 So now we are going to start the first lecture on CMB by Blake Sherby from Cambridge. Alright, great. So thanks very much for the opportunity to speak here. So I'm really excited to be back. I was sitting in the audience for one of these cosmology schools. I don't know when it was 40 years ago or whenever I was a student. And I really enjoyed it, so I'm really excited to be back. No, I'm kidding. Not that old. But it took a disturbingly long amount of time for people to realize that that's magic. Right, so I'm going to start this off by talking about the cosmic microwave background. And let me start by giving just a brief motivation for why you should pay attention during these lectures and why you should want to learn about the physics of the cosmic microwave background. So there's a huge list of regions why it's important to know about the CMB and I'll maybe just sketch some of those. So in short, the CMB is our most powerful arguably and certainly our most robust probe of really interesting cosmology and really interesting physics. So the CMB is a beautiful probe of the history of the universe. It really shows you effectively the baby picture of the fluctuations in density and potential as they were early on and you know from these sorts of fluctuations all of the structure that we see around us today grew. So if you want to understand the history of the universe, the CMB is crucial. It's an image of the way the universe looked very, very early on, right at the beginning. The CMB is also arguably the best way of figuring out what the universe is made of. So I'm sure you all know that 25% of the universe is dark matter, 70% is dark energy and 5% is baryonic matter and these high precision numbers to a large extent come from the CMB. So it's an amazing tool for figuring out what the universe is made of. It's also almost certainly currently our most powerful tool for understanding the physics of the very early universe and testing models such as inflation. And finally, the CMB is not just a great tool, a robust tool to understand cosmology, but it's also just a very, it's an amazing probe of fundamental physics. We've already talked about how it can probe the early universe and as we'll talk about when we discuss B-modes, you can access scales that would be impossible to probe with terrestrial experiments, probing inflationary scales that are a trillion times higher than the LHC or particles that are so weakly interacting that you could never see them on a realistic experiment on Earth. So in short, the CMB is an amazing and robust probe of both cosmology and fundamental physics and you should all learn how it works. So I will try to just start this topic in these few lectures that I have here. So given that the audience is quite mixed, I should apologize to those of you who are real experts in the field of CMB because I'm not going to assume that you've taken a course on CMB before and especially the first few lectures will be very introductory. Especially this first one. But hopefully even if you do work on CMB towards the end we'll cover some more advanced topics such as gravitational lensing of the CMB and CMB polarization so hopefully those will also be interesting. And I should note please feel free to just interrupt and ask questions. I might not be able to answer them all but it's more fun that way. Alright, so what are we going to discuss in this course? Today I'll talk just about the very basics. What is the background CMB? Why is it amazing evidence that the hot big bang happened and how do CMB photons propagate in our universe? And then in later lectures I'll talk about the CMB power spectrum, what it means, how we can use it to learn about cosmology and later on I'll discuss some more advanced topics. And I've drawn from a very nice list of references that I've noted here and hopefully I'll make these slides available to you and if you want to know more you can look at Wayne Hu's amazing website, Anthony Chaliner's lecture notes if you want to know a huge amount of detail and Daniel Baumann's new book has a nice summary of CMB physics as well. Alright, let's get started today with the very basics. What is the CMB and why is it a clear and unavoidable prediction that sort of proved that the hot big bang is indeed the beginning of our universe. So I'm sure you all know basic cosmology, you all know that we've observed an enormous number of galaxies and you all know also that Hubble discovered that these galaxies are all running away from us and the further away these galaxies are the faster they seem to be moving away from us. And that's not because we're unusually repulsive hopefully but is instead interpreted today as the fact that space itself is expanding the more space that we have between two objects the more it expands and then the faster they seem to move away. Now, quantitatively we can describe this observation as explained fully by assuming that the universe has the most general homogeneous and isotropic metric, the FLRW metric that I'm sure you're all familiar with here. And the distances here in this FLRW metric are rescaled by this scale factor A of t as I'm sure you all know and I will often be using conformal time eta because then I can draw nice space time diagrams where light rays travel at 45 degrees, right? So dt is AD eta, that's the definition of the conformal time. Okay, now just reminding you further of some basic cosmology I'm sure you all know this, you can plug this metric into the Einstein equations and connect the geometry of the universe with the matter and energy content of the universe and that gives you this connection between the scale factor A and what's in the universe. The density parameter is omega lambda, omega matter, and omega radiation. Okay, and so that's a differential equation that tells you how quickly the universe is expanding as a function of what's in it, right? And so if you're just to remind you, the scale factor is growing as t to the one-half during radiation domination, t to the two-thirds during matter domination, and later exponentially. And as I said, we'll often use conformal time and I'll, when I don't say anything, I'm going to be assuming flat lambda CDM and I'm going to set, I use natural units for c equals 1. In any case, what we see is that the scale factor grows with time, all distances grow with time. It's a clear prediction of the standard hot big bang model with an FLRW metric that at early times, if we run that picture in reverse, the scale factor has to have been very small and every, you know, all distances were really small and everything must have been really close together. The universe must be extremely dense at early times. Now, you can easily show, and we will actually show this later, that the temperature of a photon is proportional to the energy to a photon is proportional to one over the scale factor. So at early times, when the scale factor was really small, therefore the temperature was huge. So the early universe was incredibly dense and incredibly hot and that's crucial for the emission of the CMB. So just the standard