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.
