1 00:00:00,090 --> 00:00:01,670 The following content is provided 2 00:00:01,670 --> 00:00:03,820 under a Creative Commons license. 3 00:00:03,820 --> 00:00:06,550 Your support will help MIT OpenCourseWare continue 4 00:00:06,550 --> 00:00:10,160 to offer high quality educational resources for free. 5 00:00:10,160 --> 00:00:12,700 To make a donation or to view additional materials 6 00:00:12,700 --> 00:00:16,620 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,620 --> 00:00:17,280 at ocw.mit.edu. 8 00:00:20,424 --> 00:00:24,830 PROFESSOR: A quick review of what we talked about last time. 9 00:00:24,830 --> 00:00:26,760 So the first thing we did last time 10 00:00:26,760 --> 00:00:29,350 was to discuss the age of universe, 11 00:00:29,350 --> 00:00:31,280 considering so far only at this point, 12 00:00:31,280 --> 00:00:34,950 flat matter-dominated universes where the scale factor goes 13 00:00:34,950 --> 00:00:36,540 like t to the 2/3. 14 00:00:36,540 --> 00:00:40,110 And we were easily able to see that the age of such a universe 15 00:00:40,110 --> 00:00:43,320 was 2/3 times h inverse. 16 00:00:43,320 --> 00:00:45,720 And we did discuss what happens if you plug-in numbers 17 00:00:45,720 --> 00:00:50,610 into that formula using the best current value of h, 18 00:00:50,610 --> 00:00:54,986 the value obtained by the Planck team last March. 19 00:00:54,986 --> 00:01:00,260 The age turns out to be about 9.5 to 9.9 billion years. 20 00:01:00,260 --> 00:01:03,530 And that can't be the real age of the universe, we think, 21 00:01:03,530 --> 00:01:06,410 because there are stars that are older than that. 22 00:01:06,410 --> 00:01:08,120 The age of the stars seems to indicate 23 00:01:08,120 --> 00:01:10,240 that the university should be at least, 24 00:01:10,240 --> 00:01:13,640 according to one paper I cited last time, 11.2 billion years 25 00:01:13,640 --> 00:01:16,030 old, and this is younger. 26 00:01:16,030 --> 00:01:18,880 So the conclusion is that our universe is not 27 00:01:18,880 --> 00:01:21,284 a flat, moderate, matter-dominated universe. 28 00:01:21,284 --> 00:01:23,700 We do in fact have good evidence that the universe is very 29 00:01:23,700 --> 00:01:26,500 nearly flat, so it's the matter-dominated part that 30 00:01:26,500 --> 00:01:28,600 has to fail, and it does fail. 31 00:01:28,600 --> 00:01:33,400 We also have good evidence that the universe is dominated today 32 00:01:33,400 --> 00:01:36,510 by dark energy, which we'll be talking about later. 33 00:01:36,510 --> 00:01:40,430 But one of the pieces of evidence for this dark energy 34 00:01:40,430 --> 00:01:41,900 is this age calculation. 35 00:01:41,900 --> 00:01:43,500 The age calculation just does not 36 00:01:43,500 --> 00:01:46,930 work unless you assume that the universe has 37 00:01:46,930 --> 00:01:50,167 a significant component of this dark energy, which 38 00:01:50,167 --> 00:01:51,250 we'll be discussing later. 39 00:01:54,220 --> 00:01:56,310 We then talked about the big bang singularity, 40 00:01:56,310 --> 00:01:58,780 which is an important part of understanding, when you talk 41 00:01:58,780 --> 00:02:02,470 about the age, what exactly you mean by the age, age 42 00:02:02,470 --> 00:02:04,290 since when. 43 00:02:04,290 --> 00:02:05,860 And the point that I tried to make 44 00:02:05,860 --> 00:02:10,470 there is that the big bang singularity, which gives us 45 00:02:10,470 --> 00:02:12,990 mathematically the statement that the scale 46 00:02:12,990 --> 00:02:16,500 factor at some time which we call zero was equal to zero, 47 00:02:16,500 --> 00:02:19,810 and if you put back that formula into other formulas, 48 00:02:19,810 --> 00:02:22,350 you discover that the mass density, for example, 49 00:02:22,350 --> 00:02:26,810 was infinite at this magical time that we call zero. 50 00:02:26,810 --> 00:02:30,350 That singularity is certainly part of our mathematical model 51 00:02:30,350 --> 00:02:32,590 and doesn't go away even when we make 52 00:02:32,590 --> 00:02:35,010 changes in the mathematical model. 53 00:02:35,010 --> 00:02:37,410 But we don't really know if it's an actual feature 54 00:02:37,410 --> 00:02:39,760 of our universe, because there's certainly no reason 55 00:02:39,760 --> 00:02:42,870 to trust this mathematical model all the way back to t 56 00:02:42,870 --> 00:02:45,820 equals zero, where the densities become infinite. 57 00:02:45,820 --> 00:02:47,400 We know a lot about matter and we 58 00:02:47,400 --> 00:02:49,300 think we can predict how matter will behave, 59 00:02:49,300 --> 00:02:52,260 even the temperatures and energy densities somewhat 60 00:02:52,260 --> 00:02:54,230 beyond what we measure in laboratory. 61 00:02:54,230 --> 00:02:56,080 But we don't think we can't necessarily 62 00:02:56,080 --> 00:02:59,780 extrapolate all the way to infinite matter density. 63 00:02:59,780 --> 00:03:01,730 So these equations do break down when 64 00:03:01,730 --> 00:03:03,220 you get very close to t equals zero 65 00:03:03,220 --> 00:03:06,240 and nobody really knows exactly what should be said about t 66 00:03:06,240 --> 00:03:08,170 equals zero. 67 00:03:08,170 --> 00:03:09,990 When we talk about the age, we're 68 00:03:09,990 --> 00:03:12,950 really talking about the age since the extrapolated time 69 00:03:12,950 --> 00:03:16,050 at which a would have been zero in this model, 70 00:03:16,050 --> 00:03:18,500 but we don't really know that it ever actually was zero. 71 00:03:21,350 --> 00:03:25,570 We next discussed the concept of the horizon distance. 72 00:03:25,570 --> 00:03:27,850 If the universe, at least the universe 73 00:03:27,850 --> 00:03:32,220 as we know it, we want to be agnostic about what happened 74 00:03:32,220 --> 00:03:35,435 before t equals zero, but certainly the universe as we 75 00:03:35,435 --> 00:03:37,560 know it, really began at t equals zero in the sense 76 00:03:37,560 --> 00:03:42,220 that that's when structure and complicated things 77 00:03:42,220 --> 00:03:44,330 started to develop. 78 00:03:44,330 --> 00:03:47,090 So since t equals zero there's been only a finite amount 79 00:03:47,090 --> 00:03:48,660 of time elapsed. 80 00:03:48,660 --> 00:03:51,560 And since light travels at a finite speed, 81 00:03:51,560 --> 00:03:53,690 that means that light could only have traveled 82 00:03:53,690 --> 00:03:58,380 some finite distance since the big bang, since t equals zero. 83 00:03:58,380 --> 00:04:00,850 And that means that there's some object which 84 00:04:00,850 --> 00:04:03,490 is the furthest possible object that we could see, 85 00:04:03,490 --> 00:04:07,070 and any object further than that would be in a situation 86 00:04:07,070 --> 00:04:08,930 where light from that object would not yet 87 00:04:08,930 --> 00:04:10,980 have had time to reach us. 88 00:04:10,980 --> 00:04:14,780 And that leads to this notion of our horizon distance, 89 00:04:14,780 --> 00:04:17,500 where the definition of the horizon distance 90 00:04:17,500 --> 00:04:21,490 is that it is defined to be the present distance of the most 91 00:04:21,490 --> 00:04:25,200 distant objects which we are capable of seeing, 92 00:04:25,200 --> 00:04:28,380 limited only by the speed of light. 93 00:04:28,380 --> 00:04:32,170 And once able to calculate that, and in particular for the model 94 00:04:32,170 --> 00:04:34,660 that we so far understood at this point, 95 00:04:34,660 --> 00:04:37,160 the matter-dominated flat universe, 96 00:04:37,160 --> 00:04:41,225 the horizon distance turned out to be three times c t. 97 00:04:41,225 --> 00:04:42,720 Tt 98 00:04:42,720 --> 00:04:44,850 Now remember, if the universe were just static 99 00:04:44,850 --> 00:04:48,750 and appeared at time t ago, then the horizon distance 100 00:04:48,750 --> 00:04:50,500 would just be c t, the distance that light 101 00:04:50,500 --> 00:04:52,630 could travel in time t. 102 00:04:52,630 --> 00:04:55,607 What makes it larger is the fact that the universe is expanding, 103 00:04:55,607 --> 00:04:57,690 and that means that everything was closer together 104 00:04:57,690 --> 00:04:59,670 in the early time and light could make more 105 00:04:59,670 --> 00:05:02,690 progress at the early time, and then these objects have since 106 00:05:02,690 --> 00:05:05,380 moved out to much larger distances. 107 00:05:05,380 --> 00:05:09,040 So that allows the horizon distance to be larger than c t, 108 00:05:09,040 --> 00:05:12,470 and in this particular model, it's three times c t. 109 00:05:16,094 --> 00:05:17,510 Next we began a calculation, which 110 00:05:17,510 --> 00:05:20,790 is what we're going to pick up to continue on now. 111 00:05:20,790 --> 00:05:25,210 We were calculating how to extend our understanding of a 112 00:05:25,210 --> 00:05:27,990 of t, the behavior of the scale factor, 113 00:05:27,990 --> 00:05:31,560 away from the flat case, to ultimately discuss 114 00:05:31,560 --> 00:05:35,000 the two other cases, the open and closed universe. 115 00:05:35,000 --> 00:05:37,310 And we decided on a flip of a coin-- 116 00:05:37,310 --> 00:05:39,504 I promise you I flipped a coin at some point-- 117 00:05:39,504 --> 00:05:40,920 to start with the closed universe. 118 00:05:40,920 --> 00:05:43,761 We could have done either one first. 119 00:05:43,761 --> 00:05:46,010 The equation that we start with is basically the same, 120 00:05:46,010 --> 00:05:48,160 it's the sine of k that makes the difference. 121 00:05:48,160 --> 00:05:51,150 This is the so-called Friedmann equation, 122 00:05:51,150 --> 00:05:54,440 and for a closed universe k is positive. 123 00:05:54,440 --> 00:05:56,700 This is the evolution equation, but it 124 00:05:56,700 --> 00:05:58,600 has to be coupled with an equation that 125 00:05:58,600 --> 00:06:01,340 describes how rho behaves with time. 126 00:06:01,340 --> 00:06:04,920 And for a matter-dominated universe, 127 00:06:04,920 --> 00:06:07,860 rho is just representing non-relativistic matter, 128 00:06:07,860 --> 00:06:11,800 which is nothing but spread as the universe expands. 129 00:06:11,800 --> 00:06:14,740 And the spreading gives a factor of one 130 00:06:14,740 --> 00:06:19,310 over a cubed as the volume grows as a cubed. 131 00:06:19,310 --> 00:06:22,430 And that means that rho times a cubed is a constant, 132 00:06:22,430 --> 00:06:24,890 and that expresses everything that there 133 00:06:24,890 --> 00:06:27,460 is to know about how rho behaves with time. 134 00:06:30,820 --> 00:06:33,430 OK, then after writing these equations, 135 00:06:33,430 --> 00:06:36,260 we said that things will simplify a little bit, not 136 00:06:36,260 --> 00:06:39,756 a lot, but a little bit if we redefine variables basically 137 00:06:39,756 --> 00:06:42,380 to incorporate all the constants that appear in these equations 138 00:06:42,380 --> 00:06:45,590 into one overall constant. 139 00:06:45,590 --> 00:06:50,520 And we decided, or I claimed, that a good way 140 00:06:50,520 --> 00:06:53,040 to do that, an economical way to do that, 141 00:06:53,040 --> 00:06:56,470 is to define things so that the variables all 142 00:06:56,470 --> 00:06:59,120 have units which are easily understood. 143 00:06:59,120 --> 00:07:01,620 And in this case the units of length 144 00:07:01,620 --> 00:07:03,460 can describe everything that we need, 145 00:07:03,460 --> 00:07:05,490 so we chose to express everything 146 00:07:05,490 --> 00:07:08,890 in terms of variables that have units of length. 