1 00:00:00,012 --> 00:00:06,039 Finally let us address the subject of the power spectrum of large scale structure. 2 00:00:06,039 --> 00:00:11,635 You may recall that to introduce this concept when we talked about premordial 3 00:00:11,635 --> 00:00:15,679 density field. And now just like then some familiarity 4 00:00:15,679 --> 00:00:19,086 with free analysis is absolutely essential. 5 00:00:19,087 --> 00:00:24,174 So if you are familiar with Fourier Analysis, the main subject of today's 6 00:00:24,174 --> 00:00:29,238 lecture, will be almost trivial. And if you are not, it will make no sense 7 00:00:29,238 --> 00:00:32,892 whatsoever. So this would be like really good time to 8 00:00:32,892 --> 00:00:36,047 refresh your knowledge of Fourier Analysis. 9 00:00:36,047 --> 00:00:38,274 Done? Okay, well that was fast. 10 00:00:38,274 --> 00:00:43,404 So let's get going. So we can characterize the density field 11 00:00:43,404 --> 00:00:48,281 of large scale structure in the same way we did before. 12 00:00:48,281 --> 00:00:53,271 So lets take this Fourier transform here is the formula. 13 00:00:53,271 --> 00:00:59,595 It's essentially almost the and the inverse transform is given by this. 14 00:00:59,595 --> 00:01:06,257 So in this way spacial scales, a wave number scale are connected and the power 15 00:01:06,257 --> 00:01:13,017 spectrum of that defluctuation is obviously a complex product of spectrum by 16 00:01:13,017 --> 00:01:16,841 itself. Now recall what was the value of two point 17 00:01:16,841 --> 00:01:22,746 correlation function expressed from density field of the expectation value. 18 00:01:22,746 --> 00:01:25,646 In other words, this is exactly what it is. 19 00:01:25,646 --> 00:01:29,921 So, correlation function is a transform of the power spectrum. 20 00:01:29,921 --> 00:01:34,024 Power spectrum and correlation function are a Fourier pair. 21 00:01:34,024 --> 00:01:38,291 They are completely equivalent to each other mathematically. 22 00:01:38,291 --> 00:01:43,144 And here is an example for what they look like from Las Campanas Redshift Survey 23 00:01:43,144 --> 00:01:48,040 that you may recall, what is shown here is both 3 dimensional 2 point correlations 24 00:01:48,040 --> 00:01:52,991 functions and power spectrum corresponding to the same large scale structure. 25 00:01:52,991 --> 00:01:57,599 You may recall that 2 point new point correlation function was fairly easy to 26 00:01:57,599 --> 00:02:02,412 evaluate in any one of several ways. And, but power spectrum is really what 27 00:02:02,412 --> 00:02:06,544 theory produces. So, one can be translated into the other. 28 00:02:06,544 --> 00:02:11,945 Now, we usually express the power spectrum on a log lock axis, and theoretical 29 00:02:11,945 --> 00:02:16,417 models, fitted against observations, give us the shape of it. 30 00:02:16,417 --> 00:02:19,631 However, the amplitude has not been defined. 31 00:02:19,631 --> 00:02:22,606 There are two ways in which that can be done. 32 00:02:22,606 --> 00:02:28,931 One is to measure fluctuations so that the density field at the particular scale. 33 00:02:28,931 --> 00:02:34,364 And typically, sphere of radius of H, H to the minus 1 megaparsecs is used because 34 00:02:34,364 --> 00:02:37,731 that gives the RMS of fluctuations close to unit. 35 00:02:37,731 --> 00:02:43,843 Turns out it's a little less than that for modern measurements but traditionally 8, 8 36 00:02:43,843 --> 00:02:48,439 megaparsecs is used. So, whatever the value of that is gives us 37 00:02:48,439 --> 00:02:53,249 unit normalization of the power spectrum at that spatial scale. 