1 00:00:00,012 --> 00:00:06,377 >> We have seen how redshift surveys can be used to describe large scale structure, 2 00:00:06,377 --> 00:00:12,514 at least in the cosmographic sense. But how do we quantify the distributional 3 00:00:12,514 --> 00:00:17,254 galaxies such that we can compare with theoretical models? 4 00:00:17,255 --> 00:00:22,375 The first way in which people have done this is to use the so called two-point 5 00:00:22,375 --> 00:00:27,007 correlation function. Nowadays, power spectrum is more often 6 00:00:27,007 --> 00:00:31,287 used and the two are actually related in a very simple fashion. 7 00:00:31,287 --> 00:00:36,059 We will go about that next time. So, let's talk about the galaxy two-point 8 00:00:36,059 --> 00:00:40,908 correlation function. What it means is that, if galaxies are 9 00:00:40,908 --> 00:00:48,334 clustered together, they're correlated. Each galaxy is somehow more likely to be 10 00:00:48,334 --> 00:00:54,196 found next to another galaxy. And one way to quantify this is to ask the 11 00:00:54,196 --> 00:00:58,154 question. Assume that galaxies are actually 12 00:00:58,154 --> 00:01:04,336 uniformly, randomly distributed in space. Then, at a distance from any given galaxy, 13 00:01:04,336 --> 00:01:08,867 there will be a certain probability of finding another galaxy. 14 00:01:08,867 --> 00:01:13,088 If you actually measure this, you'll find out, there is an axis. 15 00:01:13,088 --> 00:01:18,523 There are more galaxies near other galaxies and you'd expect from purely 16 00:01:18,523 --> 00:01:23,135 random distribution. And that access above the random is what 17 00:01:23,135 --> 00:01:28,106 correlation function is. So, one simply does the counting of galaxy 18 00:01:28,106 --> 00:01:34,040 pairs for each galaxy and then normalizes by what the random distribution with the 19 00:01:34,040 --> 00:01:40,106 same number of data points would be. As it turns out, the two-point correlation 20 00:01:40,106 --> 00:01:46,379 function is well-represented by power-law, and it's usually written in this form. 21 00:01:46,379 --> 00:01:52,081 That, radius, divided by some scaling radius, to some power which is close to 22 00:01:52,081 --> 00:01:55,826 minus 1.8. And typically, for normal galaxies in this 23 00:01:55,826 --> 00:02:00,028 neck of the universe, the scaling length is about 5 megaparsec. 24 00:02:00,028 --> 00:02:04,868 This, however, is not universal. Different kinds of galaxies have different 25 00:02:04,868 --> 00:02:07,994 clustering properties, as you'll see shortly. 26 00:02:07,995 --> 00:02:12,743 So, here is an example of a modern, well measured, two-point correlation function 27 00:02:12,743 --> 00:02:15,903 from, from the galaxies from, to the F redshift survey. 28 00:02:15,903 --> 00:02:18,533 It does look pretty close to the power-law. 29 00:02:18,533 --> 00:02:22,961 Although, if you subtract the best fit power-law, you'll see that there are some 30 00:02:22,961 --> 00:02:27,908 significant deviations from it. Now, if you don't have redshifts, you can 31 00:02:27,908 --> 00:02:32,836 measure the angular correlation fraction just projected in the sky. 32 00:02:32,836 --> 00:02:38,512 And that this two-dimensional projected correlation function usually [unknown] 33 00:02:38,512 --> 00:02:43,858 looks like little w, not to be confused with the equation of state parameter, is 34 00:02:43,858 --> 00:02:49,042 related to the three-dimensional correlation function, xi of r, in a fairly 35 00:02:49,042 --> 00:02:53,017 simple fashion. The parallax--point is differed exactly by 36 00:02:53,017 --> 00:02:57,087 one because we reduced the dimensionality of problem from 3 to 2. 37 00:02:57,087 --> 00:03:01,944 Another important point is that if galaxies are more likely to be found near 38 00:03:01,944 --> 00:03:05,156 other galaxies, there is that axis probability. 39 00:03:05,156 --> 00:03:10,060 Then, in order to keep the average constant, it has to turn negative at some 40 00:03:10,060 --> 00:03:14,341 point, and it does. Scales that are roughly corresponding to 41 00:03:14,341 --> 00:03:17,238 those of voids seen in galaxy distribution. 42 00:03:17,238 --> 00:03:22,061 If there is a void in distribution, in some sense, that's anti-correlation. 43 00:03:22,061 --> 00:03:26,825 It's less likely to find galaxy there, then if it would be, if the space was 44 00:03:26,825 --> 00:03:31,348 uniformly populated with galaxies. So, how do we do this in practice? 45 00:03:31,348 --> 00:03:34,935 Suppose we have a catalog of galaxies from some survey. 46 00:03:34,935 --> 00:03:38,403 Then, you can create an, a random catalog of galaxies. 47 00:03:38,403 --> 00:03:43,401 Same number of galaxies, but randomly distributed in a Poissonian fashion. 48 00:03:43,401 --> 00:03:49,632 Then, you can do the simple counts. Count galaxy pairs, one against the other, 49 00:03:49,632 --> 00:03:55,178 and count the fake galaxy pairs, the random catalog, divide the two and 50 00:03:55,178 --> 00:03:59,253 subtract to one because its the axis probability. 