Nonderivative Modified Gravity: a Classification
  We analyze the theories of gravity modified by a generic nonderivative
potential built from the metric, under the minimal requirement of unbroken
spatial rotations. Using the canonical analysis, we classify the potentials $V$
according to the number of degrees of freedom (DoF) that propagate at the
nonperturbative level. We then compare the nonperturbative results with the
perturbative DoF propagating around Minkowski and FRW backgrounds. A generic
$V$ implies 6 propagating DoF at the non-perturbative level, with a ghost on
Minkowski background. There exist potentials which propagate 5 DoF, as already
studied in previous works. Here, no $V$ with unbroken rotational invariance
admitting 4 DoF is found. Theories with 3 DoF turn out to be strongly coupled
on Minkowski background. Finally, potentials with only the 2 DoF of a massive
graviton exist. Their effect on cosmology is simply equivalent to a
cosmological constant. Potentials with 2 or 5 DoF and explicit time dependence
appear to be a further viable possibility.
