 Okay, it's a pleasure to introduce the first lecturer with Professor Barbara Raden and she's going to talk about introduction to cosmology. Thank you. It's an honor to be the first lecturer of the entire school, but before I begin I have one very important question. Can you hear me in the back? Yes, I hear nods and waves. The other important question is can you see my slides? For this talk I'm going to be using PowerPoint. I'll reserve the blackboard in case I need it for any questions that you have. I try to use large font in my PowerPoint primarily to avoid the temptation to shove too much in. Because of course this is a lecture that's going to have to cover a great deal of ground. It's customary at this point for the speaker to thank the organizers for inviting them to give lectures. But I realize that the organizers have asked me to do an impossible task. I'm supposed to give you an introduction to cosmology. However I'm sure that all of you have been introduced to cosmology at some level. Some of you are very well acquainted with cosmology and some of you I'm sure have long term very close friendships with cosmology. However I want to make sure that so for the rest of the two weeks that the school lasts that we all have the same common background knowledge about cosmology. So I'm sure that all of you have heard some of what I'm going to be talking about today. And I would bet that some of you have heard all that I'm going to be talking about today. However it's necessary that we go through this material and to keep things interesting I'll try to throw in little historical tidbits now and then. I'm assuming I was asked to give a course or a series of lectures on introduction to cosmology because I've written the textbook entitled Introduction to Cosmology. They probably googled Introduction to Cosmology and found the cover of the textbook. Anyway this invitation was very well timed since I am in the process of producing a second edition of my textbook. So my head is jammed full of various cosmological bits. Second edition published December of this year Cambridge University Press makes excellent presents for all of your friends. Okay after that commercial announcements there's the public service announcements. What I'm going to be talking about for the course of the four lectures today I'll be talking about what we might call the standard model of cosmology. So I put standard model in quotation marks because there's not a standard model of cosmology the way there's a standard model of particle physics. However in recent decades cosmologists have converged upon a common or standard model of how the universe evolves how it behaves. And I would argue this is a standard model that consists of two parts. The first larger part I'll be talking about this morning the hot big bang model which stated simply says the universe began a finite time ago in an extremely dense extremely hot state and has been expanding continuously since then. However over the past couple of decades astronomers have developed a specialized subset of hot big bang models. And now the standard way in which we describe the universe is the Lambda CDM model. Lambda stands for the cosmological constants a form of dark energy. CDM is cold dark matter. So the universe began expanding and today we look around the universe and the evidence that I'll be going into this afternoon indicates that most of the energy density of the universe is in the form of dark energy possibly a cosmological constant. And the remainder of the universe consists of matter that is dark doesn't absorb emit or scatter photons and is cold. It's been non-relativistic since very early in the history of the universe. In a hot big bang model it's continuously expanding the mean particle energy decreases with time. And Tuesday tomorrow I'll be talking about those moments in the universe that are special because the energy drops to an interesting level. So when the energy drops sufficiently low below 13.6 electron volts the ionization energy of hydrogen you have recombination. And by looking at the cosmic microwave background we learn a lot about that special epic. Going back further in time when the energy drops sufficiently low between sufficiently low below 2.2 MeV the dissociation energy of deuterium then you had an era of big bang nucleosynthesis and the primordial abundances of light elements like deuterium and lithium tell us what the universe was like back then. And finally just a few words about inflation. There's an entire series of lectures on inflation but I'll just give you the background of what inflation is and why people thought it was a really good idea. And finally Wednesday at last I'll be abandoning the fiction of cosmologists that the universe is homogeneous and isotropic. And I'll be talking about the formation of structure, the formation of super clusters, voids, galaxies and little teeny tiny things like stars. Warning I am an astronomer. Don't panic and bolt for the exits. You just need to know that astronomers are like everyone else only they use a very strange language. So in my talks I'll be taking the point of view of an astronomer not a theoretical particle physicist and also I'll be using some of the units commonly used by astronomers. So in cosmology sometimes we have to talk about very large length scales and very long time scales and very large masses. So the basic time unit I'll be using is the giga year, 10 to the 9 years, about 3 times 10 to the 16 seconds. Or to use fundamental units of time it's about 10 to the 60 times the Planck time. So in Planck units we'll be talking about very long time scales indeed. The giga year is useful to astronomers because many astronomical objects are giga years old for instance the sun about 4.57 giga years. And the age of the universe that is the time that's elapsed since the beginning of the expansion in the Big Bang about 13.7 giga years. The unit of distance frequently used in cosmology is the megaparsecs 10 to the 6 parsecs about 3 times 10 to the 6 light years or 3 times 10 to the 22 meters. And again in Planck units it's pretty big about 10 to the 57 times the Planck distance. Examples of the megaparsec in action from here to the Andromeda galaxy the nearest galaxy neighbor. The nearest neighboring galaxy comparable in size to our own about three quarters of a megaparsec. The distance from here to the Coma cluster of galaxies about 100 megaparsecs. So fix in your mind a megaparsec is a useful distance for measuring the distance between galaxies 100 megaparsecs. That's about the scale of distances between rich clusters like the Coma