Barrow cosmology and big-bang nucleosynthesis

Using thermodynamics-gravity conjecture, we present the formal derivation of the modified Friedmann equations inspired by the Barrow entropy, $S\ensuremath{\sim}{A}^{1+\ensuremath{\delta}/2}$, where $0\ensuremath{\le}\ensuremath{\delta}\ensuremath{\le}1$ is the Barrow exponent and $A$ is the horizon area. We then constrain the exponent $\ensuremath{\delta}$ by using big-bang nucleosynthesis (BBN) observational data. In order to impose the upper bound on the Barrow exponent $\ensuremath{\delta}$, we set the observational bound on $|\frac{\ensuremath{\delta}{T}_{f}}{{T}_{f}}|$. We find that the Barrow parameter $\ensuremath{\delta}$ should be around $\ensuremath{\delta}\ensuremath{\simeq}0.01$ in order not to spoil the BBN era. Next we derive the bound on the Barrow exponent $\ensuremath{\delta}$ in a different approach in which we analyze the effects of Barrow cosmology on the primordial abundances of light elements, i.e., helium $^{4}\mathrm{He}$, deuterium D, and lithium $^{7}\mathrm{Li}$. We observe that the deviation from standard Bekenstein-Hawking expression is small as expected. Additionally, we present the relation between cosmic time $t$ and temperature $T$ in the context of modified Barrow cosmology. We confirm that the temperature of the early Universe increases as the Barrow exponent $\ensuremath{\delta}$ (fractal structure of the horizon) increases, too.