Threedimensional and fourdimensional scalar, vector, tensor cosmological fluctuations and the cosmological decomposition theorem
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 3dimensional rotations. In attempting to solve the fluctuation equations that are automatically written in terms of gaugeinvariant 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 gaugeinvariant combinations only separate at a higherderivative level. To achieve separation at the level of the fluctuation equations themselves one has to assume boundary conditions for the higherderivative 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 wellbehaved at the origin. We show that in certain cases the gaugeinvariant combinations themselves involve both scalars and vectors. For such cases there is no decomposition theorem for the individual SVT components themselves, but for the gaugeinvariant combinations there still can be. Given the lack of manifest covariance in defining a basis with respect to 3dimensional rotations, we introduce an alternate SVT basis whose components are defined according to how they transform under 4dimensional general coordinate transformations. With this basis the fluctuation equations greatly simplify, and while one can again break them up into separate gaugeinvariant sectors at the higherderivative 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.
 Publication:

arXiv eprints
 Pub Date:
 December 2019
 arXiv:
 arXiv:1912.10448
 Bibcode:
 2019arXiv191210448P
 Keywords:

 General Relativity and Quantum Cosmology
 EPrint:
 61 pages