AI 中文总结
研究均匀各向异性背景下线性扰动长波长演化，利用空间梯度展开推导出超视界解并建立对应关系，确定守恒量，揭示曲率扰动演化更丰富，还建立了相关形式主义及关系，为预测统计各向异性提供基础。
AI 中文摘要
我们研究了在一个与矢量场耦合的标量场的均匀各向异性背景下线性扰动的长波长演化。利用在均匀-$\mathcal{N}$规范下的空间梯度展开，其中$e$-折叠数不受扰动，我们推导出了超视界解的完整集合，并建立了它们与均匀各向异性背景的无穷小变化之间的对应关系。这将之前在各向同性FLRW宇宙学中已知的独立宇宙图景扩展到了各向异性时空，尽管由于旋转对称性破缺会引起标量、矢量和张量扰动的混合。我们表明长波长方程形成了一个自洽系统，并确定了一个守恒量，它推广了各向同性宇宙学中的守恒朗斯基行列式。与各向同性情况不同，产生曲率扰动的超视界模式由与标量场、背景切变和规范场倾斜相关的三个独立守恒通道以及源自各向异性几何的额外动态切变贡献所支配。这表明各向异性背景周围曲率扰动的演化本质上比各向同性多场模型更丰富。我们的公式提供了一个直接从视界穿越涨落计算最终曲率扰动的实用方法，从而建立了$\delta N$形式主义的各向异性推广。我们进一步推导了曲率扰动与原初引力波之间的明确关系，展示了各向异性膨胀如何在超视界尺度上耦合标量和张量部分。我们的框架为预测原初标量和张量扰动中的统计各向异性提供了一个实用基础。
英文摘要
We investigate the long-wavelength evolution of linear perturbations in a homogeneous and anisotropic background with a scalar field coupled to a vector field. Using the spatial gradient expansion in the uniform-$\mathcal{N}$ gauge in which the number of $e$-folds is unperturbed, we derive the complete set of superhorizon solutions and establish their correspondence with infinitesimal variations of the homogeneous anisotropic background. This extends the separate-universe picture, previously known for isotropic FLRW cosmology, to anisotropic spacetimes despite the mixing of scalar, vector, and tensor perturbations induced by broken rotational symmetry. We show that the long-wavelength equations form a self-consistent system and identify a conserved quantity that generalizes the conserved Wronskian of isotropic cosmology. Unlike the isotropic case, the superhorizon modes sourcing the curvature perturbation are governed by three independent conserved channels associated with the scalar field, the background shear, and the gauge-field tilt, together with an additional dynamical shear contribution originating from the anisotropic geometry. This reveals that the evolution of curvature perturbations around anisotropic background is intrinsically richer than in isotropic multi-field models. Our formulation provides a practical prescription for computing the final curvature perturbation directly from horizon-crossing fluctuations, thereby establishing the anisotropic generalization of the $δN$ formalism. We further derive an explicit relation between curvature perturbations and primordial gravitational waves, demonstrating how anisotropic expansion couples scalar and tensor sectors on superhorizon scales. Our framework provides a practical basis for predicting statistical anisotropies in primordial scalar and tensor perturbations.
Comments33 pages
