Cosmology Lecture 1

(January 14, 2013) Leonard Susskind introduces the study of Cosmology and derives the classical physics formulas that describe our expanding universe.

Originally presented in the Stanford Continuing Studies Program.

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This quarter's subject is Cosmology. Cosmology is of course a very old subject.

It goes back thousands of years, but I am not going to tell you about thousands of years of cosmology. But when I say thousands of years I am talking about the Greeks of course.

But we're not going to go back here thousands of years.

We're going to go back at most to some time in the second quarter of the twentieth 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. At least to 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 a remnant of the Big Bang.

That happened some time in the sixties. I was a young student.

Before that Cosmology, was in a certain sense less like Physics and more like . . .

a natural science like what a naturalist does . . . studies this kinda thing studies that kinda 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 were always involved,

but they were involved because many of the things you see, many of these strange creatures, funny stars, galaxies and so forth are of course physical systems

and to describe them properly, they have angular momentum, they have all the things physical systems have,

there's chemicals out there, so physical chemists are involved - but thinking of the 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 fifty years, something like that.

And that's what we're gonna study, we're gonna study the Universe as a system, in other words the Universe as a system that we can study with equations.

So if you don't like equations, you're in the wrong place. All right, 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 physics is 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

-now of course if you look right at a star it looks a little different from 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, the 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 over there, and you looked around from sixteen galaxies over and you looked around what you would see, you would see about the same things 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's very simple. Imagine there's some distribution of galaxies.

You know, incidentally, at least in the first part of this study here, it's not gonna matter very much whether what we're talking about,

whether we call them galaxies or whether we just call them particles. They're just effectively mass points,

distributed throughout space. For the moment, I might even lapse into calling them particles from time to time.

Now you must read me, when I say particles I mean litteraly galaxies, unless I otherwise specify.

So the universe has a lot of them.

Anybody know how many galaxies are within visible...?

About a hundred billion: 10 to the11

Theres' 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 with telescopes, out to as far as astronomy takes us,

about 10 to the 11 galaxies, each galaxy of about 10 to the 11 stars, altogether 10 to the 22 stars

If each star has roughly ten planets that 10 to the 23 number of planets out there.

Imagine that we're over here and every direction we can look in it looks petty 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 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...

... yeah shells, I think somebody said shells. We'll have the geometry of some sort of shell-like structure.

Why? It doesn't litterally mean shells, it just means, yeah...

So, if that were the case and you went some place 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 centre of the universe

- if we happened to be at the very centre where everything just accidentally, or not accidentally, maybe by design,

happens to be nice in rotation with everything symmetric around us-

if we don't want to believe that, then we have to believe it's pretty much the same everywhere 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, okay.

That's called the cosmological principle. Why is it true?

Well how can it not be true it's the cosmological principle? And sometimes people argue like that.

It's true because it's been observed to be true, to some degree of approximation.

As was mentionned in some media that I don't know how to evaluate,

some astronmomers apparently claim to see structures out there

which are so big if the blackboard 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 to say that 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 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.

So that's the basic fact that we're gonna begin with.

Now what's the first step in formulating a physics problem?

Yeah know your variables, usually it's sharpen your pencil

After you've sharpened your pencil and you're an expert who knows your variables,

a good step, I'm not sure if it comes before that or after that, is ...

Oh you bet, you bet you bet, but we're going back, purposefully going back a few decades

to some time around the sixties or something like that. Fifties, sixties, forties...

The idea of a 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'll call it a 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, 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 are electrically neutral

but they are not gravitationnally 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.

So gravity is either pulling all the stuff together or is doing something to it, but it's a little bit confusing

What happens to this point over here?

Does it accelerate towards the centre, because at the centre 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.

It ought to just stay there because there's as much on one side as on the other side.

It ought to 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, there's nothing pulling it one way or another.

That's wrong. We're gonna 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, 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.

So the first thing, know your variables for sure,

but the first step is usually to 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 gonna draw two

In other words: introduce a fictitious grid of coordinates

What shall we take for the distance between neighbouring lattice points on this grid?

We could take it to be one metre, ten metres, a million metres, we could take 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, the galaxies here provide a grid.

They provide a grid in such a way that no matter what happens,

since the galaxies are nice and uniform,

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...

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.

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

sort of a 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, 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 we can choose cooordinates. 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 because the length of a grid cell may change with time.

So we've labelled the galaxies by where they are in a grid, and now we can ask the question:

Let's say the distance...

Let's start with two points separated by an x-distance here. Let's call that x-distance, delta-x.

How far apart are they?

Well I don't know how far they are yet, but I am now going to postulate that the distance between them,

-the actual distance, in metres, or in some physical unit that you measure with a ruler,-

it could be a light year on the side

it could be a million lightyear on the 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,

and a third of 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(t) times delta-x ab, where delta-x is the coordinate-distance between them.

