$\psi$ is the only gravitational potential that nonrelativistic particles can sense. So the value of $\zeta(k,z)$ as a function of scale and redshift, along with the initial condition and the expansion rate of the universe, completely fixes the evolution of the matter density and velocity fields. This is the major reason that we adopt this single parameter parametrization on modified gravity N-body simulations. For more detailed discussion on this parametrization, please refer to paper I.

Neverthless, such parameterization is highly simplified in the sense that it lacks any screening mechanism [@Jain10] required to pass the solar system tests. For this reason, these simulations lose the capability to explore the rich consequences induced by the MG environmental dependence. These features have been explored in advanced simulations on $f(R)$ and DGP [@Schmidt2009a; @Oyaizu08a; @Oyaizu08b; @ZhaoGB2011; @Schmidt2009b; @Chan09; @Li12]. Nevertheless, our simulations benefit from costing no extra time than the ordinary CDM simulations. So in principle one can run a large number of such simulations within reasonable amount of time to fairly sample relevant parameter space. One then interpolates/extrapolates the simulation results to explore the whole relevant parameter space and understand the nonlinear evolution for $\zeta$ of general spatial and time dependence. The 