# Non relativistic Neutrinos

1. ### ChrisVer

How can someone think of the neutrinos as non relativistic?
OK I understand for example that the neutrino temperature is very small even compared to their masses... but at the same time I find it non trivial to think of very light particles with energies:
$E≥1eV$
non-relativistic.... How can the two pictures be compatible?

6,395
3. ### Trifis

148
I think the normal active neutrino is always ultra-relativistic. It is some hypothetical heavier neutrino (e.g. sterile, Majorana neutrinos) that would have to be treated differently, I think.

### Staff: Mentor

Primordial neutrinos (T=2K) could be non-relativistic, if their mass is close to the current exclusion limits.

5. ### phinds

8,122
Uh ... what is the "temperature" of a single neutrino?

6. ### TumblingDice

463
Bee Hossenfelder did a bit on her blog about the temperature of the CvB, expansion, and the velocity of these neutrinos here:
The Cosmic Neutrino Background

### Staff: Mentor

Just to add a bit to what has been said: The lower bound on the heaviest neutrino mass from oscillation experiments is around 0.05 eV. Since the C##\nu##B has a temperature significantly lower than this, it means that C##\nu##B neutrinos will typically have a kinetic energy significantly lower than their mass (or at least a third of it will). While this background is predicted, the low energy of the C##\nu##B makes it essentially impossible to detect and current experiments are far from sensitive enough to do so. Its existence thus remains inferred by our current knowledge of particle physics and cosmology. However, there are a few ideas floating around on how such a measurement could (theoretically) be performed.

8. ### nikkkom

857
As long as its mass is nonzero, its velocity is less than c and there is a reference frame where it is not only not relativistic, but where it is stationary.

### Staff: Mentor

PTOLEMY is an actual project.

### Staff: Mentor

Their prototype does not have sufficient energy resolution to discover the CNB. I would consider this to be an idea that is floating around and the prospects of this type of experiment depend on several factors, such as gravitational clustering. I also do not think their keV sterile neutrino search will be successful. Such a neutrino would necessarily have a very small mixing, resulting in a highly suppressed event rate.

### Staff: Mentor

I didn't see results from the prototype.
But on the other hand: it is a prototype. A few years of development can have an amazing impact. I see this at the LHC. You start with some design goals, develop and build the machine/detector and while you exceed the design values (at least many of them) you plan the next upgrade that can do an order of magnitude better.

### Staff: Mentor

I dont think the prototype has results, it is just what they claim it will be able to do.

We are talking about measuring the kinetic energy of an electron at a sub-eV level. There are physical limitations at play here. By design the experiment will also measure the background of normal tritium decay and resolution is essential, in particular the high end tail of the resolution function - if you do not control it extremely well you get swamped. I don't remember PTOLEMY's strategy at the moment, but this typically involves large spectrometers. KATRIN has a pretty large spectrometer and its resolution is still in the O(eV) range. To guarantee a good signal/background ratio if the neutrino mass ordering is inverted you would need something like 0.01 eV resolution. In normal ordering it is typically worse. Things lighten up a bit in the degenerate regime, both because you get more clustering and less need for energy resolution, but cosmology seems about to challenge this possibility.

That said, nobody would be happier than me if they actually could perform this measurement.

### Staff: Mentor

Sure.
One of the main limitations of KATRIN is the source - molecular tritium where the remaining tritium atom can get some (variable) part of the decay energy. PTOLEMY avoids this problem with a (demonstrated) lower binding energy.
The heaviest mass eigenstate has to be above that value, I don't know how many standard deviations separation you need there. Regular beta decay electrons so close to the threshold are rare, too - even if you do not get a clear gap between primordial neutrinos and the endpoint of the beta decay spectrum, data could be sufficient to find a signal. If not, it would at least promise to give a very good upper limit on neutrino masses.

### Staff: Mentor

I remember checking this at some point. For the lowest possible neutrino masses there should be essentially no gravitational clustering. Even if the end-point decays are rare, convoluting the normal beta decay distribution with a Gaussian resolution function results in a very steep growth of the background.

In the case of normal mass ordering the ##\nu_e## would contain mostly a nearly massless state which would be overwhelmed by the background. The already low event rate for the ##\nu_3## part would be further suppressed by ##\sin^2\theta_{13}##.

In the inverted ordering things lighten up a bit as most of the ##\nu_e## would be contained in states with a mass of around 0.05 eV. However, here ##\theta_{13} \neq 0## plays a negative role as there is a part of the background spectrum which extends to the ##Q## value. For ##\theta_{13} = 10^\circ##, which is relatively close to the actual value, the background spectrum convoluted with a Gaussian energy resolution of 0.03 eV (FWHM) grows steeply (in comparison to the signal) around 0.04 eV.

Of course, things would look better if neutrino masses are in the degenerate regime as this would add more separation between the signal and background.

No arguments here.