Estimating the galaxy two-point correlation function using a split random catalog

¹ Department of Physics and Helsinki Institute of Physics, University of Helsinki, Gustaf Hällströmin katu 2, 00014 Helsinki, Finland
e-mail: elina.keihanen@helsinki.fi
² Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy
³ Department of Mathematics and Physics, Roma Tre University, Via della Vasca Navale 84, 00146 Rome, Italy
⁴ Dipartimento di Fisica e Astronomia – Alma Mater Studiorum Università di Bologna, Via Piero Gobetti 93/2, 40129 Bologna, Italy
⁵ INAF – Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, Via Piero Gobetti 93/3, 40129 Bologna, Italy
⁶ INFN – Sezione di Bologna, Viale Berti Pichat 6/2, 40127 Bologna, Italy
⁷ ICC & CEA, Department of Physics, Durham University, South Road, Durham DH1 3LE, UK
⁸ INAF, Osservatorio Astronomico di Trieste, Via Tiepolo 11, 34131 Trieste, Italy
⁹ Aix Marseille Univ., CNRS, CNES, LAM, Marseille, France
¹⁰ SISSA, International School for Advanced Studies, Via Bonomea 265, 34136 Trieste, TS, Italy
¹¹ INFN, Sezione di Trieste, Via Valerio 2, 34127 Trieste, TS, Italy
¹² Aix Marseille Univ., Université de Toulon, CNRS, CPT, Marseille, France
¹³ Max-Planck-Institut für Extraterrestrische Physik, Postfach 1312, Giessenbachstr., 85741 Garching, Germany
¹⁴ IFPU – Institute for Fundamental Physics of the Universe, Via Beirut 2, 34014 Trieste, Italy

Received: 3 May 2019
Accepted: 24 July 2019

Abstract

The two-point correlation function of the galaxy distribution is a key cosmological observable that allows us to constrain the dynamical and geometrical state of our Universe. To measure the correlation function we need to know both the galaxy positions and the expected galaxy density field. The expected field is commonly specified using a Monte-Carlo sampling of the volume covered by the survey and, to minimize additional sampling errors, this random catalog has to be much larger than the data catalog. Correlation function estimators compare data–data pair counts to data–random and random–random pair counts, where random–random pairs usually dominate the computational cost. Future redshift surveys will deliver spectroscopic catalogs of tens of millions of galaxies. Given the large number of random objects required to guarantee sub-percent accuracy, it is of paramount importance to improve the efficiency of the algorithm without degrading its precision. We show both analytically and numerically that splitting the random catalog into a number of subcatalogs of the same size as the data catalog when calculating random–random pairs and excluding pairs across different subcatalogs provides the optimal error at fixed computational cost. For a random catalog fifty times larger than the data catalog, this reduces the computation time by a factor of more than ten without affecting estimator variance or bias.