Let's move on, then, and look in a little more detail how the universe expands or evolves in time. So, as soon as general relativity was recognized to be, essentially, a theory of the universe at large scales, Einstein himself came up with some cosmological models. He realized that, left to itself, space filled with matter will either expand or contract. It has to do one or the other, depending on how much mass energy there is. So he introduced a quantity called cosmological constant, which can think with is integration constant, that can have any value whatsoever. And so if he chose the value just so that it balances the gravity of regular stuff and that universe was static, because in 1917, people thought the universe was static. All right, turn out not to be the case, so Einstein miss-predicting the expansion of the universe, and he regretted it later. Now this cosmological constant term in his equations, turn out it's actually there. But that required much, much more sophisticated measurements which we now have. And it is probably one of the most authentic mysteries of science. At the same time, the Dutch physicist Willem De Sitter, also computed similar things, and in fact when Hubble discovered expansion of the universe it was called the De Sitter effect because it was recognized that space can expand itself. And then two of them later had a joint model called Einstein De Sitter model which is neither one of these, it's something else. Now, the real pioneers of these ones were two scientists one in Soviet Union, Alexander Friedmann and one in Belgium, Georges Lemaitre. Friedmann was a meteorologist and he didn't live very long. Maybe because of conditions in Leningrad or wherever it was, or maybe Soviet Union was the way it was. But in 1922, he looked at models of the universe based on general relativity. He used assumptions of symmetry that we've mentioned. If things are homogeneous and isotropic, then the only coordinate that matters is just the radio coordinate between any two points. That simplifies his life enormously. The original Einstein equations, are 16 equations, Stanzer equation four by four. And so if there is only one coordinate that's important and their is isotropy that reduces the single equation, which people call the Friedmann equation. Independent of him, because people didn't realize Friedmann's models right up front, Georges Lemaitre, the Jesuit in Belgium, developed essentially the same model. And so sometimes we call it Friedmann-Lemaitre model, Friedmann-Lemaitre equation, but sometimes just Friedmann to give him priority. And so these are the cosmological models we still use today cuz they seem to describe the universe at large scale perfectly fine. Essentially, they're a direct consequence of general relativity and in the simplest possible case, homogeneity and isotropy. So if you want to quantify what's going to happen with the universe, again, you use any two distant points that are not gravitationally bound. They're kind of riding on the expanding space. And measure how's the separation of them changing as a function of time. So we call it scale factors, and usually it's called R(t) and sometimes a(t). And so this will apply for any two unbound points, and so therefore, you don't have to worry about units, and you can just look at qualitative behavior. And if you can solve for that, then you can make predictions about the universe, what it's going to do, and that's exactly what was done with Friedmann, or Friedmann-Lemaitre equation. The solution of that equation gives you one of these curves, and parameters that determine how that curve looks like include matter and energy density content of the universe. Now, remember the presence of matter and density of anything is related to geometry. And so it turns out that there are three distinct cases. There is euclidean case, which is called spatially flat. This is spaces you use to think about. And it's denoted curvature constant to zero. There is closed space, three dimensional equivalent of two dimensional sphere. And that's positive curvature. And there is the opposite one, sort of like hyperbolic surface, like saddle, but again generalizing one extra dimension's it's called open negative or ope curvature. So those with positive curvatures will be closed, therefore have high net volume, right? Just like a sphere. Those with the negative curvature go on to infinity. And whether our universe is infinite in extent depends on the value of the curvature. Now, what determines that, is the matter and energy content of the universe. Now, let's ignore cosmological constant for the moment, and so think there is just regular matter in the universe. The only quantity that matters, then, is the mean density because what else can you possibly have, right? And if there is too much gravity, if the density is more than certain value, universe will expand for a while, turn around, and collapse back into reverse of Big Bang. The so called Big Crunch. On the other hand, if the density is less than that particular value, it's just going to go expanding forever [LAUGH]. I mean, it gets slowed down. This is why these curves are all bent a little bit, depending on how much mass there is in the universe. And the case between them, the dividing case, is the flat one. Now if you remember orbits, this is the equivalent of elliptical orbits for a gravitationally bound pair of mass points, critical case, the parabolic orbit, and then hyperbolic orbit when they're completely unbound. So, this is four dimensional space time equivalent of that three dimensional consideration of Newtonian orbits. So, let's define some parameters that will help us quantify these things. The first one is the Hubble constant, which can be written