1 00:00:00,012 --> 00:00:04,042 Hello. So far, we've found out how to compute 2 00:00:04,042 --> 00:00:10,457 different cosmological models, but what good are they? Basic goal of 3 00:00:10,457 --> 00:00:17,228 cosmology is to figure out in what model universe do we live and you will recall 4 00:00:17,228 --> 00:00:21,579 that they are basically distinguished by their history of the expansion rate. 5 00:00:21,579 --> 00:00:25,786 How does the scale factor changes as a function of time? If we can figure out 6 00:00:25,786 --> 00:00:29,760 which curve of those we live on, we know, we'll know about cosmological 7 00:00:29,760 --> 00:00:32,683 parameters. The expansion factor, R(t), is simply 8 00:00:32,683 --> 00:00:36,302 related to red shift, that is an observable quantity and that's 9 00:00:36,302 --> 00:00:39,560 an easy part. The other axis is the time axis. 10 00:00:39,560 --> 00:00:45,058 Now, unfortunately, distant galaxies do not carry gigantic clocks on them, so we, 11 00:00:45,058 --> 00:00:50,138 it's very hard to figure out what is the look back time between us and some 12 00:00:50,138 --> 00:00:53,196 distant point in a way that can be measured. 13 00:00:53,196 --> 00:00:57,952 So instead of that, what we do is, we do, we transform coordinates. 14 00:00:57,952 --> 00:01:07,872 Instead of the look back time, we can use distance which is simply time multiplied 15 00:01:07,872 --> 00:01:14,497 by the speed of light. And distance is in principle, we can 16 00:01:14,497 --> 00:01:17,382 measure. So we flip this diagram, and instead of 17 00:01:17,382 --> 00:01:18,382 expansion R(t), we use the red shift, which is an observable quantity. 18 00:01:18,382 --> 00:01:21,692 And instead of the time, we use a distance, which we can figure, figure out 19 00:01:21,692 --> 00:01:27,147 how to measure in some way. So centrally, all cosmological tests boil 20 00:01:27,147 --> 00:01:30,957 down to this. We have to somehow measure a set of 21 00:01:30,957 --> 00:01:37,452 distances to points of the function of redshift and because the whole thing just 22 00:01:37,452 --> 00:01:43,432 scales with Hubble constant, we only need to determine the shape of that curve. 23 00:01:43,432 --> 00:01:47,547 So let's figure out how to measure distances in cosmology. 24 00:01:47,547 --> 00:01:53,122 A convenient unit of the distance is Hubble distance, which is simply speed of 25 00:01:53,122 --> 00:01:58,352 the light divided by the hubble constant. We call it hubble constant, it has 26 00:01:58,352 --> 00:02:03,272 dimensions one over time, and obviously, one over Hubble constant 27 00:02:03,272 --> 00:02:07,419 is called the Hubble time. In the units of Hubble constant of 70 28 00:02:07,419 --> 00:02:12,633 kilometers per second per megaparsec, which is close to its actual measured 29 00:02:12,633 --> 00:02:17,844 value, Hubble length is about 4.3 gigaparsecs and Hubble time is a little 30 00:02:17,844 --> 00:02:22,147 shy of 14 billion years, which is actually pretty close to the 31 00:02:22,147 --> 00:02:26,490 actual age of years. Now, at low redshifts, the expansion is 32 00:02:26,490 --> 00:02:29,954 linear. Hubble's law applies and the distance is 33 00:02:29,954 --> 00:02:33,052 simply redshift times the Hubble distance, 34 00:02:33,052 --> 00:02:38,508 but the further out we go, the realistic effects come into play and things are a 35 00:02:38,508 --> 00:02:43,128 little more complicated. Generally speaking, we can compute the 36 00:02:43,128 --> 00:02:47,404 comoving distance, distance in coordinates that do expand 37 00:02:47,404 --> 00:02:51,792 with the expanding space. As follows, we can integrate the element 38 00:02:51,792 --> 00:02:56,937 of redshift divided by the function, which is really Hubble constant at that 39 00:02:56,937 --> 00:03:01,667 given time and which is given here. You may actually remember similiar 40 00:03:01,667 --> 00:03:05,522 expression inside a Friedmann equation. You'll see three terms, 41 00:03:05,522 --> 00:03:10,307 the density of matter multiplied by the cube of one plus redshift. Well, that's 42 00:03:10,307 --> 00:03:15,067 probably because density of the matter scales as the cube of the expansion, 43 00:03:15,067 --> 00:03:20,252 then there is a curvature term and there is a constant term, which corresponds to 44 00:03:20,252 --> 00:03:24,447 cosmological constant. There is actually one more term missing 45 00:03:24,447 --> 00:03:29,137 which is the relativistic matter, which would be omega of radiation times one 46 00:03:29,137 --> 00:03:33,513 plus redshift to the fourth power. But that's only important in the early 47 00:03:33,513 --> 00:03:37,129 universe, so it's generally neglected in this computation. 