$\zeta=1.2$ and $\zeta=1.0$ is about $10\%$. We find that the difference of $\zeta=1.2$ is about twice that of $\zeta=0.8$, although they have the same $20\%$ deviation from GR. For a $10\%$ deviation from GR($\zeta=0.9$, or $\zeta=1.1$), the measured mass functions can differ by about $5-10\%$. Interestingly, the difference of $\zeta=0.9$ is also about twice that of $\zeta=1.1$. In general, for the identical deviation of $\zeta$ from unity, the deviation of halo mass function of $\zeta<1$ is about twice that of $\zeta>1$, which implies that the decrease of gravity can lead to more difference of halo mass function. This behavior can also be found in the bottom-right panel of Fig. $4$ in Paper [slowromancap1@]{} for the comparison of the nonlinear power spectra. For the same deviation of $\zeta$ from unity, the deviation of nonlinear power spectra of $\zeta<1$ are about twice that of $\zeta>1$ at $\Delta^2\sim10$.

If the halo abundance is completely determined by the shape and amplitude of the linear power spectrum, the halo mass functions for different $\zeta$ simulation should have the identical form. However, based on the measured halo mass functions from N-body simulations for different $\zeta$ values at the same linear power spectrum, we find that this 