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, D_B, in the SSS model. Using the D_B 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 = 1/8πα_f⁶, 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.