We're now ready to start evaluating some 
cosmological models. 
We'll consider several special cases. 
The first one would be a flat, spatially 
flat matter-dominated model, 
which is also known as Einstein-de Sitter 
model. 
We will then look at this version when 
it's dominated by radiation and not 
matter. 
Well then consider what looks like, at 
first, superficial model called Milne 
model, and it contains neither matter nor 
energy, but it is very instructive in 
some ways and then finally we'll look at 
the behavior of more general models. 
So let's first consider a spatially flat 
model, with exactly critical density. 
This is also known as Einstein-de Sitter 
model, because that's the one that two of 
them developed together in 1930s, and 
it's been used quite a bit, even though 
the universe we live in does not really 
correspond to it, 
but it is very useful as a description of 
an early universe. 
We'll start with the Friedmann Equation, 
which we can then rewrite in the 
following form. 
From that, we can see the value of the 
time derivative for the scale factor a 
dot or as expressed as explicitly as time 
derivative. 
It is plus or minus, square root of the 
quantity shown there, 
this is very easy to integrate as shown 
here, 
and, the solution is shown here. 
We can just rewrite that and see that in 
the matter-dominated phase, 
universe expanse as a 2/3 power of the 
time. 
Now, lets look at similar model but, this 
time dominated by the radiation. 
Recall that the radiation energy density, 
changes with the steeper power of, scale 
factor, 4 rather than 3. 
You get 3, for the expansion of the 
space, and the evolution of the number 
density of photons. 
You get an extra power for the energy 
lost due to the stretching of the 
wavelengths. 
All right, so here is its Friedman 
Equation, which again, we can rewrite in 
the following form, and from that we see 
the time derivative of scale factor is 
inversely proportional to the scale 
factor itself. 
That too is an easy one to integrate. 
And the solution is that the universe 
expands according to a square root of the 
time. 
Let's now consider what's called, Milne 
Cosmology. 
This is spatially flat model, good old 
Euclidean geometry, it follows the 
spacial relativity and it is realized in 
Sanskrit there no matter at all, it's any 
empty universe, just specs, so you can 
think of it as galaxies having no mass of 
their own just being test particles to 
show us how the coordinates are expanding 
as you will see how, Hubble's law follows 
immediate and directly from it. 
Special relativity still holds, Lorentz 
contractions are still at play, 
and again, there is no special location, 
because cosmological principle applies. 
In fact, Milne was the first one to 
introduce cosmological principle. So this 
was spatially often negative curvature 
model with a zero density. 
Its Friedmann equation is shown here and 
since curvature constant is less than 
zero, we can see it here, therefore, the 
time derivative of scale factor is 
constant. 
If it's constant, that implies a linear 
expansion. 
The universe expands is the first power 
of time or it can be contracting, 
seeing as the square root can be of 
either sign. 
Alright, now, let's consider slightly 
different version of that. 
Again, curvature constant is negative, 
but this time density is finite and 
positive. 
The Friedmann Equation is now shown here, 
and this time, the square of the time 
derivative scale factor is positive. 
Well, as the universe expands, time goes 
to infinity, the scale factor will go to 
infinity, and one over a scale factor 
will go to zero. 
And since the obvious inequality applies 
here, we see that in asymptotic case, 
either matter or radiation-dominated 
universe, will in the end behave just 
like the Milne model. 
Now, lets consider a more general case, 
models with, in which matter, radiation, 
cosmological constant, or dark energy can 
be present. 
First, let us look at the model of the 
positive curvature. 
Here, we can write the Friedmann equation 
again and transform it as shown here. 
Now if the density declines as the third 
power of scale factor as it would be 
appropriate for the matter field universe 
or as fourth power as it is appropriate 
for radiation, then in any case the 
product of the density and square of the 
scale factor declines as scale factor on 
the power side is -1 or -2. 
Since in the right-hand side, we have a 
term that looks like that, minus a 
constant. 
Sooner or later, that whole side, the 
right side of equation has to be zero. 
When the time derivative of scale factor 
is zero, 
that means that it's either at the 
maximum or at the minimum. 
However, look at the expression for the 
acceleration. 
