In cosmological perturbation theory it is convenient to use the scalar, vector, tensor (SVT) basis as defined according to how these components transform under 3-dimensional rotations. In attempting to solve the fluctuation equations that are automatically written in terms of gauge-invariant combinations of these components, the equations are taken to break up into separate SVT sectors, the decomposition theorem. Here, without needing to specify a gauge, we solve the fluctuation equations exactly for some standard cosmologies, to show that in general the various gauge-invariant combinations only separate at a higher-derivative level. To achieve separation at the level of the fluctuation equations themselves one has to assume boundary conditions for the higher-derivative equations. While asymptotic conditions suffice for fluctuations around a dS background or a $k=0$ RW background, for fluctuations around a $k\neq 0$ RW background one additionally has to require that the fluctuations be well-behaved at the origin. We show that in certain cases the gauge-invariant combinations themselves involve both scalars and vectors. For such cases there is no decomposition theorem for the individual SVT components themselves, but for the gauge-invariant combinations there still can be. Given the lack of manifest covariance in defining a basis with respect to 3-dimensional rotations, we introduce an alternate SVT basis whose components are defined according to how they transform under 4-dimensional general coordinate transformations. With this basis the fluctuation equations greatly simplify, and while one can again break them up into separate gauge-invariant sectors at the higher-derivative level, in general we find that even with boundary conditions we do not obtain a decomposition theorem in which the fluctuations separate at the level of the fluctuation equations themselves.

中文翻译：

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三维和四维标量、矢量、张量宇宙涨落和宇宙分解定理

在宇宙学微扰理论中，根据这些分量在 3 维旋转下的变换方式定义的标量、矢量、张量 (SVT) 基很方便。在尝试求解根据这些分量的规范不变组合自动编写的波动方程时，将方程分解为单独的 SVT 扇区，即分解定理。在这里，无需指定规范，我们精确求解某些标准宇宙学的涨落方程，以表明通常各种规范不变的组合仅在更高的导数水平上分离。为了在波动方程本身的层次上实现分离，必须为高阶微分方程假设边界条件。虽然渐近条件足以满足围绕 dS 背景或 $k=0$ RW 背景的波动，但对于围绕 $k\neq 0$ RW 背景的波动，还必须要求波动在原点表现良好。我们表明，在某些情况下，规范不变组合本身涉及标量和向量。对于这种情况，对于单独的 SVT 分量本身没有分解定理，但对于规范不变的组合，仍然可以有。鉴于在定义关于 3 维旋转的基础时缺乏明显的协方差，我们引入了一个替代的 SVT 基础，其分量是根据它们在 4 维一般坐标变换下的变换方式来定义的。在此基础上，波动方程大大简化，