Stanford University. Okay, let's start. This quarter's subject is cosmology.
Cosmology is, of course, a very old subject. It goes... Back thousands of years, but I'm not going to tell you about thousands of years of cosmology, but I say thousands of years, I'm talking about the Greeks of course, but we're not going to go here back thousands of years. We're going to go back at most to sometime in the second quarter of the 20th century. when Hubble discovered that the universe is expanding but let's just say a few words about the science of cosmology the science of cosmology is new or at least what we know about it a minute ago I said it was very old yes, in a sense but the modern subject of cosmology is very new it really dates To well after Hubble, it dates to the discovery of the Big Bang, the three degree microwave radiation that was discovered as the remnant of the Big Bang.
And that happened. sometime in the 60s so I was a student, I was a young student and before that cosmology was in a certain sense less like physics and more like natural science, like what a naturalist does studies this kind of thing, studies that kind of thing, you find a funny star over there you find a galaxy over there that looks a little weird you classify, you name things, you measure things to be sure but the accuracy with which things were known was so poor that it was extremely difficult to be precise about it And it's only fairly recently that physicists, physicists are always involved, but they were involved because many of the things that you see, many of these strange creatures, funny stars, galaxies and so forth, of course are physical. systems and to describe them properly they have angular momentum they have all the things that physical systems have there's chemicals out there and so physical chemists were involved but thinking of the universe universe as a physical system, as a system to study mathematically, and with a set of physical principles and a set of equations. Of course, there were always sets of equations way back, but wrong equations, right equations. And accurate equations, things which agreed with observation, that's relatively new.
More or less over the history of my career in physics, which is 50 years, some... Something like that. And that's what we're going to study.
We're going to study the universe as a system. In other words, a universe as a system that we can study with equations. That's if you don't like equations, you're in the wrong place.
Alright, so where do you start? You start with some observations. Now the first observation, which may not really turn out to be absolutely true for reasons that it's not absolutely true, but it looks like it's approximately true, is that the universe is what is called isotropic. Isotropic means that when you look in that direction or that direction or that direction or that direction now of course if you look right at a star it looks a little different than if you look away from the star but on the whole, averaging over patches in the sky and looking out far enough so that you get away from the immediate foreground of our own galaxy the universe looks pretty much the same in every direction That's called isotropic. Same in every direction.
Now if the universe is isotropic, with one exception that I'll describe in a moment, if it's isotropic around us, then you can bet with a high degree of confidence that it's also pretty close to being homogeneous. Homogeneous doesn't mean it's the same in every direction, it means it's the same in every place. If you went out there and you saw a planet, out over there and you looked around from 16 galaxies over and you looked around what you would see, you would see about the same thing you saw here. So first of all what's the argument for that? Why does being isotropic, which means the same in every direction, tell you anything about why it would be the same if you moved away to a very distant place?
And the argument is very simple. Imagine that there's some distribution of Galaxies. You know incidentally, at least in the first part of this study here, it's not going to matter very much whether what we're talking about, whether we call them galaxies or whether we just call them particles. They're just a... Collectively mass points distributed throughout the space.
For the moment, I might even lapse into calling them particles from time to time. Now you must mean when I say particles, I mean literal galaxies, but unless I otherwise specify. Okay, so the universe has a lot of them.
Anybody know how many galaxies are within the visible? About a hundred billion, 10 to the 11th. Just there's some nice numbers to keep track of, incidentally.
It's a good idea to keep track of a few numbers. Within what we can see, within what we can see with telescopes, out to as far as astronomy takes us, about 10 to the 11th galaxies. Each galaxy about 10 to the 11th stars.
Altogether 10 to the 22 stars. If each Star has roughly 10 planets, that's 10 to the 23 Avogadro's number of planets out there Mole, right, planetary mole, right, alright, now Imagine that we're over here and every direction that you look in, it looks pretty much the same. Well, then I maintain that not only must it be the same in every direction, but it must be the same from place to place.
What would it mean for it not to be the same from place to place? Well, if it's isotropic, the only way it could not be homogeneous is if it somehow formed rings of some sort. It's got to be such that it looks the same in every direction, but it's not the same...
Yeah, shells. I think somebody said shells. It would have the geometry of some sort of shell-like structure. Why?
It doesn't literally mean shells, it just means... Yeah. So, if that were the case...
If that were the case and you went someplace else and you looked around, clearly it wouldn't look isotropic anymore. So for it to look isotropic, unless by accident we just happen to be at the center of the universe, if we happen to be at the very center where everything just accidentally or not accidentally, maybe by design, happens to be nice and rotationally symmetric about us, if we don't want to believe that, Then we have to believe it's pretty much the same everywheres and that it's homogeneous. So homogeneous means that as far as we can see, space is uniformly filled on the average with particles. Uniformly filled.
That's called the cosmological principle. Now, you can't... why is it true? Well, how can it not be true?
It's the cosmological principle. Right? And sometimes people argue like that.
It's true because it's been observed to be true to some degree of approximation. Now, as was mentioned, in some media that I don't know how to evaluate, Some astronomers apparently claim to see structures out there which are so big, if the blackboard here was the whole visible universe, they would stretch across great big patches of it, and that seems to be a little bit counter to this idea of complete uniformity. And of course, certainly the idea of complete uniformity is not exact.
Just the fact that there are galaxies means it's not the same over here and over here. And in fact, there are clusters of galaxies. And super clusters of galaxies. So it, it appears it's not really homogeneous, but it tends to come in sort of clusters. ...which on some big enough scale, like a billion light years roughly, maybe a little bit less, if you average over that much, it looks homogeneous.
Okay, so that's the basic fact that we're going to begin with. Now, what's the first step in formulating a physics problem? Yep, know your variables. Usually the answer is sharpen your pencil. After you've sharpened your pencil, the next part is know your variables.
But a good step, I'm not sure whether it comes after that or before that, is... Oh, you bet, you bet, you bet. But we're going back, I'm purposefully going back a few decades to sometime around the 60s or something like that, 50s, 60s, 40s The idea of the cosmological principle was put forward before people had any real right to put it forward They just said, oh, well, let's just say it's homogeneous. We call it the cosmological principle, and if people ask us why it's true, it's because it's a principle. But then, with more and more astronomical investigation, and then finally the cosmic microwave background really nailed it.
And in some sense, the primordial distribution of matter was extremely smooth. Extremely smooth, but we'll get to that. All right, so here we have a uniform gas, if you like.
