 Hello, hello Okay Hello, hello everyone This is is it working? Hello Can you hear me? Yes. Yes. Yes, sir. So Welcome everyone. This is ICTP summer school on cosmology 2022 So I haven't prepared the welcome speech They just told me to start so Now I guess we can start with the first lecture by asking you are invited to ask questions during the lecture and Especially those who are on zoom, please raise hand or write your questions in the chat And then we will I will read them to us Is there anything else? No, I just want to add Both for people here and people in the in zoom that there is a discussion session in the afternoon so if there are too many questions you can just Take a note and you can ask in the afternoon so all the lecturers will be around and That's it. I'm Paolo Kriminelli by the way. Welcome to to Trieste Thank you. Yeah, I'm Erdogan here by the way Okay Okay, my mic is on Yeah, all right, so hello everyone and thanks to Paolo Mirdad and Ravi for Asking me to speak here. It's very exciting for me I've been associated with ICTP for several years and these schools have always been a highlight of the events that happen here So it's really great for me to be here sharing the excitement of cosmology with you all and I for one Yeah, I think I'll do that because otherwise it becomes a bit problematic right so Like I was saying so me and I'm looking forward to this next couple of weeks with all the other speakers as well and I hope everybody here is Getting over the the COVID exhaustion of being online You know, it's unfortunate that we cannot have this in the big lecture hall where everyone is here But for those who are here, I hope you take this opportunity to participate and for those on zoom I will try my best to keep you involved as well So maybe the people on zoom can already just ping Mirdad with a thumbs up If you can hear me properly and if you can see the slides and everything and if there's a problem Please don't hesitate to to interrupt. Okay. All right so I was asked to give a few lectures on large-scale structure and structure formation and Also, this is a very diverse audience So there are people who are already advanced PhD students There are people who are beginners and then you know different parts of the world have different backgrounds in the way You learn cosmology. So what we thought we would do in the first lecture since this is the first lecture of the school as well Is to be very elementary. Okay, so for most of you, this will be extremely familiar You may think that you know, this is all things that you learned in high school or something But please bear with me this lecture is meant to bring everybody to the same page And from the next lecture onwards in my course, at least we'll start doing a little bit more advanced topics And this will also help to set up some language, which I think will be useful for the subsequent speakers Okay, so this is today's lecture. I'm told that if I point this anywhere. Yes, I can see a pointer there Very cool so I will spend some time talking about the expanding universe and What is sometimes called the standard model of cosmology? So before we begin I wanted to Flash a slide showing some useful literature on the subject of cosmology. This is a very biased list There are many books available These are just some of the books that I have found useful Growing up and I've highlighted in bold some of the books which have you know been very influential for me as well So the book by Narellika was my first book on cosmology. It's quite dated now But it is extremely concisely written and it's a fat book So you can imagine that it has you know a huge range of topics covered in it The book by Dodelson is excellent for learning linear perturbation theory many of the things I will talk about when we come to that part of the the course will be taken from this book and There is a more recent book called galaxy formation and evolution by Moe van den Bosch and white This is I mean it may not be very interesting as a topic for you know People who do early universe physics, but it covers almost all of cosmology as well And most of the topics we discuss here are also covered here, and it's it's much more up to date. This was written in 2010 Okay, so let's start with something really basic distances in the universe We live on earth the earth is in the solar system Typical sizes of solar systems are measured in astronomical units or parsecs And then beyond this what are we going to talk about? Okay? So this is our experience as as travelers in the cosmos around us But the kind of distances that we deal with in cosmology are very different So this slide is just in the next couple of slides help you to get a feel for these distances Although it's very difficult to appreciate really how vast these distances are so what I've tried to do here is to show you a comparison comparison between solar system scales which are measured in parsecs to Scales over which the entire galaxy can be viewed which are measured in tens of kilo parsecs a kilo parsec is a thousand parsecs Okay, and I guess everybody knows what a parsec is so the conversions are there for you to see if you're if you're not familiar At scales where galaxies start to conglomerate and become Parts of groups and clusters we reach scales that are tens of mega parsecs Okay, a mega parsec is a thousand kilo parsecs. So we've gone up by factors of a hundred million or so already This is also not the scales that we are interested in for cosmology at late times where structure formation is interesting For those you have to go to much larger scales So this is one of the earliest large-scale images that that we know of of our local universe So don't worry about the word wreck ship for now if you're not familiar I will discuss this a couple of slides down the line What you can try to take away from this slide is that what is shown here is an angular scale So what has been done is a survey of galaxies around us and in the sky you have two locations two angles Right ascension and over to a declination and a right ascension. So one of the angles has been collapsed Okay, so you just ignore that angle. So then the right ascension is left free for you to plot. So that is done here and The radial direction is radial distance from us for the time being just take it to be real radial distance in some strange units So this is a map of the universe around us with us sent centered here right at them at the middle and The interesting thing here is that in previous slides you saw Images it is an artist impression of the milky way then even here although you can't really see the individual structures of galaxies You can make out that there are some galaxies in the next slide every black dot on the image is one galaxy So this is the scale at which we are talking about okay, and this is already now hundreds of mega parsecs in in in scale so these are the kind of scales that we deal with in cosmology at late times and what you can see is that the Distribution of galaxies this is these are the real positions of galaxies around us these are not random Okay, it's not as if you just randomly through down points There is structure here for example There are places where you can basically just see a black blob which means there are many many many objects there And then there are places where there is almost nothing okay There are completely white regions which are completely devoid of galaxies. They are in fact called voids This is another image of slightly larger distances So in the previous slide the distance scale this redshift axis went up to 0.2 Imagine that this is a linear distance from us in this next slide This is taken from a different survey that one was from the two degree field galaxy redshift survey This is from the wiper survey which happened About 10 to 12 years later now you've gone out to larger distances now the redshift axis has gone out to 0.8 So four times higher in distance compared to previously and here they've also conveniently Provided a dictionary into mega parsecs Okay So you can clearly see here that now you're talking about distances that are thousands of mega parsec and the scales that you're that you're dealing with are several hundreds of mega parsec and What you can see here is that even at these large distances this structure that is apparent here It's a filamentary network It's called the cosmic web and this cosmic web is already in place or also at these large distances for example I first saw this slide in a talk by gg good So and he emphasized something which I would like to emphasize here look at this region here. Okay. It's completely black Sorry, it's completely white and it's not because you were not able to see galaxies right because you have seen galaxies here Which are further away so your survey was capable of imaging galaxies here if they they existed So the fact that you're not seeing anything here means that there are no galaxies there So now try to appreciate the scale right so look at these tick marks the distance between the tick marks is a hundred mega parsec So this diameter here is about 30 mega parsec or so So there is a 30 mega parsec wide region which has absolutely nothing in it as far as galaxies are concerned It's a huge void. Okay, so this is something interesting that these are the kind of interesting things that come up when you when you study these observations In addition to the large-scale structure of galaxies that we see around us There is another very important Observation probe of cosmology that has been around since the 60s and theoretically it has been predicted since even earlier This is a cosmic microwave background. So this is a uniform radiation bath Which is which has a photon spectrum the energy spectrum of the photons is almost a perfect blackbody a plank spectrum Which is characterized