cluster. The standard unit of mass is the solar mass. The circle with the dot in the center that's the astronomers symbol for the sun. It's of great antiquity. Historically speaking it's the ancient Egyptian hieroglyph for the sun god Ra. So astronomy goes back really really far into human history. And a lot of the terminology used by astronomers is kind of archaic. The mass of the sun about 2 times 10 to the 33 kilograms. Notice that in Planck units it's not as humongously big. The sun's mass is about 10 to the 38 times the Planck mass. And so when we're talking about reasonably big objects like galaxies and clusters of galaxies. The mass of the Milky Way galaxy about 10 to the 12 solar masses. The mass of the Coma cluster of galaxies contains many thousands of galaxies. And it has a mass of about 10 to the 15 solar masses. So these are some of the time scales, length scales and mass scales. You should have in the back of your head when I go off into my astronomical tangents. It's commonly stated that the hot big bang model has a foundation made up of three observations. Olbers paradox, Hubble's law and the cosmic microwave background. So I'm going to go through all three of those in turn very briefly. Olbers paradox is simply the statement that the night sky is dark. It's named after Friedrich Olbers who wrote quite a long paper about the darkness of the night sky near 1823. However, Olbers was neither the first person to worry about the darkness of the night sky nor the last person. He wasn't even the first person to give a correct explanation for the darkness of the night sky. But you know, science is like that. Names sometimes get arbitrarily attached. Now when I say the night sky is dark, I can quantify that. Go above the Earth's atmosphere, pick out a seemingly black patch of sky where there are no bright stars. And the flux of light that you receive from that patch comes to about 5 times 10 to the minus 17 watts per square meter of your telescope per square arc second of the sky. Compare that to the brightness of the sun. The surface brightness of the sun is about 5 times 10 to the minus 3 watts per square meter per square arc second of the sun's disk. So saying that the night sky is dark is saying, oh, it's about 14 orders of magnitude lower than the surface brightness of the sun. Most of human history, people did not think that the darkness of the sky was paradoxical at all. In the geocentric model of the ancient Greeks, illustrated here on the left in a 16th century engraving, the night sky was dark because, well, there's this celestial sphere, a thin spherical shell centered on the Earth, and that celestial sphere is dark. Aside from some thousands of little bright points of light attached to it, the lights that we call stars. So no paradox there. We only see a few thousand stars in the night sky, the ancient Greeks said, because there are only a few thousand stars out there, and they're all stuck down or somehow embedded within the celestial sphere. Things changed, however, in the heliocentric model of Copernicus. One of the followers of Copernicus, an English astronomer called Thomas Diggs, realized that, oh, wait, we don't need a celestial sphere anymore. When you look up at stars at night and they go in circles around the celestial poles, that's not the rotation of a celestial sphere. That's the reflection of the rotation of the Earth. And therefore, Diggs said the stars can all be at different distances, the brighter stars are the ones closest to us, the dimmer stars are the ones further away, and Diggs concluded we can have an infinite universe containing an infinite number of stars. He wrote it down in this wonderful Elizabethan language. Let me read it to you. This orb of stars fixed infinitely up extended itself in altitudes, spherically, and therefore, immovable the palace of felicity, garnished with perfect shining, oh, I can't even read it, glorious lights innumerable. So the stars are glorious lights, but they are innumerable. Diggs means this literally. You cannot number them because they are infinite in number. And when you have an infinite universe filled with an infinite number of stars, that's where the paradox comes in. Let's do a little calculation, just a simple one to get us warmed up on Monday morning. If you're in an infinite universe filled with an infinite number of stars, how far can you see on average before your line of sight intersects a star? Okay, first bit of astronomical information, sorry I'm slightly jet lagged this morning. A star is an opaque sphere that glows in the dark. Okay, got that? Now, my astronomical colleagues who study stars will point out, of course, stars come in all sizes, from little dwarf stars to enormous super-giant stars, but let's take the Sun as a typical middle-sized star. The Sun's radius, 700,000 kilometers, in my favorite units of length, the megaparsec, that comes at 2 times 10 to the minus 14 megaparsecs. The number of stars per cubic megaparsec, we don't know exactly, not the sort of thing you can do counting stars one by one, but if you take a big cube of space, a few hundred megaparsecs on the side, count up the galaxies, estimate the number of stars per galaxy, it comes to an average of about a billion stars per cubic megaparsec today. There you are, you're looking outward along a line of sight. And of course, if you draw a cylinder whose radius is equal to the Sun's radius around that line of sight, if there exists a star whose center is inside that cylinder, then it's going to block your view of more distant stars. Of course, it's just the question, how big does this cylinder have to be? What does it volume have to be in order for it to contain a star on average? Well, it's just a mean-free path problem. The distance lambda that you can see typically before your line of sight is stopped by this opaque star, one over the number density of stars times the cross-section of a star, so 10 to the 9 per cubic megaparsec times 10 to the minus 27 square megaparsecs, so it's a cross-sectional area of a star, and it comes out to 10 to the 18 megaparsecs, 10 to the 18 megaparsecs. Even to an astronomer, that's a very long distance. However, it is a finite distance. And so Ulbers and other people who studied this problem after Thomas Diggs realized that in an infinite universe, or one that extends at least 10 to the 18 megaparsecs in all directions, the sky is going to be paved with stars. Every direction you look, you're going to see a star. And of course, since Euclidean geometry tells us the flux we receive from a star falls off as one over the square of distance, and since the