# The Flatness Problem and the Variable Physical Constants

## Abstract

**:**

## 1. Introduction

## 2. Theory

## 3. Flatness Problem

## 4. Resolution of Flatness Problem

## 5. Estimation of the Curvature Parameter $u$

## 6. Results and Discussion

_{0}is km s

^{−1}Mpc

^{−1}. The table includes the data fit results for the standard $\Lambda $CDM model for comparison with the varying constant model, the latter being identified as the VcGU model (varying $c,G,$ and $U$ model). The table shows that the goodness of fit is slightly in favor of the varying constant model. The results are also compared graphically in Figure 1.

## 7. Conclusions

- The variable physical constant approach can naturally eliminate the flatness problem that has been pervasive in most cosmological models.
- The universe is open type, was strongly curved in the past, and is substantially curved at present with a curvature $1.64{c}_{0}/{H}_{0}$.
- The scaling of the curvature can be reliably determined by fitting the most recently available supernovae 1a data; the new model fits the data better than the standard $\Lambda $CDM model.
- The radius of curvature of the universe evolves differently than assumed in the standard model; it evolves proportional to the ${a}^{3.3}$.
- The cosmological constant, and consequently the dark energy, is no longer required to save the cosmos from collapsing.

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Supernovae Ia redshift $z$ vs. distance modulus $\mu $ data fit with the ΛCDM model and the fit with the new model using the varying speed of light $c,$ varying gravitational constant $G$, and variable curvature constant $U$ (VcGU) model.

**Figure 3.**Curvature in units of ${c}_{0}/{H}_{0}$ plotted against $a\equiv {\left(1+z\right)}^{-1}$.

**Table 1.**Model parameters and goodness-of-fit parameters for the $\Lambda $ CDM model the varying c, G, and U (VcGU) model. The unit of H

_{0}is km s

^{−1}Mpc

^{−1}. P% is the χ

^{2}probability in percent that is used to assess the better fit model; the higher the χ

^{2}probability $P,$ the better the model fits to the data. R

^{2}is the square of the correlation between the response values and the predicted response values. RMSE is the root mean square error. DOF: degrees of freedom.

Parameter/Model | ΛCDM | VcGU |
---|---|---|

H_{0} | 70.18 ± 0.43 | 70.65 ± 0.60 |

Ω_{m,0} | 0.2845 ± 0.0245 | 0.6309 ± 0.1103 |

u | NA | −2.938 ± 0.549 |

χ^{2} | 1036 | 1032 |

DOF | 1046 | 1045 |

P% | 58 | 61 |

R^{2} | 0.9970 | 0.9970 |

RMSE | 0.995 | 0.994 |

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Gupta, R.P.
The Flatness Problem and the Variable Physical Constants. *Galaxies* **2019**, *7*, 77.
https://doi.org/10.3390/galaxies7030077

**AMA Style**

Gupta RP.
The Flatness Problem and the Variable Physical Constants. *Galaxies*. 2019; 7(3):77.
https://doi.org/10.3390/galaxies7030077

**Chicago/Turabian Style**

Gupta, Rajendra P.
2019. "The Flatness Problem and the Variable Physical Constants" *Galaxies* 7, no. 3: 77.
https://doi.org/10.3390/galaxies7030077