It's a uniform gas. And that gas is interacting, it's a gas of particles, it's interacting, each particle is interacting with the other particles Now, galaxies on the whole are not electrically charged, they're electrically neutral But they're not gravitationally neutral They interact through Newtonian gravity, and that's the only important force on big enough scales. On big enough scales where matter tends to be electrically neutral, the only really important force is gravity. And so gravity is pulling all this stuff together, or it's doing something to it, but it's a little bit confusing.
What happens... To this point over here, does it accelerate toward the center? Because at the center there's a whole bunch of matter there?
Or does it accelerate out to here? Because after all, there's as much matter out there as there is on this side. In fact, it sort of looks like it oughtn't to move anywhere.
They ought to just stay there because there's as much on one side as on the other side, right? So it'll just stay there Well, what about this one over here? Same thing, because every place is the same as every other place So the natural thing to guess is that the universe must be just static It must just sit there because nothing has any net force on it And so there's nothing pulling it one way or another That's wrong We're going to work out tonight The actual Newtonian equations of cosmology But you may have heard that the expanding universe somehow fit together especially well and wasn't really understood until general relativity, until Einstein That is simply false It may be so historically, I mean in terms of years, yes, it is true That the expanding universe was not understood until after Einstein had created the general theory of relativity. That is a fact about dates.
It's not a fact at all about logic. Newton could have done the expanding universe. Since Newton didn't do it, we are going to do it here the way Newton should have done it if only Newton was a little bit smarter. Alright, so the first thing...
Know your variables for sure, but the first step is usually to introduce a set of coordinates. Introduce a set of coordinates into a problem, and that means exactly what it always means. Take space and rule it into coordinates.
Three dimensions for sure, but I'm only going to draw two. In other words, introduce fictitious... ...dictitious grid of coordinates.
Now, what shall we take for the distance between neighboring lattice points on this grid? We could take it to be one meter, we could take it to be ten meters, we could take it to be a million meters, we could take it whatever we like, but there's a smarter thing to do than to just fix the distance between the points. The smarter thing to do... is to imagine these points have been chosen so that the grid points always pass through the same galaxies in other words that the galaxies here provide a grid provide a grid In such a way that no matter what happens, since the galaxies are nice and uniform, no matter what happens, this galaxy over here will always be at that point on the grid.
That galaxy over here will always be at that point on the grid. And that means that if the universe indeed either expands or contracts, the grid has to expand or, let me say it differently, if the galaxies are moving relative to each other, perhaps away from each other or closer to each other, then the grid moves with them. Let's choose coordinates so that the galaxies are sort of frozen in the grid.
Now it's not obvious you can do that. It's not obvious you can do that. If the galaxies were such that some were moving this way over here, some were moving that way over here, some were moving that way over here, a sort of random kind of motion, then there would be no way to fix the coordinates by attaching them to the galaxies, because even at a point, the different ones would be moving in different ways. But that's not what you see when you look out.
At the heavens, what you see is that they're moving very coherently, exactly as if they were embedded in a grid, with the grid perhaps expanding, perhaps contracting, we'll come to that, but the whole grid being sort of frozen. Any motion that takes place is because the grid is either expanding in size or contracting in size. That's an observation about the relative motion of nearby galaxies. Galaxies over here and over here, which are relatively nearby, are not moving with tremendous velocity relative to each other. They're moving in a nice coherent way, as I said, so that, uh, so we can choose coordinates, we'll call them x, y, and z, standard names for coordinates, x, y, and z, but x, y, and z are not measured in length.
They're not measured in length because the length of a grid cell may change with time. Okay, so we've labeled the galaxies by where they are in a grid, and now we can ask the question, let's say the distance, let's take two points, let's start with two points separated by an x distance over here. Let's call that x distance delta x. How far apart are they? Well, I don't know how far apart they are yet, but I'm now going to postulate that the distance between them, the actual distance in meters or in some physical unit that you measure with a ruler, could be a light year on a side, it could be a million light years on a side, but a ruler, that the actual distance is proportional to delta x The distance between these two people over here is half the distance between these two is a third the distance between these two so it's proportional to Delta X times a Parameter that's called the scale parameter the scale parameter may or may not be just a constant It may just be a constant if it were just constant Then the distance between galaxies fixed in the grid would stay constant with time But it may also be time dependent, so let's allow that That would say the distance between two galaxies, let's say this is galaxy A, this is galaxy B The distance from A to B is A of t times delta x AB Where delta x is the x distance, is the coordinate distance between them Let me write it more generally.
If we have two galaxies, arbitrary positions on the grid, then the distance between them, dA b, is equal to A of t, the same A of t, and then the square root of Pythagoras'theorem, delta x squared plus delta y squared plus delta z squared. In other words, you measure the distance along the grid in grid units, and then multiply it by A to find the actual physical distance between the two points. As I said, A may or may not be constant in time.
Well, of course it's not. If it was constant in time, that would mean literally the galaxies were just frozen in space and they didn't move. And that's not what we see. We see them moving apart from each other. Okay, so let's calculate now the velocity.
Between galaxy A and galaxy B Here's the distance between galaxy A and this of course should be Delta A B the distance The coordinate grid distance Let's just use the simpler formula up here. Let's forget Pythagoras and just take them to be along the x-axis It doesn't really matter Okay, here's DAB What's the velocity between, what's the relative velocity of the A-B galaxies? It's just the time derivative of this, right?
Just the time derivative of the distance is the velocity So the velocity between A and B is just equal to the time derivative And the only thing that's changing for A and B a and b are fixed in the grid, so delta x is not changing, that's fixed the only thing that's changing perhaps is a, so the velocity is just the time derivative of a a dot means the time derivative of a a dot times delta x All I've done is differentiate this formula with respect to time. Now I can write that the ratio of the velocity to the distance, I'll leave out the a, well let's put them in, ab, the ratio of the velocity to the distance is just the ratio of a dot to a. Notice that delta x cancelled out. Well that's interesting. It means that the ratio of the velocity to the distance doesn't depend on which pair of galaxies we're talking about.
Every pair of galaxies, no matter how far apart, no matter how close, no matter what angle they're oriented in, the relative velocity between the two of them Relative either separation or the opposite of separation, the ratio of the velocity to the distance is a dot over a. What's the name for this thing? Anybody know? The Hubble constant.
It's called the Hubble constant. Let's call it h. Now, is there any reason why it should be a constant? What do we mean when we say it's a constant? There's no reason for it to be independent of the time, and in fact it's not.
