1 00:00:00,012 --> 00:00:05,000 We now turn to cosmological tests, which in some sense, are at the very heart of 2 00:00:05,000 --> 00:00:07,765 cosmology. First, let's make some general 3 00:00:07,765 --> 00:00:11,320 considerations. The goal of cosmological tests is to 4 00:00:11,320 --> 00:00:16,455 determine the global geometry, and therefore ultimate fate of the universe, 5 00:00:16,455 --> 00:00:21,368 which particular universe we live in. And since the behavior of different 6 00:00:21,368 --> 00:00:26,490 cosmological models is expressed as expansion curve R of T, we somehow have 7 00:00:26,490 --> 00:00:31,354 to compare that with the observations. So we assume a family of models, such as 8 00:00:31,354 --> 00:00:35,852 solutions of the Friedmann equation, or Friedmann-Lemaitre models. 9 00:00:35,852 --> 00:00:40,907 There are others, but these are pretty generic, and they are characterized by a 10 00:00:40,907 --> 00:00:45,792 number of cosmological parameters that we defined earlier They create a whole 11 00:00:45,792 --> 00:00:50,632 family of these curves of R of T, and somehow, we have to find out on which one 12 00:00:50,632 --> 00:00:54,112 we are on. we can observe only in the past, but from 13 00:00:54,112 --> 00:00:58,432 that, we can predict what the universe will do in the future as well. 14 00:00:58,432 --> 00:01:03,657 We integrate the equations, producing observable quantities, such as distances, 15 00:01:03,657 --> 00:01:06,702 and then we compare that to the observations. 16 00:01:06,702 --> 00:01:10,478 The basis of all cosmological tests is as follows. 17 00:01:10,478 --> 00:01:15,264 We have R of T curve. R of T is relatively easily observable as a 18 00:01:15,264 --> 00:01:19,004 redshift. The look-back time, T, is not. So instead 19 00:01:19,004 --> 00:01:24,428 of that, we multiply that with speed of light, converting it into a distance. 20 00:01:24,428 --> 00:01:29,741 And distances, at least the relative distances, we can measure, at least in 21 00:01:29,741 --> 00:01:33,628 principle. So we somehow invert this diagram. So 22 00:01:33,628 --> 00:01:39,582 instead of R of T versus time, we plot a distance of some sort, versus redshift. 23 00:01:39,582 --> 00:01:45,117 Note that the whole thing does not depend on the exact value of the Hubble 24 00:01:45,117 --> 00:01:50,935 constant, that just scales the whole diagram up or down, but the curvature, 25 00:01:50,935 --> 00:01:54,542 the shape of the curves, does not depend on it. 26 00:01:54,542 --> 00:01:58,029 Let's see. What is the generic behavior that we 27 00:01:58,029 --> 00:02:01,740 expect? Consider just the models where the matter 28 00:02:01,740 --> 00:02:05,418 dominates. If there is more matter, there is more 29 00:02:05,418 --> 00:02:09,771 gravity and so the, the, the expansion slows faster. 30 00:02:09,771 --> 00:02:14,795 It will always be faster at the beginning, slowing down, depending on how 31 00:02:14,795 --> 00:02:19,650 much gravity is there to slow it down. And so, if there is more gravity, or 32 00:02:19,650 --> 00:02:24,702 higher density, it will reach a certain value of Hubble constant sooner. 33 00:02:24,702 --> 00:02:30,041 So if you take two models, one of high density, one of low density and pinch 34 00:02:30,041 --> 00:02:34,900 them at the same value of the Hubble constant, which is what defines today, 35 00:02:34,900 --> 00:02:38,443 Hubble time, well, then, the one with higher density 36 00:02:38,443 --> 00:02:43,885 actually spends shorter time getting there, and its intercept on the time axis 37 00:02:43,885 --> 00:02:47,736 is shorter. So models with the high density would 38 00:02:47,736 --> 00:02:53,470 tend to have shorter life times, therefore smaller distances. 