is not the case, which means that the halo abundance also depends on the structure growth history and the underlying gravity. The deviation of halo abundance becomes larger if the deviation of $\zeta$ from unity is larger. The halo mass function is therefore a sensitive probe for our modified gravity model.

Halo concentration
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![Measured halo concentrations from N-body simulations for different $\zeta$ values at the baseline redshift $z_S=1.2$. The lower panel shows the ratio of the measured halo concentrations of MG models to the one in $\Lambda$CDM. For clarity, we plot the error bars of the $\zeta=1$ result. As explained in Fig. \[fig:massfunction\], these errors are conservative upper limit. Hence the observed differences in $c$ are statistically significant. Difference in the nature of gravity contributes to these differences. However, since $z_S\neq z_\zeta$, difference in $c$ also arises from difference in the background matter density. More systematical studies are required to disentangle the two and to infer the nature of gravity from the halo concentration. []{data-label="fig:concentration"}](f3.eps){width="40.00000%"}

The concentration parameter has been widely used to describe the internal structure of halos[@Jing2000; @Bullock2001; @Zhao2003; @Neto2007; @Zhao2009]. Recently, @Jeeson2011 investigated the correlation between nine different dark matter halo properties, and they claimed that 