Holographic dark energy in a universe with spatial curvature and massive neutrinos: a full Markov Chain Monte Carlo exploration

TL;DR

This study tests holographic dark energy in a non-flat universe with massive neutrinos by performing a full MCMC global fit to WMAP7, Union2.1, BAO, and $H_0$ data, using the PPF framework to handle dark-energy perturbations. The analysis shows that in the simplest HDE case $c<1$ at $>4\sigma$, implying a Big Rip future, but including spatial curvature relaxes this to about $2.5\sigma$; the inclusion of massive neutrinos does not significantly alter the constraint on $c$. Across the four models, the data imply tight limits on $\Omega_{k0}$ (near zero with $2\sigma$ bounds at the $10^{-2}$ level) and bound the total neutrino mass $\sum m_\nu$ (e.g., $<0.48$ eV at $2\sigma$ in VHDE), with clear degeneracies such as between $\Omega_{k0}$ and $c$ and between $\Omega_{k0}$ and $\sum m_\nu$ that broaden the allowed neutrino mass range when both are included. The study also demonstrates that using the full WMAP7 data yields stronger constraints than WMAP distance priors. These results underscore the importance of accounting for curvature and neutrinos in holographic dark-energy analyses and validate the PPF approach for dynamical dark energy perturbations.

Abstract

In this paper, we report the results of constraining the holographic dark energy model with spatial curvature and massive neutrinos, based on a Markov Chain Monte Carlo global fit technique. The cosmic observational data include the full WMAP 7-yr temperature and polarization data, the type Ia supernova data from Union2.1 sample, the baryon acoustic oscillation data from SDSS DR7 and WiggleZ Dark Energy Survey, and the latest measurements of $H_0$ from HST. To deal with the perturbations of dark energy, we adopt the parameterized post-Friedmann method. We find that, for the simplest holographic dark energy model without spatial curvature and massive neutrinos, the phenomenological parameter $c<1$ at more than $4σ$ confidence level. The inclusion of spatial curvature enlarges the error bars and leads to $c<1$ only in about $2.5σ$ range; in contrast, the inclusion of massive neutrinos does not have significant influence on $c$. We also find that, for the holographic dark energy model with spatial curvature but without massive neutrinos, the $3σ$ error bars of the current fractional curvature density $Ω_{k0}$ are still in order of $10^{-2}$; for the model with massive neutrinos but without spatial curvature, the $2σ$ upper bound of the total mass of neutrinos is $\sum m_ν < 0.48$ eV. Moreover, there exists clear degeneracy between spatial curvature and massive neutrinos in the holographic dark energy model, which enlarges the upper bound of $\sum m_ν$ by more than 2 times. In addition, we demonstrate that, making use of the full WMAP data can give better constraints on the holographic dark energy model, compared with the case using the WMAP ``distance priors''.

Figures (10)

• Figure 1: The evolution of the dimensionless Hubble expansion rate $H(z)/H_0$ in the holographic dark energy model. As an example, we show the cases with $c=0.7$, $c=1.0$, and $c=1.5$. For a comparison, the case in the $\Lambda$CDM model is also plotted. For the other model parameters, we adopt their best-fit values given by the WMAP 7-yr observations.
• Figure 2: The CMB $C_l^{TT}$ power spectrum in the holographic dark energy model. The black dots with error bars denote the observed data with their corresponding uncertainties from WMAP 7-yr results. The solid lines denote the holographic dark energy models with different $c$, and the dotted lines denote the XCDM models with different $w$. For the other model parameters, we adopt their best-fit values given by the WMAP 7-yr observations.
• Figure 3: Evolution of the density fields in the synchronous gauge for $k=0.1$ Mpc$^{-1}$ in the holographic dark energy model with $c=1$. The pink solid line, orange short-dashed line, and cyan dashed line denote the density perturbations of dark matter, baryon, and holograph dark energy, respectively. For the other model parameters, we adopt their best-fit values given by the WMAP 7-yr observations.
• Figure 4: Left panel: the evolution of the density perturbations of cold dark matter for $k=0.1$ Mpc$^{-1}$. Right panel: the matter power spectra at $z=0$. The cases of $c=0.7$, $c=1.0$, and $c=1.5$ in the holographic dark energy model are shown. For the other model parameters, we adopt their best-fit values given by the WMAP 7-yr observations.
• Figure 5: The 1D marginalized distributions of individual parameters and 2D marginalized 1--3$\sigma$ CL contours, for the HDE model.