An Exploration of Symmetries in the Friedmann Equation

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.