1 00:00:00,870 --> 00:00:06,400 [Auto-generated transcript. Edits may have been applied for clarity.] So I, I would say. 2 00:00:10,360 --> 00:00:13,659 Uh. Thank you. Thank you, everyone, for, uh, coming to the event. 3 00:00:13,660 --> 00:00:18,850 And, uh, I thought I'd tell you about what is called a crisis in cosmology. 4 00:00:19,210 --> 00:00:23,770 And it goes by the name of the Hubble tension. And, uh, maybe introduce you to what? 5 00:00:23,770 --> 00:00:30,850 What people are talking about when when you see these words reported in the media and also, uh, show you what, 6 00:00:31,150 --> 00:00:37,300 what it would take to solve the Hubble tension, what we might learn from it, uh, and really do spoil the punch line. 7 00:00:37,780 --> 00:00:41,530 Uh, we don't actually have any satisfactory solutions, so we don't really know what's going on. 8 00:00:41,530 --> 00:00:46,000 I give you a few solutions and show you why they're not really satisfactory in any sense. 9 00:00:48,460 --> 00:00:54,250 So this is a version of, uh, of the of the timeline, Edward, showing us. 10 00:00:54,570 --> 00:00:58,360 Uh, this is, uh, beginning of the universe on the left. 11 00:00:58,840 --> 00:01:01,910 And maybe I have a pointer here somewhere. Um. 12 00:01:04,730 --> 00:01:07,970 Uh. And as time goes on, the universe is expanding. 13 00:01:08,420 --> 00:01:15,620 And the epoch we are going to focus on, uh, is, uh, relatively late in cosmology compared to what Ed was talking about. 14 00:01:16,040 --> 00:01:19,519 And so we're going to, uh, focus on the epochs after recombination. 15 00:01:19,520 --> 00:01:27,170 So this is depicted in this picture as, uh, the surface here, uh, and all the way to today, uh, in a, in the cosmology, 16 00:01:27,410 --> 00:01:35,090 the epoch of recombination is the epoch where, uh, as Joe was describing, protons and electrons came together to form neutral hydrogen. 17 00:01:35,390 --> 00:01:40,910 And so the universe became transparent to photons. And so that's, uh, that's sometimes called the surface of last scattering. 18 00:01:41,300 --> 00:01:46,070 Most photons encountered the last scattering at that surface, and the universe became transparent. 19 00:01:46,070 --> 00:01:50,450 So we can actually see, uh, down, uh, all the way back to that surface. 20 00:01:51,170 --> 00:01:57,080 And really, we'd actually be focusing on most of the, um, focusing mostly on the physics of the surface of last scattering, 21 00:01:57,530 --> 00:02:04,640 uh, the CMB, uh, itself and physics around, uh, current times and not so much about the intervening, uh, dark edges. 22 00:02:07,160 --> 00:02:15,640 Um, so, uh, uh, before we get into the details, let's just, uh, say what the hub of tension is. 23 00:02:16,090 --> 00:02:20,860 Uh, uh, there's this parameter in cosmology called the Hubble parameter. 24 00:02:21,310 --> 00:02:25,280 Uh, and, uh, the thing that we are focussed on is the Hubble parameter today. 25 00:02:25,300 --> 00:02:32,970 So this is called h. Not usually not denotes, uh, things that we're talking that, uh uh uh, not denote the time today. 26 00:02:32,980 --> 00:02:36,670 So denote as time today H nought is the Hubble constant Hubble parameter today. 27 00:02:37,390 --> 00:02:45,630 And uh, this blue uh, uh, this blue point here represents, uh, the Hubble constant today. 28 00:02:45,640 --> 00:02:50,050 It's not as deduced from measurements of the cosmic microwave background by the Planck satellite. 29 00:02:50,410 --> 00:02:56,170 And it's a really nice measurement. You can see it's, uh, uh, it's about, uh, a percent level measurement of the Hubble constant. 30 00:02:57,100 --> 00:03:05,110 And this red, uh, here is a measurement what is often called a local measurement of the Hubble constant, which is we look out to, uh, 31 00:03:05,110 --> 00:03:17,140 nearby or, uh, relatively nearby galaxies and, uh, and supernovae and, and, uh, calculate the Hubble uh, parameter, uh, uh, using that method. 32 00:03:17,620 --> 00:03:21,730 Uh, and that has been, uh, that has been an industry for a very long time. 33 00:03:22,060 --> 00:03:28,840 That was indeed the first way we discovered that the universe, uh, that's expansion of the universe was accelerating. 34 00:03:29,020 --> 00:03:36,460 So this is also a very mature science, and that gives us a value that is insignificant discrepancy with the value that we deduce from the CMB. 35 00:03:37,960 --> 00:03:45,970 Uh, these yellow, uh, points are relatively recent analysis, uh, using James Webb data. 36 00:03:46,330 --> 00:03:52,900 And then you can see sort of they're not quite there yet in terms of precision to really, uh, say what's going on. 37 00:03:53,410 --> 00:03:58,450 Uh, and they themselves have a little bit of a spread between them. So, so the situation is a bit murky. 38 00:03:58,450 --> 00:04:04,659 Certainly these two measurements, which, uh, which have been really thoroughly vetted and investigated over, over many, 39 00:04:04,660 --> 00:04:11,260 many years by many, many different groups are incomplete, uh, statistical, uh, tension with each other with very high significance. 40 00:04:11,740 --> 00:04:15,220 And we don't really know, uh, who's right, so to speak. 41 00:04:15,400 --> 00:04:21,940 Okay. And, uh, but I introduce more elements of what we mean by the Hubble constant and so on as we go with them. 42 00:04:23,530 --> 00:04:33,129 So, uh, this so at one glance, it might seem that there's sure there's some there's some disagreement, uh, in some experimental data. 43 00:04:33,130 --> 00:04:40,690 So why is that really so significant? So one point that is significant is that this is one of the six parameters in the standard model of cosmology, 44 00:04:40,990 --> 00:04:46,149 and we don't have that many opportunities to measure it. And so people have really spent a long time measuring the this quantity. 45 00:04:46,150 --> 00:04:51,460 And the discrepancy in this really, uh, might be pointing out that something is going on in our understanding. 46 00:04:52,240 --> 00:04:59,979 So my interest in the subject, uh, relates to this aspect, uh, from the era of recombination until today, uh, 47 00:04:59,980 --> 00:05:09,070 our universe is, uh, has been dominated by, uh, the so-called dark components, uh, of the, of the universe. 48 00:05:09,520 --> 00:05:15,100 Uh, most of the matter density that we know of, uh, we know that it's in the form of a dark matter. 49 00:05:15,520 --> 00:05:18,129 And today, if you look at the energy budget of the universe, 50 00:05:18,130 --> 00:05:22,990 most of the energy is actually in form of dark energy that causes the accelerated expansion of the universe. 51 00:05:24,340 --> 00:05:31,750 So, um, we have very we have relatively firmly established the existence of these objects, 52 00:05:32,140 --> 00:05:35,710 the dark some some kind of dark energy is required to drive the accelerated expansion. 53 00:05:35,980 --> 00:05:43,960 Some kind of dark matter is required to, to ensure that, uh, matter clumps at the right, uh, right time and right edge, uh, to form galaxies. 54 00:05:44,350 --> 00:05:50,380 But we don't know very much about them. Uh, as may be, uh, hinted at by the word dark in front of them. 55 00:05:50,680 --> 00:05:53,740 And so we don't we really don't know any properties about dark matter and dark energy. 56 00:05:53,920 --> 00:05:59,590 And so it's very plausible that as we increase the precision of cosmology, uh, observations, 57 00:05:59,590 --> 00:06:06,250 that we start learning about these quantities and they don't behave exactly like, uh, what we our first naive estimate is. 58 00:06:06,790 --> 00:06:13,119 And so, uh, what we know about dark matter is that it should have small interactions with us. 59 00:06:13,120 --> 00:06:17,139 It shouldn't interact with light very strongly. Uh, and it should be non-relativistic. 