147 00:07:08,890 --> 00:07:12,450 So the scale factor itself is units of meters per notch, 148 00:07:12,450 --> 00:07:13,724 and that's not a length. 149 00:07:13,724 --> 00:07:15,640 And notches we'd like to get rid of because we 150 00:07:15,640 --> 00:07:17,740 know they're un-physical. 151 00:07:17,740 --> 00:07:23,220 That is, there's no standard for what the notch should be. 152 00:07:23,220 --> 00:07:25,930 So if we divide a of t by the square root of k, 153 00:07:25,930 --> 00:07:27,336 the notches disappear, and we get 154 00:07:27,336 --> 00:07:29,210 something which just has the units of meters, 155 00:07:29,210 --> 00:07:36,160 or units of length, and I call that a twiddle of t. 156 00:07:36,160 --> 00:07:40,810 Similarly, but more obviously, t can be turned into a length 157 00:07:40,810 --> 00:07:43,130 by just multiplying by c, the speed of light. 158 00:07:43,130 --> 00:07:46,310 So I defined a variable t tilde which is just c times t. 159 00:07:46,310 --> 00:07:49,650 So both of these new variables, with the tildes, 160 00:07:49,650 --> 00:07:51,830 have units of length. 161 00:07:51,830 --> 00:07:53,600 And then the Friedmann equation can 162 00:07:53,600 --> 00:07:55,500 be rewritten, just reshuffling things 163 00:07:55,500 --> 00:08:00,560 according to these new definitions, in this way where 164 00:08:00,560 --> 00:08:05,060 all the constants are lumped into this variable alpha, where 165 00:08:05,060 --> 00:08:07,550 alpha is this complicated expression, which absorbs all 166 00:08:07,550 --> 00:08:09,910 the constants from the earlier equation. 167 00:08:09,910 --> 00:08:13,090 Alpha also has units of length, and even 168 00:08:13,090 --> 00:08:15,880 though it has a rho times a tilde cubed in it, 169 00:08:15,880 --> 00:08:19,650 and both rho and a tilde each depend on time, 170 00:08:19,650 --> 00:08:24,420 the product of rho times a tilde cubed is a constant because rho 171 00:08:24,420 --> 00:08:27,930 times a cubed is a constant and a tilde differs only 172 00:08:27,930 --> 00:08:31,080 by the square root of k, which is also a constant. 173 00:08:31,080 --> 00:08:32,659 So alpha is a constant even though it 174 00:08:32,659 --> 00:08:35,140 has time dependent factors. 175 00:08:35,140 --> 00:08:37,100 The time dependence of those two factors 176 00:08:37,100 --> 00:08:38,980 cancel each other out to give something 177 00:08:38,980 --> 00:08:42,890 which is time independent and can be evaluated any old time. 178 00:08:42,890 --> 00:08:49,510 So this now is our equation, and we proceeded to manipulate it. 179 00:08:49,510 --> 00:08:54,010 So the first thing we did was to take it's square root, 180 00:08:54,010 --> 00:08:57,670 and to rearrange it so that d t tilde appeared on one side, 181 00:08:57,670 --> 00:08:59,670 and everything else was on the other side. 182 00:08:59,670 --> 00:09:02,140 And everything else depends on a tilde 183 00:09:02,140 --> 00:09:05,330 but not explicitly on time. 184 00:09:05,330 --> 00:09:07,720 So this completely separates everything 185 00:09:07,720 --> 00:09:09,540 that depends on t tilde on the left, 186 00:09:09,540 --> 00:09:13,430 and everything that depends on a tilde on the right. 187 00:09:13,430 --> 00:09:16,110 And now we can just integrate both sides of that equation, 188 00:09:16,110 --> 00:09:19,200 and and they point to that, that there are basically 189 00:09:19,200 --> 00:09:22,000 two ways of proceeding here, one of which 190 00:09:22,000 --> 00:09:24,810 we already did when we did the flat case. 191 00:09:24,810 --> 00:09:27,900 When we did the flat case, we integrated both sides 192 00:09:27,900 --> 00:09:29,820 as indefinite integrals. 193 00:09:29,820 --> 00:09:33,750 And when you carry out an indefinite integration, 194 00:09:33,750 --> 00:09:35,600 you get a constant of integration, 195 00:09:35,600 --> 00:09:38,960 which then becomes a constant in your solution. 196 00:09:38,960 --> 00:09:43,470 And in that case we discovered that the constant really 197 00:09:43,470 --> 00:09:45,660 just shifted the origin of time. 198 00:09:45,660 --> 00:09:48,480 And since we had not said anything previously that 199 00:09:48,480 --> 00:09:51,190 in any way determined the origin of time, 200 00:09:51,190 --> 00:09:54,260 we used that constant to arrange the origin of time 201 00:09:54,260 --> 00:09:56,880 so that a of zero would equal zero, 202 00:09:56,880 --> 00:09:59,400 and that eliminated the constant. 203 00:09:59,400 --> 00:10:03,950 Just for variety, I am going to do it another way this time. 204 00:10:03,950 --> 00:10:05,860 Instead of doing an indefinite integral, 205 00:10:05,860 --> 00:10:07,734 I will do a definite integral. 206 00:10:07,734 --> 00:10:09,150 And if you do a definite integral, 207 00:10:09,150 --> 00:10:10,290 you have to make sure you're integrating 208 00:10:10,290 --> 00:10:12,540 both sides over the same range, or at least 209 00:10:12,540 --> 00:10:13,880 corresponding ranges. 210 00:10:13,880 --> 00:10:15,960 We have different names for the variables, 211 00:10:15,960 --> 00:10:18,520 but the range of integration of the two sides 212 00:10:18,520 --> 00:10:21,560 has to match in order to maintain 213 00:10:21,560 --> 00:10:24,050 the equality between the two sides. 214 00:10:24,050 --> 00:10:26,850 So we're going to integrate the left side from zero 215 00:10:26,850 --> 00:10:28,580 to some final time. 216 00:10:28,580 --> 00:10:34,470 And I'm going to call the final time t tilde variable sub 217 00:10:34,470 --> 00:10:36,700 f, for f to stand for final. 218 00:10:36,700 --> 00:10:38,200 And there's no real final time here. 219 00:10:38,200 --> 00:10:40,116 It could be any time, it's just the final time 220 00:10:40,116 --> 00:10:42,290 for the integration, and in the end 221 00:10:42,290 --> 00:10:46,290 discover then how things behave at time t tilde sub f. 222 00:10:46,290 --> 00:10:48,940 And then once we've figured that out we can drop the effort. 223 00:10:48,940 --> 00:10:53,790 It will be a formula that will be valid for any time. 224 00:10:53,790 --> 00:10:56,740 So the left hand side is integrated from zero 225 00:10:56,740 --> 00:10:58,910 to time t tilde f, the right hand side 226 00:10:58,910 --> 00:11:02,650 has to be integrated over the corresponding time interval. 227 00:11:02,650 --> 00:11:08,040 And we would like to define a tilde and the origin of time 228 00:11:08,040 --> 00:11:10,725 so that a tilde equals zero at time zero, 229 00:11:10,725 --> 00:11:14,570 the same convention we used for the other case. 230 00:11:14,570 --> 00:11:17,110 The standard convention in cosmology, t equals zero, 231 00:11:17,110 --> 00:11:18,750 is the instance of the Big Bang. 232 00:11:18,750 --> 00:11:20,740 And the instance of the Big Bang is the time 233 00:11:20,740 --> 00:11:23,600 at which the scale factor vanished. 234 00:11:23,600 --> 00:11:28,160 So that means when we have a lower limit of integration 235 00:11:28,160 --> 00:11:30,390 of t tilde equals zero on the left hand side, 236 00:11:30,390 --> 00:11:32,280 we should have a lower limit of integration 237 00:11:32,280 --> 00:11:36,620 of a tilde equals zero on the right hand side. 238 00:11:36,620 --> 00:11:39,190 And similarly the upper limits of integration 239 00:11:39,190 --> 00:11:41,540 should correspond to each other. 240 00:11:41,540 --> 00:11:44,120 So the upper limit of integration on the left hand 241 00:11:44,120 --> 00:11:48,680 side was t tilde sub f, really just an arbitrary time 242 00:11:48,680 --> 00:11:52,120 that we designated by the subscript f. 243 00:11:52,120 --> 00:11:54,020 So the right hand side the limit should 244 00:11:54,020 --> 00:11:56,760 be the value of a tilde at that time. 245 00:11:56,760 --> 00:12:00,300 And I'll call that a tilde sub f. 246 00:12:00,300 --> 00:12:02,710 So a tilde sub f is just defined to be 247 00:12:02,710 --> 00:12:08,020 the value of a tilde at the time t tilde sub f. 248 00:12:08,020 --> 00:12:11,130 And in this way the limits of integration on the two sides 249 00:12:11,130 --> 00:12:13,150 correspond and now we can integrate them 250 00:12:13,150 --> 00:12:16,630 and we don't need any new integration constants. 251 00:12:16,630 --> 00:12:19,100 These definite integrals have definite values, 252 00:12:19,100 --> 00:12:21,929 which is why they're called definite integrals, I suppose. 253 00:12:21,929 --> 00:12:23,720 OK any questions about that, because that's 254 00:12:23,720 --> 00:12:26,590 where we're ready to take off and start doing new material. 255 00:12:26,590 --> 00:12:27,090 Yes. 256 00:12:27,090 --> 00:12:29,230 AUDIENCE: I have a general question. 257 00:12:29,230 --> 00:12:33,720 So regarding the issue of how we're not really sure how 258 00:12:33,720 --> 00:12:36,054 to extrapolate up to t equals zero, 259 00:12:36,054 --> 00:12:37,720 this is just kind of a general question. 260 00:12:37,720 --> 00:12:40,220 I was wondering how we're kind of assuming 261 00:12:40,220 --> 00:12:42,930 throughout that time kind of flows uniformly, 262 00:12:42,930 --> 00:12:44,400 and so I've heard about something 263 00:12:44,400 --> 00:12:46,420 like gravitational time dilation. 264 00:12:46,420 --> 00:12:48,750 So at the beginning especially when there's 265 00:12:48,750 --> 00:12:52,420 such a high density of matter or radiation, 266 00:12:52,420 --> 00:12:56,240 then wouldn't that affect how, I guess, time flows? 267 00:12:56,240 --> 00:12:58,100 PROFESSOR: OK good question, good question. 268 00:12:58,100 --> 00:13:02,300 The question was, does things like general realistic time 269 00:13:02,300 --> 00:13:05,970 dilation affect how time flows, and are we 270 00:13:05,970 --> 00:13:09,750 perhaps being overly simplistic and assuming that time just 271 00:13:09,750 --> 00:13:13,210 flows smoothly from time zero onward. 272 00:13:13,210 --> 00:13:16,130 The answer to that is that general relativity does 273 00:13:16,130 --> 00:13:18,250 predict an extra time dilation, which 274 00:13:18,250 --> 00:13:21,560 in fact is built into the Doppler shift calculation 275 00:13:21,560 --> 00:13:23,690 that we already did. 276 00:13:23,690 --> 00:13:26,760 And there are other instances where similar things happen. 277 00:13:26,760 --> 00:13:28,920 If a photon travels from the floor of this room 278 00:13:28,920 --> 00:13:30,490 to the ceiling of this room, there's 279 00:13:30,490 --> 00:13:35,080 a small Doppler shift, a small shift in the timing. 280 00:13:35,080 --> 00:13:37,460 And you could see it in principle with clocks as well. 281 00:13:37,460 --> 00:13:39,960 If you had a clock on the floor, and a clock on the ceiling, 282 00:13:39,960 --> 00:13:42,260 they would not run at quite the same rate. 283 00:13:42,260 --> 00:13:44,690 But to talk about time dilation, you always 284 00:13:44,690 --> 00:13:47,340 have to have two clocks to compare. 285 00:13:47,340 --> 00:13:50,220 In the case of the universe, we have this thing 286 00:13:50,220 --> 00:13:55,130 that we call cosmic time, which can be measured on any clock. 287 00:13:55,130 --> 00:13:56,890 The homogeneity assumption implies 288 00:13:56,890 --> 00:13:58,680 that all clocks will do the same thing, 289 00:13:58,680 --> 00:14:04,580 so so the issue of time dilation really does not arise. 