First order phase transitions in a Bianchi type-I universe

Considering the theory of induced gravity coupled to matter fields, taking the ${\phi }^6$ interaction potential model we evaluate the one-loop effective potential in a (3+1)-dimensional Bianchi type-I spacetime. It is proved that the ${\phi }^6$ theory can be regularized in (3+1)-dimensional curved spacetime. We evaluate the finite temperature effective potential and study the temperature dependence of phase transitions. The nature of phase transitions in the early universe is clarified to be of first order. The effects of spacetime curvature and arbitrary field coupling on the phase transitions in the early universe are also discussed.