1 00:00:01,170 --> 00:00:17,000 They? Welcome to Hillary each morning of theoretical physics. 2 00:00:18,200 --> 00:00:28,790 This one is a bit special because rather than focusing on a subject, we're focusing on some wonderful young scientists that we have in the department. 3 00:00:29,930 --> 00:00:37,760 Shivaji Sundy joined us from Princeton. As the African professor next to The Bachelor. 4 00:00:38,060 --> 00:00:48,900 And he was able to bring something which isn't a Buddhist to a point which implies photos. 5 00:00:49,970 --> 00:00:54,110 Which we were able to advertise across the whole of theoretical physics. 6 00:00:54,770 --> 00:00:58,130 And we've been lucky enough to end up with George. 7 00:00:58,430 --> 00:01:02,060 I'm trying to just go in and talk to you today. 8 00:01:02,270 --> 00:01:07,340 And so it's going to be a whistle stop tour through through modern theoretical physics. 9 00:01:09,200 --> 00:01:17,480 Nurse in the department. We have appointed a new faculty member, Dr. Edward Hardy, 10 00:01:18,080 --> 00:01:24,740 who did his dphil with John Wheater and is currently at Liverpool, will be joining us in September. 11 00:01:26,360 --> 00:01:33,220 Ed is an astro particle physicist and so he's going to be working on things like the sun and the sky, 12 00:01:33,440 --> 00:01:43,009 looking at all the new astrophysical experiments which are due to give us information about about particle physics and also CERN in 13 00:01:43,010 --> 00:01:53,150 the basement of the Beecroft where quantum detectors are pushing the boundaries to when they can look at single elementary particles. 14 00:01:53,570 --> 00:01:57,770 So that's going to be exciting. New Frontier in Astroparticle Physics. 15 00:01:59,840 --> 00:02:03,440 Other things to say if anybody wants something to do this afternoon. 16 00:02:03,440 --> 00:02:09,170 Topix is going on at the moment, so if you want to wander down to the river, rowing up and down the river. 17 00:02:13,190 --> 00:02:18,770 And indeed they posted on the same thing the next morning of theoretical physics would be the end the fifth week, 18 00:02:19,130 --> 00:02:24,350 which is the 27th of May, and that would be on Fusion. 19 00:02:25,220 --> 00:02:35,120 So you can learn about the new advances in Fusion, in particular the net energy gain that was much advertised a few months ago, 20 00:02:35,420 --> 00:02:44,030 and that will be led by Michael Barnes along with Joe Direct and Archie bots and they'll be telling you about the theory behind Fusion. 21 00:02:44,690 --> 00:02:53,600 And that coincides with eight weeks, if you want to make a weekend of it, starting there in the morning and down the river in the afternoon. 22 00:02:53,880 --> 00:02:56,680 By then, the weather might be a bit better. Good. 23 00:02:58,770 --> 00:03:07,830 Anyway, this morning we're going to start with George, Dr. George Obit, who did his Ph.D. at Harvard and is a particle physicist. 24 00:03:08,130 --> 00:03:13,800 And George is going to tell us about cosmic inflation and the very early universe. 25 00:03:15,910 --> 00:03:20,050 Okay. Thank you, Julia. Is this Mike working better now, or do I have to shout louder? 26 00:03:21,810 --> 00:03:26,760 You might as well. Oh, it's not. Okay. So I'll try to shout. I'm not usually very loud, but I'll try my best. 27 00:03:27,640 --> 00:03:31,090 Uh, okay. So thank you, Julie, for the invitation. 28 00:03:31,110 --> 00:03:37,349 And thank you all for coming. Good morning to you all. So today, as advertised, I'll be talking about inflation. 29 00:03:37,350 --> 00:03:41,760 And there has been a lot of talk about inflation lately, and we're going to talk just about that. 30 00:03:42,360 --> 00:03:46,500 And by that, of course, I mean cosmic inflation and. 31 00:03:47,280 --> 00:03:55,530 Yeah. All right. So let me start by giving some motivation of why we study inflation and then overview of where the stock is going. 32 00:03:56,700 --> 00:04:01,590 So everything I say in the next few slides will be covered again in more detail throughout the talk. 33 00:04:02,070 --> 00:04:09,810 But this overview is meant to be as a roadmap so that you can follow the flow of logic easily so through the presentation. 34 00:04:13,680 --> 00:04:18,810 All right. So inflation is a period of accelerated expansion in the very early universe. 35 00:04:19,260 --> 00:04:23,220 And the reason we're interested in inflation is because it can explain some puzzles in data. 36 00:04:24,380 --> 00:04:25,800 I will tell you what puzzles later on. 37 00:04:26,510 --> 00:04:32,540 And yeah, we'll make you know, we'll look at one puzzle in particular and will make it very precise and why it's so puzzling. 38 00:04:33,860 --> 00:04:41,780 And so this year I'm showing what I'm showing is a cartoon of the universe and time runs from from left to right. 39 00:04:42,350 --> 00:04:45,440 So this is the universe today. And this is the very the very early universe. 40 00:04:46,400 --> 00:04:50,420 And as we go back, the universe was hotter and denser and smaller. 41 00:04:51,830 --> 00:04:58,790 And also as we go back, the universe behind before about 380,000 years in age, it was a peak. 42 00:04:58,910 --> 00:05:03,300 So light in this universe could not travel that far. Okay. 43 00:05:03,520 --> 00:05:09,099 380,000 years, the universe became transparent. And from then onwards, we could do cosmology. 44 00:05:09,100 --> 00:05:14,050 So we could look very far at galaxies and we can see things back in the history of the universe. 45 00:05:14,230 --> 00:05:16,780 So the farther away we look, the farther back in time we see. 46 00:05:17,860 --> 00:05:23,740 But of course, we cannot see back in time, farther than this surface, because light doesn't go behind here. 47 00:05:25,090 --> 00:05:29,320 Okay. So this surface here is called the surface of the scattering. 48 00:05:29,410 --> 00:05:32,980 And the light that we get from the surface is called the cosmic microwave background. 49 00:05:34,210 --> 00:05:39,610 Or simply for short. Okay. So this is the light that we can receive. 50 00:05:39,610 --> 00:05:45,429 This is the farthest light we can receive today and we live here and so like goes here and we catch that 51 00:05:45,430 --> 00:05:50,110 light and we're going to see that a puzzle on the surface actually teaches us about something really, 52 00:05:50,110 --> 00:05:53,500 really early that happened in the early universe, which is inflation. 53 00:05:54,250 --> 00:05:57,490 So in this talk I'll be talking about the accelerated expansion that happens here. 54 00:05:58,240 --> 00:06:04,420 The universe is also accelerating today, but this is not relevant for for you know, for the talk today. 