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2022•Astronomy and AstrophysicsOpen access

Euclid: Fast two-point correlation function covariance through linear construction

Elina Keihänen, V. Lindholm, Pierluigi Monaco, L Blot, C. Carbone, K. Kiiveri, Ariel G. Sánchez, A. Viitanen, J. Väliviita, Adam Amara, N. Auricchio, Marco Baldi, D. Bonino, E. Branchini, M. Brescia, J. Brinchmann, S. Camera, V. Capobianco, J. Carretero, M. Castellano, S. Cavuoti, A. Cimatti, R. Clédassou, G. Congedo, L. Conversi, Y. Copin, L. Corcione, Mark S. Cropper, A. Da Silva, Hubert Degaudenzi, Marian Douspis, F. Dubath, C. A. J. Duncan, X. Dupac, S. Dusini, A. Ealet, S. Farrens, S. Ferriol, M. Frailis, Enrico Franceschi, M. Fumana, Bryan R. Gillis, C. Giocoli, A. Grazian, Frank U. Grupp, L. Guzzo, S. V. H. Haugan, Henk Hoekstra, W. A. Holmes, F. Hormuth, K. Jahnkę, M Kümmel, Smaïn Kermiche, A. Kiessling, T. Kitching, Martin Kunz, Hannu Kurki-Suonio, S. Ligori, Per B. Lilje, Ivan Lloro, E. Maiorano, O. Mansutti, O. Marggraf, F. Marulli, R. Massey, M. Melchior, Massimo Meneghetti, George L. Meylan, M. Moresco, Benjamin R. Morin, Lauro Moscardini, Emiliano Munari, S.-M Niemi, C. Padilla, S. Paltani, FABIO PASIAN, Kevin R. Pedersen, V. Pettorino, S. Pires, Gianluca Polenta, Maurice Poncet, Lucia Aurelia Popa, F. Raison, Alessandro Renzi, Jason D. Rhodes, E. Romelli, Roberto P. Saglia, Barbara Sartoris, P. Christian Schneider, T. Schrabback, Aurelia Secroun, Gregor Seidel, C. Sirignano, Gabriele Sirri, Luca Stanco, Christian Surace, Pau Tallada-Crespí, Daniele Tavagnacco, A. N. Taylor, Ismael Tereno, R. Toledo-Moreo, F. Torradeflot, Edwin A. Valentijn, Luca Valenziano, T. Vassallo, Yun Wang, J. Weller, G. Zamorani, J. Zoubian, Stefano Andreon, D. Maino, S. de la Torre

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Abstract

We present a method for fast evaluation of the covariance matrix for a two-point galaxy correlation function (2PCF) measured with the Landy–Szalay estimator. The standard way of evaluating the covariance matrix consists in running the estimator on a large number of mock catalogs, and evaluating their sample covariance. With large random catalog sizes (random-to-data objects’ ratio M ≫ 1) the computational cost of the standard method is dominated by that of counting the data-random and random-random pairs, while the uncertainty of the estimate is dominated by that of data-data pairs. We present a method called Linear Construction (LC), where the covariance is estimated for small random catalogs with a size of M = 1 and M = 2, and the covariance for arbitrary M is constructed as a linear combination of the two. We show that the LC covariance estimate is unbiased. We validated the method with PINOCCHIO simulations in the range r = 20 − 200 h−1 Mpc. With M = 50 and with 2 h−1 Mpc bins, the theoretical speedup of the method is a factor of 14. We discuss the impact on the precision matrix and parameter estimation, and present a formula for the covariance of covariance.

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We present a method for fast evaluation of the covariance matrix for a two-point galaxy correlation function (2PCF) measured with the Landy–Szalay estimator. The standard way of evaluating the covariance matrix consists in running the estimator on a large number of mock catalogs, and evaluating their sample covariance. With large random catalog sizes (random-to-data objects’ ratio M ≫ 1) the computational cost of the standard method is dominated by that of counting the data-random and random-random pairs, while the uncertainty of the estimate is dominated by that of data-data pairs. We present a method called Linear Construction (LC), where the covariance is estimated for small random catalogs with a size of M = 1 and M = 2, and the covariance for arbitrary M is constructed as a linear combination of the two. We show that the LC covariance estimate is unbiased. We validated the method with PINOCCHIO simulations in the range r = 20 − 200 h−1 Mpc. With M = 50 and with 2 h−1 Mpc bins, the theoretical speedup of the method is a factor of 14. We discuss the impact on the precision matrix and parameter estimation, and present a formula for the covariance of covariance.

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Available abstract

We present a method for fast evaluation of the covariance matrix for a two-point galaxy correlation function (2PCF) measured with the Landy–Szalay estimator. The standard way of evaluating the covariance matrix consists in running the estimator on a large number of mock catalogs, and evaluating their sample covariance. With large random catalog sizes (random-to-data objects’ ratio M ≫ 1) the computational cost of the standard method is dominated by that of counting the data-random and random-random pairs, while the uncertainty of the estimate is dominated by that of data-data pairs. We present a method called Linear Construction (LC), where the covariance is estimated for small random catalogs with a size of M = 1 and M = 2, and the covariance for arbitrary M is constructed as a linear combination of the two. We show that the LC covariance estimate is unbiased. We validated the method with PINOCCHIO simulations in the range r = 20 − 200 h−1 Mpc. With M = 50 and with 2 h−1 Mpc bins, the theoretical speedup of the method is a factor of 14. We discuss the impact on the precision matrix and parameter estimation, and present a formula for the covariance of covariance.

Key concepts: Covariance, Rational quadratic covariance function, Matérn covariance function, Law of total covariance, Covariance mapping, Estimation of covariance matrices, Covariance matrix, Covariance function

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