[Paper Review] Likelihood Functions for Galaxy Cluster Surveys

G. P. Holder|arXiv (Cornell University)|Feb 11, 2006

Galaxies: Formation, Evolution, Phenomena3 citations

TL;DR

This paper develops exact likelihood functions for galaxy cluster surveys using hierarchical correlation functions, showing that sample variance effects—often neglected in Poisson-based analyses—can be accurately modeled. It demonstrates that for massive clusters (>10¹⁴h⁻¹M⊙ at z=0), the derived probability distributions are highly accurate, significantly improving cosmological parameter estimation beyond Poisson assumptions.

ABSTRACT

Galaxy cluster surveys offer great promise for measuring cosmological parameters, but survey analysis methods have not been widely studied. Using methods developed decades ago for galaxy clustering studies, it is shown that nearly exact likelihood functions can be written down for galaxy cluster surveys. The sparse sampling of the density field by galaxy clusters allows simplifications that are not possible for galaxy surveys. An application to counts in cells is explicitly tested using cluster catalogs from numerical simulations and it is found that the calculated probability distributions are very accurate at masses above several times 10^{14}h^{-1} solar masses at z=0 and lower masses at higher redshift.

Motivation & Objective

• To address the lack of rigorous statistical analysis methods for galaxy cluster surveys in cosmology.
• To correct the common assumption of Poisson statistics in cluster abundance studies, which underestimates sample variance effects.
• To derive exact likelihood functions using hierarchical correlation functions, applicable to binned cluster counts.
• To test the accuracy of these likelihoods using simulated cluster catalogs from the Hubble Volume simulation.
• To demonstrate that sample variance significantly broadens probability distributions and must be included for precise cosmological constraints.

Proposed method

• Uses the formalism of White (1979) and DHJ97 to derive likelihoods based on n-point correlation functions for cluster distributions.
• Replaces standard weights $w_i$ with $W_i$, which incorporate the absence of clusters outside observed positions via infinite series over correlation functions.
• Applies the Gaussian limit to simplify the hierarchy, focusing on two-point and three-point functions, with higher-order terms truncated.
• Derives the counts-in-cells likelihood using the generating function approach, expressing it as a series expansion in terms of $\alpha = \bar{w}_2/2$.
• Truncates the series at second order in $\alpha$ for computational feasibility, assuming $\alpha x$ and $\alpha x^2$ are small.
• Validates the method using cluster catalogs from the Hubble Volume simulation, comparing theoretical probability distributions to observed counts.

Experimental results

Research questions

• RQ1 Can exact likelihood functions be derived for galaxy cluster surveys that properly account for sample variance and clustering?
• RQ2 How does the inclusion of sample variance via correlation functions improve upon Poisson-based likelihoods in cluster abundance analysis?
• RQ3 What is the regime of validity for the derived likelihoods, particularly in terms of mass and redshift?
• RQ4 How accurate are the theoretical probability distributions when compared to simulated cluster catalogs?
• RQ5 What conditions must be met for the Gaussian approximation of the cluster distribution to remain valid?

Key findings

• The derived likelihood functions accurately reproduce the probability distributions for cluster counts in cells at masses above several times 10¹⁴h⁻¹M⊙ at z=0.
• For lower masses at higher redshifts, the method remains accurate, though with increasing uncertainty.
• The Poisson assumption underestimates the width of the probability distribution due to unaccounted sample variance, which the new method captures exactly.
• The method is computationally feasible for sparse surveys where the cluster distribution is approximately Gaussian and clustering is weak.
• The series expansion converges well when $\alpha x$ and $\alpha x^2$ are small, indicating a regime where the approximation is reliable.
• The formalism breaks down when $W_0$ becomes positive, signaling a physical inconsistency in Poisson sampling of a Gaussian random field, implying non-Gaussianity must emerge at high densities.