The Rise of Cosmological Complexity: Saturation of Growth and Chaos

TL;DR

The paper quantifies circuit complexity for cosmological perturbations in FLRW universes by modeling curvature perturbations as two-mode squeezed states and applying Nielsen’s geometric method. It derives the squeezing parameter dynamics for expanding and contracting backgrounds with fixed w, revealing chaotic-like growth of complexity after horizon exit and a universal bound on the growth rate proportional to |H|. The bound is saturated for expanding backgrounds with w <= -5/3 and contracting backgrounds with w >= 1, while de Sitter space among NEC-satisfying backgrounds exhibits maximal chaos and a horizon-exit–driven scrambling time. These results link quantum-chaos diagnostics to cosmological evolution and horizon-entanglement considerations, suggesting new avenues for understanding information processing in the early and late-time universe.

Abstract

We compute the circuit complexity of scalar curvature perturbations on FLRW cosmological backgrounds with fixed equation of state $w$ using the language of squeezed vacuum states. Backgrounds that are accelerating and expanding, or decelerating and contracting, exhibit features consistent with chaotic behavior, including linearly growing complexity. Remarkably, we uncover a bound on the growth of complexity for both expanding and contracting backgrounds $λ\leq \sqrt{2} \ |H|$, similar to other bounds proposed independently in the literature. The bound is saturated for expanding backgrounds with an equation of state more negative than $w = -5/3$, and for contracting backgrounds with an equation of state larger than $w = 1$. For expanding backgrounds that preserve the null energy condition, de Sitter space has the largest rate of growth of complexity (identified as the Lyapunov exponent), and we find a scrambling time that is similar to other estimates up to order one factors.

Figures (9)

• Figure 1: (Left) The squeezing parameter $r_k$ for expanding accelerating solutions grows at late times as $\ln a$ in a universal way, so that all equations of state have the same slope in the growth of the squeezing parameter. (Right) The squeezing angle, shown as $\sin(2\phi_k)$, for expanding accelerating solutions approaches $\phi_k \rightarrow -\pi/2 + {\mathcal{O}}(1/a^n)$ at late times, for some power $n$ determined in (\ref{['AccelLargeScaleSolnAngle']}), so that $\sin(2\phi_k)$ approaches zero at late times. As discussed in the text, the power $n$ depends on the equation of state $w$, and determines the slope of the complexity growth, as discussed in Section \ref{['sec:Complexity']}.
• Figure 2: (Left) The squeezing parameter $r_k$ for expanding decelerating solutions grows at early times (when the mode is outside the horizon) as $\ln a$ in a universal way, so that all equations of state have the same slope in the growth of the squeezing parameter. The squeezing then "freezes in" when the mode crosses the horizon, which depends on the equation of state. (Right) The squeezing angle, shown as $\sin(2\phi_k)$, for expanding decelerating solutions is approximately $\phi_k \rightarrow -\pi/2 + {\mathcal{O}}(a^n)$ at early times, for some power $n$, so that $\sin(2\phi_k)\approx$ zero when the mode is superhorizon. Once the mode crosses the horizon $\phi_k$ begins to grow, leading to oscillations in $\sin(2\phi_k)$. As discussed in the text, the power $n$ depends on the equation of state $w$, and determines the slope of the complexity decay, as discussed in Section \ref{['sec:Complexity']}.
• Figure 3: (Left) The squeezing parameter $r_k$ for an accelerating contracting universe grows until the mode exits the horizon, after which it "freezes in". Note that since the universe is contracting, the scale factor evolves from large values to small values, thus the flow of time is to the right. (Right) The squeezing parameter $r_k$ for a decelerating contracting universe is approximately zero until the mode exits the horizon, after which the squeezing grows as $\sim \ln a$, universally for all equations of state $w$.
• Figure 4: The complexity for accelerating expanding solutions grows linearly with $\ln a$ at late times. The slope grows with decreasing equation of state until $w = -5/3$, after which the slope saturates, so that the growth of complexity is bounded by $d{\mathcal{C}}/dt \leq \sqrt{2} H$.
• Figure 5: The slope of complexity growth $\lambda = d{\mathcal{C}}/dt$, evaluated in terms of the Hubble parameter $H = da/dt$, for an expanding (left) and contracting (right) FRLW background with equation of state $w$.