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Register a new userDisconnected pseudo-$C_ell$ covariances for projected large-scale structure data
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Added by
Carlos Garc\\'ia-Garc\\'ia
Publication date
2019
fields
Physics
and research's language is
English
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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 $O(ell_{rm max}^6)$ 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 $O(ell_{rm max}^3)$ 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).
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The covariance matrix $boldsymbol{Sigma}$ of non-linear clustering statistics that are measured in current and upcoming surveys is of fundamental interest for comparing cosmological theory and data and a crucial ingredient for the likelihood approximations underlying widely used parameter inference and forecasting methods. The extreme number of simulations needed to estimate $boldsymbol{Sigma}$ to sufficient accuracy poses a severe challenge. Approximating $boldsymbol{Sigma}$ using inexpensive but biased surrogates introduces model error with respect to full simulations, especially in the non-linear regime of structure growth. To address this problem we develop a matrix generalization of Convergence Acceleration by Regression and Pooling (CARPool) to combine a small number of simulations with fast surrogates and obtain low-noise estimates of $boldsymbol{Sigma}$ that are unbiased by construction. Our numerical examples use CARPool to combine GADGET-III $N$-body simulations with fast surrogates computed using COmoving Lagrangian Acceleration (COLA). Even at the challenging redshift $z=0.5$, we find variance reductions of at least $mathcal{O}(10^1)$ and up to $mathcal{O}(10^4)$ for the elements of the matter power spectrum covariance matrix on scales $8.9times 10^{-3}<k_mathrm{max} <1.0$ $h {rm Mpc^{-1}}$. We demonstrate comparable performance for the covariance of the matter bispectrum, the matter correlation function and probability density function of the matter density field. We compare eigenvalues, likelihoods, and Fisher matrices computed using the CARPool covariance estimate with the standard sample covariance estimators and generally find considerable improvement except in cases where $Sigma$ is severely ill-conditioned.
This is the second paper in a series where we propose a method of indirectly measuring large scale structure using information from small scale perturbations. The idea is to build a quadratic estimator from small scale modes that provides a map of structure on large scales. We demonstrated in the first paper that the quadratic estimator works well on a dark-matter-only N-body simulation at a snapshot of $z=0$. Here we generalize the theory to the case of a light cone halo catalog with a non-cubic region taken into consideration. We successfully apply the generalized version of the quadratic estimator to the light cone halo catalog based on an N-body simulation of volume $sim15.03,(h^{-1},rm Gpc)^3$. The most distant point in the light cone is at a redshift of $1.42$, indicating the applicability of our method to next generation of galaxy surveys.
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