  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 $\bf5$ 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 with certain non-isometric, conformal Killing forms
that locally correspond to a rescaling of the volume element $dV_4.$
  For convenience, we frame our discussion in the context of renormalizable
gravity, but the conclusions apply equally to the corresponding zero modes in
Einstein gravity. 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.
