not sabotage the success of BBN, a minimal requisite that we need to impose over the assumed expansion history is that the stiff epoch must end before the beginning of BBN, i.e. $\tau_{\text{RD}}<\tau_{\text{BBN}}$.

![$\Lambda$CDM+inflation expansion history with a stiff epoch.[]{data-label="fig:setup"}](setup.pdf){width="15cm"}

In this section we will solve for the aforementioned cosmological scenario, propagate the solution to the present era when it can be detected, and express the result in terms of the GW energy density spectrum today . Our focus is mainly on the modes that re-enter the horizon during the stiff phase (SD-reentering modes), as they constitute the part of the GW spectrum that is enhanced and hence potentially detectable. The expression for the present-day GW energy spectrum depends on the assumption on how the transition from the SD to RD era takes place. There are two natural cases to consider:

1.  **Instant transition.** Here the transition from SD to RD is modeled as ’instantaneous’, i.e. it occurs in a very short time interval compared to the Hubble timescale at the moment of the transition. The scale factor $a(\tau)$ and Hubble rate $H(\tau)$ are continuous during the transition, but we consider a sudden jump in the effective EoS from ${\mkern 