Let me write it more generally.

If we have two galaxies at arbitrary positions on the grid

Then the distance between them...

D ab is equal to a(t) - the same a(t), then square root - pythagoras theorem -

delta-x squared, plus delta-y squared, plus delta-z squared.

In other words, you measure your distance along the grid, in grid units, and then multiply it by a of t

to find the actual physical distance between two points.

As I said, a(t) 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 frozen in space, and they didn't move.

And that's not what we see, we see them moving apart from eachother.

Ok, so let's calculate now, the velocity between galaxy a and galaxy b.

Here's the distance between galaxy a - and this of course this should be delta-a-b... the distance... coordinate for distance...

Let's just use the simpler formula up here. Let's forget pythagoras and just pick them to be along the x axis. It doesn't really matter.

Here's D a b. 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, 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... no, no let's put it in... a b...

the ratio of the velocity to the distance is just the ratio of a-dot to a.

Notice that delta-x canceled out. We'll 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.

Have a look at it, 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 found 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...

the Hubble parameter, the Hubble function is independent of position,

but depends on time.

And now I just write this in a standard form.

That the velocity between any two ...

galaxies in the universe

is equal to the same Hubble parameter

times a distance between them

that is the derivation of the Hubble law.

student question

yes indeed. Absolutely.

ja

you would never write these

if Hubble hadn't discovered that...

that Hubble law is right

but on the other hand the Hubble law

in some sense is not all that surprising

some w... person said, you shouldn't be surprised

that the fastest horse go the further

ok right

the faster you move, the faster further you go

that is all the thing says. However

it is interesting... the connection between this formula

and the Hubble formula, as you pointed out, is close one

but what it says is every thing is moving

on a grid and it is 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.

Ok that are the facts. That are the facts

as Hubble discovered them and

as theoretical cosmologist has something to work with.

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 the... I don't know what happen to my universe

I have my universe here

big enough so that we can average over the,

the small scale structure

How much mass is in there

well, the amount of mass in there

is going to be proportional to Dx, Dy Dz

the bigger the region you take, the more mass

and so just the amount of mass is, we will call it nu

nu is nothing but the amount of mass per

unit volume of the grid,

but the volume not being measured by meters

but measured by x

and so let's say that's the mass

in a given region of coordinate volume, Dx Dy Dz

on the other hand, what's the actual volume of that region

let's say this, the volume of the same region

the volume of the same region is not Dx Dy Dz

why?

because the distance along the x-axis, y-axis and z-axis

is not Dx, is a times Dx

so, that means the volume of the same cell

the same cell is a^3 times Dx Dy Dz, right

right?

that's because the way along the x-axis

is a times Dx, a times Dy and a times Dz

and so now let's write the formula for the density of the mass

for the density, I mean the physical density of the mass now

How much mass is the per cubic km,or cubic ly,

or whatever unit

we haven't specified unit yet

later on we will specify unit

ah. meters are fine.

meters, second and kilogram are fine

mass measured in kilogramme

volume measured in cubic meters

what's the density?

that's right. let's call the density the standard terminology for density is rho

I don't know where it comes from.

rho stands for density

let's write over here, density.

and density means the number of kilogrammes per cubic meters, if you like

It is the ratio of mass to the volume.

it's the ratio of mass to the volume,

and it's just nu here divided by a^3.

that's formula we have, nu divided by a^3

now, the amount of mass,

in each cell here, stays fixed.

why stay fixed? because galaxies move with the grid.

So the amount of mass for given region of grid stays the same

that's just something called nu, the

and divided by the volume to get the density

and of course if a changes with time

the density changes with time.

that's obvious, the universe grows, the density decreases.

if the universe collapse, the density increases.

so this is the formula that we will use from time to time

all right, so far...

we have done nothing that Euclid himself could have done.

right, we didn't even need Newton yet.

Now, enters Newton

And Newton says, look,

Let's not play games

let's forget all these....

and take into account the universe is homogeneous, and all that stuff

but Newton was a very very self-center person

he always believe that he was at the center of the universe

and so that's nature for him to take the perspective

that I, so Isaac Newton

am at the origin.

Now of course we know, and Newton would also know,

that if he is clever

he will get the same equations,

no matter where he places himself

there is nothing wrong with choosing the grid,

such that Newton and we are at the center of the universe

all right.

then surrounding Newton, and moreover

Newton also say I am not moving.

I am not moving. well I am staying still.

so Newton rests at the center of the universe

as far as...for mathematical propose,

and now you want to, and of course

we're talking about on a scale,

so that everything is uniformly distributed.

Now let's look out to a distance galaxy,

it looks out a galaxy over here.

anyone knows how that galaxy moves?

