# How to calculate the density of relic neutrinos?

May be not neutrinos, but antineutrinos? Or both types? In the last case, why they didn't annihilate and what is the ratio of relic neutrinos to relic antineutrinos? Is that ratio somehow related to the barion asymmetry?

For reference: Relic neutrinos or cosmic neutrino background

Like the cosmic microwave background radiation (CMB), the cosmic neutrino background is a relic of the big bang

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What are "relic neutrinos"? I'm not familiar with the term, so if you could add a link to some more information, that would be quite helpful. – David Zaslavsky♦ Jan 16 '11 at 9:22
@David, I have added the link – voix Jan 16 '11 at 9:57
Do anti-neutrinos exist? I thought the general belief was that neutrinos were Majorana particles, and so are their own anti-particle. – Peter Shor Jan 16 '11 at 15:56
"Relic neutrinos" are those that were coupled to the hot early universe (before temperatures dropped below the Z mass) and have since cooled to ridiculously low energies. They have very low cross-sections for interaction with ordinary matter by neutrino physics standards, and such interactions would be lost in the thermal noise anyway. – dmckee♦ Jan 16 '11 at 18:12
– voix Jan 16 '11 at 20:12
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## 1 Answer

Back when I was in graduate school in the 1990s, the standard reference for this sort of thing was Kolb and Turner's book The Early Universe. Even after all these years, that book's treatment of this subject is probably still a good place to look.

Even if there's no asymmetry-producing process for neutrinos (like baryogenesis), you still expect a relic neutrino background that's a thermal (Fermi-Dirac) distribution of both neutrinos and antineutrinos, with a temperature of about 2 K. The reason is that, at a certain time in the evolution of the Universe, the density dropped low enough that the neutrino number "froze out": interactions that could change the number of neutrinos (such as primarily $e^- e^+ \leftrightarrow \nu_e\ \bar\nu_e$) became so rare that the time for any given particle to undergo such a reaction grew much longer than a Hubble time.

It's been a long time since I looked at baryogenesis models with any care, but as I recall some models would be expected to produce an asymmetry in the neutrino sector as well. But in practice I don't think that would change the prediction much. The reason is that baryogenesis only has to produce a one part in $10^9$ asymmetry (a billion and one protons for every billion antiprotons). That produces very noticeable effects today, because there was essentially complete annihilation of the antiprotons. But neutrino freeze-out occurs much earlier, while neutrinos are still relativistic, so we don't think that that massive annihilation happened for neutrinos. So even if there is a neutrino-antineutrino asymmetry comparable to the asymmetry produced by baryogenesis, it should only result in a tiny difference in the number of neutrinos over antineutrinos.

Let me put that another way. At early times, (temperature much greater than the proton mass), there were comparable numbers of photons, neutrinos, and protons. Baryogenesis resulted in an asymmetry of protons over antiprotons at that time. After that, nearly all of the protons and antiprotons annihilated, leaving the observed result that today there are a billion photons for every proton. But we expect the number of relic neutrinos to be of the same order as the number of photons, not protons, so a baryogenesis-level neutrino asymmetry won't be noticeable.

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It's interesting to know what is the experimental cross-section of the collider reaction "electron-positron --> neutrino-antineutrino" when electron and positron just "disappear". – voix Jan 16 '11 at 20:33

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@voix: How would you distinguish that case from non-interaction? You would have to be able to measure the individual particles removed from the beams at a precision sufficient to detect weak interactions. That's not in the cards at this time. – dmckee♦ Jan 16 '11 at 21:02
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