1 00:00:00,012 --> 00:00:05,207 So far, we have seen how the nature of the initial conditions, as specified by the 2 00:00:05,219 --> 00:00:10,360 density perturbation spectrum, and the nature of the dark matter that determines 3 00:00:10,372 --> 00:00:15,133 the damping mechanisms, govern the formation of the large scale structure. 4 00:00:15,241 --> 00:00:20,432 Now let us look in a little more detail what happens to those density fluctuations 5 00:00:20,444 --> 00:00:25,850 in the early universe. During the radiation dominated era, fluctuations do 6 00:00:25,862 --> 00:00:31,760 not grow. And their, the density field is dominated by the radiation, and the matter 7 00:00:31,772 --> 00:00:37,464 is tightly coupled to it. However, after the matter domination begin, begins, the 8 00:00:37,476 --> 00:00:43,221 fluctuations could grow. And the crucial quantity here is the Jeans Length, which 9 00:00:43,233 --> 00:00:49,043 you may recall from learning about star formation. Essentially, the pressure 10 00:00:49,055 --> 00:00:55,038 suppresses collapse of fluctuations at all scales smaller than genes length. This 11 00:00:55,050 --> 00:01:01,021 formula is given here. Physically, the way you can think of it is, the genes length 12 00:01:01,033 --> 00:01:06,692 is the size of the collapsing cloud, where the sound waves can cross before it 13 00:01:06,704 --> 00:01:12,246 collapses. In a radiation dominated universe, Jeans Length is very close to 14 00:01:12,258 --> 00:01:18,456 the horizon size itself. However, after the matter domination begins, it peels off 15 00:01:18,468 --> 00:01:25,077 and becomes smaller. The physical reason for this is the speed of sound drops, 16 00:01:25,193 --> 00:01:30,745 because it's dominated by matter, not radiation, and fluctuation can start to 17 00:01:30,757 --> 00:01:37,174 grow. This is schematically shown here. The diagonal line shows the mass enclosed 18 00:01:37,186 --> 00:01:42,892 within the horizon. It grows in time, because the horizon grows in time. And the 19 00:01:42,904 --> 00:01:48,798 Jeans Mass is very close to it. At the time of radiation matter equality, the 20 00:01:48,810 --> 00:01:54,169 curves begin to split, and Jeans Mass stays almost constant until the 21 00:01:54,181 --> 00:01:59,633 recombination, whereas the horizon mass keep growing. At the time of the 22 00:01:59,645 --> 00:02:04,469 recombination, the Jeans Mass drops precipitously by many orders of magnitude, 23 00:02:04,578 --> 00:02:10,078 roughly from super clusters of galaxies to scale of globular clusters. And so the key 24 00:02:10,090 --> 00:02:15,108 points about the growth and fluctuation is that always the horizon scale determines 25 00:02:15,602 --> 00:02:20,550 the characteristic length of the fluctuations. After the radiation matter 26 00:02:20,562 --> 00:02:25,890 equality, barriers are still in teracting with radiation field and interacting with 27 00:02:25,902 --> 00:02:31,045 it through Thompson scattering. The density perturbations show us the baryonic 28 00:02:31,057 --> 00:02:36,245 acoustic oscillations that we discussed earlier. Those are of course responsible 29 00:02:36,257 --> 00:02:40,955 for the Doppler peaks in the power spectrum with a cosmic micro-background. 30 00:02:41,162 --> 00:02:47,247 However, after recombination fluctuations can grow and essentially variance fall 31 00:02:47,259 --> 00:02:53,205 into the potential wells defined by the dark matter fluctuations. Schematically, 32 00:02:53,324 --> 00:02:59,273 this is shown here. The density contrast does not grow until, roughly, matter and 33 00:02:59,285 --> 00:03:04,688 radiation equality. At that point the dark matter fluctuations can begin to grow in 34 00:03:04,700 --> 00:03:10,294 contrast, whereas the baryons stay coupled to the radiation. After the recombination, 35 00:03:10,403 --> 00:03:15,518 the baryons are free to fall into the potential wells defined by the dark matter 36 00:03:15,530 --> 00:03:20,822 perturbations. And soon enough, they become equal. A more detailed computation 37 00:03:20,834 --> 00:03:25,746 is shown here. This shows the actual Baryonic Acoustic Oscillation as they 38 00:03:25,758 --> 00:03:30,916 would appear, and those will map into the peaks of the upper peaks in the cosmic 39 00:03:30,928 --> 00:03:36,349 micro-background. So here is the power spectrum of advanced defluctuations. At 40 00:03:36,361 --> 00:03:41,439 the very large scales, it is normalized by using the cosmic micro background. Those 41 00:03:41,451 --> 00:03:46,486 are fluctuations that go beyond the horizon, and here's the spectrum. 