COPING WITH SCATTER IN GALAXY CLUSTER SCALING RELATIONSHIPS
By
David A. Ventimiglia

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Astrophysics and Astronomy
2012

ABSTRACT
COPING WITH SCATTER IN GALAXY CLUSTER SCALING
RELATIONSHIPS
By
David A. Ventimiglia
Galaxy clusters exhibit regular scaling relations among their bulk properties, establishing vital links between halo mass and cluster observables. Therefore, precision cosmology
studies depending on these links benefit from a better understanding of scatter in the massobservable scaling relations. Here, we study the role of merger processes in introducing
scatter into the mass-temperature (M-TX ) relation, using a sample of 121 galaxy clusters
simulated with radiative cooling and supernova feedback, along with three morphological
statistics previously proposed to measure X-ray surface brightness substructure. These are
the centroid variation (w), the axial ratio (η), and the power ratios (P20 ) and P30 ). We
also examine spectral statistics which do not require imaging. In particular we look at the
ratio of hardband to broadband spectral-fit temperatures THBR , which is a measure of temperature inhomogeneity inferred from X-ray spectral properties. We find that in this set of
simulated clusters, each substructure measure is correlated with a cluster’s departures δ ln T
and δ ln M from the mean M-TX relation, both for emission-weighted temperatures TEW
and for spectroscopic-like temperatures TSL , in the sense that clusters with more substructure tend to be cooler at a given halo mass. In all cases, a three-parameter fit to the M-TX
relation that includes substructure information has less scatter than a two-parameter fit to
the basic M-TX relation. In this work we also test the effectiveness of THBR as a measure
of scatter using the 118 galaxy clusters from our sample for which we have simulated X-ray
images using the X-MAS software. We find that, while THBR is correlated with clusters’

departures δ ln M from the mean M-TX relation, the effect is modest. Finally, we present
the CosmoSurvey software for efficiently evaluating simple flux-limited galaxy cluster survey
models on commodity hardware, and discuss possible extensions.

To my parents, Patricia Jane and Phillip Anthony Ventimiglia, whose love and sacrifice
are the source of their children’s accomplishments.

iv

ACKNOWLEDGMENTS

I would like to thank my advisor, Dr. G. Mark Voit, for his wisdom, his humor, and his
incredible patience. I could not have asked for a better mentor, and whatever credit derives
from this small accomplishment goes to him. Thanks, Mark.

v

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Substructure and Scatter in the Mass-Temperature Relations
of Simulated Clusters . . . . . . . . . . . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Mass-Temperature Relation in Simulated Clusters . . . . . . . . . . . . . . .
Quantifying Substructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Axial Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Power Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3 Centroid Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.4 Substructure and Scaling Relationships . . . . . . . . . . . . . . . . .
2.3.5 Comparisons with Other Substructure Studies . . . . . . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6
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Chapter 1
Chapter 2
2.1
2.2
2.3

2.4

Chapter 3
3.1
3.2

3.3

3.4

3.5

Temperature Structure and Mass-Temperature Scatter in Galaxy
Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 X-Ray Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.3 M-TX Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Cool Lump Excision . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.3 Spectral Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.4 Quantifying Temperature Structure . . . . . . . . . . . . . . . . . . .
Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 THBR from Simulations. . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.2 Temperature Structure and Scaling Relations . . . . . . . . . . . . .
3.4.3 Effects of Masking Strategy . . . . . . . . . . . . . . . . . . . . . . .
3.4.4 Substructure Measures and THBR . . . . . . . . . . . . . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vi

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Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

98

Chapter 4 Modeling Galaxy Cluster Surveys
4.1 Introduction . . . . . . . . . . . . . . . . . .
4.2 CosmoSurvey . . . . . . . . . . . . . . . . .
4.2.1 Design . . . . . . . . . . . . . . . . .
4.2.1.1 Goals . . . . . . . . . . . .
4.2.1.2 Principles . . . . . . . . . .
4.2.1.3 Architecture . . . . . . . . .
4.2.1.4 Modules . . . . . . . . . . .
4.2.1.5 Trade-offs . . . . . . . . . .
4.2.1.6 Cosmological Model . . . .
4.2.1.7 Cluster Model . . . . . . . .
4.2.1.8 Survey Model . . . . . . . .
4.3 Summary . . . . . . . . . . . . . . . . . . .
Chapter 5

APPENDICES . . . . . . . . . . . . . . .
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Appendix B CosmoFisher Installation Guide
B.1 CosmoFisher Quick Start . . . . . . . . . .
B.1.1 Requirements . . . . . . . . . . . .
B.1.2 Download . . . . . . . . . . . . . .
B.1.3 Install . . . . . . . . . . . . . . . .

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vii

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Appendix A CosmoSurvey User Guide
A.1 Quick Start . . . . . . . . . . . . .
A.1.1 Requirements . . . . . . . .
A.1.2 Download . . . . . . . . . .
A.1.3 Install . . . . . . . . . . . .
A.1.4 Run . . . . . . . . . . . . .
A.2 Simulating A Survey . . . . . . . .
A.3 Visualizing A Survey . . . . . . . .
A.4 Fortran API . . . . . . . . . . . . .
A.4.1 quadpack . . . . . . . . . .
A.4.2 quadrature . . . . . . . . . .
A.4.3 utils . . . . . . . . . . . . .
A.4.3.1 Subprograms . . .
A.4.4 constants . . . . . . . . . .
A.4.4.1 Data . . . . . . . .
A.4.5 cosmology . . . . . . . . . .
A.4.5.1 Subprograms . . .
A.4.6 structure . . . . . . . . . . .

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Bibliography . . . . . . . . . . . . . . .

viii

. . . . . . . . . . . . . . . . . . . 117

LIST OF TABLES

Table 3.1

THBR & σT
with a All Clusters and b Clusters with kB T2.0−7 > 2
HBR
keV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

71

LIST OF FIGURES

Figure 2.1

Figure 2.2

Figure 2.3

Figure 2.4

Figure 2.5

Figure 2.6
Figure 2.7

Mass-temperature (M-TX ) relationships for the 121 clusters in our
sample. Spectroscopic-like temperature TSL is plotted in the top
panel, while emission-weighted temperature TEW is plotted in the
bottom panel. The solid lines show the best-fit power-law relation
M = M0 (TX /3 keV)α over the whole sample, while the dashed lines
show the best-fit relation for systems with TX > 2 keV. Open circles
represent the clusters that have the greatest positive temperature offset from the mean relationship, and filled circles represent the clusters
with the greatest negative temperature offset. . . . . . . . . . . . . .

12

Surface-brightness contour maps for sixteen of the clusters in our
sample, overlaid with a circular aperture of radius R500 . The top
panel has eight maps representing the clusters that have the largest
positive deviation δ ln TX from the mean M-TX relation, calculated
using the spectroscopic-like temperature TSL . . . . . . . . . . . . . .

14

Surface-brightness contour plot of an asymmetric cluster which, for
certain choices of aperture placement, yields an axial ratio close to 1.
The circle represents an aperture of R500 . . . . . . . . . . . . . . .

18

Surface of axial ratio η as a two-dimensional function of the coordinates of the aperture center. The axial ratio statistic appears to be
ill-defined for this cluster. . . . . . . . . . . . . . . . . . . . . . . . .

19

Abstract surface of axial ratio η as a two-dimensional function of the
coordinates of the aperture center. This is a relaxed cluster, for which
the axial ratio is better-defined. . . . . . . . . . . . . . . . . . . . .

20

Relationship between a cluster’s deviation δ ln TX from the mean MTX relationship and four measures of substructure. . . . . . . . . . .
Comparison of mass offset δ ln M between true mass and predicted
mass, based on the M(TX ) relation. Plus signs indicate residuals for
masses estimated from the M-TX relation derived from all 121 clusters, while black filled circles are for masses estimated from the M-TX
relation for clusters above 2 keV. Upper panels are for spectroscopiclike temperature and lower panels are for emission-weighted temperature. The standard deviations σ in the residuals are given in each
plot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
x

24

27

Figure 2.8

Figure 2.9

Figure 2.10

Figure 3.1

Figure 3.2

Figure 3.3

Relationship between substructure and mass offset δ ln M(TX ) from
the mean M-TX relationship for the same substructure measures as
in the previous figure. The lower panels are for TX = TSL , and the
upper panels are for TX = TEW . The solid lines indicate the bestfitting linear relationships to the above 2 keV sub-sample denoted by
black filled circles, and the gray bands and vertical dashed lines mark
the extremes in substructure between which 80% of our clusters lie.
The plus signs correspond to systems below 2 keV, while the black
filled circles correspond to systems above 2 keV. . . . . . . . . . . .
Comparison of mass offset δ ln M between true mass and predicted
mass, based on the M(TX ) relation (open circles) and the M(TX , S)
relation (filled circles). Upper panels are for TSL and lower panels
are for TEW . The standard deviations σ in the residuals are given in
each plot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Comparison of mass offset δ ln M between true mass and predicted
mass, based on the M(TX ) relation (open circles) and the M(TX , S)
relation (filled circles). These results are for are for TEW , with the
central 0.15 R500 region removed both from the average temperature
and from the substructure measures. The standard deviations σ in
the residuals are given in each plot. . . . . . . . . . . . . . . . . . .
X-MAS simulated counts spectrum for a simulated cluster that appears to have significant temperature structure. A single-temperature
MEKAL model fit to the [0.7–7.0] keV broad band is over-plotted as
the solid line, with kB T = 2.42 ± 0.03 keV. Fit residuals appear in
the bottom panel. This figure illustrates qualitatively the effect that
additional cool components have on a single-temperature fit. . . . .
X-MAS simulated counts spectrum for the same simulated cluster as
in the previous figure. A single-temperature MEKAL model fit to
the [2.0–7.0] keV hard band is over-plotted as the solid line, with
kB T = 3.52 ± 0.2 keV. Fit residuals appear in the bottom panel. . .
Mass-temperature (M-TX ) relation for the clusters in our sample.
Clusters with kB T < 2 keV are plotted in purple open triangles, while
clusters with kB T ≥ 2 keV are plotted in blue open squares, and the
four massive D09 clusters in green open circles. Mean relations are
plotted with solid lines for the whole sample, and dashed lines for
clusters with kB T ≥ 2keV. Finally, note that average temperatures
are taken within a core-excised annulus whose outer radius is placed
at R2500. For interpretation of the references to color in this and all
other figures, the reader is referred to the electronic version of this
dissertation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

28

35

35

40

41

46

Figure 3.4

Figure 3.5

Figure 3.6

Figure 3.7

X-MAS simulated X-ray surface-brightness images for the same cluster whose spectrum is presented in the previous figure. The aperture
is set at R2500 and a core region of size 0.15R2500 is excised. Note
the surface-brightness peaks in the image. In this study these are
detected automatically by the CIAO tool wavdetect, which generates
source regions identified in this figure by red ellipses. Labeled “cool
lumps” in this analysis, they are subjected to varying degrees of excision, from no masking (left) to full masking (right), as described in
the text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
THBR plotted against T0.7−7 for simulated X-MAS observations that
have approximately 120K counts. Hydrodynamic simulations may
produce spurious over-condensations of cool gas. These cool lumps
can be excised using the CIAO tool wavdetect, whose aggressiveness
can be controlled via its ellsigma parameter. Squares correspond to
single-temperature MEKAL fits whose underlying observations are
processed by wavdetect with ellsigma set to 3, and circles correspond
to those with ellsigma set to 0. Finally, note that the 4 massive D09
clusters are denoted by filled symbols rather than by open symbols.
The solid line represents the mean when cool lumps are not removed,
while the dashed line represents the mean when they are removed.
The dotted line indicates THBR = 1. . . . . . . . . . . . . . . . . . .
REScool plotted against T0.7−7 for simulated X-MAS observations
that have approximately 120K counts. This statistic tracks the percent excess count rate of the actual observation with respect to count
rate for a MEKAL model fit to just the hard band. Squares correspond to single-temperature MEKAL fits whose underlying observations are processed by wavdetect with ellsigma set to 3, and circles
correspond to those with ellsigma set to 0. As in the previous plot,
the 4 massive D09 clusters are denoted by filled symbols rather than
by open symbols. The solid line represents the mean when cool lumps
are not removed, the dashed line represents the mean when they are
removed, and the dotted line indicates zero residual. . . . . . . . . .
Relationship between THBR and mass offset δ ln M(TX ) from the
mean M-TX relationship for the clusters in combined B04+D09 simulation sample. The temperatures in this relation are our broadband
spectral-fit temperatures. Squares correspond to simulated X-MAS
observations that are processed by wavdetect with ellsigma set to
3 (full masking), while circles correspond to observations that have
ellsigma set to 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xii

49

54

56

58

Figure 3.8

Figure 3.9

Figure 3.10

Figure 3.11

Figure 3.12

Relationship between REScool and mass offset δ ln M(TX ) from the
mean M-TX relationship for the clusters in combined B04+D09 simulation sample. The temperatures in this relation are our broadband
spectral-fit temperatures. Squares correspond to simulated X-MAS
observations that are processed by wavdetect with ellsigma set to 3,
while circles correspond to observations that have ellsigma set to 0. .

