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.