55 00:06:04,900 --> 00:06:09,700 So there's an acceleration, an accelerated phase here and an X and we believe also an accelerated phase here. 56 00:06:10,000 --> 00:06:14,450 What I'm talking about is the very early part, not today. Okay. 57 00:06:18,000 --> 00:06:23,940 So we can collect light that comes from the CMB and it looks pretty boring to zero. 58 00:06:23,940 --> 00:06:25,290 Or there it looks just like this. 59 00:06:26,010 --> 00:06:33,270 Okay, we can look at the CMB and it looks exactly like a blackbody and it has a temperature of 2.7 Kelvin, whichever direction you look. 60 00:06:34,890 --> 00:06:41,430 Now this is really puzzling because two antipodal points on the CMB say one that comes from the North Pole and one that comes from the South Pole. 61 00:06:42,080 --> 00:06:47,070 I have not had time to be in causal contact with each other, even in the age of the universe. 62 00:06:47,820 --> 00:06:51,600 All right. And so how could it be that they are in thermal and thermal equilibrium, then? 63 00:06:52,720 --> 00:07:00,010 So this is exactly the question that we want to answer is why is the CMB so uniform and how can inflation, you know, explain this? 64 00:07:04,140 --> 00:07:07,710 All right. So with this in mind, this is the outline of my talk. 65 00:07:08,160 --> 00:07:10,170 We'll start by learning a little bit of cosmology. 66 00:07:10,200 --> 00:07:16,680 So I'll tell you about expanding universes, the Friedman equations, and how we can solve for scale factors, how we describe expanding universes. 67 00:07:17,910 --> 00:07:24,510 Then I will introduce a concept called the Particle Horizon, which tells us how far signals can travel in the history of the universe. 68 00:07:25,110 --> 00:07:29,100 And we'll calculate the particle horizon for our universe and the usual hot big bang picture. 69 00:07:30,280 --> 00:07:37,000 Okay. Once we do that we're going to realise that there's a horizon problem, which is that the CMB is made up of many disconnected touches. 70 00:07:38,040 --> 00:07:41,700 Okay. So that have not been a cause of contact. And then. 71 00:07:42,120 --> 00:07:44,729 But, you know, if you paid attention during the first few parts, 72 00:07:44,730 --> 00:07:50,370 then you would have been able to guess where the solution is going to be, which is inflation. 73 00:07:50,370 --> 00:07:52,950 And I will tell you that it's actually very easy to guess the solution. 74 00:07:54,030 --> 00:07:59,490 And then very briefly, I will say maybe one slide or two about quantum perturbations and how they generate the seeds for structure. 75 00:08:01,190 --> 00:08:05,030 And then I'll conclude. All right. 76 00:08:05,190 --> 00:08:09,960 So let's start by discussing the expanding universe. So special activity. 77 00:08:10,470 --> 00:08:18,780 You might remember that the interval between two points separated by date and time and the x, y, z in space is calculated like this. 78 00:08:18,780 --> 00:08:22,140 It's minus D squared, plus the x squared, plus d y squared, plus DC squared. 79 00:08:22,890 --> 00:08:30,420 And this is similar to the the Pythagoras theorem with a with a sign that has been flipped. 80 00:08:32,290 --> 00:08:38,500 Now this is the space time of special activity and it's all well and good, but it's actually boring and doesn't actually end up describing cosmology. 81 00:08:39,130 --> 00:08:42,390 The reason is because this space time doesn't expand in time. 82 00:08:42,430 --> 00:08:48,310 So if you put two galaxies at some fixed point, there will always be at the same distance at every time. 83 00:08:48,910 --> 00:08:50,890 So you put two galaxies initially at one distance. 84 00:08:51,760 --> 00:08:56,620 This universe is not going to expand, and the two galaxies will always remain at that same distance. 85 00:08:57,680 --> 00:09:01,370 What we want to do in cosmology is is describe an expanding universe. 86 00:09:01,760 --> 00:09:06,290 And so what we do is we introduce a scale factor, A of T, which is an increasing function of time. 87 00:09:07,930 --> 00:09:13,240 Okay. So in this picture now, if we put two galaxies at two fixed coordinates, 88 00:09:14,200 --> 00:09:18,760 then at a later time that the the physical distance between the galaxies would have increased. 89 00:09:19,970 --> 00:09:24,530 So notice now the interpretation of X, Y and Z is actually slightly different from the interpretation here. 90 00:09:26,360 --> 00:09:33,340 This X, Y and Z are just coordinates. And in order to find the physical distance between objects, you have to multiply by the scale factor. 91 00:09:35,720 --> 00:09:37,459 Oh. So. 92 00:09:37,460 --> 00:09:44,030 So two galaxies at fixed coordinates are not actually at fixed proper distance because there's a factor of a which depends on time multiplying them. 93 00:09:45,350 --> 00:09:50,450 All right. So now once so we have introduced a and the name of the game is determining a. 94 00:09:50,630 --> 00:09:54,170 All right. So once we determine a, then we know a lot about our cosmology. 95 00:09:54,260 --> 00:09:58,880 Okay? We can we can calculate how far signals travel. We can calculate the expansion rate of the universe, everything. 96 00:09:59,840 --> 00:10:05,940 All right, so how do we find a. We find a by using the Friedman equation. 97 00:10:06,660 --> 00:10:09,940 So the evolution of A is dictated by the contents of the universe. 98 00:10:09,960 --> 00:10:14,940 So whatever matter fills the universe, that will tell you how it evolves dynamically. 99 00:10:15,630 --> 00:10:19,900 All right. And the dynamical equations to solve is the Friedman equation, which is just this one. 100 00:10:19,920 --> 00:10:24,900 You can drive this from from general activity just by plugging in the metric that we saw on the previous slide. 101 00:10:27,020 --> 00:10:31,850 But I won't do that here. And this is a simple differential equation. 102 00:10:32,240 --> 00:10:38,059 What what it tells you is that the first derivative of a saw a dot squared divided by a 103 00:10:38,060 --> 00:10:42,320 squared normalised by a squared is just proportional to the energy density in the universe. 104 00:10:43,690 --> 00:10:46,840 Okay. So all we have to do is measure the energy density and then we can solve for a. 105 00:10:47,890 --> 00:10:51,760 This quantity on the left is also called the Hubble Wraith and is denoted H. 106 00:10:52,480 --> 00:10:55,840 Okay. I won't talk much about the Hubble. I was talking about this a lot. 107 00:10:56,020 --> 00:11:01,690 But if you've heard the Hubble race in the in popular literature, then this is what they are talking about. 108 00:11:01,840 --> 00:11:05,320 It's a lot of a. All right. 109 00:11:06,290 --> 00:11:13,010 Notice one thing. I have written the energy density as a function of a. And this is because we're working in an expanding universe. 