# Estimated Age of the Universe in Fractional Cosmology

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## Abstract

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## 1. Introduction

## 2. Fractional Quantum Cosmology

## 3. Fractional FLRW Cosmology

## 4. Fractional Cosmology Versus Data

## 5. The Synchronicity Problem

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

FLRW | Friedmann–Lemaître–Robertson–Walker |

FQC | Fractional Quantum Cosmology |

GR | General Relativity |

SE | Schrödinger equation |

WDW | Wheeler–DeWitt |

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**Figure 1.**The evolution of the deceleration parameter, q, as a function of redshift, z, for $D=2.2$ and ${\tilde{\Omega}}_{0}^{\left(m\right)}=0.2$ (solid line), ${\tilde{\Omega}}_{0}^{\left(m\right)}=0.26$ (dashed), ${\tilde{\Omega}}_{0}^{\left(m\right)}=0.3$ (dot-dashed), and ${\tilde{\Omega}}_{0}^{\left(m\right)}=0.4$ (doted).

**Figure 2.**The evolution of the deceleration parameter, q, as a function of redshift, z, for ${\tilde{\Omega}}_{0}^{\left(m\right)}=0.264$ and $D=2$ (solid line), $D=2.1$ (dashed), $D=2.4$ (dot-dashed), and $D=2.8$ (doted).

**Figure 3.**The evolution of the deceleration parameter $q\left(z\right)$ is shown against redshift. The dotted line represents the $\Lambda $CDM model, while the dashed line represents the fractional $\Lambda $CDM model.

**Figure 4.**The two-dimensional contours and one-dimensional marginalized distributions represent the fractional $\Lambda $CDM model parameters with $68\%$ CL and $95\%$ CL, respectively.

**Figure 5.**The $68\%$ and $95\%$ constraint contours on the cold dark matter density parameter ${\Omega}_{0}^{\left(cdm\right)}$, baryon density parameter ${\Omega}_{0}^{\left(b\right)}$, matter density parameter ${\Omega}_{0}^{\left(m\right)}$, curvature density parameter ${\Omega}_{0}^{\left(k\right)}$, and Hubble parameter ${H}_{0}$ when using data from the BAO+SN+CMB+BBN+OHD.

**Figure 6.**The distance modulus of supernovae (data points) is compared with the theoretically predicted distance modulus (red line) in a fractional $\Lambda $CDM model.

**Figure 7.**The graph displays the evolution of the Hubble parameter, $H\left(z\right)$, in units of $\mathrm{km}\xb7{\mathrm{sec}}^{-1}\xb7{\mathrm{Mpc}}^{-1}$ versus redshift z, with error bars. The red line represents the dynamics of the Hubble parameter obtained from Equation (40). The values of the density parameters, fractional density parameters, and Hubble constant at the present epoch are the best-fit values from Table 2 and Table 1, respectively.

**Figure 9.**The dependence of ${H}_{0}{t}_{0}$ to the fractional parameter, D. The best-fit value of the fractional density parameter of dust is substituted from Table 2.

**Table 1.**The best-fit parameters with $1\sigma $ and $2\sigma $ confidence levels (CLs) for the FLRW model.

Parameter | 68% CL | 95% CL | Best-Fit Value |
---|---|---|---|

${\Omega}_{0}^{\left(cdm\right)}$ | ${0.226}_{-0.018}^{+0.020}$ | ${0.226}_{-0.036}^{+0.033}$ | ${0.226}_{-0.048}^{+0.044}$ |

${\Omega}_{0}^{\left(b\right)}$ | ${0.0476}_{-0.0018}^{+0.0015}$ | ${0.0476}_{-0.0032}^{+0.0034}$ | ${0.0476}_{-0.0037}^{+0.0045}$ |

${\Omega}_{0}^{\left(m\right)}$ | ${0.274}_{-0.018}^{+0.021}$ | ${0.274}_{-0.037}^{+0.034}$ | ${0.274}_{-0.050}^{+0.040}$ |

${\Omega}_{0}^{\left(k\right)}$ | $0.052\pm 0.045$ | ${0.052}_{-0.089}^{+0.089}$ | ${0.05}_{-0.11}^{+0.11}$ |

${H}_{0}$ | $68.9\pm 1.2$ | ${68.9}_{-2.3}^{+2.3}$ | ${68.9}_{-3.0}^{+2.7}$ |

${\chi}_{min}^{2}$ | ${68.6}_{-3.3}^{+1.3}$ | ${68.6}_{-4.2}^{+5.6}$ | ${69}_{-5}^{+9}$ |

**Table 2.**The best-fit parameters with $1\sigma $ and $2\sigma $ confidence levels (CLs) for fractional FLRW model.

Parameter | 68% CL | 95% CL | Best-Fit Value |
---|---|---|---|

${\tilde{\Omega}}_{0}^{\left(cdm\right)}$ | $0.226\pm 0.018$ | ${0.226}_{-0.038}^{+0.036}$ | ${0.226}_{-0.050}^{+0.046}$ |

${\tilde{\Omega}}_{0}^{\left(b\right)}$ | $0.0480\pm 0.0018$ | ${0.0480}_{-0.0036}^{+0.0036}$ | ${0.0480}_{-0.0045}^{+0.0052}$ |

${\tilde{\Omega}}_{0}^{\left(m\right)}$ | $0.274\pm 0.018$ | ${0.274}_{-0.036}^{+0.032}$ | ${0.274}_{-0.049}^{+0.045}$ |

${\tilde{\Omega}}_{0}^{\left(k\right)}$ | ${0.049}_{-0.060}^{+0.050}$ | ${0.05}_{-0.10}^{+0.11}$ | ${0.05}_{-0.13}^{+0.14}$ |

${H}_{0}$ | ${68.6}_{-1.4}^{+1.2}$ | ${68.6}_{-2.4}^{+2.6}$ | ${68.6}_{-3.4}^{+3.5}$ |

D | ${2.0069}_{-0.0061}^{+0.0036}$ | ${2.007}_{-0.0087}^{+0.011}$ | ${2.007}_{-0.010}^{+0.015}$ |

${\chi}_{min}^{2}$ | ${68.4}_{-3.3}^{+1.5}$ | ${68.4}_{-4.4}^{+5.8}$ | ${68}_{-5}^{+9}$ |

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Costa, E.W.d.O.; Jalalzadeh, R.; da Silva Júnior, P.F.; Rasouli, S.M.M.; Jalalzadeh, S.
Estimated Age of the Universe in Fractional Cosmology. *Fractal Fract.* **2023**, *7*, 854.
https://doi.org/10.3390/fractalfract7120854

**AMA Style**

Costa EWdO, Jalalzadeh R, da Silva Júnior PF, Rasouli SMM, Jalalzadeh S.
Estimated Age of the Universe in Fractional Cosmology. *Fractal and Fractional*. 2023; 7(12):854.
https://doi.org/10.3390/fractalfract7120854

**Chicago/Turabian Style**

Costa, Emanuel Wallison de Oliveira, Raheleh Jalalzadeh, Pedro Felix da Silva Júnior, Seyed Meraj Mousavi Rasouli, and Shahram Jalalzadeh.
2023. "Estimated Age of the Universe in Fractional Cosmology" *Fractal and Fractional* 7, no. 12: 854.
https://doi.org/10.3390/fractalfract7120854