Reconstructions of f(𝒫)Lg and f(𝒬)Lg gravity models from (m,n)Lg-type Barrow Holographic Dark Energy: Analysis and Observational Constraints

In this research, we have reconstructed the extended [Formula: see text] cubic gravity and symmetric [Formula: see text] teleparallel gravity from the [Formula: see text]-type Barrow Holographic Dark Energy (BHDE) model. We have derived the unknown functions [Formula: see text] and [Formula: see text] in terms of [Formula: see text] and [Formula: see text], assuming a flat, homogeneous, and isotropic universe. To constrain our model parameters, we employed cosmic chronometer datasets and Baryon Acoustic Oscillation (BAO) datasets, utilizing Markov Chain Monte Carlo (MCMC) method. We analyzed the behavior and stability of each model throughout the universe’s evolution by studying crucial parameters such as the deceleration parameter, equation of state (EoS) parameter [Formula: see text], density parameter [Formula: see text] and the square of the speed of sound [Formula: see text]. Additionally, we explored the cosmographic behavior by plotting the jerk parameter, snap parameter, and lerk parameter against the redshift. Furthermore, we examined the [Formula: see text] phase plane, the [Formula: see text], [Formula: see text] statefinder parameters, and the [Formula: see text] parameter offers profound revelations about the dynamics of the universe and the distinctive features of dark energy. Our analyses indicated that our model could produce a universe undergoing accelerated expansion with quintessence-type dark energy. These findings contribute to our understanding of the nature of dark energy and the evolution of the cosmos.