  Modifications to the gravitational potential affect the nonlinear
gravitational evolution of large scale structures in the Universe. To
illustrate some generic features of such changes, we study the evolution of
spherically symmetric perturbations when the modification is of Yukawa type;
this is non-trivial, because we should not and do not assume that Birkhoff's
theorem applies. We then show how to estimate the abundance of virialized
objects in such models. Comparison with numerical simulations shows reasonable
agreement: When normalized to have the same fluctuations at early times, weaker
large scale gravity produces fewer massive halos. However, the opposite can be
true for models that are normalized to have the same linear theory power
spectrum today, so the abundance of rich clusters potentially places
interesting constraints on such models. Our analysis also indicates that the
formation histories and abundances of sufficiently low mass objects are
unchanged from standard gravity. This explains why simulations have found that
the nonlinear power-spectrum at large k is unaffected by such modifications to
the gravitational potential. In addition, the most massive objects in
CMB-normalized models with weaker gravity are expected to be similar to the
high-redshift progenitors of the most massive objects in models with stronger
gravity. Thus, the difference between the cluster and field galaxy populations
is expected to be larger in models with stronger large-scale gravity.