290 00:14:04,580 --> 00:14:06,940 Our definition of cosmic time defines 291 00:14:06,940 --> 00:14:10,290 time that is the time variable that we will use. 292 00:14:10,290 --> 00:14:13,510 And when we say it flows uniformly from zero up 293 00:14:13,510 --> 00:14:17,610 to the present time, that word uniformly sounds sensible. 294 00:14:17,610 --> 00:14:20,170 But if you think about it, I don't know what it means. 295 00:14:20,170 --> 00:14:23,880 So I don't even know how to ask if it's really uniform or not. 296 00:14:23,880 --> 00:14:27,080 It's certainly true that our time variable 297 00:14:27,080 --> 00:14:29,440 evolves from zero up to some final time, 298 00:14:29,440 --> 00:14:33,030 but smoothly or uniformly is not really a question we 299 00:14:33,030 --> 00:14:36,685 can ask until one has some other clock that one can compare it 300 00:14:36,685 --> 00:14:37,525 to. 301 00:14:37,525 --> 00:14:38,150 PROFESSOR: Yes. 302 00:14:38,150 --> 00:14:41,024 AUDIENCE: So you said at the beginning 303 00:14:41,024 --> 00:14:43,419 that something like an infinite density 304 00:14:43,419 --> 00:14:46,772 is just an effect of our equation. 305 00:14:46,772 --> 00:14:50,010 Aren't a lot of theories like inflation come out 306 00:14:50,010 --> 00:14:53,300 of assuming that the universe had gotten 307 00:14:53,300 --> 00:14:57,310 some infinite, dense, small area? 308 00:14:57,310 --> 00:15:02,053 So what effect does this assumption that may or may not 309 00:15:02,053 --> 00:15:04,920 be correct have on theories? 310 00:15:04,920 --> 00:15:08,180 PROFESSOR: OK, the question is, if we are not 311 00:15:08,180 --> 00:15:11,180 sure we should believe the t equals zero singularity, 312 00:15:11,180 --> 00:15:13,660 how does that affect other theories like inflation, which 313 00:15:13,660 --> 00:15:17,740 are based on extrapolating backwards to very early times. 314 00:15:17,740 --> 00:15:20,110 And there is an answer to that. 315 00:15:20,110 --> 00:15:22,275 The answer may or may not sound sensible, 316 00:15:22,275 --> 00:15:24,030 and it may or may not be sensible. 317 00:15:24,030 --> 00:15:26,240 It's hard to know for sure. 318 00:15:26,240 --> 00:15:29,440 But one can be more detailed and ask 319 00:15:29,440 --> 00:15:32,590 how far back do we think we can trust our equations? 320 00:15:32,590 --> 00:15:34,090 And nobody really knows the answer 321 00:15:34,090 --> 00:15:37,380 to that of course, that's part of the uncertainty here. 322 00:15:37,380 --> 00:15:42,050 But a plausible answer which is kind of the working hypothesis 323 00:15:42,050 --> 00:15:45,880 for many of us, is that the only obstacle to extrapolating 324 00:15:45,880 --> 00:15:48,320 backwards is our lack of knowledge 325 00:15:48,320 --> 00:15:51,125 of the quantum effects of gravity, 326 00:15:51,125 --> 00:15:52,500 and therefore the quantum effects 327 00:15:52,500 --> 00:15:55,340 of what space time looks like. 328 00:15:55,340 --> 00:15:57,930 And we can estimate where that sets in. 329 00:15:57,930 --> 00:16:01,870 And it's at a time called the Planck time, which is about 10 330 00:16:01,870 --> 00:16:06,010 to the minus 43 seconds, and inflation, 331 00:16:06,010 --> 00:16:08,880 which we'll see later as sort of a natural timescale 332 00:16:08,880 --> 00:16:12,640 of about probably 10 to the minus 37 seconds. 333 00:16:12,640 --> 00:16:14,976 So although that's incredibly small, 334 00:16:14,976 --> 00:16:16,600 it's actually incredibly large compared 335 00:16:16,600 --> 00:16:19,390 to 10 to the minus 43 seconds. 336 00:16:19,390 --> 00:16:23,810 So we think there is at least a basis for believing 337 00:16:23,810 --> 00:16:26,450 that things like our discussions of inflation which we'll 338 00:16:26,450 --> 00:16:30,050 be talking about later, are valid even though we don't 339 00:16:30,050 --> 00:16:32,510 think we can extrapolate back to t equals zero. 340 00:16:35,410 --> 00:16:35,910 Yes. 341 00:16:35,910 --> 00:16:38,368 AUDIENCE: I have a question about use 342 00:16:38,368 --> 00:16:39,600 of the definite integrals. 343 00:16:39,600 --> 00:16:40,225 PROFESSOR: Yes. 344 00:16:40,225 --> 00:16:42,872 AUDIENCE: So we have a twiddle defined 345 00:16:42,872 --> 00:16:44,655 as a over square root of k. 346 00:16:44,655 --> 00:16:47,075 And we noticed in our equation that a 347 00:16:47,075 --> 00:16:53,180 goes to zero as t goes to zero, so a twiddle also goes to zero. 348 00:16:53,180 --> 00:16:55,692 How do we know then that that definite integral 349 00:16:55,692 --> 00:16:57,150 is convergent, because then we have 350 00:16:57,150 --> 00:17:01,640 zero over zero competing case. [INAUDIBLE] 351 00:17:01,640 --> 00:17:06,390 PROFESSOR: Let me think. 352 00:17:09,676 --> 00:17:11,300 Yeah, we'll see, I think is the answer. 353 00:17:11,300 --> 00:17:12,349 How do we know it's convergent? 354 00:17:12,349 --> 00:17:13,940 Well, we're going to actually do the integral, 355 00:17:13,940 --> 00:17:15,599 and then the integral does converge. 356 00:17:15,599 --> 00:17:16,310 You are right. 357 00:17:16,310 --> 00:17:20,030 The integrand does become zero over zero, but that in fact 358 00:17:20,030 --> 00:17:24,430 means the integrand is some finite number actually. 359 00:17:24,430 --> 00:17:27,839 Both numerator and denominator go to zero, 360 00:17:27,839 --> 00:17:29,770 I mean let me think about this. 361 00:17:29,770 --> 00:17:31,570 I guess a tilde squared becomes negligible, 362 00:17:31,570 --> 00:17:34,760 so the denominator goes like one over the square root 363 00:17:34,760 --> 00:17:35,650 of a tilde. 364 00:17:35,650 --> 00:17:38,570 So in fact the numerator divided by the denominator 365 00:17:38,570 --> 00:17:41,567 goes like the square root of a tilde as time goes to zero. 366 00:17:41,567 --> 00:17:43,650 Because you have a square root in the denominator, 367 00:17:43,650 --> 00:17:46,130 the a tilde squared becomes negligible, 368 00:17:46,130 --> 00:17:48,710 so it's manifestly convergent. 369 00:17:48,710 --> 00:17:51,210 The integrand does not even blow up, 370 00:17:51,210 --> 00:17:54,564 even though a tilde does go to zero. 371 00:17:54,564 --> 00:17:56,480 It's certainly worth looking at, you're right. 372 00:17:56,480 --> 00:17:58,880 One should always check to make sure integrals are convergent. 373 00:17:58,880 --> 00:18:01,379 But since we will actually be explicitly doing the interval, 374 00:18:01,379 --> 00:18:03,890 if it was divergent we would get a divergent answer, 375 00:18:03,890 --> 00:18:08,357 and we will not as you'll see in a couple minutes. 376 00:18:08,357 --> 00:18:09,190 Any other questions? 377 00:18:11,960 --> 00:18:15,525 OK, so in that case, to the blackboard. 378 00:18:36,770 --> 00:18:38,340 OK, so I will write on the blackboard 379 00:18:38,340 --> 00:18:42,150 the same equation we have up there, so we can continue. 380 00:18:42,150 --> 00:18:48,240 t twiddle f is equal to the integral from zero 381 00:18:48,240 --> 00:18:56,940 to a twiddle sub f, a twiddle d a twiddle 382 00:18:56,940 --> 00:19:02,870 over the square root of two alpha 383 00:19:02,870 --> 00:19:07,655 a twiddle minus a twiddle squared. 384 00:19:12,480 --> 00:19:14,290 OK, so what we'd like to do now is 385 00:19:14,290 --> 00:19:17,246 to carry out the integral on the right hand side. 386 00:19:17,246 --> 00:19:19,620 Ideally what we'd like to do is to carry out the integral 387 00:19:19,620 --> 00:19:26,486 on the right hand side and get some function of a tilde f 388 00:19:26,486 --> 00:19:28,490 and then we'd like to convert that function 389 00:19:28,490 --> 00:19:31,240 to be able to write a tilde f as a function of time. 390 00:19:31,240 --> 00:19:33,550 We actually won't quite be able to do that. 391 00:19:33,550 --> 00:19:36,130 We'll end up with what's called the parametric solution, 392 00:19:36,130 --> 00:19:38,850 and you'll see how that arises and what that means. 393 00:19:38,850 --> 00:19:40,790 I don't need to try to describe exactly 394 00:19:40,790 --> 00:19:43,580 in advance what that means. 395 00:19:43,580 --> 00:19:46,980 Doing the integral can be done by some tricks, 396 00:19:46,980 --> 00:19:48,480 some substitutions. 397 00:19:48,480 --> 00:19:51,990 And the first substitution is based 398 00:19:51,990 --> 00:19:56,610 on completing the square in the denominator, 399 00:19:56,610 --> 00:19:59,180 and that motivates the substitution that we will make. 400 00:19:59,180 --> 00:20:03,070 So we can rewrite this just by doing some algebra 401 00:20:03,070 --> 00:20:05,210 on the denominator, which is called completing 402 00:20:05,210 --> 00:20:06,820 the square for reasons that you'll 403 00:20:06,820 --> 00:20:09,270 see when I write down what it is. 404 00:20:09,270 --> 00:20:13,840 The numerator will stay a tilde d a tilde. 405 00:20:13,840 --> 00:20:18,370 And the denominator can be written 406 00:20:18,370 --> 00:20:28,572 as alpha squared minus a tilde minus alpha quantity squared. 407 00:20:28,572 --> 00:20:30,030 So completing the square just means 408 00:20:30,030 --> 00:20:34,040 to put the a tilde inside a perfect square. 409 00:20:34,040 --> 00:20:35,980 And the nice thing about this is that now we 410 00:20:35,980 --> 00:20:38,160 can shift our variable of integration, 411 00:20:38,160 --> 00:20:40,790 and turn this into just a single variable instead 412 00:20:40,790 --> 00:20:42,090 of the sum of the two. 413 00:20:42,090 --> 00:20:43,673 And then you have an expression, which 414 00:20:43,673 --> 00:20:45,900 is clearly simpler looking than this one, which 415 00:20:45,900 --> 00:20:48,130 had a tildes in both places. 416 00:20:48,130 --> 00:20:50,760 Now the variable integration will appear only there. 417 00:20:50,760 --> 00:20:54,360 And substitution which does that obviously enough, is we're 418 00:20:54,360 --> 00:20:57,240 going to let something, we can choose whatever we want, 419 00:20:57,240 --> 00:20:58,830 and then once I call it x, we're going 420 00:20:58,830 --> 00:21:03,910 to let x equal a tilde minus alpha. 421 00:21:07,200 --> 00:21:11,560 And we're just going to rewrite that integral in terms of x. 422 00:21:11,560 --> 00:21:15,070 So what we get when we do that is t 423 00:21:15,070 --> 00:21:21,030 sub f tilde is equal to the rewriting of that integral, 424 00:21:21,030 --> 00:21:28,110 and just substituting the a tilde becomes x plus alpha. 425 00:21:30,900 --> 00:21:34,240 So we get x plus alpha where a tilde was, 426 00:21:34,240 --> 00:21:36,795 and d a tilde becomes just d x. 427 00:21:40,060 --> 00:21:42,520 And the denominator, which was our motivation 428 00:21:42,520 --> 00:21:44,850 for making the substitution in the first place, 429 00:21:44,850 --> 00:21:47,890 becomes just alpha squared minus x squared. 430 00:21:55,950 --> 00:21:57,806 So this is perfectly straight forward. 431 00:21:57,806 --> 00:21:59,180 The next step which is important, 432 00:21:59,180 --> 00:22:01,200 is to get the limits of integration right, 433 00:22:01,200 --> 00:22:03,040 because with this definite integral method 434 00:22:03,040 --> 00:22:05,456 we really have to make sure that our limits of integration 435 00:22:05,456 --> 00:22:06,250 are correct. 