Now let us take a look at the Friedmann 
equation. 
This is the basic equation of cosmology, 
and it's solution tells how universe will 
evolve on large scales. 
We can derive it, actually, in a fairly 
simple Newtonian approximation, if we 
accept a couple results. 
First, there is, so called, Birkhoff's 
theorem, 
which says that, for a spherically 
symmetric system, the force due to 
gravity at some radius is determined only 
by the mass interior to this radius, 
which makes sense. 
And then also, the energy contributes to 
gravitational mass density. 
So you really need to add them up 
together. 
So, total energy or mass density can be 
expressed as one or the other modules, 
square of speed of light. 
So how do we go about this. 
Imagine a sphere with a mean density of 
row and radius of R. 
And acceleration to a particle in 
surface, which is the second derivative 
of the radius, is given by the familiar 
formula. 
Now, if, if the density changes, it 
changes according to the inverse cube of 
the radius. 
So if we, again, use subscript zero for 
present moment, 
then you can derive the density at any 
given time through the cube of the radius 
divided the present one, 
which we can assume to be unity for 
simplicity. 
So now, plug that in the, in the first 
equation. 
And we have the following equation for 
the second derivative. 
Well, right away, this is telling us 
something interesting. 
If the density is not zero, the universe 
must be either expanding or contracting, 
because the second derivative is finite. 
And this sign depends, obviously, on the 
first derivative. 
But right away, this tells us that 
relativistic universe cannot be static. 
Well, we can integrate that in the 
following fashion. 
We multiply the whole thing with first 
derivative of r, r dot. 
And now, we remember that, from your 
calculus, that derivative of the square 
of the first derivative is given by this 
expression. 
So we can plug it in, 
and the equation now looks a little more 
complicated. 
But there is a reason for this. 
And we also remember that, the following 
will apply. 
That's the derivative of one over radius, 
divided by time as an independent 
coordinate, is given by default in 
formula. 
So now we plug that in, and we have an 
interesting, simple differential 
equation, 
which says that derivative time 
derivative of this thing in brackets is 
actually zero. 
And therefore, the thing in brackets must 
be constant, so it's called K. 
Turns out that's curvature constant. 
Now, if we replace present day density 
row zero with general density at any 
given time, 
rho, scaling by, appropriate scale 
factor, and then divide by r2. 
squared, we get the following equation. 
That looks simpler. 
That, in fact, is the Friedmann equation 
in the absence of the cosmological 
constant. 
So, finally we get this result, 
and let's take a, just, inspection of 
this. 
If K0 as it could be, then R dot must be 
always positive, 
because you move the gravity turn to the 
right side and the expansion goes 
forever. 
But it slows down, because the density 
goes down. 
This is so called critical or flat 
universe. 
If curvature constant is greater than 
zero, then initially R dot will be 
positive, and that means the universe 
will be expanding, but then at some 
point, we'll flip the sign and start 
contracting, so that closed the universe. 
And if curvature constant is less than 
zero then r dot must be always positive. 
It's never zero and universe expands 
forever, so that's the open universe. 
Alright, so this is the Friedmann 
equation, 
and we've neglected details including 
what happens with cosmological constant. 
But if cosmological constant is included, 
that is just yet another term. 
You can think of it as another density 
term, although it behaves differently 
from other kinds of density. 
So Friedmann equation is the equation of 
motion for a universe, for homogeneous, 
isotropic, relativistic universe. And 
solving this equation for a given 
parameters gives good cosmological 
models. 
But in order to do this we need one more 
equation, which is the equation of state, 
which tells us how is the density of 
whatever is filling up the universe 
changing as the universe expands. 
So next time we'll talk about 
cosmological parameters, and express 
Friedman equation in those terms.