angular area subtended by a star falls off as one over the square of distance, then the surface brightness of a star in watts per square meter per square arc second should be independent of distance. So the sky by this analysis should have a surface brightness as great as that of the sun everywhere. Let's imagine paving the entire sky with suns. Obviously, the sky brightness in our own universe is smaller than this value by 14 orders of magnitude. I frequently make calculations and get the wrong answer, but being off by 14 orders of magnitude, yeah, that's really bad. And so the calculation is correct. So it's a question of garbage in, garbage out. One of my assumptions that went into that calculation has to be wrong. So which of the assumptions that I made was incorrect? Of course, we could sit around and brainstorm wondering what the actual resolution of Ober's paradox could be. I've just put up four of the most obvious resolutions of the paradox. Answer number one, that was the one given by Ober's himself in the 1820s. He said, well, if there were some sort of opaque screen between us and the more distant stars, that would hide distant stars from our view. And therefore, that could be a resolution of Ober's paradox. Unfortunately, this is a resolution that doesn't work in the long run. If you put an opaque screen between us and distant stars, the reason the screen is opaque is because it absorbs lights and the lights heats up the screen and it's the equilibrium state. It has a surface temperature equal to that of the surface of a star. And it's two glows at 5 times 10 to the minus 3 watts per square meter per square arc second. So, well, if you try to screen off distant stars, the screen itself heats up until it becomes as bright as a star. However, you can resolve it using method number two. You can just say, OK, so the universe isn't infinite. If the universe comes to a halt at some distance much smaller than 10 to the minus 18 megaparsecs, then the night sky will be dark. Notice this resolution also works if the universe is infinitely large, but for some reason stars only occupy a small finite value. A third possible resolution, slightly more subtle than the previous one, is that perhaps the universe is of finite age. Since light travels at a finite speed, if the age of the universe is some time t, then if c times t is much smaller than 10 to the 18 megaparsecs, then stars beyond this distance won't have had time to send light to us. This resolution also works if the universe is infinitely old, but for some reason stars only have existed for a finite length of time. Fourth possible resolution, well, maybe distant stars just have lower surface brightness. Maybe we're looking out there, we're seeing the surface of stars, but they're really, really dark for some reason. Well, in fact, the reason why distant stars have a lower surface brightness is just, well, first of all, you can have non-Euclidean geometry. Also, you could have an expanding universe, and the light from distant stars is redshifted to lower photon energy. Well, in our particular universe, well described by a hot big bang, the resolution is mostly number three. It's mostly due to the fact that the universe is of finite age, so stars have existed for less than 14 billion years. And it's a little bit of four. More distant stars have a lower surface brightness because of tolman fading, an effect you see in an expanding universe. But primarily it's just the fact that we live in a universe of finite age, unlike the eternal universe that was assumed by Ulbers in computing, oh yeah, you know, the sky should be bright at night. So Ulbers' paradox is putting limits on our universe. It can't be infinitely large and infinitely old and obey Euclidean geometry all at the same time. Something's got to give. Well, the thing that gave became apparent in the 1920s with the discovery of Hubble's law, the fact that distant galaxies have red shifts in their spectrum and that red shift is proportional to our distance from us. I told you all you need to know about stars is that they are opaque spheres that glow in the dark. You also need to know that they have absorption lines in their spectra from the hydrogen and other elements in their atmospheres. So galaxies contain stars, galaxies have absorption lines in their spectra if they're active galactic nuclei then they also have emission lines from the hot gas that the central black hole is gobbling. So it is, in most cases, relatively easy to measure the wavelength of the absorption and emission lines from galaxies. So suppose an emission line or absorption line has a rest wavelength lambda sub e. For instance, Lyman alpha has a rest wavelength of, gosh, now I've forgotten, I am jet lag, lambda sub e. And you measure that same absorption or emission line in the spectrum of a distant galaxy and it's some value lambda sub zero, which in general is not equal to lambda sub e. And the red shift is just wavelength that you observe minus the wavelength in the rest frame of the light source emitting the light normalized by dividing by lambda sub e. Now technically this number z is only a red shift if z is greater than zero. That means you are shifted towards longer wavelength. So for lines in the visible range of the spectrum the longest wavelength of visible light is red. If z is less than zero you're shifted to shorter wavelengths which logically should be called a violet shift but astronomers inevitably call it a blue shift probably because it's easier to say. The first person to measure the red shift of a galaxy was Vesto Schleifer and this is his spectrum of the Andromeda galaxy. He was doing something really difficult with the technology of the time. That's smudge running horizontally. That's the spectrum of the galaxy. This is a photographic plate so that's pale vertical line. That's an absorption line there. The more easily seen spectra above and below those are emission line spectra used to calibrate the actual spectrum of the galaxy. So Vesto Schleifer was doing something really difficult and he was interested in taking the spectrum of galaxies from 1912 onward in part because people weren't sure exactly what they were. The Andromeda galaxy wasn't called the Andromeda galaxy it was called the Andromeda nebula there were other spiral nebulae the Andromeda galaxy out there and by 1923 Arthur Eddington in his influential textbook The Mathematical Theory of Relativity was able to make up a list of 41 spiral nebulae galaxies and here's their catalogue number their right ascension and declination where they are in the sky and here's their red shift