They? Welcome to Hillary each morning of theoretical physics. 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. Shivaji Sundy joined us from Princeton. As the African professor next to The Bachelor. And he was able to bring something which isn't a Buddhist to a point which implies photos. Which we were able to advertise across the whole of theoretical physics.
And we've been lucky enough to end up with George. I'm trying to just go in and talk to you today. And so it's going to be a whistle stop tour through through modern theoretical physics. Nurse in the department. We have appointed a new faculty member, Dr. Edward Hardy, who did his dphil with John Wheater and is currently at Liverpool, will be joining us in September.
Ed is an astro particle physicist and so he's going to be working on things like the sun and the sky, looking at all the new astrophysical experiments which are due to give us information about about particle physics and also CERN in the basement of the Beecroft where quantum detectors are pushing the boundaries to when they can look at single elementary particles. So that's going to be exciting. New Frontier in Astroparticle Physics.
Other things to say if anybody wants something to do this afternoon. Topix is going on at the moment, so if you want to wander down to the river, rowing up and down the river. And indeed they posted on the same thing the next morning of theoretical physics would be the end the fifth week, which is the 27th of May, and that would be on Fusion.
So you can learn about the new advances in Fusion, in particular the net energy gain that was much advertised a few months ago, 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. 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. By then, the weather might be a bit better. Good.
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. And George is going to tell us about cosmic inflation and the very early universe. Okay. Thank you, Julia. Is this Mike working better now, or do I have to shout louder? 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. Uh, okay. So thank you, Julie, for the invitation.
And thank you all for coming. Good morning to you all. So today, as advertised, I'll be talking about inflation. And there has been a lot of talk about inflation lately, and we're going to talk just about that. And by that, of course, I mean cosmic inflation and. 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. So everything I say in the next few slides will be covered again in more detail throughout the talk.
But this overview is meant to be as a roadmap so that you can follow the flow of logic easily so through the presentation. All right. So inflation is a period of accelerated expansion in the very early universe. And the reason we're interested in inflation is because it can explain some puzzles in data. I will tell you what puzzles later on. 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.
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. So this is the universe today. And this is the very the very early universe. And as we go back, the universe was hotter and denser and smaller. And also as we go back, the universe behind before about 380,000 years in age, it was a peak. So light in this universe could not travel that far. Okay.
380,000 years, the universe became transparent. And from then onwards, we could do cosmology. So we could look very far at galaxies and we can see things back in the history of the universe. So the farther away we look, the farther back in time we see. But of course, we cannot see back in time, farther than this surface, because light doesn't go behind here. Okay. So this surface here is called the surface of the scattering.
And the light that we get from the surface is called the cosmic microwave background. Or simply for short. Okay. So this is the light that we can receive. This is the farthest light we can receive today and we live here and so like goes here and we catch that light and we're going to see that a puzzle on the surface actually teaches us about something really, really early that happened in the early universe, which is inflation.
So in this talk I'll be talking about the accelerated expansion that happens here. The universe is also accelerating today, but this is not relevant for for you know, for the talk today. So there's an acceleration, an accelerated phase here and an X and we believe also an accelerated phase here. What I'm talking about is the very early part, not today. Okay. So we can collect light that comes from the CMB and it looks pretty boring to zero. Or there it looks just like this.
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. 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. I have not had time to be in causal contact with each other, even in the age of the universe. All right. And so how could it be that they are in thermal and thermal equilibrium, then?
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? All right. So with this in mind, this is the outline of my talk. We'll start by learning a little bit of cosmology. So I'll tell you about expanding universes, the Friedman equations, and how we can solve for scale factors, how we describe expanding universes.
Then I will introduce a concept called the Particle Horizon, which tells us how far signals can travel in the history of the universe. And we'll calculate the particle horizon for our universe and the usual hot big bang picture. 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. Okay. So that have not been a cause of contact. And then.
But, you know, if you paid attention during the first few parts, then you would have been able to guess where the solution is going to be, which is inflation. And I will tell you that it's actually very easy to guess the solution. And then very briefly, I will say maybe one slide or two about quantum perturbations and how they generate the seeds for structure. And then I'll conclude. All right. So let's start by discussing the expanding universe. So special activity.
