The best fit for the observed galaxy counts-in-cell distribution function

¹ Departamento de Matemática Aplicada y Estadística, Universidad CEU San Pablo, C/ Julian Romea 23, 28003 Madrid, Spain
e-mail: lluis.hurtadogil@ceu.es
² Observatori Astronòmic, Universitat de València, C/ Catedrático José Beltran, 2, 46980 Paterna (València), Spain
³ Departament d’Astronomia i Astrofísica, Universitat de València, 46100 Burjassot, València, Spain
⁴ Departamento de Matemática Aplicada, Facultad de Estudios Estadísticos, Universidad Complutense de Madrid, 28040 Madrid, Spain
⁵ Departamento de Sistemas Informáticos y Computación, Facultad de Estudios Estadísticos, Universidad Complutense de Madrid, 28040 Madrid, Spain
⁶ Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, C/Dr. Fleming s/n, 30202 Cartagena, Spain

Received: 10 June 2016
Accepted: 28 February 2017

Abstract

Context. The Sloan Digital Sky Survey (SDSS) is the first dense redshift survey encompassing a volume large enough to find the best analytic probability density function that fits the galaxy counts-in-cells distribution f_V(N), the frequency distribution of galaxy counts in a volume V.

Aims. Different analytic functions have been proposed that can account for some of the observed features of the observed frequency counts, but fail to provide an overall good fit to this important statistical descriptor of the galaxy large-scale distribution. Our goal is to find the probability density function that best fits the observed counts-in-cells distribution f_V(N).

Methods. We have made a systematic study of this function applied to several samples drawn from the SDSS. We show the effective ways to deal with incompleteness of the sample (masked data) in the calculation of f_V(N). We used LasDamas simulations to estimate the errors in the calculation. We tested four different distribution functions to find the best fit: the gravitational quasi-equilibrium distribution, the negative binomial distribution, the log normal distribution, and the log normal distribution including a bias parameter. In the two latter cases, we apply a shot-noise correction to the distributions assuming the local Poisson model.

Results. We show that the best fit for the counts-in-cells distribution function is provided by the negative binomial distribution. In addition, at large scales the log normal distribution modified with the inclusion of the bias term also performs a satisfactory fit of the empirical values of f_V(N). Our results demonstrate that the inclusion of a bias term in the log normal distribution is necessary to fit the observed galaxy count-in-cells distribution function.