Well, that galaxy moves under the assumption of Newton's equations.

Newton's equations say that everything gravitate everything else.

but there is something special about Newton's theorem

Newton knows this theorem,

In fact, it called Newton's theorem.

What Newton's theorem says?

is that you want, what gravitational force, on a system is,

given that everything is isotropic,

doesn't have to be homogeneous.

Given that everything is isotropic,

You want to know,

the gravitational force in a frame of references,

that I drawing 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 take all the mass within that sphere.

and pretend that sitting at the origin.

to pretend, we are not literally move in.

just pretend, that the only mass within the sphere

is at the origin

and what about the outside?

the masses on the outside?

ignore it,

Newtons' theorem says, that the force on a particle

in an isotropic world like this,

all comes from the sphere inside the radii of the particle.

and nothing from the outside.

I think me prove that in a previous class,

the classical mechanics, I don't remember, but it's ture. It's a true theorem.

It's a true theorem, and it's the reason that,

We here in evaluate the gravitational field, on this pen here.

while we pretend that all of the mass of the Earth is at the center of the Earth.

While I evaluate the gravitational field here,

ahhh, keep in mind, that the Earth is a phere.

keep in mind that it's pretty uniform.

So forth, I can just pretend that all of the mass at the center of the Earth.

until of course, the pen is falling

they will say no,

until that fall, pretend all the mass concentrated at the center.

and furthermore, the mass outside, beyond this,

even though, there are a lot more out there, which is exact,

there are a lot of matter, I am not talking about the ceiling of the building.

I am talking about the galaxy out there.

There is a lot more, but the pen doesn't feel anything

Only feeling the thing inside this sphere.

So, Newton says what I am going to do

is, I am going to take this galaxy, which is,

at a certain distance away, what's the distance here,

It is the distance d,

this distance is

the square root of x^2 +y^2+z^2.

x^2, y^2, z^2, that's the coordinate of this point over here.

times a .

the distance from the center.

Can you read this, it is written of red. I don't know why it starts to red.

It's just labels, is red readable? okay.

square root of x^2+y^2+z^2, that's Pythagoras.

And you multiple by a to find the actual distance.

And call that, let's call that D equals to a(t), and let's call all these things here R. captical R.

R is not measured in meters,

it's just square root of x^2+y^2+z^2.

that's the distance from the center to the galaxy in question.

Now, the 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, relative to the origin.

Well, first of all, the velocity,

the velocity is V, is equals to a dot of t, times R.

What about the acceleration? The acceleration is just differential again.

Acceleration equals to a double dot of t, times R.

Do we have to worry about what R changing with time?

No, because the galaxy is at the fixed point in these expanding lattices.

R is fixed for that galaxy,

So this is the acceleration

We can multiply by the mass of the galaxy if we wanted to

but we don't need to. It's just acceleration.

and what we going to say that equal to,

we're going to say that equal to the acceleration what we get from all of the gravitating material inside here.

let's see how much, first question, how much mass is in there?

first call them mass, the mass inside this sphere.

the formula that we are going to compare this with

is Newton's gravitational formula, force equals to mass times mass,

which mass is the little one here?

that's the galaxy

the mass of big one, which one is that?

that's all the mass on the inside

and, the distance between, or the distance squared,

and I missing a couple of things, two things are missing.

Newton's gravitational constant, 6.67x10^-11 plus some units

I missing one more thing, anybody knows what it is?

the minus sign, the minus sign indicates that the force is attractive, pulling in

all right, that's the convention: force pulling in is counted as negative;

force pushing out is counted as positive

all right, 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.

drop it out.

forget the mass here, the acceleration is the force per unit mass

all right, this is the acceleration.

and minus M G divided by D^2

that's acceleration of the ...

what's that

no, no, divided that by small m.

so that's the acceleration due to the present of all the mass and material here

and that equal to a double dot of t, times R

God knows where this is going,

but we're just following on knows, writing equations

and you know that's always how you do it, you start out with a physical principle,

you written down the equations,

and then you blindly follow them for a way, until

until you need to think again.

so, we are autopilot now, we just doing equations.

that's the virial, let's write it down here.

a double dot, R, is equal to minus M G,

oh, the D^2. Let's plug in this guy over here.

distance is a times R

So maybe we can, maybe who knows, some point maybe actually discover something looks interesting

But the moment, just blind

a of t squared, or just a^2.

a^2 times D^2, no, a^2 times R^2, right.

ok...... now

excuse me, but I am just going to divide by R here.

I secretly know what I am going. right.

maybe you do too. that's right.

get R^3, and divide it

Remember that divide by another a

that's a cube.

ok.

Now, this is great. this will do.

but, next question, what's the volume of this sphere.

let's write the volume of this sphere.

this is Newton's equation.

