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 two
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 one-third
corresponds to radiation photons or relativistic particles of some kind. And w
of -one 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 three simple cases, matter, radiation, and cosmological
constant. In reality, the universe has all three 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 one, two, three, four 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 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 -three*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, zero for the regular
matter, we do get, the density goes as scale factor and -one-third power. If we
assume that it's one-third, then indeed we recover the pendance of radiation energy
density as scale factor two^-4. And finally, if it's -one, 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 -one.
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.