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We present a semi-analytical method to investigate the systematic effects and statistical uncertainties of the calculated angular power spectrum when incomplete spherical maps are used. The computed power spectrum suffers in particular a loss of angular frequency resolution, which can be written as , where is the effective maximum extent of the partial spherical maps. We propose a correction algorithm to reduce systematic effects on the estimated , as obtained from the partial map projection on the spherical harmonic basis. We have derived near-optimal bands and weighting functions in -space for the power spectrum calculation using small maps, and a correction algorithm for partially masked spherical maps that contain information on the angular correlations on all scales. [ABSTRACT FROM AUTHOR]

The form of the primordial power spectrum has the potential to differentiate strongly between competing models of perturbation generation in the early universe and so is of considerable importance. The recent release of five years of Wilkinson Microwave Anisotropy Probe observations have confirmed the general picture of the primordial power spectrum as deviating slightly from scale invariance with a spectral tilt parameter of . None the less, many attempts have been made to isolate further features such as breaks and cut-offs using a variety of methods, some employing more than ∼10 varying parameters. In this paper, we apply the robust technique of the Bayesian model selection to reconstruct the optimal degree of structure in the spectrum. We model the spectrum simply and generically as piecewise linear in between ‘nodes’ in k space whose amplitudes are allowed to vary. The number of nodes and their k-space positions are chosen by the Bayesian evidence so that we can identify both the complexity and location of any detected features. Our optimal reconstruction contains, perhaps, surprisingly few features, the data preferring just three nodes. This reconstruction allows for a degree of scale dependence of the tilt with the ‘turn-over’ scale occurring around . More structure is penalized by the evidence as overfitting the data, so there is currently little point in attempting reconstructions that are more complex. [ABSTRACT FROM AUTHOR]

It has been long recognized that, besides being a formidable experiment to observe the primordial cosmic microwave background (CMB) anisotropies, Planck will also have the capability to detect galaxy clusters via their Sunyaev–Zel'dovich (SZ) imprint. In this paper constraints on cosmological parameters derivable from the Planck cluster candidate sample are examined for the first time as a function of cluster sample selection and purity obtained from realistic simulations of the microwave sky at the Planck observing frequency bands, observation process modelling and a cluster extraction pipeline. In particular, we employ a multifrequency matched filtering (MFMF) method to recover clusters from mock simulations of Planck observations. Obtainable cosmological constraints under realistic assumptions of priors and knowledge about cluster redshifts are discussed. Just relying on cluster redshift abundances without making use of recovered cluster fluxes, it is shown that from the Planck cluster catalogue cosmological constraints comparable to the ones derived from recent primordial CMB power spectrum measurements can be achieved. For example, for a concordance Λ cold dark matter (ΛCDM) model and a redshift binning of , the 1σ uncertainties on the values of Ωm and σ8 are and , respectively. Furthermore, we find that the constraint of the matter density depends strongly on the prior which can be imposed on the Hubble parameter by other observational means. [ABSTRACT FROM AUTHOR]

Bayesian model selection is a tool for deciding whether the introduction of a new parameter is warranted by the data. I argue that the usual sampling statistic significance tests for a null hypothesis can be misleading, since they do not take into account the information gained through the data, when updating the prior distribution to the posterior. In contrast, Bayesian model selection offers a quantitative implementation of Occam's razor. I introduce the Savage–Dickey density ratio, a computationally quick method to determine the Bayes factor of two nested models and hence perform model selection. As an illustration, I consider three key parameters for our understanding of the cosmological concordance model. By using Wilkinson Microwave Anisotropy Probe ( WMAP) 3-year data complemented by other cosmological measurements, I show that a non-scale-invariant spectral index of perturbations is favoured for any sensible choice of prior. It is also found that a flat universe is favoured with odds of 29:1 over non-flat models, and that there is strong evidence against a cold dark matter isocurvature component to the initial conditions which is totally (anti)correlated with the adiabatic mode (odds of about 2000:1), but that this is strongly dependent on the prior adopted. These results are contrasted with the analysis of WMAP 1-year data, which were not informative enough to allow a conclusion as to the status of the spectral index. In a companion paper, a new technique to forecast the Bayes factor of a future observation is presented. [ABSTRACT FROM AUTHOR]

We develop the pseudo- Cℓ method for reconstructing the cosmic microwave background (CMB) temperature and polarization auto- and cross-power spectra, and estimate the pseudo- Cℓ covariance matrix for a realistic experiment on the cut sky. We calculate the full coupling equations for all six possible CMB power spectra, relating the observed pseudo- Cℓ values to the underlying all-sky Cℓ values, and test the reconstruction on both full-sky and cut-sky simulated CMB data sets. In particular we consider the reconstruction of the Cℓ from upcoming ground-based polarization experiments covering areas of a few hundred deg2 and find that the method is fast, unbiased and performs well over a wide range of multipoles from to . We then calculate the full covariance matrix between the modes of the pseudo-temperature and polarization power spectra, assuming that the underlying CMB fields are Gaussian randomly distributed. The complexity of the covariance matrix prohibits its rapid calculation, required for parameter estimation. Hence we present an approximation for the covariance matrix in terms of convolutions of the underlying power spectra. The coupling matrices in these expressions can be estimated by fitting to numerical simulations, circumventing direct and slow calculation, and further, inaccurate analytic approximations. We show that these coupling matrices are mostly independent of cosmology, and that the full covariance matrix for all six pseudo- Cℓ power spectra can be quickly and accurately calculated for any given cosmological model using this method. We compare these semi-analytic covariance matrices against simulations and find good agreement, the accuracy of which depends mainly on survey area and the range of cosmological parameters considered. [ABSTRACT FROM AUTHOR]

Destriping methods for constructing maps of the cosmic microwave background (CMB) anisotropies have been investigated extensively in the literature. However, their error properties have been studied in less detail. Here we present an analysis of the effects of destriping errors on CMB power spectrum estimates forPlanck-like scanning strategies. Analytic formulae are derived for certain simple scanning geometries that can be rescaled to account for different detector noise. AssumingPlanck-like low-frequency noise, the noise power spectrum is accurately white at high multipoles. Destriping errors, though dominant at lower multipoles, are small in comparison to the cosmic variance. These results show that simple destriping map-making methods should be perfectly adequate for the analysis ofPlanckdata and support the arguments given in an earlier paper in favour of applying a fast hybrid power spectrum estimator to CMB data with realistic‘1/ f’ noise. [ABSTRACT FROM AUTHOR]