Conformal Invariance, Dark Energy, and CMB Non-Gaussianity
  In addition to simple scale invariance, a universe dominated by dark energy
naturally gives rise to correlation functions possessing full conformal
invariance. This is due to the mathematical isomorphism between the conformal
group of certain 3 dimensional slices of de Sitter space and the de Sitter
isometry group SO(4,1). In the standard homogeneous isotropic cosmological
model in which primordial density perturbations are generated during a long
vacuum energy dominated de Sitter phase, the embedding of flat spatial sections
in de Sitter space induces a conformal invariant perturbation spectrum and
definite prediction for the shape of the non-Gaussian CMB bispectrum. In the
case in which the density fluctuations are generated instead on the de Sitter
horizon, conformal invariance of the horizon embedding implies a different but
also quite definite prediction for the angular correlations of CMB
non-Gaussianity on the sky. Each of these forms for the bispectrum is intrinsic
to the symmetries of de Sitter space and in that sense, independent of specific
model assumptions. Each is different from the predictions of single field slow
roll inflation models which rely on the breaking of de Sitter invariance. We
propose a quantum origin for the CMB fluctuations in the scalar gravitational
sector from the conformal anomaly that could give rise to these
non-Gaussianities without a slow roll inflaton field, and argue that conformal
invariance also leads to the expectation for the relation n_S-1=n_T between the
spectral indices of the scalar and tensor power spectrum. Confirmation of this
prediction or detection of non-Gaussian correlations in the CMB of one of the
bispectral shape functions predicted by conformal invariance can be used both
to establish the physical origins of primordial density fluctuations and
distinguish between different dynamical models of cosmological vacuum dark
energy.