42 00:03:46,491 --> 00:03:51,033 Then there is the rollover due to the damping, as predicted by the cold dark 43 00:03:51,045 --> 00:03:56,417 matter theory. And there, of course, the baryonic acoustic oscillations. The data 44 00:03:56,429 --> 00:04:02,144 points here show the galaxy clustering power spectrum scaled to the apropriate 45 00:04:02,156 --> 00:04:07,996 size. It does not follow exactly the theoretical prediction, and this is due to 46 00:04:08,008 --> 00:04:13,490 the non-linear effects of merging hierchical assembly of galaxies. As you 47 00:04:13,502 --> 00:04:17,807 recall, the observations of micro-background confirm this theoretical 48 00:04:17,819 --> 00:04:22,827 picture to exquisite precision. So we know that this is roughly what happened in the 49 00:04:22,839 --> 00:04:27,904 early universe. So what happens after the recombination? The density perturbations 50 00:04:27,916 --> 00:04:32,676 continue to grow. They're already imprinted in the dark matter. The baryons 51 00:04:32,688 --> 00:04:37,544 soon follow. Baryons falls in those potential wells, and as t hey do so, they 52 00:04:37,556 --> 00:04:41,845 will start dissipating energy. Essentially, gas clouds keep colliding, 53 00:04:41,948 --> 00:04:46,683 that will inevitably cause some shock heating and energy dissipation. Later on, 54 00:04:46,786 --> 00:04:51,687 there will be things like start formation and energy input by stars. Or by coals in 55 00:04:51,699 --> 00:04:56,135 form of active galactic nuclear. Simulating these in detail is at the 56 00:04:56,147 --> 00:05:00,983 cutting edge of our ability to model universe in computers. But we can still 57 00:05:00,995 --> 00:05:05,622 make some progress on that. Now the key point here, is as you recall, the 58 00:05:05,634 --> 00:05:10,867 transition from simply gravitationally bound fluctuation to the one that's fully 59 00:05:10,879 --> 00:05:15,786 virialized, meaning matter is just rearanged due to the gravity. There is no 60 00:05:15,798 --> 00:05:20,909 energy dissipation per se. Leads to collapse by factor of 2, because the ratio 61 00:05:20,921 --> 00:05:26,036 of kinetic and potential energy changes by a factor of 2 and potential energy is 62 00:05:26,048 --> 00:05:30,390 inversely proportional to the characteristic size. So the density 63 00:05:30,402 --> 00:05:35,395 contrast that can be achieved due to dissipationless collapse is on the order 64 00:05:35,407 --> 00:05:41,031 of magnitude. Cube of 2, 8 or close to 10. In order for density fluctuation to 65 00:05:41,043 --> 00:05:46,727 achieve higher density contrasts, energy must be dissipated in some way. This 66 00:05:46,739 --> 00:05:52,666 process is sometimes called cooling. A good physical mechanism for this to happen 67 00:05:52,678 --> 00:05:58,362 is inverse Compton cooling of hot gas on cosmic micro background photons. This 68 00:05:58,374 --> 00:06:04,220 really only becomes effective at z less than 100. Because before then, the photons 69 00:06:04,232 --> 00:06:09,220 are too hot for them to act as an effective coolant for the gas. To quantify 70 00:06:09,232 --> 00:06:14,970 this, we will introduce the concept of the cooling time, and that's simply the energy 71 00:06:14,982 --> 00:06:20,245 that needs to be dissipated divided by the energy dissipation rate. So, it has 72 00:06:20,257 --> 00:06:25,534 dimensions of seconds. Plasma physics gives us the formula for this which is 73 00:06:25,546 --> 00:06:30,570 shown here. And there is a function lambda which is not to be confused with 74 00:06:30,582 --> 00:06:36,139 cosmological constants called the, the cooling function and its value depends on 75 