Observational%20Test%20of%20Halo%20Model:

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Observational Test of Halo Model an empirical
approach
Mehri Torki
Bob Nichol
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The Halo-model of clustering
(lahmu.phyast.pitt.edu/sheth/courses/allahabad/ha
lomodel.ppt)

• Two types of pairs both particles in same halo,
  or particles in different halos
• ?dm(r) ?1h(r) ?2h(r)
• All physics can be decomposed similarly
  influences from within halo, versus from outside
  (Sheth 1996)

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Halo Model as a tool to extract Cosmology
Galaxies
Mass Size
n(M)
,
Cosmology (
)
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The SDSS-C4 Galaxy Cluster Catalogue
http//www.ctio.noao.edu/chrism/current/research/
C4/dr3

• Largest spectroscopic cluster catalogue ever
  made.
• Contains galaxy clusters found in
• the SDSS DR3 spectroscopic database.
• 1106 clusters.
• Clusters are found in a seven
• dimensional space.
• Galaxies within clusters are co-evolving.
• Thus, galaxies will not only cluster in position
  but also in colour.

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Group membership

• We examine 94795 galaxies.
• Redshift range of 0.03 lt z lt 0.13
• Using all the galaxies projected
• within
• And of the cluster centres.
• Absolute magnitude range of -24 lt lt -21.2
• Colour-cut
• we look at the radial profile of
  all the galaxies
• within
• of the red sequence
  for each cluster.

Z0.07
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Mass estimation

• Mass comes from the scaling relationship
  determined from the simulation presented in
  Miller et al. 2005
• (summed optical r-band luminosity ) is a
  powerful tool
• -superior to the galaxy line-of-sight
  velocity dispersion
• -or the richness
  

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Determination

• It is not possible to measure directly the radius
  at which
• cluster has a mass over density of
  
• measure space over
  density of
• is radius where mean number
  density of galaxies
  
• 200
  critical density
• Calculate for 1106 clusters in C4 by
  building the radial density profile of a certain
  mass and a certain r-band.
• Stack all the galaxies in 4 bins of mass.
• Determine for each bin of mass.

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Tests of

• Check value of the mean space density of field
  
• Check the effect of misidentifying the cluster
  centre
• Check values by using only good
  centres (von der Linden et al. in prep)
• Check for X-ray detection
• Check the colour constraint
• Check the fit to NFW profile

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Compare results

• Sheldon et al. (in prep.) have derived lensing
  profiles for clusters of galaxies in SDSS.

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Finn et al. (in prep.) have determined
as
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Halo Occupation Distribution(HOD)
Total galaxy occupancy of C4
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Halo Occupation Distribution
Collister Lahav 2004, Berlind Weinberg 2002
0
)
Red galaxies All galaxies
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Investigating HOD as a function of galaxy
properties
Red
All
Faint
Bright
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Summary Conclusion

• Motivated by halo model, we use C4 to make a
  direct and empirical determination of HOD from
  the known halos (clusters).
• Compared to recent lensing work by
  Sheldon et al. (in prep.) found remarkable
  agreement in size of radii.

• Found a good fit to our galaxy radial
  distribution provided by NFW.
• We have a stable HOD with respect to the colour
  luminosity.

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Future work

• Try to find an analytic equation for our mass
  function.
• Combine our HOD parameters with galaxy clustering
  measurements to better constrain cosmological
  parameters as and .
• Study HOD as a function of local environment.
• Compare HOD with other measurements of a cluster
  and group mass like X-ray parameters.
• Compare our results with the mock SDSS catalogue
  to ensure that the catalogues are a fair
  representation of the SDSS.
• Improve our results with latest SDSS and C4
  catalogues.
• Compare properties of galaxies as colour,
  luminosity, morphology for different HODs to see
  which properties of galaxies in a halo change?

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The Holy Grail
The Halo Grail
(phrase coined by Jasjeet Bagla!)
Halo model provides natural framework within
which to discuss, interpret most measures of
clustering it is the natural language of galaxy
bias
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Mass dependence
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Halo Model as a tool to extract cosmology
an empirical approach

• The model come up recently is the best used for
  the statistical analysis and understanding the
  large datasets as SDSS survey.
• All the mass in the universe is assumed to be
  allocated in individual units called haloes.
• Specifically provides the Halo Occupation
  Distribution (HOD) which is a function telling
  us how dark matter halos populate with galaxies.
• In contrast with the previous work which used the
  galaxy correlation functions to constrain HOD, we
  use known halos clusters of galaxies to
  determine HOD.
• Matter distribution can be studied in two steps
  the distribution of the mass within every halo
  and the spatial distribution of the haloes

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Luminosity Function
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Testing Luminosity Function

• We used the r-band LF of Blanton et al. (2003a)
  in order to derive the mean space density of
  field.
• We test if we get the same distribution in that
  band pass for our sample.
• We make this distribution for the absolute
  magnitude range of
• of the whole
  SDSS database.
• We find that for the galaxies in the absolute
  magnitude between -21 and -18 (as we go toward
  fainter galaxies), the number density of galaxies
  decrease.
• which is exactly where we are not complete!
• Our data is in redshift
  and
• Having considered -21.2 as our limit of
  completeness, there is no disagreement in the
  distribution we have achieved.
  
