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We study anisotropic cosmologies of a scalar field interacting with an SU(2)gauge field via a gauge-kinetic coupling. We analyze Bianchi class A models,which include Bianchi type I, II, VI0, VII0, VIII and IX. The linear stabilityof isotropic inflationary solution with background magnetic field is shown,which generalizes the known results for U(1) gauge fields. We also studyanisotropic inflationary solutions, all of which turn out to be unstable. Thennonlinear stability for the isotropic inflationary solution is examined bynumerically investigating the dependence of the late-time behaviour on theinitial conditions. We present a number of novel features that may well affectphysical predictions and viability of the models. First, in the absence ofspatial curvature, strong initial anisotropy leads to a rapid oscillation ofgauge field, thwarting convergence to the inflationary attractor. Secondly, theinclusion of spatial curvature destabilizes the oscillatory attractor and theglobal stability of the isotropic inflation with gauge field is restored.Finally, based on the numerical evidence combined with the knowledge of theeigenvalues for various inflationary solutions, we give a generic lower-boundfor the duration of transient anisotropic inflation, which is inverselyproportional to the slow-roll parameter.

Stability analysis of inflation with an SU(2) gauge field