Parameter estimation with Sandage-Loeb test
  The Sandage-Loeb (SL) test directly measures the expansion rate of the
universe in the redshift range of $2\lesssim z\lesssim 5$ by detecting redshift
drift in the spectra of Lyman-$\alpha$ forest of distant quasars. We discuss
the impact of the future SL test data on parameter estimation for the
$\Lambda$CDM, the $w$CDM, and the $w_0w_a$CDM models. To avoid the potential
inconsistency with other observational data, we take the best-fitting dark
energy model constrained by the current observations as the fiducial model to
produce 30 mock SL test data. The SL test data provide an important supplement
to the other dark energy probes, since they are extremely helpful in breaking
the existing parameter degeneracies. We show that the strong degeneracy between
$\Omega_m$ and $H_0$ in all the three dark energy models is well broken by the
SL test. Compared to the current combined data of type Ia supernovae, baryon
acoustic oscillation, cosmic microwave background, and Hubble constant, the
30-yr observation of SL test could improve the constraints on $\Omega_m$ and
$H_0$ by more than 60\% for all the three models. But the SL test can only
moderately improve the constraint on the equation of state of dark energy. We
show that a 30-yr observation of SL test could help improve the constraint on
constant $w$ by about 25\%, and improve the constraints on $w_0$ and $w_a$ by
about 20\% and 15\%, respectively. We also quantify the constraining power of
the SL test in the future high-precision joint geometric constraints on dark
energy. The mock future supernova and baryon acoustic oscillation data are
simulated based on the space-based project JDEM. We find that the 30-yr
observation of SL test would help improve the measurement precision of
$\Omega_m$, $H_0$, and $w_a$ by more than 70\%, 20\%, and 60\%, respectively,
for the $w_0w_a$CDM model.
