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Register a new userNotes on bias and covariance matrix of the angular power spectrum on small sky maps
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Added by
Christophe Magneville Dr
Publication date
2007
fields
Physics
and research's language is
English
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We compute the effects induced by the use of small CMB maps on the measurement of the $cl{l}$ coefficients of the angular power spectrum and show that small systematic effects have to be taken into account. We also compute numerically the cosmic variance and covariance of the $cl{l}$ spectrum for various spherical cap like maps. Comparisons with simulations are presented. The calculations are done using the standard method based on the spherical harmonic transform or using the temperature angular correlation spectrum.
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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 delta_l ~ pi/gamma_max, where gamma_max is the effective maximum extent of the partial spherical maps. We propose a correction algorithm to reduce systematic effects on the estimated C_l, as obtained from the partial map projection on the spherical harmonic Ylm(l,m) basis. We have derived near optimal bands and weighting functions in l-space for 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.
We show how to estimate the covariance of the power spectrum of a statistically homogeneous and isotropic density field from a single periodic simulation, by applying a set of weightings to the density field, and by measuring the scatter in power spectra between different weightings. We recommend a specific set of 52 weightings containing only combinations of fundamental modes, constructed to yield a minimum variance estimate of the covariance of power. Numerical tests reveal that at nonlinear scales the variance of power estimated by the weightings method substantially exceeds that estimated from a simple ensemble method. We argue that the discrepancy is caused by beat-coupling, in which products of closely spaced Fourier modes couple by nonlinear gravitational growth to the beat mode between them. Beat-coupling appears whenever nonlinear power is measured from Fourier modes with a finite spread of wavevector, and is therefore present in the weightings method but not the ensemble method. Beat-coupling inevitably affects real galaxy surveys, whose Fourier modes have finite width. Surprisingly, the beat-coupling contribution dominates the covariance of power at nonlinear scales, so that, counter-intuitively, it is expected that the covariance of nonlinear power in galaxy surveys is dominated not by small scale structure, but rather by beat-coupling to the largest scales of the survey.
We seek to improve estimates of the power spectrum covariance matrix from a limited number of simulations by employing a novel statistical technique known as shrinkage estimation. The shrinkage technique optimally combines an empirical estimate of the covariance with a model (the target) to minimize the total mean squared error compared to the true underlying covariance. We test this technique on N-body simulations and evaluate its performance by estimating cosmological parameters. Using a simple diagonal target, we show that the shrinkage estimator significantly outperforms both the empirical covariance and the target individually when using a small number of simulations. We find that reducing noise in the covariance estimate is essential for properly estimating the values of cosmological parameters as well as their confidence intervals. We extend our method to the jackknife covariance estimator and again find significant improvement, though simulations give better results. Even for thousands of simulations we still find evidence that our method improves estimation of the covariance matrix. Because our method is simple, requires negligible additional numerical effort, and produces superior results, we always advocate shrinkage estimation for the covariance of the power spectrum and other large-scale structure measurements when purely theoretical modeling of the covariance is insufficient.
As the determination of density fluctuations becomes more precise with larger surveys, it becomes more important to account for the increased covariance due to the non-linearity of the field. Here I have focussed on the galaxy density, with analytical prediction of the non-Gaussianity using the halo model coupled with standard perturbation theory in real space. I carried out an exact and exhaustive derivation of all tree-level terms of the non-Gaussian covariance of the galaxy $C_ell$, with the computation developed up to the third order in perturbation theory and local halo bias, including the non-local tidal tensor effect. A diagrammatic method was used to derive the involved galaxy 3D trispectra, including shot-noise contributions. The projection to the angular covariance was derived in all trispectra cases with and without Limbers approximation, with the formulae being of potential interest for other observables than galaxies. The effect of subtracting shot-noise from the measured spectrum is also discussed, and does simplify the covariance, though some non-Gaussian shot-noise terms still remain. I make the link between this complete derivation and partial terms which have been used previously in the literature, including super-sample covariance (SSC). I uncover a wealth of additional terms which were not previously considered, including a whole new class which I dub braiding terms as it contains multipole-mixing kernels. The importance of all these new terms is discussed with analytical arguments. I find that they become comparable to, if not bigger than, SSC if the survey is large or deep enough to probe scales comparable with the matter-radiation equality $k_mathrm{eq}$. A short self-contained summary of the equations is provided in Section 9 for the busy reader, ready to be implemented numerically for analysis of current and future galaxy surveys.
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