Constraints on the interacting holographic dark energy models: implications from background and perturbations data

TL;DR

This work evaluates interacting holographic dark energy (IHDE) models with three DE–DM coupling forms against background and perturbation data, using a two-step approach: first constraining background parameters via MCMC with Pantheon+, BAO, CMB, and H(z) data, then incorporating growth-rate data through fσ_8(z) to assess structure formation. The IHDE framework ties holographic DE to a future event horizon cutoff and introduces Q_I to model energy transfer, yielding a background evolution for Ω_de and H, as well as a perturbation system that depends on DE clustering via the effective sound speed c_eff. Across models, the EoS w_de remains in quintessence at high redshift but tends toward phantom near the present, with IHDE-II and IHDE-III crossing phantom earlier than IHDE-I and HDE; growth data show HDE generally aligns best with ΛCDM in fσ_8(z) while IHDE variants diverge. Model comparison via AIC/BIC reveals that the data choice (background vs background+growth) and the assumed DE homogeneity vs clustering substantially affect which IHDE variant is favored, with homogeneous DE generally providing better agreement for several datasets. Overall, the study demonstrates that combining background and growth information, while accounting for DE perturbations, can meaningfully constrain IHDE scenarios and reveal the nuanced impact of DE clustering on cosmic expansion and structure growth.

Abstract

In this study, we employ a two-step method to analyze models of holographic dark energy (HDE) and interacting holographic dark energy (IHDE), incorporating three distinct dark energy (DE)-dark matter (DM) interaction terms. First, using the latest background dataset, we conduct a Markov chain Monte Carlo (MCMC) analysis to constrain the free parameters of the models. Then, we assess the models against each other using the key background parameters and compare them to the $Λ$CDM standard model. Our results show that at high redshifts, the equation of state (EoS) parameter related to the models for both homogeneous and clustered DE cases falls within the quintessence region. However, as we approach the present time, all models except HDE transition into the phantom region, and two models cross the phantom line earlier than others. In the next step, we focus on the evolution of perturbations in DE and DM. Using background and growth rate data, we constrain parameters including $σ_8$. We then investigate the evolution of the growth rate of matter perturbations, $fσ_8(z)$, and its deviation, $Δf σ_8$, from the $Λ$CDM model. The HDE model shows the best agreement with observational data, while other models predict varying growth rates compared to $Λ$CDM. Finally, we demonstrate through Akaike and Bayesian information criteria (AIC and BIC) analysis that the compatibility of models with observational data depends on the type of data used, the DE-DM interaction term, and the assumptions regarding DE homogeneity and clustering. Our results suggest that homogeneous DE models yield more agreement with observational data than clustered DE models.

Figures (4)

• Figure 1: Top left: evolution of the EoS parameter for the HDE and different IHDE models as a function of redshift z investigated in this work. Bottom left : evolution of the deceleration parameter for various models as a function of $z$ [see Eq. (\ref{['q2zzz']})]. Top right: comparison of cosmic chronometer data points from Table \ref{['tabHdata']} with the theoretical Hubble parameter evolution for HDE and IHDE models with respect to redshift $z.$ Bottom right: $\Delta E(\%)$ of the models compared to the $\Lambda$CDM model. The various models have been specified by different colors and line styles in the inner panels of the figure. The dashed (solid) line represents the homogeneous (clustered) case of DE.
• Figure 2: Top: a comparison of observational growth rate data points with the theoretical prediction of the growth rate $f\sigma_{8}(z)$ as a function of redshift $z$ [refer to Sec \ref{['growth']}]. Bottom: deviations in theoretical predictions of $f(z) \sigma_8(z)$ among $\Lambda$CDM and models in question, normalized to $\Lambda$CDM values [see Eq. (\ref{['dels8']})].
• Figure 3: Confidence levels of the $1\sigma$ and $2\sigma$ limits for the IHDE-I (left panel), IHDE-II (middle panel), and IHDE-III (right panel) models. These confidence levels are determined using the background dataset alone for clustered (blue) and homogeneous DE (brown) scenarios. In addition, the combined background and growth rate dataset is utilized for clustered (red) and homogeneous DE (green) scenarios.
• Figure 4: Confidence levels illustrated for the $1\sigma$ and $2\sigma$ limits concerning the HDE and IHDE models. The upper panels display the HDE (left) and IHDE-I (right) models, while the lower panels feature the IHDE-II (left) and IHDE-III (right) models. These confidence levels have been established using solely the background dataset for both clustered (blue) and homogeneous DE (brown) scenarios. Furthermore, a combined background and growth rate dataset has been employed for both clustered (red) and homogeneous DE (green) scenarios. For more details, refer to Eqs. (\ref{['xi22']}) and (\ref{['xi222']}), along with Tables \ref{['tabbacnu']} and \ref{['tabclus']} for numerical values.