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