48 00:03:37,129 --> 00:03:41,882 Now, the really useful quantity in computing physical parameters, is not 49 00:03:41,882 --> 00:03:47,366 comoving distance per say, but a quantity that is adjusted for the, for the 50 00:03:47,366 --> 00:03:52,832 curvature of the space, which is called the transverse comoving distance, 51 00:03:52,832 --> 00:03:56,412 and it's in the formula for that is given here. 52 00:03:56,412 --> 00:04:00,105 In the case of a flat universe, those are the same. 53 00:04:00,105 --> 00:04:05,004 In the case of positively or negatively curved ones, there is a crtuch. 54 00:04:05,004 --> 00:04:10,321 Now remember, omega curvature, so called curvature density does not really 55 00:04:10,321 --> 00:04:15,702 correspond to density of anything. It is just a quantity that makes the sum 56 00:04:15,702 --> 00:04:21,147 of all densities equal to one, always. So, omega curvature is really one minus 57 00:04:21,147 --> 00:04:25,970 the omega of matter, omega radiation, and omega of cosmological constant. 58 00:04:25,970 --> 00:04:30,463 So how can we derive this? We'll start with Robertson-Walker metric, 59 00:04:30,463 --> 00:04:34,012 the basis of all relativistic cosmological models. 60 00:04:34,012 --> 00:04:39,159 Since, only the radial coordinate matters by large, we can then simplify that get 61 00:04:39,159 --> 00:04:43,232 rid of all the, both the angular terms and just have the radial term. 62 00:04:43,232 --> 00:04:46,131 So take square root of both sides and integrate. 63 00:04:46,131 --> 00:04:50,029 Generally speaking, this integral cannot be solved analytically. 64 00:04:50,029 --> 00:04:54,833 It has to be solved numerically. In some special cases, such, as when the 65 00:04:54,833 --> 00:05:00,444 density of the vacuum, cosmology constant is equal to zero, there are analytical 66 00:05:00,444 --> 00:05:03,543 solutions. In that case, the formula for the 67 00:05:03,543 --> 00:05:07,693 distance is shown here. It's expressed in the older units of q 68 00:05:07,693 --> 00:05:11,306 naught, which remember is deceleration parameter, 69 00:05:11,306 --> 00:05:16,199 and in the absence of cosmological constant, it is equal to the density 70 00:05:16,199 --> 00:05:20,112 parameter divided by two. Now, for universe with non-zero 71 00:05:20,112 --> 00:05:26,892 cosmological constant, things are little more complicated and generally speaking, 72 00:05:26,892 --> 00:05:33,502 infigural has to be computed numerically. So here is a plot of distances for 73 00:05:33,502 --> 00:05:37,489 several different cosmological models expressed in the units of the Hubble 74 00:05:37,489 --> 00:05:41,659 distance as a function of redshift. It's in units of Hubble distance because 75 00:05:41,659 --> 00:05:45,414 the whole thing scales directly, linearly, proportionately to Hubble 76 00:05:45,414 --> 00:05:49,592 constant, and we can separate measurement of the scale of the universe through 77 00:05:49,592 --> 00:05:53,947 Hubble's constant, in the shape of the universe if you will, which is the outer 78 00:05:53,947 --> 00:05:57,973 cosmological parameters. But how do we relate this to the things 79 00:05:57,973 --> 00:06:02,024 that are actually measured? We can measure, roughly speaking, two 80 00:06:02,024 --> 00:06:05,730 kinds of things, brightness, from some source far away or 81 00:06:05,730 --> 00:06:09,342 its angular size. In cases where we can use brightness, you 82 00:06:09,342 --> 00:06:12,534 can imagine using source of a standard luminosity, 83 00:06:12,534 --> 00:06:17,206 also called standard candle, viewed at different distances and using the 84 00:06:17,206 --> 00:06:22,168 relativistic version of universe square law to find out how far it is. So in 85 00:06:22,168 --> 00:06:27,352 simple, Euclidean, non-relativistic universe, the flux would be luminosity 86 00:06:27,352 --> 00:06:31,822 divide, divided by the area of the sphere with a radius from here to there. 