Cosmological Pseudo-Entropy

We study pseudo entropy mathcal{S}, a recent generalization of entanglement entropy, for scalar cosmological perturbations in de Sitter space with sound speed 0.024 leq c_s leq 1, and in expanding and contracting FLRW backgrounds with varying equation-of-state parameter w. In de Sitter space, Re\(mathcal{S}\) grows after horizon exit while c_s controls its onset and saturates at late times. A similar saturation occurs in expanding-accelerating and contracting-decelerating backgrounds. In contrast, expanding-decelerating and contracting-accelerating backgrounds show large early-time Re\(mathcal{S}\) followed by oscillations after horizon re-entry. This happens because while the squeezing freezes, the squeezing angle doesn't. Unlike entanglement entropy, pseudo entropy possesses an imaginary part, Im\(mathcal{S}\), as well, which can encode the relative phase. Im\(mathcal{S}\) decays to zero in de Sitter and expanding-accelerating cases, but forms dense sub-Hubble oscillation bands in expanding-decelerating and contracting-accelerating backgrounds. Compared with entanglement entropy, Krylov complexity, and Nielsen circuit complexity, pseudo entropy captures otherwise hidden phase information; in the unsaturated regime, its slope is sqrt{2} times that of Nielsen complexity. Unlike circuit complexity, whose saturation bound is w-independent, pseudo entropy is sensitive to w during the transition regime, making it a finer information theoretic diagnostic of cosmological dynamics.