hot big bang picture predicts clearly a extremely hot and extremely dense universe and in fact, if I go to early enough times, the universe has to be made up of a plasma, an incredibly dense and hot plasma. So at the beginning of the universe, that's what you have in this standard hot big bang picture. What do I mean by that? Let's go into a little bit more details. So just a few hundred thousand years after the big bang, very close to the beginning, the universe is so hot that you have, you know, that atoms are not bound and I have a plasma made up of protons, electrons and photons. So the electrons and protons are not bound and everything is just flying around. Now the photons rapidly scatter and they rapidly scatter in particular off of the electrons. Does anyone know why they mainly scatter off the electrons and not the protons? That's right. So if you look at the Thompson cross-section, it was a factor of 1 over mass squared and so the scattering with protons will be suppressed quite heavily. So the photons mainly scatter off of the electrons but the protons obviously are affected by this as well because through sort of Coulomb interactions they scatter with the electrons in turn. So that state persists for some time. Early universe is this glowing hot plasma of photons, electrons and protons. I'm neglecting some helium, which is a small number of... But then what happens is the universe, of course, expands and it cools off and eventually a terribly named process called recombination happens which is that the universe becomes so cool that electrons and protons can come together and form hydrogen. Now, because the photons were scattering so rapidly off the free electrons in the early universe, the early universe was basically opaque. You couldn't see through it, the mean free path was really short. But now after recombination, I said it's a terrible name because this is the first time protons and electrons combine to form hydrogen. After recombination, the universe is neutral, the scattering is much, much, much weaker, maybe you have a tiny bit of Rayleigh scattering but you can neglect that. And the photons will just free stream through the universe all the way until they reach our telescopes. Okay? So that's sort of the basic picture. Again, to summarize this, hydrogen forms very quickly at a temperature of around 0.3 electron volts from free electrons and protons and before recombination, you have lots of Tonson scattering, the mean free path is short, and then the universe is neutral, photons free stream to us. All right? Just to show you a really terrible illustration of this process, you can see these photons in blue bouncing off the electrons in green and the protons are doing very little. But then the universe cools, neutral, the hydrogen forms, and then the photons travel through the whole picture where we can fly along with the photons. So you all know that you cannot do this, right? But the photons then travel through the universe to us. So I like how, you know, when people try to make PR images, they sort of tell you as many wrong things as correct things. Okay, are there any questions about the very basic picture? And then we'll get a little bit more quantitative. Yeah? I think it's made by the sort of Planck PR team, so you should just look at, like, Planck PR videos or something. I've cut out some other funny things that the photon does in the middle, which are also not correct. Yeah. Okay, so let's now get a little bit more quantitative in terms of when recombination happens and when the photons get released and stream through the universe, okay? So you can do a pretty good job by just assuming that this process of electrons and protons forming a neutral hydrogen is in thermal equilibrium. That's not going to give you an exact answer to get an exact answer, so you have to use the Boltzmann equation for this process and, you know, consider the details where you have a three-level atom and it's actually quite complicated to get this exactly right. But you get a good answer if you just assume everything is in equilibrium and you probably know that in equilibrium, in thermal equilibrium, the equilibrium number densities of the electrons, the protons, and the hydrogen. Okay? And so I will have you do this as an exercise, but if you write down those expressions for the number density and equilibrium, those just depend on a sort of Boltzmann factor which contains the mass and the chemical potential over the temperature. And if you want to know how ionized the universe is, in other words, how many free electrons there are versus electrons that are bound into neutral hydrogen, you can write down the free electron fraction N e over N b, or N e over N p plus N h. Okay? So when that free electron fraction is one, everything is a plasma. Everything is fully ionized and when it's zero, then all the electrons are in hydrogen and the universe is neutral. Okay? So as you will show, you can sort of plug in these expressions for assuming thermal equilibrium and you can rearrange to get a nice equation for the free electron fraction. So you know how ionized and how versus how neutral the universe is as a function of a very small number of quantities. That's the amazing thing about thermodynamics when things are in thermal equilibrium, everything just pops out from nothing. The first one is the temperature and the binding energy, mass of the electron, and this factor eta, which is the barion-to-photon ratio. How many barions do you have per photon? Okay, and then what you can do is you can solve this. You can say, when does the universe recombine? When does this ionization fraction become really low? Okay? The binding energy of hydrogen is 13.6EV. You know everything else. We know the barion-to-photon ratio is 10 to the minus 9. And we can just plug in and solve and naively you would think, okay, the binding energy is 13.6EV. Recombination has to happen roughly at a temperature of 13.6EV. But instead it gives the temperature of 0.3EV. So does anyone know why? It's so much lower than the naive answer. Exactly. So there's so many photons that even those rare photons that are way in the tail of the Planck distribution, rare rare exceptionally high energy photons are enough to ionize hydrogen and keep it ionized until the number density drops way way way below 13.6EV. So all these photons push that reaction to the left. Now that's just when, that was a calculation of when the universe becomes neutral and you've seen that we've done the SAHA equation calculation in the dotted line, but a proper calculation agrees pretty well, at least for the onset of recombination with this very approximate thermal equilibrium expression. Now what we're interested though is when the photons get released because that's sort of the cosmic microwave background as we'll talk about in a second that happens when the Thompson scattering of these photons off the free electrons becomes inefficient. And as you probably know you can compute when a process when a sort of scattering process becomes inefficient and when this particle freezes