Neutrinos in the holographic dark energy model: constraints from latest measurements of expansion history and growth of structure

TL;DR

The paper investigates holographic dark energy (HDE) models that include massive neutrinos and/or dark radiation, aiming to constrain them with current expansion-history and growth-of-structure data. It computes both background and perturbation evolutions, employing the parametrized post-Friedmann (PPF) framework to handle the $w$ crossing of the phantom divide for $c<1$, and analyzes how the neutrino sector and extra relativistic degrees of freedom modify the CMB and matter power spectra. Using Planck CMB, BAO, JLA SN, direct $H_0$, weak lensing, CMB lensing, and redshift-space distortions, it constrains three model variants: HDE+$ extstyle extstyle extstyle extstyle extstyle nu$, HDE+$N_{ m eff}$, and HDE+$ nu$+$N_{ m eff}$. The results favor a nonzero $N_{ m eff}$ around $3.75$ and place tight upper limits on $ extstyle extstyle extstyle extstyle$ $ extstyle nu$, with $ extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle$ $ extstyle nu<0.186$ eV (95% CL) and $N_{ m eff}=3.75^{+0.28}_{-0.32}$ for the complete data combination, while the HDE parameter $c$ settles near $0.67$–$0.75$ depending on the dataset. These findings demonstrate the power of combining geometric and growth measurements to tighten constraints on dynamical dark energy models with extra relativistic components.

Abstract

The model of holographic dark energy (HDE) with massive neutrinos and/or dark radiation is investigated in detail. The background and perturbation evolutions in the HDE model are calculated. We employ the PPF approach to overcome the gravity instability difficulty (perturbation divergence of dark energy) led by the equation-of-state parameter $w$ evolving across the phantom divide $w=-1$ in the HDE model with $c<1$. We thus derive the evolutions of density perturbations of various components and metric fluctuations in the HDE model. The impacts of massive neutrino and dark radiation on the CMB anisotropy power spectrum and the matter power spectrum in the HDE scenario are discussed. Furthermore, we constrain the models of HDE with massive neutrinos and/or dark radiation by using the latest measurements of expansion history and growth of structure, including the Planck CMB temperature data, the baryon acoustic oscillation data, the JLA supernova data, the Hubble constant direct measurement, the cosmic shear data of weak lensing, the Planck CMB lensing data, and the redshift space distortions data. We find that $\sum m_ν<0.186$ eV (95\% CL) and $N_{\rm eff}=3.75^{+0.28}_{-0.32}$ in the HDE model from the constraints of these data.

Figures (5)

• Figure 1: The evolutions of matter and metric perturbations in the HDE model at $k$= 0.01$\rm\, Mpc^{-1}$, 0.1$\rm \, Mpc^{-1}$, and 1.0$\rm\, Mpc^{-1}$. Here the matter density perturbations are calculated in the synchronous gauge, and the metric perturbations $\Phi$ and $\Psi$ are the gauge-invariant variables. We fix $\sum m_\nu=0.06$ eV and $N_{\rm eff}=3.046$, and other parameters are fixed to be the best-fit values from Planck.
• Figure 2: The CMB anisotropy power spectrum $C_\ell^{TT}$ and the matter power spectrum $P(k)$ in the HDE model with $c=0.8$. In the upper panels, the parameter $\sum m_\nu$ is varied and other parameters are fixed; we choose $\sum m_\nu=0$, 0.5 eV, and 1.0 eV. In the lower panels, the parameter $N_{\rm eff}$ is varied and other parameters are fixed; we choose $\Delta N_{\rm eff}=0$, 0.5, and 1.0.
• Figure 3: The CMB+BAO constraints on the models of HDE+$\sum m_\nu$, HDE+$N_{\rm eff}$, and HDE+$\sum m_\nu$+$N_{\rm eff}$.
• Figure 4: The CMB+BAO+SN+$H_0$ constraints on the models of HDE+$\sum m_\nu$, HDE+$N_{\rm eff}$, and HDE+$\sum m_\nu$+$N_{\rm eff}$.
• Figure 5: The CMB+BAO+SN+$H_0$+WL+RSD constraints on the models of HDE+$\sum m_\nu$, HDE+$N_{\rm eff}$, and HDE+$\sum m_\nu$+$N_{\rm eff}$.