Ambiguous Tests of General Relativity on Cosmological Scales
  There are a number of approaches to testing General Relativity (GR) on linear
scales using parameterized frameworks for modifying cosmological perturbation
theory. It is sometimes assumed that the details of any given parameterization
are unimportant if one uses it as a diagnostic for deviations from GR. In this
brief report we argue that this is not necessarily so. First we show that
adopting alternative combinations of modifications to the field equations
significantly changes the constraints that one obtains. In addition, we show
that using a parameterization with insufficient freedom significantly tightens
the apparent theoretical constraints. Fundamentally we argue that it is almost
never appropriate to consider modifications to the perturbed Einstein equations
as being constraints on the effective gravitational constant, for example, in
the same sense that solar system constraints are. The only consistent
modifications are either those that grant near-total freedom, as in
decomposition methods, or ones which map directly to a particular part of
theory space.
