 Lectures and we shall first hear the second lecture by Barbara Reiden on Introduction to Cosmology. Thank you. Welcome back. Lectures on cosmology very frequently feature a pie chart like this one, illustrating what the universe is made of at the present day. In many ways, this pie chart is quite useful. It graphically illustrates, for instance, that the baryonic matter surrounding us in this room, for that matter, the baryonic matter of what of which we are made, when you look at cosmological scales makes up less than 5% of the energy density of the universe. Dark matter provides about 26% of the energy density and the remainder of the pie, this big 69% section of the pie, is provided by dark energy. So I promise this afternoon that I would give a little bit of the background of lambda CDM and the observational evidence for it. And so lambda, the cosmological constant, that's one particular form of dark energy. And it's the form of dark energy that was assumed in computing these numbers from the Planck 2015 data. And the dark matter, as Tracy mentioned in her lecture this morning, is predominantly in the form of cold dark matter, CDM. That is not only are the dark matter particles non-relativistic today, but they have been since very early in the history of the universe. So that's free streaming of the matter particles, hasn't wiped out any of the structure. So lambda plus CDM is the lambda CDM model for the contents of the universe. As I mentioned, this pie chart can be quite useful in our thinking. There are some limitations to the pie chart, for instance. For instance, the cosmic microwave background provides a sliver of pie so narrow as to be invisible. It provides only about 0.005% of the total energy density of the universe today. So pie charts are not very good at depicting small fractions. Another problem with this pie chart is it doesn't tell you how big the pie is. That is, this is how the energy density today is allocated among the different components, but what's the overall energy density? Well, we have a hint. Because spoiler alerts, the curvature of the universe today, is very close to flat on large scales. And if you go back to the Friedman equation and erase the curvature term on the right-hand side, it becomes very simple. The square of the Hubble parameter is proportional to the energy density. And since we know the values of big G and C, and since we know the values of h, pi, and 3, we can compute for the known value of the Hubble constant today. Choose your favorite value. I choose 68. We can compute the critical density, epsilon, epsilon sub c, c for critical, at which space is perfectly flat. And for a Hubble constant of 68 kilometers per second per megaparsec, it comes out to a density of around 5 GeV per cubic meter. This can also be expressed as a mass density, just divide by the square of the speed of light, and expressed in solar masses per cubic megaparsec. I have to remind you, I'm still in a seronomer after all. About 1.3 times tenth of the 11 solar masses per cubic megaparsec. By terrestrial standards, this is not a very high density, 5 GeV per cubic meter. For instance, consider a 200 liter drum, or a 55 gallon drum, if you've wandered in from a non-metric community. The critical density of 5 GeV per cubic meter is equivalent to one hydrogen atom within that drum. If the drum is full of water, by contrast, it contains about 10 to the 29 protons and neutrons. So the critical density is extraordinarily low by everyday earthly standards. However, most of space consists of extremely low density cosmic voids. And so the density of the universe today is in fact very close to this critical density, once you average over very large scales. So reverting to the Friedman equation with the curvature term on the right-hand side, it's a very important equation in cosmology, but it is one equation with two unknowns. So even if you know the boundary values perfectly, if you know exactly the energy density in the Hubble constant and the radius of curvature today, you can't uniquely solve this for both the scale factor and the energy density as a function of time. So we need another equation to close our systems of equations so that we can get a solution for how scale factor and energy density change with time. And that additional equation is the fluid equation. Consider a big box, hundreds of megaparsecs on the side, and it's a co-moving box. It's expanding along with the assumed homogeneous and isotropic expansion of the universe, so that its volume goes as the cube of the scale factor. The total energy of all the stuff inside this box, the matter, the radiation, the dark energy, that's just the volume of the box times energy density. Well, so far so good. What equation shall we use? Well, a very familiar equation. First law of thermodynamics. The change in the energy of the box plus the PDV term is equal to DX, the flow of heat through the walls of the box. However, if the universe is homogeneous and isotropic, there's no reason for heat to flow in or out. So we can take the box as expanding in an adiabatic way, no heat flow through its boundaries. And so if the energy in the box changes with time, this would say it's just because of the PDV term. Now I'm writing this down as if we were 19th century engineers and working with cylinders and pistons. The interpretation of this equation is a little bit different cosmologically, but it's still a valid equation. And so we have a equation for the fluid equation, how the fluid in the box changes its energy with time as it expands. And since the volume goes as a cubed and since the energy goes as the energy density times a cubed, we can write this down as an equation linking together, oh, the rate of change of the energy density, the energy and the pressure, don't forget the pressure term, and oh look, it's the Hubble parameter again. That keeps popping up over and over again when you're talking about an expanding universe. Well, we'd also show up if you were talking about a contracting or shrinking universe, since this is an adiabatic process, it's time reversible. Yay, we've got another equation involving the scale factor in the energy density, hurrah, hurrah, oh wait, we've just introduced another variable, pressure. So now the Friedmann equation plus the fluid equation, we've got two equations and we've got three unknowns. Now, it looks like we're not getting any more advanced adding equations, but adding unknowns at the same rate. However, we can now close