Implications of Cosmic Microwave Background Anisotropies for Large Scale Variations in Hubble's Constant

Low amplitude (linear regime) cosmic density fluctuations lead to spatial variations in the locally measurable value of $H_0$ (denoted as $H_L$), $\delta_H \equiv (H_L-H_0)/H_0$, which are of order 3-6% (95% confidence interval) in a sphere of 200$h^{-1}$Mpc in diameter, and of order 1-2% in a sphere of 400$h^{-1}$Mpc in diameter, for three currently viable structure formation models (tilted CDM, $\Lambda$CDM, and open CDM) as normalized by the 4 year COBE DMR data. However, the true matter distribution power spectrum may differ from any of the currently viable models. For example, it may contain sharp features which have escaped detection so far. The measured CMB dipole velocity (the Galaxy's peculiar velocity with respect to the CMB rest frame) provides additional constraints on the probability distribution of $\delta_H$ that supplement our limited knowledge of the power spectrum. For a matter power spectrum which consists of the smooth power spectrum of a viable cosmological model plus a delta-function bump, we find that given the CMB dipole velocity, the 95% CL upper limit of $|\delta_H|$ increases approximately by a factor of two, but the probability distribution of $\delta_H$ is non-Gaussian, with increased probability at small $\delta_H$ compared to Gaussian. Abandoning model power spectra entirely, we find that the observed CMB dipole velocity alone provides a very robust limit, $\sqrt{< \delta_H^2 >_R} <10.5 h^{-1}{Mpc}/R$ at 95% CL, in a sphere of radius $R$, for an arbitrary power spectrum. Thus, variations between currently available local measures of $H_0$ and its true global value of a few to several percent are to be expected and differences as large as 10% are possible based on our current knowledge of the CMB anisotropies.