1 00:00:00,012 --> 00:00:05,456 We're now ready to start evaluating some cosmological models. 2 00:00:05,456 --> 00:00:12,299 We'll consider several special cases. The first one would be a flat, spatially 3 00:00:12,299 --> 00:00:18,315 flat matter-dominated model, which is also known as Einstein-de Sitter 4 00:00:18,315 --> 00:00:21,451 model. We will then look at this version when 5 00:00:21,451 --> 00:00:24,537 it's dominated by radiation and not matter. 6 00:00:24,537 --> 00:00:29,620 Well then consider what looks like, at first, superficial model called Milne 7 00:00:29,620 --> 00:00:35,168 model, and it contains neither matter nor energy, but it is very instructive in 8 00:00:35,168 --> 00:00:40,862 some ways and then finally we'll look at the behavior of more general models. 9 00:00:40,862 --> 00:00:45,567 So let's first consider a spatially flat model, with exactly critical density. 10 00:00:45,567 --> 00:00:50,347 This is also known as Einstein-de Sitter model, because that's the one that two of 11 00:00:50,347 --> 00:00:54,907 them developed together in 1930s, and it's been used quite a bit, even though 12 00:00:54,907 --> 00:00:58,432 the universe we live in does not really correspond to it, 13 00:00:58,432 --> 00:01:02,530 but it is very useful as a description of an early universe. 14 00:01:02,530 --> 00:01:07,500 We'll start with the Friedmann Equation, which we can then rewrite in the 15 00:01:07,500 --> 00:01:11,213 following form. From that, we can see the value of the 16 00:01:11,213 --> 00:01:17,254 time derivative for the scale factor a dot or as expressed as explicitly as time 17 00:01:17,254 --> 00:01:21,277 derivative. It is plus or minus, square root of the 18 00:01:21,277 --> 00:01:26,074 quantity shown there, this is very easy to integrate as shown 19 00:01:26,074 --> 00:01:29,187 here, and, the solution is shown here. 20 00:01:29,187 --> 00:01:34,712 We can just rewrite that and see that in the matter-dominated phase, 21 00:01:34,712 --> 00:01:38,572 universe expanse as a 2/3 power of the time. 22 00:01:38,572 --> 00:01:44,376 Now, lets look at similar model but, this time dominated by the radiation. 23 00:01:44,376 --> 00:01:50,981 Recall that the radiation energy density, changes with the steeper power of, scale 24 00:01:50,981 --> 00:01:55,653 factor, 4 rather than 3. You get 3, for the expansion of the 25 00:01:55,653 --> 00:02:00,532 space, and the evolution of the number density of photons. 26 00:02:00,532 --> 00:02:05,011 You get an extra power for the energy lost due to the stretching of the 27 00:02:05,011 --> 00:02:08,141 wavelengths. All right, so here is its Friedman 28 00:02:08,141 --> 00:02:13,415 Equation, which again, we can rewrite in the following form, and from that we see 29 00:02:13,415 --> 00:02:18,251 the time derivative of scale factor is inversely proportional to the scale 30 00:02:18,251 --> 00:02:22,657 factor itself. That too is an easy one to integrate. 31 00:02:22,657 --> 00:02:29,727 And the solution is that the universe expands according to a square root of the 32 00:02:29,727 --> 00:02:33,969 time. Let's now consider what's called, Milne 33 00:02:33,969 --> 00:02:37,616 Cosmology. This is spatially flat model, good old 34 00:02:37,616 --> 00:02:42,549 Euclidean geometry, it follows the spacial relativity and it is realized in 35 00:02:42,549 --> 00:02:47,751 Sanskrit there no matter at all, it's any empty universe, just specs, so you can 36 00:02:47,751 --> 00:02:52,831 think of it as galaxies having no mass of their own just being test particles to 37 00:02:52,831 --> 00:02:58,429 show us how the coordinates are expanding as you will see how, Hubble's law follows 38 00:02:58,429 --> 00:03:03,317 immediate and directly from it. Special relativity still holds, Lorentz 39 00:03:03,317 --> 00:03:08,206 contractions are still at play, and again, there is no special location, 40 00:03:08,206 --> 00:03:13,465 because cosmological principle applies. In fact, Milne was the first one to 41 00:03:13,465 --> 00:03:19,534 introduce cosmological principle. So this was spatially often negative curvature 42 00:03:19,534 --> 00:03:24,729 model with a zero density. Its Friedmann equation is shown here and 43 00:03:24,729 --> 00:03:30,606 since curvature constant is less than zero, we can see it here, therefore, the 44 00:03:30,606 --> 00:03:33,288 time derivative of scale factor is constant. 45 00:03:33,288 --> 00:03:36,247 If it's constant, that implies a linear expansion. 