Complexity of non-trivial sound speed in inflation

Abstract: In this paper, we study the impact of non-trivial sound on the evolution of cosmological complexity in inflationary period. The vacuum state of curvature perturbation could be treated as squeezed states with two modes, characterized by the two most essential parameters: angle parameter $\phi_k$ and squeezing parameter $r_k$. Through $Schr\ddot{o}dinger$ equation, one can obtain the corresponding evolution equation of $\phi_k$ and $r_k$. Subsequently, the quantum circuit complexity between a squeezed vacuum state and squeezed states are evaluated in scalar curvature perturbation with a type of non-trivial sound speed. Our results reveal that the evolution of complexity at early times shows the rapid solution comparing with $c_S=1$, in which we implement the resonant sound speed with various values of $\xi$. In these cases, it shows that the scrambling time will be lagged with non-vanishing $\xi$. Further, our methodology sheds a new way of distinguishing various cosmological models.