436 00:22:06,250 --> 00:22:10,570 So straightforward to do that, the lower limit of integration 437 00:22:10,570 --> 00:22:16,130 was a tilde equals zero, and if a tilde equals zero, 438 00:22:16,130 --> 00:22:18,560 x is equal to minus alpha. 439 00:22:18,560 --> 00:22:20,180 So the lower limit of integration 440 00:22:20,180 --> 00:22:22,770 expressed as a value of x is minus alpha. 441 00:22:25,600 --> 00:22:28,770 And the upper limit expressed as x 442 00:22:28,770 --> 00:22:31,890 was a tilde f, and that means x is 443 00:22:31,890 --> 00:22:35,390 equal to a tilde f minus alpha. 444 00:22:35,390 --> 00:22:39,167 So the limit here is a tilde sub f minus alpha 445 00:22:39,167 --> 00:22:40,625 for the upper limit of integration. 446 00:22:47,680 --> 00:22:51,980 OK, now this integral is still not easy, 447 00:22:51,980 --> 00:22:56,360 but it can be made easy by one more substitution. 448 00:22:56,360 --> 00:22:59,500 And the one more substitution is a trigonometric substitution 449 00:22:59,500 --> 00:23:02,620 to simplify the denominator. 450 00:23:02,620 --> 00:23:09,970 We can let x equal minus alpha cosine 451 00:23:09,970 --> 00:23:15,410 of theta, where theta is our new variable of integration. 452 00:23:15,410 --> 00:23:17,550 And then alpha squared minus x squared 453 00:23:17,550 --> 00:23:19,250 becomes alpha squared minus alpha 454 00:23:19,250 --> 00:23:20,890 squared cosine squared theta. 455 00:23:20,890 --> 00:23:23,348 And the alphas factor out and you have the square root of 1 456 00:23:23,348 --> 00:23:24,560 minus cosine squared theta. 457 00:23:24,560 --> 00:23:27,559 1 minus cosine squared theta is sine squared theta, which 458 00:23:27,559 --> 00:23:29,600 is a convenient thing to take the square root of, 459 00:23:29,600 --> 00:23:30,860 you just get sine theta. 460 00:23:30,860 --> 00:23:33,190 And everything else also simplifies. 461 00:23:33,190 --> 00:23:36,230 And the bottom line, which I will just give you, 462 00:23:36,230 --> 00:23:49,380 is that now we find that t sub s tilde is just the integral of 1 463 00:23:49,380 --> 00:23:55,040 minus cosine theta d theta. 464 00:23:57,890 --> 00:23:58,490 That's it. 465 00:23:58,490 --> 00:24:01,930 Everything simplifies to that which is easy to integrate. 466 00:24:01,930 --> 00:24:05,880 We also have to keep track of our limits of integration. 467 00:24:05,880 --> 00:24:09,850 The limit of integration, the lower limit of integration, 468 00:24:09,850 --> 00:24:12,790 is where x equals minus alpha. 469 00:24:12,790 --> 00:24:15,410 And if x equals minus alpha, that 470 00:24:15,410 --> 00:24:18,400 means cosine theta equals 1. 471 00:24:18,400 --> 00:24:22,060 And cosine theta equals 1 means theta equals 0. 472 00:24:22,060 --> 00:24:27,710 So the limit of integration on theta is easy, it starts at 0. 473 00:24:27,710 --> 00:24:30,370 And the final value is really just a value 474 00:24:30,370 --> 00:24:35,130 that corresponds to the final value of x 475 00:24:35,130 --> 00:24:39,350 which is a twiddle f minus alpha. 476 00:24:39,350 --> 00:24:41,730 For now I'm just going to call it theta sub f. 477 00:24:49,732 --> 00:24:51,190 Things that we have just determined 478 00:24:51,190 --> 00:24:53,560 is the value of theta that goes with the upper limit there. 479 00:24:53,560 --> 00:24:56,101 We'll figure out in a second how to write it more explicitly. 480 00:24:56,101 --> 00:24:58,810 Let me first do the integral to just get that out of the way. 481 00:24:58,810 --> 00:25:06,410 Doing the integral, you get alpha times 1 482 00:25:06,410 --> 00:25:09,935 minus cosine theta sub f. 483 00:25:19,100 --> 00:25:22,560 So this in fact becomes half of our solution. 484 00:25:22,560 --> 00:25:27,062 It expresses t sub f in terms of theta sub f. 485 00:25:27,062 --> 00:25:29,020 And now that we're done with the whole problem, 486 00:25:29,020 --> 00:25:30,880 I'm going to drop the subscript f. 487 00:25:30,880 --> 00:25:32,520 There's just some time that we're 488 00:25:32,520 --> 00:25:35,852 interested in which is called t and the value of a at that time 489 00:25:35,852 --> 00:25:38,060 will be called a, and the value of theta at that time 490 00:25:38,060 --> 00:25:39,010 will be called theta. 491 00:25:39,010 --> 00:25:41,430 So I'm just going to drop the subscript everywhere 492 00:25:41,430 --> 00:25:44,740 because we are now in a situation were the subscript is 493 00:25:44,740 --> 00:25:46,580 everywhere, so dropping everywhere 494 00:25:46,580 --> 00:25:48,950 does not lose any information. 495 00:25:48,950 --> 00:25:50,640 So one of our equations is going to be 496 00:25:50,640 --> 00:25:56,040 simply t is equal to alpha times 1 minus cosine theta. 497 00:25:59,850 --> 00:26:02,660 And another equation will come from figuring out 498 00:26:02,660 --> 00:26:04,297 what theta sub f really is, which 499 00:26:04,297 --> 00:26:06,380 I said comes from making sure that the upper limit 500 00:26:06,380 --> 00:26:08,620 of integration here corresponds to the upper limit 501 00:26:08,620 --> 00:26:11,970 of integration in the previous integral. 502 00:26:11,970 --> 00:26:14,480 So theta sub f, I had to determine 503 00:26:14,480 --> 00:26:17,755 this on another blackboard and then put the final equations 504 00:26:17,755 --> 00:26:18,255 together. 505 00:26:18,255 --> 00:26:19,129 AUDIENCE: [INAUDIBLE] 506 00:26:25,010 --> 00:26:28,460 PROFESSOR: Yes, that's right I don't want to drop twiddles. 507 00:26:28,460 --> 00:26:32,490 T twiddle, thank you, is equal to alpha times 1 508 00:26:32,490 --> 00:26:44,390 minus cosine theta, and and then we have the final value of x. 509 00:26:44,390 --> 00:26:50,790 Xx x is equal to a tilde minus alpha. 510 00:26:50,790 --> 00:26:54,540 So the final value of x is equal to the final value 511 00:26:54,540 --> 00:26:56,765 of a tilde minus alpha. 512 00:26:59,720 --> 00:27:04,600 And the final value of x is also related to theta 513 00:27:04,600 --> 00:27:06,825 by this equation, which is what we're trying to get, 514 00:27:06,825 --> 00:27:10,090 how theta is related to the other variables. 515 00:27:10,090 --> 00:27:17,800 So this is equal to minus alpha cosine of theta sub f. 516 00:27:23,010 --> 00:27:25,920 And now we might want to, for example, 517 00:27:25,920 --> 00:27:30,060 solve this for a tilde sub f, which just involves looking 518 00:27:30,060 --> 00:27:32,700 at the right hand half of the equation, 519 00:27:32,700 --> 00:27:36,690 bringing this alpha to the other side making a plus alpha. 520 00:27:36,690 --> 00:27:42,300 So this implies that a tilde sub f is equal to alpha times 1 521 00:27:42,300 --> 00:27:46,442 minus cosine theta sub f. 522 00:27:46,442 --> 00:27:48,150 So this equation now just says that theta 523 00:27:48,150 --> 00:27:50,110 sub f means what it should mean to give us 524 00:27:50,110 --> 00:27:52,510 a final limit of integration that corresponds 525 00:27:52,510 --> 00:27:53,885 to the final limit of integration 526 00:27:53,885 --> 00:27:57,560 on our original integral, which we called a tilde sub f. 527 00:27:57,560 --> 00:27:58,060 Yes. 528 00:27:58,060 --> 00:27:59,976 AUDIENCE: Sorry, perhaps I missed it. 529 00:27:59,976 --> 00:28:01,892 But when you are doing the integral of 1 530 00:28:01,892 --> 00:28:04,121 minus cosine theta, how come-- 531 00:28:04,121 --> 00:28:05,370 PROFESSOR: Oh, I got it wrong. 532 00:28:08,110 --> 00:28:08,710 Good point. 533 00:28:12,920 --> 00:28:14,430 The integral of cosine theta d theta 534 00:28:14,430 --> 00:28:16,855 is sine theta, not cosine theta. 535 00:28:16,855 --> 00:28:17,737 AUDIENCE: [INAUDIBLE] 536 00:28:22,189 --> 00:28:23,230 PROFESSOR: Wait a minute. 537 00:28:23,230 --> 00:28:23,730 Hold on. 538 00:28:23,730 --> 00:28:25,950 Do I still have it wrong? 539 00:28:25,950 --> 00:28:28,500 AUDIENCE: [INAUDIBLE] 540 00:28:28,500 --> 00:28:29,977 PROFESSOR: OK hold on. 541 00:28:29,977 --> 00:28:31,310 There was an alpha missing here. 542 00:28:31,310 --> 00:28:34,080 That's part of the problem, coming 543 00:28:34,080 --> 00:28:38,610 from the original equation here. 544 00:28:38,610 --> 00:28:40,360 Let's see if I have this right now. 545 00:28:55,767 --> 00:28:58,380 This is copying from a long line altogether. 546 00:28:58,380 --> 00:29:00,020 So I got both factors wrong. 547 00:29:03,000 --> 00:29:04,950 And then wait a minute. 548 00:29:12,640 --> 00:29:15,689 I think this is right now. 549 00:29:15,689 --> 00:29:17,105 There's still a wrong [INAUDIBLE]? 550 00:29:17,105 --> 00:29:17,979 AUDIENCE: [INAUDIBLE] 551 00:29:27,860 --> 00:29:30,220 PROFESSOR: If I differentiate sine, I get cosine. 552 00:29:30,220 --> 00:29:33,200 So if I differentiate minus sine, I get minus cosine. 553 00:29:33,200 --> 00:29:34,890 So differentiating this, I get this. 554 00:29:34,890 --> 00:29:36,973 That should mean that integrating this I get that. 555 00:29:36,973 --> 00:29:38,872 I think I have it right. 556 00:29:38,872 --> 00:29:40,080 Sometimes I get things right. 557 00:29:40,080 --> 00:29:41,486 It's a surprise, but-- 558 00:29:41,486 --> 00:29:42,360 AUDIENCE: [INAUDIBLE] 559 00:29:42,360 --> 00:29:43,026 PROFESSOR: What? 560 00:29:43,026 --> 00:29:45,100 AUDIENCE: The last equation you wrote. 561 00:29:45,100 --> 00:29:47,433 PROFESSOR: The last equation needs to be changed, right. 562 00:29:47,433 --> 00:29:49,880 This was copied from that, right. 563 00:29:49,880 --> 00:29:52,580 Thanks for reminding me. 564 00:29:52,580 --> 00:29:59,520 So t tilde is equal to alpha times theta minus sine theta. 565 00:30:05,280 --> 00:30:08,290 OK, and now I was working out the relationship 566 00:30:08,290 --> 00:30:13,390 between theta and a tilde. 567 00:30:13,390 --> 00:30:15,177 And let me just remind you that all I did 568 00:30:15,177 --> 00:30:17,260 was make sure that the upper limits of integration 569 00:30:17,260 --> 00:30:18,770 correspond to each other. 570 00:30:18,770 --> 00:30:21,380 So I'm just basically rewriting the upper limit of integration 571 00:30:21,380 --> 00:30:23,690 in terms of the new variable each time 572 00:30:23,690 --> 00:30:26,810 when we change variables starting from a tilde going 573 00:30:26,810 --> 00:30:30,460 to a tilde to x, and from x to theta. 574 00:30:30,460 --> 00:30:32,265 And then I use the equality of these two 575 00:30:32,265 --> 00:30:35,270 to write a tilde in terms of theta. 576 00:30:38,750 --> 00:30:43,480 And these equations now hold with the subscript f present. 577 00:30:43,480 --> 00:30:45,600 When subscript f's appear everywhere we can just 578 00:30:45,600 --> 00:30:48,620 drop them and say that the time that we called 579 00:30:48,620 --> 00:30:51,890 t sub f now we're just going to call t. 580 00:30:51,890 --> 00:30:54,400 And now we could put together our two final results 581 00:30:54,400 --> 00:30:56,830 which are maybe right over here. 582 00:30:56,830 --> 00:30:58,570 We have ct. 583 00:30:58,570 --> 00:31:01,780 I'll eliminate my tildes altogether now. 584 00:31:01,780 --> 00:31:04,840 ct, which is t tilde, is equal to alpha times theta 585 00:31:04,840 --> 00:31:06,000 minus sine theta. 586 00:31:16,540 --> 00:31:21,380 And a divided by the square root of k, 587 00:31:21,380 --> 00:31:27,040 which is a tilde and its sub f, but we're dropping the sub f, 588 00:31:27,040 --> 00:31:32,580 is equal to alpha times 1 minus cosine theta. 