or blue shift just as the equivalent radial velocity notice most of these numbers in the right hand column are positive numbers of the 41 galaxies whose spectra had been measured mostly by Vesto Schleifer 36 red shifts, 35 blue shifts an interesting problem in combinatorics if you toss a fair coin 41 times how likely are you to get 36 or more heads that's something like a chance of one in a million so if this were just a random selection from galaxies some with red shifts and some with blue shifts you would not expect to see what Eddington called the great preponderance of positive receding velocities it's very striking he said not merely that red shifts were so very common compared to blue shifts but also if you take the average radial velocity of these 41 galaxies over 500 kilometers per second by contrast if you look at stars in the solar neighborhood they're typically moving around 30 or 40 kilometers per second relative to the sun so the velocities you deduce radial velocity equals c times z for these galaxies one mostly positive two very high in velocity compared to motion of stars within a galaxy like the Milky Way now in 1923 the distances to galaxies wasn't well known it wasn't known whether they were large distant objects comparable in size to our Milky Way galaxy or if there were small satellite galaxies around the Milky Way galaxy but Edwin Hummel decided oh I'm going to determine the distance to galaxies using a Cepheid variable star as a standard candle a standard candle more about them later they're just objects whose luminosity you know and Edwin Hummel was looking at the Andromeda galaxy again this is a photographic negative so the black blob in the middle that's the bright central bulge of the galaxy on this photographic plates taken on the 6th of October 1923 he marked a couple of Novi and Fornova these are new stars that weren't there before notice up at the top however first he wrote down N he thought it was a Nova but then he wrote down V-A-R exclamation point really excited this is a variable star it's not a one-shot Nova becomes bright and then disappears but it's a variable Cepheid star and by comparing its flux to the flux of similar Cepheid's within our own galaxy he was able to make an estimate of the distance to the Andromeda galaxy his conclusion was the corresponding distance that is the distance the Andromeda galaxy about points 285 mega parsecs well it wasn't quite right the best estimate today is about three times but at least he was within an order of magnitude so Hubble now has a tool he has a standard candle he can use to estimate the distance to galaxies and of course you're an astronomer you have a tool you start using it and by the year 1929 Hubble published a famous paper in which he had not merely the measured red shifts and in a few cases blue shifts of nearby galaxies but he also had distance estimates and off on the left you see the plot from his 1929 paper the axis labels aren't very legible horizontal that's distance 0, 1 and 2 mega parsecs vertical axis that's the radial velocity c times z 0, 500, 1,000 kilometers per second and the two diagonal lines that's his best fit to the data using different weighting schemes for the individual galaxies so he says not only is there an excess of red shifts over blue shifts however also the blue shifted galaxies are relatively close to us and as you go further away you find that the red shifts are larger and larger and larger there's a lot of scatter in this plot due to the peculiar velocities of galaxies they're being tugged to and fro by their neighbors the reason why the Andromeda galaxy actually has a blue shift rather than a red shift is that our galaxy and the Andromeda galaxy are falling towards each other so random gravitational tugs of this kind there's inevitably going to be scatter in this plot however flash forward to the 21st century using better distance estimators Hubble was chronically underestimating distances and using measured red shifts you still see a similar linear relationship between velocity c times z and distance Hubble's law is important in a hot big bang model because if you write down the relation the estimated radial velocity c times red shift linearly proportional distance the proportionality constant h sub zero sometimes called h naught called the Hubble constant a tribute to Edwin Hubble and I'm just going to say 68 plus or minus 2 kilometers per second per megaparsec of distance you can get into some interesting debates about so the actual value of h naught but most values come to about 68 so I'll assume that value if you take the inverse of this well kilometers and megaparsecs both have dimensionality of distance so 1 over h is just going to be a time scale which turns out to be about 14.4 giga years an interesting time scale because well if a galaxy is moving away from you with a velocity that's proportional to its distance you work out if its speed has been constant how long has it taken to reach that distance from you and you find out oh it's a time that is equal to 1 over h naught and it's independence of the current distance between two galaxies if you think of running the film of the universe backwards at a time about equal to the Hubble time ago all the galaxies were crammed together to a small volume this is an estimate of course that assumes the relative velocity is constant which isn't actually but it turns out to be a pretty good approximation for our universe if you have a time span built into your expanding universe you also have distance built into your expanding universe c times the Hubble time 4,400 mega parsecs Edwin Hummel he's an astronomer he observes redshift proportional to distance what does this imply for our universe? well Hummel's law is the result of a uniform expansion well that's homogeneous and isotropic so if for instance you take three points in space one two three connect them by geodesics to make a triangle uniform expansion tells you that the shape of the triangle remains constant as it expands the fact that it's pure expansion with no rotation means the orientation of the triangle remains constant as it expands and so this kind of uniform homogeneous isotropic expansion tells you that the distance between any pair of galaxies increases as a scalar function a of t called by cosmologists the scale factor it's the same for every pair and it's a function only of time and not of position so every triangle everywhere in the galaxy undergoes homogeneous expansion it's isotropic means that a is a scalar it's not a tensor it doesn't imply expansion rights different in different directions and again