Three-dimensional and four-dimensional scalar, vector, tensor cosmological fluctuations and the cosmological decomposition theorem

TL;DR

This paper investigates the scalar, vector, tensor (SVT) decomposition in cosmological perturbation theory, showing that the standard decomposition theorem generally holds only at a higher-derivative level and introducing an alternative basis for simplification.

Contribution

It introduces an alternative SVT basis based on 4D transformations and analyzes the conditions under which the decomposition theorem applies in various cosmological backgrounds.

Findings

Decomposition theorem holds at higher-derivative level with boundary conditions.

Standard SVT components may involve mixed scalar and vector parts.

Alternative basis simplifies fluctuation equations but does not guarantee decomposition at the equation level.

Abstract

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.