We perform a forecast analysis on how well a Euclid-like photometric galaxy cluster survey will constrain the total neutrino mass and effective number of neutrino species .
We base our analysis on the Monte Carlo Markov Chains technique by combining information from cluster number counts and cluster power spectrum .
We find that combining cluster data with Cosmic Microwave Background ( CMB ) measurements from Planck improves by more than an order of magnitude the constraint on neutrino masses compared to each probe used independently .
For the \Lambda CDM+ m _ { \nu } model the 2 \sigma upper limit on total neutrino mass shifts from \sum m _ { \nu } < 0.35 \text { eV } using cluster data alone to \sum m _ { \nu } < 0.031 \text { eV } when combined with Planck data .
When a non-standard scenario with N _ { \text { eff } } \neq 3.046 number of neutrino species is considered , we estimate an upper limit of N _ { \text { eff } } < 3.14 ( 95 \% CL ) , while the bounds on neutrino mass are relaxed to \sum m _ { \nu } < 0.040 \text { eV } .
This accuracy would be sufficient for a 2 \sigma detection of neutrino mass even in the minimal normal hierarchy scenario ( \sum m _ { \nu } \simeq 0.05 \text { eV } ) .
In addition to the extended \Lambda CDM+ m _ { \nu } + N _ { \text { eff } } model we also consider scenarios with a constant dark energy equation of state and a non-vanishing curvature .
When these models are considered the error on \sum m _ { \nu } is only slightly affected , while there is a larger impact of the order of \sim 15 \% and \sim 20 \% respectively on the 2 \sigma error bar of N _ { \text { eff } } with respect to the standard case .
To assess the effect of an uncertain knowledge of the relation between cluster mass and optical richness , we also treat the \Lambda CDM+ m _ { \nu } + N _ { \text { eff } } case with free nuisance parameters , which parameterize the uncertainties on the cluster mass determination .
Adopting the over-conservative assumption of no prior knowledge on the nuisance parameter the loss of information from cluster number counts leads to a large degradation of neutrino constraints .
In particular , the upper bounds for \sum m _ { \nu } are relaxed by a factor larger than two , \sum m _ { \nu } < 0.083 \text { eV } ( 95 \% CL ) , hence compromising the possibility of detecting the total neutrino mass with good significance .
We thus confirm the potential that a large optical/near-IR cluster survey , like that to be carried out by Euclid , could have in constraining neutrino properties , and we stress the importance of a robust measurement of masses , e.g .
from weak lensing within the Euclid survey , in order to full exploit the cosmological information carried by such survey .