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

TL;DR

This paper addresses constraints on an interacting dark energy model with $Q=3βHρ_{de}$ by applying a generalized parametrized post-Friedmann (PPF) framework to resolve large-scale perturbation instabilities. It develops the IDE perturbation formalism and implements a PPF construction with a function $f_ζ(a)$ and a transition scale $c_Γ$, enabling stable evolution and full exploration of the $(w,β)$ parameter space. By combining Planck, BAO, JLA, $H_0$, and redshift-space distortions (RSD) data, the authors find a preference for $β<0$ and show that RSD significantly tighten the constraints on $β$ and $Ω_m$. Allowing $w$ and $β$ to vary freely within the PPF IDE framework yields different results than previous analyses that fixed $w>-1$ and $β>0$, underscoring the importance of the full parameter space. The study demonstrates a viable path to test IDE models with current observations and motivates future growth measurements to further constrain dark sector interactions.

Abstract

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 (RSD), can provide a powerful tool to constrain the interacting dark energy (IDE) models. For the widely studied $Q=3βHρ_{de}$ model, previous works showed that only a very small coupling ($β\sim\mathcal{O}(10^{-3})$) can survive in current RSD data. However, all these analyses had to assume $w>-1$ and $β>0$ 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 (PPF) 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 reexamine the observational constraints on the $Q=3βHρ_{de}$ model within the PPF framework. By using the Planck data, the baryon acoustic oscillation data, the JLA sample of supernovae, and the Hubble constant measurement, we get $β=-0.010^{+0.037}_{-0.033}$ ($1σ$). The fit result becomes $β=-0.0148^{+0.0100}_{-0.0089}$ ($1σ$) once we further incorporate the RSD data in the analysis. The error of $β$ is substantially reduced with the help of the RSD data. Compared with the previous results, our results show that a negative $β$ is favored by current observations, and a relatively larger interaction rate is permitted by current RSD data.

Figures (3)

• Figure 1: The CMB temperature power spectrum for the $Q^{\mu}=3\beta H\rho_{de}u^{\mu}_c$ model. The red curve denotes the case with $f_\zeta(a)$ taken to be the calibrated function in Eq. (\ref{['eq:fzetafinal']}), 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 $Q^{\mu}=3\beta H\rho_{de}u^{\mu}_c$ model at $k=0.01\,\rm{Mpc^{-1}}$, $k=0.1\,\rm{Mpc^{-1}}$ and $k=1.0\,\rm{Mpc^{-1}}$. Here, the matter perturbations are the corresponding quantities in the synchronous gauge and the metric perturbations $\Phi$ and $\Psi$ are the gauge invariant variables of Kodama and Sasaki Kodama:1985bj. 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 $Q^{\mu}=3\beta H\rho_{de}u^{\mu}_c$ model. Within the PPF framework, the full parameter space can be explored. The degeneracy between $\beta$ and $\Omega_m$ is substantially reduced with the help of the RSD data.