Now, volume of the sphere

what's the volume

4/3 pi,

now is it there R cube? no, D cube,

which means a^3 times R^3. right.

Because distance is really a times R.

that's the actual physical volume.

I say the volume, I mean volume as measured in some standard unit, like meters.

that's the volume

look here we have a^3 times R^3 here.

let me write that, volume,

3 over 4 pi volume is equal to a^3R^3

or maybe I miss that, maybe I shouldn't.

yeah, let's start. let's not do that.

let's look at this formula here. notice that we have a^3 R^3 downstairs.

let's multiply by 4 over 3 pi, or divide by 4 over 3 pi

and multiplied 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 the M over volume?

the density, wow

something nice maybe happening.

a double dot over a is equal to

minus 4/3 pi, Newton's constant, times

the ratio of the mass in that sphere to the volume of that sphere.

that is the density

now, all that's equations.

notice, that really doesn't depend on R anymore.

if we know the density of the universe.

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 drops out. the right-hand side, no memory of R

It means that this equation is true for every galaxy,

no matter how far away.

same equation had for different galaxies 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 drops out of the equation.

that's, of course, a good thing.

because we want to think of a, that something which doesn't depend on where you are.

then it had been, that drops out.

so, Newton e... confirms what it might be expected

that the equation of a is a universal equation for all galaxies.

(student's question)

it was, again, we would get the exact same thing no matter what do R we get.

that's right.

well, it has it. It depends on what....

no, no, the point is you have to do the transformations carefully

you have to do the transformation carefully,

you go to another origin, and in your frame, Newton could have said,

"let me work this out for my frame of reference, which could be myself on the origin,

but let me study now the relative motion of the galaxy relative to some galaxy which is moving

he would find exactly the same equations, but it would have to do the transformation carefully.

so we, the next step we get away from it, by just putting off self-center.

but you can see the final formula doesn't care where you are.

it confirms the fact that nothing really depends on which galaxy we thought as our home.

(student's question: about the direction of the gravitational force is always towards to origin )

it's relative force. the right way to think about it is really a relative force.

no, ja.....

in this way of thinking about it, the force is always towards to the origin. right.

but have we station ourselves on some other galaxy that is 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 being toward here,

plus a fake force, the fake force being the inertial force due to its acceleration.

but we get around that by just saying let's position ourselves at the center, no acceleration, no velocity, we just at the center.

so, the only test question, the only question is, do we get an answer which does depend on who we are, and which galaxy we're on.

ok, that's part of the meaning message this,

the answer doesn't depend on which galaxy you on,

so it really didn't depend on Newton's assumption that he was the center.

(student's question )

Oh, yes.

yes, don't we think of whatever change, know it was exact constant.

(student's question)

really, a constant in space

yes, to say that it is a constant in space is the principle of the universe is homogeneous

absolutely, everything engines on the homogeneity of the universe

right, that the number of the mass per unit volume is the same everywhere, in space. ok.

yes, everything engines on that

and, okay. so here is one equation,

the central fundamental equation of cosmology

and it is a differential equation, the equation of how a changes with time

there're a number of things to look at,

the first interesting thing to look at,

is it's impossible to have the universe, which is static

static means a doesn't change with time

unless it's empty. empty means rho equals to 0.

only if it is empty, so this side is 0,

can the time derivative of a, or the second time derivative, in this case be 0.

so, we derive the fact that the universe is not static.

all right, one more thing we could do, to make this sort of equation we could solve

is to replace rho, by the constant nu divided by a^3.

know nu is literally a constant

it's the number of galaxies times the mass of the galaxy in a unit coordinate volume

it doesn't change with time because the galaxies are frozen in the grid,

so we can write this equation, one more step

a double dot, not surprising it's a double dot, why is double dot.

because Newton's equation is about acceleration.

and, not surprisingly it's 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^3 is not, because a is changing with time.

so we'd better put that in here. nu divided by a^3.

ok, so there's lots of constants here. the minus sign is constant,

4 pi over 3, G is Newton's constant

and we can pick nu also to be a constant.

so everything here is constant, a is not a constant.

so we have a kind of differential equation,

it is a differential equation, it's kind of equation of motion,

in terms of one constant, 4piG nu over 3,

We have a equation of motion for the scale factor, for the scale factor a is a function of time.

ah, who is the first to discover this equation?

it was actually discovered in the context of GR

it was discovered, I think Friednmann, Alexamder Friedmann,

and for example, was killed at World War I, I think.

using the general theory of Relativity.

it's consistent with, Einstein should have done.

but, it's perfectly possible with nothing in it, there was just good all the Newton mechanics.

ja, (student's question )

all right, multiple it, if you like

sure, you can do that, just tradition provides this way, it's just tradition.

