1 00:00:00,012 --> 00:00:05,350 Let us now take a closer look at the collapse of density fluctuations. 2 00:00:05,350 --> 00:00:10,749 The toy model that's often used is called the spherical top-hat model. 3 00:00:10,749 --> 00:00:16,886 What it means is simply that there is a uniform overdense sphere embedded in the 4 00:00:16,886 --> 00:00:22,456 constant density background. As you recall from the last time, you can 5 00:00:22,456 --> 00:00:27,986 consider the background universe expanding according to the critical 6 00:00:27,986 --> 00:00:32,523 universe line. And then the fluctuation itself, which is 7 00:00:32,523 --> 00:00:36,293 overdense, acts like a little closed universe. 8 00:00:36,293 --> 00:00:41,520 So, it splits from its background. And if it were perfect model that can 9 00:00:41,520 --> 00:00:46,290 recollapse to a point, then it will be simply shown as the curve here. 10 00:00:46,290 --> 00:00:51,107 In reality, it does not collapse back in a point, it collapses to a certain 11 00:00:51,107 --> 00:00:55,602 radius, finite radius in which system is in a variable equilibrium. 12 00:00:55,602 --> 00:01:01,064 But qualitatively, the density contrast, which is the difference between that line 13 00:01:01,064 --> 00:01:04,723 and the background line, grows as the universe expands. 14 00:01:04,723 --> 00:01:07,853 There is a turnover radius at the turnover time. 15 00:01:07,853 --> 00:01:12,620 And then, there is a collapse that caught, that takes about same amount of 16 00:01:12,620 --> 00:01:18,108 time as it took to get to the turn-around time, and the system ends up at some 17 00:01:18,108 --> 00:01:23,911 finite radius of equilibrium state. A simple argument to understand the final 18 00:01:23,911 --> 00:01:29,465 virial radius is that the system is just bound, so kinetic energy and potential 19 00:01:29,465 --> 00:01:35,636 energy are equal at the beginning. When it's in the equilibrium state, the 20 00:01:35,636 --> 00:01:41,301 Virial theorem state, tells you kinetic energy will be 1/2 the potential 21 00:01:41,301 --> 00:01:45,544 [UNKNOWN]. Now, if you'll look at simple formulas 22 00:01:45,544 --> 00:01:49,072 for combining energy [UNKNOWN] gravitational system, 23 00:01:49,072 --> 00:01:54,069 you can see the potential energy is inversely proportional to the radius. 24 00:01:54,069 --> 00:01:59,089 And so, if we're going to achieve a factor of 2 difference, that means that 25 00:01:59,089 --> 00:02:04,007 the radius of the final system will be 1/2 exactly of the initial one and 26 00:02:04,007 --> 00:02:08,694 therefore, the density will increase by cube of that or a factor of 8. 27 00:02:08,694 --> 00:02:11,768 Now, let's go back to the Friedmann equation. 28 00:02:11,768 --> 00:02:16,757 We can break it down as follows. And it has the well-familiar solution of 29 00:02:16,757 --> 00:02:20,432 universe expanding at time, it's time to the third, to the third power. 30 00:02:20,432 --> 00:02:25,932 Now, for the overdense perturbation, we can express the solution in a parametric 31 00:02:25,932 --> 00:02:29,857 form shown here. The derivation of those is beyond this 32 00:02:29,857 --> 00:02:33,882 class, but there are places we can look it up such as Barbara Ryden's book. 33 00:02:35,713 --> 00:02:39,396 For now, assume that this is a view of what's happening. 34 00:02:39,396 --> 00:02:44,566 So, we can find out the turnaround time or radius by asking where is the maximum 35 00:02:44,566 --> 00:02:48,812 of this curve, which means that its first [UNKNOWN] is zero. 36 00:02:48,812 --> 00:02:53,849 Obviously, since this is a periodic solution, even though universe will not 37 00:02:53,849 --> 00:02:59,391 necessarily bounce back, it tells us that for the first turn around, it will be at 38 00:02:59,391 --> 00:03:02,984 the value of pi. And so, the solution for this is given 39 00:03:02,984 --> 00:03:05,894 here. The turn-around time is given by this 40 00:03:05,894 --> 00:03:08,703 formula. Now, if we look at the background 41 00:03:08,703 --> 00:03:13,722 universe, which continues expanding, its density go, keeps going down. 42 00:03:13,722 --> 00:03:18,876 And the ratio of the densities of the spheres at the turnaround time and 43 00:03:18,876 --> 00:03:24,304 surrounding the universe, is given as the cube of the expansion factors that 44 00:03:24,304 --> 00:03:30,044 correspond to them, which is a simple numerical factor of 9pi^2/16, for a very 45 00:03:30,044 --> 00:03:33,852 idealized case that we are talking. But remember, 46 00:03:33,852 --> 00:03:39,642 that by the time the universe collapses, again, to the Virial equilibrium, its 47 00:03:39,642 --> 00:03:45,187 physical scale will go down by a factor of two, and that means the density will 48 00:03:45,187 --> 00:03:49,672 go by the factor of eight. In the meantime, the background will get 49 00:03:49,672 --> 00:03:55,667 further diluted by a factor of 2^3, to the cube, and that means four. 50 00:03:55,667 --> 00:04:01,642 So, there is an additional gain of a factor of 32 in density contrast. 