Cosmological Complexity

Abstract

We compute the quantum circuit complexity of the evolution of scalar curvature perturbations on expanding backgrounds, using the language of squeezed vacuum states. In particular, we construct a simple cosmological model consisting of an early-time period of de Sitter expansion followed by a radiation-dominated era and track the evolution of complexity throughout this history. During early-time de Sitter expansion the complexity grows linearly with the number of e-folds for modes outside the horizon. The evolution of complexity also suggests that the Universe behaves like a chaotic system during this era, for which we propose a scrambling time and Lyapunov exponent. During the radiation-dominated era, however, the complexity decreases until it "freezes in" after horizon re-entry, leading to a "de-complexification" of the Universe.

Figures (8)

• Figure 1: In this qualitative plot, we follow the growth of the squeezing parameter $r_k$ as a function of the scale factor $a$ for fixed $k$ as it starts small inside the horizon, then grows larger than one after horizon exit, then "freezes out" upon horizon re-entry with a decaying oscillation, as described in the text. Notice that while outside of the horizon the squeezing parameter grows as the number of e-folds spent super-horizon $r_k \sim \log a \sim N_e^{(k)}$.
• Figure 2: (Left) The squeezing parameter $r_k$ as a function of the scale factor $a$ for de Sitter space for the exact solution (\ref{['exactdSSqueezing']}) and numerical solutions to the squeezing equations (\ref{['martin22']}) for $k = 0.001$ in units of $\eta_0$, defined by $a(\eta_0) = 1$ (color online). The squeezing parameter grows appreciably -- and logarithmically -- only on super-horizon scales $k < 1/|\eta|$. (Right) The same graph shown with a linear scale for $r_k$ demonstrates that the growth on super-horizon scales is proportional to the number of e-folds of expansion since mode $k$ exited the horizon $r_k \sim N_e^{(k)}$.
• Figure 3: (Left) The squeezing angle $\phi_k$ for a dS background (for the same $k$ as Figure \ref{['fig:dSSqueezing']}) oscillates around $\phi_k = -\pi/4$ when the mode is inside the horizon, and then transitions to $\phi = -\pi/2$ after the mode exits the horizon, in accordance with our qualitative results from the text and the exact solution (\ref{['exactdSSqueezing']}). (Right) The squeezing angle for a radiation-dominated background with $k = 0.1$ (again in units of $\eta_0$), plotted as $\cos(2\phi_k)$. Notice that at early times while the mode is super-horizon we have $\phi_k \approx -\pi/2$, while after the mode re-enters the horizon we have $\phi_k \sim k \eta$ increasing with time leading to oscillations in $\cos(2\phi_k)$ which cuts off further growth in $r_k$, in agreement with our qualitative analysis in the text.
• Figure 4: (Left) The squeezing parameter $r_k$ for a radiation background with $k = 0.1$ in units of $\eta_0$ is plotted against the scale factor $a$. Since modes start outside the horizon in a radiation background, the squeezing is large and growing at early times. Once the mode re-enters the horizon, however, the squeezing "freezes in" with a damped oscillation about the value at horizon crossing. (Right) Different wavenumbers (again in units of $\eta_0$) lead to different times of horizon re-entry, and thus different "freeze in" values of the squeezing.
• Figure 5: The squeezing parameter $r_k$ for a cosmological background consisting of de Sitter followed by radiation shows the features already seen in the de Sitter and radiation plots separately ($k=0.01$ in units of $\eta_0$). The squeezing, initially small, grows upon horizon exit and continues growing through the transition to radiation. Eventually the mode re-enters the horizon during the radiation era and "freezes out" at its value at horizon crossing.