our series, set of equations by finding an equation of states in this context, just an equation relating the pressure of each component inside your box, the matter or the radiation or the dark energy or the cosmic strings or the domain walls or whatever is in that box. You just need a relation that gives you the pressure in terms of the energy density. If this were a summer school in condensed matter physics, the equation of state could be quite complicated. Fortunately, however, this is a summer school in cosmology and all the components that are of cosmological interest have a very simple equation of state in which the pressure is just linearly proportional to the energy density. In fact, although we usually express pressure as force per unit area and an energy density is energy per unit volume, they have the same dimensionality and so the proportionality constants, conventionally written as w, is a dimensionless number. To see an example of an equation of state of this kind, consider a gas of massive particles for which the thermal velocities are non-relativistic. So the individual particles are moving in random directions at speeds much lower than the speed of light. Now cosmologists conventionally refer to a gas of this kind as being made of matter. So a gas of non-relativistic particles is always referred to as matter. In the older literature, you sometimes hear it referred to as dust. This is an obsolete term, thank goodness, since it can easily be confused with real honest-to-god interstellar dust. So matter is just my shorthand term for a gas of particles that are massive and are moving such that their thermal speed is non-relativistic. You can see why we use the word matter instead. It's a lot easier to say. So a gas of particles highly dilutes, okay. Obviously our equation of state is the ideal gas law, P equals NKT. But how are you going to express that as an energy density? Well, you can express it in terms of the mass density, rho, number of gas particles is just rho divided by M, the mass per particle. And if the particles thermal speeds have a Maxwell-Boltzmann distribution, integrate overall velocities, and you find that KT over M is just the mean square thermal velocity divided by 3. How are we going to express this in terms of the energy density? Well, it's a non-relativistic gas. So the bulk of its energy is given by the rest energy of the particles, E epsilon equals rho C squared. The leading term, if you do the expansion, goes as V squared. But let's say it's a very quite a cool gas. And so in that case we can approximate the energy density epsilon as being, you know, all due to the rest energy of the particles. In that case we have P, we have epsilon, and our equation of state parameter W, P over epsilon, is the mean square velocity divided by 3 times the speed of light, which by our initial assumption of non-relativistic particles, is a number that is very much less than 1. For an example, the air around you, all of those nitrogen molecules, are moving around at speeds comparable to the sound speed a few hundred meters per second. Speed of light is a few hundred million meters per second, and so W turns out to be 10 to the minus 12 or so. Now a lot of interstellar intergalactic gas is in fact much hotter than the gas in this room. But still it's sufficiently low in temperature that the individual molecules, atoms, protons, and electrons are non-relativistic. And so when you look at the list of things that we can describe as matter, well dark matter particles, axions, or wimps, free electrons, as long as the thermal energy KT is much less than the rest energy of an electron. Free protons, as long as the energy density is less than the rest energy of the proton. Atoms, molecules, you can even think of the stars in a galaxy as being a gas of individual particles in which each particle is a star. On larger scales, you can think of a cluster of galaxies as being a gas of galaxies. Each gas particle is an individual galaxy, but in all of these cases the random velocities of the particles are sufficiently small that you could say W is effectively equal to zero for these massive non-relativistic particles. Another example, take the other extreme, a gas of particles that are highly relativistic, either a gas of photons or a gas of particles that have thermal speeds very close to the speed of light. In this case, if you have a gas of photons, for instance, the energy density, just the number density and of photons times the energy per particle, for a black body, that goes as the temperature to the fourth power. Alpha here is the radiation constant. It's usually written as A, but for me the letter A is sacred for the scale factor, so I'm just going to adopt the Greek alpha as the radiation constant. The pressure of a gas of photons or other relativistic particles is just a third the energy density, and so if you have highly relativistic particles, here the equation of state parameter W is equal to a third. What counts as radiation? Well, obviously, photons, neutrinos fall into the radiation category, as long as KT is much greater than the rest energy of an electron, which is mentioned this morning is in the sub-electron volt range, although we don't know the neutrino masses exactly, we know they have a little bit of mass. Electrons, so as long as it's hotter than half an MEV, you might think that in the early universe when the temperature is greater than one GEV, you would have highly relativistic protons, but in fact protons and neutrons dissolve into quark soup at temperatures greater than about 150 MEV, so in the early universe you would have had relativistic quark soup. Matter, W is very small, usually taken to be zero, since the pressure is not significant in any cosmological sense. Radiation, W is a third, but what's the value of W for dark energy? After all, if the dark energy makes up about 70% of the energy density today, we'd like to know a little bit about its equation of state. First out, we need to have a definition for dark energy, and my functional seat-of-the-pants definition is dark energy is a component of the universe that makes the acceleration speed up. Gravity working on matter or on radiation will cause the expansion to slow down, so dark energy has to be something well pretty unusual, again talking about terrestrial standards. If we take the Friedman equation and the Fluid equation, we can do a little bit of mathematical magic on it and combine them into the acceleration equation. I'm