How Do Perturbation Equations Affect FRW Cosmology Metrics?

  • #1
ergospherical
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Homework Statement
##T^{00} = a^{-2} \bar{\rho}(1+\delta)##
##T^{0i} = a^{-2} \bar{\rho}(1+w)v^i##
##T^{ij} = a^{-2} \bar{\rho} [(1+\delta)\delta^{ij} - h^{ij}]##
Relevant Equations
##\nabla_{\mu} T^{\mu \nu} = 0##
The perturbed line element: ##g = a(\tau)^2[-d\tau^2 + (\delta_{ij} + h_{ij})dx^i dx^j]##
Expanding the covariant derivative with ##\nu = 0##, you get a few pieces. Here on keeping only terms linear in the perturbations,

##\partial_{\mu} T^{\mu 0} = a^{-2} \bar{\rho} \left[ \delta' - 2\mathcal{H} (1+\delta) + (1+w) i\mathbf{k} \cdot \mathbf{v} \right]##

here ##\mathcal{H} = a'/a## and ##i \mathbf{k} \cdot \mathbf{v} = \partial_i v^i##. Then

##\Gamma^{\mu}_{\mu \rho} T^{\rho 0} = a^{-2} \bar{\rho} \left[ 4\mathcal{H}(1+\delta) + \frac{1}{2} h' \right]##

##\Gamma^{0}_{\mu \rho} T^{\mu \rho} = a^{-2} \bar{\rho} \left[ \mathcal{H}(1+\delta)(1+3w) + \frac{1}{2} w h' \right]##

Overall,
##0 = a^{-2} \bar{\rho} \left[ \delta' + 3(1+w) \mathcal{H}(1+\delta) + (1+w) i \mathbf{k} \cdot \mathbf{v} + \frac{1}{2}(1+w)h'\right]##

but the term ##3(1+w) \mathcal{H}(1+\delta)## shouldn't be there. I can't see why not? For reference, the connection coefficients

1707826896803.png
 

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