Autonomous λ φ4 theory in a time-dependent space-time

After performing a special type of wave-function renormalization procedure, we obtain the autonomous \ensuremath{\lambda}${\ensuremath{\varphi}}^{4}$ Gaussian effective potential in time-dependent curved space. The bare coupling constant ${\ensuremath{\lambda}}_{B}$ is allowed to be positive but infinitesimal (${\ensuremath{\lambda}}_{B}$\ensuremath{\rightarrow}${0}^{+}$) for the occurrence of spontaneous symmetry breaking. Some properties different from those for flat space-time are also discussed.