Application of Self-Similar Symmetry Model to Dark Energy

Recent observations of the dark energy density demonstrates the fine-tuning problem and challenges in theoretical modelling. In this study, we apply the self-similar symmetry (SSS) model, describing the hierarchical structure of the universe based on the Dirac large numbers hypothesis, to Einstein’s cosmological term. We introduce a new similarity dimension, DB, in the SSS model. Using the DB SSS model, the cosmological constant, vacuum energy density, and Hubble parameter can be simply expressed as a function of the cosmic microwave background (CMB) temperature. We show that the initial value of the vacuum energy density at the creation of the universe is ρ0 = 1/8παf , where α f is the fine structure constant. The results indicate that the CMB is the primary factor for the evolution of the universe, providing a unified understanding of the problems of naturalness.


Introduction
The cosmological constant problem, i.e., the dark energy problem, poses a formidable challenge in physics.In 1998, observations of distant supernovae provided evidence for the acceleration of the expansion of the universe [1,2].Einstein's cosmological term emerged as the simplest candidate to explain the mechanism of the accelerating universe.However, the inconsistencies between theoretical expectations and observations are extremely problematic, despite many attempts to provide a proper explanation [3][4][5][6][7][8].In order to provide insights into this issue, the axiomatic approach has been proposed by Beck [9].Beck formulated a description of the cosmological constant, Λ, using four statistical axioms: fundamentality (Λ depends only on the fundamental constants of the nature), boundedness (Λ has a lower bound, 0 < Λ), simplicity (Λ is given by the simplest possible formula, consistent with the other axioms), and invariance (Λ values obtained using potentially different values of the fundamental parameters preserve the scale-invariance of the large-scale physics of the universe).Using the four axioms, Beck showed that Λ is given by: where G is the gravitational constant, h is the reduced Planck constant, α f is the fine structure constant, and m e is the electron mass.The same formula has been proposed using different approaches [10,11], and recently discussed in several reports [12][13][14][15].
In this study, we applied the self-similar symmetry (SSS) model [16], that explains the hierarchical structure of the universe based on the Dirac large numbers hypothesis (LNH) [17,18], to Beck's formula.We show that the values of the cosmological constant, vacuum energy density, and Hubble parameter can be simply expressed as a function of the cosmic microwave background (CMB) temperature, and that the initial vacuum energy density is uniquely determined by ρ 0 = 1/8πα 6  f .These results indicate that the CMB is the primary factor responsible for the evolution of the universe, revealing novel insights into the outstanding challenges.

D B SSS model
The SSS model [16] describes the CMB with a symmetrical self-similar structure.The model consists of dimensionless values because a physical constant with a dimension would not have universality.Therefore, we define the fundamental dimensionless mass ratios of the proton mass m pr , electron mass m e , and Planck mass m Pl as follows: We also defined the fundamental dimensionless time and length ratios as follows: where t and l are the time and length scales of the objects, respectively, and t Pl and l Pl are the Planck time and length, respectively.Using these dimensionless values, we define the similarity dimension D A as: A new similarity dimension, D B , is then introduced: The hierarchical structures of the D B SSS model are constructed according to the following sequences: where n and m are the natural numbers that represent the hierarchical levels.The time scales of each hierarchy are also calculated using Eq.(4).

