1.2$ in the $\Lambda$CDM and the corresponding redshifts in the MG universe (10th row of Table [slowromancap1@]{} in Paper [slowromancap1@]{}).

![Comparison of halo distributions from N-body simulations for different $\zeta$ values at the baseline redshift $z_S=1.2$, which is that of $\zeta=1$ ($\Lambda$CDM). The panels show the dark matter halos with the mass $M_{200}\ge10^{11.5}\msun$ in slices $100\times100\times8h^{-1}{\rm
    Mpc}$, and the sizes of the dots are proportional to the radii $R_{200}$ of the halos.[]{data-label="fig:slice"}](f1.eps){width="45.00000%"}

Dark matter halos were identified from the simulation at each snapshot using the standard friends-of-friends (FOF) algorithm[@Davis1985] with a linking length of $b=0.2$ times the mean interparticle separation. For currently favored cosmologies, $15$-$20\%$ of $b=0.2$ FOF halos have irregular substructure or have two or more major halo components linked together[@Lukic2009]. Only isolated, relaxed halos are well-fit by the Navarro-Frenk-White (NFW) profile[@Navarro1997], therefore, we use the virial halo mass to fit NFW profile in the simulations. The mass $M_{200}$ is defined as the interior mass within a sphere of radius $R_{200}$ centered on the most bound particle of the halo, where $R_{200}$ is the radius within which the over-density is $\Delta_{200}=200$ times the mean density $\rho_{\rm mean}$ of the universe. Thus, the mass and radius of a halo are 