  The Cardassian universe is a proposed modification to the Friedmann Robertson
Walker (FRW) equation in which the universe is flat, matter dominated, and
accelerating. Here we generalize the original Cardassian proposal to include
additional variants on the FRW equation. Specific examples are presented.
  In the ordinary FRW equation, the right hand side is a linear function of the
energy density, $H^2 \sim \rho$. Here, instead, the right hand side of the FRW
equation is a different function of the energy density, $H^2 \sim g(\rho)$.
This function returns to ordinary FRW at early times, but modifies the
expansion at a late epoch of the universe. The only ingredients in this
universe are matter and radiation: in particular, there is {\it no} vacuum
contribution. Currently the modification of the FRW equation is such that the
universe accelerates. The universe can be flat and yet consist of only matter
and radiation, and still be compatible with observations. The energy density
required to close the universe is much smaller than in a standard cosmology, so
that matter can be sufficient to provide a flat geometry. The modifications may
arise, e.g., as a consequence of our observable universe living as a
3-dimensional brane in a higher dimensional universe. The Cardassian model
survives several observational tests, including the cosmic background
radiation, the age of the universe, the cluster baryon fraction, and structure
formation. As will be shown in future work, the predictions for observational
tests of the generalized Cardassian models can be very different from generic
quintessence models, whether the equation of state is constant or time
dependent.
