||We present a method to link phenomenological parameterizations of dark energy on cosmological scales with theory.
We will use the phenomenological parameterization adopted by the Planck collaboration , parameterizing the phenomenological functions $\mu$ and $\eta$ as functions of scale $k$, and time using the scale factor $a$.
When linking the phenomenological functions with theory, we restrict ourselves to the Horndeski class of theories  and neglect time derivatives, i.e. we work with the general and model independent quasi static approximation (QSA), to be able to find analytical expressions for our phenomenological parameterizations.
Despite its generality, the QSA forces the phenomenological parameterization adopted by the Planck collaboration into a scale-independent one; which motivated us to propose an alternative phenomenological parameterization using a parameterization of the phenomenological functions $\mu(a,k)$ and $\Sigma(a,k)$.
This phenomenological parameterization does allow for scale dependence in $\mu$ under our conditions.
The effect of imposing extra conditions on our models, e.g. imposing the speed of gravitational waves to equal the speed of sound, is investigated by searching the parameter space of the phenomenological parameterizations for points that yield physically viable models.
The viability here is evaluated by means of ghost and gradient stability conditions.