# A Confront between Amati and Combo Correlations at Intermediate and Early Redshifts

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## Abstract

**:**

## 1. Introduction

## 2. The OHD Model-Independent Calibration Method

## 3. The Calibrated Amati and Combo Correlations

## 4. Results from Cosmological Fits

#### 4.1. The $\mathsf{\Lambda}$CDM Model

#### 4.2. The wCDM Model

#### 4.3. Results

## 5. Conclusions and Discussions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**OHD best-fit (solid thick blue) line (black points) with its 1–$\sigma $ (light blue shaded area) and 3–$\sigma $ (blue dashed curves) confidence regions. Reproduced from Ref. [32].

**Figure 2.**Left: the calibrated ${E}_{\mathrm{p}}$–${E}_{\mathrm{iso}}$ correlation (black data), the best-fitting function (blue solid line) and the $1{\sigma}_{\mathrm{A}}$ and $3{\sigma}_{\mathrm{A}}$ limits (dark-gray and light-gray shaded regions, respectively). Right: the distribution of the GRB distance moduli ${\mu}_{\mathrm{A}}$ as obtained from the calibrated Amati correlation. Reproduced from Ref. [32].

**Figure 3.**Left: the calibrated Combo correlation (black data), the best-fit (red solid line) and the $1{\sigma}_{\mathrm{C}}$ and $3{\sigma}_{\mathrm{C}}$ limits (dark-orange and light-orange shaded regions, respectively). Right: the distribution of the GRB distance moduli ${\mu}_{\mathrm{C}}$ as obtained from the calibrated Combo correlation.

**Figure 4.**Plots of the ${\chi}_{\mathrm{A}/\mathrm{C}}^{2}$ distribution (red points) from the MCMC simulation for the $\mathsf{\Lambda}$CDM model. Left: the ${\chi}_{\mathrm{A}}^{2}$ of the Amati correlation; right: the ${\chi}_{\mathrm{C}}^{2}$ of the Combo correlation. The blue points represent the starting point of the MCMC simulation.

**Figure 5.**Plots of the ${\chi}^{2}$ distribution (red points) from the MCMC simulation for the $\mathsf{\Lambda}$CDM model. Left: Amati GRB+SN dataset; right: Combo GRB+SN dataset. The blue points represent the starting point of the MCMC simulation.

**Figure 6.**${\mathsf{\Omega}}_{\mathrm{m}}$–w contour plots for the wCDM model obtained from SNe+Amati GRBs (left), and SNe+Combo GRBs (right). Black dots represent the best-fit values; the 1- and 2–$\sigma $ contours are displayed, from The inner/darker to the outer/lighter areas.

**Table 1.**Best-fit results and errors at 1-$\sigma $ (3–$\sigma $) confidence level of the MCMC simulation for the $\mathsf{\Lambda}$CDM model, with both GRB and GRB+SN samples, and The wCDM model, with GRB+SN sample. The ${\chi}^{2}/$DoF ratios are also indicated.

Sample | Amati | Combo | ||||
---|---|---|---|---|---|---|

w | ${\mathbf{\Omega}}_{\mathit{m}}$ | ${\mathit{\chi}}^{2}/$DoF | w | ${\mathbf{\Omega}}_{\mathit{m}}$ | ${\mathit{\chi}}^{2}/$DoF | |

ΛCDM | ||||||

GRB | $-1$ | ${0.43}_{-0.03\phantom{\rule{0.166667em}{0ex}}(-0.07)}^{+0.03\phantom{\rule{0.166667em}{0ex}}(+0.09)}$ | $1126.9/192$ | $-1$ | ${0.37}_{-0.08\phantom{\rule{0.166667em}{0ex}}(-0.19)}^{+0.08\phantom{\rule{0.166667em}{0ex}}(+0.22)}$ | $49.1/59$ |

GRB+SN | $-1$ | ${0.36}_{-0.01\phantom{\rule{0.166667em}{0ex}}(-0.05)}^{+0.02\phantom{\rule{0.166667em}{0ex}}(+0.05)}$ | $2177.0/1240$ | $-1$ | ${0.30}_{-0.02\phantom{\rule{0.166667em}{0ex}}(-0.06)}^{+0.02\phantom{\rule{0.166667em}{0ex}}(+0.07)}$ | $1084.5/1107$ |

wCDM | ||||||

GRB+SN | $-{1.15}_{-0.20\phantom{\rule{0.166667em}{0ex}}(-0.60)}^{+0.16\phantom{\rule{0.166667em}{0ex}}(+0.45)}$ | ${0.40}_{-0.04\phantom{\rule{0.166667em}{0ex}}(-0.15)}^{+0.04\phantom{\rule{0.166667em}{0ex}}(+0.10)}$ | $2176.3/1239$ | $-{1.12}_{-0.26\phantom{\rule{0.166667em}{0ex}}(-0.77)}^{+0.15\phantom{\rule{0.166667em}{0ex}}(+0.38)}$ | ${0.34}_{-0.04\phantom{\rule{0.166667em}{0ex}}(-0.16)}^{+0.06\phantom{\rule{0.166667em}{0ex}}(+0.16)}$ | $1084.2/1106$ |

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Muccino, M.
A Confront between *Amati* and *Combo* Correlations at Intermediate and Early Redshifts. *Symmetry* **2020**, *12*, 1118.
https://doi.org/10.3390/sym12071118

**AMA Style**

Muccino M.
A Confront between *Amati* and *Combo* Correlations at Intermediate and Early Redshifts. *Symmetry*. 2020; 12(7):1118.
https://doi.org/10.3390/sym12071118

**Chicago/Turabian Style**

Muccino, Marco.
2020. "A Confront between *Amati* and *Combo* Correlations at Intermediate and Early Redshifts" *Symmetry* 12, no. 7: 1118.
https://doi.org/10.3390/sym12071118