Integrable Scalar Cosmologies I. Foundations and links with String Theory

We build a number of integrable one-scalar spatially flat cosmologies, which play a natural role in inflationary scenarios, examine their behavior in several cases and draw from them some general lessons on this type of systems, whose potentials involve combinations of exponential functions, and on similar non-integrable ones. These include the impossibility for the scalar to emerge from the initial singularity descending along asymptotically exponential potentials with logarithmic slopes exceeding a critical value (“climbing phenomenon”) and the inevitable collapse in a Big Crunch whenever the scalar tries to settle at negative extrema of the potential. We also elaborate on the links between these types of potentials and “brane supersymmetry breaking”, a mechanism that ties together string scale and scale of supersymmetry breaking in a class of orientifold models.