59

(top) Decline in the THBR standard deviation σTHBR as wavdetect’s
ellsigma parameter ranges from 0 to 3. These are plotted for a family
of simulated observations of increasingly higher quality, from approximately 15k counts to approximately 120k counts. (bottom) Decline
in σTHBR as simulated X-MAS observations go from lowest-quality
(approximately 15k counts) to highest-quality (approximately 120k
counts). These are plotted for a family of simulated observations
with wavdetect’s ellsigma parameter ranges from 0 to 3. . . . . . . .

61

Relative change in spectral fit temperatures as the cool lumps are removed from the simulated clusters for high-quality simulated observations. (top) Observations with approximately 120K counts. (bottom)
Observations with approximately 15K counts. Note that these are
for observations in an aperture corresponding to R2500 . . . . . . . .

63

(Top) Shift in the T0.7−7-THBR plane as wavdetect’s ellsigma parameter ranges from 0 to 3. Clusters from the B04 sample are depicted
with solid arrows, while clusters from the D09 sample are depicted
with long-dashed arrows. broadband spectral-fit temperatures are
estimated within an aperture set at R2500. Notice that the distribution of most of the points shifts significantly to higher spectral
fit temperatures and smaller temperature ratios as cool lumps are
excised. Notice also that some simulated clusters are unafflicted by
cool lumps, so that their net shift is 0. (Bottom) The THBR component (y-axis) of the shift presented in the top panel. Note that these
are for simulated X-MAS observations of maximum quality, having
approximately 120K counts. . . . . . . . . . . . . . . . . . . . . . .

65

Relationship between THBR and four measures of substructure: the
axial ratio η, the centroid variation w, and the power ratios P20 and
P20 . Squares correspond to simulated X-MAS observations that are
processed by wavdetect with ellsigma set to 3 (full masking), while
circles correspond to observations that have ellsigma set to 0. . . . .

67

xiii

Figure 3.13

Surface-brightness contours for four clusters drawn from our simulation sample, illustrating that temperature ratio and centroid shift are
both imperfect measures of relaxation. All four clusters are at nearly
the same temperature, with kBT 3.2. The two in the left column
exhibit little centroid shift w, while the two in the right column have
centroid shift near the maximum for the sample. The two clusters
in the bottom row have low THBR while the two in the top row
have higher THBR suggesting the presence of multiple temperature
components. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

Figure 4.1

Growth factor G(z) for the growth of density fluctuations. . . . . . .

87

Figure 4.2

Differential mass function for the number density of collapsed Dark
Matter halos, at three different redshifts . . . . . . . . . . . . . . . .

89

Flux limit calculation used to cut off the integration of galaxy cluster
number counts over a luminosity bin. . . . . . . . . . . . . . . . . .

95

CosmoSurvey simulated cluster survey counts in observable bins, obtained using the cosmoplot program. . . . . . . . . . . . . . . . . . .

96

Figure 4.3

Figure 4.4

Figure A.1

CosmoSurvey simulated cluster survey counts in observable bins, obtained using the cosmoplot program. . . . . . . . . . . . . . . . . . . 108

xiv

Chapter 1
Introduction
Astronomers have known about clusters of galaxies for almost as long as they have known
about galaxies. Though of course its roots stretch much deeper into the past, one could
defensibly locate the birth of modern astronomy, like much of the physical sciences, in the
Enlightenment of the eighteenth century. And by the end of that century astronomers like
Charles Messier and William Herschel had noticed that galaxies, whatever they were, were
concentrated in particular regions of the sky. As astronomy was placed on firm observational
grounds in the nineteenth and twentieth centuries, galaxy clusters were added to the list of
exotic celestial beasts to be cataloged, indexed, and studied. In the modern era, George
Abell [Abell(1958)] defined what it means to identify a galaxy cluster, how to catalog them,
and how to describe their relevant properties. Of course this was the era when “optical astronomy” was astronomy, so understandably the key observational property Abell identified
for galaxy clusters was their “optical richness”, i.e. simple a number count of the galaxies
in a cluster, within certain constraints. Since then, galaxy clusters have been observed and
cataloged in a variety of other ways, and especially in the X-ray band.
With the first generation of space-based astronomical observing platforms in the late
1960s and early 1970s, it became possible to see astrophysical phenomena in wave-bands
that are absorbed by the Earth’s atmosphere. The UHURU satellite (1970) was the first
X-ray observing satellite, and quickly it was used to identify luminous diffuse X-ray emission
from galaxy clusters [Forman et al.(1972)]. Since then, space-based X-ray observations have
1

become one of the most productive ways to measure the important properties of galaxy
clusters. This is especially true since the X-ray emission now is identified with thermal
emission from a hot diffuse plasma that envelopes a galaxy cluster and dominates its nonDark Matter mass.
Nowadays, galaxy clusters are being observed in greater and greater detail in the X-ray
and other bands, and their numbers are being counted in surveys with greater and greater
reach. They now provide one of the pillars of modern observational cosmology and will be
critical in understanding Dark Energy. Dark Energy is the mysterious form of energy that
dominates the contents of the Universe at the present epoch and is driving an accelerating
Universal expansion. The characteristics of Dark Energy—its abundance, its evolution,
and its equation of state, for instance—modulate the growth of structure in the Universe,
and in particular, the Dark Matter halos that are the sites of galaxy and galaxy cluster
formation. In this way, the population characteristics of galaxy clusters, especially their
distribution in mass and that distribution’s evolution with cosmic time, are critically linked
to global properties of the Universe. This is why observing and measuring galaxy clusters is
so important and exciting to modern cosmology.
A key feature in any such endeavors is measuring the mass of galaxy clusters. Like
distance, measuring the mass of any astrophysical object is often a dicey proposition, and it
is especially true of galaxy clusters. Like we do with other objects, we must infer a galaxy
cluster’s mass by relating it to its observable properties, such as its overall luminosity, the
velocity dispersion of its galaxies, etc. While the theoretical understanding of the structure
and dynamics of all the components of a galaxy cluster is indeed an exciting and important
area of research, the corollary is that our understanding of the ways their properties relate
is still evolving. Typically, we rely on simple scaling relationships between the observable
2

properties of a galaxy cluster and its mass, bolstered by a theoretical foundation which, while
incomplete, supports and explains the existence of these scaling relationships. The details
we elide in this process inevitably introduce the notion of “scatter”.
At any given mass, observed clusters will be scattered across a distribution in any particular observable property, such as luminosity. Naturally, some of that is introduced in the
act of observing. What this work focuses on is intrinsic scatter, which arises in part from the
deficit in our understanding of how galaxy formation occurs, how clusters are assembled, and
what role feedback processes play. As our models increase in sophistication and accuracy,
intrinsic scatter simply becomes a feature of those models, but some always remains.
In this work, we look at two related ways of coping with scatter in galaxy cluster scaling
relationships. We also present software to aid in the modeling of scatter in these scaling
relationships and the effects it has on the kinds of large surveys of galaxy clusters that are
being developed.
Chapter 2 of this work presents the results of a research program designed to evaluate the efficacy of galaxy cluster substructure in measuring scatter. Galaxy clusters come
in a wide variety of shapes, ranging from relaxed spheres to very irregular aggregations of
knots, lumps, and filaments, exhibiting the effects of energetic merging events and feedback processes. We adopt several methods of quantifying the morphological irregularity—or
“substructure”—of galaxy clusters and apply it to a sample selected from a large-volume
hydrodynamical simulation of a portion of the Universe. The particular component of the
cluster whose substructure we are observing is the surface-brightness of the X-ray emission
from the luminous intra-cluster medium (ICM). The virtue of simulations in this context is
that we know the mass of the galaxy clusters precisely. In this controlled environment, we
then measure the effectiveness of substructure as an approach to obtain more accurate mass
3

estimates in cluster surveys. Chapter 2 is adapted from [Ventimiglia et al.(2008)], where the
results of this research program first were published by this author.
Chapter 3 of this work presents the results of another, similar research program, designed
to evaluate the efficacy of galaxy cluster temperature inhomogeneity in measuring scatter.
As introduced above and discussed in subsequent chapters, the intra-cluster medium (ICM)
is the diffuse, hot plasma that permeates a galaxy cluster and dominates its non-Dark Matter
mass. While in chapter 2 we measure the morphological substructure of this gas, via imaging of its X-ray emission, in chapter 3 we measure the thermal substructure of this gas, via
spectroscopy of that X-ray emission. This provides another, complementary way of, in principle, measuring and correcting for scatter in mass-observable scaling relationships. Again,
we use the same sample of simulated clusters as in chapter 2, adopting several methods of
quantifying thermal irregularity—or “inhomogeneity”—of galaxy clusters. We then measure
their effectiveness in obtaining more accurate galaxy cluster mass estimates. Chapter 3 also
is adapted from a journal article by this author [Ventimiglia et al.(2012)].
Chapter 4 of this work presents software for modeling the kinds of flux-limited large-area
galaxy cluster surveys upon which the corrections discussed in the previous chapters would
be brought to bear. Techniques for measuring and correcting for scatter are important, but
it also is important to understand the effect that scatter ultimately will have on the cosmological goals of the planned surveys. After all, with an appropriate model for scatter and
with large enough number statistics, scaling-relationship scatter could be “self-calibrated” in
the model without resorting to other observables like substructure or temperature inhomogeneity. Making those judgments will require extensive modeling, numerical simulations, the
application of Monte Carlo Markov Chain (MCMC) methods, etc. And that, in turn, will
require fast, efficient, lightweight galaxy cluster survey models that are easily adapted. We
4

present one first step in this direction, with the CosmoSurvey program and library. Chapter
4 largely deals with the theoretical underpinnings of CosmoSurvey’s cosmological, structure,
and survey models, and can serve in someways as a general introduction to these concepts.
Finally, chapter 5 summarizes the results of this work and introduces possible next steps.