60 00:06:17,140 --> 00:06:23,180 It should be what is called, uh, cold, uh, in, in cosmological parlance, uh, for dark energy, 61 00:06:23,180 --> 00:06:26,860 we what the only thing we know about it is it should be relatively constant. 62 00:06:27,310 --> 00:06:32,230 Should behave like, uh, almost like a cosmological constant. But we don't know anything about it beyond that. 63 00:06:32,950 --> 00:06:36,940 We don't know whether these components interact with each other with themselves. 64 00:06:37,540 --> 00:06:43,059 Uh, whether the cosmological, uh, whether the dark energy is a cosmological constant or itself an evolving field. 65 00:06:43,060 --> 00:06:44,950 Energy density is slowly changing with time. 66 00:06:46,390 --> 00:06:54,640 Um, and there may be other contributions, uh, beyond just these kinds of contributions that, that we might not have, uh, yet seen. 67 00:06:54,640 --> 00:06:59,920 But we are starting to see, starting to see now as we crank up the precision in our cosmological measurements. 68 00:07:01,030 --> 00:07:08,559 So, uh, some of these questions I've explored in these papers, I won't have, uh, time to go through much of the details of these particular papers. 69 00:07:08,560 --> 00:07:11,380 Maybe I'll come back to one of the ideas in these papers. 70 00:07:11,680 --> 00:07:16,960 But if you're interested in following this thread, uh, I'd encourage you to look at these, uh, these papers. 71 00:07:23,180 --> 00:07:29,690 Okay, so I wanted to just give you, uh, at least a cartoon version of what are these two kinds of measurements, 72 00:07:30,110 --> 00:07:36,020 uh, that are made and how how do they come about. And so to see what are the moving parts of this discrepancy? 73 00:07:37,550 --> 00:07:45,620 Um, good. So, uh, to set the stage, uh, the background cosmology, we we have very good. 74 00:07:46,040 --> 00:07:53,600 We have very good evidence, uh, that it's dictated by, uh, that, uh, the universe will live as in homogeneous and isotropic. 75 00:07:54,290 --> 00:07:58,100 Um, we also have pretty strong evidence that it's spatially flat. 76 00:07:58,430 --> 00:08:05,990 So if you take any given time, slice, uh, the universe, uh, looks like ordinary Euclidean three, uh, three dimensional space. 77 00:08:06,770 --> 00:08:11,239 Uh, and such a universe is described by the FLW metric, where the line element, 78 00:08:11,240 --> 00:08:17,360 the distance between any two points in space time, uh, given by something that looks very similar to the Minkowski metric, 79 00:08:17,780 --> 00:08:24,410 uh, except for this, uh, scale factor of T, and the Hubble parameter that we have been talking about is nothing but just, 80 00:08:25,040 --> 00:08:28,640 uh, the logarithmic derivative of the scale factor of T over E. 81 00:08:29,810 --> 00:08:35,540 And so in some sense, uh, uh, the expanding universe that that cartoon I was showing you at the beginning of the talk, 82 00:08:36,020 --> 00:08:41,090 uh, is, uh, encapsulated by just a function f of t, which is growing with time. 83 00:08:42,560 --> 00:08:47,690 Uh, and all of cosmology in some sense can be encoded all of at least the background cosmology can be encoded, 84 00:08:48,200 --> 00:08:55,310 uh, in this function, one function of time, uh, and Hubble then becomes the rate of expansion of the universe. 85 00:08:57,560 --> 00:09:02,300 Uh, an intuitive way to think about. Uh, think about the scale factor is the following. 86 00:09:02,630 --> 00:09:07,490 Imagine two galaxies that are very far separated from each other so that they have negligible pull on each other. 87 00:09:07,490 --> 00:09:12,320 So they would just follow geodesics in spacetime in these coordinates T and x. 88 00:09:12,740 --> 00:09:19,640 They will just they will not move. So if you put the put first galaxy at some coordinate x1 and the other galaxy at some coordinate x2, 89 00:09:19,970 --> 00:09:23,410 uh, they're just they just retain their coordinates. They're going on geodesics. 90 00:09:23,420 --> 00:09:30,920 That's what these geodesics look like. Uh, and the fact that the universe is expanding tells you that, uh, 91 00:09:30,920 --> 00:09:36,260 the physical distance between them is changing, and that's changing due to this presence of a scale factor. 92 00:09:36,500 --> 00:09:41,180 So to go from coordinate distance, which in this case is fixed to physical distance, 93 00:09:41,480 --> 00:09:45,530 which in this case is given by eight times delta uh given by eight times delta x. 94 00:09:45,770 --> 00:09:49,790 We see that these galaxies would appear to be receding from each other in terms of physical distance. 95 00:09:50,480 --> 00:09:59,110 Okay. Well, how do we how do we find out what this function have? 96 00:09:59,160 --> 00:10:03,690 It looks like. Um, we we have a metric. 97 00:10:03,750 --> 00:10:07,000 If we have some energy density in the universe that we know about. 98 00:10:07,020 --> 00:10:14,610 We can solve Einstein's equation. Einstein's equation famously tells us that, uh, geometry, the metric follows from energy density. 99 00:10:15,210 --> 00:10:20,720 And so, uh, in this context, these ancient equation look like, uh, look like this. 100 00:10:20,730 --> 00:10:27,180 These are called Friedmann equations in the context of cosmology. Uh, and this is one of the Friedmann equations. 101 00:10:30,850 --> 00:10:35,200 Um, on the left hand side, you essentially have this graph of the Hubble parameter, 102 00:10:36,490 --> 00:10:39,580 and that is proportional to the total energy density of the universe. Rho. 103 00:10:39,580 --> 00:10:47,200 Here is the energy density of the universe, uh, and the energy density of the universe, as we said, can be modelled as ideal fluids. 104 00:10:47,490 --> 00:10:53,670 Um, um, and these fluids are essentially characterised by the equation of state. 105 00:10:53,680 --> 00:10:57,720 So what kind of fluids they are, what is their pressure relative to their energy density? 106 00:10:57,730 --> 00:11:03,370 And equivalently, uh, they're characterised by their, their scaling of energy with respect to air. 107 00:11:04,600 --> 00:11:08,680 Uh, so let's take an example. Let's think about uh, matter. 108 00:11:09,070 --> 00:11:17,620 So, uh, again, as I was showing this, the energy density matter, uh, the total energy density rho is uh, the mass. 109 00:11:18,700 --> 00:11:30,010 Uh, I have it, uh. Look, I have it here as it would be, satisfy this equation that energy and matter was the mass of the dark matter. 110 00:11:30,230 --> 00:11:32,930 A mass of matter times the number density. 111 00:11:40,080 --> 00:11:46,229 And again, if this if these matter particles were on geodesics, they would just stand there on their coordinates wherever you put them. 112 00:11:46,230 --> 00:11:48,930 They're non-relativistic, remember. So they're essentially at rest. 113 00:11:49,350 --> 00:11:58,140 So in a comoving volume, uh, the number density remains constant, which means that in the physical volume the number density decreases as a cubed. 114 00:11:59,220 --> 00:12:04,320 So these matter particles are valued as a cubed. And that means that energy density dilutes as a cubed. 115 00:12:04,950 --> 00:12:12,440 And so that tells us that there's a contribution to rho which is proportional to uh e to the minus three times some proportionality constant. 116 00:12:12,450 --> 00:12:14,970 That depends on your initial conditions that you set up. 117 00:12:15,930 --> 00:12:21,299 Uh, a similar kind of argument can be made for radiation, where the number density of radiation, 118 00:12:21,300 --> 00:12:26,010 number density of photons would dilute as a cubed as well. Uh, but the wavelength also redshifts. 