We explore the cosmological signals of theories in which the neutrinos decay into invisible dark radiation after becoming non-relativistic. We show that in this scenario, near-future large scale structure measurements from the Euclid satellite, when combined with cosmic microwave background data from Planck, may allow an independent determination of both the lifetime of the neutrinos and the sum of their masses. These parameters can be independently determined because the Euclid data will cover a range of redshifts, allowing the growth of structure over time to be tracked. If neutrinos are stable on cosmological timescales, these observations can improve the lower limit on the neutrino lifetime by seven orders of magnitude, from $mathcal{O}(10)$ years to $2times 10^8$ years ($95%$ C.L.), without significantly affecting the measurement of neutrino mass. On the other hand, if neutrinos decay after becoming non-relativistic but on timescales less than $mathcal{O}(100)$ million years, these observations may allow, not just the first measurement of the sum of neutrino masses, but also the determination of the neutrino lifetime from cosmology.
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