39 00:02:53,470 --> 00:03:01,402 Therefore, things would look brighter and bigger than they would be in models with 40 00:03:01,402 --> 00:03:07,212 low density. Adding cosmoligical constant can boost the effect of gravity, or 41 00:03:07,212 --> 00:03:12,888 counterveil it, if it's a positive cosmoligical constant that essentially 42 00:03:12,888 --> 00:03:18,816 acts as anti-gravity. And so, low density and or positive cosmlogical constant work 43 00:03:18,816 --> 00:03:22,822 in the same way. High density and or negative cosmological 44 00:03:22,822 --> 00:03:27,758 constant will work the other way. There are several types of cosmological 45 00:03:27,758 --> 00:03:30,998 tests. Ultimately we have to measure distances 46 00:03:30,998 --> 00:03:34,364 somehow. The first one was introduced by Hubble, 47 00:03:34,364 --> 00:03:39,347 and that's Hubble diagram. The slope of Hubble diagram is the Hubble 48 00:03:39,347 --> 00:03:43,117 constant. Deviations of the trend with redshift are 49 00:03:43,117 --> 00:03:48,297 dependent on cosmological parameters. So its the curvature of the Hubble 50 00:03:48,297 --> 00:03:51,352 diagram that is really a cosmological test. 51 00:03:51,352 --> 00:03:56,767 In order to perform this test, we need a family of sources that are bright enough 52 00:03:56,767 --> 00:04:02,187 to be seen far away, and that have constant intrinsic luminosity or standard 53 00:04:02,187 --> 00:04:06,592 candles. The other classical cosmological test is 54 00:04:06,592 --> 00:04:12,357 the angular diameter test. There, if we had a family of objects big 55 00:04:12,357 --> 00:04:19,452 enough to see far away, say clusters of galaxies, and of constant intrinsic size, 56 00:04:19,452 --> 00:04:23,501 standard rulers, we can then perform angular diameter test. 57 00:04:23,501 --> 00:04:28,562 So this is where luminosity distance and angular diameter distance come in. 58 00:04:28,562 --> 00:04:33,747 Finally we could see, in principle, how the volume of the universe is changing as 59 00:04:33,747 --> 00:04:38,710 a function of time. And so if we had a universe populated uniformly with some 60 00:04:38,710 --> 00:04:43,852 sort of test particles, say, galaxies, and if we can count them as a function of 61 00:04:43,852 --> 00:04:49,252 redshift, or alternatively, as a function of flux, because flux would get dimmer 62 00:04:49,252 --> 00:04:53,727 with redshift, then we could constrain the models in which we are. 63 00:04:53,727 --> 00:04:58,327 So that is the basis of the source counts, and that is another type of 64 00:04:58,327 --> 00:05:03,161 cosmological test that people do. And in principle, if we could somehow 65 00:05:03,161 --> 00:05:08,431 estimate the look-back time to a family of objects, say, galaxies, through the 66 00:05:08,431 --> 00:05:13,937 age of their stellar populations, we could perform direct tests of look-back 67 00:05:13,937 --> 00:05:18,113 time versus redshift. But that turns out to be extremely 68 00:05:18,113 --> 00:05:22,822 difficult and model dependent, and essentially, has never been done. 69 00:05:22,822 --> 00:05:28,214 What can be done is to measure the local matter density of the universe, using 70 00:05:28,214 --> 00:05:31,974 dynamics of galaxies or clusters of galaxies near us. 71 00:05:31,974 --> 00:05:37,110 That produces measurement of the omega matter parameter, a fraction of the 72 00:05:37,110 --> 00:05:41,677 critical density in regular matter, regardless of all of the distant 73 00:05:41,677 --> 00:05:44,632 measurements. And finally, if you can somehow 74 00:05:44,632 --> 00:05:49,647 independently measure Hubble constant and age of the universe, as we discussed 75 00:05:49,647 --> 00:05:54,872 earlier, then that combination constrains a combination of omega matter and omega 76 00:05:54,872 --> 00:06:00,092 lack in density of cosmological constant, but does not determine it uniquely. 