not going to stop to do the derivation, but you can see if you take the time derivative of the Friedman equation, you have something involving the second derivative of A with respect to T, you have something involving the first derivative of epsilon with respect to T, and you can substitute from the fluid equation for epsilon dot. The acceleration equation, the second time derivative of the scale factor goes as the energy density plus three times the pressure. So, when I said that matter and radiation make the acceleration slow down, I'm saying that's for these components epsilon is a positive number and the pressure is either zero for the matter or equal to one-third the energy density for radiation. However, if we write down the acceleration equation now in terms of the equation of state parameter w, my initial definition of dark energy, oh, it's something that makes the acceleration speed up. Now, if we assume it's some component of the universe with a simple equation of state, now we have something that's now restriction on the combination of energy density, epsilon, and equation of state parameter w. So, if dark energy has a positive energy density, then it must have an equation of states that is less than minus one-third. So, it must have not only a negative value for w, but it has to have w less than minus a third as long as the energy density is greater than zero. So, the term dark energy was coined by Mike Turner around the year 1998. It's proved to be very popular with makers of beer and of coffee. It's a catchy phrase, but notice the definition of dark energy does not include the fact that it's dark, it does not include the fact that it has energy, but rather the definition implies something about the relationship between its energy density and its pressure. So, dark energy is the more all-encompassing term. The cosmological constant lambda is first introduced by Einstein is one particular sub-variety of dark energy. So, slightly twisted history here, story of lambda, it's kind of a love-hate relationship, but the idea of the cosmological constant lambda goes back to 1917. After Einstein published his field equation in 1915, he wanted to apply them to the universe as a whole. And in 1917, it was generally thought that the universe was static, neither expanding nor contracting. Einstein knew that the mass density of the universe times c squared was much greater than the energy density of light. He didn't know about the cosmic microwave background, but even if you toss in all of those microwaves out there, it is true that's the mass today has a much greater energy density than the photons. So, he said, I want a universe that is filled with matter that is static, neither expanding nor contracting. Well, look at the acceleration equation for a universe containing only matter. A double dot over a equals minus 4 pi g over 3 times the mass density rho. Since the mass density rho is always non-negative, these come to the conclusion that's, okay, I start with a universe filled with matter that's not expanding or contracting. If I wanted to remain static, I want a double dots to be zero, but that requires a density of zero. So, Einstein has concluded that the only static universe is an empty universe. This is the universe described by Minkowski space. But, you know, there does exist matter in the universe. And so, he was trying to reconcile the existence of matter with what he thought was the correct model, a model that was static. And so, when you introduce the cosmological constant lambda into the field equations of general relativity, you then re-derive the Friedman equation and the acceleration equation, you find there's an additional term on the right-hand side, a term involving the cosmological constant lambda. Now, you start out with a universe that contains matter, that has a cosmological constant lambda that's not zero, and now, huh, you can have it remain static, the second derivative of the scale factor equal to zero. If you have a cosmological constant that is exactly equal to 4 pi g times rho, the mass density of your universe. Einstein published this result. He thought it was publishable. It did technically achieve his goal of having a matter-filled universe that was static, but he wasn't happy with it. He thought that adding the cosmological constant lambda to his field occasions detracted from the beauty of his theory, as he later said, and there's also the more practical objection that a universe containing matter and lambda, although it's in equilibrium, neither expanding nor contracting, it's an unstable equilibrium. It's only static if the mass density is equal to lambda everywhere in the universe, but matter is movable. So take a hunk of the universe, scoop out the matter, pile it up on one side, and the region from which you've excavated the matter now is lambda-dominated, starts to expand, the region where you dumped the excess matter, it's now matter-dominated, and starts to contract. Now, there's nothing wrong with gravitational instability. It's how structure is formed. Well, that plus baryonic physics. However, once lambda dominates, things become, well, exponentially unstable. So here's the Friedman equation. When you add the cosmological constant lambda to the field equations, you get, once again, an extra term involving one-third lambda on the right-hand side. Although Einstein was concerned with the case where the matter and the cosmological constant were exactly balancing out, in the same year, the Dutch cosmologist, Willem de Sitter, said, well, okay, let's take a universe where there's no matter, there's no curvature, there's nothing but a cosmological constant. And as you see here, you have a time derivative of something that's equal to a constant times that something, again, exponential expansion, or in this case, exponential contraction. Since this is an adiabatic process, expansion is just a time reversal of the contraction. So you have exponential growth. Again, as with the steady-states model, where the mass density of the universe was constant, here you have the Hubble constant, constant with time, and so the universe just keeps expanding exponentially. In 1917, you have two models. Einstein has introduced the cosmological constants to keep the universe from collapsing. However, de Sitter says, well, if you take out the matter, suddenly you have exponential expansion. Now, it's easy to say the phrase cosmological constant, it kind of trips off the tongue, but what is the cosmological