there's no shear no rotation just pure expansion so if every distance is proportional to this scale factor a of t we can normalize it in my talks I'm going to be using one of the possible standard normalizations that the scale factor is equal to one at a reference time t sub zero which equals now so you have homogeneous isotropic expansion it is described by this simple scalar function of time what's the relative velocity of any two points well you take the time derivative and you find out the velocity of galaxy one relative to galaxy two is proportional to the distance between those two galaxies and the proportionality constants let's call it h for Hubble is just the time derivative of the scale factor divided by the scale factor itself so a little bit more cosmological jargon this function defined as a dot over a is called the Hubble parameter generally it will vary with time in an expanding universe and the value of the Hubble parameter is measured right now by Edwin Hubble or more accurately by Wendy Friedman and her collaborators this is called the Hubble constant so the Hubble constant is the value of a time varying parameter as measured at a particular time now when Hubble published Hubble's law it did that immediately result in people saying oh hot big bang theory because although Hubble's law is consistent with the big bang model for the universe it doesn't demand it if the universe is expanding today that does not necessarily imply that it began a finite time ago in an extremely dense state and in fact in the mid 20th century there were two competing models for the universe people who supported the hot big bang model embraced what's called the cosmological principle it's called the cosmological principle it's one that's very important to modern cosmology and the cosmological principle is just the assumption that once you get to sufficiently large scales about 100 megaparsecs or so the universe is homogeneous and isotropic however in a hot big bang model things can change with time in fact they do change with time the universe started out a lot hotter and denser than it is now however the competing theory the steady state theory as proposed by Bondi and Gold Fred Hoyle in 1948 went a step further and embraced not merely the cosmological principle but the perfect cosmological principle that not only is the universe homogeneous and isotropic in space but it's also unchanging in time so that global properties of the universe like the Hubble constants and the average density and the temperature of the universe are consistent as a function of time which is an interesting principle could be philosophically attractive but is it consistent with observations over the next two weeks we're going to learn a lot of wonderful theories some of it a little bit speculative but in the end if your theory contradicts observations nobody's going to be particularly interested however in 1948 the steady state theory wasn't entirely crazy if you assume the perfect cosmological principle you have to assume that the Hubble constant is constant with time really in that case the relative velocity of any two points goes as H0 times r where H0 is constant time derivative of something is a constant time something that's exponential growth which is interesting also the steady state model assumed that the average density of the universe is constant with time and that implies since volume increases exponentially that you have to be creating matter as a volume of the universe expands massive particles pop out of nowhere in order to keep the density constant so notice that exponential growth are the distance between two galaxies approaches zero only asymptotically as you go to t equals minus infinity so there is no instance when the universe began expanding and if you plug in the observed Hubble constant and the observed mass density of our universe means you have to take you have to create about 6 times 10 to the minus 28 kilograms per cubic meter per giga year a sufficiently low rate that you can't just sit around and stare at a cubic meter of space in the hopes of seeing, I don't know a hydrogen atom pop out of nowhere so you couldn't observationally verify the steady state model in that way just by looking for the creation of matter in 1963 the famous cosmologist, Malcolm Longer was just a small graduate student starting out and his thesis advisor told him there are only two and a half facts in cosmology so only a little more than half a century ago cosmology had only two and a half facts so cosmology schools back in 1963 would have been really really short the two and a half facts that were known then is, you know, number one the sky is dark at night, Ulber's Paradox number two, galaxies have a redshift proportional to their distance, Hubble's Law and the additional half a facts known in 1963 was that contents of the universe have probably changed as the universe grows older so the perfect cosmological principle that things don't change was probably false this was only a half a fact in 1963 because at that time radio surveys of active galactic nuclei and quasars were only just beginning to get sensitive enough to measure high redshift galaxies in 1963 the record holder for the highest redshift galaxy known was that little fellow right there the radio galaxy 3C295 with a redshift of a little less than a half so that's a light travel time of, you know, less than 7 giga years and so people were beginning to realize, you know there are a lot more radio galaxies and quasars in the past than there are now but the statistics were not yet very firm and so this half a fact things have changed in the universe at the time wasn't elevated to a full fledged fact until the discovery of the cosmic microwave background by Arno Penzias and Bob Wilson in the year 1965 so first of all they discovered that the cosmic microwave background was isotropic and in order to detect microwaves well you need to get above the earth's damp atmosphere water molecules are very good at absorbing most wavelengths of microwaves but the COBE satellites measured the spectrum of the cosmic microwave background in a wide range of frequencies and found that it was extremely well fitted by a black body spectrum that is the distribution of photon energies had a Planck distribution or equivalently it had a Bose-Einstein distribution for massless bosons so here I've expressed it as the number density of photons as a function of frequency and now this is just a gorgeous beautiful black body curve and the temperature is 2.7255 plus or minus 0.0006 Kelvin very well-mentured temperature black body microwaves why