Post-Inflationary Constraints on Nonminimally Coupled Quintessential Inflation

We investigate quintessential inflation in a nonminimally coupled scalar-tensor theory, parameterizing the post-inflationary radiation abundance independently of the reheating mechanism. The nonadiabatic inflation-kination transition generates a stochastic gravitational-wave background whose contribution to $\Delta N_{\textrm{eff}}$ imposes a lower limit on the reheating temperature. Because this temperature dictates the duration of kination and the available scalar-field excursion, it directly constrains the present-day dark-energy equation of state. While a single-exponential coupling achieves the required post-inflationary potential drop, the same constant slope does not provide viable late-time acceleration. A double-exponential deformation resolves this tension by decoupling the average slope governing the total potential drop from the asymptotic slope driving cosmic acceleration. Full numerical solutions confirm this picture, yielding a thawing quintessence regime with $w_{\varphi,0}\simeq (-0.90, -0.95)$ for our benchmarks. Our results demonstrate that future dark-energy measurements can directly probe the post-inflationary reheating history of the Universe.