So, we now have a description of how in 
the context of gravity as geometry, we 
can describe universe in which space at 
any time is isotropic and homogeneous. 
it's a space of constant curvature and 
all that changes is a scale. 
And we have a natural set of coordinates 
to describe this. 
These are coordinates and which are, what 
we call co-moving. 
You pick where something will be at a 
given time, say the present, if it is 
freely moving and take a collection of 
freely falling observers that start out 
addressed at some point and use their 
positions today. 
If they're freely falling througout the 
history of the universe, to denote the 
name of a point. 
And that observer, is as far as I'm 
concerned, always at the same position. 
So these are observers are at rest. 
And then relative to them you can have 
peculiar velocities. 
It is in these coordinates that the space 
looks, nice and isotropic and homogenous. 
It would be nice if we had such 
observers, we do! By and large galaxies, 
we claim, are such observers. 
They are at rest, they only, their 
average motion is the Hubble motion so 
galaxies are precisely such an object. 
And now, having described the kinematics, 
we now turn to the dynamics. 
In other words, we have to understand the 
solutions to Einstein's equations and to 
we've studied sort of a left hand side, 
the geometry. 
Now we need to study what matter looks 
like. 
You would not expect an isotropic 
homogeneous universe to contain, for 
example, stars, because certainly if you 
have a star then the place where the star 
is, is not the place where the star 
isn't, so matter in an isotropic 
homogeneous universe has to be itself 
isotropic and homogeneous. 
Now one way to do that is to have nothing 
at all in the universe. 
That does not describe our universe, and 
so what we mean is, imagine, averaging 
out over large distances, ignoring little 
fluctuations, like galaxies and clusters, 
on average, we should be able to 
describe, at large distances, the 
distribution of matter in the universe, 
as completely uniform. 
So that means there is a constant density 
of energy, because mass is energy, so 
we're talking relativistically. 
There's a constant energy density, rho, 
or mass, if you want, density that is the 
same everywhere. Otherwise the universe 
wouldn't be homogeneous. 
And if there is a gas with a constant 
density, then we know enough about 
thermodynamics to assume that there will 
be a constant pressure. 
And these are constants everywhere in 
space, 
but of course they can depend on time. 
So all of the crazy right-hand side of 
Einstein's equation about energy and 
momentum is contained in two functions of 
only time. There's no variation with 
position, the equations are going to 
simplify. 
in further, and furthermore, we have 
density and we have pressure. Density and 
pressure are related by the properties of 
whatever the stuff is that you fill you 
space with. 
for example, if you have an ideal gas, 
then we know that pressure and density 
are related to temperature. 
in particular, we will deal with, two 
extreme cases of equations of state, 
relations between pressure and density. 
We have one case, which in cosmology 
lingo is called dust, and so we will be 
cosmologists and call it dust. 
What is, dust has nothing to do with the 
kind of dust that causes extinction. 
the particles, if you will, of this dust 
are galaxy clusters. 
dust is what describes a collection of 
uniform distribution of slow massive 
particles moving freely and interacting 
only through gravitationally. 
Each with each other or more importantly 
with a whole collection of uniform 
distribution because they're moving very 
slowly we're going to imagine that this 
is a 0 temperature object. 
The pressure is essentially 0. 
so this is the canonical example of dust 
of course is galaxies, slowly moving 
objects 
A collection of dust is isotropic in a 
frame that moves with it. 
In other words, in a frame in which you 
measure the velocity zero. 
This is not something fancy, every pilot 
knows that your air speed is constant, 
whether you're going upwind or downwind. 
In other words, the air, the moving air, 
is isotropic in a frame that's moving 
with the air. 
It's only when you're measuring ground 
speed that you realize that you might not 
be making head, any headway trying to go 
upwind. 
Your air speed of your craft is the same. 
The airplane flies the same way upwind as 
downwind. 
It's only relative to the ground that 
things change, so in a frame, where the 
velocity of the fluid is zero, at any 
given time, at any given place. So at any 
event, 
flew this dust has a preferred rest 
frame. 
That's the rest frame in which the dust 
at that event appears to be addressed. 
