measurement uncertainty. The lower panel shows the ratio of the measured mass functions of MG models to the one in $\Lambda$CDM when they have the same linear power spectrum. The plotted errors are the same Poisson errors showing in the upper panel. Since the Poisson errors of different simulations have strong positive correlations, they largely cancel when taking the ratio. So the plotted errors represent a conservative upper limit of the halo mass function ratio. The lower panel then shows that the observed deviations from a universal mass function are significant. []{data-label="fig:massfunction"}](f2.eps){width="40.00000%"}

One of the long-standing efforts in precision cosmology is to determine the mass function of dark matter halos ${\rm d}n/{\rm d}M$, which is the number of halos per unit volume per unit mass. It is a sensitive probe of gravity and has been used to put strong constraints on modified gravity models (e.g. [@Schmidt2009b] on $f(R)$ gravity).

Although the number density of halos of a given mass depends on the shape and amplitude of the power spectrum, analytical work has suggested that the halo abundance can be expressed by a universal functional form when expressed in terms of suitable variables [@Press1974; @Bond1991; @Sheth1999]. A convenient form to describe 