We can also derive some basic facts about the big bang cosmology. Let us assume the universe is not only expanding but also homogeneous and isotropic. The expansion of the universe is vouched for by the redshifts of distant galaxies. The other assumptions also seem to be approximately correct, at least when we average over small-scale inhomogeneities such as stars and galaxies. For simplicity, we will imagine the universe is homogeneous and isotropic even on small scales.

An observer at any point in such a universe would see all objects receding from her. Suppose that, at some time , she identifies a small ball of test particles centered on her. Suppose this ball expands with the universe, remaining spherical as time passes because the universe is isotropic. Let stand for the radius of this ball as a function of time. The Einstein equation will give us an equation of motion for . In other words, it will say how the expansion rate of the universe changes with time.

It is tempting to apply equation (2) to the ball , but we must take care. This equation applies to a ball of particles that are initially at rest relative to one another -- that is, one whose radius is not changing at . However, the ball is expanding at . Thus, to apply our formulation of Einstein's equation, we must introduce a second small ball of test particles that are at rest relative to each other at .

Let us call this second ball , and call its radius as a function
of time . Since the particles in this ball begin at rest
relative to one another, we have

To keep things simple, let us also assume that at both balls have the exact same size:

Equation (2) applies to the ball , since
the particles in *this* ball are initially at rest relative to
each other. Since the volume of this ball is proportional to ,
and since at , the left-hand side of
equation (2) is simply

Since we are assuming the universe is isotropic, we know that the various components of pressure are equal: . Einstein's equation (2) thus says that

We would much prefer to rewrite this expression in terms of
rather than . Fortunately, we can do this.
At , the two spheres have the same radius: . Furthermore,
the second derivatives are the same:
. This
follows from the equivalence principle, which says that, at any given
location, particles in free fall do not accelerate with respect to each
other. At the moment , each test particle on the surface of
the ball is right next to a corresponding test particle in .
Since they are not accelerating with respect to each other, the observer
at the origin must see both particles accelerating in the same way,
so
.
It follows that we can replace with in the above equation,
obtaining

We derived this equation for a very small ball, but
in fact it applies to a ball of any size. This is because, in a
homogeneous expanding universe, the balls of all radii
must be expanding at the same fractional rate. In other words,
is independent of the radius , although
it can depend on time. Also, there is nothing special in
this equation about the moment , so the equation must
apply at all times. In summary, therefore, the basic equation
describing the big bang cosmology is

To go further, we must make more assumptions about the nature of
the matter filling the universe. One simple model is a universe
filled with pressureless matter. Until recently, this
was thought to be an accurate model of our universe. Setting , we
obtain

If the energy density of the universe is mainly due to the mass in galaxies, `conservation of galaxies' implies that for some constant . This gives

or

Amusingly, this is the same as the equation of motion for a particle in an attractive force field. In other words, the equation governing this simplified cosmology is the same as the Newtonian equation for what happens when you throw a ball vertically upwards from the earth! This is a nice example of the unity of physics. Since ``whatever goes up must come down -- unless it exceeds escape velocity,'' the solutions of this equation look roughly like this:

So, the universe started out with a big bang! It will expand forever if its current rate of expansion is sufficiently high compared to its current density, but it will recollapse in a `big crunch' otherwise.

* © 2006 John Baez and Emory Bunn*