Phenomenological reconstruction of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>f</mml:mi><mml:mo stretchy="false" mathvariant="bold">(</mml:mo><mml:mi>T</mml:mi><mml:mo stretchy="false" mathvariant="bold">)</mml:mo></mml:math> teleparallel gravity

We present a novel reconstruction method of $f(T)$ teleparallel gravity from phenomenological parametrizations of the deceleration parameter or other alternatives. This can be used as a toolkit to produce viable modified gravity scenarios directly related to cosmological observations. We test two parametrizations of the deceleration parameter considered in recent literatures in addition to one parametrization of the effective (total) equation of state (EoS) of the universe. We use the asymptotic behavior of the matter density parameter as an extra constraint to identify the viable range of the model parameters. One of the tested models shows how tiny modification can produce viable cosmic scenarios quantitatively similar to $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ but qualitatively different whereas the dark energy (DE) sector becomes dynamical and fully explained by modified gravity not by a cosmological constant.

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