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.