Detect the cosmic neutrino background
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It's definitely something we're interested in, but there are some significant difficulties:
1. The energies of these neutrinos are very, very low, meaning many of the ways that current neutrino detectors work just can't detect them. For example, many current detectors operate by waiting for the incoming neutrino to smack something and create a fast-moving electron that emits Cherenkov radiation. That detection method isn't going to work with neutrinos that have energies far below the mass of the electron.
2. Many of the neutrinos will, by now, have slowed to the point that they will have been captured by massive objects like galaxies and galaxy clusters. This will muck up their distribution, making it more difficult to detect their origin, even if we can detect them.
Personally, I would tend to suspect that we'll be capable of detecting the cosmic gravitational wave background before we detect the cosmic neutrino background. But I'll admit I'm not close enough to those fields to know for sure.
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