Anisotropic cosmological solutions in massive vector theories

In beyond-generalized Proca theories including the extension to theories higher than second order, we study the role of a spatial component v of a massive vector field on the anisotropic cosmological background. We show that, as in the case of the isotropic cosmological background, there is no additional ghostly degrees of freedom associated with the Ostrogradski instability. In second-order generalized Proca theories we find the existence of anisotropic solutions on which the ratio between the anisotropic expansion rate Σ and the isotropic expansion rate H remains nearly constant in the radiation-dominated epoch. In the regime where Σ/H is constant, the spatial vector component v works as a dark radiation with the equation of state close to 1/3. During the matter era, the ratio Σ/H decreases with the decrease of v. As long as the conditions |Σ| ≪ H and v2 ≪ ϕ2 are satisfied around the onset of late-time cosmic acceleration, where ϕ is the temporal vector component, we find that the solutions approach the isotropic de Sitter fixed point (Σ = 0 = v) in accordance with the cosmic no-hair conjecture. In the presence of v and Σ the early evolution of the dark energy equation of state wDE in the radiation era is different from that in the isotropic case, but the approach to the isotropic value wDE(iso) typically occurs at redshifts z much larger than 1. Thus, apart from the existence of dark radiation, the anisotropic cosmological dynamics at low redshifts is similar to that in isotropic generalized Proca theories. In beyond-generalized Proca theories the only consistent solution to avoid the divergence of a determinant of the dynamical system corresponds to v = 0, so Σ always decreases in time.

在超广义普罗卡理论（包括对二阶以上理论的扩展）中，我们研究了有质量矢量场的空间分量\(v\)在各向异性宇宙学背景下的作用。我们表明，如同在各向同性宇宙学背景的情况一样，不存在与奥斯特罗格拉茨基不稳定性相关的额外幽灵自由度。在二阶广义普罗卡理论中，我们发现存在各向异性解，在辐射主导时期，各向异性膨胀率\(\Sigma\)与各向同性膨胀率\(H\)之间的比率几乎保持恒定。在\(\Sigma/H\)恒定的区域，空间矢量分量\(v\)作为一种暗辐射，其状态方程接近\(1/3\)。在物质时代，\(\Sigma/H\)随着\(v\)的减小而减小。只要在晚期宇宙加速开始时满足\(\vert\Sigma\vert\ll H\)和\(v^{2}\ll\phi^{2}\)的条件（其中\(\phi\)是时间矢量分量），我们发现解会根据宇宙无毛猜想趋近于各向同性的德西特不动点（\(\Sigma = 0 = v\)）。在存在\(v\)和\(\Sigma\)的情况下，辐射时代暗能量状态方程\(w_{DE}\)的早期演化与各向同性情况不同，但趋近于各向同性值\(w_{DE}(iso)\)通常发生在红移\(z\)远大于\(1\)的时候。因此，除了暗辐射的存在，低红移下的各向异性宇宙动力学与各向同性广义普罗卡理论中的情况相似。在超广义普罗卡理论中，避免动力系统行列式发散的唯一一致解对应于\(v = 0\)，所以\(\Sigma\)总是随时间减小。