The possibility of inflationary models in which inflation is driven by a vector field rather than a scalar field is discussed. The vector field $A^{\rho }$ is taken to be self-coupled through a ‘‘potential’’ V(ξ), where ξ=$A^{\rho }$$A_{\rho }$. If V has a flat region, where ‖ξV’‖≪V, then the Universe can undergo a period of isotropic inflation in which the space is approximately de Sitter. Because the vector field’s stress tensor is not isotropic, the Universe will exit inflation into an anisotropic expansion. If the stable minimum of V occurs at ξ=${\xi }_0$=0, then this anisotropy will damp away during the reheating period. If this minimum occurs at a nonzero value of ${\xi }_0$, then the anisotropy can be small at late times if collisionless particles, such as gravitons, are generated during reheating. In this case, the observed limits on anisotropy of the cosmic microwave background require that ${\xi }_0$≲${(10\mathrm{}}^{15}$ GeV${)}^2$. Finally, even if V does not have the flat region needed for de Sitter inflation, it is possible to have anisotropic inflation in which the Universe expands at different exponential rates in different directions. The conditions under which this can occur are discussed, and the stability of the resulting solutions is analyzed.

• Received 1 February 1989

DOI:https://doi.org/10.1103/PhysRevD.40.967