  
  
  

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Luminosity Function
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Testing value of the mean space density of field
We determine simply the value for N (number of
galaxies in DR3 in spectroscopic area of 4188
sq. deg in the )
divided by the volume of our chosen sample. N
94795 S Total sky area
F Fraction of sky covered 0.1 V1
Volume of the sphere in redshift 0.13 V2 Volume
of the sphere in redshift 0.03 V (V1-V2). F N
/ V 0.0042
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Test the effect of misidentifying the cluster
centre

• Check if we are in the right centre, otherwise it
  cause different radial profile and hence
  different value for .
• There are three methods for finding the cluster
  centre BCG, MEAN and GEOM cluster centroid
  measurements.
• BCG position of the brightest galaxy in the
  cluster, we think is best to use because this
  method is relied on observations that clusters
  host a population of early type galaxies with
  small dispersion in colour.
• MEAN coordinates of the galaxy with the highest
  density.
• GEOM luminosity weighted mean centroids,
  theses are cluster centres using all galaxies
  within 1 calculating a luminosity
  weighted average (in r-band) for RA and DEC of
  them.
• We find that by using other measurements of the
  cluster centroid there is no significant change
  in values of .

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• We also recalculate our estimates using
  the clusters with only
• good centres.
• For good centres we use the list of C4
  clusters with corrected BCG centres (von der
  Linden et al. in prep), they claimed that SDSS
  photometry of BCGs underestimates the flux and
  they correct for it.
• We use this list to remove the bad BCGs from
  C4.
• We find that there is no significant difference
  in our estimates of

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Test the colour constraint
In the algorithm used to identify the galaxies
around each cluster, we add this constraint in
the sense that galaxies are clustered in colour
space. We looked at the radial profile of all
galaxies within of the red sequence
for each cluster. We may miss some galaxies. By
relaxing this colour-cut to and
we evaluate the impact on the value of We
also vary -21.2 (limit of completeness) to
brighter fainter galaxies.
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for X-ray detections

• Calculating the virial radius is crucial for our
  work.
• The X-ray detection is very accurate to measure
  the radii.
• We match NORAS to C4 in order to find which
  cluster has X-ray detection, the X-ray selected
  clusters are taken from Bohringer et al. (2000).
• We find 40 overlapped clusters.
• With the same formalism explained before we
  derive

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Radial distribution of galaxies in groups

• We determine the projected galaxy density profile
  given mass from stacking groups scaled by their
  virial radius.
• Calculate the distance from cluster centre to
  each galaxy.
• Express them in units of (divide each
  distance to virial radius of each cluster).
• Stack them once in 4 bins of mass and then for
  the whole sample.
• Calculate the number of galaxies in radial bins
  divided by surface of each bin.
• Correct for the effect of fibre collision.

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Profile fitting

• NFW profile is described as the universal
  density profile expressed in terms of
  by the formula
• Best-fitting NFW concentration parameters are
• This means that the criteria used in C4
  clusters provides a good definition for the
  member galaxies and the clusters have the same
  shape with and without the colour-cut.

2.9 0.1
All galaxies
2.6 0.1
Red
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Z0.07
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0.38 0.44
0.46 0.49
0.55 0.56 0.61
0.71 0.73
0.79 0.94 0.96
0.42 0.46
0.47 0.54
0.59 0.60 0.69
0.75 0.76
0.84 0.97 0.99
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Scatter about HOD
Is it Poisson as has been assumed?
Expression by Gehrels (1986)
Mean value of the observed sigma over the
predicted one
for
for
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Summary Conclusion

• We found a good fit to our galaxy radial
  distribution provided by NFW profile and obtain c
  almost the same for galaxies with without the
  colour-cut.
• This makes us feel confident with criteria used
  in C4 clusters keep their shape.
• Analysing our HOD for all, red, bright faint
  galaxies shows that does not depend on the type
  of the galaxies
• Thus we have a stable HOD with respect to the
  colour luminosity.

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Take as empirical an approach as
possible.Directly measure the radial and HOD of
galaxies in the C4 catalogue.For investigating
our HOD , we need to calculate
mass and
sizeDirectly determine size-mass for clusters
with a model independent method.Stack systems in
bins measure the distribution of
galaxies in clusters over a wide range of masses

virial size
Measure HOD with SDSSC4
Provides the Halo Occupation Distribution (HOD)
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Tests of HOD

• Determine the projected galaxy density profile.

• NFW profile is described as the universal
  density profile expressed in terms of
  by
• Best-fitting NFW concentration parameters are
• This means the criteria used in C4 clusters
  provides a good definition for the member
  galaxies
• clusters have the same shape with and without
  the colour-cut.

All galaxies
2.9 0.1
Red galaxies
2.6 0.1
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Measuring mass distribution

• An important cosmological aim is to constrain
• , its average density
• , amplitude of its power spectrum
  p(k)
  
• More formally we want to know what the halo mass
  function looks like in cosmology.
• Following halo model formalism, apply it to the
  C4 Catalogue using SDSS data set.

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X-ray
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X-ray
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Tests of

• Check luminosity function
• Check value of the mean space density of field
• Check the effect of misidentifying the cluster
  centre
• Check values by using only good
  centres (von der Linden et al. in prep)
• Check for X-ray detection
• Check the colour constraint
• Check the fit to NFW profile

44
Summary Conclusion

• Motivated by halo model, we use C4 to make a
  direct and empirical
  determination of
  HOD from the known halos (clusters).
• We have tested this extensively.
• Compared to recent lensing work by
  Sheldon et al. (2006) found remarkable
  agreement in size of radii.

• Found a good fit to our galaxy radial
  distribution provided by NFW.
• This makes us feel confident with criteria used
  in C4.
• We have a stable HOD with respect to the colour
  luminosity.

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