug it into here to evaluate all these, 00:52:17.610 --> 00:52:19.860 this gives me a remarkably simple form. 00:52:40.070 --> 00:52:45.195 So rho is the pressure of my perfect fluid-- 00:52:45.195 --> 00:52:47.070 excuse me, the density of my perfect fluid, P 00:52:47.070 --> 00:52:50.870 is the pressure of my perfect fluid, a is my scale factor. 00:52:57.170 --> 00:53:01.030 If you like, you can put the factor of R0 back in there, 00:53:01.030 --> 00:53:07.700 and an equivalent way of writing this, which I think 00:53:07.700 --> 00:53:10.910 is very useful for giving some physical insight as to what 00:53:10.910 --> 00:53:11.570 this means-- 00:53:30.266 --> 00:53:31.980 so put that R back in. 00:53:48.708 --> 00:53:51.360 OK, so let's look at what this is saying. 00:53:51.360 --> 00:53:54.990 R cubed is modulo numerical factor, 00:53:54.990 --> 00:53:58.140 that is the volume of a spacial slice. 00:53:58.140 --> 00:54:01.380 And so this is saying, the rate of change 00:54:01.380 --> 00:54:02.670 of energy in a volume-- 00:54:11.830 --> 00:54:14.740 so a volume describing my spatial slice 00:54:14.740 --> 00:54:20.260 is equal to negative pressure times the rate of change 00:54:20.260 --> 00:54:21.340 of that volume. 00:54:33.330 --> 00:54:36.790 I hope this looks familiar. 00:54:36.790 --> 00:54:41.080 This, in somewhat more convoluted notation, 00:54:41.080 --> 00:54:43.330 is negative dp-- 00:54:43.330 --> 00:54:47.450 du equals negative P dv. 00:54:47.450 --> 00:54:49.686 It's just the first law of thermodynamics. 00:54:52.910 --> 00:54:53.410 All right. 00:54:53.410 --> 00:54:56.980 So this relationship, whether written in this form 00:54:56.980 --> 00:54:58.810 or in that form, is something that we 00:54:58.810 --> 00:55:00.310 will exploit moving forward. 00:55:05.360 --> 00:55:10.047 Let's now solve the Einstein field equations. 00:55:25.970 --> 00:55:34.640 So we'll begin with g mu nu equals 8 pi g t mu nu. 00:55:34.640 --> 00:55:37.040 The equations that are traditionally 00:55:37.040 --> 00:55:42.300 used to describe cosmology are a little bit more naturally 00:55:42.300 --> 00:55:42.800 written. 00:55:42.800 --> 00:55:47.330 If I change this into the form that uses the Ricci tensor-- 00:55:47.330 --> 00:55:54.240 so let me rewrite this as R mu nu equals 8 pi g t mu nu. 00:55:57.720 --> 00:55:58.220 OK? 00:55:58.220 --> 00:56:03.850 So this is equivalent where t is just the usual trace 00:56:03.850 --> 00:56:06.890 of the stress energy tensor. 00:56:06.890 --> 00:56:10.330 And what you find, there are two-- 00:56:10.330 --> 00:56:14.890 if you just look at the 0, 0 components of this equation, 00:56:14.890 --> 00:56:30.890 it tells you the acceleration of the scale factor a divided by a 00:56:30.890 --> 00:56:31.490 is-- 00:56:31.490 --> 00:56:38.930 it is simply related to the density and 3 times 00:56:38.930 --> 00:56:41.100 the pressure. 00:56:41.100 --> 00:56:44.850 If you evaluate Rii-- 00:56:44.850 --> 00:56:45.907 in other words, any-- 00:56:45.907 --> 00:56:47.490 this is-- there's no sum implied here, 00:56:47.490 --> 00:56:54.070 just take any spatial component of this guy, 00:56:54.070 --> 00:56:56.570 and add on R0,0 because it's a valid equation, 00:56:56.570 --> 00:57:11.820 it helps you to clear out some stuff, 00:57:11.820 --> 00:57:15.570 you get the following relationship between the rate 00:57:15.570 --> 00:57:20.100 of change to the scale factor, the density, and remember, 00:57:20.100 --> 00:57:24.010 kappa is your rescaled k. 00:57:24.010 --> 00:57:28.540 So I'm going to call this equation F1, 00:57:28.540 --> 00:57:32.020 I'm going to call this one F2. 00:57:32.020 --> 00:57:34.680 These are known as the Friedmann equations. 00:57:41.180 --> 00:57:47.250 When one uses them to solve to describe your line element, 00:57:47.250 --> 00:57:59.480 you get Friedmann-Robertson-Walker 00:57:59.480 --> 00:57:59.980 metrics. 00:58:17.408 --> 00:58:18.950 So just a little bit of nomenclature. 