Purely scalar perturbations are characterized by the fact that vectors and tensors are derived from scalar potentials, i.e.,
Scalar perturbative quantities are formed from the potentials via the (3D) Laplacian, e.g.,The KK energy density produces a scalar mode (which is present even if
in the
background). The KK momentum density carries scalar and vector
modes, and the KK anisotropic stress carries scalar, vector, and
tensor modes:
These equations are the basis for a -covariant analysis of cosmological perturbations
from the brane observer’s viewpoint, following the approach
developed in 4D general relativity (for a review, see [92]). The equations contain
scalar, vector, and tensor modes, which can be separated out if
desired. They are not a closed system of equations until
is determined by a 5D analysis of the bulk
perturbations. An extension of the
-covariant
perturbation formalism to
dimensions would require a
decomposition of the 5D geometric quantities along a timelike
extension
into the bulk of the brane 4-velocity
field
, and this remains to be done. The
-covariant perturbation formalism is incomplete until
such a 5D extension is performed. The metric-based approach does
not have this drawback.