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 H0 (denoted as HL), δH ≡ (HL - H0)/H0, 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 cold dark matter [CDM], ΛCDM, and open CDM) as normalized by the 4 yr COBE DMR data. However, the true matter distribution power spectrum may differ from any of the current viable models. For example, it may contain sharp features which have escaped detection so far. The measured cosmic microwave background (CMB) dipole velocity (the Galaxy's peculiar velocity with respect to the CMB rest frame) provides additional constraints on the probability distribution of δH that supplement our limited knowledge of the power spectrum. For a matter power spectrum that consists of the smooth power spectrum of a viable cosmological model plus a δ-function bump, we find that given the CMB dipole velocity, the 95% confidence level (CL) upper limit of |δH| increases approximately by a factor of 2, but the probability distribution of δH is non-Gaussian, with increased probability at small δH compared to Gaussian. Abandoning model power spectra entirely, we find that the observed CMB dipole velocity alone provides a very robust limit, (δ2HR)-1/2 < 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 H0 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.