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We study inflationary universes with an SU(3) gauge field coupled to aninflaton through a gauge kinetic function. Although the SU(3) gauge field growsat the initial stage of inflation due to the interaction with the inflaton,nonlinear self-couplings in the kinetic term of the gauge field becomesignificant and cause nontrivial dynamics after sufficient growth. Weinvestigate the evolution of the SU(3) gauge field numerically and revealattractor solutions in the Bianchi type I spacetime. In general cases where allthe components of the SU(3) gauge field have the same magnitude initially, theyall tend to decay eventually because of the nonlinear self-couplings.Therefore, the cosmic no-hair conjecture generically holds in a mathematicalsense. Practically, however, the anisotropy can be generated transiently in theearly universe, even for an isotropic initial condition. Moreover, we findparticular cases for which several components of the SU(3) gauge field surviveagainst the nonlinear self-couplings. It occurs due to flat directions in thepotential of a gauge field for Lie groups whose rank is higher than one. Thus,an SU(2) gauge field has a specialty among general non-Abelian gauge fields.

Cosmic No-hair Conjecture and Inflation with an SU(3) Gauge Field