Cluster abundance constraints for cosmological models with a time-varying, spatially inhomogeneous energy component with negative pressure

The abundance of rich clusters is a strong constraint on the mass power spectrum. The current constraint can be expressed in the form σ₈Ω^γ_m = 0.5 ± 0.1, where σ₈ is the rms mass fluctuation on 8 h^-1 Mpc scales, Ω_m is the ratio of matter density to the critical density, and γ is model dependent. In this paper, we determine a general expression for γ that applies to any models with a mixture of cold dark matter plus cosmological constant or quintessence (a time-evolving, spatially inhomogeneous component with negative pressure) including dependence on the spectral index n, the Hubble constant h, and the equation of state of the quintessence component w. The cluster constraint is combined with COBE measurements to identify a range of best-fitting models. The constraint from the evolution of rich clusters is also discussed.