110 00:11:13,430 --> 00:11:20,990 And so the energy density will depend on time. But it's going to be much easier to talk about the energy density as a function of a not time. 111 00:11:21,200 --> 00:11:23,000 And we will see why in the next slide. 112 00:11:25,170 --> 00:11:33,170 This is because, you know, certain types of matter that we're very familiar with, they have very nice scaling with with a. 113 00:11:33,840 --> 00:11:39,150 So let's do an example. Let's consider a box with side length equal to one coordinate unit. 114 00:11:40,100 --> 00:11:44,030 Okay. And let's start at time zero where the scale factor has value is zero. 115 00:11:45,780 --> 00:11:49,080 So then the physical length of the side is just a zero. 116 00:11:50,590 --> 00:11:55,930 And of the energy density at this time was rose zero then the energy density at a different time. 117 00:11:56,230 --> 00:11:59,230 Is this related to zero by this very simple quantity? 118 00:12:00,100 --> 00:12:04,329 So if at a later time the scale factor is just a not a zero, 119 00:12:04,330 --> 00:12:08,890 then all you have to do is multiply by a zero over a cubed to get the energy density here. 120 00:12:10,210 --> 00:12:14,380 Okay. There's this because we went through the motions and, you know, the the number density decreases. 121 00:12:15,580 --> 00:12:21,280 So this is a story for matter. If you have radiation like photons, it's a little bit I mean, 122 00:12:21,280 --> 00:12:27,850 a little bit more complicated in the sense that not just the number density decreases, but also the photons get stretched. 123 00:12:28,090 --> 00:12:33,580 So there are some redshift. Right. So we don't only multiply by a factor of zero over a cubed, 124 00:12:33,790 --> 00:12:37,900 but we multiply by another factor of A0 over a and this is to take into account the redshift. 125 00:12:39,800 --> 00:12:43,640 Okay. So the photons have lower energy here at when the scale factor is bigger. 126 00:12:45,270 --> 00:12:51,060 Good. So but we still end up with some very simple expression, which is zero zero times one over A to the four. 127 00:12:51,840 --> 00:12:58,530 We had 1 to 3 on the previous slide. So this is most of the matter that we are familiar with. 128 00:12:59,440 --> 00:13:03,100 And and but in cosmology, there is one more type of, you know, 129 00:13:03,370 --> 00:13:08,280 material or substance or it's an energy density that we don't actually we're not familiar with in everyday life. 130 00:13:08,620 --> 00:13:16,040 And this is this is dark energy. And dark energy is a bit strange in that if you expand the box, it doesn't actually dilute. 131 00:13:16,340 --> 00:13:21,980 So it's the energy density in space. Just because there is space there, there's no other reason why that energy density is there. 132 00:13:22,280 --> 00:13:27,590 Just whenever you have space, there is an energy density in it. And this is called dark energy. 133 00:13:27,600 --> 00:13:30,770 So if you imagine a box that is filled with dark energy, whatever that means, 134 00:13:32,000 --> 00:13:37,990 at some later time, at some later time, then this box will just have the same energy density. 135 00:13:38,000 --> 00:13:43,070 It will not dilute even if you stretch the box. Good. 136 00:13:44,570 --> 00:13:48,920 So these are the three the three main forms of energy density that we have and 137 00:13:49,160 --> 00:13:52,910 we can summarise the behaviour of the three energy densities by writing this. 138 00:13:53,510 --> 00:13:57,049 So row is just row zero times a zero over a two, 139 00:13:57,050 --> 00:14:03,290 the power three one plus W where W takes on different values depending on on the type of energy density we're talking about. 140 00:14:03,710 --> 00:14:07,010 So we're talking about matter. It's zero for talking about radiation as a third. 141 00:14:07,010 --> 00:14:10,970 If we're talking about dark energy, it's minus one and that's it. 142 00:14:11,790 --> 00:14:19,489 That's all there is to it. In general, in some universe, we're going to have a you know, we're going to have all these energy densities. 143 00:14:19,490 --> 00:14:23,809 For example, in our universe, we have all of them. And to find the total energy density, you just have to add up all of them. 144 00:14:23,810 --> 00:14:26,960 So our universe looks more like this. Okay. 145 00:14:29,220 --> 00:14:37,370 Good. So let's do it, for example, just to get, you know, our hands a bit dirty, working out, working things out. 146 00:14:37,820 --> 00:14:46,770 So. We have found an expression for Roffey on the previous slide and a few slides ago I showed you the Friedman equation, 147 00:14:46,770 --> 00:14:50,090 which was a differential equation for a as long as we know of a. 148 00:14:50,700 --> 00:14:54,570 All right. So here I've copied the Friedman equation again, but taken a square root, 149 00:14:55,260 --> 00:14:58,560 and I have plugged in the expression for off a that we found on the previous slide. 150 00:15:00,270 --> 00:15:03,360 Now this is a differential equation for a and it's separable. 151 00:15:03,390 --> 00:15:08,520 It's easy to solve. You can just move to that side and take an integral to it. 152 00:15:08,790 --> 00:15:12,930 And so you end up with some equation like this. So I will leave it up to you to do this integral. 153 00:15:13,170 --> 00:15:19,380 And then. And then. And then separate and then move all A's to the left, all to the right. 154 00:15:19,860 --> 00:15:24,120 And what you end up with is something like this. Okay. 155 00:15:24,930 --> 00:15:28,770 So we find out that the scale factory has some distinct behaviour, 156 00:15:28,770 --> 00:15:35,700 whether the universe contains matter radiation or dark energy and for matter it goes like t to the two thirds. 157 00:15:36,690 --> 00:15:42,690 For radiation. It goes like two to the one half. And for dark energy, it goes like, is it the HD where H is a constant? 158 00:15:44,540 --> 00:15:50,400 Okay. So this is some, some behaviour and now in our universe we can't actually solve, you know, 159 00:15:50,480 --> 00:15:57,440 solve the equation analytically because we have more components, but we can do it numerically and this is what the answer looks like. 160 00:15:57,950 --> 00:16:02,600 So if you try to solve for the scale factor of our universe as a function of time, then you'll get this black curve. 161 00:16:04,870 --> 00:16:09,819 And on this floor, I have also overlaid, you know, two curves that we saw last time. 162 00:16:09,820 --> 00:16:18,340 So this is from the from the toy examples that we did. The the cyan one is a is a revision curve and the magenta one is the macro curve. 163 00:16:19,120 --> 00:16:22,510 And we see that our universe in the past looks like it's a radiation universe. 164 00:16:23,490 --> 00:16:28,260 At some point it becomes the mass of the universe and then it starts deviating. 