What we find here is that it's independent of x. It doesn't matter where you are, it doesn't matter which two galaxies you're talking about, the same Hubble constant at a given time. So the Hubble constant is a kind of misnomer.
The Hubble... Hubble. The Hubble parameter. The Hubble function. The Hubble function is independent of position but depends on time, and now we just write this in the standard form that the velocity between any two galaxies in the universe is equal to the same Hubble parameter times the distance between them.
And that's the derivation of the Hubble law. Excuse me, question? Don't you sort of start out almost assuming what you're trying to show by saying that DAV equals one function a of t that's independent?
Yes, yes, yes indeed. Absolutely. Yeah, I mean...
You never would have written this if Hubble hadn't discovered that the Hubble Law was right But on the other hand, the Hubble Law is in some sense not all that surprising All it's, you know, as some witty person said said you shouldn't be surprised that the fastest horse goes the furthest okay right the faster you move the far far farther you go so that's all this thing says however It's interesting, the connection between this formula and the Hubble formula is, as you point out, a close one, but what it says is everything is moving on a grid and it's the grid itself. whose size scale may or may not be changing with time, but of course it is changing with time and the Hubble constant is just the ratio of the time derivative of A to A itself Okay, that's the facts, those are the facts, those are the facts as Hubble discovered them and as theoretical cosmologists then had something to work with Well, let's say a few more things about this. What about the mass within a region, let's take a region of size delta x, delta y, delta z.
And now I mean a region which is big enough so that, I don't know what happened to my universe, I have my universe here, but... Big enough so that we can average over the small scale structure. How much mass is in there? Well, that, the amount of mass that's in there is going to be proportional to delta x, delta y, delta z. The bigger the region that you take, the more mass.
And so, let's just say the amount of mass, we'll call it nu. Nu is nothing but the amount of mass per unit volume of the grid, but volume not being measured in meters, but being measured in x. So let's say that's the mass in a given region of coordinate volume delta x, delta y, delta z.
On the other hand, what's the actual volume of that region? Let's take this volume of the same region. The volume of the same region is not delta x, delta y, delta z.
Why? Because the distance along the x-axis and the y-axis and the z-axis is not delta x. It's a times delta x. So, that means the volume of that same cell, that same cell, is a cubed times delta x, delta y, delta z, right? That's because the length along the x-axis is a times delta x, a times delta y, a times delta z And so now let's write a formula for the density of mass The density, I mean the physical density of mass now How much mass is there per cubic kilometer or cubic light year or whatever units We haven't specified units yet Later on we'll specify units Meters are fine.
Meters, seconds, and kilograms are fine. Mass measured in kilograms, volume measured in cubic meters. What's the density? So let's write, let's call the density, a standard terminology for density is rho. I don't know where it came from.
Rho stands for density. Let's write it over here, density. And density means the number of kilograms per cubic meter, if you like It's the ratio of the mass to the volume It's the ratio of the mass to the volume, and it's just as nu here, divided by a cubed That's a formula that we have, nu divided by a cubed Now, the amount of mass In each cell in here stays fixed. Why does it stay fixed?
Because the galaxies move with the grid. So the amount of mass in a given region in the grid stays the same. That's just something I've called nu, the Greek letter nu, and I divided it by the volume to get the density, and of course if A changes with time, the density changes with time.
That's obvious. If the universe grows, the density decreases. If the universe collapses, the density increases, and so this is a formula that we will use from time to time.
All right. So far, we've done nothing. Something that Euclid himself couldn't have done.
Alright, we didn't even need Newton yet. Now enters Newton. Newton says, look, let's not play games, let's forget all this, we'll take into account that the universe is homogeneous and all that stuff But Newton was a very, very self-centered person, he always believed that he was at the center of the universe And so it was very natural for him to take the perspective that I, Sir Isaac Newton I'm at the origin Now of course we know, and Newton also would have known, that if he's clever He'll get the same equations no matter where he places himself, but there's nothing wrong with choosing the grid such that Newton and we are at the center of the grid then surrounding Newton and moreover Newton will also say, I'm not moving. I'm not moving, I'm standing still. So Newton is at rest at the center of the universe as far as for mathematical purposes.
And now he wants to, and of course we're talking about on a scale so that everything is nice and uniformly distributed. Now he looks out to a distant galaxy. He looks out at the galaxy over here.
And he wants to know how that galaxy moves. Well, that galaxy moves under the assumption of Newton's equations. Newton's equations say that everything gravitates with everything else.
But there's something special about Newton's theorem. Newton knew this theorem. In fact, it's called Newton's theorem. What Newton's theorem says... Is that if you want to know what the gravitational force on a system is, given that everything is isotropic, doesn't even have to be homogeneous for this, given that everything is isotropic, you want to know the gravitational force in a frame of reference, like I've drawn here, you want to know the gravitational force on that particle, then draw a sphere.
With that particle on the sphere, centered at the origin, and then take all of the mass within that sphere and pretend that it's just sitting at the origin. Just pretend. We're not literally moving it.
Just pretend that the only mass in the universe within the sphere is at the origin. And what about the outside? The mass is on the outside. Ignore it.
Newton's theorem says that the force on a particle in a isotropic world like this all comes from the sphere inside the radius of a particle and nothing from the outside. I think we may have proved that in previous classes in classical mechanics I don't remember, but it's a true, it's a true theorem It's a true theorem, and it's the reason that We here, in evaluating the gravitational field on this pen here Why we can pretend that all of the mass of the Earth is at the center of the Earth When I evaluate the gravitational field here, keeping in mind that the Earth is a sphere, keeping in mind that it's pretty uniform and so forth, I can just pretend that all of the mass was at the center of the Earth. Until of course the pen hits the floor.
Then it'll say, no, the mass wasn't the... but until it hits the floor, pretend that all the mass was concentrated at the center, and furthermore, the mass is outside, beyond... the Earth. Even though there's a lot more out there, incidentally, there's a lot of mass out there. I'm not talking about the ceiling of the building, I'm talking about all the galaxies out there.
There's a lot more, but the pen doesn't feel them. It only feels the things on the inside of the sphere. So, Newton says, what I'm going to do is I'm going to take this galaxy, which is...
At a certain distance away, what's its distance here? Its distance is d. Its distance is the square root of x squared plus y squared plus z squared. x squared and y squared and z squared, that's the coordinates of this point over here, times a, the distance from the center. Can you read the, is it easy to read the red?
