At present , the strongest upper limit on \sum m _ { \nu } , the sum of neutrino masses , is from cosmological measurements .
However , this bound assumes that the neutrinos are stable on cosmological timescales , and is not valid if the neutrino lifetime is less than the age of the universe .
In this paper , we explore the cosmological signals of theories in which the neutrinos decay into invisible dark radiation on timescales of order the age of the universe , and determine the bound on the sum of neutrino masses in this scenario .
We focus on the case in which the neutrinos decay after becoming non-relativistic .
We derive the Boltzmann equations that govern the cosmological evolution of density perturbations in the case of unstable neutrinos , and solve them numerically to determine the effects on the matter power spectrum and lensing of the cosmic microwave background .
We find that the results admit a simple analytic understanding .
We then use these results to perform a Monte Carlo analysis based on the current data to determine the limit on the sum of neutrino masses as a function of the neutrino lifetime .
We show that in the case of decaying neutrinos , values of \sum m _ { \nu } as large as 0.9 eV are still allowed by the data .
Our results have important implications for laboratory experiments that have been designed to detect neutrino masses , such as KATRIN and KamLAND-ZEN .