165 00:16:28,320 --> 00:16:32,940 You know, it's very hard to see, but it starts deviating just today from being a massive universe. 166 00:16:34,000 --> 00:16:39,810 Okay. Um. So is what I said so in the past. 167 00:16:40,620 --> 00:16:44,880 So I will go back to of the one half and then at some later time this will go like Pisa two thirds. 168 00:16:45,510 --> 00:16:48,660 And what happens is that there's also an acceleration. 169 00:16:49,440 --> 00:16:53,910 There's also today the universe begins accelerating just very close to today. 170 00:16:55,260 --> 00:17:02,700 And this is due to this term here, which is role of dark energy, which does not it doesn't scale with the scale factor like we saw. 171 00:17:04,010 --> 00:17:07,190 So I'm going to emphasise that this is responsible for late time acceleration. 172 00:17:07,280 --> 00:17:11,410 This is different from inflation. So the talk is not about this dark energy. 173 00:17:11,420 --> 00:17:14,870 It's about the dark energy that is, you know, in the very, very early universe. 174 00:17:19,350 --> 00:17:26,310 Good. So now that we can compute a let's actually find how far signals can travel in the universe. 175 00:17:26,880 --> 00:17:29,390 All right. So this is this concept is the particle horizon. 176 00:17:29,400 --> 00:17:34,559 So remember, what we're doing is we're going to compute the particle horizon and we're going to ask, is this C? 177 00:17:34,560 --> 00:17:40,030 And because we connected so our points on the good points of the C and we have communicated in the past. 178 00:17:40,770 --> 00:17:45,630 Okay. And that's why we're heading this way. So we are finding the particle horizon to ask that question. 179 00:17:47,460 --> 00:17:52,730 And the political horizon is the largest distance that a light signal could have travelled in the history of the universe. 180 00:17:52,740 --> 00:17:58,560 So ever since the universe began. What's the largest distance that a photon can travel? 181 00:17:59,640 --> 00:18:04,540 And this is easy to calculate. Is this an integral, you know, of recording a distance? 182 00:18:04,560 --> 00:18:10,270 So we have to find first the coordinate distance that the photon travels and then we multiply by by t, 183 00:18:10,410 --> 00:18:15,270 which is the scale factor to get the physical distance. Okay. 184 00:18:15,990 --> 00:18:20,969 So the horizon, the particle horizon at some time, t f is just a scale factor at the time, 185 00:18:20,970 --> 00:18:26,129 t f multiplied by the coordinate distance that the photon would have travelled by time. 186 00:18:26,130 --> 00:18:33,590 T. F. And to get this in business, we can just add up all the little corner distances that the photon travels in a short time. 187 00:18:34,460 --> 00:18:41,900 So in a short time the photon travels a distance C times delta T in the physical space, but in coordinates space travel. 188 00:18:41,900 --> 00:18:47,090 C. D. T. Divided by a. And so this is the integral that we have to do and then multiplied by a. 189 00:18:48,400 --> 00:18:51,630 Okay. Oh. Good. 190 00:18:51,640 --> 00:18:57,420 So now we can just do some rearranging, which is to say just multiply and divide by the air and then we get something like this. 191 00:18:57,430 --> 00:19:00,250 So we get an a dot in the denominator. 192 00:19:00,460 --> 00:19:07,180 And that is nice to have here because a dot we can get from the Friedman equation, which we can just express in terms of a right. 193 00:19:07,180 --> 00:19:12,639 So, so this whole integral is actually an integral only over a and you can easily write it from the Friedman equation. 194 00:19:12,640 --> 00:19:16,650 Just substitute here for a dot. Oh, good. 195 00:19:16,680 --> 00:19:23,700 So just like cleaning up, writing it in a nicer way. So the at some scale facts are a is just given by this integral. 196 00:19:25,600 --> 00:19:28,930 And now notice the quantity we're integrating is actually one over a dot. 197 00:19:30,370 --> 00:19:35,830 And what's special about one of that is that for all normal substances, one over a dot is always increasing. 198 00:19:37,800 --> 00:19:43,680 Okay. So we're going to see this later. But yeah, for normal substances, this quantity is increasing. 199 00:19:44,400 --> 00:19:53,070 The integrated is increasing. And so what happens is that this horizon distance gets its most important contributions from late times. 200 00:19:53,580 --> 00:20:01,410 If one over a dot is always increasing, then you can pretty much ignore the early time tail and just integrate over late times. 201 00:20:01,530 --> 00:20:06,690 And that will give you a very good approximation to the to the to the particle horizon. 202 00:20:10,020 --> 00:20:18,150 Now what this is important because yeah, if this gets it, if this gets all the contributions from late time, we know the late universe very well. 203 00:20:18,510 --> 00:20:22,260 And so we can compute exactly what this is in the late universe. 204 00:20:22,860 --> 00:20:24,390 Right. And there is no wiggle room there. 205 00:20:24,420 --> 00:20:30,710 So if this gets its contributions only from late times, then we know exactly what the is that we cannot you know, 206 00:20:31,680 --> 00:20:39,120 nobody can change that because we can measure the universe and there's data and that we cannot argue with, oh, good. 207 00:20:39,930 --> 00:20:43,680 So let's do this integral. So remember, this is the horizon distance. 208 00:20:44,190 --> 00:20:47,339 And so the quantity we're integrating is I'm showing here in the red line. 209 00:20:47,340 --> 00:20:51,360 So here on the x axis, I'm showing a the value of the scale factor. 210 00:20:51,990 --> 00:20:56,580 And on the y axis I'm just showing distance and megaparsec there's also a time axis up here, 211 00:20:56,850 --> 00:21:00,780 if you like, thinking in terms of time instead of a but they're the same thing. 212 00:21:02,040 --> 00:21:05,250 Okay. So the red line here is this is this a lot inverse? 213 00:21:06,240 --> 00:21:14,460 This is the quantity that we are integrating. This is an integral and D is this what you get after you integrate and multiply by A at each point? 214 00:21:15,680 --> 00:21:18,680 Okay. And again because I thought versus increasing. 215 00:21:19,940 --> 00:21:25,340 Up until maybe today where it starts decreasing. Uh, so this quantity, 216 00:21:25,460 --> 00:21:29,900 the capital DFA will get most of its contributions from lead times so we can even not 217 00:21:29,900 --> 00:21:34,070 worry about what happens earlier here and just use the late time cosmology that we know. 218 00:21:35,420 --> 00:21:42,310 Okay. Oh. So every so all the rage that we know here that I'm showing you here, we have actually measured very well. 219 00:21:42,320 --> 00:21:45,650 We have actually measured before this up to like ten to the minus ten or something. 