Relationship between the CMB, Sunyaev-Zel’dovich cluster counts, and local Hubble parameter measurements in a simple void model

The discrepancy between the amplitudes of matter fluctuations inferred from Sunyaev-Zel'dovich (SZ) cluster number counts and the measurement of temperature and polarization anisotropies of the cosmic microwave background (CMB) measured by the Planck satellite can be reconciled if the local universe is embedded in an underdense region as shown by Lee, 2014. Here using a simple void model assuming the open Friedmann-Robertson-Walker geometry and a Markov Chain Monte Carlo technique, we investigate how deep the local underdense region needs to be to resolve this discrepancy. Such local void, if it exists, predicts the local Hubble parameter value that is different from the global Hubble constant. We derive the posterior distribution of the local Hubble parameter from a joint fitting of the Planck CMB data and SZ cluster number counts assuming the simple void model. We show that the predicted local Hubble parameter value of ${H}_{\text{loc}}=70.1\ifmmode\pm\else\textpm\fi{}0.34\text{ }\text{ }\mathrm{km}\text{ }{\mathrm{s}}^{\ensuremath{-}1}\text{ }{\mathrm{Mpc}}^{\ensuremath{-}1}$ is in better agreement with direct local Hubble parameter measurements, indicating that the local void model may provide provide a consistent solution to the cluster number counts and Hubble parameter discrepancies.