constant? I am fairly, but not entirely, humorous in introducing this newspaper cartoon from the year 1930. This is Willem de Sitter, he of the exponentially expanding universe. Notice he's drawn as the Greek letter lambda. Little joke from the newspaper cartoonist there. And if like me, you are ignorant of Dutch, the translated caption says, what, however, blows up the ball? What makes the universe expand or swell up? That is done by the lambda. Another answer cannot be given. So now, first of all, it's not the lambda that makes the universe expand. It's what makes the expansion speed up, but you can't expect strict scientific accuracy from a cartoonist. But what is the lambda? In Einstein's initial 1917 paper, you can look it up, even if you don't speak German. All the equations are there, and he puts the term involved in the cosmological constant lambda on the left-hand side of the equation. So he was thinking of it in effect as modified general relativity. That is, you have in his thinking some energy and pressure and then bulk flow and then sheer stresses on the right-hand side, and on the left-hand side you have the curvature, and this extra lambda term, it's on the curvature side of the equation. So it affects how much curvature you get out from a given density of matter and radiation. However, in recent decades, it has been found to be more useful, in our thinking, to take the lambda term and move it to the right-hand side of the equation. To say that it is part of the energy density and pressure of the universe. If you think of the cosmological constant as being something, some substance, some component of the universe adding to its energy density and pressure, what are the energy density and pressure going to be? Well, again, here's the Friedman equation with the cosmological constant term lambda over 3, and if you equate the term lambda over 3 to 8 pi g over 3c squared times the energy density of something, you can think of lambda as a component of the universe that has a constant energy density. It's just proportional to the original constant lambda that Einstein introduced into his equations. Moreover, if you look at the fluid equation, the energy density of whatever's in your universe must be proportional to the energy density plus the pressure. So, in order for epsilon dots to be zero, in order for the energy density to be a constant with time, we need the pressure associated with the cosmological constant to be minus the energy density, or w equals minus 1. So if dark energy consists of the sets, of things that have w less than minus a third, cosmological constants can be thought of as the subset where w is exactly equal to minus 1. There's the cosmological constant. Einstein introduced it to keep the universe from expanding or collapsing. He didn't like it, but so, hey, everybody loves lambda now. Well, it's not an unconditional love, and I'll talk about that later. So, let's model the universe as containing matter, as a equation of state parameter w equal to zero, radiation, w equals a third, and a cosmological constant lambda defined as having w equals minus one. So, here's the warning sign. It is possible, it might even be likely, that the acceleration we detect today is due to some sort of dark energy that has w not equal to minus one. Could be minus 0.9. Could be minus 1.1. It could have w changing with time, or it could be due to modifications of gravity, and there'll be a series of lectures at the end of this week all about dark energy and modified gravity. However, the cosmological constant lambda, in addition to being familiar, the cosmologists, also is a very simple way of parameterizing the dark energy. So, although lambda CDM is kind of this standard, it's still a provisional standard. It hasn't been ruled out yet, but if convincing evidence comes in that w is not exactly equal to minus one, we can change our models. So, you have these three components, matter, radiation, lambda. The energy and the pressure are additive. The total energy that you have in a co-moving volume is equal to the energy of the matter, of the radiation, and of the cosmological constant. Same to the pressure, it's additive. This means that the fluid equation must apply to every different components separately. And so, oh, look, epsilon dot over epsilon equals a dot over a times a constant, and this tells you that the energy density of any component is a power law in the scale factor. With a power being minus three times one plus w of whatever component it is that you're looking at. Another reason why we like components with constant equation of state parameter w, it makes the integration easy. So, I mean, how the energy density of each of these components depends upon scale factor as the universe expands. So, the larger the value of w, the more deeply the energy density drops off as the universe expands. Now, this is fairly easy to see intuitively if you look at, for instance, a co-moving volume that contains matter. In this case, it's a co-moving volume that contains nine particles. As it expands, well, you still have the same number of particles. Since they're moving very, very slowly, their relative position is essentially unchanged. And so, the energy density, well, it's the rest energy of each particle, which doesn't change as a function of time times the number density, which falls off as one over a cubed as our volume expands. For radiation, things are a little different. Here, the volume contains nine photons. As the volume expands, it still contains nine photons. Probably not the same photons since they're zipping around at the speed of light, but, you know, on average, the same number of photons. However, they've been redshifted to longer wavelength and lower photon energy. So, for radiation, energy density is equal to energy per particle. It goes as one over lambda times the number density. Lambda goes as one over scale factor. Number density of photons goes as one over the cube of scale factor. That gives you the steeper A to the minus four power for radiation. There's a little slight of hand in this calculation, as it implicitly assumes that photon number is conserved. Which it isn't. It's the easiest thing in the world to create photons. You just flip a switch and they pour down from the lights overhead. However, although photon number is not strictly conserved, the energy density of starlight, even though stars have been working away for over 13 billion years, even today is only about 10% the energy