as the ratio of the time derivative of the scale factor, because it [INAUDIBLE] expansion, divided by the scale factor itself, because it doesn't matter on what scale you're measuring it. And if you think about units, this is velocities, which is length over time, is the Hubble Constant times length, and so therefore Hubble constant has to have dimensions of one over time, and as you can see, that's certainly the case. So, R a dimensionalist number but it's boundary width is not. And so essentially Hubble constant is the normalized slope of these curves at any given time, because at any given time we can declare R to be one in [INAUDIBLE]. And today its value is somewhere around 70 km/s/Mpc distance. So the galaxy is exactly 1 Mpc, well, that's not a good idea. Let's say it's ten Mpc away. Then it's moving away with 700 kilometers per second, give or take a little peculiar velocity. So this is what Hubble's Constant does. It sets the scale of the universe because it's a particular value of the slope in one over time units. And so you draw a tangential line on the curve that intersects the time axis. That length is called Hubble time. And multiplied by speed of light gives you so-called Hubble length. Those turn out to be useful units in which to measure scale of the universe. Now, because these lines have some curvature, it's not exactly Hubble time. And if it was perfectly flat, might be, but it's of that same order. And so that's what sets the scale of the universe, both in space and in time. How fast is it expanding at any given moment, right? Now, let's look at the matter content of it. And this is what the Friedmann equation with just plain matter looks like, and you don't have to know that. But if universe is flat, the curvature constant little k = 0, then there is a critical density, which is given by this formula, 3 times Hubble constant squared divided 8 Pi G. G is gravitational constant. And then you divide the actual density of the universe, whatever it is, with this unit of density, the critical density. And if they're exactly equal, then this is a flat universe. If it's less than 1, then there isn't enough matter to close it. It's an open or hyperbolic universe. If it's more than 1, then there is more than enough matter to close it. It's a closed universe with positive curvature that eventually recollapses unless you do something special to it. So cosmologists use notation of omega, capital omega with subscript if they were talking with M for matter, to denote ratio of the density to the critical density. Because what matters for those curves qualitatively is the value of this dimensionless number, and units you can scale as you wish. Likewise, this cosmological constant actually corresponds to uniform energy density in space. And energy being equivalent to matter, for relativity equals e equals mc squared, it has gravity. So whatever the physical nature of this uniform energy is, it has some gravity. Now it can have either sign. This is the interesting part. Unlike regular matter and gravity that is only attractive, this one could be attractive or repulsive. Just like in electromagnetism, there are two components. There's electrostatic, and there is magnetism. And you can have inverse two different signs for electrostatic. So this is sort of gravitation like we want thereof. So there is energy density that you can convert by dividing it by c squared, due to kind of mass like density. And you can divide that with critical density and get this omega sub lambda. Lambda is traditional notation for cosmological constant. So then the total density parameter is the sum of these two, all right? And to answer an earlier question, we can define dimensionless number called deceleration parameter, which is circled now second derivative of those curves. The Hubble constant was giving you the first derivative slope. And this will be curvature. So if the universe is slowing down its expansion, then this number will be positive the way it's defined here. And if the universe is accelerating, it will be negative. So this is what is these cosmological parameters do. They define the shape of these curves. The Hubble constant sets the scale, gives you the basic unit of the unit result at any given time, but the values of these omegas define on which curve it will sit. And so the job of cosmology was to measure these parameters. And it turns out this is a very difficult thing to do. Because we're talking about scales that are much, much bigger than anything we can probe in other ways. So, couple useful numbers to know. That sometimes you will see Hubble constant expressed in dimensionless units like a little h. And that means it's divided, it's in units of 100 kilometers per second per megaparsec. Or sometimes h with subscript 70 because use correct value, have a constant about 70 kilometers per second per megaparsec. The reason we do this is to take it out of the equations, it's just confusion factor. So you choose units, you can set it up, but qualitative behavior does not depend on the actual value of Hubble constant. The critical density, remember it's proportional to the square of Hubble constant in a given time. I should have pointed out the subscript of 0 means today. When there is no subscript, it could be at any time in history of the universe. If the substitute is 0, like H0 or Q0 or R0, it's today. And so today, the value of critical density is, roughly speaking, 10 to the -29 grams per cubic centimeter. And that sounds like a real small number, and it is. But there are many cubic centimeters in the universe. And so this gets to be very important when you start going on many hundreds of megaparsecs scale. And so just to make it all look nice, you can say that some of the