Power of the redshift drift on cosmological models and expansion history

We investigate the power of the velocity drift ($\Delta v$) on cosmological parameters and expansion history with observational Hubble data (OHD), type Ia supernova (SNIa). We estimate the constraints of $\Delta v$ using the Fisher information matrix based on the model by \citet{pasquini2005codex,whitelock2006scientific}. We find that $\Delta v$ with 20 years can reduce the uncertainty of $\Omega_m$ by more than 42% than available observations. Based on the statistical figures of merit (FoM), we find that in order to match the constraint power of OHD and SNIa, we need 21 and 26 future measurements, respectively. We also quantitatively estimate for the first time the number of years required for the velocity drift to become comparable with current observations on the equation of state $w$. The statistical FoM indicate that we need at least 12 years to cover current observations. Physically, we could monitor 30 quasars for 30 years to obtain the same accuracy of $w$. Considering two parameterized deceleration factor $q(z)$, we find that the available observations give an estimation on current value $-0.9 \lesssim q_0 \lesssim -0.3$. Difference between the two types of $q(z)$ is the precise determination of variation rate $dq/dz$. For the first model with constant $dq/dz$, $\Delta v$ with only 10 years provides a much better constraint on it, especially when compared with SNIa. However, we need $\Delta v$ for more years in the variable $dq/dz$ model. We find that $\Delta v$ with 30 years reduces the uncertainty of transition redshift to approximately three times better than those of OHD and SNIa.