1 00:00:00,500 --> 00:00:01,940 [SQUEAKING] 2 00:00:01,940 --> 00:00:04,365 [RUSTLING] 3 00:00:04,365 --> 00:00:07,275 [CLICKING] 4 00:00:10,680 --> 00:00:13,320 SCOTT HUGHES: So in the lectures I'm going to record today, 5 00:00:13,320 --> 00:00:15,570 we're going to conclude cosmology. 6 00:00:15,570 --> 00:00:19,710 And then we're going to begin talking about a another system 7 00:00:19,710 --> 00:00:22,140 in which we saw the Einstein field equations 8 00:00:22,140 --> 00:00:22,860 using asymmetry. 9 00:00:22,860 --> 00:00:25,320 So this is one where we will again 10 00:00:25,320 --> 00:00:27,622 consider systems that are spherically symmetric 11 00:00:27,622 --> 00:00:29,580 but we're going to consider them to be compact. 12 00:00:29,580 --> 00:00:31,080 In other words, things that are sort 13 00:00:31,080 --> 00:00:33,003 of, rather than filling the whole universe, 14 00:00:33,003 --> 00:00:34,920 the spacetime that arises from a source that's 15 00:00:34,920 --> 00:00:38,220 localized in some particular region in space. 16 00:00:38,220 --> 00:00:39,990 So let me just do a quick recap of what 17 00:00:39,990 --> 00:00:45,270 we did in the previous lecture that I recorded. 18 00:00:45,270 --> 00:00:49,920 So by arguing solely from the basis of how 19 00:00:49,920 --> 00:00:56,010 to make a spacetime that is as symmetric as possible in space 20 00:00:56,010 --> 00:00:59,340 but has a time asymmetry, so the past and the future 21 00:00:59,340 --> 00:01:03,840 look different, we came up with the Robertson-Walker metric, 22 00:01:03,840 --> 00:01:06,777 which I will call the RW metric, which has the form I've written 23 00:01:06,777 --> 00:01:07,860 there on the top one line. 24 00:01:07,860 --> 00:01:09,900 There were actually several forms for this. 25 00:01:09,900 --> 00:01:13,320 I've actually written down two variants of it on the board 26 00:01:13,320 --> 00:01:15,180 here. 27 00:01:15,180 --> 00:01:17,100 Key thing which I want to highlight 28 00:01:17,100 --> 00:01:19,500 is that there is, in the way I've written it 29 00:01:19,500 --> 00:01:23,000 on the first line, there's a hidden scale in there, r zero, 30 00:01:23,000 --> 00:01:25,800 which you can think of as, essentially, 31 00:01:25,800 --> 00:01:27,900 setting an overall length scale to everything 32 00:01:27,900 --> 00:01:29,850 that could be measured. 33 00:01:29,850 --> 00:01:33,210 There is a parameter, either k or kappa. 34 00:01:33,210 --> 00:01:38,730 k is either minus 1, 0, or 1. 35 00:01:38,730 --> 00:01:43,170 Kappa is just k with a factor of that length scale squared 36 00:01:43,170 --> 00:01:44,470 thrown in there. 37 00:01:44,470 --> 00:01:48,180 We define the overall scale factor to be-- 38 00:01:48,180 --> 00:01:49,360 it's a number. 39 00:01:49,360 --> 00:01:53,100 And we define it so that its value is 1 right now. 40 00:01:53,100 --> 00:01:56,790 And then everything is scaled to the way distances look 41 00:01:56,790 --> 00:01:59,730 at the present time. 42 00:01:59,730 --> 00:02:01,117 A second form of that-- 43 00:02:01,117 --> 00:02:03,450 excuse me-- which is essentially the change of variables 44 00:02:03,450 --> 00:02:07,620 is you change your radial coordinates 45 00:02:07,620 --> 00:02:10,880 to this parameter chi. 46 00:02:10,880 --> 00:02:13,830 And depending on the value of k, what 47 00:02:13,830 --> 00:02:16,200 you find then is that the relationship 48 00:02:16,200 --> 00:02:23,340 between the radius r and the coordinate chi, if k equals 1, 49 00:02:23,340 --> 00:02:25,440 you have what we call a closed universe. 50 00:02:25,440 --> 00:02:31,080 And the radius is equal to the sign of the parameter chi. 51 00:02:31,080 --> 00:02:36,540 If you have an open universe, k equals minus 1, 52 00:02:36,540 --> 00:02:40,881 then the radius is the sinh of that parameter chi. 53 00:02:40,881 --> 00:02:43,890 And for a flat universe, k equals 0, 54 00:02:43,890 --> 00:02:48,050 they are simply 1 is just the other modulo, a choice of units 55 00:02:48,050 --> 00:02:49,980 with the R of 0 there. 56 00:02:49,980 --> 00:02:51,740 All this is just geometry. 57 00:02:51,740 --> 00:02:52,240 OK. 58 00:02:52,240 --> 00:02:55,980 So when you write this down, you are 59 00:02:55,980 --> 00:03:00,060 agnostic about the form of a and you have no information 60 00:03:00,060 --> 00:03:03,720 about the value of k. 61 00:03:03,720 --> 00:03:07,170 So to get more insight into what's going on, 62 00:03:07,170 --> 00:03:08,800 you need to couple this to your source. 63 00:03:08,800 --> 00:03:10,650 And so we take these things. 64 00:03:10,650 --> 00:03:12,660 And using the Einstein field equations, 65 00:03:12,660 --> 00:03:18,540 you equate these to a perfect fluid stress energy tensor. 66 00:03:18,540 --> 00:03:23,130 What pops out of that are a pair of equations 67 00:03:23,130 --> 00:03:25,470 that arise in the Einstein field equations. 68 00:03:25,470 --> 00:03:26,880 I call these F1 and F2. 69 00:03:26,880 --> 00:03:29,100 These are the two Friedmann equations. 70 00:03:29,100 --> 00:03:32,455 F1 tells me about the velocity associated with that expansion 71 00:03:32,455 --> 00:03:32,955 premise. 72 00:03:32,955 --> 00:03:35,490 There's an a dot divided by a squared. 73 00:03:35,490 --> 00:03:38,550 We call that the Hubble parameter, h of a squared. 74 00:03:38,550 --> 00:03:41,880 And it's related to the density of the source of your universe 75 00:03:41,880 --> 00:03:43,580 as well as this kappa term. 76 00:03:43,580 --> 00:03:44,100 OK. 77 00:03:44,100 --> 00:03:48,660 And as we saw last time, we can get some information 78 00:03:48,660 --> 00:03:51,670 about kappa from this equation. 79 00:03:51,670 --> 00:03:55,110 The second Friedmann equation relates the acceleration 80 00:03:55,110 --> 00:03:57,300 of this expansion term, a double dot-- 81 00:03:57,300 --> 00:03:59,430 dot, by the way, d by dy. 82 00:03:59,430 --> 00:04:02,490 A double dot divided by a is simply 83 00:04:02,490 --> 00:04:05,730 related to a quantity that is the density plus 3 84 00:04:05,730 --> 00:04:08,430 times the pressure of this perfect fluid that makes up 85 00:04:08,430 --> 00:04:10,110 our universe. 86 00:04:10,110 --> 00:04:14,220 We also find by requiring that local energy conservation be 87 00:04:14,220 --> 00:04:16,470 held, in other words that your stress energy tensor be 88 00:04:16,470 --> 00:04:19,500 divergence free, we have a constraint that 89 00:04:19,500 --> 00:04:21,899 relates the amount of energy-- 90 00:04:21,899 --> 00:04:24,450 the rate of change of energy in a fiducial volume-- 91 00:04:24,450 --> 00:04:28,200 to the negative pressure times the rate of change 92 00:04:28,200 --> 00:04:29,910 of that fiducial volume. 93 00:04:29,910 --> 00:04:32,220 And this, as I discussed in the last lecture, 94 00:04:32,220 --> 00:04:34,830 is essentially nothing more than the first law 95 00:04:34,830 --> 00:04:36,030 of thermodynamics. 96 00:04:36,030 --> 00:04:37,920 It's written up in fancy language 97 00:04:37,920 --> 00:04:41,055 appropriate to a cosmological spacetime. 98 00:04:41,055 --> 00:04:42,930 As we move forward, we find it useful to make 99 00:04:42,930 --> 00:04:45,090 a couple of definitions. 100 00:04:45,090 --> 00:04:48,480 So if you divide the Hubble parameter squared 101 00:04:48,480 --> 00:04:50,160 by Newton's gravitational constant, 102 00:04:50,160 --> 00:04:53,320 that's got the dimensions of density. 103 00:04:53,320 --> 00:04:56,030 And so we're going to define a critical density to be 3h 104 00:04:56,030 --> 00:04:57,840 squared over 8 pi g. 105 00:04:57,840 --> 00:05:02,850 And we're going to define density parameters, omega, 106 00:05:02,850 --> 00:05:05,350 as the actual physical density is normalized 107 00:05:05,350 --> 00:05:06,900 to that critical density. 108 00:05:06,900 --> 00:05:10,530 And when you do this, you find that the critical-- 109 00:05:10,530 --> 00:05:13,700 the first Friedmann equation can be written as omega-- 110 00:05:13,700 --> 00:05:15,470 oh, that's a typo-- 111 00:05:15,470 --> 00:05:21,070 omega plus omega curvature equals 112 00:05:21,070 --> 00:05:26,410 1 where omega curvature-- pardon the typo here. 113 00:05:26,410 --> 00:05:31,420 Omega curvature is not actually related to identity 114 00:05:31,420 --> 00:05:34,630 but it sort of plays one in this equation. 115 00:05:34,630 --> 00:05:38,170 It is just a parameter that has the proper dimensions 116 00:05:38,170 --> 00:05:42,160 to do what is necessary to fit this equation. 117 00:05:42,160 --> 00:05:46,540 And it only depends on what the curvature parameter is. 118 00:05:46,540 --> 00:05:49,840 So remember that this kappa is essentially 119 00:05:49,840 --> 00:05:53,480 minus 1, 0, or 1 module of factor of my overall scale. 120 00:05:53,480 --> 00:05:55,840 So this is either a positive number, 0, 121 00:05:55,840 --> 00:05:58,880 or a negative number. 122 00:05:58,880 --> 00:05:59,650 All right. 123 00:05:59,650 --> 00:06:02,470 So let's carry things forward from here. 124 00:06:02,470 --> 00:06:06,520 Mute my computer so I'm not distracted by things coming in. 125 00:06:11,830 --> 00:06:15,250 We now have everything we need using this framework 126 00:06:15,250 --> 00:06:18,130 to build a universe. 127 00:06:18,130 --> 00:06:21,420 Let's write down the recipe to build a universe, 128 00:06:21,420 --> 00:06:23,934 or I should say to build a model of the universe. 129 00:06:32,830 --> 00:06:37,823 So first thing you do is pick your spatial curvature. 130 00:06:50,880 --> 00:06:57,740 So pick the parameter k to be minus 1, 0, or 1. 131 00:07:02,940 --> 00:07:19,590 Pick a mixture of species that contribute 132 00:07:19,590 --> 00:07:24,250 to the energy density budget of your model universe. 133 00:07:24,250 --> 00:07:27,480 So what you would say is that the total density of things 134 00:07:27,480 --> 00:07:31,650 in your universe is a sum over whatever mixture of stuff 135 00:07:31,650 --> 00:07:33,098 is in your universe. 136 00:07:37,490 --> 00:07:40,340 You will find it helpful to specify an equation of state 137 00:07:40,340 --> 00:07:41,750 for each species. 138 00:07:41,750 --> 00:07:43,823 Cosmologists typically choose equation of state 139 00:07:43,823 --> 00:07:44,990 that has the following form. 140 00:08:00,890 --> 00:08:01,390 OK. 141 00:08:01,390 --> 00:08:03,070 So you require your species-- 142 00:08:03,070 --> 00:08:07,060 if you follow this model that most cosmologists use, 143 00:08:07,060 --> 00:08:08,800 each species will have a pressure 144 00:08:08,800 --> 00:08:10,590 that is linear in the density. 145 00:08:21,220 --> 00:08:25,950 If you do choose that form, then when 146 00:08:25,950 --> 00:08:31,590 you enforce local conservation of energy, what you will then 147 00:08:31,590 --> 00:08:45,860 find is that for every one of your species, 148 00:08:45,860 --> 00:08:50,567 there is a simple relationship. 149 00:08:50,567 --> 00:08:52,150 There's a simple differential equation 150 00:08:52,150 --> 00:08:54,280 that governs how that species evolves 151 00:08:54,280 --> 00:08:56,680 as the scale factor changes. 152 00:08:56,680 --> 00:09:02,920 This can be immediately integrated up 153 00:09:02,920 --> 00:09:06,010 to find that the amount of, at some particular moment, 154 00:09:06,010 --> 00:09:09,400 the density of species i depends on how it looks. 155 00:09:09,400 --> 00:09:12,440 So 0 again denotes now. 156 00:09:12,440 --> 00:09:16,930 It is simply proportional to some power of the scale factor 157 00:09:16,930 --> 00:09:27,270 where the power that enters here can be simply calculated 158 00:09:27,270 --> 00:09:29,740 given that equation of state parameter. 159 00:09:29,740 --> 00:09:38,210 Once you have these things together, you're ready to roll. 160 00:09:38,210 --> 00:09:39,710 You've got your Friedmann equations. 161 00:09:39,710 --> 00:09:42,230 You've got all the constraints and information 162 00:09:42,230 --> 00:09:45,680 you need to dig into these equations. 163 00:09:45,680 --> 00:09:48,820 Sit down, make your models, have a party. 164 00:09:48,820 --> 00:09:52,180 Now what we really want to do-- 165 00:09:52,180 --> 00:09:52,680 OK. 166 00:09:52,680 --> 00:09:55,050 We are physicists. 