# On the Evolution of the Hubble Constant with the SNe Ia Pantheon Sample and Baryon Acoustic Oscillations: A Feasibility Study for GRB-Cosmology in 2030

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## Abstract

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## 1. Introduction

## 2. SNe Ia Cosmology

## 3. The Contribution of BAOs

## 4. Multidimensional Binned Analysis with SNe Ia and BAOs

## 5. Perspective of the Future Contribution of GRB-Cosmology in 2030

## 6. Discussions on the Results

#### 6.1. Astrophysical Effects

#### 6.2. Theoretical Interpretations

#### 6.2.1. The Scalar Tensor Theory of Gravity

#### 6.2.2. Metric f(R) Gravity in the Jordan Frame

## 7. The Binned Analysis with Modified f(R) Gravity

#### Hu–Sawicki Model

## 8. Requirements for a Suitable f(R) Model

#### 8.1. An Example for Low Redshifts

#### 8.2. Discussion

## 9. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Notes

1 | https://github.com/dscolnic/Pantheon (accessed on 21 December 2020). |

2 | The code is available upon request. |

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**Figure 1.**

**Left panel**The ${H}_{0}\left(z\right)$ vs. z wit varying also ${\Omega}_{0}M$. The red color indicates the case with only SNe Ia as probes, while the blue refers to the case of SNe + 1 BAO per bin. This color-coded will be applied also in the right panel.

**Right**. The same plot for the ${w}_{0}{w}_{a}$CDM model, considering the local fiducial value ${H}_{0}=70$, where both ${H}_{0}$ and ${w}_{a}$ are left free to vary.

**Figure 2.**

**Upper left.**An example of 1000 simulated GRBs with the posterior distribution of the fundamental plane parameters a, b, c, and its intrinsic scatter $\sigma $ together with the total matter density parameter ${\Omega}_{0m}$. In this case, the steps of the simulation are 9500 and the errors on the variables of the fundamental plane have not been divided by any factor ($n=1$).

**Upper right.**The same case of the upper left panel, but considering 2000 GRBs and a number of steps of 13,000.

**Lower left.**The results of 1500 simulated GRBs, dividing by two the errors on the fundamental plane variables (halved errors, $n=2$): here the steps are 11,100.

**Lower right.**The same result of the lower-left panel, considering 2000 GRBs and 14,600 steps instead.

**Figure 3.**Profile of the scalar field potential $V\left(\varphi \right)$ in the JF equivalent scalar-tensor formalism of the f(R) modified gravity. The form of $V\left(\varphi \right)$) is inferred from the behavior of ${H}_{0}\left(z\right)$ (Equation (9)). Note that $V\left(\varphi \right)/{m}^{2}$ is a dimensionless quantity. A flat profile of $V\left(\varphi \right)$ occurs only at low redshifts, for $0<z\lesssim 0.3$ or equivalently $\varphi \lesssim 1.005$. Note, also, the non-linearity of the scale for the redshift axis on top, considering the relation (22) between $\varphi $ and z. In this plot, $\eta =0.009$.

**Figure 4.**The numerical solution for Equation (35) (blue dashed curve) plotted together with its polynomial fitting (green continuous curve) in the case of ${F}_{R0}=-{10}^{-7}$. The assumption of a function of redshift in the form of a order-8 polynomial allows an accurate fit for the numerical values. The same fitting procedure has been used in the ${F}_{R0}=-{10}^{-4}$ case.

**Figure 5.**The first four panels deal with ${H}_{0}$ vs. z for SNe, the four bottom panels include BAO measurements for the H-S model. The upper 4 panels show from the left to the right ${F}_{R0}=-{10}^{-7},-{10}^{-6},-{10}^{-5},-{10}^{-4}$, respectively. The standard $\Lambda $CDM cosmology is shown in red and the Hu–Sawicki model in blue. Analogously, the bottom panels have the same notation about the values of ${F}_{R0}$.

**Figure 6.**The Hubble constant versus redshift plots for the three bins of SNe Ia only, considering the Hu–Sawicki model.

**Upper left panel.**The condition of ${F}_{R0}=-{10}^{-7}$ is applied to the case of SNe only, with the different values of ${\Omega}_{0m}=0.301,0.303,0.305$.

**Upper right panel.**The same of the upper left, but with the contribution of BAOs.

**Lower left panel.**The SNe only case with the ${F}_{R0}=-{10}^{-4}$ condition, considering the different values of ${\Omega}_{0m}=0.301,0.303,0.305$.

**Lower right panel.**The same as the lower left, but with the contribution of BAOs. The orange color refers to ${\Omega}_{0m}=0.301$, the red to ${\Omega}_{0m}=0.303$, the magenta to ${\Omega}_{0m}=0.305$, and the blue to ${\Omega}_{0m}=0.298$.