And of course since we're going to make 
this dust out of galaxies that are 
comoving in the Robertson Walker 
coordinates then what we're going to say 
is that the preferred Robertson Walker 
frame at any point is the prefer, is the 
preferred, is the rest frame of the dust. 
This is be, that way, they will both be 
isotropic. 
Both space and matter will be isotropic 
in the same frame. 
And then the other extreme case is very 
relativistic particles. 
In other words the extreme case is 
particles moving at the speed of light 
and we'll call such particles radiation. 
This can include photons. 
It can include gravitons. 
It can include essentially neutrinos. 
Which even if they're not exactly 
mass-less are very light and move, under 
most circumstances, at very near the 
speed of light or highly relativistic 
speeds, and you can make a calculation 
for an ideal gas of relativistic 
particles. 
Some of you have followed through in the 
context of electron degeneracy learning 
this and it turns out that for a 
relativistic gas, since the speed with 
which the particles are moving is the 
speed of light. 
density determines the pressure, and even 
dimensional analysis will tell you that 
pressure is the density times the square 
of the speed of light, and it turns out, 
the, the coefficient is 1/3, so the 
pressure of a relativistic gas is, 
related to its density by the square of 
the speed light over 3. 
There's no, in, temperature dependence, 
because a relativistic gas, by 
definition, everything is moving at the 
speed of light. 
And again, 
you cannot go to the rest frame of 
relativistic gas in which the speed of 
the particles is zero, because they're 
always moving at the speed of light in 
all different directions. 
But there will be a frame, a specific 
frame, in which the radiation field looks 
isotropic. 
This is the frame in which, if you look 
to the right and look to the left, you 
see exactly the same spectrum. 
And of course, if you are in that frame 
and someone is at the same place you are 
in the same radiation field but they are 
moving, they will observe a blueshift, 
Dop-, blue Doppler shift in one 
direction, a red Doppler shift in the 
other direction, and then say aha, the 
spectrum is not isotropic. 
Only one observer in any event sees an 
isotropic spectrum. 
That is the instantaneous rest frame. 
So we have our description of matter and 
we're going to assert that the matter is 
at rest in the Friedmann coordinates, so 
that space and matter are isotropic and 
homogeneous in one system of coordinates. 
And with that in mind, we write down our 
friend the Einstein's equations, 
this time written correctly. These, the, 
latex error that ruined things, so right 
here the Einstein equation correctly, and 
we insert what we know. 
On the right hand side we insert that 
the, geometry, remember these R's 
represent curvature, and the geometry is 
the geometry of a space of constant 
curvature with curvature R zero which 
scales with when the distance is 
increased, like 1 over a squared. 
And then this coefficient, k, is simply 
given by 0 or plus or minus 1. 
0 means, we're in the flat space case. 
1 means we're in the positive curvature 
case. 
-1, negative curvature case, 
and in all cases I can give the same 
Parametrization in the curved cases our 0 
represents the curvature at T equals 0. 
At the presence at T equals T0, when A is 
1. 
At present in the case that K equals 0, 
you can put whatever you want for R 
equals 0, cause it's multiplying 0. 
That's the simplification on the left 
hand, on the right, left hand side. 
Whereas on the right hand side, we insert 
what we know about matter. 
And what we know about matter again is 
that it's the same everywhere and it's 
determined by this one function rho, and 
then the pressure is which appears inside 
the energy momentum tensor is a, some 
given function of rho, depending on what 
you put in. 
Okay, so you take all this data, 
you plug it into Einstein's equations, 
you do the mathematics. This is one of 
the cases in this class where I say, you 
do the calculation. 
And the equations that you are, you find 
are two equations for 1, whatever 
variables here. 
The variables are the time dependance of 
the scale constant. 
There's no space dependance of anything. 
So we're trying to determine the time 
dependance of the scale factor The time 
dependence of the energy density, and 
that's it, we have 2 functions, and we're 
trying to find the time dependence of 
these 2. 