5

Chapter 2
Substructure and Scatter in the
Mass-Temperature Relations of
Simulated Clusters

2.1

Introduction

Clusters of galaxies play a critical role in our understanding of the Universe and its history and are potentially powerful tools for conducting precision cosmology. For example,
large cluster surveys can discriminate between cosmological models with different darkenergy equations of state by providing complementary observations of the shape of the
cluster mass function, evolution in the number density of clusters with redshift, and bias in
the spatial distribution of clusters [Wang & Steinhardt(1998), Levine et al.(2002), Hu(2003),
Majumdar & Mohr(2004), Voit(2005)]. However, this potential to put tight constraints on
the properties of dark energy will be realized only if we can accurately measure the masses
of clusters and can precisely characterize the scatter in our mass measurements.
Scatter in X-ray cluster properties is directly related to substructure in the intracluster
medium. If clusters were all structurally similar, then there would be a one-to-one relationship between halo mass and any given observable property. Generally speaking, deviations

6

from a mean mass-observable relationship are attributed to structural differences among
clusters. One kind of structural difference is the presence or absence of a cool core, in
which the central cooling time is less than the Hubble time at the cluster’s redshift, and
the prominence of a cool core is observed to be a source of scatter in scaling relationships
[Fabian et al.(1994)]. We also expect structural differences to arise from substructure in
the dark matter, galaxy, and gas distributions. For instance, there may be a spread in
halo concentration at a given mass, variations in the incidence of gas clumps, differences in
the level of AGN feedback, or various effects due to mergers. All of these deviations can
be considered forms of substructure that produce scatter in the mass-observable relations
one would like to use for cosmological purposes. While it may ultimately be possible to
constrain the amount of scatter and its evolution with redshift using self-calibration techniques [Lima & Hu(2005)], such constraints would be improved by prior knowledge about
the relationship between scatter and substructure.
Traditionally, the most worrisome form of substructure has been that due to the effects
of merger events. Clusters are often identified as “relaxed” or “unrelaxed”, with the former
assumed to be nearly in hydrostatic equilibrium and the latter suspected of being far from
equilibrium. Cosmological simulations of clusters indicate that the truth is somewhere in
between. The cluster population as a whole appears to follow well-defined virial relations
with log-normal scatter around the mean, showing that clusters do not cleanly separate into
relaxed and unrelaxed systems [Evrard et al.(2008)]. Even the most relaxed-looking clusters
are not quite in hydrostatic equilibrium [Kravtsov et al.(2006)]. Instead of simply being
“relaxed” or “unrelaxed,” clusters occupy a continuum of relaxation levels determined by
their recent mass-accretion history.
Quantifying this continuum of relaxation offers opportunities for reducing scatter in the
7

mass-observable relations. If mergers are indeed responsible for much of the observed scatter
around a given scaling relation, then there may be correlations between a cluster’s morphology and its degree of deviation from the mean relation. Once one identifies a morphological parameter that correlates with the degree of deviation, one can construct a new
mass-observable relation with less scatter by including the morphological parameter in the
relation. Such an approach would be analogous to the improvement of Type Ia supernovae
as distance indicators by using light-curve stretch as a second parameter to indicate the
supernova’s luminosity [Phillips(1993), Riess et al.(1996)].
Here we investigate how merger-related substructure in simulated clusters affects the relationship between a simulated cluster’s mass and the temperature of its intracluster medium,
building upon [Buote & Tsai(1995)] and [O’Hara et al.(2006a)]. [Buote & Tsai(1995)] quantified the morphologies and dynamical states of observed clusters and found structure measures to be an indicator of the dynamical state of a cluster. [O’Hara et al.(2006a)] also
examined morphological measurements, for both observed and simulated clusters, and found
that simulations without cooling showed no correlation between substructure and scaling
relation scatter. In this work we examine substructure for simulated clusters with radiative
cooling and focus on the idea that merger processes introduce intrinsic scatter into the M-TX
relationship by displacing clusters in the M-TX plane away from the mean X-ray temperature
TX |M at a given mass M, either to higher or lower average ICM temperature. We then
adopt a set of statistics [Buote & Tsai(1995), O’Hara et al.(2006a)] for quantifying galaxy
cluster substructure and merger activity in order to investigate this hypothesis. Section 2.2
discusses the M-TX scaling relationship in our sample of simulated clusters and shows that
disrupted-looking clusters in this sample tend to be cooler at a given cluster mass. In Section
2.3 we attempt to quantify the relationship between morphology and temperature using four
8

different substructure statistics and compare it to similar studies. We then show that substructure in these simulated clusters indeed correlates with scatter in the M-TX relationship
and assess the prospects for using that correlation to reduce scatter in the M-TX plane.
Section 2.4 summarizes our results.

2.2

Mass-Temperature Relation in Simulated Clusters

This study is based on an analysis of 121 clusters simulated using the cosmological hydrodynamics TREE+SPH code GADGET-2 [Springel(2005)], which were simulated in a standard
Dark Energy + cold dark matter (ΛCDM) universe with matter density ΩM = 0.3, h = 0.7,
Ωb = 0.04, and σ8 = 0.8. These are key cosmological parameters that bear directly on the
growth of structure. The parameter h maps directly to the Hubble Constant H0 , and merely
is the scaling of that value to 100 km/s/Mpc. The dimensionless parameter ΩM , which
accounts for the abundance of matter (both Dark Matter and ordinary baryonic matter) in
the Universe, is the ratio of the matter density ρM to the critical density ρc . The critical
density is the density of mass-energy that is needed geometrically to close the Universe via
General Relativity and in this setting mainly provides a convenient scale factor for comparing
the relative abundance of its various ingredients. Similarly, Ωb is the density parameter for
ordinary baryonic matter. Finally, σ8 provides the normalization for the scale-free spectrum
of in density fluctuations imprinted in the early Universe and that were the seeds of structure
growth. Specifically, σ8 is the variance in density contrast δρ/ρ for density fluctutions with a
Gaussian distribution, sampled at a length scale of 8 Mpc (projected to the present epoch).
The simulation treats radiative cooling with an optically-thin gas of primordial composition, includes a time-dependent UV background from a population of quasars, and handles

9

star formation and supernova feedback using a two-phase fluid model with cold star-forming
clouds embedded in a hot medium. All but four of the clusters are from the simulation
described in [Borgani et al.(2004)], who simulated a box 192 h−1 Mpc on a side, with 4803
dark matter particles and an equal number of gas particles. The present analysis considers
the 117 most massive clusters within this box at z = 0, which all have M200 greater than
5 × 1013 h−1 M⊙ . By convention, M∆ refers to the mass contained in a sphere which has a
mean density of ∆ times the critical density ρc , and whose radius is denoted by R∆
That cluster set covers the ∼1.5-5 keV temperature range, but the 192 h−1 Mpc box is
too small to contain significantly hotter clusters. We therefore supplemented it with four
clusters with masses > 1015 h−1 M⊙ and temperatures > 5 keV drawn from a dark-matteronly simulation in a larger 479 h−1 Mpc box [Dolag & Stasyszyn(2009)]. The cosmology
for this simulation also was ΛCDM, but with σ8 = 0.9. These were then re-simulated
including hydrodynamics, radiative cooling, and star formation, again with GADGET-2 and
using the zoomed-initial-conditions technique of [Tormen(1997)], with a fourfold increase in
resolution. This is comparable to the resolution of the clusters in the smaller box. Adding
these four massive clusters to our sample gives a total of 121 clusters with M200 in the
interval 5 × 1013 h−1 M⊙ to 2 × 1015 h−1 M⊙ .
We first need to specify our definitions for mass and temperature. In this chapter,
cluster mass refers to M200 . For temperature, we use two definitions. The first is the
emission-weighted temperature

TEW =

T [n2 Λ(T )] d3 x
,
n2 Λ(T ) d3x

(2.1)

where n is the electron number density and Λ(T ) is the usual cooling function for intracluster

10

plasma. The second is the spectroscopic-like temperature of [Mazzotta et al.(2004)],

TSL =

T [n2 T −3/4] d3 x
,
n2 T −3/4 d3 x

(2.2)

where the power-law weighting function replacing Λ(T ) is chosen so that TSL approximates
as closely as possible the temperature that would be determined from fitting a singletemperature plasma emission model to the integrated spectrum of the intracluster medium.
The presence of metals in the ICM of real clusters introduces line emission that complicates the computation of TSL for clusters <3 keV [Vikhlinin(2006)]. However, the simulated
spectra for the clusters in our sample are modeled with zero metallicity, which eases this
restriction in our analysis.
Figure 2.1 shows the mass-temperature relations based on these definitions for our sample
of simulated clusters. The best fits to the power-law form

M = M0

TX α
3 keV

(2.3)

have the coefficients M0 ≃ 5.9×1013 h−1 M⊙ , α ≃ 1.5 for TX corresponding to TSL and M0 ≃
4.4 × 1013 h−1 M⊙ , α ≃ 1.4 for TX corresponding to TEW . As is generally the case for simulated clusters, the power-law indices of the mass-temperature relations found here are consistent with cluster self-similarity and the virial theorem [Kaiser(1986), Navarro et al.(1995)].
These relationships have scatter, which we characterize by the standard deviation in log
space σln M about the best-fit mass at fixed temperature TX . When relating M to the
emission-weighted temperature TEW , we find σln M = 0.102. When relating cluster mass M
to the spectroscopic-like temperature TSL , the scatter is σln M = 0.127. That the scatter is

11

larger for the spectroscopic-like temperature is not surprising, given the sensitivity of TSL to
the thermal complexity of the ICM.
Figure 2.1 also highlights two sub-samples for each definition of temperature, selected
based on the clusters’ deviations in ln TX space from the mean mass-temperature relation. In
each panel, open circles represent the eight clusters that have the largest positive deviations
and are therefore “hotter” than other clusters of the same mass, while filled circles represent
the eight with the largest negative deviation and are “cooler” than other clusters of the same
mass. In general, these temperature estimates are well correlated, so that hotter outliers in
TEW are also hotter outliers in TSL and cooler outliers in TEW are also cooler outliers in TSL .
Since Figure 2.1 distinguishes the most extreme outliers for the two temperature estimates,
this distinction may define slightly different sets, though they still overlap.
Figure 2.2 presents a gallery of surface brightness maps for two sets of eight clusters with
the most extreme offsets from the mean M-TSL relation. The eight unusually hot clusters
are in the top panel, and the eight cooler clusters are in the bottom panel. In these plots the
hotter clusters appear more symmetric, and are seemingly “more relaxed,” and the cooler
clusters appear less symmetric and seemingly “less relaxed.” The gallery as a whole therefore
suggests that relaxed clusters tend to be hot for their mass and unrelaxed clusters tend to
be cool for their mass.
At first glance, the result that disrupted-looking clusters in cosmological simulations
tend to be cooler than other clusters of the same mass may seem counter-intuitive, since
one might expect that mergers ought to produce shocks that raise the mean temperature of
the intracluster medium. This finding has also been noted by [Mathiesen & Evrard(2001)]
and [Kravtsov et al.(2006)]. Cluster systems in the process of merging tend to be cool for
their total halo mass because much of the kinetic energy of the merger has not yet been
12

Figure 2.1: Mass-temperature (M-TX ) relationships for the 121 clusters in our sample.
Spectroscopic-like temperature TSL is plotted in the top panel, while emission-weighted temperature TEW is plotted in the bottom panel. The solid lines show the best-fit power-law
relation M = M0 (TX /3 keV)α over the whole sample, while the dashed lines show the bestfit relation for systems with TX > 2 keV. Open circles represent the clusters that have the
greatest positive temperature offset from the mean relationship, and filled circles represent
the clusters with the greatest negative temperature offset.

13

Figure 2.2: Surface-brightness contour maps for sixteen of the clusters in our sample, overlaid with a circular aperture of radius R500 . The top panel has eight maps representing
the clusters that have the largest positive deviation δ ln TX from the mean M-TX relation,
calculated using the spectroscopic-like temperature TSL .

14

thermalized.
The idealized simulations of [Poole et al.(2007)] illustrate what may happen to the ICM
temperature during a single merger. Before the cores of the two merging systems collide, the
mean temperature is cool for the overall halo mass because it is still approximately equal
to the pre-merger temperature of the two individual merging halos. There is a brief upward
spike in temperature when the cores of the merging halos collide, after which the system is
again cool for its mass. Then, as the remaining kinetic energy of the merger thermalizes over
a period of a few billion years, the temperature gradually rises to its equilibrium value. The
merging system therefore spends a considerably longer time at relatively cool values of mean
temperature for its halo mass than at relatively hot values. Hence, such simulations suggest
a possible explanation for why more relaxed systems would tend to lie on the hot side of
the M-TX relation, while disrupted systems would tend to lie on the cool side. A caveat,
however, is that the current generation of hydrodynamic cluster simulations tend to produce
relaxed clusters whose temperature profiles continue to rise to smaller radii than is observed
in real clusters [Tornatore et al.(2003), Nagai et al.(2007)], potentially enhancing average
temperatures for such systems. As a separate test of this effect, we excise the core regions
from our sample clusters, calculate new substructure measures and new emission-weighted
temperatures for the core-excised clusters, and repeat our analysis.

2.3

Quantifying Substructure

The question we would like to address in this study is whether the surface-brightness substructure evident in Figure 2.2 is well enough correlated with deviations from the mean masstemperature relation to yield useful corrections to that relation. In order to answer that ques-

15

tion, we need to quantify the surface-brightness substructure in each cluster image, so that
we can determine the degree of correlation across the entire sample. [O’Hara et al.(2006a)]
explored the relationship between cluster structure and X-ray scaling relations in both observed and simulated clusters, and we adopt their suite of substructure measures in this study.
These include centroid variation, axial ratio, and the power ratios of [Buote & Tsai(1995)].
In this section we define and discuss those statistics and apply them to surface-brightness
maps made from three orthogonal projections of each cluster. Then we assess how well these
statistics correlate with offsets from the mean mass-temperature relation.

2.3.1

Axial Ratio

The axial ratio η for a cluster surface-brightness map is a measure of its elongation, which is of
interest because it has been found from simulations that the ICM is often highly elongated
during merger events [Evrard et al.(1993), Pearce et al.(1994)]. It is computed from the
second moments of the surface brightness,

Mij =

IX xi xj .