589 00:31:43,130 --> 00:31:46,700 And this is as good as it gets. 590 00:31:46,700 --> 00:31:49,560 Ideally, it would be nice if we could 591 00:31:49,560 --> 00:31:53,040 solve the top equation for theta as a function of t, 592 00:31:53,040 --> 00:31:55,120 and plug that into the bottom equation, 593 00:31:55,120 --> 00:31:57,330 and then we would get a as a function of t. 594 00:31:57,330 --> 00:32:00,030 That's what ideally we would love to have. 595 00:32:00,030 --> 00:32:02,460 But there's no way to do that analytically. 596 00:32:02,460 --> 00:32:05,790 In principle of course, this equation can be inverted. 597 00:32:05,790 --> 00:32:07,460 You could do it numerically. 598 00:32:07,460 --> 00:32:09,432 For any particular value of t you 599 00:32:09,432 --> 00:32:11,640 could figure out what value of theta makes this work, 600 00:32:11,640 --> 00:32:13,940 and then plug that value of theta into here. 601 00:32:13,940 --> 00:32:15,440 But there's no analytic way that you 602 00:32:15,440 --> 00:32:16,898 can write theta as a function of t. 603 00:32:16,898 --> 00:32:19,470 It's not a soluble equation. 604 00:32:19,470 --> 00:32:21,760 So this is called a parametric solution in the sense 605 00:32:21,760 --> 00:32:24,930 that theta is a parameter. 606 00:32:24,930 --> 00:32:29,100 And as theta varies, both t and a vary in just the right way 607 00:32:29,100 --> 00:32:31,970 so that a is always related to t in the correct way 608 00:32:31,970 --> 00:32:35,930 to solve our original differential equation. 609 00:32:35,930 --> 00:32:39,190 That's what's meant by a parametric solution. 610 00:32:39,190 --> 00:32:40,937 We can also see from this equation, 611 00:32:40,937 --> 00:32:43,270 or maybe from thinking back about how things are defined 612 00:32:43,270 --> 00:32:45,510 in the first place, how theta varies 613 00:32:45,510 --> 00:32:49,230 over the lifetime of our model universe. 614 00:32:49,230 --> 00:32:52,190 Theta we discovered starts at 0. 615 00:32:52,190 --> 00:32:59,000 We discovered that when we wrote our integral over there. 616 00:32:59,000 --> 00:33:02,550 And we could also probably see it from here. 617 00:33:02,550 --> 00:33:04,900 We start our universe at a equals 0. 618 00:33:04,900 --> 00:33:08,560 And at a equals 0, we want 1 minus cosine theta to be 0 619 00:33:08,560 --> 00:33:10,400 and theta equals 0 does that. 620 00:33:10,400 --> 00:33:13,650 So theta starts at 0, which corresponds to a equals 0, 621 00:33:13,650 --> 00:33:16,470 and it also corresponds putting theta equals 0 here 622 00:33:16,470 --> 00:33:17,414 to t equals 0. 623 00:33:17,414 --> 00:33:18,830 So we have arranged things the way 624 00:33:18,830 --> 00:33:21,180 we intended so that a equals 0 happens 625 00:33:21,180 --> 00:33:24,260 the same time t equals 0 happens. 626 00:33:24,260 --> 00:33:27,090 And then theta starts to grow. 627 00:33:27,090 --> 00:33:30,700 As theta grows, a increases, the universe 628 00:33:30,700 --> 00:33:34,270 gets bigger until theta reaches pi. 629 00:33:34,270 --> 00:33:37,930 And when theta reaches pi, cosine of pi is minus 1, 630 00:33:37,930 --> 00:33:40,630 you get a factor of 2 here, 2 alpha. 631 00:33:40,630 --> 00:33:42,740 That's as big as our universe gets. 632 00:33:42,740 --> 00:33:46,360 And then as theta continues beyond that, 633 00:33:46,360 --> 00:33:49,620 1 minus cosine theta starts getting smaller again. 634 00:33:49,620 --> 00:33:51,330 So our universe reaches a maximum size 635 00:33:51,330 --> 00:33:54,470 when theta equals pi, and then starts to contract. 636 00:33:54,470 --> 00:33:56,900 And then by the time theta equals 2 pi, 637 00:33:56,900 --> 00:33:58,930 you are back to where you started from, 638 00:33:58,930 --> 00:34:00,530 a is again equal to 0. 639 00:34:00,530 --> 00:34:03,760 We have universe which starts at 0 size, goes to a maximum size, 640 00:34:03,760 --> 00:34:08,179 goes back to 0 size, giving a Big Crunch at the end. 641 00:34:08,179 --> 00:34:10,555 And that's the way this closed universe behaves. 642 00:34:13,480 --> 00:34:17,739 It turns out that these equations actually 643 00:34:17,739 --> 00:34:19,343 correspond to some simple geometry. 644 00:34:21,850 --> 00:34:24,929 It corresponds to a cycloid. 645 00:34:24,929 --> 00:34:30,480 And you may or may not remember what a cycloid is. 646 00:34:30,480 --> 00:34:33,510 There are the equations written on the screen, which 647 00:34:33,510 --> 00:34:37,050 are hopefully the same equations that I have on the blackboard. 648 00:34:37,050 --> 00:34:39,130 Can't always count on these things it turns out. 649 00:34:39,130 --> 00:34:42,820 But yeah, they are the same equations, that's healthy. 650 00:34:42,820 --> 00:34:45,290 And I have a diagram here which explains 651 00:34:45,290 --> 00:34:49,909 this cycloid correspondence. 652 00:34:49,909 --> 00:34:58,600 A cycloid is defined as what happens in this picture. 653 00:34:58,600 --> 00:35:00,000 Let me explain the picture. 654 00:35:00,000 --> 00:35:03,660 We have a disk shown in the upper left 655 00:35:03,660 --> 00:35:07,210 in its original position with a dot on the disk, which 656 00:35:07,210 --> 00:35:09,040 is initially at the bottom. 657 00:35:09,040 --> 00:35:11,060 And initially we're going to put this disk 658 00:35:11,060 --> 00:35:12,720 at the origin of our coordinate system 659 00:35:12,720 --> 00:35:15,440 to make things as simple as possible. 660 00:35:15,440 --> 00:35:18,450 And then we're going to imagine that this disk rolls 661 00:35:18,450 --> 00:35:21,440 without slipping to the right. 662 00:35:21,440 --> 00:35:25,900 And the path that this dot traces out, 663 00:35:25,900 --> 00:35:29,780 which is shown along that line, is a cycloid by definition. 664 00:35:29,780 --> 00:35:31,850 That's what defines a cycloid. 665 00:35:31,850 --> 00:35:41,000 It's the path that a point on a rolling disk evolved through. 666 00:35:41,000 --> 00:35:42,980 So what I would like to do is convince you 667 00:35:42,980 --> 00:35:47,470 that this geometric picture corresponds to those equations. 668 00:35:47,470 --> 00:35:49,720 Let me put the equations higher to make sure everybody 669 00:35:49,720 --> 00:35:50,261 can see them. 670 00:35:54,560 --> 00:35:56,580 And it's actually not so complicated 671 00:35:56,580 --> 00:36:00,190 once you figure out how to parse the pictures. 672 00:36:00,190 --> 00:36:03,960 So what I've drawn in the upper right is a blow up of this disk 673 00:36:03,960 --> 00:36:07,010 after it's rolled through some angle theta. 674 00:36:07,010 --> 00:36:09,810 And I even made it the same angle theta in the two cases. 675 00:36:09,810 --> 00:36:11,830 But this is just a bigger version 676 00:36:11,830 --> 00:36:15,405 of what's in the corner there showing the disk after it 677 00:36:15,405 --> 00:36:16,800 rolled a little bit. 678 00:36:16,800 --> 00:36:19,010 So after it's rolled through an angle theta, what 679 00:36:19,010 --> 00:36:22,070 we want to verify is that the horizontal and vertical 680 00:36:22,070 --> 00:36:24,840 coordinates here correspond to these two equations. 681 00:36:24,840 --> 00:36:27,150 And if they do, it means that we're tracing out 682 00:36:27,150 --> 00:36:30,250 the behavior of those two equations. 683 00:36:30,250 --> 00:36:32,460 So look first at the horizontal component. 684 00:36:32,460 --> 00:36:35,270 The horizontal component is the ct axis. 685 00:36:35,270 --> 00:36:37,340 So that should correspond to alpha times theta 686 00:36:37,340 --> 00:36:40,650 minus alpha times sine theta. 687 00:36:40,650 --> 00:36:42,150 And you can see that in the picture. 688 00:36:42,150 --> 00:36:44,200 We're talking about the horizontal component 689 00:36:44,200 --> 00:36:47,550 of the coordinates of this point P. 690 00:36:47,550 --> 00:36:49,310 And the point is that you can get 691 00:36:49,310 --> 00:36:51,864 to P starting at the origin. 692 00:36:51,864 --> 00:36:54,030 Now remember we're only looking at horizontal motion 693 00:36:54,030 --> 00:36:57,120 so we could start anywhere on that line. 694 00:36:57,120 --> 00:37:00,540 We could start at the origin, go to the right by alpha theta, 695 00:37:00,540 --> 00:37:03,960 and to the left by alpha times sine theta, and we get there. 696 00:37:03,960 --> 00:37:06,920 And that's exactly what this formula says. 697 00:37:06,920 --> 00:37:09,740 So if we just understand the alpha theta and the alpha sine 698 00:37:09,740 --> 00:37:14,210 theta of those two lines we have it made. 699 00:37:14,210 --> 00:37:17,180 So let's look first at what happens where 700 00:37:17,180 --> 00:37:20,900 this first arrow comes from, the alpha theta line. 701 00:37:20,900 --> 00:37:26,030 That's just the total distance that the point of contact 702 00:37:26,030 --> 00:37:29,060 has moved during the rolling process. 703 00:37:29,060 --> 00:37:31,690 And the claim is that as something rolls, 704 00:37:31,690 --> 00:37:34,770 really the definition of rolling without slipping, 705 00:37:34,770 --> 00:37:38,590 is that the arc length that is swept out by the rolling 706 00:37:38,590 --> 00:37:41,510 is the same as the length along the surface on which it's 707 00:37:41,510 --> 00:37:42,810 rolling. 708 00:37:42,810 --> 00:37:45,390 You could imagine, if you like, that as it 709 00:37:45,390 --> 00:37:47,010 rolls there's a tape measure that's 710 00:37:47,010 --> 00:37:50,840 wrapped around it that gets left on the ground as it rolls. 711 00:37:50,840 --> 00:37:52,620 And if you could picture that happening, 712 00:37:52,620 --> 00:37:54,750 the existence of that movie really 713 00:37:54,750 --> 00:37:57,260 guarantees that the length on the ground 714 00:37:57,260 --> 00:38:01,170 is the same as what the length was when the tape measure was 715 00:38:01,170 --> 00:38:04,410 wrapped around the cylinder. 716 00:38:04,410 --> 00:38:09,740 So the length that the point of contact has moved 717 00:38:09,740 --> 00:38:11,930 is just alpha times the angle through which 718 00:38:11,930 --> 00:38:13,470 the disk is rolled. 719 00:38:13,470 --> 00:38:17,840 So that explains the alpha theta label on that line. 720 00:38:17,840 --> 00:38:22,170 To get the alpha sine theta on the line above, 721 00:38:22,170 --> 00:38:24,790 that's the distance between the point P 722 00:38:24,790 --> 00:38:29,500 and a vertical line that goes through the center of the disk. 723 00:38:29,500 --> 00:38:33,820 And that's just simple trigonometry on this triangle. 724 00:38:33,820 --> 00:38:35,725 The radius of our circle is alpha. 725 00:38:39,250 --> 00:38:42,940 And then by simple trigonometry, this length 726 00:38:42,940 --> 00:38:47,650 is alpha times sine theta, which is what the label says. 727 00:38:47,650 --> 00:38:49,650 So to summarize what we've got here 728 00:38:49,650 --> 00:38:51,690 we can find the x-coordinate of the horizontal 729 00:38:51,690 --> 00:38:53,510 according to the point P by going 730 00:38:53,510 --> 00:38:55,220 to the right a distance alpha theta, 731 00:38:55,220 --> 00:38:59,770 and then back to the left the distance of minus alpha theta. 732 00:38:59,770 --> 00:39:02,170 And that gives us an x-coordinate, 733 00:39:02,170 --> 00:39:05,430 which is exactly the formula that appears in the ct formula. 734 00:39:05,430 --> 00:39:07,190 So ct works. 