Primordial Neutrinos: Density & Temperatures

[Buzz Bloom]

The folowing quote is from a Wikpedia article https://en.wikipedia.org/wiki/Neutrino .

"From cosmological arguments, relic background neutrinos are estimated to have density of 56 of each type per cubic centimeter and temperature 1.9 K (1.7×10−4 eV) if they are massless, much colder if their mass exceeds 0.001 eV."

There is no reference given for the "facts" presented in the quote. I entered a request for such a reference on the Talk page, but I expect it will take some time for this reference to become included. Can anyone help me find a reference about this that will include an explanation about assumptions and the method used to calculate the numbers?

 

[wabbit]

Did you check http://en.wikipedia.org/wiki/Cosmic_neutrino_background ?

 

[Buzz Bloom]

Hi wabbit:

Thanks for responding. Yes, I looked at that also, and it also has "facts' without any reference:

"Like the cosmic microwave background radiation (CMB), the CνB is a relic of the big bang; while the CMB dates from when the universe was 379,000 years old, the CνB decoupled from matter when the universe was two seconds old. It is estimated that today, the CνB has a temperature of roughly 1.95 K."

 

[wabbit]

It does provide a calculation of the temperature. Not in the introduction you quote but in the first paragraph. I don't understand what you re saying.

Which fact are you seeking an explanation about ?

 

[Buzz Bloom]

Hi wabbit:

I just added a comment the talk page for that Wiki article. The article does provide the derivation for the temperature, but not the assumptions. My guess is that one assumption is that temperature varies inversely with a(t), but that is so only if the neutrinos are in a state of equilibrium with other stuff, and that isn't so after nuclear synthesis ends. After synthesis ends, temperature would vary inversely with a². (See https://www.physicsforums.com/threa...ergy-vary-with-a-t.811627/page-2#post-5096390 .)

 

[wabbit]

OK I wasn't aware of this 1/a^2 scaling of temperature, thought it was 1/a.
This gives more details I think
http://eagle.phys.utk.edu/guidry/astro421/lectures/lecture490_ch20.pdf

 

[PeterDonis]

After synthesis ends, temperature would vary inversely with a2.

That's not quite correct. The correct statement is that temperature varies inversely with $a$ for relativistic particles (which includes radiation), and inversely as $a^2$ for non-relativistic particles (which includes all the ordinary matter and dark matter in our present universe). At the time of neutrino decoupling, the neutrinos are still highly relativistic, so their temperature will still vary as $1 / a$. Whether or not the neutrinos become non-relativistic at some point after decoupling depends on the neutrino masses; for small enough neutrino masses, they could still be relativistic even today.

 

[Buzz Bloom]

Hi PeterDonis:

Thank you for your post. I do undestand the distinction you make here.

The origin of my post about a² is the quote "if they are massless, much colder if their mass exceeds 0.001 eV." I would like a reference that explains the assumptions and the calulation for the 0.001 eV threshold energy for the neutrino's currently still behaving like a massless particle. Also, it seems strange to me that there should be such a threshold value (rather than a range) between behaving like a massless particle or berhaving like a particle with mass. There should be a range when the particle makes a continuous transition between these behaviors.

The era of nuclear synthesis is given as about 10 secs (some say 2 seconds, others say several minutes). The value of a is not given. but the temperatrure for synthesis has been given as 10,000,000 K. That would correspond to an a of about 3,000,000 for photons. I have not seen a relativistic calculation for a near light speed particle, but I would very much like to undestand the calculation for a 0.001 eV mass particle with a 10,000,000 K temperature when a = 3,000,000.

Can you recommend a reference?

 

[Chalnoth]

Hi wabbit:

I just added a comment the talk page for that Wiki article. The article does provide the derivation for the temperature, but not the assumptions. My guess is that one assumption is that temperature varies inversely with a(t), but that is so only if the neutrinos are in a state of equilibrium with other stuff, and that isn't so after nuclear synthesis ends. After synthesis ends, temperature would vary inversely with a². (See https://www.physicsforums.com/threa...ergy-vary-with-a-t.811627/page-2#post-5096390 .)

See the following paragraph:

The above discussion is valid for massless neutrinos, which are always relativistic. For neutrinos with a non-zero rest mass, the description in terms of a temperature is no longer appropriate after they become non-relativistic; i.e., when their thermal energy $3/2 kT_\nu$ falls below the rest mass energy $m_\nu c^2$. Instead, in this case one should rather track their energy density, which remains well-defined.

You'd get at their energy density by looking at their number density while they're still relativistic. The energy density then becomes a function of the neutrino masses.

 

[Chalnoth]

Note that neutrino masses are likely in the $eV$ range, which is equivalent to a temperature of over $10,000K$, so their temperature is almost certainly small enough to be negligible today.

 

[Chalnoth]

Also, let me add one other thing. This statement isn't true:

My guess is that one assumption is that temperature varies inversely with a(t), but that is so only if the neutrinos are in a state of equilibrium with other stuff, and that isn't so after nuclear synthesis ends.

Temperature varies inversely as $1/a(t)$ as long as the neutrinos remain relativistic. They don't need to interact with anything to maintain this temperature scaling because they'll retain their thermal distribution without interactions.

 

[mfb]

Note that neutrino masses are likely in the $eV$ range, which is equivalent to a temperature of over $10,000K$, so their temperature is almost certainly small enough to be negligible today.

eV range sounds a bit high with upper limits of <0.23 eV on the sum of all three types. The squared mass differences from mixing suggest masses in the range of a few meV to tens of meV. That still makes them likely to be non-relativistic, at least the heavier type(s).

 

[Buzz Bloom]

I have found a site about current efforts to measure the rest mass of neutrinos: http://www.katrin.kit.edu/ . There is a presentation dated September 8, 2014 giving the followng upper limit for the mass: 0.2 eV (90% CL). Does anyone know what the "CL" means? My guess is "Confidence Level".

The site says that they expect to have actual values later this year with a much better error range than anything previously done.

 

[Buzz Bloom]

I have been trying to understand how the effective mass (energy/c²) of a neutrino varies with a(t) from the time neutrinos stopped being in equilibrium with other stuff, up to the present. I did not realize how complicated the math is to do this calcualtion. Here is what I have learned so far.

The Maxwell-Juttner relativistic distribution function:
(Please excuse the layout. I have issues with the tooI use to create equations.)

What I would like to calculate is the the average value of γ for this distibution for values of ϑ. From that I can then calculate how the temperature of the primordial neutrinos varies with a(t). Does anyone know a reference that can help me?​

 

[Mordred]

I have been trying to understand how the effective mass (energy/c²) of a neutrino varies with a(t) from the time neutrinos stopped being in equilibrium with other stuff, up to the present. I did not realize how complicated the math is to do this calcualtion. Here is what I have learned so far.

The Maxwell-Juttner relativistic distribution function:
(Please excuse the layout. I have issues with the tooI use to create equations.)

View attachment 83090
https://www.physicsforums.com/attachments/83094

What I would like to calculate is the the average value of γ for this distibution for values of ϑ. From that I can then calculate how the temperature of the primordial neutrinos varies with a(t). Does anyone know a reference that can help me?​

http://arxiv.org/pdf/1212.6154

also
http://www.wiese.itp.unibe.ch/lectures/universe.pdf:" Particle Physics of the Early universe" by Uwe-Jens Wiese Thermodynamics, Big bang Nucleosynthesis

see chapter 3

 

[Buzz Bloom]

Hi Mordred:

Thank you very much for the references.

 

[Chalnoth]

View attachment 83090

View attachment 83097​

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