out by setting the rate of interaction to be equal to the Hubble rate. Because when that rate falls below the Hubble rate then I don't even have one interaction per Hubble time this process has basically stopped. So if we write that down for the photons scattering off electrons what we find is that because of this exponential cutoff basically decoupling happens right after recombination. So the electron density drops so quickly recombination happens so fast that decoupling happens very soon afterwards. And that occurs and if you'd like I can walk you through the calculation at redshift 1100. So that is basically when these C and B photons no longer scatter because all the electrons are bound into hydrogen and the photons can escape. One key point I want to emphasize is that this process is set by the local temperature this is all just effectively a thermodynamic argument we get 0.3EV just from the binding energy of hydrogen and some sort of basic quantities like this. So at a fixed temperature of 0.3EV recombination happens and then photons are released. So as an exercise if you haven't done this already and I suspect many of you will have please just go through and derive the SAHA equation it's worth your while. And then I have a quick question for those of you who thought a little bit about C and B so you always hear that C and B temperature anisotropies we measure in the sky they probe the fluctuations and the radiation density that were there primordially. How can I square this with the fact that photons are emitted at a fixed local C and B temperature independent of the radiation density. So it's always emitted at 0.3EV. So how can I see the radiation density how can I see the fluctuations in the radiation density in the C and B. Does anyone know? That could be the answer but it's not. There's some physics there also, yeah? Yeah, so you're right that there is and we will talk about this there is an effect where it climbs out of the potential and that changes the that changes the the temperature but that's not the only effect something else. So even if the potentials were 0 you would see the variations in radiation density even though the local temperature of emission is the same which is a very good approximation. Alright, we'll think about it and we'll come back to that later. Alright, I'm glad. Okay, so in any case the key point is there's a clear prediction of the hot big bang model and this is extremely, maybe not of importance now but historically this was a huge debate, right? Is the hot big bang cosmology correct or do you have some steady state model and this is the proof. If you have the hot big bang model you have a clear prediction that the afterglow radiation from this primordial plasma should be released and should travel through the universe to our telescopes, okay? So we should see this radiation every time we look out in the night sky we should go out and we should look in the sky at a burning sea of fire because that's radiation from the sort of big bang plasma. Okay, so if I look in the sky I should see the glow of the primordial plasma because the light has to be released. So this is an obvious question but why, hopefully when you go out at night you don't see a burning sea of fire in the sky, why do I not see that? Obvious question. That's right. So we talked about how the energy of the photon is proportional to 1 over A. The wavelength stretches from redshift 1100 to today by a factor of a thousand and when the CME is emitted it looked very much like the surface of the sun, right? That was a similar kind of temperature and similar black body spectrum but that gets now redshifted that gets redshifted to microwave frequencies. Okay, so you need to build specialized microwave telescopes with lots of effort to be able to see this cosmic microwave background or you can just get lucky like these people did. That's completely unfair. They did an amazing job. I'm sure you've all heard these stories of Penzias and Wilson, how they built this telescope to do astronomy and test communication equipment and you've all heard the story of how just by being really careful experimenters they, you know, they trapped this down. They found an excess signal that eventually they concluded has to be real, right? And then the Princeton group helped them to interpret this as the cosmic microwave background, the prediction of the hot Big Bang. So I'm sure you've all heard these stories where they tried everything they could possibly try and at some point they even climbed in and removed the pigeon droppings from their experiment because they couldn't explain this mysterious, you know, thermal radiation and that didn't get rid of it and eventually they were convinced this is real. So I always heard the story but at some point I came across what I think is kind of a dark side to this story and it probably shows just how upset they were getting if their telescope not working which is they didn't just clean the pigeon droppings one of them went out and shot all the pigeons which I think is just kind of illustrates just how frustrated they must have been getting if their telescope not working. In any case, this was the proof that indeed the hot Big Bang happened. We can see its radiation. We can see its afterglow and that killed these steady-state models almost entirely. So that is, well, I should say though that obviously it's not enough to just say oh I found some radiation you need to prove it is cosmic microwave background radiation and one really good way of doing this is showing that it is a black body a nearly perfect black body because that's what you'd expect from a glowing plasma. So if you measure this that is then the definitive proof that this is radiation coming from the hot Big Bang and I'm sure you're all familiar with the Kobe fire ass measurement one of the most amazing experiments in cosmology where the error bar has to be magnified by a factor of around 200 so that you can even see them on this plot. So this Kobe fire ass experiment showed that the CMB is actually I think to my knowledge the most perfect black body that you can find in nature. Now if people know something else you can feel free to correct me on that and you know you can ask how can they measure it the most perfect black body so it's kind of an interesting experimental question but unless people are interested I won't go into that. In any case that was the confirmation that this is the CMB we found this background radiation. Are there any questions about what the CMB is and the sort of background properties of the CMB before I move on to the fluctuations? So our universe today is not perfectly uniform but the CMB is not almost perfectly uniform. And for some time experimenters struggled to find variations in the CMB. In fact they had already ruled out the levels that we find today due to some errors. So you can increase the contrast by a factor of a thousand maybe you see a dipole but basically you cannot find fluctuations in the CMB temperature until you increase this contrast but then you see them. Then you see these beautiful