I don't know why I started the red, I just, uh, elapsed. Is the red readable? Okay. Square root of x squared plus y squared plus e squared, that's Pythagoras, and you multiply by a to find the actual distance.
I can call that, let's call that d equals a of t, and let's just call this thing here r. r. r is not measured in meters, it's just square root of x squared plus y squared plus z squared.
That's the distance from the center to the galaxy in question. Now, Newton's equations are about forces and accelerations. So the first thing is let's calculate the acceleration of X of the point X of the galaxy at the point X relative to the origin.
Well first the velocity. The velocity is V is equal to A dot of T times R And what about the acceleration? The acceleration is just differentiate again Acceleration is equal to a double dot of t times r Do we have to worry about whether r is changing with time?
No, because the galaxy is at a fixed point in this expanding lattice. r is fixed for that galaxy and so this is the acceleration. We could multiply it by the mass of the galaxy if we wanted to, but I don't need to. It's just the acceleration and what are we going to set that equal to? We're going to set that equal to the acceleration that we would get from all the gravitating material inside here.
Okay, so let's see what how much First question is how much mass is in there? Well, let's just call it the mass. This is the mass that's inside this sphere.
The formula that we're going to compare this with is Newton's gravitational formula. Force is equal to mass times mass. Which mass is this little one here? That's the galaxy. The mass of the big one?
Which one is that? That's all the mass on the inside and the distance between, well the distance squared and I'm missing a couple of things, two things I'm missing Newton's gravitational constant, 6.7 times 10 to the minus 11th in some units. I'm missing one more thing.
Anybody know what it is? My minus sign. The minus sign indicates that the force is attractive, pulling in.
Alright, that's the convention, force pulling in is counted as negative, force pushing out is counted as positive. Alright this is the force of gravity on a particle of mass m, what is the acceleration of gravity? The acceleration of gravity is just drop the mass, just drop this out, forget the mass here.
The acceleration is the force per unit mass, and write that this is the acceleration M minus mg divided by c squared, that's the acceleration of that, of that, what's that? Which one are you using, the small one? No, no, no, I divided out the small m Good.
So that's the acceleration due to the presence of all this mass in the interior here. And that had better be equal to a double dot of t times r. God knows where this is going, but we're just following our nose, writing equations.
And you know, that's always the way you do it. You start out with some physical principles. You write down the equations and then you blindly follow them for a way until you need to think again So we're on autopilot now, we're just doing equations Let me rewrite it down here A double dot r is equal to minus mg over d squared Okay, let's plug in this guy over here, the distance is A times R.
So maybe we can, maybe who knows, at some point we might have actually discovered something that looks interesting. At the moment that's just a blind aft squared or just a squared, let's just call it a squared. A squared times d squared?
No, a squared times R squared, right? Okay, now excuse me but I'm just going to divide by r here. I secretly know where I'm going. Right.
Maybe you do too, but that's alright. r cubed, I divide it, and I'm going to divide by another a. Makes this a cubed. Okay. Now...
This is good, this will do. But, next question, what's the volume of the sphere? Let's write the volume of the sphere.
This is Newton's equations. Now, volume of the sphere, what's the volume? 4 thirds, 4 thirds, pi, now is it r cubed?
No, d cubed. which means a cubed times r cubed, right? because distance is really a times r that's the actual physical volume, when I say the volume, I mean the volume is measured in some standard unit like meters that's the volume, now look here, we have a cubed times r cubed here Let me write that as volume 3 over 4 pi volume is equal to a cubed r cubed. Or maybe I'm being dumb. Maybe I shouldn't.
Yeah, that's dumb. Let's not do that. Let's just look at this formula here.
Notice that we have a cubed r cubed downstairs. Let's multiply by 4 over 3 pi, or divide by 4 over 3 pi, and multiply by 4 over 3 pi. 4 thirds pi.
What I did here, I undid here. But now, I have m over the volume, what's m over the volume? The density, ooh, something nice may be happening a double dot over a is equal to minus four thirds pi Newton's constant times the ratio of the mass in that sphere to the volume in that sphere and that is the density Now, that's a nice equation.
Notice that it really doesn't depend on R anymore. If we know what the density of the universe is, and the density of the universe does not depend on where you are, the density of the universe does not depend on R. The left-hand side, R has dropped out. The right hand side, no memory of r, it means this equation is true for every galaxy no matter how far away Same equation. Had we done a different galaxy, we would have gotten the same equation.
The only way that this equation had any memory of which galaxy we were talking about was because of r, but r dropped out of the equation. That's of course a good thing, because if we want to think of A as something which doesn't depend on where you are, Then it had better be that it drops out. So Newton confirms what he might have expected, that the equation for A is a universal equation for all galaxies. It seems like something seems a little off because we picked the origin and it was arbitrary, I suppose.
It was. Again, we would have gotten exactly the same thing no matter what origin we picked. But I mean, it says whichever origin we pick, then we get the force going toward that origin, right?
That's right. And so something doesn't add up there. Well, it has added up, but the answer doesn't depend on which… No, you have to… No, no, no, the point is you have to do the transformation carefully.
You have to do the transformation carefully, you go to another origin, and in your… Newton could have said, let me work this out from my frame of reference, which I will put myself at the origin, but let me study now the relative motion of some galaxies relative to some moving, some galaxy which is moving. He would have found exactly the same equations, but he would have had to do the transformation carefully. So we finessed that and got away from it by just putting ourselves at the center But as you can see, the final formula doesn't care where you are It confirms the fact that nothing really depended on which galaxy we thought of as our home But the direction of the force, if we're really looking at the direction of the gravitational force, that's always toward the origin, right?
There's a relative force. The right way to think about it is really a relative force. No. Yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah. Yeah, yeah, yeah, yeah.
In this way of thinking about it, the force is always toward the origin. Right. But had we stationed ourselves on some other galaxy that was moving, and did all the transformations... Remember, when you go to a moving frame, there are fake forces. Inertial forces, fake forces that you have to add in.
So from the point of view of this guy over here, this galaxy over here... Has a force which could be thought of as being toward here, plus a fake force, the fake force being an inertial force due to his acceleration. But we got around that by just saying, let's position ourselves at the center. No acceleration, no velocity, we just sit at the center. So the only test question, the only question is do we get an answer which doesn't depend on who we are and which galaxy we're on, right?
Okay, so that's part of the main message of this. The answer doesn't depend on which galaxy you're on. And so it really didn't depend on Newton's assumption that he was at the center. Can we go back to the equation for mass?
Would anything have changed if we had assumed that mu was not a constant? Oh, yes. Yes, yes, nu.
Things would have changed if nu was not a constant. Could we say we've kind of made it happen by making that assumption? Made what happen? The end result.
A constant in space? Yes, to say that it's a constant in space is the principle that the universe is homogeneous Absolutely, everything hinges on the homogeneity of the universe Right, and that the number of the mass per unit volume is the same everywhere, in space Okay, yes, everything hinged on that and okay so here's one equation it's a central fundamental equation of cosmology and it's a differential equation it's an equation for how a changes with time there's a number of things to look at but the first interesting thing to look at, is it's impossible to have a universe which is static. Static means that A doesn't change with time, unless it's empty. Empty means rho equals zero. Only if it is empty, so that this side is zero, can the time derivative of A, or the second time derivative in this case, be zero.
So. We derive the fact that the universe is not static. Alright, one more thing we could do to make this an equation that we could solve is to replace rho by the constant nu divided by a cubed. Now nu is literally a constant.
It's the number of galaxies times the mass of a galaxy in a unit coordinate volume. It doesn't change with time because the galaxies are frozen in the grid. And so we could write this equation one more step. A double dot.
Not surprising that there's an a double dot. Why is there an a double dot? Because Newton's equations are about acceleration.
And not surprising that there's an a double dot. Equals minus 4 over 3 pi times g. Times the density, but the point is now the density is not a constant.
Nu is a constant, but nu over a cubed is not, because a is changing with time. And so we better put that in here, nu divided by a cubed. Okay, so there's a lot of constants. Here, minus, the minus sign is a constant, 4 pi over 3, g newton is a constant, and we can pick nu also to be a constant. So everything out here is constant.
A is not constant, but we have a kind of differential equation. It is a differential equation, a kind of equation of motion. In terms of one constant, 4 pi g nu over 3, we have an equation of motion for the scale factor.
for the scale factor a is a function of time. Who was the first to discover this equation? It was actually discovered in the context of the general theory of relativity.
It was discovered, I think, Friedman, Alexander Friedman, before he got himself killed in World War I, I think. using the general theory of relativity consistent with what Einstein should have done but it's perfectly possible there's nothing in it that wasn't just good old Newtonian mechanics Do you need to keep the A and the denominator on the left hand side? Well, I can multiply it if I like. A double dot equals? Hmm?
A double dot equals a constant over a squared? Sure, you can do that. It's just, it's traditional to write it this way. It's just a tradition. Yeah.
So, does that negative sign not tell us anything about whether we're expanding or contracting? It doesn't tell us whether we're expanding or contracting. Okay, so let me explain why.
Let me write... Forget that now. Now we just have the Earth.
Let's compare this with something else. We have the Earth, and we have a particle over here. Let's put it on the x-axis, on the x-axis.
All right, there's an equation for that particle. It's the same equation. Let's call it now, let's call it x, but x doesn't stay. for these coordinates now, it just stands for the standard position coordinate, or the height above the earth. That satisfies some equation, x double dot is equal to the gravitational force, whatever the gravitational force is, mg over x squared minus That's it, something like that.
Okay, does this equation, what this equation tells us is that the particle is accelerating toward the earth. The minus sign tells us that the acceleration is toward the earth. But whether it's moving away from the earth or toward the earth is a question of velocity, not acceleration. Is the velocity that way, or is it that way?
Well, you can imagine somebody over here taking this particle and ejecting it that way It will have a positive velocity, it will be moving away from the Earth You could also imagine the same person pushing it that way It will be moving toward the Earth, x is decreasing But the acceleration will be the same In either case, the velocity will have a negative acceleration which means if it's going this way, it'll turn around or may turn around. If it's going this way, it'll increase the inward velocity. Whether it turns around or not depends on what?
Initial conditions. And the initial conditions, or whether it's above or below the escape velocity. But in either case, the acceleration is toward the Earth. So knowing that the acceleration is toward the Earth, as it is for this pen, does not tell me whether it's moving up or moving down. We can move up and then move down, and you get the point.
Okay, so no, this equation doesn't tell us whether the universe is expanding or contracting, but it tells us that the second derivative is negative. So it means that even if it's expanding, it's tending to slow down. If it's expanding, it's tending to slow down. If it's contracting, it's tending to speed up its contraction. There is an analog here of whether you are above or below the escape velocity and we'll come to it All right, so I was asked a question which I Will just point out Art, sorry Art, but I'm gonna use your name Art asked me well, he looked at this and said this is negative and He looked at what is this?
He looked at This, and said, this is positive. If the universe is expanding, H is positive. How come this one's negative?
But that's because he didn't read carefully. There's two dots here and only one dot here. This is velocity, this is acceleration.
Not hard for acceleration to be negative. You know, you're in your Ferrari and you're going down the Autobahn or whatever. And you press down on the brake. Your acceleration is negative, but your velocity is positive.
You're slowing down, but you're still going ahead. Now, in fact, the universe is not slowing down. This, we're really doing what Newton would have done and what all cosmologists thought the right thing to do was until about 15 years ago. So 15... It's Newton's model of the universe, and it is the model that would have been called the standard model, or close to it, the standard model of the universe, until the accelerated universe was discovered.
This is the decelerated universe, you see. But the universe accelerates, so there's got to be something else in this equation. Well, there is.
There are several other things in that equation, but we'll come to them. Some parts do have to do with Einstein. Okay, let's talk about... Not cosmology, but just particles, rocks, stones thrown upward from the surface of the Earth. Equations are very similar.
Let's just examine them for a minute and take home a couple of lessons about it Here's the earth and we might as well think of it as a point because Newton Newton proved the theorem that said we can think of it as a point. We're outside. We're above the surface of the earth So here's that here's the earth Here's our particle over here No, put it over here, x-axis. Put it on the x-axis. And what are its equations?
The equations of Newton's equations. But there's actually a more useful version of Newton's equations, which is just energy conservation. Just energy conservation. Let's write down the energy of this particle over here, and write down that it's conserved.
In fact, it's a more useful equation than this one over here, the energy equation, or more useful than F equals mA. What is the energy of this particle here? It's moving outward, it has some velocity, the velocity could be negative, it could be moving inward.
But what is the total energy of this particle? The total energy of it is its kinetic energy plus its potential energy. Kinetic energy, one half. The mass of the particle, not the mass of the Earth, the mass of the particle times its velocity squared, which we could call x dot squared if we wanted.