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. It's minus D squared, plus the x squared, plus d y squared, plus DC squared. And this is similar to the the Pythagoras theorem with a with a sign that has been flipped. 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.
The reason is because this space time doesn't expand in time. So if you put two galaxies at some fixed point, there will always be at the same distance at every time. So you put two galaxies initially at one distance. This universe is not going to expand, and the two galaxies will always remain at that same distance. What we want to do in cosmology is is describe an expanding universe. And so what we do is we introduce a scale factor, A of T, which is an increasing function of time.
Okay. So in this picture now, if we put two galaxies at two fixed coordinates, then at a later time that the the physical distance between the galaxies would have increased. So notice now the interpretation of X, Y and Z is actually slightly different from the interpretation here. 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.
Oh. So. 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. All right. So now once so we have introduced a and the name of the game is determining a. All right. So once we determine a, then we know a lot about our cosmology. Okay? We can we can calculate how far signals travel. We can calculate the expansion rate of the universe, everything.
All right, so how do we find a. We find a by using the Friedman equation. So the evolution of A is dictated by the contents of the universe. So whatever matter fills the universe, that will tell you how it evolves dynamically. All right. And the dynamical equations to solve is the Friedman equation, which is just this one. You can drive this from from general activity just by plugging in the metric that we saw on the previous slide.
But I won't do that here. And this is a simple differential equation. What what it tells you is that the first derivative of a saw a dot squared divided by a squared normalised by a squared is just proportional to the energy density in the universe. Okay. So all we have to do is measure the energy density and then we can solve for a. This quantity on the left is also called the Hubble Wraith and is denoted H. Okay. I won't talk much about the Hubble. I was talking about this a lot.
But if you've heard the Hubble race in the in popular literature, then this is what they are talking about. It's a lot of a. All right. 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. 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. And we will see why in the next slide.
This is because, you know, certain types of matter that we're very familiar with, they have very nice scaling with with a. So let's do an example. Let's consider a box with side length equal to one coordinate unit. Okay. And let's start at time zero where the scale factor has value is zero. So then the physical length of the side is just a zero. And of the energy density at this time was rose zero then the energy density at a different time. Is this related to zero by this very simple quantity?
So if at a later time the scale factor is just a not a zero, then all you have to do is multiply by a zero over a cubed to get the energy density here. Okay. There's this because we went through the motions and, you know, the the number density decreases. So this is a story for matter. If you have radiation like photons, it's a little bit I mean, a little bit more complicated in the sense that not just the number density decreases, but also the photons get stretched.
So there are some redshift. Right. So we don't only multiply by a factor of zero over a cubed, but we multiply by another factor of A0 over a and this is to take into account the redshift. Okay. So the photons have lower energy here at when the scale factor is bigger. Good. So but we still end up with some very simple expression, which is zero zero times one over A to the four. We had 1 to 3 on the previous slide. So this is most of the matter that we are familiar with.
And and but in cosmology, there is one more type of, you know, material or substance or it's an energy density that we don't actually we're not familiar with in everyday life. 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. 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.
Just whenever you have space, there is an energy density in it. And this is called dark energy. So if you imagine a box that is filled with dark energy, whatever that means, at some later time, at some later time, then this box will just have the same energy density. It will not dilute even if you stretch the box. Good. So these are the three the three main forms of energy density that we have and we can summarise the behaviour of the three energy densities by writing this.
So row is just row zero times a zero over a two, the power three one plus W where W takes on different values depending on on the type of energy density we're talking about. So we're talking about matter. It's zero for talking about radiation as a third. If we're talking about dark energy, it's minus one and that's it. 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.
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. So our universe looks more like this. Okay. Good. So let's do it, for example, just to get, you know, our hands a bit dirty, working out, working things out. So. We have found an expression for Roffey on the previous slide and a few slides ago I showed you the Friedman equation, which was a differential equation for a as long as we know of a.
