Entanglement Entropy Near Cosmological Singularities

TL;DR

The paper investigates how entanglement entropy in a confining gauge theory responds to cosmological singularities using holography. By modeling the boundary theory on Kasner$_{3+1}\times S^{1}$ and employing a Kasner-AdS soliton bulk, the authors compute the covariant holographic entanglement entropy for a strip and analyze its UV-divergent structure and universal finite part as functions of time and the Kasner exponent $p_1$. A key finding is a confinement/deconfinement transition signaled by a topology change of the extremal surface when it reaches the soliton cap, with the transition timing dependent on $p_1$. The time-dependent finite part $c_0(t_b)$ reveals rich behavior beyond static extrapolations, showing that the energy scale of confinement is controlled more by the evolving entangling-region width than by the area of the entangling surface, and highlighting how singular cosmological dynamics imprint on quantum entanglement in strongly coupled gauge theories. These results extend our understanding of holographic entanglement in time-dependent backgrounds and potentially illuminate early-universe quark-gluon plasma physics.

Abstract

We investigate the behavior of the entanglement entropy of a confining gauge theory near cosmological singularities using gauge/gravity duality. As expected, the coefficients of the UV divergent terms are given by simple geometric properties of the entangling surface in the time-dependent background. The finite (universal) part of the entanglement entropy either grows without bound or remains bounded depending on the nature of the singularity and entangling region. We also discuss a confinement/deconfinement phase transition as signaled by the entanglement entropy.

Figures (5)

• Figure 1: For small proper width $\mathcal{L}$ in the Kasner-AdS soliton background, the extremal surfaces remain close to the boundary and resemble surfaces in pure AdS. The area of the surface depends on $\mathcal{L}$, so the entanglement entropy is a function of $\mathcal{L}$ and the modes which contribute are deconfined. As the length of the entangling region becomes larger, extremal surfaces extend further into the bulk. The soliton cap begins to affect their geometry, and they start leveling out. For some critical length, the surface splits into two separate surfaces (displayed above in dashed blue). At this point, the area of the extremal surface is independent of $\mathcal{L}$, effectively signaling that $1/\mathcal{L}$ is below the mass gap of the confined gauge theory.
• Figure 2: (a): The maximum value of $z$ as a function of $t_{b}$, the boundary time. Each line corresponds to a different value of $p_{1} \in [-0.3, 1]$: surfaces with positive $p_{1}$ shrink towards the boundary as $t_{b}\rightarrow 0$ while surfaces with $p_{1}<0$ approach the cap. Surfaces with $p_{1}=0$ maintain constant $z_{*}$. (b): A plot of $z_{*}(t_{b})= 1.48 \mathcal{L}$ for small $t_{b}$ and $p_{1} \in [- \frac{1}{3}, 1]$. This describes the radial extent of minimal surfaces until $z_* \approx 1$ when the surface splits into two, signaling a confinement/deconfinement transition.
• Figure 3: A plot of $\zeta(p_{1})$, where $c_{0}(t) \propto t_{b}^{\zeta(p_{1})}$, for $\frac{1}{2}<p_{1}<1$. The best-fit line is $\zeta(p_{1})= 1-4p_{1}$.
• Figure 4: $c_{0}$ as a function of time for $0<p_{1}<1/2$. As $p_{1}$ decreases, the $c_{0}$ curves level out, while still apparently diverging at early times, in contrast with the static prediction, which would imply that they go to zero for $p_{1}< 1/4$.
• Figure 5: $c_{0}$ as a function of time for $-0.3<p_{1}<0$. The curves correspond to $p_{1}=-0.03$, $-0.11$, $-0.17,$, $-0.22$, $-0.26$, and $-0.30$.