In Section \[sec:GWspectrum\], we derive the energy density spectrum of the inflationary GW background in the presence of an SD epoch, considering two possible modelings for the SD-to-RD transition: an ’instantaneous’ transition with a sharp jump in the EoS, and a ’smooth’ transition modeled by the evolution of two fluid components, one made of radiation and another by a scalar field dominated by its own kinetic energy. In Section \[sec:Results\], we first quantify the full parameter space $\lbrace w, H_{\rm inf}, f_{\rm RD}\rbrace$ that can be probed by LISA and aLIGO. We then restrict such parameter space to be consistent with upper bounds on stochastic GW backgrounds from BBN and CMB considerations. We also extend our analysis to include possible changes in the number of relativistic degrees of freedom ($dof$), and the presence of a small red tensor tilt, as motivated in slow-roll inflation. In Section \[sec:Discussion\], we summarize our findings and discuss briefly the implications for some early Universe scenarios resting upon the presence of an SD epoch.

From now on, [$m_p = {1/\sqrt{8\pi G}} \simeq 2.44\cdot10^{18}$ GeV]{} is the reduced Planck mass, [$a(\tau)$]{} is the scale factor, [$\tau \equiv \int {dt\over a(t)}$]{} is the conformal time, and we 