Constraining dark energy with Hubble parameter measurements: an analysis
  including future redshift-drift observations
  Dark energy affects the Hubble expansion rate (namely, the expansion history)
$H(z)$ by an integral over $w(z)$. However, the usual observables are the
luminosity distances or the angular diameter distances, which measure the
distance-redshift relation. Actually, dark energy affects the distances (and
the growth factor) by a further integration over functions of $H(z)$. Thus, the
direct measurements of the Hubble parameter $H(z)$ at different redshifts are
of great importance for constraining the properties of dark energy. In this
paper, we show how the typical dark energy models, for example, the
$\Lambda$CDM, $w$CDM, CPL, and holographic dark energy (HDE) models, can be
constrained by the current direct measurements of $H(z)$ (31 data in total,
covering the redshift range of $z\in [0.07,2.34]$). In fact, the future
redshift-drift observations (also referred to as the Sandage-Loeb test) can
also directly measure $H(z)$ at higher redshifts, covering the range of $z\in
[2,5]$. We thus discuss what role the redshift-drift observations can play in
constraining dark energy with the Hubble parameter measurements. We show that
the constraints on dark energy can be improved greatly with the $H(z)$ data
from only a 10-year observation of redshift drift.
