We study the cluster mass function in mixed dark matter ( MDM ) models , using two COBE normalized simulations with \Omega _ { h } = 0.26 and n = 1.2 , and \Omega _ { h } = 0.14 and n = 1.05 , both with 2 massive \nu ’ s ( MDM1 and MDM2 , respectively ) .
For the sake of comparison , we also simulate a CDM model with spectral index n = 0.8 ( TCDM ) , also COBE normalized .
We argue that , in our non–hydro simulations , where CDM particles describe both actual CDM and baryons , the galaxy distribution traces CDM particles .
Therefore , we use them to define clusters and their velocities to work out cluster masses .
As CDM particles are more clustered than HDM and therefore have , in average , greater velocities , this leads to significant differences from PS predictions .

Such predictions agree with simulations if both HDM and CDM are used to define clusters .
If this criterion is adopted , however , we see that : ( i ) MDM corresponds \delta _ { c } values slightly but systematically greater than CDM ; ( ii ) such \delta _ { c } exhibit a scale dependence : on scales \sim 10 ^ { 14 } M _ { \odot } , we find \delta _ { c } \sim 1.7 or 1.8 for CDM or MDM , respectively ; at greater scales the required \delta _ { c } decreases and a substantial cluster excess is found , at the large mass end ( M > 10 ^ { 15 } M _ { \odot } ) .

Clusters defined through CDM in MDM models , instead , are less numerous than PS estimates , by a factor \sim 0.3 , at the low mass end ; the factor becomes \sim 0.6 –0.8 , depending on the mix , on intermediate mass scales ( \sim 4 – 5 h ^ { -1 } 10 ^ { 14 } M _ { \odot } ) and almost vanishes on the high mass end .
Therefore : ( i ) MDM models expected to overproduce clusters over intermediate scales are viable ; ( ii ) the greater reduction factor at small scales agrees with the observational data dependence on the cluster mass M ( which , however , may be partially due to sample incompleteness ) ; ( iii ) the higher spectral normalization is felt at large scales , where MDM models produce more objects ( hence , large clusters ) than CDM .
MDM1 even exceeds Donahue et al .
( 1998 ) findings , while MDM2 is consistent with them .

Simulations are performed using a parallel algorithm worked out from Couchman AP3M serial code , but allowing for different particle masses and used with variable time steps .
This allowed to simulate a cubic box with side of 360 h ^ { -1 } Mpc , reaching a Plummer resolution of 40.6 h ^ { -1 } kpc , and using ( 3 \times ) 180 ^ { 3 } particles .

PACS : 95.35 ; 98.80 ; 98.65.Cw