On the emergence of fractal cosmic space from fractional quantum gravity

Abstract

This dissertation investigates a cosmological model that explains the observational data on the matter content of the Universe using Padmanabhan’s theory of emergent cos- mology and insights from fractional quantum gravity applied to the Schwarzschild black hole. Two main directions lead to this model. On the one hand, we start with the Hamil- tonian formalism of general relativity and the canonical quantization of the theory leading to the Wheeler-DeWitt equation. A spherically symmetric spacetime then simplifies the application of the Wheeler-DeWitt equation and we can investigate the quantization of the Schwarzschild black hole, its mass spectrum, and thermodynamics, in the semi-classical limit. The study of fractals and the use of the Riesz fractional derivative via fractional quantum gravity show that the surface area of the event horizon of the Schwarzschild black hole has a random fractal structure, whose description is possible by fractional quanti- ties. On the other hand, we show that the apparent cosmological horizon provides both a Hawking temperature associated with the horizon of an FLRW spacetime and is the most suitable horizon for obtaining the Friedmann equations with Padmanabhan’s theory in which cosmic space and its expansion emerge due to the tendency to satisfy the holo- graphic principle. Finally, due to the results indicated by fractional quantum cosmology, we argue the following proposition: the cosmological apparent horizon of the Universe has the same structure of a random fractal as the event horizon of the Schwarzschild black hole. This leads to modified Friedmann equations that reveal an effect of fractal geometry that amplifies the content of baryonic matter already existing in the Universe and thus simulates the additional content of matter that we currently call dark matter.