Vector perturbations

6.5 Vector perturbations

The vorticity propagation equation on the brane is the same as in general relativity,

Taking the $curl$ of the conservation equation (106

) (for the case of a perfect fluid, $qμ = 0 = π μν$ ), and using the identity in Equation (249

), one obtains

as in general relativity, so that Equation (323

) becomes

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 [219

]. This can be seen via the divergence equation for the magnetic part $H μν$ of the 4D Weyl tensor on the brane,

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:

On large scales, we can find a closed system for these vector perturbations on the brane [219

˙ ( 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 ].