1 00:00:00,012 --> 00:00:05,666 Okay, we worked, we learned about how GR describes homogeneous isotropic spaces 2 00:00:05,666 --> 00:00:11,058 like ours, how they can evolve and what the equations are that describe them. 3 00:00:11,058 --> 00:00:16,394 What has all that got to do with us? Can we see how it applies to our universe? 4 00:00:16,394 --> 00:00:21,449 Yes, we can actually link it to astronomy, so here's how we parameterize 5 00:00:21,449 --> 00:00:25,702 our universe. here the 2 freeman equations as we wrote 6 00:00:25,702 --> 00:00:31,937 them the one for velocity and the one for the deceleration parameter and what do we 7 00:00:31,937 --> 00:00:36,935 know? Well, we measure the Hubble constant at present, remember we 8 00:00:36,935 --> 00:00:42,492 parameterize it as a 100 times H traditionally but we will use H as 0.71 9 00:00:42,492 --> 00:00:47,656 There are constant refinements of the precision of our understanding of the 10 00:00:47,656 --> 00:00:51,528 Hubble Constant. It is certainly not 110, it will settle 11 00:00:51,528 --> 00:00:56,691 down somewhere in the vicinity of 70 but we're going to use 71 kilometers per 12 00:00:56,691 --> 00:01:02,108 second per megaparsec for the remainder of this class unless they discover that 13 00:01:02,108 --> 00:01:04,962 it's 30, so. So this is what is known here. 14 00:01:04,962 --> 00:01:08,202 We know this. Can we measure anything else? Well we 15 00:01:08,202 --> 00:01:13,382 certainly can't measure the scale factor. That's something that we know the scale 16 00:01:13,382 --> 00:01:17,772 factor by construction today as 1. What you want is the history of it. 17 00:01:17,772 --> 00:01:21,762 What do you know about. The density, well we can measure the 18 00:01:21,762 --> 00:01:26,994 density of stuff of energy of mass in the universe by our we talked about it the 19 00:01:26,994 --> 00:01:30,013 way we measure the density of the solar system. 20 00:01:30,013 --> 00:01:32,489 We count dust. we can count stars. 21 00:01:32,489 --> 00:01:37,032 we can do better than that. We can actually estimate the mass say of 22 00:01:37,032 --> 00:01:40,720 the Milky Way. Beyond the stars, and the dust, and the 23 00:01:40,720 --> 00:01:43,906 gas, we discover that most of it is dark matter. 24 00:01:43,906 --> 00:01:49,190 We do the same thing for a galaxy cluster by doing the Newtonian orbits of things 25 00:01:49,190 --> 00:01:53,512 and find that most of the mass of a cluster is made of dark matter. 26 00:01:53,512 --> 00:01:58,106 and so we can get some idea about the total density of matter. 27 00:01:58,106 --> 00:02:03,904 And it will turn out that the density of the, the, the, most of the energy density 28 00:02:03,904 --> 00:02:08,852 of matter in the universe is going to be in the form of dust, of slowly moving 29 00:02:08,852 --> 00:02:12,155 heavy particles. There is radiation density of 30 00:02:12,155 --> 00:02:16,218 relativistic particles, but it'll turn out to be much smaller. 31 00:02:16,218 --> 00:02:21,434 Now It turns out that it pays to parameterize the density by, instead of 32 00:02:21,434 --> 00:02:27,417 describing [UNKNOWN] as a function of t, describing something called omega which 33 00:02:27,417 --> 00:02:31,792 is [UNKNOWN] basically scaled at any time by this quantity. 34 00:02:31,792 --> 00:02:37,102 This is called the critical density for a reason that will become clear. 35 00:02:37,102 --> 00:02:42,794 And the reason this is called the critical density is very simply that if 36 00:02:42,794 --> 00:02:48,990 you imagine, ignoring for a moment the cosmological constant term which we are 37 00:02:48,990 --> 00:02:52,986 going to interpret as though it were a kind of energy. 38 00:02:52,986 --> 00:02:56,562 So, we are going to imagine that we put lambda. 