inf}\simeq H_*$ in deriving the numerical results. In general the EoS is determined by the inflaton potential and is a function of time. However we expect it to change only adiabatically during SD, and in any case we can always describe the scaling of the energy density in terms of an effective (logarithmic-averaged) value of the EoS during the stiff period, $3\int {da\over a}(1+w(a)) \equiv 3(1+{\mkern 1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern 1.5mu})\log (a/a_*)$, so that [$\rho_{\rm tot} = \rho_*(a/a_*)^{-3({\mkern 1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern 1.5mu}+ 1)}$]{}.

Let us consider therefore an SD period between the end of inflation and the onset of RD, with effective EoS ${\mkern 1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern 1.5mu} \equiv w_{\rm s} > 1/3$ deep inside the SD (i.e. way before reaching RD). Once the Universe enters into RD, we match the expansion history with that of a Universe with energy budget dominated by the SM radiation $dof$, according to the standard hot Big Bang picture. We sketch the different epochs of the expansion history we consider in Figure \[fig:setup\]. From now on, the subscripts $_{\text{*}}$, $_{\text{RD}}$, and $_{\text{BBN}}$, stand for “evaluation at" or “corresponding to" the end of inflation/beginning of SD epoch, end of SD epoch/beginning of RD period, and onset of BBN, respectively. In order to 