neglect the radiation component.

-   $\Lambda$CDM model: Since the cosmological constant $\Lambda$ can explain the various cosmological observations quite well, it has nowadays become the preferred and simplest candidate for dark energy, although it has been suffering the severe theoretical puzzles. The EoS of the cosmological constant is $w=-1$, and thus we have $$E^2(z)=\Omega_{\rm{m}}(1+z)^{3}+(1-\Omega_{\rm{m}}).
    $$

-   $w$CDM model: In this model, the EoS of dark energy is assumed to be a constant, i.e., $w={\rm constant}$, and thus it is the simplest case for the dynamical dark energy. For this model, the expression of $E(z)$ is given by $$E^2(z)=\Omega_{\rm{m}}(1+z)^{3}+(1-\Omega_{\rm{m}})(1+z)^{3(1+w)}.$$

-   CPL model: In this model, the form of the EoS of dark energy $w(a)$ is parameterized as $w(a) = w_{0}+ w_{a}(1 - a)$ with two free parameters $w_{0}$ and $w_{a}$. Thus, we have $$\begin{aligned}
    E^{2}(z)&=\Omega_{\rm{m}}(1+z)^{3}+(1-\Omega_{\rm{m}})\\
    &\times(1+z)^{3(1+w_{\rm{0}}+w_{\rm{a}})}\exp\left(-\frac{3w_{\rm{a}}z}{1+z}\right).
     \end{aligned}$$

-   HDE model: In this model, the dark energy density is assumed to be of the form $\rho_{{\rm de}}=3c^{2}M^{2}_{\rm{pl}}R_{\rm{eh}}^{-2}$ [@Li:2004rb], where $c$ is a dimensionless parameter, $M_{\rm pl}$ is the reduced Planck mass, and $R_{\rm eh}$ is the future event horizon defined as $R_{\rm{eh}}(t)=ar_{\rm{max}}(t)=a(t)\int_t^\infty{dt'}/{a(t')}$. The evolution of the universe in this model is determined by the following two differential equations, $$\begin{aligned}
    \frac{1}{E(z)}\frac{dE(z)}{dz}=-\frac{\Omega_{\rm{de}}(z)}{1+z}\left(\frac{1}{2}+\frac{\sqrt{\Omega_{\rm{de}}(z)}}{c}-\frac{3}{2\Omega_{\rm{de}}(z)}\right),
    \end{aligned}$$ $$\begin{aligned}
    