Well, let's just leave it as velocity squared for a moment. But what about the potential energy? Remember the potential energy?
Potential energy is minus little m, big M, Newton's constant, divided by what? R. Not r squared, just r.
Say it again. X, x, x, yeah. Now, this can be positive or negative, believe it or not. The energy does not have to be positive.
For example, supposing this particle over here is at rest. I don't know how it got there. It got there, it's an initial condition. It got here at some time t.
It's at rest. But at a positive value of x. x is really always positive.
It really stands for the distance from the Earth, not the x coordinate. So x is always positive. This is always negative. This can be zero if the particle is at rest, and so the energy is negative in that case.
The energy can also be positive. Supposing we now take the same particle at the same position, but give it a velocity. If the velocity is big enough, then this can outweigh that.
This can outweigh that simply when, if I write the equation down, I'll write it down in a minute, when this is bigger than this, when the kinetic energy is bigger than the potential energy, and then the total energy is positive. Now, if the total energy is positive, this thing cannot turn around. Cannot, you might say, well, let's see, this particle could go out and turn around.
Why can't it turn around if the total energy is positive? Incidentally, energy of course is conserved, so whatever the energy is at one instant, it's the energy at every instant. Energy is conserved.
Let's suppose it turned around at that point. What would its velocity be at that point? Zero. So what would its energy be?
Negative. Right? So therefore, if it turns around, It's negative. The energy is negative.
If it doesn't turn around, the energy is positive. Energy equals zero is a sort of edge of parameter space. If the energy is positive, the particle just keeps going and going and going, it escapes. If the energy is zero, that's exactly the escape velocity.
We'll ask later whether it escapes or not if it's exactly at zero. What is the escape velocity? The escape velocity is the solution of the equation that this is equal to zero.
So let's write it out. 1 half v squared, I'm dropping the m because it cancels from both sides, the little m. 1 half v squared is equal to big M, big G, divided by x, and now we can just multiply by 2. And that gives us a formula for the escape velocity. That's the formula for the escape velocity when the energy is exactly equal to zero. In exactly the same manner, the universe can be above the escape velocity, below the escape velocity, or at the escape velocity.
We're going to work that out in a minute. But all it means is... If it's above the escape velocity, it means that initially, at some point, the outward expansion was large enough that it doesn't turn around. If it's below the escape velocity, then the universe turns around and recontracts.
That's the main reason for showing you this, and the escape velocity is kind of the edge. The escape velocity is also the velocity at which the energy is equal to zero. Keep that in mind. Escape velocity, same thing as energy equal to zero.
Alright, now let's concentrate on this particle over here. Now for all practical purposes, this particle over here, all it knows is that it's moving in the gravitational field of a point mass at the center where the point mass is capital M. So for all practical purposes, we can replace this problem over here by this one over here.
In fact, it's exactly the same problem. So let's work out the energetics, the kinetic, the potential energy of this particle, and keep in mind that it's constant. It's constant because for all practical purposes This particle is moving exactly as it would be if all there was was a mass at the center And in that case energy would be constant so we can just lift the things that I wrote before But let's let's work them out. Yeah No No, the whole thing is growing, but remember the grid, everything moves with the grid Everything moves with the grid The only thing that's changing is A The amount of mass in this sphere stays fixed In other words, the number of galaxies that this fellow over here sees within this sphere stays fixed. Right.
Good. Okay, so, no, so we don't have to worry about the mass changing. Alright, let's work out now the energy, the kinetic, well, the kinetic plus potential energy in Newton's frame.
In Newton's frame, we'll work out the kinetic. Alright, so what is it? It's one-half mv squared again. One-half the mass... of this galaxy times the velocity squared, but that's a dot squared r squared, right?
Same r. Where is it? Same r, same, yeah, same r.
d is equal to a times r. Distance is a times r. Velocity is a dot times r. This is one half mv squared.
And then minus little m Big M G divided by distance, right? Just divided by distance. That's the potential energy, M, M, M, G. And what is the distance? The distance is A times R, right?
Let's do the... and that's equal to the energy of this... Now for simplicity, and because it's, because of simplicity and also because I'm getting a little tired, I think I will just do tonight the case where the energy is exactly equal to zero.
What does that correspond to? Exactly the critical escape velocity. That case, the other case is just as easy, but let's, let's do that case. All right, so that's the case where The universe is sort of just on the edge Not clear whether it's going to turn around and fall back or whether it's going to keep going And it's on the edge of the cusp of doing one or the other all right, so we're going to set this equal to zero Let's work out that equation. Let's work out that equation using the various things We know okay first thing to do is to get rid of the little M here Why should we get rid of the little m?
Because it appears in both terms here and the whole thing is equal to zero, so I divide it out. I'll also multiply by 2. Next I'm going to divide by r squared. You know why am I dividing by r squared? I want to get r cubed down here because I know That r cubed has to do with the volume, and the volume, I'm going to get a density. I'm trying to get this thing in terms of density.
Alright, so I divide by r squared, and that gives me an r cubed downstairs. That's nice, because there's a mass here, and an r cubed downstairs. It looks like I'm getting the density, but not quite, because the volume of the sphere is a cubed times r cubed, not a times r cubed.
So what do I do? I just divide the equation by another a squared, a cubed. Okay, that's good. Ah, a cubed times r cubed.
a cubed times r cubed. What do I do next? Well, if I'm smart, I will multiply this by 4 over 3 times pi.
That will literally make this a volume, but I'm doing something illegal unless I multiply it here also. 4 over 3 times pi equals 0, right? Equals 0. We're almost there. Let me rewrite it a dot over a squared Remember what a dot over a is?
It's the Hubble constant So this is the square of the... I take it back It's not a constant, the Hubble thingy a dot over a squared, that's the Hubble thingy squared And that's equal to I'm just transposing this to the right hand side There's an 8 pi over 3, famous 8, 2 times 4 is 8, 8 pi over 3. There's a G, and now there's an M divided by the volume of the sphere That's why I went to this effort here to put another couple of factors of A and R downstairs So that I would get A divided by the volume of the sphere and that's rho That's the mass density rho, the actual mass density a dot over a squared equals 8 pi over 3 g times rho That is the Friedmann equation That's the Friedmann equation The way that it's usually written It's equivalent to this equation This one over here is energy conservation, also set the energy equal to zero. This one over here is Newton's equations, but the same physics. The same physics, the Newton version of it, the conservation of energy version of it.
This one is more useful. As I said, it's called the Friedman equation. It's not completely general because we did set the energy to zero.