(student's question: the negative sign, was there anything about whether expanding or contracting)

it doesn't tell us whether expanding or contracting.

so, let me explain why.

let me write.

forget that, now we just have the Erath

let's compare this with something else.

there the Earth, and we have a particle over here.

let's put it on the x-axis, on the x-axis.

all right, that equation for the particle is the same equation

let's call it now, let's call it x, but x doesn't stand for the coordinate now.

it just stands for the standard position coordinate or the height from the Earth.

let's satisfy some equation, x double dot is equal to

the gravitational force, what was the gravitational forces?

M G over x^2, minus

that is. something like that.

Okay. does this equation tell us,

is that the particle accelerating towards the Earth,

the minus sign tells us the acceleration is towards the Earth

but, whether it's moving away from the Earth or towards the Earth,

is a question of velocity, not acceleration.

is the velocity that way or it is that way.

well, you can image somebody over here, taking this particle, and ejecting it in that way.

it will have a positive velocity, it will be moving away from the Earth.

you could also image the same person, pushing it that way. so it moves toward the Earth.

that's the decreasing. But the acceleration would be the same.

in either case, the velocity will have a negative acceleration, which means if it's going this way,

it will turn around or may turn around.

if it's going this way, it will increase the velocity.

ah, whether it turns around or not, depends on what?

initial condition, or whether it's above or below the escape velocity.

but in either case, the acceleration is towards the Earth.

So knowing the acceleration towards the Earth, as it is for this pen, does not tell me whether it's moving up or moving down.

it can move up and move down, and you get the point. okay.

so, no, this equation doesn't tell us if the universe is expanding or contracting,

but it tells us that the second derivative is negative.

so it means even if it is expanding, it's tending slow down.

So expanding tending to slow down.

but if it's contracting, it's tending to speed up the contraction.

The reason analogue here, of whether you are above or below the escape velocity.

We will come to it.

All right, so I was asked a question which,

I will point out, all right.

If you look at this, this is negative.

and... look that, this is positive.

The universe is expanding. it's positive.

how come this to negative? well that's because you didn't read carefully.

it'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 are in your Ferrari, you are going down,

and you press down the brake,

your acceleration is negative, but the velocity is positive.

you are slowing down but still go ahead.

Now, in fact the universe is not slowing down.

This will make, we're really doing what Newton would have done, and when all cosmologists thought the right thing to do was,

until about 15 years ago.

so 15

this's Newton's model of the universe.

and it is the model that would have been for standard model.

or close to it, the standard model of universe,

until the accelerated universe was discovered

this is the decelerated universe per se.

but the universe accelerates, so it gonna be something else in this equation.

whereas there is several things matter this equation, we will come to that.

ah, some parts, note, do have to do with Einstein

Okay.

let's talk about the, not cosmology,

but just particles, rocks, stones, from the upwards of the surface of the Earth.

equations are very similar.

let's exam for a minute, and take home for a couple of lessons about it.

Here is the Earth, and we might think of it as a point.

because Newton proves that the theorem says, we could think it of a point.

we are outside, we are above the surface of the Earth.

So, here's the Earth.

Here's a particle over here. I don't know. No.

put it over here, x-axis. put it on the x-axis.

and, what are equations, the equations of Newton equations.

but there's actually a useful version of Newton's equations. just the energy conservation

let's write down the energy of this particle over here.

and write down the conserve. In fact, it's a

it's a more useful equation than this one over here.

energy, the energy equation is more useful

what is the energy of this particle over here

it's moving out, so is has some velocity. The velocity could be negative. it cold be moving inward.

and what is the total energy of this particle. the total energy is the kinetic energy plus potential energy.

kinetic energy, one half, the mass of the particle, not the mass of the Earth.

the mass of the particle, times the velocity squared,

which we could call x dot squared if you wanted.

Well, I will just leave it velocity squared for the moment.

but what about the potential energy. remember the potential energy.

the potential energy is, minus little m big M, Newton's constant,

divided by what? R, not R squared, just R

say it again. x, yes

Now, this can be positive or negative, believe it 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 the initial condition. It got there at some time t, and rests.

but at a positive value of x, x is really always positive

even stand over this side, from the Earth, not from the x coordiante.

So x is always positive. This is always negative.

This can be 0, if the particle is at rest.

So the energy is negative, in that case.

The energy can also be positive.

supposing we now pick the same particle, at the same position.

but given a velocity.

if the velocity is big enough, then this can out weight that.

this can out weight that, simply when for equation went down to infinity.

this is bigger than this. the kinetic energy is bigger than the potential energy.

and then, the total energy is positive.

now the total energy is positive, this thing cannot turn around

cannot, you might say, well, let's see, this particle go out and turn around

why it can't turn around, if the total energy is positive

accidentally, energy, of course, is conserved. so whatever the energy is,

in one instant, it's the energy in every instant. The energy is conserved.

let's suppose it turn around at that point.

what would be velocity at that point? zero

so what would 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 to zero is sort of edge of the parameter space.

