  We study the crossover between equilibrium and off-equilibrium dynamical
universality classes in the Vicsek model near its ordering transition. Starting
from the incompressible hydrodynamic theory of Chen et al
\cite{chen2015critical}, we show that increasing the activity leads to a
renormalization group (RG) crossover between the equilibrium ferromagnetic
fixed point, with dynamical critical exponent $z = 2$, and the off-equilibrium
active fixed point, with $z = 1.7$ (in $d=3$). We run simulations of the
classic Vicsek model in the near-ordering regime and find that critical slowing
down indeed changes with activity, displaying two exponents that are in
remarkable agreement with the RG prediction. The equilibrium-to-off-equilibrium
crossover is ruled by a characteristic length scale beyond which active
dynamics takes over. Such length scale is smaller the larger the activity,
suggesting the existence of a general trade-off between activity and system's
size in determining the dynamical universality class of active matter.
