Cosmological gravity on all scales II: Model independent modified
  gravity $N$-body simulations
  Model-independent constraints on modified gravity models hitherto exist
mainly on linear scales. A recently developed formalism presented a consistent
parameterisation that is valid on all scales. Using this approach, we perform
model-independent modified gravity $N$-body simulations on all cosmological
scales with a time-dependent $\mu$. We present convergence tests of our
simulations, and we examine how well existing fitting functions reproduce the
non-linear matter power spectrum of the simulations. We find that although
there is a significant variation in the accuracy of all of the fitting
functions over the parameter space of our simulations, the ReACT framework
delivers the most consistent performance for the matter power spectrum. We
comment on how this might be improved to the level required for future surveys
such as Euclid and the Vera Rubin Telescope (LSST). We also show how to compute
weak-lensing observables consistently from the simulated matter power spectra
in our approach, and show that ReACT also performs best when fitting the
weak-lensing observables. This paves the way for a full model-independent test
of modified gravity using all of the data from such upcoming surveys.