by a temperature and that temperature is almost identical in all directions Okay, and this is something very very interesting for theoretical purposes, which I believe Marco will talk about in his in his lectures So for the time being we start with this very uniform temperature bath around us Temperature is approximately three Kelvin and this is not it because if you if you Find tune your eyes and you are able to tell differences which are very very tiny The next thing that you see here is a dipole like structure Where now I have indicated the temperature of the the associated anisotropy using Dimensionless variable. So this is the temperature difference from the mean value of three Kelvin That's delta t and I divided by the mean value of three Kelvin So I got a dimensionless quantity and the variation that you see here from this hot region to the cold region has a Magnitude of approximately one part in the thousand. Yeah, and this is generally attributed to our motion Relative to the large-scale structure around us So we of course are moving around the Sun the Sun is moving in the galaxy on an orbit around the galaxy and the galaxy itself Is moving in the gravitational potential of the surrounding large-scale structure? so all of these motions add up and they create a dipole and this dipole it can be Can be understood in terms of our velocity and that velocity magnitude turns out to be around a few hundred six hundred kilometers per second So this is the anisotropy in the CMB that you see at the level of one part in ten to the three What is even more interesting for cosmology is that if you believe that? The universe evolved in a certain way starting from a hot dense phase Expanding and cooling and becoming the universe that it that it appears to be like today Which we saw in the previous graphs with the galaxy redshift surveys then in the CMB You expect some level of anisotropies which are all pervasive they should be there everywhere that you look Okay, and just to remind you this kind of ellipsoidal image is basically a map of the CMB all around us Okay, so it's like the mole white projection that you use for the earth you project the earth onto onto an ellipse It's exactly what has been done except that now you're not on the surface of anything You are inside the ball and you're looking at it all around you. Okay, so this was This theoretical expectation was was you know, it was verified by observations in 1993 These are not observations from 93. These are much more recent observations from the plank instrument But what one knows now is that there are tiny primordial anisotropies So these are tiny anisotropies tiny because they are even a hundred times smaller than the dipole anisotropy that I showed You before okay, so these are one part in ten to the five a few hundred micro Kelvin and They are there everywhere in the in the CMB sky and they look basically like random distribution Okay, so this aspect of these tiny anisotropies will be very important for us in coming slides Its origin will be discussed in the in the top in the lectures on inflation and its consequences will be discussed in these lectures Okay, so I've now given you a broad overview of yes Okay, so let me just repeat quickly so this ellipse that you see here is not It's nothing real. It's because I want to display for you the spherical CMB sky around us on a two-dimensional surface So this is just a projection trick It's the same projection trick that is used when you want to display the map of the earth Which you think of as a sphere on to a two-dimensional surface, okay? So it's just a drawing tool. It has nothing to do with the cosmos What is real is the variation that you see from orange to blue in different patches Okay, it's from red to blue. Those are real variations and they are the real temperature anisotropies Which have a primordial origin which is very interesting for studying the physics of of structure formation Okay So just to pause this is the place where we ask how did all this happen? How did this come to be so the structure formation lectures are centered around this thing, okay? And in fact you can use this theme to more or less Understand all of cosmology as well because this is what you want to understand It is possible to do cosmology from a purely theoretical perspective In fact, this is how Einstein and his contemporaries started the field when they did not have any observations whatsoever Okay, so I there's a very interesting human history, which you can ask me about maybe in the discussion session but I just want to emphasize here that Cosmology has transformed itself over the last century from being a very theoretical exercise to being completely observationally driven today It's the theorists who go to the observers to ask is my theory correct Okay, and whereas it was the other probably the other way round where it was the theorists in the mid-20s or so who are the who are the dominant players in this game Okay, so these lectures and I believe most lectures in this in this school will focus around the The framework of the standard model of cosmology which itself lives in the hot big big bang framework Okay, so there are alternative ways of trying to understand cosmology. These are Less mainstream now mostly because all of the observational evidence as I said It's an observationally driven field all of the observational evidence suggests that the hot big bang framework is Completely consistent with everything that we know about in the data so this is a slide stolen from NASA and it's probably something you've seen many times before it's a cartoon summary of the evolution of the universe so time is shown on the horizontal axis and I've chosen this image because it emphasizes the kind of things I want to discuss then other people Discuss their topics inflation in particular or or the cosmic microwave background more emphasis would be late to this early part So then maybe you want to you know You change the scale on the x-axis and show things logarithmically there and so on okay So I will not spend too much time on this. It's more interesting to actually do the things that we want to do Okay, good. So let's become a little more technical now We want to work in the framework of homogeneous and isotropic cosmologies The universe that you see around you is not homogeneous and isotropic Okay, I showed you the image from the two-degree field galaxy redshift survey This is definitely not a homogeneous distribution. There are more galaxies in other in some places less in other places but What we believe or what we can see in fact is that if you consider Smoothing out the distribution of these galaxies on some sufficiently large scales Then what emerges is a distribution which is reasonably homogeneous in the sense that it does not change much from place to place So let me show you what I mean. Let's go back to this galaxy image Imagine that I I take little circles and I place them randomly and then inside every circle I count up the number of galaxies and divide by the volume of each circle or every sphere and I call this the number density of Galaxies at that location smooth over some very large scale. Let's take a scale of a hundred megaparsec or so If I do this in different parts of the sky What I will in fact find is that the number density fluctuations are not that dramatic Okay, and as I increase the size of my spheres I will find that in fact around a hundred megaparsec or so for the radius of the sphere this distribution starts looking pretty homogeneous So this is the sense in which the large-scale distribution of structure around us is homogeneous The CMB is isotropic around us Okay, and these two facts that the distribution close to you is homogeneous and the CMB is isotropic Combined with what's called the Copernican principle that you are not special observers Allows you to claim that more or less everywhere in the universe such a situation should exist and this then tells you that the overall distribution of Matter around you is homogeneous and isotropic. So it's worth discussing within the context of general relativity where we will work What kind of metrics will describe this structure and the metric that emerges is called the Friedman Lumetre Robertson Walker or FLR W metric which I have written down here Okay, so this is this is written down using certain language which this slide describes So what one says is that? Imagine a universe in which there is a set of observers who has been picked out as being special these observers they see a uniform CMB sky around them and They carry clocks with them such that at the same tick of the clock on each of these observers All of them see the same TM CMB temperature Okay, so such observers if they exist what metric will they will can you write down? So the metric that you write down for this space time in coordinates carried by these fundamental observers is written down here So these are theta phi are coordinates which are Comoving with these fundamental observers in the sense that if I am a fundamental observer my r theta phi relative to any origin in space Will be fixed for all time Okay, and similarly if somebody else is a co-moving observer they will have some other r theta phi relative to the same origin So these are the spatial coordinates the time t is the time measured on the clocks of these fundamental observers And these clocks are supposed to be synchronized at some initial time slice and thereafter they stay synchronized because of homogeneity and isotropy the metric apart from the coordinates of course is Characterized by two quantities one is this constant k. It's a constant in both space and time and The other is this quantity a of t which is which does not depend on space, but it depends only on time so this constant k is Related to the constant curvature of each of