All right. So here I've copied the Friedman equation again, but taken a square root, and I have plugged in the expression for off a that we found on the previous slide. Now this is a differential equation for a and it's separable. It's easy to solve. You can just move to that side and take an integral to it. And so you end up with some equation like this. So I will leave it up to you to do this integral. And then. And then. And then separate and then move all A's to the left, all to the right.
And what you end up with is something like this. Okay. So we find out that the scale factory has some distinct behaviour, whether the universe contains matter radiation or dark energy and for matter it goes like t to the two thirds. 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?
Okay. So this is some, some behaviour and now in our universe we can't actually solve, you know, solve the equation analytically because we have more components, but we can do it numerically and this is what the answer looks like. 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. And on this floor, I have also overlaid, you know, two curves that we saw last time.
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. And we see that our universe in the past looks like it's a radiation universe. At some point it becomes the mass of the universe and then it starts deviating. You know, it's very hard to see, but it starts deviating just today from being a massive universe. Okay. Um. So is what I said so in the past.
So I will go back to of the one half and then at some later time this will go like Pisa two thirds. And what happens is that there's also an acceleration. There's also today the universe begins accelerating just very close to today. 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. So I'm going to emphasise that this is responsible for late time acceleration.
This is different from inflation. So the talk is not about this dark energy. It's about the dark energy that is, you know, in the very, very early universe. Good. So now that we can compute a let's actually find how far signals can travel in the universe. All right. So this is this concept is the particle horizon. So remember, what we're doing is we're going to compute the particle horizon and we're going to ask, is this C?
And because we connected so our points on the good points of the C and we have communicated in the past. Okay. And that's why we're heading this way. So we are finding the particle horizon to ask that question. And the political horizon is the largest distance that a light signal could have travelled in the history of the universe. So ever since the universe began. What's the largest distance that a photon can travel?
And this is easy to calculate. Is this an integral, you know, of recording a distance? So we have to find first the coordinate distance that the photon travels and then we multiply by by t, which is the scale factor to get the physical distance. Okay. So the horizon, the particle horizon at some time, t f is just a scale factor at the time, t f multiplied by the coordinate distance that the photon would have travelled by time.
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. So in a short time the photon travels a distance C times delta T in the physical space, but in coordinates space travel. C. D. T. Divided by a. And so this is the integral that we have to do and then multiplied by a. Okay. Oh. Good. 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.
So we get an a dot in the denominator. 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. So, so this whole integral is actually an integral only over a and you can easily write it from the Friedman equation. Just substitute here for a dot. Oh, good. 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.
And now notice the quantity we're integrating is actually one over a dot. And what's special about one of that is that for all normal substances, one over a dot is always increasing. Okay. So we're going to see this later. But yeah, for normal substances, this quantity is increasing. The integrated is increasing. And so what happens is that this horizon distance gets its most important contributions from late times.
If one over a dot is always increasing, then you can pretty much ignore the early time tail and just integrate over late times. And that will give you a very good approximation to the to the to the particle horizon. 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. And so we can compute exactly what this is in the late universe. Right. And there is no wiggle room there.
So if this gets its contributions only from late times, then we know exactly what the is that we cannot you know, nobody can change that because we can measure the universe and there's data and that we cannot argue with, oh, good. So let's do this integral. So remember, this is the horizon distance. And so the quantity we're integrating is I'm showing here in the red line. So here on the x axis, I'm showing a the value of the scale factor.
And on the y axis I'm just showing distance and megaparsec there's also a time axis up here, if you like, thinking in terms of time instead of a but they're the same thing. Okay. So the red line here is this is this a lot inverse? 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? Okay. And again because I thought versus increasing.
Up until maybe today where it starts decreasing. Uh, so this quantity, the capital DFA will get most of its contributions from lead times so we can even not worry about what happens earlier here and just use the late time cosmology that we know. Okay. Oh. So every so all the rage that we know here that I'm showing you here, we have actually measured very well. We have actually measured before this up to like ten to the minus ten or something. Okay. And there's no wiggle room here.