(2.4)

The summation is conducted over the coordinates (x1 , x2 ) of the pixels that lie within an
aperture centered at the origin of the coordinate system to which (x1 , x2 ) refer. Following
the work of [O’Hara et al.(2006a)], we use an aperture of radius R500 centered on the brightness peak. We then compute η from the ratio of the non-zero elements that result from
diagonalizing the matrix M. That is,

D = U T MU,

16

(2.5)

where U is a diagonalizing matrix for M, and

η=


 D
 12


D21 ,
 D
 21


D12 ,

D12 ≤
D12 >





D21 



D21 

.

(2.6)

The axial ratio is therefore defined to be in the range η ∈ [0, 1], with η = 1 for a circular
cluster. Of course there are other choices for the origin of the coordinate system, besides
using the brightness peak. For instance, in order to avoid misplaced apertures yielding
artificially low axial ratios for nearly circular distributions, one could adjust the position of
the aperture to seek a maximum in η. Doing this, we sometimes find that η ≈ 1 even for
non-circular clusters, as is evident in Figure 2.3. This figure depicts the surface-brightness
map of what appears to be a disturbed cluster, chosen from among those in our sample that
appear by eye to be the most unrelaxed. Yet, it happens to have an axial ratio very close
to 1 for an aperture placed so as to maximize η. This example demonstrates that, while
the axial ratio statistic may yield results consistent with a visual interpretation of cluster
substructure, it is also capable of unexpected results for some clusters.
To further illustrate this point, we have computed an axial ratio value for this cluster
for every possible choice of aperture placement. Apertures of radius R500 were centered on
each and every pixel within the surface-brightness map, provided the aperture so placed does
not reach the edge of the map. This procedure generated an axial-ratio “surface” mapping
all of the aperture placements to a value of η. Figure 2.4 shows the axial-ratio surface
for the cluster in Figure 2.3. For comparison purposes, Figure 2.5 presents an axial-ratio
surface map for a very symmetric, uniform, and apparently relaxed cluster, in which the
cluster’s brightness peak reassuringly corresponds to the aperture location that maximizes
η. In contrast, the presence of two peaks in the axial-ratio surface for the asymmetric cluster
17

Figure 2.3: Surface-brightness contour plot of an asymmetric cluster which, for certain
choices of aperture placement, yields an axial ratio close to 1. The circle represents an
aperture of R500
.

18

shows that η can sometimes depend strongly on aperture placement. Ideally, we would like
to place the aperture on the “center” of this cluster, but the center of an unrelaxed cluster
can be difficult to define, meaning that the axial ratio statistic may be likewise ill-defined
for such clusters.

19

Figure 2.4: Surface of axial ratio η as a two-dimensional function of the coordinates of the
aperture center. The axial ratio statistic appears to be ill-defined for this cluster.

20

Figure 2.5: Abstract surface of axial ratio η as a two-dimensional function of the coordinates
of the aperture center. This is a relaxed cluster, for which the axial ratio is better-defined.

21

2.3.2

Power Ratio

The power-ratio statistics [Buote & Tsai(1995), O’Hara et al.(2006a)] quantify substructure
by decomposing the surface-brightness image into a two-dimensional multi-pole expansion,
the terms of which are calculated from the moments of the image, computed within an
aperture of radius Rap :

bm (Rap ) =

Σ(x′ )(R′ )m cos mφ′ d2 x′

(2.7)

Σ(x′ )(R′ )m sin mφ′ d2 x′ .

(2.8)

(a2 + b2 )
m
m
2m .
2m2 Rap

(2.9)

P0 = [a0 ln(Rap )]2 .

am (Rap ) =

(2.10)

R′ ≤Rap

R′ ≤Rap

The power in terms of order m is then

Pm =

For m = 0, the power is given by

The power ratios Pm0 ≡ Pm /P0 are then dimensionless measures of substructure which have
differing interpretations. For instance, P10 quantifies the degree of balance about some origin
and can be used to find the image centroid, P20 is related to the ellipticity of the image,
and P30 is related to the triangularity of the photon distribution. As in the case of the axial
ratio computations, we set the aperture radius Rap equal to R500 . The most appropriate
place to center the aperture is at the set of pixel coordinates that minimizes P10 , which we
achieve using a self-annealing algorithm.

22

2.3.3

Centroid Variation

The centroid variation statistic w is a measure of the center shift, or “skewness”, of a
two-dimensional photon distribution. It is measured for a cluster surface-brightness map
in the following way. For a set of surface-brightness levels one finds the centroids of the
corresponding isophotal contours and computes the variance in the coordinates of those
centroids, scaled to R500 . Here we select 10 isophotes evenly spaced in log IX between the
minimum and maximum of IX within an aperture of radius R500 centered on the brightness
peak, so as to adapt to the full dynamic range of surface brightness for different clusters.
We employed this adaptive scheme because using one set of isophotes for all clusters tended
to ignore important substructure in less massive clusters when they had surface brightness
substructure inside R500 but outside of the lowest isophote.

2.3.4

Substructure and Scaling Relationships

Using the quantitative measures of substructure described in the previous section, we can
test the significance of the relationship between substructure and temperature offset hinted
at in Figure 2.2. We begin by treating four of our substructure statistics—centroid variation
w, axial ratio η, and power ratios P20 and P30 —as different imperfect measurements of an
intrinsic degree of substructure S. Figure 2.6 shows the relationship between substructure
and a cluster’s deviation δ ln TX from the mean M-TX relation for each substructure measure.
In each case we present results for both the spectroscopic-like temperature TSL and the
emission-weighted temperature TEW . Note that centroid variation w and the power ratios
P20 and P30 have large dynamic ranges, whereas the axial ratio η is always of order unity.
We therefore attempt to fit the relationships between δ ln TX and the different substructure

23

measures with the following forms:

δ ln TX =





























Aw α
B + βη
γ
CP20

(2.11)

λ
DP30

To visually indicate where the bulk of our substructure measures lie, Figure 2.6 has light gray
bands covering the extremes, so that 80% of our sample clusters have substructure measures
lying between the extremes. The power ratios in our study generally span two decades (in
units of 10−7 ), from ∼2—300 for P20 and from ∼0.01—10 for P30 . These ranges are consistent with those of [Buote & Tsai(1995)], [O’Hara et al.(2006a)], and [Jeltema et al.(2008)].
The measurements of axial ratio in our sample, with 80% of clusters having η∼0.4—0.95,
cover a slightly wider range than do the simulated clusters of [O’Hara et al.(2006a)]. Finally,
our measurements of centroid variation, with 80% of clusters having w[R500 ]∼0.01—0.1, are
again similar to those of [O’Hara et al.(2006a)].

24

Figure 2.6: Relationship between a cluster’s deviation δ ln TX from the mean M-TX relationship and four measures of substructure.

25

As denoted in Figure 2.6 by black filled circles, the systems with TX above 2 keV occupy
a slightly narrower range of substructure values than the systems below 2 keV, which are
denoted by plus signs. For the axial ratio and the power ratios, the variance is 15 to 25 percent
larger among the low-temperature systems when compared to the systems with TX > 2 keV.
For centroid variation the variance among the low-temperature systems is approximately the
same as it is among the high-temperature systems. However, it is not clear that there is a
significant correlation between substructure and mass, since the mean substructure values
are generally very similar between the low-temperature and high-temperature sub-samples.
The mean value of the power ratio P30 is significantly larger for the low-temperature subsample, however this measure also has the weakest correlation with offsets from the mean
M-TX relation.
To test whether the low-mass clusters in our sample significantly boost the overall scatter
in the M-TX relation, we perform a cut at 2 keV and fit this relation both to the whole sample
and to the sub-sample above 2 keV. Figure 2.7 shows the residuals in mass, actual minus
predicted, where the predicted mass derives only from the M-TX relation. The plus signs
indicate clusters whose mass is predicted from an M-TX relation derived from all 121 clusters.
The black filled circles indicate clusters that are above 2 keV in X-ray temperature, with
the mass estimated using the sub-sample M-TX relation. There is a negligible reduction in
scatter, from 0.127 to 0.124 for TSL and from 0.102 to 0.094 for TEW , suggesting that at best
only a modest improvement is found in our sample if we remove the low-mass systems. In
order to test the degree to which incorporating substructure measures adds to this modest
improvement, when we compare mass estimates derived using substructure to those derived
only from the M-TX relation, we focus on clusters above 2 keV in the rest of our analysis.
Figure 2.6 shows that for our simulated clusters, a greater amount of measured sub26

structure tends to be associated with “cooler” clusters while less substructure tends to be
associated with “hotter” clusters. Also, the centroid variations w are more highly-correlated
with δ ln TX than are the other substructure parameters. We interpret this to mean that
the centroid variation is a better predictor of the offset in the M-TX relationship than are
the power ratios and the axial ratio, though all four measures appear to be related to the
temperature offset. Again, in this figure we denote systems above 2 keV by black filled
circles, and systems below 2 keV by plus signs.
Correlations between substructure and δ ln TX can potentially be exploited to improve
on mass estimates of real clusters derived from the mass-temperature relation. Instead of
computing the temperature offset at fixed mass, we can determine a substructure-dependent
mass offset at fixed temperature and then apply it as a correction to the predicted mass
Mpred (TX ) one would derive from the mean M-TX relation alone. To assess the prospects
for such a correction, based on this sample of simulated clusters, we first define the mass
offset from the mean mass-temperature relation to be

δ ln M(TX ) = ln

M
Mpred (TX )

,

(2.12)

where M is the cluster’s actual mass, and examine the correlations between substructure
measures and δ ln M. Figure 2.8 shows the results. These plots show mass predictions
from both the M-TSL relation and the M-TEW relation. Consistent with our analysis of
δ ln TX , the centroid variation w appears to be a more effective predictor of the mass offset
δ ln M(TX ). Nonetheless, all four measures of substructure appear to be correlated with
mass offset.
In order to incorporate a substructure correction into the mass-temperature relation,

27

Figure 2.7: Comparison of mass offset δ ln M between true mass and predicted mass, based
on the M(TX ) relation. Plus signs indicate residuals for masses estimated from the M-TX
relation derived from all 121 clusters, while black filled circles are for masses estimated from
the M-TX relation for clusters above 2 keV. Upper panels are for spectroscopic-like temperature and lower panels are for emission-weighted temperature. The standard deviations σ
in the residuals are given in each plot.

28

Figure 2.8: Relationship between substructure and mass offset δ ln M(TX ) from the mean
M-TX relationship for the same substructure measures as in the previous figure. The lower
panels are for TX = TSL , and the upper panels are for TX = TEW . The solid lines indicate
the best-fitting linear relationships to the above 2 keV sub-sample denoted by black filled
circles, and the gray bands and vertical dashed lines mark the extremes in substructure
between which 80% of our clusters lie. The plus signs correspond to systems below 2 keV,
while the black filled circles correspond to systems above 2 keV.

29

we perform a multiple regression, fitting our simulated clusters’ mass, temperature, and
substructure data to the form,

log Mpred (TX , S) = log M0 + α log TX + βS,

(2.13)

where S represents one of the following substructure measures: η, log w, log P20 , or log P30 .
This fit gives us a substructure-corrected mass prediction Mpred (TX , S) for each substructure
measure, and we can assess the effectiveness of that correction by measuring the dispersion
of the substructure-corrected mass offset

δ ln M(TX , S) = ln

M
Mpred (TX , S)

,

(2.14)

between the revised prediction and the true cluster mass.
Figure 2.9 shows the results of that test. Open circles in each panel indicate mass
offsets δ ln M(TX ) without substructure corrections, which have a standard deviation σM (T ) .
Filled circles indicate mass offsets δ ln M(TX , S) with substructure corrections, which have a
standard deviation σM (T,S) . The upper set of panels shows results for TSL , and the lower set
is for TEW . In each case, incorporating a substructure correction to the mass-temperature
relation reduces the scatter, yielding more accurate mass estimates. The centroid variation
corrections are the most effective, reducing the scatter in mass from 0.124 to 0.085 in the
M-TSL relation and from 0.094 to 0.072 in the M-TEW relation, though admittedly this
is again a modest improvement. Although non-negligible structure correlates significantly
with offsets in the M-TX plane, apparently it does so with substantial scatter. This scatter
may be partly due to projection effects, in which line-of-sight mergers are discounted by the

30

measures of substructure and may dilute their corrective power [Jeltema et al.(2008)].
Lastly, Figure 2.10 shows the results for a similar analysis to that of Figure 2.9, except
that in this case we have excised a region of radius 0.15R500 around the center of each cluster
and recomputed TEW . We do this to test whether offset in temperature, whose correlation
with substructure is the basis of our correction scheme, stems from a potentially unrealistic
feature, which is that the cores of many real clusters have temperature profiles that decline
at larger radii than occurs in simulated clusters. As in Figure 2.9, we restrict our analysis
to clusters above 2 keV. After doing this test, for TEW excising the core actually increases
the scatter in M-TX from 0.094 to 0.106. It may be that by removing the bright central
region, the average temperature becomes more sensitive to structure outside the core. Also,
this figure shows that the effect of incorporating substructure measurements into the massestimates is still present. The scatter is reduced to 0.075 for w, 0.093 for η, 0.090 for P20 and
0.094 for P30 . Figure 2.10 summarizes the results of this test, which support the conclusion
that the reduction in scatter we realize using substructure is a real effect and not an artifact
of known defects in the simulations.

