The topology and size of the universe from CMB temperature and polarization data

Abstract

We analyze seven year and nine year WMAP temperature maps for signatures of three finite flat topologies ℳ0 = \U0001d54b3, ℳ1 = \U0001d54b2 × ℝ1, and ℳ2 = S1 × ℝ2. We use Monte-Carlo simulations with the Feldman-Cousins method to obtain confidence intervals for the size of the topologies considered. We analyze the V, W, and Q frequency bands along with the ILC map and find no significant difference in the results. The 95.5% confidence level lower bound on the size of the topology is 1.5L0 for ℳ0, 1.4L0 for ℳ1, and 1.1L0 for ℳ2, where L0 is the radius of the last scattering surface. Our results agree very well with the recently released results from the Planck temperature data. We show that the likelihood function is not Gaussian in the size, and therefore simulations are important for obtaining accurate bounds on the size. We then introduce the formalism for including polarization data in the analysis. The improvement that we find from WMAP polarization maps is small because of the high level of instrumental noise, but our forecast for Planck maps shows a much better improvement on the lower bound for L. For the ℳ0 topology we expect an improvement on the lower bound of L from 1.7L0 to 1.9L0 at 95.5% confidence level. Using both polarization and temperature data is important because it tests the hypothesis that deviations in the TT spectrum at small l originate in the primordial perturbation spectrum.