### 8.2 The simplest model

The simplest model is the one in which
in the background, with . The regulator brane is assumed to be far enough away that its
effects on the physical brane can be neglected over the timescales of interest. By Equation (372) it follows
that
i.e., the matter on the regulator brane must have fine-tuned and negative energy density to prevent the
regulator brane from moving in the background. With these assumptions, and further assuming adiabatic
perturbations for the matter, there is only one independent brane-world parameter, i.e., the parameter
measuring dark radiation fluctuations:
This assumption has a remarkable consequence on large scales: The Weyl anisotropic stress terms
in the Sachs–Wolfe formula (321) cancel the entropy perturbation from dark radiation fluctuations, so that
there is no difference on the largest scales from the standard general relativity power spectrum. On small
scales, beyond the first acoustic peak, the brane-world corrections are negligible. On scales up to the first
acoustic peak, brane-world effects can be significant, changing the height and the location of the first peak.
These features are apparent in Figure 12. However, it is not clear to what extent these features are
general brane-world features (within the low-energy approximation), and to what extent they are
consequences of the simple assumptions imposed on the background. Further work remains to be
done.

A related low-energy approximation, using the moduli space approximation, has been developed for
certain 2-brane models with bulk scalar field [38, 268]. The effective gravitational action on the physical
brane, in the Einstein frame, is

where is a coupling constant, and and are moduli fields (determined by the zero-mode of the
bulk scalar field and the radion). Figure 13 shows how the CMB anisotropies are affected by the
-field.