38 00:02:53,249 --> 00:02:58,887 Essentially what's been done here is you're convolving what's technically 39 00:02:58,887 --> 00:03:04,959 called a spherical top-hat filter but it's really nothing but a sphere with a power 40 00:03:04,959 --> 00:03:08,587 spectrum itself. An alternative way is to simply use 41 00:03:08,587 --> 00:03:13,462 observations of cosmic microwave background which give us power spectrum on 42 00:03:13,462 --> 00:03:17,005 very large scales. So if you have a model like cold dark 43 00:03:17,005 --> 00:03:22,087 matter model, you can fit it on those scales, evolve for the appropriate linear 44 00:03:22,087 --> 00:03:25,550 growth of fluctuations, and that works just as well. 45 00:03:25,551 --> 00:03:30,675 So, here is one of the modern renderings of the power spectrum of large scale 46 00:03:30,675 --> 00:03:34,494 structures. Spanning the range all the way from super 47 00:03:34,494 --> 00:03:37,776 horizon sizes, down to the scales of galaxies. 48 00:03:37,776 --> 00:03:42,081 The plot assembles several different kind of measurements. 49 00:03:42,081 --> 00:03:46,146 They all brought to the same redshift by scaling in linear regime. 50 00:03:46,146 --> 00:03:50,476 At large scales cosmic microwave background is what's done. 51 00:03:50,476 --> 00:03:55,708 That overlaps with measurements of clustering from large scale structure from 52 00:03:55,708 --> 00:03:59,860 rigid surveys and such. And then going to smaller scales, we used 53 00:03:59,860 --> 00:04:04,892 things like abundance of clusters of galaxies, that's essentially, statistics 54 00:04:04,892 --> 00:04:09,998 of high density peaks and clustering of Lyman alpha clouds which are sub-galactic 55 00:04:09,998 --> 00:04:13,588 fragments. Weak gravitational lensing reflects the 56 00:04:13,588 --> 00:04:18,604 nasty distribution of the dark matter and so that too can be used to probe the 57 00:04:18,604 --> 00:04:21,513 distribution of CDM on intermediate scales. 58 00:04:21,513 --> 00:04:26,172 And the line going through at points is the standard cold dark matter model. 59 00:04:26,172 --> 00:04:28,827 As you can see, it's a remarkable good fit. 60 00:04:28,827 --> 00:04:33,441 And also an excellent agreement between these completely different ways of 61 00:04:33,441 --> 00:04:36,492 observing it, in the regions where they overlap. 62 00:04:36,492 --> 00:04:41,601 So this gives us a real confidence that we actually, you know at some level of 63 00:04:41,601 --> 00:04:47,190 certainty what's going on and one of the major reasons why people believe CDM model 64 00:04:47,190 --> 00:04:50,960 is correct one. Just as an aside numerical called baryonic 65 00:04:50,960 --> 00:04:54,737 acoustic oscillations. And this is actually how they were 66 00:04:54,737 --> 00:04:58,177 detected. By looking at the two point correlation 67 00:04:58,177 --> 00:05:03,612 function or power spectrum at scales in the vicinity of hundred megaparsecs. 68 00:05:03,613 --> 00:05:10,156 So this an absolutely essential use of this uh,, concept as a cosmological test 69 00:05:10,156 --> 00:05:16,892 that can be, be used to improve our knowledge of cosmological parameters. 70 00:05:16,892 --> 00:05:23,217 So far we talked about powerless spectrum, that is the distribution of amplitudes of 71 00:05:23,217 --> 00:05:28,102 dense defluctuations. But in that process, as in computing power 72 00:05:28,102 --> 00:05:32,299 spectrum, we'll completely neglect phase information. 73 00:05:32,299 --> 00:05:38,493 Now here is a dramatic illustration of why phase is information is important, 74 00:05:38,493 --> 00:05:41,856 produced by [unknown] And it's a toy model. 75 00:05:41,856 --> 00:05:47,030 The density model that is really [unknown] form kind of mimicking the filamentary 76 00:05:47,030 --> 00:05:50,168 structure that we see in large scale structure. 77 00:05:50,168 --> 00:05:55,250 What was done here is free transform was taken and then the phases were randomly 78 00:05:55,250 --> 00:05:58,611 scrambled, leaving the power spectrum identical. 79 00:05:58,611 --> 00:06:03,863 Then transformed back, with scrambled phases, and you see the plot on the 80 00:06:03,863 --> 00:06:06,546 bottom. Looks like a complete noise. 81 00:06:06,546 --> 00:06:11,206 The coherence is completely gone. So whereas the distribution of 82 00:06:11,206 --> 00:06:16,260 fluctuations is the same. Their arrangement is completely different. 