220 00:21:46,310 --> 00:21:48,050 Okay. And there's no wiggle room here. 221 00:21:48,470 --> 00:21:55,340 So, so so everything that you do here will have to be confronted by data, which will have to be you know, it will have to fit data. 222 00:21:58,500 --> 00:22:03,299 There's one more line, one vertical line that I'm showing here, which is the surface of last scattering. 223 00:22:03,300 --> 00:22:07,950 So a lot of scattering happens at a time, which is where the universe was about 380,000 years old. 224 00:22:08,490 --> 00:22:11,780 Like like we saw, you know, on the second or third slide. 225 00:22:12,090 --> 00:22:15,210 And this corresponds to a scale factor of about ten to the minus three. 226 00:22:16,360 --> 00:22:19,810 Okay. So the last gathering happens is an event. It happens here. 227 00:22:20,530 --> 00:22:24,250 And from this plot, we can actually read what the horizon was at last scattering. 228 00:22:25,590 --> 00:22:29,880 Okay. So we can just know this value and this value ends up being point three megaparsec. 229 00:22:32,820 --> 00:22:39,240 Okay. So pictorially, this is what we have. We have a service of the scattering and the horizon size. 230 00:22:39,450 --> 00:22:45,839 At the time of our scattering is point three megaparsec. Now our distance to the last scattering surface. 231 00:22:45,840 --> 00:22:50,729 You can also do by performing an integral intervals similar to the one that I showed you. 232 00:22:50,730 --> 00:22:56,700 But we won't do that here. And the distance is 13 megaparsec so you can compute the angle. 233 00:22:56,970 --> 00:23:02,610 That's one horizon, you know, that's surrounded by one horizon on the scattering surface. 234 00:23:03,840 --> 00:23:08,760 Okay. So the angle is easy to compute this point three divided by 13 and radians and it ends up being a point or two. 235 00:23:10,070 --> 00:23:13,370 And now if you want to find the number of positive disconnected batches in the C and B, 236 00:23:13,370 --> 00:23:21,410 all you have to do is divide the full solid angle by this number squared and you get about 24,000 possibly disconnected touches. 237 00:23:22,870 --> 00:23:30,170 Okay. So the CMB is actually, even though it looks uniform, it's made up of 24,000 little causally disconnected touches. 238 00:23:30,190 --> 00:23:34,030 They would have been causally disconnected patches in the Big Bang model. 239 00:23:34,720 --> 00:23:36,460 So this is the Big Bang model. 240 00:23:38,290 --> 00:23:45,880 This is a very, very big puzzle as to why, you know, 20,000 positive, disconnected touches seem like they are in equilibrium. 241 00:23:46,780 --> 00:23:50,920 Now, let me. So let me tell you a bit more about this, using a football analogy. 242 00:23:53,410 --> 00:23:59,110 So let's say so let's say I go to London, to the stadium, to to to to watch a football match. 243 00:23:59,830 --> 00:24:07,960 Okay. And if I go there and then if I find Andy and Francesco, then I would be a little bit surprised, you know? 244 00:24:08,620 --> 00:24:12,250 I'll ask them, why did you guys not invite me to come see this match with you? 245 00:24:12,640 --> 00:24:15,730 And they'll be like, No, no, it's a coincidence. We actually did not planned this. 246 00:24:15,970 --> 00:24:21,660 We just happened to be here at the same time. And you know, again, they're my really good friends, so I'm inclined to believe them. 247 00:24:22,620 --> 00:24:26,490 And so that's fine. And, you know, we'll just sit there, watch the match and enjoy it. 248 00:24:28,590 --> 00:24:33,840 But if you've actually had a little bit more funds, then the situation could have been slightly different. 249 00:24:34,770 --> 00:24:40,800 Which is to say, I could have went to the stadium and the the state of the stadium could have been something like this, 250 00:24:42,930 --> 00:24:47,520 where we have 20,000 level umpires, fellows, and they all fill all the seats. 251 00:24:48,420 --> 00:24:54,300 All right. And then I would ask my very good friends, I would like, why did you not invite me to this game, you know, to watch this match? 252 00:24:54,420 --> 00:25:00,440 And then, you know, no matter how much they promised that this was not planned, it was a coincidence, etc., etc., I was like, No, I don't believe you. 253 00:25:00,450 --> 00:25:03,570 How come, you know, you all happened to be at the same time, at the same place? 254 00:25:04,080 --> 00:25:07,740 This was definitely planned. There's, you know, a group chat or something on which I'm not. 255 00:25:10,220 --> 00:25:13,350 And so, you know, I just won't believe them. 256 00:25:13,370 --> 00:25:16,940 I will be like, no matter how much you promise, I just won't leave you guys. 257 00:25:18,140 --> 00:25:19,910 And this is the same story with the CMB. 258 00:25:19,940 --> 00:25:28,340 Okay, so the CMB has 20,000, you know, causally disconnected patches in the in the, you know, in the big bang picture. 259 00:25:28,700 --> 00:25:34,790 And yet they all choose to go to the same temperature. And how come that that's just not something believable? 260 00:25:35,490 --> 00:25:38,830 Okay, good. 261 00:25:38,900 --> 00:25:44,060 So so to reiterate, the CMB should have different temperatures and, you know, in each of these circles. 262 00:25:44,330 --> 00:25:52,130 But it doesn't. And we should really be suspicious that the that that the Big Bang model is underestimating the true size of the horizon. 263 00:25:52,190 --> 00:25:55,849 The horizon cannot be the small, because otherwise how would you know? 264 00:25:55,850 --> 00:25:57,800 How would the CMB have had the same temperature? 265 00:25:58,250 --> 00:26:04,640 So there has to have been communication between all these, uh, you know, just like I suspect my friends are communicating behind my back. 266 00:26:04,880 --> 00:26:09,110 This, these, these are also, these patches should also be communicating with each other. 267 00:26:11,970 --> 00:26:18,450 All right. So what we do is we need to find a way to make sure that there's that there are no calls, disconnected patches in the CMB. 268 00:26:18,720 --> 00:26:21,990 In other words, we need to make the horizon a lot, lot bigger. Okay. 269 00:26:22,800 --> 00:26:28,820 But for those paying attention, this is really hard because the horizon and the big picture. 270 00:26:29,220 --> 00:26:35,680 Sorry. This is really hard because the horizon and the big picture receives this contribution 271 00:26:35,680 --> 00:26:39,130 from the late universe and the late universe we have measured very well. 272 00:26:39,310 --> 00:26:42,780 So you cannot change that. There is nothing to do. Right. 