spectrum is (quasi-)scale invariant, with the same tilt as the original super-horizon inflationary tensor power spectrum. For periods of expansion when the energy budget of the Universe is not dominated by relativistic species, the (quasi-)scale invariance is broken, and the resulting GW spectrum becomes significantly tilted in the frequency range corresponding to the modes crossing the horizon during such period(s). The expansion history of the Universe is imprinted in this way in the power spectrum of the freely-lingering primordial GW background that we can observe today. In other words, we might detect or constrain the post-inflation expansion history by attempting to detect the relic GWs of inflationary origin, see e.g. [@Giovannini:1998bp; @Giovannini:1999bh; @Boyle:2005se; @Watanabe:2006qe; @Boyle:2007zx; @Kuroyanagi:2008ye; @Kuroyanagi:2010mm; @Kuroyanagi:2014qza]. In turn, the expansion history will inform us about the matter fields driving the expansion.

To characterize the post-inflation pre-BBN expansion history, we consider an intermediate epoch taking place between the end of inflation and the onset of RD, with an EoS parameter $w \neq 1/3$. In scenarios where the inflaton oscillates around the minimum of its potential after inflation, an effective EoS (averaged over inflaton oscillations) emerges [@Turner:1983he; @Lozanov:2016hid], lying within the range $0 < w < 1/3$ for $V(\phi) \propto 