does this tell us that the universe has evolved with time well black body spectra are produced by opaque objects in which the photons and the absorbers and scatterers come to the same temperature kinetic equilibrium and so if you want a black body spectrum you have to have an opaque source stars are opaque objects they glow in the dark they're pretty good black bodies although they're chopped up by absorption lines and the cosmic microwave background is a beautiful black body which means that the early universe was opaque now the universe at most wavelengths is highly transparent otherwise you wouldn't be able to see galaxies at a redshift of a half why was the early universe opaque it's because the baryonic matter in the universe in the early universe it was hot and dense so the baryonic stuff was ionized jargon alert baryonic matter is matter that's made up of protons, neutrons, and electrons so you're made up of baryonic matter only the protons and neutrons of course are baryons the electrons are leptons however since a baryon a proton or neutron is over 1800 times the mass of the electron cosmologists refer to it as baryonic matter for its most massive component however the electrons cannot be forgotten because they're the ones that interact most readily with photons free electrons can scatter photons of any frequency with a Thompson cross section and so in the early universe it was ionized it was dense, you had a high density of free electrons and so go far back enough in time the rate at which photons scattered from free electrons was greater than the expansion rate of the universe as expressed by the Hubble parameter now, equivalently that means the mean free path of the photons before scattering was very short compared to the Hubble distance back then so saying the early universe was opaque is a statement about the relative distance of the mean free path and the Hubble distance things have changed back then the universe was opaque now it's mostly transparent this of course is a severe violation of the perfect cosmological principle and it was the discovery of the cosmic microwave background in the mid 1960s that really made the hot big bang model the preferred model for cosmology the steady state model in which everything changes in which the average density is constant as a function of time because hydrogen atoms pop out of nowhere and it was no longer favored after the mid 1960s cosmic microwave background a very rich source of information worthy of oh I don't know a series of four lectures say I'll just run through the basics if you look at the entire sky at microwave wavelengths as the W map satellite and the Planck satellite have done you can make a map of the sky this is a map of the sky in galactic coordinates the Milky Way runs horizontally the center of our galaxy and the constellation Sagittarius is at the center the blobby Planck line going from side to side that's foreground emission from gas within our own Milky Way galaxy it's mostly synchrotron emission which has a spectrum very different from a black body so it's relatively easy to track the foreground emission in the background you see it's white on one half of the sky black on the other half this grayscale map is a map of temperature although the average temperature of the cosmic microwave background is 2.7255 kelvin on half of the sky it's about three and a half millikelvin hotter than average the other half it's about cooler than average so you have a dipole anisotropy in the cosmic microwave background one hemisphere of the sky is about one part per thousand hotter than average and the other hemisphere is a part per thousand cooler than average this dipole is interesting it's telling us something interesting about the universe but it's telling us something interesting about the local universe the cmb dipole anisotropy is due mainly or perhaps almost entirely due to the Doppler shift from our motion through space the w-map and Planck satellites are at the L2 Lagrangian points on the far side of the earth from the sun and so w-map was still is for that matter going around the sun about once per year that requires a velocity of about 30 km per seconds the sun is going around the center of our galaxy about 220 maybe 240 km per seconds and the reason why the Andromeda galaxy is blue-shifted relative to us is that our galaxy and Andromeda are falling towards each other and so our galaxy's motion relative to the center of mass of the Andromeda Milky Way system is about 80 km per second so subtract all these well-known motions in vector space and you find out that the local group of galaxies a little cluster of galaxies containing the Milky Way and the Andromeda galaxy is moving in the direction of the constellation Hydra with a speed of about two thousandths of the speed of the lights about 630 km per second so why Hydra what's so great about that constellation here's another view of the sky same map center of our galaxy at the center again and the red circle here that's the CMP dipole that's the direction in which the local group is moving all of these other blobs, splotches these are galaxies whose redshift is known and you ask what's around here oh it's the Hydra cluster a rich cluster of galaxies the Centaurus cluster together they're part of the Hydra Centaurus super cluster of galaxies all these clusters are labeled by their redshift now Hydra redshift of 0.01 Centaurus 0.02 these are two of the nearest which clusters of galaxies they represent a nearby concentration of mass no wonder we're being gravitationally accelerated towards them so CMP dipole but it's telling us about those clusters of galaxies over there not really telling us what the universe was like at the moment it became transparent however if you subtract away the dipole anisotropy of the cosmic microwave background you get this famous map of the sky at microwaves this is from the Planck satellites the most recent highest resolution map of the cosmic microwave background notice the color scale red equals higher temperature which should equal shorter wavelength so why is it red I don't know it's very irritating when they do this scale blue is lower temperature I know it's because red is psychologically hot and blue is psychologically cold but it's still deeply irritating okay it's blotchy it's blobby but notice now the dipole the scale was plus minus 3.5 millikelvin here the scale is plus minus a half a millikelvin so these small angular scale anisotropies are also lower in amplitude than the dipole so the CMB anisotropy on small scales once you subtract away the dipole is telling us what the universe