density of the cosmic microwave background. So, even today, it's a fair approximation that all the energy density of photons in the universe comes from the CMB. And the further back in time you go, the better that approximation becomes. And my laptop is not happy. Excuse me for a moment. Well, I tend to its care and feeding. My, very unhappy. Need to find where we were. Voila. Ah, yes. More cosmology jargon. Frequently, what cosmologists want is not the density of some components in MEV per cubic meter, but rather the dimensionless density parameter. Called by the Greek letter omega. Omega is just defined for each component as the energy density of that particular components, divided by the critical density, which you'll recall, is that special density at which the universe is completely Euclidean or flat on large scales. And for the universe in which we find ourselves with a Hubble constant of 68 kilometers per second per megaparsec, remember it's about 5G EV or 5000 MEV. So keep that number in mind while we start to do an accounting of the contents of our universe. So what's in our universe today? Let's start with the easy one. Let's start by looking at the density parameter in the background radiation in our universe. We know very well the energy density of the background photons, the CMB. Remember CMB, very good approximation to a black body spectrum. Here expressed as number density of photons as a function of photon energy. The temperature of the photons in the cosmic microwave background has been measured quite accurately, 2.7255 Kelvin. I can recite that in my sleep. And so the energy density of photons, remember we can ignore star light and other forms of photons, it's predominantly the CMB, radiation constant times temperature to the fourth power, only a little more than a quarter of an MEV per cubic meter. Remember the critical density is nearly 5000 MEV per cubic meter. So the cosmic microwave background, cosmologists love it, it's full of information about the time of last scattering, but from an accountant's point of view it only contributes about 5.35 times 10 to the minus 5 of the critical density today. Just as the cosmic microwave background is a relic of the time when the universe was opaque to photons, there must be a cosmic neutrino background in our universe, which is a relic of the time when the universe was opaque to neutrinos, back in its first second or so. I'm going to assume for the moments the useful fiction that's neutrinos are completely massless. Neutrinos are fermions rather than bosons, so you have the fermi-derac distribution rather than the Bose-Einstein distribution. So there if you integrate up to find the total energy density of neutrinos, it's 7 eighths times the radiation constant times temperature to the fourth power, and that holds for each of the three species of neutrinos. The temperature of the cosmic neutrino background is not identical to the temperature of the cosmic microwave background, since neutrinos decoupled at T twiddle one second before positron electron annihilation. When the temperature dropped below the rest energy of positrons and electrons, they annihilate each other with one last glorious burst of photons. None of that energy went into the neutrinos, they had already decoupled at this point. So detailed calculation, details left for the reader tells us that there's this wonderful factor of four elements to the one third power, and therefore that the cosmic neutrino background should have a temperature of just a little bit under two kelvin. Now, this is all theory. The neutrinos of the cosmic neutrino background have, well, at a temperature of two kelvin, that's energies in the milli-electron volt range, and it's hard enough to detect neutrinos that have an energy of big MEV, mega-electron volts, much less neutrinos with energies of milli-electron volts, little m. However, no, it's well-established physics, and so take three species of neutrinos, each with an energy density 7 eighths alpha t to the fourth power, temperature 2.945 kelvin, and you find out that the density parameter neutrinos, if they were massless today, would be about 68 percent of the energy density of photons, not much by way of radiation compared to the critical density at least. Even if all types of neutrinos were completely massless today, radiation would still have omega of 0.0009, excuse me, 0,0009, a number that is much less than one. So, the recent expansion of the universe, since radiation doesn't contribute much, can be usefully expressed in terms of omega in lambda, the cosmological constant, and omega in matter. So, omega, lambda, omega, matter, these are the useful parameters in describing the recent acceleration of the universe, for instance. So, go back to the acceleration equation, put in the energy density of matter, the energy density of lambda, the pressure of the cosmological constant, remember it's minus its energy density, and converted everything into terms of omega and h, rather than epsilon and p. And so, the universe would be coasting today. That is, its acceleration would be 0 if the universe had a matter density that is equal to twice its energy density in a cosmological constant. And in general, if you know the value of omega, matter, and omega, lambda, you can compute, not merely what the acceleration is today, but what it was in the past, what it will be in the future. Because, describing the expansion of the universe in terms of omega in matter and omega, lambda is so useful, it's also a common place for talks on cosmology, to contain a picture of the omega, lambda, omega, matter plane. So, horizontally, you have the matter, excuse me, the density parameter in matter, it's always non-negative, vertical axis, you have the density parameter in lambda, the cosmological constant, which at least in theory land, we're letting to be negative. After all, Einstein introduced lambda as, so this number in his field equations, this constants, it doesn't have to be a positive number. The acceleration today is 0, if omega, matter equals twice omega, lambda, that's this dotted diagonal line rising to the right, above the dotted line, the universe is speeding up today. Below the dotted line, it's slowing down today. Also, if you ignore the contribution of radiation, and why not, it's less than 1 part and 10 to the 4, the curvature today is 0, if the sum of omega, matter and omega, lambda is equal to, excuse me, 1, looks like a 0, but it's a 1, believe me. That's this diagonal