Zero modes in de Sitter background

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 S4 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 S4 corresponding to certain coherent fluctuations. When dS space is embedded in flat five dimensions E5, they may be seen as a legacy of translation of the center of the S4 sphere. Rigid translations of the S4-sphere on E5 leave the classical action invariant but are unobservable displacements from the point of view of gravitational dynamics on S4. 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 S4 but can be identified with certain non-isometric, conformal Killing forms that locally correspond to a rescaling of the volume element dV4. 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 H4 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.