  Within the framework of modified gravity (MG), the quasi-static (QS) and
sub-horizon (SH) approximations are widely used in analyses aiming to identify
departures from the concordance model at late-times. In general, it is assumed
that time derivatives are subdominant with respect to spatial derivatives given
that the relevant physical modes are those well inside the Hubble radius. In
practice, the perturbation equations under these approximations are reduced to
a tractable algebraic system in terms of the gravitational potentials and the
perturbations of involved matter fields. Here, in the framework of $f(R)$
theories, we revisit standard results when these approximations are invoked
using a new parameterization scheme that allows us to track the relevance of
each time-derivative term in the perturbation equations. This new approach
unveils terms which are neglected in the standard procedure. We assess the
relevance of these differences by comparing results from both approaches
against full numerical solutions for two well-known toy-models: the designer
$f(R)$ model and the Hu-Sawicki model. We find that: i) the SH approximation
can be safely applied to linear perturbation equations for scales $0.06 h/$Mpc
$\lesssim k \lesssim 0.2 h/$Mpc, ii) in this "safety region", the QS
approximation provides a very accurate description of the late-time
cosmological dynamics even when dark energy significantly contribute to the
cosmic budget, and iii) our new methodology performs better than the standard
procedure, even for several orders of magnitude in some cases. Although, the
impact of this major improvement on the linear observables is minimal for the
studied cases, this does not represent an invalidation for our approach.
Instead, our findings indicate that the perturbation expressions derived under
these approximations in more general MG theories, such as Horndeski, should be
also revisited.