00:58:18.950 --> 00:58:21.860 Robertson-Walker tells you about the geometry, 00:58:21.860 --> 00:58:25.640 you then equate these guys to a source, 00:58:25.640 --> 00:58:29.640 and that gives you Friedmann-Robertson-Walker line 00:58:29.640 --> 00:58:30.140 elements. 00:58:37.340 --> 00:58:39.678 One other bit of information-- so 00:58:39.678 --> 00:58:41.720 let's introduce a little bit of terminology here. 00:58:54.300 --> 00:59:02.580 So a dot over a, this tells me how the overall length 00:59:02.580 --> 00:59:05.370 scale associated with my spatial slices 00:59:05.370 --> 00:59:07.020 is evolving as a function of time. 00:59:10.480 --> 00:59:16.542 This is denoted H and it's known as the Hubble parameter. 00:59:24.740 --> 00:59:30.950 H0 is the value of H that we measure 00:59:30.950 --> 00:59:35.180 in our universe corresponding to its expansion right now, OK? 00:59:35.180 --> 00:59:39.050 And the notes that I have scanned and placed online 00:59:39.050 --> 00:59:42.020 claim a best value for this of 73 00:59:42.020 --> 00:59:45.650 plus or minus 3 kilometers per second per megaparsec. 00:59:45.650 --> 00:59:47.360 These notes were originally hand written 00:59:47.360 --> 00:59:50.390 about 11 or 12 years ago, that number is already out of date, 00:59:50.390 --> 00:59:51.750 OK? 00:59:51.750 --> 00:59:55.790 If you went to Adam Reese's colloquium 00:59:55.790 --> 00:59:59.223 shortly before MIT went into its COVID shutdown, 00:59:59.223 --> 01:00:00.890 you will have seen that there's actually 01:00:00.890 --> 01:00:02.848 little bit of controversy about this right now. 01:00:02.848 --> 01:00:04.760 So our best measurements of this thing, 01:00:04.760 --> 01:00:08.578 indeed, they are clustering around 72 or 73 in these units, 01:00:08.578 --> 01:00:11.120 but they're inconsistent with some other measures by which we 01:00:11.120 --> 01:00:15.200 can infer to be the-- what the Hubble parameter should be. 01:00:15.200 --> 01:00:16.310 And it's a very-- 01:00:18.767 --> 01:00:19.850 very interesting problems. 01:00:19.850 --> 01:00:21.530 Unclear sort of-- it sort of smells 01:00:21.530 --> 01:00:22.970 like something might be a little bit off 01:00:22.970 --> 01:00:25.387 in our cosmological models, but we're not quite there yet. 01:00:25.387 --> 01:00:27.470 Let's consider-- let's proceed with sort 01:00:27.470 --> 01:00:29.780 of the standard picture, and just bear in mind 01:00:29.780 --> 01:00:32.300 that this is an evolving field. 01:00:32.300 --> 01:00:41.600 The one thing I will note is that H has the dimensions 01:00:41.600 --> 01:00:43.910 of inverse time, OK? 01:00:43.910 --> 01:00:45.680 The way one actually measures it. 01:00:45.680 --> 01:00:48.770 So the dimensions in which most astronomers quote its value, 01:00:48.770 --> 01:00:51.590 it looks like a velocity over a length, which is, of course, 01:00:51.590 --> 01:00:53.360 also an inverse time. 01:00:53.360 --> 01:00:57.830 And that is because objects that are at rest with respect 01:00:57.830 --> 01:01:00.860 to the-- that are at rest in these co-moving coordinates, 01:01:00.860 --> 01:01:06.050 as this fluid is meant to be, if the universe is expanding, 01:01:06.050 --> 01:01:08.810 we see them moving away from us. 01:01:08.810 --> 01:01:10.790 OK, I'm going to make a few definitions. 01:01:14.640 --> 01:01:29.390 Let us define rho crit to be 3H squared over 8 pi j. 01:01:29.390 --> 01:01:38.520 So the way I got that was take F1-- 01:01:38.520 --> 01:01:42.900 imagine kappa is equal to 0, just ignore kappa for a second. 01:01:42.900 --> 01:01:46.109 Left-hand side as H squared, solve for rho, OK? 01:02:10.590 --> 01:02:15.098 Notice, rho quit-- rho crit is a parameter that you can measure. 01:02:15.098 --> 01:02:16.640 You can measure the Hubble parameter, 01:02:16.640 --> 01:02:19.015 I'll describe to you how that is done in my next lecture, 01:02:19.015 --> 01:02:21.390 but it's a number that can be measured. 01:02:21.390 --> 01:02:25.070 And then 3 and 8 pi are just exact numbers, 01:02:25.070 --> 01:02:26.962 g is a fundamental constant. 01:02:26.962 --> 01:02:28.670 So that's something that can be measured. 