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.
