special relativity - Is it possible that all "spontaneous nuclear decay" is actually "slow neutrino" induced?

This thought was inspired by a comment from the current leading answer, by @Sentry, to the question Where are all the slow neutrinos?

This [slow-neutrino induced nuclear decay] will still be an extremely rare process and the big problem is to distinguish it from normal spontaneous nuclear decay.

Questions which would need to be addressed as corollaries to the main question, I believe, include:

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  Is such a possibility self-consistent as a theory?

  
  


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  How does the required energy density of slow neutrinos compare to the postulated energy density of dark matter?

  
  

Answer

Consider the average beta decay, which at nucleon level looks like $$ n \longrightarrow p + e^- + \bar{\nu} \,. \tag{1}$$ The distribution of electron energies (as measured in neutron's frame) is controlled by the phase space of the products. We observe an electron energy spectrum consistent with these physics.

What you propose is essentially that this reaction is properly described by $$ n + \nu \longrightarrow p + e^- \,. \tag{2}$$ with a very low energy neutrino. (As an aside, that reaction with high energy neutrinos is seen in accelerator and atmospheric neutrino experiments.)

However, the energy distribution of the electron in the final state of EQN (2) (again, measured in the neuton's rest frame) would be controlled by the incident neutrino's momentum. It could only look like the observed spectrum if the energy-spectrum of the incident neutrinos were like those predicted for the outgoing neutrino in EQN (1). But as those neutrinos have energies of many MeV (depending on the particular decay) they are in no way slow.

Worse, weak universality works using the same effective coupling constant (the Fermi constant) for reactions involving a neutrino in the initial state as for those involving an anti-neutrino in the initial state. (And likewise for (anti-)neutrinos in the final state). So now you need not only a conspiracy to get the right energy spectrum for the neutrinos, but the conspiracy must insure the same number and spectrum for anti-neutrinos as well, despite the nearly factor of two difference in the abundance of quarks for these two kinds to interact with at low energy.

Short answer: No, it's not possible. Not even for weak decays.