273 00:26:43,350 --> 00:26:49,500 And the reason it receives this contribution from the late universe is because adult and verse is an increasing function of time. 274 00:26:50,160 --> 00:26:57,360 And so if you do this integral, it will be dominated by late times, and then you can ignore early times. 275 00:26:58,230 --> 00:27:07,080 All right. The catch, though, is that, you know, the inverse is increasing is an increasing function of time in the usual hot, big picture. 276 00:27:07,350 --> 00:27:12,960 But we can change that picture. We can change that picture at early times where we have no data yet. 277 00:27:13,500 --> 00:27:16,800 I mean, we have some some some some some indirect probes. 278 00:27:17,310 --> 00:27:22,590 But if we are able to change the picture at early times, so that adult inverse is actually decreasing, 279 00:27:23,190 --> 00:27:30,300 so that day receives a large contribution from very early times, then we might be able to increase the size of the horizon. 280 00:27:31,860 --> 00:27:35,649 Okay. So this slide is what I just said the words. 281 00:27:35,650 --> 00:27:38,900 So the. Yeah. 282 00:27:40,610 --> 00:27:46,099 All right. So, so, so. And here. So, yeah. Like I was telling you, this is about one second after the big bang. 283 00:27:46,100 --> 00:27:51,380 We know pretty much everything up to about one second after the Big Bang, you know, after that. 284 00:27:51,920 --> 00:27:57,060 So data here is very constraining. But before here, we only have some indirect probes of which I will show you one. 285 00:27:57,080 --> 00:28:01,850 One of them is as the perturbations in the CMB, and I will talk more about that later. 286 00:28:02,210 --> 00:28:06,890 But these are all indirect probes. We never actually we don't actually see things here directly. 287 00:28:07,770 --> 00:28:13,980 Okay. Um. Good. All right. 288 00:28:14,010 --> 00:28:19,320 So because these are here is not as constraining. We can play with the functionality initially. 289 00:28:19,950 --> 00:28:25,559 So what we want is we want a dot inverse to decrease, which means one over eight dot decreases, 290 00:28:25,560 --> 00:28:29,470 which means a dot increases, which means the universe has to be accelerating. 291 00:28:29,490 --> 00:28:34,440 So that's why inflation is an accelerating period, because it has to be an increasing function of time. 292 00:28:35,070 --> 00:28:40,230 Okay. So a double dot is positive. And in other words, like the acceleration is positive. 293 00:28:41,920 --> 00:28:46,600 So what we do is that we can just change what it looks like, you know, at earlier times by. 294 00:28:46,990 --> 00:28:51,790 Yeah. Either by hand or by posturing. Some different energy density like dark energy at early times. 295 00:28:52,480 --> 00:28:56,740 But that's again, independent of this here. Um, yeah. 296 00:28:56,740 --> 00:29:04,870 So the scale factor and the usual big bang model is this black line. And here I've just tacked on a, you know, a little patch of inflation. 297 00:29:04,870 --> 00:29:08,290 Right? And the very, very early universe at around ten to the -30 6 seconds. 298 00:29:09,520 --> 00:29:15,490 Okay. So all it takes is for the universe to inflate from ten to the -30 6 seconds to ten to the -30 4 seconds. 299 00:29:15,700 --> 00:29:19,330 So this is really fast acceleration. Yeah. And you can see that. 300 00:29:20,020 --> 00:29:25,540 Yeah. So the expansion here is really, really fast. It's accelerating. If you compute the derivative, like a double dose here will be positive. 301 00:29:26,350 --> 00:29:30,480 And the scale factor increases by a factor of either 100. Okay. 302 00:29:31,110 --> 00:29:34,680 So this is one very particular model that, you know, that we have picked. 303 00:29:34,890 --> 00:29:38,459 You can change. You can play with these parameters. You can move this period here. 304 00:29:38,460 --> 00:29:43,050 You can move it here. You can decrease the, uh, you know, the e to the 100th factor. 305 00:29:43,770 --> 00:29:48,990 And all these things are for free parameters. There are some constraints on them, but, but not, not that many. 306 00:29:49,920 --> 00:29:56,490 Oh, good. And so with this fear of this inflation, what does the horizon look like then? 307 00:29:57,900 --> 00:30:02,390 Okay, so we can compute the same plot again. So this is. 308 00:30:03,720 --> 00:30:08,700 This is a dot inverse of the usual hot big bang. And this is the horizon of the usual hot big bang. 309 00:30:09,790 --> 00:30:15,790 I thought inverse inflation is going to decrease. So this is what the inverse in inflation looks like, this red line. 310 00:30:15,790 --> 00:30:18,879 And then it catches on with this one and carries on. It doesn't do anything after. 311 00:30:18,880 --> 00:30:24,730 That's different. But now, if you can view the integral, you will get this green curve, 312 00:30:24,940 --> 00:30:28,600 because most of the contribution will come from early times, not from late times. 313 00:30:29,780 --> 00:30:34,360 Right. So the area under this curve is dominated now by early times, not by late times. 314 00:30:35,570 --> 00:30:42,890 Okay. So intuitively what's happening is that inflation is an expansion that is, you know, very, very fast. 315 00:30:42,900 --> 00:30:46,400 And so what's happening is that two points that are very far away. 316 00:30:46,410 --> 00:30:54,350 Now, if you try to extrapolate in the past and the big bank picture, they would not get that close because the expansion is not that fast. 317 00:30:54,560 --> 00:30:59,570 So if you try to extrapolate back, these two points will still be far away and the in the big picture. 318 00:31:00,080 --> 00:31:07,190 But if you have a period of expansion that's really exponential, that's accelerating and and just like inflation, 319 00:31:07,670 --> 00:31:11,150 then these two points, even though they're far away now, they could have been really close in the past. 320 00:31:11,570 --> 00:31:15,350 And if they are really close in the past, then they could have talked in the past. Okay. 321 00:31:15,680 --> 00:31:23,989 So in the Big Bang model, they cannot because the expansion rate is not as is not as, you know, it's it's not accelerating. 322 00:31:23,990 --> 00:31:27,150 So what we see here, the important part is that it has to be accelerating, right? 323 00:31:27,260 --> 00:31:35,060 So it has to be, uh, yeah, that inverse has to be decreasing, which means a double dot is positive. 324 00:31:36,890 --> 00:31:44,320 Um. Okay. And in particular, there's one thing to point out here that this here is the surface of the scattering. 325 00:31:44,340 --> 00:31:49,920 So this here is the line where we have no data beforehand. This here is the surface of the scattering. 326 00:31:50,460 --> 00:31:55,590 And remember, the horizon was here was the intersection of the blue line with the with the green line. 327 00:31:55,590 --> 00:31:57,450 And it was about 0.3 megaparsec. 