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. There's one more line, one vertical line that I'm showing here, which is the surface of last scattering. So a lot of scattering happens at a time, which is where the universe was about 380,000 years old. Like like we saw, you know, on the second or third slide. And this corresponds to a scale factor of about ten to the minus three.
Okay. So the last gathering happens is an event. It happens here. And from this plot, we can actually read what the horizon was at last scattering. Okay. So we can just know this value and this value ends up being point three megaparsec. Okay. So pictorially, this is what we have. We have a service of the scattering and the horizon size. At the time of our scattering is point three megaparsec. Now our distance to the last scattering surface.
You can also do by performing an integral intervals similar to the one that I showed you. But we won't do that here. And the distance is 13 megaparsec so you can compute the angle. That's one horizon, you know, that's surrounded by one horizon on the scattering surface. 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.
And now if you want to find the number of positive disconnected batches in the C and B, all you have to do is divide the full solid angle by this number squared and you get about 24,000 possibly disconnected touches. Okay. So the CMB is actually, even though it looks uniform, it's made up of 24,000 little causally disconnected touches. They would have been causally disconnected patches in the Big Bang model. So this is the Big Bang model.
This is a very, very big puzzle as to why, you know, 20,000 positive, disconnected touches seem like they are in equilibrium. Now, let me. So let me tell you a bit more about this, using a football analogy. So let's say so let's say I go to London, to the stadium, to to to to watch a football match. Okay. And if I go there and then if I find Andy and Francesco, then I would be a little bit surprised, you know? I'll ask them, why did you guys not invite me to come see this match with you?
And they'll be like, No, no, it's a coincidence. We actually did not planned this. 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. And so that's fine. And, you know, we'll just sit there, watch the match and enjoy it. But if you've actually had a little bit more funds, then the situation could have been slightly different.
Which is to say, I could have went to the stadium and the the state of the stadium could have been something like this, where we have 20,000 level umpires, fellows, and they all fill all the seats. 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? 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.
How come, you know, you all happened to be at the same time, at the same place? This was definitely planned. There's, you know, a group chat or something on which I'm not. And so, you know, I just won't believe them. I will be like, no matter how much you promise, I just won't leave you guys. And this is the same story with the CMB. Okay, so the CMB has 20,000, you know, causally disconnected patches in the in the, you know, in the big bang picture.
And yet they all choose to go to the same temperature. And how come that that's just not something believable? Okay, good. So so to reiterate, the CMB should have different temperatures and, you know, in each of these circles. 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. The horizon cannot be the small, because otherwise how would you know? How would the CMB have had the same temperature?
So there has to have been communication between all these, uh, you know, just like I suspect my friends are communicating behind my back. This, these, these are also, these patches should also be communicating with each other. 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. In other words, we need to make the horizon a lot, lot bigger. Okay.
But for those paying attention, this is really hard because the horizon and the big picture. Sorry. This is really hard because the horizon and the big picture receives this contribution from the late universe and the late universe we have measured very well. So you cannot change that. There is nothing to do. Right. And the reason it receives this contribution from the late universe is because adult and verse is an increasing function of time.
And so if you do this integral, it will be dominated by late times, and then you can ignore early times. 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. But we can change that picture. We can change that picture at early times where we have no data yet. I mean, we have some some some some some indirect probes.
But if we are able to change the picture at early times, so that adult inverse is actually decreasing, so that day receives a large contribution from very early times, then we might be able to increase the size of the horizon. Okay. So this slide is what I just said the words. So the. Yeah. All right. So, so, so. And here. So, yeah. Like I was telling you, this is about one second after the big bang. We know pretty much everything up to about one second after the Big Bang, you know, after that.
So data here is very constraining. But before here, we only have some indirect probes of which I will show you one. One of them is as the perturbations in the CMB, and I will talk more about that later. But these are all indirect probes. We never actually we don't actually see things here directly. Okay. Um. Good. All right. So because these are here is not as constraining. We can play with the functionality initially.
So what we want is we want a dot inverse to decrease, which means one over eight dot decreases, which means a dot increases, which means the universe has to be accelerating. So that's why inflation is an accelerating period, because it has to be an increasing function of time. Okay. So a double dot is positive. And in other words, like the acceleration is positive. So what we do is that we can just change what it looks like, you know, at earlier times by.
Yeah. Either by hand or by posturing. Some different energy density like dark energy at early times. But that's again, independent of this here. Um, yeah. 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. Right? And the very, very early universe at around ten to the -30 6 seconds. 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.
So this is really fast acceleration. Yeah. And you can see that. 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. And the scale factor increases by a factor of either 100. Okay. So this is one very particular model that, you know, that we have picked. You can change. You can play with these parameters. You can move this period here.
You can move it here. You can decrease the, uh, you know, the e to the 100th factor. And all these things are for free parameters. There are some constraints on them, but, but not, not that many. Oh, good. And so with this fear of this inflation, what does the horizon look like then? Okay, so we can compute the same plot again. So this is. This is a dot inverse of the usual hot big bang. And this is the horizon of the usual hot big bang.
I thought inverse inflation is going to decrease. So this is what the inverse in inflation looks like, this red line. And then it catches on with this one and carries on. It doesn't do anything after. That's different. But now, if you can view the integral, you will get this green curve, because most of the contribution will come from early times, not from late times. Right. So the area under this curve is dominated now by early times, not by late times.
Okay. So intuitively what's happening is that inflation is an expansion that is, you know, very, very fast. And so what's happening is that two points that are very far away. 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. So if you try to extrapolate back, these two points will still be far away and the in the big picture.
But if you have a period of expansion that's really exponential, that's accelerating and and just like inflation, then these two points, even though they're far away now, they could have been really close in the past. And if they are really close in the past, then they could have talked in the past. Okay. 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.
So what we see here, the important part is that it has to be accelerating, right? So it has to be, uh, yeah, that inverse has to be decreasing, which means a double dot is positive. Um. Okay. And in particular, there's one thing to point out here that this here is the surface of the scattering. So this here is the line where we have no data beforehand. This here is the surface of the scattering.
And remember, the horizon was here was the intersection of the blue line with the with the green line. And it was about 0.3 megaparsec. Now the horizon is each of the hundredth bigger than that because we have this factor of, of, of of feature 100 and expansion. So the horizon here is going to be easily 100 times bigger than that. And that's big enough to actually include all of the whole the the whole surface of of of of less scattering.
And so there is no no disconnected patches on the surface of scattering. If we have something like inflation. Okay. So so I think a period of accelerated expansion. Oh, sorry. Typo. So I didn't hear the accelerated expansion, which is inflation. In the early universe, we see that the horizon can be much bigger than what the Big Bang predicts, and that's because it gets most of most of its contributions from the early times.
Okay. So, yeah, equivalently, points could have been much closer than what the Big Bang predicts. Okay. But that is that's actually not the only thing that inflation can do. One other thing that inflation can do, which is that it can explain these perturbations. Okay. So it can explain the pattern of perturbations that we see on the cosmic microwave background. 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.
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. So these little tiny mountains and valleys are ten to the minus five times 2.7. Right. And we wouldn't know, for example, where. 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.
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. And in order to see that, let's review first the physics of the quantum simple harmonic oscillator. 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.
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. It's the blue line. And what we imagine. We imagine a particle that sits in this potential and we ask, what is the wave function of that particle? Okay, we can solve the Schrödinger equation and then we find the wave function of that particle. And if you remember, the wave function of the particle in the ground state is a Gaussian.
Okay. 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. That is the square of the wave function, which is discussion. 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. This is one manifestation of the uncertainty principle where the particle cannot exactly be localised at x equals zero.
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. So it's just a quantum mechanical process. So while the expectation value of the variable X which denotes the position of the particle is zero, its variance is not so different. Measurements will give different values of x and typically if you do a measurement, you won't get zero.
So the set zero here, getting zero is, oh yeah, because the point is as measure zero. And so you probably won't get zero if you do a number of random measurements. All right. So typical values of the position are actually not zero, even though the expectation values is zero. Now an inflation. Something similar happens. Okay. And the simplest models of inflation. We need some energy density that will drive this accelerated expansion.
