C${\cal C}$osmological K${\cal K}$rylov C${\cal C}$omplexity. Issue 12 (13th September 2022) - Adhikari, Kiran; Choudhury, Sayantan
Abstract: 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 c s $c_s$ . From our analysis, we obtain the explicit behavior of Krylov complexity ( K ) and lancoz coefficients ( b n $b_n$ ) with respect to the conformal time scale and scale factor in the presence of effective sound speed c s $c_s$ . Since lancoz coefficients ( b n $b_n$ ) grow linearly with integer n, this suggests that universe acts like a chaotic system during this period. We also obtain the corresponding Lyapunov exponent λ in presence of effective sound speed c s $c_s$ . 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. Abstract : Here a study is carried out about 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 . The explicit behavior of Krylov complexity ( K ) and Lancoz coefficients ( b n $b_n$ ) is obtained with respect to the conformal time scale and scale factor in the presence of effective sound speed cs . Since Lancoz coefficients ( b n $b_n$ ) grow linearly with integer n, this suggests that universe acts like a chaotic system during this period. The authors also obtain the corresponding Lyapunov exponent λ in presence of effective sound speed cs . It is shown that the Krylov complexity ( K ) for this system is equal to average particle numbers suggesting it's relation to the volume. Finally, a comparison is given of Krylov complexity ( K ) with entanglement entropy (Von‐Neumann) where it is 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.