Universality of the halo mass function in screened gravity theories
  We examine the efficacy of the halo mass function as a probe of $f(R)$,
symmetron and DGP gravity. We develop an excursion-set method to generalise a
range of popular HMF fitting functions from GR to screened MG, considering the
HMF dependence on the critical density parameter $\delta_{c}$. In particular,
we propose a variety of new methods to account for the environmental dependence
of chameleon screening and compare their accuracy to existing ones via $N$-body
data. Finally, using the nested sampling routine MultiNest, we examine two
possible interpretations of the MG $N$-body results: whether they can be
interpreted as $\Lambda$CDM HMFs and whether the MG HMFs display any
universality.
  We find that the effects of MG can be interpreted as a change in best-fit
parameters in the $\Lambda$CDM HMF---i.e. can be mistaken for GR---for all of
the fitting functions. Alternatively, the relation can be inverted to judge the
universality of the HMF---not in terms of its purported invariance to $z$,
$H_{0}$, $\Omega_{m}$, $\Omega_{\Lambda}$ as in GR, but its independence on the
underlying theory of gravity. Although we found no completely universal HMF,
the parameter values did cluster according to the type of screening mechanism.
The results suggest that a single, best-fit HMF might be used for each type of
screening, independent of the parameters in the MG model. This demonstrates
that the additional complexity of the gravitational collapse in screened MG
theories cannot always be accounted for using the techniques developed in GR.
However, the variations are small enough that it is unnecessary to develop new
fitting functions and calibrate them on a case-by-case basis.