77 00:06:00,092 --> 00:06:04,524 And that's all it can do. So cosmological tests where developed, at least in 78 00:06:04,524 --> 00:06:08,447 principle, immediately after the discovery of the expansion of the 79 00:06:08,447 --> 00:06:13,199 universe, thus the Hubble diagram bears his name, and Hubble and his successors, 80 00:06:13,199 --> 00:06:16,290 notably Alan Sandage, tried to put them in practice. 81 00:06:16,290 --> 00:06:20,313 There was a lot of work done one this from the 1950s through 1970s at Mount 82 00:06:20,313 --> 00:06:24,813 Palomar and elsewhere. Various types of luminous, and or large 83 00:06:24,813 --> 00:06:29,423 objects, we use in lieu of standard candles, or, standard rulers. 84 00:06:29,423 --> 00:06:35,059 Most notably, Sandage and collaborators, and others, have worked on the use of the 85 00:06:35,059 --> 00:06:39,390 brightest cluster galaxies, which tend to be giant elliptical 86 00:06:39,390 --> 00:06:44,855 galaxies of standard candles. Standard rulers were things like typical 87 00:06:44,855 --> 00:06:50,202 sizes of galaxy clusters or radio sources, and all of these objects evolved 88 00:06:50,202 --> 00:06:54,496 in some way or other. Galaxies are composed of stars. stars 89 00:06:54,496 --> 00:07:00,112 evolve, their brightness will change in time collectively. Galaxies merge. 90 00:07:00,112 --> 00:07:05,676 Radio sources expand. Clusters collapse. And so, there were no really standard 91 00:07:05,676 --> 00:07:10,321 candles or standard rulers. So galaxy evolution, or other forms of 92 00:07:10,321 --> 00:07:15,864 evolution, basically prevented us from doing cosmological tests as originally 93 00:07:15,864 --> 00:07:19,182 envisioned. The revival of this came in the 1990s 94 00:07:19,182 --> 00:07:24,755 through Supernova Hubble diagram, and also through a revival of the angular 95 00:07:24,755 --> 00:07:28,349 diameter test through cosmic micro-background. 96 00:07:28,349 --> 00:07:33,815 This, essentially, completely revived the subject, and the original goal of 97 00:07:33,815 --> 00:07:37,311 cosmology, that, to determine what kind of universe we 98 00:07:37,311 --> 00:07:41,896 live in, was finally achieved, at the level of precision that people, in prior 99 00:07:41,896 --> 00:07:46,544 decades, just couldn't even dream about. But there are some general issues to be 100 00:07:46,544 --> 00:07:49,266 aware of. We are always observing some sort of 101 00:07:49,266 --> 00:07:51,603 sources, say, whether they're supernovae, or 102 00:07:51,603 --> 00:07:53,272 galaxies. It doesn't matter. 103 00:07:53,272 --> 00:07:58,788 Our observations have a flux limit. We usually cannot see fainter than some 104 00:07:58,788 --> 00:08:04,297 given threshold, and so if you look deeper and deeper, at some point you are 105 00:08:04,297 --> 00:08:09,788 going to start losing objects that are just below the detection threshold. 