**Table 1.**

**Upper half.**Fit parameters of ${H}_{0}\left(z\right)$ for three bins (flat $\Lambda $CDM model, varying ${H}_{0}$ and ${\Omega}_{0m}$) in the cases with SNe only and with the SNe + BAOs contribution. The columns are: (1) the number of bins; (2) ${\mathcal{H}}_{0}$, (3) $\eta $; (4) how many $\sigma $s the evolutionary parameter $\eta $ is compatible with zero (namely, $\eta /{\sigma}_{\eta}$).

**Lower half.**Similarly to the upper half, the lower half shows the fit parameters of ${H}_{0}\left(z\right)$ (flat ${w}_{0}{w}_{a}$CDM model, varying ${H}_{0}$ and ${w}_{a}$) without and with the BAOs.

Flat $\Lambda $CDM model, without BAOs, varying ${\mathit{H}}_{0}$ and ${\Omega}_{0\mathit{m}}$ | |||

Bins | ${\mathcal{H}}_{0}$ | $\eta $ | $\frac{\eta}{{\sigma}_{\eta}}$ |

3 | $70.093\pm 0.102$ | $0.009\pm 0.004$ | $2.0$ |

Flat $\Lambda $CDM model, including BAOs, varying ${H}_{0}$ and ${\Omega}_{0m}$ | |||

Bins | ${\mathcal{H}}_{0}$ | $\eta $ | $\frac{\eta}{{\sigma}_{\eta}}$ |

3 | $70.084\pm 0.148$ | $0.008\pm 0.006$ | $1.2$ |

Flat ${w}_{0}{w}_{a}$CDM model, without BAOs, varying ${H}_{0}$ and ${w}_{a}$ | |||

Bins | ${\mathcal{H}}_{0}$ | $\eta $ | $\frac{\eta}{{\sigma}_{\eta}}$ |

3 | $69.847\pm 0.119$ | $0.034\pm 0.006$ | $5.7$ |

Flat ${w}_{0}{w}_{a}$CDM model, including BAOs, varying ${H}_{0}$ and ${w}_{a}$ | |||

Bins | ${\mathcal{H}}_{0}$ | $\eta $ | $\frac{\eta}{{\sigma}_{\eta}}$ |

3 | $69.821\pm 0.126$ | $0.033\pm 0.005$ | $5.8$ |

**Table 2.**Fitting parameters of ${H}_{0}\left(z\right)$ for three bins within the Hu–Sawicki model, with SNe only and SNe + BAOs with a fixed value of ${\Omega}_{0m}=0.298$ and with several values of ${F}_{R0}:-{10}^{-4},-{10}^{-5},-{10}^{-6},-{10}^{-7}$. The columns contains: (1) the number of bins; (2) ${\mathcal{H}}_{0}$, (3) is $\eta $, according to Equation (9); (4) how many $\sigma $s $\eta $ is compatible with zero (namely, the ratio $\eta /{\sigma}_{\eta}$); (5) ${F}_{R0}$ values; (6) the sample used.

Hu–Sawicki Model, Results of the Redshift Binned Analysis | |||||
---|---|---|---|---|---|

Bins | ${\mathcal{H}}_{\mathbf{0}}$ | $\mathbf{\eta}$ | $\frac{\mathbf{\eta}}{{\mathbf{\sigma}}_{\mathbf{\eta}}}$ | ${\mathbf{F}}_{\mathbf{R}\mathbf{0}}$ | Sample |

3 | $70.089\pm 0.144$ | $0.008\pm 0.006$ | $1.2$ | $-{10}^{-4}$ | SNe |

3 | $70.127\pm 0.128$ | $0.008\pm 0.006$ | $1.4$ | $-{10}^{-4}$ | SNe + BAOs |

3 | $70.045\pm 0.052$ | $0.007\pm 0.002$ | $3.0$ | $-{10}^{-5}$ | SNe |

3 | $70.062\pm 0.132$ | $0.007\pm 0.005$ | $1.3$ | $-{10}^{-5}$ | SNe + BAOs |

3 | $70.125\pm 0.046$ | $0.010\pm 0.002$ | $5.4$ | $-{10}^{-6}$ | SNe |

3 | $70.115\pm 0.153$ | $0.008\pm 0.007$ | $12.1$ | $-{10}^{-6}$ | SNe + BAOs |

3 | $70.118\pm 0.131$ | $0.011\pm 0.006$ | $1.9$ | $-{10}^{-7}$ | SNe |

3 | $70.053\pm 0.150$ | $0.007\pm 0.007$ | $1.1$ | $-{10}^{-7}$ | SNe + BAOs |

**Table 3.**Fitting parameters of ${H}_{0}\left(z\right)$ for three bins within the Hu–Sawicki model, with SNe and SNe + BAOs by fixing several values of ${\Omega}_{M}=0.298,0.303,0.301,0.305$ and values of ${F}_{R0}=-{10}^{-4}$ and ${F}_{R0}=-{10}^{-7}$. The columns are as follows: (1) the ${\Omega}_{0m}$ value; (2) ${\mathcal{H}}_{0}$, (3) $\eta $, according to Equation (9); (4) how many $\sigma $s the evolutionary parameter $\eta $ is compatible with zero (namely, $\eta /{\sigma}_{\eta}$); (5) ${F}_{R}0$; (6) the sample used.

