  In a system of noisy self-propelled particles with interactions that favor
directional alignment, collective motion will appear if the density of
particles is beyond a critical density. Starting with a reduced model for
collective motion, we determine how the critical density depends on the form of
the initial perturbation. Specifically, we employ a renormalization-group
improved perturbative method to analyze the model equations, and show
analytically, up to first order in the perturbation parameter, how the critical
density is modified by the strength of the initial angular perturbation in the
system.