39 00:02:56,562 --> 00:03:00,574 As one of the kinds of energy in the universe. 40 00:03:00,574 --> 00:03:05,444 In other words, in the right hand side of the equation. 41 00:03:05,444 --> 00:03:10,112 then if the density is the critical density. 42 00:03:10,112 --> 00:03:14,341 In other words, if A Pi G Rho/3 is H^2. The k is 0. 43 00:03:14,341 --> 00:03:17,687 So if the density is critical, space is flat. 44 00:03:17,687 --> 00:03:22,843 If rho is bigger, s- the curvature is positive, if rho is less than the 45 00:03:22,843 --> 00:03:27,102 critical density the curvature of space is negative. 46 00:03:27,102 --> 00:03:32,482 So critical density, which means density given by this or omega = 1. 47 00:03:32,482 --> 00:03:37,337 Is the condition for space to be flat,okay? So this is why it pays to 48 00:03:37,337 --> 00:03:43,170 scale density to this critical density that is a significant standard density to 49 00:03:43,170 --> 00:03:48,617 compare to and we can evaluate it, since we measure H 0 and you know Newton's 50 00:03:48,617 --> 00:03:54,126 Constant, we can evaluate this in today's universe and the critical density 51 00:03:54,126 --> 00:03:58,332 corresponds. To 10^-28 approximately kilo / m^3 , 52 00:03:58,332 --> 00:04:04,258 that's about six protons per m^3. This is far smaller than say the density 53 00:04:04,258 --> 00:04:09,961 that we found for the Milky Way. Obviously, most of space is not part of 54 00:04:09,961 --> 00:04:14,340 the Milky Way. And so, if the density of matter and 55 00:04:14,340 --> 00:04:18,302 space is this, then space is of energy and space. 56 00:04:18,302 --> 00:04:24,359 Is this, that space is flat, if it's larger than this it's positively curved, 57 00:04:24,359 --> 00:04:29,976 if it's less it's negatively curved. And as I said, because this, we are 58 00:04:29,976 --> 00:04:36,046 accounting for the cosmological term as part of our energy and momentum, then 59 00:04:36,046 --> 00:04:42,173 this will allow us to write the total. energy momentum or the total scaled 60 00:04:42,173 --> 00:04:47,631 energy momentum dens- energy density of space as a contribution to, to 61 00:04:47,631 --> 00:04:53,474 nonrelativistic objects today plus a contribution due to relativistic 62 00:04:53,474 --> 00:04:59,827 particles, say photons and neutrinos, and a contribution due to the cosmological 63 00:04:59,827 --> 00:05:03,046 term. Which because we've moved it on the right 64 00:05:03,046 --> 00:05:08,111 hand side is a form of energy and because we don't understand what it is, is this 65 00:05:08,111 --> 00:05:10,892 contribution, is what we term dark energy. 66 00:05:10,892 --> 00:05:16,003 It's dark only in the sense that it does not correspond to any form of matter that 67 00:05:16,003 --> 00:05:18,838 we know. We do not see it, it does, like dark 68 00:05:18,838 --> 00:05:23,924 matter does not interact with light, but it's certainly not dark matter, it has 69 00:05:23,924 --> 00:05:28,905 these very bizarre This bizarre equation of state that is not like anything that 70 00:05:28,905 --> 00:05:31,950 we know. Almost at the end of the class we might 71 00:05:31,950 --> 00:05:36,498 discuss things that it could be. And so, Hubble's three months first 72 00:05:36,498 --> 00:05:41,268 equation becomes in terms of this, the equation that says omega just dividing 73 00:05:41,268 --> 00:05:44,852 everything by H-squared I get 1 on the right-hand side. 74 00:05:44,852 --> 00:05:49,327 And here I get omega here I get kR^2, R0c^2 over H s-. 