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 $b_v(k)=1+b(k\mathrm{/}{k}_0{)}^n$, 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 $b_v(k)=1+b(k\mathrm{/}{k}_0{)}^2$ 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$y{0}=(12.39^{+3.65}_{-3.66})\times 10^{-9}$($68%$confidencelevel)\mathrm{.}Theprobabilityof$b>0$isonly$3.12 \times 10^{-8}$fortheparameter$b$, which clearly suggests a detection of scale-dependent velocity bias. The fitting results indicate that the large-scale ($k\le 0.1h{\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$) 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.