00:06:36,151 --> 00:06:41,024 the density and temperature and composition of the material. So the key 76 00:06:41,036 --> 00:06:46,430 question here is the relationship between characteristic cooling time. The 77 00:06:46,442 --> 00:06:51,280 gravitational free fall time and the Hubble time. If the cooling ti me is 78 00:06:51,292 --> 00:06:56,355 shorter than the free fall time which we defined earlier, then objects will 79 00:06:56,367 --> 00:07:01,405 condense faster than they would just through gravitational collapse alone, 80 00:07:01,517 --> 00:07:06,513 reaching higher densities. If, on the other hand, both cooling time and 81 00:07:06,525 --> 00:07:12,192 free-fall time are greater than Hubble time, objects simply cannot form. Here is 82 00:07:12,204 --> 00:07:17,356 what cooling curves for plasmas of appropriate chemical composition look 83 00:07:17,368 --> 00:07:21,918 like. To show as a function of temperature. The relevant range of. 84 00:07:22,162 --> 00:07:27,417 Chemical compositions is from pure hydrogen and helium which is primordial 85 00:07:27,429 --> 00:07:32,394 gas, through all the way to solar composition gas, which is produced by 86 00:07:32,406 --> 00:07:37,605 stellar evolution. So, somewhere in between is all the relevant range. So, 87 00:07:37,720 --> 00:07:43,568 objects inside the cooling curve can cool faster than they can fall together, due to 88 00:07:43,580 --> 00:07:49,274 gravitation alone, and outside of cooling curve they cannot cool as fast and on the 89 00:07:49,286 --> 00:07:54,613 gravitational collapse of matter. Now we can recast the cooling curve to be shown 90 00:07:54,625 --> 00:08:00,267 in the temperature density diagram. Like most thermodynamical quantities, it can be 91 00:08:00,279 --> 00:08:05,986 expressed as a function of these two variables. And since the Jeans mass is 92 00:08:05,998 --> 00:08:12,251 also a function of density and temperature, the lines of equal Jeans mass 93 00:08:12,263 --> 00:08:18,789 correspond to diagonal lines in this log, log block shown here. You may notice that 94 00:08:18,801 --> 00:08:25,045 characteristic mass galaxies, 10 to the 12 solar masses, goes right through the 95 00:08:25,057 --> 00:08:30,050 region where cooling would be dominant, whereas characteristic masses of clusters 96 00:08:30,062 --> 00:08:34,765 of galaxies, something like 10 to the 15 solar masses, are outside of its region. 97 00:08:34,867 --> 00:08:39,350 So, to bring this point again. As a density fluctuation collapses, 2 things 98 00:08:39,362 --> 00:08:43,845 can happen. If it cannot cool faster than collapses, then it'll be just pure 99 00:08:43,857 --> 00:08:49,361 gravitational collapse on a free fall time scale. If however it does cool faster than 100 00:08:49,373 --> 00:08:54,669 it collapses, then it will inevitably fragment into smaller pieces, which then 101 00:08:54,681 --> 00:08:59,756 later may merge. So here is the cooling diagram again. And this time, shaded 102 00:08:59,768 --> 00:09:04,693 regions indicate where objects of different kinds are. Remarkably enough, 103 00:09:04,804 --> 00:09:10,273 galaxies of different travel types, occupy a region inside the cooling curve. And 104 00:09:10,285 --> 00:09:15,944 groups and clusters of galaxies occupy region outside. You may recall that 105 00:09:15,956 --> 00:09:22,243 galaxies are over dense by a factor of a million, whereas a venial collapse in an 106 00:09:22,255 --> 00:09:28,532 expanding universe.can only achieve contrast in the order of 100. And indeed, 107 00:09:28,532 --> 00:09:33,976 we do believe the galaxies formed through dissipative fra-, processes, and where as 108 00:09:33,988 --> 00:09:39,135 groups and clusters and large scale structure are product of dissipationless, 109 00:09:39,246 --> 00:09:44,271 purely gravity driven collapse. So, let us recap the key ideas of structure 110 00:09:44,283 --> 00:09:49,455 formation. Structure formation grows out of dense defluxuations in the early 111 00:09:49,467 --> 00:09:55,314 universe, which can grow under their own gravity, accreting more material, and 112 00:09:55,326 --> 00:10:01,074 through merging. The initial conditions are often described using the power 113 00:10:01,086 --> 00:10:07,295 spectrum of the density field. And the one that is often used is power law dependent 114 00:10:07,307 --> 00:10:13,911 power spectrum, with an exponent of one. Also called, Harrison-Zeldovich spectrum. 