Abstract
SU(2) gauge fields coupled to an axion field can acquire an isotropic background solution during inflation. We study homogeneous but anisotropic inflationary solutions in the presence of such (massless) gauge fields. A gauge field in the cosmological background may pose a threat to spatial isotropy. We show, however, that such models generally isotropize in Bianchi type-I geometry, and the isotropic solution is the attractor. Restricting the setup by adding an axial symmetry, we revisited the numerical analysis presented in [1]. We find that the reported numerical breakdown in the previous analysis is an artifact of parametrization singularity. We use a new parametrization that is well-defined all over the phase space. We show that the system respects the cosmic no-hair conjecture and the anisotropies always dilute away within a few e-folds.
Highlights
That such models generally isotropize in Bianchi type-I geometry, and the isotropic solution is the attractor
The SU(2) gauge field and its spatial isotropy have a number of compelling phenomenological and observational consequences. Is this isotropic gauge field’s VEV the attractor solution? Do the SU(2)-axion models respect the cosmic no-hair conjecture? Embedding the gauge-flation and chromo-natural inflation models in Bianchi type-I geometry, the above questions have been addressed in [36] and [37], respectively, and the case of massive SU(2) gauge field has been studied in [38]. All these studies were based on assuming i) an axial symmetry in Bianchi type-I geometry and ii) that the SU(2) VEV is diagonal in the same frame as the metric
Based on these restrictive assumptions, it was shown that the massless SU(2) gauge fields coupled to the axion field by a Chern-Simons interaction do respect the cosmic no-hair conjecture in Bianchi type-I geometry
Summary
Where R is the Ricci scalar, μ is the axion energy scale, λ is the Chern-Simons coupling constant, and Fμaν is the field strength tensor of the SU (2) gauge field given by. Where εσρμν is the totally anti-symmetric tensor with ε0123 = 1 We embed this system in a Bianchi type-I geometry with axial symmetry in x-direction such that ds2 = −dt2 + e2α(t) e−4σ(t)dx2 + e2σ(t) dy2 + dz. We embed this system in a Bianchi type-I geometry with axial symmetry in x-direction such that ds2 = −dt2 + e2α(t) e−4σ(t)dx2 + e2σ(t) dy2 + dz2 Upon introducing this geometry, we have the spatial triads for the SU(2) gauge group given by ea1(t) = eα−2σδ1a, ea2(t) = eα+σδ2a, ea3(t) = eα+σδ3a. Where the last part is the Chern-Simons term, which does not contribute to the energy density
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