Euclid Preparation. Sensitivity to Neutrino Parameters

The Euclid mission of the European Space Agency will deliver weakgravitational lensing and galaxy clustering surveys that can be used toconstrain the standard cosmological model and extensions thereof. We presentforecasts from the combination of these surveys on the sensitivity tocosmological parameters including the summed neutrino mass $M_\nu$ and theeffective number of relativistic species $N_{\rm eff}$ in the standard$\Lambda$CDM scenario and in a scenario with dynamical dark energy ($w_0w_a$CDM). We compare the accuracy of different algorithms predicting thenonlinear matter power spectrum for such models. We then validate severalpipelines for Fisher matrix and MCMC forecasts, using different theory codes,algorithms for numerical derivatives, and assumptions concerning the non-linearcut-off scale. The Euclid primary probes alone will reach a sensitivity of$\sigma(M_\nu)=$56meV in the $\Lambda$CDM+$M_\nu$ model, whereas thecombination with CMB data from Planck is expected to achieve$\sigma(M_\nu)=$23meV and raise the evidence for a non-zero neutrino mass to atleast the $2.6\sigma$ level. This can be pushed to a $4\sigma$ detection iffuture CMB data from LiteBIRD and CMB Stage-IV are included. In combinationwith Planck, Euclid will also deliver tight constraints on $\Delta N_{\rm eff}<0.144$ (95%CL) in the $\Lambda$CDM+$M_\nu$+$N_{\rm eff}$ model, or $\DeltaN_{\rm eff}< 0.063$ when future CMB data are included. When floating $(w_0,w_a)$, we find that the sensitivity to $N_{\rm eff}$ remains stable, while thatto $M_\nu$ degrades at most by a factor 2. This work illustrates thecomplementarity between the Euclid spectroscopic and imaging/photometricsurveys and between Euclid and CMB constraints. Euclid will have a greatpotential for measuring the neutrino mass and excluding well-motivatedscenarios with additional relativistic particles.