Effective Equation of State Oscillations at Matter-Radiation Equality and Primordial Gravitational Waves

The theory controlling the Universe's evolution in the classical regime has to be motivated by particle physics reasoning and should also generate inflation and dark energy eras in a unified way. One such framework is $F(R)$ gravity. In this work we examine a class of exponential deformations of $R^2$ gravity motivated by fundamental physics of scalaron evolution in a de Sitter background. As we show this class of models describe both inflation and the dark energy era in a viable way compatible with the Planck constraints on inflation and the cosmological parameters. Regarding the inflationary era, the exponentially deformed $R^2$ model also yields a rescaled Einstein-Hilbert term which remarkably does not affect the dynamics and the inflationary evolution is identical to that of an $R^2$ model. The dark energy era is also found to be viable and mimics the $\Lambda$-Cold-Dark-Matter model. More importantly, this class of $F(R)$ gravity exponential $R^2$ deformations also has an important characteristic, and specifically it yields total equation of state oscillations deeply in the matter domination era, for redshifts $z\sim 3400$, so near the matter-radiation equality. These total equation of state deformations at such a large redshift may directly affect the energy spectrum of the primordial gravitational waves. Indeed as we show, the effect is measurable and it leads to an enhancement of the tensor perturbations energy spectrum for low frequencies probed by the future LiteBIRD mission. This enhancement might have a measurable effect on the $B$-modes of the Cosmic Microwave Background radiation and thus may be detectable by the LiteBIRD mission. Only a handful of theoretical frameworks can generate the gravitational wave pattern generated by the class of exponentially deformed $R^2$ models we presented.