735 00:39:07,190 --> 00:39:08,930 The horizontal component of that dot 736 00:39:08,930 --> 00:39:12,890 is just where it should be to trace out the equations that 737 00:39:12,890 --> 00:39:15,650 describe the evolution of a closed universe. 738 00:39:15,650 --> 00:39:18,880 Similarly we can now look at the vertical components 739 00:39:18,880 --> 00:39:20,330 of that dot. 740 00:39:20,330 --> 00:39:22,250 Again it's most easily seen as the difference 741 00:39:22,250 --> 00:39:23,830 of two contributions. 742 00:39:23,830 --> 00:39:27,180 We're trying to reproduce this formula that says that's alpha 743 00:39:27,180 --> 00:39:31,100 minus alpha cosine theta. 744 00:39:31,100 --> 00:39:35,600 So if we start at vertical coordinate 0, which 745 00:39:35,600 --> 00:39:38,970 means on the x-axis, we can begin 746 00:39:38,970 --> 00:39:41,370 by going up to the center of the disk. 747 00:39:41,370 --> 00:39:43,650 And the disk has radius alpha, so that's 748 00:39:43,650 --> 00:39:46,490 going up the distance alpha. 749 00:39:46,490 --> 00:39:51,120 And then we go down the distance of this piece of the triangle, 750 00:39:51,120 --> 00:39:53,960 going from the center of the disk 751 00:39:53,960 --> 00:39:57,540 to the point which is parallel to the point P. 752 00:39:57,540 --> 00:40:00,730 And that again is just trigonometry on that triangle, 753 00:40:00,730 --> 00:40:05,160 and it's alpha cosine theta by simple trigonometry. 754 00:40:05,160 --> 00:40:07,450 So we can get to the elevation of point P 755 00:40:07,450 --> 00:40:11,280 by going up by alpha and down by alpha times cosine theta. 756 00:40:11,280 --> 00:40:14,520 And that's exactly that formula. 757 00:40:14,520 --> 00:40:16,490 So it works. 758 00:40:16,490 --> 00:40:19,150 The x and y components, the horizontal 759 00:40:19,150 --> 00:40:20,720 and vertical components of that dot, 760 00:40:20,720 --> 00:40:23,330 are exactly the two formulas here. 761 00:40:23,330 --> 00:40:26,223 So one of them can be thought of as the x-axis, and one of them 762 00:40:26,223 --> 00:40:27,700 can be thought of as the y-axis. 763 00:40:27,700 --> 00:40:29,550 And the rolling of the disk just traces out 764 00:40:29,550 --> 00:40:33,550 the evolution of our closed universe. 765 00:40:33,550 --> 00:40:36,699 So closed universes evolve like a cycloid. 766 00:40:36,699 --> 00:40:37,740 Any questions about that? 767 00:40:41,570 --> 00:40:44,560 OK, great. 768 00:40:44,560 --> 00:40:50,600 OK, Let me just mention that this angle theta is sometimes 769 00:40:50,600 --> 00:40:51,560 given a name. 770 00:40:51,560 --> 00:40:54,240 It's called the development angle of the universe 771 00:40:54,240 --> 00:40:56,280 or of the solution. 772 00:41:08,620 --> 00:41:11,840 And that is just intended to have the connotation that theta 773 00:41:11,840 --> 00:41:15,210 describes how developed the universe is 774 00:41:15,210 --> 00:41:17,390 and theta has a fixed scale. 775 00:41:17,390 --> 00:41:20,340 It always goes from 0 to 2 pi over the lifetime 776 00:41:20,340 --> 00:41:21,890 of this closed universe, no matter 777 00:41:21,890 --> 00:41:25,170 how big the closed universe might be. 778 00:41:25,170 --> 00:41:30,070 That brings me to my next question I want to mention. 779 00:41:30,070 --> 00:41:34,280 How many parameters do we have in this solution? 780 00:41:34,280 --> 00:41:36,190 The way we've written it, it looks 781 00:41:36,190 --> 00:41:52,680 like it can depend on both alpha and k because both of them 782 00:41:52,680 --> 00:41:54,229 appear in the answer. 783 00:41:54,229 --> 00:41:56,020 And k is positive for our closed universes. 784 00:41:56,020 --> 00:41:58,860 These formulas will not make sense if k were negative, 785 00:41:58,860 --> 00:42:00,230 square root of k appears there. 786 00:42:00,230 --> 00:42:03,200 We don't want anything to be imaginary. 787 00:42:03,200 --> 00:42:04,830 But k can have any value in principle 788 00:42:04,830 --> 00:42:07,740 and these equations would still be valid. 789 00:42:07,740 --> 00:42:10,910 So on the surface it appears like there's a two parameter 790 00:42:10,910 --> 00:42:13,710 class of closed universe solutions. 791 00:42:13,710 --> 00:42:16,640 But that's actually not true. 792 00:42:16,640 --> 00:42:19,971 Can somebody tell me why it's not true? 793 00:42:19,971 --> 00:42:20,470 Yes. 794 00:42:20,470 --> 00:42:23,750 AUDIENCE: [INAUDIBLE] 795 00:42:23,750 --> 00:42:25,490 PROFESSOR: Exactly. 796 00:42:25,490 --> 00:42:28,770 Yes, since k has units of 1 over notch squared, 797 00:42:28,770 --> 00:42:31,370 you can change k to anything you want 798 00:42:31,370 --> 00:42:33,610 by changing your definition of a notch. 799 00:42:33,610 --> 00:42:37,110 And there's nothing fixed about the definition of a notch. 800 00:42:37,110 --> 00:42:39,210 So k is an irrelevant parameter. 801 00:42:39,210 --> 00:42:42,310 If we change k we're just rescaling the same solution, 802 00:42:42,310 --> 00:42:45,170 and not actually creating a new solution. 803 00:42:45,170 --> 00:42:47,810 So there's really only one parameter class of solutions. 804 00:42:47,810 --> 00:42:50,464 One could, for example, fix k to always be 1, 805 00:42:50,464 --> 00:42:51,880 and then we'd have a one parameter 806 00:42:51,880 --> 00:42:55,560 class of solutions indicated by alpha. 807 00:42:55,560 --> 00:43:02,180 Alpha, unlike k, really does have a clear, physical meaning 808 00:43:02,180 --> 00:43:04,135 related to the behavior of the universe. 809 00:43:06,890 --> 00:43:21,970 And we could see what that is if we 810 00:43:21,970 --> 00:43:25,220 ask, what is the total lifetime of this universe from beginning 811 00:43:25,220 --> 00:43:28,330 to end, from Big Bang to Big Crunch. 812 00:43:28,330 --> 00:43:32,990 We can answer that by just looking at the ct equation. 813 00:43:32,990 --> 00:43:38,370 From Big Bang to Big Crunch, we know that theta evolves from 0 814 00:43:38,370 --> 00:43:42,270 to pi back to 2 pi, which is the same as 0. 815 00:43:42,270 --> 00:43:44,220 So theta goes through one cycle of 2 pi 816 00:43:44,220 --> 00:43:47,780 during a lifetime of our model universe. 817 00:43:47,780 --> 00:43:52,210 As theta goes from 0 to 2 pi, sine theta starts at 0 818 00:43:52,210 --> 00:43:55,450 and eventually comes back to 0 when theta equals 2 pi. 819 00:43:55,450 --> 00:44:00,960 But theta increases from 0 to 2 pi over one cycle. 820 00:44:00,960 --> 00:44:04,060 So over one cycle of our universe, 821 00:44:04,060 --> 00:44:07,950 ct increases by alpha times 2 pi. 822 00:44:07,950 --> 00:44:10,470 So that tells us what the total lifetime of the universe 823 00:44:10,470 --> 00:44:23,035 is, I'll call it t total. 824 00:44:28,910 --> 00:44:37,930 And we get it by just writing c times t total equals 2 pi 825 00:44:37,930 --> 00:44:38,950 times alpha. 826 00:44:44,380 --> 00:44:49,950 And I think I can do this one without making a mistake. t 827 00:44:49,950 --> 00:44:55,190 total is then 2 pi alpha divided by c. 828 00:44:59,030 --> 00:45:00,660 And we can even check our units there. 829 00:45:00,660 --> 00:45:05,500 Alpha has units of length, so length divided by c 830 00:45:05,500 --> 00:45:07,960 becomes a time, c being a velocity. 831 00:45:07,960 --> 00:45:09,450 So it has the right units. 832 00:45:09,450 --> 00:45:10,790 And that's the total lifetime of the universe. 833 00:45:10,790 --> 00:45:12,140 It's just determined by alpha. 834 00:45:12,140 --> 00:45:14,101 So alpha can be viewed as just the measure 835 00:45:14,101 --> 00:45:16,600 of the total lifetime of the universe, which can be anything 836 00:45:16,600 --> 00:45:18,141 for different sized closed universes. 837 00:45:21,260 --> 00:45:25,330 Alpha is also related to the maximum value of a over root k. 838 00:45:25,330 --> 00:45:30,680 And a has no fixed meaning, this is meters per notch. 839 00:45:30,680 --> 00:45:33,430 But a over root k does have units of meters. 840 00:45:33,430 --> 00:45:35,965 We haven't yet really seen what it means physically, 841 00:45:35,965 --> 00:45:38,940 because that's related to the geometry of the closed universe 842 00:45:38,940 --> 00:45:41,010 which we'll be discussing later. 843 00:45:41,010 --> 00:45:43,470 But in any case as a mathematical fact, 844 00:45:43,470 --> 00:45:47,490 we could always say that the maximum value of a over root k 845 00:45:47,490 --> 00:45:48,765 is determined by alpha. 846 00:45:53,660 --> 00:46:01,070 So a max over root k is equal to the maximum value 847 00:46:01,070 --> 00:46:02,576 of this expression. 848 00:46:02,576 --> 00:46:04,200 And this expression has a maximum value 849 00:46:04,200 --> 00:46:06,760 when theta equals pi, which gives cosine theta 850 00:46:06,760 --> 00:46:08,530 equals minus 1, which gives a 2 here. 851 00:46:08,530 --> 00:46:10,237 And that's as big as it ever gets, 852 00:46:10,237 --> 00:46:11,570 so that's just equal to 2 alpha. 853 00:46:18,430 --> 00:46:22,650 So alpha is related in a very clear way to the total lifetime 854 00:46:22,650 --> 00:46:25,890 of our universe, and is also related to a over root 855 00:46:25,890 --> 00:46:27,770 k, although we haven't really given 856 00:46:27,770 --> 00:46:29,510 a physical meaning to a over root k yet. 857 00:46:29,510 --> 00:46:31,280 But we know it has dimensions of meters. 858 00:46:42,080 --> 00:46:44,850 OK, the next calculation I want to do 859 00:46:44,850 --> 00:46:48,080 is to calculate the age of the universe 860 00:46:48,080 --> 00:46:50,740 as a function of measurable things. 861 00:46:50,740 --> 00:46:52,815 We learned for the flat matter dominated case 862 00:46:52,815 --> 00:46:54,440 that there was a simple answer to that. 863 00:46:54,440 --> 00:46:59,170 The age was just 2/3 times the inverse Hubble parameter. 864 00:46:59,170 --> 00:47:02,680 So what you do now is get the analogous formula here, 865 00:47:02,680 --> 00:47:05,620 it follows in principle immediately 866 00:47:05,620 --> 00:47:08,070 from our description of the evolution. 867 00:47:08,070 --> 00:47:10,580 But we have to do a fair number of substitutions 868 00:47:10,580 --> 00:47:13,360 before we can really see how to express the age in terms 869 00:47:13,360 --> 00:47:16,580 of things that we're interested in. 870 00:47:16,580 --> 00:47:20,100 The formula here tells us directly 871 00:47:20,100 --> 00:47:22,100 how to express the age of the universe. t 872 00:47:22,100 --> 00:47:26,720 is the age of the universe as a function of alpha and theta. 873 00:47:26,720 --> 00:47:29,670 But if you tell an astronomer to go out and measure alpha 874 00:47:29,670 --> 00:47:32,280 and theta so I could calculate the age, 875 00:47:32,280 --> 00:47:34,810 he says what in the world are alpha and theta. 876 00:47:34,810 --> 00:47:37,330 So what we'd like to do is to express the age 877 00:47:37,330 --> 00:47:39,750 in terms of things that astronomers know about. 878 00:47:39,750 --> 00:47:42,090 And the characterizations of the universe 879 00:47:42,090 --> 00:47:44,090 like this that an astronomer would know about 880 00:47:44,090 --> 00:47:46,830 would be the Hubble expansion rate, 881 00:47:46,830 --> 00:47:49,540 and some notion of the mass density. 882 00:47:49,540 --> 00:47:52,170 And the easiest way to talk about the mass density 883 00:47:52,170 --> 00:47:54,490 is in terms of omega, the fraction 884 00:47:54,490 --> 00:47:57,430 of the critical density that the actual mass density has. 885 00:47:57,430 --> 00:48:00,169 So our goal is going to be to manipulate these equations. 