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 N-dimensional space with constant positive curvature is an N-dimensional 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 N-dimensional space with constant negative curvature is an N-dimensional pseudosphere. To compare this funny word with something more familiar, a hyperboloid is a two-dimensional 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 r₀. Let W = r/r₀. 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 10,000-100,000 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.