Cluster mass functions in the quintessential Universe

We use $N$-body simulations to measure mass functions in flat cosmological models with quintessence characterized by constant $w$ with $w=-1$, -2/3 and -1/2. The results are compared to the predictions of the formula proposed by Jenkins et al. at different redshifts, in terms of FOF masses as well as Abell masses appropriate for direct comparison to observations. The formula reproduces quite well the mass functions of simulated haloes in models with quintessence. We use the cluster mass function data at a number of redshifts from Carlberg et al. to constrain $\Omega_0$, $\sigma_8$ and $w$. The best fit is obtained in the limit $w \to 0$, but none of the values of $w$ in the considered range $-1 \le w < 0$ can actually be excluded. However, the adopted value of $w$ affects significantly the constraints in the $\Omega_0-\sigma_8$ plane. Taking into account the dependence on $w$ we find $\Omega_0=0.32 \pm 0.15$ and $\sigma_8=0.85_{-0.12}^{+0.38}$ (68% c.l.). Since less negative $w$ push the confidence regions toward higher $\Omega_0$ and lower $\sigma_8$ we conclude that relaxing the assumption of $w=-1$ typically made in such comparisons may resolve the discrepancy between recent cluster mass function results (yielding rather low $\Omega_0$ and high $\sigma_8$) and most other estimates. The fact that high $w$ values are preferred may however also point towards some unknown systematics in the data or the model with constant $w$ being inadequate.