Hello, again. 
So far, we introduced Friedmann Equation, 
which is the basic equation of 
relativistic cosmology. 
Its solutions describe how is universe 
expanding as a function of time, and 
therefore how a distance between any 2 
different objects can be computed. 
So here is Friedmann's equation in its 
newly expressed form, as a function of 
cosmological parameters. 
It's a differential equation. 
However, in order to solve it, we need 
something else. 
The expansion of the universe at any 
given moment is determined by the mean 
density of matter, matter and energy in 
it. 
As the universe expands, 
that dynasty changes in different way for 
different components. 
The equation that specifices how that is 
changing is called the equation of state. 
And it is often expressed as a relation 
between pressure and density. 
In its simplest form, it is just a 
multiplicative constant, usually denoted 
as little w. 
So, the equation of state parameter 
itself is sometimes called the equation 
of state. 
And, note that this is a fairly arbitrary 
functional form. 
In, in principle, the equation of state 
could be much more complicated. 
However, this is the simplest one and so 
it's a good start in assumption unless 
it's proven that something else happens. 
Also, 
we know that some of the major 
components, like ordinary matter, energy, 
density, or cosmological constant are 
behaving in this fashion. 
So, there's some special values. 
w0 means that pressure is zero. 
And so, that's for, that's is a non 
relativistic matter. 
Just galaxies or atoms that do not 
interact other than gravitationally. 
w of 1/3 corresponds to radiation photons 
or relativistic particles of some kind. 
And w of -1 corresponds to a constant 
energy density that doesn't change, even 
though the universe does expand. And that 
is suitable for cosmological constant. 
So, let us consider these 3 simple cases, 
matter, radiation, and cosmological 
constant. 
In reality, the universe has all 3 of 
those, plus maybe other things. 
And there is a mixture, so all three have 
to be included in Friedmann's equation. 
So, each of those will evolve in a 
different fashion because each one of 
them implies a different behavior of 
density as the universe expands. 
And so, as the, as the universe expands, 
its density changes, that affects the 
expansion, and so on. 
Alright, so how does the density of the 
matter and energy change as the universe 
expands? Let's start with the three 
simple cases that we just introduced 
earlier. 
Ordinary matter, radiation, or 
relativistic matter, and the cosmological 
constant. 
We'll just number them 1, 2, 3, 4 for 
convenience. 
So, if we put those in the fluid 
equation, which is essentially the 
continuity equation. And hopefully, you 
have encountered it when you learning 
about fluid dynamics. But if not, we'll 
refresh that knowledge. 

But, in addition to that, the energy will 
also go down by one power of density 
because of the cosmic expansion, the 
[UNKNOWN] stretched. 
And therefore, energy goes down. 
So, in that case, the density goes as 
inverse fourth power of size. 
And finally, in the third case, the 
derivative is zero and therefore, the 
density must be constant. 
So, in general, we can express behavior 
of density as a, an expansion factor R or 
A, 
on power of -3*w+1. 
And let's see how well, and let's see how 
this works with the three cases we've 
seen. 
If we substitute values that we define 
before, 
0 for the regular matter, 
we do get, the density goes as scale 
factor and -1/3 power. 
If we assume that it's 1/3, then indeed 
we recover the pendance of radiation 
energy density as scale factor 2^-4. 
And finally, if it's -1, then density 
stays constant. 
You may remember that energy is not 
conserved in an expanding universe. 
Well, here is an obvious case. 
The universe has a constant energy 
density in terms of say, ergs per cubic 
centimeter. But there are more cubic 
centimeters as it expands so there is 
more and more energy in the universe. 
An interesting case is if w is less than 
-1. 
In that case, the more universe expands, 
the more energy it gets, and the 
expansion keeps accelerating even more. 
And, in fact, this leads to what's called 
a big rip or universe that expands so 
dramatically that atoms will be torn 
apart as well as everything else. 
But rest easy. 
Very likely, we do not live in such a 
universe. 
And even if we did, it would take many 
billions of years. 
In the realistic universe, there'll be a 
mixture of these components. 
And the expansion will change. 
At different times, different components 
will determine how the universe is 
expanding because some of them decline is 
faster than the others. 
Also remember that we have assumed that w 
is constant, but it need not be. 
In itself, it can be changing as a 
function of time. 
But we have no theory whatsoever which 
could indicate why would that be and how. 
So, this is how we solve Friedmann 
equation. The output of it is really how 
the density of the universe changes as a 
function of time. 
Next time, we will consider some examples 
of actual cosmological models.