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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,
2
  
   c 
  ( m) 
•
We have different mass functions:
 Sheth-Tormen:
 Press-Schechter:
 Normalizable Tinker:
 a
a
p
F ( |  )  A
(1  (a ) )e 2

2

1
F ( ) 

1

F ( |  ) 
1

 2
e
2
B
a
(( a ) b  c(a ) t )e a / 2
2
• 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.
Nc


 
Nd
p( |  , M )  pdust ( , M )  p( j |  , M )
j
• [p(being in dust)] x [p(being in clusters)]
Bayesian Statistics Methodology
• To determine which theoretical framework or model is
preferred.
• Bayesian Evidence
asserts the merit of a model:



p( D | M )   p( | M )  p( D |  , M )d
Evidence 
• Bayes Factor
model:
 prior  likelihood
is the ratio of evidences for each
B01 
p( D | M 0 )
p( D | M 1 )
The Bayesian Razor
• Is proportional to the expectation of the Evidence
RN ( M )   prior  e  ND  Evidence
• ‘D’ is the Kullback-Liebler distance.

D  D(t ( ) || u ( ,  ))   t ( ) ln
t ( )
 d
u ( ,  )
• It is a measurement of the difference between a fiducial
distribution t, and a given distribution u.
• Two types of prior distributions:
• Flat prior:

1
p( | M )  

• Jeffreys prior: 






det J ij ( | M )


det J ij ( | M ) d
 2L
J ij ( | M ) 
 i  j
• Where
,is the Fisher Information
Matrix and describes the behaviour of the
likelihood about its peak in the parameter
space.
Findings
We 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 ShethTormen.
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 limits
The 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.
Fiducial model: PressSchechter
Changing the prior
distributions
We 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: ShethTormen
Tinker against ShethTormen
The 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
V  N
m
dz
 ( z)
• Ignoring evolution effects,
V 
N m

• Number of clusters is given by,



1
N h  V  F ( )d  N m  F ( )d
d m
d m