three density parameters, the one of matter, the one of cosmological constant, or dark energy, need not be constant actually. Plus some fictitious number called omega subcurvature, add up to 1. The job of this omega k is just to make this equation equal to 1. And so when cosmologists look at deviations from flatness in the universe with precision measurements, using micro background, they sometimes use with little omega sub k, which is not the physical quantity itself. It's just telling you how different is the global geometry from being flat. Okay, so those are the numbers that we will see occasionally. And what we need to look at now is how is the density of matter and energy content of the universe changing in time. Because deceleration depends on the density and the nature of the gravitational force, and that is related to kinematics. There is intimate relationship between scale or changing scale of the universe and curvature and all that with its energy density content. Now, this may sound silly, because of course if you expand something by factor, then the density goes down by cube of that factor. This is true in Euclidean case for regular matter. But, Not if it is radiation. So suppose you have a box full of photons, there is some amount of energy there, it can divide by c squared to get mass equivalent. You expand the box to show you they loop the number density of photons according to the cube, but they're also stretched by that factor. They lose energy, right, so that absolute value of density was not disturbed by the fourth power of the scale factor. And in different times in history of universe, the dominant component was radiation and matter, and comes to be cosmological constant. Cosmological constant, as the name implies, is density stays constant. The whole space is filled with constant energy density field, and even though space expands, it always stays the same. Now there is another aspect how energy is not conserved in expanding space. And other different ways you can parameterize this, but what's usually done is to express, density behaves as scale factor to some power, which is -3 for ordinary matter, and then this little w plus one, called equation of state parameter. And turns out if it's less than minus one, then universe goes into super exponential expansion, so called Big Rip. That's unlikely to happen. Now we know that in real universe, we have mixture of radiation and regular matter, now we know there is also cosmological constant, or direct energy. And so how's this going to play out? Well, let's first look just at the matter and radiation. In the early universe, the density of matter, and for that matter, radiation, were so high, that value of cosmological constant was just trivial, completely negligible, and so let's ignore it for now. Now energy density of radiation declines as a fourth power of expansion, matter is the third power, so those two curves are bound to cross, and that turns out to be at the right redshift of several thousand. Before then, universe is dominated by their radiation density. After that, it's dominated by the matter density. Just as you call it in stellar structure in very massive stars, pressure has two components. There is gas pressure, and radiation pressure, and for really hot stellar interiors, radiation pressure is larger than gas pressure. So, depending then on how this goes, recall here how density scales is third or fourth power, the expansion will actually have different behavior. And in case of just pure radiation dominated universe, just photons, nothing else, it goes as a square root of time. And if it's matter dominate, it goes as two-thirds power of time. Okay, so there is a more general formula, I'm showing it to you just for completeness, but you don't have to know that. And so the history of the universe is then dominated by three different components. It's going to be first dominated by the one that's declining fastest of the space expanse and gets diluted, and that will be radiation. Then it will be dominated by the one that's next fastest, which will be matter, it goes as a cube of scale matter. And eventually, they'll drop down so low that this constant level of cosmological constant dark energy, whatever it is, is bigger than either one of the two. So today, cosmic micro background, which is remnant of the photons from the early universe, is so diluted that its density parameter is something 10 to the minus 4, 10 to the minus 5, whereas that one for the regular matter, is more like a little shy of 0.3. And that one of dark energy is about 0.7, and so we now live in a universe which is dominated, its dynamics is dominated, by the dark energy, and it works in the sense of accelerated expansion. So there is transition from radiated dominated, to matter dominated, to cosmological constant or dark energy dominated, and the expansion law changes depending upon how you do it. So these are examples of some of the models. Their normalized scale factor is one today, and plotted in billions of years. And so highest density models bend most, and lower, they get flatter. But if you have pure cosmological constant, which is positive in sign, corresponds to positive energy density, It works as a repulsive force, sort of like elastic force, proportional to the separation, and in principle, drives exponential expansion. So it turns out there's a real mixture, and in the early days, the universe was dominated by the matter and radiation, it was decelerating expansion. At some point those become equal, comparable to the energy density of the dark matter, I'm sorry, dark energy. And, at that point, the universe started accelerating again. And now it's in accelerated expansion phase, it's been so for several billion years.