Hu–Sawicki Model, Results of the 3 Bins Analysis | |||||
---|---|---|---|---|---|

${\Omega}_{\mathbf{0}\mathbf{m}}$ | ${\mathcal{H}}_{\mathbf{0}}$ | $\mathbf{\eta}$ | $\frac{\mathbf{\eta}}{{\mathbf{\sigma}}_{\mathbf{\eta}}}$ | ${\mathbf{F}}_{\mathbf{R}\mathbf{0}}$ | Sample |

0.298 | $70.140\pm 0.045$ | $0.011\pm 0.002$ | $5.1$ | $-{10}^{-7}$ | SNe |

0.298 | $70.050\pm 0.126$ | $0.007\pm 0.006$ | $1.2$ | $-{10}^{-7}$ | SNe + BAOs |

0.303 | $70.088\pm 0.075$ | $0.012\pm 0.004$ | $3.0$ | $-{10}^{-7}$ | SNe |

0.303 | $70.004\pm 0.139$ | $0.009\pm 0.007$ | $1.3$ | $-{10}^{-7}$ | SNe + BAOs |

0.301 | $70.054\pm 0.056$ | $0.009\pm 0.003$ | $3.0$ | $-{10}^{-7}$ | SNe |

0.301 | $70.072\pm 0.170$ | $0.010\pm 0.008$ | $1.2$ | $-{10}^{-7}$ | SNe + BAOs |

0.305 | $70.048\pm 0.034$ | $0.012\pm 0.002$ | $6.0$ | $-{10}^{-7}$ | SNe |

0.305 | $70.004\pm 0.140$ | $0.010\pm 0.007$ | $1.4$ | $-{10}^{-7}$ | SNe + BAOs |

0.298 | $70.135\pm 0.080$ | $0.009\pm 0.004$ | $2.2$ | $-{10}^{-4}$ | SNe |

0.298 | $70.087\pm 0.155$ | $0.009\pm 0.007$ | $1.2$ | $-{10}^{-4}$ | SNe + BAOs |

0.303 | $70.096\pm 0.146$ | $0.012\pm 0.007$ | $1.7$ | $-{10}^{-4}$ | SNe |

0.303 | $70.044\pm 0.129$ | $0.009\pm 0.006$ | $1.5$ | $-{10}^{-4}$ | SNe + BAOs |

0.301 | $70.111\pm 0.158$ | $0.012\pm 0.008$ | $1.5$ | $-{10}^{-4}$ | SNe |

0.301 | $70.038\pm 0.170$ | $0.009\pm 0.008$ | $1.1$ | $-{10}^{-4}$ | SNe + BAOs |

0.305 | $70.074\pm 0.026$ | $0.016\pm 0.001$ | $16.0$ | $-{10}^{-4}$ | SNe |

0.305 | $70.028\pm 0.090$ | $0.011\pm 0.004$ | $2.4$ | $-{10}^{-4}$ | SNe + BAOs |

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Dainotti, M.G.; De Simone, B.; Schiavone, T.; Montani, G.; Rinaldi, E.; Lambiase, G.; Bogdan, M.; Ugale, S.
On the Evolution of the Hubble Constant with the SNe Ia Pantheon Sample and Baryon Acoustic Oscillations: A Feasibility Study for GRB-Cosmology in 2030. *Galaxies* **2022**, *10*, 24.
https://doi.org/10.3390/galaxies10010024

**AMA Style**

Dainotti MG, De Simone B, Schiavone T, Montani G, Rinaldi E, Lambiase G, Bogdan M, Ugale S.
On the Evolution of the Hubble Constant with the SNe Ia Pantheon Sample and Baryon Acoustic Oscillations: A Feasibility Study for GRB-Cosmology in 2030. *Galaxies*. 2022; 10(1):24.
https://doi.org/10.3390/galaxies10010024

**Chicago/Turabian Style**

Dainotti, Maria Giovanna, Biagio De Simone, Tiziano Schiavone, Giovanni Montani, Enrico Rinaldi, Gaetano Lambiase, Malgorzata Bogdan, and Sahil Ugale.
2022. "On the Evolution of the Hubble Constant with the SNe Ia Pantheon Sample and Baryon Acoustic Oscillations: A Feasibility Study for GRB-Cosmology in 2030" *Galaxies* 10, no. 1: 24.
https://doi.org/10.3390/galaxies10010024