Coexistence of early-time inflation and late-time acceleration with scalar fields in theory of F(G, T) gravity

In this study, we discuss coexistence of the early-time inflation and the late-time acceleration of the universe in the context of the theory of F(G, T) gravity with scalar field which is minimally coupled with the gravity, where G is the gauss bonnet invariant and T is the trace of energy-momentum tensor (EMT). We reconstruct the Friedmann equation (FE) and then search for the real value of a particular model F(G, T) = f(G)+beta T-n, where beta and n are real constants. A Gauss-Bonnet system (GBS) for viable cosmologies arising from the matter-source term beta T-n and the scalar field, is obtained. We find that the case n < 0 together with f(G) similar to Gn in the system gives the late-time cosmic acceleration while the source term beta T-n acts as a quintessence type of dark energy. On the other hand, the general entropy expression of the universe is obtained by making use of the first law of thermodynamics (FLT) method. After theoretically analyzing the inflation in the entropy frame, we find a new condition 0 < sigma < 1/2 with f(G) similar to G(sigma) in the system. Then, from the observational analysis of inflation, the spectral index parameter and the scalar-tensor ratio are calculated under the new condition. In brief, we obtain a viable cosmological system arising from some modifications such as the scalar field and the source term, which can unify the early inflation and the late-time cosmic acceleration besides the deceleration regions of the universe.