And so we indeed find 2 equations, they 
are physics equations, they are 
differential equations, so watch out, I 
said we would skate near calculus, so 
there's this little dot here, this dot 
means the rate of change of a, with 
respect to time and we already said, if a 
were a position this would be it's 
velocity and we already noted that the 
rate of change of a divided by either 
relative change of a is what we call the 
Hubble constant, only we no longer expect 
it to be constant. 
So in general this will be a function of 
time, and this is given by Einsteins 
equation tells us that this is given by 
the right hand side 8 pi G. 
The c squares end up cancelling, 
times rho of t. It's the same everywhere 
in space, but it varies with time. 
And then, this is actually not a part of 
the right hand side of it, obviously. 
This is a curvature piece, it's a another 
piece of the left hand side, and we've 
put it on the right hand side to get an 
equation for the Hubble Constant. 
And so the curvature of space and the 
density of energy both together determine 
the rate at which the scale constant, the 
scale factor changes. 
That's one equation. 
We need another equation, and so again 
this tells us that the curvature of space 
if you want. 
Another way to think about it, is if you 
take the density and the velocity of 
everything. 
So that gives you the Hubble Constant, 
that will determine the curvature of 
space. 
Space curves in response to the density 
of energy and momentum. And so, the 
density of energy rho and the rate at 
which everything is moving, which is 
determined by a dot. 
put both of those in, and you can 
determine the curvature of space. 
But of course, like all equations, if you 
know one thing, you can determine 
another. 
Everything ends up determined. 
What's the second equation? Second 
equation looks even scarier. 
I have here an a with two dots. 
if A were a position, this would be the 
velocity, 
this would be the acceleration. 
It's the rate of change of the rate of 
change of a. 
And it's related to the Hubble Constant 
by this expression, where this is the 
rate of change of the Hubble Constant, 
and this its square. 
And this is given in term of the matter 
distribution by a combination of the 
matter, the energy density and the 
pressure. 
notice that for a relativistic gas, both 
of these are in fact equal because for a 
relativistic gas, p is equal to rho c 
squared over 3, whereas for the case of 
dust, of slow-moving matter, p was 
actually 0 so this is just rho. 
And now This is really a lot of calculus. 
So let's try to understand what it is 
that we're trying to say. 
So imagine that I have some time t* and I 
expand a times t, and express it for 
times near t*, and the first term in the 
expansion was the Hubble expansion term 
that we talked about before. 
The next term in the order will be a 
quadratic term proportional to t minus t* 
squared, and of course in general there, 
you can expand and get higher order 
terms. 
And I will, for dimensional reasons, 
factor out the factor of H, H squared and 
then the coefficient is called q over 2 
and notice I put a minus sign here. 
Because I understand that gravity will 
decelerate things and indeed, if I write 
the expression in this way. 
So a of t for t near t star is a of t 
star times 1, which is the value when t 
is equal to t star plus the 1st order 
term which, who's coefficient I called H, 
and then the 2nd order term whose 
coefficient I call minus H squared, q 
over 2. 
q is called the deceleration parameter 
for good reason, because what it 
indicates is that if you take since 
you're subtracting a positive quantity, 
it means that if you start as a function 
of T and you plot a. 
It starts at t0, at the value a0. 
The line determined by the Hubble 
expression would be this constant 
increase, and look what the deceleration 
parameter does is it slows it down. 
It decreases below the straight line, 
and in terms of this deceleration 
parameter, you can rewrite this crazy a 
double dot thing as just minus H squared 
q. It's just twice this coefficient. 
And notice that the sign here is well 
justified, because pressure being 
positive, density being positive, this 
sign being negative, gravity, [SOUND] 
decelerates expansion. 
This is not a surprise, but this 
non-surprising result is derivable with 
great work by following through 
Einstein's equations. 
So Einstein's equations not surprisingly 
tell us that both density and pressure 
which is, the, remember, the pressure 
results from the velocity The momentum 
that the particles are carrying. 
This momentum contributes to 
gravitational attraction because momentum 
gravitates just in the same way that 
energy does. 
So remember, notice that the pressure of 
the relativistic gas actually increases 
the deceleration, contributes to 
deceleration, as opposed to one would 
think of pressure leads to, makes thing 
try to expand. 