83 00:06:16,260 --> 00:06:21,531 The phase coherence is been lost in this computation, and obviously we have to 84 00:06:21,531 --> 00:06:27,282 somehow learn how to use statistics of the phases of the density field power spectrum 85 00:06:27,282 --> 00:06:32,871 or rather the Fourier spectrum in order to characterize it fully, because when you 86 00:06:32,871 --> 00:06:37,878 look at large scale structure, you don't see just lumpy density field. 87 00:06:37,878 --> 00:06:41,418 You do see a measure of elements and voids and what not. 88 00:06:41,418 --> 00:06:46,394 Amazingly enough, so far nobody has come up with a really simple, elegant, complete 89 00:06:46,394 --> 00:06:49,205 way of describing this so it might be a little. 90 00:06:49,205 --> 00:06:54,248 So we discussed clustering of galaxies in some detail but what about clustering of 91 00:06:54,248 --> 00:06:58,175 clusters of galaxies? Or in fact you can discuss clustering of 92 00:06:58,175 --> 00:07:02,763 any elements of large scale structure galaxies, groups, clusters, super 93 00:07:02,763 --> 00:07:06,944 clusters, and so on. As it turns out clusters also cluster and 94 00:07:06,944 --> 00:07:10,405 their clustering is stronger than that of galaxies. 95 00:07:10,405 --> 00:07:15,878 Actually there is an excellent correlation between the strength of clustering, say 96 00:07:15,878 --> 00:07:21,268 measured through clustering length, and the richness of the system at which you're 97 00:07:21,268 --> 00:07:24,554 looking. The richer clusters are clustered more 98 00:07:24,554 --> 00:07:27,652 strongly. This is actually a very interesting 99 00:07:27,652 --> 00:07:31,429 result, and at first it was puzzling why should that be so. 100 00:07:31,429 --> 00:07:36,357 Until Nick Keiser came up with the explanation that this is due to phenomenon 101 00:07:36,357 --> 00:07:39,990 now called bias. You can consider density field, and so 102 00:07:39,990 --> 00:07:45,110 there are some high peaks and low peaks. Remember it can be also be composed in 103 00:07:45,110 --> 00:07:50,537 waves of large wavelengths and small wavelengths fluctuations riding on top of 104 00:07:50,537 --> 00:07:54,218 large wavelength. So if you impose a threshold for a 105 00:07:54,218 --> 00:07:58,836 formation of certain amount, a kind of object say five sigma cut. 106 00:07:58,836 --> 00:08:04,806 Then the highest peaks will be strongly clustered because the smaller fluctuations 107 00:08:04,806 --> 00:08:10,408 are carried together by the large waves. And those that happen to be a crest of a 108 00:08:10,408 --> 00:08:15,782 big wave will be all clumped together. This is in same sense as if you were to 109 00:08:15,782 --> 00:08:20,019 ask what are the highest points on the surface of planet Earth. 110 00:08:20,019 --> 00:08:25,081 And if you put a cap a few kilometers elevation above the sea level, you'll see 111 00:08:25,081 --> 00:08:29,377 they are very strongly clustered where the high mountains are. 112 00:08:29,377 --> 00:08:32,760 In this case say Himalayas and[UNKNOWN] and so on. 113 00:08:32,760 --> 00:08:38,471 Whereas if you lower down the threshold. To the ground level, then everything will 114 00:08:38,471 --> 00:08:43,340 be more or less uniformly distributed. So this is a simple explenation. 115 00:08:43,340 --> 00:08:48,244 At the highest density peaks, always cluster more strongly than the lower 116 00:08:48,244 --> 00:08:51,707 density peaks And this applies to galaxies as well. 117 00:08:51,707 --> 00:08:56,425 We find out the denser galaxies, elliptical galaxies say, plus their more 118 00:08:56,425 --> 00:09:01,125 strongly than spiral galaxies. And I've shown you part of that we've seen 119 00:09:01,125 --> 00:09:03,612 earlier. This is the explanation why. 120 00:09:03,612 --> 00:09:08,001 It also has some implications. The highest density peaks will be the 121 00:09:08,001 --> 00:09:13,032 first ones to form, at small scales. This we'll come back to this phenomenon 122 00:09:13,032 --> 00:09:16,486 when we talk about galaxy formation and evolution. 