About Analogous Gravitational Atoms

Axions, along with other ultra-light bosons, have been the focus of theoretical research in recent years as potential candidates for dark matter. Unraveling the composition of dark matter is of great significance for understanding the evolution of the universe and completing the Standard Model. Considering the ultra-light bosons bound by the gravitational potential of a black hole, they form a long-lasting "boson cloud". This system, due to its high analogy with atomic nuclei and their electron clouds, is also referred to as the "gravitational atom". This paper primarily investigates the energy level structure of various gravitational atom systems. This paper will outline the solutions to two stable black holes under general relativity: the Schwarzschild black hole (with mass as its key physical quantity) and the Kerr black hole (with mass and spin as its key physical quantities). These two black holes will be the central object of the gravitational atom model in this paper. Starting from the basic Schwarzschild black hole and scalar ultra-light boson, through this simple example, this paper will elaborate on the process of constructing the gravitational atom model, provide a general method for more complex models in the following text of this paper, and derive the general properties of gravitational atoms. In this process, we will find that the gravitational atom is highly analogous to the hydrogen-like atom system in terms of energy spectrum structure. The paper will explore methods to construct the spectra of more complex gravitational atom systems for scalar and vector bosons. During the process, this paper will, through appropriate approximation, analytically derive the solutions to the field equations of gravitational atoms for scalar and vector fields. The solution process will use the method of matched asymptotic expansion to ensure the consistency of solutions between different regions. Solutions for scalar and vector fields have similarities and differences, and the gravitational atom spectra from these solutions will determine the superradiant instability of the gravitational atom system. This paper will also outline possible future research preferences and provide a brief update on the latest research status of gravitational atoms.

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