87 00:06:31,822 --> 00:06:36,810 In the case of an expanding universe, there are two terms of one plus redshift 88 00:06:36,810 --> 00:06:41,314 that come into play. One of them accounts for the energy loss 89 00:06:41,314 --> 00:06:46,931 of photons, because their wavelength is stretched as one plus redshift. But the 90 00:06:46,931 --> 00:06:52,639 thing is moving far away from us and fast, and so, we have to account for the 91 00:06:52,639 --> 00:06:57,761 relativistic time dilation. The clocks tick slower there by a factor 92 00:06:57,761 --> 00:07:02,847 of one plus z from our point of view, and so, the rate of the photon emission is 93 00:07:02,847 --> 00:07:08,022 affected by exactly the same factor, so we have 1 + ^2 in denominator. 94 00:07:08,022 --> 00:07:13,227 In other words, in an expanding relativistic universe, objects appear 95 00:07:13,227 --> 00:07:17,200 dimmer than they would if the universe wasn't expanding. 96 00:07:18,342 --> 00:07:24,082 So for convenience, luminosity distance is defined as the actual radial distance 97 00:07:24,082 --> 00:07:27,152 times one plus z. So when you take the square of it, you 98 00:07:27,152 --> 00:07:30,432 recover a familiar universe square of formula. 99 00:07:30,432 --> 00:07:35,350 That is if we can measure the total luminosity, but, we usually measure 100 00:07:35,350 --> 00:07:40,412 specific flux, which is power per unit wavelength or unit frequency. 101 00:07:40,412 --> 00:07:47,132 In that case, we recover one power of 1 + z, because the angstroms are stretched by 102 00:07:47,132 --> 00:07:53,207 exactly same factor or the hertz are also stretched by one of the factor. 103 00:07:53,207 --> 00:07:58,554 So in case of specific fluxes there are, no two powers of 1 + z, 104 00:07:58,554 --> 00:08:01,658 they're one. So here is a plot of luminosity distance 105 00:08:01,658 --> 00:08:05,960 in several cosmological models, again, expressed in units of Hubble 106 00:08:05,960 --> 00:08:10,174 length as a function of redshift. You see, the typical redshifts of 107 00:08:10,174 --> 00:08:14,432 interest in modern cosmology, it's of the order of ten Hubble lengths. 108 00:08:14,432 --> 00:08:18,972 The other kind of thing that we can measure is the angular size of things. 109 00:08:18,972 --> 00:08:24,907 Imagine if you had standard gigantic ruler that you can view a different size, 110 00:08:24,907 --> 00:08:30,079 different [INAUDIBLE] from us, then its angular diameter will tell us 111 00:08:30,079 --> 00:08:35,612 how far we are for a given cosmology. In a simple Euclidean non-expanding 112 00:08:35,612 --> 00:08:40,438 space, that angle will be the size of the ruler divided by the distance. 113 00:08:40,438 --> 00:08:43,793 That is as far as comoving coordinates are concerned, 114 00:08:43,793 --> 00:08:47,267 but, what if the object is fixed in proper coordinates, 115 00:08:47,267 --> 00:08:50,879 say like a galaxy? Galaxies do not expand with expanding 116 00:08:50,879 --> 00:08:55,592 space at any given redshift. Proper size is larger than the comoving 117 00:08:55,592 --> 00:09:00,942 size, because comoving coordinates have expanded since then and it's larger by 118 00:09:00,942 --> 00:09:05,067 exactly power of 1 + z. So here we have the opposite scenario. 119 00:09:05,067 --> 00:09:10,560 The objects in an expanding universe are actually bigger than they would be in a 120 00:09:10,560 --> 00:09:15,975 non-expanding case. And we divide the distance by 1 + z so that we can use the, 121 00:09:15,975 --> 00:09:20,759 a familiar formula and that's called the angular diameter distance. 122 00:09:20,759 --> 00:09:25,770 Here is a plot of angular diameter distance as a function of redshift for 123 00:09:25,770 --> 00:09:31,032 several popular cosmological models. You will notice that, in many of them, 124 00:09:31,032 --> 00:09:34,368 there is a maximum. At some distance from us, things no 125 00:09:34,368 --> 00:09:37,836 longer appear smaller, this is specific to relativistic 126 00:09:37,836 --> 00:09:40,777 cosmology. In a Euclidean space, 127 00:09:40,777 --> 00:09:44,690 the further away something is, the smaller it's going to look, 128 00:09:44,690 --> 00:09:47,907 always. Here, past certain depth, things actually 129 00:09:47,907 --> 00:09:52,600 start getting larger again. Finally, we can imagine measuring volumes 130 00:09:52,600 --> 00:09:56,005 out to some redshift. If say, universe was populated by 131 