# Status on Bidimensional Dark Energy Parameterizations Using SNe Ia JLA and BAO Datasets

## Abstract

**:**

## 1. Introduction

## 2. Modeling Dark Energy

## 3. Bidimensional Dark Energy Models

#### 3.1. Lambda Cold Dark Matter-Redshift Parameterization (ΛCDM)

#### 3.2. Linear-Redshift Parameterization

#### 3.3. Chevallier-Polarski-Linder Parameterization (CPL)

#### 3.4. Barboza-Alcaniz Parameterization (BA)

#### 3.5. Low Correlation Parameterization (LC)

#### 3.6. Jassal-Bagla-Padmanabhan Parameterization (JBP)

#### 3.7. Wetterich-Redshift Parameterization (WP)

## 4. Observational Data

#### 4.1. Analysis Using SNe Ia Data

#### 4.2. Analysis Using BAO Data

## 5. Bayesian Evidence

## 6. Results

#### 6.1. About the Likelihood and Tension

#### 6.2. About the Figure of Merit (FoM)

#### 6.3. About the Bayesian Evidence

## 7. Conclusions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**1 and 2σ confidence contours for dark energy parameterizations. Ia supernovae Joint Lightcurve Analysis (SNe Ia JLA) is represented by the green region, the baryon acoustic oscillations (BAO) by the purple region and SNe Ia JLA+BAO by the red region. The best fits are indicated by the points for each sample, respectively. The point where the dashed line cross indicates the concordance model (ΛCDM).

**Figure 2.**$E{\left(z\right)}^{2}$ evolution function for each dark energy parameterizations. We use the best fit obtained in each parameterizations with the SNe Ia JLA+BAO joined dataset. Left: Evolution of Equation (3) and the bidimensional dark energy parameterizations Equations (5), (7) and (15). Right: Evolution of the Equations (3) and the bidimensional dark energy parameterizations (9)–(13) (with ${z}^{2}$-terms in $w\left(z\right)$).

**Table 1.**Dark energy parameterizations with best fits and $\sigma -$distances values using SNe Ia JLA data.

Model | Parameterization | ${\mathit{d}}_{\mathit{\sigma}}^{\mathbf{\Lambda}\mathit{C}\mathit{D}\mathit{M}}$ | Best Fit Parameters using SNe Ia JLA data |
---|---|---|---|

LCDM | ${H}^{2}\left(z\right)={H}_{0}^{2}[{\mathrm{\Omega}}_{m}{(1+z)}^{3}+(1-{\mathrm{\Omega}}_{m})]$ | − | ${\mathrm{\Omega}}_{m}=0.295\pm 0.034$ |

Linear | ${H}^{2}\left(z\right)={H}_{0}^{2}[{\mathrm{\Omega}}_{m}{(1+z)}^{3}+(1-{\mathrm{\Omega}}_{m}){(1+z)}^{3(1+{w}_{0}+{w}_{1})}{e}^{-3{w}_{1}z}]$ | $0.285$ | ${w}_{0}=-0.991\pm 0.036,\phantom{\rule{0.277778em}{0ex}}{w}_{1}=0.297\pm 0.779$ |

CPL | ${H}^{2}\left(z\right)={H}_{0}^{2}[{\mathrm{\Omega}}_{m}{(1+z)}^{3}+(1-{\mathrm{\Omega}}_{m}){(1+z)}^{3(1+{w}_{0}+{w}_{1})}$ | ||

$\times {e}^{\frac{-3{w}_{1}z}{1+z}}]$ | $0.258$ | ${w}_{0}=-0.997\pm 0.049,\phantom{\rule{0.277778em}{0ex}}{w}_{1}=-0.337\pm 1.822$ | |

BA | ${H}^{2}\left(z\right)={H}_{0}^{2}[{\mathrm{\Omega}}_{m}{(1+z)}^{3}+(1-{\mathrm{\Omega}}_{m}){(1+z)}^{3(1+{w}_{0})}$ | ||

$\times {(1+{z}^{2})}^{3{w}_{1}/2}$ | $0.243$ | ${w}_{0}=-0.993\pm 0.034,\phantom{\rule{0.277778em}{0ex}}{w}_{1}=-0.245\pm 0.545$ | |

LC | ${H}^{2}\left(z\right)={H}_{0}^{2}[{\mathrm{\Omega}}_{m}{(1+z)}^{3}+(1-{\mathrm{\Omega}}_{m}){(1+z)}^{3(1-2{w}_{0}+3{w}_{0.5})}$ | ||

$\times {e}^{\left(\right)}$ | $0.258$ | ${w}_{0}=-0.997\pm 0.049,\phantom{\rule{0.277778em}{0ex}}{w}_{0.5}=-1.109\pm 0.066$ | |

JBP | ${H}^{2}\left(z\right)={H}_{0}^{2}[{\mathrm{\Omega}}_{m}{(1+z)}^{3}+(1-{\mathrm{\Omega}}_{m}){(1+z)}^{3(1+{w}_{0})}$ | ||

$\times {e}^{\frac{3{w}_{1}{z}^{2}}{2{(1+z)}^{2}}}]$ | $0.236$ | ${w}_{0}=-1.013\pm 0.070,\phantom{\rule{0.277778em}{0ex}}{w}_{1}=-0.295\pm 4.306$ | |