167 00:09:55,050 --> 00:09:57,690 And our goal in doing these models 168 00:09:57,690 --> 00:10:02,310 is to come up with some kind of a description of the universe 169 00:10:02,310 --> 00:10:04,530 and compare it with our data so that we 170 00:10:04,530 --> 00:10:07,020 can see what is the nature of the universe 171 00:10:07,020 --> 00:10:09,220 that we actually live in. 172 00:10:09,220 --> 00:10:09,720 OK. 173 00:10:09,720 --> 00:10:18,490 So what is of interest to us is how does varying all 174 00:10:18,490 --> 00:10:28,313 these different terms change-- 175 00:10:28,313 --> 00:10:30,480 well, we're going to talk about various observables. 176 00:10:30,480 --> 00:10:32,850 But really, if you think about this model, 177 00:10:32,850 --> 00:10:35,250 how does it change the evolution of the scale factor? 178 00:10:43,240 --> 00:10:44,605 Everything is bound up in that. 179 00:10:44,605 --> 00:10:46,340 That is the key thing. 180 00:10:46,340 --> 00:10:46,840 OK. 181 00:10:46,840 --> 00:10:49,872 So if I can make a mathematical model that 182 00:10:49,872 --> 00:10:52,330 describes a universe with all these various different kinds 183 00:10:52,330 --> 00:10:56,968 of ingredients, yeah, I can sit down and make-- 184 00:10:56,968 --> 00:10:58,510 if I make a really complicated thing, 185 00:10:58,510 --> 00:11:01,150 I probably can't do this analytically so that's fine. 186 00:11:01,150 --> 00:11:02,740 I will make a differential equation 187 00:11:02,740 --> 00:11:05,530 integrator that solves all these coupled differential equations. 188 00:11:05,530 --> 00:11:10,030 And I will then make predictions for how AFT evolves depending 189 00:11:10,030 --> 00:11:12,190 upon what the different mixtures of species are, 190 00:11:12,190 --> 00:11:14,890 what the curvature term is equal to, 191 00:11:14,890 --> 00:11:18,140 all those together, and make my model. 192 00:11:18,140 --> 00:11:20,810 But my goal as a physicist is then to compare this to data. 193 00:11:20,810 --> 00:11:25,240 And so what I need to do is to come up with some kind of a way 194 00:11:25,240 --> 00:11:28,750 this will then be useful. 195 00:11:28,750 --> 00:11:47,630 We then need some kind of an observational surrogate 196 00:11:47,630 --> 00:11:49,210 for a of t. 197 00:11:49,210 --> 00:11:50,960 What I would like to be able to do is say, 198 00:11:50,960 --> 00:11:54,230 great, model A predicts the following evolution 199 00:11:54,230 --> 00:11:55,070 of the scale factor. 200 00:11:55,070 --> 00:11:58,370 Model B predicts this evolution of the scale factor. 201 00:11:58,370 --> 00:12:00,560 Can I look at the universe and deduce 202 00:12:00,560 --> 00:12:04,110 whether we are closer to model A or closer to Model B? 203 00:12:04,110 --> 00:12:05,960 And in order to do that, I need to know 204 00:12:05,960 --> 00:12:08,007 how do I measure a of t. 205 00:12:08,007 --> 00:12:09,590 And as we're going to see, this really 206 00:12:09,590 --> 00:12:13,500 boils down to two things. 207 00:12:13,500 --> 00:12:16,860 I need to be able to deduce if I look 208 00:12:16,860 --> 00:12:19,710 at some event in the universe, if I look at something, 209 00:12:19,710 --> 00:12:23,010 I want to know what scale factor to associate with that. 210 00:12:23,010 --> 00:12:24,540 I need to measure a. 211 00:12:24,540 --> 00:12:28,580 And I need to know what t to label that a with. 212 00:12:28,580 --> 00:12:33,240 So this kind of sounds like I'm reading 213 00:12:33,240 --> 00:12:34,850 from the journal of duh. 214 00:12:34,850 --> 00:12:37,350 But if I want to do this, what it basically boils down to is 215 00:12:37,350 --> 00:12:40,980 I need to know how to associate an a with things that I measure 216 00:12:40,980 --> 00:12:43,747 and how to associate the t with the things that I measure. 217 00:12:43,747 --> 00:12:45,580 That's what we're going to talk about today. 218 00:12:45,580 --> 00:12:48,360 What are the actual observational surrogates, 219 00:12:48,360 --> 00:12:52,590 the ways in which we can go out, point telescopes 220 00:12:52,590 --> 00:12:54,960 and other instruments at things in the sky, 221 00:12:54,960 --> 00:12:58,530 and deduce what a of t is for the kind of events, 222 00:12:58,530 --> 00:13:00,690 the kind of things that we are going to measure? 223 00:13:05,890 --> 00:13:08,550 Let's talk about how I can measure the scale factor first. 224 00:13:08,550 --> 00:13:08,730 OK. 225 00:13:08,730 --> 00:13:10,605 So let's ignore the fact there's a t on here. 226 00:13:10,605 --> 00:13:12,060 How can I measure a? 227 00:13:12,060 --> 00:13:16,440 We saw a hint of this in the previous lecture 228 00:13:16,440 --> 00:13:18,150 that I recorded. 229 00:13:18,150 --> 00:13:21,770 So recall-- pardon me. 230 00:13:21,770 --> 00:13:27,410 I've got a bit of extra junk here in my notes. 231 00:13:27,410 --> 00:13:35,010 Recall that in my previous lecture, 232 00:13:35,010 --> 00:13:38,890 we looked at the way different kinds of densities behaved. 233 00:13:38,890 --> 00:13:40,600 But I've got the results right here. 234 00:13:40,600 --> 00:13:46,100 For matter which had an equation state parameter of 0, 235 00:13:46,100 --> 00:13:52,030 what we found was that the density associated 236 00:13:52,030 --> 00:13:54,850 with that matter, it just fell as the scale factor 237 00:13:54,850 --> 00:13:56,440 to the inverse third power. 238 00:13:56,440 --> 00:13:59,020 That's essentially saying that the number of particles 239 00:13:59,020 --> 00:14:01,360 of stuff is constant. 240 00:14:01,360 --> 00:14:04,780 And so as the universe expands, it just 241 00:14:04,780 --> 00:14:07,030 goes with the volume of the universe. 242 00:14:07,030 --> 00:14:10,420 If it was radiation, we found it went as a scale factor 243 00:14:10,420 --> 00:14:12,460 to the inverse fourth power. 244 00:14:12,460 --> 00:14:14,980 And that's consistent with diluting the density 245 00:14:14,980 --> 00:14:19,600 as the volume gets larger provided we also decrease 246 00:14:19,600 --> 00:14:23,530 the energy per particle of radiation per photon, 247 00:14:23,530 --> 00:14:26,980 per graviton, per whateveron. 248 00:14:26,980 --> 00:14:32,650 If I require that the energy per quantum of radiation 249 00:14:32,650 --> 00:14:40,150 is redshifted with this thing, that explains the density flaw 250 00:14:40,150 --> 00:14:42,100 that we found for radiation. 251 00:14:42,100 --> 00:14:45,250 And so that sort of suggests that what we're going to find 252 00:14:45,250 --> 00:14:47,080 is that the scale factor is directly 253 00:14:47,080 --> 00:14:49,330 tied to a redshift measure. 254 00:14:49,330 --> 00:14:50,737 OK. 255 00:14:50,737 --> 00:14:52,570 I just realized what this page of notes was. 256 00:14:52,570 --> 00:14:53,200 My apologies. 257 00:14:53,200 --> 00:14:54,617 I'm getting myself organized here. 258 00:14:59,642 --> 00:15:01,600 Let's make that a little bit more rigorous now. 259 00:15:01,600 --> 00:15:02,100 OK. 260 00:15:02,100 --> 00:15:06,120 So that argument on the basis of how the density of radiation 261 00:15:06,120 --> 00:15:06,620 behaves. 262 00:15:06,620 --> 00:15:08,200 It's not a bad one as a first pass. 263 00:15:08,200 --> 00:15:10,270 It is quite indicative. 264 00:15:10,270 --> 00:15:12,400 But let's come at it from another point of view. 265 00:15:12,400 --> 00:15:18,840 And this allows me to introduce in a brief 266 00:15:18,840 --> 00:15:25,690 aside a topic that is quite useful here. 267 00:15:25,690 --> 00:15:31,210 So we talked about Killing vectors a couple of weeks ago. 268 00:15:31,210 --> 00:15:33,330 Let's now talk about a generalization of this 269 00:15:33,330 --> 00:15:35,760 known as Killing tensors. 270 00:15:35,760 --> 00:15:38,130 So recall that a Killing vector was 271 00:15:38,130 --> 00:15:42,210 defined as a particular vector in my space time 272 00:15:42,210 --> 00:15:45,810 manifold such that if I Lie transport 273 00:15:45,810 --> 00:15:50,040 the metric along that Killing vector, I get 0. 274 00:15:50,040 --> 00:15:55,110 This then leads to the statement that if I put together 275 00:15:55,110 --> 00:16:03,870 the symmetrized covariant gradient of the Killing vector, 276 00:16:03,870 --> 00:16:06,820 I get 0. 277 00:16:06,820 --> 00:16:09,300 Another way to write this is to use this notation. 278 00:16:09,300 --> 00:16:09,800 Whoops. 279 00:16:17,122 --> 00:16:18,130 OK. 280 00:16:18,130 --> 00:16:20,230 So these are equations that tell us about the way 281 00:16:20,230 --> 00:16:23,480 that Killing vectors behave. 282 00:16:23,480 --> 00:16:26,020 A Killing tensor is just a generalization 283 00:16:26,020 --> 00:16:31,090 of this idea to an object that has more than one index 284 00:16:31,090 --> 00:16:37,090 line, to a higher rank tensorial object. 285 00:16:37,090 --> 00:16:41,720 So we consider this to be a rank one Killing tensor. 286 00:16:41,720 --> 00:16:54,610 A rank n Killing tensor satisfies-- 287 00:16:59,310 --> 00:17:01,230 so let's say k is my Killing tensor. 288 00:17:06,150 --> 00:17:12,359 Imagine I have n indices here if I 289 00:17:12,359 --> 00:17:16,020 take the covariant gradient of that Killing tensor 290 00:17:16,020 --> 00:17:19,079 and I symmetrize over all n indices. 291 00:17:19,079 --> 00:17:20,250 That gives me 0. 292 00:17:20,250 --> 00:17:22,020 This defines a Killing tensor. 293 00:17:28,410 --> 00:17:35,920 Starting with this definition, it's not at all 294 00:17:35,920 --> 00:17:58,980 hard to show that if I define a parameter, k, 295 00:17:58,980 --> 00:18:01,680 which is what I get when I contract the Killing 296 00:18:01,680 --> 00:18:06,240 tensor, every one of its indices with the four-velocity 297 00:18:06,240 --> 00:18:07,170 of a geodesic. 298 00:18:23,030 --> 00:18:26,923 If my u satisfies the geodesic equation-- or this could be-- 299 00:18:26,923 --> 00:18:28,215 let's write this as a momentum. 300 00:18:33,660 --> 00:18:37,380 Which you say is the tangent to a world line. 301 00:18:37,380 --> 00:18:39,270 Could be either a velocity or a momentum. 302 00:18:43,900 --> 00:18:47,910 So if I define the scalar k by contracting my Killing 303 00:18:47,910 --> 00:18:52,560 tensor with n copies of tangent to the world line, 304 00:18:52,560 --> 00:19:02,920 and that thing satisfies the geodesic equation, 305 00:19:02,920 --> 00:19:05,320 then the following is true. 306 00:19:05,320 --> 00:19:07,870 You guys did this on a homework exercise 307 00:19:07,870 --> 00:19:12,340 for when we thought about a spacetime-- you did something 308 00:19:12,340 --> 00:19:16,040 similar to this, I should say, for a spacetime containing 309 00:19:16,040 --> 00:19:17,110 an electromagnetic field. 310 00:19:17,110 --> 00:19:19,690 We talked about how this works for the case of a Killing 311 00:19:19,690 --> 00:19:20,908 vector. 312 00:19:20,908 --> 00:19:22,450 Hopefully you can kind of see the way 313 00:19:22,450 --> 00:19:25,480 you would do this calculation at this point. 314 00:19:25,480 --> 00:19:28,060 Now the reason I'm doing this aside 315 00:19:28,060 --> 00:19:31,630 is that if you have a Friedmann-Robertson-Walker 316 00:19:31,630 --> 00:19:37,350 spacetime, search spacetimes actually 317 00:19:37,350 --> 00:19:39,360 have a very useful Killing tensor. 318 00:19:48,300 --> 00:19:53,250 So let's define k with two indices, mu nu. 319 00:19:53,250 --> 00:19:55,710 And this is just given by the scale factor. 320 00:19:58,640 --> 00:20:05,780 Multiplying the metric u mu, u mu, u nu. 321 00:20:05,780 --> 00:20:15,780 Where this u comes from the four-velocity 322 00:20:15,780 --> 00:20:17,190 of a co-moving fluid element. 323 00:20:24,860 --> 00:20:26,440 So this is the four-velocity that we 324 00:20:26,440 --> 00:20:27,970 use to construct the stress energy 325 00:20:27,970 --> 00:20:33,013 tensor that is the source of our Friedmann equations. 326 00:20:42,510 --> 00:20:45,140 So here's how we're going to use this. 327 00:20:45,140 --> 00:20:50,020 Let's look at what we get for this Killing vector. 328 00:20:50,020 --> 00:20:54,710 Excuse me, this Killing tensor when I consider 329 00:20:54,710 --> 00:20:58,130 it's a long a null geodesic. 330 00:20:58,130 --> 00:21:00,890 We're going to want to think about null geodesics a lot, 331 00:21:00,890 --> 00:21:06,980 because the way that we are going to probe our universe 332 00:21:06,980 --> 00:21:07,820 is with radiation. 333 00:21:07,820 --> 00:21:10,070 We're going to look at it with things like telescopes. 334 00:21:15,540 --> 00:21:17,290 These days people are starting to probe it 335 00:21:17,290 --> 00:21:19,207 with things like gravitational wave detectors. 