51 00:03:59,253 --> 00:04:05,434 For large extensive catalog of galaxies with uniform borders, this will work 52 00:04:05,434 --> 00:04:09,438 fairly well. But in reality catalogs do have some 53 00:04:09,438 --> 00:04:13,058 incompleteness or uneven borders, and so on. 54 00:04:13,058 --> 00:04:19,312 So, there is a better estimator, which is called the Landy-Szalay estimator, and its 55 00:04:19,312 --> 00:04:23,969 formula is given here. We make a correction by counting galaxy 56 00:04:23,969 --> 00:04:28,551 versus random catalog pairs. This takes care of the boundary 57 00:04:28,551 --> 00:04:32,275 conditions. Now, if you're not counting individual 58 00:04:32,275 --> 00:04:37,540 galaxies, but have, essentially, a galaxy density field where you can divide 59 00:04:37,540 --> 00:04:42,967 galaxies in, in boxes or, or pixels, then numerical galaxy density can be used in 60 00:04:42,967 --> 00:04:46,922 the same fashion. Just subtract the expected average from 61 00:04:46,922 --> 00:04:50,058 the actual count n, and divide by the average. 62 00:04:50,059 --> 00:04:56,086 So, if you do this, for any kinds of pairs or density pairs, then xi of r is really 63 00:04:56,086 --> 00:05:00,674 the expectation value of this probabilistic distribution. 64 00:05:00,674 --> 00:05:06,857 Now, strictly speaking, because we are correlating galaxies with themselves, this 65 00:05:06,857 --> 00:05:10,498 should be called the autocorrelation function. 66 00:05:10,498 --> 00:05:14,381 But common usage is just to call it correlation function. 67 00:05:14,381 --> 00:05:19,826 Note, also, that you can correlate sample of one kind of objects versus the others. 68 00:05:19,826 --> 00:05:24,319 So, for example, you could ask, are galaxies clustered around quasars? 69 00:05:24,319 --> 00:05:28,890 And, and we can evaluate the cross-correlation function between 70 00:05:28,890 --> 00:05:33,381 galaxies and quasars, and so on. By analogy, we can expand this to 71 00:05:33,381 --> 00:05:37,879 higher-order terms. The three-point correlation function, now 72 00:05:37,879 --> 00:05:43,225 asks, given a galaxy, given a probability of finding another galaxy at certain 73 00:05:43,225 --> 00:05:48,166 distance, what is now probability of finding a third galaxy at some other 74 00:05:48,166 --> 00:05:51,427 distance? This obviously gets lot more complicated 75 00:05:51,427 --> 00:05:56,021 and numerically tedious very fast. However, there is some useful information 76 00:05:56,021 --> 00:06:00,676 in these high-order correlation functions, and sometimes they're evaluated. 77 00:06:00,676 --> 00:06:05,372 I mentioned that different kinds of galaxies can cluster differently, and here 78 00:06:05,372 --> 00:06:09,369 is a simple thing you can do. You can ask, how are galaxies clustered, 79 00:06:09,369 --> 00:06:13,647 say, bright ones versus faint ones? And it turns out the bright ones are 80 00:06:13,647 --> 00:06:17,652 clustered more strongly. The amplitude is higher and the slope is 81 00:06:17,652 --> 00:06:20,396 steeper. The steepness of the slope obviously 82 00:06:20,396 --> 00:06:23,314 correlates with how strongly they are clustered. 83 00:06:23,314 --> 00:06:27,961 If the correlation function was perfectly flat, there would be no correlation. 84 00:06:27,961 --> 00:06:33,339 Understanding effects like this contains some useful information about galaxy 85 00:06:33,339 --> 00:06:37,792 formation mechanisms. And we'll talk more about those later in 86 00:06:37,792 --> 00:06:40,580 the class. Or, you can divide galaxies by 87 00:06:40,580 --> 00:06:44,346 morphological type, say, ellipticals versus spirals. 88 00:06:44,346 --> 00:06:49,152 It turns out that elliptical galaxies or redder galaxies are clustered more 89 00:06:49,152 --> 00:06:52,103 strongly than the blue ones, the disk galaxy. 90 00:06:52,103 --> 00:06:56,543 That, too, has an interesting clue about galaxy formation and evolution. 91 00:06:56,543 --> 00:07:00,017 So, it's a power-law. Does that mean it's a fractal? 92 00:07:00,017 --> 00:07:05,336 Remember, for any distribution of points, the probability of finding another one 93 00:07:05,336 --> 00:07:09,605 from the same set increases the sum dimensionality of the space. 94 00:07:09,605 --> 00:07:13,342 In normal tedious space, this would be like cube of radius. 95 00:07:13,342 --> 00:07:17,211 If that number, is not an integer, the set is called fractal. 96 00:07:17,211 --> 00:07:22,294 So, we can write this formula, which is resembling correlation function. 97 00:07:22,294 --> 00:07:27,250 And if, indeed, it was pure power law, then you could say, universe was fractal, 98 00:07:27,250 --> 00:07:31,645 if it was a pure power-law extending to infinitely large distances. 99 00:07:31,645 --> 00:07:37,318 But in reality, that's not the case. There are significant deviations from 100 00:07:37,318 --> 00:07:43,121 power-law, it's slightly bent. And, therefore, universe is not fractal or 101 00:07:43,121 --> 00:07:48,526 large-scale structure is not fractal, although pretty close to it. 102 00:07:48,526 --> 00:07:54,317 Next time, we'll talk about power spectrum of galaxy clustering, which can be 103 00:07:54,317 --> 00:07:57,970 directly related to theoretical predictions.