119 00:12:26,010 --> 00:12:30,540 And so the total energy density of photons actually redshifts says uh at minus four. 120 00:12:31,710 --> 00:12:35,760 And then there's components like this dark energy component which is essentially constant. 121 00:12:36,360 --> 00:12:42,989 So various components of the universe, the particles that we know and love, um, either fall into the category of radiation, 122 00:12:42,990 --> 00:12:51,510 like photons and neutrinos when they're relativistic matter, like baryons, protons, uh, or the cold dark matter or this unknown dark energy component. 123 00:12:53,250 --> 00:13:02,969 So this, um. This sets up some, uh, equations that we can figure out what the Hubble rate of the universe is, 124 00:13:02,970 --> 00:13:07,680 given what the relative fraction of the universe is in radiation matter dark. 125 00:13:08,980 --> 00:13:12,570 And we can see that if we, uh, if we choose some epoch. 126 00:13:13,410 --> 00:13:20,880 Uh, so, so if you choose some epoch where a is really small, remember the, um, the convention is that we set a equals one today. 127 00:13:21,210 --> 00:13:24,120 And so in the past, in the past it was really, really small. 128 00:13:24,480 --> 00:13:29,820 And so you can see that in the past the radiation becomes a much more important component, 129 00:13:30,600 --> 00:13:35,520 uh, and eventually, uh, as it increases, that becomes less important. 130 00:13:35,970 --> 00:13:41,250 And that gives away to matter becoming a more important component because it it scales slower than, uh, 131 00:13:41,250 --> 00:13:49,380 radiation with respect to a and eventually, uh, uh, for today, uh, dark energy becomes important as well. 132 00:13:49,650 --> 00:13:56,640 So this is this this equation also gives, uh, gives us this idea that in the early universe, the universe was radiation dominated. 133 00:13:57,420 --> 00:14:02,430 Uh, eventually it became matter dominated. And today it's dark energy dominated. 134 00:14:08,420 --> 00:14:12,530 If this was a question at possible lectures. Uh, possible question, but, uh, maybe. 135 00:14:15,030 --> 00:14:21,980 Okay. Another thing that will be. So one other ingredient that would be useful for us to, uh, set up the measurement of the Hubble constant is, 136 00:14:22,580 --> 00:14:28,760 uh, different, uh, uh, different notions of time as the related question that was asked earlier. 137 00:14:29,260 --> 00:14:34,309 Uh, uh, the time coordinate that we chose could, uh, could, uh, is not unique. 138 00:14:34,310 --> 00:14:37,730 We could have chosen other parameterisations of the FLW metric. 139 00:14:37,730 --> 00:14:43,730 And one thing that one particular, uh parameterisation that would be really useful to us is the so-called conformal time, 140 00:14:44,060 --> 00:14:49,010 which is defined, uh, just as such, uh, that DT is just eight times the ETA. 141 00:14:49,520 --> 00:14:57,070 But the advantage of that is that, uh, the metric, uh, the line element in this coordinate looks something like this. 142 00:14:57,080 --> 00:15:02,870 So it looks basically like a Minkowski metric, uh, multiplied by some overall factor squared. 143 00:15:04,070 --> 00:15:09,380 And, uh, but we're going to be studying photons travelling in the universe for photons. 144 00:15:09,440 --> 00:15:14,570 Uh, DS squared is zero. Geodesics of photons have, uh, photons travel along null rays. 145 00:15:15,020 --> 00:15:19,610 And so the photons don't see in these coordinates don't see this factor squared error at all. 146 00:15:20,000 --> 00:15:24,890 As you can see that if I said d squared equals zero, the pre factor uh factors out. 147 00:15:25,340 --> 00:15:30,110 And so for photons in these coordinates uh the universe looks like Minkowski space. 148 00:15:30,530 --> 00:15:37,280 And the photons literally travel along straight lines in these coordinates. And so that hence the hence the uh utility of these coordinate system. 149 00:15:38,510 --> 00:15:45,950 Um, another, another uh, kind of uh, clock that appears uh, often in cosmology is called redshift. 150 00:15:46,400 --> 00:15:54,980 Uh, and it's, uh, it's related to the fact that the photon wavelength, uh, um, is proportional to, uh, is proportional to the scale factor. 151 00:15:55,400 --> 00:16:01,940 And so, uh, this is a definition that's often used in cosmology where, uh, one plus z is defined as one over a. 152 00:16:05,790 --> 00:16:08,399 For part, partially for historical reasons, really. 153 00:16:08,400 --> 00:16:15,180 But um, so this is again related to a question about, uh, time that was asked earlier, which time really it is. 154 00:16:15,600 --> 00:16:19,740 Uh, and there's a time in the early universe is the time in the early universe the same as time today. 155 00:16:20,160 --> 00:16:29,219 And you see from from at least from, uh, this aspect, you can see that, um, the frequency of a given photon in the early universe, uh, with redshift. 156 00:16:29,220 --> 00:16:34,320 And so if you, uh, if you emit some photon at a particular known frequency in the universe today, 157 00:16:34,650 --> 00:16:38,040 we measure its frequency to be, uh, redshifted by a certain amount. 158 00:16:38,340 --> 00:16:43,919 So certainly, if you're using atomic clocks to keep time, the notion of time in the early universe, 159 00:16:43,920 --> 00:16:49,530 the notion of time to day would be redshifted to each other, similar to gravitational redshift or relativistic redshift. 160 00:16:51,120 --> 00:16:59,220 Um, so in the end, there's many, many, uh, equivalent descriptions of the cosmological clock and could use t the time coordinate we started with. 161 00:16:59,760 --> 00:17:07,500 We could uh, use the conformal time, or equivalently, the scale factor itself, or the redshift or the temperature of the of the photons, 162 00:17:07,830 --> 00:17:14,610 or as, uh, cosmological clocks, uh, basically because each of them have, uh, monotonic relation to the other. 163 00:17:14,610 --> 00:17:18,210 And so we could equivalently convert between any one of them as our clock. 164 00:17:24,730 --> 00:17:31,960 Okay, so, uh, the way we calculate the Hubble constant is using different distance measures in cosmology. 165 00:17:32,530 --> 00:17:39,790 Um, and the two, the two set of distance measures that come to play in these two ways of calculating the Hubble constant. 166 00:17:40,180 --> 00:17:45,160 I relate it to the standard ruler way of doing it in the standard, uh, candle way of doing it. 167 00:17:45,760 --> 00:17:51,069 Uh, and so here's how that, uh, here's why these things are, uh, these things. 168 00:17:51,070 --> 00:17:56,260 Let us, uh, calculate the Hubble constant. Uh, so the first thing is the standard ruler method. 169 00:17:56,680 --> 00:18:06,009 Uh, and it goes by the name of angular diameter distance. The idea is that if you know, if, you know, uh, um, at least in Euclidean geometry, 170 00:18:06,010 --> 00:18:08,379 we are familiar with the fact that if you have something that, you know, 171 00:18:08,380 --> 00:18:14,950 the length of, like there's a metre stick and I know the length of that with a stick, I look at it and I see a certain angle that is certain to me. 172 00:18:15,190 --> 00:18:22,840 I can figure out how far it is to from me, just from geometry. And so, uh, we want to use the same idea but apply to this evolving cosmology. 173 00:18:23,290 --> 00:18:30,000 Luckily, we have, uh, at our disposal this conformal time coordinate in which light rays really do travel that straight line. 174 00:18:30,010 --> 00:18:33,070 So we can apply this Euclidean geometry logic. Right. 175 00:18:33,100 --> 00:18:35,170 Uh, directly to this, uh, to this problem. 176 00:18:35,830 --> 00:18:43,120 So imagine, uh, there's some, uh, imagine a picture in which there's an object for which, you know, that physical distance. 177 00:18:43,870 --> 00:18:45,610 Uh, physical size. Sorry. 