01:02:31.480 --> 01:02:40.750 Let's define omega to be any density divided by rho crit. 01:02:40.750 --> 01:02:43.805 Putting all these together, I can rewrite the first Friedmann 01:02:43.805 --> 01:02:44.305 equation. 01:02:49.190 --> 01:02:59.840 This guy can be written as omega minus 1 equals kappa over H 01:02:59.840 --> 01:03:04.230 squared a squared. 01:03:04.230 --> 01:03:08.950 Now notice, H and a, they are real numbers. 01:03:08.950 --> 01:03:14.060 H squared and a squared are positive definite. 01:03:14.060 --> 01:03:19.580 We at last can now see how the large-scale distribution 01:03:19.580 --> 01:03:21.920 of matter in our universe allows us 01:03:21.920 --> 01:03:26.150 to constrain one of the parameters that 01:03:26.150 --> 01:03:29.420 sets our Robertson-Walker line element. 01:03:29.420 --> 01:03:35.150 If omega is less than 1-- 01:03:35.150 --> 01:03:41.350 in other words, if rho is less than rho crit, 01:03:41.350 --> 01:03:46.960 then it must be the case that kappa is negative, 01:03:46.960 --> 01:03:51.243 k equals minus 1, and we have an open universe. 01:03:56.920 --> 01:04:00.090 If a omega equals 1 such that rho is exactly 01:04:00.090 --> 01:04:07.350 equal to rho crit, kappa must equal 0, k must equal 0, 01:04:07.350 --> 01:04:10.432 and we have a Euclidean spatially flat universe. 01:04:17.040 --> 01:04:23.250 If omega is greater than 1, kappa is greater than 1, 01:04:23.250 --> 01:04:28.496 k equals 1, and we have a closed universe. 01:04:36.473 --> 01:04:38.223 Clean up my handwriting a little bit here. 01:04:52.027 --> 01:04:54.220 OK, this is really interesting. 01:04:54.220 --> 01:04:59.760 This is telling us if we can determine 01:04:59.760 --> 01:05:04.710 whether the density of stuff in our universe 01:05:04.710 --> 01:05:11.170 exceeds, is equal to, or is less than that critical value, 01:05:11.170 --> 01:05:12.990 we know something pretty profound 01:05:12.990 --> 01:05:16.500 about the spatial geometry of our universe. 01:05:16.500 --> 01:05:20.620 Either it's finite, sort of simply infinite, 01:05:20.620 --> 01:05:23.025 or ridiculously infinite. 01:05:25.382 --> 01:05:27.840 Let me do a few more things before I conclude this lecture. 01:05:44.150 --> 01:05:47.000 First, this isn't that important for our purposes, 01:05:47.000 --> 01:05:50.305 but it's something that some of you students will see. 01:05:50.305 --> 01:05:51.805 A little bit of notational trickery. 01:05:56.160 --> 01:05:58.980 It's not uncommon in the literature 01:05:58.980 --> 01:06:01.440 to see people define what's called a curvature density. 01:06:06.240 --> 01:06:11.180 And what this is, is you just combine 01:06:11.180 --> 01:06:16.880 factors of kappa, g, and the scale factor in such a way 01:06:16.880 --> 01:06:20.060 that this has the dimensions of density. 01:06:20.060 --> 01:06:26.690 You can then find an omega associated with curvature 01:06:26.690 --> 01:06:32.150 to be rho with curvature over the critical density. 01:06:32.150 --> 01:06:37.650 And when you do this, F1, the first Friedmann equation, 01:06:37.650 --> 01:06:45.540 becomes simply omega plus omega curvature equals 1, OK? 01:06:45.540 --> 01:06:47.910 Just bear in mind, that is not really a density, 01:06:47.910 --> 01:06:49.390 it's just a concept-- 01:06:49.390 --> 01:06:52.110 it's a useful auxiliary concept. 01:06:52.110 --> 01:06:55.180 This is often for certain kinds of calculations, 01:06:55.180 --> 01:06:58.750 a nice constraint to bear in mind, OK? 01:06:58.750 --> 01:07:01.290 People are very interested in understanding 01:07:01.290 --> 01:07:04.620 the geometry of our universe, and this 01:07:04.620 --> 01:07:12.090 is a way of formulating it that sort of puts the term involving 01:07:12.090 --> 01:07:15.600 the k parameter or the kappa parameter on the same footing 01:07:15.600 --> 01:07:18.180 as other densities that contribute to the energy 01:07:18.180 --> 01:07:19.555 budget of our universe. 