measurements made by the COBE DMR experiment and this is sort of the first image of the almost primordial fluctuations that we can see in the CMB. And this is again showing the full sky unfurled into this lozzinge. This is a map as a function of direction of the sky so some directions the CMB is a little bit brighter has a higher temperature and some directions it's a little less bright has a lower temperature. And so this is a really really amazing observation and very much it's one of the better noble prizes I think. Okay it is, isn't it? I don't know. There's some other ones that are not so cool. Okay but of course luckily we didn't stop at COBE. There is a question in the chat. Okay. So it's asking if the universe were still not expanding would we expect to have received a different picture of the CMB? Well, right. So if it weren't expanding follow and if the universe keeps expanding in larger scale times will we have different picture of the CMB? Yeah. So if the universe was not expanding in other words if the scale factor wasn't increasing with time there's no reason let's say it was static then there's no reason to think the past was dramatically different from today. You wouldn't expect it to be at a temperature of many thousands of degrees Kelvin when it's at a temperature of 2.7 Kelvin today. So indeed you would not expect this. You only expect that if the universe is evolving and has a very hot early phase because it's so dense and small. And then the other question was is the CMB evolving? And the answer is yes. So it's as the scale factor continues to increase the CMB temperature will continue to fall and in fact we're kind of lucky because if we lived I don't know how many hundreds of billions of years in the future dark energy will have blown up the universe a lot and the energy density will be a lot lower so it would be harder to observe have a much lower temperature and the CMB patterns will also change with time but on time scales of billions of years that depends on scales. Anyway someone made a video how the CMB will evolve if you want to see that and try to dig that out. In any case, okay so after this Koby discovery we didn't stop fortunately but we built a successive series of better and better experiments going from Koby to WMAP to Planck. Okay and the current state of the art along with ground based CMB experiments that I work on that I won't talk about quite as much this is our current best image of the Cosmic Microwave background fluctuations the variations in the brightness or the temperature across the sky which I'll call theta the fractional difference in the CMB temperature is a function of where I looked and obviously orange is showing you regions that are a little bit higher in temperature and blue regions directions in which the CMB is a little bit colder and you know this is I think an amazing amazing image of the universe as it looked 400,000 years after the Big Bang okay so I'm going to start discussing a little bit what exactly it is we're looking at and I'll finish that at the end of this lecture okay so what are we looking at exactly right so just in terms of the background now you probably are familiar with the concept of optical depth which is an integral over any sigma t it sort of tells you the probability of scattering and in particular e to the minus tau is the probability that a photon has not scattered from a time eta to today okay so if e to the minus tau is the probability of a photon not scattering we can also use that to derive the probability of a photon last scattering at a certain time and with a little bit of thought you can convince yourself that that's basically the derivative of e to the minus tau and therefore is given by this quantity minus tau dot e to the minus tau which is called the visibility function that's the probability that a photon last scattered at a time eta now here on the bottom is a plot of the visibility function versus time or versus temperature and you can see on the bottom is the visibility function what's the probability of a photon last scattering at a certain time or redshift and on the top you can see the ionization fraction that we talked about with the sa-ha equation and you see there's a big step right a big step here and that corresponds to recombination and decoupling and so there's a big derivative in tau there and that explains why I have this spike in the visibility function so from thinking about a little bit you conclude that the most of the last scattering happens in a very narrow range of redshifts and times so basically photons last scatter right after recombination, right near decoupling okay there's a big spike and usually we assume that to be approximate treatment of the CMB you can assume this is basically a delta function right it's a very sharp peak in the visibility function so it's called the visibility function because generally where photons last scatter that's sort of how far you can see into a medium you can see maybe one optical depth okay so what that means is that the when we're looking at the CMB we're effectively seeing a surface right we're seeing a surface in both time and space 14 billion years in the past and I think it's worth just trying to visualize this properly because it'll come in handy later when we're talking about CMB fluctuation okay so just make sure you have a picture of what exactly it is we're seeing so here's sort of a space time diagram but I've suppressed one spatial dimension instead I'm plotting time okay so these are equal time slices through the universe and I'm using conformal time so photons travel at 45 degrees so this is early on the universe is in opaque plasma photons are rapidly scattering and then the universe cools and cools and cools and at one time at one temperature redshift around 1100 suddenly from everywhere these photons can stream free stream okay so they're released all at the same time in every direction but the photons we see define a surface right so we only see photons that were released from you know 14 billion years ago basically 14 billion light years away right so the ones that are gonna end up exactly at our point in space time when we're observing today uh yeah that is actually gonna be the answer so we'll come back to that okay so this is the picture you should have effectively we are uh you should have this picture that we're sort of seeing a surface when the C and B photons were released that's very far away but it's also of course 14 billion years in the past right so you're seeing back along the light cone and uh what you're seeing is sort of the intersection of that constant time or constant temperature surface with that light cone okay does that make sense it's a little confusing in cosmology always to think about uh you know to think about us visualizing things along the light cone but we're seeing things far away therefore we have to be seeing them far into the past this is what we're seeing when we're looking at the C and B it's a shell around us 14 billion years in the past okay yep yep yep yep mm-hmm yeah it does right so in fact as you're