00:09:56,005 --> 00:09:59,567 particles such as galaxies or alignment of a clouds, 132 00:09:59,567 --> 00:10:04,941 and we can count, then their number per unit volume is something that will be 133 00:10:04,941 --> 00:10:08,760 also dependent on the expansion rate. So the volume element, 134 00:10:08,760 --> 00:10:13,507 its increment, radial increment as a function of redshift is given by this 135 00:10:13,507 --> 00:10:17,012 formula. Generally speaking, that has to be always 136 00:10:17,012 --> 00:10:21,619 evaluated numerically. And the total volume integrated from here 137 00:10:21,619 --> 00:10:25,035 to some redshift of z is given by these formulas. 138 00:10:25,035 --> 00:10:30,198 So here are plots of the volume element as a function of redshift for several 139 00:10:30,198 --> 00:10:33,656 models. As you can see, those curves can differ 140 00:10:33,656 --> 00:10:40,125 very substantially at high redshifts, and that's why this is potentially an 141 00:10:40,125 --> 00:10:46,043 interesting comological test. Here as well as in the other plots, all 142 00:10:46,043 --> 00:10:52,057 of the curves are really close together at very low redshifts, when things that 143 00:10:52,057 --> 00:10:53,082 are asymptotically Euclidean. So that's the distances, what about the 144 00:10:53,082 --> 00:10:55,628 look-back time? Whereas we cannot directly measure it as 145 00:10:55,628 --> 00:11:00,463 a function of redshift. It is a useful quantity when considering things like 146 00:11:00,463 --> 00:11:04,257 galaxy evolution. The time elapsed, since some redshift of 147 00:11:04,257 --> 00:11:09,311 z, which is the look-back time is given by the same formula as we've seen before, 148 00:11:09,311 --> 00:11:13,057 except corrected by 1 + z. So then, again, this also has to be 149 00:11:13,057 --> 00:11:17,487 evaluated numerically, except for some special cases, like with zero 150 00:11:17,487 --> 00:11:21,587 cosmological constant, when an elliptical solution does exist. 151 00:11:21,587 --> 00:11:26,352 If we're to in, integrate this all the way out to infinity redshift, that is to 152 00:11:26,352 --> 00:11:30,042 the Big Bang, we will get the total age of the universe. 153 00:11:30,042 --> 00:11:35,772 So here is the look-back time as the function of redshift or alternatively, 154 00:11:35,772 --> 00:11:41,502 age of the universe as a function of redshift measured in units of the Hubble 155 00:11:41,502 --> 00:11:47,856 time for a variety of different models. They all behave qualitatively in similar 156 00:11:47,856 --> 00:11:52,822 fashion, but the curves are separate. So this is what will lead us into 157 00:11:52,822 --> 00:11:56,772 cosmological tests. We're trying to determine which R(t) 158 00:11:56,772 --> 00:12:00,147 curve we live on. This is really measured, done by 159 00:12:00,147 --> 00:12:03,372 measuring redshifts and some form of distance. 160 00:12:03,372 --> 00:12:08,447 But, take these curves, and pinch them together when the slope is the same. 161 00:12:08,447 --> 00:12:12,412 In other words, when they have the same Hubble constant. 162 00:12:12,412 --> 00:12:17,737 Well then, the intercept on the time axis would be different for different curves. 163 00:12:17,737 --> 00:12:23,102 Cosmological models with high densities and or zero cosmological constant will 164 00:12:23,102 --> 00:12:28,377 tend to have smaller distances, smaller look-back times, and smaller volume, than 165 00:12:28,377 --> 00:12:32,619 models with lower densities and positive cosmological constant. 166 00:12:32,619 --> 00:12:37,887 Thus, if you pinch these curves at the same value, Hubble constant, which is the 167 00:12:37,887 --> 00:12:42,911 same slope, and see where the intercepts are on the time axis, that the lower 168 00:12:42,911 --> 00:12:48,587 density or positive cosmological constant models will tend to be larger and both 169 00:12:48,587 --> 00:12:51,682 longer duration. So the objects in lower density models, 170 00:12:51,682 --> 00:12:57,542 or, models with positive cosmological constant will appear smaller and fainter 171 00:12:57,542 --> 00:13:02,067 than they would in the same models with high density in all cosmological 172 00:13:02,067 --> 00:13:04,882 constant. So next, we will start looking into 173 00:13:04,882 --> 00:13:07,722 measuring the scale of the universe or the Hubble constant.