Probing modified gravity theories with the Sandage-Loeb test

In this paper, we quantify the ability of a future measurement of the Sandage-Loeb test signal from the Cosmic Dynamic Experiment-like spectrograph to constrain some popular modified gravity theories, including the Dvali-Gabadadze-Porrati braneworld scenario, $f(R)$ modified gravity, and $f(T)$ gravity theory. We find that the Sandage-Loeb test measurements are able to markedly break degeneracies between model parameters and thus greatly improve cosmological constraints for all concerned modified gravity theories when combined with the latest observations of the cosmic microwave background shift parameter. However, they yield almost the same degeneracy directions between model parameters as that from the distance ratio data derived from the latest observations of the cosmic microwave background and baryon acoustic oscillations. Moreover, for $f(R)$ modified gravity, the Sandage-Loeb test could provide completely different bounded regions in model-parameter space as compared to cosmic microwave background and baryon acoustic oscillations and thus supplement strong complementary constraints.