line, decreasing, going down as you go to the right, above the line, the total density is greater than the critical density, so it's positively curved below the line, it's less than the critical density, so it's below, so it's negatively curved below the line. The other labels, big crunch, big chill, big bounce and loitering, well, see the meaning of these particular labels, we need to extrapolate back into the past and back into the future. If you have a value of omega, matter and omega, lambda today, and if you say that matter doesn't decay, and that the cosmological constant really is constant, you can extrapolate backward to find the scale factor in the past, and what the scale factor will be in the future. So, here on the left, scale factor is a function of time, normalized to be 1 at t equals t sub 0. All of these universes have omega, a matter of 0.3. They all have the same value of the Hubble constant today, so if all you could measure was today's Hubble constant and today's mass density, these four universes would be indistinguishable from each other. Nevertheless, if you extrapolate into the future or back into the past, you see that they have different paths and different futures. First of all, consider this dotted line. This is a universe that has omega and matter in 0.3 and omega and lambda of 0.7, so it's spatially flat and initially it's decelerating because it's matter dominated, but then the acceleration picks up and becomes faster and faster and faster with time. This dashed line that reaches a maximum scale factor and then decreases again, that's omega lambda of 0.3 today. This is a big crunch universe since it's going to expand to a maximum and then re-collapse at some time in the future. By comparison, omega lambda of 0.7, that's a big chill universe, it's just going to keep on expanding forever and the cosmic background radiation will become cooler and cooler and cooler and cooler. Big crunch universe, sometimes known as the ganab gib. Since the collapse phase is just the expansions phase in reverse, so if it starts with a big bang, it's got to end, of course, with a ganab gib. What about this universe? The one that starts out contracting reaches a minimum scale factor and then bounces back. Well, that's a very high value of lambda 1.8 and this universe is called a big bounce universe. It starts out really big, collapses down and then bounces back. Not very realistic, but it is a mathematically valid solution to the Friedman equation. Finally, this interesting line, the one that's nearly horizontal for a long stretch of time, that's with omega lambda today of 1.7134. If you fine-tune the value of lambda just right, you have this long stretch of time in the past where the scale factor is very nearly constant and where the universe is very, very close to Einstein's ideal static universe. However, eventually lambda gets the upper hand and it expands upward. So this is a loitering universe, sometimes also called a Lometra universe after Georges Lometra who speculated about the possibility of its existence. So very different in the past, very different in the future, but in the immediate past and the immediate future, these different universes are difficult to distinguish from each other. So I suppose it's an illustration of the very great importance of knowing exactly what the value of lambda, the cosmological constant is. So it might be amusing to live in a loitering universe or a big crunch universe, but the evidence indicates that, well, you are here. Omega in matter about 0.3, omega in lambda, the cosmological constant about 0.7. But how do we know that? What evidence is there that we live in a lambda CDM universe with over twice as much lambda as CDM? Well, let's set the ground rules. We know that the universe contains matter, right? No arguments there. We know it contains radiation. If it didn't, you wouldn't be able to see me. There would be no photons bouncing off me. But there's not much radiation. It's easy to count up photons because, well, we're astronomers, we're experts at counting up photons. And we will graciously permit the universe to have a cosmological constant. Again, it might be some other form of dark energy, but let's try first with a cosmological constant and see if that works. The total density parameter, well, we just add together the lambda, the matter, the radiation, although radiation doesn't contribute much. And, well, this is interesting. If the total density parameter omega sub-zero is not equal to 1, then the Friedman equation tells us space is curved. Moreover, if you knew the exact value of omega, you would know both the sign of the constant kappa, and you would know the value of the radius of curvature in units of the Hubble distance. Or conversely, if you knew the radius of curvature in units of the Hubble distance, you'd be able to pin down what omega the total density parameter is. The question I posed was more or less, how can we use observations of the universe to constrain the values of omega, matter, and omega, lambda? And the Friedman equation suggests one way. Oh, if we could only measure the curvature of the universe exactly, that would tell us the sum of omega, matter, and omega, lambda. And a little bit of radiation. But, you know, we don't want just the sum of the two. We need some other linear combination of omega, matter, and omega, lambda in order to pin down where you are on the omega, matter, omega, lambda plane. Okay. Here we are. Astronomy. We're going to be looking at distant objects that emit light. Love it. You can learn about omega, matter, and omega, lambda from observing standard candles. The term standard candle is quite ancient in astronomy. It dates back to the era where people actually used candles every day. And so a standard candle is just an object whose luminosity you know. Don't ask how you know it. You just know it from some other set of observations. So there you are looking off into space. Let's place you at the origin. It seems egotistical, but it's in a homogeneous and isotropic universe. Any place is as good as any other for the origin. You're looking at a standard candle, not a literal candle, but it makes a cute picture, at coordinate location r, theta, phi. Remember I'm using a Roberts and Walker metric in which the radial coordinate is defined as the current proper distance to the standard candle. The distance you would measure if you could stretch a tape measure out at the current time and pull it tight. Unfortunately, you