was like when the photons of the cosmic microwave background last scattered from a free electron and if you look at angular scales between 5 degrees and 180 degrees so smaller than the dipole bigger than 5 degrees you find out that the temperature fluctuations on those angular scales come to about one part in 100,000 I'm ignoring the smaller scale stuff here because as we'll see later on on small scales you get interesting baryonic physics interaction of photons with electrons and through them through the protons but on these angular scales between 5 and 180 degrees this is telling you about the density fluctuations and hence the potential fluctuations in the dark matter at the time of last scattering of photons so the fact that the temperature fluctuations are low in amplitude about a part in 100,000 is also telling you that these potential gravitational potential fluctuations were low in amplitude about one part in 100,000 so back then the universe was very smooth the only density perturbations you had were very low in amplitude today the universe is very lumpy it's got very dense lumps like stars and people it's got bigger slightly less dense lumps like galaxies and you've got structure on all scales so it's extremely over dense indeed your density is about 10 to the 30 times the average baryonic density of the universe so the story of the universe then it was hot dense now it's cool and lower in density then it was smooth now it's lumpy it's all badly in violation of the perfect cosmological principle so the cosmic microwave background provides very strong evidence that the universe is in fact evolving with time it's evolving in the direction of becoming less dense on average the background temperature is dropping as the photons of the cosmic microwave background cool as the universe expands and although the average density of matter is dropping as the universe expands you're getting higher amplitude density fluctuations so to sum up the observational evidence over as paradox darkness of the night sky plus Hubble's law plus the existence of a cosmic microwave background led to the adoption of a standard model a universe that's very well described by a hot big bang model started out hot dense a finite time ago around 14 giga years so the cosmological principle the assumption that things are homogeneous and isotropic today really works on large scales scales of around 100 megaparsecs or more or about 2% of the Hubble distance or more in the past however you look at the cosmic microwave background it gives evidence for a universe that was more nearly homogeneous and isotropic than it is now so thanks to the cosmological principle cosmologists use as their starting assumption that okay we'll start with a basic model that's homogeneous and isotropic and then later on we'll add on the bells and whistles the super clusters and voids and clusters and galaxies and little lumpy cosmologists the cosmological principle that things are homogeneous and isotropic is so very very very useful and in fact it was adopted long before there was any evidence that the universe is actually observably homogeneous and isotropic on large scales it just makes the mathematics so much easier in the context of general relativity the expansion of a homogeneous and isotropic universe is described by a Robertson Walker metric and by the Friedman equation so very brief run through of the Robertson Walker metric and the Friedman equation in the context of general relativity you want to know what the curvature of a four-dimensional space time is this is sort of key to the whole project of general relativity and you can describe the curvature of space time with a metric just the mathematical relation that tells you the length of a geodesic between two points the locally shortest distance between two points obviously a simple space is a two-dimensional Euclidean space like the floor of this room if I pick one point over here and another point over there what's the shortest distance between them you can pull a string in between them or you can use the laws of Euclide you have one point x y the other a little bit further away x plus dx y plus dy and so I know the answer to this one it's just the Pythagorean theorem square of the hypotenuse is equal to the sum of the square of the other two sides so this is the metric for Euclidean space and a little reminder if your gr is rusty that I'm leaving out the parentheses dx squared is the square of the infinitesimal quantity dx and not a infinitesimal change in the quantity x squared you can change coordinates you can write down this small distance ds in polar coordinates looks different written on the page but it's a simple coordinate transformation we can expand into three dimensions again in Euclidean otherwise known as flat space the distance between two points is just the three-dimensional extension of the Pythagorean theorem and again you can make a coordinate change it's frequently more useful to use spherical coordinates so here theta is your polar angle phi is your azimuthal angle and in space time well in a universe without gravity in the universe of special relativity the distance between two events in a four-dimensional space time is just given by the Minkowski metric ds squared equals minus c squared dt squared plus the spatial component and here I'm using the convention that the spatial term is positive time is negative there are people who use the opposite sign convention but it's just essential that the dt squared term is of opposite sign to the spatial terms and again can be expressed in spherical coordinates okay so all very simple stuff Euclidean space is mathematically simple very elegant but the universe of course is not Euclidean and in general space time curvature can be very complicated imagine you're doing an animated film and you wanted to show the adventures of a Sharpay dog that's the kind of dog that has very wrinkly skin oh my god to metal that very wrinkly skin requires a lot of computer time then you realize oh god the next shot is a close up I'm going to have to do every individual here then you realize oh god the next shot the dog's going to be running along so you have to do the time dependence of this very complicated curvature on the surface of your Sharpay dog however vindication if the universe is homogeneous and isotropic there are only three possible curvatures for a three dimensional space it can be flat or Euclidean it can have unit form a positive curvature the three-dimensional equivalent of the surface of a sphere or it can have uniform negative curvature the three-dimensional equivalent of