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
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
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:
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,
It follows that we can replace with in the above equation,
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
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