I am stuck varying an action, trying to get an equation of motion. (Going from eq. 91 to eq. 92 in the image.) This is the action
S = ∫d4xa2(t)2(˙h2−(∇h)2).
And this is the solution,
¨h+2˙aa˙h−∇2h = 0.
This is what I get
∂0(a2∂0h)−∂0(a2∇h)−∇(a2∂0h)+∇2(ha2) = 0.
I don't really see my mistake, perhaps I am missing something. (dot represents ∂0)
It is this problem (see Lectures on the Theory of Cosmological Perturbations, by Brandenburger):
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