The effective field theory approach of teleparallel gravity, $f(T)$
  gravity and beyond
  We develop the effective field theory approach to torsional modified
gravities, a formalism that allows for the systematic investigation of the
background and perturbation levels separately. Starting from the usual
effective field theory approach to curvature-based gravity, we suitably
generalize it at the background level by including terms of the contracted
torsion tensor, and at the perturbation level by including pure torsion
perturbative terms and mixed perturbative terms of torsion and curvature.
Having constructed the effective field theory action of general torsional
modified gravity, amongst others we focus on $f(T)$ gravity and we perform a
cosmological application. We investigate the scalar perturbations up to second
order, and we derive the expressions of the Newtonian constant and the post
Newtonian parameter $\gamma$. Finally, we apply this procedure to two specific
and viable $f(T)$ models, namely the power-law and the exponential ones,
introducing a new parameter that quantifies the deviation from general
relativity and depends on the model parameters. Since this parameter can be
expressed in terms of the scalar perturbation mode, a precise measurement of
its evolution could be used as an alternative way to impose constraints on
$f(T)$ gravity and break possible degeneracies between different $f(T)$ models.
