Now let us take a look at the Friedmann equation.
This is the basic equation of cosmology, and it's solution tells how universe will
evolve on large scales. We can derive it, actually, in a fairly
simple Newtonian approximation, if we accept a couple results.
First, there is, so called, Birkhoff's theorem,
which says that, for a spherically symmetric system, the force due to
gravity at some radius is determined only by the mass interior to this radius,
which makes sense. And then also, the energy contributes to
gravitational mass density. So you really need to add them up
together. So, total energy or mass density can be
expressed as one or the other modules, square of speed of light.
So how do we go about this. Imagine a sphere with a mean density of
row and radius of R. And acceleration to a particle in
surface, which is the second derivative of the radius, is given by the familiar
formula. Now, if, if the density changes, it
changes according to the inverse cube of the radius.
So if we, again, use subscript zero for present moment,
then you can derive the density at any given time through the cube of the radius
divided the present one, which we can assume to be unity for
simplicity. So now, plug that in the, in the first
equation. And we have the following equation for
the second derivative. Well, right away, this is telling us
something interesting. If the density is not zero, the universe
must be either expanding or contracting, because the second derivative is finite.
And this sign depends, obviously, on the first derivative.
But right away, this tells us that relativistic universe cannot be static.
Well, we can integrate that in the following fashion.
We multiply the whole thing with first derivative of r, r dot.
And now, we remember that, from your calculus, that derivative of the square
of the first derivative is given by this expression.
So we can plug it in, and the equation now looks a little more
complicated. But there is a reason for this.
And we also remember that, the following will apply.
That's the derivative of one over radius, divided by time as an independent
coordinate, is given by default in formula.
So now we plug that in, and we have an interesting, simple differential
equation, which says that derivative time
derivative of this thing in brackets is actually zero.
And therefore, the thing in brackets must be constant, so it's called K.
Turns out that's curvature constant. Now, if we replace present day density
row zero with general density at any given time,
rho, scaling by, appropriate scale factor, and then divide by r2.
squared, we get the following equation. That looks simpler.
That, in fact, is the Friedmann equation in the absence of the cosmological
constant. So, finally we get this result,
and let's take a, just, inspection of this.
If K0 as it could be, then R dot must be always positive,
because you move the gravity turn to the right side and the expansion goes
forever. But it slows down, because the density
goes down. This is so called critical or flat
universe. If curvature constant is greater than
zero, then initially R dot will be positive, and that means the universe
will be expanding, but then at some point, we'll flip the sign and start
contracting, so that closed the universe. And if curvature constant is less than
zero then r dot must be always positive. It's never zero and universe expands
forever, so that's the open universe. Alright, so this is the Friedmann
equation, and we've neglected details including
what happens with cosmological constant. But if cosmological constant is included,
that is just yet another term. You can think of it as another density
term, although it behaves differently from other kinds of density.
So Friedmann equation is the equation of motion for a universe, for homogeneous,
isotropic, relativistic universe. And solving this equation for a given
parameters gives good cosmological models.
But in order to do this we need one more equation, which is the equation of state,
which tells us how is the density of whatever is filling up the universe
changing as the universe expands. So next time we'll talk about
cosmological parameters, and express Friedman equation in those terms.