FRW pertubations in overdense and underdense regions

[center o bass]

I'm currently reading Dodelson's "Modern Cosmology" where he in Chapter 4.2 discuss the Boltzmann equation for photons and consider a perturbed FRW spacetime for which $g_{00} = -1 - 2 \Psi$ and $g_{ij} = a^2 \delta_{ij}(1+2\Phi)$. At page 90 he states that "in an overdense region we have $\Psi <0$ and $\Phi >0$. Why is this true?

 

[PeterDonis]

"in an overdense region we have Ψ<0\Psi 0. Why is this true?

I'm not an expert in this area, so I can't give the "standard" answer for why it's true, but I can explain why it seems reasonable to me. Consider the Schwarzschild metric, for which $g_{00} = - \left( 1 - 2M / r \right) = -1 + 2M / r$ and $g_{rr} = 1 / \left( 1 - 2M / r \right) \approx 1 + 2M / r$ (where we are using an approximation in which $M / r$ is small so we can ignore quadratic and higher terms). This obviously looks a lot like the perturbed FRW metric; in fact, if we put the FRW metric in spherical coordinates, the two are the same (except for the scale factor $a^2$) if we set $\Psi = - M / r$ and $\Phi = M / r$. So basically, what Dodelson is saying is that in an overdense region, the extra mass (due to the overdensity) acts like an ordinary gravitating body.