Measuring Cold Dark Matter Power Spectrum from Variations of Hubble Flows

When Cold Dark Matter (CDM) power spectrum normalized by COBE results, its amplitude at smaller scales can be parametrized by $\Gamma\sim\Omega_0h$. The expected variations of Hubble flows in two samples, the sample of 36 clusters in the Mark III catalogue, and the sample of 20 Type Ia supernovae (SNe), are calculated for the power spectrum in critical-density CDM models (including tilted CDM models and vacuum-dominated CDM models). The comparison between the expectations and the real variations in the data offers a bias-free way to constrain the power spectrum. The cluster sample yields $\Gamma\le 0.30-0.88(n_{ps}-1)+1.9 (n_{ps}-1)^2$ at 95% C.L., with best fits $\Gamma = 0.15-0.39(n_{ps}-1)+0.37(n_{ps}-1)^2$, where $n_{ps}$ is the spectral index of the power spectrum. The Type Ia SN sample yields $\Gamma\le 0.25-0.80(n_{ps}-1)+1.6 (n_{ps}-1)^2$ at 95% C.L., strongly favoring lower $\Gamma$'s. The results are inconsistent with a critical-density matter-dominated universe with $n_{ps}\ga 0.8$ and $H_0\ga 50$ km/sec/Mpc.