Challenges in connecting modified gravity theory and observations

Cosmic acceleration may be due to modifications of cosmic gravity and to test this we need robust connections between theory and observations. However, in a model independent approach like effective field theory or a broad class like Horndeski gravity, several free functions of time enter the theory. We show that simple parametrizations of these functions are unlikely to be successful; in particular the approximation $\alpha_i(t)\propto\Omega_{\rm de}(t)$ drastically misestimates the observables. This holds even in simple modified gravity theories like $f(R)$. Indeed, oversimplified approximations to the property functions $\alpha_i(t)$ can even miss the signature of modified gravity. We also consider the question of consistency relations and the role of tensor (gravitational wave) perturbations.
Comment: 10 pages, 5 figures; v2 new Appendix on f(R) with new figure; v3 accepted to PRD, new figure, reordering and clarification, conclusions unchanged