336 00:21:19,207 --> 00:21:22,590 All things that involve radiation that moves on null 337 00:21:22,590 --> 00:21:23,230 geodesics. 338 00:21:28,220 --> 00:21:42,130 So let's examine the associated conserved quantity 339 00:21:42,130 --> 00:21:44,054 that is associated with a null geodesic. 340 00:21:48,700 --> 00:21:53,510 So let's say v-- 341 00:21:53,510 --> 00:21:56,070 let's make it a p, actually. 342 00:21:56,070 --> 00:21:57,650 So it's going to be a null geodesic, 343 00:21:57,650 --> 00:22:01,820 so we're going to imagine it's radiation that is following. 344 00:22:01,820 --> 00:22:04,060 It has a four-momentum, pu. 345 00:22:07,150 --> 00:22:14,860 And let's define k, case of ng, from my null geodesic. 346 00:22:14,860 --> 00:22:20,620 That is going to be k mu nu, p mu, p nu. 347 00:22:20,620 --> 00:22:24,160 Let's plug-in the definition of my Killing tensor. 348 00:22:24,160 --> 00:22:34,240 So this is a square root of t, g mu nu, p nu, p nu. 349 00:22:34,240 --> 00:22:34,825 This is zero. 350 00:22:34,825 --> 00:22:35,700 It's a null geodesic. 351 00:22:48,410 --> 00:22:52,630 Then I get u mu p nu, u nu p mu. 352 00:22:52,630 --> 00:22:54,170 Now remind you of something. 353 00:23:02,060 --> 00:23:04,120 Go back to a nice little Easter egg, 354 00:23:04,120 --> 00:23:08,330 an exercise you guys did a long time ago. 355 00:23:08,330 --> 00:23:13,520 If I look at the dot product of a four-momentum 356 00:23:13,520 --> 00:23:24,850 and a four-velocity, what I get is the energy associated 357 00:23:24,850 --> 00:23:29,500 with that four-momentum as measured by the observer whose 358 00:23:29,500 --> 00:23:32,730 four-velocity is u. 359 00:23:32,730 --> 00:23:34,890 So what we get here is two copies 360 00:23:34,890 --> 00:23:39,210 of the energy of that null geodesic measured 361 00:23:39,210 --> 00:23:41,430 by the observer who is co-moving. 362 00:23:47,190 --> 00:23:50,070 So what this null geodesic-- 363 00:23:50,070 --> 00:23:52,370 what this quantity associated with this null geodesic 364 00:23:52,370 --> 00:23:57,800 is two powers of the scale factor times the energy that 365 00:23:57,800 --> 00:24:02,600 would be measured by someone who is co-moving 366 00:24:02,600 --> 00:24:04,877 with the fluid that fills my universe. 367 00:24:25,290 --> 00:24:28,010 Energy of p mu, as measured by u mu. 368 00:24:37,070 --> 00:24:39,590 And remember, this is a constant. 369 00:24:45,600 --> 00:24:48,890 So as this radiation travels across the universe-- 370 00:25:13,370 --> 00:25:15,830 as this radiation travels across the universe, 371 00:25:15,830 --> 00:25:19,700 the product of the scale factor and the energy 372 00:25:19,700 --> 00:25:22,730 associated with that radiation as measured 373 00:25:22,730 --> 00:25:25,265 by co-moving observers is a constant. 374 00:26:02,630 --> 00:26:05,140 So this is telling us that the energy, 375 00:26:05,140 --> 00:26:06,910 as measured by a co-moving observer, 376 00:26:06,910 --> 00:26:11,560 let's say it is emitted at some time with a scale factor is a. 377 00:26:11,560 --> 00:26:16,930 When it propagates to us, we define our scale factor as 1, 378 00:26:16,930 --> 00:26:21,402 the energy will have fallen down by a factor of 1 over a. 379 00:26:21,402 --> 00:26:23,860 So this makes it a little bit more rigorous, this intuitive 380 00:26:23,860 --> 00:26:26,200 argument that we saw from considering 381 00:26:26,200 --> 00:26:28,450 how the density of radiation fell off. 382 00:26:28,450 --> 00:26:33,160 What we see is the energy is indeed redshifting 383 00:26:33,160 --> 00:26:35,810 with the scale factor. 384 00:26:35,810 --> 00:26:41,350 So if I use the fact that the energy that I observe-- 385 00:26:45,150 --> 00:26:48,470 if I'm measuring light, light has a frequency of omega-- 386 00:26:51,470 --> 00:26:57,050 what I see is the omega that I observe at my scale 387 00:26:57,050 --> 00:27:00,770 factor, which I define to be 1, normalized to that 388 00:27:00,770 --> 00:27:09,660 when it was emitted, it looks like the scale factor 389 00:27:09,660 --> 00:27:12,240 when it was admitted. 390 00:27:12,240 --> 00:27:16,740 Divided by a now, a observed. 391 00:27:16,740 --> 00:27:17,490 I call this 1. 392 00:27:23,700 --> 00:27:25,180 I can flip this over, another way 393 00:27:25,180 --> 00:27:28,920 of saying this is that, if I write it 394 00:27:28,920 --> 00:27:37,240 in terms of wavelengths of the radiation, the wavelength 395 00:27:37,240 --> 00:27:41,710 of the radiation and when it was emitted versus the wavelength 396 00:27:41,710 --> 00:27:45,850 that we observe it tells me about the scale factor 397 00:27:45,850 --> 00:27:48,550 when the radiation was emitted. 398 00:28:02,010 --> 00:28:04,215 Astronomers like to work with redshift. 399 00:28:06,780 --> 00:28:08,310 They like to work with wavelength 400 00:28:08,310 --> 00:28:12,450 when they study things like the spectra 401 00:28:12,450 --> 00:28:13,920 of distant astronomical objects. 402 00:28:17,890 --> 00:28:22,600 And they use it to define a notion of redshift. 403 00:28:22,600 --> 00:28:27,730 So we define the redshift z to be the wavelength 404 00:28:27,730 --> 00:28:32,020 that we observe, minus the wavelength at the radiation 405 00:28:32,020 --> 00:28:36,010 that when it is emitted divided by the wavelength when 406 00:28:36,010 --> 00:28:38,170 it was emitted. 407 00:28:38,170 --> 00:28:41,650 Put all of these definitions together, 408 00:28:41,650 --> 00:28:43,990 and what this tells me is that the scale 409 00:28:43,990 --> 00:28:50,340 factor at which the radiation was emitted 410 00:28:50,340 --> 00:28:55,630 is simply related to the redshift that we observe. 411 00:28:55,630 --> 00:29:00,690 So this at last gives us a direct and not terribly 412 00:29:00,690 --> 00:29:02,910 difficult to use observational proxy 413 00:29:02,910 --> 00:29:06,960 that directly encodes the scale factor of our universe. 414 00:29:06,960 --> 00:29:23,060 Suppose we measure the spectrum of radiation from some source, 415 00:29:23,060 --> 00:29:27,710 and we see the distinct fingerprint 416 00:29:27,710 --> 00:29:30,740 associated with emission from a particular set 417 00:29:30,740 --> 00:29:32,399 of atomic transitions. 418 00:29:38,390 --> 00:29:45,370 What we generally find is some well-known fingerprints 419 00:29:45,370 --> 00:29:58,040 of well-characterized transitions, 420 00:29:58,040 --> 00:30:17,938 but in general they are stretched by some factor 421 00:30:17,938 --> 00:30:20,180 that we call the redshift z. 422 00:30:20,180 --> 00:30:22,680 Actually, you usually stretch by-- when you go through this, 423 00:30:22,680 --> 00:30:23,720 you'll find that what you measure 424 00:30:23,720 --> 00:30:25,300 is actually stretched by 1 plus z. 425 00:30:30,840 --> 00:30:32,430 You measure that, you have measured 426 00:30:32,430 --> 00:30:35,170 the scale factor at which this radiation was measured-- 427 00:30:35,170 --> 00:30:35,670 was emitted. 428 00:30:51,300 --> 00:30:52,360 So this is beautiful. 429 00:30:52,360 --> 00:30:54,450 This is a way in which the universe hands 430 00:30:54,450 --> 00:31:02,010 us the tool by which we can directly characterize some 431 00:31:02,010 --> 00:31:04,200 of the geometry of the universe at which light 432 00:31:04,200 --> 00:31:05,772 has been emitted. 433 00:31:05,772 --> 00:31:07,230 This is actually one of the reasons 434 00:31:07,230 --> 00:31:13,770 why a lot of people who do observational cosmology also 435 00:31:13,770 --> 00:31:16,470 happen to be expert atomic spectroscopists. 436 00:31:16,470 --> 00:31:19,260 Because you want to know to very high precision what 437 00:31:19,260 --> 00:31:22,127 is the characteristics of the hydrogen Balmer lines. 438 00:31:22,127 --> 00:31:24,210 Some of the most important sources for doing these 439 00:31:24,210 --> 00:31:26,520 tend to be galaxies in which there's 440 00:31:26,520 --> 00:31:29,520 a lot of matter falling onto black holes, some of the topics 441 00:31:29,520 --> 00:31:32,520 we'll be talking about in an upcoming video. 442 00:31:32,520 --> 00:31:34,830 As that material falls in, it gets hot, 443 00:31:34,830 --> 00:31:36,780 it generates a lot of radiation, and you'll 444 00:31:36,780 --> 00:31:39,990 see things like transition lines associated 445 00:31:39,990 --> 00:31:42,180 with carbon and iron. 446 00:31:42,180 --> 00:31:45,455 But often all reddened by a factor of several. 447 00:31:45,455 --> 00:31:46,830 You sort of go, oh, look at that, 448 00:31:46,830 --> 00:31:51,450 carbon falling onto a black hole at redshift 4.8. 449 00:31:51,450 --> 00:31:53,280 That is happening at a time when the scale 450 00:31:53,280 --> 00:31:57,990 factor of the universe was 1 over 4.8-- 451 00:31:57,990 --> 00:32:00,560 or 1 over 5.8, forgot my factor of 1 plus there. 452 00:32:03,890 --> 00:32:08,870 So you measure the redshift, and you 453 00:32:08,870 --> 00:32:10,240 have measured the scale factor. 454 00:32:15,040 --> 00:32:17,710 But you don't know when that light was emitted. 455 00:32:21,730 --> 00:32:24,400 We need to connect the scale factor 456 00:32:24,400 --> 00:32:30,970 that we can measure so directly and so beautifully 457 00:32:30,970 --> 00:32:34,870 to the time at which it was emitted. 458 00:32:34,870 --> 00:32:37,490 We now have a way of determining a, 459 00:32:37,490 --> 00:32:39,640 but we need a as a function of t. 460 00:33:07,950 --> 00:33:11,100 And in truth, we do this kind of via a surrogate. 461 00:33:11,100 --> 00:33:15,240 Because we are using radiation as our tool 462 00:33:15,240 --> 00:33:18,000 for probing the scale factor, we really 463 00:33:18,000 --> 00:33:20,430 don't measure t directly. 464 00:33:20,430 --> 00:33:22,650 When we look at light and it's coming to us, 465 00:33:22,650 --> 00:33:26,190 it doesn't say I was emitted on March 466 00:33:26,190 --> 00:33:31,590 27th of the year negative 6.8 billion 467 00:33:31,590 --> 00:33:34,380 BC, or something like that. 468 00:33:34,380 --> 00:33:37,140 We do know, though, that it traveled 469 00:33:37,140 --> 00:33:39,970 towards us at the speed of light on a null geodesic. 470 00:33:39,970 --> 00:33:42,120 And because it's a null geodesic, 471 00:33:42,120 --> 00:33:43,730 there's a very simple-- 472 00:33:43,730 --> 00:33:45,480 simple's a little bit of an overstatement, 473 00:33:45,480 --> 00:33:49,800 but there is at least a calculable connection. 474 00:33:49,800 --> 00:33:52,755 Because it's moving at the speed of light, we can simply-- 475 00:33:52,755 --> 00:33:54,150 I should stop using that word-- 476 00:33:54,150 --> 00:33:58,440 we can connect time to space. 477 00:33:58,440 --> 00:34:00,360 And so rather than directly determining 478 00:34:00,360 --> 00:34:02,940 the time at which the radiation was emitted, 479 00:34:02,940 --> 00:34:07,463 we want to calculate the distance of the source from us 480 00:34:07,463 --> 00:34:08,130 that emitted it. 481 00:34:41,170 --> 00:34:44,580 So rather than directly building up a of t, 482 00:34:44,580 --> 00:34:47,520 we're going to build up an a of d, where d 483 00:34:47,520 --> 00:34:50,540 is the distance of the source. 484 00:34:50,540 --> 00:34:52,739 And if you're used to working in Euclidean geometry, 485 00:34:52,739 --> 00:34:54,540 you sort of go, ah, OK, great. 486 00:34:54,540 --> 00:34:57,570 I know that light travels at the speed of light, 487 00:34:57,570 --> 00:35:00,180 so all I need to do is divide the distance by c, 488 00:35:00,180 --> 00:35:03,900 and I've got the time, and I build a of t. 489 00:35:03,900 --> 00:35:05,513 Conceptually, that is roughly right, 490 00:35:05,513 --> 00:35:07,680 and that gives at least a cartoon of the idea that's 491 00:35:07,680 --> 00:35:08,800 going on here. 492 00:35:08,800 --> 00:35:11,890 But we have to be a little bit careful. 493 00:35:11,890 --> 00:35:15,330 Because it turns out when you are making measurements 494 00:35:15,330 --> 00:35:21,900 in a curved space time, the notion of distance that you use 495 00:35:21,900 --> 00:35:26,650 depends on how you make the distance measurement. 496 00:35:26,650 --> 00:35:30,780 So this leads us now to our discussion of distance measures 497 00:35:30,780 --> 00:35:32,661 in cosmological spacetime. 