Rise of cosmological complexity: Saturation of growth and chaos

We compute the circuit complexity of scalar curvature perturbations on Friedmann-Lemaître-Robertson-Walker cosmological backgrounds with a 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 $\lambda \le \sqrt{2}\left|H\right|$, 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, and we find a scrambling time that is similar to other estimates up to order 1 factors.