Open, closed or flat?
If at every time, space at every point looks
the same in every direction, then space has to have constant
curvature. If the curvature was different at any point, then
space would look different in that direction from every other point.
Therefore if space is maximally symmetric, the curvature
has to be the same at every point.
So that narrows us down to three options for
the geometry of space: positive, negative or zero curvature. When there
is no vacuum energy present, just matter or radiation, the curvature
of space also tells us the time evolution of the spacetime in question:

A sphere has constant positive curvature. 
Positive: The unique Ndimensional
space with constant positive curvature is an Ndimensional sphere. The
cosmological scenario where space has positive constant curvature is
called a closed Universe. In this spacetime,
space expands from zero volume in a Big Bang but then reaches a maximum
volume and starts to contract back to zero volume in a Big Crunch.
Zero: A space with zero curvature is
called (no surprise here) a flat space.
A flat space is noncompact, space extends infinitely far in any direction,
so this option also represents an open
Universe. This spacetime has space expanding forever in time.

A hyperboloid has constant negative curvature. 
Negative: The unique Ndimensional
space with constant negative curvature is an Ndimensional pseudosphere.
To compare this funny word with something more familiar, a hyperboloid
is a twodimensional pseudosphere. With negative curvature, space has
infinite volume. The negative curvature option represents an open
Universe. This spacetime also has space expanding forever in time.
What determines whether a Universe is open
or closed? For a closed Universe, the total energy density r
in the Universe has to be greater than the value that gives a flat Universe,
called the critical density r_{0}.
Let W = r/r_{0}.
So a closed Universe has W
> 1, a flat Universe has W
= 1 and an open Universe has W
< 1.
The above analysis only takes into account energy
from matter, and neglects any vacuum energy
that might be present. Vacuum energy leads to a constant energy density
that is called the cosmological constant.
Which behavior represents our observed Universe? To
discuss the most recent observations, first we need to look at dark
matter and the cosmological constant.
Where does dark matter come in?
The matter in the Universe that we can see mainly
consists of stars and hot gas or other stuff that emits light of some
wavelength that can be detected by either our eyes, telescopes or complicated
instrumentation. But for the last two decades, astronomers have been
seeing evidence of vast amounts of invisible matter in the Universe.
For example, there doesn't seem to be enough visible
matter in the form of stars and interstellar gas to hold most galaxies
together gravitationally. According to estimates of how much mass would
actually be needed to keep the average galaxy from flying apart, it
is now widely believed by physicists and astronomers that most
of the matter in the Universe is invisible. This matter is called
dark matter, and it's important for
cosmology.
If there is dark matter, then what could it be made
of? If it were made of quarks like ordinary matter, then in the early
Universe, more helium and deuterium would have been produced than could
exist in the Universe today. Particle physicists tend to think that
dark matter could consist of supersymmetric
particles that are very heavy but couple very weakly to the particles
observed in accelerators now.
The visible matter in the Universe is much less than
closure density_{}, therefore, if there were nothing else, our
Universe should be open. But is the dark matter enough to close the
Universe? In other words, if W_{B}
is the density of ordinary matter and W_{D}
is the density of dark matter in the Universe today, does W_{B}
+ W_{D} = 1? Studies
of galactic motion show that even including dark matter, the total only
adds up to about 30% of closure density, with W_{B}
making up 5% and W_{D}
accounting for as much as 25%.
But that's not the end of the story. There's another
possible source of energy in the Universe: the cosmological constant.
What about the cosmological constant?
Einstein didn't always like the conclusions of his
own work. His equation of motion for spacetime predicted that a Universe
filled with ordinary matter would expand. Einstein wanted a theory where
the Universe stayed the same size forever. To fix the Einstein equation,
he added a term now called the cosmological
constant, that balanced the energy density of matter and radiation
to make a Universe that neither expanded nor contracted, but stayed
the same for eternity.
Once everyone accepted Hubble's evidence that the
Universe was expanding, Einstein's cosmological constant theory was
abandoned. However, it was resurrected by relativistic quantum theories
where a cosmological constant arises naturally and dynamically from
the quantum oscillations of virtual particles and antiparticles. This
is called the quantum zero point energy, which is a possible source
of the vacuum energy of spacetime. The
challenge in quantum theory is to avoid producing too much vacuum energy,
and that's one reason why physicists study supersymmetric
theories.
A cosmological constant can act to speed up
or slow down the expansion of the Universe, depending on whether it
is positive or negative. When a cosmological constant is added to a
spacetime with matter and radiation, the story gets more complicated
than the simple open or closed scenarios described above.
What's the final answer?
The Big Bang began with a radiation
dominated era, which 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. The latest measurements
from astronomers tell us:
1. Our Universe is pretty
flat: The cosmic microwave background is the relic of Big Bang
thermal radiation, cooled to the temperature of 2.73° Kelvin. But
it didn't cool perfectly smoothly, and after the radiation cooled, there
were some lumps left over. The angular size of those lumps as observed
from our present location in spacetime depends on the spatial curvature
of the Universe. The currently observed lumpiness in the temperature
of the cosmic microwave background is just right for a flat
Universe that expands forever.
2. There is a cosmological
constant: There is vacuum energy, or something that acts just
like IT, to make the expansion of the Universe accelerate in time. The
acceleration of the Universe can be seen in the redshifts of distant
supernovae.
3. Most of the matter in
the Universe is dark matter: Studies of galatic motion show that
ordinary visible matter in stars, galaxies, planets, and interstellar
gas only makes up a small fraction of the total energy density of the
Universe.
The Universe at our
current epoch has (approximately)
So right now the density of vacuum energy in our Universe is only about
twice as large as the energy density from dark matter, with the contribution
from visible baryonic matter almost negligible. The total adds up to
a flat universe which should expand forever.
