Conformal Invariance, Dark Energy, and CMB Non-Gaussianity

TL;DR

The paper argues that a dark-energy–dominated de Sitter phase endows primordial fluctuations with full conformal invariance through the de Sitter isometry group SO(4,1). It develops two complementary realizations—conformal invariance on flat ${ m R}^3$ slices embedded in de Sitter space and conformal invariance on the cosmological ${ m S^2}$ horizon boundary—that yield distinct, symmetry-fixed forms for the CMB non-Gaussian bispectrum, independent of specific dynamical models. It links the Harrison–Zel'dovich power spectrum to conformal weights of density perturbations and proposes a trace-anomaly–driven quantum origin for fluctuations that could obviate the need for an inflaton field, while predicting a direct relation $n_S-1=n_T$ under these conformal assumptions. The work emphasizes that observed CMB non-Gaussianities can discriminate between different dark-energy cosmologies and horizon-based realizations, offering a robust, symmetry-based path to understanding the primordial universe and the role of dark energy.

Abstract

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.

Figures (6)

• Figure 1: The bispectral shape function (\ref{['bispectrum']}) with pole contribution subtracted as in (\ref{['Ssub']}), for conformal weight $w=1.98$, corresponding to a CMB spectral index $n=0.96$, as a function of $X = \frac{k_2^2}{k_1^2}$ and $Y=\frac{k_2^2}{k_1^2}$.
• Figure 2: The bispectral shape function (\ref{['bispectrum']}), for conformal weight $w=2.02$, corresponding to a CMB spectral index $n=1.04$, as a function of $X = \frac{k_2^2}{k_1^2}$ and $Y=\frac{k_2^2}{k_1^2}$. As in the previous figure, the pole contribution at $w=2$ has been subtracted: (\ref{['Ssub']})
• Figure 3: The subtracted bispectral shape function (\ref{['S2']}) for conformal weight $w=2.00$ corresponding to a CMB spectral index $n=1.00$, as a function of $X = \frac{k_2^2}{k_1^2}$ and $Y=\frac{k_2^2}{k_1^2}$.
• Figure 4: Contour plot for the bispectral shape function (\ref{['S2']}) corresponding to Figs. \ref{['Fig:BispectrumHZ']} for a CMB spectral index of its classical HZ value $n=1.00$, as a function of $X = \frac{k_2^2}{k_1^2}$ and $Y=\frac{k_2^2}{k_1^2}$. The larger values of $S(X,Y;2)$ are for smaller $X,Y$ in the red region at the upper left, gradually decreasing toward larger $X$ and $Y$ at the lower right.
• Figure 5: The de Sitter manifold represented as a single sheeted hyperboloid of revolution about the $T$ axis, in which the $X^1$, $X^2$ coordinates are suppressed. The hypersurfaces at constant $T$ are three-spheres, $\mathbb{S}^3$. The three-spheres at $T= \pm \infty$ are denoted by $I_{\pm}$.