328 00:31:59,630 --> 00:32:06,650 Now the horizon is each of the hundredth bigger than that because we have this factor of, of, of of feature 100 and expansion. 329 00:32:07,070 --> 00:32:09,620 So the horizon here is going to be easily 100 times bigger than that. 330 00:32:09,950 --> 00:32:18,290 And that's big enough to actually include all of the whole the the whole surface of of of of less scattering. 331 00:32:18,530 --> 00:32:22,040 And so there is no no disconnected patches on the surface of scattering. 332 00:32:22,990 --> 00:32:28,120 If we have something like inflation. Okay. 333 00:32:28,150 --> 00:32:32,650 So so I think a period of accelerated expansion. Oh, sorry. 334 00:32:32,680 --> 00:32:35,889 Typo. So I didn't hear the accelerated expansion, which is inflation. 335 00:32:35,890 --> 00:32:40,270 In the early universe, we see that the horizon can be much bigger than what the Big Bang predicts, 336 00:32:40,270 --> 00:32:43,840 and that's because it gets most of most of its contributions from the early times. 337 00:32:44,260 --> 00:32:50,710 Okay. So, yeah, equivalently, points could have been much closer than what the Big Bang predicts. 338 00:32:54,130 --> 00:32:59,470 Okay. But that is that's actually not the only thing that inflation can do. 339 00:33:00,280 --> 00:33:05,020 One other thing that inflation can do, which is that it can explain these perturbations. 340 00:33:05,890 --> 00:33:10,720 Okay. So it can explain the pattern of perturbations that we see on the cosmic microwave background. 341 00:33:11,990 --> 00:33:19,950 So the picture that I showed you before is actually a zero order picture of, of of the CMB and its temperature is 2.7 Kelvin. 342 00:33:21,020 --> 00:33:28,220 But what happens is that they're actually very tiny fluctuations that are one that are at the level of 1%, ten to the five. 343 00:33:29,060 --> 00:33:34,970 So these little tiny mountains and valleys are ten to the minus five times 2.7. 344 00:33:39,320 --> 00:33:42,470 Right. And we wouldn't know, for example, where. 345 00:33:43,490 --> 00:33:51,410 Yeah. We wouldn't have a very good explanation of why we see these. So these have been detected, you know, recently as, uh, ten, 12 years ago. 346 00:33:52,520 --> 00:34:03,260 But, but, uh, yeah. So we wouldn't exactly know how to explain this pattern, but inflation actually produces a way of, of, of giving this pattern. 347 00:34:04,170 --> 00:34:09,660 And in order to see that, let's review first the physics of the quantum simple harmonic oscillator. 348 00:34:10,810 --> 00:34:17,440 So everything in physics is a simple oscillator and this is even the C and B pattern is a simple harmonic oscillator, which is crazy. 349 00:34:19,120 --> 00:34:25,840 All right. So let's look at the quantum simple harmonic oscillator. There's a potential and this is the potential as a function of position. 350 00:34:26,170 --> 00:34:29,209 It's the blue line. And what we imagine. 351 00:34:29,210 --> 00:34:34,880 We imagine a particle that sits in this potential and we ask, what is the wave function of that particle? 352 00:34:35,210 --> 00:34:39,290 Okay, we can solve the Schrödinger equation and then we find the wave function of that particle. 353 00:34:41,010 --> 00:34:45,600 And if you remember, the wave function of the particle in the ground state is a Gaussian. 354 00:34:47,740 --> 00:34:47,970 Okay. 355 00:34:48,010 --> 00:34:56,290 So the so which means that the position of the particle X, if you try to measure it at some random time, will be drawn from probability distribution. 356 00:34:56,320 --> 00:34:59,440 That is the square of the wave function, which is discussion. 357 00:35:00,960 --> 00:35:09,540 Okay. So if you measure the position of the particle, even though the particle is is in its ground state, you will not get zero quantum. 358 00:35:09,810 --> 00:35:15,030 This is one manifestation of the uncertainty principle where the particle cannot exactly be localised at x equals zero. 359 00:35:15,630 --> 00:35:23,580 So if you measure the position, you won't get zero. You will get some some number with a spread depending on when the measurement is done, etc. 360 00:35:23,580 --> 00:35:25,530 So it's just a quantum mechanical process. 361 00:35:27,720 --> 00:35:35,670 So while the expectation value of the variable X which denotes the position of the particle is zero, its variance is not so different. 362 00:35:35,670 --> 00:35:39,600 Measurements will give different values of x and typically if you do a measurement, you won't get zero. 363 00:35:39,840 --> 00:35:46,829 So the set zero here, getting zero is, oh yeah, because the point is as measure zero. 364 00:35:46,830 --> 00:35:50,550 And so you probably won't get zero if you do a number of random measurements. 365 00:35:52,420 --> 00:35:57,160 All right. So typical values of the position are actually not zero, even though the expectation values is zero. 366 00:36:00,250 --> 00:36:04,060 Now an inflation. Something similar happens. Okay. 367 00:36:05,080 --> 00:36:10,930 And the simplest models of inflation. We need some energy density that will drive this accelerated expansion. 368 00:36:12,170 --> 00:36:15,770 Okay. And this energy density is provided by some scalar field usually. 369 00:36:16,100 --> 00:36:19,670 So there's a scalar field which has a potential. So it has it has an energy. 370 00:36:20,240 --> 00:36:25,580 And the energy density to drive a solid expansion is provided just by the scalar field. 371 00:36:27,620 --> 00:36:32,960 During inflation, the scalar field will have some background value, some average value, which I'm calling fibre. 372 00:36:33,170 --> 00:36:38,510 That depends only on time. Okay. So it can only change as a function of time, but not in space. 373 00:36:40,790 --> 00:36:46,040 But in addition to this, there could be a perturbation that depends on both space and time. 374 00:36:46,440 --> 00:36:54,060 Okay. So this perturbation is usually normalised with this factor of a but the fact of fate is not so important for the story. 375 00:36:54,080 --> 00:36:58,220 The important part is the Delta five, which depends on both space and time. 376 00:37:01,150 --> 00:37:07,750 So just like the original picture that I showed you of the expanding universe, space during inflation is actually flat. 377 00:37:08,890 --> 00:37:15,100 So what we can do. So the spatial slices are actually flat. So what we can do is we can expand this Delta PHI in four modes. 378 00:37:15,910 --> 00:37:23,170 So we can take, you know, a four year transform on the variable X and we'll get Delta Phi of T and K. 379 00:37:23,500 --> 00:37:26,650 So we want for you transform the T variable only the X variable. 