The effective field theory approach to the strong coupling issue in $f(T)$ gravity

We investigate the scalar perturbations and the possible strong coupling issues of $f(T)$ around a cosmological background, applying the effective field theory (EFT) approach. We revisit the generalized EFT framework of modified teleparallel gravity and apply it by considering both linear and second-order perturbations for $f(T)$ theory. No new scalar mode is present in linear and second-order perturbations in $f(T)$ gravity, which suggests a strong coupling problem. However, based on the ratio of cubic to quadratic Lagrangians, we provide a simple estimation of the strong coupling scale, a result which shows that the strong coupling problem can be avoided at least for some modes. In conclusion, perturbation behaviors that at first appear problematic may not inevitably lead to a strong coupling problem, as long as the relevant scale is comparable with the cutoff scale $M$ of the applicability of the theory.