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In this work, we shall incorporate a stiff era in the Universe's evolution inthe context of $F(R)$ gravity. After deriving the vacuum $F(R)$ gravity, whichmay realize a stiff evolution, we combine the stiff $F(R)$ gravity with an$R^2$ model, and we construct a qualitative model for the inflationary andstiff era, with the latter commencing after the end of the inflationary era. Weassume that the baryogenesis occurs during the stiff era, and we calculate thebaryon to entropy ratio, which effectively constraints the functional form ofthe stiff $F(R)$ gravity. Further constraints on the stiff $F(R)$ gravity maycome from the primordial gravitational waves, and particularly their scalarmode, which is characteristic of the $F(R)$ gravity theory. The stiff erapresence does not contradict the standard cosmology era, namely, inflation, andthe radiation-matter domination eras. Furthermore, we investigate which $F(R)$gravity may realize a dust and stiff matter dominated Einstein-Hilbertevolution.

The Early-time Cosmology with Stiff Era from Modified Gravity