380 00:37:29,030 --> 00:37:35,360 And now we can talk about each freedom or delta fi k separately, and that is only a function of time. 381 00:37:36,470 --> 00:37:40,850 All right. Much like the position of the simple harmonic oscillator as a function of time. 382 00:37:41,510 --> 00:37:46,470 Delta Facebook is only a function of time. But the analogy goes even deeper. 383 00:37:46,520 --> 00:37:49,999 It obeys the same equations of motion. It obeys an equation of motion. 384 00:37:50,000 --> 00:37:52,490 That is an equation of motion of the simple harmonic oscillator. 385 00:37:52,820 --> 00:37:58,340 So each mode of a is a, you know, an equation of motion that is very similar to the simple harmonic oscillator. 386 00:38:00,130 --> 00:38:03,310 The difference is that this depends on time, but that's yeah, that's a technicality. 387 00:38:03,310 --> 00:38:08,140 So the equation is very similar. Oh, good. 388 00:38:08,920 --> 00:38:11,020 And so when we consider the system quantum mechanically. 389 00:38:12,060 --> 00:38:17,580 Then the physics of the system will be the same as the physics of the simple harmonic oscillator in quantum mechanics. 390 00:38:18,200 --> 00:38:23,790 Okay, so the expectation value of Delta five will be zero, but its variance would not be zero. 391 00:38:25,180 --> 00:38:29,049 Okay. And if it's various would not be zero. 392 00:38:29,050 --> 00:38:35,950 And so if you measure that office up at some random time, you're more you know, you typically will get some non-zero value. 393 00:38:37,530 --> 00:38:41,250 Okay. And. Yeah, so. So I remind you. 394 00:38:41,310 --> 00:38:45,210 So what does this mean? This means that. So yeah. 395 00:38:45,340 --> 00:38:48,850 Remember that Delta Phi K is the amplitude of the k wave mode. 396 00:38:49,630 --> 00:38:54,910 Okay. So this means that different k wave modes have a have an amplitude that is 397 00:38:54,910 --> 00:38:59,590 drawn from a random distribution and that random distribution is a Gaussian, 398 00:38:59,590 --> 00:39:02,750 just like the simple harmonic oscillator. Okay. 399 00:39:04,040 --> 00:39:11,329 So just like the simple harmonic oscillator. So these keyboards that if I k are typically non-zero, they would be drawn from a random distribution. 400 00:39:11,330 --> 00:39:23,399 And this tells you about the amplitude of the case mode. So the picture we have is that this for each came on we have a so this is a came so 401 00:39:23,400 --> 00:39:28,650 there's a vector that the trans orthogonal to these waves so for each came mode. 402 00:39:30,140 --> 00:39:34,070 We have like a wave. It's Delta five and it's amplitude. 403 00:39:35,060 --> 00:39:39,200 So we have a wave for each came out, so it'll be easy to make out. 404 00:39:39,590 --> 00:39:47,720 And it's amplitude is delta force of K. And what we have to do is is draw it as draw from a random distribution, all these amplitudes. 405 00:39:49,880 --> 00:39:54,260 And for each game. Also, we have to sum up with each game what we draw around the season. 406 00:39:54,260 --> 00:40:00,050 We sum up all the game modes to get exactly what Delta Fi looks like in the end. 407 00:40:00,950 --> 00:40:05,990 All right, so what I'm showing here are the yeah, this is the x, this is x, y spatial directions. 408 00:40:06,000 --> 00:40:09,680 I haven't drawn z because then we wouldn't see the wave. 409 00:40:10,520 --> 00:40:18,320 Oh, yeah. But basically just summing up. So, yeah, summing up all these fluctuations will give us some picture that looks like this. 410 00:40:19,130 --> 00:40:25,370 Yeah. So this is called a Gaussian on the field. And this picture is exactly this. 411 00:40:27,210 --> 00:40:30,350 Okay. Even the colour scheme matches. Yeah. 412 00:40:31,230 --> 00:40:38,070 Yeah. So, yeah. So the statistics of the fluctuations that you get just by summing up all these delta files. 413 00:40:38,070 --> 00:40:43,170 Okay. Exactly. Match the statistics of the fluctuations that we get from the CMB. 414 00:40:44,150 --> 00:40:44,420 Okay. 415 00:40:44,660 --> 00:40:55,400 So so inflation doesn't just produce, you know, like a homogeneous CMB, but it can also produce the the 1%, kind of the five fluctuations that we see. 416 00:40:56,270 --> 00:40:59,390 And this actually is the only probe of the early universe that we have. 417 00:40:59,900 --> 00:41:03,740 So this can probe things as early as ten to the -30 6 seconds I can show you. 418 00:41:04,730 --> 00:41:09,080 So depending depending on what inflation happened, this this will probe things really, really early. 419 00:41:09,500 --> 00:41:13,310 But this is the only probe of the early universe that we have at the moment. 420 00:41:13,670 --> 00:41:19,310 And so studying these these fluctuations, we might be able to uncover more details about inflation or, 421 00:41:19,760 --> 00:41:24,470 you know, like when it happened, how it happened, which fields are responsible, etc. 422 00:41:27,150 --> 00:41:29,430 Okay. So I have two more minutes. So let me conclude. 423 00:41:31,320 --> 00:41:39,990 So we discussed the puzzle called the horizon problem that stems from observing a uniform CMB Despite predictions of the standard hot Big Bang model. 424 00:41:39,990 --> 00:41:45,690 So the predictions of the Big Bang models say that the CMB should be made up of 20,000 causally disconnected patches, 425 00:41:45,840 --> 00:41:54,120 and yet it looks uniform when we look at it. Okay, so something must give you the the observation is right, and so we must alter our model. 426 00:41:55,710 --> 00:42:01,230 We saw that inflation can actually remedy this issue by allowing these patches to come into causal contact in the early universe. 427 00:42:02,650 --> 00:42:07,660 Okay. So the horizon at the time of inflation is calculated slightly differently and gets very, 428 00:42:07,660 --> 00:42:13,810 very big and important contributions from the early universe, which are absent in the standard hot big bang model. 429 00:42:15,430 --> 00:42:18,580 And then as a bonus, if you study inflation in the context of quantum mechanics, 430 00:42:19,030 --> 00:42:25,150 you get the perturbations that we see in the sand below ten to the minus five level perturbations that we see in the CMB. 431 00:42:27,040 --> 00:42:31,090 So all this is well and good. And despite all the success of the theory, 432 00:42:31,780 --> 00:42:35,319 really pinning down the microscopic realisation of inflation is still a very 433 00:42:35,320 --> 00:42:40,770 important open problem with we don't exactly know which fields are responsible, 434 00:42:40,780 --> 00:42:44,470 whether it's one or multiple fields. Yeah, there are various possibilities there. 435 00:42:45,760 --> 00:42:49,510 And so it's this problem is important in both theory and observations. 436 00:42:51,390 --> 00:42:55,830 Oh, good. So that's all I have to say. Thank you so much for your attention.