# Revisiting the Cosmological Constant Problem within Quantum Cosmology

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## Abstract

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## 1. Introduction

## 2. The Three Main Model Groups and the Seven Principal Model Categories

## 3. Cosmological Constant within Quantum Cosmology

## 4. Multiverse Partition Function and the Typical Size of a General $\zeta $-Universe

## 5. Discussions and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Spergel, D.N.; Bean, R.; Doré, O.; Nolta, M.R.; Bennett, C.L.; Dunkley, J.; Hinshaw, G.; Jarosik, N.; Komatsu, E.; Page, L.; et al. Three-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations:Implications for Cosmology. Astrophys. J. Suppl. Ser.
**2007**, 170, 377–408. [Google Scholar] [CrossRef] [Green Version] - Komatsu, E.; Smith, K.M.; Dunkley, J.; Bennett, C.L.; Gold, B.; Hinshaw, G.; Jarosik, N.; Larson, D.; Nolta, M.R.; Page, L.; et al. Seven-year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Interpretation. Astrophys. J. Suppl. Ser.
**2011**, 192, 18. [Google Scholar] [CrossRef] [Green Version] - Ade, P.A.R.; Aghanim, N.; Arnaud, M.; Ashdown, M.; Aumont, J.; Baccigalupi, C.; Banday, A.J.; Barreiro, R.B.; Bartlett, J.G.; Bartolo, N.; et al. Planck 2015 results. XIII. Cosmological parameters. Astron. Astrophys.
**2016**, 594, A13. [Google Scholar] - Hawking, S.W. The cosmological constant. Philos. Trans. R. Soc. Lond. Ser. A
**1983**, 310, 303–309. [Google Scholar] - Weinberg, S. The cosmological constant problem. Rev. Mod. Phys.
**1989**, 61, 1–23. [Google Scholar] [CrossRef] - Coleman, S. Why there is nothing rather than something: A theory of the cosmological constant. Nucl. Phys. B
**1988**, 310, 643–668. [Google Scholar] [CrossRef] - Hinshaw, G.; Larson, D.; Komatsu, E.; Spergel, D.N.; Bennett, C.L.; Dunkley, J.; Nolta, M.R.; Halpern, M.; Hill, R.S.; Odegard, N.; et al. Nine-year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Parameter Results. Astrophys. J. Suppl. Ser.
**2013**, 208, 19. [Google Scholar] [CrossRef] [Green Version] - Di Valentino, E.; Melchiorri, A.; Silk, J. Planck evidence for a closed Universe and a possible crisis for cosmology. Nat. Astron.
**2020**, 4, 196. [Google Scholar] [CrossRef] [Green Version] - Carroll, S.M.; Press, W.H.; Turner, E.L. The cosmological constant. Annu. Rev. Astron. Astrophys.
**1992**, 30, 499–542. [Google Scholar] [CrossRef] - Smolin, L. Quantization of unimodular gravity and the cosmological constant problems. Phys. Rev. D
**2009**, 80, 084003. [Google Scholar] [CrossRef] [Green Version] - Ali, A.F.; Das, S. Cosmology from quantum potential. Phys. Lett. B
**2015**, 741, 276. [Google Scholar] - Chiarelli, P. The Gravity of the Classical Klein-Gordon Field. Symmetry
**2019**, 11, 322. [Google Scholar] [CrossRef] [Green Version] - Chiarelli, P. The Spinor-Tensor Gravity of the Classical Dirac Field. Symmetry
**2020**, 12, 1124. [Google Scholar] [CrossRef] - Antoniadis, I.; Mottola, E. Four-dimensional quantum gravity in the conformal sector. Phys. Rev. D
**1992**, 45, 2013. [Google Scholar] [CrossRef] [PubMed] - Mottola, E. Functional integration over geometries. J. Math. Phys.
**1995**, 36, 2470. [Google Scholar] [CrossRef] [Green Version] - Ambjørn, J.; Jurkiewicz, J.; Loll, R. Emergence of a 4D World from Causal Quantum Gravity. Phys. Rev. Lett.
**2004**, 93, 131301. [Google Scholar] [CrossRef] [Green Version] - Nottale, L. Scale relativity and fractal space-time: Theory and applications. Found. Sci.
**2010**, 15, 101–152. [Google Scholar] [CrossRef] [Green Version] - Lucat, S.; Prokopec, T.; Swiezewska, B. Conformal symmetry and the cosmological constant problem. Int. J. Mod. Phys. D
**2018**, 27, 1847014. [Google Scholar] [CrossRef] - Pathria, R.K. The Universe as a Black Hole. Nature
**1972**, 240, 298–299. [Google Scholar] [CrossRef] - Perelman, C.C. Is Dark Matter and Black-Hole Cosmology an Effect of Born’s Reciprocal Relativity Theory? Can. J. Phys.
