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

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