Two wave functions and dS/CFT on S1 × S2

We evaluate the tunneling and Hartle-Hawking wave functions on S1 × S2 boundaries in Einstein gravity with a positive cosmological constant. In the large overall volume limit the classical predictions of both wave functions include an ensemble of Schwarzschild-de Sitter black holes. We show that the Hartle-Hawking tree level measure on the classical ensemble converges in the small S1 limit. A divergence in this regime can be identified in the tunneling state. However we trace this to the contribution of an unphysical branch of saddle points associated with negative mass black holes. Using a representation in which all saddle points have an interior Euclidean anti-de Sitter region we also derive a holographic form of both semiclassical wave functions on S1 × S2 boundaries.