\phi^p$, $p < 4$. There is however $a~priori$ no reason, neither theoretical nor observational, to exclude a [*stiff*]{} case [$1/3 < w \leq 1$]{}. A period with a stiff EoS can actually be achieved naturally in a universe dominated by the kinetic energy of a scalar field after inflation, either through inflaton oscillations [@Turner:1983he] under a steep potential (e.g. $V(\phi) \propto \phi^p$ with $p > 4$), or simply by an abrupt drop of the potential for large values of the inflaton. In the latter case it is particularly easy and natural to obtain an EoS close to unity $w\simeq 1$. We note that $w \leq 1$ emerges as a natural upper bound from the requirement that the sound speed $c_{\text{s}}^2=\partial p/\partial \rho \equiv w \leq 1$ of a fluid does not exceed the speed of light. Nevertheless, we expect that in general field theory constructions $w$ can approach unity only from below, so we will consider only $w < 1$. Scenarios with a stiff dominated (SD) epoch between inflation and RD are particularly appealing from observational point of view: the stiff period induces a blue-tilt in the GW energy spectrum at large frequencies [@Giovannini:1998bp; @Giovannini:1999bh; @Riazuelo:2000fc; @Sahni:2001qp; @Tashiro:2003qp; @Boyle:2007zx; @Giovannini:2008zg; 