these spatial slices So these are these are space times which have three-dimensional slices of constant curvature This is another mathematical way of understanding them a of t therefore is the only dynamical quantity in the problem It's the main function of time that we want to understand if we want to study cosmology This is called the scale factor and it connects lengths Across different times. So if I take two fundamental observers, let's say me and the gentleman in the first row and I measure the distance between us at some fixed time using some technique, okay? The quantity a of t will tell me at a later time. How has the distance between us changed? So for example if a of t is an increasing function of time then the distance between us at some later time will have increased and in fact this is the Situation that we want to discuss where a of t is almost always an increasing function of time So the way one thinks about it colloquially is that the universe in such models is expanding This is the idea of the expanding universe. Okay a of t increases with time So this is the metric of an FLRW space time Now if I want to do general relativity with a metric I have to calculate I have to write down the Einstein equations So the left-hand side of the Einstein equations have the Einstein tensor Which can be constructed by taking derivatives of this metric and I have to supply a right-hand side Which is the energy momentum tensor of the matter content of this universe So if I have a homogeneous and isotropic space time it turns out that the structure of the energy momentum tensor has is very restricted and It must take this diagonal form where the first quantity the zero zero element is Related to the energy density of the fluid that you're talking about the diagonal three elements on the spatial coordinates are Related to the they are equal to the pressure of this fluid and both the energy density and the pressure have to be independent of the spatial coordinates in these in this coordinate system Okay, so this is the restriction imposed by homogeneity and isotropy So rho is the energy density p is the pressure The relation between energy density and pressure is not known a priori it has to be supplied So this is called an equation of state and an equation of state has to be given in order to be able to solve equations Okay, so this is the broad framework which in which we will work Okay, so let's do a little bit more with the with this space time what you can mathematically prove is that Something very interesting happens to photons which travel in this space time Okay, so in order to do this. I actually don't need Einstein's equations once I write down this metric I can ask how do photons propagate what happens if I have a source of photons which you know I switch on a light bulb at a particular time at a particular location And I view the photons that are traveling from this light bulb at some later time at some later position And now I can do this mathematically using this metric and something very interesting happens It turns out that the wavelength of these photons changes according to the scale factor It changes proportionally to the scale factor. So in an expanding universe the wavelength increases with time Okay, so this is very interesting because Wavelengths of photons are something that you can observe in spectrometers in the lab sitting on earth So if I have a very distant Let's say I have a distant galaxy and it is emitting photons or some distant object the wavelength And if I know that it is supposed to be emitting photons of a particular wavelength because I understand the physics which generates these photons Okay, so my knowledge of basic physics of atomic physics might be able to tell me that the wavelength at emission must be something With this information if I now observe photons from this object and I see that they have a different wavelength I get an observational handle on How much the universe has expanded from the time that those photons were emitted to the time at which they were received? And the way this is conventionally done is to define a quantity z which is the relative shift in the photon wavelength It is the wavelength at that I observe minus the expected wavelength divided by the expected wavelength and This in the calculation that I have not shown you but can be done using this metric using photon propagation Is related to the scale factor using this relation here So this z which is the redshift is just one over a minus one Okay, it's a very simple relation and you can prove this very rigorously So this is extremely nice because it gives you an observational handle on cosmology Okay, so this is the key idea that is used in studying distances in cosmology The other idea of course is your usual special relativity notion that time and distance are related to each other because the speed of light Is finite okay, so if if I have an object which is further away Then it will obviously take more time for light to be sent to me as compared to an object that isn't that that is relatively nearby so in this sense cosmological distance redshift and Epoch of cosmology the time at which certain events have happened are all related to each other Okay, so I can use tracers which are bright enough that I can see them at large distances to Tell me not only how far away things are but also how much time has evolved since those tracers were emitting light Okay, so now With this idea of redshift FLRW space time and the Einstein equations You can simplify things because the Einstein equations as you know in general are very complicated Nonlinear partial differential equations, but we are dealing with a very very simple metric in which there is almost no I mean There's no non-trivial spatial dependence Okay, the spatial part of the metric is very easy the only thing that we don't know is the scale factor So the Einstein equations simply become ordinary differential equations They're still non-linear but ordinary differential equations for the only dynamical quantity, which is the scale factor Okay, and the main equation so these are the equations that that one deals with this is They're called by different names. The first one is always called the Friedman equation Okay after Alexander Friedman and the second one is sometimes called the acceleration equation or the second Friedman equation Okay, so I will just refer to both of these as the Friedman equations. They were used by Alexander Friedman For almost the first time actually Einstein used them before but discarded the solutions that he got it was Friedman Who actually published these solutions that came from analyzing these equations? Okay, so let's look at the Friedman equations for a second Here I have defined a quantity H H in honor of Edwin Hubble Which is defined as the logarithmic derivative of the scale factor with respect to this cosmic time variable This is called the Hubble parameter and the Friedman equation is a Differential equation which relates this Hubble parameter to the energy density content of the universe In this energy density row bar I have added up the elements of all the components that we believe exist in the standard model of cosmology But if you want to study non standard models of cosmology, you want to invent Matter energy content of your own that is where you have to put it in. Okay. Is there a question? Yes Come again Yes, I have put a bar to indicate that these are homogeneous and isotropic quantities later on We will study the inhomogeneous universe where I hope I will self-consistently remove the bar. Okay, that was the idea here All right, so this is the energy density content of various components, which I will discuss as I go along and Additionally, there is a contribution due to the constant curvature of this space time Which appears in this term k divided by a squared It is also conventional in the literature to non-dimensionalize these energy densities by Defining a critical density row crit, which is just 3h squared divided by 8 pi g Okay, and I will tell you why this is called a critical density. I think in the next slide But just take just understand here that if I take 3h squared divided by 8 pi g in units Where the speed of light is one this quantity row crit also has the dimensions of an energy density Okay, so I have just non-dimensionalized these equations by taking 8 squared on to this side That's all I did and Then I get these quantities omega and I define one omega for every energy density component that I have here Okay, so there'll be an omega c dm and omega barion and omega radiation and an omega de which is dark energy and I similarly define a quantity omega k which is a non-dimensional version of the curvature constant here and In this language the Friedman equation the first equation on the left here Just turns out to be the summation of all the omegas has to be unity So it's a very simple non-dimensional equation the acceleration equation is also interesting and I will talk about it as I go along Okay, so these equations were being developed in the in the early 1920s and Around the same time there was observational evidence that something interesting is happening when I look at distances of galaxies Away from me and I compare them with these wavelength shifts or red shifts So when Hubble and LeMathre were doing this At least prior to that the notion of redshift was always thought of in terms of the Doppler shift okay, and what Hubble and LeMathre realized is that the Doppler shifts that you infer from these delta lambda by lambda measurements So you take a delta lambda by lambda measurement for an object and I just multiply that quantity with the speed of light I get something which has units of velocity. Okay, so that velocity is plotted here It's actually just the redshift being plotted and I compare these