can't stretch out a tape measure to a distant quasar or other distant luminous object. Instead, you have to sit at home and collect up the photons that are traveling from that object. So the photons that you observe today at t sub zero were emitted at some earlier time, t sub e. Photons travel on geodesics through space, but photons, since they travel at the speed of light, follow null geodesics through space. So the small distance ds that they travel in a time dt on coordinates distance dr is equal to zero. So space time separation is zero between any two paths on a photon's trajectory through four-dimensional space time. Since space is homogeneous and isotropic by definition, the coordinates theta and phi do not change. They do if you have gravitational lensing, but that requires an inhomogeneous universe. So in the homogeneous and isotropic approximation, the time it takes a photon to travel a coordinate distance dr is just c dt divided by the scale factor. The entire proper distance from u to the light source as measured today, you just do the integral over time on the right hand side of that equation from the time the photon was emitted in the past to the time t sub zero that you observe the photon. So what do you say? Well, now first of all, if we could actually measure the proper distance, we wouldn't have to fuss with this, we'd just measure it. However, since we can't measure the proper distance, it is useful to know that the proper distance to an observed light source, it's sort of an encoding of the expansion history of the universe between the time the photon was emitted and the time it is observed by u. In our hot Big Bang universe, the scale factor a has been continuously increasing with time. So we can change our variable of integration from t to a. Since we know that a dot over a is the Hubble parameter h, so we can rewrite this relationship between the proper distance and the expansion history of the universe in terms of the scale factor a and the Hubble parameter h. However, I'm going to do something even further. I'm going to change the variable of integration from the scale factor a to the redshift z. Remember, if you see a distant light source, you can measure the redshift of its light and you can map that to the scale factor a at the time the light was emitted. So since a is equal to 1 over 1 plus z, the substitution is simple. And now it's the same, of course, information content, the proper distance to that very distant galaxy over there. It's just a history of how the universe has expanded, but now the parameter is the redshift z. The limits of integration are from z equals 0 now to z corresponding to the time the light we observe was emitted. So this is an interesting equation because the redshift z, that's something that is observable in cosmology and in science in general. If there's something you can observe, then you should go out and observe it and you should cling to it and see what use you can make out of it. The Hubble parameter, well, it's given by the Friedman equation. You can write it as the Hubble parameter as a function of time or of scale factor. And here, just written as a function of redshift because of the simple mapping between expansion factor and redshift. Ah, this is nice. It's a relatively simple equation. A term from the matter density, which goes as 1 over a cubed. A term involving the curvature, if any, which goes as 1 over a squared and a constant term due to the cosmological constant. In the general case where you have all three terms on the right hand side, it doesn't integrate to something that's simple and analytic. But who cares? Computers are cheap. So you choose values of omega matter and omega lambda and then you integrate. So you can plot up here on the left hand side the proper distance at the time of observation to a distant light source as a function of the redshift of the light from the source. Still not quite over that jet lag. There are three different lines. The top line, the dot dash line, that's the lambda line. That's a universe that is Euclidean, spatially flat, containing nothing but lambda. So that's the Einstein decider, excuse me, that's the decider exponentially expanding universe. Notice that the decider universe in which you have exponential expansion. That's the only case in which proper distance is linearly proportional to z for arbitrarily large redshift. For other universes, for instance, this is bottom line, the dotted line on the bottom. That's also a spatially flat universe, but a universe in which the energy density is provided entirely by matter. In this case, the deceleration of the universe causes the proper distance redshift relation to start leveling off at high z. The solid black line between them, that's the combination of lambda and matter that best describes the observations today. Omega lambda of about 0.69, omega matter of about 0.31. So left hand side, proper distance is a function of redshift for three different model universes. This is the proper distance at the time of observation. Remember though, the distance to the light source at the time the light was emitted was smaller by a factor of, well, the scale factor, or 1 over z. So take the left plot, divide everything by 1 plus z, and that's the distance to the light source at the time the light you're observing right now was emitted. So for the exponentially expanding decider universe, it levels off at a proper distance equal to the present day Hubble distance. For the matter-dominated universe and for our benchmark, matter plus lambda universe, it has a maximum at some relatively mild redshift around one or two. So if you look at an object with an extremely high redshift, z much greater than 1 in our benchmark model, then when the light was emitted, it was very close to you, in your face as it were. And the light has been slowly moving towards you as the spatial distance between you and the light source has increased. Although this log-log plot only goes up to redshift of 500, you find out that for universes other than the decider universe, as you go to a redshift of infinity corresponding to t equals 0, scale factor of 0, the proper distance to the object that you're observing levels off at a constant value. It's called the particle horizon distance. And it's the maximum proper distance as measured at the present day that light can have traveled since the beginning of the expansion. So it's a horizon distance because you cannot see beyond it. And for the benchmark model, it's