$\sigma$. Besides $\delta_c$ is weakly dependent on the redshift, thus the difference of the halo bias for different $\zeta$ values is very small. We can conclude that the halo bias is a weak statistics to detect the modified gravity model.

![Same as Fig. \[fig:massfunction\] but for the virial halo mass $M_{\rm  vir}$ defined by a value $\Delta_{\rm vir}$ from the spherical collapse model. []{data-label="fig:ms_vir"}](f5.eps){width="45.00000%"}

Discussion and Conclusion {#sec:conclusion}
=========================

![Same as Fig. \[fig:concentration\] but for the virial halo mass $M_{\rm  vir}$ defined by a value $\Delta_{\rm vir}$ from the spherical collapse model. []{data-label="fig:concen_vir"}](f6.eps){width="45.00000%"}

In this paper we have compared the halo mass function, concentration and bias based on a series of modified gravity N-body simulations. The modified gravity model is characterized by a single parameter $\zeta$, which determines the enhancement of particle acceleration with respect to GR. All the simulations for different $\zeta$ values are started from the same initial condition. The redshifts for different $\zeta$ of the comparison of halo properties are selected when they have the same linear power spectrum as $\Lambda$CDM simulations at $z_S=1.2$.

![Same as Fig. \[fig:bias\] but for the virial halo mass $M_{\rm
    vir}$ defined by a value $\Delta_{\rm vir}$ from the spherical collapse model. 