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 DGP brane-world
scenario, $f(R)$ modified gravity and $f(T)$ gravity theory. We find that the
Sandage-Loeb test measurements are able to break degeneracies between model
parameters markedly and thus greatly improve cosmological constraints for all
concerned modified gravity theories when combined with the latest observations
of CMB--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
baryonic acoustic oscillations (CMB/BAO). Moreover, for the $f(R)$ modified
gravity, the Sandage-Loeb test could provide completely different bounded
regions in model parameters space as compared to CMB/BAO and thus supplement
strong complementary constraints.
