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Problem in Dodelson Modern Cosmology

  1. Jul 11, 2014 #1

    ChrisVer

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    Could you please confirm my idea of how to deal with the problem question in the attachment?
    In case of a non-relativistic neutrino, the energy density will be given by:
    [itex] \rho = m_{\nu} n [/itex]
    And if we adopt a perturbation, then [itex]n \rightarrow n_{0} [1+ x ] [/itex]
    So in general what he asks from us is to calculate:
    [itex] m_{\nu} n_{0} x [/itex]
    ?
     

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  2. jcsd
  3. Jul 12, 2014 #2

    Chalnoth

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    It's been a while so I don't know offhand, but let me just state that the problem is talking about a neutrino with non-zero mass, not a non-relativistic neutrino. In the early universe, neutrinos were still highly relativistic.
     
  4. Jul 12, 2014 #3

    ChrisVer

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    it's the last phrase- assume it non relativistic...
    It's OK I don't seek for a solution, I just want to see if I have grasped the idea. Because I don't have a closed form for that [itex]x[/itex] which happens to be the monopole expansion of the neutrinos' temperature fluctuation.
    [itex]x= \frac{3}{4 \pi} \int dΩ N(x,\hat{p}^{i},t ) \equiv 3 N_{0} (x,t) [/itex]

    and on the other hand, the equation for [itex]N(x,\hat{p}^{i},t )[/itex] is not solvable without having again any information about the fluctuations of the metric...
    in Fourier Space ([itex]\mu[/itex] the cosine between k,p ... [itex]\Psi,\Phi[/itex] the time and spatial fluctuations of metric, dots for the conformal time derivatives) :
    [itex] \dot{N} + \frac{p}{E} i k \mu N + \dot{\Phi} + \frac{E}{p} i k \mu \Psi =0 [/itex]
    which I don't think is solvable without knowing anything about the variables appearing.
     
    Last edited: Jul 12, 2014
  5. Jul 12, 2014 #4

    Chalnoth

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    Hmm, maybe you're right. I'm still unsure whether or not they can be assumed as non-relativistic for a), but clearly they are non-relativistic for b) (which makes sense, as they are definitely non-relativistic today).

    I could probably figure it out if I had my copy of Dodelson in front of me (sadly, it's packed away at the moment). But hopefully somebody else here can help.
     
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