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We calculate the cross-correlation function $\langle (\DeltaT/T)(\mathbf{v}\cdot \mathbf{n}/\sigma_{v}) \rangle$ between the kineticSunyaev-Zeldovich (kSZ) effect and the reconstructed peculiar velocity fieldusing linear perturbation theory, to constrain the optical depth $\tau$ andpeculiar velocity bias of central galaxies with Planck data. We vary theoptical 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 withoutvarying the calibration parameter $y_{0}$ that controls the vertical shift ofthe correlation function. By constructing a likelihood function andconstraining $\tau$, $b$ and $n$ parameters, we find that the quadraticpower-law model of velocity bias $b_{v}(k)=1+b(k/k_{0})^{2}$ provides thebest-fit to the data. The best-fit values are $\tau=(1.18 \pm 0.24) \times10^{-4}$, $b=-0.84^{+0.16}_{-0.20}$ and $y_{0}=(12.39^{+3.65}_{-3.66})\times10^{-9}$ ($68\%$ confidence level). The probability of $b>0$ is only $3.12\times 10^{-8}$ for the parameter $b$, which clearly suggests a detection ofscale-dependent velocity bias. The fitting results indicate that thelarge-scale ($k \leq 0.1\,h\,{\rm Mpc}^{-1}$) velocity bias is unity, while onsmall scales the bias tends to become negative. The value of $\tau$ isconsistent with the stellar mass--halo mass and optical depth relation proposedin the previous literatures, and the negative velocity bias on small scales isconsistent with the peak background-split theory. Our method provides a directtool to study the gaseous and kinematic properties of galaxies.

monthly notices of the royal astronomical society

Constraining the optical depth of galaxies and velocity bias with  cross-correlation between kinetic Sunyaev-Zeldovich effect and peculiar  velocity field