01:07:28.110 --> 01:07:28.650 OK. 01:07:28.650 --> 01:07:32.910 So the equations that we are working with here 01:07:32.910 --> 01:07:34.380 involve these-- 01:07:37.790 --> 01:07:41.300 it involves the pressure and density here. 01:07:41.300 --> 01:07:44.350 I haven't said too much about them so far. 01:07:44.350 --> 01:07:48.070 If I want to make further progress, 01:07:48.070 --> 01:07:50.210 I've got to know a little bit about the matter that 01:07:50.210 --> 01:07:51.639 fills my universe. 01:08:02.630 --> 01:08:09.480 So to make more progress, I need to choose 01:08:09.480 --> 01:08:12.355 what is called an equation of state that 01:08:12.355 --> 01:08:14.480 relates the pressure and the density to each other. 01:08:23.649 --> 01:08:27.819 What I really need is to know that my pressure is 01:08:27.819 --> 01:08:30.550 some function of the density, OK? 01:08:30.550 --> 01:08:33.279 This can be written down for just about all kinds of matter 01:08:33.279 --> 01:08:35.170 that we care about. 01:08:35.170 --> 01:08:49.970 In cosmology, one usually take and assumes 01:08:49.970 --> 01:08:55.029 that the pressure is a linear function-- it's just linearly 01:08:55.029 --> 01:08:57.340 related to the energy density. 01:08:57.340 --> 01:09:00.899 Let me emphasize that as a very restrictive form. 01:09:00.899 --> 01:09:02.899 When we finish cosmology, one of the next things 01:09:02.899 --> 01:09:05.080 we're going to talk about are spherically symmetric 01:09:05.080 --> 01:09:07.420 compact objects-- 01:09:07.420 --> 01:09:10.580 stars-- and we want to describe them as a fluid, 01:09:10.580 --> 01:09:13.149 and we'll need an equation of state to make progress there, 01:09:13.149 --> 01:09:15.565 we do not use a form like that for stars. 01:09:18.250 --> 01:09:22.330 As we'll see, though, for the kind of matter that 01:09:22.330 --> 01:09:25.823 dominates the behavior of our universe on the largest scales, 01:09:25.823 --> 01:09:27.490 this is actually a very reasonable form. 01:09:31.359 --> 01:09:33.819 So if I were to write down my thoughts on that, 01:09:33.819 --> 01:09:39.399 I would say restrictive but useful 01:09:39.399 --> 01:09:41.859 on the large scales appropriate to cosmology. 01:09:54.570 --> 01:09:57.950 Pardon me just one moment. 01:09:57.950 --> 01:10:08.668 Let's imagine that-- yeah, sorry. 01:10:08.668 --> 01:10:10.460 Let's imagine that I have a universe that's 01:10:10.460 --> 01:10:14.660 dominated by a single species of some kind of stuff, OK? 01:10:14.660 --> 01:10:16.490 So in reality, what you will generally 01:10:16.490 --> 01:10:19.168 have is a universe in which there 01:10:19.168 --> 01:10:20.710 are several different things present. 01:10:20.710 --> 01:10:22.730 So you might have a W corresponding 01:10:22.730 --> 01:10:26.742 to one form of matter, another W for a different form of matter, 01:10:26.742 --> 01:10:28.700 and you'll sort of have a superposition of them 01:10:28.700 --> 01:10:32.400 all present at one given moment. 01:10:32.400 --> 01:10:47.110 So to start, start by imagining a universe dominated 01:10:47.110 --> 01:11:01.520 by a particular what I will call a species rho i, 01:11:01.520 --> 01:11:04.970 and the pressure will be related to this 01:11:04.970 --> 01:11:08.770 by a particular W for whatever that rho happens to be. 01:11:13.380 --> 01:11:21.620 So before I even hook this up to the Friedmann equations, 01:11:21.620 --> 01:11:26.540 let's require that this form of matter respects 01:11:26.540 --> 01:11:29.250 stress energy conservation. 01:11:29.250 --> 01:11:31.028 OK, so the equation I just wrote, 01:11:31.028 --> 01:11:33.070 let me rewrite that in a slightly different form. 01:11:33.070 --> 01:11:35.305 I can divide both sides by R0 cubed. 01:11:51.664 --> 01:11:53.780 OK, that looks like so. 01:11:53.780 --> 01:11:59.040 Now it's not too hard to show using this assumed form here-- 01:11:59.040 --> 01:12:07.000 so if I plug in that my p is Wi rho i, 01:12:07.000 --> 01:12:09.790 in a line or two of algebra, you can turn this into-- 01:12:22.350 --> 01:12:28.625 and using-- well, I didn't even really do 01:12:28.625 --> 01:12:30.250 anything that sophisticated, I can just 01:12:30.250 --> 01:12:32.560 integrate up both sides. 