Viable<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>T</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>models are practically indistinguishable from<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>Λ</mml:mi><mml:mi>CDM</mml:mi></mml:math>

We investigate the cosmological predictions of several $f(T)$ models, with up to two parameters, at both the background and the perturbation levels. Using current cosmological observations (geometric supernovae type Ia, cosmic microwave background and baryonic acoustic oscillation and dynamical growth data) we impose constraints on the distortion parameter, which quantifies the deviation of these models from the concordance $\ensuremath{\Lambda}$ cosmology at the background level. In addition we constrain the growth index $\ensuremath{\gamma}$ predicted in the context of these models using the latest perturbation growth data in the context of three parametrizations for $\ensuremath{\gamma}$. The evolution of the best fit effective Newton constant, which incorporates the $f(T)$-gravity effects, is also obtained along with the corresponding $1\ensuremath{\sigma}$ error regions. We show that all the viable parameter sectors of the $f(T)$ gravity models considered practically reduce these models to $\ensuremath{\Lambda}\mathrm{CDM}$. Thus, the degrees of freedom that open up to $\ensuremath{\Lambda}\mathrm{CDM}$ in the context of $f(T)$ gravity models are not utilized by the cosmological data leading to an overall disfavor of these models.

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