The starting point of a theoretical exploration of
cosmology is the Einstein equation
with a metric of the form
The spacetime being modeled by this equation can be neatly separated
into time and space, so we can talk of this spacetime as representing
the evolution of space in time.
The space part of this spacetime is homogeneous (looks
the same at any point in a given direction) and isotropic (looks the
same in any direction from a given point). This is an abstract ideal
approximation to the Universe, but it's one that has worked extremely
well from an observational point of view, as will be shown below.
There are three options for the spatial geometry of
a spacetime with the above metric, represented by three choices for
the value of the parameter k: k=1, k=0 or k=1. The condition of being
spatially homogeneous and isotropic means that the surfaces of constant
time t have constant curvature, which can be either positive, zero or
negative.


A sphere has constant positive curvature. 
A hyperboloid has constant negative curvature. 
To solve the Einstein equation, we need to postulate
some "stuff" in the spacetime, such as matter, radiation or
vacuum energy, with energy momentum tensor T_{mn}
whose components are the energy density r
and pressure p of the "stuff" in question. The equations for
the scale factor a(t) are
Here we have truncated Newton's constant G_{N} to plain G.
The top equation contains the condition for the closure density of the
Universe explained below.
There are many different kinds of "stuff"
that can be a part of the energy density r,
with different equations of state relating r
to the pressure p. For bookkeeping purposes, let's label each different
r by an index i, so that r_{i}
refers to the energy density from the ith type of "stuff"
in this spacetime. Let's also set something called the critical density
r_{crit} and then make
the above equation dimensionless by dividing everything by the critical
density:
Let's call W the density parameter.
The equation that will tell us the curvature of space from the stuff
content of the spacetime becomes
The three possibilities for the value of the parameter k correspond
the three different possibilities for the curvature of space in this
spacetime. A value of k=1 corresponds to constant positive curvature,
k=0 to zero curvature and k=1 to constant negative curvature.
The time evolution of space is more complicated because
it depends on the equation of state of the stuff in spacetime. The equation
of state is the relationship between pressure and density in the stuff.
Energy conservation plus the equation of state determine how the energy
density changes as space evolves in time.
This is where vacuum energy becomes important. The
energy densities for matter, radiation and vacuum energy change with
the size of space (the scale factor a(t)) like
So as the Universe is getting bigger, the energy density from matter
and radiation would be getting smaller, but vacuum energy density would
remain the same. Another name for vacuum energy is the cosmological
constant. A cosmological constant eventually controls the time
evolution of an expanding universe, because its energy density stays
the same while those of matter and radiation are getting smaller.
In a spacetime with all three forms of energy present,
the radiation part of the mix will dominate the dynamics when the scale
factor a(t)<<1. In the Big Bang model this is called the radiation
dominated era, and accounted for the first 10,000100,000 years
of the evolution of our Universe. Right now the dominant forms of energy
in our Universe are matter and vacuum energy.
That being said, we will avoid dealing with any vacuum
energy right now and consider a spacetime with only matter, with no
radiation or cosmological constant. In this case, the time evolution
of space is related to the curvature of space as follows:
k 
W 
Topology 
Time Evolution 
1 
>1 
Closed 
Space is positively curved and
finite, expands from zero size to a maximum size and then shrinks
back to zero again 
0 
=1 
Open 
Space is flat and infinite,
and expands forever 
1 
<1 
Open 
Space is negatively
curved and infinite, and expands forever 
If the amount of energy density in the spacetime
is over the critical density, so that W
> 1, then the fate of the Universe is to expand in a Big Bang but
then eventually contract back into a Big Crunch. Despite the fact that
this would take place on a time scale of billions of years, humans today
find this possibility philosophically undesirable. More importantly,
the data do not support it.
The visible matter in the Universe observed by humans
today has barely a fraction of closure density. In fact, the Universe
as observed today seems to have barely a fraction of the mass needed
to keep galaxies from flying apart, based on the rotations of the stars
in the galaxy about the galactic center.
What keeps the galaxies from flying apart? It must
be a lot of mass that we can't see. Which brings us to the subject of
dark matter.
Next:
Dark matter and the cosmological constant>>
