Energy Constraints and Phenomenon of Cosmic Evolution in $f(T,B)$ Framework

We investigate the cosmological evolution in a new modified teleparallel theory, called $f(T,B)$ gravity, which is formulated by connecting both $f(T)$ and $f(R)$ theories with a boundary term $B$. Here, $T$ is the torsion scalar in teleparallel gravity and $R$ is the scalar curvature. For this purpose, we assume flat Friedmann-Robertson-Walker (FRW) geometry filled with perfect fluid matter contents. We study two cases in this gravity: One is for a general function of $f(T,B)$, and the other is for a particular form of it given by the term of $-T+F(B)$. We also formulate the general energy constraints for these cases. Furthermore, we explore the validity of the bounds on the energy conditions by specifying different forms of$f(T,B)$ and $F(B)$ function obtained by the reconstruction scheme for de Sitter, power-law, the $\Lambda$CDM and Phantom cosmological models. Moreover, the possible constraints on the free model parameters are examined with the help of region graphs. In addition, we explore the evolution of the effective equation of state (EoS) $\omega_{eff}$ for the universe and compare theoretical results with the observational data. It is found that the effective EoS represents the phantom phase or the quintessence one in the accelerating universe in all of the cases consistent with the observational data.