C${\cal C}$osmological K${\cal K}$rylov C${\cal C}$omplexity

In this paper, we study the Krylov complexity (K) from the planar/inflationary patch of the de Sitter space using the two mode squeezed state formalism in the presence of an effective field having sound speed cs$c_s$. From our analysis, we obtain the explicit behavior of Krylov complexity (K) and lancoz coefficients (bn$b_n$) with respect to the conformal time scale and scale factor in the presence of effective sound speed cs$c_s$. Since lancoz coefficients (bn$b_n$) grow linearly with integer n, this suggests that universe acts like a chaotic system during this period. We also obtain the corresponding Lyapunov exponent lambda in presence of effective sound speed cs$c_s$. We show that the Krylov complexity (K) for this system is equal to average particle numbers suggesting it's relation to the volume. Finally, we give a comparison of Krylov complexity (K) with entanglement entropy (Von-Neumann) where we found that there is a large difference between Krylov complexity (K) and entanglement entropy for large values of squeezing amplitude. This suggests that Krylov complexity (K) can be a significant probe for studying the dynamics of the cosmological system even after the saturation of entanglement entropy.

Automated summary

In this paper, the Krylov complexity (K) of the de Sitter space is studied using the two mode squeezed state formalism in the presence of an effective sound speed. The explicit behavior of Krylov complexity (K) and Lancoz coefficients (bn) with respect to the conformal time scale and scale factor are obtained. The results suggest that the universe acts like a chaotic system during this period and that Krylov complexity (K) is related to the volume.