We show that the number and redshifts distribution of galaxy clusters in future deep cluster surveys can place strong constraints on the matter density , \Omega _ { m } , the vacuum energy density , \Omega _ { \Lambda } , and the normalization of the matter power spectrum , \sigma _ { 8 } .
Degeneracies between these parameters are different from those in studies of either high–redshift type Ia Supernovae ( SNe ) , or cosmic microwave background ( CMB ) anisotropies .
Using a mass threshold for cluster detection expected to be typical for upcoming Sunyaev-Zel ’ dovich effect ( SZE ) surveys , we find that constraints on \Omega _ { m } and \sigma _ { 8 } at the level of roughly 5 % or better can be expected , assuming redshift information is known at least to z \sim 0.5 and in the absence of significant systematic errors .
Without information past this redshift , \Omega _ { \Lambda } is constrained to 25 \% .
With complete redshift information , deep ( M _ { lim } \sim 10 ^ { 14 } h ^ { -1 } { M _ { \odot } } ) , relatively small solid angle ( \sim 12 { deg } ^ { 2 } ) surveys , can further constrain \Omega _ { \Lambda } to an accuracy of \sim 15 \% , while large solid angle surveys with ground-based , large-format bolometer arrays could measure \Omega _ { \Lambda } to a precision of \sim 4 \% or better .