Well pressure makes things expand, if you 
put something in a container, but in this 
case we're talking about a universe where 
the pressure is the same everywhere. 
There's no pressure gradient between 
inside and outside for the pressure to 
put us, push on. 
There's no container for the universe, no 
edge of the universe for the pressure to 
push on. 
Every bit of the universe experiences the 
same pressure on both sides. 
What is going on is that pressure 
contributes to the gravitational 
attraction, because remember, momentum as 
well as energy is a source for gravity, 
couldn't be otherwise consistent with 
relativity. 
So these are our equations, and they are 
going to, in the end, determine the 
evolution both of the density and of the 
scale constant. 
However in general, we will have 
differing sources of density. 
Remember, we can have a universe that 
contains both dust, slow moving things, 
and, say, photons, relativistic things. 
And one can imagine a universe in which 
these don't interact, and then that means 
that the energy in dust is conserved. 
Because no dust is created, the total 
mass, remember the energy of the dust is 
just is mass, it's addressed, and so the 
total mass of the dust is conserved. 
And separately the total energy in 
photons is conserved. 
And if this is the situation when they do 
not interact, then I know the time 
evolution of rho of t if I know the time 
evolution of a of t. 
and this is because, if you take the 
given volume, 
it has some dust in it, and then as this 
volume expands, it has the same amount of 
dust in it. But if the distances have 
grown by a factor of 2, the volume of 
this region has grown by a factor of 3. 
So when distance by a factor of 8, and so 
the density, mass conservation says 
density decreases like the third power of 
the scale factor, 
when space doubles in length and all 
lengths double, the density decreases by 
a factor of 8. 
How does this affect relativistic 
particles? Well for relativistic 
particles, think again about photons, but 
of course the theory is universal. 
In a, the case of a relativistic particle 
moving at the speed of light, the same 
property of Thames, if I take the volume, 
containing a certain number of them, 
their number is conserved when I, that 
volume increases, it carries the same 
amount, number of particles in twice the, 
eight times the volume. 
However, they're all redshifted, so the 
energy carried by each of them is also 
decreased by a factor of 8. 
And so the behavior of the density of 
nonrelativist-, the density of 
nonrelativistic matter is that as a 
function of time, if there is no 
interaction and no creation of matter or 
loss of matter, like say the loss of 
stars, if you wish, convert mass dust 
energy into radiation energy, 
but stars are a small perturbation in the 
large scale of things. 
Most of the matter of the universe is 
inter cluster of gas, which is not 
participating in stellar processes, and 
even when there are, when dust isn't a 
star, 
only a few percent of the mass of a star 
over 10 billion years will be converted 
to energy. But, so this is a small 
violation of this conservation of energy. 
Ignoring little effects like stars, then 
the energy density of dust scales with 
the scale factor, like the negative 3rd 
power, whereas the energy density of 
radiation scales like the negative 4th 
power. 
Remember, three of those are because of 
the volume, and then one extra factor 
because each photon's energy is decreased 
by the redshift. 
Dense slide. 
Now, Einstein had understood all of these 
equations. 
it's the usual case that the, what you 
get out an equation is what you put into 
it. 
You put in a lot of symmetry, the 
equation simplifies. 
This complicated set of coupled partial 
differential equations becomes this 
relatively simple set of ordinary 
differential equations for functions of 
one variable, time. 
And Einstein was trying to write the 
solution that describes our universe, and 
what he knew about our universe was that 
it's static. 
So what he wanted was a universe in which 
the skill factor didn't change, in which 
nothing essentially changed. 
He had no reason to imagine that 
distances were changing. 
He did not know at the time of Hubble's 
result, which by the way had previously 
been, it turns out, measured by [FOREIGN] 
and understood in a deep way. 
And so he tried to set this to 0, and 
it's possible to set the rate of change 
of A to 0 by making balancing the energy 
density against curvatures. So you need 
to have a curved space, if you have any 
matter in it, but if you have positive 
curvature, and positive energy density, 
you can balance them to get zero 
velocity. 