123 00:09:16,486 --> 00:09:20,986 So let us recap everything we learned about large scale structure. 124 00:09:20,986 --> 00:09:26,432 So we see a range of scales from galaxies which are scales of a kiloparsecs through 125 00:09:26,432 --> 00:09:31,472 groups and clusters in scales of megaparsecs to superclusters on scales of 126 00:09:31,472 --> 00:09:35,732 maybe hundred parsecs. We measure it, largely, by using Redshift 127 00:09:35,732 --> 00:09:40,340 series of galaxies, and today we have Redshift for couple million galaxies, 128 00:09:40,340 --> 00:09:45,452 which gives us really excellent insights into what large scale structure looks like 129 00:09:45,452 --> 00:09:49,372 as a function of Redshift. But clustering itself is not enough. 130 00:09:49,372 --> 00:09:52,478 There's also lot of interesting topology going on. 131 00:09:52,478 --> 00:09:57,448 Coherent structures like filaments and voids and sheets that separate them in We 132 00:09:57,448 --> 00:10:02,039 understand why this happens. You may recall that aspherical fluctuation 133 00:10:02,039 --> 00:10:07,149 will first turn around and collapse along one axis while expanding in the other two, 134 00:10:07,149 --> 00:10:12,040 making things like sheets or, or walls, then will turn around on an intermediate 135 00:10:12,040 --> 00:10:16,931 axis, collapse into a filament-like structure, and eventually all drain into, 136 00:10:16,931 --> 00:10:20,248 again, to. So this is the origin of the large scale 137 00:10:20,248 --> 00:10:25,012 topology, although we still don't have good ways of characterizing it. 138 00:10:25,012 --> 00:10:30,110 Clustering itself is customarily characterized through 2-point correlation 139 00:10:30,110 --> 00:10:33,394 function or equivalently the power spectrum. 140 00:10:33,394 --> 00:10:38,049 The 2-point correlation function has a fairly robust functional form. 141 00:10:38,049 --> 00:10:42,728 It is well represented. Not perfectly, but well over large range 142 00:10:42,728 --> 00:10:48,335 of scales as a parallel with slope of minus 1.8 in three dimensional space. 143 00:10:48,335 --> 00:10:54,393 For galaxies, the normalization given through clustering length is of the order 144 00:10:54,393 --> 00:10:59,372 of five Megaparsec or so. For clusters, it's larger, responding to 145 00:10:59,372 --> 00:11:04,082 larger amplitude of clustering. And whereas we can easily measure a 2 146 00:11:04,082 --> 00:11:07,255 point correlation function for any kind of objects. 147 00:11:07,255 --> 00:11:12,160 To compare them with theory, we actually have to turn 2 point correlation function 148 00:11:12,160 --> 00:11:16,017 into a power spectrum. And fortunately the two are equivalent, 149 00:11:16,017 --> 00:11:19,225 and therefore a pair, so we know how to compute that. 150 00:11:19,226 --> 00:11:24,212 Overall, the cold dark matter model, fits the data remarkably well over a full range 151 00:11:24,212 --> 00:11:28,376 of scales that we can measure through a variety of different methods. 152 00:11:28,376 --> 00:11:32,871 Which gives us confidence that cold dark matter model is, in fact, correct. 153 00:11:32,871 --> 00:11:37,492 And I should know to this point, that would be very difficult to achieve this if 154 00:11:37,492 --> 00:11:42,562 there were, were no dark matter at all. Generally speaking, objects of different 155 00:11:42,562 --> 00:11:47,182 kinds have different clustering strengths that applies to different kinds of 156 00:11:47,182 --> 00:11:49,744 galaxies. Luminous versus less luminous. 157 00:11:49,744 --> 00:11:54,481 Early versus late types, and so on. Carrying over to the clusters of different 158 00:11:54,481 --> 00:11:58,972 regions, and so on. Next we will talk about large-scale 159 00:11:58,972 --> 00:12:05,587 peculiar velocity field, which is a direct consequence of the existence of 160 00:12:05,587 --> 00:12:08,202 large-scale density field.