Study of a flat universe using the FRW model

We explore the Friedmann–Robertson–Walker cosmological model within the context of Lyra geometry, with a particular focus on the deceleration parameter expressed in terms of the Hubble parameter, $q = - \frac{R\ddot{R}}{\dot{R^{2}}} = - \Bigl( \frac{\dot{H} + H^{2}}{H^{2}} \Bigr) = b\ (\mathrm{const})$, and the equation of state in the form $P = \gamma\rho$. The Lyra geometry, an extension of Riemannian geometry, introduces a gauge function that modifies the conventional metric, potentially offering novel insights into cosmological phenomena. We investigate the impact of this geometry on the dynamics of the universe by analyzing the behavior of the deceleration parameter, which indicates the rate of change of the universe's expansion. Our findings demonstrate that the incorporation of Lyra geometry significantly influences the deceleration parameter, suggesting new possibilities for understanding cosmic acceleration and the transition from deceleration to acceleration phases. This work enhances our understanding of the universe's evolution and provides a platform for future research into alternative geometric frameworks in cosmology.