Constraints on the topology of the universe from the 2-yr COBE data

The cosmic microwave background (CMB) is a unique probe of cosmological parameters and conditions. There is a connection between anisotropy in the CMB and the topology of the Universe. Adopting a universe with the topology of a 3-Torus, or a universe where only harmonics of the fundamental mode are allowed, and using 2-years of COBE/DMR data, we obtain constraints on the topology of the Universe. Previous work constrained the topology using the slope information and the correlation function of the CMB. We obtain more accurate results by using all multipole moments, avoiding approximations by computing their full covariance matrix. We obtain the best fit for a cubic toroidal universe of scale 7200h^{-1} Mpc for n=1. The data set a lower limit on the cell size of 4320h^{-1} Mpc at 95% confidence and 5880h^{-1} Mpc at 68% confidence. These results show that the most probable cell size would be around 1.2 times larger than the horizon scale, implying that the 3-Torus topology is no longer an interesting cosmological model.Comment: Minor revisions to match published version. 14 pages, with 4 figures included. Color figures and links at http://www.sns.ias.edu/~angelica/topology.htm

Mapping the CMB III: combined analysis of QMAP flights

We present results from the QMAP balloon experiment, which maps the Cosmic Microwave Background (CMB) and probes its angular power spectrum on degree scales. In two separate flights, data were taken in six channels at two frequency bands between 26 to 46 GHz. We describe our method for mapmaking (removal of 1/f-noise and scan-synchronous offsets) and power spectrum estimation, as well as the results of a joint analysis of the data from both flights. This produces a 527 square degree map of the CMB around the North Celestial Pole, allowing a wide variety of systematic cross-checks. The frequency dependence of the fluctuations is consistent with CMB and inconsistent with Galactic foreground emission. The anisotropy is measured in three multipole bands from l~40 to l~200, and the angular power spectrum shows a distinct rise which is consistent with the Saskatoon results.Comment: 4 pages, with 3 figures included. Submitted to ApJL. Window functions are available at http://pupgg.princeton.edu/~cmb/welcome.html and color figures and links at http://www.sns.ias.edu/~angelica/skymap.html#qma

E/B Decomposition of Finite Pixelized CMB Maps

Separation of the E and B components of a microwave background polarization map or a weak lensing map is an essential step in extracting science from it, but when the map covers only part of the sky and/or is pixelized, this decomposition cannot be done perfectly. We present a method for decomposing an arbitrary sky map into a sum of three orthogonal components that we term ‘‘pure E,’’ ‘‘pure B,’’ and ‘‘ambiguous.’’ The fluctuations in the pure E and B maps are due only to the E and B power spectra, respectively, whereas the source of those in the ambiguous map is completely indeterminate. This method is useful both for providing intuition for experimental design and for analyzing data sets in practice. We show how to find orthonormal bases for all three components in terms of bi-Laplacian eigenfunctions, thus providing a type of polarized signal-to-noise eigenmodes that simultaneously separate both angular scale and polarization type. The number of pure and ambiguous modes probing a characteristic angular scale u scales as the map area over u 2 and as the map boundary length over u, respectively. This implies that fairly round maps (with short perimeters for a given area) will yield the most efficient E/B decomposition and also that the fraction of the information lost to ambiguous modes grows towards larger angular scales. For real-world data analysis, we present a simple matrix eigenvalue method for calculating nearly pure E and B modes in pixelized maps. We find that the dominant source of leakage between E and B is aliasing of small-scale power caused by the pixelization, essentially since derivatives are involved. This problem can be eliminated by heavily oversampling the map, but is exacerbated by the fact that the E power spectrum is expected to be much larger than the B power spectrum and by the extremely blue power spectrum that cosmic microwave background polarization is expected to have. We found that a factor of 2 to 3 more pixels are needed in a polarization map to achieve the same level of contamination by aliased power than in a temperature map. Oversampling is therefore much more important for the polarized case than for the unpolarized case, which should be reflected in experimental design