Holographic dark energy described at the Hubble length

Ivan Duran and Luca Parisi

Phys. Rev. D 85, 123538 – Published 26 June 2012

Abstract

We consider holographic cosmological models of dark energy in which the infrared cutoff is set by the Hubble’s radius. We show that any interacting dark energy model with a matterlike term able to alleviate the coincidence problem (i.e., with a positive interaction term, regardless of its detailed form) can be recast as a noninteracting model in which the holographic parameter $c^{2}$ evolves slowly with time. Two specific cases are analyzed. First, the interacting model presented by Duran, Pavón, and Zimdahl [J. Cosmol. Astropart. Phys. 07 (2010) 018.] is considered, and its corresponding noninteracting version found. Then, a new noninteracting model, with a specific $c^{2} (z)$ expression, is proposed and analyzed along with its corresponding interacting version. We constrain the parameters of both models using observational data and show that they can be distinguished at the perturbative level.

Received 31 January 2012

DOI:https://doi.org/10.1103/PhysRevD.85.123538

Authors & Affiliations

Ivan Duran^1,* and Luca Parisi^2,†

¹Departament de Física, Universitat Autònoma de Barcelona, Bellaterra, Spain
²Dipartimento di Fisica “E.R. Caianiello,” Università di Salerno, Fisciano (Sa), Italy, and INFN, Sezione di Napoli, GC di Salerno, Fisciano (Sa), Italy

^*ivan.duran@uab.cat
^†parisi@sa.infn.it

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Vol. 85, Iss. 12 — 15 June 2012

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Images

Figure 1
$1 σ$ and $2 σ$ confidence regions for the parameters $A$ and $H_{0}$ of model 1. The plot holds for both scenarios, the interacting and the ${\tilde{c}}^{2} (t)$ (noninteracting). The dot indicates the best fit values.Reuse & Permissions
Figure 2
Left panel: EoS parameter for the interacting ( $w$ thin line and $w_{eff}$ thick line), the ${\tilde{c}}^{2}$ , and the $Λ CDM$ models. Right panel: energy densities ratios, $r \equiv ρ_{M} / ρ_{X}$ , versus $1 + z$ for the $Λ CDM$ , the interacting, and the ${\tilde{c}}^{2}$ models. All the graphs were plotted using the best fit values of the parameters, shown in Table . Solid (green) lines are used for the interacting case, thin dot-dashed (red) lines for the ${\tilde{c}}^{2}$ model, and thick short dashed (blue) for $Λ CDM$ . The $1 σ$ region of the parameters is also plotted, but due to the very small errors, it results nearly inappreciable.Reuse & Permissions
Figure 3
Left panel: EoS parameter for the interacting ( $w$ with thin line and $w_{eff}$ with thick line), the ${\tilde{c}}^{2}$ , and $Λ CDM$ models. Right panel: energy densities ratios, $r \equiv ρ_{M} / ρ_{X}$ , versus $1 + z$ for the $Λ CDM$ , the interacting, and the ${\tilde{c}}^{2}$ models. Notice that the energy densities ratio of the ${\tilde{c}}^{2}$ and $Λ CDM$ models practically overlap. All graphs were drawn using the best fit values of the respective parameters, shown in Table . Solid (green) lines are used for the interacting case, thin dot-dashed (red) lines for the ${\tilde{c}}^{2}$ , and thick short dashed blue for the $Λ CDM$ . The $1 σ$ region of the parameters is also plotted.Reuse & Permissions
Figure 4
Left panel: The $1 σ$ and $2 σ$ confidence regions for the parameters ${\tilde{r}}_{0}$ and $H_{0}$ of model 2. Right panel: The same regions for the parameters $w_{0}$ (equivalent to ${\tilde{w}}_{0}$ and derived from $ϵ = - 3 w_{0}$ ) and ${\tilde{r}}_{0}$ . Notice that ${\tilde{r}}_{0}$ is the ratio of the energy densities just for the ${\tilde{c}}^{2} (t)$ (noninteracting) case; for the interacting case it has no straightforward physical meaning. Central dots indicate the best fit values.Reuse & Permissions
Figure 5
Left panel: the evolution of DM density perturbations versus redshift in model 1. Right panel: The same for model 2. In plotting the graphs the dashed (green) lines describe the matter density perturbations of the interacting scenario, the dot-dashed (red) lines the ${\tilde{c}}^{2}$ , and the solid (blue) line $Λ CDM$ for comparison purposes. Different initial conditions are used for the interacting versions; from top to bottom at the right, $α = 0$ and $β = 0$ , $α = 1$ and $β = 1$ , and $α = 1$ and $β = 10$ . For the ${\tilde{c}}^{2}$ , two different sets of initial conditions are used, but the corresponding graphs practically overlap each other ( $α = 0$ and $β = 0$ , and $α = 1$ and $β = 10$ ). Notice that in the right panel, the ${\tilde{c}}^{2}$ scenario overlaps $Λ CDM$ , as it behaves as a $w CDM$ model with $w = 0.99$ .Reuse & Permissions
Figure 6
Left panel: the evolution of the growth function, $f \equiv \frac{d \ln δ_{M}}{d \ln a}$ , versus redshift for model 1. Right panel: The same for model 2. In plotting the graphs the dashed (green) lines describe the growth function of the interacting scenario, the dot-dashed (red) lines the ${\tilde{c}}^{2}$ , and the solid (blue) line $Λ CDM$ for comparison purposes. Different initial conditions are used for the interacting versions; from top to bottom $α = 0$ and $β = 0$ , $α = 1$ and $β = 1$ , and $α = 1$ and $β = 10$ . For the ${\tilde{c}}^{2}$ , two different sets of initial conditions are used, but they practically overlap each other ( $α = 0$ and $β = 0$ , and $α = 1$ and $β = 10$ ). Notice that in the right panel, the ${\tilde{c}}^{2}$ scenario overlaps $Λ CDM$ , as it behaves as a $w CDM$ model with $w = 0.99$ . Observational data are borrowed from [47].Reuse & Permissions

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