Fractal Dimensions and the Expansion History of the Universe

We present a cosmological framework in which the large-scale evolution of the universe is governed by the interplay between the fractal dimensions of matter ($D_m$) and spacetime ($D_v$). Motivated by the hypothesis that the universe originates from a fundamentally entangled state, we propose a fractal model for the Hubble parameter $H(z)$, partitioned into three redshift regimes and constrained using observational data from Baryon Acoustic Oscillations (BAO), Supernovae Type Ia (SNe Ia), and Cosmic Chronometers (CCT). We show that this model offers an improved fit over the standard $¥Lambda$CDM cosmology, particularly near the transitional redshift range $z ¥sim 2.7$. The relation $H(z)^2 ¥propto a^{D_m - D_v}$, where $a$ is the scale factor, encapsulates the dynamical effects of fractal geometry. This formulation naturally accounts for late-time acceleration as a result of $D_m - D_v < 0$ across most of cosmic history. We further explore implications for structure formation, entanglement-modulated dynamics, and observational tests.