Cosmological constant decaying with CMB temperature

Recent observations of the dark energy density have demonstrated the fine-tuning problem, and the challenges faced by theoretical modeling. 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 simi-larity dimension, DB , in the SSS model. Using the DB SSS model, the cosmological constant Λ is simply expressed as a function of the cosmic microwave background (CMB) temperature. The result shows that both the gravitational constant G and Λ are coupled with the CMB temper-ature, which simplifies the solution of Einstein’s field equations for the variable Λ-G model.


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 of the acceleration of the expansion of the universe [1,2].Einstein's cosmological term came to be recognized as the simplest candidate to explain the mechanism behind the accelerating universe.However, the inconsistencies between theoretical expectations and observations are extremely problematic, despite many theoretical models being proposed to provide a suitable explanation [3,4,5,6,7,8,9].To approach this issue from a different aspect, an axiomatic method has been proposed by Beck [10].Beck formulated a description of the cosmological constant, Λ, using four statistical axioms: fundamentality (Λ depends only on the fundamental constants of 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 these four axioms, Beck showed that Λ is given as follows: 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 [11,12], and has recently been discussed in several reports [13,14,15,16].
The result in our earlier study [40], which explains the hierarchical structure of the universe based on the Dirac large numbers hypothesis (LNH) [41,42], indicates that both G and m e are coupled with the cosmic microwave background (CMB) temperature.In the present study, we applied the previous result to Eq. (1).The new result indicates that Λ can be expressed as a function of the CMB temperature.It is well known that Λ and G are both important parameters in Einstein's field equations.Our result indicates that Λ and G are not independent functions, but are coupled with the CMB temperature.This means that we can simplify the solution of Einstein's field equations for the variable Λ-G model, because we can unify the two independent variables Λ and G into one simple function of the CMB temperature.

D B SSS model
The self-similar symmetry (SSS) model [40] 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 [29].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 define the fundamental dimensionless time and length ratios as follows: where t and l are the time and length scales of 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 as follows: The values of D A and D B correspond to the fractal dimensions for large-scale structures in the universe, reported by several authors (D = 1.2 [43,44,45,46] and D = 1.4 [47]).
The hierarchical structures of the D B SSS model are constructed according to the following sequences: where n and m are natural numbers that represent the hierarchical levels.The time scales of each hierarchy are also calculated using Eq. ( 4).Eqs. ( 7) and ( 8) indicate that L n and L m are self-similar to L 0 , which corresponds to the CMB temperature [40].

Verification of the D B SSS model
To verify the proposed D B SSS model, we compared the model values with reference values.Tables 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 closely agree with the reference values.Figure 1 shows the hierarchy time scale as a function of the length scale.The coincidences observed in the figure confirm the validity of the SSS model.

Discussion
Using the gravitational coupling constant α G = Gm 2 pr /hc and Eq. ( 2), it is obtained that 2A = − log α G .The following relations are satisfied: d Time scale of the universe derived from the D A SSS model [40].
Therefore, α G and β are important in forming the hierarchical structure of the universe.Regarding the similarity dimension, ) are the ratios of the length scales of the hierarchies [40]) 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 ratios1 .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), then 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 [40] in Eq. ( 14): where τ CMB = T CMB /T Pl , with T CMB being the CMB temperature and T Pl the Planck temperature.Then, we obtain where λ is the cosmological constant in reduced Planck units, λ = l 2 Pl Λ, and we define ξ ≡ α −2 f α 17) is based on the LNH.If we employ T CMB = 2.725K as the current parameter in Eq. ( 17), then we obtain λ 0 ≈ 3.07 × 10 −122 , which is consistent with the latest observational data [52].
If we adopt T CMB = T Pl as the initial condition of the universe, and substitute this into Eqs.( 15), (16), and ( 17), then we obtain α = β = 1 and λ TCMB=T Pl = α −6 f , which implies that the entire hierarchy was contained in a single point, and that a large λ can trigger cosmic inflation.The value of λ 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 λ → 0. This indicates that the universe falls into an inactive state as it expands to infinity.

Conclusions
We showed that the similarity dimension D B based on the SSS model is valid for describing the hierarchy structure of the universe.Using D B , Λ is expressed as a function of the CMB temperature.Linde [17] proposed that Λ is a function of temperature, and related it to the process of broken symmetries.Gasperini [20,21] also argued that Λ can be interpreted as a measure of the temperature of the vacuum.Our result supports their arguments in this respect.The cosmological scenario of a dynamically decaying Λ gives a natural interpretation for sufficiently small Λ in the present epoch.Taking Eq. ( 17) together with Eq. ( 15), we can unify the two parameters Λ and G in Einstein's field equations as a single parameter of the CMB temperature.We should apply this result to Einstein's field equations to analyze further details in the future.

Figure 1 :
Figure 1: The time scale as a function of the length scale for the SSS model and reference values.The reference values for the D A SSS model are taken from Ref. [40].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 :
[40]th scales of the hierarchies of the universe.Experimental results show that the range of the weak interaction is r w ≤ 10 −16 m[48].Diameter of the sun, based on the nominal solar radius defined by the International Astronomical Union[49].dUpperbound of the universe derived from the D A SSS model[40].
a Planck length l Pl = hG/c 3 , where c is the speed of light in vacuum.b c
a Planck time t