Non-linear matter power spectrum without screening dynamics modelling in $f(R)$ gravity

Halo model is a physically intuitive method for modelling the non-linear power spectrum, especially for the alternatives to the standard $\mathrm{\Lambda }$CDM models. In this paper, we exam the Sheth-Tormen barrier formula adopted in the previous $\mathtt{\text{CHAM}}$ method \citep{2018MNRAS.476L…65H}. As an example, we model the ellipsoidal collapse of top-hat dark matter haloes in $f(R)$ gravity. A good agreement between Sheth-Tormen formula and our result is achieved. The relative difference in the ellipsoidal collapse barrier is less than or equal to $1.6\mathrm{\%}$. Furthermore, we verify that, for F4 and F5 cases of Hu-Sawicki $f(R)$ gravity, the screening mechanism do not play a crucial role in the non-linear power spectrum modelling up to $k\sim 1[h\mathrm{/}\mathrm{M}\mathrm{p}\mathrm{c}]$. We compare two versions of modified gravity modelling, namely with/without screening. We find that by treating the effective Newton constant as constant number ($G_{\rm eff}=4/3G_N$) is acceptable. The scale dependence of the gravitational coupling is sub-relevant. The resulting spectra in F4 and F5, are in $0.1\mathrm{\%}$ agreement with the previous $\mathtt{\text{CHAM}}$ results. The published code is accelerated significantly. Finally, we compare our halo model prediction with N-body simulation. We find that the general spectrum profile agree, qualitatively. However, via the halo model approach, there exists a systematic under-estimation of the matter power spectrum in the co-moving wavenumber range between $0.3h\mathrm{/}\mathrm{M}\mathrm{p}\mathrm{c}$ and $3h\mathrm{/}\mathrm{M}\mathrm{p}\mathrm{c}$. These scales are overlapping with the transition scales from two halo term dominated regimes to those of one halo term dominated.