1 00:00:00,012 --> 00:00:03,212 We started the class by talking about classical Astronomy. 2 00:00:03,212 --> 00:00:07,935 We moved on to slowly expanding our point of view on the universe starting with the 3 00:00:07,935 --> 00:00:12,179 solar system, proceeding to stars and star clusters and galaxy's last week and 4 00:00:12,179 --> 00:00:16,044 that seems only fitting to end our journey by talking about the [UNKNOWN] 5 00:00:16,044 --> 00:00:18,384 scene. What is it that we can say about the 6 00:00:18,384 --> 00:00:22,678 universe as a whole? And principle that guides us and being able to say anything 7 00:00:22,678 --> 00:00:26,559 and all about the universe is the cosmological principle, the assertion 8 00:00:26,559 --> 00:00:28,679 that the universe is homogenous and isotropic. 9 00:00:28,679 --> 00:00:31,980 Homogeneous, recall, means that they're in no preferred locations in the 10 00:00:31,980 --> 00:00:34,100 universe. Everywhere is the same as everywhere 11 00:00:34,100 --> 00:00:36,076 else. In particular, this implies there's no 12 00:00:36,076 --> 00:00:39,598 point that is the center of the universe nor is there an edge of the universe, 13 00:00:39,598 --> 00:00:42,462 those will be distinguished points and there are none. 14 00:00:42,462 --> 00:00:46,532 Furthermore, the universe is isotropic. It's the same in all directions. 15 00:00:46,532 --> 00:00:50,857 It's the same in all directions no matter where you are looking from, and so the 16 00:00:50,857 --> 00:00:55,457 existence of a homogeneous isotropic universe, all of this symmetry, is what's 17 00:00:55,457 --> 00:00:58,672 going to allow is to say something useful about the universe. 18 00:00:58,672 --> 00:01:03,115 Now clearly, the universe is neither homogeneous nor isotropic at small 19 00:01:03,115 --> 00:01:07,379 distances, so the best we can hope to imagine is that, in fact, at large 20 00:01:07,379 --> 00:01:12,396 distances, when you average out little perturbations like stars and galaxies and 21 00:01:12,396 --> 00:01:17,394 clusters and super clusters at really large distances, there is no leftover 22 00:01:17,394 --> 00:01:20,404 structure. And we saw some evidence for that last 23 00:01:20,404 --> 00:01:25,515 week in the decrease of the correlation function and galaxy counts at distances 24 00:01:25,515 --> 00:01:29,503 of over a 100 Mpc or more. we will see a lot more evidence this week 25 00:01:29,503 --> 00:01:33,933 and I want to say that we already saw another indirect piece of evidence, which 26 00:01:33,933 --> 00:01:36,601 is the Hubble Flow that we discussed last week. 27 00:01:36,601 --> 00:01:40,671 So, at the end of last week we mentioned Hubble's result that galaxies are 28 00:01:40,671 --> 00:01:44,790 receding from us and the average recessional velocity is proportional to 29 00:01:44,790 --> 00:01:49,058 the distance of galaxies, the constant of proportionality being the Hubble 30 00:01:49,058 --> 00:01:53,752 Constant. And so, here's a cartoon of what that could look like, here's a 31 00:01:53,752 --> 00:01:58,352 one-dimensional universe. Of course, I've only drawn a piece of it 32 00:01:58,352 --> 00:02:04,052 these arrows are meant to indicate that the universe goes on in all directions. 33 00:02:04,052 --> 00:02:08,177 This is the universe, let's say, t=0. There are three galaxies here. 34 00:02:08,177 --> 00:02:13,247 These are the same three galaxies over here at some later time t and we can 35 00:02:13,247 --> 00:02:17,335 label them, we'll label this galaxy number 1, 0, 1, 36 00:02:17,335 --> 00:02:22,995 -1, and if I wasn't so stingy, I would have added galaxy number 2 over here and 37 00:02:22,995 --> 00:02:27,794 galaxy number -2 over here. And what we see is that the father the 38 00:02:27,794 --> 00:02:34,457 galaxy is from galaxy 0, the faster it's motion, the velocities of these galaxies 39 00:02:34,457 --> 00:02:39,582 are attempting to represent the Hubble Law and in particular, I'm going to erase 40 00:02:39,582 --> 00:02:45,382 all my figures when I move on but here we go if the I label these galaxies, i. 41 00:02:45,382 --> 00:02:49,707 So, i counts the name, [UNKNOWN] is the name of a galaxy and the position of a 42 00:02:49,707 --> 00:02:53,885 galaxy at time t is its position at time 0 times 1 plus H0t. 43 00:02:53,885 --> 00:03:00,138 This, if you look at the speed that this, or the velocity that this position 44 00:03:00,138 --> 00:03:03,866 implies, then this is just a constant shift. 