178 00:18:46,360 --> 00:18:56,560 Um, and, uh, so the physical size that, you know, is some quantity called D, and you look at that object and it turns on your detector an angle theta. 179 00:18:56,620 --> 00:19:01,030 You measure that the it's some angle. In cosmology we're always just measuring angles. 180 00:19:01,270 --> 00:19:03,370 The angle that you measure it sometimes is theta. 181 00:19:04,210 --> 00:19:09,610 Uh and so the angular diameter distance is defined as the quantity such that this equation is correct. 182 00:19:10,020 --> 00:19:17,830 In other words the physical distance, physical size of that object, uh, physical size of the object divided by the angle is substance d r detector. 183 00:19:19,210 --> 00:19:23,800 So here's a picture where the picture is now drawn in conformal time coordinates. 184 00:19:23,800 --> 00:19:29,100 So this is a coordinate space picture that's remember that's the that's the space in which light travels in a straight line. 185 00:19:29,110 --> 00:19:35,110 So if you want to draw this picture and take it literally we should we should work in coordinate space and convert it to physical space. 186 00:19:35,740 --> 00:19:43,990 Uh, so let's say in coordinate space the size of the object is x, uh, and that is the distance to the object from us is r. 187 00:19:44,410 --> 00:19:50,410 And so here we are allowed to use Euclidean geometry. So r is just uh so theta is just x over r. 188 00:19:51,280 --> 00:19:58,990 And now we want to convert it into this form to actually figure out what the angular diameter distance would be related to the physical distances. 189 00:19:59,320 --> 00:20:04,020 So the physical size of the object, uh, would be the scale factor at the time of emission. 190 00:20:04,060 --> 00:20:10,990 So a at ETA being the conformal time uh times uh the quarter distance x. 191 00:20:12,610 --> 00:20:15,909 Um, since uh, I've set C to one here. 192 00:20:15,910 --> 00:20:25,420 So the time light takes to travel from a coordinate distance R to us is the same as the conformal time, uh, at the conformal time today zero. 193 00:20:25,750 --> 00:20:29,020 So the total time taken is the same as the total distance when c is one. 194 00:20:29,650 --> 00:20:33,070 And so this uh the coordinate distance r is the same as that. 195 00:20:34,300 --> 00:20:43,780 So so we can we can use the simple geometric picture and actually get to a, uh, an equation for what the angular diameter distance is. 196 00:20:44,440 --> 00:20:48,060 Uh, x over r is nothing but d over eight times eta e. 197 00:20:48,490 --> 00:20:52,420 And so that tells us that the angular diameter distance is, uh, 198 00:20:52,420 --> 00:20:59,710 the scale factor at the time of admission times the total conformal time to the time, uh, between us and the time of admission. 199 00:21:02,800 --> 00:21:09,610 This can be converted using that using the definition of the conformal time into an integral over time. 200 00:21:10,030 --> 00:21:13,270 Remember data was uh ada eta was dt. 201 00:21:13,510 --> 00:21:20,140 So I've just written eta as integral d over a and you can again convert use any other clock that you'd like. 202 00:21:20,320 --> 00:21:26,920 If you want to convert this to uh, convert this integral over time to integral over a, you can use uh, 203 00:21:26,920 --> 00:21:33,760 multiply and divide by a dot over a, uh, and convert into integral over a, or you can uh, convert into a integral over redshift. 204 00:21:34,240 --> 00:21:41,649 So this is just coming again from definitions, uh, of what we, what we said a wasn't relative to uh, 205 00:21:41,650 --> 00:21:47,830 ETA was whether it, relative to time or H uh or Z was relative to uh, an instance. 206 00:21:49,240 --> 00:22:00,250 Uh, so the upshot is, uh, that this angular diameter distance, uh, is given by an integral over, say, here redshift, uh, from the red uh, 207 00:22:00,250 --> 00:22:10,840 from the redshift information uh, to today, redshift uh zero is today, uh, or equivalently the scale factor dimension to uh the scale factor today. 208 00:22:11,440 --> 00:22:16,150 So this angular diameter distance is something that's sensitive to the, to the entire cosmology, 209 00:22:16,360 --> 00:22:22,570 cosmological evolution of the Hubble constant from the time of uh, emission, uh, to the time of observation. 210 00:22:24,280 --> 00:22:29,710 Usually we would find something, uh, in cosmology for which we know the physical size or some, 211 00:22:29,740 --> 00:22:33,070 some standard ruler, and we would measure theta very precisely. 212 00:22:33,520 --> 00:22:40,810 So that would give us, uh, the angular diameter distance to that object, which in turn gives us this integral over, 213 00:22:41,080 --> 00:22:44,920 uh, the Hubble parameter, at least from the time of emission till today. 214 00:22:45,550 --> 00:22:49,720 And so in this way, we would be sensitive, uh, to the Hubble parameter. 215 00:22:49,720 --> 00:22:53,820 Today particularly this integral is actually dominated by redshifts close to zero. 216 00:22:53,830 --> 00:22:58,080 So this integral is actually quite sensitive to the, uh, Hubble parameter today. 217 00:22:58,090 --> 00:23:02,200 But it's also sensitive to what's happening between the emission and, uh, emission. 218 00:23:02,200 --> 00:23:07,000 And today okay. Uh, so this is one way. 219 00:23:07,420 --> 00:23:12,820 Uh, and we'll come back to this. This is the way CMB actually measures, uh, the, uh, Hubble parameter. 220 00:23:16,000 --> 00:23:21,040 The other distance measure is luminosity distance, which is two with which again in Euclidean space. 221 00:23:21,040 --> 00:23:28,680 You would say, if I knew there was 100 watt bulb, uh, if I if I see it, it's brighter and I reduce that, I'm closer to it. 222 00:23:28,690 --> 00:23:31,150 If I think it's dimmer and I'm far further from it. 223 00:23:31,390 --> 00:23:37,960 So if I know the intrinsic luminosity of an object, and I know the apparent luminosity of an object, I can figure out how far that is. 224 00:23:39,430 --> 00:23:49,389 And again, we want to do this in, uh, in the context of uh, uh, uh, expanding cosmology, but the same, same idea and applying the same idea. 225 00:23:49,390 --> 00:23:55,300 The definition of luminosity distance is the following. There's some intrinsic luminosity of an object, and there's a measured flux. 226 00:23:55,780 --> 00:23:59,980 But luminosity distance is defined to be the, uh, to be such that this equation is true. 227 00:24:02,890 --> 00:24:09,130 Um, we can again run the same kind of argument I, uh, I was telling you about for, uh, 228 00:24:09,220 --> 00:24:16,600 the angular diameter distance, um, in, in conformal time coordinates, again, light travels and on straight lines. 229 00:24:16,750 --> 00:24:22,960 So the area in in these coordinates, the area of a sphere at distance r is just four pi r squared. 230 00:24:23,740 --> 00:24:28,870 And so the dilution of flux as you go to a distance r is just one over four pi r squared. 231 00:24:29,080 --> 00:24:33,130 R is the coordinate distance. And we can convert it to a physical distance. 232 00:24:33,520 --> 00:24:38,169 And the upshot is that we can get another expression for uh this so-called luminosity distance, 233 00:24:38,170 --> 00:24:45,990 which uh uh, looks similar, but it's not quite exactly the same as the angular diameter distance. 234 00:24:46,000 --> 00:24:52,300 But again, it's given by an integral over the cosmology from the time of emission until the time of observation. 235 00:24:52,870 --> 00:25:00,880 So both these distance measures are, um, uh, sensitive to h nought the Hubble parameter today through an integral over, 236 00:25:01,150 --> 00:25:07,510 uh, uh, over the Hubble parameter between emission and uh, um, and today. 