46 00:03:36,247 --> 00:03:40,490 The universe expands is the first power of time or it can be contracting, 47 00:03:40,490 --> 00:03:43,222 seeing as the square root can be of either sign. 48 00:03:43,222 --> 00:03:46,962 Alright, now, let's consider slightly different version of that. 49 00:03:46,962 --> 00:03:51,193 Again, curvature constant is negative, but this time density is finite and 50 00:03:51,193 --> 00:03:54,282 positive. The Friedmann Equation is now shown here, 51 00:03:54,282 --> 00:04:00,733 and this time, the square of the time derivative scale factor is positive. 52 00:04:00,733 --> 00:04:07,749 Well, as the universe expands, time goes to infinity, the scale factor will go to 53 00:04:07,749 --> 00:04:12,190 infinity, and one over a scale factor will go to zero. 54 00:04:12,190 --> 00:04:18,510 And since the obvious inequality applies here, we see that in asymptotic case, 55 00:04:18,510 --> 00:04:23,596 either matter or radiation-dominated universe, will in the end behave just 56 00:04:23,596 --> 00:04:27,621 like the Milne model. Now, lets consider a more general case, 57 00:04:27,621 --> 00:04:33,201 models with, in which matter, radiation, cosmological constant, or dark energy can 58 00:04:33,201 --> 00:04:36,942 be present. First, let us look at the model of the 59 00:04:36,942 --> 00:04:41,680 positive curvature. Here, we can write the Friedmann equation 60 00:04:41,680 --> 00:04:47,888 again and transform it as shown here. Now if the density declines as the third 61 00:04:47,888 --> 00:04:54,580 power of scale factor as it would be appropriate for the matter field universe 62 00:04:54,580 --> 00:04:59,612 or as fourth power as it is appropriate for radiation, then in any case the 63 00:04:59,612 --> 00:05:06,084 product of the density and square of the scale factor declines as scale factor on 64 00:05:06,084 --> 00:05:12,292 the power side is -1 or -2. Since in the right-hand side, we have a 65 00:05:12,292 --> 00:05:15,677 term that looks like that, minus a constant. 66 00:05:15,677 --> 00:05:21,026 Sooner or later, that whole side, the right side of equation has to be zero. 67 00:05:21,026 --> 00:05:23,898 When the time derivative of scale factor is zero, 68 00:05:23,898 --> 00:05:27,698 that means that it's either at the maximum or at the minimum. 69 00:05:27,698 --> 00:05:31,028 However, look at the expression for the acceleration. 70 00:05:31,028 --> 00:05:35,976 The right-hand side must be always negative, because both density pressure 71 00:05:35,976 --> 00:05:40,232 are positive quantities. And therefore, the second derivative must 72 00:05:40,232 --> 00:05:43,254 be negative, which means that the scale factor has 73 00:05:43,254 --> 00:05:46,210 reached maximum, then it can only collapse back. 74 00:05:46,210 --> 00:05:51,214 So in a model like this, recollapse back to the initial singularity or reverse of 75 00:05:51,214 --> 00:05:55,194 the big bang is inevitable. Now, let's consider a model which only 76 00:05:55,194 --> 00:05:58,722 has dark energy in the form of a cosmological constant. 77 00:05:58,722 --> 00:06:03,440 Its Friedmann equation is show here, which we can rewrite in a fairly 78 00:06:03,440 --> 00:06:07,718 straightforward fashion. And so now, if we assume that scale 79 00:06:07,718 --> 00:06:11,141 factory is allowed to increase, as long as needed, 80 00:06:11,141 --> 00:06:16,068 eventually, the first term on the right-hand side dominates the second 81 00:06:16,068 --> 00:06:19,101 term, regardless of the actual value of the 82 00:06:19,101 --> 00:06:21,117 curvature constant. Okay. 83 00:06:21,117 --> 00:06:27,301 So if the energy density of the physical vacuum or cosmological constant is 84 00:06:27,301 --> 00:06:33,882 positive, then the first time derivative scale factor will be positive, and that 85 00:06:33,882 --> 00:06:37,827 means that that universe will expand forever. 86 00:06:37,827 --> 00:06:45,231 Now, if the energy density of dark energy is negative and that's actually possible, 87 00:06:45,231 --> 00:06:51,265 things will get to be little more complicated, but we need not concern 88 00:06:51,265 --> 00:06:57,766 ourselves with that at the moment. Now, let's look at models in which both 89 00:06:57,766 --> 00:07:03,143 radiation and matter are present. And, if their energy density is much 90 00:07:03,143 --> 00:07:08,254 higher, then the energy density of the dark energy, which is certainly true in 91 00:07:08,254 --> 00:07:13,192 the early universe, then this is a perfectly good description of the early 92 00:07:13,192 --> 00:07:16,599 universe. The rigorous solution for the density as 93 00:07:16,599 --> 00:07:21,760 a function of time will be a little more difficult, but we can do the following 94 00:07:21,760 --> 00:07:26,342 fairly good approximation. We can divide the history in the time 95 00:07:26,342 --> 00:07:30,646 where radiation is dominate and when the matter is dominate. 