red shifts or velocities with the actual distances Which I can measure using some other techniques which is for example I can have the variable stars whose properties tell me something about how far away they are from me So this is what was done in the 20s And what one realized is that there is a nearly linear relationship between the observed redshift and the distance to these objects So this redshift distance relation is called the Hubble LeMathre law It used to be called the Hubble law, but LeMathre's contributions to this field were recently formally recognized So this is the Hubble LeMathre law And this is one of the cornerstones of observational cosmology starting from our end Of our advantage point at the late universe. The other cornerstone is the cosmic microwave background Which I will talk about in a little bit Okay, so this these are the things I want you to take away from here. There are the Friedman equations Which can be written in this way. There are several components. We'll discuss each of them one by one and With the along with this there is this observational constraint that there is an almost linear relationship between nearby galaxies In terms of their distance and their redshift Okay, and again some human history will follow later if I have some time So in this mathematical framework one can now go about Discussing what happens to the evolution of different energy components in the universe, which might be interesting for us So for example, if there is non relativistic matter filling the universe in a homogeneous and isotropic manner You can show that its energy density falls like the cube of the scale factor one over a cube And I will not derive any of this here. So for these lectures, I will just give you hand-waving arguments for why this is the case Yeah, so there are two questions in the chat one is that For objects on the around 200 mega-parcy, what do the cluster points signify? I think the previous yeah, okay, and the other one the second one What does the pressure on this just energy tensor act on? Okay, so the answer to the first one is These clusters are are basically responses of the matter distribution to the evolving Gravitational potential. This is the topic of the next three lectures in my course So I will not say anything more about that here The pressure that you're talking about here is the pressure of a fluid So this is a the fluid dynamic way of understanding the distribution of matter any fluid will be described by an And as by a density a pressure a temperature and other thermodynamic variables when you have a Simple system like a homogeneous and isotropic fluid The only two relevant quantities are the energy density and pressure and an equation of state governing them So this is the pressure that you know the particles of the fluid feel because of the rest of the fluid There's the usual statistical mechanical way of understanding the pressure Okay, so this is the wave unusually approaches the pressure in these in these situations And for describing non relativistic matter in fact where I can think of even the distribution of galaxies on large enough scales as As a as a effective fluid with some energy density and some pressure That pressure is actually just going to be very close to zero because these Particles or galaxies can be treated as being collision less and for a collision less fluid even in statistical mechanics You can show that the pressure is exponentially suppressed at finite temperature So this is the way one usually understands these things Okay, so this is what a non relativistic matter does and you can just think of this as mass conservation or number conservation Because you know if I take a co-moving volume which is fixed and I fill it up with some mass and this mass is non relativistic In an expanding universe the volume will increase like the cube of the scale factor and the mass will be conserved So the density will fall like one over the scale factor cubed. Okay. It's a very simple thing here You can do this of course much more formally and rigorously using the energy momentum tensor and so on The radiation So if there is a uniform radiation bath We want to describe such a component because we know that the cosmic microwave background exists Okay, so let's keep track of a radiation bath, which is also homogeneous and isotropic You can show that the energy density now falls like one over eight of the power four So in this case you could heuristically think about this as number conservation Which already you did here so that gives you three powers of the scale factor and secondly I told you that photons propagating in an expanding universe will have their wavelengths increasing with time According to the scale factor the wavelength of a photon is related to the momentum of a photon The momentum will therefore fall like one over the scale factor So the energy density which is the energy per unit time will therefore fall like one over a to the power four With energy per unit time per unit volume Sorry, not I said something wrong the energy density is the energy per unit volume The energy falls like one over the scale factor volume falls like for increases like a cube and that's what gives you the eight of the power four the curvature Behaves like one over a squared because so for example In this term here, you can clearly see this. Okay. This just follows from the Einstein equations. So this This if you think of this as something like an energy density, which is adding to other energy densities this Component falls like one over a squared And then there is also a component that I have reserved for dark energy Which I will in these lectures always treat as a cosmological constant But there are many models of this. This is one of the primary uncertainties in in late-time cosmology This if it is a cosmological constant its energy density will simply be a constant. Okay in brackets here I have also indicated what happens in such a universe to the scale factor if These individual components are the only components that dominate the energy budget of that particular model Okay So in a matter dominated universe the scale factor you will have to go back here and Solve the Friedman equation by saying this one over a dA by dt the whole squared is equal to something proportional to one over a cubed Okay so now this is not a very difficult equation to solve actually and you can do it and What turns out is that a will be proportional to t to the power two-thirds? Similarly in the radiation dominated universe the power law here with the index will change It will become t to the power half if my universe is curvature dominated and my k has a negative sign Then this a of t will simply be proportional to t and In a universe dominated by the cosmological constant my a of t because the right-hand side will be constant My a of t will increase exponentially Okay, so these are interesting asymptotic behaviors of scale factor in different situations each of these is interesting for different epochs of the universe and In this last bit here. I've shown you what happens when there is curvature in the problem So something interesting happens in the presence of curvature So imagine a universe in which I don't have radiation and I don't have this dark energy for simplicity I just have non-relativistic matter and curvature then my first Friedman equation is just omega matter plus omega curvature equal to 1 So now you can see that okay Actually, you can't see but I'm telling you that if I take a situation in which Omega curvature is negative which means that quantity k is positive Okay, so this omega k was defined with a negative sign here just to make the equation look nice But this quantity this omega k being negative is a universe with positive curvature If you analyze the Friedman equation in such a universe You will find a solution in which the scale factor initially increases with time But then it reaches a maximum and it falls back it starts decreasing after that and falls back to zero Okay, so this is a universe in which the universe expands In which the scale factor expands and then turns around and then the universe collapses. This is called a closed model on the other side if I have omega curvature positive and Then correspondingly omega matter less than one then the universe always expands Okay, and again you can write down a solution for this and the Case omega k equal to zero a spatially flat universe which has zero constant curvature is the boundary between these two Where it the turnaround happens, but it happens after an infinite time Okay, and the solutions that you see here are actually I mean that you will get here are actually very related to something that we will discuss later in the context of of Evolution of perturbations when we study spherical collapse. Okay, so I will come back to this topic at that time There was a question It will actually not plateau it will That's true. Yes at infinite time the derivative of the scale factor will reach zero the first derivative. That's true But at any finite time the universe is always expanding in that model Yes, there is a question in the chat Which is what do you mean by a smooth? Expansion Should have explained yes smooth in the sense of homogeneous my apologies. So when I say smooth I mean homogeneous. So this is the expansion of a homogeneous and isotropic universe In the presence of a homogeneous and isotropic set of energy density components Another good Is there does anything special happen if omega k is equal to omega m? This could be an exit this cannot happen because omega matter plus omega k has well Okay, it could happen if both of them are equal to a half. This would be still valid That's an interesting thing. I've never thought about it. I don't think anything interesting happens It is a universe with With which always