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

Abstract: In this research, we have reconstructed the extended $f(\mathcal{P})$ cubic gravity and symmetric $f(\mathcal{Q})$ teleparallel gravity from the $(m,n)$-type Barrow Holographic Dark Energy (BHDE) model. We have derived the unknown functions $f(\mathcal{P})$ and $f(\mathcal{Q})$ in terms of $\mathcal{P}$ and $\mathcal{Q}$, assuming a flat, homogeneous, and isotropic universe. To constrain our model parameters, we employed cosmic chronometer datasets and Baryon Acoustic Oscillation datasets, utilizing Markov Chain Monte Carlo (MCMC) method. We analysed the behaviour and stability of each model throughout the universe's evolution by studying crucial parameters such as the deceleration parameter, equation of state (EoS) parameter $\omega_{DE}$, density parameter $\Omega(z)$ and the square of the speed of sound $v_s<sup>2$. Additionally, we explored the cosmographic behaviour by plotting the jerk parameter, snap parameter, and lerk parameter against the redshift. Furthermore, we examined the $\omega&#39;<em>{DE}-\omega</em>{DE}$ phase plane, the $(r,s<sup>*)$, $(r,q)$ statefinder parameters, and the $Om(z)$ 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.