this hyperbolic space and that's it so you've suddenly made things incredibly simple instead of the four-dimensional space-time adventures of a Sharpay dog you have three possible curvatures for space so space is homogeneous and isotropic that's the cosmological principle but it is allowed to expand in a homogeneous and isotropic manner or contract and in that case the metric of four-dimensional space-time with these restrictions is the Robertson-Walker metric important footnote here also known as the Friedman-Robertson-Walker metric FRW or the Friedman-Lometra Robertson-Walker metric FLRW and I'm guessing that every lecturer at the school will have a different choice but it's all the same metric so time-space opposite sign you have the square of the scale factor and notice now you have this curvature function S sub kappa which is different for a positively curved space a flat or euclidean space and a negative space so in general general relativity gives you a very complicated field equation of a very complicated metric but with the great power of homogeneity and isotropy everything you need to know about the curvature of space is boiled down to just a few bits of information the assumption of homogeneity is extremely powerful with the assumption of homogeneity and isotropy everything you need to know about space-time curvature well you need a curvature constant which can have one of three values positive negative or zero if kappa is not zero if you have a curved space you need to know the radius of curvature at the present day R sub zero and you need to know the scale factor as a function of time this is just a simple scalar function of time starts out at some very small number one today however as the cartoon character spider-man points out with great power comes great responsibility and if you want to use the very powerful assumptions underlying the Robertson-Walker metric you have to first of all know what you mean when you talk about time and space so here's the Robertson-Walker metric again this coordinates t I've been referring to it as time as if there were some absolute time in the Newtonian sense I'll agree on and the Robertson-Walker metric the coordinates t is the cosmological proper time sometimes also known as the cosmic time for short this is time as measured by an observer who sees the universe expanding uniformly isotropically around him or her so you and I we're not perfect cosmic observers in this sense we see a dipole in the cosmic microwave background that reflects our motion through space remember that's something at parts per thousand level so we're pretty good approximations to a cosmic observer the radial coordinate are that's the proper distance from you to a distant galaxy at our reference time t sub zero the current moment in cosmic time so it's just the distance from us to the galaxy as measured right now this is our normalization the proper distance oh that's just the length of a geodesic from you to the light source as measured right now so distances that increase as the universe expands the proper distance from you to a distant galaxy that increases along with the scale factor the radius of curvature of the universe yep that's a distance that expands along with the scale factor and the wavelength of lights moving freely through space that also expands along with the scale factor so the cosmic microwave background is shifting to longer peak lengths and hence lower temperatures the coordinates are theta and phi these are what cosmologists call the co-moving coordinates those are coordinates that remain constants as the universe expands so you're using a coordinate system that expands at the same rate of the universe very convenient the scale factor how can you tie the scale factor to observable quantities well again you're looking at a distant light source a galaxy say it was the light was emitted with a wavelength lambda sub e at some time in the past t sub e you observe it now with a longer wavelength lambda sub 0 at a time t sub 0 so what's the redshift that you observe oh ok you plug it into the formula for redshift and you see that the redshift z this is something you can observe by looking at the spectrum of a galaxy it's just 1 over the scale factor at the time the light was emitted minus 1 when you look at light from a distant galaxy unfortunately it isn't stamped with the time that the light was emitted however it is stamped with something that is equally interesting the scale factor of the universe at the time the light was emitted so here's the most distant galaxy known in the year 1963 its redshift is measured and you say oh at the time the light we're now observing was emitted the scale factor of the universe was 0.68 a little over two thirds of its current value here's the current record holder at least it was the last time I checked Wikipedia a redshift greater than 8 and so the scale factor at the time the light was emitted was less than 1 ninth so in a monotonically expanding big bag model a larger redshift you're looking at a galaxy that admitted its light when the scale factor was smaller than it is today when the cosmic time was earlier than it is today and you're looking out to larger and larger proper distances I just wanted to point out measuring the redshift of a galaxy when it's bigger than 8 really really difficult of course if it were easier now people would have done it already and so in conclusion the curvature of space time related to its energy content by Einstein's field equation in general very complicated but once again the power of homogeneity and isotropy Flings comes to the fore and Alexander Friedmann in the 1920s realized that I can take the complexity of Einstein's field equation and using the power of homogeneity and isotropy reduce it to a very simple equation that links together the scale factor A the curvature of the universe remember Kappa tells you whether curvature is positive, negative, or zero R sub zero that's the radius of curvature today and it's all tied into the energy content of the universe not a few problems here notice also you can express this as an equation linking together the Hubble parameter the energy density of the universe and the curvature of the universe today so much of cosmology and much of what we'll be focusing our attention on in the next two weeks is how can we measure some of these parameters and use them to determine the others however I think now that I've introduced Dr. Friedmann along with Drs. Robertson and Walker we have enough to go on for the next lecture after the break about dark matter so thank you and please ask me any questions