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8.2 The simplest model

The simplest model is the one in which
˙ ρℰ = 0 = d¯ (382 )
in the background, with p ∕ρ = p∕ρ − −. 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 (372View Equation) it follows that
¯ ρ− = − ρe2d, (383 )
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:
δρ ℰ δC ∗ = ----. (384 ) ρrad
View Image

Figure 12: The CMB power spectrum with brane-world effects, encoded in the dark radiation fluctuation parameter δC∗ as a proportion of the large-scale curvature perturbation for matter (denoted ζ∗ in the plot). (Figure taken from [177Jump To The Next Citation Point].)

This assumption has a remarkable consequence on large scales: The Weyl anisotropic stress δπ ℰ terms in the Sachs–Wolfe formula (321View Equation) 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 12View Image. 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 [38Jump To The Next Citation Point, 268Jump To The Next Citation Point]. The effective gravitational action on the physical brane, in the Einstein frame, is

∫ [ 2 ] S = -1-- d4x√ −-g R − -12-α---(∂φ)2 − ---6----(∂ χ)2 − V(φ, χ) , (385 ) eff 2κ2 1 + 2α2 1 + 2α2
where α is a coupling constant, and φ and χ are moduli fields (determined by the zero-mode of the bulk scalar field and the radion). Figure 13View Image shows how the CMB anisotropies are affected by the χ-field.
View Image

Figure 13: The CMB power spectrum with brane-world moduli effects from the field χ in Equation (385View Equation). The curves are labelled with the initial value of χ. (Figure taken from [38Jump To The Next Citation Point, 268Jump To The Next Citation Point].)

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