saying there's some yep can you repeat the question oh sorry yeah of course uh so the question is I talked about the visibility function being strongly peaked but it's not exactly a delta function it has some width does that do anything and indeed it will do things to the observables that we're measuring so probably the most prominent of fact of the fact that this kind of this surface is not exactly flat it has some width is the fact that very small fluctuations kind of get blurred out okay so if I have a very small fluctuation that gets blurred out and that's one of the reasons the C and B power spectrum is cut off the other one which is diffusion but that's the contribute similar amounts roughly so you know it's hard to see tiny fluctuations because I'm sort of integrating over a finite width what's that uh so at some point maybe it's not seeing the C and B yeah I think that that is effectively true right it'll be increasingly difficult just the energy will fall so low and yeah so that is certainly true in fact if you have W less than minus one that will happen in a finite time you won't be able to anything will be thrown apart so the question was okay yep as long as I can go over a little bit I probably take some questions can you repeat this oh sorry the question is if I have dark energy do I see sort of well will it be increasingly hard to see the C and B that was sort of the question um yeah I think that's right so the sort of co-moving horizon will sort of shrink right and so we won't be able to see as uh yeah as far just like inflation right you're sort of zooming into the patch yeah so sorry say that again I didn't click yeah so you have to be careful about distinguishing then between the sort of the particle horizon versus the the co-moving Hubble radius and those will give you a different answer so if you think about the co-moving Hubble radius it does look like it's shrinking but of course actually we have information you know we don't lose information that we had in the past and so we can actually proceed so there's some subtleties there I have to think about this yeah okay uh good alright let's move on but let's look at this now from the top I'm going to stop plotting uh this time coordinate I'm always just going to plot sort of x y and you should remember that this surface is 14 billion years in the past and we are in the middle looking at this today okay so there's a shell around us and it's looking far away but it's also looking far into the past okay now I want to try to describe quantitatively what uh we are seeing and then I want us to be able to understand and derive the CMB power spectrum but before we can make things quantitative we need to introduce the relevant machinery that I'm sure you're all familiar with but I'll just remind you about the machinery used to compute a power spectrum what is the power spectrum etc so let's quantify the information in this CMB anisotropy field so again we have a pattern of hot and cold spots and currently our theories and there's good reasons for this do not predict the exact realization they don't predict the exact distribution of hot and cold spots you could say that's due to symmetries or because our theories are quantum mechanical but instead what we need to do to quantitatively describe this CMB anisotropy pattern is describe it statistically in terms of variances and correlation functions and things like this okay and that's what our theories can robustly predict now before I talk about spherical harmonics I want to just consider a small cut out a small two dimensional cut out of the CMB because then I can assume what's called the flat sky approximation a small cut out of the CMB last scattering surface I can just use Fourier transforms and then everything is really simple so let's start by looking at this small 2D cut out of the CMB and assume this is flat and then I can describe this field using the point correlation function psi which is the average of the temperature at a point x multiplied by the temperature at a point x prime averaged over realizations of the universe okay now one thing that we have to assume just due to homogeneity and anisotropy for our correlators our two point correlator for example is that there's no difference between the correlator in this direction or another direction so basically there's a translation in variance in this statistical quantity the CMB should be the same statistically in that direction and in this other direction and so what that means is that two point correlator has to have a translation in variance they have to be able to move it by an arbitrary vector a and still get the same answer now what does that mean for the Fourier modes of this field so I'm going to take this temperature map theta of x or theta as a function of where I'm looking and I'm going to Fourier transform it and I'm just going to write that as theta of l I'm not going to write any tildes on it whenever there's a Fourier index you should just be clear this is the Fourier mode so I'm going to Fourier transform that and ask what does this translation in variance imply for the Fourier coefficients one second so what I can do is I can take that correlation function write it in terms of these Fourier modes using the two Fourier transforms and then I can say I want this to be unchanged by translating by a vector a so how do I translate by a vector a I add a to x and x prime now in general that's going to change the answer for general theta of l, theta of l prime that's going to change the answer the only way that translation is not going to introduce an additional phase factor is if I impose that l prime is equal to minus l in other words the only way I can have this symmetry is if these two Fourier modes are proportional to this delta function so translation in variance requires this delta function and the pre-factor is the C and B power factor so effectively it tells you the variance as a function of l as a function of which Fourier mode you're looking at so there is a question no so I guess it works sort of both ways we sort of assume one of the foundational assumptions in building up quantitative cosmology is that we're not special there's no special point in no special direction so we assume a cosmological principle of homogeneity and isotropy and we just sort of assume that has to hold for correlators that we can write down obviously though for isotropy at least this is something we can test extremely well and so far all tests with a few asterisks have passed extremely well so we have tested this and it does work sorry I should repeat the question there was a question about why can we assume this translation in variance to some extent it's a foundational question you heard my answer good let's move on this is the power spectrum it's basically just the square of a Fourier mode but now Fourier modes are not good to describe a sphere they're good to describe a flat surface but if we want to describe the full sphere we need to use better basis functions we can't use Fourier modes but the ideas are exactly the same just like I can