Integrable Scalar Cosmologies II. Can they fit into Gauged Extended Supergavity or be encoded in N=1 superpotentials?

The question whether the integrable one-field cosmologies classified in a previous paper by Fre, Sagnotti and Sorin can be embedded as consistent one-field truncations into Extended Gauged Supergravity or in N=1 supergravity gauged by a superpotential without the use of D-terms is addressed in this paper. The answer is that such an embedding is very difficult and rare but not impossible. Indeed we were able to find two examples of integrable models embedded in Supergravity in this way. Both examples are fitted into N=1 Supergravity by means of a very specific and interesting choice of the superpotential W(z). The question whether there are examples of such an embedding in extended Gauged Supergravity remains open. In the present paper, relying on the embedding tensor formalism we classified all gaugings of the N=2 STU model, confirming, in the absence on hypermultiplets, the uniqueness of the stable de Sitter vacuum found several years ago by Fre, Trigiante and Van Proeyen and excluding the embedding of any integrable cosmological model. A detailed analysis of the space of exact solutions of the first Supergravity embedded integrable cosmological model revealed several new features worth an in depth consideration. When the scalar potential has an extremum at a negative value, the universe necessarily collapses into a Big Crunch notwithstanding its spatial flatness. The causal structure of these universes is quite different from that of the closed, positive curved, universe: indeed in this case the particle and event horizons do not coincide and develop complicated patterns. The cosmological consequences of this unexpected mechanism deserve careful consideration.