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We focus on viable $f(T)$ teleparallel cosmological models, namely power law,exponential and square-root exponential, carrying out a detailed study of theirevolution at all scales. Indeed, these models were extensively analysed in thelight of late time measurements, while it is possible to find only upper limitslooking at the very early time behavior, i.e. satisfying the Big BangNucleosynthesis (BBN) data on primordial abundance of ${}^4He$. Starting fromthese indications, we perform our analysis considering both background andlinear perturbations evolution and constrain, beyond the standard sixcosmological parameters, the free parameters of $f(T)$ models in both caseswhether the BBN consistency relation is considered or not. We use a combinationof Cosmic Microwave Background, Baryon Acoustic Oscillation, Supernovae Ia andgalaxy clustering measurements, and find that very narrow constraints on thefree parameters of specific $f(T)$ cosmology can be obtained, beyond anyprevious precision. While no degeneration is found between the helium fraction,$Y_P$, and the free parameter of $f(T)$, we note that these models constrainthe current Hubble parameter, $H_0$, {higher extent than the standard model}one, fully compatible with the Riess et al. measurement in the case of powerlaw $f(T)$ model. Moreover, the free parameters are constrained at non-zerovalues in more than 3-$\sigma$, showing a preference of the observations forextended gravity models.

Updating constraints on f(T) teleparallel cosmology and the consistency with Big Bang Nucleosynthesis