Limits of Quasi-Static Approximation in Modified-Gravity Cosmologies

We investigate the limits of applicability of the quasi-static approximation in cosmologies featuring general models of dark energy or modified gravity. We show that, at best, the quasi-static approximation breaks down outside of the sound horizon of the dark-energy, rather than the cosmological horizon as is frequently assumed. When the sound speed of dark energy is significantly below that of light, the quasi-static limit is only valid in a limited range of observable scales and this must be taken into account when computing effects on observations in such models. As an order of magnitude estimate, in the analysis of data from today's weak-lensing and peculiar-velocity surveys, dark energy can be modelled as quasi-static only if the sound speed is larger than order 1% of that of light. In upcoming surveys, such as Euclid, it should only be used when the sound speed exceeds around 10% of the speed of light. In the analysis of the cosmic microwave background, the quasi-static limit should never be used for the integrated Sachs-Wolf effect and for lensing only when the sound speed exceeds 10% of the speed of light.