### Abstract

The errors on statistics measured in finite galaxy catalogues are exhaustively investigated. The theory of errors on factorial moments by Szapudi & Colombi is applied to cumulants via a series expansion method. All results are subsequently extended to the weakly non-linear regime. Together with previous investigations this yields an analytic theory of the errors for moments and connected moments of counts in cells from highly non-linear to weakly non-linear scales. For non-linear functions of unbiased estimators, such as the cumulants, the phenomenon of cosmic bias is identified and computed. Since it is subdued by the cosmic errors in the range of applicability of the theory, correction for it is inconsequential. In addition, the method of Colombi, Szapudi & Szalay concerning sampling effects is generalized, adapting the theory for inhomogeneous galaxy catalogues. While previous work focused on the variance only, the present article calculates the cross-correlations between moments and connected moments as well for a statistically complete description. The final analytic formulae representing the full theory are explicit but somewhat complicated. Therefore we have made available a FORTRAN program capable of calculating the described quantities numerically (for further details e-mail SC at colombi@iap.fr). An important special case is the evaluation of the errors on the two-point correlation function, for which this should be more accurate than any method put forward previously. This tool will be immensely useful in the future for assessing the precision of measurements from existing catalogues, as well as aiding the design of new galaxy surveys. To illustrate the applicability of the results and to explore the numerical aspects of the theory qualitatively and quantitatively, the errors and cross-correlations are predicted under a wide range of assumptions for the future Sloan Digital Sky Survey. The principal results concerning the cumulants ξ̄, Q_{3}, and Q_{4} is that the relative error is expected to be smaller than 3, 5 and 15 per cent, respectively, in the scale range of 1-10 h^{-1} Mpc; the cosmic bias will be negligible.

Original language | English |
---|---|

Pages (from-to) | 428-444 |

Number of pages | 17 |

Journal | Monthly Notices of the Royal Astronomical Society |

Volume | 310 |

Issue number | 2 |

Publication status | Published - Dec 1 1999 |

### Fingerprint

### Keywords

- Galaxies: clusters: general
- Large-scale structure of Universe
- Methods: numerical
- Methods: statistical

### ASJC Scopus subject areas

- Space and Planetary Science

### Cite this

*Monthly Notices of the Royal Astronomical Society*,

*310*(2), 428-444.

**Cosmic statistics of statistics.** / Szapudi, I.; Colombi, Stéphane; Bernardeau, Francis.

Research output: Contribution to journal › Article

*Monthly Notices of the Royal Astronomical Society*, vol. 310, no. 2, pp. 428-444.

}

TY - JOUR

T1 - Cosmic statistics of statistics

AU - Szapudi, I.

AU - Colombi, Stéphane

AU - Bernardeau, Francis

PY - 1999/12/1

Y1 - 1999/12/1

N2 - The errors on statistics measured in finite galaxy catalogues are exhaustively investigated. The theory of errors on factorial moments by Szapudi & Colombi is applied to cumulants via a series expansion method. All results are subsequently extended to the weakly non-linear regime. Together with previous investigations this yields an analytic theory of the errors for moments and connected moments of counts in cells from highly non-linear to weakly non-linear scales. For non-linear functions of unbiased estimators, such as the cumulants, the phenomenon of cosmic bias is identified and computed. Since it is subdued by the cosmic errors in the range of applicability of the theory, correction for it is inconsequential. In addition, the method of Colombi, Szapudi & Szalay concerning sampling effects is generalized, adapting the theory for inhomogeneous galaxy catalogues. While previous work focused on the variance only, the present article calculates the cross-correlations between moments and connected moments as well for a statistically complete description. The final analytic formulae representing the full theory are explicit but somewhat complicated. Therefore we have made available a FORTRAN program capable of calculating the described quantities numerically (for further details e-mail SC at colombi@iap.fr). An important special case is the evaluation of the errors on the two-point correlation function, for which this should be more accurate than any method put forward previously. This tool will be immensely useful in the future for assessing the precision of measurements from existing catalogues, as well as aiding the design of new galaxy surveys. To illustrate the applicability of the results and to explore the numerical aspects of the theory qualitatively and quantitatively, the errors and cross-correlations are predicted under a wide range of assumptions for the future Sloan Digital Sky Survey. The principal results concerning the cumulants ξ̄, Q3, and Q4 is that the relative error is expected to be smaller than 3, 5 and 15 per cent, respectively, in the scale range of 1-10 h-1 Mpc; the cosmic bias will be negligible.

AB - The errors on statistics measured in finite galaxy catalogues are exhaustively investigated. The theory of errors on factorial moments by Szapudi & Colombi is applied to cumulants via a series expansion method. All results are subsequently extended to the weakly non-linear regime. Together with previous investigations this yields an analytic theory of the errors for moments and connected moments of counts in cells from highly non-linear to weakly non-linear scales. For non-linear functions of unbiased estimators, such as the cumulants, the phenomenon of cosmic bias is identified and computed. Since it is subdued by the cosmic errors in the range of applicability of the theory, correction for it is inconsequential. In addition, the method of Colombi, Szapudi & Szalay concerning sampling effects is generalized, adapting the theory for inhomogeneous galaxy catalogues. While previous work focused on the variance only, the present article calculates the cross-correlations between moments and connected moments as well for a statistically complete description. The final analytic formulae representing the full theory are explicit but somewhat complicated. Therefore we have made available a FORTRAN program capable of calculating the described quantities numerically (for further details e-mail SC at colombi@iap.fr). An important special case is the evaluation of the errors on the two-point correlation function, for which this should be more accurate than any method put forward previously. This tool will be immensely useful in the future for assessing the precision of measurements from existing catalogues, as well as aiding the design of new galaxy surveys. To illustrate the applicability of the results and to explore the numerical aspects of the theory qualitatively and quantitatively, the errors and cross-correlations are predicted under a wide range of assumptions for the future Sloan Digital Sky Survey. The principal results concerning the cumulants ξ̄, Q3, and Q4 is that the relative error is expected to be smaller than 3, 5 and 15 per cent, respectively, in the scale range of 1-10 h-1 Mpc; the cosmic bias will be negligible.

KW - Galaxies: clusters: general

KW - Large-scale structure of Universe

KW - Methods: numerical

KW - Methods: statistical

UR - http://www.scopus.com/inward/record.url?scp=0001456509&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0001456509&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0001456509

VL - 310

SP - 428

EP - 444

JO - Monthly Notices of the Royal Astronomical Society

JF - Monthly Notices of the Royal Astronomical Society

SN - 0035-8711

IS - 2

ER -