Okay. And this energy density is provided by some scalar field usually. So there's a scalar field which has a potential. So it has it has an energy. And the energy density to drive a solid expansion is provided just by the scalar field. During inflation, the scalar field will have some background value, some average value, which I'm calling fibre. That depends only on time. Okay. So it can only change as a function of time, but not in space.
But in addition to this, there could be a perturbation that depends on both space and time. Okay. So this perturbation is usually normalised with this factor of a but the fact of fate is not so important for the story. The important part is the Delta five, which depends on both space and time. So just like the original picture that I showed you of the expanding universe, space during inflation is actually flat.
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. So we can take, you know, a four year transform on the variable X and we'll get Delta Phi of T and K. So we want for you transform the T variable only the X variable. And now we can talk about each freedom or delta fi k separately, and that is only a function of time. All right. Much like the position of the simple harmonic oscillator as a function of time.
Delta Facebook is only a function of time. But the analogy goes even deeper. It obeys the same equations of motion. It obeys an equation of motion. That is an equation of motion of the simple harmonic oscillator. So each mode of a is a, you know, an equation of motion that is very similar to the simple harmonic oscillator. The difference is that this depends on time, but that's yeah, that's a technicality. So the equation is very similar. Oh, good.
And so when we consider the system quantum mechanically. Then the physics of the system will be the same as the physics of the simple harmonic oscillator in quantum mechanics. Okay, so the expectation value of Delta five will be zero, but its variance would not be zero. Okay. And if it's various would not be zero. 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. Okay. And. Yeah, so. So I remind you.
So what does this mean? This means that. So yeah. Remember that Delta Phi K is the amplitude of the k wave mode. Okay. So this means that different k wave modes have a have an amplitude that is drawn from a random distribution and that random distribution is a Gaussian, just like the simple harmonic oscillator. Okay. 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.
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 there's a vector that the trans orthogonal to these waves so for each came mode. We have like a wave. It's Delta five and it's amplitude. So we have a wave for each came out, so it'll be easy to make out. 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.
And for each game. Also, we have to sum up with each game what we draw around the season. We sum up all the game modes to get exactly what Delta Fi looks like in the end. All right, so what I'm showing here are the yeah, this is the x, this is x, y spatial directions. I haven't drawn z because then we wouldn't see the wave. Oh, yeah. But basically just summing up. So, yeah, summing up all these fluctuations will give us some picture that looks like this.
Yeah. So this is called a Gaussian on the field. And this picture is exactly this. Okay. Even the colour scheme matches. Yeah. Yeah. So, yeah. So the statistics of the fluctuations that you get just by summing up all these delta files. Okay. Exactly. Match the statistics of the fluctuations that we get from the CMB. Okay. 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.
And this actually is the only probe of the early universe that we have. So this can probe things as early as ten to the -30 6 seconds I can show you. So depending depending on what inflation happened, this this will probe things really, really early. But this is the only probe of the early universe that we have at the moment.
And so studying these these fluctuations, we might be able to uncover more details about inflation or, you know, like when it happened, how it happened, which fields are responsible, etc. Okay. So I have two more minutes. So let me conclude. 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.
So the predictions of the Big Bang models say that the CMB should be made up of 20,000 causally disconnected patches, 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. We saw that inflation can actually remedy this issue by allowing these patches to come into causal contact in the early universe.
Okay. So the horizon at the time of inflation is calculated slightly differently and gets very, very big and important contributions from the early universe, which are absent in the standard hot big bang model. And then as a bonus, if you study inflation in the context of quantum mechanics, you get the perturbations that we see in the sand below ten to the minus five level perturbations that we see in the CMB.
So all this is well and good. And despite all the success of the theory, really pinning down the microscopic realisation of inflation is still a very important open problem with we don't exactly know which fields are responsible, whether it's one or multiple fields. Yeah, there are various possibilities there. And so it's this problem is important in both theory and observations. Oh, good. So that's all I have to say. Thank you so much for your attention.