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INTRODUCTION {#sec_intro}
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Dark matter halos are prominent structures in the dark universe. Their abundance, density profile and clustering (halo bias) contain valuable cosmological information. In particular, dark matter halos form and grow under gravitational instability. Hence halo properties contain rich information on the nature of gravity at $\sim$ Mpc scales and above, and can provide strong gravity constraints [@Narikawa2012; @Thomas2011a; @Thomas2011b; @Sartoris2011; @LiB2011; @Lombriser2011; @Mak2011; @LiYin2011; @Jain2011; @LiMiao2011; @Allen2011; @Clifton2011; @Ferraro2011; @Schmidt2009a; @Schmidt2009c; @ZhaoG2011; @Jain10]. These properties also provide a powerful tool to understand the matter clustering through the halo model [@Jing98; @Cooray02], which is also a sensitive measure of gravity.

In [@Cui10] (hereafter paper I) we run a controlled set of N-body simulations with identical initial condition to study the impact of modified gravity (MG) on the matter power spectrum. These simulations adopt a MG parameterization with a single parameter $\zeta$. $\zeta$ is the relative enhancement of nonrelativistic particle acceleration with respect to general relativity (GR), given the identical mass distribution. More specifically, it is the quantity that enters into the $\psi$-$\rho$ relation in Fourier space, $$\label{eqn:psi}
k^2\psi=-\zeta4\pi G a^2\delta\rho\ .$$ Here, $\delta \rho\equiv \rho-\bar{\rho}$ and $\rho$ is the matter density. $\psi$ is defined through $ds^2=-(1+2\psi)dt^2+a^2(1+2\phi)d{\bf
  x}^2$. 