We calculate the cross-correlation function \langle ( \Delta T / T ) ( \mathbf { v } \cdot \hat { \mathbf { n } } / \sigma _ { v } ) \rangle 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 \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 { 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 .