How to derive the first-order perturbed Klein-Gordon equation:
$$ \square \phi=\left[\frac{1}{\sqrt{-g}} \partial_{\mu}\left(\sqrt{-g}g^{\mu\nu} \partial_{\nu} \right) \right]\phi=0$$
For a first-order perturbed metric:
$$ g_{00} = -a^2 ( 1+ 2 \Phi), ~~~ g_{0i} = 0 , ~~~~ g_{ij} = a^2 (1-2 \Psi). $$
Here is my trial:
First : $ \sqrt{-g}= \sqrt{a^2 (1 +2 \Phi) (1 -2 \Psi)^3 }$
Also the scalar field: $\phi$ is perturbed : $\phi(x,t) \to \phi_0(t) + \phi_1(x,t) $. And both $\Phi$ and $\Psi$ are functions of (x,t).
The KG equation becomes:
$$\partial_0 \sqrt{-g} g^{00} \partial_0 \phi + \partial_i \sqrt{-g} g^{ij} \partial_j \phi=0 $$
Perturbe $g^{\mu\nu}$ and $\phi$, the equation becomes:
$$\partial_0 \sqrt{-g} (g^{00}+ \delta^{(1)} g^{00} ) \partial_0 (\phi_0 + \phi_1 ) + \partial_i \sqrt{-g} (g^{ij}+ \delta^{(1)} g^{ij} ) \partial_j (\phi_0 + \phi_1 ) =0 $$
Yields:
$$ \partial_0 \sqrt{-g} (-a^{-2} + -a^{-2}/2 \Phi^{-1} ) \partial_0 (\phi_0 + \phi_1 ) + \partial_i \sqrt{-g} (a^{-2} - a^{-2}/2 \Psi^{-1}) \partial_j (\phi_0 + \phi_1 ) =0 $$
Now I’m confuced how to substite by $\sqrt{-g}$ and get only the first order terms .
So any help to deal with the terms under the square root and complete the derivation.
I should get a result similar to equation (3.4) paper.
Any help is appreciated!
