continuous smooth functions, but we model the EoS as a discrete function $$\begin{aligned}
 \bar w = w_{\rm s}\Theta(\tau_{\rm RD}-\tau) + {1\over3}\Theta(\tau-\tau_{\rm RD})\,,\end{aligned}$$ where $\Theta(x)$ is the step-function. The energy density of the background is therefore continuous, and scales as $$\begin{aligned}
\rho_{\rm tot} = \left\lbrace 
\begin{array}{ll}
 \rho_*(a/a_*)^{-3(1+w_{\rm s})} & ,\, \tau \leq \tau_{\rm RD}~ ({\rm Stiff~Domination})\vspace*{4mm}\\
\rho_{\rm RD}(a/a_{\rm RD})^{-4} & ,\, \tau \geq \tau_{\rm RD} ~({\rm Radiation~Domination})
\end{array}
\right .\end{aligned}$$ where $\rho_{\rm RD} \equiv \rho_*(a_{\rm RD}/a_*)^{-3(1+w_{\rm s})}$. From here the scale factor during each period can then be solved exactly as $$\begin{aligned}
a(\tau) &=& a_{*}\left[1+{a_*H_*\over \alpha_{\rm s}}(\tau - \tau_*)\right]^{\alpha_{\rm s}} = a_{*}\left({a_*H_*\over \alpha_{\rm s}}\right)^{\alpha_{\rm s}}[{\tilde\tau}_s(\tau)]^{\alpha_{\rm s}}\,,~~~~ 
\tau_{*}\leq\tau\leq\tau_{\text{RD}} \label{astiff}\\
a(\tau) &=& a_{\rm RD}\left[1+a_{\rm RD}H_{\rm RD}(\tau - \tau_{\rm RD})\right] = a_{\text{RD}}^2H_{\text{RD}}{\tilde\tau}_r(\tau)\,,~~~~ 
\tau_{\text{RD}}\leq \tau \ll\tau_{\text{eq}} \label{arad}    \end{aligned}$$ and, correspondingly, the Hubble rate $aH(\tau) \equiv {d\log a\over d\tau}$ as $$\begin{aligned}
aH(\tau) &=& {a_*H_*\over 1+{a_*H_*\over \alpha_{\rm s}}(\tau-\tau_*)} = {\alpha_{\rm s}\over \tilde{\tau}_s(\tau)}\,,~~~~ \tau_{*}\leq\tau\leq\tau_{\text{RD}}\label{eq:Hstiff}\\
aH(\tau) &=& {a_{\rm RD}H_{\rm RD}\over 1+a_{\rm RD}H_{\rm RD}(\tau-\tau_{\rm RD})} =  {1\over \tilde{\tau}_r(\tau)}\,,~~~~ \tau_{\rm RD}\leq\tau\ll \tau_{\text{eq}}\,,\label{eq:Hrad}\end{aligned}$$ where $\tau_{\text{eq}}$ denotes the time at the radiation-matter equality, and we have defined $$\alpha_{\rm s} \equiv \frac{2}{1+3w_{\rm s}}\,.$$ For convenience, we have introduced the time variables $\tilde{\tau}_{\rm s}$ and $\tilde{\tau}_{\rm r}$ via $$\begin{aligned}
{a_*H_*\over \alpha_{\rm s}}\tilde{\tau}_s(\tau) \equiv 1+{a_*H_*\over \alpha_{\rm 