886 00:48:00,169 --> 00:48:01,710 All the information is already there. 887 00:48:01,710 --> 00:48:05,020 But our goal will be to manipulate these equations 888 00:48:05,020 --> 00:48:09,310 to be able to express the age t in terms of h and omega. 889 00:48:41,260 --> 00:48:46,900 OK, so first we need to remind ourselves what omega is. 890 00:48:46,900 --> 00:48:49,380 The critical density is defined as that density 891 00:48:49,380 --> 00:48:51,060 which makes the universe flat. 892 00:48:51,060 --> 00:48:53,830 And we've calculated that the critical density is 893 00:48:53,830 --> 00:48:58,300 equal to 3 h squared over 8 pi times newtons 894 00:48:58,300 --> 00:49:09,530 constant, capital G. We can then write the mass density rho 895 00:49:09,530 --> 00:49:13,210 as omega times the critical density, 896 00:49:13,210 --> 00:49:14,840 which is just the definition of omega. 897 00:49:14,840 --> 00:49:19,130 Omega is rho divided by rho c, the actual mass density divided 898 00:49:19,130 --> 00:49:20,780 by the critical density. 899 00:49:20,780 --> 00:49:25,800 And putting what rho c is, we can express rho 900 00:49:25,800 --> 00:49:46,650 as 3h squared omega divided by 8 pi G. And being very pedantic, 901 00:49:46,650 --> 00:49:48,410 I'm just going to rewrite that in the form 902 00:49:48,410 --> 00:49:52,010 that we're actually going to use it by multiplying through. 903 00:49:52,010 --> 00:49:57,290 8 pi over 3 G rho, taking these factors 904 00:49:57,290 --> 00:49:59,470 and bringing them to the other side, 905 00:49:59,470 --> 00:50:02,270 becomes just equal to h squared times omega. 906 00:50:06,020 --> 00:50:08,650 And you might recognize this particular combination 907 00:50:08,650 --> 00:50:11,370 as what appears in the Friedmann equation. 908 00:50:11,370 --> 00:50:17,250 The Friedmann equation told us that a dot over a squared 909 00:50:17,250 --> 00:50:27,100 is equal to 8 pi over 3 G rho, minus kc 910 00:50:27,100 --> 00:50:29,500 squared over a squared. 911 00:50:40,580 --> 00:50:44,410 And in order to get the substitutions that I want, 912 00:50:44,410 --> 00:50:47,030 I'm going to just rewrite this putting 913 00:50:47,030 --> 00:50:51,120 h squared for a dot over a squared. 914 00:50:51,120 --> 00:50:52,900 8 pi over 3 G rho we said we could 915 00:50:52,900 --> 00:50:54,310 write as 8 squared times omega. 916 00:50:58,080 --> 00:51:01,620 And then we have minus kc squared over a squared. 917 00:51:01,620 --> 00:51:04,380 Note that we really have here a tilde squared. 918 00:51:04,380 --> 00:51:07,660 This is a squared divided by k if I put them together. 919 00:51:07,660 --> 00:51:14,740 So this term can be written as minus c squared 920 00:51:14,740 --> 00:51:15,760 over a tilde squared. 921 00:51:22,766 --> 00:51:24,390 And this accomplishes one of our goals. 922 00:51:24,390 --> 00:51:27,580 It allows us to express a tilde in terms of the quantities 923 00:51:27,580 --> 00:51:30,960 that we want in our answers, h and omega. 924 00:51:30,960 --> 00:51:32,840 And if we can do the same for theta, 925 00:51:32,840 --> 00:51:36,530 we have everything we need to express the age. 926 00:51:36,530 --> 00:51:43,960 So the implication here is that a tilde squared 927 00:51:43,960 --> 00:51:53,775 is equal to c squared divided by h squared times omega minus 1. 928 00:52:23,062 --> 00:52:24,770 To take the square root of that equation, 929 00:52:24,770 --> 00:52:26,310 to find out what a tilde is, we need 930 00:52:26,310 --> 00:52:29,970 to think a little bit about signs and things like that. 931 00:52:29,970 --> 00:52:33,040 A tilde is always positive. 932 00:52:33,040 --> 00:52:35,510 This is the scale factor divided by the square root of k, 933 00:52:35,510 --> 00:52:38,070 square root of k is positive, scale factors 934 00:52:38,070 --> 00:52:40,340 are always positive the way we defined it. 935 00:52:40,340 --> 00:52:41,960 So we can take the square root of that 936 00:52:41,960 --> 00:52:44,490 taking the positive square root of the right hand side. 937 00:52:44,490 --> 00:52:48,430 Omega is bigger than 1 for our case, so omega minus 1 938 00:52:48,430 --> 00:52:51,780 is a positive number, 8 squared is a positive number. 939 00:52:51,780 --> 00:52:56,120 So taking the square root there offers no real problem. 940 00:52:56,120 --> 00:53:00,260 We can write a tilde is equal to c over, 941 00:53:00,260 --> 00:53:03,155 I guess this point I might not notice until later. 942 00:53:03,155 --> 00:53:05,290 h h can be positive or negative over the course 943 00:53:05,290 --> 00:53:06,240 of our calculation. 944 00:53:06,240 --> 00:53:07,823 We're going to talk about an expanding 945 00:53:07,823 --> 00:53:10,190 phase and a contracting phase. 946 00:53:10,190 --> 00:53:13,250 So when we take the square root of h squared, 947 00:53:13,250 --> 00:53:16,250 we want the positive number to give us a positive a tilde. 948 00:53:16,250 --> 00:53:18,250 So it's the positive square root that we want, 949 00:53:18,250 --> 00:53:21,090 which is the magnitude of h, not necessarily h. 950 00:53:21,090 --> 00:53:23,237 When h is positive, the magnitude of h is h. 951 00:53:23,237 --> 00:53:25,320 h could be negative though, and the magnitude of h 952 00:53:25,320 --> 00:53:28,130 is still positive, and then the square root of omega 953 00:53:28,130 --> 00:53:32,420 minus 1, which is always positive. 954 00:53:32,420 --> 00:53:35,370 So that's our formula for a tilde in terms of h and omega. 955 00:53:45,590 --> 00:53:48,790 OK, now we want to evaluate alpha, 956 00:53:48,790 --> 00:53:55,750 and I guess I did not keep the formula for alpha quite as 957 00:53:55,750 --> 00:53:58,290 long as we needed it. 958 00:53:58,290 --> 00:54:00,070 When we defined alpha in the first place, 959 00:54:00,070 --> 00:54:26,670 let me remind you how it was defined, 4 pi over 3 G 960 00:54:26,670 --> 00:54:35,000 rho a tilde cubed over c squared. 961 00:55:01,510 --> 00:55:12,040 And that can be evaluated using our formula for rho 962 00:55:12,040 --> 00:55:21,950 and we'll put that in for rho, and what we get 963 00:55:21,950 --> 00:55:29,490 is c over 2 times the magnitude of h 964 00:55:29,490 --> 00:55:31,116 using this formula for a tilde as well. 965 00:55:35,220 --> 00:55:43,820 And then omega over omega minus 1 to the 3/2 power. 966 00:55:43,820 --> 00:55:45,380 Just using this formula, and we know 967 00:55:45,380 --> 00:55:47,670 how to express rho from the right hand side, 968 00:55:47,670 --> 00:55:50,550 and we know how to express a tilde from this formula here. 969 00:55:50,550 --> 00:55:53,580 So everything's straightforward, and this is what we get. 970 00:55:53,580 --> 00:56:08,550 And now I want to use these to rewrite this equation, 971 00:56:08,550 --> 00:56:14,320 a over the square root of k is equal to alpha times 1 972 00:56:14,320 --> 00:56:15,630 minus cosine theta. 973 00:56:22,030 --> 00:56:25,040 I'm going to replace a over root k by this formula. 974 00:56:25,040 --> 00:56:27,855 I'm going to replace alpha by that formula. 975 00:56:32,360 --> 00:56:36,430 So this implies, rewriting it, that c 976 00:56:36,430 --> 00:56:41,640 over the magnitude of h times the square root of omega 977 00:56:41,640 --> 00:56:51,300 minus 1 is equal to c over twice the magnitude of h times omega 978 00:56:51,300 --> 00:56:57,930 minus 1 to the 3/2 power times 1 minus cosine theta. 979 00:57:21,420 --> 00:57:23,530 And now we've had to survive some boring algebra. 980 00:57:23,530 --> 00:57:26,750 But notice that now most things cancel away here. 981 00:57:26,750 --> 00:57:30,160 We get a very simple relationship between theta 982 00:57:30,160 --> 00:57:34,840 and h and omega, actually just omega . 983 00:57:34,840 --> 00:57:58,840 In particular, when we solve that, 984 00:57:58,840 --> 00:58:01,920 we get simply that cosine theta is 985 00:58:01,920 --> 00:58:08,770 equal to two minus omega over omega. 986 00:58:08,770 --> 00:58:10,800 So theta is directly linked to omega. 987 00:58:10,800 --> 00:58:13,008 If you know omega, you know theta, If you know theta, 988 00:58:13,008 --> 00:58:15,169 you know omega, by that formula. 989 00:58:15,169 --> 00:58:16,710 And we can rewrite this the other way 990 00:58:16,710 --> 00:58:19,530 around by solving for omega if we want. 991 00:58:19,530 --> 00:58:24,745 Omega is equal to 2 over 1 plus cosine theta. 992 00:58:33,860 --> 00:58:35,600 Now we can look at this qualitatively 993 00:58:35,600 --> 00:58:38,420 to understand how omega is going to behave. 994 00:58:38,420 --> 00:58:46,460 At the very beginning cosine theta is equal to 1, 995 00:58:46,460 --> 00:58:51,650 theta is equal to 0, so omega is 2 over 1 plus 1, which is 1. 996 00:58:51,650 --> 00:58:53,810 So at very early times omega is driven to 1 even 997 00:58:53,810 --> 00:58:56,630 in a closed universe. 998 00:58:56,630 --> 00:59:01,710 As theta gets larger, cosine theta gets less than 1. 999 00:59:01,710 --> 00:59:04,060 This then becomes more than 1. 1000 00:59:04,060 --> 00:59:09,490 So omega starts to grow as the universe starts to evolve. 1001 00:59:09,490 --> 00:59:12,215 At the turning point, when the universe 1002 00:59:12,215 --> 00:59:16,140 has reached its maximum size, theta is pi, 1003 00:59:16,140 --> 00:59:19,410 cosine theta is minus 1, omega is 1004 00:59:19,410 --> 00:59:20,850 infinite at the turning point. 1005 00:59:20,850 --> 00:59:23,220 That may or may not be a surprise. 1006 00:59:23,220 --> 00:59:25,530 But it you think about it, it's obvious. 1007 00:59:25,530 --> 00:59:28,710 At the turning point h is 0, therefore the critical density 1008 00:59:28,710 --> 00:59:31,690 is 0, but the actual density is not 0. 1009 00:59:31,690 --> 00:59:33,920 And the only way the actual density can be 0 1010 00:59:33,920 --> 00:59:36,500 while the critical density is 0 is for omega 1011 00:59:36,500 --> 00:59:39,570 to be infinite, so we should have expected that. 1012 00:59:39,570 --> 00:59:42,620 And then the return trip, the collapsing phases, 1013 00:59:42,620 --> 00:59:45,940 a mirror image of the expanding phase, 1014 00:59:45,940 --> 00:59:48,980 omega goes from infinity at the turning point back to 1 1015 00:59:48,980 --> 00:59:51,360 at the moment of the Big Crunch. 1016 00:59:51,360 --> 00:59:51,860 Yes. 1017 00:59:51,860 --> 00:59:54,320 AUDIENCE: I'm confused with what the universe would 1018 00:59:54,320 --> 00:59:56,937 look like when it gets to infinity. 1019 00:59:56,937 --> 00:59:58,270 PROFESSOR: It would look static. 1020 00:59:58,270 --> 00:59:59,294 It's temporarily static. 1021 00:59:59,294 --> 01:00:00,710 It's reached a maximum size and is 1022 01:00:00,710 --> 01:00:03,245 about to turn around and collapse, so h is 0. 1023 01:00:03,245 --> 01:00:06,822 AUDIENCE: OK, but like with the density. 1024 01:00:06,822 --> 01:00:08,780 PROFESSOR: Well we could calculate the density. 1025 01:00:08,780 --> 01:00:10,446 It's some number which depends on alpha. 1026 01:00:10,446 --> 01:00:13,394 AUDIENCE: OK, but that doesn't diverge or anything. 1027 01:00:13,394 --> 01:00:15,310 PROFESSOR: It doesn't diverge or anything, no. 1028 01:00:15,310 --> 01:00:17,470 It's just some finite density. 