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

Abstract

Low amplitude (linear regime) cosmic density fluctuations lead to spatial variations in the locally measurable value of $H_0$ (denoted as $H_L$), $δ_H \equiv (H_L-H_0)/H_0$, 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, $Λ$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 $δ_H$ 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 $|δ_H|$ increases approximately by a factor of two, but the probability distribution of $δ_H$ is non-Gaussian, with increased probability at small $δ_H$ compared to Gaussian. Abandoning model power spectra entirely, we find that the observed CMB dipole velocity alone provides a very robust limit, $\sqrt{< δ_H^2 >_R} <10.5 h^{-1}{Mpc}/R$ 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.

Final version with very minor changes in wording. To appear in ApJ on May 1, 1998