75 00:05:49,327 --> 00:05:55,012 And if I remember that this is included in the total density then this is 76 00:05:55,012 --> 00:06:01,212 Fremont's first equation evaluated today if you val- if you remove the 0's you 77 00:06:01,212 --> 00:06:05,929 just get another H^2. Down here, I've just set a to 1 at the 78 00:06:05,929 --> 00:06:12,175 present and so this is now the current density determines the curvature and then 79 00:06:12,175 --> 00:06:17,692 if I plug in the relation between pressure and density for matter where 80 00:06:17,692 --> 00:06:23,110 pressure is 0 and for radiation. Where, these two terms have the same 81 00:06:23,110 --> 00:06:26,527 magnitude. I find that the second equation simply 82 00:06:26,527 --> 00:06:32,246 reads that the deceleration parameter is 1/2 * , remember I have to divide by h^2 83 00:06:32,246 --> 00:06:37,840 to get q, the deceleration parameter is 1/2 * the, Density, normalized density of 84 00:06:37,840 --> 00:06:41,032 a non-relativistic matter, sometimes I call slip. 85 00:06:41,032 --> 00:06:45,083 We sometimes call dust matter, so you have matter and radiation. 86 00:06:45,083 --> 00:06:47,930 I'll try to remember it's dust and radiation. 87 00:06:47,930 --> 00:06:53,022 Plus, the contribution of radiation does not get a factor of 1/2 because there are 88 00:06:53,022 --> 00:06:58,089 2 equal terms, minus the contribution of the cosmological constant which, as you 89 00:06:58,089 --> 00:07:04,652 recall, shows up with an opposite sign. And so the parameters of the universe are 90 00:07:04,652 --> 00:07:11,377 given if you know this number, that number and that number and age 0, then 91 00:07:11,377 --> 00:07:18,787 you can determine the entire evolution of the universe and the pie chart that you 92 00:07:18,787 --> 00:07:23,258 saw in the . Introductory could, told you, that the 93 00:07:23,258 --> 00:07:27,589 best data that we have, is what we have over here. 94 00:07:27,589 --> 00:07:34,763 The total, fraction of the critical density that is dust is, about a quarter 95 00:07:34,763 --> 00:07:38,534 of this. A smal fraction is what we call bionic 96 00:07:38,534 --> 00:07:42,127 dust. Atoms and the rest, dark matter, which 97 00:07:42,127 --> 00:07:48,967 has you will recall a much larger mass than the bar-ionic mass of the universe. 98 00:07:48,967 --> 00:07:54,577 radiation as I said, is a negligible part of the energy density today. 99 00:07:54,577 --> 00:07:58,087 And About 74%. Again, the pie chart, this, pie chart 100 00:07:58,087 --> 00:08:01,385 didn't say 74%. The numbers are, known to a better 101 00:08:01,385 --> 00:08:06,383 accuracy than what I'm representing, but it's the principle that I'm trying to get 102 00:08:06,383 --> 00:08:09,112 across. You can look up better numbers than I 103 00:08:09,112 --> 00:08:11,690 would have, than I have used, if you want. 104 00:08:11,690 --> 00:08:16,361 About 74% of the energy density Let's just do this crazy dark energy.This is 105 00:08:16,361 --> 00:08:20,513 the meaning of that pie chart. However, what all this tells us is not 106 00:08:20,513 --> 00:08:23,602 that clear. And we already got these numbers is far 107 00:08:23,602 --> 00:08:26,557 from clear. And we're going to spend the rest of the 108 00:08:26,557 --> 00:08:30,123 week figuring that out. But, can we do anything with this. 109 00:08:30,123 --> 00:08:34,878 So, our universe today, ignoring for a moment the cosmological constant. 110 00:08:34,878 --> 00:08:39,592 Before, people discovered that the cosmological constant was nonzero, 111 00:08:39,592 --> 00:08:42,222 When I was a graduate student 20 years ago. 112 00:08:42,222 --> 00:08:45,464 It was assumed that the cosmological constant was 0. 113 00:08:45,464 --> 00:08:49,275 For theoretical reasons that we'll discuss later in the week. 114 00:08:49,275 --> 00:08:54,249 But, so people assumed that the energy density was all non-relativistic matter. 115 00:08:54,249 --> 00:08:58,812 We knew a little bit about dark matter and so imagine a universe in which. 