01:12:32.560 --> 01:12:36.550 And what you see is that rho normalized 01:12:36.550 --> 01:12:38.770 to some initial time, some initial value. 01:12:47.820 --> 01:12:51.360 It is very simply related to the behavior of the scale factor. 01:12:54.198 --> 01:12:55.150 OK? 01:12:55.150 --> 01:12:57.330 But if you like, you can set a0 equal to 1, 01:12:57.330 --> 01:13:00.330 if you make that your stuff now, and this gives you 01:13:00.330 --> 01:13:02.640 a simple relationship that allows 01:13:02.640 --> 01:13:06.390 you to see how matter behaves as the large-scale structure 01:13:06.390 --> 01:13:08.010 of the universe changes. 01:13:08.010 --> 01:13:10.810 Let's look at a couple examples of how this behaves. 01:13:44.890 --> 01:13:48.655 So I'm going to call my first category matter. 01:13:51.874 --> 01:13:53.970 OK, that's pretty broad. 01:13:53.970 --> 01:13:57.100 When a cosmologist speaks of matter, generally what 01:13:57.100 --> 01:14:02.260 they are thinking of, this is stuff for which W equals 0. 01:14:02.260 --> 01:14:06.690 So this is something that is pressureless. 01:14:06.690 --> 01:14:10.320 And we talked about pressureless stuff very early in this class. 01:14:10.320 --> 01:14:11.945 This is what we call dust. 01:14:15.423 --> 01:14:17.340 So what we're talking about here is a universe 01:14:17.340 --> 01:14:21.330 that is filled with dust, which seems kind of stupid 01:14:21.330 --> 01:14:22.560 at first approximation, OK? 01:14:22.560 --> 01:14:25.290 Our universe sure as hell doesn't look like dust. 01:14:25.290 --> 01:14:27.982 But bear in mind, what we really mean about this is, 01:14:27.982 --> 01:14:29.190 go back to this pressureless. 01:14:29.190 --> 01:14:30.960 We're just referring to something 01:14:30.960 --> 01:14:36.480 that is sufficiently non-interactive, that when 01:14:36.480 --> 01:14:40.750 particles basically do not interact with each other. 01:14:40.750 --> 01:14:43.330 Our typical dust particle is going 01:14:43.330 --> 01:14:48.250 to actually be something on the scale of a galaxy. 01:14:48.250 --> 01:14:54.660 On cosmological scales, matter-matter interactions 01:14:54.660 --> 01:14:58.120 are, in fact, quite weak. 01:14:58.120 --> 01:15:00.663 So this is a very, very good description. 01:15:09.970 --> 01:15:11.470 Sort of imagine the universe is kind 01:15:11.470 --> 01:15:13.870 of a gas of galaxies and galaxy clusters, 01:15:13.870 --> 01:15:17.140 it's a pressureless gas of galaxy and galaxy clusters. 01:15:17.140 --> 01:15:19.270 When you put all this together-- so let's 01:15:19.270 --> 01:15:22.900 take a look at this form here. 01:15:22.900 --> 01:15:29.830 The density of matter looks like I'm going to set a0 equal to 1, 01:15:29.830 --> 01:15:38.600 it looks like the density now times a to the minus 3. 01:15:38.600 --> 01:15:39.100 OK? 01:15:39.100 --> 01:15:40.475 What I've done is I've just taken 01:15:40.475 --> 01:15:45.070 that evolution law there and I have plugged in Wi equals 0. 01:15:49.550 --> 01:15:55.580 What this is basically saying is that the conservation of stress 01:15:55.580 --> 01:15:59.760 energy demands that the-- 01:15:59.760 --> 01:16:01.970 excuse me-- that the density of this matter 01:16:01.970 --> 01:16:11.170 changes in such a way that the number of dust particles 01:16:11.170 --> 01:16:23.600 is constant, but their density varies as a to the minus 3. 01:16:23.600 --> 01:16:25.490 a sets all of my length scales. 01:16:25.490 --> 01:16:27.050 If I make the universe twice as big, 01:16:27.050 --> 01:16:29.180 the density will be 1/8 as large. 01:16:33.500 --> 01:16:36.310 Your second species of matter that your cosmologist often 01:16:36.310 --> 01:16:39.190 likes to worry-- or second species of stuff 01:16:39.190 --> 01:16:42.560 that your cosmologist likes to worry about is radiation. 