51 00:04:01,642 --> 00:04:07,417 Multiplying these two together, the final virialized state of structured 52 00:04:07,417 --> 00:04:12,919 equilibrium, would be 180 times denser than surrounding background. 53 00:04:12,919 --> 00:04:17,504 Remarkably, that's exactly what we see in large scale structure today. 54 00:04:17,504 --> 00:04:21,446 This is a treatment of single idealized density fluctuation. 55 00:04:21,446 --> 00:04:26,010 Now, of course, what happens later, such density fluctuations, what happen for 56 00:04:26,010 --> 00:04:30,691 that matter halos, keep merging, building larger ones and that is called a 57 00:04:30,691 --> 00:04:36,206 hierarchy confirmation setup Or or the bottom up structure information because 58 00:04:36,206 --> 00:04:39,071 you build bigger ones starting from small ones. 59 00:04:39,071 --> 00:04:43,422 Now in reality, the density fluctuations are unlikely to be perfect spheres. 60 00:04:43,422 --> 00:04:47,418 More likely, they can be approximated, say, as triaxial ellipsoids. 61 00:04:47,418 --> 00:04:51,926 They might have some very irregular shapes, but the triaxial ellipsoid is a 62 00:04:51,926 --> 00:04:55,933 better approximation than a sphere. Now, look what happens. 63 00:04:55,933 --> 00:05:00,006 At first, the perturbation expands within surrounding areas. 64 00:05:00,006 --> 00:05:05,068 But then, it turns around and it will turn around first at its shortest axis 65 00:05:05,068 --> 00:05:10,104 and it'll start collapsing along the shortest axis while still expanding in 66 00:05:10,104 --> 00:05:13,882 the other two. So, a three-dimensional blob will flatten 67 00:05:13,882 --> 00:05:19,169 itself into a pancake like structure. And then, we'll turn around on the second 68 00:05:19,169 --> 00:05:23,985 shortest axis and it will collapse along two axis and expanding a third one. 69 00:05:23,985 --> 00:05:29,171 The structure will turn into a filament. Finally, the third axis turns around and 70 00:05:29,171 --> 00:05:32,562 the, that's really [UNKNOWN] cluster collapses along that. 71 00:05:32,562 --> 00:05:36,863 and it will become, again, a quasi-spherical object in the end. 72 00:05:36,863 --> 00:05:41,980 This is the origin of the topology of the large scale structure that we see walls, 73 00:05:41,980 --> 00:05:46,707 and filaments, and the intersecting things which really are clusters, and 74 00:05:46,707 --> 00:05:51,509 this is the simple reason why it happens. So, let's figure out how long it will 75 00:05:51,509 --> 00:05:55,714 take for this to happen. Now, let's divide initial density 76 00:05:55,714 --> 00:05:59,289 perturbation, that's collapsing into a lot of thin layers. 77 00:05:59,289 --> 00:06:01,956 They will be all falling towards the middle. 78 00:06:01,956 --> 00:06:06,043 Let's say, two adjacent ones are, are essentially subject to the same 79 00:06:06,043 --> 00:06:10,460 acceleration, the inner one slightly higher acceleration, so it would be 80 00:06:10,460 --> 00:06:13,722 falling a little faster than the one just behind it. 81 00:06:13,722 --> 00:06:18,835 Therefore, the layers will never cross, the inner parts falling the fastest and 82 00:06:18,835 --> 00:06:22,833 the outer one is the slowest. So, if we take the point of the outermost 83 00:06:22,833 --> 00:06:27,540 shell and ask how long before a point collapses to the middle, that is 84 00:06:27,540 --> 00:06:32,754 basically the time for the collapse of the fluctuation, that's called a free 85 00:06:32,754 --> 00:06:36,581 fall time. And since all of the mass is enclosed to 86 00:06:36,581 --> 00:06:41,819 that radius at all times for all practical purposes, it is a free fall in 87 00:06:41,819 --> 00:06:45,374 the gravitational field of a point mass in the middle. 88 00:06:45,374 --> 00:06:49,635 And the solution for this is very simple and is shown here. 89 00:06:49,635 --> 00:06:54,666 You can also immediately see that intuitively, it is also true, not 90 00:06:54,666 --> 00:06:59,581 formally, that the lowest density fluctuations would collapse slowest and 91 00:06:59,581 --> 00:07:04,191 the highest density ones fastest. Now, also the largest ones would take 92 00:07:04,191 --> 00:07:07,281 longest time, the smaller ones would take less. 93 00:07:07,281 --> 00:07:12,547 Now, if we plug in typical numbers, say, for extent of dark halos of galaxies and 94 00:07:12,547 --> 00:07:17,872 sizes of clusters of galaxies, we find out that the relative collapse times are 95 00:07:17,872 --> 00:07:23,122 a few hundred million years for galaxies and few billion years for clusters. 96 00:07:23,122 --> 00:07:28,422 Which means that galaxies will collapse first in the early universe, and large 97 00:07:28,422 --> 00:07:33,272 constructions will be collapsing even today, and that's exactly what's 98 00:07:33,272 --> 00:07:36,577 observed. Next, we will talk about primordial 99 00:07:36,577 --> 00:07:42,274 density spectrum of fluctuations, which is a descriptor of how many fluctuations, 100 00:07:42,274 --> 00:07:43,730 of what size they are.