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6.5 Vector perturbations

The vorticity propagation equation on the brane is the same as in general relativity,
1 ω˙μ + 2H ωμ = − --curlA μ. (323 ) 2
Taking the curl of the conservation equation (106View Equation) (for the case of a perfect fluid, qμ = 0 = π μν), and using the identity in Equation (249View Equation), one obtains
2 curl Aμ = − 6Hc sω μ (324 )
as in general relativity, so that Equation (323View Equation) becomes
˙ω + (2 − 3c2) H ω = 0, (325 ) μ s μ
which expresses the conservation of angular momentum. In general relativity, vector perturbations vanish when the vorticity is zero. By contrast, in brane-world cosmology, bulk KK effects can source vector perturbations even in the absence of vorticity [219Jump To The Next Citation Point]. This can be seen via the divergence equation for the magnetic part H μν of the 4D Weyl tensor on the brane,
⃗2 ¯ 2 [ ρ] 4- 2 1- 2 ℰ ∇ H μ = 2κ (ρ + p) 1 + λ ωμ + 3 κ ρℰωμ − 2 κ curl ¯qμ, (326 )
where ⃗ ¯ H μν = ∇⟨μH ν⟩. Even when ωμ = 0, there is a source for gravimagnetic terms on the brane from the KK quantity curlq¯ℰμ.

We define covariant dimensionless vector perturbation quantities for the vorticity and the KK gravi-vector term:

a α¯μ = aω μ, β¯μ = --curl ¯qℰμ. (327 ) ρ
On large scales, we can find a closed system for these vector perturbations on the brane [219Jump To The Next Citation Point]:
˙ ( 2) ¯αμ + 1 − 3cs H ¯αμ = 0, [ ] (328 ) ¯˙ ¯ 2- ( 2 ) ρ-ℰ 2ρ- βμ + (1 − 3w )H βμ = 3 H 4 3cs − 1 ρ − 9(1 + w ) λ ¯α μ. (329 )
Thus we can solve for ¯αμ and ¯βμ on super-Hubble scales, as for density perturbations. Vorticity in the brane matter is a source for the KK vector perturbation ¯βμ on large scales. Vorticity decays unless the matter is ultra-relativistic or stiffer (w ≥ 1 3), and this source term typically provides a decaying mode. There is another pure KK mode, independent of vorticity, but this mode decays like vorticity. For w ≡ p∕ρ = const., the solutions are
( )3w−1 α¯ = b -a- , (330 ) μ μ a0 ( )3w− 1 [ ( )2 (3w− 1) ( ) −4] ¯β = c -a- + b 8ρℰ-0 a-- + 2(1 + w)ρ0- a-- , (331 ) μ μ a0 μ 3ρ0 a0 λ a0
where ˙bμ = 0 = ˙cμ.

Inflation will redshift away the vorticity and the KK mode. Indeed, the massive KK vector modes are not excited during slow-roll inflation [40, 269].


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