237 00:25:08,020 --> 00:25:15,520 So maybe I should say, uh, for the CMB, the relevant, uh, redshift of emission would be about 1000. 238 00:25:16,000 --> 00:25:20,260 So that's when the at the surface of last scattering, when the CMB was generated, 239 00:25:20,770 --> 00:25:24,560 um, the universe was about a thousand times smaller than it is today. 240 00:25:24,580 --> 00:25:27,760 So this integral would go from about a thousand to, to zero. 241 00:25:28,630 --> 00:25:39,370 Uh, and um, for, for the luminosity distance, which would be the relevant, uh, quantity that supernovae, uh, the local measurement measurements use. 242 00:25:39,820 --> 00:25:43,090 Uh, these measurements go out to redshift of order one, 1 or 2. 243 00:25:43,150 --> 00:25:51,490 So this integral would go um from say 2 to 0 or something like that, whatever that, uh, redshift of uh, the supernovae happen to be. 244 00:25:52,030 --> 00:25:56,650 Um, and so, so it's not exactly entirely the same quantity they're measuring. 245 00:25:56,830 --> 00:26:00,820 One is measuring the universe at really late times between redshift of two and one, 246 00:26:01,120 --> 00:26:05,139 and the other one is measuring the integral of the Hubble constant from redshift to to today. 247 00:26:05,140 --> 00:26:09,040 So we might wonder whether that that's a hint towards what the solution could be. 248 00:26:11,050 --> 00:26:19,480 Okay. So this let's first look at how the, uh, how the supernovae measurement measurements are done, uh, using the so-called distance ladder method, 249 00:26:20,110 --> 00:26:25,660 uh, to calculate, uh, essentially calculate the, this luminosity distance and the Hubble constant. 250 00:26:26,920 --> 00:26:33,430 And so the idea is to build up, um, the idea is of use, known distance measurement, measurement techniques, 251 00:26:33,940 --> 00:26:40,210 uh, to calibrate and then use uh, the calibration to then move that distance measurement technique further. 252 00:26:40,630 --> 00:26:47,530 So how does that work? Uh, the the best way we know in cosmology and astrophysics to measure distances is using parallax. 253 00:26:48,070 --> 00:26:53,110 Uh, we well, uh, we we can we can wait for it for half a year. 254 00:26:53,650 --> 00:26:56,950 Uh, as the earth goes around the sun and we can look at a certain star. 255 00:26:57,340 --> 00:27:01,720 And in this particular case, the star that is looked, looked at is a separate variable star. 256 00:27:02,170 --> 00:27:06,580 And we can look at how that star moves in the field of some very background, distant stars. 257 00:27:07,090 --> 00:27:12,760 And uh, assuming the distances are right, the actual that, that motion of that, uh, 258 00:27:12,760 --> 00:27:17,080 separate star in the background of the distant stars is visible and measurable. 259 00:27:17,320 --> 00:27:23,740 And so that, that, that this geometry gives you a measurement of, uh, of the distance to the star. 260 00:27:24,190 --> 00:27:28,989 So close for close by things, parallax is the best way to measure distances. 261 00:27:28,990 --> 00:27:32,020 And that's what we use the in some region around. 262 00:27:32,020 --> 00:27:38,380 So this is, this is us and some region around us about, say, uh, 10,000 light years or so. 263 00:27:38,680 --> 00:27:45,579 Anything around, uh, to that distance we measure, use parallax to measure distances so we can measure distances of field stars. 264 00:27:45,580 --> 00:27:53,770 And what what we find is actually that, um, the reason we use stars is because variable stars have a certain periodicity, and, 265 00:27:53,770 --> 00:28:00,340 and it's empirically seen that their luminosity, their intrinsic luminosity has a higher degree of correlation with their periodicity. 266 00:28:01,390 --> 00:28:06,730 We can also measure the distance using this parallax method. So if you see a spade star we can measure its periodicity. 267 00:28:07,090 --> 00:28:13,330 We can measure its luminosity. But we know how far it is because of this, uh, just this parallax method. 268 00:28:13,780 --> 00:28:16,930 Uh, and so we can figure out what its intrinsic luminosity is. 269 00:28:20,390 --> 00:28:26,750 I should say that. And on distances of such a small, uh, the distances that are this small compared to cosmology, 270 00:28:26,750 --> 00:28:28,940 the fact that we are living in an expanding universe are irrelevant. 271 00:28:28,940 --> 00:28:35,030 Just like when we when we calculate anything about the solar system, we don't worry about the fact that we live in an expanding universe. 272 00:28:35,030 --> 00:28:42,379 So this is still, uh, small enough distances that we can work in the approximation that we live in a flat, uh, flat space time. 273 00:28:42,380 --> 00:28:43,400 And we don't worry about that. 274 00:28:44,090 --> 00:28:50,150 We can include it, but it would be a small correction to the to to to anything we calculate using just a flat surface limit. 275 00:28:50,840 --> 00:28:57,110 So using this parallax we can figure out the distances field which tells us what the intrinsic luminosity of the surface is. 276 00:28:57,560 --> 00:29:04,430 And then we find galaxies which host both, uh, separate variable stars as well as, uh, supernovae. 277 00:29:05,300 --> 00:29:12,110 We don't have very many supernovae around us, but we do have galaxies further away which whole host would and supernovae, the surface. 278 00:29:12,110 --> 00:29:15,589 We know the intrinsic luminosity for we know their apparent luminosity. 279 00:29:15,590 --> 00:29:20,780 So we can figure out their the luminosity distance just using the thing that we just went through. 280 00:29:21,410 --> 00:29:24,620 Uh, so that gives us a calibration of the distance to these galaxies. 281 00:29:25,190 --> 00:29:31,430 Uh, and then we look at the supernovae and, uh, look at their apparent, uh, luminosity, uh, 282 00:29:31,790 --> 00:29:36,710 and using the distance inferred from Cepheids, we can figure out their intrinsic luminosity as well. 283 00:29:37,460 --> 00:29:40,460 And so these, uh, supernovae are called standard candles. 284 00:29:40,460 --> 00:29:45,680 They have, uh, they have a very strong relationship between their so-called light curve and their intrinsic luminosity. 285 00:29:46,130 --> 00:29:50,690 So this gives us a way to calculate the, uh, intrinsic luminosity of the supernovae. 286 00:29:50,930 --> 00:29:57,890 And then very far away, we don't see speeds anymore, but we do see it supernovae, uh, for which we have deduced the intrinsic luminosity. 287 00:29:58,160 --> 00:30:07,610 So now using these supernovae, we can figure out, uh, distances, uh, to these very, very far, uh, far off, uh uh, um, um, cosmological objects. 288 00:30:08,990 --> 00:30:13,910 Um, so this is this this is to give you a sense of how this calculation is actually done, 289 00:30:13,910 --> 00:30:18,170 but also to see this is a rather intricate thing which relies on a bunch of astrophysics, 290 00:30:18,170 --> 00:30:24,469 some, some empirical, uh, estimation of these standardised lines that we don't have first principle and the first principle, 291 00:30:24,470 --> 00:30:26,530 understanding of the variability of surface stars. 292 00:30:26,540 --> 00:30:34,190 We don't have first principle, uh, truly first principle, uh, analysis of why, uh, supernovae are standard candles. 293 00:30:34,610 --> 00:30:38,810 Uh, but we can build this run of, uh, rungs up the distance ladder. 294 00:30:39,260 --> 00:30:45,110 And again, this is a this is a very mature, uh, piece of experimental physics that has been going on for a while. 295 00:30:45,440 --> 00:30:51,800 And this is the measurement by which we find the Hubble constant today is 72km per second per megaparsec. 