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.