expands. So I don't think there's anything much more special than this But I may be wrong. I will check and if there is something interesting. I'll get back to you Okay, so now let's also just for completeness and also to Get something down in you know in terms of language discuss distances in cosmology Okay, this is an important topic especially for for doing what is called cosmography Which I will mention at the end Distances in cosmology are non-trivial because of the expansion of the universe Okay, so even if I imagine a spatially flat universe in which the spatial part of the metric just looks like Euclidean space Which you know you're used to dealing with in this room for example The presence of a scale factor, which is changing with time means that you cannot apply the same logic As you would you know you cannot apply the same formulae for distances and how distances change Between different objects etc because things are changing as a function of time So you have to be a little bit careful So what one does conventionally is to say that let's focus on what one can observe This is Einstein's way of thinking okay He used to he say that define all distances in terms of the propagation of light That's a very clean way of defining distances. So that's the same thing that we will do here So imagine that there is a source of light and it is being seen by us at some later time So it emits at some time and we see it at some later time How should I interpret the photons that I receive from this source in terms of a distance to this source? This is the question so now this can be done in a few different ways. There are two that I will talk about here So this is what I've said here the expansion means that Euclidean notions need to be revised So what one does is to say that Let's use the Euclidean notions but Interpret them using observational quantities okay observational and theoretically well understood quantities So for example, okay So for example one theoretically well understood quantity would be photon counting and the second one that we will apply is the propagation of light on an FLW background so DS squared equal to zero is the propagation of null rays Which is what we used to describe photons and then we'll count photons that arise from arrive from different parts of the of the universe Okay, so mathematically it's interesting to first define this quantity Chi source Which is this integral CDT by a which comes by just saying that you know So imagine universe with omega k equal to zero or k equal to zero. So the FLRW metric is just So I've not written the angular part because I want to describe a ray of light in which I have taken the origin to be me for example and The ray of light extends from the source to me or I can take the origin to be the source and the rays extending radially to me There's a question. Yes. Yes, I think I mentioned that Yes, yes, you can set it to one if you want, okay, so I will be a little bit loose about powers of C so DR squared I have kept because I want to describe radial distance and the remaining things I can throw away because you can actually show That geodesics photon geodesics obey theta equal to theta equal to constant and phi equal to constant in any Coordinate system which is tied to fundamental observers in an FLRW space time Okay, so my geodesic will follow from setting DS squared equal to zero and Then what you will see is that this quantity DR in this case will just be related to this integral CDT by a Someone needs to be muted This is comic relief provided by zoom All right, so this integral is the co-moving distance to the source And the reason it is written here is because it is in it is implicitly a function of redshift You see because the source emits light at a certain time, which is T source This light takes a certain this amount of time T source to T naught in order to reach me But I know that in the time that the light has come from the source to me The wavelength of the photons has increased by some amount that is captured by redshift And I know that redshift is related also to the scale factor okay, so I can use all of this knowledge and In and think of this co-moving distance to the source as a function of the observed redshift of the source Redshift is an observed quantity. Okay, that's the reason it is so important And here all I've done is to change variables instead of T I am using the redshift of different different sources along the line of sight as a time variable Okay, so remember that slide where I told you that redshift distances and epochs in cosmology are all the same thing you can just you know freely Move from one concept to the other. So I'm just thinking of this so T here is Related to the scale factor because if I know the evolution of the scale factor a is a function only of T So T is a function of a okay, so from the variable T I can go to a as a clock and A is related to the redshift in a very simple way in that expression that I gave you before so from a I can change also to redshift as a variable and Inside the integral. I have kept it as z prime. It's a dummy integration variable And then the actual redshift of the source is sitting here in the integration limits Okay, so although it looks simple. There is some thought that has to be put into and Unpacking this kind of expression But once you get your head around this things become very interesting because Okay, so here I have just shown you what happens when k is non-zero don't worry about this You can look at these slides later and understand what is happening But once I have this co-moving distance to the source I can now define Different kinds of distances to sources this co-moving distance is a theoretical quantity It has it is chi as a function of z Okay, so suppose you said that I want to use co-moving distance as my definition of distance fantastic For a given source, what can I observe? For co-moving distance the only thing I can observe is the redshift So I so then in order to convert the the redshift into co-moving distance I need to know the cosmological model that I'm talking about and maybe I don't know this model Okay, maybe I want to infer what the model is what is omega lambda? What is omega matter? I don't know this so what do I do so instead of using co-moving distance I will define something let us say call the luminosity distance in the following way Suppose I have a class of sources where in addition to the redshift, which I will observe from them I also know what is their intrinsic luminosity, which is the amount of energy that they output per unit time At their location in their rest frame Okay, if I have such a class of objects these are called standard candles and I know what their intrinsic luminosity is Then what I can go and do is this exercise I will I will use this knowledge of the intrinsic luminosity and I will observe the flux that I get from these objects The flux is just energy that I receive from the objects in Terms of photons per unit time per unit area at my detector Okay, so if I combine if I take this luminosity and flux in a Euclidean universe They would be related to distance through this expression here So I will declare that even in my FLRW universe. Let me define this quantity wherever I see three lines like this It's a definition and you're not allowed to ask any questions. Okay, that's the way this quantity is defined so I can define this quantity DL and Now the theoretically interesting part comes in trying to understand this quantity DL as a function of redshift and this is a exercise in photon counting and Null geodesics which you can find in let's say in our liquor's book and you get this expression Okay, one plus z times our source you can for the time being just think of our source as being equal to the sky source so now for a given cosmological model I have a conversion between the luminosity distance and the redshift and The luminosity distance crucially is also an observed quantity because I know the intrinsic luminosity and I observe the flux Okay, so this is how you break this degeneracy and you get rid of this dependence on the cosmological Parameters or the cosmological model and you can now go and plot you can do what Hubble and LeMath reddit And you can plot distances on one axis and red shifts on the other axis everything coming from observations And this allows you to start constraining cosmological models Okay, so this is one key thing which happened in the 90s which led to for example a Nobel Prize eventually for the discovery of the accelerating expansion of the universe Another such distance that you can define is the angular diameter distance Which now in instead of luminosity suppose I have objects which have fixed sizes known sizes Okay, so imagine if there was a class of galaxies whose size in the in physical in physical space is Some fixed number and I knew this number. Let's say I had a class of galaxies each of them is 10 kiloparsec in diameter What will happen now if I take a 10 kiloparsec size galaxy and I place it at 1 megaparsec from me It will subtend some small angle at my eyes If I take the same galaxy and I place it a hundred megaparsec from me It will subtend a smaller angle at my eyes hundred times smaller So I can use the angle subtended by this finite object, which is not a point anymore I can use this angle delta theta and the knowledge of this intrinsic size of the of this object If I have it to define what is called the angular diameter distance same logic as before in Euclidean geometry This would be the unique distance to the object if the angle is very small. I've ignored a sign or tangent And now in FLRW I simply define this to be the angular diameter distance I do a photon counting exercise and Along with geodesics and I find that this DA is also related to this our source or chi source But now with a