Glorious, however, you cannot make the 
acceleration zero, so the zero velocity 
because of course both of these terms are 
all negative And so the acceleration is 
always negative. So the 0 velocity 
solution you found out is sort of like 
the 0 velocity of my little rubber ball 
at the top of its flight. 
It stops for a minute but it's 
acceleration is still negative, it won't 
stay at 0 velocity, it will fall. 
So, indeed if you have 0a., 
then the negative acceleration means that 
you have universe that is not expanding 
but is about to start contracting. 
So Einstein was very saddened by his 
inability to find a description of what 
he thought to be the universe, namely a 
static universe. 
But Einstein was nothing if not creative. 
He looked at his equation, and he 
realized that without ruining any of the 
axioms or the logic that led him, to the 
equations of general relativity, he could 
make a modification that would have 
absolutely no impact on the connection, 
the agreement with Newtonian physics, 
but would fix this, would allow a static 
universe. 
And the correction, in terms of 
Einstein's equation, involved adding one 
more term to the left hand side. 
So, modifying the way that gravity, the, 
metric function G is determined by the 
distribution of energy momentum tensor. 
He called this the cosmological term, we 
today call it the cosmological constant. 
the properties of lambda is that for this 
to be a consistent modification of 
Einstein's equation lambda is just a 
number. 
A constant, it has the dimensions if you 
work through of inverse length squared. 
So lamba is measured in 1 over meters 
squared, 
and this is the modified version of 
Einstein's equation. 
When, of course you can alternatively 
think of taking this same blue thing, 
flipping its sign and putting it on the 
right-hand side of the equation, and 
thinking of good old Einstein's equation 
in the presence of a different kind of 
energy momentum distribution. 
So what kind of crazy energy momentum 
distribution is lambda? Well it's a 
constant, so it does not depend on space, 
so it satisfies our isotropic and 
homogeneous relations. So I should be 
able to describe it by a constant energy 
density. 
Indeed, lambda corresponds to a constant 
energy. 
Related to the value of lambda by this, 
this was what follows from placing lambda 
in the position where rho sits here, and 
then the cosmological constant 
corresponds to a pressure which is 
negative rho times c squared. 
So if you have a positive, and the 
cosmological constant can have either 
sign. If it's positive, then it has a 
positive corresponds to positive energy 
density and a negative pressure, 
whereas a negative cosmological constant 
the roles are reversed and then if you 
now re-do the calculation that led us to 
Friedman's equations in the presence of 
this cosmological term Einstein finds 
that the first equation for H squared is 
modified and the 2nd equation for minus 
QH squared. 
Remember, here we're determining Q, the 
acceleration, the deceleration parameter 
is also modified, 
but it is essentially by exactly the same 
term. 
This has to do with the fact that 
pressure and density are essentially the 
same thing. 
And so, you plug this thing into that. 
And what do you discover, what you find 
is that now I can find a solution where 
the left hand side, the rate of change of 
the scale factor, and its acceleration, 
are both 0. 
How do I do this? 
Well, I'm going to pick a universal which 
p is equal to 0. 
So my matter is going to be dust. 
I'm going to of course need positive 
curvature so that the energy density is 
balanced against curvature. And then, 
it's balanced by, curvature, and I'm 
going to also have a cosmological 
constant, which will allow the negative 
term here, this is going to be zero, to 
be balanced by this term. 
And it turns out that if you set our 0 
equal to lambda. 
Remember both of these have dimensions of 
1 over length squared, so that even makes 
sense, 
equal to this concoction, proportional to 
the energy density, both of these 
equation vanish. 
This is called the static Einstein 
universe, and Einstein thought, well, 
since this is the homogeneous, isotropic, 
static universe, this must describe our 
universe. 
He then was informed about Hubble's 
measurements and realized that our 
universe perhaps is not static. 
And this cosmological term was 
unnecessary. 
Einstein then regretted introducing this 
uglification of what was a beautifully 
elegant equation before. He called it his 
greatest blunder. 
As we shall see, it might not have been a 
blunder, Einstein, even when he 
blundered, had foresight.