Observational constraints of the modified cosmology through Barrow entropy

Taking into account a fractal structure for the black hole horizon, Barrow argued that the area law of entropy get modified due to quantum-gravitational effects [Phys. Lett. B $\mathbf{808}$ (2020) 135643]. Accordingly, the corrected entropy takes the form $S \sim A^{1+\mathrm{\Delta}/2}$, where $0\leq\mathrm{\Delta}\leq1$, indicates the amount of the quantum-gravitational deformation effects. In this paper, based on Barrow entropy, we first derive the modified gravitational field equations through the Clausius relation. We then consider the Friedmann-Lemaitre-Robertson-Walker (FLRW) as the background metric and derive the modified Friedmann equations inspired by Barrow entropy. In order to explore observational constraints on the modified Barrow cosmology, we employ two different combinations of available datasets, mainly "Planck + Pantheon + BAO" and "Planck + Planck-SZ + CFHTLenS + Pantheon + BAO + BAORSD" datasets. According to numerical results, we observe that "Planck + Pantheon + BAO" dataset predicts higher values of $H_0$ in Barrow cosmology with a phantom dark energy compared to $\mathrm{\Lambda}$CDM, so tensions between low redshift determinations of Hubble constant and cosmic microwave background (CMB) results is slightly reduced. On the other hand, in case of dataset "Planck + Planck-SZ + CFHTLenS + Pantheon + BAO + BAORSD" there is a slight amelioration in $\sigma_8$ tension in Barrow cosmology with a quintessential dark energy.