Exploring the full parameter space for an interacting dark energy model with recent observations including redshift-space distortions: Application of the parametrized post-Friedmann approach

Dark energy can modify the dynamics of dark matter if there exists a direct interaction between them. Thus, a measurement of the structure growth, e.g., redshift-space distortions (RSDs), can provide a powerful tool to constrain the interacting dark energy (IDE) models. For the widely studied $Q=3\beta H{\rho }_{\mathrm{de}}$ model, previous works showed that only a very small coupling [$\beta \sim \mathcal{O}({10}^{-3})$] can survive in current RSD data. However, all of these analyses had to assume $w>-1$ and $\beta >0$ due to the existence of the large-scale instability in the IDE scenario. In our recent work [Phys. Rev. D 90, 063005 (2014)], we successfully solved this large-scale instability problem by establishing a parametrized post-Friedmann framework for the IDE scenario. So we, for the first time, have the ability to explore the full parameter space of the IDE models. In this work, we re-examine the observational constraints on the $Q=3\beta H{\rho }_{\mathrm{de}}$ model within the parametrized post-Friedmann framework. By using the Planck data, the baryon acoustic oscillation data, the JLA sample of supernovae, and the Hubble constant measurement, we get $\beta =-0.010_{-0.033}^{+0.037}$ ($1\sigma$). The fit result becomes $\beta =-0.0148_{-0.0089}^{+0.0100}$ ($1\sigma$) once we further incorporate the RSD data in the analysis. The error of $\beta$ is substantially reduced with the help of the RSD data. Compared with the previous results, our results show that a negative $\beta$ is favored by current observations, and a relatively larger interaction rate is permitted by current RSD data.

Figure 1

The CMB temperature power spectrum for the $Q^{\mu }=3\beta H{\rho }_{\mathrm{de}}{u}_c^{\mu }$ model. The red curve denotes the case with $f_{\zeta }(a)$ taken to be the calibrated function in Eq. (a26), while the black curve is the case with $f_{\zeta }(a)=0$. We fix $w=-1.05$, $\beta =-0.01$, and other parameters at the best-fit values from Planck. The overlap of these two curves indicates that the CMB temperature power spectrum does not have the ability to detect the effect of the calibrated $f_{\zeta }(a)$.

Figure 2

The evolutions of matter and metric perturbations for the $Q^{\mu }=3\beta H{\rho }_{\mathrm{de}}{u}_c^{\mu }$ model at $k=0.01{\mathrm{Mpc}}^{-1}$, $k=0.1{\mathrm{Mpc}}^{-1}$, and $k=1.0{\mathrm{Mpc}}^{-1}$. Here, the matter perturbations are the corresponding quantities in the synchronous gauge, and the metric perturbations $\mathrm{\Phi }$ and $\mathrm{\Psi }$ are the gauge-invariant variables of Kodama and Sasaki [43]. We fix $w=-1.05$, $\beta =-0.01$, and other parameters at the best-fit values from Planck. Clearly, all the perturbation evolutions are well behaved.

Figure 3

The one-dimensional marginalized distributions and two-dimensional marginalized 68.3% and 95.4% contours for the parameters in the $Q^{\mu }=3\beta H{\rho }_{\mathrm{de}}{u}_c^{\mu }$ model. Within the PPF framework, the full parameter space can be explored. The degeneracy between $\beta$ and ${\mathrm{\Omega }}_m$ is substantially reduced with the help of the RSD data.