01:16:42.560 --> 01:16:44.680 Here, just go back to Stat Mech. 01:16:44.680 --> 01:16:50.350 If you have a gas of photons, it exerts photon pressure, 01:16:50.350 --> 01:16:51.880 and that is of the form-- 01:16:54.990 --> 01:17:01.730 the radiation pressure is 1/3 of the energy density. 01:17:01.730 --> 01:17:04.170 Factors of speed of light are being omitted here. 01:17:04.170 --> 01:17:07.796 So this corresponds to a law in which-- 01:17:07.796 --> 01:17:09.210 here, let me put it this way. 01:17:09.210 --> 01:17:10.400 I should've make this an m. 01:17:10.400 --> 01:17:13.460 So this is my i equals m for matter. 01:17:13.460 --> 01:17:17.190 So my W for radiation is 1/3. 01:17:17.190 --> 01:17:21.950 And what you find in this case is that rho 01:17:21.950 --> 01:17:25.430 of radiation scales with the scale 01:17:25.430 --> 01:17:29.210 factor to the fourth power. 01:17:29.210 --> 01:17:31.070 What's going on here? 01:17:31.070 --> 01:17:33.590 Well let's imagine that the scale factor increases 01:17:33.590 --> 01:17:36.500 by a factor of 2, OK? 01:17:36.500 --> 01:17:39.720 Imagine that the number of photons is not changing. 01:17:39.720 --> 01:17:41.720 So what this is basically saying is, 01:17:41.720 --> 01:17:44.270 OK, I get my factor of 8 corresponding 01:17:44.270 --> 01:17:47.110 to the volume increasing by a factor of 8, 01:17:47.110 --> 01:17:50.210 but I have an additional factor of 2, where's that coming from? 01:17:50.210 --> 01:17:52.140 Well remember, that's an energy density. 01:17:52.140 --> 01:17:55.310 So this is saying that not only is the density being diluted 01:17:55.310 --> 01:17:58.670 by the volume growing, but each packet of energy 01:17:58.670 --> 01:18:03.380 is also getting smaller as the universe increases in size. 01:18:06.060 --> 01:18:09.230 Each quantum of radiation is redshifting. 01:18:29.740 --> 01:18:31.690 It's redshifting with a scale factor a. 01:18:34.610 --> 01:18:37.690 We are going to revisit that in the next lecture. 01:18:37.690 --> 01:18:39.150 That's an important point and we're 01:18:39.150 --> 01:18:49.060 going to re-derive that result somewhat more rigorously as we 01:18:49.060 --> 01:18:51.970 began exploring how it is that we can observational it probe 01:18:51.970 --> 01:18:54.118 the properties of our universe. 01:18:57.190 --> 01:19:00.760 Just for fun, there's another form of-- 01:19:00.760 --> 01:19:02.620 there's another kind of perfect fluid 01:19:02.620 --> 01:19:06.650 that cosmologists worry about, and that's 01:19:06.650 --> 01:19:07.760 the cosmological constant. 01:19:17.013 --> 01:19:21.710 So a cosmological constant has pressure 01:19:21.710 --> 01:19:26.880 equal to minus the density. 01:19:26.880 --> 01:19:28.890 This corresponds to an equation of state 01:19:28.890 --> 01:19:33.540 parameter W equal to minus 1. 01:19:38.980 --> 01:19:43.100 If I go to my form here, I plug in W equals minus 1, 01:19:43.100 --> 01:19:48.855 rho goes as a to the 0th power, a constant. 01:19:53.232 --> 01:19:55.190 Well, it is a cosmological constant, after all, 01:19:55.190 --> 01:19:57.970 so that shouldn't be too surprising. 01:19:57.970 --> 01:20:01.170 This is a very interesting one because it is basically 01:20:01.170 --> 01:20:07.430 telling us that the amount of energy in spacial slices-- 01:20:07.430 --> 01:20:08.970 the energy density does not change. 01:20:08.970 --> 01:20:12.288 The amount of energy appears to be Increasing. 01:20:12.288 --> 01:20:14.580 Now bear in mind, it's hard to define the total energy, 01:20:14.580 --> 01:20:17.670 we cannot really in a covariant way add up energy at various 01:20:17.670 --> 01:20:19.440 different kinds of points. 01:20:19.440 --> 01:20:21.967 Local energy is still being conserved, 01:20:21.967 --> 01:20:24.300 but there's no question that this guy is a little weird. 01:20:28.610 --> 01:20:32.440 So one of the next things that we want to do 01:20:32.440 --> 01:20:36.970 is take this stuff, run it through Einstein's equations. 01:20:36.970 --> 01:20:38.680 Einstein's equations, of course, give us 01:20:38.680 --> 01:20:40.540 the Friedmann equations. 01:20:40.540 --> 01:20:43.420 And solve to see what the expansion the universe 01:20:43.420 --> 01:20:43.960 looks like. 01:20:43.960 --> 01:20:48.040 We saw already that if the density of the universe 01:20:48.040 --> 01:20:51.640 relative the critical density is either higher, the same, 01:20:51.640 --> 01:20:54.880 or lower, that tells us about the value of this k parameter, 01:20:54.880 --> 01:20:56.650 or rather, the kappa parameter. 01:20:56.650 --> 01:21:01.480 We haven't yet seen how to solve for the scale factor a. 01:21:05.030 --> 01:21:07.430 However, we have the two Friedmann equations, 01:21:07.430 --> 01:21:12.860 and if nothing else, write them down, write out your stuff, 01:21:12.860 --> 01:21:15.620 you got yourself a system of equations, 01:21:15.620 --> 01:21:19.370 Odin gave us mathematica-- attack. 01:21:19.370 --> 01:21:21.590 To give you some intuition as to what you end up 01:21:21.590 --> 01:21:24.970 seeing when you look at these kind of solutions, 01:21:24.970 --> 01:21:26.845 let me look at the simplest kind of universes 01:21:26.845 --> 01:21:28.820 that we can solve this way. 01:21:28.820 --> 01:21:33.710 So let's examine what I will call a monospecies universe-- 01:21:33.710 --> 01:21:36.650 in other words, a universe that only contains 01:21:36.650 --> 01:21:40.060 one of these forms of matter that I have described here, 01:21:40.060 --> 01:21:42.560 one of these sources of stress energy that I described here. 01:21:46.180 --> 01:21:51.370 And for simplicity, I'm going to take it to be spatially flat. 01:21:58.860 --> 01:22:02.340 Neither of these two conditions are true in general, 01:22:02.340 --> 01:22:03.700 but they are-- 01:22:03.700 --> 01:22:05.760 they're fine for us to wrap our heads 01:22:05.760 --> 01:22:09.310 around what the characteristics of the solutions look like. 01:22:09.310 --> 01:22:20.240 So in this limit, Friedmann 1 becomes a dot over a squared 01:22:20.240 --> 01:22:26.930 equals 8 pi g rho over 3. 01:22:26.930 --> 01:22:33.800 I can borrow this form that I've got here to write this as 8 01:22:33.800 --> 01:22:40.280 pi over 3 rho at some particular moment times scale 01:22:40.280 --> 01:22:42.600 factor to the minus n. 01:22:42.600 --> 01:22:45.230 So what I'm doing here is I'm assuming a0 now. 01:22:45.230 --> 01:22:48.260 My a right now is 1, and I'm defining n 01:22:48.260 --> 01:23:01.265 to be 3 times 1 plus W. 01:23:01.265 --> 01:23:02.640 This is easy enough to solve for. 01:23:18.590 --> 01:23:19.530 OK. 01:23:19.530 --> 01:23:20.450 Take the square root. 01:23:26.630 --> 01:23:28.382 What you find is-- 01:23:28.382 --> 01:23:29.840 I'm going to just sort of-- there's 01:23:29.840 --> 01:23:32.510 a constant you guys can work out on your own if you like. 01:23:32.510 --> 01:23:42.960 a dot must be proportional to a 1 minus n over 2, 01:23:42.960 --> 01:23:50.190 or a is proportional to t to the 2 over n. 01:23:57.360 --> 01:23:59.250 n equals 0 is a special case that we'll 01:23:59.250 --> 01:24:01.092 talk about in just a moment. 01:24:01.092 --> 01:24:02.550 If we are dealing with what we call 01:24:02.550 --> 01:24:13.620 a matter-dominated universe, well, 01:24:13.620 --> 01:24:17.820 in this sort of monospecies, spatially flat form, 01:24:17.820 --> 01:24:21.070 this would have W equal 0, n equals 01:24:21.070 --> 01:24:28.110 3, and a scale factor that grows as t to the 2/3 power. 01:24:28.110 --> 01:24:31.530 A matter-dominated universe is one that expands, 01:24:31.530 --> 01:24:34.050 but it expands with this kind of a loss. 01:24:34.050 --> 01:24:37.120 So it slows down with time. 01:24:37.120 --> 01:24:50.140 A radiation-dominated universe, W equals 1/3, n 01:24:50.140 --> 01:25:02.720 equals 4, that is a universe that expands 01:25:02.720 --> 01:25:03.720 as the square root of t. 01:25:09.030 --> 01:25:10.760 What about my cosmological constant? 01:25:10.760 --> 01:25:11.260 Ah. 01:25:11.260 --> 01:25:12.385 OK, well that's a problem-- 01:25:15.030 --> 01:25:19.050 n equals 0, and my solution doesn't work for that one. 01:25:36.820 --> 01:25:38.350 So what you do is just-- 01:25:38.350 --> 01:25:44.762 let's just go back to our F or W equations. 