is different in the two cases, the GWs that have entered the horizon during SD will evolve differently. In turn, this means that the GW energy spectrum for the modes that re-entered the horizon before and around the SD-to-RD transition will differ in the two cases. The GW energy spectrum can be computed fully analytically in the instant transition case. However, in the smooth transition case we can only compute analytically the asymptotic high and low frequency branches of the GW spectrum, corresponding to the modes that entered the horizon deep inside SD and RD, respectively. From an observational point of view, this is not a problem, as only the high frequency branch of the spectrum can be potentially probed by GW detectors. For completeness, in any case, we will provide a numerical computation of the full GW spectrum in the smooth transition case.

Instant transition
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We consider in this subsection the expansion history shown in Figure \[fig:setup\], assuming the transition from the SD to RD epoch occurs instantaneously, that is, in a time much shorter than the instantaneous Hubble time at the moment of transition. We consider the scale factor and the Hubble rate around the transition as 