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If detecting neutrinos is so difficult, how do we have an estimate of how many of them pass through a cubic centimeter per second?

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The number density of neutrinos in the universe is totally dominated by primordial neutrinos created in the big bang when leptons and photons were in thermal equilibrium. The density of these neutrinos now can be estimated in the context of the big bang model by assuming they cooled in the same way as photons (because they would have been ultra-relativistic and effectively massless particles in the early universe). There is thus a fixed relationship between the temperature and number density of the neutrinos and the photons that are present in the cosmic microwave background (CMB). This ratio depends a little bit on some details of the number of neutrino species and the precise value of the non-zero neutrino mass.

The number density and temperature of photons in the CMB is well known and measured by various experiments and indirectly, this yields the predicted neutrino number density and temperature (actually a temperature may be meaningless if the neutrino mass is sufficiently high that they are currently not ultrarelativistic).

Thus it's a theoretical prediction. The primordial neutrinos have such low energies that they are almost impossible to detect. There is some indirect evidence that the prediction is in the right ballpark based on detailed analysis of the small anisotropies in the CMB.

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Here is a list of a large number of neutrino experiments and on how they detect the neutrino. In mainstream physics there exists the standard model of particle physics (SM) that in general is very good in calculating the interactions expected in the accelerator experiments.

The cosmic experiments can measure neutrinos coming from the cosmos and can fit for the energy spectrum passing through the earth and the number of neutrinos per meter-squared per second .

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