Abstract
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.
1. Introduction
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 1039 [–]. The Dirac large numbers hypothesis (LNH) tackles this problem. It claims that the gravitational constant G is inversely proportional to the age of the universe. The LNH is based on the coincidences between three very large dimensionless numbers . N1, the ratio of the radius of the observable universe to the radius of the electron, is approximately 1039; N2, the ratio of the electromagnetic and gravitational forces between a proton and an electron, is also approximately 1039; and N, the number of protons in the observable universe, is approximately 1078. 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.
2. The self-similar symmetry model
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: where mpr is the proton mass, me is the electron mass, and mpl is the Planck mass, and the fundamental dimensionless time and length ratios are defined as where t and l are the time and length scales of the hierarchies and tpl and lpl are the Planck time and Planck length, respectively. The similarity dimension D is defined using these dimensionless parameters: We then assume that hierarchical structures are constructed according to the following sequences: where n and m are natural numbers that represent the hierarchical level. In addition, the time scales of each hierarchy are calculated using Equation (3).
3. Verification of the SSS model
To verify the SSS model, we compared values obtained with it against reference values. Tables 1, 2 summarize the length and time scales, respectively, of the grand unification, electroweak unification, atom, pulsar, solar system, and galactic system hierarchies of the universe. The SSS model values agree well with the reference values. Figure 1 shows the hierarchy time scale as a function of the length scale. The coincidences in the figure confirm the validity of the SSS model.
Table 1
| Hierarchy | l (m) | L | SSS model | Error (%) |
|---|---|---|---|---|
| Plancka | 1.6 × 10−35 | 0 | – | – |
| Grand unificationb | 10−27 | 7.79 | 7.81 (m = 3) | 0.2 |
| Electroweak unificationb | 10−17 | 17.79 | 17.30 (m = 2) | −2.7 |
| Atomc | 2.4 × 10−10 | 25.17 | 25.17 (m = 1) | 0.0 |
| Pulsard | 2.4 × 104 | 39.17 | 38.23 (n = 1) | −2.4 |
| Solar systeme | 3.0 × 1011 | 46.27 | 46.10 (n = 2) | −0.3 |
| Galaxyf | 5.3 × 1020 | 55.52 | 55.59 (n = 3) | 0.1 |
Length scales of the hierarchies of the universe.
The Planck length lpl is defined as , where h- is the Dirac's constant, G is the gravitational constant, and c is the speed of light in a vacuum.
Values are taken from a magnetic monopole structure in grand unified theories (GUTs) [].
Assumed to be twice the van der Waals radius of a hydrogen atom [].
Estimated to be 1.5 times the mass of the Sun with a radius of 12 km [].
Based on the average diameter of the Earth's orbit around the Sun; 2 astronomical units (AUs) [].
Taken as twice the distance from the center of the galaxy to the solar system, which is 28,000 light years [].
Table 2
| Hierarchy | t (s) | T | SSS model | Error (%) |
|---|---|---|---|---|
| Planckb | 5.4 × 10−44 | 0 | – | – |
| Grand unificationc | 2.2 × 10−35 | 8.61 | 8.31 (m = 3) | −3.5 |
| Electroweak unificationd | 6.6 × 10−27 | 17.09 | 18.42 (m = 2) | 7.7 |
| Atome | 4.8 × 10−17 | 26.95 | 26.79 (m = 1) | −0.6 |
| Pulsarf | 2.9 × 10−2 | 41.72 | 40.69 (n = 1) | −2.5 |
| Solar systemg | 3.2 × 107 | 50.77 | 49.07 (n = 2) | −3.3 |
| Galaxyh | 7.6 × 1015 | 59.15 | 59.17 (n = 3) | 0.0 |
Time Scales of the Hierarchies of the Universea.
The orbital motion and magnetic fields of the Earth and Sun cause emitted light to have a long period. Therefore, although the physical structures are different in each hierarchy, their time scales can be compared in a unifying manner using the period of light.
Planck time .
Grand unification at the intermediate mass scale GeV proposed by Dienes et al. []; t = h-/(3 × 1019) s.
Electromagnetic force and weak force unify at 102 GeV []; t = h-/1011 s.
Based on the first ionization energy of hydrogen []; t = h-/13.6 s.
Inverse of the average observed frequency of fpulsar≈35Hz(N = 2, 307pulsars) determined from [].
Period of the Earth's revolution around the Sun [].
The revolution of the Sun around the center of the Milky Way, i.e., 1 galactic year ≈240 million years, based on a galactic rotational speed of approximately 220 km/s [].
Figure 1
4. Discussion
From Equation (1), 2A = −logαG, where is the gravitational coupling constant, the following coincidences occurs:
Equation (7) shows that αG plays an important role in forming the hierarchical structure of the universe. In addition,
Thus, if Lu 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 which we get
From Equation (12), we see that ra ≠ rb ≠ 1.
With respect to the first term of the geometric sequence, L0, we find that where Tpl is the Planck temperature. The value in Equation (13) is consistent with the CMB radiation temperature TCMB[]. Assuming the right-hand side of Equation (13) represents the TCMB, if LNH is applied to Equation (13) and we define the dimensionless temperature ratio τCMB = TCMB/Tpl, we get Similarly, Thus, both αG and β are power functions of TCMB.
Substituting TCMB = Tpl, 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 me ≪ mpr ≪ mpl, in response to the changing TCMB, where TCMB ≪ Tpl. Assuming that TCMB → 0 is the ultimate fate of the universe, then α → ∞, β → 0, and L0 → ∞, indicating that me → 0 and G → 0 as the universe expands to infinity.
5. Conclusions
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 me and G vary with TCMB. Any errors arising from the SSS model are problems to be tackled in the future.
Statements
Author contributions
TS conceived the study and prepared the manuscript.
Acknowledgments
The author thanks M. B. Greenfield and K. Kitahara for helpful discussions.
Conflict of interest
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.
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Summary
Keywords
self-similarity, symmetry, cosmic microwave background, large numbers hypothesis, large numbers coincidences
Citation
Sonoda T (2016) Self-Similar Symmetry Model and Cosmic Microwave Background. Front. Appl. Math. Stat. 2:5. doi: 10.3389/fams.2016.00005
Received
01 January 2016
Accepted
26 April 2016
Published
11 May 2016
Volume
2 - 2016
Edited by
Peng Gao, Harvard University, USA
Reviewed by
Veselin Filev, Dublin Institute for Advanced Studies, Ireland; Kazuharu Bamba, Fukushima University, Japan
Updates
Copyright
© 2016 Sonoda.
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.
*Correspondence: Tomohide Sonoda tomo@alm.icu.ac.jp
This article was submitted to Mathematical Physics, a section of the journal Frontiers in Applied Mathematics and Statistics
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