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6.1 1 + 3-covariant perturbation equations on the brane

In the 1 + 3-covariant approach [218Jump To The Next Citation Point201Jump To The Next Citation Point220], perturbative quantities are projected vectors, Vm = V<m>, and projected symmetric tracefree tensors, Wmn = W <mn>, which are gauge-invariant since they vanish in the background. These are decomposed into (3D) scalar, vector, and tensor modes as
Vm = \~/ mV + Vm, (245) W = \~/ \~/ W + \~/ W + W , (246) mn &lt;m n&gt; &lt;m n&gt; mn
where Wmn = W <mn> and an overbar denotes a (3D) transverse quantity,
\~/ mVm = 0 = \~/ nWmn. (247)
In a local inertial frame comoving with um, i.e., um = (1,0), all time components may be set to zero: Vm = (0,Vi), W0m = 0, \~/ m = (0, \~/ i).

Purely scalar perturbations are characterized by the fact that vectors and tensors are derived from scalar potentials, i.e.,

Vm = Wm = Wmn = 0. (248)
Scalar perturbative quantities are formed from the potentials via the (3D) Laplacian, e.g., V = \~/ m\ ~/ mV =_ \~/ 2V. Purely vector perturbations are characterized by
Vm = Vm, Wmn = \~/ &lt;mWn &gt;, curl \~/ mf = - 2f wm, (249)
where wm is the vorticity, and purely tensor by
\~/ mf = 0 = Vm, Wmn = Wmn. (250)

The KK energy density produces a scalar mode \~/ mrE (which is present even if rE = 0 in the background). The KK momentum density carries scalar and vector modes, and the KK anisotropic stress carries scalar, vector, and tensor modes:

qEm = \~/ mqE + qEm, (251) E E E E pmn = \~/ &lt;m \~/ n&gt;p + \~/ &lt;mpn&gt; + pmn. (252)
Linearizing the conservation equations for a single adiabatic fluid, and the nonlocal conservation equations, we obtain
r + Q(r + p) = 0, (253) c2 \~/ + (r + p)A = 0, (254) s m m r + 4Qr + \~/ mqE = 0, (255) E 3 E m E E 1 4 n E (r + p) qm + 4Hq m + -- \~/ mrE + -rE Am + \~/ pmn = - ------- \~/ mr. (256) 3 3 c
Linearizing the remaining propagation and constraint equations leads to
1 1 k2 r Q + -Q2 - \~/ mAm + -k2(r + 3p) - /\ = - ---(2r + 3p)--- k2rE , (257) 3 2 2 c w + 2Hw + 1-curlA = 0, (258) m m 2 m k2 E smn + 2Hsmn + Emn - \~/ &lt;mAn &gt; =--pmn, (259) 2 2 2 E + 3HE - curlH + k-(r + p)s = - k--(r + p)rs mn mn mn 2 mn 2 c mn k2 [ E E E] - --- 4rE smn + 3pmn + 3Hp mn + 3 \~/ &lt;mqn&gt; , (260) 26 Hmn + 3HHmn + curlEmn = k-curl pE , (261) 2 mn \~/ mwm = 0, (262) \~/ nsmn - curlwm - 2- \~/ mQ = - qEm, (263) 3 curlsmn + \~/ &lt;mwn&gt;- Hmn = 0, (264) 2 2 2 [ ] \~/ nE - k- \~/ r = k-r- \~/ r + k-- 2 \~/ r - 4HqE - 3 \~/ npE , (265) mn 3 m 3 c m 6 m E m mn n 2 2 r k2[ E] \~/ Hmn - k (r + p)wm = k (r + p) -wm + ---8rE wm - 3curl qm . (266) c 6
Equations (253View Equation), (255View Equation), and (257View Equation) do not provide gauge-invariant equations for perturbed quantities, but their spatial gradients do.

These equations are the basis for a 1 + 3-covariant analysis of cosmological perturbations from the brane observer’s viewpoint, following the approach developed in 4D general relativity (for a review, see [92]). The equations contain scalar, vector, and tensor modes, which can be separated out if desired. They are not a closed system of equations until pEmn is determined by a 5D analysis of the bulk perturbations. An extension of the 1 + 3-covariant perturbation formalism to 1 + 4 dimensions would require a decomposition of the 5D geometric quantities along a timelike extension uA into the bulk of the brane 4-velocity field um, and this remains to be done. The 1 + 3-covariant perturbation formalism is incomplete until such a 5D extension is performed. The metric-based approach does not have this drawback.



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