The Shape of Bouncing Universes
  What happens to the most general closed oscillating universes in general
relativity? We sketch the development of interest in cyclic universes from the
early work of Friedmann and Tolman to modern variations introduced by the
presence of a cosmological constant. Then we show what happens in the cyclic
evolution of the most general closed anisotropic universes provided by the
Mixmaster universe. We show that in the presence of entropy increase its cycles
grow in size and age, increasingly approaching flatness. But these cycles also
grow increasingly anisotropic at their expansion maxima. If there is a positive
cosmological constant, or dark energy, present then these oscillations always
end and the last cycle evolves from an anisotropic inflexion point towards a de
Sitter future of everlasting expansion.
