Barrow Entropy Corrections to Friedmann Equations

Inspired by the Covid-$19$ virus structure, Barrow argued that quantum-gravitational effects may introduce intricate, fractal features on the black hole horizon [Phys. Lett. B {\bf808} (2020) 135643]. In this viewpoint, black hole entropy no longer obeys the area law and instead it can be given by $S\sim A^{1+\delta/2}$, where the exponent $\delta$ ranges $0\leq\delta\leq1$, and indicates the amount of the quantum-gravitational deformation effects. Based on this, and using the deep connection between gravity and thermodynamics, we disclose the effects of the Barrow entropy on the cosmological equations. For this purpose, we start from the first law of thermodynamics, $dE=TdS+WdV$, on the apparent horizon of the Friedmann-Robertson-Walker (FRW) Universe, and derive the corresponding modified Friedmann equations by assuming the entropy associated with the apparent horizon has the form of Barrow entropy. We also examine the validity of the generalized second law of thermodynamics for the Universe enclosed by the apparent horizon. Finally, we employ the emergence scenario of gravity and extract the modified Friedmann equation in the presence of Barrow entropy which coincide with one obtained from the first law of thermodynamics. When $\delta=0$, the results of standard cosmology are deduced.