Modelling Time-varying Dark Energy with Constraints from Latest Observations

We introduce a set of two-parameter models for the dark energy equation of state (EOS) $w(z)$ to investigate time-varying dark energy. The models are classified into two types according to their boundary behaviors at the redshift $z=(0,\infty)$ and their local extremum properties. A joint analysis based on four observations (SNe + BAO + CMB + $H_0$) is carried out to constrain all the models. It is shown that all models get almost the same $\chi^2_{min}\simeq 469$ and the cosmological parameters $(\Omega_M, h, \Omega_bh^2)$ with the best-fit results $(0.28, 0.70, 2.24)$, although the constraint results on two parameters $(w_0, w_1)$ and the allowed regions for the EOS $w(z)$ are sensitive to different models and a given extra model parameter. For three of Type I models which have similar functional behaviors with the so-called CPL model, the constrained two parameters $w_0$ and $w_1$ have negative correlation and are compatible with the ones in CPL model, and the allowed regions of $w(z)$ get a narrow node at $z\sim 0.2$. The best-fit results from the most stringent constraints in Model Ia give $(w_0,w_1) = (-0.96^{+0.26}_{-0.21}, -0.12^{+0.61}_{-0.89})$ which may compare with the best-fit results $(w_0,w_1) = (-0.97^{+0.22}_{-0.18}, -0.15^{+0.85}_{-1.33})$ in the CPL model. For four of Type II models which have logarithmic function forms and an extremum point, the allowed regions of $w(z)$ are found to be sensitive to different models and a given extra parameter. It is interesting to obtain two models in which two parameters $w_0$ and $w_1$ are strongly correlative and appropriately reduced to one parameter by a linear relation $w_1 \propto (1+w_0)$.