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

Abstract

We calculate the cross-correlation function $⟨(\Delta T/T)(\mathbf{v}\cdot \mathbf{n}/{\sigma }_v)⟩$ 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 $\tau =(1.18\pm 0.24)\times 10^{-4}$, $b=-0.84_{-0.20}^{+0.16}$ and $y_0=(12.39_{-3.66}^{+3.65})\times 10^{-9}$ ($68\%$ 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 ($k\le 0.1h{\mathrm{Mpc}}^{-1}$) 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.

arXiv comment13 pages, 13 figures, 3 tables

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