Conformal vacua and entropy in de Sitter space

The de Sitter/conformal field theory (dS/CFT) correspondence is illuminated through an analysis of massive scalar field theory in d-dimensional de Sitter space. We consider a one-parameter family of dS-invariant vacua related by Bogolyubov transformations and compute the corresponding Green functions. It is shown that none of these Green functions correspond to the one obtained by analytic continuation from AdS. Among this family of vacua are in (out) vacua which have no incoming (outgoing) particles on ${\mathcal{I}}^{-}$ $\left({\mathcal{I}}^{+}\right).\mathrm{}$ Surprisingly, it is shown that in odd spacetime dimensions the in and out vacua are the same, implying the absence of particle production for this state. The correlators of the boundary CFT, as defined by the dS/CFT correspondence, are shown to depend on the choice of vacuum state—the correlators with all points on ${\mathcal{I}}^{-}$ vanish in the in vacuum. For ${\mathrm{dS}}_3$ we argue that this bulk vacuum dependence of the correlators is dual to a deformation of the boundary ${\mathrm{CFT}}_2$ by a specific marginal operator. It is also shown that Witten’s nonstandard de Sitter inner product (slightly modified) reduces to the standard inner product of the boundary field theory. Next we consider a scalar field in the Kerr-${\mathrm{dS}}_3$ Euclidean vacuum. A density matrix is constructed by tracing out over modes which are causally inaccessible to a single geodesic observer. This is shown to be a thermal state at the Kerr-${\mathrm{dS}}_3$ temperature and angular potential. It is further shown that, assuming Cardy’s formula, the microscopic entropy of such a thermal state in the boundary CFT precisely equals the Bekenstein-Hawking value of one-quarter the area of the Kerr-${\mathrm{dS}}_3$ horizon.