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:
[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]
?

 

[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:
[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]
?

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 [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.

 

[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 [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.

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.

 

[sardar]

Your idea seems to be on the right track. In order to fully address the problem question, we need to consider the effects of a non-relativistic neutrino on the energy density of the universe. As you mentioned, the energy density can be expressed as a product of the neutrino mass (m_{\nu}) and the number density of neutrinos (n). This is a good starting point.

However, in the case of a perturbation, the number density of neutrinos will also be affected. As you suggested, we can express the perturbed number density (n_{0}[1+x]) as a function of the unperturbed number density (n_{0}) and the perturbation (x). This perturbation can represent changes in the number of neutrinos due to various factors, such as gravitational interactions or cosmological expansion.

Therefore, in order to fully address the problem question, we need to calculate the product of the neutrino mass, the unperturbed number density, and the perturbation. This will give us the contribution of the non-relativistic neutrino to the energy density of the universe. I would also suggest considering the effects of other particles and factors on the energy density, in order to get a complete understanding of the problem.

In summary, your idea is a good starting point, but in order to fully address the problem question, we need to consider the effects of a perturbation on the number density of neutrinos and calculate the resulting contribution to the energy density. I hope this helps and good luck with your research!