2.3.5

Comparisons with Other Substructure Studies

[O’Hara et al.(2006a)] examined the relationship between galaxy cluster substructure and
X-ray scaling relationships, including the M-TX relation, using both a flux-limited sample
of nearby clusters and a sample of simulated clusters, and found a greater amount of scatter
among the more relaxed clusters in their observed sample. Contrasting that result they also
found a greater amount of scatter among the more disrupted clusters in their simulation
sample, though they characterize the evidence for this second result to be weak. Finally,
they see no evidence in either sample for more disrupted clusters to be below the mean, and
31

the more relaxed clusters to be above. One difference between our study and theirs is the
presence of radiative cooling and supernova feedback in the simulation that produced our
cluster sample. Also, the focus of our work is different from theirs in that we concentrate on
the degree of correlation between the amount of substructure and the size and direction of
the offset from the mean relation. We do find significant evidence of this correlation, such
that relaxed clusters are hotter than expected given their mass. We also test, as best we can
given our simulation sample, the hypothesis that substructure can be used to improve mass
estimates derived from the ICM X-ray temperature. It is possible that our detection of a
correlation between substructure and temperature offset arises from the additional physics
in our simulated clusters, since when radiative cooling is included, cool lumps may be better
preserved than in simulations that don’t include cooling.
Our results are in agreement with [Valdarnini(2006)], who examined substructure in
clusters simulated with cooling and feedback and found that unrelaxed clusters, identified
with a larger power ratio P30 , have spectral-fit temperatures biased low relative to the
mass-weighted temperatures. This trend aligns with our finding that the spectroscopic-like
temperature TSL is lower than the best-fit temperature at fixed mass for clusters with larger
power ratios P20 and P30 . However, [Valdarnini(2006)] did not investigate the effectiveness
of substructure measures in reducing scatter in the mass-temperature relation.
Our results are also in agreement with some of the results of [Jeltema et al.(2008)], who
have recently investigated correlations between substructure and offsets in mass predictions
in simulated clusters. They found that measuring cluster structure is an effective way to
correct masses estimated using the assumption of hydrostatic equilibrium, which tend to
be underestimates. Our findings support these results, given that we find substructure can
be used to correct masses estimated directly from the M-TX relationship. There also are
32

differences between our findings and theirs. They report that the M-TX relation for their
simulation sample shows no dependence on structure, whereas the clusters in our sample
exhibit offsets that correlate with the degree of substructure. One possibility is that these
differences stem from differences in the simulations’ feedback mechanisms. Another possibility is that some of the offset we observe derives from enhanced temperatures in simulations
with radiative cooling. As we describe in section 3, we perform a test in which we estimate
TEW using projected surface-brightness and temperature maps, in order to remove the core
regions from our analysis, but this may be less effective than properly excising the cores in
the simulations, as [Jeltema et al.(2008)] have done.
[Kravtsov et al.(2006)] also looked at the relationship that cluster structure has to the
M-TX relation in simulated clusters, to show that the sensitivity of mass proxies YX and
YSZ to substructure is not very strong. They divided their sample into unrelaxed and
relaxed sub-samples, based on the presence or absence of multiple peaks in the surfacebrightness maps of clusters, and found the normalization of the M-TX relation to be biased
to cooler temperatures for the unrelaxed systems. Other workers also have looked at the
relationship between the M-TX relation and substructure, as reflected in the X-ray spectral
properties. [Mathiesen & Evrard(2001)] have examined the ratio of X-ray spectral-fit temperatures in hard and full bandpasses for an ensemble of simulated clusters, and found it to
be a way of quantifying the dynamical state of a cluster. We consider our approach of using
surface-brightness morphology information to be complementary to theirs. More recently,
[Kay et al.(2007)] performed an interesting analysis on another large-volume simulation sample, using as substructure metrics the centroid variation and measures of concentration to
report evolution in the luminosity-temperature relationship. Specifically, they report that
the more irregular clusters in their sample lie above the mean M-TX relation (i.e., they are
33

cooler than average), for the spectroscopic-like temperature TSL .

2.4

Summary

Using a sample of galaxy clusters simulated with cooling and feedback, we investigated three
substructure statistics and their correlations with temperature and mass offsets from mean
scaling relations in the M-TX plane. First, we showed that the substructure statistics w, η,
P20 and P30 all correlate significantly with δ ln TX , though with non-negligible scatter. In
all cases this scatter is larger for δ ln TSL than it is for δ ln TEW . Next, we considered the
possibility that M-TX scatter is driven by low-mass clusters. We tested the degree to which
scatter can be reduced by filtering out these systems. This consisted of performing a cut
at 2 keV, for which we saw that it yielded a modest improvement in mass estimates. To
see whether incorporating substructure could refine these mass estimates, we first showed
that w, η, P20 , and P30 correlate significantly with the difference δ ln M between masses
predicted from the mean M(TX ) relation and the true cluster masses, with non-negligible
scatter that again is less for M(TEW ) than it is for M(TSL ). Then we adopted a full threeparameter model, M-TX -S, which includes substructure information S estimated using w, η,
P20 , and P30 . Scatter about the basic two-parameter M-TEW relation was 0.094. Including
substructure as a third parameter reduced the scatter to 0.072 for centroid variation, 0.084
for axial ratio, 0.081 for P20 , and 0.084 for P30 . Scatter about the basic two-parameter MTSL relation was 0.124, and including substructure as a third parameter reduced the scatter
to 0.085 for centroid variation, 0.112 for axial ratio, 0.110 for P20 , and 0.108 for P30 . As one
last test, and to increase our confidence that our substructure measures are not relying on
potentially non-physical core structure in the simulations, we also repeated the comparison

34

of mass-estimates for TEW , with the core regions of the clusters excised. First, removing the
core slightly increased the scatter in M-TX possibly by making the average temperature more
sensitive to structure outside the core. Second, even with the cores removed the improvement
in mass-estimates obtained using substructure information remains. Based on these results,
it appears that centroid variation is the best substructure statistic to use when including a
substructure correction in the M-TEW relation. However, the correlations we have found in
this sample of simulated clusters might not hold in samples of real clusters, because relaxed
clusters in the real universe tend to have cooler cores than our simulated clusters do.

35

Figure 2.9: Comparison of mass offset δ ln M between true mass and predicted mass, based
on the M(TX ) relation (open circles) and the M(TX , S) relation (filled circles). Upper panels
are for TSL and lower panels are for TEW . The standard deviations σ in the residuals are
given in each plot.

Figure 2.10: Comparison of mass offset δ ln M between true mass and predicted mass, based
on the M(TX ) relation (open circles) and the M(TX , S) relation (filled circles). These results
are for are for TEW , with the central 0.15 R500 region removed both from the average
temperature and from the substructure measures. The standard deviations σ in the residuals
are given in each plot.

36

Chapter 3
Temperature Structure and
Mass-Temperature Scatter in Galaxy
Clusters

3.1

Introduction

Galaxy clusters play an important role in precision cosmology that complements other techniques like Type Ia supernovae luminosity-distance relation measurements [Perlmutter et al.(1999),
Riess et al.(1998)], baryonic acoustic oscillations (BAOs) angular-distance relation measurements [Eisenstein et al.(2005)], and observations of the cosmic microwave background radiation (see [Frieman et al.(2008)] for a review). For example, [Vikhlinin et al.(2009)] exploit
dark energy’s influence on the growth of structure by using 37 moderate-redshift and 49 lowredshift clusters to measure the shape of the galaxy-cluster mass function and its redshift
evolution, which constrain the dark-energy density parameter ΩΛ to 0.83 ± 0.15 in a non-flat
ΛCDM cosmology and the dark energy equation of state parameter w0 to −1.14 ± 0.21 in a
flat cosmology. [Mantz et al.(2010)] have obtained similar results from measurements of the
evolving number density of the largest clusters in order to constrain w0 to −1.01 ± 0.20.
Strategies like these that compare model predictions to galaxy-cluster sample statistics

37

inevitably confront sample error and sample bias. Future surveys expected to gather samples
of 10–40 thousand galaxy clusters1 will maximize survey reach while maintaining sufficient
observation quality in order to minimize sample error, but they still must grapple with a
major source of sample bias, which is scatter in the relationship used to infer cluster mass
from an observable mass proxy. An important mass-observable relation for galaxy cluster
studies connects dark matter halo mass to the temperature of the intracluster medium (ICM)
inferred from its X-ray spectrum (its “X-ray temperature” TX ). In this chapter we investigate
the possibility of correcting for scatter in this relation using temperature-inhomogeneity in
the ICM, and discuss challenges that may exist in such a program.
A significant amount of uncertainty in the dark-energy constraints obtainable from large
cluster surveys derives from uncertainty in scatter about the mean scaling relations obeyed by
galaxy clusters’ bulk properties [Lima & Hu(2005), Cunha & Evrard(2010)]. The key galaxy
cluster property to measure when trying to constrain dark energy with clusters is the cluster’s mass, which cannot be directly observed. Theoretical considerations predict correlations
among halo mass and more readily observed cluster properties, like its galaxy richness, the
velocity dispersion of its galaxies, TX , the Sunyaev-Zel’dovich decrement, the gas mass, and
YX parameter, which is the product of TX and the gas mass inferred from X-ray observations
[Kravtsov et al.(2006)]. Theory also predicts intrinsic scatter in these relations owing to variation in cluster dynamical state [Stanek et al.(2010), Fabjan et al.(2011), Rasia et al.(2012)].
One way to deal with intrinsic scatter is to join a cluster model to a cosmological model
and simultaneously fit for the parameters of both, leveraging the statistical power of large
surveys and “self-calibrating” the mass-observable relations. Another approach is to combine
observables that tend to depart from the expected scaling relations in opposite ways, yielding
1 http://www.mpe.mpg.de/heg/www/Projects/EROSITA/main.html

