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The Friedmann Equation
for the
Expanding Universe

The Friedmann equation relates various cosmological parameters within the context of Einstein’s general relativity. Alexander Friedmann derived them in 1922. This famous equation states that the energy of expansion is proportional to the sum of matter energy, cosmological (vacuum) energy and curvature energy.

As stated before, the model proposes three different types of space-time curvature for the universe: positive, zero or negative curvature:

  • A closed universe is one where the space-time curvature is positive, somewhat like the surface of a sphere. It will first expand, then reverse the expansion and contract back to a singularity.
  • A flat universe has zero curvature, somewhat like a tabletop. It will expand forever, but at a rate that eventually approaches zero.
  • An open universe is one with negative space-time curvature, which can be more or less described by the form of a saddle (in three dimensions, where space-time has four). Such a universe will expand forever with a rate that will eventually approach some positive constant value.
  • There is in fact a fourth possibility, where the expansion rate will first decline, then become more or less constant for a while and then increase ad infinitum. The cosmological constant can, if nonzero, give rise to this situation. This is what present observations tell us about the real universe.

    The attached pdf document gives an engineering overview of what this model “tells” us. It leads to the standard big-bang model of the universe, where our observable universe originated out of a single point containing all the matter and radiation energy of the present observable universe.


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