Velocity bias in the distribution of dark matter halos

The standard formalism for the co-evolution of halos and dark matter predicts that any initial halo velocity bias rapidly decays to zero. We argue that, when the purpose is to compute statistics like power spectra etc., the coupling in the momentum conservation equation for the biased tracers must be modified. Our new formulation predicts the constancy in time of any statistical halo velocity bias present in the initial conditions, in agreement with peak theory. We test this prediction by studying the evolution of a conserved halo population in N-body simulations. We establish that the initial simulated halo density and velocity statistics show distinct features of the peak model and, thus, deviate from the simple local Lagrangian bias. We demonstrate, for the first time, that the time evolution of their velocity is in tension with the rapid decay expected in the standard approach.