# Cosmological Perturbations in Phantom Dark Energy Models

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## Abstract

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## 1. Introduction

- i
- ii
- iii

## 2. Background Models

- i
- The BR singularity model: a singularity of the type BR [8,9,10] can be induced by a perfect fluid whose EoS parameter is constant and smaller than $-1$.$$\begin{array}{cccccc}\hfill {p}_{\mathrm{d}}& ={w}_{\mathrm{d}}{\rho}_{\mathrm{d}}\phantom{\rule{0.166667em}{0ex}},\hfill & \hfill {\rho}_{\mathrm{d}}\left(a\right)& ={\rho}_{\mathrm{d}0}\phantom{\rule{0.166667em}{0ex}}{a}^{-3\left(1+{w}_{\mathrm{d}}\right)}\phantom{\rule{0.166667em}{0ex}},\hfill & \hfill {w}_{\mathrm{d}}& <-1\phantom{\rule{0.166667em}{0ex}}.\hfill \end{array}$$
- ii
- The LR abrupt event model: an event of the type LR [11,12,13] can be caused by a perfect fluid whose EoS and energy density can be written as [12]$$\begin{array}{cccccc}\hfill {p}_{\mathrm{d}}& =-\left({\rho}_{\mathrm{d}}+B{\rho}_{\mathrm{d}}^{1/2}\right)\phantom{\rule{0.166667em}{0ex}},\hfill & \hfill {\rho}_{\mathrm{d}}\left(a\right)& ={\rho}_{\mathrm{d}0}{\left[\frac{3}{2}\sqrt{\frac{{\mathrm{\Omega}}_{\mathrm{d}0}}{{\mathrm{\Omega}}_{*}}}ln\left(\frac{a}{{a}_{0}}\right)+1\right]}^{2}\phantom{\rule{0.166667em}{0ex}},\hfill & \hfill B& >0\phantom{\rule{0.166667em}{0ex}},\hfill \end{array}$$
- iii
- The LSBR abrupt event model: the event denominated as LSBR can be induced by a perfect fluid whose EoS deviates from that of a cosmological constant by adding a constant parameter [14]. Therefore, the corresponding EoS and energy density in terms of the scale factor can be written as$$\begin{array}{cccccc}\hfill {p}_{\mathrm{d}}& =-\left({\rho}_{\mathrm{d}}+\frac{A}{3}\right)\phantom{\rule{0.166667em}{0ex}},\hfill & \hfill {\rho}_{\mathrm{d}}\left(a\right)& ={\rho}_{\mathrm{d}0}+Aln\left(\frac{a}{{a}_{0}}\right)\phantom{\rule{0.166667em}{0ex}},\hfill & \hfill A& >0\phantom{\rule{0.166667em}{0ex}}.\hfill \end{array}$$Note that ${\rho}_{\mathrm{d}0}$ plays the role of a cosmological constant. In this case, we have chosen the numerical value taken in [14]: $A={10}^{-3}\left(3{H}_{0}^{2}/8\pi G\right)$, while for ${\mathrm{\Omega}}_{\mathrm{m}0}$ and ${H}_{0}$, we have used the same values as in the case for the BR.

## 3. Perturbed Equations

## 4. Results

## 5. Discussion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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^{1.}In this work we disregard any anisotropy at the linear level of the scalar perturbations, therefore, from this point onward we will set $\mathrm{\Psi}=\mathrm{\Phi}$ [27].^{2.}From now on, all perturbed quantities referred to in the manuscript represent the Fourier transform action of such perturbations, even though no specific notation is used, e.g., $\mathrm{\Psi}=\mathrm{\Psi}(x,k)$, where k is the wave-number.

**Figure 1.**The first, second, and third row of this figure presents the evolution of dark matter (DM), dark energy (DE), and the gravitational potential perturbations, respectively. Each column belongs to a particular model, while each colour represents a mode where the values of the corresponding wave-number are: $k=\phantom{\rule{3.33333pt}{0ex}}3.33\times {10}^{-4}\phantom{\rule{4pt}{0ex}}\mathrm{h}\xb7{\mathrm{Mpc}}^{-1}$ (purple); $k=7.93\times {10}^{-4}\phantom{\rule{4pt}{0ex}}\mathrm{h}\xb7{\mathrm{Mpc}}^{-1}$ (dark blue); $k=3.50\times {10}^{-3}\phantom{\rule{4pt}{0ex}}\mathrm{h}\xb7{\mathrm{Mpc}}^{-1}$ (light blue); $k=1.54\times {10}^{-2}\phantom{\rule{4pt}{0ex}}\mathrm{h}\xb7{\mathrm{Mpc}}^{-1}$ (green); $k=6.80\times {10}^{-2}\phantom{\rule{4pt}{0ex}}\mathrm{h}\xb7{\mathrm{Mpc}}^{-1}$ (orange); $k=0.30\phantom{\rule{4pt}{0ex}}\mathrm{h}\xb7{\mathrm{Mpc}}^{-1}$ (red).

**Figure 2.**The relative deviation of the matter power spectrum of: model (i) in red; model (ii) in blue; and model (iii) in black; with regards to the predictions of ΛCDM (CDM: cold dark matter). The model that induces a Big Rip (BR) singularity shows the highest deviation, while the model that induces a Little Sibling of the Big Rip (LSBR) abrupt event presents the smallest deviation with respect to the ΛCDM model.

**Figure 3.**The evolution of $f{\sigma}_{8}$ in terms of the redshift for each model (black curve) together with the corresponding ΛCDM model (red curve). The blue dots and corresponding error bars indicate observational values ( please, see [33] and references therein).

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**MDPI and ACS Style**

Albarran, I.; Bouhmadi-López, M.; Morais, J.
Cosmological Perturbations in Phantom Dark Energy Models. *Universe* **2017**, *3*, 22.
https://doi.org/10.3390/universe3010022

**AMA Style**

Albarran I, Bouhmadi-López M, Morais J.
Cosmological Perturbations in Phantom Dark Energy Models. *Universe*. 2017; 3(1):22.
https://doi.org/10.3390/universe3010022

**Chicago/Turabian Style**

Albarran, Imanol, Mariam Bouhmadi-López, and João Morais.
2017. "Cosmological Perturbations in Phantom Dark Energy Models" *Universe* 3, no. 1: 22.
https://doi.org/10.3390/universe3010022