Quantifying the CMB Degeneracy Between the Matter Density and Hubble Constant in Current Experiments

We revisit the degeneracy between the Hubble constant, $H_0$, and matter density, $\Omega_m$, for current cosmic microwave background (CMB) observations within the standard $\Lambda CDM$ model. We show that Planck, Wilkinson Microwave Anisotropy Probe (WMAP), South Pole Telescope (SPT), and Atacama Cosmology Telescope Polarimeter (ACTPol) temperature power spectra produce different values of the exponent $x$ from minimizing the variance of the product $\Omega_mH_0^x$. The distribution of $x$ from the different data sets does not follow the Markov Chain Monte Carlo (MCMC) best-fit values for $H_0$ or $\Omega_m$. Particularly striking is the difference between Planck multipoles $\ell\leq800$ ($x=2.81$), and WMAP ($x = 2.94$), despite very similar best-fit cosmologies. We use a Fisher matrix analysis to show that, in fact, this range in exponent values is exactly as expected in $\Lambda CDM$ given the multipole coverage and power spectrum uncertainties for each experiment. We show that the difference in $x$ from the Planck $\ell \leq 800$ and WMAP data is explained by a turning point in the relationship between $x$ and the maximum effective multipole, at around $\ell=700$. The value of $x$ is determined by several physical effects, and we highlight the significant impact of gravitational lensing for the high-multipole measurements. Despite the spread of $H_0$ values from different CMB experiments, the experiments are consistent with their sampling of the $\Omega_m-H_0$ degeneracy and do not show evidence for the need for new physics or for the presence of significant underestimated systematics according to these tests. The Fisher calculations can be used to predict the $\Omega_m-H_0$ degeneracy of future experiments.