498 00:35:55,262 --> 00:35:56,970 So just to give a little bit of intuition 499 00:35:56,970 --> 00:35:59,280 as to what's the kind of calculation 500 00:35:59,280 --> 00:36:01,890 we're going to need to do, let me describe one distance 501 00:36:01,890 --> 00:36:06,240 measure that is observation, which is about useless, 502 00:36:06,240 --> 00:36:09,390 but not a bad thing to at least begin 503 00:36:09,390 --> 00:36:15,540 to get a handle on, the way different parameters 504 00:36:15,540 --> 00:36:17,400 of the spacetime come in and influence 505 00:36:17,400 --> 00:36:19,450 what the distance measure is. 506 00:36:19,450 --> 00:36:22,800 So let's just think about the proper distance from us 507 00:36:22,800 --> 00:36:23,460 to a source. 508 00:36:36,350 --> 00:36:40,770 So let's imagine that-- 509 00:36:40,770 --> 00:36:44,110 well, let's just begin by first, let's write down 510 00:36:44,110 --> 00:36:44,940 my line element. 511 00:36:44,940 --> 00:36:46,690 And here's the form that I'm going to use. 512 00:36:57,640 --> 00:36:59,330 OK, so here's my line element. 513 00:36:59,330 --> 00:37:00,830 This is my differential connection 514 00:37:00,830 --> 00:37:07,300 between two events spaced between one another by dt, 515 00:37:07,300 --> 00:37:12,570 d chi, d theta, d phi, all hidden in that angular element, 516 00:37:12,570 --> 00:37:13,070 the omega. 517 00:37:16,280 --> 00:37:19,970 Let's imagine that we want to consider 518 00:37:19,970 --> 00:37:23,438 two sources that are separated purely in the radial direction. 519 00:37:34,900 --> 00:37:37,422 So my angular displacement between the two events 520 00:37:37,422 --> 00:37:38,130 is going to be 0. 521 00:37:40,860 --> 00:37:42,860 So the only thing I need to care about is d chi, 522 00:37:42,860 --> 00:37:44,900 and let's imagine that I determine 523 00:37:44,900 --> 00:37:48,560 the distance between these two at some instant-- 524 00:38:02,080 --> 00:38:04,810 So then you just get ds squared equals 525 00:38:04,810 --> 00:38:08,200 a squared or zero squared d chi squared, and you can integrate 526 00:38:08,200 --> 00:38:12,910 this up and you get our first distance measure, 527 00:38:12,910 --> 00:38:18,760 d sub p equals scale factor. 528 00:38:18,760 --> 00:38:22,780 Your overall distance scale are zero and chi. 529 00:38:26,370 --> 00:38:29,870 So Carroll's textbook calls this the instantaneous physical 530 00:38:29,870 --> 00:38:30,370 distance. 531 00:38:43,950 --> 00:38:47,633 Let's think about what this means if you do this. 532 00:38:47,633 --> 00:38:49,050 This is basically the distance you 533 00:38:49,050 --> 00:38:52,140 would get if you took a yardstick, 534 00:38:52,140 --> 00:38:54,150 you put one end at yourself, you're 535 00:38:54,150 --> 00:38:56,100 going to call yourself chi equals 0, 536 00:38:56,100 --> 00:38:57,960 you put the other end of the yardstick 537 00:38:57,960 --> 00:39:01,200 at the object in your universe at some distance 538 00:39:01,200 --> 00:39:04,260 chi in these coordinates, and that's the distance. 539 00:39:04,260 --> 00:39:06,240 d sub p is the distance that you measure. 540 00:39:13,370 --> 00:39:14,912 It is done, you're sort of imagining 541 00:39:14,912 --> 00:39:17,120 that both of the events at the end of this yardstick. 542 00:39:17,120 --> 00:39:19,328 You're sort of ascertaining their position at exactly 543 00:39:19,328 --> 00:39:22,490 the same instant, hence the term instantaneous, 544 00:39:22,490 --> 00:39:26,420 and you get something out of it that 545 00:39:26,420 --> 00:39:30,050 encodes some important aspects of how we think 546 00:39:30,050 --> 00:39:32,600 about distances in cosmology. 547 00:39:32,600 --> 00:39:37,490 So notice everything scales with the overall length scale 548 00:39:37,490 --> 00:39:40,430 that we associated with our spatial slices 549 00:39:40,430 --> 00:39:44,270 with our spatial sector of this metric, the r0. 550 00:39:44,270 --> 00:39:49,350 Notice that whatever is going on your scale factor, your a, 551 00:39:49,350 --> 00:39:51,380 your distance is going to track that. 552 00:39:51,380 --> 00:39:53,818 As a consequence of this, two objects 553 00:39:53,818 --> 00:39:55,610 that are sitting in what we call the Hubble 554 00:39:55,610 --> 00:39:57,290 flow, in other words, two objects that 555 00:39:57,290 --> 00:40:00,560 are co-moving with the fluid that makes up 556 00:40:00,560 --> 00:40:03,150 the source of our universe. 557 00:40:03,150 --> 00:40:05,590 They have an apparent motion with respect to each other. 558 00:40:19,420 --> 00:40:33,900 If I take the time derivative of this, 559 00:40:33,900 --> 00:40:40,290 the apparent is just a dot, or 0 chi, 560 00:40:40,290 --> 00:40:48,860 which is equal to the Hubble parameter times dp. 561 00:40:48,860 --> 00:40:53,205 Recall that the Hubble parameter is a dot over a, 562 00:40:53,205 --> 00:40:54,830 and if I'm doing this right now, that's 563 00:40:54,830 --> 00:40:59,680 the value of the Hubble parameter now. 564 00:40:59,680 --> 00:41:02,230 So this is the Hubble Expansion Law, the very famous Hubble 565 00:41:02,230 --> 00:41:03,880 Expansion Law. 566 00:41:03,880 --> 00:41:05,613 So we can see it hidden in this-- 567 00:41:05,613 --> 00:41:07,030 not even really hidden, it's quite 568 00:41:07,030 --> 00:41:09,550 apparent in this notion of an instantaneous physical 569 00:41:09,550 --> 00:41:10,690 distance. 570 00:41:10,690 --> 00:41:12,620 Let me just finally emphasize, though, 571 00:41:12,620 --> 00:41:15,070 that the instantanaeity that is part 572 00:41:15,070 --> 00:41:25,860 of this object's name, instantaneous measurements, 573 00:41:25,860 --> 00:41:27,210 are not done. 574 00:41:29,895 --> 00:41:31,630 As I said-- I mean, this is it sounds 575 00:41:31,630 --> 00:41:33,630 like I'm being slightly facetious, but it's not. 576 00:41:33,630 --> 00:41:36,195 The meaning of this distance measure 577 00:41:36,195 --> 00:41:37,820 is, like I said, it's a yardstick where 578 00:41:37,820 --> 00:41:41,070 I have an event at me, the other end of my yardstick 579 00:41:41,070 --> 00:41:44,210 is at my cosmological event. 580 00:41:44,210 --> 00:41:46,900 Those are typically separated by millions, 581 00:41:46,900 --> 00:41:49,410 billions of light years. 582 00:41:49,410 --> 00:41:50,490 Even if you could-- 583 00:41:50,490 --> 00:41:52,740 OK, the facetious bit was imagining it as a yardstick. 584 00:41:52,740 --> 00:41:54,630 But the non-facetious point I want to make 585 00:41:54,630 --> 00:41:57,360 is we do not make instantaneous measurements with that. 586 00:41:57,360 --> 00:42:00,420 When I measure an event that is billions of light years away, 587 00:42:00,420 --> 00:42:02,790 I am of course measuring it using light 588 00:42:02,790 --> 00:42:07,240 and I'm seeing light that was emitted billions of years ago. 589 00:42:07,240 --> 00:42:08,880 So we need to think a little bit more 590 00:42:08,880 --> 00:42:19,310 carefully about how to define distance 591 00:42:19,310 --> 00:42:22,220 in terms of quantities that really correspond 592 00:42:22,220 --> 00:42:23,930 to measurements we can make. 593 00:42:43,630 --> 00:42:49,120 And to get a little intuition, here 594 00:42:49,120 --> 00:42:57,400 are three ways where, if you were living in Euclidean space 595 00:42:57,400 --> 00:43:07,540 and you were looking at light from distant objects, 596 00:43:07,540 --> 00:43:09,730 here are three ways that you could define distance. 597 00:43:18,160 --> 00:43:23,560 So if spacetime were that of special relativity-- 598 00:43:23,560 --> 00:43:25,800 well, let's just say if space were purely Euclidean. 599 00:43:25,800 --> 00:43:27,050 Let's just leave it like that. 600 00:43:45,350 --> 00:43:47,930 Here are three notions that we could use. 601 00:43:47,930 --> 00:43:53,390 One, imagine there was some source of radiation 602 00:43:53,390 --> 00:43:57,620 in the universe that you understood so well that you 603 00:43:57,620 --> 00:44:02,950 knew its intrinsic luminosity. 604 00:44:02,950 --> 00:44:14,100 What you could do is compare the intrinsic luminosity 605 00:44:14,100 --> 00:44:16,419 of a source to its apparent brightness. 606 00:44:30,120 --> 00:44:34,125 So let's let f be the flux we measure from the source. 607 00:44:41,790 --> 00:44:46,970 This will be related to l, the luminosity, which-- 608 00:44:50,300 --> 00:44:51,846 suspend disbelief for a moment, we 609 00:44:51,846 --> 00:44:55,090 want to imagine that we know it for some reason. 610 00:44:55,090 --> 00:44:59,000 And if we imagine this is an isotropic emitter, 611 00:44:59,000 --> 00:45:03,110 this will be related by a factor of 4 pi, 612 00:45:03,110 --> 00:45:06,440 and the distance between us and that source. 613 00:45:06,440 --> 00:45:09,020 Let's call this d sub l. 614 00:45:09,020 --> 00:45:12,035 This is a luminosity distance. 615 00:45:18,480 --> 00:45:22,140 It is a distance that we measure by inferring 616 00:45:22,140 --> 00:45:25,350 the behavior of luminosity of a distant object. 617 00:45:28,710 --> 00:45:31,330 Now it turns out, and this is a subject for a different class, 618 00:45:31,330 --> 00:45:33,540 but nature actually gives us some objects 619 00:45:33,540 --> 00:45:38,820 whose luminosity is known or at least can be calibrated. 620 00:45:38,820 --> 00:45:43,590 In the case of much of what is done in cosmology today, 621 00:45:43,590 --> 00:45:47,430 we can take advantage of the behavior of certain stars 622 00:45:47,430 --> 00:45:52,520 whose luminosity is strongly correlated to the way that-- 623 00:45:52,520 --> 00:45:54,870 these are stars whose luminosity is variable, 624 00:45:54,870 --> 00:45:57,585 and we can use the fact that their variability is correlated 625 00:45:57,585 --> 00:46:02,310 to their luminosity to infer what their absolute luminosity 626 00:46:02,310 --> 00:46:03,720 actually is. 627 00:46:03,720 --> 00:46:06,690 There are other supernova events whose luminosity 628 00:46:06,690 --> 00:46:09,830 likewise appears to follow a universal law. 629 00:46:09,830 --> 00:46:14,310 It's related to the fact that the properties 630 00:46:14,310 --> 00:46:16,740 of those explosions are actually set by the microphysics 631 00:46:16,740 --> 00:46:18,750 of the stars that set them. 632 00:46:18,750 --> 00:46:22,920 More recently, we've been able to exploit the fact 633 00:46:22,920 --> 00:46:27,570 that gravitational wave sources have an intrinsic luminosity 634 00:46:27,570 --> 00:46:29,953 in gravitational waves, the dedt associated 635 00:46:29,953 --> 00:46:32,220 with the gravitational waves that they emit, 636 00:46:32,220 --> 00:46:35,400 which depends on the source gravitational physics in a very 637 00:46:35,400 --> 00:46:37,650 simple and predictable way that doesn't 638 00:46:37,650 --> 00:46:39,522 depend on very many parameters. 639 00:46:39,522 --> 00:46:40,980 I actually did a little bit of work 640 00:46:40,980 --> 00:46:43,830 on that over the course of my career, 641 00:46:43,830 --> 00:46:46,170 and it's a very exciting development 642 00:46:46,170 --> 00:46:48,300 that we can now use these as a way of setting 643 00:46:48,300 --> 00:46:51,170 the intrinsic luminosity of certain sources. 644 00:46:51,170 --> 00:46:54,630 At any rate, if you can take advantage of these objects that 645 00:46:54,630 --> 00:46:56,880 have a known luminosity and you can then 646 00:46:56,880 --> 00:46:59,160 measure the flux of radiation in your detector 647 00:46:59,160 --> 00:47:02,910 from these things, you have learned the distance. 648 00:47:02,910 --> 00:47:05,483 At least you have learned this particular measure 649 00:47:05,483 --> 00:47:06,150 of the distance. 650 00:47:13,348 --> 00:47:14,140 That's measure one. 651 00:47:26,990 --> 00:47:32,270 Measure two is imagine you have some object 652 00:47:32,270 --> 00:47:37,100 in the sky that has a particular intrinsic size associated 653 00:47:37,100 --> 00:47:38,955 with it. 654 00:47:38,955 --> 00:47:41,330 You can sort of think of the objects whose luminosity you 655 00:47:41,330 --> 00:47:43,910 know about as standard candles. 656 00:47:43,910 --> 00:47:46,550 Imagine if nature builds standard yardsticks, 657 00:47:46,550 --> 00:47:49,220 there's some object whose size you always know. 658 00:47:52,670 --> 00:47:59,750 Well, let's compare that physical size to the angular 659 00:47:59,750 --> 00:48:00,800 size that you measure. 660 00:48:08,240 --> 00:48:21,680 The angle that the object sub tends in the sky 661 00:48:21,680 --> 00:48:31,410 is going to be that intrinsic size, delta l divided 662 00:48:31,410 --> 00:48:33,190 by the distance. 