well motivated but also phenomenologically interesting. As we have mentioned in the Introduction, and as we will show in detail, an SD period with equation of state ${\mkern 1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern 1.5mu} > 1/3$ induces a large blue-tilt in the high frequency branch of the inflationary GW energy density spectrum [@Giovannini:1998bp], making this signal possibly observable with direct detection GW experiments. The aim of this paper is to quantify precisely our observational ability to measure such GW background, i.e. to determine the observable parameter space (we will also present a brief discussion on the implications for particle physics models). For our purposes, the details of the SD model implementation are unimportant. Hence, from now on we will focus on the phenomenology of SD assuming that for some unknown reason there is such a phase following the end of inflation. The background energy density of the inflationary sector evolves after inflation as [$\rho_{\rm tot} =\rho_*\exp\lbrace-3\int {da\over a} (w(a)+ 1)\rbrace$]{}, with $\rho_* = 3m_p^2H_{*}^2$ the initial energy density at the end of inflation. We will distinguish the Hubble rate at the end of inflation $H_*$ from that during inflation $H_{\rm inf}$ throughout the text, except in Section \[subsec:BBNandCMBbounds\] and \[subsec:TiltAndDof\] where we take $H_{\rm 