296 00:30:55,240 --> 00:31:01,000 Uh, the other, uh, the the other way, the, the other half of the measurement, 297 00:31:01,000 --> 00:31:06,760 which was the CMB measurement comes from, uh, comes from observations from, 298 00:31:06,760 --> 00:31:14,080 say, Planck, uh, where the Planck satellite observes the, the sky and sees the cosmic microwave background that looks like this. 299 00:31:14,590 --> 00:31:20,140 Uh, if you look at if you Google Planck CMB, then this is the picture that they throw up on their website. 300 00:31:20,440 --> 00:31:22,480 But this picture is false for many reasons. 301 00:31:22,840 --> 00:31:28,780 First of all, if you actually look at observation, there should be a huge mosque in the middle for the galaxy that they don't actually have data. 302 00:31:29,290 --> 00:31:32,590 There's also mosques elsewhere because of that, uh, instrumental reasons. 303 00:31:32,860 --> 00:31:35,169 So this is and, uh, yeah. 304 00:31:35,170 --> 00:31:42,550 So this is an image that's actually generated, uh, generated and filled in these regions of, uh, space, which we actually don't observe, 305 00:31:43,090 --> 00:31:49,960 but we observe enough of the, of the whole sky to actually be able to reliably, um, reliably fill in this, uh, picture. 306 00:31:51,490 --> 00:31:58,630 Uh, sorry, I should say, uh, with this, uh, I assume everybody's seen this picture before, but what this is showing is, uh, temperature on the sky. 307 00:31:59,170 --> 00:32:03,819 Uh, and, uh, more precisely, what we have taken out is, uh, uniform temperature in the sky. 308 00:32:03,820 --> 00:32:05,139 So if you look out in the sky, you actually. 309 00:32:05,140 --> 00:32:12,070 What would you see is, uh, uniform three Kelvin, black body radiation coming from everywhere, which is equal from every direction. 310 00:32:12,610 --> 00:32:16,960 You subtract the average out, what you would see is a huge ten to the minus three, 311 00:32:17,080 --> 00:32:23,500 huge ten to the minus three dipole, uh, that is associated with us moving through the rest frame of the CMB. 312 00:32:24,010 --> 00:32:25,089 So you take that out as well. 313 00:32:25,090 --> 00:32:32,570 And then what you're left with, uh, is a temperature contrast of order ten to the minus five, uh, on top of the, the background levels. 314 00:32:32,620 --> 00:32:37,270 So this is that subtracted picture and then filled in where we don't actually have observations. 315 00:32:37,750 --> 00:32:39,010 And this is what this picture looks like. 316 00:32:39,160 --> 00:32:43,900 And the idea is that the cosmic microwave background actually provides us with, uh, kind of a standard ruler. 317 00:32:44,140 --> 00:32:47,980 And the standard ruler is one thing we can use to measure distances, uh, as well. 318 00:32:52,920 --> 00:32:57,370 So, um. So what? Where do these cosmological fluctuations come from? 319 00:32:57,390 --> 00:33:07,140 There was, uh. Um. As Joe was, as Joe was telling us, if there were these these cosmological fluctuations are the reasons. 320 00:33:07,140 --> 00:33:10,920 Eventually, uh, galaxies form and stars form and planets form. 321 00:33:10,920 --> 00:33:18,329 And so the existence of interesting stretches of the universe to this initial perturbations, even though they start off as ten to the minus five, 322 00:33:18,330 --> 00:33:22,469 eventually gravitational wells make them grow and uh, and, 323 00:33:22,470 --> 00:33:27,630 and do interesting things like have star formation and then nuclear fusion inside the star, inside those stars. 324 00:33:28,350 --> 00:33:33,720 Um, the initial conditions, uh, for these things are set by inflation, as Joe was mentioning. 325 00:33:34,140 --> 00:33:39,959 Uh, and there's some initial condition that inflation gives us that there was some fluctuation in the universe at some epoch, 326 00:33:39,960 --> 00:33:43,500 depending on their wavelength, they start, uh, they start oscillating. 327 00:33:43,650 --> 00:33:49,799 And further, for the for our purposes, what is relevant is that in the CMB epoch, when the, uh, 328 00:33:49,800 --> 00:33:54,870 when the photons and uh, uh, the photons and protons and electrons are tightly bound to each other. 329 00:33:55,380 --> 00:33:58,950 So this is right before recombination, right before decoupling. Uh. 330 00:34:00,570 --> 00:34:08,670 And then eventually the crossed the temperature wet foot where dominantly all the, uh, protons electrons bound together in neutral hydrogen. 331 00:34:09,060 --> 00:34:15,900 Uh, and then this, uh, the photons can scatter. So that is the epoch we're interested in, where we go from a tightly coupled baryon photon fluid, 332 00:34:16,140 --> 00:34:21,270 plasma, uh, to a transparent universe with photons streaming to us from the surface of last scattering. 333 00:34:21,900 --> 00:34:29,970 And so, uh, for, uh, for our purposes, what is relevant is these fluctuations lead to fluctuations in these plasma and sound waves in this plasma. 334 00:34:30,300 --> 00:34:32,970 So what we see in the CMB is really these sound waves. 335 00:34:33,060 --> 00:34:37,530 We have some you take a plasma and you set up some initial fluctuations and you let it oscillate. 336 00:34:37,530 --> 00:34:41,040 And these waves are just oscillate. Uh, these are just sound waves. 337 00:34:41,040 --> 00:34:44,390 And the, uh, peaks and troughs of these sound waves. 338 00:34:44,400 --> 00:34:48,480 Is that what you see as, uh, peaks and troughs of temperature in the in the CMB? 339 00:34:49,800 --> 00:34:58,780 Um. And so if you, uh, if you look at that CMB map that I showed you, but instead of looking at in real space, 340 00:34:58,780 --> 00:35:03,010 we look at in Fourier space are more, more, more precisely spherical harmonics space. 341 00:35:03,430 --> 00:35:07,149 Uh, you see this feature rather rather rather clearly that, uh, 342 00:35:07,150 --> 00:35:12,600 you see these peaks and troughs which are, which are associated with, uh, uh, which are associated with, 343 00:35:12,610 --> 00:35:17,860 uh, fluctuations, uh, that, uh, for example, 344 00:35:17,860 --> 00:35:22,300 a peak would be associated with a fluctuation that has just reaches extrema as the surface of last scattering happens. 345 00:35:22,590 --> 00:35:29,709 So so again, let's imagine this plasma where there's a bunch of sound waves and various fluctuations at various wavelengths are, 346 00:35:29,710 --> 00:35:38,740 uh, are, uh, are oscillating, uh, and at some point, therefore this, uh, uh, plasma decouples and the photons can stream through. 347 00:35:38,770 --> 00:35:43,030 So it's sort of like the universe taking a photo of that oscillating plasma. 348 00:35:43,690 --> 00:35:48,819 And at that instance of taking of the photograph, whichever fluctuations were at the, 349 00:35:48,820 --> 00:35:52,930 uh, whether at the extrema, they would appear as peaks in the spectrum, 350 00:35:52,930 --> 00:36:02,080 they would have most, uh, density contrast, uh, and, uh, the fluctuations that were sort of in the middle of the oscillation would appear as troughs. 351 00:36:02,800 --> 00:36:09,280 So, so, so this picture tells us properties of, uh, sound waves, particularly, uh, 352 00:36:10,000 --> 00:36:15,760 which wavelengths had reached their maximum, uh, by the time that decoupling happened. 353 00:36:19,930 --> 00:36:22,180 You could spend an hour talking about the fluctuations CMB. 354 00:36:22,180 --> 00:36:29,649 But but the rough picture is the rough picture is that this understanding of the sound waves in in the CMB plasma, 355 00:36:29,650 --> 00:36:38,320 as the photons are decoupling, give us a standard ruler. We can we can figure out what is the distance scale at which, uh, this peak should occur. 356 00:36:38,890 --> 00:36:44,290 And we look again, in cosmology, we measure angles and we measure this angle, the position of this peak rather precisely. 