Covariant and gauge-invariant approach to cosmological density fluctuations

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Abstract
It has been known for a long time that the gauge problem plagues the study of density perturbations in cosmology. The quantity δμ/μ (the fractional variation in density along a world line) usually determined in perturbation calculations is completely dependent on the gauge chosen. Even the fully covariant approach of Hawking (1966) is not immune. Bardeen’s major paper (1980) determines a set of gauge-invariant quantities that are related to density perturbations but are not those perturbations themselves. We give a simple alternative representation of density fluctuations. This representation is both fully covariant and gauge invariant; thus it sidesteps the usual problems. The basic quantity used to represent density inhomogeneities is the comoving fractional gradient of the energy density orthogonal to the fluid flow. Our description does not make the usual assumption that this gradient is small. Exact (fully nonlinear) propagation equations are derived for this quantity. They are then linearized to give propagation equations appropriate to the case of an almost-Robertson-Walker universe. Their solutions are obtained in a simple case which can be compared with the standard theory; we recover the usual growing and decaying modes. Thus the result is standard, but its derivation is completely transparent. We give an interpretation of the Bardeen variables in terms of our formalism.