[Paper Review] Conformal Invariance, Dynamical Dark Energy and the CMB

Emil Mottola|arXiv (Cornell University)|Mar 8, 2011

Cosmology and Gravitation Theories2 references3 citations

TL;DR

This paper proposes that conformal invariance, arising from the trace anomaly of quantum fields in curved spacetime, dynamically generates a time- and space-dependent cosmological constant, leading to a dynamical dark energy component. By promoting scale invariance to full conformal invariance, the model uniquely fixes the form of the CMB bispectrum and constrains the trispectrum, predicting non-Gaussianities distinct from those in standard slow-roll inflation models.

ABSTRACT

General Relativity receives quantum corrections relevant at cosmological distance scales from the conformal scalar degrees of freedom required by the trace anomaly of the quantum stress tensor in curved space. In the theory including the trace anomaly terms, the cosmological "constant" becomes dynamical and hence potentially dependent upon both space and time. The fluctuations of these anomaly scalars may also influence the spectrum and statistics of the Cosmic Microwave Background. Under the hypothesis that scale invariance should be promoted to full conformal invariance, an hypothesis supported by the exact equivalence of the conformal group of three dimensions with the de Sitter group SO(4,1), the form of the CMB bispectrum can be fixed, and the trispectrum constrained. The non-Gaussianities predicted by conformal invariance differ from those suggested by simple models of inflation.

Motivation & Objective

• To address the cosmological constant problem by incorporating quantum corrections from the trace anomaly in curved spacetime.
• To explore how conformal invariance—supported by the isomorphism between the de Sitter group SO(4,1) and the conformal group of 3D Euclidean space—can constrain CMB statistics.
• To derive the form of the CMB bispectrum and trispectrum under the hypothesis of conformal invariance, independent of inflaton dynamics.
• To distinguish the predicted non-Gaussian signatures from those in standard slow-roll inflation models.

Proposed method

• Formulate a local effective action for the trace anomaly using two scalar fields φ and ψ to represent the non-local quantum corrections.
• Apply the effective field theory (EFT) framework to include quantum corrections to Einstein's equations, with the cosmological constant becoming dynamical via the trace anomaly.
• Use the conformal symmetry of de Sitter space (SO(4,1)) to constrain correlation functions of the CMB, leveraging the fact that this group is isomorphic to the conformal group of 3D flat space.
• Derive the form of the 4-point function (trispectrum) and 3-point function (bispectrum) in momentum space, showing they depend only on cross-ratios, not on scale or time.
• Utilize the de Sitter geometry in flat slicing and static coordinates to analyze the behavior of quantum fluctuations near the cosmological horizon.
• Demonstrate that the resulting non-Gaussianities are fully determined by conformal invariance, without dependence on slow-roll parameters or inflaton fields.

Experimental results

Research questions

• RQ1 How does the inclusion of the quantum trace anomaly lead to a dynamical cosmological constant in a semi-classical gravity framework?
• RQ2 What constraints does conformal invariance impose on the form of the CMB bispectrum and trispectrum?
• RQ3 How do the predicted non-Gaussianities from conformal invariance differ from those in standard slow-roll inflation models?
• RQ4 Can the de Sitter group SO(4,1) be used to derive universal properties of CMB correlation functions via its isomorphism with the 3D conformal group?
• RQ5 What is the role of the anomaly scalar fields in generating observable signatures in the CMB beyond the power spectrum?

Key findings

• The cosmological constant becomes dynamical due to quantum trace anomalies, with the effective action including non-local terms that can be made local via two scalar fields φ and ψ.
• The CMB bispectrum is fully fixed by conformal invariance, with its form depending only on cross-ratios of momenta, not on scale or time.
• The trispectrum is constrained by conformal symmetry, with the 4-point function amplitude A₄ depending only on two cross-ratios, indicating universal structure independent of inflationary dynamics.
• Non-Gaussianities predicted by conformal invariance differ significantly from those in simple slow-roll inflation models, offering a distinct observational signature.
• The existence of special conformal transformations and dilatations in de Sitter space implies that correlation functions decompose into representations of the 3D conformal group, fixing their functional form.
• Fluctuations near the cosmological horizon in static coordinates may lead to a different form of the bispectrum, suggesting a second, horizon-localized realization of conformal invariance.