different power of the redshift appearing here Okay, so the same logic that I could apply to luminosity distances I can also apply to angular diameter distances It's also interesting that the angular diameter distance is actually related to the luminosity distance Without knowing about our source. Okay, both of them are linearly related to our source So I can eliminate our source between them and then DL is just 1 plus z squared times DA This is a very interesting relation. It is called I forget what it's called cosmic duality or something like this and it is something that is intrinsic to light propagation It does not even care about which FLRW model you're using. Okay, so people try to test this relation as well Okay, I'm going a bit slow. I have about what 20 minutes Okay So this is my last slide on distances The reason is okay, so I wanted to point out a couple of interesting things here one is that So now you may think okay, you know luminosity distance angular diameter distance and then who knows how many other distances one can define Depends on how many properties of objects you are clever enough to extract from from your data These are typically the ones that people use but it turns out that if I look at nearby objects You have to also ask what happened to the Hubble-Lometre law. Okay, theoretically you can very easily show that at Small distances or at small red shifts z much less than one any definition of distance that you come up with Will reduce to the linear relation. Okay, and it says as here It's as simple as saying that you can tailor expand any smooth function of dl of z or da of z and there will be a linear term And it will turn out that the proportionality constant of the linear term is always the same in each of these cases There's a nicer way of doing this where you don't have to think about you know Which distance why should the proportionality constant be the same you can analyze photon geodesics for nearby objects directly Without asking whether I'm talking about luminosity distance or angular diameter distance just think about nearby objects and do photon propagation the Hubble-Lometre law will emerge naturally and in fact This is the calculation that Lometre proposed in his 1927 paper now if I choose to think of this as in terms of Taylor expansion what I can also do is to go to the next order of Power in z which is the quadratic power and now I will have some second constant appearing here So what I've shown you is a conventional way of writing the the co-moving distance as a function of redshift at second order in z And this new constant q0 it is related to the scale factor through the second time derivative of the scale factor So it's called the deceleration parameter. This is again a historical you know quirk where People were not expecting this second derivative of the scale factor to be positive ever Because in any universe where I have ordinary matter non-relativistic or a relativistic bath of radiation I expect that because gravity is attractive the scale factor of the universe through the acceleration equation will always remain negative Okay, so that's why the original definition of this q was defined with a negative sign here thinking that a double dot is Anyway going to be negative. So you expect q to be positive It turned out in the mid 90s that observations of type 1 a supernovae showed that this q0 Which is the value of q at the current epoch turns out to be negative So it indicates a positive acceleration of the of the late-time universe Okay, and this exercise where notice that I just have two constants the Hubble constant It's 0 which defines the linear relation and this q0 which is an additional constant related now to the second derivative Nothing needs the knowledge of omega matter omega radiation omega now I don't need to tell you what is the energy density component of such a universe I am just using propagation of light and doing Taylor expansions So this is a very interesting exercise called cosmography and it is this exercise Which if you interpret in a lambda CDM framework, you will infer the presence of a positive cosmological constant in the cosmography framework You would simply say that the discovery that q0 is is negative Says that the acceleration of the late-time universe is so the second derivative is positive Okay, so let me take Like a brief pause and see if there are any questions because now it's the second part of the of the talk Nothing so far anything from anyone in the room Okay All right, so with this setup. Let me now go through Rapid review of what is called a standard model of cosmology Since this is firstly the first lecture and also I want to give you a feel for everything that is there without going into technical details I will mostly hand wave my way through the remaining part of the talk. Okay, so this is my first hand wave It's a pie chart and it shows you the distribution of Matter in the late-time universe in the standard model of cosmology So the stuff that we are made of is the berions The this forms about four to five percent of the energy density of the of the current universe Radiation which is in the CMB. It has such a small energy density today that you cannot even see it But if you go back to the first slide of this talk I showed you again an image stolen from NASA's website where they show you this pie chart at two different epochs Okay, so it gives you a feel for how energy densities evolved over time as well a large chunk of non relativistic matter is sitting in this CD this omega c for cold dark matter I will talk about this in some detail as we go along and Most of the energy budget of the late universe is believed to sit in a dark energy, which nobody really understands It is for all practical purposes as far as observations are concerned It is still consistent with being a cosmological constant Okay Neutrinos are also very interesting Maybe not for the kind of background cosmology that is indicated in this image But for structure formation and for some interesting neutrino physics coming from cosmology I will unfortunately not be discussing neutrinos here, but maybe someone has Questions during the discussion one can bring it up Okay, so this is again just in words and also in terms of the Friedman equation the components that we want to understand and the theme of the remaining slides is to tell you Observationally, what do we know about these components? Okay, so I will emphasize the observational aspect here I've just rewritten the Friedman equation here and I've introduced these constants now So earlier I had defined the omegas as the time dependent energy density of a particular component Divided by the time dependent Hubble parameter and the Newton's constant, etc now I have used my knowledge of the Evolution of each individual energy density component in terms of the scale factor and I have converted from scale factor into powers of 1 plus z Okay, and then I still have some unknowns and these unknowns are now actually constants They are dimensionless and they are constants So I will put the subscript zero in front of all of them to tell you that they have been evaluated at time T zero, which is current epoch Okay, and I think I have been consistent with this notation in this and the coming slides if I have not and if you find that there is a place Where you think I have written an omega matter, but it should be omega matter zero You please raise your hand and immediately ask me because there may be interesting places where you think it should be omega M Zero, but it actually has to be the time dependent quantity. Okay, so just pay attention All right So there are what one two three four constants in this equation plus one on the left hand side Which is the Hubble constant. Okay, so these five constants. We will try to track now The Hubble constant has had a very interesting history starting from Hubble's and LaMetre's original papers Down to through the 60s and 70s where it was the prime focus of cosmology and now even down to today Where in the last five years or so the problems of the Hubble constant have been revived revived again And there's a Hubble tension that we'll read about in the literature But I don't want to spend too much time on this So I will just say that the value is Conventionally parametrized in these very strange looking units They are strange until you realize the people who first wrote down this linear relationship between distances and redshifts They placed velocities on the vertical axis and distances in megaparsec on the horizontal axis So obviously the proportionality constant will be in kilometers per second per megaparsec okay, so that's the conventional unit for the Hubble constant and What one typically does is to characterize the uncertainty in the Hubble constant is to write it in units of 100 kilometers per second per megaparsec times a quantity little h which is dimensionless and its value is Approximately 0.7 plus or minus 10 percent or 20 percent or so and now with precision cosmology The errors are going down to 4 percent 3 percent So variations of 10 to 20 percent are statistically very interesting. Okay, so but for for me. I will just leave it at this Okay Before I talk about the other four constants Let me give you a crash course on what is called the thermal history of the universe Okay, this is a one-slide thermal history It's truly unfortunate because the thermal history of the universe is something that should be given an entire course of four or five lectures But we cannot do this. So I let me hand wave my way through this What one is dealing with in the hot big bang framework is a early phase of the universe where If the universe is expanding and it has a large size today an expanding universe would have been smaller and denser in the Past okay a dense universe is also hot therefore