96 00:07:30,646 --> 00:07:35,915 In either case, early on, the model is very close to flat and so we can assume 97 00:07:35,915 --> 00:07:41,163 curvature of zero to begin with. Since the density of the radiation always 98 00:07:41,163 --> 00:07:46,831 decreases faster than that one of matter sooner or later, those two curves have to 99 00:07:46,831 --> 00:07:50,162 cross, and so, at earlier times, radiation 100 00:07:50,162 --> 00:07:54,465 dominates and a later time, latter times, latter dominates. 101 00:07:54,465 --> 00:08:00,066 And then each one drives the expansion according to the different power of time. 102 00:08:00,066 --> 00:08:05,667 So, in general, we can rewrite Friedmann equation in terms of described behavior 103 00:08:05,667 --> 00:08:10,800 of three densities as follows. This is actually a useful form, which we 104 00:08:10,800 --> 00:08:16,312 will use later on, to derive expression for distances in cosmology. 105 00:08:16,312 --> 00:08:21,642 So, let's recap, again. Which component of the universe dominates 106 00:08:21,642 --> 00:08:27,787 its dynamics at what time? Since the fastest declining one is more important 107 00:08:27,787 --> 00:08:31,422 at the earlier times and that would be radiation. 108 00:08:31,422 --> 00:08:35,112 Early on, universe must be dominated by radiation. 109 00:08:35,112 --> 00:08:41,642 Sooner or later, energy radiation drops below that one of matter and then becomes 110 00:08:41,642 --> 00:08:45,202 matter-dominated. Eventually, the density of both matter 111 00:08:45,202 --> 00:08:50,710 and energy drop sufficiently to reach the level of the constant energy filling the 112 00:08:50,710 --> 00:08:54,864 universe, the cosmological constant, and after that cosmological constant must 113 00:08:54,864 --> 00:08:57,457 dominate. So generally, it's always true that 114 00:08:57,457 --> 00:09:02,159 radiation dominates the early universe no matter what present in the future 115 00:09:02,159 --> 00:09:06,555 universe will be dominated by the cosmological constant or dark energy that 116 00:09:06,555 --> 00:09:10,722 behaves in that way. So we can express dynamics of the uni, 117 00:09:10,722 --> 00:09:17,317 universe in a very general fashion here showing how scale factor depends on time 118 00:09:17,317 --> 00:09:23,882 and certain power, which is a function of the equation of state parameter w. 119 00:09:23,882 --> 00:09:27,101 During the matter dominated phase, w is zero. 120 00:09:27,101 --> 00:09:30,078 Remember, when we introduced the equation of state, 121 00:09:30,078 --> 00:09:34,880 we said that for regular matter, there is no pressure, just test particles as far 122 00:09:34,880 --> 00:09:39,696 as cosmology is concern and the universe expands, according to time to the thirds 123 00:09:39,696 --> 00:09:42,339 power. Since that power is less than one, the 124 00:09:42,339 --> 00:09:46,142 universe is decelerating. It's slowing down its expansion. 125 00:09:46,142 --> 00:09:51,558 This was intuitively obvious, because the gravity of the matter pulls everything 126 00:09:51,558 --> 00:09:56,080 back, just as if you were to throw an object up in the air, it will reach 127 00:09:56,080 --> 00:10:01,354 certain maximum height while decelerating all the way there, and then of course, 128 00:10:01,354 --> 00:10:05,229 will turn and fall down. In the case of radiation-dominated 129 00:10:05,229 --> 00:10:09,816 universe, the expansion will be according to the time to the 1/2 power and the 130 00:10:09,816 --> 00:10:14,833 universe will be decelerating. Yes, its matter and energy are 131 00:10:14,833 --> 00:10:18,885 equivalent. Energy in the form of radiation exerts 132 00:10:18,885 --> 00:10:23,051 gravitational pull, so it slows down the expansion. 133 00:10:23,051 --> 00:10:30,144 If the universe has a constant energy density like cosmological constant of w = 134 00:10:30,144 --> 00:10:36,964 -1, then, when that term is dominant, it will drive expansion according to the 135 00:10:36,964 --> 00:10:41,623 exponential law. The dividing line between accelerating 136 00:10:41,623 --> 00:10:47,590 and decelerating models is at the value of the equation of state parameter w - 137 00:10:47,590 --> 00:10:53,614 1/3, since -1 < -1/3, this is the cosmological constant model, acceleration 138 00:10:53,614 --> 00:10:57,126 will dominate. And so here is just the plot that shows 139 00:10:57,126 --> 00:11:01,612 what some expansion laws look like. They are for different values of 140 00:11:01,612 --> 00:11:05,170 cosmological parameters that are written in caption. 141 00:11:05,170 --> 00:11:08,864 And we will actually address those in more detail later.