1029 01:00:17,470 --> 01:00:20,454 At the turning point, it's just a finite density 1030 01:00:20,454 --> 01:00:22,120 that can be expressed in terms of alpha. 1031 01:00:25,371 --> 01:00:26,870 Sounds like a good homework problem. 1032 01:00:26,870 --> 01:00:29,750 I think maybe I'll do that. 1033 01:00:34,530 --> 01:00:39,530 OK, so this now allows us to express theta 1034 01:00:39,530 --> 01:00:42,730 as a function of omega, which is what we wanted. 1035 01:00:42,730 --> 01:00:45,000 If we express theta as a function of omega, 1036 01:00:45,000 --> 01:00:48,720 and alpha as a function of omega and h, we have our answer. 1037 01:00:48,720 --> 01:00:52,910 We have t expressed as a function of h and theta. 1038 01:00:52,910 --> 01:00:59,470 So there are choices about how exactly to express omega 1039 01:00:59,470 --> 01:01:01,390 in terms of theta that will involve 1040 01:01:01,390 --> 01:01:03,100 inverse trigonometric functions. 1041 01:01:03,100 --> 01:01:04,620 And anything that can be expressed 1042 01:01:04,620 --> 01:01:06,590 in terms of an inverse cosine, can also 1043 01:01:06,590 --> 01:01:08,640 be expressed as an inverse sine by doing 1044 01:01:08,640 --> 01:01:10,510 a little bit of manipulations. 1045 01:01:10,510 --> 01:01:13,390 We have our choice here about what we want to do. 1046 01:01:13,390 --> 01:01:15,910 But in any case, the answer already 1047 01:01:15,910 --> 01:01:19,860 has a factor of sine theta in it. 1048 01:01:19,860 --> 01:01:22,020 So it's most useful to express theta 1049 01:01:22,020 --> 01:01:29,710 as the inverse sine of what we get from that formula, 1050 01:01:29,710 --> 01:01:31,920 to express the answer in what at least to me 1051 01:01:31,920 --> 01:01:33,507 is the simplest form. 1052 01:01:33,507 --> 01:01:35,340 So I'm going to manipulate this a little bit 1053 01:01:35,340 --> 01:01:37,570 to find out what sine theta is in terms of omega, 1054 01:01:37,570 --> 01:01:41,130 and then invert that to express theta in terms of omega. 1055 01:01:41,130 --> 01:01:49,240 So sine theta is of course plus or minus 1056 01:01:49,240 --> 01:01:53,330 the square root of 1 minus cosine squared theta. 1057 01:01:53,330 --> 01:01:55,280 And cosine theta we know in terms of omega, 1058 01:01:55,280 --> 01:01:58,080 so we can express this in terms of omega. 1059 01:01:58,080 --> 01:02:02,110 And if you do that it's straightforward enough algebra. 1060 01:02:02,110 --> 01:02:05,310 It's plus or minus, depending on which sign of the square root 1061 01:02:05,310 --> 01:02:08,570 is relevant, and we'll talk about that in a minute. 1062 01:02:08,570 --> 01:02:11,640 It's plus or minus the square root of 2 times omega 1063 01:02:11,640 --> 01:02:13,395 minus 1 over omega. 1064 01:02:17,880 --> 01:02:20,634 So we can express theta as the inverse sine 1065 01:02:20,634 --> 01:02:21,425 of this expression. 1066 01:02:24,720 --> 01:02:27,900 And now what I want to do is to make use of this 1067 01:02:27,900 --> 01:02:32,650 to put into this expression using the value for alpha 1068 01:02:32,650 --> 01:02:37,220 that we've already calculated and wrote over there. 1069 01:02:37,220 --> 01:02:41,440 So we get our final answer, which 1070 01:02:41,440 --> 01:02:48,350 is that t is equal to omega over twice 1071 01:02:48,350 --> 01:02:54,050 the magnitude of the Hubble expansion rate times omega 1072 01:02:54,050 --> 01:03:06,990 minus 1 to the 3/2 power times the arc sine or inverse 1073 01:03:06,990 --> 01:03:17,890 sine of twice the square root of omega minus 1 over omega. 1074 01:03:17,890 --> 01:03:21,210 That's just the theta that appears in this formula written 1075 01:03:21,210 --> 01:03:25,320 in terms of sine theta or the inverse sine of the quantity 1076 01:03:25,320 --> 01:03:28,010 that we determined was the sine of theta. 1077 01:03:28,010 --> 01:03:33,520 And then I see here I should have a plus or minus 1078 01:03:33,520 --> 01:03:35,467 because we haven't figured out our signs yet. 1079 01:03:35,467 --> 01:03:37,050 Either is actually possible, depending 1080 01:03:37,050 --> 01:03:38,740 on where we are in the evolution. 1081 01:03:38,740 --> 01:03:45,870 And then minus or plus twice the square root of omega 1082 01:03:45,870 --> 01:03:49,430 minus 1 over omega. 1083 01:03:54,740 --> 01:03:57,330 That's a plus or minus, this is a minus or plus. 1084 01:03:57,330 --> 01:04:00,300 And the reason I wrote one upstairs and one downstairs 1085 01:04:00,300 --> 01:04:05,760 is that we don't yet really know how to evaluate theta, 1086 01:04:05,760 --> 01:04:08,689 but the sign of this term is always 1087 01:04:08,689 --> 01:04:10,980 going to be the negative of the sign of that term, this 1088 01:04:10,980 --> 01:04:11,900 or that minus sign. 1089 01:04:16,190 --> 01:04:18,680 So theta can be the inverse sine of this expression 1090 01:04:18,680 --> 01:04:22,070 with either sign of the sine. 1091 01:04:22,070 --> 01:04:24,500 Sorry for the puns. 1092 01:04:24,500 --> 01:04:26,380 But whichever it is, it's the same on there 1093 01:04:26,380 --> 01:04:30,170 as it is here but with a minus sign in front, that minus sign. 1094 01:04:33,100 --> 01:04:36,200 OK, so this is our final formula for the age. 1095 01:04:36,200 --> 01:04:39,530 But we still need to think a little bit about the s-i-g-n 1096 01:04:39,530 --> 01:04:44,950 signs of these inverse functions that appear here, 1097 01:04:44,950 --> 01:04:48,050 that and that, and straightforward if you just 1098 01:04:48,050 --> 01:04:50,560 take it case by case. 1099 01:04:50,560 --> 01:04:52,910 And in the notes we have a table which 1100 01:04:52,910 --> 01:04:54,880 I'll put shortly on the screen. 1101 01:04:54,880 --> 01:04:58,710 But let's start by just talking about the earliest phase where 1102 01:04:58,710 --> 01:05:01,850 the universe is shortly after the Big Bang, 1103 01:05:01,850 --> 01:05:05,315 so that the development angle is a small angle. 1104 01:05:05,315 --> 01:05:06,690 We know that theta is going to go 1105 01:05:06,690 --> 01:05:09,760 from 0 to 2 pi over the lifetime of our universe. 1106 01:05:09,760 --> 01:05:12,030 So I now want to think of theta being small. 1107 01:05:12,030 --> 01:05:15,800 And small means small compared to any number you think about. 1108 01:05:15,800 --> 01:05:19,450 So when theta is small the sine theta is nearly theta. 1109 01:05:19,450 --> 01:05:21,640 Both are positive. 1110 01:05:21,640 --> 01:05:28,360 And in that case the sine theta being positive 1111 01:05:28,360 --> 01:05:31,650 means this is the positive root in this equation, 1112 01:05:31,650 --> 01:05:34,570 and therefore the positive root in that equation. 1113 01:05:34,570 --> 01:05:37,870 So for early times this would be the plus sign, 1114 01:05:37,870 --> 01:05:40,960 and that would mean that this would be the minus sign. 1115 01:05:40,960 --> 01:05:43,280 Again the minus sign just coming from there 1116 01:05:43,280 --> 01:05:45,460 and theta being positive. 1117 01:05:45,460 --> 01:05:49,420 So for early times it's a plus and minus. 1118 01:05:49,420 --> 01:05:52,540 And the arc sine is itself ambiguous. 1119 01:05:52,540 --> 01:05:55,270 For early times the angle we know 1120 01:05:55,270 --> 01:05:57,400 is going to be just a little bit bigger than 0. 1121 01:05:57,400 --> 01:06:03,026 So that's the evaluation that we make of the arc sine function. 1122 01:06:03,026 --> 01:06:05,154 Pi plus that would also give us the same sine. 1123 01:06:05,154 --> 01:06:07,320 It would be another possible value for the arc sine. 1124 01:06:07,320 --> 01:06:09,450 And of course 2 pi plus that would also 1125 01:06:09,450 --> 01:06:11,010 be another possible root. 1126 01:06:11,010 --> 01:06:13,560 So you have to know which root to take 1127 01:06:13,560 --> 01:06:18,450 to know the right answer here because as an angle, 1128 01:06:18,450 --> 01:06:21,737 0 is equivalent to 2 pi, but as a time, 0 is not 1129 01:06:21,737 --> 01:06:22,820 at all equivalent to 2 pi. 1130 01:06:22,820 --> 01:06:25,450 So you do have to know the right one to take. 1131 01:06:25,450 --> 01:06:28,955 We'll continue doing that on a case by case basis. 1132 01:06:28,955 --> 01:06:34,040 Those are the equations, that's the formula for the age, 1133 01:06:34,040 --> 01:06:38,970 and that's the formula for the age with a description of which 1134 01:06:38,970 --> 01:06:42,560 roots to take for each case, which just comes out 1135 01:06:42,560 --> 01:06:45,580 by following the evolution, we know 1136 01:06:45,580 --> 01:06:47,770 that theta is going from 0 to 2 pi, 1137 01:06:47,770 --> 01:06:51,800 and this last column, the inverse sine of the expression 1138 01:06:51,800 --> 01:06:54,520 which means the expression that appears here. 1139 01:06:58,220 --> 01:07:00,400 For the smallest angle is 0 to pi over 2. 1140 01:07:00,400 --> 01:07:03,720 We can think of this actually as defining our columns. 1141 01:07:03,720 --> 01:07:06,240 Theta starts at 0 so the time lengths between 0 1142 01:07:06,240 --> 01:07:08,960 and pi over 2, a time length between pi over 2, 1143 01:07:08,960 --> 01:07:12,520 a time length between pi and 3 pi over 2, 1144 01:07:12,520 --> 01:07:19,384 and a final time length between 3 pi over 2 and 2 pi. 1145 01:07:19,384 --> 01:07:22,680 And the first two correspond to the expanding phase, second two 1146 01:07:22,680 --> 01:07:25,570 correspond to the contracting phase. 1147 01:07:25,570 --> 01:07:27,540 We can easily see what values of omega 1148 01:07:27,540 --> 01:07:30,190 are relevant in those cases. 1149 01:07:30,190 --> 01:07:31,860 Omega we said starts at 1 and gets 1150 01:07:31,860 --> 01:07:36,772 larger, the borderline where the angle is pi over 2 one 1151 01:07:36,772 --> 01:07:38,230 could just plug into these formulas 1152 01:07:38,230 --> 01:07:41,980 and see amounts to omega equals 2. 1153 01:07:41,980 --> 01:07:46,390 So that is a division line between these first two 1154 01:07:46,390 --> 01:07:52,186 quadrants just calculated from the value of theta. 1155 01:07:52,186 --> 01:07:53,560 Omega then goes to infinity as we 1156 01:07:53,560 --> 01:07:57,310 said, comes backwards and back to 1 in the end. 1157 01:07:57,310 --> 01:08:00,340 And in this column we just figure 1158 01:08:00,340 --> 01:08:02,480 out which sign choice corresponds 1159 01:08:02,480 --> 01:08:05,870 to getting the right value for omega 1160 01:08:05,870 --> 01:08:08,400 and the angle that appears in the arc 1161 01:08:08,400 --> 01:08:12,192 sine of our formula for the age, the formula there. 1162 01:08:12,192 --> 01:08:14,025 So any one of these I claim is very obvious. 1163 01:08:14,025 --> 01:08:16,907 Seeing the whole picture takes time 1164 01:08:16,907 --> 01:08:19,240 because I think you really have to look at each case one 1165 01:08:19,240 --> 01:08:22,130 at a time to make sure you understand it in detail. 1166 01:08:22,130 --> 01:08:25,029 But if you understand the initial expanding phase 1167 01:08:25,029 --> 01:08:27,069 that's what corresponds to our universe 1168 01:08:27,069 --> 01:08:29,300 if our universe were closed. 1169 01:08:29,300 --> 01:08:31,359 And the others are just as easy. 1170 01:08:31,359 --> 01:08:34,120 You just have to take them one at a time I think. 1171 01:08:34,120 --> 01:08:35,790 OK, we're going to end there. 1172 01:08:35,790 --> 01:08:39,020 We will continue on Thursday.