45 00:03:03,866 --> 00:03:10,793 This is xi(0)+xi(0)*H0t, and that tells you that the velocity of the ith galaxy 46 00:03:10,793 --> 00:03:15,742 is just H0*xi(0). The farther a galaxy is, the faster it's 47 00:03:15,742 --> 00:03:19,559 moving. This is the Hubble Expansion Law and this 48 00:03:19,559 --> 00:03:23,252 is all from the point of view of galaxy number 0. 49 00:03:23,252 --> 00:03:28,652 In particular, xi is the distance if the ith galaxy from galaxy number 0, 50 00:03:28,652 --> 00:03:32,903 vi is the velocity as drawn in this graph relative to galaxy number 0. 51 00:03:32,903 --> 00:03:37,680 What if you tried to represent the same thing but from a point of view of galaxy 52 00:03:37,680 --> 00:03:42,185 number 2? Well, in that case, what goes on, of course, is that the velocity you 53 00:03:42,185 --> 00:03:46,511 measure, if you said j=2 here., the relative velocity, this is all 54 00:03:46,511 --> 00:03:51,114 non-relativistic of galaxy i, is the difference of velocity between i and j, 55 00:03:51,114 --> 00:03:53,401 plugging the values from the Hubble Law in. 56 00:03:53,401 --> 00:03:57,936 I see that indeed, this is the Hubble Constant times the relative position of 57 00:03:57,936 --> 00:04:01,959 galaxy i relative to galaxy j. In other words, Hubble's Law looks the 58 00:04:01,959 --> 00:04:05,637 same as observed from galaxy j as it did observed from galaxy 0. 59 00:04:05,637 --> 00:04:10,272 Now, let me draw, erase all my scribbling so we can see what we're doing and make a 60 00:04:10,272 --> 00:04:13,614 point. this is true for Hubble's Law because it 61 00:04:13,614 --> 00:04:16,519 is a linear dependence of velocity on distance. 62 00:04:16,519 --> 00:04:20,948 Imagine, for example, that Hubble had discovered a quadratic dependence of 63 00:04:20,948 --> 00:04:24,598 velocity on distance. In other words, galaxies twice as far 64 00:04:24,598 --> 00:04:28,273 would be moving four times as fast rather than twice as fast. 65 00:04:28,273 --> 00:04:31,546 if this were true, then imagine that that's fine. 66 00:04:31,546 --> 00:04:36,325 Each of these galaxies is moving with a velocity proportional to the square of 67 00:04:36,325 --> 00:04:41,090 its distance, however, if you now observe the relative velocity of the ith galaxy 68 00:04:41,090 --> 00:04:44,118 relative to the j. So, if you're now observing with from the 69 00:04:44,118 --> 00:04:48,275 point of view of galaxy j, then the relative velocity is the difference of 70 00:04:48,275 --> 00:04:53,599 their velocity and that is not the same as the quadratic velocity law you would 71 00:04:53,599 --> 00:04:59,546 have obtained by simply imagining that this law just as it applies viewed from 72 00:04:59,546 --> 00:05:05,132 galaxy 0, applies viewed from galaxy j. In other words a quadratic velocity law 73 00:05:05,132 --> 00:05:09,187 would not be consistent with the homogeneous and isotropic universe, but 74 00:05:09,187 --> 00:05:12,109 the Hubble Flow is. So, the Hubble Flow is another subtle 75 00:05:12,109 --> 00:05:16,107 hint about, that we may consider the universe isotropic and homogeneous. 76 00:05:16,107 --> 00:05:19,699 We'll get many more of those, but for now, we're just going to assert it. 77 00:05:19,699 --> 00:05:24,127 Now, the Hubble Flow tells us a few other things that I want to emphasize. 78 00:05:24,127 --> 00:05:28,827 One is at every location, at every event, at every location at every given time in 79 00:05:28,827 --> 00:05:32,657 the universe, the Hubble Flow selects, in fact, a preferred rest frame. 