Abstract This research delves into the approximate relationship between the energy function (E) and the Hubble function (H) within cosmological. Utilizing the Friedmann equation it establishes a link between the universe s scale factor and the Hubble function. Through Taylor series approximation the study derives an approximation of the energy function under specific assumptions and approximations. Asymptotic analysis investigates the behavior of variables y and s shedding light on function limits and behaviors. The study incorporates an interactive 3D scatter plot visualization to elucidate the relationship between cosmological parameters and physical systems aiding in a comprehensive understanding of dynamics. Practical recommendations emphasize increasing data points for accuracy and validating with observational data while theoretical suggestions advocate exploring higher-order terms and considering additional physical factors. Keywords Energy Hubble Approximate relationship Asymptotic analysis Exact solution Cosmology