**2019**, 97, 198–209. [Google Scholar] [CrossRef] - Brans, C.; Dicke, R.H. Mach’s Principle and a Relativistic Theory of Gravitation. Phys. Rev.
**1961**, 124, 925–935. [Google Scholar] [CrossRef] - Li, M.; Li, X.-D.; Wang, S.; Wang, Y. Dark Energy. Commun. Theor. Phys.
**2011**, 56, 525–604. [Google Scholar] [CrossRef] [Green Version] - Popławski, N. Universe in a Black Hole in Einstein-Cartan Gravity. Astrophys. J.
**2017**, 832, 158. [Google Scholar] - Gibbons, G.W.; Hawking, S.W. Action integrals and partition functions in quantum gravity. Phys. Rev. D
**1977**, 15, 2752. [Google Scholar] [CrossRef] - De Sitter, W. Einstein’s theory of gravitation and its astronomical consequences. Third paper. Mon. Not. R. Astron. Soc.
**1917**, 78, 3. [Google Scholar] [CrossRef] [Green Version] - Maeder, A. An Alternative to the ΛCDM Model: The Case of Scale Invariance. Astrophys. J. Suppl. Ser.
**2017**, 834, 194. [Google Scholar] [CrossRef] [Green Version] - Maeder, A. Dynamical Effects of the Scale Invariance of the Empty Space: The Fall of Dark Matter? Astrophys. J.
**2017**, 849, 158. [Google Scholar] [CrossRef] [Green Version] - Mazur, P.O.; Mottola, E. The path integral measure, conformal factor problem and stability of the ground state of quantum gravity. Nucl. Phys. B
**1990**, 341, 187. [Google Scholar] [CrossRef] - Maeder, A.; Gueorguiev, V.G. The growth of the density fluctuations in the scale-invariant vacuum theory. Phys. Dark Universe
**2019**, 25, 100315. [Google Scholar] [CrossRef] - Vilenkin, A. Cosmological constant problems and their solutions. Astron. Cosmol. Fundam. Phys.
**2003**, 70–78. [Google Scholar] [CrossRef] [Green Version] - Kawai, H.; Okada, T. Asymptotically Vanishing Cosmological Constant in the Multiverse. Int. J. Mod. Phys. A
**2011**, 26, 3107. [Google Scholar] [CrossRef] [Green Version] - Pejhan, H.; Bamba, K.; Enayati, M.; Rahbardehghan, S. A small non-vanishing cosmological constant from the Krein-Gupta-Bleuler vacuum. Phys. Lett. B
**2018**, 785, 567–569. [Google Scholar] [CrossRef] - Sahni, V.; Starobinsky, A. The Case for a Positive Cosmological Λ-Term. Int. J. Mod. Phys. D
**2000**, 9, 373–443. [Google Scholar] [CrossRef] - Hawking, S.W. The cosmological constant is probably zero. Phys. Lett. B
**1984**, 134, 403. [Google Scholar] [CrossRef] - Vilenkin, A.; Yamada, M. Four-volume cutoff measure of the multiverse. Phys. Rev. D
**2020**, 101, 043520. [Google Scholar] [CrossRef] [Green Version] - Vilenkin, A. Quantum creation of universes. Phys. Rev. D
**1984**, 30, 509. [Google Scholar] [CrossRef]

1. | The negative sign is for consistency of moving $\Lambda $ from the RHS to the LHS of the equation, as well as for convenience in relating $\Lambda $ to the scalar curvature R. |

2. | The ratio is even smaller ($\sigma /\rho =5/4=1.25$) if one is to consider equal energy partition per degree of freedom ($1/2$) with dimension degree of freedom n deduced from the scaling $\rho \sim {r}^{-n}$. |

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**MDPI and ACS Style**

Gueorguiev, V.G.; Maeder, A.
Revisiting the Cosmological Constant Problem within Quantum Cosmology. *Universe* **2020**, *6*, 108.
https://doi.org/10.3390/universe6080108

**AMA Style**

Gueorguiev VG, Maeder A.
Revisiting the Cosmological Constant Problem within Quantum Cosmology. *Universe*. 2020; 6(8):108.
https://doi.org/10.3390/universe6080108

**Chicago/Turabian Style**

Gueorguiev, Vesselin G., and Andre Maeder.
2020. "Revisiting the Cosmological Constant Problem within Quantum Cosmology" *Universe* 6, no. 8: 108.
https://doi.org/10.3390/universe6080108