If the energy is positive, the particle just keeps going and going and going, and escapes.

if the energy is 0, that's exact the escape velocity.

we will ask later, whether it escapes or not, that if it's exactly 0.

what's the escape velocity, the escape velocity is the solution of this equation, that this is equal to 0.

so let's write it out. one half v squared.

I drop the m, because it cancels from both sides,

one half v squared is equal to big M, big G, divided by x.

and now just multipled by 2

and that gives the formula for escape velocity

that's the formula for escape velocity when the energy is exactly equal to 0.

in the exactly the same manner,

the universe can be above the escape velocity, below the escape velocity,

or at the escape velocity.

well, we're going to work out that in a minute

but all that means,is, if it above the escape velocity

it means, that initially at some point,

the outward expansion was large enough

that doesn't turn around

if below the escape velocity, than the universe turns around and contracts

so, that's the main reason for showing you this.

and the escape velocity is kind of edge,

the escape velocity is also the velocity of which the energy is equal to 0.

keep that in mind. escape velocity, same thing as energy equals to 0.

and now let's concentrate on this particle over here.

now for all practical purposes, this particle over here,

always knows is that's moving in gravitational field of point mass in the center,

with a point mass of capital M

so for all practical purposes, we can replace this problem over here,

by this one over here

that is exactly the same problem

so, let's work out the energetics

the connect of the potential energy of this particle

and keep in mind, that is a constant.

it's constant because for all practical purposes,

this particle is moving exactly as it would be,

if all it was the mass in the center, and in that case the energy would be constant

so we can just left the things that I wrote before,

and let's work them out.

(student's question )

no, the whole thing is growing, but remember it's a grid

everything moves in a grid

the only thing is changing is a

the amount of mass in this sphere is fixed.

in other words, the number of galaxies, that this fellow over here sees in this sphere is fixed.

all right.

Okay, so. no, we don't have to worry about the mass changing

let's work out now, the energy.

kinetic energy, or the kinetic potential energy in Newton's frame.

In Newton's frame, work out the kinetic energy,

from here the v square again, one half the mass of this galaxy,

times velocity squared, that's a dot, squared, R squared.

right, same R. where is it

same R

D is equal to a times R

distance is a times R

velocity is a dot times R, this is one-half m, v squared

and then minus, little m, big M, G

divided by distance, right? just divided by distance.

that's the potential energy. mMG, and what's the distance?

the distance is a, times R, right.

let's do the. and that's equal to the energy of this galaxy here. so the energy.

now for simplicity, and because of simplicity, and also I am getting a little tired.

I think I will just do tonight, the case which the energy is exactly equal to 0,

what's that correspond to? exactly the critical escape velocity.

that's case. the other case is just easy.

but let's do that case. all right.

that's the case, where

the universe is just on the edge

not clear whether it going to turn around, and fall back or just keep going.

the edge of cusp of one doing the other

all right, we're going to set this equal to 0.

let's work out that equation, work out that equation using the very things we known

Ok, the first thing to do is, to get rid of little m here.

why should we get rid of little m, because up here about this term here

and hoping it equals to 0, so I divide out

I also multiple by 2

let them get divided by R^2, why I am dividing by R^2

Why I am getting R^3 down here, because I know

that R^3 has to do with the volume, and the volume is, it's instinct

I am trying to get the single term of density

all right, I divide by R^2,

and that's the R^3 downstairs, that's nice,

because there's a mass here, and R^3 downstairs

looks like that I'm getting the density, but not quite

because the volume of the sphere is a^3 times R^3, not a times R^3

So what I do, I just divide the equation by another a^2.

O.K.

a^3 times R^3

What do I do next? well, if I am smart, I will multiply this by 4 over 3, times pi

that will literally make this volume.

I am doing some illegal, unless I multiply here also 4 over 3 pi

equals 0

All right, I almost there.

We re-write it.

a dot over a, squared. Remember what a dot over a is?

it's the Hubble constant. So this is the square of,... I take it back. it's not constant.

the Hubble thing.

a dot over a, squared. that's the Hubble squared.

and that's equal to, just transport this thing to the right hand side

it's 8 pi over 3, famous 8, 2 times 4 over 3

there is a G

and now there is M divided by the volume of the sphere

that's why I went to this effort here,

to put another couple of factor a and R downstairs

so that I will get M 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 to 8 pi over 3 G times rho.

that is the Friedmann equation

that's the Friedmann equation

the way it's usually written.

it's equivalent, that this equation.

this one over here is energy conservation, also I set the energy equal to 0.

this one over here is the Newton's equation.

but the same physics, with the Newton's version of it, the conservation of energy version of it.

this one is more useful

and let's call it the Friedmann equation

it's not completely general because we did set the energy to 0.

did set the just the exactly critical, the escape velocity

So this universe is not going to recollapse

but it's gonna.

what does happen, if you shoot some out at exactly the escape velocity

what happens to its motion as time goes on?

yes, it just at some asymptotically get slower and slower, but it never turns around

this universe will asympototically get slower slower and slower of its expansion,

but never turns around, for the same reason.

