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

We calculate the cross-correlation function < (Δ T/T)({v}\cdot \hat{n}/σ _v) > between the kinetic Sunyaev-Zeldovich (kSZ) effect and the reconstructed peculiar velocity field using linear perturbation theory, with the aim of constraining the optical depth τ and peculiar velocity bias of central galaxies with Planck data. We vary the optical depth τ and the velocity bias function b_v(k) = 1 + b(k/k₀)ⁿ, and fit the model to the data, with and without varying the calibration parameter y₀ that controls the vertical shift of the correlation function. By constructing a likelihood function and constraining the τ, b and n parameters, we find that the quadratic power-law model of velocity bias, b_v(k) = 1 + b(k/k₀)², provides the best fit to the data. The best-fit values are τ = (1.18 ± 0.24) × 10^-4, b=-0.84^{+0.16}_{-0.20} and y₀=(12.39^{+3.65}_{-3.66})× 10^{-9} (68 per cent confidence level). The probability of b > 0 is only 3.12 × 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 ≤ 0.1 h Mpc^-1) velocity bias is unity, while on small scales the bias tends to become negative. The value of τ is consistent with the stellar mass-halo mass and optical depth relationship proposed in the literature, and the negative velocity bias on small scales is consistent with the peak background split theory. Our method provides a direct tool for studying the gaseous and kinematic properties of galaxies.