  In string theory, the traditional picture of a Universe that emerges from the
inflation of a very small and highly curved space-time patch is a possibility,
not a necessity: quite different initial conditions are possible, and not
necessarily unlikely. In particular, the duality symmetries of string theory
suggest scenarios in which the Universe starts inflating from an initial state
characterized by very small curvature and interactions. Such a state, being
gravitationally unstable, will evolve towards higher curvature and coupling,
until string-size effects and loop corrections make the Universe "bounce" into
a standard, decreasing-curvature regime. In such a context, the hot big bang of
conventional cosmology is replaced by a "hot big bounce" in which the bouncing
and heating mechanisms originate from the quantum production of particles in
the high-curvature, large-coupling pre-bounce phase. Here we briefly summarize
the main features of this inflationary scenario, proposed a quarter century
ago. In its simplest version (where it represents an alternative and not a
complement to standard slow-roll inflation) it can produce a viable spectrum of
density perturbations, together with a tensor component characterized by a
"blue" spectral index with a peak in the GHz frequency range. That means,
phenomenologically, a very small contribution to a primordial B-mode in the CMB
polarization, and the possibility of a large enough stochastic background of
gravitational waves to be measurable by present or future gravitational wave
detectors.