expand a field in terms of Fourier coefficients I can expand a field theta in terms of spherical harmonics and instead of Fourier modes I have spherical multiple coefficients ALM and I can use the fact that these spherical harmonics which hopefully are familiar with for example quantum mechanics are orthonormal over the sphere to figure out what all of the spherical multiple coefficients are just by multiplying by a spherical harmonic and integrating over all directions I do what's called a spherical harmonic transform and again just like in flat sky this sort of translation or rotation in variance imposes these delta functions just like I had for translation in variance on the flat sky this is the definition then of the power spectrum effectively it's just the square of a mode or the variance as a function of which multiple I'm looking at now unfortunately we don't have access to an infinite ensemble of universes that we can run over and over and over again in average ALM and A star LM instead we have one universe and we can compute an estimate of the power spectrum by just taking the ALMs that we have assuming that due to rotational in variance all the M modes are statistically the same and averaging over all of these M modes so this is an estimator for the power spectrum and that's the best we can do because we just have one universe but it's an estimate of a variance with a finite number of samples there's some error I just can't get around so that's what's called the cosmic variance error alright let me give you a little bit more intuition about the power spectrum like if I show you a power spectrum you should be able to have some feeling for what it means and basically the power spectrum is telling you how strong the fluctuations are as a function of the angular scale that I'm considering so the standard C and B power spectrum again it plots how strong the fluctuations are as a function of multiple or what scale I'm considering the scale in the left at low L and small scales on the right at high L and as I'm sure you know the standard C and B power spectrum has a peak and that peak reflects the fact that in the map there's a characteristic size of the spots these spots are around a degree typically and that's the peak of this power spectrum but if I had a map with a lot smaller structures in it then that power spectrum would have a peak at higher L on smaller scales and if I have a map with huge blobs in it very large scale features then that power spectrum will peak at low L which corresponds to larger scales ok so does that intuition make sense I'm quantifying how strong the fluctuations are as a function of scale and that's how I can describe this map and because the C and B is very close to Gaussian it remains nearly all of the information any questions about that ok hopefully you've seen this all before and what happens if you go out and measure it you get this beautiful plot that will be our goal to explain this is one of the current best measurements of the C and B power spectrum with the Planck experiment and it's amazing again just how good these measurements are you can't even see the error bars for these these band powers of the C and B power spectrum so first of all this is amazing number one because the measurements are fantastic but the other reason this is an amazing plot is because of this red line and what this red line is basically our current theoretical model for what the C and B power spectrum should look like and it contains only a small number of free parameters this isn't a function huge number of parameters that we can fit it only contains the matter it contains basically a few density parameters and a few initial conditions six parameters and you're able to fit thousands of degrees of freedom at sub-sub precision accuracy so this is a real triumph for theoretical cosmology as well as experiment and so our goal will be to explain and at least give you a reasonable understanding of why this red line and why this power spectrum looks the way it does okay so there are several steps that we need to go through in order to understand this C and B power spectrum first we're going to try to understand how photons propagate in a perturbed universe then we're going to talk about the initial conditions and finally we'll talk about the evolution of these initial conditions forward in this primordial plasma to give us the perturbations in the density of radiation and potential etc that we see in the causing microwave background so let's start we're going to start today by discussing how photons propagate in the C and B to us okay there are any questions about this plan propagation, initial conditions, evolution that's what we need to understand and then we should completely know why the C and B looks the way it does alright any questions? alright let's talk about C and B photon propagation and the basics of the anisotropy there is a question so the question is why the measurements do not explain so well the predictions in the power spectrum at large is good yeah that's a really good question so the question is I think basically about this dip here right and the answer is we don't know statistically this is a totally acceptable fit you know you get a very good chi-squared to lambda-cdm and so you know it's tempting to just wait to just say well this is a posteriori statistics there's some fluctuation and you're looking by eye for features that look weird but when you have so many data points there's going to be a few things that look strange so you know overall this is a very good fit and you know this is not this is maybe not too surprising but I kind of agree it does look intriguing so it might be worth thinking about more I think there have been some proposals for interesting phenomenology that could explain this the issue with these low L-modes is we can't get we only have a small number of low L-modes the number of modes the number of M-modes is basically the numbers of samples of that variance that we have in our universe and we only have a small number of samples for these low multiples so there's not much better we can't measure these much better than we can today potentially there's some room for you know improving measurements with large scale structure but that's very hard so yeah it's a good question how you could test theories that just modify the low L's yeah so that's a good question I talked earlier about oh sorry why do the air bars depend so strongly on L so I think there are a few factors here one of them is we're trying to estimate the true variance from a finite number of samples and the number of samples I have is the number of M-modes per L so naturally at higher L I have more samples and so I get a smaller cosmic variance error that's one reason and that's actually the main reason here that the air bars shrink so I just have more modes for each L and that's I have more samples I get a better estimate of the variance now there are also other effects for example here you see they get bigger again that is because the plonk noise in the plonk detectors and there's a finite beam