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This statement is repeated so often that it has become somewhat of a cliche: 'billions of neutrinos pass through your body every second'. For example see 1, 2, 3, 4, 5, 6.

What is the evidence for it, especially considering that we have never detected even a hundred neutrinos in a second through one detector?

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As others have noted, the neutrinos come from the sun. Given that, there are two broad ways of estimating the flux of neutrinos: one is theoretical, and the other is experimental.

The theoretical way is based on the Standard Solar Model. This is a well understood model with solid experimental validation, and astronomers and astrophysicists therefore have great faith in it. According to this model, the solar neutrino flux is dominated by proton-proton fusion reactions, which generate an electron neutrino flux of approximately $6 \times 10^{10}\ \mathrm{cm^{-2}\ s^{-1}}$ at $1\ \mathrm{AU}$.

The experimental way is to build neutrino detectors and measure the flux. Because of the extremely small cross section, it is difficult to build a detector that can collect enough data to reduce the statistical and systematic uncertainty to qualify for a precision measurement. Nevertheless, a lot of work has been performed in this area and results have been obtained with uncertainties to within a percent or so.

The experimental results and theoretical predictions did not agree with each other; they were off by a factor of three, which was the so-called Solar Neutrino Problem. This was resolved by hypothesizing, and then experimentally verifying, that the electron neutrinos produced in the sun "oscillated" into other flavors of neutrinos (muon, tau) by the time they were detected on earth, so now the experimental neutrino flux measurements agree with the theoretical predictions.

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    $\begingroup$ To play devil's advocate for a moment: how can you experimentally tell the difference between, say, 1 billion neutrinos / second of which 10/million are detected, and 10 billion neutrinos / second of which 1/million are detected? $\endgroup$ – TLW Feb 26 at 3:04
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    $\begingroup$ @TLW: Just from the detected neutrino flux alone, you can’t tell. But if theoretical calculation A, about the sun’s output, says we should expect 1 billion neutrinos/sec, theoretical calculation B, about the detector, says we should expect to detect 10 per million of these, and we do indeed detect 10,000/s, then it’s less likely that both the theoretical calculations are wrong in ways that cancel out perfectly — especially when there are also other theoretical calculations C, D, and E that also agree with this, and experiments X, Y, Z that test other predictions of the theory. $\endgroup$ – PLL Feb 26 at 8:51
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    $\begingroup$ This is a great answer that abides by scientific principles rather than being sprinkled through with phrases like "we know that...". $\endgroup$ – Asteroids With Wings Feb 27 at 1:07
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    $\begingroup$ Thank you @Asteroids! @TLW, to follow up on what PLL said (which was correct), there are various ways of experimentally measuring the fraction of neutrinos that get detected when they pass through a detector. The modern and most direct way is to create a neutrino beam of known intensity using a particle accelerator, and measuring the fraction of the beam that gets detected. A summary of these measurements can be found in the "bible of particle physics", the PDG: pdg.lbl.gov/2019/reviews/rpp2018-rev-nu-cross-sections.pdf $\endgroup$ – Richter65 Feb 27 at 14:40
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Those neutrinos come from the Sun. Fusion converts protons to neutrons, so that must produce neutrinos. One can calculate the number of nuclear reactions necessary for the power output, and get a number for the neutrino flux.

One can also estimated the flux from the cross section of the detector.

The two rates differ by a factor of about three. That was resolved by the neutrino oscillations between the three flavors (electron, muon and tauon neutrinos).

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    $\begingroup$ To expand a bit on @Pieter : If we know, for example, that 99.9999999% of any neutrinos that pass through a detector won't be detected, and our detector actually detects one per day, then we know that approximately $1/(1-0.999999999)$ neutrinos per day have passed through the detector. "Cross section" relates to the fraction of neutrinos passing throuth the detector will be detected. $\endgroup$ – S. McGrew Feb 23 at 17:51
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    $\begingroup$ @S.McGrew Indeed. And different detectors (with different nuclei and different detection probabilities) measured the same flux. Which was about a factor three smaller than expected. So the experimental uncertainties are not large at all. $\endgroup$ – Pieter Feb 23 at 18:00
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    $\begingroup$ Since they differ from each other, "The two rates differ by a factor of about three" is more clear. "Both differ" makes it sound like there is some third value that they both differ from. $\endgroup$ – Acccumulation Feb 24 at 5:54
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    $\begingroup$ @S.McGrew - being the devil's advocate for a moment: how can we know that "99.9999999% of any neutrinos that pass through a detector won't be detected"? $\endgroup$ – TLW Feb 26 at 3:01
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    $\begingroup$ @TLW I don't actually know, but my best guess would be that cross sections were experimentally determined using particle accelerator experiments. You can generate neutrinos in well controlled conditions so you know exactly how many to expect, and then see how many pass through your detector undetected. Edit: see Anna's answer for more about particle accelerators $\endgroup$ – craq Feb 26 at 9:21
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The existence of the neutrinos was established using energy and momentum conservation in neutron decays. There have been experiments with neutrino and antineutrino beams both at Cern and Brookhaven and have established their interaction crossection with matter. To get one neutrino interacting in the detector it means that thousands have passed without interacting, according to the theoretical calculations. Your "that we have never detected even a hundred neutrinos in a second through one detector? " is misleading, because the one we do detect, mathematically means that the calculated beam flux is correct according to the theory

There exists a solid theory that can estimate the number of neutrinos given certain assumptions of what the cosmic charged particle background is.

For example, we measure the muon flux at sea level, and muons decay into electrons and a muon neutrino and an electron antineutrino, so we know from the kinematics what the muon induced flux of neutrinos is at sea level. ( an average flux of about 1 muon per square centimeter per minute. far from billions)

There are detectors detecting solar neutrinos and those also agree with the mainstream theory of weak interactions. Those fulfill the billions recipe,

The flux of solar neutrinos at the earth's surface is on the order of $10^{11}$ per square centimeter per second.

Theory says that there should be cosmic relic neutrinos, coming from their decoupling in the Big Bang model, similar to the Cosmic Microwave Background, this would add orders of magnitude to very low energy background neutrinos, but this is still now a theoretical prediction.

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    $\begingroup$ Technically .. our sun alone produces enough neutrinos that PER second per square CENTIMETER about 60 billion neutrinos reach earth (roughly the area covered by your thumbnail) .. [american billion] $\endgroup$ – eagle275 Feb 24 at 8:31
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    $\begingroup$ and they keep going at night, passing through the planet with practically no losses. $\endgroup$ – dlatikay Feb 24 at 19:01
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    $\begingroup$ This is a helpful explanation; +1. 'Your "that we have never detected even a hundred neutrinos in a second through one detector?" is misleading, because the one we do detect, mathematically means that the calculated beam flux is correct according to the theory.' I would disagree that the statement is misleading. It's a legitimate question, with a good answer. $\endgroup$ – LarsH Feb 25 at 14:32

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