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.