Cosmological brick walls & quantum chaotic dynamics of de Sitter horizons

Abstract

Originally proposed by 't Hooft, the brick wall model has recently reemerged as a useful framework for probing quantum aspects of horizon physics, particularly in the context of holography. In this paper, we apply it to asymptotically de Sitter spacetimes. We compute the normal modes of a massless scalar field in pure de Sitter space and in the Schwarzschild-de Sitter black hole, and analyze the resulting single-particle spectra using the level-spacing distribution, the spectral form factor, and Krylov complexity. In pure de Sitter, the spectrum exhibits clear long-range signatures of chaos despite not obeying a conventional Wigner-Dyson level-spacing distribution. The Schwarzschild-de Sitter case is qualitatively richer: in the WKB regime, where tunneling between the two classically allowed regions is exponentially suppressed, the presence of both an event horizon and a cosmological horizon gives rise to two independent near-horizon sectors, so that the full spectrum is the superposition of two subsequences. As a result, the combined level-spacing distribution develops a nonzero value at $s=0$ even when spectral correlations remain. Nevertheless, for sufficiently small stretched-horizon fluctuations, the superposed spectrum still exhibits an approximately linear ramp in the spectral form factor and a pronounced peak in Krylov complexity. Our results show that the absence of strict level repulsion should not, by itself, be taken as evidence against chaos, and that the spectral form factor and Krylov complexity provide sharper diagnostics of the underlying chaotic dynamics.

Figures (27)

• Figure 1: Normal modes of a massless scalar field in the static patch of dS$_4$, for $n=0$, with brick wall boundary condition $\Phi(r_0)=\mu_l e^{i\lambda_l\omega}$. The parameter $\lambda_l$ is drawn from a Gaussian distribution with mean $\langle\lambda_l\rangle=\frac{1}{2}\log\left(2-2r_0\right)\approx 10^{-4}$ and variance $\sigma^2=\sigma_0^2/l$.
• Figure 2: Behavior of the level-spacing distributions of the normal modes as a function of the variance $\sigma_0^2$, averaged over 200 realizations. The solid lines correspond to the conventional Wigner--Dyson distributions of the standard "$\beta$-ensemble," (GSE/GUE/GOE) as indicated in parentheses.
• Figure 3: Ensemble-averaged spectral form factor for the scalar field, with the orange lines indicating a unit-slope ramp on the log-log scale.
• Figure 4: Ensemble-averaged Krylov complexity of the scalar field normal modes when $\sigma_0=\{0,~ 0.017,~ 0.024,~ 0.030,~ 0.5\}$ for black, red, blue, green and black, respectively.
• Figure 5: Potential for the Schwarzschild--de Sitter black hole with $r_e=1$, $r_c=\sqrt{5}$, $\omega=0.1$, and $l=2$. The blue curve represents the full potential, while the orange and green curves show the leading-order expansions around the two horizons.