WP | ${H}^{2}\left(z\right)={H}_{0}^{2}\left(\right)open="\{"\; close="\}">{\mathrm{\Omega}}_{m}{(1+z)}^{3}+(1-{\mathrm{\Omega}}_{m}){(1+z)}^{3\left(\right)open="["\; close="]">1+\frac{{w}_{0}}{1+{w}_{1}ln(1+z)}}$ | $0.278$ | ${w}_{0}=-0.987\pm 0.040,\phantom{\rule{0.277778em}{0ex}}{w}_{1}=-0.169\pm 0.258$ |

**Table 2.**Dark energy parameterizations with best fits and σ-distances values using BAO and the combining samples.

Model | Best Fit Parameters using BAO data | ${\mathit{d}}_{\mathit{\sigma}}^{\mathbf{\Lambda}\mathit{C}\mathit{D}\mathit{M}}$ | Best Fit Parameters using SNe Ia JLA+BAO data | ${\mathit{d}}_{{\mathit{\sigma}}_{\mathit{T}\mathit{o}\mathit{t}\mathit{a}\mathit{l}}}^{\mathbf{\Lambda}\mathit{C}\mathit{D}\mathit{M}}$ |
---|---|---|---|---|

Linear | ${w}_{0}=-0.605\pm 0.130,\phantom{\rule{0.277778em}{0ex}}{w}_{1}=2.078\pm 4.063$ | $0.610$ | ${w}_{0}=-0.888\pm 0.025,\phantom{\rule{0.277778em}{0ex}}{w}_{1}=0.645\pm 0.650$ | $0.380$ |

CPL | ${w}_{0}=-0.540\pm 0.184,\phantom{\rule{0.277778em}{0ex}}{w}_{1}=-3.105\pm 9.327$ | $0.594$ | ${w}_{0}=-0.878\pm 0.034,\phantom{\rule{0.277778em}{0ex}}{w}_{1}=-0.894\pm 1.487$ | $0.323$ |

BA | ${w}_{0}=-0.621\pm 0.119,\phantom{\rule{0.277778em}{0ex}}{w}_{1}=-1.707\pm 2.731$ | $0.600$ | ${w}_{0}=-0.892\pm 0.024,\phantom{\rule{0.277778em}{0ex}}{w}_{1}=-0.535\pm 0.450$ | $0.316$ |

LC | ${w}_{0}=-0.540\pm 0.184,\phantom{\rule{0.277778em}{0ex}}{w}_{0.5}=-1.575\pm 0.359$ | $0.594$ | ${w}_{0}=-0.878\pm 0.034,\phantom{\rule{0.277778em}{0ex}}{w}_{0.5}=-1.175\pm 0.054$ | >1 |

JBP | ${w}_{0}=-0.456\pm 0.274,\phantom{\rule{0.277778em}{0ex}}{w}_{1}=-4.653\pm 21.910$ | $0.569$ | ${w}_{0}=-0.869\pm 0.049,\phantom{\rule{0.277778em}{0ex}}{w}_{1}=-1.196\pm 3.441$ | $0.257$ |

WP | ${w}_{0}=-0.670\pm 0.046,\phantom{\rule{0.277778em}{0ex}}{w}_{1}=-0.941\pm 0.363$ | $0.626$ | ${w}_{0}=-0.882\pm 0.022,\phantom{\rule{0.277778em}{0ex}}{w}_{1}=-0.375\pm 0.165$ | $0.386$ |

$\mathbf{Model}$ | FoM | ||
---|---|---|---|

SNe Ia JLA | BAO | SNe Ia+BAO | |

Linear | $14.203$ | $7.015$ | $23.657$ |

CPL | $9.312$ | $4.631$ | $15.681$ |

LC | $27.936$ | $13.893$ | $47.043$ |

BA | $16.981$ | $8.555$ | $28.437$ |

JBP | $6.076$ | $3.024$ | $10.337$ |

WP | $26.208$ | $33.996$ | $53.677$ |

$\mathbf{Model}$ | Bayes Factor $\mathbf{ln}{\mathit{B}}_{\mathit{i}\mathit{j}}$ | ||
---|---|---|---|

SNe Ia JLA | BAO | SNe Ia JLA+BAO | |

Linear | $1.904$ | $1.897$ | $1.857$ |

CPL | $1.912$ | $1.903$ | $1.875$ |

LC | $1.912$ | $1.903$ | $1.875$ |

BA | $1.921$ | $1.921$ | $1.921$ |

JBP | $1.918$ | $1.912$ | $1.854$ |

WP | $1.906$ | $1.891$ | $1.855$ |

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**MDPI and ACS Style**

Escamilla-Rivera, C.
Status on Bidimensional Dark Energy Parameterizations Using SNe Ia JLA and BAO Datasets. *Galaxies* **2016**, *4*, 8.
https://doi.org/10.3390/galaxies4030008

**AMA Style**

Escamilla-Rivera C.
Status on Bidimensional Dark Energy Parameterizations Using SNe Ia JLA and BAO Datasets. *Galaxies*. 2016; 4(3):8.
https://doi.org/10.3390/galaxies4030008

**Chicago/Turabian Style**

Escamilla-Rivera, Celia.
2016. "Status on Bidimensional Dark Energy Parameterizations Using SNe Ia JLA and BAO Datasets" *Galaxies* 4, no. 3: 8.
https://doi.org/10.3390/galaxies4030008