Velocity bias and the nonlinear perturbation theory of peaks

The biasing in the large-scale structure of the universe is a crucial problem in cosmological applications. The peaks model of biasing predicts a linear velocity bias of halos, which is not present in a simple model of local bias. We investigate the origin of the velocity bias in the peaks model from the viewpoint of the integrated perturbation theory, which is a nonlinear perturbation theory in the presence of general Lagrangian bias. The presence of the velocity bias in the peaks model is a consequence of the "flat constraint," ${\nabla}\delta = 0$; i.e., all the first spatial derivatives should vanish at the locations of peaks. We show that the velocity bias in the peaks model is systematically derived in the framework of the integrated perturbation theory, and then develop a formal theory to perturbatively trace the nonlinear evolution of biased objects with the flat constraint. A formula for the nonlinear velocity dispersion of peaks with the one-loop approximation is also derived.