116 00:08:58,812 --> 00:09:05,552 omega r is 0, omega lambda is 0, and we only have a density of dust which with 117 00:09:05,552 --> 00:09:12,092 expansion scales is diluted by the factor of the volume and which has pressure 0. 118 00:09:12,092 --> 00:09:18,072 Plug that into these equations, it turns out that solving these differential 119 00:09:18,072 --> 00:09:24,525 equations Is aa, rather straight forward and here are some solutions so aa, i have 120 00:09:24,525 --> 00:09:30,244 written 3 sort of characteristic solutions.This is a plot over here on the 121 00:09:30,244 --> 00:09:36,063 left of the scale factor as a function of time in billions of years and i have 122 00:09:36,063 --> 00:09:43,126 written the Description of universes with essentially the same hubble constant h 0. 123 00:09:43,126 --> 00:09:49,215 Though to be fair the present time is not exactly the same in all of these graphs. 124 00:09:49,215 --> 00:09:54,643 So there's, these are not exactly extrapolations from our universe but the 125 00:09:54,643 --> 00:09:59,345 blue curve shows a universe. With omega 0 = 0.5, since omega is less 126 00:09:59,345 --> 00:10:02,492 than 1, the curvature of space is negative. 127 00:10:02,492 --> 00:10:07,856 This is one of those hyperbolic space in this universe, is one of those hyperbolic 128 00:10:07,856 --> 00:10:10,928 spaces. The red curve is the critical universe 129 00:10:10,928 --> 00:10:16,212 omega = 1, this is a flat universe. the black curve is the case omega equals 130 00:10:16,212 --> 00:10:21,642 to 2 and if omega equals to 2 notice that the scale factor grows and then shrinks 131 00:10:21,642 --> 00:10:27,097 the universe expands from a big bang and ends in what we used to call a big crunch 132 00:10:27,097 --> 00:10:32,477 these were the kinds of diagrams we drew they assume that the cosmological 133 00:10:32,477 --> 00:10:36,032 constant is 0, the universe is full of dust and 134 00:10:36,032 --> 00:10:38,924 For the case, k=0, in other words, omega=1. 135 00:10:38,924 --> 00:10:44,007 you can actually write, the relevant, which, which will turn out to be the 136 00:10:44,007 --> 00:10:47,356 relevant case as you saw. In the data I gave before. 137 00:10:47,356 --> 00:10:51,788 I didn't actually add them up. But if you add the numbers I provided, 138 00:10:51,788 --> 00:10:55,209 they add up to omega=1. When you add up all together. 139 00:10:55,209 --> 00:11:01,269 And indeed The, our universe we know to a very good approximation is flat, how we 140 00:11:01,269 --> 00:11:04,273 know that we'll figure out as we go along. 141 00:11:04,273 --> 00:11:09,608 But for the relevant case K equal 0, so for the red curve this is in fact the 142 00:11:09,608 --> 00:11:15,061 expression for the scale factor, the scale factor grows with time since the 143 00:11:15,061 --> 00:11:18,262 Big Bang, as t to the 2/3rd and in fact the. 144 00:11:18,262 --> 00:11:24,177 Present time, the time when the scale factor is 1, is related to the Hubble 145 00:11:24,177 --> 00:11:30,307 factor measured today by not T is 0. The age of the universe is not A 0, but 146 00:11:30,307 --> 00:11:33,577 only 2/3 of A 0 inverse and so A 0 inverse. 147 00:11:33,577 --> 00:11:39,777 So, a dust filled universe predicts a slightly younger universe that the 13.8 148 00:11:39,777 --> 00:11:43,167 billion. That we got by just plugging in t zero 149 00:11:43,167 --> 00:11:48,532 equals h zero inverse, but we see that our order of magnitude estimate was is 150 00:11:48,532 --> 00:11:53,240 borne out by these more precise solutions of Einstein's equation. 151 00:11:53,240 --> 00:11:57,622 on the right hand side what we have is a plot of the red shift. 