Abstract: In the context of inflationary cosmology a natural question arises on  what precedes inflation itself. We review aspects of the Hartle-Hawking (no boundary) and Vilenkin (tunneling) proposals, and their relation to the Wheeler-DeWitt equation. These raise some paradoxes that are  complementary in nature and in clash with observations. On the other hand theories of quantum gravity are better defined in the presence of a negative cosmological constant. I will propose a new type of wavefunction of the universe with asymptotically AdS boundary conditions in the far (Euclidean) past. In the semiclassical limit, it describes a Euclidean (half)-wormhole geometry with properties that result in an expanding universe upon analytic continuation to Lorentzian signature. In this context some of the aforementioned issues can be resolved.