Varying an action (cosmological perturbation theory)

Question

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

$$S~=~\int d^{4}x \frac{a^{2}(t)}{2}(\dot{h}^{2}-(\nabla h)^2).$$

And this is the solution,

$$\ddot{h} + 2 \frac{\dot{a}}{a}\dot{h} - \nabla^{2}h~=~0. $$

This is what I get

$$\partial_{0}(a^{2}\partial_{0}h)-\partial_{0}(a^{2}\nabla h)-\nabla(a^{2}\partial_{0}h)+\nabla^{2}(ha^{2})~=~0.$$

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):

Answer 1

Hints:

1. The Lagrangian density in the $(+,-,-,-)$ convention is $$ {\cal L}~=~\frac{a^2}{2}d_{\mu}h ~d^{\mu}h. $$

2. The corresponding Euler-Lagrange equation (by varying the action $S[h]=\int \!d^4x ~{\cal L}$ wrt. the field $h$) is $$ d_{\mu}(a^2 ~d^{\mu}h)~=~0. $$

3. Or equivalently, under the assumption that $a=a(t)$, $$ \frac{2\dot{a}\dot{h}}{a} + d_{\mu}d^{\mu}h~=~0. $$

4. Finally, Fourier transform the three spatial directions to get eq. (92).