Parametrization of Hubble parameter in ((f)over cap)(T, TG) gravity model with tachyon field

We look into the cosmological uses of the new modified gravitational theory (f) over cap (T, TG), which is based on the torsion invariant T and the teleparallel equivalent of the GaussBonnet term T G. The corresponding cosmology of <^> f(T, T G) gravity is very intriguing because it varies from both f(T) theories and the f(R, G) class of curvature-modified gravity. In this paper, we have investigated the applicability of an alternative gravitational theory called (f) over cap (T, T G) by applying the energy conditions. Here, we assumed the form of (f) over cap (T, TG) = -T + alpha ln [ss(1)(T (2) + ss(2) TG)(n) ], where ss 1 and ss 2 are arbitrary constants. We have examined the cosmological applications of (f) over cap (T, T G) gravity. Here, we consider two particular expressions of the time- dependent Hubble parameter. Both the model parameters are constrained jointly by the Observation Hubble dataset (OHD), the Pantheon dataset and the BAO data to constrain the model parameters using the Markov Chain Monte Carlo (MCMC) minimization technique. We determine the present values of Hubble constant for model zeta: H (0) = 70(+0.34) (- 0.34),zeta = 1.2(+0.016) (- 0.016) and H (0) = 1.8(+0.013) (-0.013); and model A(2): H (0) = 72(+0.41) (-0.41), zeta = 1.5+(0.00767) (-0.007) and epsilon= 1.3(+0.012) (-0.012) for the OHD+BAO+Pantheon dataset. Using the most recent estimated values of Hubble parameter, we properly analyzed the energy conditions and looked at the final behavior of our model. Additionally, we have explored the omega D -omega ' D plane and also we have extracted the scalar field as well as potential of (f) over cap (T, TG) gravity and its relationship to the tachyon field. In addition, we find that, depending on the model parameters, the dark energy (DE) equation-of-state (EoS) parameter may be quintessence or phantomlike, or it may undergo phantom-divide crossing.Finally, we find that all the essential features of the model are satisfied successfully.