Spectral form factors for curved spacetimes with horizon

The spectral form factor is believed to exhibit a special type of behavior called ``dip-ramp-plateau'' in chaotic quantum systems that originates from random matrix theory. This suggests that the shape of the spectral form factor could serve as an indicator of chaos in various quantum systems. It has been shown recently that the dip-ramp-plateau structure appears in the spectral form factor when the normal modes of a massless scalar field theory in the brick-wall model of the BTZ black hole are treated as eigenvalues of a quantum Hamiltonian. At the same time, the level spacing distribution of these normal modes differs from that associated with random matrix theory ensembles. In this paper, we extend the results for BTZ background to the case of non-zero mass of the field, study the generalized spectral form-factor, and consider the same context for another non-trivial background -- de Sitter space. We compare the generalized spectral form factor for simple integrable quantum systems and for backgrounds with a horizon to the behavior predicted by random matrix theory. As a result, we confirm that BTZ and de Sitter brick-wall models are highly distinct integrable systems that exhibit the dip-ramp-plateau structure of the SFF but differ in the structure of the three-level generalized spectral form factor from the one predicted by random matrix theory. This raises the question on whether the DRP structure is an indicator of what is known as quantum chaos.