and those sort of together cause the error bars to increase again at higher L so if I went to even higher L just the experiment limitation would give me a blow up in error bars okay there is another question in the chat asking about the relation between L and scales why do we say low L corresponds to larger scales right so I mean if you're familiar with Fourier transforms sorry the question you read the question sorry I have to think about all these technology so yeah basically you're familiar with Fourier transforms how there's an inverse relationship between sort of wave number and wavelength right it's like 2 pi over lambda and it's exactly the same in the correspondence between multiple like wave number and angle which is like wavelength okay so there's a sort of a Fourier inverse relationship if you will that's one way to imagine you could say why don't we do this correlation function in real space the translation variance has a nice property that means that these different measurements are independent that delta function means different L's are independent and I don't have to have a measurement where all the data points are correlated in some funny way that's very hard to understand okay so again we want to understand this power spectrum why does it look the way it does we'll talk today about photon propagation and really this is a nice calculation and it's worth paying attention to this because it tells you what exactly we're seeing in the cmp when you look at that cmp picture what exactly are you looking at so how can we compute the propagation of the photons from the last scattering surface to today well we need to relate the cmp to the conditions on the last scattering surface and the conditions on the way and so how we do this is the following we first start off by writing down the metric in a perturbed universe so the differences in density in the perturbed universe cause changes to the space time distances and you can see the metric is then modified if I choose a Newtonian gauge of perturbation and I assume that anisotropic stress is negligible I can describe these perturbations to the metric by just a Newtonian perturbation phi okay that affects both the spatial and the time part of this metric okay so that's how I'm going to describe using perturbation theory in my perturbed universe which is clumpy what is the trajectory of a photon to first order perturbations through that perturbed universe okay so just to remind you of how we do this you want to describe your trajectory in terms of a parameter lambda the way this is done in general relativity is that for a massive particle the particle path is determined such that the proper space time distance is minimized and this gives you the geodesic equation I'm sure you're all familiar with this with these Christoffel symbols that make everything coordinate and variant and for photons it's convenient to rewrite this in terms of momenta and this is the geodesic equation that we can then apply to momenta there's obviously much more vigorous ways of deriving this okay so if we want to compute the photons path in GR we need to evaluate the geodesic equation that describes the photons trajectory now this is written in terms of momentum components but we would like to relate this to sort of physical quantities that we can actually measure like the energy that we measure using our detectors and in special relativity I can I note that the energy is given by p new u new where u new is the four velocity that's an invariant and I can just carry that invariant over to general relativity except now my velocity is not given by just one zero zero zero but instead that derivative of the the zero component with respect to proper time is given by this slightly modified expression and so you know you don't need to know the details but basically the energy is given by the zero component of the four momentum and there's a slight complication due to the perturbation okay so if we want to know how the energy of the photon evolves which is what's relevant for determining the temperature we just need to evaluate this geodesic equation for the zero component of the four momentum okay now I'm not going to walk you through this if you want to go through this I can sort of show you how to do it and I'll point you to some hints etc so if you really like calculations and you really like evaluating the crystals and symbols let me know you're welcome to do that I'm just going to not crunch through this this calculation I will just give you the answer once you plug in all the crystals and symbols etc you get this differential equation that describes how the energy evolves along the photons path so time derivative this is a total derivative of the log of AE with respect to conformal time is given by the total derivative of the potential perturbation with respect to conformal time plus two times the partial derivative of phi with respect to conformal time okay so now what I'm going to do is I'm going to integrate along the photons path the photon is emitted and it travels to us so I integrate from emission to us and what I get just doing this integral from emission to us is this expression I can then tailor expand this exponential because these perturbations are small and I get this expression relating the energy of the photon today to the energy of the photon when it's emitted and the scale factor of the photon when it's emitted and these perturbations I'm just going to add by hand but it's easy to justify this the fact that I've assumed that everything is emitted at rest if the last scattering surface is moving towards us energy is affected by a factor by this additional term minus if it's traveling towards me it gets a boost in energy and if it's traveling away it gets a decrement in energy okay I told you earlier we discussed earlier that the energy of a photon and the temperature evolve in the same way and so I can just take a ratio of the energy today divided by the energy at emission and that's just given by the temperature I observe divided by the temperature at emission so now I know almost everything I know how the temperature I observe depends on the temperature at emission which is always fixed it's always 0.3 electron volts and it's all written in terms of things I can I know what they are I know what the potentials at emission are I know what the potential today is I know what the velocity means and I'll talk about this term in a second the one thing I don't know in this expression is what the scale factor is when the C and B photon is emitted so we said it's emitted at a fixed temperature but what scale factor does that correspond to what is the scale factor what is the redshift when this photon is emitted and it's not enough to take the average one well we said earlier this was the quiz question that I asked so it's emitted at the same local temperature of 0.3 electron volts but if I have an over density then the universe has to cool a little bit longer to reach that 0.3 electron volts and so it's emitted at a slightly bigger scale factor okay so the time when it's emitted