hot big bang big bang because if you You know extrapolate all the way down to very early times you will find a place where formally the scale factor becomes zero and Curvatures blow up Einstein's equations fail and you don't know what to do there So then that becomes the realm of quantum gravity so the and the The epoch at which classically a of t becomes zero is called the big bang Okay, there's a very interesting history for why the name was invented as well But at later times when we do understand some physics of of atoms and photons and atomic interactions with photons and nuclear Interactions, this is the epochs that we will discuss and at this time there are There are you know There are components like free electrons protons Hanging around in this cosmic bath There are photons which are interacting with these electrons and protons and everything is interacting with gravity because gravity Obviously affects matter gravity also affects photons through lensing and by changing the wavelengths of light Okay, so and in addition to this stuff which we understand We also think that there is a dark matter why we think there's a dark matter. I will talk about in the coming slide So in such a universe Which is expanding the temperature of this hot bath falls as a function of time and again There is a very simple theory that will tell you that if I track the distribution of photons Which is described very well by a black body the temperature of this black body in an expanding universe will behave like one over a Approximately all the time there are interesting times when it will not behave like one over a which I will not discuss here All these species are interacting with each other Okay, so here I'm showing you atomic interactions the same thing at earlier times you can apply to nuclear interactions as well Proton and a proton combining proton and a neutron combining to form dieterium for example Okay, similar kind of things will happen there so species which are interacting with each other will have some interaction rates And if these rates are very rapid relative to the expansion of the universe So if the rates gamma are much larger than the expansion factor H Then these interactions will remain in equilibrium Okay, so this is the key idea that that one follows when studying the thermal history And after this point that that particular reaction will in this jargon decouple from the rest of the bath So the thermal history of the universe is marked by several interesting epochs Where one after the other certain interactions fall out of equilibrium and decouple from the rest of the bath neutrinos are a classic example for example and electrons and positrons Annihilating is another example So for example, so for this particular case with electrons and positrons annihilating or other other species which might annihilate Such species their number density can completely disappear from the rest of the bath okay positrons for example will behave like this and This will also do interesting things to the remaining bath because once I have annihilations Annihilation means that for example an electron and a positron come together and release photons These photons now become part of the photon bath But now there are no more photons left to create positrons and electrons because the reaction rates have fallen to Substantial amounts so these photons are now excess photons and they create they create an excess amount of entropy Which is then shared with the rest of the bath. In fact, this is these are the epochs where he will not exactly fall like Okay, so I know I'm going fast it's deliberate you can ask me questions on this later the other interesting set of things that happens in this Plasma or fluid of photons and baryons is that this fluid supports sound waves because these Species electrons protons and photons are very tightly coupled to each other okay, so this whole combination behaves like a relativistic fluid and Relativistic fluid can support sound waves with a speed approximately C by root 3 Okay, so these sound waves for some reason in a homogeneous and expanding universe homogenous and isotropic universe if for some reason Sound waves are created at some early epoch These sound waves will propagate okay And why they are created and what causes that is the subject of the initial conditions Which will be handled in the lectures on inflation, but for us Let's just assume that there is some cause for creating for pinging this fluid at some initial time Because it can support sound waves these sound waves will propagate and Now physics tells us that this propagation will happen up to a certain epoch It turns out to be around 400,000 years after the big bang The reason this epoch is interesting is because by this time the universe has cooled to such an amount that electrons and protons which become a hydrogen atom Don't see any high energy photons which have energies more than 13.6 CV to come and ionize them Okay, so I have this particular reaction at very high temperatures can go in both directions I can have electrons and protons becoming a hydrogen atom and if the universe is very hot then there are many high energy photons available and Eventually a hydrogen atom will meet a high enough energy photon that it will get ionized and again release electrons and protons So this reaction remains in equilibrium at early times Around 400,000 years after the big bang the temperature of the universe is so low that there are very very few high energy Photons available now. Okay, only in the tail of the distribution and even that tail is rapidly Disappearing so now when an electron and proton come together, they'll form a hydrogen atom which is neutral and it will not get Ionized anytime soon so now the universe starts filling up with these neutral atoms and Electrons so and photons cannot talk to neutral objects, okay to neutral neutral quantities They can only talk to charge quantities So the photons that are now hanging around all these low-energy photons will simply go along the geodesics on which they're traveling and They will keep going and keep going until they meet your telescope or your instrument or your eyes Okay, so these are free streaming photons, which will go as far as they can This is the bath of photons that you see around you This is the cosmic microwave background and therefore the cosmic microwave background is a Snapshot of the universe as it was 400,000 years 380,000 years after the big bang, okay, so it's a very very important physical Point to remember so this particular epoch goes by several names It is called recombination because electrons and protons combine actually they combine for the first time into hydrogen It is called photon decoupling because these photons don't see any charge carriers anymore and they're free stream Sometimes it is also called the epoch of last scattering because this is the last time that photons scattered with with electrons Also Before this epoch there was a photon baryon fluid so the baryons were also connected to the photons Okay, and they were participating in these oscillations or the sound waves once the photons free stream There is nothing to support the the oscillations anymore Or the sound waves and these baryons are now just going to respond to gravity by this point the electrons and protons are Completely non relativistic so this is a non relativistic bath of matter which is going to respond to gravity and whatever the local gravitational potential is doing These baryons will go and follow it Okay, so I wanted to spend some time here because this aspect will be important later when we study structure formation so just to summarize these freed photons they free stream to us with a Black body shape of their spectrum even though they are not in equilibrium You can prove that their spectrum remains a black body with a temperature that falls like 1 over a okay So this is not a equilibrium temperature It is now just a parameter describing the distribution of these photons It's like 1 over a and it becomes 2.7 Kelvin today This is the CMB and these baryons locally will eventually condense and form galaxies But the Oscillations that happened before they will leave imprints in the spatial distribution of these galaxies and these are now observed in The late-time universe as what is called baryon acoustic oscillations All right, so I think I'm running out of time, but maybe I will five minutes, okay So I will go a little bit rapidly over Some of these aspects, but I will spend a little bit of time on dark matter because I wanted to do that in a little bit properly so We are now discussing the individual constants omega r 0 omega k 0 etc Okay, so the first one is radiation this can be understood in terms of a black body spectrum and You can see here observations of this black body spectrum from the Kobe Satellite the fire ass instrument aboard Kobe and the interesting thing here is that these error bars are 400 times magnified Okay So it is the most perfect black body known to human beings. You cannot produce such a black body in the lab And the temperature of this black body is what was measured with very high precision And it gives you a direct handle through conversion into the energy density on the constant omega r 0 Okay, so these numbers are given here. You can take a look at them later Barons until very recently were primarily probed by what is called predictions of Big Bang nucleosynthesis So these are very interesting calculations in the framework of the thermal history of the universe which were performed in the 50s and 60s by several people including George Gamov and his collaborators and These so there are these theoretical predictions for what the primordial densities of Certain nuclear species should look like and the reason there must be non-zero values of these species is because