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We aim to constrain mass-varying neutrino models using large scale structureobservations and produce forecast for the Euclid survey. We investigate twomodels with different scalar field potential and both positive and negativecoupling parameters \beta. These parameters correspond to growing or decreasingneutrino mass, respectively. We explore couplings up to |\beta|<5. In the caseof the exponential potential, we find an upper limit on $\Omega_\nu h^2$<0.004at 2-$\sigma$ level. In the case of the inverse power law potential the nullcoupling can be excluded with more than 2-\sigma significance; the limits onthe coupling are \beta>3 for the growing neutrino mass and \beta<-1.5 for thedecreasing mass case. This is a clear sign for a preference of highercouplings. When including a prior on the present neutrino mass the upper limiton the coupling becomes |\beta|<3 at 2-$\sigma$ level for the exponentialpotential. Finally, we present a Fisher forecast using the tomographic weaklensing from an Euclid-like experiment and we also consider the combinationwith the cosmic microwave background (CMB) temperature and polarisation spectrafrom a Planck-like mission. If considered alone, lensing data is more efficientin constraining $\Omega_\nu$ with respect to CMB data alone. There is, however,a strong degeneracy in the \beta-$\Omega_\nu h^2$ plane. When the two data setsare combined, the latter degeneracy remains, but the errors are reduced by afactor ~2 for both parameters.

Mass-varying neutrino in light of cosmic microwave background and weak lensing