Exploring the limitless potential of the hyperbolic scale factor in a spherically symmetric spacetime

The research centered around exploring five-dimensional spherically symmetric metrics within the conceptual framework of Lyra geometry. The study aimed to address potential disparities by integrating a hyperbolic scale factor and a barotropic equation of state while assuming that the shear scalar is directly proportional to the expansion scalar. The physical and geometrical parameters were analyzed and examined with cosmic time and redshift. The study displayed isotropy within the universe for a specific value. Remarkably, the current estimates for the Hubble parameter, deceleration parameter, and the age of the universe are calculated as [Formula: see text] [Formula: see text]km[Formula: see text]s[Formula: see text]Mpc[Formula: see text], [Formula: see text], and 14.3 GYrs, respectively. The conclusions closely matched with the results derived from the [Formula: see text]CDM model.