663 00:48:33,190 --> 00:48:37,720 We'll call this distance d sub a, for the angular diameter 664 00:48:37,720 --> 00:48:38,220 distance. 665 00:48:57,032 --> 00:48:58,490 Believe it or not, nature, in fact, 666 00:48:58,490 --> 00:49:00,860 provides standard yardsticks type 667 00:49:00,860 --> 00:49:02,360 so that we can actually do this. 668 00:49:05,070 --> 00:49:06,900 Finally, at least as a matter of principle, 669 00:49:06,900 --> 00:49:08,370 imagine you had some object that's 670 00:49:08,370 --> 00:49:11,670 moving across the sky with a speed that you know. 671 00:49:11,670 --> 00:49:22,150 You could compare that transverse speed 672 00:49:22,150 --> 00:49:24,700 to an apparent angular speed. 673 00:49:37,610 --> 00:49:39,490 So the theta dot, the angular speed 674 00:49:39,490 --> 00:50:02,130 that you measure, that would be the velocity 675 00:50:02,130 --> 00:50:10,520 perpendicular to your line of sight divided by distance. 676 00:50:10,520 --> 00:50:17,720 We'll call this d sub m, the proper motion distance. 677 00:50:30,720 --> 00:50:32,330 So if our universe were Euclidean, 678 00:50:32,330 --> 00:50:34,635 not only would it be easy for us to use these three 679 00:50:34,635 --> 00:50:36,260 measures of distance, all three of them 680 00:50:36,260 --> 00:50:41,807 would give the same result. Because this is all 681 00:50:41,807 --> 00:50:42,390 just geometry. 682 00:50:50,370 --> 00:50:54,370 Turns out when you study these notions of distance, 683 00:50:54,370 --> 00:51:00,250 in an FRW spacetime, there's some variation that enters. 684 00:51:00,250 --> 00:51:06,740 Let me just emphasize here that there is an excellent summary 685 00:51:06,740 --> 00:51:07,590 on this stuff. 686 00:51:07,590 --> 00:51:09,320 I can't remember if I linked this to the course website 687 00:51:09,320 --> 00:51:09,820 or not. 688 00:51:09,820 --> 00:51:10,820 I should and I shall. 689 00:51:14,300 --> 00:51:16,910 An excellent summary of all this, 690 00:51:16,910 --> 00:51:22,010 really emphasizing observationally significant 691 00:51:22,010 --> 00:51:26,240 aspects of these things come from the article that 692 00:51:26,240 --> 00:51:43,620 is on the archive called Distance Measures in Cosmology 693 00:51:43,620 --> 00:51:49,996 by David Hogg, a colleague at New York University. 694 00:51:55,460 --> 00:52:02,660 You can find this on the Astro PH archive, 9905116. 695 00:52:02,660 --> 00:52:06,010 It's hard for me to believe this is almost 21 years old now. 696 00:52:06,010 --> 00:52:07,835 This is a gem of a paper. 697 00:52:07,835 --> 00:52:09,460 Hogg never submitted it to any journal, 698 00:52:09,460 --> 00:52:11,250 just posted it on the archive so that the community 699 00:52:11,250 --> 00:52:12,375 could take advantage of it. 700 00:52:16,700 --> 00:52:21,370 So the textbook by Carroll goes through the calculation of d 701 00:52:21,370 --> 00:52:24,270 sub l. 702 00:52:24,270 --> 00:52:27,480 On a problem set you will do d sub m. 703 00:52:27,480 --> 00:52:29,490 We're going to go through d sub a, 704 00:52:29,490 --> 00:52:32,490 just so you can see some of the way that this works. 705 00:52:32,490 --> 00:52:42,890 Let me emphasize one thing, all of these measures 706 00:52:42,890 --> 00:52:44,440 use the first Friedmann equation. 707 00:52:57,860 --> 00:53:00,830 So writing your Friedmann equation like so. 708 00:53:00,830 --> 00:53:03,830 i is a sum of all the different species 709 00:53:03,830 --> 00:53:07,010 of things that can contribute including curvature. 710 00:53:07,010 --> 00:53:09,500 So recall that even though curvature isn't really 711 00:53:09,500 --> 00:53:13,010 a density, you can combine enough factors 712 00:53:13,010 --> 00:53:15,170 to make it act as though it were a density. 713 00:53:19,600 --> 00:53:21,692 You assume a power law. 714 00:53:31,340 --> 00:53:34,390 So this n sub i is related to the equation of state parameter 715 00:53:34,390 --> 00:53:36,023 for each one of these species. 716 00:53:36,023 --> 00:53:37,690 And let's now take advantage of the fact 717 00:53:37,690 --> 00:53:42,070 that we know the scale factor directly ties to redshift. 718 00:53:42,070 --> 00:53:44,500 I can rewrite this as how the density evolves 719 00:53:44,500 --> 00:53:46,960 as a function of redshift. 720 00:53:46,960 --> 00:53:49,540 So when you put all of this together, 721 00:53:49,540 --> 00:53:56,635 this allows us to write h of a as h of z. 722 00:54:02,260 --> 00:54:04,840 This is given by the Hubble parameter now 723 00:54:04,840 --> 00:54:06,650 times some function e of z. 724 00:54:23,280 --> 00:54:24,650 And that is simply-- 725 00:54:24,650 --> 00:54:26,680 you divide everything out to normalize 726 00:54:26,680 --> 00:54:28,660 to the critical density, and that 727 00:54:28,660 --> 00:54:34,410 is a sum over all these densities with the redshift 728 00:54:34,410 --> 00:54:35,160 waiting like so. 729 00:54:42,120 --> 00:54:43,810 Let's really do an example, like I said. 730 00:54:43,810 --> 00:54:48,190 So if you read Carroll, you will see the calculation 731 00:54:48,190 --> 00:54:49,840 of the luminosity distance. 732 00:54:49,840 --> 00:54:53,585 If you do the cosmology problem set that will be-- 733 00:54:53,585 --> 00:54:59,420 I believe its p set 8, you will explore the proper motion 734 00:54:59,420 --> 00:54:59,920 distance. 735 00:55:02,580 --> 00:55:05,950 So let's do the angular diameter distance. 736 00:55:05,950 --> 00:55:07,700 Seeing someone work through this, I think, 737 00:55:07,700 --> 00:55:10,070 is probably useful for helping to solidify 738 00:55:10,070 --> 00:55:12,530 the way in which these distance measures work 739 00:55:12,530 --> 00:55:20,430 and how it is that one can tie together important observables. 740 00:55:20,430 --> 00:55:24,740 So let's consider some source that we 741 00:55:24,740 --> 00:55:31,400 observe that, according to us, subtends an angle, delta phi. 742 00:55:47,110 --> 00:55:50,670 Every spacial sector is spherically symmetric, 743 00:55:50,670 --> 00:55:53,140 and so we can orient our coordinate system 744 00:55:53,140 --> 00:55:56,080 so that this thing, it would be sort of a standard ruler-- 745 00:55:56,080 --> 00:55:58,690 what we're going to do is orient our coordinate system 746 00:55:58,690 --> 00:56:02,940 so that object lies in the theta equals pi over 2 plane. 747 00:56:23,820 --> 00:56:26,610 The proper size of that source-- 748 00:56:26,610 --> 00:56:28,793 so the thing is just sitting in the sky there-- 749 00:56:28,793 --> 00:56:30,210 the proper size of the source, you 750 00:56:30,210 --> 00:56:31,585 can get this in the line element. 751 00:56:39,250 --> 00:56:42,140 The delta l of the source will be 752 00:56:42,140 --> 00:56:46,820 the scale factor at the time at which it is emitted, r0. 753 00:56:56,350 --> 00:57:00,310 So this is using one of the forms of the FRW line elements 754 00:57:00,310 --> 00:57:04,050 I wrote down at the beginning of this lecture. 755 00:57:04,050 --> 00:57:08,924 And so the angular diameter distance, 756 00:57:08,924 --> 00:57:11,780 that's a quantity that I've defined over here, 757 00:57:11,780 --> 00:57:17,910 it's just the ratio of this length to that angle. 758 00:57:31,617 --> 00:57:33,200 Let's rewrite this using the redshift. 759 00:57:33,200 --> 00:57:47,075 Redshift is something that I actually directly observe, 760 00:57:47,075 --> 00:57:47,700 so there we go. 761 00:57:52,190 --> 00:57:57,350 This is not wrong, but it's flawed. 762 00:57:57,350 --> 00:57:58,220 So this is true. 763 00:57:58,220 --> 00:57:59,250 This is absolutely true. 764 00:57:59,250 --> 00:58:02,360 Here's the problem, I don't know the overall scale 765 00:58:02,360 --> 00:58:08,570 of my universe and this coordinate chi doesn't really 766 00:58:08,570 --> 00:58:10,580 have an observable meaning to it. 767 00:58:10,580 --> 00:58:14,300 It's how I label events, but I look at some quasar in the sky 768 00:58:14,300 --> 00:58:17,540 and I'm like, what's your chi? 769 00:58:17,540 --> 00:58:23,660 So what we need to do is reformulate the numerator 770 00:58:23,660 --> 00:58:27,050 of this expression in such a way as to get rid of that chi 771 00:58:27,050 --> 00:58:29,180 and then see what happens, see if we have a way 772 00:58:29,180 --> 00:58:30,458 to get rid of that r0. 773 00:58:30,458 --> 00:58:31,625 Let's worry about chi first. 774 00:58:39,210 --> 00:58:42,720 We're going to eliminate it by taking advantage of the fact 775 00:58:42,720 --> 00:58:45,480 that the radiation I am measuring 776 00:58:45,480 --> 00:58:46,995 comes to me on a null path. 777 00:59:01,150 --> 00:59:02,150 Not just any null path. 778 00:59:02,150 --> 00:59:04,210 I'm going imagine it's a radial one. 779 00:59:04,210 --> 00:59:08,170 We are allowed to be somewhat self-centered in defining 780 00:59:08,170 --> 00:59:10,810 FRW cosmology, we put ourselves at the origin. 781 00:59:10,810 --> 00:59:13,510 So any light that reaches us moves 782 00:59:13,510 --> 00:59:16,480 on a purely radial trajectory from its source to us. 783 00:59:22,780 --> 00:59:32,380 So looking at how the time and the radial coordinate chi 784 00:59:32,380 --> 00:59:40,105 are related for a radial null path we go into our FRW metric. 785 00:59:51,530 --> 00:59:54,500 I get this. 786 00:59:54,500 --> 00:59:58,490 So I can integrate this up to figure out what chi is. 787 01:00:13,090 --> 01:00:13,805 So this is right. 788 01:00:13,805 --> 01:00:15,430 Let's massage it a little bit to put it 789 01:00:15,430 --> 01:00:17,710 in a form that's a little bit more useful to us. 790 01:00:17,710 --> 01:00:19,870 Let's change our variable-- 791 01:00:19,870 --> 01:00:22,390 change our variable of integration from time to a. 792 01:00:27,010 --> 01:00:31,810 So this will be an integral from the scale factor at which 793 01:00:31,810 --> 01:00:36,040 the radiation is emitted to the scale factor 794 01:00:36,040 --> 01:00:38,180 which we observe it, i.e. now. 795 01:00:38,180 --> 01:00:43,420 And when you do that change of variables 796 01:00:43,420 --> 01:00:46,590 your integral changes like so. 797 01:00:46,590 --> 01:00:53,513 Let's rewrite this once more to insert my Hubble parameter. 798 01:01:29,995 --> 01:01:32,140 Now let's change variables once more. 799 01:01:32,140 --> 01:01:35,080 We're going to use the fact that our direct measurable is 800 01:01:35,080 --> 01:01:36,460 redshift. 801 01:01:36,460 --> 01:01:45,532 And so if we use a equals 1 over 1 plus z, 802 01:01:45,532 --> 01:02:01,636 I can further write this as an integral over redshift like so. 803 01:02:01,636 --> 01:02:03,630 And that h0 can come out of my integral. 804 01:02:12,070 --> 01:02:15,360 So this is in a form that is now finally formulated 805 01:02:15,360 --> 01:02:19,740 in terms of an observable redshift 806 01:02:19,740 --> 01:02:22,470 and my model dependent parameters. 807 01:02:22,470 --> 01:02:26,460 The various omegas that, when I construct my universe model, 808 01:02:26,460 --> 01:02:27,570 I am free to set. 809 01:02:27,570 --> 01:02:30,035 Or if I am a phenomenologist, that 810 01:02:30,035 --> 01:02:31,410 are going to be knobs that I turn 811 01:02:31,410 --> 01:02:34,860 to try to design a model universe that matches 812 01:02:34,860 --> 01:02:38,760 the data that I am measuring. 813 01:02:38,760 --> 01:02:41,800 r0, though, is still kind of annoying. 814 01:02:41,800 --> 01:02:44,820 We don't know what this guy is, so what 815 01:02:44,820 --> 01:03:00,490 we do is eliminate r0 in favor of a curvature density 816 01:03:00,490 --> 01:03:01,336 parameter. 817 01:03:07,660 --> 01:03:10,590 So using the fact that omega curvature-- 818 01:03:15,400 --> 01:03:17,950 go back to how this was originally defined-- 819 01:03:17,950 --> 01:03:22,570 it was negative kappa over h0 squared. 820 01:03:22,570 --> 01:03:26,590 That's negative k over r0 squared h0 squared. 821 01:03:30,700 --> 01:03:39,650 That tells me that r0 is the Hubble constant now, 822 01:03:39,650 --> 01:03:44,660 divided by the square root of the absolute value 823 01:03:44,660 --> 01:03:50,550 of the curvature, at least if k equals plus or minus 1. 824 01:03:50,550 --> 01:03:53,460 What happens when it's not plus or minus 1, if it's equal to 0? 825 01:03:53,460 --> 01:03:54,460 Well, hold that thought. 826 01:04:11,720 --> 01:04:13,400 So let's put all these pieces together. 827 01:04:35,720 --> 01:04:37,220 So assembling all the ingredients 828 01:04:37,220 --> 01:04:43,180 I have here, what we find is the angular diameter distance. 829 01:04:43,180 --> 01:04:47,570 There's a factor of 1 over 1 plus z, 830 01:04:47,570 --> 01:04:49,280 1 over the Hubble constant. 831 01:04:49,280 --> 01:04:53,670 Remember, Hubble has units of 1 over length-- 832 01:04:53,670 --> 01:04:56,215 excuse me, 1 over time, and with the factor 833 01:04:56,215 --> 01:04:57,590 of the speed of light that is a 1 834 01:04:57,590 --> 01:05:01,130 over length, so one over the Hubble parameter now 835 01:05:01,130 --> 01:05:03,380 is essentially a kind of fiducial overall distance 836 01:05:03,380 --> 01:05:04,800 scale. 837 01:05:04,800 --> 01:05:13,040 And then our solution breaks up into three branches, 838 01:05:13,040 --> 01:05:18,140 depending upon whether k equals minus 1, 0, or 1. 839 01:05:20,710 --> 01:05:26,460 So you get one term where it involves minus square root 840 01:05:26,460 --> 01:05:32,025 the absolute value of the curvature parameter times sine. 841 01:05:35,980 --> 01:05:38,620 That same absolute value of the curvature parameters square 842 01:05:38,620 --> 01:05:39,120 root. 843 01:05:55,780 --> 01:05:59,220 So here's your k equals plus 1 branch. 844 01:05:59,220 --> 01:06:02,800 For your k equals 0 branch, basically what 845 01:06:02,800 --> 01:06:07,300 you'll find when you plug-in your s of k 846 01:06:07,300 --> 01:06:09,400 is that r0 cancels out. 847 01:06:12,460 --> 01:06:15,100 So that ends up being a parameter 848 01:06:15,100 --> 01:06:17,920 you do not need to worry about, and I suggest you just 849 01:06:17,920 --> 01:06:19,450 work through the algebra and you'll 850 01:06:19,450 --> 01:06:28,450 find that for k equals 0, it simply looks like this. 851 01:06:31,980 --> 01:06:36,659 And then finally, if you are in an open universe-- 852 01:06:46,450 --> 01:06:48,135 that is supposed to be curvature-- 853 01:07:11,430 --> 01:07:12,480 what we get is this. 854 01:07:22,003 --> 01:07:23,420 So this is a distance measure that 855 01:07:23,420 --> 01:07:27,740 tells me how angular diameter distance depends 856 01:07:27,740 --> 01:07:29,540 on observable parameters. 857 01:07:29,540 --> 01:07:31,520 Hubble is something that we can measure. 858 01:07:31,520 --> 01:07:33,830 Redshift is something we can measure. 859 01:07:33,830 --> 01:07:36,840 And it depends on model parameters, 860 01:07:36,840 --> 01:07:40,317 the different densities that go into e of z, and-- 861 01:07:40,317 --> 01:07:41,900 which I have on the board right here-- 862 01:07:44,940 --> 01:07:47,000 the different densities that go into e of z. 863 01:07:47,000 --> 01:07:49,880 My apologies, I left out that h0 there-- 864 01:07:49,880 --> 01:07:55,170 and your choice of the curvature. 865 01:07:55,170 --> 01:07:59,660 When you analyze these three distances 866 01:07:59,660 --> 01:08:00,765 here is what you find. 867 01:08:21,910 --> 01:08:25,770 You find that the luminosity distance 868 01:08:25,770 --> 01:08:29,990 is related to the proper motion distance by a factor of 1 869 01:08:29,990 --> 01:08:36,210 plus z, and that's related to the angular diameter distance 870 01:08:36,210 --> 01:08:38,439 by a factor of 1 plus z squared. 871 01:08:38,439 --> 01:08:42,000 So when you read Carroll, you will find that 1 plus z factor 872 01:08:42,000 --> 01:08:43,260 there-- 873 01:08:43,260 --> 01:08:47,637 excuse me, 1 plus z to the minus 1 power turns into a 1 plus z-- 874 01:08:47,637 --> 01:08:48,970 is the camera not looking at me? 875 01:08:48,970 --> 01:08:49,470 Hello? 876 01:08:49,470 --> 01:08:50,279 There we go. 877 01:08:50,279 --> 01:08:56,955 So that 1 over 1 plus z turns into a 1 plus z. 878 01:08:56,955 --> 01:08:59,580 When you do it on the p set, you do the proper motion distance, 879 01:08:59,580 --> 01:09:03,359 so it will just be no 1 plus z factor in front of everything. 880 01:09:03,359 --> 01:09:07,290 So the name of the game when one is doing cosmology 881 01:09:07,290 --> 01:09:11,910 as a physicist is to find quantities 882 01:09:11,910 --> 01:09:14,729 that you can measure that allow you to determine 883 01:09:14,729 --> 01:09:18,479 luminosity distances, angular diameter distances, 884 01:09:18,479 --> 01:09:20,189 proper motion distances. 885 01:09:20,189 --> 01:09:23,220 Now it turns out that the proper motion distance 886 01:09:23,220 --> 01:09:26,880 is not a very practical one for basically 887 01:09:26,880 --> 01:09:29,310 any cosmologically interesting source. 888 01:09:29,310 --> 01:09:33,270 They are simply so far away that even for a source 889 01:09:33,270 --> 01:09:35,910 moving essentially at the speed of light, 890 01:09:35,910 --> 01:09:37,710 the amount of angular motion that 891 01:09:37,710 --> 01:09:42,819 can be seen over essentially a human lifetime is negligible. 892 01:09:42,819 --> 01:09:46,000 So this turns into something that-- 893 01:09:46,000 --> 01:09:46,632 hi. 894 01:09:46,632 --> 01:09:47,590 MIT POLICE: [INAUDIBLE] 895 01:09:47,590 --> 01:09:48,060 SCOTT HUGHES: That's OK. 896 01:09:48,060 --> 01:09:50,194 Yeah, I'm doing some pre-recording of lectures. 897 01:09:50,194 --> 01:09:53,133 [LAUGHS] I was warned you guys might come by. 898 01:09:53,133 --> 01:09:55,050 I have my ID with me and things like that, so. 899 01:09:55,050 --> 01:09:55,515 MIT POLICE: That's fine. 900 01:09:55,515 --> 01:09:55,980 Take care. 901 01:09:55,980 --> 01:09:57,230 MIT POLICE: You look official. 902 01:09:57,230 --> 01:09:59,132 SCOTT HUGHES: [LAUGHS] I appreciate it. 903 01:09:59,132 --> 01:10:01,590 So those of you watching the video at home, as you can see, 904 01:10:01,590 --> 01:10:03,270 the MIT police is keeping us safe. 905 01:10:06,360 --> 01:10:08,040 Scared the crap out of me for a second 906 01:10:08,040 --> 01:10:10,090 there, but it's all good. 907 01:10:10,090 --> 01:10:12,390 All right, so let's go back to this for a second. 908 01:10:12,390 --> 01:10:15,372 So the proper motion distance is something 909 01:10:15,372 --> 01:10:16,830 that is not particularly practical, 910 01:10:16,830 --> 01:10:19,330 because as I said, even if you have an object that is moving 911 01:10:19,330 --> 01:10:24,450 close to the speed of light this is not something 912 01:10:24,450 --> 01:10:26,790 that even over the course of a human lifetime 913 01:10:26,790 --> 01:10:30,550 you are likely to see significant angular motion. 914 01:10:30,550 --> 01:10:32,160 So this is generally not used. 915 01:10:32,160 --> 01:10:35,160 But luminosity distances and angular diameter distances, 916 01:10:35,160 --> 01:10:38,340 that is, in fact, extremely important, 917 01:10:38,340 --> 01:10:45,990 and a lot of cosmology is based on looking for well understood 918 01:10:45,990 --> 01:10:49,440 objects where we can calibrate the physical size 919 01:10:49,440 --> 01:10:51,330 and infer the angular diameter distance, 920 01:10:51,330 --> 01:10:53,820 or we know the intrinsic brightness 921 01:10:53,820 --> 01:10:56,700 and we can determine the luminosity distance. 922 01:10:56,700 --> 01:10:59,670 So let me just give a quick snapshot 923 01:10:59,670 --> 01:11:02,580 of where the measurements come from in modern cosmology that 924 01:11:02,580 --> 01:11:04,650 are driving our cosmological model. 925 01:11:17,470 --> 01:11:20,498 One of the most important is the cosmic microwave background. 926 01:11:28,630 --> 01:11:32,130 So, vastly oversimplifying, when we look at the cosmic microwave 927 01:11:32,130 --> 01:11:38,910 background after removing things like the flow 928 01:11:38,910 --> 01:11:43,490 of our solar system with respect to the rest frame of-- excuse 929 01:11:43,490 --> 01:11:45,000 me, the co-moving reference frame 930 01:11:45,000 --> 01:11:46,710 of the cause of the fluid that makes up 931 01:11:46,710 --> 01:11:51,240 our universe, the size of hot and cold spots 932 01:11:51,240 --> 01:11:52,140 is a standard ruler. 933 01:12:07,200 --> 01:12:08,970 By looking at the distribution of sizes 934 01:12:08,970 --> 01:12:11,040 that we see from these things, we 935 01:12:11,040 --> 01:12:14,003 can determine the angular diameter distance 936 01:12:14,003 --> 01:12:15,420 to the cosmic microwave background 937 01:12:15,420 --> 01:12:17,130 with very high precision. 938 01:12:17,130 --> 01:12:20,310 This ends up being one of the most important constraints 939 01:12:20,310 --> 01:12:23,070 on determining what the curvature parameter actually 940 01:12:23,070 --> 01:12:24,450 is. 941 01:12:24,450 --> 01:12:27,300 And it is largely thanks to the cosmic microwave background 942 01:12:27,300 --> 01:12:33,500 that current prejudice, I would say, the current best wisdom-- 943 01:12:33,500 --> 01:12:36,800 choose your descriptor as you wish-- 944 01:12:36,800 --> 01:12:39,080 is that k equals 0 and our universe 945 01:12:39,080 --> 01:12:40,340 is in fact spatially flat. 946 01:12:43,370 --> 01:12:48,453 Second one is what are called type 1a supernova. 947 01:12:57,330 --> 01:13:03,150 These are essentially the thermonuclear detonations 948 01:13:03,150 --> 01:13:04,140 of white dwarfs. 949 01:13:11,320 --> 01:13:13,070 Not even really thermonuclear, it's just-- 950 01:13:17,725 --> 01:13:19,600 my apologies, I'm confusing a different event 951 01:13:19,600 --> 01:13:21,275 that involves white dwarfs. 952 01:13:21,275 --> 01:13:22,900 The type 1a's are not the thermonuclear 953 01:13:22,900 --> 01:13:24,025 explosions of these things. 954 01:13:24,025 --> 01:13:26,900 This is actually the core collapse of a white dwarf star. 955 01:13:26,900 --> 01:13:30,610 So this is what happens when a white dwarf it creates 956 01:13:30,610 --> 01:13:33,220 or in some way accumulates enough mass 957 01:13:33,220 --> 01:13:37,300 such that electron degeneracy pressure is 958 01:13:37,300 --> 01:13:40,780 no longer sufficient to hold it against gravitational collapse, 959 01:13:40,780 --> 01:13:45,790 and the whole thing basically collapses into a neutron star. 960 01:13:45,790 --> 01:13:49,270 That happens at a defined mass, the Chandrasekhar mass, 961 01:13:49,270 --> 01:13:52,180 named after one of my scientific heroes, Subramanian 962 01:13:52,180 --> 01:13:54,910 Chandrasekhar. 963 01:13:54,910 --> 01:13:59,830 And because it has a defined mass associated with it, 964 01:13:59,830 --> 01:14:01,600 every event basically has the same amount 965 01:14:01,600 --> 01:14:03,040 of matter participating. 966 01:14:03,040 --> 01:14:04,810 This is a standard candle. 967 01:14:22,035 --> 01:14:23,660 So there's a couple others that I'm not 968 01:14:23,660 --> 01:14:26,150 going to talk about in too much detail here. 969 01:14:26,150 --> 01:14:28,500 While I'm erasing the board I will just mention them. 970 01:14:28,500 --> 01:14:32,150 So by looking at things like the clustering of galaxies 971 01:14:32,150 --> 01:14:38,510 we can measure the distribution of mass 972 01:14:38,510 --> 01:14:41,360 in the universe that allows us to determine the omega m 973 01:14:41,360 --> 01:14:42,560 parameter. 974 01:14:42,560 --> 01:14:44,120 That's one of the bits of information 975 01:14:44,120 --> 01:14:47,000 that tells us that much of the universe 976 01:14:47,000 --> 01:14:49,790 is made of matter that apparently does not 977 01:14:49,790 --> 01:14:51,650 participate in standard model processes 978 01:14:51,650 --> 01:14:52,890 as we know them today-- 979 01:14:52,890 --> 01:14:55,430 the so-called dark matter problem. 980 01:14:55,430 --> 01:14:58,490 We can look at chemical abundances, which tells us 981 01:14:58,490 --> 01:15:01,340 about the behavior of nuclear processes in the very 982 01:15:01,340 --> 01:15:02,800 early universe. 983 01:15:02,800 --> 01:15:04,550 And the last one which I will mention here 984 01:15:04,550 --> 01:15:10,355 is we can look at nearby standard candles. 985 01:15:16,760 --> 01:15:18,890 And nearby standard candles allow 986 01:15:18,890 --> 01:15:22,350 us to probe the local Hubble law and determine h0. 987 01:15:26,392 --> 01:15:27,600 "Probble," that's not a word. 988 01:15:30,410 --> 01:15:33,570 If you combine "probe" and "Hubble" you get "probble." 989 01:15:40,150 --> 01:15:42,120 And when I say nearby, that usually 990 01:15:42,120 --> 01:15:45,157 means events that are merely a few tens of millions 991 01:15:45,157 --> 01:15:45,990 of light years away. 992 01:15:48,720 --> 01:15:53,343 It's worth noting that all of these various techniques, 993 01:15:53,343 --> 01:15:55,760 all of these different things, you can kind of even see it 994 01:15:55,760 --> 01:15:58,093 when you think about the mathematical form of everything 995 01:15:58,093 --> 01:16:00,530 that went into our distance measures, 996 01:16:00,530 --> 01:16:02,970 they're all highly entangled with each other. 997 01:16:02,970 --> 01:16:05,660 And so to do this kind of thing properly, 998 01:16:05,660 --> 01:16:08,000 you need to take just a crap load of data, 999 01:16:08,000 --> 01:16:10,580 combine all of your data sets, and do 1000 01:16:10,580 --> 01:16:14,240 a joint analysis of everything, looking at the way 1001 01:16:14,240 --> 01:16:17,480 varying the parameters and all the different models 1002 01:16:17,480 --> 01:16:19,913 affects the outcome of your observables. 1003 01:16:24,080 --> 01:16:26,510 You also have to carefully take into account the fact 1004 01:16:26,510 --> 01:16:29,900 that when you measure something, you measure it with errors. 1005 01:16:29,900 --> 01:16:31,920 And so many of these things are not known. 1006 01:16:31,920 --> 01:16:33,920 Turns out you can usually measure redshift quite 1007 01:16:33,920 --> 01:16:36,203 precisely, but these distances always 1008 01:16:36,203 --> 01:16:38,120 come with some error bar associated with them. 1009 01:16:38,120 --> 01:16:40,220 And so that means that the distance you associate 1010 01:16:40,220 --> 01:16:42,170 with a particular redshift, which 1011 01:16:42,170 --> 01:16:45,048 is equivalent to associating a time with a redshift, 1012 01:16:45,048 --> 01:16:46,340 there's some error bar on that. 1013 01:16:46,340 --> 01:16:49,610 And that can lead to significant skew in what you 1014 01:16:49,610 --> 01:16:50,570 determine from things. 1015 01:16:56,090 --> 01:16:58,922 There's a lot more we could say, but time is finite 1016 01:16:58,922 --> 01:17:00,380 and we need to change topic, so I'm 1017 01:17:00,380 --> 01:17:05,840 going to conclude this lecture by talking about two mysteries 1018 01:17:05,840 --> 01:17:08,270 in the cosmological model that have been 1019 01:17:08,270 --> 01:17:11,780 the focus of a lot of research attention 1020 01:17:11,780 --> 01:17:15,220 over the past several decades. 1021 01:17:15,220 --> 01:17:15,970 Two mysteries. 1022 01:17:15,970 --> 01:17:20,870 One, why is it that our universe appears 1023 01:17:20,870 --> 01:17:23,950 to be flat, spatially flat? 1024 01:17:45,220 --> 01:17:47,040 So to frame why this is a bit of a mystery, 1025 01:17:47,040 --> 01:17:49,510 you did just sort of go, eh, come on. 1026 01:17:49,510 --> 01:17:52,450 You've got three choices for the parameter, you got 0. 1027 01:17:58,650 --> 01:18:01,170 Let's begin by thinking about the first Friedmann equation. 1028 01:18:16,350 --> 01:18:22,790 I can write this like so, or I can 1029 01:18:22,790 --> 01:18:28,070 use this form, where I say omega plus omega curvature-- 1030 01:18:28,070 --> 01:18:31,220 I'm going to call that omega c for now-- 1031 01:18:31,220 --> 01:18:32,060 that equals 1. 1032 01:18:37,040 --> 01:18:41,750 The expectation had long been that our universe would 1033 01:18:41,750 --> 01:18:45,230 basically be dominated by various species of matter 1034 01:18:45,230 --> 01:18:49,020 and radiation for much of its history, 1035 01:18:49,020 --> 01:18:50,785 especially in the early universe. 1036 01:19:23,240 --> 01:19:26,430 If it was radiation dominated, you'd 1037 01:19:26,430 --> 01:19:31,800 expect the density to go as a to the minus 4. 1038 01:19:35,030 --> 01:19:36,530 If it's matter dominated, you expect 1039 01:19:36,530 --> 01:19:37,880 it to go as a to the minus 3. 1040 01:19:44,620 --> 01:19:48,280 Now, your curvature density goes as a to the minus 2. 1041 01:20:22,680 --> 01:20:24,610 And so what this means is that if you 1042 01:20:24,610 --> 01:20:33,980 look at the ratio of omega curvature to omega, 1043 01:20:33,980 --> 01:20:41,360 this will be proportional to a, for matter, 1044 01:20:41,360 --> 01:20:50,180 a squared for radiation. 1045 01:20:50,180 --> 01:20:56,820 If your universe-- in some sense, 1046 01:20:56,820 --> 01:21:00,768 looking at these parameter k of the minus 1, 0, 1, 1047 01:21:00,768 --> 01:21:02,060 that's a little bit misleading. 1048 01:21:02,060 --> 01:21:03,330 It's probably a little bit more useful to think 1049 01:21:03,330 --> 01:21:05,430 about things in terms of the kappa parameter. 1050 01:21:05,430 --> 01:21:08,820 And when you look at that, your flat universe 1051 01:21:08,820 --> 01:21:13,350 is a set of measure 0 in the set of all possible curvature 1052 01:21:13,350 --> 01:21:15,330 parameters that you could have. 1053 01:21:15,330 --> 01:21:19,050 And physicists tend to get suspicious 1054 01:21:19,050 --> 01:21:24,060 when something that could take on any range 1055 01:21:24,060 --> 01:21:26,340 of possible random values between minus infinity 1056 01:21:26,340 --> 01:21:29,160 and infinity picks out zero. 1057 01:21:29,160 --> 01:21:30,660 That tends to tell us that there may 1058 01:21:30,660 --> 01:21:32,970 be some principle at play that actually derives things 1059 01:21:32,970 --> 01:21:33,630 to being 0. 1060 01:21:36,210 --> 01:21:37,950 Looking at it this way, imagine you 1061 01:21:37,950 --> 01:21:40,830 have a universe that at early times 1062 01:21:40,830 --> 01:21:47,160 is very close to being flat, but not quite. 1063 01:21:51,360 --> 01:22:07,200 Any slight deviation from flatness 1064 01:22:07,200 --> 01:22:10,620 grows as the universe expands. 1065 01:22:16,990 --> 01:22:19,560 That's mystery one. 1066 01:22:19,560 --> 01:22:30,090 Mystery two, why is the cosmic microwave background 1067 01:22:30,090 --> 01:22:32,420 so homogeneous? 1068 01:22:35,330 --> 01:22:38,640 So when we look at the cosmic microwave background, 1069 01:22:38,640 --> 01:22:44,120 we see that it has the same properties. 1070 01:22:44,120 --> 01:22:45,650 The light has the same brightness, 1071 01:22:45,650 --> 01:22:47,780 it has the same temperature associated with it, 1072 01:22:47,780 --> 01:22:50,217 to within in a part in 100,000. 1073 01:22:50,217 --> 01:22:51,800 Now the standard model of our universe 1074 01:22:51,800 --> 01:22:54,740 tells us that at very early times 1075 01:22:54,740 --> 01:22:57,020 the universe was essentially a dense hot plasma. 1076 01:23:18,990 --> 01:23:27,080 This thing cooled as the universe expanded, 1077 01:23:27,080 --> 01:23:31,070 much the same way that if you have a bag of gas, 1078 01:23:31,070 --> 01:23:33,650 you squeeze it very rapidly, it will get hot, 1079 01:23:33,650 --> 01:23:37,520 you stretch it very rapidly, it will cool. 1080 01:23:42,440 --> 01:23:44,792 There's a few more details of this in my notes, 1081 01:23:44,792 --> 01:23:46,750 but when we look at this one the things that we 1082 01:23:46,750 --> 01:23:53,530 see is that in a universe that is only driven by matter or by 1083 01:23:53,530 --> 01:23:54,330 radiation-- 1084 01:24:01,860 --> 01:24:04,670 so the matter dominated and radiation dominated picture-- 1085 01:24:16,510 --> 01:24:31,260 it shows us that points on opposite sides of the sky 1086 01:24:31,260 --> 01:24:34,320 were actually out of causal contact with each other 1087 01:24:34,320 --> 01:24:36,195 in the earliest moments of the universe. 1088 01:24:58,980 --> 01:25:01,800 In other words, I look at the sky, 1089 01:25:01,800 --> 01:25:07,650 and the patch of sky over here was out of causal contact 1090 01:25:07,650 --> 01:25:10,530 with the patch of sky over here in the earliest 1091 01:25:10,530 --> 01:25:12,960 days of the universe. 1092 01:25:12,960 --> 01:25:14,900 And yet they had the same temperature, 1093 01:25:14,900 --> 01:25:19,490 which suggests that they were in thermal equilibrium. 1094 01:25:19,490 --> 01:25:23,930 How can two disparate, unconnected patch of the sky 1095 01:25:23,930 --> 01:25:26,510 have the same temperature if they cannot exchange 1096 01:25:26,510 --> 01:25:27,620 information? 1097 01:25:30,093 --> 01:25:31,760 You could imagine it being a coincidence 1098 01:25:31,760 --> 01:25:33,968 if one little bit of the sky has the same temperature 1099 01:25:33,968 --> 01:25:37,350 as a bit of another piece, but in fact, 1100 01:25:37,350 --> 01:25:39,950 when you do this calculation, you find huge patch of the sky 1101 01:25:39,950 --> 01:25:42,690 could not communicate with any other. 1102 01:25:42,690 --> 01:25:49,550 And so how then is it that the entire sky that we can observe 1103 01:25:49,550 --> 01:25:52,520 has the same temperature at the earliest 1104 01:25:52,520 --> 01:25:56,020 times within a part in 100,000. 1105 01:25:56,020 --> 01:25:58,580 You guys will explore this on-- 1106 01:25:58,580 --> 01:26:01,930 I believe it's problem set eight. 1107 01:26:01,930 --> 01:26:21,490 The solution to both of these problems that has been proposed 1108 01:26:21,490 --> 01:26:23,205 is cosmic inflation. 1109 01:26:30,490 --> 01:26:34,250 So what you do is imagine that at some earlier 1110 01:26:34,250 --> 01:26:38,420 moment in the universe, our universe 1111 01:26:38,420 --> 01:26:41,423 was filled with some strange field, 1112 01:26:41,423 --> 01:26:43,340 and I'll describe the properties of that field 1113 01:26:43,340 --> 01:26:46,370 in just a moment, such that it acted like it 1114 01:26:46,370 --> 01:26:47,750 had a cosmological constant. 1115 01:27:15,900 --> 01:27:18,990 In such an epic, the scale factor of the universe 1116 01:27:18,990 --> 01:27:24,382 goes as exponentially with the square root of the size 1117 01:27:24,382 --> 01:27:25,590 of the cosmological constant. 1118 01:27:30,140 --> 01:27:33,740 What you find when you look at this is that this-- it still, 1119 01:27:33,740 --> 01:27:36,410 of course, goes as a to the minus 2-- 1120 01:27:36,410 --> 01:27:37,520 but a to the minus-- 1121 01:27:37,520 --> 01:27:38,760 well, sorry. 1122 01:27:38,760 --> 01:27:40,040 Let me back up for a second. 1123 01:27:40,040 --> 01:27:43,370 So my scale factor in this case you'll find 1124 01:27:43,370 --> 01:27:48,360 goes are a to the minus 2, and that's 1125 01:27:48,360 --> 01:27:51,030 because the density associated with the cosmological constant 1126 01:27:51,030 --> 01:27:51,900 remains constant. 1127 01:28:01,860 --> 01:28:03,830 So even if you start in the early universe 1128 01:28:03,830 --> 01:28:06,150 with some random value for the curvature, 1129 01:28:06,150 --> 01:28:10,070 if you are in this epic of exponential inflation, 1130 01:28:10,070 --> 01:28:12,530 just for-- you know, you have to worry about timescales 1131 01:28:12,530 --> 01:28:14,330 a little bit, but if you do it long enough 1132 01:28:14,330 --> 01:28:17,630 you can drive this very, very close to 0. 1133 01:28:17,630 --> 01:28:20,960 So much so that when you then move forward, 1134 01:28:20,960 --> 01:28:25,250 let's say you come out of this period of cosmic inflation 1135 01:28:25,250 --> 01:28:28,160 and you enter a universe that is radiation dominated or matter 1136 01:28:28,160 --> 01:28:31,160 dominated, it will then begin to grow, 1137 01:28:31,160 --> 01:28:34,800 but if you drive it sufficiently close to 0 it doesn't matter. 1138 01:28:34,800 --> 01:28:39,170 You're never going to catch up with what inflation did to you. 1139 01:28:39,170 --> 01:28:42,410 On the P set you will also show that if you 1140 01:28:42,410 --> 01:28:51,140 have a period of inflation like this, then 1141 01:28:51,140 --> 01:28:53,630 that also cures the problem of piece of the sky being 1142 01:28:53,630 --> 01:28:55,170 out of causal contact. 1143 01:28:55,170 --> 01:28:56,870 So when you do that, what you find 1144 01:28:56,870 --> 01:29:03,470 is that essentially everything is in causal contact early on. 1145 01:29:03,470 --> 01:29:05,510 It may sort of come out of causal contact 1146 01:29:05,510 --> 01:29:09,480 after inflation has ended, more on that in just a moment, 1147 01:29:09,480 --> 01:29:11,660 and then things sort of change as the universe 1148 01:29:11,660 --> 01:29:12,500 continues to evolve. 1149 01:29:15,320 --> 01:29:20,320 OK so it looks like recording is back on. 1150 01:29:20,320 --> 01:29:21,783 My apologies, everyone. 