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FRW pertubations in overdense and underdense regions

  1. Oct 22, 2014 #1
    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?
     
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  3. Oct 26, 2014 #2

    PeterDonis

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    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.
     
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