  We study the evolution of cooperation under the assumption that the
collective benefits of group membership can only be harvested if the fraction
of cooperators within the group, i.e. their critical mass, exceeds a threshold
value. Considering structured populations, we show that a moderate fraction of
cooperators can prevail even at very low multiplication factors if the critical
mass is minimal. For larger multiplication factors, however, the level of
cooperation is highest at an intermediate value of the critical mass. The
latter is robust to variations of the group size and the interaction network
topology. Applying the optimal critical mass threshold, we show that the
fraction of cooperators in public goods games is significantly larger than in
the traditional linear model, where the produced public good is proportional to
the fraction of cooperators within the group.