We did set it to just exactly the critical escape velocity. So this universe is not going to re-collapse, but it's going to... What does happen if you shoot something out at exactly the escape velocity?
What happens to its motion as time goes on? Doesn't it slow to zero at infinity? Yeah.
It just asymptotically gets slower and slower and slower, but it never turns around. This universe will asymptotically get slower and slower and slower in its expansion, but never turn around for the same reasons. Okay, so that's our Friedmann equation.
I'd like to solve it, but I don't know enough yet. The reason I don't know enough is because there's rho over here, and I don't know what to do with rho, except we do know what to do with rho. Remember the equation that rho is equal to the constant nu.
Incidentally, the constant nu can be set to be anything you want. It doesn't, it's not, it's, yeah, okay. It's the mass per unit coordinate volume. By changing your coordinates, you can change the amount of mass that's in your, in a coordinate volume.
So actually, nu never really comes into anything important. What? Rho is equal to nu divided by a cubed.
Remember that? Okay, so we can now write an even more useful version of this. a dot over a squared is equal to 8 pi over 3 g nu.
And nu is a constant. Nu does not change with time, divided by a cubed. We have it right. All of this junk here is just a constant. 8 pi nu over 3 times g is just a constant.
In fact, I could if I like have chosen nu so that 8 pi g o nu over 3 is just the number 1. There's nothing interesting in it. The basic equation, the basic equation, the basic form of the equation It's just that a dot over a squared is equal to some constant, but let's just choose that constant to be 1, just for simplicity, is 1 over a cubed. If we can solve this equation, we can solve this one. It's not hard to go from one to the other.
So, we'd like to see how to solve this equation. Now notice, first of all, that the right-hand side is always positive. In fact, it never quite goes to zero.
No matter how big A gets, it's always positive. As A gets really, really big, it gets smaller and smaller. So that tells us that A dot over A never becomes equal to zero. a dot equals, a dot equals zero would be the universe turning around.
That would be the place where the universe turned around when the, when the expansion rate increa, they went to zero. So this tells us the expansion rate never goes to zero. Hubble constant never changes sign, or at least the square of the Hubble constant never goes to zero.
So if it doesn't go to zero, it can't change sign. And, but, it does slow down. The Hubble constant gets smaller and smaller and smaller with time. So it's as if the universe just gets tired of expanding.
But it never gets tired enough to stop. Okay, we can try to solve this. I think I will just take the, it's getting late and I get tired about this time. So I will take the easy way of solving it, but we will come back to these kind of equations We'll come back to the kind this this type of equation.
This is absolutely when I say that this type of equation this type of equation is Absolutely central to all of cosmology and we can solve them. You can solve them quite easily Let's just look for a solution of a particular type, okay? I'll look for a solution rather than to solve the equations. Let's look to see if we can find a solution where a is some constant times time to some power. We don't know that we can solve it this way, but we can try.
We can take a trial solution. a proportional to t would just, what would a proportional to t mean? That would just mean a grows in proportion to time. A very simple way. We don't expect that to be right.
And it's not because the thing slows down. But we can look for a solution of this type. So let's try it out.
Let's see if we can if we can use the equation to see what whether we can solve for C and P. Okay, so what's A dot? A dot is C, P, T to the P minus 1, right?
That's just differentiation. Now, a dot over a, that's easy. We just have to divide by a, so we have to divide this by c, t to the p.
C's cancel. Neat. The constant here cancels.
And what's t to the p minus 1 over t to the p? p over t, right? That's the left-hand side. P over T... Oh, sorry, we have to square it.
That's P squared over T squared. That's the left side. P squared...
Sorry, P squared over T squared. Now, what about 1 over a cubed? Let's see what that is. 1 over a cubed, it's 1, divided by c cubed, t to the 3p. Do I have that right?
I do. Now we can read off how to match the two sides. Let's get rid of this over here, and let's match the two sides. In the denominator we have a power.
We also have a power over here. This is 1 over t squared. This is 1 over t to the 3p. But I haven't told you what p is yet.
So if we want to match, I've just said let's look for a solution of the form ct to the p and see if we can figure out what c and p have to be. Well, the first thing we learn is that 3p had better equal 2. Otherwise these things can't match. There's no way that t to the fourth here can match...
t squared here, so the first thing we learn is that 3p has to equal 2. We'll come back to it in a minute. Alright, that will guarantee that the t squared and the t squared agree on this side. On the other hand, we also have to match the constant, and the constant tells us that p squared equals 1 over c cubed.
So that tells us there was really only one constant that we had to worry about, either P or C. Once we know P, and we do know P, we know P from here, and therefore we know the constant. The constant is not so interesting.
What's interesting is P, because what does P say? It says that A expands like T. to the two-thirds. T is equal to two-thirds.
Some constant times T to the two-thirds power. That's the way a Newtonian universe would expand if it had, if it was right at the critical escape velocity. It would expand with a scale factor and everything, all galaxies separating as Time to the two-thirds power. That's a quite a remarkable derivation, I think.
I think, you know, Newton... I don't know why he didn't do it. It really...
it annoys me that he didn't do it. He should have done it. I think he went to the mint at this point or something. I'm not sure what happened to him.
Oh, that was supposed to be the year of the plague? No, it was the year of the tulips, when he lost his shirt on tulips. What?
He did lose his shirt on tulips, you know. Yeah, yeah, he was one of the suckers who got suckered by the tulip bubble. It's true. So, you know, smart... Hmm?
Yeah, but I think he got stung. Did Newton know there was a universe? No, but he should have predicted that there's a universe.
Oh, yes he did, and he worried about the fact that a homogeneous universe... Oh, yes. He most certainly had speculated enough that he was right on the threshold of doing this. He had asked all the questions about it and didn't quite carry it out. So, yeah.
Yes, this is... actually not, but... Yes, that's a good question.
We've done a completely Newtonian theory. In Newtonian theory, space is flat. If space is flat, it just goes on and on forever.
So yes, the Newtonian universe would have been infinite, it would have been spatially flat, It wouldn't have had an interesting Einsteinian geometry of any kind, although it would have been expanding or contracting, and it would have been entirely Newtonian. All right, so I did this just first of all because it's easy, second of all because it contains a lot of the physics that we're going to be dealing with in a simple form. And it gives us a model universe. It gives us a model universe with a scale factor that increases like the two-thirds power of the time. Excuse me.