357 00:36:44,590 --> 00:36:47,890 So again, we're in the situation where we know a physical distance scale, the standard ruler, 358 00:36:48,160 --> 00:36:51,820 which is some wavelength that we that is supposed to have reached its maximum. 359 00:36:52,300 --> 00:37:00,940 And we also, um, um, and we also know that the angle that, uh, that this uh, corresponds to. 360 00:37:01,150 --> 00:37:06,100 So we can figure out the distance, this angular diameter, distance to the CMB using these two parameters. 361 00:37:09,800 --> 00:37:14,600 So this is, this is this is that statement again in uh, uh, in equation, uh, 362 00:37:14,600 --> 00:37:20,720 there's an angle associated with the associated with the first peak in the CMB that we measure to exquisite precision. 363 00:37:21,380 --> 00:37:27,740 Uh, remember when we're talking about the Hubble tangent, we're talking about quantities with a discrepant to order, uh, 7 to 10%. 364 00:37:28,190 --> 00:37:31,760 So that's that's sort of the, uh, going rate of things we are tracking. 365 00:37:32,180 --> 00:37:37,220 And I think that's better. Much better. Much more precisely measured is, is is just almost exact. 366 00:37:37,880 --> 00:37:46,240 Uh, and uh, this angle again is given by some quantity, which is related to the wavelength of the, 367 00:37:46,310 --> 00:37:50,360 these, uh, oscillations in the plasma, uh, that we can predict. 368 00:37:52,110 --> 00:37:58,740 Divided by some, uh, divided by the distance, uh, the angular diameter distance between us and the CMB. 369 00:37:59,070 --> 00:38:05,070 And that distance is again an integration over the redshift, uh, at the CMB, which is about 1000 till today. 370 00:38:05,640 --> 00:38:10,110 Off this parameter, uh, the of this, uh, one over the Hubble parameter. 371 00:38:11,460 --> 00:38:19,620 Okay. Uh, so we measure this, uh, we have a good prediction of this depending on, uh, our model model of early cosmology. 372 00:38:19,650 --> 00:38:25,260 And so we can deduce this. And this is how CMB reduces the value of the Hubble constant that it measures. 373 00:38:29,790 --> 00:38:35,029 So, um. Right. 374 00:38:35,030 --> 00:38:40,489 So. So imagine that we said we took the I think this is the sort of two paths one could take. 375 00:38:40,490 --> 00:38:44,660 One could take say can we can we investigate what is wrong with maybe something wrong, 376 00:38:44,660 --> 00:38:47,770 or maybe there's something, something weird going on with local measurements. 377 00:38:47,770 --> 00:38:52,400 So can we take the supernovae measurements, which are 72km per second per megaparsec, 378 00:38:52,730 --> 00:38:57,620 and investigate what is, uh, what can be made to make them move from their value? 379 00:38:57,620 --> 00:39:02,270 And this is a whole industry. This industry involves figuring out how the distance ladder was actually built. 380 00:39:02,600 --> 00:39:07,790 Is there a systematic error that were underestimated if there was some biases introduced, uh, in any of those things? 381 00:39:08,260 --> 00:39:14,600 Another, as another half of the community is focussed around thinking about, well, let's take the supernova measurement, 382 00:39:14,600 --> 00:39:22,159 the 72km per second per megaparsec at face value and see can can there be some new physics in that in the dark sector? 383 00:39:22,160 --> 00:39:25,400 Can there be some new physics in the CMB that can change the CMB prediction? 384 00:39:25,880 --> 00:39:32,540 Uh, from 67 to 72. So that's the, uh, that's the path I'm going to present very quickly. 385 00:39:32,990 --> 00:39:36,410 Uh, just to show you what kind of solutions uh, I presented. 386 00:39:36,410 --> 00:39:39,890 And so this is something that we measured very precisely that cannot be changed. 387 00:39:40,460 --> 00:39:44,900 And so this ratio, we better keep fix whatever we're doing to whatever we're doing to cosmology. 388 00:39:45,320 --> 00:39:50,420 So there are two classes of solutions. People have proposed the so-called late solutions or early solutions. 389 00:39:51,170 --> 00:39:53,750 In the late solution you don't change the value of Rs. 390 00:39:54,440 --> 00:40:03,710 So this, uh, um, but uh, this physical distance scale and the CMB, uh, the standard ruler is left unchanged, 391 00:40:04,040 --> 00:40:10,700 which must mean that you should also leave, uh, DM unchanged because we've measured the ratio very precisely. 392 00:40:11,030 --> 00:40:19,880 And so the only way to change it's not the value of Hubble today without changing the M is to make sure somehow that this integral stays the same, 393 00:40:20,180 --> 00:40:24,950 but its shape changes dramatically so that it's endpoint changes, uh, from what it was before. 394 00:40:25,970 --> 00:40:32,540 And so those are the so-called solutions where you have something happening in cosmology such that the integral of this over this, 395 00:40:32,540 --> 00:40:38,989 uh, whole epoch is left unchanged. Uh, but the actual final value, h nought today is different. 396 00:40:38,990 --> 00:40:43,280 And then in better agreement, uh, I'll show you what better agreement means. 397 00:40:43,280 --> 00:40:49,669 Better and better agreement with with with uh uh with uh, uh, 72km per second per megaparsec are the early, 398 00:40:49,670 --> 00:40:58,280 early solutions are uh, for say that we can keep that as fixed but change both R's and M by the same ratio about 10%. 399 00:40:58,820 --> 00:41:04,130 Uh and don't change the shape of Z. Just scales everything up and you get uh, you would get H not to be larger. 400 00:41:06,340 --> 00:41:10,810 So, um. So the so called solutions, which are not really solutions. 401 00:41:11,050 --> 00:41:15,280 Uh, look, look, look like the following. I'll, I'll spare the details of the solution itself, 402 00:41:15,280 --> 00:41:19,720 but I just wanted to give you an example of the flavour of what the solutions proposed of AI look like. 403 00:41:20,170 --> 00:41:26,920 Uh, so the so-called solution, uh, I remind you, was that we keep this integral fixed. 404 00:41:27,580 --> 00:41:36,370 Uh, but we want to change the shape of h of Z such that the integral these are over h of Z is the same, but it's not as it is different. 405 00:41:37,000 --> 00:41:45,340 So there's there is some, uh, there's some model where you can imagine there's some new the dark energy and dark matter and not just some cold, 406 00:41:45,340 --> 00:41:49,780 some cold dark matter particle and some cosmological constant, but they actually interact with each other. 407 00:41:50,200 --> 00:41:57,700 And that has some, some motivation from, uh, from theoretical constructions, uh, elsewhere. 408 00:41:58,090 --> 00:42:01,330 Uh, but for our purposes, I just want to show you what that kind of looks like. 409 00:42:01,730 --> 00:42:03,880 Uh, what I've plotted here is a function of redshift. 410 00:42:04,510 --> 00:42:10,930 Uh, as the Hubble in this model that we get and the Hubble that you get in the standard cosmology, Lambda CDM. 411 00:42:11,200 --> 00:42:18,490 So one would be just the lambda CDM value, uh, and uh, and you can see sort of in this value because dark energy and dark matter interact, 412 00:42:18,910 --> 00:42:22,060 the Hubble that we have a Lambda CDM has a different shape. 413 00:42:22,450 --> 00:42:24,849 And indeed it has some dip and then it has a rise. 414 00:42:24,850 --> 00:42:30,879 So that actually turns out that this integral one over H is left rather is left reasonably constant, 415 00:42:30,880 --> 00:42:36,190 except you end up with a much somewhat larger value of Hubble today, which is Z of zero. 416 00:42:37,900 --> 00:42:44,130 The larger value, uh, here that you actually find is, uh, not much larger than the CMB value. 