Dark matter

   Something becomes visible when it interacts with light in such a way that we can see it. Astronomers studying the motions of stars in spiral galaxies noticed that the mean star velocity did not drop off with radius from the galactic center as rapidly as the falloff in luminous mass in the galaxy dictated according to Newtonian gravity. The stars far from the center were rotating too fast to be balanced by the gravitational force from the luminous mass contained within that radius. This led to the proposition that most of the mass in a galaxy was low luminosity mass of some kind, and this invisible mass was called dark matter.
   Dark matter is probably not baryonic matter, because the abundance of primordial elements such as hydrogen, helium and deuterium would be much higher if the Big Bang had produced enough baryon density to account for the dark matter in galaxies.
   The amount of dark matter present in the Universe has been estimated using various techniques, including observing the velocities of galaxies in clusters and calculating the gravitational mass of galactic clusters by their gravitational lensing effects on surrounding spacetime. The end result is that the baryonic density W_B is about 5% and the dark matter density W_D is about 30%
   The leading candidate for dark matter right now comes from supersymmetry. Supersymmetric versions of the Standard Model of elementary particle physics contain heavy supersymmetric partners of the electroweak gauge bosons and the Higgs field that are electrically neutral and hence don't interact with electromagnetic radiation, aka light. These neutralinos, as they are called, are fermionic partners of the neutral gauge bosons and the Higgs field. They would have high mass, yet interact very weakly, and those two qualities make them a good candidate for dark matter.

The Cosmological Constant

   The observational evidence that the Universe was expanding didn't come around until 1929, which was 14 years after the Einstein's General Theory of Relativity was first published. The Einstein equations predicted an expanding Universe for any kind of ordinary matter or radiation in existence.
   There being no evidence yet to make people believe that the expanding solutions to the Einstein equations represented observed physics, Einstein postulated a new kind of energy density that could balance the matter density in the Universe and prevent the Universe from expanding. This new theoretical energy density is called the cosmological constant, known by the symbol L. The energy density and pressure for L are

   The Einstein equations with a matter density r_m and cosmological constant L become

A static solution has a(t) = constant = a₀, which means that k=+1 and the matter density, cosmological constant L₀ and scale factor are related by

   A cosmological constant alters the time evolution that is associated with a given spatial curvature. The k=+1 spacetime with only matter expands and then recollapses, but the k=+1 spacetime with matter and a cosmological constant can either expand forever (for L > L₀), stay the same forever (L = L₀) or expand and recontracts (0 < L < L₀).
   If L > 0 and k= 0 or -1, then space expands forever. If L < 0, then k=-1. When k=-1 with matter and no cosmological constant, the Universe is open and expands forever. But for L < 0, even though k=1 and the topology of space is open, this spacetime expands and then recontracts like the k=+1 model with matter and no cosmological constant.

What's the final answer?

   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 a vacuum energy, or something that acts just like one, to make 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 galactic 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.