The right-hand side must be always 
negative, because both density pressure 
are positive quantities. 
And therefore, the second derivative must 
be negative, 
which means that the scale factor has 
reached maximum, 
then it can only collapse back. 
So in a model like this, recollapse back 
to the initial singularity or reverse of 
the big bang is inevitable. 
Now, let's consider a model which only 
has dark energy in the form of a 
cosmological constant. 
Its Friedmann equation is show here, 
which we can rewrite in a fairly 
straightforward fashion. 
And so now, if we assume that scale 
factory is allowed to increase, as long 
as needed, 
eventually, the first term on the 
right-hand side dominates the second 
term, 
regardless of the actual value of the 
curvature constant. 
Okay. 
So if the energy density of the physical 
vacuum or cosmological constant is 
positive, then the first time derivative 
scale factor will be positive, and that 
means that that universe will expand 
forever. 
Now, if the energy density of dark energy 
is negative and that's actually possible, 
things will get to be little more 
complicated, but we need not concern 
ourselves with that at the moment. 
Now, let's look at models in which both 
radiation and matter are present. 
And, if their energy density is much 
higher, then the energy density of the 
dark energy, which is certainly true in 
the early universe, then this is a 
perfectly good description of the early 
universe. 
The rigorous solution for the density as 
a function of time will be a little more 
difficult, but we can do the following 
fairly good approximation. 
We can divide the history in the time 
where radiation is dominate and when the 
matter is dominate. 
In either case, early on, the model is 
very close to flat and so we can assume 
curvature of zero to begin with. 
Since the density of the radiation always 
decreases faster than that one of matter 
sooner or later, those two curves have to 
cross, 
and so, at earlier times, radiation 
dominates and a later time, latter times, 
latter dominates. 
And then each one drives the expansion 
according to the different power of time. 
So, in general, we can rewrite Friedmann 
equation in terms of described behavior 
of three densities as follows. 
This is actually a useful form, which we 
will use later on, to derive expression 
for distances in cosmology. 
So, let's recap, again. 
Which component of the universe dominates 
its dynamics at what time? Since the 
fastest declining one is more important 
at the earlier times and that would be 
radiation. 
Early on, universe must be dominated by 
radiation. 
Sooner or later, energy radiation drops 
below that one of matter and then becomes 
matter-dominated. 
Eventually, the density of both matter 
and energy drop sufficiently to reach the 
level of the constant energy filling the 
universe, the cosmological constant, and 
after that cosmological constant must 
dominate. 
So generally, it's always true that 
radiation dominates the early universe no 
matter what present in the future 
universe will be dominated by the 
cosmological constant or dark energy that 
behaves in that way. 
So we can express dynamics of the uni, 
universe in a very general fashion here 
showing how scale factor depends on time 
and certain power, which is a function of 
the equation of state parameter w. 
During the matter dominated phase, w is 
zero. 
Remember, when we introduced the equation 
of state, 
we said that for regular matter, there is 
no pressure, just test particles as far 
as cosmology is concern and the universe 
expands, according to time to the thirds 
power. 
Since that power is less than one, the 
universe is decelerating. 
It's slowing down its expansion. 
This was intuitively obvious, because the 
gravity of the matter pulls everything 
back, just as if you were to throw an 
object up in the air, it will reach 
certain maximum height while decelerating 
all the way there, and then of course, 
will turn and fall down. 
In the case of radiation-dominated 
universe, the expansion will be according 
to the time to the 1/2 power and the 
universe will be decelerating. 
Yes, its matter and energy are 
equivalent. 
Energy in the form of radiation exerts 
gravitational pull, so it slows down the 
expansion. 
If the universe has a constant energy 
density like cosmological constant of w = 
-1, then, when that term is dominant, it 
will drive expansion according to the 
exponential law. 
The dividing line between accelerating 
and decelerating models is at the value 
of the equation of state parameter w - 
1/3, since -1 < -1/3, this is the 
cosmological constant model, acceleration 
will dominate. 
And so here is just the plot that shows 
what some expansion laws look like. 
They are for different values of 
cosmological parameters that are written 
in caption. 
And we will actually address those in 
more detail later.