Bridging between reheating and late-time observations in quintessential inflation

We propose an idea to build a bridge between reheating and late-time observations in quintessential inflation by backtracking the evolution of the inflaton field from the present time to the end of reheating. This idea is implemented when the potential gradient is negligible compared to the Hubble friction, rendering the inflaton field frozen, till the present time. We find a simple analytic relation between the reheating temperature and the observational parameters for dark energy, and numerically confirm its validity for typical models of quintessential inflation. This relation is universal and can apply to all quintessential inflation models with any reheating mechanism. It also implies that any quintessential inflation model with a successful reheating with the reheating temperature $1\textrm{MeV}\lesssim T_\textrm{re}\lesssim 10^{15}\textrm{GeV}$ predicts the equation of state of dark energy today extremely close to $-1$, i.e. $-1+10^{-60}\lesssim w_0\lesssim -1+10^{-24}$, unless the inflaton field unfreezes before the present time.