115 00:10:13,911 --> 00:10:17,609 Spectrum. The role of dark matter is essential. Dark 116 00:10:17,621 --> 00:10:22,620 matter fluctuations can grow before baryons can actually follow. Which is why 117 00:10:22,632 --> 00:10:27,397 it's possible to actually have the structure that we can see. In some sense, 118 00:10:27,505 --> 00:10:32,195 dark matter seeds this large scale structure prior to recombination. And 119 00:10:32,207 --> 00:10:36,993 after the recombination, baryons, the visible material, falls into those 120 00:10:37,005 --> 00:10:42,420 potential wells. There is damping mechanisms, say photon diffusion or sound 121 00:10:42,432 --> 00:10:47,507 waves. They raise small scale fluctuations. The nature of dark matter 122 00:10:47,519 --> 00:10:52,645 determines which mechanism will be at play, and therefore how many of small 123 00:10:52,657 --> 00:10:57,505 fluctuations will be left. The observations show that cold dark matter, 124 00:10:57,618 --> 00:11:02,794 composed of massive particles. Does fit all of the data, whereas the hot dark 125 00:11:02,806 --> 00:11:07,424 matter composed of light relativistically moving particles, say as massive 126 00:11:07,436 --> 00:11:12,489 neutrinos, simply does not reproduce the observed picture. And the topology of the 127 00:11:12,501 --> 00:11:17,077 density fluctuations will always go from there are actual blobs which first 128 00:11:17,089 --> 00:11:21,916 collapse into the sheets, which then collapse into filaments, which then drain 129 00:11:21,928 --> 00:11:27,425 into an quasi-spherical blobs like clusters. An important thing to remember 130 00:11:27,437 --> 00:11:33,600 is that, as shown throug h spherical top-hat model. The simple virialization, 131 00:11:33,727 --> 00:11:40,160 dissipationless collapse of a fluctuation in an expanding universe, can reach a 132 00:11:40,172 --> 00:11:46,081 density contrast of 200. This is more than what will happen in non-expanding universe 133 00:11:46,093 --> 00:11:50,499 where only a factor of eight will be achieved because the surrounding 134 00:11:50,511 --> 00:11:55,435 background also expands while the perturbation collapses. Free-fall time 135 00:11:55,447 --> 00:12:00,120 scales, that is, in fall just due to the gravity with nothing to oppose it, 136 00:12:00,228 --> 00:12:04,943 indicate that for galaxy scale fluctuations, the formation will be on the 137 00:12:04,955 --> 00:12:10,165 scale of hundreds of millions of years. Where some scales of clusters of galaxies 138 00:12:10,177 --> 00:12:14,962 and large scale structure formation would be on scales of billions of years, and 139 00:12:14,974 --> 00:12:19,850 that scale responds to observations. Just as in the case of star formation here in 140 00:12:19,862 --> 00:12:24,490 galaxy formation, genus length or genus mass are key concept to determine the 141 00:12:24,502 --> 00:12:29,485 scale of fluctuations that can actually grow or fragment in the smaller one. The 142 00:12:29,497 --> 00:12:34,265 Jeans mass is essentially within, mass within horizon prior to the 143 00:12:34,567 --> 00:12:39,790 radiation-matter equality. Becomes constant after that, and then drops 144 00:12:39,802 --> 00:12:45,155 precipitously at the time of the recombination. Cooling is the key concept 145 00:12:45,167 --> 00:12:50,120 because, in order to achieve higher density contrast, energy must be 146 00:12:50,132 --> 00:12:56,690 dissipated. Which is known as cooling. And galaxies form largely through dissipate 147 00:12:56,702 --> 00:13:03,453 processes where as large scale structure is formed dissipation less leak, purely 148 00:13:03,465 --> 00:13:08,956 through gravitational collapse and assembly. Next we will see how the 149 00:13:08,968 --> 00:13:14,109 nonlinear processes of structure formation are simulated using computers.