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While COBE's detection of large-angle microwave background anisotropy fixes the amplitude of density fluctuations on length scales $k^{-1} \sim (300-6000) {\rm\, h^{-1}Mpc}$, the quantity that detemines the level of large-scale clustering is the amplitude of fluctuations on scales $(5-50) {\rm\, h^{-1}Mpc}$. The amount of dynamical clustering is parameterized by $\sigma_8$, the rms amplitude of the linear mass fluctuations in $8 {\rm\, h^{-1}Mpc}$ spheres. For the standard Cold Dark Matter model, the COBE result indicates $\sigma_8 \sim 1$, while models with extra large scale power require $\sigma_8 \sim 1/2$. The most massive clusters of galaxies (M $\gta 8 \times 10^{14} {\rm\,M_{\sun}}$, $\sigma_v \gta 700 {\rm\, km\ s^{-1}}$) form in rare `peak patches' found in the initial mass density distribution. The massive cluster abundance as a function of redshift is a sensitive probe of the wavenumber band $k^{-1} \sim (3-8) {\rm\, h^{-1}Mpc}$, hence of $\sigma_8$, and so cluster evolution can discriminate among models allowed by the COBE results. We use our Hierarchical Peaks Method, which accurately reproduces the results of $P^3M$ $N$-body simulations, to calculate the evolution of cluster X-ray flux counts, luminosity and temperature functions as a function of $\sigma_8$ for CDM models and those with more large scale power. We find that the EMSS and Edge et al. cluster samples support $\sigma_8$ in the range from $\sim 0.6-0.9$, and that models with more large scale power (and hence flatter fluctuation spectra in the cluster regime) fit the X-ray bright end better. Uncertainties, pitfalls, and limitations of the method are discussed. Implications for X-Ray cluster samples and predictions for observations of the Sunyaev-Zeldovich effect are given.
