and is easy to be implemented both in simulations and in observations. The impact of using different $\Delta$ has been discussed using high resolution N-body simulations[@Tinker2008].

One might argue that the differences between MG models and the GR shown above are mainly due to the definition of the halos. To test to what extent halo definition impacts our results, we have also compared the halo mass function, concentration and bias using the virial halo mass $M_{\rm vir}$ (defined by a value $\Delta_{\rm  vir}$ [@Eke1996] from the spherical model) instead of $M_{200}$. According to the virial mass definition from GR, we do see in Fig. \[fig:ms\_vir\] that for a $20\%$ deviation from ($|\zeta-1|<0.2$), the deviation of halo mass function differs by less than $10\%$, which is smaller than that of the halo mass $M_{200}$. However, we can still detect the observed differences of the halo mass functions for different $\zeta$ values.

In Fig. \[fig:concen\_vir\], for $\zeta>1.0$, the difference of $c_{\rm
  vir}(\zeta)$ with respect to that of GR is roughly the same as the case of $c_{200}(\zeta)$. Since at redshift $z_S  =1.2$, the universe is dominated by dark matter and $R_{200}$ is approximately equal to $R_{\rm vir}$, which is $195.7$ for our 