Relativistic scalar fields and the quasi-static approximation in
  theories of modified gravity
  Relativistic scalar fields are ubiquitous in modified theories of gravity. An
important tool in understanding their impact on structure formation, especially
in the context of N-body simulations, is the quasi-static approximation in
which the time evolution of perturbations in the scalar fields is discarded. We
show that this approximation must be used with some care by studying linearly
perturbed scalar field cosmologies and quantifying the errors that arise from
taking the quasi-static limit. We focus on f(R) and chameleon models and link
the accuracy of the quasi-static approximation to the fast/slow-roll behaviour
of the background and its proximity to {\Lambda}CDM. Investigating a large
range of scales, from super- to sub-horizon, we find that slow-rolling
({\Lambda}CDM-like) backgrounds generically result in good quasi-static
behaviour, even on (super-)horizon scales. We also discuss how the
approximation might affect studying the non-linear growth of structure in
numerical N-body simulations.
