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The first class of models falling into this category still describes Dark Energy as an homogeneous field in a universe described by the Friedmann-Lemaitre-Robertson-Walker metric [ ]:  \begin{equation} ds^2 = -c^2dt^2 + a(t)\{ \delta_{ij}dx^i dx^j \} \end{equation}  whose time dependence is all described by the scale factor $a(t)$. The background evolution so it’s usually described by the Hubble function $H(a) \equiv \dot{a}/a$ that describes how the expansion rate changes as a function of time. This in turn is related to the abundance of different constituents in the universe through the Friedmann equation:  \begin{equation} \frac{H^2(a)}{H_0^2} = \Omega_M a^{-3} + \Omega_r a^{-4} + \Omega_K a^{-2} + \Omega_{DE} exp \right\{ \text{exp} \left \{  -3 \int_1^a \frac{1+w(a')}{a'}da' \left\} \right \}  \end{equation} Where the OMEGA’s are the energy densities of respectively matter, radiation, curvature and dark energy. The equation of state parameter w(a) quantifies the ratio between pressure and energy density of the DE component.  For a cosmological constant this parameter is constant and has a value w = -1 but we can readily see how just by allowing this parameter to be time dependant we can obtain a completely new expansion history.   Common phenomenological parametrizations of the equation of state parameter are the CPL parametrization [ ] :