Constraining the optical depth of galaxies and velocity bias with
  cross-correlation between kinetic Sunyaev-Zeldovich effect and peculiar
  velocity field
  We calculate the cross-correlation function $\langle (\Delta
T/T)(\mathbf{v}\cdot \mathbf{n}/\sigma_{v}) \rangle$ between the kinetic
Sunyaev-Zeldovich (kSZ) effect and the reconstructed peculiar velocity field
using linear perturbation theory, to constrain the optical depth $\tau$ and
peculiar velocity bias of central galaxies with Planck data. We vary the
optical depth $\tau$ and the velocity bias function
$b_{v}(k)=1+b(k/k_{0})^{n}$, and fit the model to the data, with and without
varying the calibration parameter $y_{0}$ that controls the vertical shift of
the correlation function. By constructing a likelihood function and
constraining $\tau$, $b$ and $n$ parameters, we find that the quadratic
power-law model of velocity bias $b_{v}(k)=1+b(k/k_{0})^{2}$ provides the
best-fit to the data. The best-fit values are $\tau=(1.18 \pm 0.24) \times
10^{-4}$, $b=-0.84^{+0.16}_{-0.20}$ and $y_{0}=(12.39^{+3.65}_{-3.66})\times
10^{-9}$ ($68\%$ confidence level). The probability of $b>0$ is only $3.12
\times 10^{-8}$ for the parameter $b$, which clearly suggests a detection of
scale-dependent velocity bias. The fitting results indicate that the
large-scale ($k \leq 0.1\,h\,{\rm Mpc}^{-1}$) velocity bias is unity, while on
small scales the bias tends to become negative. The value of $\tau$ is
consistent with the stellar mass--halo mass and optical depth relation proposed
in the previous literatures, and the negative velocity bias on small scales is
consistent with the peak background-split theory. Our method provides a direct
tool to study the gaseous and kinematic properties of galaxies.
