Selecting a mass function by way of the Bayesian Razor

Darell Moodley (UKZN), Dr. Kavilan Moodley (UKZN), Dr. Carolyn Sealfon

(WCU)

Overview

• Importance of galaxy clusters• The cluster mass function• Bayesian Statistics approach• Findings

What are Galaxy Clusters?

• Galaxy clusters are the largest known gravitationally bound objects to have arisen in the process of cosmic structure formation.

• Densest part of the large scale structure.• They provide an insight into structure

formation during the early universe and may also help us understand dark matter.

What is the Galaxy Cluster Mass Function?

• The cluster mass function n(m|θ) is the number density of galaxy clusters with mass greater than m.

• Assume constant redshift for all clusters.

• Transform n(m|θ) →F(ν|θ), where ν is dimensionless and θ is a set of parameters. The variable ν is given by,

• We have different mass functions:

Sheth-Tormen:

Press-Schechter:

Normalizable Tinker:

2))(1(2

1)|(

ap ea

aAF

2

2

1)(

eF

2/))()((2

1)|(

atb eaca

aBF

2

)(

mc

• Mass that is not detected in clusters is referred to as ‘dust.’

• Define a lower mass limit md with corresponding dimensionless νd for clusters.

• We wish to assign a probability distribution function for cluster masses.

• [p(being in dust)] x [p(being in clusters)]

c

d

N

jj

Ndust MpMpMp ),|(),(),|(

Bayesian Statistics Methodology

• To determine which theoretical framework or model is preferred.

• Bayesian Evidence asserts the merit of a model:

• Bayes Factor is the ratio of evidences for each model:

dMDpMpMDp ),|()|()|(

)|(

)|(

1

001 MDp

MDpB

likelihoodpriorEvidence

The Bayesian Razor• Is proportional to the expectation of the Evidence

• ‘D’ is the Kullback-Liebler distance.

• It is a measurement of the difference between a fiducial distribution t, and a given distribution u.

• Two types of prior distributions:

• Flat prior:

EvidenceepriorMR NDN )(

du

ttutDD

),(

)(ln)()),(||)((

1

)|( Mp

• Jeffreys prior:

• Where ,is the Fisher Information Matrix and describes the behaviour of the likelihood about its peak in the parameter space.

dMJ

MJ

ij

ij

)|(det

)|(det

jiij

LMJ

2

)|(

FindingsWe examine two scenarios. One in which we use the Sheth-Tormen model as our fiducial model and the other is when we use the Press-Schechter model.

We require at least 27 particles to discriminate strongly between models when the fiducial model is Sheth-Tormen.

More than 100 particles is required when Press-Schechter is the fiducial model.

Including dust also increases the number of particles required to sufficiently discriminate models from each other.

Changing the dust limitsThe General trend is that as we increase our dust limit, then we would need more particles in order to discriminate one model from the other.

Higher dust limit means less clusters. Hence not sufficient number of clusters for our mass function.

Changing the prior distributionsWe compare the razor ratio using a flat prior against a Jeffreys prior.

In both cases the razor ratio for a Jeffreys prior is less than the case of a flat prior.

Consider the Press-Schechter model as our fiducial model scenario. This implies the razor for the Sheth-Tormen model is greater for the Flat Prior case.

A prior distribution that is more informative is one in which the likelihood uses more of its volume.

In this case the likelihood uses more of the Flat prior volume than the Jeffreys prior which results in a higher razor.

Fiducial model: Press-Schechter

Fiducial model: Sheth-Tormen

Tinker against Sheth-TormenThe normalizable Tinker mass function has become quite popular in the cosmology community. It has been universally successful in describing simulations.

Dashed line represents the razor ratio for the Jeffreys prior case whereas the solid line is for the Flat prior.

The razor ratio favours the simpler model for small N which is mainly due to the extra parameters in the Tinker model down-weighting the razor. This is the Occam’s Razor effect that serves to penalise a more complicated model for the prior space that is not utilized.

The End

Thank you

References

• V. Balasubramanian. arxiv:adap-org/9601001, 1996

• M. Manera et al. arxiv:0906.1314, 2009• W. H. Press and P. Schechter. Astrophys. J.,

187:425-438, 1974.• Tinker et al. arxiv:0803.2706, 2008

Going from number of particles to number of clusters

• For a given survey volume

• Ignoring evolution effects,

• Number of clusters is given by,

dzz

NV m

)(

dd

dFm

NdFm

VN mh

)(

1)(

mN

V