Cosmological coevolution of Yang-Mills fields and perfect fluids

We study the coevolution of Yang-Mills fields and perfect fluids in Bianchi type I universes. We investigate numerically the evolution of the universe and the Yang-Mills fields during the radiation and dust eras of a universe that is almost isotropic. The Yang-Mills field undergoes small amplitude chaotic oscillations, as do the three expansion scale factors which are also displayed by the expansion scale factors of the universe. The results of the numerical simulations are interpreted analytically and compared with past studies of the cosmological evolution of magnetic fields in radiation and dust universes. We find that, whereas magnetic universes are strongly constrained by the microwave background anisotropy, Yang-Mills universes are principally constrained by primordial nucleosynthesis but the bound is comparatively weak with ${\Omega }_{\mathrm{YM}}<0.105{\Omega }_{\mathrm{rad}}$.

Figure 1

The typical behaviors of ${\Sigma }^2$ ${\Omega }_{\mathrm{YM}}$, and ${\Omega }_{\mathrm{m}}$ with increasing time. This figure shows how ${\Sigma }^2\to 0$, and the spacetime approaches the isotropic Friedmann model. It is remarkable that ${\Omega }_{\mathrm{YM}}$ does not damp.

Figure 2

Typical behaviors of the Yang-Mills field. (a) $a$, $b$, and $c$ damp while undergoing chaotic oscillations of increasing frequency. (b) The averaged damping rate of $a$, $b$, and $c$ has the approximate form $e^{-\tau }$.

Figure 3

Typical behaviors of the time derivative of the principal components of the Yang-Mills field. The mean amplitudes of this shows amplitudes of $a^{\prime }$, $b^{\prime }$, and $c^{\prime }$ remain nearly constant.

Figure 4

The typical behaviors of the anisotropic pressure components ${\Pi }_{\pm }$. This shows that ${\Pi }_{\pm }$ oscillate in a complicated way with nearly constant mean amplitudes.

Figure 5

The typical behaviors of the $⟨{\mathcal{A}}^2+{\tilde{\mathcal{A}}}^2⟩$, $⟨{\mathcal{B}}^2+{\tilde{\mathcal{B}}}^2⟩$, and $⟨{\mathcal{C}}^2+{\tilde{\mathcal{C}}}^2⟩$, where $⟨\cdots ⟩$ means time average per $0.1\tau$. This shows that ${\mathcal{A}}^2+{\tilde{\mathcal{A}}}^2$, ${\mathcal{B}}^2+{\tilde{\mathcal{B}}}^2$, and ${\mathcal{C}}^2+{\tilde{\mathcal{C}}}^2$ are nearly equal after the shear is significantly reduced, i.e. $\tau >2$, although ${\mathcal{A}}^2$, ${\mathcal{B}}^2$, ${\mathcal{C}}^2$, ${\tilde{\mathcal{A}}}^2$, ${\tilde{\mathcal{B}}}^2$, and ${\tilde{\mathcal{C}}}^2$ oscillate strongly.