Autonomous λφ4theory in a time-dependent space-time

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