If I've understood it correctly, the energy of an object depends on the reference frame, so when you accelerate, the CMB and CNB appear to gain energy (relative to your rest frame).
So if you are inside a lab that moves at 99.999% of the speed of light, will the CNB (Cosmic Neutrino Background) gain enough energy to be detectable?
Answer
I think the answer has to be yes, it would be much easier to detect the cosmic neutrino background if you could arrange for your laboratory to be travelling at close to the speed of light in the local, co-moving reference frame defined by the cosmic microwave background.
The current energies of neutrinos in the cooled cosmic neutrino background are expected to be of order $10^{-4}$ to $10^{-3}$ eV (depending a little bit on what the neutrino masses are) and are almost certainly moving at a small fraction of the speed of light.
Whilst there are experiments under way to try detecting these very low energy neutrinos (for example the Ptolemy experiment), the technology for detecting higher energy neutrinos is much better established. For example the Super Kamiokande detector is sensitive to neutrinos from the Sun with energies around 5-20 MeV. To boost the cosmic background neutrinos up to these kinds of energies would require Lorentz factors of $\gamma = (1 - v^2/c^2)^{-1/2} \simeq 10^{10}$ and thus velocities of $0.999999999999999999995c$. So that's a bit faster than the figure hypothesised in your question.
The other advantage to doing this is of course that the flux of neutrinos would be greatly increased coming from the direction of motion and I guess that knowing the direction that the neutrinos would be coming from would aid the detector and experiment design considerably.
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