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 $\lambda \leq \sqrt{2} \ |H|$, 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.
Highlights
The relatively new concept called circuit complexity— conceptually defined as the minimum number of quantum gates necessary to construct a desired target state from a given reference state—has found several interesting applications in recent years
We have used the language of squeezed states to compute the circuit complexity of scalar cosmological perturbations, as outlined in [28], for expanding and contracting FLRW backgrounds with a fixed equation of state w
The cosmological complexity grows linearly with log a when modes are outside the horizon for accelerating expanding backgrounds and decelerating contracting backgrounds
Summary
The relatively new concept called circuit complexity— conceptually defined as the minimum number of quantum gates necessary to construct a desired target state from a given reference state—has found several interesting applications in recent years. While the mode is outside the horizon, the complexity is decreasing linearly with the logarithm of the scale factor, and it “freezes-in” once the mode reenters the horizon at late times This suggests that cosmological perturbations in an expanding radiation background are not chaotic, in contrast with the de Sitter case. We find that the squeezing and complexity for accelerating and decelerating (expanding) backgrounds behave qualitatively similar to their de Sitter and radiation counterparts from [28]: accelerating solutions show signs of quantum chaos, with complexity growing as modes exit the horizon, while the complexity for decelerating solutions decays until the mode reenters the horizon. The last term in (4) is usually removed by an integration by parts, but we will retain this form for our analysis This action represents perturbations of a scalar field coupled to an external time-varying source..
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