Modified Friedmann equations and fractal Black Hole thermodynamics

The general relativity unification and quantum theory is a significant open problem in theoretical physics. This problem arises from the fact that these two fundamental theories, which describe gravity and the behavior of particles at the microscopic level, respectively, are currently incompatible. The unification of these theories is crucial for a complete comprehension of the fundamental forces and the nature of the universe. In this regard, the quantum properties of a Black Hole result in fundamental importance. By analyzing such properties in quantum field theory, in the first step, the gravity enters as a classical background. In semi-classical approximation, Black Holes will emit Hawking radiation with an almost thermal spectrum, while Black Hole’s entropy is proportional to the Black Hole’s horizon. Besides, Hawking’s temperature and Black Hole entropy should follow the first law of Black Hole thermodynamics. Also, Jacobson [Thermodynamics of spacetime: The Einstein equation of state, Phys. Rev. Lett.75 (1995) 1260, https://doi.org/10.1103/PhysRevLett.75.1260] showed shown that there is a connection between Black Hole thermodynamics and Einstein’s equation that opens the root of a potential thermodynamic nature of gravity. This issue opened a new impressive research framework in which the Einstein field equation can be expressed as a form of the first law of thermodynamics and vice versa. In this study, it is assumed that the universe has a fractal structure. Accordingly, modified Friedmann’s equations and the Black Holes thermodynamics in a fractal universe have been examined. The fractal framework shows what features and changes occur in the description of the universe, particularly in studying the thermodynamics of a Black Hole. However, the paper strategy is organized as follows: in the beginning, we consider the first thermodynamic law in a fractal universe. Then, we investigate the Friedmann equation of the fractal universe in the form of the entropy balance, this means $dQ=T_{h}d{S}_h$, where $dQ$ and $T_h$ are the thermal energy and horizon temperature. We consider the entropy $S_h$ have two terms; (1) obeys the usual area law and (2) the entropy production term due to the non-equilibrium thermodynamics of a fractal universe. Therefore, in a fractal universe, a term with non-equilibrium thermodynamics of spacetime may be needed. Also, we study the generalized second law of thermodynamics in a fractal universe. When the temperature of the apparent horizon and the temperature of the matter fields inside the horizon are equal, i.e. $T=T_h$, the second law of generalized thermodynamics can be obtained according to the state parameter range equation, which is consistent with the recent observations. Finally, in Sec. 6, based on the mathematical calculations, we study the various cosmological parameters such as the Hubble parameter, scale factor, deceleration parameter and equation of state parameter.