Cluster abundance in chameleon $f(R)$ gravity I: toward an accurate halo
  mass function prediction
  We refine the mass and environment dependent spherical collapse model of
chameleon $f(R)$ gravity by calibrating a phenomenological correction inspired
by the parameterized post-Friedmann framework against high-resolution $N$-body
simulations. We employ our method to predict the corresponding modified halo
mass function, and provide fitting formulas to calculate the fractional
enhancement of the $f(R)$ halo abundance with respect to that of General
Relativity (GR) within a precision of $\lesssim 5\%$ from the results obtained
in the simulations. Similar accuracy can be achieved for the full $f(R)$ mass
function on the condition that the modeling of the reference GR abundance of
halos is accurate at the percent level. We use our fits to forecast constraints
on the additional scalar degree of freedom of the theory, finding that upper
bounds competitive with current Solar System tests are within reach of cluster
number count analyses from ongoing and upcoming surveys at much larger scales.
Importantly, the flexibility of our method allows also for this to be applied
to other scalar-tensor theories characterized by a mass and environment
dependent spherical collapse.
