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Hello, again. 
So far, we introduced Friedmann Equation, 

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which is the basic equation of 
relativistic cosmology. 

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Its solutions describe how is universe 
expanding as a function of time, and 

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therefore how a distance between any 2 
different objects can be computed. 

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So here is Friedmann's equation in its 
newly expressed form, as a function of 

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cosmological parameters. 
It's a differential equation. 

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However, in order to solve it, we need 
something else. 

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The expansion of the universe at any 
given moment is determined by the mean 

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density of matter, matter and energy in 
it. 

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As the universe expands, 
that dynasty changes in different way for 

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different components. 
The equation that specifices how that is 

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changing is called the equation of state. 
And it is often expressed as a relation 

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between pressure and density. 
In its simplest form, it is just a 

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multiplicative constant, usually denoted 
as little w. 

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So, the equation of state parameter 
itself is sometimes called the equation 

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of state. 
And, note that this is a fairly arbitrary 

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functional form. 
In, in principle, the equation of state 

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could be much more complicated. 
However, this is the simplest one and so 

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it's a good start in assumption unless 
it's proven that something else happens. 

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Also, 
we know that some of the major 

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components, like ordinary matter, energy, 
density, or cosmological constant are 

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behaving in this fashion. 
So, there's some special values. 

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w0 means that pressure is zero. 
And so, that's for, that's is a non 

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relativistic matter. 
Just galaxies or atoms that do not 

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interact other than gravitationally. 
w of 1/3 corresponds to radiation photons 

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or relativistic particles of some kind. 
And w of -1 corresponds to a constant 

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energy density that doesn't change, even 
though the universe does expand. And that 

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is suitable for cosmological constant. 
So, let us consider these 3 simple cases, 

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matter, radiation, and cosmological 
constant. 

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In reality, the universe has all 3 of 
those, plus maybe other things. 

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And there is a mixture, so all three have 
to be included in Friedmann's equation. 

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So, each of those will evolve in a 
different fashion because each one of 

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them implies a different behavior of 
density as the universe expands. 

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And so, as the, as the universe expands, 
its density changes, that affects the 

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expansion, and so on. 
Alright, so how does the density of the 

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matter and energy change as the universe 
expands? Let's start with the three 

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simple cases that we just introduced 
earlier. 

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Ordinary matter, radiation, or 
relativistic matter, and the cosmological 

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constant. 
We'll just number them 1, 2, 3, 4 for 

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convenience. 
So, if we put those in the fluid 

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equation, which is essentially the 
continuity equation. And hopefully, you 

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have encountered it when you learning 
about fluid dynamics. But if not, we'll 

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refresh that knowledge. 

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But, in addition to that, the energy will 
also go down by one power of density 

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because of the cosmic expansion, the 
[UNKNOWN] stretched. 

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And therefore, energy goes down. 
So, in that case, the density goes as 

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inverse fourth power of size. 
And finally, in the third case, the 

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derivative is zero and therefore, the 
density must be constant. 

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So, in general, we can express behavior 
of density as a, an expansion factor R or 

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A, 
on power of -3*w+1. 

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And let's see how well, and let's see how 
this works with the three cases we've 

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seen. 
If we substitute values that we define 

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before, 
0 for the regular matter, 

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we do get, the density goes as scale 
factor and -1/3 power. 

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If we assume that it's 1/3, then indeed 
we recover the pendance of radiation 

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energy density as scale factor 2^-4. 
And finally, if it's -1, then density 

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stays constant. 
You may remember that energy is not 

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conserved in an expanding universe. 
Well, here is an obvious case. 

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The universe has a constant energy 
density in terms of say, ergs per cubic 

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centimeter. But there are more cubic 
centimeters as it expands so there is 

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more and more energy in the universe. 
An interesting case is if w is less than 

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-1. 
In that case, the more universe expands, 

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the more energy it gets, and the 
expansion keeps accelerating even more. 

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And, in fact, this leads to what's called 
a big rip or universe that expands so 

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dramatically that atoms will be torn 
apart as well as everything else. 

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But rest easy. 
Very likely, we do not live in such a 

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universe. 
And even if we did, it would take many 

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billions of years. 
In the realistic universe, there'll be a 

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mixture of these components. 
And the expansion will change. 

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At different times, different components 
will determine how the universe is 

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expanding because some of them decline is 
faster than the others. 

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Also remember that we have assumed that w 
is constant, but it need not be. 

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In itself, it can be changing as a 
function of time. 

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But we have no theory whatsoever which 
could indicate why would that be and how. 

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So, this is how we solve Friedmann 
equation. The output of it is really how 

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the density of the universe changes as a 
function of time. 

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Next time, we will consider some examples 
of actual cosmological models.