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We investigate the inflationary consequences of the oscillating dark energymodel proposed by Ti\'an[\href{https://doi.org/10.1103/PhysRevD.101.063531}{Phys. Rev. D {\bf 101},063531 (2020)}], which aims to solve the cosmological coincidence problem withmulti-accelerating Universe (MAU). We point out that the inflationary dynamicsbelong to slow-roll inflation. The spectral index of scalar perturbations andthe tensor-to-scalar ratio $r$ are shown to be consistent with current\textit{Planck} measurements. Especially, this model predicts $r\sim10^{-7}$,which is far below the observation limits. This result motivates us to explorethe smallness of $r$ in the general MAU. We propose a quintessentialgeneralization of the original model and prove $r<0.01$ in general. The nulldetection to date of primordial gravitational waves provides a circumstantialevidence for the MAU. After the end of inflation, the scalar field rolls towardinfinity instead of a local minimum, and meanwhile its equation of state isoscillating with an average value larger than $1/3$. In this framework, we showthat gravitational particle creation at the end of inflation is capable ofreheating the Universe.

physical review. d/physical review. d.

Cosmological consequences of a scalar field with oscillating equation of  state. III. Unifying inflation with dark energy and small tensor-to-scalar  ratio