106 00:08:09,788 --> 00:08:15,120 So if you try to fit a model curve only to those that you do see, you're going to 107 00:08:15,120 --> 00:08:18,851 get a biased result. If this is understood, and if you can 108 00:08:18,851 --> 00:08:22,576 somehow make a statistical correction for it, that's fine. 109 00:08:22,576 --> 00:08:27,693 But generally, you don't know which part of the population are you missing, and so 110 00:08:27,693 --> 00:08:32,510 that can cause significant bias. Likewise, if you are measuring angular 111 00:08:32,510 --> 00:08:37,212 diameter test, there is usually a limit to angular resolution, maybe because of 112 00:08:37,212 --> 00:08:42,344 the seeing through the atmosphere, and so you're not going to measure anything 113 00:08:42,344 --> 00:08:47,313 smaller than a certain level, which will bias your result in that way as well. 114 00:08:47,313 --> 00:08:52,652 So lets now consider Hubble diagram. There, in classical rendering, we compare 115 00:08:52,652 --> 00:08:55,602 magnitudes, which, remember, are inversely 116 00:08:55,602 --> 00:09:00,877 proprotional to the log of the flux, so a higher magnitude means further away, 117 00:09:00,877 --> 00:09:04,952 versus redshift. And, different models will have different 118 00:09:04,952 --> 00:09:09,027 curves in that diagram. In the past, things like brightest 119 00:09:09,027 --> 00:09:14,842 cluster galaxies were considered, but more modern the renderings include 120 00:09:14,842 --> 00:09:19,287 supernovae or even gamma ray bursts. In principle, anything else that you can 121 00:09:19,287 --> 00:09:23,487 convincingly state can be standardized to a given luminosity can be used. 122 00:09:23,487 --> 00:09:28,177 But before you do this, there is an important correction that usually has to 123 00:09:28,177 --> 00:09:32,035 be made. We observe, typically, in a given filter, 124 00:09:32,035 --> 00:09:35,843 a red filter, say, or a visual filter and so on. 125 00:09:35,843 --> 00:09:40,395 And here are the spectra of galaxies of different types. 126 00:09:40,395 --> 00:09:47,074 The reddish band indicates roughly where the rest frame filter, like a red filter, 127 00:09:47,074 --> 00:09:51,631 would cover the spectrum, as observed in the universe. 128 00:09:51,631 --> 00:09:56,941 But, if you observe them say, at redshift of 1, then the wavelength shifts by a 129 00:09:56,941 --> 00:10:00,152 factor of 2, and so does the width of the filter. 130 00:10:00,152 --> 00:10:05,510 So you're sampling a completely different part of the spectrum of the galaxy, and 131 00:10:05,510 --> 00:10:09,572 not in quite the same width of the, of the measurement. 132 00:10:09,572 --> 00:10:14,637 That has to be compensated for, because galaxies do have different spectra, and 133 00:10:14,637 --> 00:10:18,362 different types will have different amount of correction. 134 00:10:18,362 --> 00:10:22,897 If you take the ratio, if you integrate the flux over the two filters, the rest 135 00:10:22,897 --> 00:10:27,757 frame 1 and, its red shifted version, take the ratio of that, that is called 136 00:10:27,757 --> 00:10:31,312 the K-correction, usually expressed in magnitudes. 137 00:10:31,312 --> 00:10:36,787 And since most galaxies tend to be redder, redder than bluer, K-corrections 138 00:10:36,787 --> 00:10:40,237 are positive. So here is a set of curves that were 139 00:10:40,237 --> 00:10:45,112 computed for non-evolving galaxies, galaxies that are observed here and now. 140 00:10:45,112 --> 00:10:50,162 How much would you have to correct, brightness in a given filter, if you were 141 00:10:50,162 --> 00:10:54,351 to put them at some redshift? So that way, you, uniform, make a uniform 142 00:10:54,351 --> 00:10:57,600 set of measurement, comparing apples and apples, 143 00:10:57,600 --> 00:11:02,514 and you can say, if this wasn't an evolving population, then, this is what I 144 00:11:02,514 --> 00:11:07,935 should see to that given redshift. It doesn't have to be galaxies, you can 145 00:11:07,935 --> 00:11:11,257 do the exact same exercise for supernovae. 146 00:11:11,257 --> 00:11:16,407 So next time, we will see how we actually use Hubble diagram as a cosmological 147 00:11:16,407 --> 00:11:16,765 tool.