  We show that any accelerating Friedmann-Robertson-Walker (FRW) cosmology with
equation of state w < -1/3 (and therefore not only a de Sitter stage with w
=-1) exhibits three-dimensional conformal symmetry on future constant-time
hypersurfaces. We also offer an alternative derivation of this result in terms
of conformal Killing vectors and show that long wavelength comoving curvature
perturbations of the perturbed FRW metric are just conformal Killing motions of
the FRW background. We then extend theb boundary conformal symmetry to the bulk
for accelerating cosmologies. Our findings indicate that one can easily
generate perturbations of scalar fields which are not only scale invariant, but
also fully conformally invariant on super-Hubble scales. Measuring a
scale-invariant power spectrum for the cosmological perturbation does not
automatically imply that the universe went through a de Sitter stage.