1151 01:29:21,783 --> 01:29:23,200 So as I was in the middle of talk, 1152 01:29:23,200 --> 01:29:26,500 I talked a little bit too long in this particular lecture 1153 01:29:26,500 --> 01:29:29,260 so we're going to spill over into a little bit 1154 01:29:29,260 --> 01:29:32,730 of an addendum, just a five-ish minute piece that 1155 01:29:32,730 --> 01:29:35,500 goes a bit beyond this. 1156 01:29:35,500 --> 01:29:38,260 Doing this by myself is a little bit weird, I'm tired, 1157 01:29:38,260 --> 01:29:40,060 and I will confess, I got a little bit 1158 01:29:40,060 --> 01:29:42,745 rattled when the police came in to check in on me. 1159 01:29:42,745 --> 01:29:43,870 Let's back up for a second. 1160 01:29:43,870 --> 01:29:47,140 So I was talking about two mysteries 1161 01:29:47,140 --> 01:29:49,600 of the modern cosmological model. 1162 01:29:49,600 --> 01:29:52,270 One of them is this question of why the universe is 1163 01:29:52,270 --> 01:29:54,520 so apparently flat. 1164 01:29:54,520 --> 01:29:58,180 The spatial sector of the universe appears to be flat. 1165 01:29:58,180 --> 01:30:00,055 And we had this expectation that the universe 1166 01:30:00,055 --> 01:30:05,400 is either radiation dominated or matter dominated, 1167 01:30:05,400 --> 01:30:09,420 which would give us the density associated with radiation. 1168 01:30:09,420 --> 01:30:11,462 If it was radiation dominated, then the density 1169 01:30:11,462 --> 01:30:13,920 of stuff in our universe would fall off as the scale factor 1170 01:30:13,920 --> 01:30:14,820 to the fourth power. 1171 01:30:14,820 --> 01:30:16,570 If it's matter dominant, it's scale factor 1172 01:30:16,570 --> 01:30:18,067 to the third power. 1173 01:30:18,067 --> 01:30:20,400 When you define a density associated with the curvature, 1174 01:30:20,400 --> 01:30:24,550 it falls off as scale factor to the second power. 1175 01:30:24,550 --> 01:30:28,110 And so if we look at the ratio of the curvature density 1176 01:30:28,110 --> 01:30:34,440 to any other kind of density, it grows as the universe expands. 1177 01:30:34,440 --> 01:30:37,530 So any slight deviation from flatness we 1178 01:30:37,530 --> 01:30:39,300 would expect to grow. 1179 01:30:39,300 --> 01:30:41,430 And that's just confusing. 1180 01:30:41,430 --> 01:30:44,430 Why is it when we make the various measurements that we 1181 01:30:44,430 --> 01:30:46,912 have been making for the past several decades, 1182 01:30:46,912 --> 01:30:48,870 all the evidence is pointing to a universe that 1183 01:30:48,870 --> 01:30:52,590 has a flatness of 0? 1184 01:30:52,590 --> 01:30:55,980 If you sort of imagine that the parameter kappa can 1185 01:30:55,980 --> 01:30:58,360 be any number between minus infinity to infinity, 1186 01:30:58,360 --> 01:31:00,760 why is nature picking out 0? 1187 01:31:00,760 --> 01:31:04,000 Another mystery is why is the cosmic microwave background 1188 01:31:04,000 --> 01:31:05,880 so homogeneous? 1189 01:31:05,880 --> 01:31:08,460 We believe that the universe was a very hot dense plasma 1190 01:31:08,460 --> 01:31:09,660 at very early times. 1191 01:31:09,660 --> 01:31:14,570 It cooled as the universe expanded, 1192 01:31:14,570 --> 01:31:21,740 and when we measured the radiation from that cooling 1193 01:31:21,740 --> 01:31:23,570 expanding ball of plasma, what we find 1194 01:31:23,570 --> 01:31:25,070 is that it has the same temperature 1195 01:31:25,070 --> 01:31:28,520 at every point in the sky to within a part in 100,000. 1196 01:31:28,520 --> 01:31:34,580 But when one looks at the behavior of how light moves 1197 01:31:34,580 --> 01:31:37,760 around in the early universe, if the universe is matter 1198 01:31:37,760 --> 01:31:39,740 dominated or radiation dominated, 1199 01:31:39,740 --> 01:31:43,280 what you find is that a piece of the sky over here cannot 1200 01:31:43,280 --> 01:31:45,650 communicate with a piece of sky over here. 1201 01:31:45,650 --> 01:31:46,850 Or over here, or over here. 1202 01:31:46,850 --> 01:31:50,480 You actually find that the size of the sky 1203 01:31:50,480 --> 01:31:52,400 that, if I look at a piece of sky over 1204 01:31:52,400 --> 01:31:54,500 here, how much of the universe could it talk to, 1205 01:31:54,500 --> 01:31:56,220 it's surprisingly small. 1206 01:31:56,220 --> 01:31:58,970 So how is it that the entire universe 1207 01:31:58,970 --> 01:32:00,110 has the same temperature? 1208 01:32:00,110 --> 01:32:03,590 How is that they are apparently in thermal equilibrium, 1209 01:32:03,590 --> 01:32:08,260 even if they cannot exchange information? 1210 01:32:08,260 --> 01:32:11,380 So I spent some while talking to myself 1211 01:32:11,380 --> 01:32:13,710 after the cameras went out. 1212 01:32:13,710 --> 01:32:16,420 So I'll just sketch what I wrote down. 1213 01:32:16,420 --> 01:32:19,270 A proposed solution to this, to both of these mysteries, 1214 01:32:19,270 --> 01:32:21,310 is what is known as cosmic inflation, something 1215 01:32:21,310 --> 01:32:24,790 that our own Alan Guth shares a lot of the credit for helping 1216 01:32:24,790 --> 01:32:25,930 to develop. 1217 01:32:25,930 --> 01:32:31,130 So recall, if we have a cosmological constant, 1218 01:32:31,130 --> 01:32:33,730 then the scale factor grows exponentially 1219 01:32:33,730 --> 01:32:35,695 and the density of stuff associated 1220 01:32:35,695 --> 01:32:39,190 with that cosmological constant is constant. 1221 01:32:39,190 --> 01:32:42,540 As the universe expands, the energy density 1222 01:32:42,540 --> 01:32:44,290 associated with that cosmological constant 1223 01:32:44,290 --> 01:32:46,060 does not change. 1224 01:32:46,060 --> 01:32:49,100 If our universe is dominated by such a constant, 1225 01:32:49,100 --> 01:32:52,390 then the ratio of density is associated with curvature, 1226 01:32:52,390 --> 01:32:54,970 the density associated with cosmological constant, actually 1227 01:32:54,970 --> 01:32:59,080 falls off inversely with the curvature scale squared, 1228 01:32:59,080 --> 01:33:01,450 and the curvature scale is growing exponentially, 1229 01:33:01,450 --> 01:33:03,970 it means that omega c is being driven 1230 01:33:03,970 --> 01:33:11,410 to zero relative to the density in cosmological constant, as e 1231 01:33:11,410 --> 01:33:12,390 to a factor like this. 1232 01:33:12,390 --> 01:33:14,713 It's being exponentially driven close to 0. 1233 01:33:14,713 --> 01:33:16,630 You'd do do a little bit of work to figure out 1234 01:33:16,630 --> 01:33:18,505 what the timescales associated with this are, 1235 01:33:18,505 --> 01:33:21,280 but this suggests that if you can put the universe in a state 1236 01:33:21,280 --> 01:33:23,590 where it looks like a cosmological constant, 1237 01:33:23,590 --> 01:33:28,090 you can drive the density associated with curvature as 1238 01:33:28,090 --> 01:33:29,570 close to 0 as you want. 1239 01:33:35,350 --> 01:33:38,420 Recall that a cosmological constant is actually 1240 01:33:38,420 --> 01:33:41,300 equivalent to there being a vacuum energy. 1241 01:33:41,300 --> 01:33:43,640 If we think about the universe being filled 1242 01:33:43,640 --> 01:33:46,940 with some kind of a scalar field at early times, 1243 01:33:46,940 --> 01:33:51,410 it can play the role of such a vacuum energy. 1244 01:33:51,410 --> 01:33:53,000 Without going into the details, one 1245 01:33:53,000 --> 01:33:55,070 finds that in an expanding universe 1246 01:33:55,070 --> 01:33:58,080 there is an equation of motion for that scalar field. 1247 01:33:58,080 --> 01:34:01,520 How the field itself behaves is driven by a differential 1248 01:34:01,520 --> 01:34:02,710 equation, looks like this. 1249 01:34:02,710 --> 01:34:04,293 Here's your Hubble parameters, so this 1250 01:34:04,293 --> 01:34:06,650 has to do with a scale factor in here. 1251 01:34:06,650 --> 01:34:09,590 v is a potential for this scalar field, which I'm not 1252 01:34:09,590 --> 01:34:11,630 going to say too much about. 1253 01:34:11,630 --> 01:34:13,885 Take this guy, couple it to your Friedmann equations, 1254 01:34:13,885 --> 01:34:15,260 and the one that's most important 1255 01:34:15,260 --> 01:34:16,630 is the first Friedmann equation. 1256 01:34:16,630 --> 01:34:19,700 What you see is that v of phi is playing the role 1257 01:34:19,700 --> 01:34:22,310 of a cosmological constant. 1258 01:34:22,310 --> 01:34:24,610 It's playing the role of a density. 1259 01:34:24,610 --> 01:34:27,860 So if we can put the universe into a state where 1260 01:34:27,860 --> 01:34:31,190 it is in fact being dominated by this scalar field, 1261 01:34:31,190 --> 01:34:33,290 by the potential associated with the scalar field, 1262 01:34:33,290 --> 01:34:35,660 it will inflate. 1263 01:34:35,660 --> 01:34:38,540 A lot of the research in early universe physics 1264 01:34:38,540 --> 01:34:40,490 that has gone on over the past couple decades 1265 01:34:40,490 --> 01:34:43,670 has gone into understanding what are the consequences of such a 1266 01:34:43,670 --> 01:34:45,080 such a potential. 1267 01:34:45,080 --> 01:34:46,480 Can we make such a potential? 1268 01:34:46,480 --> 01:34:49,460 Do the laws of physics permit something like this to exist? 1269 01:34:49,460 --> 01:34:53,030 If the universe is in this state early on, what changed? 1270 01:34:53,030 --> 01:34:54,745 How is it that this thing evolves? 1271 01:34:54,745 --> 01:34:56,120 Is there is an equation of motion 1272 01:34:56,120 --> 01:34:57,745 here so that scalar field is presumably 1273 01:34:57,745 --> 01:34:59,480 evolving in some kind of a way? 1274 01:34:59,480 --> 01:35:01,163 Is there a potential, does nature 1275 01:35:01,163 --> 01:35:03,830 permit us to have a field of the sort with a potential that sort 1276 01:35:03,830 --> 01:35:05,720 of goes away after some time? 1277 01:35:05,720 --> 01:35:09,440 Once it goes away, what happens to that field? 1278 01:35:09,440 --> 01:35:12,450 Is there any smoking gun associated with this? 1279 01:35:12,450 --> 01:35:14,990 If we look at the universe and this 1280 01:35:14,990 --> 01:35:20,240 is a plausible explanation for why the universe appears 1281 01:35:20,240 --> 01:35:22,700 to be spatially flat and why the cosmic microwave 1282 01:35:22,700 --> 01:35:25,303 background is so homogeneous. 1283 01:35:25,303 --> 01:35:26,720 Is there anything else that we can 1284 01:35:26,720 --> 01:35:30,020 look at that would basically say yes, the universe did in fact 1285 01:35:30,020 --> 01:35:32,720 have this kind of an expansion? 1286 01:35:32,720 --> 01:35:34,430 Without getting into the weeds too much, 1287 01:35:34,430 --> 01:35:37,520 it turns out that if the universe expanded like this, 1288 01:35:37,520 --> 01:35:41,450 we would expect there to be a primordial background 1289 01:35:41,450 --> 01:35:44,420 of gravitational waves, very low frequency gravitational waves. 1290 01:35:44,420 --> 01:35:47,420 Sort of a moaning background filling the universe. 1291 01:35:47,420 --> 01:35:49,070 And so there's a lot of experiments 1292 01:35:49,070 --> 01:35:51,313 looking for the imprints of such gravitational waves 1293 01:35:51,313 --> 01:35:51,980 on our universe. 1294 01:35:51,980 --> 01:35:53,930 If it is measured, it would allow 1295 01:35:53,930 --> 01:35:57,060 us to directly probe what this inflationary potential actually 1296 01:35:57,060 --> 01:35:58,580 is. 1297 01:35:58,580 --> 01:36:02,430 I'm going to conclude our discussion of cosmology here. 1298 01:36:02,430 --> 01:36:05,390 So some of the quantitative details, 1299 01:36:05,390 --> 01:36:09,740 the way in which inflation can cure the flatness problem 1300 01:36:09,740 --> 01:36:13,940 and cure the homogeneity problem you will explore on problem 1301 01:36:13,940 --> 01:36:15,350 set seven. 1302 01:36:15,350 --> 01:36:18,740 Going into the weeds of how one makes it a scalar field, 1303 01:36:18,740 --> 01:36:21,020 designs a potential and does things like this 1304 01:36:21,020 --> 01:36:22,850 beyond the scope of 8.962-- 1305 01:36:22,850 --> 01:36:27,380 there are courses like this, and I would not 1306 01:36:27,380 --> 01:36:30,500 be surprised if some of the people in this class 1307 01:36:30,500 --> 01:36:32,720 spent a lot more time studying this in their futures 1308 01:36:32,720 --> 01:36:34,400 than I have in my life. 1309 01:36:34,400 --> 01:36:36,260 All right, so that is where we will conclude 1310 01:36:36,260 --> 01:36:39,190 our discussion of cosmology.