Disconnected pseudo-C_ℓ covariances for projected large-scale structure data

Abstract

The disconnected part of the power spectrum covariance matrix (also known as the "Gaussian" covariance) is the dominant contribution on large scales for galaxy clustering and weak lensing datasets. The presence of a complicated sky mask causes non-trivial correlations between different Fourier/harmonic modes, which must be accurately characterized in order to obtain reliable cosmological constraints. This is particularly relevant for galaxy survey data. Unfortunately, an exact calculation of these correlations involves O( 6 max ) operations that become computationally impractical very quickly. We present an implementation of approximate methods to estimate the Gaussian covariance matrix of power spectra involving spin-0 and spin-2 flat-and curved-sky fields, expanding on existing algorithms. These methods achieve an O( 3 max ) scaling, which makes the computation of the covariance matrix as fast as the computation of the power spectrum itself. We quantify the accuracy of these methods on large-scale structure and weak lensing data, making use of a large number of Gaussian but otherwise realistic simulations. We show that, using the approximate covariance matrix, we are able to recover the true posterior distribution of cosmological parameters to high accuracy. We also quantify the shortcomings of these methods, which become unreliable on the very largest scales, as well as for covariance matrix elements involving cosmic shear B modes. The algorithms presented here are implemented in the public code NaMaster https://github.com/LSSTDESC/NaMaster.

Tomographic galaxy clustering with the Subaru Hyper Suprime-Cam first year public data release

“…The SSC contribution accounts for the coherent shift in the amplitude of density fluctuations within the surveyed volume caused by long wavelength modes larger than the survey. We estimate the Gaussian covariance of the pseudo-C estimator as described in [85,86]. The covariance of the observed pseudo-power spectra is given by…”

“…where the quantities W x ll are coupling coefficients depending only on the mask of x (see [86] for further details). Without further approximations, computing the covariance matrix would therefore imply solving a 4-dimensional integral for each pair ( , ).…”

“…This allows us to simplify the expression above significantly, leading to 22) where M are the mode-coupling matrices described in the previous section, except now they are computed from the products of two masks. It has been shown by [86] that this approach yields a very good approximation for the power spectrum covariance, fully accounting for the effects of mode coupling due to survey geometry.…”

Cross-correlation of Planck CMB lensing with DESI-like LRGs

“…the covariance for perfectly Gaussian fields, dominates the total covariance matrix on linear and weakly nonlinear scales. While trivial to compute for full-sky fields, the exact correlations between different modes induced by a partial sky are computationally expensive to calculate, requiring O (ℓ 6 max ) operations (Efstathiou 2004b, García-García et al 2019. A common approximation assumes that the off-diagonal elements remain negligible after mode coupling and simply modifies the diagonal elements by rescaling the number of degrees of freedom,…”

Joint constraints on cosmology and the impact of baryon feedback: combining KiDS-1000 lensing with the thermal Sunyaev-Zeldovich effect from Planck and ACT

“…We compute the Gaussian covariance term C G using NA-MASTER, accounting for the effect of partial and weighted sky observations (García-García et al 2019). While this can be done exactly, the calculation scales as O( 6max ) for the sample variance term.…”