Verification of the D B SSS model
In order to verify the proposed D B SSS model, we compared the model values with reference values.Table 1 and 2 summarize the length and time scales of the Planck, weak, solar, and universe hierarchies.The values obtained using the D B SSS model agree well with the reference values.Figure 1 shows the hierarchy time scale as a function of length scale.The coincidences seen in the figure confirm the validity of the SSS model.
) are the ratios of the length scales of the hierarchies [16]) can be used to obtain a simple relation between r a and r b : (αβ) Equation ( 11) can be interpreted as the basic formula for the similarity dimension and indicates the correlation between the cosmic structure and fundamental dimensionless mass ratios 1 .Using Eq. ( 11), we obtain: However, the numerical relation between D B and r a is: Equation ( 13) indicates the validity of D B ; if the D B value is substituted into Eq.( 8), L m=2 is consistent with the Planck length.Regarding Λ, Eq. ( 1) can be written in an equivalent dimensionless form using G = hc/m 2 Pl : We employed the following formulas derived from the D A SSS model [16] in Eq. ( 14): where τ CMB = T CMB /T Pl ; T CMB is the CMB temperature and T Pl is the Planck temperature.Then, we obtain: where λ is the cosmological constant in reduced Planck units, λ = l 2 Pl Λ, and we defined ξ CMB .Equation ( 17) is based on the LNH and indicates that the CMB temperature can be considered as a cosmological scalar field.
Using the relation between the vacuum energy density ρ Λ and Λ in Einstein's field equation, we obtain: ρ Λ = c 2 Λ/8πG.Therefore, the dimensionless vacuum energy density can be expressed as: where Ω Λ is the normalized vacuum energy density with respect to the critical density.Then, we obtain: where h is the Hubble parameter in reduced Planck units, h = t Pl H.
If we employ T CMB = T Pl for the universe initial condition and substitute it into Eqs.( 15), ( 16), and ( 18), we obtain α = β = 1 and ρ 0 = 1/8πα 6  f , which implies that the entire hierarchy was contained in a single point and that a high-energy density ρ 0 can trigger the cosmic inflation.The value of ρ decreases with the decrease of T CMB T Pl , while the size of the universe L expands according to L ∼ log (T Pl /T CMB ).Assuming that T CMB → 0 is the ultimate fate of the universe, L → ∞ and ρ → 0. This indicates that the universe falls into an inactive state as it expands to infinity.
The SSS model can be evaluated by investigating the precise values of α G and β for the region that exhibits CMB anisotropy [24].The model predicts that a higher temperature region yields a larger G and m e .This can be identified as the reason for the formation of the large-scale structure in the universe.An alternative is to measure the precise CMB temperature in the region where dark matter is considered to exist [25,26].The model predicts that the CMB temperature in that region is higher than elsewhere because larger values of G and m e can be identified as dark matter.

Conclusions
We have demonstrated that the D B SSS model offers the simplest solution to the fine-tuning problem or the problems of naturalness.The dynamical vacuum energy that can be simply expressed as a function of the CMB temperature can cause inflation, and thus facilitates the evolution of the universe.We suggested a testable prediction to verify the hypothesis.Therefore, it is desired to perform observational investigations using the SSS model in the future.

1 UsingFigure 1 .
Figure 1.Time scale as a function of length scale for the SSS model and reference values.The reference values for the D A SSS model are taken from Ref. [16].The lower and upper bounds of the universe are interpolated in the D B SSS model.Note the symmetry of the first term L 0 , which corresponds to the CMB temperature.This symmetry indicates that each hierarchy is self-similar to the CMB temperature.

Table 1 .
[16]th scales of the hierarchies of the universe.Pl = √ hG/c 3 , where c is the speed of light in vacuum.bExperimentalresultsshowthat the range of the weak interaction is r w ≤ 10 −16 m[19].cDiameter of the sun, based on the nominal solar radius defined by the International Astronomical Union[20].dUpperbound of the universe derived from the D A SSS model[16].
a Planck length l

Table 2 .
Time scales of the hierarchies of the universe.
a Planck time t Pl = √ hG/c 5 .b The electromagnetic and weak forces unify at 100 GeV; [21] t = h/10 11 s.c Sun's rotational period; [22] t = 2.32 × 10 5 s.d Time scale of the universe derived from the D A SSS model

) Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 27 April 2018 doi:10.20944/preprints201804.0348.v1 where
ρ Pl is the Planck density.The solution of the Friedmann equation for a flat universe reveals the Hubble parameter H: