Redshift drift constraints on f(T) gravity

We explore the impact of the Sandage-Loeb (SL) test on the precision of cosmological constraints for $f(T)$ gravity theories. The SL test is an important supplement to current cosmological observations because it measures the redshift drift in the Lyman-$\alpha$ forest in the spectra of distant quasars, covering the "redshift desert" of $2 \lesssim z \lesssim5$. To avoid data inconsistency, we use the best-fit models based on current combined observational data as fiducial models to simulate 30 mock SL test data. We quantify the impact of these SL test data on parameter estimation for $f(T)$ gravity theories. Two typical $f(T)$ models are considered, the power-law model $f(T)_{PL}$ and the exponential-form model $f(T)_{EXP}$. The results show that the SL test can effectively break the existing strong degeneracy between the present-day matter density $\Omega_m$ and the Hubble constant $H_0$ in other cosmological observations. For the considered $f(T)$ models, a 30-year observation of the SL test can improve the constraint precision of $\Omega_m$ and $H_0$ enormously but cannot effectively improve the constraint precision of the model parameters.