# 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

## References

- Penzias, A.A.; Wilson, R.W. A Measurement of Excess Antenna Temperature at 4080 Mc/s. Astrophys. J.
**1965**, 142, 419–421. [Google Scholar] [CrossRef] - Guth, A.H. Inflationary universe: A possible solution for the horizon and flatness problem. Phys. Rev. D
**1981**, 23, 347–356. [Google Scholar] [CrossRef] - Albrecht, A.; Steinhardt, P.J. Cosmology for Grand Unified Theories with Radiatively Induced Symmetry Breaking. Phys. Rev. Lett.
**1982**, 48, 1220–1223. [Google Scholar] [CrossRef] - Linde, A. A new inflationary universe scenario: A possible solution of the horizon, flatness, homogeneity, isotropy and primordial monopole problems. Phys. Lett. B
**1982**, 108, 389–393. [Google Scholar] [CrossRef] - Gupta, R.P. Varying physical constants, astrometric anomalies, redshift and Hubble units. Galaxies
**2019**, 7, 55. [Google Scholar] [CrossRef] - Olive, K.A. Inflation. Phys. Rep.
**1990**, 190, 307–403. [Google Scholar] [CrossRef] - Ryden, B. Introduction to Cosmology; Cambridge University Press: Cambridge, UK, 2017. [Google Scholar]
- Levine, J.; Freese, K. Possible solution to the horizon problem: Modified aging in massive scalar theories of gravity. Phys. Rev. D
**1993**, 47, 4282–4291. [Google Scholar] [CrossRef] [PubMed] - Hu, Y.; Turner, M.S.; Weinberg, E.J. Dynamical solutions to the horizon and flatness problems. Phys. Rev. D
**1994**, 49, 3830–3836. [Google Scholar] [CrossRef][Green Version] - Barrow, J.D.; Tipler, F. Analysis of the generic singularity studies by Belinskii, Khalatnikov, and Lifschitz. Phys. Rep.
**1979**, 56, 371–402. [Google Scholar] [CrossRef] - Belinski, V.A.; Khalatnikov, I.M.; Lifshitz, E.M. A general solution of the Einstein equations with a time singularity. Adv. Phys.
**1982**, 31, 639–667. [Google Scholar] [CrossRef] - Singal, A.K. Horizon, homogeneity and flatness problems – do their resolutions really depend upon inflation? arXiv
**2016**, arXiv:1603.01539. [Google Scholar] - Barrow, J.D.; Magueijo, J. Solution to the quasi-flatness and quasi-lambda problems. Phys. Lett. B
**1999**, 447, 246–250. [Google Scholar] [CrossRef] - Barrow, J.D.; Magueijo, J. Solving the flatness and quasi-flatness problems in Brans-Dicke cosmologies with varying light speed. Class. Quantum Grav.
**1999**, 16, 1435–1454. [Google Scholar] [CrossRef] - Berera, A.; Gleiser, M.; Ramos, R.O. A first principle warm inflation model that solves cosmological horizon and flatness problems. Phys. Rev. Lett.
**1999**, 83, 264–267. [Google Scholar] [CrossRef] - Lake, K. The flatness problem and Λ. Phys. Rev. Lett.
**2005**, 94, 201102. [Google Scholar] [CrossRef] [PubMed] - Fathi, M.; Jalalzadeh, S.; Moniz, P.V. Classical universe emerging from quantum cosmology without horizon and flatness problems. Eur. Phys. J. C
**2016**, 76, 527. [Google Scholar] [CrossRef][Green Version] - Bramberger, S.F.; Coates, A.; Magueijo, J.; Mukohyama, S.; Namba, R.; Watanabe, Y. Solving the flatness problem with an anisotropic instanton in Horava-Lifshitz gravity. Phys. Rev. D
**2018**, 97, 043512. [Google Scholar] [CrossRef] - Narlikar, J.V. An Introduction to Cosmology, 3rd ed.; Cambridge University Press: Cambridge, UK, 2002. [Google Scholar]
- Barrow, J.D. Cosmologies with varying light speed. Phys. Rev. D
**1999**, 59, 043515. [Google Scholar] [CrossRef][Green Version] - Scolnic, D.M.; Jones, D.O.; Rest, A.; Pan, Y.C.; Chornock, R.; Foley, R.J.; Huber, M.E.; Kessler, R.; Narayan, G.; Riess, A.G.; et al. The Complete Light-curve Sample of Spectroscopically Confirmed SNe Ia from Pan-STARRS1 and Cosmological Constraints from the Combined Pantheon Sample. Astrophys. J.
**2018**, 859, 101. Available online: https://archive.stsci.edu/hlsps/ps1cosmo/scolnic/https://github.com/dscolnic/Pantheon (accessed on 10 May 2019). [CrossRef] - Peebles, P.J.E. Principles of Physical Cosmology; Princeton University Press: Princeton, NJ, USA, 1993. [Google Scholar]
- Baes, M.; Camps, P.; Van De Putte, D. Analytical expression and numerical evaluation of the luminosity distance in a flat cosmology. Mon. Not. R. Astron. Soc.
**2017**, 468, 927–930. [Google Scholar] [CrossRef] - Zaninetti, L. A new analytical solution for the distance modulus in flat cosmology. Int. J. Astron. Astrophys.
**2019**, 9, 51–62. [Google Scholar] [CrossRef] - Scolnic, D.M.; (University of Chicago). Personal Email communication, 22 November 2018.
- Press, W.H.; Teukolsky, S.A.; Vetterling, W.T.; Flannery, B.P. Numerical Recipes in C—The Art of Scientific Computing, 2nd ed.; Cambridge University Press: Cambridge, UK, 1992. [Google Scholar]
- Walker, J. Chi-Square Calculator. Available online: https://www.fourmilab.ch.rpkp/experim-ents/analysis/chiCalc.html (accessed on 3 August 2019).
- Gupta, R.P. Weighing cosmological models with SNe 1a and gamma ray burst redshift data. Universe
**2019**, 5, 102. [Google Scholar] [CrossRef] - Einstein, A. Kosmologische Betrachtungen zur allgemeinen Relativitatstheorie, Sitzungsberichte der Preussischen Akad. d. Wissenschaften
**1917**, 142–152. [Google Scholar]

**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 |

© 2019 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

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