Note that even for adiabatic matter perturbations; for example, if in the background, then

The KK effects on the brane contribute a non-adiabatic mode, although at low energies.Although the density and curvature perturbations can be found on super-Hubble scales, the Sachs–Wolfe effect requires in order to translate from density/curvature to metric perturbations. In the 4D longitudinal gauge of the metric perturbation formalism, the gauge-invariant curvature and metric perturbations on large scales are related by

where the radiation anisotropic stress on large scales is neglected, as in general relativity, and is the scalar potential for , equivalent to the covariant quantity defined in Equation (295). In 4D general relativity, the right hand side of Equation (325) is zero. The (non-integrated) Sachs–Wolfe formula has the same form as in general relativity: The brane-world corrections to the general relativistic Sachs–Wolfe effect are then given by [271] where is the KK entropy perturbation (determined by ). The KK term cannot be determined by the 4D brane equations, so that cannot be evaluated on large scales without solving the 5D equations. (Equation (327) has been generalized to a 2-brane model, in which the radion makes a contribution to the Sachs–Wolfe effect [244].)The presence of the KK (Weyl, dark) component has essentially two possible effects:

- A contribution from the KK entropy perturbation that is similar to an extra isocurvature contribution.
- The KK anisotropic stress also contributes to the CMB anisotropies. In the absence of anisotropic stresses, the curvature perturbation would be sufficient to determine the metric perturbation and hence the large-angle CMB anisotropies via Equations (324, 325, 326). However, bulk gravitons generate anisotropic stresses which, although they do not affect the large-scale curvature perturbation , can affect the relation between , , and , and hence can affect the CMB anisotropies at large angles.

A simple phenomenological approximation to on large scales is discussed in [24], and the Sachs–Wolfe effect is estimated as

where is the 4D Planck time, and is the time when the KK anisotropic stress is induced on the brane, which is expected to be of the order of the 5D Planck time.A self-consistent approximation is developed in [245], using the low-energy 2-brane approximation [398, 428, 387, 399, 400] to find an effective 4D form for and hence for . This is discussed below. In a single brane model in the AdS bulk, full numerical simulations were done to find the behaviour of [69], as will be discussed in the next subsection.

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