152 00:11:57,622 --> 00:12:04,862 Versus distance in these three universes again normalized so that they have the 153 00:12:04,862 --> 00:12:10,967 same value of H0 so what the same value of H0 of course means is that all of 154 00:12:10,967 --> 00:12:16,172 these graphs agree in the region where we have z = H0 D over c. 155 00:12:16,172 --> 00:12:20,777 For a small this is a D coordinate distance by the way, so this is D0. 156 00:12:20,777 --> 00:12:25,922 And of course at small distance it doesn't matter what distance you use 157 00:12:25,922 --> 00:12:31,272 cause they're all the same and the Hubble expression holds and we see that the 158 00:12:31,272 --> 00:12:36,272 deviation from Hubble. depends on the kind of universe in which 159 00:12:36,272 --> 00:12:40,862 you live. for example the universe with the, the 160 00:12:40,862 --> 00:12:47,112 super critical, the closed universe with positive curvature has departures where 161 00:12:47,112 --> 00:12:52,371 at large distances The red shifts are larger than you would have expected. 162 00:12:52,371 --> 00:12:57,458 What that means is that at, large distances means longs time, long times 163 00:12:57,458 --> 00:13:01,186 ago because the look back time increases with distance. 164 00:13:01,186 --> 00:13:05,702 And it means that the universe, the motion of galaxies was faster. 165 00:13:05,702 --> 00:13:10,652 in the past than one would have expected by extrapolating the Hubble expansion, 166 00:13:10,652 --> 00:13:15,172 and if you imagine that the present is here, then you realize indeed, that in 167 00:13:15,172 --> 00:13:19,342 the past the scale factor was growing faster than it is now, and so that 168 00:13:19,342 --> 00:13:23,177 explains this deviation above what would, one would expect. 169 00:13:23,177 --> 00:13:27,997 But all of these curves are tilting down, so in all of the cases I find A little 170 00:13:27,997 --> 00:13:33,447 upward curvature of the redshift distance relation, how much depends on how much 171 00:13:33,447 --> 00:13:38,022 the scale factor's slowing down. So this is sort of an example in the 172 00:13:38,022 --> 00:13:43,197 context of an easily computable dust universe as we will see, this behavior, 173 00:13:43,197 --> 00:13:47,722 this t to the 2/3rd behavior, in fact does describe our universe. 174 00:13:47,722 --> 00:13:52,963 Universe to a resonable approximation until the recent past, though it's not an 175 00:13:52,963 --> 00:13:58,037 idle academic exercise, but I wanted to get us, us to get a sense for how these 176 00:13:58,037 --> 00:14:02,181 equations determine the actual evolution of the universe. 177 00:14:02,181 --> 00:14:07,539 OK, so now, we have the parameters of our unicerse, right over here, We can use 178 00:14:07,539 --> 00:14:12,029 them to plug in to the equations the kind of calculations are little more 179 00:14:12,029 --> 00:14:15,616 complicated that what it took me to produce these graphs. 180 00:14:15,616 --> 00:14:20,595 But you can get some quality of, of ideas without doing any kind of calculation. 181 00:14:20,595 --> 00:14:26,069 In fact, the data I gave you is recent, I'm talking about what people could do in 182 00:14:26,069 --> 00:14:28,976 the 1930s. So in the 1930s here is what they 183 00:14:28,976 --> 00:14:33,905 realized, first of all that because the universe is expanding, we measure that 184 00:14:33,905 --> 00:14:38,915 it's expanding right now, that means that in the past a of t was smaller, a of t 185 00:14:38,915 --> 00:14:43,869 was smaller, conservation of mass tells us that the universe was denser, and as 186 00:14:43,869 --> 00:14:48,162 you go farther and farther into the past it gets denser and denser. 