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\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} \text{exp} \left \{ -3 \int_1^a \frac{1+w(a')}{a'}da' \right \} \end{equation}  Where the $\Omega$’s are the energy densities of respectively matter, radiation, curvature and Dark Energy. Energy.\\  The equation of state parameter w(a) $w(a)$  quantifies the ratio between pressure and energy density of the DE Dark Energy  component. For a cosmological constant this parameter is constant and has a value w $w  = -1 -1$  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 Chevallier-Polarski-Linder  parametrization [ [70 71 baldi  ] : \begin{equation} w(a) = w0 w_0  + w1 w_a  (1-a) \end{equation} based on the behavior of w(a) at low redshifts, and the early dark energy parametrization [ look at fontanot ]   \begin{equation} w(a) = blalbala s\end{equation}  that postulates \frac{w_0}{1+b ln(1/a)}\end{equation}  where $b$ is  a different parameter dependant on the the  abundance of DE Dark Energy  at early times. Another class of widely studied scenarios views dark energy as a scalar field whose dynamical evolution is driven by a parametrized potential, the representative models of this class are Quintessence [b], k-essence[b], Phantom[b], Quintom [b] and perfect fluid [] DE models.For Quintessence models the most common choices of potentials include slowly varying runaway potentials such as an inverse power law [ b]:  \begin{equation}equation\end{equation}  or an exponential [b]: