Can torsion mimic a cosmological constant? A one-parameter logarithmic $f(T)$ extension of $Λ$CDM

We study a modified teleparallel gravity model of the $f(T)$ type. We propose a new functional form for in order to obtain a geometric origin to the $ΛCDM$ behavior in an appropriate limit. A perturbative expansion around the $ΛCDM$ regime is carried out, and the model's cosmic dynamics are calculated inside the flat FLRW background. By employing a Markov Chain Monte Carlo (MCMC) analysis using multiple observational datasets, such as Cosmic Chronometers, Type Ia Supernova compilations (DES-SN5YR, Pantheon+SH0ES, and Union 3.0), DESI DR2 BAO measurements, and Planck 2018 CMB distance priors, to constrain the model parameters. The findings show that the suggested $f(T)$ model accurately replicates the observed accelerated expansion and is statistically consistent with current cosmological observations. The model stays close to the $ΛCDM$ scenario while permitting tiny dynamical adjustments, as evidenced by the extra parameter $γ$, which quantifies deviations from the conventional cosmological model, being limited to values near zero.