187 00:14:48,162 --> 00:14:52,683 Moreover because of the way peculiar velocities decrease with expansion it was 188 00:14:52,683 --> 00:14:55,755 also hotter. This is not that shocking what we're 189 00:14:55,755 --> 00:14:59,935 saying is that an expanding gas cools and the gas in question is the gas of 190 00:14:59,935 --> 00:15:02,630 galaxies. We're observing at essentially zero 191 00:15:02,630 --> 00:15:05,862 temperature but if it has any peculiar motion at all. 192 00:15:05,862 --> 00:15:11,295 Then in the past, the motion was faster, there was more random motion, and the gas 193 00:15:11,295 --> 00:15:14,876 was hotter in the past, and has cooled as it expanded. 194 00:15:14,876 --> 00:15:19,790 As you go far enough into the past, temperatures get hot enough, that when 195 00:15:19,790 --> 00:15:24,804 you get, one of the first major events, is that when you get to a red shift of 196 00:15:24,804 --> 00:15:30,034 approximately 1000, in other words When all distances in the universe were about 197 00:15:30,034 --> 00:15:35,082 1,000 times smaller than they are now this turns out to be about 380,000 years 198 00:15:35,082 --> 00:15:40,249 depending on it, in, exactly your model, but in our current model of the universe. 199 00:15:40,249 --> 00:15:45,226 About 350,000 years after the big bang At that point, before that time, 200 00:15:45,226 --> 00:15:49,245 temperatures were so high, that hydrogen atoms were ionized. 201 00:15:49,245 --> 00:15:54,231 And so, if you read the history of the universe forward, you have ionized 202 00:15:54,231 --> 00:15:58,754 hydrogen, and then as temperatures dropped down below the ionization 203 00:15:58,754 --> 00:16:04,337 temperature Hydrogen atoms becomes, atoms become stable against ionization and that 204 00:16:04,337 --> 00:16:09,207 occurs at that time and for the rest of history hydrogen is ionized of course in 205 00:16:09,207 --> 00:16:13,912 stars and in interstellar clouds. But it's not globally everywhere ionized 206 00:16:13,912 --> 00:16:18,302 by the cosmic temperature. this event we will call it ionization. 207 00:16:18,302 --> 00:16:21,582 It's often called by the confusing term recombination. 208 00:16:21,582 --> 00:16:24,762 Recombination is what happens when you have ions. 209 00:16:24,762 --> 00:16:29,115 In, you know, chemistry, in some experiment, and the ions recombine to 210 00:16:29,115 --> 00:16:34,071 form atoms, it becomes logical constant, it's a misguided term, because it's not 211 00:16:34,071 --> 00:16:36,884 recombination, it's, the first combination. 212 00:16:36,884 --> 00:16:40,358 There was not a, previously, combined neutral atom. 213 00:16:40,358 --> 00:16:45,121 These ions, were formed as ions. And then, later, electrons and protons, 214 00:16:45,121 --> 00:16:48,185 managed to produce atoms, as the universe cooled. 215 00:16:48,185 --> 00:16:54,307 before that, before, the time Or in the first 380,000 years after the Big Bang, 216 00:16:54,307 --> 00:16:59,869 baryonic matter and radiation exchange energy rapidly so our conservation 217 00:16:59,869 --> 00:17:05,722 equations might not exactly hold because as long as the universe, the matter in 218 00:17:05,722 --> 00:17:11,312 the universe, the dust, is in the form of hydrogen ions, so we have a dense. 219 00:17:11,312 --> 00:17:15,456 Plasma of charged particles. And remember than, that means the mean 220 00:17:15,456 --> 00:17:20,211 free path of the photon is very short. Remember we talked about a photon taking 221 00:17:20,211 --> 00:17:24,832 100,000 years to get out of the sun. In the process it's absorbed and emitted 222 00:17:24,832 --> 00:17:27,720 many times. So there's a, a, tight interaction 223 00:17:27,720 --> 00:17:30,466 between matter, between dust and radiation. 224 00:17:30,466 --> 00:17:36,221 until Dust forms hydrogen atoms. Once you have neutral hydrogen atoms, and 225 00:17:36,221 --> 00:17:41,958 once the, then the photons, with frequencies too low to ionize hydrogen, 226 00:17:41,958 --> 00:17:47,512 so anything below the ultraviolet energies that are required to excite 227 00:17:47,512 --> 00:17:51,065 hydrogen atoms Are basically then unimpeded. 