Conformal invariance, dark energy, and CMB non-gaussianity

Abstract

In addition to simple scale invariance, a universe dominated by dark energy naturally gives riseto correlation functions possessing full conformal invariance. This is due to the mathematicalisomorphism between the conformal group of certain three dimensional slices of de Sitter spaceand the de Sitter isometry group SO(4,1). In the standard homogeneous, isotropic cosmologicalmodel in which primordial density perturbations are generated during a long vacuum energydominated de Sitter phase, the embedding of flat spatial ℝ3 sections in de Sitterspace induces a conformal invariant perturbation spectrum and definite prediction for the shapeof the non-Gaussian CMB bispectrum. In the case in which the density fluctuations are generatedinstead on the de Sitter horizon, conformal invariance of the \U0001d54a2 horizon embeddingimplies a different but also quite definite prediction for the angular correlations of CMB non-Gaussianityon 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 predictionsof single field slow roll inflation models, which rely on the breaking of de Sitter invariance. Wepropose a quantum origin for the CMB fluctuations in the scalar gravitational sector from theconformal 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 nS−1 = nTbetween the spectral indices of the scalar and tensor power spectrum. Confirmation of this predictionor detection of non-Gaussian correlations in the CMB of one of the bispectral shape functions predictedby 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.