Is everything that you've said here still true in the case where we don't have escape velocity zero? Not quite, no, no, then there's another term in this equation There's another term in this equation and We will come to that other term No, it can't be because if you were if you had negative energy it would re-collapse Right, so there's another term and the next time we'll take up that other term and we'll talk about the three possibilities less than zero In other words, we collapse. Greater than zero, that means the universe just expands without even thinking about it.
And this, which is the critical point where it slows down in a certain way. Another diagram that people always draw for this kind of thing looks something like this. You've probably seen diagrams like this.
You plot on the vertical axis. You plot A, the scale factor, and on the horizontal axis you plot time. Now, A equals T. There's no sensible cosmology that does that, but let's just draw it in. Here's A equals T.
Now what does it mean that a decelerates? That the acceleration is negative. That it decelerates is a statement that the curve is bent over this way, as opposed to this way. The second derivative is negative.
The curve goes this way. a to the 2 thirds looks approximately like this. And of course it keeps growing What about a re-contracting universe?
What if the universe re-collapsed? A collapsing universe would look like a crash And a... This does not approach a straight line, incidentally It does not approach a straight line. It just keeps bending over slightly more and more.
And the universe of positive energy would look pretty much the same and then go off on a straight line, on a straight line. Those are the three cases that we'll describe. Did I get that right?
No, I mean, I take this back. This is not quite right. I take that back. It's a...
No, no, no, that's not, that's incorrect. Well, we'll do the case of positive energy. But in any case, in all of these cases, the tendency is to curve over because the acceleration is negative The real universe does not look like that The real universe starts out looking like that and then starts to curve upward It's accelerating, the real universe is accelerating So we got this ct to the 2 thirds because we tried solutions in a certain form.
Because what? Because we tried solutions in a certain form. Yeah.
If we would have just sit down and experiment with other solutions, would there be others that gave us a different result? We'll solve the equation in detail, though. This is it. This is it.
This is the solution. Yeah, we'll, yeah. No, this is the only solution.
Or we can change the energy. We can change the energy away from zero, and if we do, we can generate other kinds of solutions. Okay, any questions? I'm tired, but yeah. The critical density universe, does it bend over and- No, no, no, no, a, well, sorry.
Yeah, the derivative gets smaller and smaller, a to the two-thirds, all right, so let's see, what do we know? Yeah, no, we've already done it, yeah. Let's, d to the two-thirds is a.
And a dot is equal to two-thirds, one over t to the one-third, right? So the slope goes to zero, the slope goes to zero as t gets larger, right? But it's always positive Okay, so this is the sense in which it's getting tired. The slope is, uh, is... And you can see now why Einstein failed to be able to describe a static universe.
Well, we'll come to it. I'm getting ahead of myself. Yeah, I don't want to get ahead of myself.
Good. Okay, good. Let's... If it was very old and it's the freedom case, right? If it was what?
Very old, it does get sort of flat. I think that Newton was prejudiced, yeah, let's see, yeah, see, Newton had this idea that the universe was 6,000 years old, and this wasn't fitting together with this. Yeah, yeah, yeah, Newton was a believer. So I think he had some, I think the reason he probably didn't do it is because he couldn't make it. fit with his prejudice about the age of the universe.
It's true. Did he write papers or anything about his thinking? What's that? Did he write papers about his thinking and philosophy and stuff like that?
Unfortunately, he did. Yeah, sure he did. He wrote more about religion than he did about, yeah, he wrote more about, more about, more about religion, I think, and alchemy than he did about physics. He's a prolific writer. He started with Einstein's field equations.
I see that a pi g there. Yep. Is that, I know that...
Same a pi g. What's that? Yeah.
Energy, is that how you... Mm-hmm, mm-hmm, yep, yep, yep. Not surprising, since this is about energy.
Yep, yep, absolutely. Is this kind of study synonymous with what you might call a... No, this is the theory without a cosmological constant.
The cosmological constant is what has to do with... The acceleration. This is the theory without the cosmological constant. In fact, this is called a matter-dominated universe. A matter-dominated universe for reasons that I will explain later.
So, while we know that the universe is expanding overall, like the entire universe, are there some galaxies in between that could be contracting? Well, certainly, yes. Calculating our overall calculations? Yes, yes, yes.
On the average, out to the largest observable distances, it is expanding, but individual little portions, there are places. For example, our galaxy is contracting together with Andromeda. Andromeda is not receding away from us. But you know that's on large enough scales. The Hubble law is not exactly true for all possible distances.
It becomes accurate as distances get larger. It's certainly not accurate for the for things which are bound together. Things which are close enough together that they're really bound together by gravity or any other force may be being pulled together. So as it happens, it's not unique, but on the average everything is moving away from everything else. But here and there you can find galaxies which have peculiar motion.
The term peculiar motion is a technical term. It is. It's a technical term and it means, it means what it says, sort of away from the average. So in an overall calculation, we should avoid those little galaxies and just try and look straight past them?
We should average over large volumes that these little fluctuations don't matter. Right. It's the same kind of thing. You say the air in the room is uniform.
Well, that's not really true. There are places where there's a fluctuation where it's more dense and a fluctuation where it's less dense. But when averaged over a sizable region, bigger than many molecules, the room is uniform.
Same thing holds here. So you mentioned Andromeda moving toward the Milky Way. Is that just motion within an expanding universe? Yes, yes, yes. The Andromeda just happens that...
for whatever reason, I don't know if it's really, I don't know if the complete history of the Andromeda Milky Way dynamics is, probably is. However it was formed, it was formed in a pocket which was dense enough that just slightly out of the ordinary, it was dense enough that these two galaxies had enough mass to overcome. the effect of the expansion.
So it's a fluctuation away from the norm. Really batter it's just a velocity. No, no, that's it. They're identical. They mean the same you asked me that the last time I remember to the technical was it seven years ago.
I think you asked me the same question No, there's no there's no difference you see you either take the position that the galaxies are moving Away from each other or you take the position that they're embedded in this grid And the grid is expanding. It's a mathematical, yeah. And perhaps in Einstein's way of thinking about it, it's a little more natural to think of it as space expanding, but they are equivalent.
Yeah, one more question. Yeah. Is there any corroborating evidence other than the brightness of the distant Type Ia supernova for the accelerating universe? Oh, yeah.
Yes, there is from a cosmic microwave background. Yeah, there is. And we will come to it. It's a sort of network of different observations.
The supernova, mostly supernova and cosmic microwave background fit together just precisely. For more, please visit us at stanford.edu.