In this paper, using the first law of thermodynamics (FLT), we derive a modification of the Friedmann equation from the nonadditive Tsallis entropy associated with the apparent horizon of the Friedmann–Robertson–Walker (FRW) Universe, where matter inside the apparent horizon is represented by a scalar field with a potential. We suggest that the inflationary era of the universe may phenomenologically consist of the two phases. We determined that the first phase is an inflate under the slow-roll condition known as exponential accelerated expansion, and the second one is an expansion phase containing the kinetic term of the scalar field, leading to a quintessential era. After expressing the observable inflation indices for both phases in terms of e-foldings number with choosing power-law scalar potential, we have compared the results with the latest Planck observation data. With some restrictions on the nonadditive Tsallis parameter and power-term of the potential, we observe that inflation can occur with the two phases. It is determined that while the nonadditive Tsallis parameter is positive in the slow-roll phase, it takes negative values in the kinetic energy dominating phase. Nonetheless, for the apparent horizon of the universe, an increasing entropy exists in the first phase, while a decrease in entropy is observed for the second phase, which indicates that the universe is ready for the creation of matter. In addition, a correlation between slow-roll and kinetic inflation is shown by determining the kinetic energy-potential relationship via the exponential term of the potential.

There are various mechanisms that explain both the inflationary epoch of the early universe and a unification of this epoch with the other stages of the universe. In this study, we show all the expansion history of the universe and transition among of them in a single form by using the theoretical framework of F (T , T G ) gravity in the context of the FRW (Friedmann-Robertson-Walker) universe. According to a particular model we obtain the unified solutions of the field equations. Without using any scalar field description we especially present the super inflation mechanism composed of three phase regions which describes the evolution of the early universe. The mechanism begins with a vacuum state and then follows a super accelerated period where there are two regions. The first continues in a quintessential field, and the second is a region where the radiation is created. Furthermore, we verified this inflationary mechanism by using the spectral index parameter and the scalar tensor ratio, i.e., n s , r, and calculated the ratio of radiation emergent from the quintessence field. This creation should be in a certain rate in the early universe otherwise we show that the universe cannot survive and continue to expand. Also, we have obtained a phantom solution of the model that shows two regions which are compatible with the recent cosmological observations. In one respect, it is observed that the late time expansion of the universe is similar to the early time inflation.

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The purpose of this study is to explain the super inflation scenario for the early universe with scalar field which is minimally coupled with theory of [Formula: see text] gravity, where [Formula: see text] is the Ricci curvature scalar and [Formula: see text] is the trace of the energy–momentum tensor (EMT). We consider the model [Formula: see text], where the curvature is minimally coupled with matter. All the modification terms are properly compacted in [Formula: see text], by means of a unified solution. After a vacuum inflation, it is observed that the scalar field oscillations via the power-term in potential [Formula: see text] naturally produce the quintessential type of dark energy. Then, we observe that the matter is created by this type of dark energy during inflation. Also, the spectral index parameter and the tensor-to-scalar ratio are examined for this inflation mechanism.

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