Zero modes in de Sitter background

• Main Authors: Martin B. Einhorn, D. R. Timothy Jones
• Format: Article
• Language: English
• Published: SpringerOpen 2017-03-01
• Series: Journal of High Energy Physics
• Subjects: Space-Time Symmetries; Differential and Algebraic Geometry; Models of Quantum Gravity
• Online Access: http://link.springer.com/article/10.1007/JHEP03(2017)144

Abstract There are five well-known zero modes among the fluctuations of the metric of de Sitter (dS) spacetime. For Euclidean signature, they can be associated with certain spherical harmonics on the S 4 sphere, viz., the vector representation 5 of the global SO(5) isometry. They appear, for example, in the perturbative calculation of the on-shell effective action of dS space, as well as in models containing matter fields. These modes are shown to be associated with collective modes of S 4 corresponding to certain coherent fluctuations. When dS space is embedded in flat five dimensions E 5, they may be seen as a legacy of translation of the center of the S 4 sphere. Rigid translations of the S 4-sphere on E 5 leave the classical action invariant but are unobservable displacements from the point of view of gravitational dynamics on S 4. Thus, unlike similar moduli, the center of the sphere is not promoted to a dynamical degree of freedom. As a result, these zero modes do not signify the possibility of physically realizable fluctuations or flat directions for the metric of dS space. They are not associated with Killing vectors on S 4 but can be identified with certain non-isometric, conformal Killing forms that locally correspond to a rescaling of the volume element dV 4. We frame much of our discussion in the context of renormalizable gravity, but, to the extent that they only depend upon the global symmetry of the background, the conclusions should apply equally to the corresponding zero modes found in Einstein gravity. Although their existence has only been demonstrated at one-loop, we expect that these zero modes will be present to all orders in perturbation theory. They will occur for Lorentzian signature as well, so long as the hyperboloid H 4 is locally stable, but there remain certain infrared issues that need to be clarified. We conjecture that they will appear in any gravitational theory having dS background as a locally stable solution of the effective action, regardless of whether additional matter is included.