Cosmological Complexity from initial thermal state

The cosmological scalar perturbations should satisfy the thermal distribution at the beginning of inflation since the cosmic temperature is presumably very high. In this paper, we investigate, by the Fubini-study method, the effect of this thermal contribution, which is characterized by a parameter $\kappa_{0}$, on the evolution of the cosmological complexity $\mathcal{C}_{FS}$ . We find that when the thermal effect is considered, the Universe would ``decomplex" firstly with the cosmic expansion after the mode of the scalar perturbations exiting the horizon in the de Sitter (dS) phase and $\mathcal{C}_{FS}$ has a minimum about $\pi/4$. If $\mathcal{C}_{FS}$ can reach its minimum during the dS era, which requires a small $\kappa_0$ or a large e-folding number for a large $\kappa_0$, it will bounce back to increase, and after the Universe enters the radiation dominated (RD) phase from the dS one, $\mathcal{C}_{FS}$ will decrease, pass its minimum again, and then increase till the mode reenters the horizon. For the case of a large enough $\kappa_0$, $\mathcal{C}_{FS}$ decreases but does not reach its minimum during the dS era, and it begins to increase after the transition from the dS phase to the RD one. When the mode reenters the horizon during the RD era, the cosmological complexity will oscillate around about $\kappa_{0}$. These features are different from that of the initial zero-temperature case, i.e., the cosmological complexity increases during the dS phase and decreases in the RD era till the mode reenters the horizon. Our results therefore suggest that the thermal effect changes qualitatively the evolutionary behavior of the cosmological complexity.