An interacting dark energy model in a non-flat universe

Abstract

In this paper, an interacting dark energy model in a non-flat universe is studied, with taking interaction form $C=\alpha H\rho _{de}$. And in this study a property for the mysterious dark energy is aforehand assumed, i.e. its equation of state $w_{\Lambda }=-1$. After several derivations, a power-law form of dark energy density is obtained $\rho _{\Lambda } \propto a^{-\alpha }$, here $a$ is the cosmic scale factor, $\alpha$ is a constant parameter introducing to describe the interaction strength and the evolution of dark energy. By comparing with the current cosmic observations, the combined constraints on the parameter $\alpha$ is investigated in a non-flat universe. For the used data they include: the Union2 data of type Ia supernova, the Hubble data at different redshifts including several new published datapoints, the baryon acoustic oscillation data, the cosmic microwave background data, and the observational data from cluster X-ray gas mass fraction. The constraint results on model parameters are $\Omega _{K}=0.0024\,(\pm 0.0053)^{+0.0052+0.0105}_{-0.0052-0.0103}, \alpha =-0.030\,(\pm 0.042)^{+0.041+0.079}_{-0.042-0.085}$ and $\Omega _{0m}=0.282\,(\pm 0.011)^{+0.011+0.023}_{-0.011-0.022}$. According to the constraint results, it is shown that small constraint values of $\alpha$ indicate that the strength of interaction is weak, and at $1\sigma$ confidence level the non-interacting cosmological constant model can not be excluded.

Appendix A: Discussions on taking interaction form \(C=\beta H\rho _{m}\)

For taking interaction form $C=\beta H\rho _{m}$, according to Eq. (5) one can derive the energy density of matter

$$\begin{aligned} \rho _{m}/\rho _{c}=\Omega _{0m}(1+z)^{3-\beta }. \end{aligned}$$

(31)

Using Eqs. (6) and (31), the density parameter of dark energy can be expressed as

$$\begin{aligned} \rho _{de}/\rho _{c}=\frac{\beta \Omega _{0m}}{3-\beta }(1+z)^{3-\beta }+\frac{3(1-\Omega _{0m})-\beta }{3-\beta }. \end{aligned}$$

(32)

The corresponding Friedmann equation in a non-flat universe is written as

$$\begin{aligned} H^{2}=H_{0}^{2}\left[ \frac{3\Omega _{0m}(1+z)^{3-\beta }}{3-\beta }+\frac{3-\beta -3\Omega _{0m}}{3-\beta }+\Omega _{K}(1+z)^{2}\right] . \end{aligned}$$

(33)

By fitting the combined datasets of SNIa+H(z)+BAO+CMB+$f_{gas}$ to constrain this interacting model, the constraint results on cosmological parameters are plotted in Fig. 3 for the non-flat universe. From Fig. 3, we can see $\beta =-0.0050\,(\pm 0.0056)^{+0.0055+0.0133}_{-0.0056-0.0108},\,\, \Omega _{0m}\!=\!0.282\,(\pm 0.011)^{+0.011+0.023}_{-0.011-0.021}, \Omega _{K}= -0.0018\,(\pm 0.0040)^{+0.0040+0.0079}_{-0.0040-0.0080}$ and $Age(Gyr)=13.649\,(\pm 0.222)^{+0.219+0.440}_{-0.220-0.424}$, with $\chi _\mathrm{min}=602.649$ and $\chi _\mathrm{min}/dof=0.9596$. According to the constraint results, at $1\sigma$ confidence level the non-interacting CC model (interacting parameter $\beta =0$) can not be excluded.

Fig. 3

The $2D$ contours with $1\sigma$ and $2\sigma$ confidence levels and 1-D distribution of interaction model parameters with $C=\beta$H$\rho _{m}$ for using SNIa+H(z)+BAO+CMB+$f_{gas}$ data in a non-flat universe. Solid lines are mean likelihoods of samples, and dotted lines are marginalized probabilities for 1-D distribution