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.