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6.5 Vector perturbations

The vorticity propagation equation on the brane is the same as in general relativity,
1 wm + 2Hwm = - --curlAm. (323) 2
Taking the curl of the conservation equation (106View Equation) (for the case of a perfect fluid, qm = 0 = pmn), and using the identity in Equation (249View Equation), one obtains
2 curl Am = - 6Hc swm (324)
as in general relativity, so that Equation (323View Equation) becomes
w + (2 - 3c2)Hw = 0, (325) m s m
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 Hmn of the 4D Weyl tensor on the brane,
2 2 [ r] 4- 2 1- 2 E \~/ Hm = 2k (r + p) 1 + c wm + 3 k rEwm - 2 k curl qm, (326)
where Hmn = \~/ <mHn >. Even when wm = 0, there is a source for gravimagnetic terms on the brane from the KK quantity curl qEm.

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

a am = awm, bm = --curlqEm. (327) r
On large scales, we can find a closed system for these vector perturbations on the brane [219Jump To The Next Citation Point]:
( 2) am + 1 - 3cs Ham = 0, [ ] (328) 2- ( 2 )rE- 2r- bm + (1 - 3w)Hbm = 3 H 4 3cs- 1 r - 9(1 + w) c am. (329)
Thus we can solve for am and bm on super-Hubble scales, as for density perturbations. Vorticity in the brane matter is a source for the KK vector perturbation bm 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/r = const., the solutions are
( )3w-1 a = b -a- , (330) m m a0 ( )3w- 1 [ ( )2(3w- 1) ( ) -4] b = c -a- + b 8rE-0 a-- + 2(1 + w)r0- a-- , (331) m m a0 m 3r0 a0 c a0
where bm = 0 = cm.

Inflation will redshift away the vorticity and the KK mode. Indeed, the massive KK vector modes are not excited during slow-roll inflation [40269].



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