# Extending Friedmann Equations Using Fractional Derivatives Using a Last Step Modification Technique: The Case of a Matter Dominated Accelerated Expanding Universe

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## Abstract

**:**

## 1. Introduction

## 2. Standard Cosmology

## 3. Fractional Friedmann Equation

#### Matter Dominated Universe

## 4. SN Ia Fits

## 5. Results

## 6. Final Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

MDPI | Multidisciplinary Digital Publishing Institute |

DOAJ | Directory of open access journals |

TLA | Three letter acronym |

LD | Linear dichroism |

## Appendix A. Fractional Calculus: A Simple Introduction

## Appendix B. ΛCDM Standard Cosmology

**Table A1.**Best fit results for the standard cosmological model with one free parameter: the matter density parameter ${\mathsf{\Omega}}_{\mathrm{M}}$, presented with its corresponding error, SSR and p-value.

Planck | SN Ia | Local | |
---|---|---|---|

${\mathsf{\Omega}}_{M}$ | $0.185818$ | $0.3601$ | $0.027163$ |

Asymptotic Error | $\pm 0.008958$ | $\pm 0.01343$ | $\pm 0.005725$ |

SSR | $557.741$ | $822.58$ | $742.651$ |

p-value | $0.203964$ | <0.0001 | <0.0001 |

**Figure A1.**Apparent magnitude $\mu $ vs. redshift z Hubble diagram from the Union 2.1 SN Ia data (dots with their corresponding error bars) and the best fit from the $\mathsf{\Lambda}CDM$ cosmological model with ${H}_{0}^{\mathrm{Planck}}=67.36\phantom{\rule{0.166667em}{0ex}}{\mathrm{kms}}^{-1}{\mathrm{Mpc}}^{-1}$. The solid line represents the distance modulus $\mu \left(z\right)$ from the best fit to the data.

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**Figure 1.**Apparent magnitude $\mu $ vs. redshift z Hubble diagram from the Union 2.1 SNe Ia data (dots with their corresponding error bars) and the best fit from our model. The solid line represents the distance modulus $\mu \left(z\right)$ from the best fit to the data of the model.

**Figure 2.**Apparent magnitude $\mu $ vs. redshift z Hubble diagram from the Union 2.1 SNe Ia data (dots with their corresponding error bars) and the best fit from our model. The solid blue line represents the distance modulus $\mu \left(z\right)$ from the best fit to the data of the model. The other solid curves represents the envelope of the statistical errors.

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**MDPI and ACS Style**

Barrientos, E.; Mendoza, S.; Padilla, P.
Extending Friedmann Equations Using Fractional Derivatives Using a Last Step Modification Technique: The Case of a Matter Dominated Accelerated Expanding Universe. *Symmetry* **2021**, *13*, 174.
https://doi.org/10.3390/sym13020174

**AMA Style**

Barrientos E, Mendoza S, Padilla P.
Extending Friedmann Equations Using Fractional Derivatives Using a Last Step Modification Technique: The Case of a Matter Dominated Accelerated Expanding Universe. *Symmetry*. 2021; 13(2):174.
https://doi.org/10.3390/sym13020174

**Chicago/Turabian Style**

Barrientos, Ernesto, Sergio Mendoza, and Pablo Padilla.
2021. "Extending Friedmann Equations Using Fractional Derivatives Using a Last Step Modification Technique: The Case of a Matter Dominated Accelerated Expanding Universe" *Symmetry* 13, no. 2: 174.
https://doi.org/10.3390/sym13020174