Problem in Dodelson Modern Cosmology

[ChrisVer]

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:
\rho = m_{\nu} n
And if we adopt a perturbation, then n \rightarrow n_{0} [1+ x ]
So in general what he asks from us is to calculate:
m_{\nu} n_{0} x
?

 

[Chalnoth]

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:
\rho = m_{\nu} n
And if we adopt a perturbation, then n \rightarrow n_{0} [1+ x ]
So in general what he asks from us is to calculate:
m_{\nu} n_{0} x
?

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.

 

[ChrisVer]

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 x which happens to be the monopole expansion of the neutrinos' temperature fluctuation.
x= \frac{3}{4 \pi} \int dΩ N(x,\hat{p}^{i},t ) \equiv 3 N_{0} (x,t)

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

 

[Chalnoth]

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 x which happens to be the monopole expansion of the neutrinos' temperature fluctuation.
x= \frac{3}{4 \pi} \int dΩ N(x,\hat{p}^{i},t ) \equiv 3 N_{0} (x,t)

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

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.