^{*}

Edited by: Peng Gao, Harvard University, USA

Reviewed by: Veselin Filev, Dublin Institute for Advanced Studies, Ireland; Kazuharu Bamba, Fukushima University, Japan

*Correspondence: Tomohide Sonoda

This article was submitted to Mathematical Physics, a section of the journal Frontiers in Applied Mathematics and Statistics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

In this paper, we present the self-similar symmetry (SSS) model that describes the hierarchical structure of the universe. The model is based on the concept of self-similarity, which explains the symmetry of the cosmic microwave background (CMB). The approximate length and time scales of the six hierarchies of the universe—grand unification, electroweak unification, the atom, the pulsar, the solar system, and the galactic system—are derived from the SSS model. In addition, the model implies that the electron mass and gravitational constant could vary with the CMB radiation temperature.

What determines the values of the physical constants and whether they will remain constant over time are fundamental questions in physics. A long-standing conundrum associated with the physical constants is that large dimensionless numbers that are seemingly unrelated can be linked by a scale factor of 10^{39} [_{1}, the ratio of the radius of the observable universe to the radius of the electron, is approximately 10^{39}; _{2}, the ratio of the electromagnetic and gravitational forces between a proton and an electron, is also approximately 10^{39}; and ^{78}. The Dirac LNH argues that “any two of the very large dimensionless numbers occurring in Nature are connected by a simple mathematical relation, in which the coefficients are of the order of unity”[

In this paper, we present the self-similar symmetry (SSS) model in which the relationships among these seemingly unrelated physical quantities are represented using a simple geometric sequence for which the first term and the geometric ratio are given by dimensionless ratios of masses. Based on the LNH, the first term of the geometric sequence corresponds to the cosmic microwave background (CMB) radiation temperature, which points to the possibility that the values of the physical constants are determined by the CMB radiation temperature.

In the SSS model, the CMB has a symmetrical self-similar structure and the physical constants are dimensionless, otherwise they would not have universality. Therefore, the fundamental dimensionless mass ratios are defined as follows:
_{pr} is the proton mass, _{e} is the electron mass, and _{pl} is the Planck mass, and the fundamental dimensionless time and length ratios are defined as
_{pl} and _{pl} are the Planck time and Planck length, respectively. The similarity dimension

To verify the SSS model, we compared values obtained with it against reference values. Tables

Planck^{a} |
1.6 × 10^{−35} |
0 | – | – |

Grand unification^{b} |
10^{−27} |
7.79 | 7.81 ( |
0.2 |

Electroweak unification^{b} |
10^{−17} |
17.79 | 17.30 ( |
−2.7 |

Atom^{c} |
2.4 × 10^{−10} |
25.17 | 25.17 ( |
0.0 |

Pulsar^{d} |
2.4 × 10^{4} |
39.17 | 38.23 ( |
−2.4 |

Solar system^{e} |
3.0 × 10^{11} |
46.27 | 46.10 ( |
−0.3 |

Galaxy^{f} |
5.3 × 10^{20} |
55.52 | 55.59 ( |
0.1 |

_{pl} is defined as

^{a}

Planck^{b} |
5.4 × 10^{−44} |
0 | – | – |

Grand unification^{c} |
2.2 × 10^{−35} |
8.61 | 8.31 ( |
−3.5 |

Electroweak unification^{d} |
6.6 × 10^{−27} |
17.09 | 18.42 ( |
7.7 |

Atom^{e} |
4.8 × 10^{−17} |
26.95 | 26.79 ( |
−0.6 |

Pulsar^{f} |
2.9 × 10^{−2} |
41.72 | 40.69 ( |
−2.5 |

Solar system^{g} |
3.2 × 10^{7} |
50.77 | 49.07 ( |
−3.3 |

Galaxy^{h} |
7.6 × 10^{15} |
59.15 | 59.17 ( |
0.0 |

^{19}) s

^{2} GeV [^{11} s

_{pulsar}≈35Hz(N = 2, 307pulsars) determined from [

^{2} = 0.9991). Note the symmetry about the first term _{0}, which corresponds to the CMB radiation temperature. This symmetry leads to speculation about the approximate length and time scale of the universe, which are 4.1 × 10^{28} m and 5.2 × 10^{16} years, respectively.

From Equation (1), 2_{G}, where

Equation (7) shows that α_{G} plays an important role in forming the hierarchical structure of the universe. In addition,

Thus, if _{u} is the length of the universe, the following hierarchy holds:

Therefore, the ratios of the coincidences between the length scales of the hierarchies are

From Equation (12), we see that _{a} ≠ _{b} ≠ 1.

With respect to the first term of the geometric sequence, _{0}, we find that
_{pl} is the Planck temperature. The value in Equation (13) is consistent with the CMB radiation temperature _{CMB}[_{CMB}, if LNH is applied to Equation (13) and we define the dimensionless temperature ratio τ_{CMB} = _{CMB}/_{pl}, we get
_{G} and β are power functions of _{CMB}.

Substituting _{CMB} = _{pl}, an initial condition of the universe, into Equations (14) and (15) yields α = β = 1, which means that the entire hierarchy was contained in a single point and that the electron, proton, and Planck masses were equivalent. These masses have varied since that initial single point such that _{e} ≪ _{pr} ≪ _{pl}, in response to the changing _{CMB}, where _{CMB} ≪ _{pl}. Assuming that _{CMB} → 0 is the ultimate fate of the universe, then α → ∞, β → 0, and _{0} → ∞, indicating that _{e} → 0 and

Our SSS model describes the large-scale structure of the universe and shows that the six hierarchies of the universe are self-similar to the CMB, indicating that the CMB is key to unifying quantum theory with general relativity. In addition, the SSS model leads to the conclusion that _{e} and _{CMB}. Any errors arising from the SSS model are problems to be tackled in the future.

TS conceived the study and prepared the manuscript.

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The author thanks M. B. Greenfield and K. Kitahara for helpful discussions.