related by $$M_{200} = \frac{4}{3}\pi \Delta_{200} \rho_{\rm mean} R^3_{200},$$ where $\rho_{\rm mean}$ is the mean density of the universe at redshift $z$. Note that some authors use a value $\Delta_{\rm vir}$ motivated by the spherical collapse model, where the corresponding radius is specified as $R_{\rm vir}$. This value is both redshift and gravity dependent. It gives $\Delta_{\rm vir}=339.5$ at $z=0$ and $178$ at $z\gg 1$ for our adopted $\Lambda$CDM parameters [@Eke1996]. It also shows significant deviation in MG models [@Schmidt2009a]. Since we will compare the halo properties of different MG models at different redshifts (e.g. $z_s$ vs. $z_\zeta$), this latter choice could mix the impact of background cosmology with the impact of gravity. For this reason, we will not adopt this latter definition. Unless otherwise specified, throughout the paper we use a fixed value $\Delta=200$ to define the halo mass.

Fig. \[fig:slice\] shows the distributions of dark matter halos with the mass $M_{200}\ge10^{11.5}\msun$ in a slice of thickness $8h^{-1}{\rm Mpc}$ for different $\zeta$ values at the baseline redshift $z_S=1.2$, which is that of $\zeta=1$ ($\Lambda$CDM). According to visual inspection, the general appearances of the large-scale structures are remarkably similar, as a result of the same linear power spectrum for different 