228 00:17:51,065 --> 00:17:56,283 And the universe which, remember is full of hydrogen, goes from being an opaque 229 00:17:56,283 --> 00:18:00,938 strongly interacting plasma to a transparent gas of neutral hydrogen. 230 00:18:00,938 --> 00:18:04,619 This is bizzare. We often talked about ionized hydrogen 231 00:18:04,619 --> 00:18:09,285 being transparent and. Hydrogen atoms being opaque but that is 232 00:18:09,285 --> 00:18:15,189 because or excited hydrogen ions being opaque, this is true but that is not at 233 00:18:15,189 --> 00:18:21,179 the densities that here we are talking, what we are talking about is very dense 234 00:18:21,179 --> 00:18:26,442 plasma is opaque and strongly interacting photons do not propagate. 235 00:18:26,442 --> 00:18:31,635 Whereas a gas of neutral hydrogen. Of atoms is transparent to low frequency 236 00:18:31,635 --> 00:18:36,499 radiation, below the ultraviolet frequencies required to excite it, and as 237 00:18:36,499 --> 00:18:39,136 we'll see, that was the relevant question. 238 00:18:39,136 --> 00:18:43,054 So the universe becomes transparent at the age of 380,000 years. 239 00:18:43,054 --> 00:18:47,114 We'll see what that means. If you go further back, then you realize 240 00:18:47,114 --> 00:18:51,417 that if there's any radiation, and remember, we said that there's some 241 00:18:51,417 --> 00:18:54,532 radiation. some negligible energy density and 242 00:18:54,532 --> 00:18:57,732 radiation today. As you go farther and farther back in 243 00:18:57,732 --> 00:19:02,732 time d- dust becomes dense, the energy density of dust increases, but the energy 244 00:19:02,732 --> 00:19:07,152 density of radiation increases more. It increases more because in the past 245 00:19:07,152 --> 00:19:11,622 these photons were not only more tightly clumped, they were also not yet red 246 00:19:11,622 --> 00:19:14,878 shifted. And so the energy per photon was larger. 247 00:19:14,878 --> 00:19:19,984 So if you go far in, far, far enough back in time then there's value for the scale 248 00:19:19,984 --> 00:19:23,810 factor given by the appropriate ratio of the energy densities. 249 00:19:23,810 --> 00:19:28,649 Such that when the world was about 10,000 times when distances were about 10,000 250 00:19:28,649 --> 00:19:34,074 times as small as they are now. Corresponding to a redshift about 3,300 251 00:19:34,074 --> 00:19:38,162 and an age for the universe of about 55,000 years. 252 00:19:38,162 --> 00:19:43,037 Note a higher redshift because it's before recombination. 253 00:19:43,037 --> 00:19:49,382 At that point radiation energy density, energy density radiation and energy 254 00:19:49,382 --> 00:19:54,962 density in matter, in mass were about equal and before that time. 255 00:19:54,962 --> 00:19:59,952 The world was dominated, the energy density of the universe, was dominated by 256 00:19:59,952 --> 00:20:03,184 radiation. And the same kind of solution that gave 257 00:20:03,184 --> 00:20:08,245 us T to the two-thirds behavior for the scale factor in a universe containing 258 00:20:08,245 --> 00:20:11,619 only dust. If the universe contains only radiation, 259 00:20:11,619 --> 00:20:15,142 then you find that the scale factor grows with time. 260 00:20:15,142 --> 00:20:20,792 AS the square root of time, and sort of because at the time of this transition if 261 00:20:20,792 --> 00:20:26,397 you sort of imagine that rather than a continuous changeover from radiation to 262 00:20:26,397 --> 00:20:30,192 some equal mixture of radiation and dust to all dust. 263 00:20:30,192 --> 00:20:34,387 The universe went from all radiation suddenly to all dust. 264 00:20:34,387 --> 00:20:37,742 Then I basically match on the two solutions. 