Ok, that's our Friedmann equation

I like to solve it, but I don't know enough yet.

the reason I don't know enough is because this rho here

I don't know what to do with rho

except we do know what to do with rho

remember the equation rho is equal to the constant nv,

incidentally the constant nu can be set to be anything you want,

it's the mass per unit coordinate volume

by changing you coordinate, you could change the amount of mass that in you coordinate volume

so actually, Newton never really concept anything important

well rho is equal to nu divided by a^3, remember that?

Okay, so we can now write a even more useful version of this

a dot over a, squared is equal to 8 pi over 3

G, and nu

and nu is a constant, nu does not change with time

divided by a^3

all of this jump here just a constant, 8 pi nu over 3 times G, just a constant

In fact, that I could have if I like, that chosen nu, so that 8 pi G over 3 is just number of 1

nothing interesting

the basic equation, the basic form of equation

is just a dot over a, squared, is equal to some constant,

but just let's choose the constant to be 1, just for simplicity.

is one over a^3

if we can solve this equation, then 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 right hand side is always positive

In fact, it never quite goes to 0, no matter how big it gets,

so it's always positive

as a gets really really big, it's smaller and smaller

so that tells us, that a dot over a never becomes to equal to 0

a dot equals to 0, means universe turning around

that will the place where the universe turning around, or the expansion rate went to 0.

so it tells us that the expansion rate never goes to 0,

Hubble constant never changes sign, or at least the squared Hubble never goes to 0.

doesn't go to 0 and changes sign

and well, it does slow down, the Hubble constant is smaller smaller and smaller with time

so it's just the universe is tired of expanding

it never get tired and after stop

Ok, let's try to solve this

I think I will just take, It's getting late, and we're getting tired about this time

so I will take the easy way of solving it, but we will come back to this kind of equations.

we will come back to this, this type of equation is absolutely

well I said that this type of equation

is absolutely centre of all cosmology, we can solve them

we can solve them quite easily

let's just look for a solution for a particular type

we look for solution rather than solve equation

let's see if we can find a solution when a is some constant times time for some power.

we don't know that if it solved this way, but we can try

we can take a child solution, a proportional to t, what would a proportional to t means

it just means a grows and proportional to time, very simple way

we don't expect that to be right, because the things slow down

but we can look for solutions of this type, so let's try it out.

let's see if we can use this equation

to see 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, we divide this by c, t to the p

c is cancelled, the constant here is cancelled

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 the squared

p squared over t squared, that's the left side

now what about 1 over a^3, let's see what that is

1 over a^3, that's 1, divided by c^3,

t to the 3p, we have that right?

Now we can read off how to match the two sides

let's get rid of this over here, and match the two sides

in the denominator we have power, we also have power over here.

this is 1 over t^2, this is 1 over t to the 3p, but I haven't told you what p is yet.

so we want to match, let's look for solution for the form of c, t to p

and see if we can figure out what c and p have to be

well the first thing we learned is that 3p is equal to 2,

otherwise these things can't be matched.

there's no way t to the 4 can match t^2 over here.

so the first thing we learned

is that 3p has to equal to 2

we will come back to it in a minute.

all right, that will guarantee t^2 and t^2 agree on this side

on the other hand, we also have to match the constant

and the constant tells us

that p^2 equals to 1 over c^3

so that tells us there is really only one constant we have 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. the interesting is p.

because one of the p says that a

expands like t to the 2/3

p is equal to 2/3

some constant times t to the 2/3 power

that's the way of Newtonian universe would expand

if it had, if we right at the critical escape velocity

it would expand at scale factor, everything all galaxies,

separating as time to the 2/3 power

that's quite remarkable variation

Newton, en... I don't know why he didn't do it.

it annoys me that he didn't do it. He should have done it.

