Probing Dark Energy through Scale Dependence

We consider the consequences of having no prior knowledge of the true dark energy model for the interpretation of cosmological observations. The magnitude of redshift-space distortions and weak-lensing shear is determined by the metric on the geodesics of which galaxies and light propagate. We show that, given precise enough observations, we can use these data to completely reconstruct the metric on our past lightcone and therefore to measure the scale- and time-dependence of the anisotropic stress and the evolution of the gravitational potentials in a model-independent manner. Since both dark matter and dark energy affect the visible sector only through the gravitational field they produce, they are inseparable without a model for dark energy: galaxy bias cannot be measured and therefore the distribution of dark matter determined; the peculiar velocity of dark matter can be identified with that of the galaxies only when the equivalence principle holds. Given these limitations, we show how one can nonetheless build tests for classes of dark energy models which depend on making measurements at multiple scales at a particular redshift. They are null tests on the model-independent observables, do not require modeling evolution in time and do not require any parametrization of the free functions of these models, such as the sound speed. We show how one can rule out or constrain the whole class of the most-general scalar-tensor theories even without assuming the quasi-static limit.