265 00:20:37,742 --> 00:20:43,064 By saying that until this time, I had this function and when I got to this 266 00:20:43,064 --> 00:20:48,552 time, I had this value of the scale factor and thereafter, I had the growth 267 00:20:48,552 --> 00:20:53,849 as t^2/3 matched on at that point. Now, I keep talking about radiation. 268 00:20:53,849 --> 00:20:57,552 I said radiation is any relativistic particle. 269 00:20:57,552 --> 00:21:03,202 What's a relativistic particle? Well any particle will become relativistic if it's 270 00:21:03,202 --> 00:21:08,027 moving at the speed of light. The characteristic velocity of a particle 271 00:21:08,027 --> 00:21:13,027 in a thermal gas is determined by the temperature and if the temperature is 272 00:21:13,027 --> 00:21:18,102 such that KB times T is bigger than the rest energy of the particle, then the 273 00:21:18,102 --> 00:21:21,183 average. Energy of a particle being, k b t, this 274 00:21:21,183 --> 00:21:24,644 particle will be relativistic. So at high enough temperatures, 275 00:21:24,644 --> 00:21:28,627 everything becomes relativistic. when you have a gas of relativistic 276 00:21:28,627 --> 00:21:31,035 particles, then, they behave like photons. 277 00:21:31,035 --> 00:21:35,272 Their energy spectrum is determined by the Plank blackbody distribution. 278 00:21:35,272 --> 00:21:40,224 It's a property of thermodynamics. The mean energy of course, of every 279 00:21:40,224 --> 00:21:43,956 particle is kBT. And the enrgy density, well it's the 280 00:21:43,956 --> 00:21:49,036 energy density of a gas of photons. And so, there's some constant G which 281 00:21:49,036 --> 00:21:52,072 we'll describe in a moment. G for a photon. 282 00:21:52,072 --> 00:21:55,628 Is 2. And then, you're not surprised it's 283 00:21:55,628 --> 00:21:59,976 related to the Stefan Boltzmann Law, the sigma T^4 law. 284 00:21:59,976 --> 00:22:06,777 so, and then how do you take a flux and change it, it into an energy density you 285 00:22:06,777 --> 00:22:11,114 need something with the dimensions of velocity, the relevant velocity of course 286 00:22:11,114 --> 00:22:14,321 is the speed of light. Basically, this is the same as the 287 00:22:14,321 --> 00:22:18,472 calculation of the number of winds going through a square meter, you take the 288 00:22:18,472 --> 00:22:22,387 density of wind and multiply by their velocity, you want to know how much 289 00:22:22,387 --> 00:22:26,547 radiation traverses a square meter that's the flux with which the universe is 290 00:22:26,547 --> 00:22:31,018 radiating, you take the energy density of radiation multiply by its speed, you get 291 00:22:31,018 --> 00:22:36,097 the flux and the only Tricky business is this geometric factor 292 00:22:36,097 --> 00:22:39,742 of 2. We're used to our order of magnitudes 293 00:22:39,742 --> 00:22:44,529 missing geometric factors of order 2. Remember, G is 2. 294 00:22:44,529 --> 00:22:51,137 So, for photons the equation reads the energy density of photons is sigma 295 00:22:51,137 --> 00:22:54,142 T^4/2C. Now, of course as the 296 00:22:54,142 --> 00:22:59,573 Universe expands the energies of all relativistic particles are redshifted, 297 00:22:59,573 --> 00:23:04,437 this as I said, preserves the blackbody spectrum but the temperature of a 298 00:23:04,437 --> 00:23:09,770 relativistic gas decreases as the scale factor grows as the inverse power or 299 00:23:09,770 --> 00:23:11,550 increases. Into the past. 300 00:23:11,550 --> 00:23:16,485 If you go far enough into the past, a gets small enough, temperatures get high. 301 00:23:16,485 --> 00:23:19,815 Far enough into the past everything is relativistic. 302 00:23:19,815 --> 00:23:22,349 We'll see where that comes into play soon.