Revisiting the matter power spectra in $f(R)$ gravity

In this paper, we study the non-linear matter power spectrum in a specific family of $f(R)$ models that can reproduce the $\Lambda$CDM background expansion history, using high resolution $N$-body simulations based on the {\sc ecosmog} code. We measure the matter power spectrum in the range of $0.05h{\rm Mpc}^{-1}<k<10h{\rm Mpc}^{-1}$ from simulations for our $f(R)$ models and give theoretical explanations to their behaviour and evolution patterns. We also examine the chameleon mechanism for our models and find that it works throughout the cosmic history in dense regions, for our $f(R)$ models with $|f_{R0}|<10^{-4}$. On the other hand, for models with $|f_{R0}|>10^{-3}$, we find no chameleon screening in dense regions at late times ($z<3$), which means that those models could be ruled out due to the factor-of-$1/3$ enhancement to the strength of Newtonian gravity. We also give the best-fit parameters for a generalised PPF fitting formula which works well for the models studied here.