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.