38

a new, low-scatter composite observable. An example low-scatter composite observable is
YX , since at a given halo mass, offsets in the measured gas mass at fixed total mass tend to
anti-correlate with offsets in the measured temperature [Kravtsov et al.(2006)].
Another family of low-scatter composite observables attempt to measure structural variation directly. Mergers, relaxation, and non-adiabatic processes like radiative cooling, star
formation, and feedback ought to leave a visible imprint that may allow us to measure and
correct for scatter. For example, one might use imaging to quantify resolved morphological
substructure. [Jeltema et al.(2008)] apply two observationally-motivated structure measures,
the power ratios [Buote & Tsai(1995)] and the centroid shift [Mohr et al.(1993)], to a sample
of galaxy clusters simulated with Enzo [Norman & Bryan(1999), O’Shea et al.(2004)]. They
find that cluster structure correlates strongly with bias in mass estimates derived from TX
under the assumption of hydrostatic equilibrium and accounting for cluster structure can be
used to correct some of the bias. Similarly, [Ventimiglia et al.(2008)] apply the power ratios, centroid shift, and axial ratio [O’Hara et al.(2006b)] substructure measures to the same
sample of simulated clusters used in this chapter, and find that cluster substructure correlates with departures from the mean M-TX relationship in the sense that clusters with more
substructure tend to have a lower temperature at a given mass, and can be used to refine
mass estimates derived from the ICM X-ray temperature. [Piffaretti & Valdarnini(2008)]
likewise find that greater substructure, as quantified with power ratios, correlates with lower
temperature at a given mass. [Yang et al.(2009)] find a strong correlation between masstemperature scatter and halo concentration in their sample of simulated clusters, with cooler
clusters appearing more concentrated than warmer clusters at similar mass.
[Jeltema et al.(2008)] observe, however, that line-of-sight projection effects lead to significant uncertainties in morphologically-derived substructure measures. Substructure also
39

becomes more difficult to resolve at high redshift. Spectral signatures of dynamical state are
therefore attractive because they are aspect-independent and redshift-independent. One such
spectral signature of dynamical state is the “temperature ratio” THBR [Mathiesen & Evrard(2001),
Cavagnolo et al.(2008)], which divides a “hardband” spectral-fit temperature by a “broadband” spectral-fit temperature. An energy cut applied to a broadband spectrum produces
a hardband spectrum and serves to filter out cooler line-emitting components of the ICM.
[Mathiesen & Evrard(2001)] studied the effects of relaxation on the observable properties
of galaxy clusters, using a sample of numerically-simulated galaxy clusters generated by
[Mohr & Evrard(1997)]. They found that hardband (2.0–9.0 keV) X-ray spectral fit temperatures average ∼20% higher than broadband (0.5–9.0 keV) temperatures and suggested that
this effect may signal the presence of cool, luminous sub-clusters lowering the broadband
temperature. [Valdarnini(2006)] corroborated these findings in simulations that included
radiative cooling.
Figures 3.1 and 3.2 illustrate the effect. Both show a counts spectrum and singletemperature fit for a typical, unrelaxed cluster in our sample, with an aperture set at R2500
and a core region out to 0.15R2500 excised. Figure 3.1 is for a single-temperature model
fit to the broad band, while figure 3.2 is for the hard band. Note the excess emission relative to the model above 4.0 keV and the deficit below 2.5 keV that arises because the
model cannot simultaneously fit both a hot component and a cooler, line-emitting component [Mazzotta et al.(2004)]. Figure 3.2 shows the same spectrum, but with the residuals
for a model fit just to the hard band. In this case the fit is much better over the range from
2.0 to 7.0 keV but under-predicts the emission below 2.0 keV.

40

Figure 3.1: X-MAS simulated counts spectrum for a simulated cluster that appears to have
significant temperature structure. A single-temperature MEKAL model fit to the [0.7–7.0]
keV broad band is over-plotted as the solid line, with kB T = 2.42 ± 0.03 keV. Fit residuals
appear in the bottom panel. This figure illustrates qualitatively the effect that additional
cool components have on a single-temperature fit.

41

Figure 3.2: X-MAS simulated counts spectrum for the same simulated cluster as in the
previous figure. A single-temperature MEKAL model fit to the [2.0–7.0] keV hard band is
over-plotted as the solid line, with kB T = 3.52 ± 0.2 keV. Fit residuals appear in the bottom
panel.

42

[Mathiesen & Evrard(2001)] suggested that the temperature skewing they observed might
indicate a real temperature skewing detectable in real clusters using Chandra. Other researchers, including [Cavagnolo et al.(2008)] fit single-temperature emission models to the
hard band (2.0–7.0 keV) and broad band (0.7–7.0 keV) for a large (N = 192) sample of clusters with observations selected from the Chandra Data Archive. These authors show that for
a large, heterogeneous sample of clusters across a broad temperature range, the distribution
of THBR has a mean of 1.16 and an rms deviation σ = ±0.10, with THBR tending to be larger
in merging systems. They also report that while this signal is significant in the aggregate,
the errors for any single THBR measurement and the scatter across all of the measurements
together pose challenges for any effort to use THBR either to select for merging systems or
to obtain more accurate mass estimates. In order to meet these challenges it is important
also to examine THBR and similar spectral signatures of dynamical state in a simulation
context. Like the original study of [Mathiesen & Evrard(2001)], this chapter examines the
temperature ratio for a sample of simulated clusters and simulated Chandra observations,
and as in the subsequent study by [Valdarnini(2006)], the simulated clusters analyzed here
were generated using a hydrodynamical code with radiative cooling included.
This chapter is organized as follows. In section 3.2, we describe the simulated clusters
in our sample along with the X-MAS code for generating their mock X-ray observations. In
section 3.3, we present our analysis methods and define the spectral signatures of dynamical state we examine. In section 3.4, we report and discuss our results, and section 3.5
summarizes our work.

43

3.2
3.2.1

Methods
Numerical Simulations

This study is based on an analysis of 118 clusters simulated using the cosmological hydrodynamics TREE+SPH code GADGET-2 [Springel(2005)], which were simulated in a standard
Λ cold dark matter (ΛCDM) universe with matter density ΩM = 0.3, h = 0.7, Ωb = 0.04, and
σ8 = 0.8. The simulation includes radiative cooling assuming an optically-thin gas of primordial composition, with a time-dependent UV background from a population of quasars,
and handles star formation and supernova feedback using a two-phase fluid model with cold
star-forming clouds embedded in a hot medium. All but four of the clusters are from the
simulation described in [Borgani(2004)], who simulated a box 192 h−1 Mpc on a side, with
4803 dark matter particles and an equal number of gas particles. The present analysis considers the 114 most massive clusters within this box at z = 0, which all have M200 greater
than 5 × 1013 h−1 M⊙ . These are referred to as the B04 Sample in the remainder of this
chapter. By convention, M∆ refers to the mass contained in a sphere which has a mean
density of ∆ times the critical density ρc , and whose radius is denoted by R∆
That cluster set covers the ∼1.5-5 keV temperature range, but the 192 h−1 Mpc box is
too small to contain significantly hotter clusters. We therefore supplemented it with four
clusters with masses > 1015 h−1 M⊙ and temperatures > 5 keV drawn from a dark-matteronly simulation in a larger 479 h−1 Mpc box [Dolag & Stasyszyn(2009)], referred to in this
chapter as the D09 sample. The cosmology for this simulation also was ΛCDM, but with
σ8 = 0.9. These were then re-simulated including hydrodynamics, radiative cooling, and
star formation, again with GADGET-2 and using the zoomed-initial-conditions technique of
[Tormen(1997)], with a fourfold increase in resolution. This is comparable to the resolution
44

of the clusters in the smaller box. Adding these four massive clusters to our sample gives
a total of 118 clusters with M200 in the range 5 × 1013 h−1 M⊙ to 2 × 1015 h−1 M⊙ . The
mean structural properties of massive clusters drawn from a sample with σ8 = 0.9 may differ
somewhat from those of similar-mass clusters in a σ8 = 0.8 universe because they reflect
a more advanced state of cosmic evolution. However, these four additional clusters carry
minimal statistical weight in the context of the overall sample. They are included primarily
to evaluate whether the mean and dispersion of their THBR values are consistent with those
of the lower-mass systems.

3.2.2

X-Ray Simulations

The simulated galaxy clusters in our sample are processed with the X-ray Map Simulator
version 2 (X-MAS) [Gardini et al.(2004), Rasia et al.(2008)] to generate X-ray images suitable for standard Chandra reduction techniques. In its first step X-MAS uses the outputs
of the hydrodynamic code to calculate the emissivity of each simulation element within the
chosen field of view and to project this onto the image plane. In its second step it convolves the resulting flux with the appropriate response of a given detector. In the case of
our simulated Chandra observations, the second step applies the response matrix file and
ancillary response file for the ACIS S3 CCD, with a 200 kilosecond exposure time. In order
to separate out potential calibration issues from the focus of this particular study, these
response matrices implement a constant response over the detector, and in generating the
simulated X-ray images, the evolved zero-redshift clusters are shifted to a redshift sufficient
to fit R500 within the 16 arcminute field of view. Because of this step, though the clusters
in the sample are distributed over a range of physical sizes, they all have approximately the
same apparent size projected onto the image plane and into the simulated observations.
45

3.2.3

M-TX Relation

In this chapter, cluster mass refers to M200 , while TX refers to our estimated “spectralfit temperatures” obtained using XSPEC [Arnaud(1996)] to fit single-temperature MEKAL
plasma models to various energy bands in simulated X-MAS X-ray observations taken from
the numerically-simulated clusters. The steps involved in the process are described in detail
in § 3.3.
Figure 3.3 shows the mass-temperature relation based on our sample of simulated clusters
and estimated spectral fit temperatures, measured in a broad 0.7–7 keV band within an
annular aperture from 0.15R500 to R2500 as in [Cavagnolo et al.(2008)]. The best fits to the
power-law form
M = M0

TX α
2 keV

(3.1)

have the coefficients M0 = 1.09 ± 0.01 × 1014 h−1 M⊙ , α = 1.57 ± 0.06 for the full sample
of clusters. For the subset of clusters with T2.0−7> 2.0kev, the best-fit parameters are
M0 = 1.06 ± 0.02 × 1014 h−1 M⊙ , α = 1.71 ± 0.07. As is generally the case for simulated
clusters, the power-law indices of the mass-temperature relations found here are consistent
with cluster self-similarity and the virial theorem [Kaiser(1986), Navarro et al.(1995)]. These
relationships have scatter, which we characterize by the standard deviation in log space σln M
about the best-fit mass at fixed temperature TX . We find σln M = 0.10.

46

T in R2500
All Clusters
kBT ≥ 2 keV

15

M200 (M⊙)

10

14

10

1

10
kBT (keV)

Figure 3.3: Mass-temperature (M-TX ) relation for the clusters in our sample. Clusters with
kB T < 2 keV are plotted in purple open triangles, while clusters with kB T ≥ 2 keV are
plotted in blue open squares, and the four massive D09 clusters in green open circles. Mean
relations are plotted with solid lines for the whole sample, and dashed lines for clusters
with kB T ≥ 2keV. Finally, note that average temperatures are taken within a core-excised
annulus whose outer radius is placed at R2500. For interpretation of the references to color in
this and all other figures, the reader is referred to the electronic version of this dissertation.

47

3.3

Analysis

3.3.1

Filtering

We use X-MAS to simulate X-ray observations—provided as standard Chandra event files—
then subject them to a series of reduction steps using the Chandra Interactive Analysis
of Observations package (CIAO) v4.1 [Fruscione et al.(2006)]. All of the event files have
at least 100K counts, while those for the most luminous clusters have over half a million
counts. Simulated observations of this quality provide us with an opportunity to address
the question of how much of the scatter in our temperature-structure statistics is intrinsic.
Consequently, our first reduction step filters each raw event file into a new set of event files
by randomly sampling the events it records. The number of counts in each file is its “count
level,” and in order to cover the space of typical Chandra archival observations, we apply
the filtering step to each of the raw event files four times, at count levels 15K, 30K, 60K, and
120K. These count levels roughly map to the range between observations of short duration
or of relatively low surface-brightness objects to observations of long duration or of relatively
high surface-brightness objects.

3.3.2

Cool Lump Excision

An example of a filtered event file appears in Figure 3.4 as a surface-brightness image. It
displays a common feature of numerical simulations of galaxy clusters, which is the presence of relatively dense, cool, and metal-rich substructures that continuously undergo mass
accretion and have not yet come into thermal equilibrium with the hot ICM surrounding
them [Borgani & Kravtsov(2009)]. These bright point-like spots, or “cool lumps”, typically
are associated with the dense cool cores of smaller halos that have merged with the primary
48

halo. Generally regarded as an unphysical artifact of numerical simulations, at least insofar
as their temperatures, densities, and concentrations are concerned, commonly they are excised before further analysis is conducted [Rasia et al.(2006), Piffaretti & Valdarnini(2008),
Nagai et al.(2007)].

49

Figure 3.4: X-MAS simulated X-ray surface-brightness images for the same cluster whose
spectrum is presented in the previous figure. The aperture is set at R2500 and a core
region of size 0.15R2500 is excised. Note the surface-brightness peaks in the image. In this
study these are detected automatically by the CIAO tool wavdetect, which generates source
regions identified in this figure by red ellipses. Labeled “cool lumps” in this analysis, they
are subjected to varying degrees of excision, from no masking (left) to full masking (right),
as described in the text.

