  The Friedmann Equation for a conformal scale factor a(t) is observed to be
invariant under a Mobius transformation. Using that freedom, a synthetic scale
factor z(t) is defined that obeys a modified Friedman equation invariant under
the replacement z(t) {\rightarrow} {\pm}1/ z(t). If this is taken this to be
the more fundamental form then the traditional Friedmann equation can be shown
to be missing a term due to a species with equation of state w = -2/3. We
investigate in detail one particular cosmology in which it is possible to
specify the contribution from this new species.
  We suggest a means of avoiding a potentially redundant copy of the
development of the universe the above implies through a cosmological spacetime
manifold that is a Mobius band closed in time and non-orientable in space.
Though it is closed, a Dirac field in such a spacetime may still possess a
global arrow of time by virtue of the twist of the Mobius band.
