Gravitational wave equation in FR sapace

1. Mar 9, 2013

I am stuck varying an action.
This is the action $S=\int d^{4}x \frac{a^{2}(t)}{2}(\dot{h}^{2}-(\nabla h)^2)$

And this is the solution, $\ddot{h} + 2 \frac{\dot{a}}{a}\dot{h} - \nabla^{2}h$.

This is what I get $\partial_{0}(a^{2}\partial_{0}h)-\partial_{0}(a^{2}\nabla h)-\nabla(a^{2}\partial_{0}h)+\nabla^{2}(ha^{2})$
I don't really see my mistake, perhaps i am missing something. (dot represents $\partial_{0}$)

It is this problem (see Lectures on the Theory of Cosmological
Perturbations, by Brandenburger)

Going from step 91 to 92