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.