I think he went to the mean of this point, I am not sure what happen to him

oh, that's the year of

no, he should have predicted the universe

oh, yes he did. he worried about the fact that the homogeneous universe

oh, yes. he most certainly had speculated enough that he was right on the threshold of doing this.

he asked all the questions about it, and didn't quite carry it out.

yes, this is.

actually not.

that's a good question. that is completely Newtonian theory.

in the Newtonian theory, space is flat.

and space is flat, it just goes on and on for ever

so, yes. the Newtonian universe would be infinity

it would be spatially flat.

it wouldn't have any interesting about Einstein geometry any time.

although it would be expanding or contracting

and it would be entirely Newtonian

all right, so I did this just, first of all, it's easy.

second of all, because it contains lots of physics

that I will be dealing with, the simple form

and it gives us a model universe

with scale factor that increases with 2/3 power of the time

not quite, no. there is another term in this equation

and we will come to that term

no, it can't be, because if you have negative energy, it will recollapse. right

another term. and next time we will pick up that other term,

and we will talk about three possibilities:

less than 0, other words it will collapse

greater than 0, that means the universe just expand without even thinking about it.

and this, which is the critical point

that slows down in a certain way

a diagram people always draw for this kind of thing

looks something like this, you probably have seen diagram like this

it plots on the vertical axis, it plot a, scale factor

and on the horizontal axis, you plot time

now, a equals to t, there's no sensible cosmology does that.

we just draw it in. here is a equals to t.

now what does it mean that a decelerates? that the acceleration is negative.

that the decelerate statement the curve is bent over this way, not goes this way

the second derivative is negative

the curve goes this way

a to the 2/3, looks approximately like this

and of course it keeps growing

what about, a re-contracting universe?

what if universe recollapse

a collapsing universe would look like

crash

this does not approach to straight lines, incidentally

it just keep bending over, slightly more and more

the universe of positive energy would look pretty much the same,

and go off on the straight line

those are the three cases that I will describe

did I get that right? No I take this back, this is not quite right.

no, no that's incorrect.

we will do the case of positive energy

but in any case, in all of these cases

the tendency is 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

we will solve the equation, this is the solution

no this's the only solution

but we can change the energy away from 0

we can generate other kind of solution

no, sorry! the derivative gets smaller and smaller. a to the 2/3

so let's see. what we know

we've already done it.

t to the 2/3, is a

and a dot is equal to 2/3, one over t to the 1/3.

so the slope goes to 0, goes to 0 with t goes to large

but it always positive, this is the sense we getting tired

the slope is ...

and you can see now why would Einstein failed to be able to describe the static universe.

well, we will come to it. I am getting ahead of myself. I don't want to get ahead myself.

I think that Newton was prejudice?

ya, let's see.

see, Newton had this idea that universe is 6000 yeas old.

this wasn't fitting together with ...

ye,ye, Newton was a believer.

so I think he has some, I think the reason is he probably didn't do it,

is because he couldn't make it fit with his prejudice about the age of the universe.

same the pi G

not surprising that this is about energy

no, this is the theory without a cosmological constant

the cosmological constant is what has to with the acceleration

this is the theory without a cosmological constant

in fact, this is called a matter dominated universe

the matter dominated universe for reasons I will explain

(student's question: what we know that the universe is expanding overall, like the entire unverse

are there some galaxies between could be attracting)

well, certainly, yes.

on the average, out to the large observable distance, it is expanding,

but individual they are of course

for example, our galaxy is contracting together with Andromeda

Andromeda is not receding away from us

but you know on large enough scales

the Hubble law is not exact true for all the possible distance

it becomes accurate as the distance get larger

it's certainly not accurate, for the

for things such as bounding together,

things are close enough together are really bounding together by gravity

or any other force, maybe pull together

so it happens, not unique, but on the average, everything is moving away from everything else

but here and there you can find that galaxies have peculiar motions

the term peculiar motion's technical term

it is, it is a technical term, means

means sort of away from the average

(student's question: so in overall calculation, we should avoid those little galaxies not interesting and trying to straight)

we should average over large enough volume

that these little fluctuations don't matter

it's the same kind of thing, saying the air in the room is uniform

well, that's not really true, there are places where the fluctuations with more dense,

fluctuations with less dense, but on average over sizeable region,

bigger than any molecules, the room is uniform

the same thing here

(student's question: the Andromeda moves toward the Milky Way, is that motion within the expanding)

the Andromeda just happens for whatever the reason

I don't know whether, I don't know the complete history of the Andromeda

and Miky way the dynamics

However, it was formed, it was formed in a packet which was dense enough

just enough, slightly out of ordinary, dense enough these two galaxies have enough mass to

to over come the effect of expansion

it's fluctuation away from the normal

no, they are identical

no there is no difference, you see

you either take the position that the galaxy are moving away from each other,

or you take the position that they embedded in this grid, and the grid is expanding.

that's a mathematical artifact

perhaps in Einstein's way of thinking about it,

it's little more nature to think of the space is expanding. but they are equivalent.

(student's question, about SNIa)

yes, there is from the CMB

we will come to it

it's sort of network of different observations

mostly SN and CMB fit together precisely