about equal to 3.2 times the present Hubble distance or 14,000 megaparsecs. You'll notice the horizon distance, if it was only matter, would be a little bit smaller. And again, in an exponentially expanding decider universe, there is no finite particle horizon distance. In our universe today, stars that are more than 14,000 megaparsecs, so I haven't had time to send us light yet. So there's the resolution of Ulber's paradox in a nutshell. Notice that 14,000 megaparsecs or about 10 to the 4 megaparsecs is smaller than 10 to the 18 megaparsecs by 14 orders of magnitude. So in fact, it does give a back of envelope properly quantitative resolution to Ulber's paradox. Hey, if we knew the proper distance to sufficiently distant light sources, and when you get to redshifts greater than 1, all these lines diverge. If you could find the proper distance to objects at redshift of 1 or greater, then just plot out proper distance versus z, compare it to that model whose values of omega-matter and omega-lambda best fits the curve, and yay, you've found the best fitting values for omega-matter and omega-lambda. Unfortunately, proper distance is not directly measurable. So I've got a problem here. How can we estimate the distance from observable properties? Now you measure the redshift z. In an ideal world, you can measure the complete volumetric flux over all frequencies. It's a little bit difficult to do in practice, but you could do it. You go above the Earth's atmosphere for one thing, and you can compute a function that's usually called the luminosity distance. It's just the luminosity divided by 4 pi times the flux. Take the square root. This is how Edwin Hubble estimated distances. So it's good enough for Hubble, it's good enough for us, right? Well, not exactly. It's called the luminosity distance. It goes well. It has units of distance, and because it's what the proper distance would be if space were static and euclidean. Ah, yeah, I know. Space is not static and euclidean. And so, again, in an expanding spatially curved universe, you have this function s sub kappa, which is not equal to the proper distance if space is curved. And you've got a factor of 1 plus z to the minus 2 power. 1 power of 1 plus z, because the photons are redshifted to longer wavelength and lower energy. And another factor of 1 plus z, because the arrival time of the photons is stretched out. And so, there's a relationship between luminosity distance, curvature, proper distance today in 1 plus z. Again, luminosity distance versus redshift for our three different euclidean models. One all lambda, one all matter, and one the best fitting mix. Notice that if space is flat, or nearly so, then the luminosity distance, this function s sub kappa reduces the proper distance. And you find that in flat space, the luminosity distance is always going to be an overestimate of the proper distance. Nevertheless, it's an overestimate that you know, since the redshift z is a measurable property. So now, aha, I will measure the redshift, I will measure the flux, I will compute the proper, excuse me, compute the luminosity distance to my standard candle, and see what's best fits. For this, we're going to need a standard candle. And the preferred standard candle of cosmologists is type 1a supernovae. One of those obscure names that astronomers love. It's just a thermonuclear supernova. If a white dwarf supported by electron degeneracy pressure goes over the Chandrasekhar limits, starts to collapse, triggers a thermonuclear runaway. These supernovae are really luminous. Here's a before-enduring pair of images of a supernova in the galaxy M101. At their brightest, type 1a supernovae can get up near 100 billion times the luminosity of the sun. So at their very peak, a single supernova can be as bright as an entire mid-sized galaxy. The top spectrum here is of a type 1a or thermonuclear supernova. It's got lots of nice absorption lines, so you can measure its redshift. Although not all type 1a supernovae have identical luminosities, which would be nice if you use them as a standard candle, they do have this well-known correlation between the rise and fall time of the supernova's luminosity and its total luminosity. The more luminous ones rise and fall more slowly, the less luminous ones rise and fall more rapidly. So if you account for this correlation between peak luminosity and rise and fall time, you see they all fall on very nearly the same curve. So they are a standardizable candle. They all be calibrated to the same, you know, recomputed luminosity. So here's a plot of luminosity distance vertically versus redshift horizontally for a sample of 580 type 1a supernovae. Oh, vertical axis, more astronomy jargon. It's the apparent magnitude minus the absolute magnitude. All you need to know is it's a logarithmic measure of the luminosity distance. So it goes from 100 megaparsecs luminosity distance to 1,000 to 10,000. And take my word for it, there are 580 data points here, although the scatter on each individual point is large. You see that as you go out to redshift 1 are three different Euclidean universes diverged from each other. And it's the middle line that gives the best fit. So back to the omega-matter, omega-lambda plane. Here's the 95% confidence interval for the type 1a supernova results. So if the universe really is described by matter plus lambda, then the true values of omega-lambda and omega-matter lie somewhere within that ellipse, 95% of the time. And 8, you'll notice, well within the region where the acceleration is positive. Even with the relatively limited sample size available in 1998, the accelerating universe was still the science breakthrough of the year for 1998, and additional data, better calibration of the supernova light curves is only giving stronger and stronger emphasis to the fact we live in an accelerating universe. I should also note the accelerating result is quite robust. Doesn't it depend on your assuming that the dark energy is a cosmological constant? Any dark energy will do it. And even if you just say the scale factor is expandable as a Taylor series, assume nothing about the underlying physics. You will still get an acceleration term that is positive. Now, supernovae, really exciting, gave you this at the time quite startling result that's the expansion the universe is speeding up. Note, however, this error ellipse straddles the kappa equals zero line. Supernovae are great, love them, accelerating universe, really exciting results.