50

In order to study the effect that excising cool lumps has on measures of temperature
substructure, we produce from the originals several new event files whose cool lumps have
been excised to an increasing degree. The CIAO wavdetect tool identifies peaks in the
photon distribution by correlating an event file’s image with a sequence of “Mexican-Hat”
wavelet functions of differing scale sizes, measured in pixels, then generates a source list
with associated region files. We use its default sequence of wavelet scale sizes in this analysis
(2 and 4 pixels), though we apply the tool four times to each filtered event file, each time
incrementing the multiplicative factor by which the source regions are scaled. In CIAO
wavdetect, the parameter governing this multiplicative factor is ellsigma and in our study
ranges between 0 to 3. A value of 0 for ellsigma is equivalent to “no masking” while a value
of 3 corresponds to what we call “full masking”. The effect is to produce versions of each
cluster observation with a range of masking. Note that the wavelet scale size is a constant
fraction of the cluster size because all the clusters have been redshifted so that R500 fits in
the field of view.
We finish the extraction phase of our analysis with the following steps. First, for every
excised file we apply an aperture of R2500 and excise the central 0.15 R2500 region, generating
new copies of the event files. This step, including the centering algorithm, matches the
procedure in [Cavagnolo et al.(2008)] so that regions measured in the simulated clusters
correspond to those in the Chandra archival observations. Next, each fully-processed event
file is extracted into a standard pulse-invariant (PI) spectral file binned so as to have at least
25 counts in each energy channel. The extraction phase is complete when each original raw
event file generates PI spectral files suitable for spectral fitting in XSPEC.

51

3.3.3

Spectral Fitting

The PI spectral files are then fed into XSPEC v12.5.0 for spectral fitting. In order to form
the temperature ratio THBR two fits are performed for each spectrum. The first fit is over
the hard band from 2.0(1 + z)−1 –7 keV, for which the (1 + z)−1 factor exists in order to shift
the 2 keV cutoff from the observer’s frame to the cluster’s rest-frame. The simulated clusters
occupy a range of redshifts because larger clusters are translated to higher redshifts in order
to fit R500 within 16 arc-minutes, when creating artificial observations. The second fit is
over the broad band, including all energy channels in the spectrum from 0.7–7 keV. Every fit
is made by minimizing the χ2 statistic for a single-temperature MEKAL model multiplied
by a warm-absorber (WABS, to account for Galactic absorption). The Galactic column is
fixed to NH = 5 × 1020 cm−2 and the metallicity to 0.3 Solar, leaving the temperature of the
emission component, its H density (although this makes no difference), and its normalization
as the only free parameters.

3.3.4

Quantifying Temperature Structure

We adopt two spectral measurements of temperature structure. The first is the temperature
ratio THBR of [Cavagnolo et al.(2008)] and is found in the following way. For each simulated
cluster in our sample, for each count level (15K, 30K, 60K, 120K), and for each cool lump
masking level (0, 1, 2, 3), we find a spectral-fit temperature in the hard band and the broad
band and form the temperature ratio THBR . Emission from cooler, line-emitting metal-rich
parts of the ICM ought to be excluded from the hard band, so we expect that a THBR value
greater than unity signals the presence of merging sub-clumps.
Our second measure of temperature, which we call the “cool residual,” (REScool ) com-

52

pares an observation’s actual broadband count rate to a model-predicted count rate for a
spectral model fit only over the hard band. Again, we determined a spectral-fit temperature
for each combination of cluster, count level, and masking level, except that in this case we
performed only a hardband fit. Since emission from cooler components should be excluded
from this band, in general we achieve good fits even when the count level is sufficiently high
that broadband fits may formally have large reduced χ2 values. From this hardband model fit
we estimate the corresponding broadband count rate and calculate its percentage deviation
from the actual broadband count rate. While single-temperature systems will have actual
count rates that are essentially the same as their model count rates, the introduction of
cooler, luminous substructure components should generate an excess broadband count rate
relative to the model. This measure of temperature substructure avoids the uncertainties
associated with a single-temperature fit to the broad band.

3.4

Results and Discussion

Having calculated spectral-fit temperatures, THBR , and the cool residual for all of the clusters
in our sample, we have the means to study how well temperature structure correlates with
departures from the mean M-TX relation, and how well these measures can be used to obtain
better mass estimates. We will also examine how our results depend on the removal of the
cool lumps.

3.4.1

THBR from Simulations.

We present in Figure 3.5 the temperature ratios THBR for our simulation sample, plotted
as a function of the broadband temperature fit T0.7−7 . This figure is similar to Figure 8 in
53

[Cavagnolo et al.(2008)], which shows that the mean value of THBR observed among clusters
in the Chandra archive is THBR = 1.16, with a standard deviation of σTHBR = 0.10. We
remind the reader of important differences between the two samples being considered. The
Chandra archive sample of [Cavagnolo et al.(2008)] contains clusters most of which have
kB T > 3 keV, whereas most of the clusters in our sample have kB T < 3 keV. Nevertheless,
even with that caveat the temperature ratios of the simulated clusters are distributed in
approximately the same way as are those in the real sample of [Cavagnolo et al.(2008)], with
THBR = 1.12 and σTHBR = 0.11, provided the cool lumps are not excised (circles in Figure
3.5). When the cool lumps are excised (squares in Figure 3.5), the mean and variance of
THBR diminish, with THBR = 1.07 and σTHBR = 0.07. These values are for the full 120K
count data, though the full set of THBR for the four ellsigma values and four count values
are tabulated in Table 3.1. This table provides mean values for THBR and its variance for
all of the clusters in the sample, and for a subset whose T2.0−7 value is greater than 2 keV.
Much of the variance derives from the lower-temperature clusters, and as they are removed
the variance in THBR drops significantly, especially when full masking is applied.

54

Figure 3.5: THBR plotted against T0.7−7 for simulated X-MAS observations that have approximately 120K counts. Hydrodynamic simulations may produce spurious over-condensations
of cool gas. These cool lumps can be excised using the CIAO tool wavdetect, whose aggressiveness can be controlled via its ellsigma parameter. Squares correspond to singletemperature MEKAL fits whose underlying observations are processed by wavdetect with
ellsigma set to 3, and circles correspond to those with ellsigma set to 0. Finally, note that the
4 massive D09 clusters are denoted by filled symbols rather than by open symbols. The solid
line represents the mean when cool lumps are not removed, while the dashed line represents
the mean when they are removed. The dotted line indicates THBR = 1.

55

Figure 3.6 presents a similar effect for REScool . Here, we see that the excision of cool
lumps again reduces the mean and variance of the temperature substructure measure, although in this case the effect is more dramatic. Evidently, the presumably unphysical cool
lumps in simulated clusters may be necessary to reproduce the distribution of quantitative
temperature substructure measures found in real clusters. This is a subject which we return
to in Section 3.4.3.

56

Figure 3.6: REScool plotted against T0.7−7 for simulated X-MAS observations that have
approximately 120K counts. This statistic tracks the percent excess count rate of the actual
observation with respect to count rate for a MEKAL model fit to just the hard band. Squares
correspond to single-temperature MEKAL fits whose underlying observations are processed
by wavdetect with ellsigma set to 3, and circles correspond to those with ellsigma set to 0.
As in the previous plot, the 4 massive D09 clusters are denoted by filled symbols rather than
by open symbols. The solid line represents the mean when cool lumps are not removed, the
dashed line represents the mean when they are removed, and the dotted line indicates zero
residual.

57

3.4.2

Temperature Structure and Scaling Relations

Our original motivation for conducting this study was to determine if temperature structure,
as quantified by the temperature ratio THBR , correlates with and can be used to correct for
departures from the mean mass-temperature relation M-TX . In order to test this idea, we
define the “mass offset” at fixed temperature by the relation

δ ln M(TX ) = ln

M
Mpred (TX )

(3.2)

where M is the cluster’s actual mass, and Mpred is the mass predicted from the mean M-TX
relation. Figure 3.7 plots the mass offsets for our simulated clusters as a function of THBR ,
for the case in which predicted masses are derived from the M-TX relation for the broadband
spectral fit temperature. Values of THBR calculated both with cool lumps excised (squares)
and without excision (circles) appear in this figure. Error bars on THBR are omitted for
clarity. While there is some correlation, such that clusters with more temperature structure
(larger THBR ) tend to be more massive than predicted by the mean M-TX relation, the trend
is weak and has substantial scatter. Excising the cool lumps weakens the trend further.
Figure 3.8 shows the same comparison for the REScool measure instead of THBR , and in this
case we again find that excising the cool lumps has an even larger effect for THBR although
in both cases accounting for temperature substructure does not greatly reduce scatter in
mass offset.

58

Figure 3.7: Relationship between THBR and mass offset δ ln M(TX ) from the mean M-TX
relationship for the clusters in combined B04+D09 simulation sample. The temperatures in
this relation are our broadband spectral-fit temperatures. Squares correspond to simulated
X-MAS observations that are processed by wavdetect with ellsigma set to 3 (full masking),
while circles correspond to observations that have ellsigma set to 0.

59

Figure 3.8: Relationship between REScool and mass offset δ ln M(TX ) from the mean M-TX
relationship for the clusters in combined B04+D09 simulation sample. The temperatures in
this relation are our broadband spectral-fit temperatures. Squares correspond to simulated
X-MAS observations that are processed by wavdetect with ellsigma set to 3, while circles
correspond to observations that have ellsigma set to 0.

60

3.4.3

Effects of Masking Strategy

We now examine the decline in THBR and its variance as cool lumps are excised, beginning
with Figure 3.9, which focuses on THBR . Here, we plot the statistic’s standard deviation
σTHBR as a function of the two dimensions along which we adjust our analysis pipeline,
with the top panel devoted to the masking strategy, and the bottom panel devoted to the
count level. Focusing on the top panel, we see that increasing the ellsigma parameter of
the CIAO wavdetect tool from 0 to 3 reduces σTHBR from 0.15 down to 0.13, when the
observations are relatively “poor” (with ∼15K counts). With higher-quality observations of
60K or 120K counts, the decline is in σTHBR greater, going from 0.12 down to 0.07 as the
ellsigma parameter rises from 0 to 3, and σTHBR for full masking ends up being significantly
less than the value of 0.10 observed in the [Cavagnolo et al.(2008)] sample.

61

Figure 3.9: (top) Decline in the THBR standard deviation σTHBR as wavdetect’s ellsigma
parameter ranges from 0 to 3. These are plotted for a family of simulated observations of
increasingly higher quality, from approximately 15k counts to approximately 120k counts.
(bottom) Decline in σTHBR as simulated X-MAS observations go from lowest-quality (approximately 15k counts) to highest-quality (approximately 120k counts). These are plotted
for a family of simulated observations with wavdetect’s ellsigma parameter ranges from 0 to
3.

62

Figure 3.10 helps show why the temperature ratio THBR and its variance decline as the
cool lumps are removed. It plots the sample average of the relative change in temperature
as the CIAO wavdetect ellsigma parameter is increased. As more of each cool lump is
excised, both the broadband spectral fit temperature, and the hardband fit temperature
increase. However, the increase is significantly larger for the broadband temperature, as the
cool lumps’ contribution to the flux is already largely excluded from the hardband fits. The
top panel in this figure shows this effect for spectra with approximately 120K counts, while
the bottom panel is for spectra with approximately 15K counts.

63

Figure 3.10: Relative change in spectral fit temperatures as the cool lumps are removed
from the simulated clusters for high-quality simulated observations. (top) Observations with
approximately 120K counts. (bottom) Observations with approximately 15K counts. Note
that these are for observations in an aperture corresponding to R2500 .

64

Another way of visualizing the effect of more aggressive masking is depicted in Figure
3.11. The top panel in this figure is similar to Figure 3.6 in that it occupies the T0.7−7–
THBR plane, with THBR plotted as a function of T0.7−7 for our simulated clusters. Arrows
illustrate the shift in THBR and in T0.7−7 , as the cool lumps are excised. Some of the clusters
experience very large shifts in both quantities, while others experience no shifts at all. The
former are associated with clusters that have many well-defined and bright cool lumps, while
the latter correspond to those clusters that are completely free of cool lumps. The bottom
panel in this figure shows just the size of this shift for the THBR statistic, from which we
can see that some clusters indeed have a shift of precisely 0. Again, these are clusters
whose simulated X-ray observations are unchanged after applying the CIAO wavdetect tool
because it finds no sources to mask out. Comparing Figure 3.11 and 3.12, we also see that
the dependence of the broadband spectral-fit temperature on the aggressiveness of masking
does not depend strongly on aperture size.

