🚀 go-pugleaf

Matt Carey <matt.carey@ieee.org> writes:

> "Gordon D. Pusch" wrote:
> 
>> This is known as a ``hypothetical'' question, i.e., a ``what if?''
>> *IF* it where possible for neutrinos to be at equilibrium at what
>> humans consider ``room temperature,'' what would their mean speed be?
>> (Note that the question is not entirely silly --- somewhere between
>> the neutrino decoupling time (T_\nu ~= 1e10 Kelvin) and the current
>> epoch (T_\nu ~= 2 Kelvin), they must have passed through T_\nu ~= 300 K...
> 
> It wasn't orginally posed as a hypothetical question, but that isn't 
> really important.
> 
> Why would the original neutrinos have cooled at all?  Another
> odd statement by the other poster mentions "thermalized" neutrinos.

Two ways: In the early Universe, the density of matter was high enough that
the mean free path of a neutrino between scatterings via the weak interaction
was small compared to the Hubble radius; during that period, neutrinos were
essentially in thermal equilibrium with the things they scattered off of. 
After the neutrino ``decoupling time,'' when their mean free path became
long compared to the Hubble radius, they ``cooled'' for the same reasons
the blackbody background-radiation ``cooled'' --- the expansion of the 
Universe caused the neutrino-equivalent of a ``red-shift'' effect.

 
> Once in the universe as we know it now, how do you thermalize a
> neutrino?

Happens all the time in supernovae --- neutronium is nearly opaque to
neutrinos. During a type-II ``core collapse'' supernova, the core matter 
is dense enough that a ``neutrinosphere'' forms, in analogy to the
``photosphere'' of stars. Within the ``neutrinosphere,'' a neutrino's
mean free path between weak interactions is much smaller than the core's
radius, so the neutrinos formed by the inverse beta-decay reaction
p + e  -->  n + \nu  that converts the core to neutronium escape 
by diffusion, on a mean timescale on the order of ~10 sec.


> Can it scatter elastically off some other particle?  

Yes --- every time it undergoes a weak interaction. 


> Even if it could the huge mass difference would require a large number 
> of such collisions.

No, AFAIK, the number of collisions require to effectively thermalize 
a particle is independent of its mass; IIRC, five to ten collisions
is good enough FAPP, as I believe someone posted earlier in this thread. 
(Do you know of a contrary reference where it _does_ depend on the relative
masses?)


> For a particle with nearly zero cross section for any interaction,
> it wouldn't seem likely that a neutrino would thermalize.

Note that the weak interaction cross-section scales as the CM energy
squared, so high-energy neutrinos scatter much more strongly than
low-energy neutrinos.


-- Gordon D. Pusch   

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