Barrow Interacting holographic dark energy cosmology with Hubble horizon as IR cutoff: A model can Alleviating the Hubble and $S_8$ Tension

In this study, we perform a comprehensive analysis of the Hubble constant ($H_0$) and matter clustering ($S_8$) tensions within the framework of non-interacting and interacting Barrow Holographic Dark Energy (BHDE) models. Utilizing a combination of observational datasets, including the Cosmic Microwave Background (CMB), Baryon Acoustic Oscillations (BAO), cosmic chronometers (CC), Pantheon, and lensing data, we assess the degree of tension relative to the Planck 2018 results and recent measurements such as the Riess et al. 2022 (R22) value for $H_0=73.04\pm 1.04{\text{km s}}^{-1}{\text{Mpc}}^{-1}$ in 68% C.L and KiDS-1000 and DES-Y3 for $S_8$. Our findings show that both BHDE models mitigate the $H_0$ and $S_8$ tensions compared to the standard $\mathrm{\Lambda }$ Cold Dark Matter ($\mathrm{\Lambda }$CDM) model. The non-interacting BHDE model achieves a moderate reduction in the $H_0$ tension, while the interacting BHDE model offers a better fit for both parameters, suggesting it is more effective in addressing the tensions. Additionally, the quantum-gravitational deformation parameter $\mathrm{\Delta }$, constrained using the CMB+All dataset, indicates significant quantum effects in both models. The interacting scenario provides tighter constraints on $\mathrm{\Delta }$ and total neutrino mass $\sum m_{\nu }$, offering a more precise representation of these effects. This study highlights the potential of BHDE models as viable alternatives to the $\mathrm{\Lambda }$CDM framework for resolving cosmological tensions.