A simple example of "non-minimal" Pre-Big Bang scenario

We give an example of non-minimal pre-big bang scenario able to produce the PTA signal considering a modified evolution of the high-curvature string phase, including the contribution of high-energy string sources. We use a fluid-dinamical model of sources and show that their effective viscosity breaks the $S$-duality symmetry of the tensor-axion perturbation spectra, as in general expected for the non-minimal scenario.