417 00:42:44,140 --> 00:42:51,640 So again, CMB says 67, uh, supernovas at 72, this sort of drops in somewhere in the middle of the, of the two. 418 00:42:51,910 --> 00:42:55,600 So by no means you would call this a resolution of the paradox of the Hubble tension. 419 00:42:56,170 --> 00:43:00,969 Uh, and indeed, uh, people have done, um, people have included more data sets, 420 00:43:00,970 --> 00:43:05,950 such as the Baryon Acoustic Oscillation data set, which is looking at clustering of galaxies. 421 00:43:06,070 --> 00:43:13,420 Uh, and these kinds of LED solutions are very, very unlikely to work because changing the shape very dramatically is very, very constrained. 422 00:43:13,840 --> 00:43:16,659 So even this sort of gets you a little bit of the way. 423 00:43:16,660 --> 00:43:21,010 But to actually get this, I mean, to do is virtually impossible with all the data sets that you might have. 424 00:43:22,390 --> 00:43:25,090 Um, in terms of early recombination. 425 00:43:25,630 --> 00:43:34,900 You would want to change this quantity, which was the standard ruler, and then you also change this denominator by the same ratio, about 7%. 426 00:43:35,380 --> 00:43:41,440 Uh, and then you, you would uh, if that, if you did that, then you would indeed get h nought uh, 7% higher. 427 00:43:41,680 --> 00:43:44,830 But the standard ruler is standard for a reason. It's very hard to change this. 428 00:43:44,840 --> 00:43:52,210 Uh, this quantity is set by the makeup of the universe with the CMB, uh, constraints rather, uh, severely. 429 00:43:52,810 --> 00:43:58,180 Uh, and one, sort of one compelling possibility would have been that if there were some new 430 00:43:58,210 --> 00:44:02,380 degrees of freedom which are relativistic of the CMB beyond the Standard Model, 431 00:44:02,860 --> 00:44:06,939 uh, and these could be something like an extra generation of neutrinos, uh, 432 00:44:06,940 --> 00:44:11,140 some new kinds of, uh, particles, like axion, but relativistic, like I was talking about. 433 00:44:11,320 --> 00:44:15,520 There could even be gravitons, but they were produced relativistic in some phase transition. 434 00:44:15,940 --> 00:44:21,610 Uh, these would actually serve to increase, uh, the deduced value of h nought from the CMB. 435 00:44:22,060 --> 00:44:27,430 And so one could try to see how much extra, how much, how many extra degrees of freedom would you need for this to be the case? 436 00:44:28,150 --> 00:44:33,810 Um, so that usually it's measured in this, uh, parameter called an effective for whatever reasons, 437 00:44:33,820 --> 00:44:38,590 again, historical reasons, where it's sort of counting the number of degrees of freedom of neutrinos. 438 00:44:38,950 --> 00:44:43,059 And there are three nutritional standard models. So the standard model is close to three. 439 00:44:43,060 --> 00:44:47,200 Again, close and not exactly three for historical definitional reasons. 440 00:44:47,560 --> 00:44:51,390 So what you would need is about, uh, any factor to be about 4.2. 441 00:44:51,400 --> 00:44:56,530 So about an one extra degree of freedom on top of the neutrinos, say for, for you to have, 442 00:44:57,070 --> 00:45:02,139 uh, for you to have something like this work, which by itself is not something which is, uh, 443 00:45:02,140 --> 00:45:06,160 which, which is uh, uh, such a hard thing to believe, 444 00:45:06,580 --> 00:45:13,570 except that once you actually add this much amount of an effective the the fit to the CMB spectrum, uh, becomes much, much worse. 445 00:45:14,200 --> 00:45:17,820 And so you you do you can satisfy all these other constraints, but you actually find that, 446 00:45:17,830 --> 00:45:25,420 uh, there's this, uh, large, uh, large l damping tail of the CMB, which I can show you here. 447 00:45:26,620 --> 00:45:31,359 Uh, here. Sorry. So for at large l values, the CMB starts damping down. 448 00:45:31,360 --> 00:45:35,830 These oscillations start damping down. And that damping has to do with how many relativistic degrees of freedom you have. 449 00:45:36,340 --> 00:45:40,030 And we have measured this damping tail be rather high precision as well as you can see. 450 00:45:40,450 --> 00:45:44,290 And if you had that many degrees of freedom, this damping there doesn't like it at all. 451 00:45:44,620 --> 00:45:46,900 So you can't actually add those degrees of freedom. 452 00:45:47,590 --> 00:45:55,090 Uh, in the early measurement, which is to say, in general, in general, it's actually very hard to change this value, which is a standard ruler, 453 00:45:55,090 --> 00:45:59,139 without messing up this very nice fit to the CMB as against people pace, 454 00:45:59,140 --> 00:46:03,640 place then to try and inject some energy, these relativistic degrees of freedom. 455 00:46:03,640 --> 00:46:09,959 But they do it in a way that. Doesn't dump the the Daylesford sort of discriminatory injection right at the right 456 00:46:09,960 --> 00:46:14,340 time of the epoch that you needed to change ours but not change the dumping tale. 457 00:46:14,670 --> 00:46:22,560 And so this goes by the name of early dark energy, which, um, if it's not clear from my tone, is not a very compelling, compelling solution at all. 458 00:46:23,070 --> 00:46:28,770 And so, uh, even this is this works to some extent, but you really need to sort of by hand, 459 00:46:28,770 --> 00:46:33,780 put in some energy and make in your model such that you actually get exactly what you want, 460 00:46:33,780 --> 00:46:37,080 and then you get some better fit with the, with the, with a suspension. 461 00:46:37,560 --> 00:46:42,060 So by no means something that, that seems like the universe would do for, for no reason at all. 462 00:46:42,960 --> 00:46:48,990 So, so these are the, there's, there's various versions of these latent early solutions that I've just sort of drawn cartoons of, 463 00:46:49,500 --> 00:46:52,950 um, but really there's no real compelling solution that you can point to. 464 00:46:53,400 --> 00:46:57,870 Uh, the experiments go back and, uh, the observational observations go back and check. 465 00:46:58,440 --> 00:47:02,999 And, uh, various groups have done cross checks and nobody can find a bug. 466 00:47:03,000 --> 00:47:07,530 And so this is just a problem that's in our hand. We don't know if there's something simple going on. 467 00:47:07,530 --> 00:47:12,450 Somebody forgot a factor. There's some population of stars that looks different from this other population of stars. 468 00:47:12,450 --> 00:47:17,160 And there's some systematic bias that we're introducing or that we're learning something deep about cosmology. 469 00:47:17,790 --> 00:47:24,030 Just because this is the first time that we've sort of gone to percent level precision, uh, in cosmology in the future. 470 00:47:24,810 --> 00:47:31,440 One one great hope was that neutron star mergers could completely break this degeneracy, because that would be a completely third new way to find, 471 00:47:31,980 --> 00:47:37,559 uh, to do to find a it's not, uh, we haven't been seeing as many neutron star mergers as we really hoped. 472 00:47:37,560 --> 00:47:42,390 So the actual statistics we would need to build up seem to say that this is not a short term proposition, 473 00:47:42,840 --> 00:47:47,790 and people are obviously pursuing constantly new ideas to look for different markers in astrophysics to, 474 00:47:48,240 --> 00:47:54,260 uh, like the, the, the yellow, uh, intermediate yellow, uh, error bars that I showed you that were, 475 00:47:54,360 --> 00:47:59,580 uh, interpolating between the, uh, CMB measurement and the and the and the supernova measurement. 476 00:47:59,760 --> 00:48:03,570 And those will get refined hopefully over time. And so maybe they will shed light on what's going on. 477 00:48:03,930 --> 00:48:07,950 But at the moment, Hubble tension remains a mystery. Uh, waiting for resolution. 478 00:48:07,950 --> 00:48:08,340 Thank you.