We study the matter and velocity divergence power spectra in a f(R) gravity theory and their time evolution measured from several large-volume N-body simulations with varying box sizes and resolution. We find that accurate prediction of the matter power spectrum in f(R) gravity places stronger requirements on the simulation than is the case with Λ cold dark matter (ΛCDM) because of the non-linear nature of the fifth force. Linear perturbation theory is shown to be a poor approximation for the f(R) models, except when the chameleon effect is very weak. We show that the relative differences from the fiducial ΛCDM model are much more pronounced in the non-linear tail of the velocity divergence power spectrum than in the matter power spectrum, which suggests that future surveys which target the collection of peculiar velocity data will open new opportunities to constrain modified gravity theories. A close investigation of the time evolution of the power spectra shows that there is a pattern in the evolution history, which can be explained by the properties of the chameleon-type fifth force in f(R) gravity. Varying the model parameter |fR0|, which quantifies the strength of the departure from standard gravity, mainly varies the epoch marking the onset of the fifth force, as a result of which the different f(R) models are in different stages of the same evolutionary path at any given time.
Li, B., Hellwing, W., Koyama, K., Zhao, G., Jennings, E., & Baugh, C. (2013). The non-linear matter and velocity power spectra in f(R) gravity. Monthly Notices of the Royal Astronomical Society, 428(1), 743-755. https://doi.org/10.1093/mnras/sts072