has to be a little bit different if I have more radiation in that in a certain region so that's kind of the physical reason why it can be emitted at the same local temperature but I'm still sensitive to the radiation perturbation it's because if I have a lot of radiation it has to, the universe has to expand for longer to cool to the right temperature and it's emitted later okay so let's make this quantitative what you can show is that if I have a radiation density perturbation delta r delta rho r over rho r bar then the scale factor of emission is not the sort of average one but it's a bit bigger it's emitted a bit later because the universe has to cool just a little bit extra now you can work out exactly why that's the case so basically what you do is you assume that the temperature at emission is the same and that has to be given by the perturbed radiation density in time just as much as it's given by the unperturbed radiation density in time so the radiation density always and the temperature has to be the same at emission and so these perturbations to the time of emission and the radiation density of emission have to cancel because it has to be emitted at the same local temperature given just by this nice recombination and decoupling business alright and if I do a Taylor expansion of this these perturbations have to cancel I can do some rearrangement to figure out the time delay this causes and plug that in to a Taylor expansion of the the scale factor with time and then I get this expression but again to emphasize the physics here if I have more radiation if I have a perturbation in radiation density the universe has to cool for longer it's emitted at a larger scale factor and therefore it experiences less red shifting along its path to us ok so where we have a higher density the CMB is emitted later with a larger scale factor giving us less red shift and that's why we see a temperature fluctuation that's the physics ok let's put this all together putting this all together this is an important expression this tells us what we're seeing in the CMB and it relates it to quantities on the last scattering surface and along the path of the photon so if you haven't you lost the plot how's the time to pay attention again ok so the CMB temperature anisotropy theta as a function of direction depends on the radiation density perturbation evaluated on the last scattering surface in the direction I'm looking so if I'm looking in that direction depends on the radiation density perturbation at that point on the last scattering surface and at the time that corresponds to emission and similarly it depends on we just explain why that is it depends on the these terms which I'll walk you through so we just talked about the fact that the density perturbation causes you to get less red shifting and the temperature increase now that second term I think makes a lot of sense if the photon is emitted on the last scattering surface from a region where there's a potential perturbation let's say there's a potential well that photon has to lose energy to climb out of the potential well and therefore it's red shifted and I have a negative temperature fluctuation on the other hand if it's emitted from maximum of the potential it actually gains energy and I see a higher temperature that makes complete sense finally second are you talking about this diagram this might not be the most ideal diagram you could just draw and then it's falling down the hill is that what you mean or did I misunderstand it's emitted from inside the potential well at a fixed energy so that's it hasn't fallen in it's emitted from the bottom of the potential or from the top so it's because it comes from the potential well it's not because it traverses it although the question was why doesn't it cancel out when it falls in it's because it's emitted from in the potential well although we will come back to your question in a second it's a very interesting point this third term is called the Doppler term and I think it makes complete sense if the last scattering surface is moving towards us then I get a sort of Doppler boost to the energies of the photons and I see a positive temperature fluctuation the last term is a little bit more subtle and it's actually related to the question the point that was just made so normally if I have a potential well along the photons path that won't do anything because the photon will fall into the potential well and it will gain some energy but then it will crawl right back out and it will lose that energy again so the blue shifting falling in is completely canceled by the redshift crawling up the only way that cannot be the case is if while the photon is traveling through the potential well that potential well changes in amplitude so it decays or increases in other words that partial time derivative of the potential is non-zero so then I can actually change the photon's energy for example if a photon falls into this potential well gains a lot of energy gets blue shifted and then the potential well decays although it has to give up less energy than it gained and so that can introduce a net boost or decrement to the C and B temperature in the direction of these decaying potentials now that's a pretty small effect usually has normally during matter domination the potentials are constant but there are some regimes where this is important and the first one is dark energy dark energy domination at late times dark energy it's a non-clustering component it causes the potential to decay and so this late ISW or integrated sex wolf effect is one way that we have of testing dark energy as we can sort of check whether the potentials decay the way dark energy says they should there's also another there's also an early ISW effect though because the C and B during matter domination but not long after radiation is stopped being important so it's not long after matter radiation equality and so there's still a little bit of radiation around and so there's also an early integrated sex wolf effect from the radiation causing the potentials to decay okay but that basically summarizes the meaning of all these terms when I look at a C and B temperature fluctuation the sum of density perturbations that cause more or less redshifting due to the delays of emission potential minima or maxima that cause me to lose or gain energy Doppler term which causes a Doppler booster to the energy if the last scattering surface is moving and this integrated sex wolf effect that we talked about so hopefully you now understand what the C and B is and what exactly it is in terms of these perturbations you're seeing when we look at these beautiful images of the cosmic microwave background and so in the next lecture we will connect these perturbations here with the initial conditions and then go all the way from the initial conditions to a prediction of the C and B power spectrum and that's all I have for today happy to take more questions though so we'll have a discussion session in half an hour so perhaps we can keep questions for them thank you and see you in half an hour bye