Effective gravitational coupling in modified teleparallel theories

In the present study, we consider an extended form of teleparallel Lagrangian $f(T,\phi,X)$, as function of a scalar field $\phi$, its kinetic term $X$ and the torsion scalar $T$. We use linear perturbations to obtain the equation of matter density perturbations on sub-Hubble scales. The gravitational coupling is modified in scalar modes with respect to the one of General Relativity, albeit vector modes decay and do not show any significant effects. We thus extend these results by involving multiple scalar field models. Further, we study conformal transformations in teleparallel gravity and we obtain the coupling as the scalar field is non-minimally coupled to both torsion and boundary terms. Finally, we propose the specific model $f(T,\phi,X)=T + \partial_\mu \phi\ \partial^\mu \phi +\xi T \phi^2$. To check its goodness, we employ the observational Hubble data, constraining the coupling constant, $\xi$, through a Monte Carlo technique based on the Metropolis-Hastings algorithm. Hence, fixing $\xi$ to its best-fit value got from our numerical analysis, we calculate the growth rate of matter perturbations and we compare our outcomes with the latest measurements and the predictions of the $\Lambda$CDM model.