01:25:44.762 --> 01:25:46.220 Or rather, our Friedmann equations. 01:25:46.220 --> 01:25:48.980 Let's write F1 down again. 01:25:48.980 --> 01:25:54.410 I have a dot over a equals 8 pi-- 01:25:54.410 --> 01:26:02.075 whoops-- 8 pi g rho over 3, and this is now a constant. 01:26:08.460 --> 01:26:11.310 Rho equals rho 0 because it's a constant. 01:26:11.310 --> 01:26:13.980 I can rewrite this in terms of the cosmological constant 01:26:13.980 --> 01:26:16.070 lambda. 01:26:16.070 --> 01:26:20.020 And so another way to write this is-- 01:26:20.020 --> 01:26:22.840 sorry about that-- a dot a-- a dot over a squared 01:26:22.840 --> 01:26:30.420 equals this, which equals lambda over 3. 01:26:33.800 --> 01:26:43.715 So this leads to an exponential solution. 01:26:48.270 --> 01:26:52.590 Now our real universe is not as simple as these three 01:26:52.590 --> 01:26:55.440 illustrative cases that I have put in here just 01:26:55.440 --> 01:26:57.660 to illustrate what the extremes look like, OK? 01:26:57.660 --> 01:27:01.110 We are not a monospecies universe, 01:27:01.110 --> 01:27:05.370 we have a mixture of matter, we have a mixture of radiation, 01:27:05.370 --> 01:27:07.770 we appear to have something that smells 01:27:07.770 --> 01:27:09.660 a lot like a cosmological constant, 01:27:09.660 --> 01:27:14.580 although the jury is still out if one is being perfectly fair. 01:27:14.580 --> 01:27:17.410 Work is ongoing. 01:27:17.410 --> 01:27:20.830 What you need to do in general is sort of model things. 01:27:20.830 --> 01:27:24.790 You try to make models of the universe that correspond 01:27:24.790 --> 01:27:28.540 to different mixtures of things that can go into it, 01:27:28.540 --> 01:27:32.770 and then you go through and ask yourself, 01:27:32.770 --> 01:27:36.280 do the observables that emerge in this universe match what 01:27:36.280 --> 01:27:38.860 we see? 01:27:38.860 --> 01:27:40.660 Generally you will see sort of trends 01:27:40.660 --> 01:27:42.580 that are similar to this that emerge, right? 01:27:42.580 --> 01:27:44.500 There might be a particular epoch where matter 01:27:44.500 --> 01:27:45.730 is more important, there might be 01:27:45.730 --> 01:27:47.500 an epoch where radiation is more important, 01:27:47.500 --> 01:27:49.750 there might be an epoch where cosmological constant is 01:27:49.750 --> 01:27:50.540 more important. 01:27:50.540 --> 01:27:52.810 And so you might see sort of you know transitions 01:27:52.810 --> 01:27:55.210 between these things where it's mostly a square root t 01:27:55.210 --> 01:27:58.240 expansion, and then something happens 01:27:58.240 --> 01:28:00.598 and the radiation becomes less important, 01:28:00.598 --> 01:28:03.140 there's an intermediate regime where both are playing a role, 01:28:03.140 --> 01:28:04.180 and then matter becomes important 01:28:04.180 --> 01:28:06.550 and it kicks over to a t to the 2/3 kind of expansion 01:28:06.550 --> 01:28:09.550 when it becomes matter-dominated. 01:28:09.550 --> 01:28:11.830 You don't want to assume the universe is flat, 01:28:11.830 --> 01:28:14.530 you need to do your analysis, including a non-zero flatness 01:28:14.530 --> 01:28:16.655 parameter in there, which makes things a little bit 01:28:16.655 --> 01:28:17.560 complicated. 01:28:17.560 --> 01:28:20.560 So in the next lecture, I am going 01:28:20.560 --> 01:28:23.470 to talk a little bit about how one extracts observables 01:28:23.470 --> 01:28:24.670 from these spacetimes. 01:28:24.670 --> 01:28:31.450 How is it that we are able to actually go into an FRW 01:28:31.450 --> 01:28:35.145 universe and measure things-- what can we measure, 01:28:35.145 --> 01:28:36.520 how can we use those measurements 01:28:36.520 --> 01:28:39.280 to learn about the energy budget of our universe 01:28:39.280 --> 01:28:42.880 and formulate cosmology as an observational and physical 01:28:42.880 --> 01:28:44.200 science? 01:28:44.200 --> 01:28:47.640 And with that, I will end this lecture.