with EoS $w \neq 1/3$, before RD is established, the resulting GW energy density spectrum today will no longer remain scale-invariant. As we will see next, the spectrum today will actually consist of two parts: a high-frequency branch, corresponding to the modes that crossed the horizon during the transient epoch, and a (quasi-)scale invariant branch corresponding to the modes that crossed the horizon during RD[^1].

Inflationary spectrum in the presence of a stiff epoch {#sec:GWspectrum}
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Let us consider now that there is a period in the early Universe, spanning from the end of inflation till the onset of RD, with EoS $w \neq 1/3$ (possibly depending on time). In standard single field slow-roll scenarios the inflaton exhibits a minimum in the potential around which it oscillates in the period following inflation. For an inflaton potential of the form $V \propto \phi^{2n}$, an effective (oscillation averaged) EoS emerges as ${\mkern 1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern 1.5mu} \simeq (n-1)/(n+1)$ [@Turner:1983he]. For $n = 2$ ($V \propto \phi^{4}$) we obtain a RD period with ${\mkern 1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern 1.5mu} \simeq 1/3$, while for $n = 1$ ($V \propto \phi^2$) we obtain instead a matter dominated (MD) era with ${\mkern 1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern 1.5mu} \simeq 0$. Interestingly, for $n \geq 3$ 