of the physics of Decoupling of reactions, okay This is the reactions involving these species would eventually decouple based on our knowledge of nuclear physics And those are the predictions that one used to write down what one expects and then So there are these theoretical curves in different colors All of them are parametrized by the overall bearing on density of the universe So the quantity omega B zero, okay, and this is the way the theory works And what is shown in the boxes are individual measurements of these primordial Components these measurements are also very interesting again You can ask me later on how each of them is done and what one Can see from this image is that more or less all of them are consistent with this blue band So all the black boxes are consistent with this blue band where they each intersect the corresponding theoretical curve So this blue band therefore gives you a prediction for omega B zero coming from Big Bang nucleosynthesis and Until very recently as in by recently I mean until like 20 years ago or so This was what one would call precision cosmology, okay, 10% errors on omega B zero Today we have much more precise measurements from CMB studies But the original BBN constraints are completely consistent with the constraints coming from CMB even though the latter are Ten times better than these okay. We are at one percent precision now In addition to the ordinary non relativistic matter, which is Bayreons We also believe that the dominant non relativistic matter today is in the form of cold arc matter So let me spend a few minutes on why this is the case Okay, so the way cold arc matter is understood is that it is a non-Mirionic component number one number two It is non relativistic for most of cosmic history All right, and it's dominant interaction with ordinary matter and radiation is through gravity And the reason we think that these things are correct is because otherwise The CMB anisotropy is which you saw earlier. I showed you that CMB anisotropies are at the level of one part in 10 to the Five if I interpret these anisotropies in terms of the density fluctuations of matter and radiation of that Photonberry on plasma these fluctuations are also at the level of one part in ten to the five What we will see in the third lecture is that the growth of structure in linear or quasi linear theory From that epoch four hundred thousand years after the Big Bang to today will only produce Evolution in the density contrasts of matter of the order of ten to the power three or ten to the power four at most At scales where we see that the CMB has ten to the minus five fluctuations Okay, so let me just write this down So I expect ten to the three to ten to the four growth At scales where the CMB tells me that there are ten to the minus five fluctuations In the initial conditions There is a factor of ten to one hundred missing in this accounting and there is no way that you can understand this factor Without introducing a species which has not been talking to the photons Okay, and it has not been talking to the baryons, but it has been gravitating at the time of the CMB Because only then can you create gravitational potentials which are deep enough that they will account for the excess that is missing From ten to the power minus five to one. Okay, so this this is the relative growth so what I should say is that this is the relative growth expected and This so ten to the minus five fluctuations and therefore ten to the five growth Expected this is Theoretically this is Okay, so as far as I understand this is the most robust evidence that there must be a species of non baryonic Matter which does not interact with photons. Therefore, it should be collision collision less Dark matter dark because you cannot see it. Otherwise if it interacted with photons, you would see it And it should also be non relativistic for most of history Otherwise it would have its fluctuations would have wiped out the fluctuations that you see in the CMB So there used to be hot dark matter models, which were very early ruled out once the spectrum of fluctuations on the CMB was discovered Okay, so there is a lot of sometimes confusion Regarding what is the most robust evidence of dark matter as far as I understand this is it There are several other pieces of evidence historically which were used to argue for the presence of dark matter The very first that I think Was done was by a Fritz wiki in the early 1930s Where he used the virial theorem to argue that observations of galaxy clusters are Inconsistent with the number of stars that you see in the galaxies forming these clusters Okay So what he did was to apply the virial theorem to velocity dispersions of these galaxies and Inferred a dynamical mass from these velocities by applying the virial theorem This mass turned out to be a factor 50 larger than the mass that he counted in the stars And he is the one who invented this term dark matter don't come out there in German. I think Later observations of these galaxy clusters showed that there is in actually a lot of mass contained in gas Which would not be visible in the optical but which is visible in x-rays This is very hot gas emitting in x-rays But even this amount of gas falls short of the total dynamical mass by a factor of 10 or so So the missing mass problem in clusters is consistent with there being dark matter in the 70s There were observations first done by Vera Rubin of Rotation curves of galaxies and this is the classic thing which we learn usually as students where rotation curves becoming flat Where it can be interpreted in terms of the presence of dark matter The problem is you with using this as the only evidence for dark matter is that there are alternate theories Which you may rule out for on other grounds Such as bond where they're where flat rotation curves will be predicted They are almost constructed to be to be produced as far as I know again Monde has a lot of trouble with with this statement here. Okay, so this is one interesting fact Scale-dependent master light. Let me not talk about Bullet like clusters is another thing which was proposed and it became very popular Two things to keep it keep track of so what is the idea here? You see the gas of a cluster you see the stars of the galaxies of a cluster now you look for clusters which have collided with each other and In this case the gas and the stars are separated from each other because the gas is collisional and the stars are collision-less Now you do weak lensing observations weak gravitational lensing and you trace out the mass distribution Inferred from these lensing observations and what you find is that this mass distribution shown by the contours is Different from the gas, but it kind of coincides with the stars So this is again consistent with saying that there is a dark component which must be collision-less There were again issues related to this firstly. This is just one object that part has gone away There are now many such objects known bullet like clusters The other problem is that there are non relativistic extensions of modern like theories which do Produce such images, okay? They may not do it as well and they may be very complicated But you can't say that this is the only way of doing it again These kind of theories will not be able to produce the CMB anisotropies and it's a very simple argument It's just an order of magnitude which is which is missed. So that's why it's so robust okay cosmological constant is Something that that has now been established Well, the cosmological constant like component has been established primarily by Cosmography of standard candles such as type 1a supernovae. So again, I will not talk about this in detail But this is a measurement which was given the Nobel Prize All right, so let me not say too much more here But things are considered observations are consistent with there being a cosmological constant with a value of omega lambda of about 0.7 Spatial curvature is very interesting It is very tightly constrained by the cosmic microwave background anisotropies So these are things that I will I think discuss a little bit later And maybe you will have a course on this next week The location of the first peak here which I have not described to you But you can just think of whatever these measurements are they are a function of angular scale Okay inverse angular scale and the location of this peak is like a standard ruler in the sky This is the typical scale over which CMB anisotropy is fluctuate in the sky So this standard ruler subtends an angle to us, which is about one degree in observations So if I apply my logic of angular diameter distances, I can try to constrain the corresponding cosmological parameters and the primary one now that can be constrained is a combination of the curvature and the Hubble parameter if you try to Infer the value of the Hubble parameter from some other probe or now also consistently with the CMB You will find that there's a very tight constraint on the value of k where if I made k very positive I would move this peak in one direction with negative k. I would move it in the other direction So the observation gives me the value of omega k. It turns out to be very close to 0 okay, so This is just a plot of The standard model and the expansion of energy density the total energy density of the universe in the standard model So I think I will just leave you with this different components are shown and you can see how they enter how they behave relative to each other I'll stop here. Okay. Thank you. I see. Yes, so Since we are a bit late, maybe we can Keep the questions for the for the discussion session No, I think we where is the break on the terrace, right? I Guess we have up. There's on the terrace. There will be a Coffee break for those who are present. Thank you. See you at 1150