80 00:05:32,657 --> 00:05:36,897 The reason is that this is the frame in which the Hubble Flow looks isotropic. 81 00:05:36,897 --> 00:05:41,332 Of course, if I see an isotropic Hubble Flow and you are moving relative to me 82 00:05:41,332 --> 00:05:45,875 and at my same position at the same time, at the same event, you are moving with 83 00:05:45,875 --> 00:05:50,360 some large velocity relative to me than in the direction of your motion, galaxies 84 00:05:50,360 --> 00:05:54,016 will appear to be receding at a smaller velocity than what I observe. 85 00:05:54,016 --> 00:05:57,879 Whereas, behind you, galaxies will be observed to be receding with larger 86 00:05:57,879 --> 00:06:01,344 velocities, all this completely non-relativistic and it is not a 87 00:06:01,344 --> 00:06:05,382 violation of special relativity. This is not a preferred reference frame 88 00:06:05,382 --> 00:06:09,759 with respect to the laws of Physics, it's a preferred reference frame with respect 89 00:06:09,759 --> 00:06:13,100 to the positions of the motions of the objects in the universe. 90 00:06:13,100 --> 00:06:17,552 And so, the observation is, that by and large, galaxies follow the Hubble Flow. 91 00:06:17,552 --> 00:06:21,614 They have peculiar velocities, they move relative to the local Hubble Flow. 92 00:06:21,614 --> 00:06:25,887 But on average, in any region of space at a given time, the average peculiar 93 00:06:25,887 --> 00:06:30,395 velocities of the galaxies average to 0 if the region of spaces large enough, by 94 00:06:30,395 --> 00:06:34,969 and large, galaxies follow the Hubble flow and then relative to that, they have 95 00:06:34,969 --> 00:06:38,833 their peculiar velocities. And so at every point in space and time, 96 00:06:38,833 --> 00:06:43,494 and at every event, there is a unique frame, unique reference frame in which 97 00:06:43,494 --> 00:06:47,448 the Hubble Flow looks isotropic. Now, we use the Hubble Flow or we 98 00:06:47,448 --> 00:06:50,572 discover the Hubble Flow by measuring the redshift. 99 00:06:50,572 --> 00:06:53,356 Again, everything here is non-relativistic. 100 00:06:53,356 --> 00:07:00,133 you remember that the redshift z is given by 1+z is lambda over lambda 0 and this 101 00:07:00,133 --> 00:07:06,027 is equal in turn to 1+V/C and V/C in turn is 1+H0D/C. 102 00:07:07,542 --> 00:07:15,225 So, the redshift was given by H0/C times D, so we have a linear redshift distance 103 00:07:15,225 --> 00:07:20,492 relationship. When is this valid? Well, this is valid 104 00:07:20,492 --> 00:07:26,332 because we used a non-relativistic version of the Doppler formula. 105 00:07:26,332 --> 00:07:32,147 We expect it to be valid for small non-relativistic velocities, and this 106 00:07:32,147 --> 00:07:37,987 corresponds to distances small compared to C times the inverse of the Hubble 107 00:07:37,987 --> 00:07:41,708 constant. In other words there's this distance, a 108 00:07:41,708 --> 00:07:47,756 typical size, characteristic size of the universe, that is and the Hubble Law, we 109 00:07:47,756 --> 00:07:53,197 expect and we found, holds for distances small to this, small relative to this. 110 00:07:53,197 --> 00:07:56,678 What happens if you want to understand larger z's? 111 00:07:56,678 --> 00:08:00,947 Well, you could try these the special relativistic Doppler Law formula 112 00:08:00,947 --> 00:08:05,241 replacing this by that square root. As we saw in the homework, it doesn't 113 00:08:05,241 --> 00:08:09,993 quite work and, in fact, in the context of an expand of dynamical expanding 114 00:08:09,993 --> 00:08:15,314 universe corrections to this beyond small z are going to be sensitive on the one 115 00:08:15,314 --> 00:08:19,212 hand to relativistic corrections because the velocities 116 00:08:19,212 --> 00:08:23,419 that you observe when z is not much less than 1 are going to be relativistic. 117 00:08:23,419 --> 00:08:27,820 On the other hand the light will have been traveling a significant fraction at 118 00:08:27,820 --> 00:08:31,844 z not much smaller than 1 of the history of the universe and so, you will be 119 00:08:31,844 --> 00:08:36,209 sensitive not only to the state of the universe now but to the entire history of 120 00:08:36,209 --> 00:08:39,379 the universe. And, in particular, if you try to extend 121 00:08:39,379 --> 00:08:44,589 an expression like this to distances D that are not small relative to the amount 122 00:08:44,589 --> 00:08:49,205 of the, the distance that light would have traveled in the age of the universe, 123 00:08:49,205 --> 00:08:54,043 then when you're talking about objects that are that far away the definition of 124 00:08:54,043 --> 00:08:58,862 distance becomes ambiguous. What do you mean by distance? You can't 125 00:08:58,862 --> 00:09:02,815 go around laying a ruler from galaxy at one side of the universe to a galaxy at 126 00:09:02,815 --> 00:09:06,830 the other side of the universe because if those galaxies are moving, then by the 127 00:09:06,830 --> 00:09:10,758 time you finish your ruler, they will not be ready, they were, when you started. 128 00:09:10,758 --> 00:09:14,499 So, you can understand what you mean by distance will become ambiguous and 129 00:09:14,499 --> 00:09:17,960 because of all of these, you would be surprised to find that the special 130 00:09:17,960 --> 00:09:22,510 relativistic expression for Doppler shift inserted into this would give an actual 131 00:09:22,510 --> 00:09:28,831 correct interpretation and understanding the redshift distance relation will tell 132 00:09:28,831 --> 00:09:32,969 us, in fact, a lot about what goes on in the universe. 133 00:09:32,969 --> 00:09:39,341 Now, naively, just extending from this, the statement that distances between 134 00:09:39,341 --> 00:09:46,027 galaxies at time t are given in terms of their distances at time zero times 1 plus 135 00:09:46,027 --> 00:09:49,607 H0t, then it's clear that going into the past 136 00:09:49,607 --> 00:09:55,972 things were closer. And you predict that when t is equal to the inverse Hubble 137 00:09:55,972 --> 00:10:02,550 constant distances were all 0, so you predict a singularity or a Big Bang with 138 00:10:02,550 --> 00:10:06,995 a characteristic time of 13.8 billion years given by the inverse of the Hubble 139 00:10:06,995 --> 00:10:09,891 Constant. Now, this assumes some sort of constant 140 00:10:09,891 --> 00:10:12,674 ongoing expansion extrapolated into the past. 141 00:10:12,674 --> 00:10:16,741 We don't really expect that. Hubble certainly knew not to expect that. 142 00:10:16,741 --> 00:10:20,959 what are we describing? We're describing a bunch of massive objects. 143 00:10:20,959 --> 00:10:25,672 Galaxies flying away from each other. These objects interact principally 144 00:10:25,672 --> 00:10:29,118 gravitationally. And what we expect is gravitation is a 145 00:10:29,118 --> 00:10:33,072 long range force. these galaxies are distant but there's a 146 00:10:33,072 --> 00:10:36,593 lot of them. And we expect that these galaxies flying 147 00:10:36,593 --> 00:10:39,870 apart will be slowed down by gravitational interactions. 148 00:10:39,870 --> 00:10:43,788 Sure, I can make something fly away from a massive object by giving it the correct 149 00:10:43,788 --> 00:10:46,434 initial conditions like I can throw a rubber ball in the air. 150 00:10:46,434 --> 00:10:48,495 But eventually, gravity will slow it down. 151 00:10:48,495 --> 00:10:52,107 And the question is, if I throw it fast enough, it has the escape velocity, it 152 00:10:52,107 --> 00:10:55,389 will slow down but keep receding. If I throw with less than the escape 153 00:10:55,389 --> 00:10:58,366 velocity, it will slow down, stop, and return. 154 00:10:58,366 --> 00:11:03,311 But Hubble certainly did not expect his law to, to extend to the entire history 155 00:11:03,311 --> 00:11:08,512 of the universe because what one expects is that expansion is slowing down. 156 00:11:08,512 --> 00:11:13,561 This is a very reasonable expectation given our understanding of gravity. 