Dmitri I. Novikov

The detection of B-mode polarization in the CMB is one of the most important outstanding tests of inflationary cosmology. One of the necessary steps for extracting polarization information in the CMB is reducing contamination from so-called "ambiguous modes" on a masked sky, which contain leakage from the larger E-mode signal. This can be achieved by utilising derivative operators on the real-space Stokes Q and U parameters. This paper presents an algorithm and a software package to perform this procedure on the nearly full sky, i.e., with projects such as the Planck Surveyor and future satellites in mind; in particular, the package can perform finite differences on masked, irregular grids and is applied to a semi-regular spherical pixellization, the HEALPix grid. The formalism reduces to the known finite-difference solutions in the case of a regular grid. We quantify full-sky improvements on the possible bounds on the CMB B-mode signal. We find that in the specific case of E and B-mode separation, there exists a "pole problem" in our formalism which produces signal contamination at very low multipoles l. Several solutions to the "pole problem" are presented; one proposed solution facilitates a calculation of a general Gaussian quadrature scheme, which finds application in calculating accurate harmonic coefficients on the HEALPix sphere. Nevertheless, on a masked sphere the software represents a considerable reduction in B-mode noise from limited sky coverage.

The first and third year data releases from the WMAP provide evidence of an anomalous Cold Spot (CS) at galactic latitude b=-57deg and longitude l=209deg. We have examined the properties of the CS in some detail in order to assess its cosmological significance. We have performed a cluster analysis of the local extrema in the CMB signal to show that the CS is actually associated with a large group of extrema rather than just one. In the light of this we have re-examined the properties of the WMAP ILC and co-added "cleaned" WCM maps, which have previously been used for the analysis of the properties of the signal in the vicinity of the CS. These two maps have remarkably similar properties on equal latitude rings for |b|>30deg, as well as in the vicinity of the CS. We have also checked the idea that the CMB signal has a non-Gaussian tail, localized in the low multipole components of the signal. For each ring we apply a linear filter with characteristic scale R, dividing the CMB signal in two parts: the filtered part, with characteristic scale above that of the filter R, and the difference between the initial and filtered signal. Using the filter scale as a variable, we can maximize the skewness and kurtosis of the smoothed signal and minimize these statistics for the difference between initial and filtered signal. We have discovered that the shape of the CS is formed primarily by the components of the CMB signal represented by multipoles between 10<=L<=20, with a corresponding angular scale about 5-10 degs. This signal leads to modulation of the whole CMB sky, clearly seen at |b|>30deg in both the ILC and WCM maps, rather than a single localized feature. After subtraction of this modulation, the remaining part of the CMB signal appears to be consistent with statistical homogeneity and Gaussianity.

The Cold Spot (CS) at galactic coordinates (b = −57°, l = 209°) was discovered in the Wilkinson Microwave Anisotropy Probe (WMAP)data as a cosmic background anomaly. In order to assess the cosmological significance of the Spot, we examine its properties using the cluster analysis of the local extrema in the cosmic microwave background (CMB) signal. We also check the hypothesis that the CMB signal has a non-Gaussian tail, localized in the low-multipole components. We constructed a linear filter, dividing the signal into two parts: non-Gaussian and Gaussian. Using the filter scale as a variable, we can maximize the skewness and kurtosis of the smoothed signal and minimize these statistics. We discovered that the shape of the CS is formed primarily by the components of the CMB signal represented by the multipoles between 10 ≤ ℓ ≤ 20, with a corresponding angular scale of about 5°–10°. This signal leads to the modulation of the CMB on the whole sky, clearly seen at |b| > 30° in both the ILC andWCM maps, rather than in a single localized feature. After subtraction of this modulation, the remaining part of the CMB signal appears to be consistent with statistical homogeneity and Gaussianity. We therefore infer that the mystery of the WMAP Cold Spot reflects directly the peculiarities of low multipoles of the CMB signal, rather than a single local (isolated) defect or the manifestations of a globally anisotropic cosmology.

We present developing of method of the numerical analysis of polarization in the Gauss--Legendre Sky Pixelization (GLESP) scheme for the CMB maps. This incorporation of the polarization transforms in the pixelization scheme GLESP completes the creation of our new method for the numerical analysis of CMB maps. The comparison of GLESP and HEALPix calculations is done.

abridged] A method to rapidly estimate the Fourier power spectrum of a point distribution is presented. This method relies on a Taylor expansion of the trigonometric functions. It yields the Fourier modes from a number of FFTs, which is controlled by the order N of the expansion and by the dimension D of the system. In three dimensions, for the practical value N=3, the number of FFTs required is 20. We apply the method to the measurement of the power spectrum of a periodic point distribution that is a local Poisson realization of an underlying stationary field. We derive explicit analytic expression for the spectrum, which allows us to quantify--and correct for--the biases induced by discreteness and by the truncation of the Taylor expansion, and to bound the unknown effects of aliasing of the power spectrum. We show that these aliasing effects decrease rapidly with the order N. The only remaining significant source of errors is reduced to the unavoidable cosmic/sample variance due to the finite size of the sample. The analytical calculations are successfully checked against a cosmological N-body experiment. We also consider the initial conditions of this simulation, which correspond to a perturbed grid. This allows us to test a case where the local Poisson assumption is incorrect. Even in that extreme situation, the third-order Fourier-Taylor estimator behaves well. We also show how to reach arbitrarily large dynamic range in Fourier space (i.e., high wavenumber), while keeping statistical errors in control, by appropriately "folding" the particle distribution.

The three Minkowski functionals and the recently defined length of the skeleton are estimated for the co-added first-year Wilkinson Microwave Anisotropy Probe (WMAP) data and compared with 5000 Monte Carlo simulations, based on Gaussian fluctuations with the a-priori best-fit running-index power spectrum and WMAP-like beam and noise properties. Several power spectrum-dependent quantities, such as the number of stationary points, the total length of the skeleton, and a spectral parameter, gamma, are also estimated. While the area and length Minkowski functionals and the length of the skeleton show no evidence for departures from the Gaussian hypothesis, the northern hemisphere genus has a chi^2 that is large at the 95% level for all scales. For the particular smoothing scale of 3.40 degrees FWHM it is larger than that found in 99.5% of the simulations. In addition, the WMAP genus for negative thresholds in the northern hemisphere has an amplitude that is larger than in the simulations with a significance of more than 3 sigma. On the smallest angular scales considered, the number of extrema in the WMAP data is high at the 3 sigma level. However, this can probably be attributed to the effect of point sources. Finally, the spectral parameter gamma is high at the 99% level in the northern Galactic hemisphere, while perfectly acceptable in the southern hemisphere. The results provide strong evidence for the presence of both non-Gaussian behavior and an unexpected power asymmetry between the northern and southern hemispheres in the WMAP data.