h_{0 μ} = 0

, so the explicit metric terms vanish. Symmetry of the covariant energy-momentum tensor and the background metric imply

δ T^{j}_{0} = - a^{- 2} δ^{ji} δ T^{0}_{i} .

Hence the two expansion terms in part (c) combine to

3 Hδ T^{0}_{j}

\partial_{0} [(\overset{ρ}{ˉ} + \overset{ˉ}{P}) \partial_{j} δ u] + 3 H (\overset{ρ}{ˉ} + \overset{ˉ}{P}) \partial_{j} δ u + \partial_{j} δ P + \partial_{j} \nabla^{2} π^{S} = 0.

Solved by gpt-5.6-sol high.

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