157 00:11:13,561 --> 00:11:18,736 Now, just this bit of the picture as a first tidbit from this week is going to 158 00:11:18,736 --> 00:11:23,516 give us the resolution to Olbers paradox that has followed us for a few weeks. 159 00:11:23,516 --> 00:11:28,497 And the resolution is precisely the one that Edgar Allen Poe intuited so many 160 00:11:28,497 --> 00:11:32,205 years ago. in a universe whose age is finite the sky 161 00:11:32,205 --> 00:11:36,399 is dark even if the universe is infinite and even if it's filled with a 162 00:11:36,399 --> 00:11:41,474 homogeneous distribution of stars. there are stars that are so far away that 163 00:11:41,474 --> 00:11:46,262 light from them has not yet had time to reach us and, in fact, if the universe is 164 00:11:46,262 --> 00:11:50,171 infinite, almost all of it is too far away for us to have seen it. 165 00:11:50,171 --> 00:11:54,273 Yet, in fact, we only see a finite distance out into the universe because 166 00:11:54,273 --> 00:11:58,559 light has only had a finite amount of time to travel from there to here. 167 00:11:58,559 --> 00:12:03,742 This means that most of the infinite sum of equal terms that made the divergence 168 00:12:03,742 --> 00:12:08,832 that caused the paradox in Olbers most of that infinite sum is not yet visible and 169 00:12:08,832 --> 00:12:12,262 so at any given time, you see out to a fit, finite distance. 170 00:12:12,262 --> 00:12:16,757 Within a finite distance, the stars take up a small fraction of the sky and the 171 00:12:16,757 --> 00:12:20,137 sky appears dark. Of course, we expect by now that it's not 172 00:12:20,137 --> 00:12:24,312 that you see all the stars up to a certain distance and then beyond that, 173 00:12:24,312 --> 00:12:26,562 it's dark. That kind of sudden transition is not 174 00:12:26,562 --> 00:12:29,012 what happens. Stars closer and closer to the maximal 175 00:12:29,012 --> 00:12:32,287 distance will be increasingly redshifted so that they will be dimmed. 176 00:12:32,287 --> 00:12:35,937 And eventually there will be sort of an infinite dimming and we will not see any 177 00:12:35,937 --> 00:12:38,612 stars past a particular distance. We'll talk about that. 178 00:12:38,612 --> 00:12:42,252 We haven't seen the entire universe yet, as we'll see by the end of this week. 179 00:12:42,252 --> 00:12:45,901 We never will, in fact. We've seen most of what we ever will see 180 00:12:45,901 --> 00:12:50,567 and understanding the meaning of that statement will be part of the joys of 181 00:12:50,567 --> 00:12:54,563 studying cosmology this week. And so, what is it that we're going to 182 00:12:54,563 --> 00:12:58,968 do? Well, we're going to start with the intellectual framework that allows us to 183 00:12:58,968 --> 00:13:01,742 discuss relativistic corrections to Hubble, 184 00:13:01,742 --> 00:13:04,924 which, of course, will lead us to general, general relativistic point of 185 00:13:04,924 --> 00:13:07,119 view. We'll apply general relativity to a 186 00:13:07,119 --> 00:13:10,447 homogeneous isotropic universe. We will understand the kinds of 187 00:13:10,447 --> 00:13:14,648 cosmological models that we construct. We will understand the parameters of the 188 00:13:14,648 --> 00:13:19,001 model and how they are determined. And in the process, understand this a 189 00:13:19,001 --> 00:13:23,661 very famous pie chart over here on the right that tells you what it is that 190 00:13:23,661 --> 00:13:27,532 comprises the total energy density of the university today. 191 00:13:27,532 --> 00:13:32,267 about 5% of that is in the form of atoms, baryons, protons, neutrons. 192 00:13:32,267 --> 00:13:37,758 24% is that dark matter of which we've had some indications and about the nature 193 00:13:37,758 --> 00:13:41,107 of which we've made some reasonable discussions. 194 00:13:41,107 --> 00:13:45,912 And then, 71.4% of the energy density of the universe is in the form of this even 195 00:13:45,912 --> 00:13:50,127 more mysterious dark energy about which we have not spoken but we will get to it, 196 00:13:50,127 --> 00:13:52,425 and about which, in fact, we know very little.