65

Figure 3.11: (Top) Shift in the T0.7−7-THBR plane as wavdetect’s ellsigma parameter ranges
from 0 to 3. Clusters from the B04 sample are depicted with solid arrows, while clusters from
the D09 sample are depicted with long-dashed arrows. broadband spectral-fit temperatures
are estimated within an aperture set at R2500. Notice that the distribution of most of
the points shifts significantly to higher spectral fit temperatures and smaller temperature
ratios as cool lumps are excised. Notice also that some simulated clusters are unafflicted
by cool lumps, so that their net shift is 0. (Bottom) The THBR component (y-axis) of the
shift presented in the top panel. Note that these are for simulated X-MAS observations of
maximum quality, having approximately 120K counts.

66

3.4.4

Substructure Measures and THBR

Various researchers have established a clear correlation between morphological measures of
cluster dynamical state and δ ln M [Jeltema et al.(2008)], a correlation which is less apparent
in our study of spectral measures of substructure. Similarly, [Cavagnolo et al.(2008)] found
a correlation between the temperature ratio and structure for their Chandra sample, in that
merging events are associated with elevated THBR . To probe this issue further we focus on
THBR and compare it to several metrics for cluster substructure. These are the centroid
variation w, the axial ratio η, and the power ratios P20 and P30 [Ventimiglia et al.(2008)].
The centroid variation w measures the skewness of a cluster’s two-dimensional photon distribution by calculating the variance in a series of isophotes for the cluster surface-brightness
map. The axial ratio η measures a cluster’s elongation, which tends to increase during merger
events. The power ratios P20 and P30 decompose a cluster’s surface-brightness image into
two-dimensional multi-pole expansions, capturing different aspects of a cluster’s geometry.
P20 relates to the ellipticity in an image and is similar to the axial ratio η, while P30 measures
the “triangularity” in an image.
In [Ventimiglia et al.(2008)] we calculated these morphological substructure metrics for
a superset of the B04 sample of simulated galaxy clusters and compared them to departures
from the M-TX relation (see Fig. 2.6 in chapter 2). Here we use the same substructure
measures for the 114 B04 clusters for which we are able to calculate THBR and compare the
results. These are presented in Figure 3.12. Whereas in [Ventimiglia et al.(2008)] there is a
clear correlation between morphological substructure and δ ln M, we find in this study that
there is little or no correlation between substructure and THBR for our B04 sample.

67

Figure 3.12: Relationship between THBR and four measures of substructure: the axial ratio η,
the centroid variation w, and the power ratios P20 and P20 . Squares correspond to simulated
X-MAS observations that are processed by wavdetect with ellsigma set to 3 (full masking),
while circles correspond to observations that have ellsigma set to 0.

68

In order to understand how temperature structure and morphological structure can be
uncorrelated, we looked at four simulated clusters having approximately the same temperature (kB T ≃ 3.2keV ) and occupying the relative extremes in THBR and in the centroid
variation w. Two were selected for relatively low THBR (≃ 1.1) but large variation in w
(≃ 0.02–1.0). Two were selected for large THBR (≃ 1.4) but again large variation in the
morphological structure parameter ω (≃ 0.1–1.0). These clusters are presented in Figure
3.13 with surface-brightness contours overplotted and with the associated values for THBR
and w. The two clusters in the left column appear more symmetric in their contours, while
the two in the right column exhibit noticeable centroid shift. However, the morphologically
apparent structure in the lower right cluster is not observed spectrally in the temperature
ratio THBR . Evidently, neither morphology nor THBR is a perfect measure of relaxation in
our simulation sample, as it contains clusters with small centroid shift ω and obvious substructure, as in the upper left of this figure. And, it contains clusters with large w that are
nearly isothermal, as in the lower right of this figure.

3.5

Summary

We used a sample of galaxy clusters simulated with radiative cooling and supernova feedback, along with simulated Chandra X-ray observations of these clusters, to study temperature inhomogeneity as a signature of cluster dynamical state. Specifically, we adopted two
methods of quantifying temperature inhomogeneity spectroscopically, the temperature ratio
THBR [Mathiesen & Evrard(2001), Cavagnolo et al.(2008)] and the cool residual REScool .
The former is the ratio of a hardband X-ray spectral-fit temperature to a broadband temperature, and becomes greater than 1 for clusters whose ICM contains cool, over-luminous

69

Figure 3.13: Surface-brightness contours for four clusters drawn from our simulation sample, illustrating that temperature ratio and centroid shift are both imperfect measures of
relaxation. All four clusters are at nearly the same temperature, with kBT 3.2. The two in
the left column exhibit little centroid shift w, while the two in the right column have centroid shift near the maximum for the sample. The two clusters in the bottom row have low
THBR while the two in the top row have higher THBR suggesting the presence of multiple
temperature components.
sub-components. The latter is the excess broadband count rate relative to the count rate
predicted by a model fit to the hard X-ray band.

Though our simulated clusters are

typically less massive and have lower temperatures than the Chandra archive clusters in
[Cavagnolo et al.(2008)], we find that their temperature ratios THBR occupy generally the
same distribution as the observed clusters.
We also looked for an opportunity to combine THBR and the cool residual with the
mean mass-temperature relation to obtain better mass estimates than are achieved just
with the scaling relation alone. We find, however, that while both THBR and the REScool
are correlated with offset from the M-TX relation, these correlations are weak, at least for
this sample. We conclude that these measures of temperature inhomogeneity are not very
70

effective at reducing scatter in the mass-temperature relation.
Finally, we note a particular difficulty that arises when trying to use clusters from hydrodynamic simulations to calibrate scatter-correction observables based on temperature
inhomogeneity, such as THBR . Historically, simulated clusters have tended to exhibit their
own kind of “over-cooling problem”, in which dense lumps of luminous, cool gas associated
with merged sub-halos appear. These cool lumps are often excised from simulated clusters before their global properties are measured, but masking them appears to make the
remaining temperature structure of the simulated clusters overly homogeneous. This finding
suggests that real clusters may have cooler sub-components, that are more diffuse and less
concentrated than in their simulated counterparts. Some physical process, perhaps thermal conduction, turbulent heat transport [Dennis & Chandran(2005), Parrish et al.(2010),
Ruszkowski & Oh(2011)], or a more aggressive form of feedback, prevents cool lumps from
forming in real clusters and might not completely eliminate those temperature inhomogeneities. Newer simulations incorporate treatments of conduction and AGN feedback, as
well as more accurate treatments of mixing, and it will be interesting to revisit these temperature inhomogeneity measures in simulated clusters when large samples of such simulated
clusters become available.

71

counts ellsigma
15000
0
15000
1
15000
2
15000
3
30000
0
30000
1
30000
2
30000
3
60000
0
60000
1
60000
2
60000
3
120000
0
120000
1
120000
2
120000
3
Table 3.1: THBR & σT

HBR

a
THBR
1.19
1.17
1.13
1.13
1.16
1.13
1.10
1.10
1.13
1.11
1.08
1.07
1.12
1.10
1.07
1.07

b
σ a THBR
0.15 1.20
0.13 1.18
0.12 1.13
0.13 1.13
0.12 1.16
0.11 1.13
0.10 1.10
0.10 1.09
0.10 1.14
0.08 1.11
0.06 1.08
0.06 1.07
0.11 1.12
0.09 1.10
0.07 1.07
0.07 1.07

σb
0.15
0.13
0.11
0.12
0.12
0.11
0.10
0.10
0.10
0.08
0.06
0.06
0.12
0.09
0.07
0.06

with a All Clusters and b Clusters with kB T2.0−7 > 2 keV

72

Chapter 4
Modeling Galaxy Cluster Surveys

4.1

Introduction

Given the importance of galaxy clusters in precision cosmology in general and in helping
unravel the mystery of Dark Energy in particular, ambitious cluster surveys are planned for
the coming decades, such as at the South Pole Telescope [Song et al.(2012)], the Dark Energy
Survey [DES(2012)], eROSITA [Merloni et al.(2012)], Pan-STARRS [PanSTARRS(2012)],
and the LSST [LSST(2012)]. These surveys will take a census of a substantial fraction
of their number in large solid-angle surveys, and will make detailed studies of important
characteristics in more narrow, targeted surveys. Naturally, numerical and semi-analytical
models will be essential to such programs. While numerical simulations are powerful, general,
and sometimes resolve details with amazing precision, it is worthwhile to complement them
with semi-analytical models, which have their own virtues. Computationally efficient, they
sometimes can be crafted to yield deep insights. Semi-analytical galaxy cluster survey models
also are often built around just a few simple, key principles. One common approach is to join
a cosmological model to a structure formation model, subject to observational constraints.
In this work we take advantage of these common characteristics by abstracting them into a
simple program and library, called CosmoSurvey, which is efficient to evaluate on commodity
computing hardware and is adaptable to a variety of applications. It is a simple, convenient,
extensible, flexible, efficient model of flux-limited galaxy cluster surveys.
73

4.2

CosmoSurvey

CosmoSurvey is a semi-analytical modeling framework for galaxy cluster surveys, which
relies on the Press-Schechter formalism [Press & Schechter(1974)] augmented with halo mass
functions derived from numerical simulations. It is both a program in that it can be used outof-the-box to model flux-limited galaxy cluster redshift surveys simply from the commandline, and also a framework in that it provides library functions for modeling other kinds of
cluster surveys. Its overall design is described in the next section, while a user guide and
reference manual appears in an appendix.

4.2.1

Design

CosmoSurvey ultimately is a very simple program and set of tools. Its utility derives from a
coherent, deliberate set of goals, principles, and architectural choices.

4.2.1.1

Goals

Broadly speaking the goal of this project is to create a general-purpose framework for modeling galaxy cluster surveys according to a small set of guiding principles.

4.2.1.2

Principles

These guiding principles for CosmoSurvey are the following.
• It should be easy to understand.
• It should be easy to use.
• It should be easy to adapt.
74

• It should be easy on the hardware.
4.2.1.3

Architecture

Probably the first thing the user notices about CosmoSurvey as soon as they inevitably wish
to extend or adapt it, is that it is written in Fortran. This may seem an unusual choice at
first. While Fortran has a strong pedigree in science and engineering, it is undeniable that
this oldest general-purpose programming language still in use grows more out-of-fashion as
it enters its seventh decade.
Nevertheless, Fortran still has considerable life within it. It is still widely-used and
widely-understood in the scientific and engineering communities. There are abundant highquality libraries available in Fortran. Fortran excels in numerical computation. Fortran
is exceptionally fast and efficient. Newer versions of Fortran, like Fortran 95 and Fortran
2003 add modern programming features. There are excellent free and commercial compilers
for Fortran. And, newer versions of Fortran perform powerful optimizations, like autovectorizing array operations on multi-core hardware. CosmoSurvey is written in Fortran 95
in order to take advantage of these features, including the new module system, dynamic
memory allocation, and array auto-vectorization.
Modeling a galaxy cluster census as it does, CosmoSurvey naturally makes abundant
use of one-dimensional and multi-dimensional numerical integration. These services are
provided by robust, well-optimized, and sometimes ancient third-party libraries written in
either Fortran or C. One-dimensional integration is performed using the excellent Netlib
libraries, which are written in Fortran 77 and in some cases Fortran 66. Multi-dimensional
integration is performed using the CUBA library, which is written in modern C but is
equipped with a Fortran interface.
75

In addition to its core function of providing quantitative estimates of galaxy cluster survey
results, CosmoSurvey also provides additional plotting tools. These functions are supplied
by the PLplot library. Like CUBA, it also is written in C but possesses a convenient Fortran
interface.
Tailored to the writing of small command-line programs as it is, CosmoSurvey has a
flexible, easy-to-use system for creating command-line interfaces. This is no small task,
given that user interactivity, I/O, and text processing are among Fortran’s weakest areas.
Command-line interface functions are performed using the Kracken library.
CosmoSurvey has just a few core modules that can be combined and adapted to create
different kinds of cluster survey models. Out of the box, they supply the cosmosurvey,
cosmoplot, and cosmotest programs. Of these, the cosmosurvey program is the main program of interest. Its basic operation is to evaluate its internal model according to commandline arguments and emit observables to standard output. One nice aspect of this design is
that the use of command-line arguments and standard output make it easily scriptable.
The other programs provided out of the box are cosmoplot and cosmotest.

The

cosmoplot program is an auxiliary tool that helps the user visualize the internal survey
model. The cosmotest program is another tool that helps the user visualize specific components of the internal model.

4.2.1.4

Modules

As introduced above, CosmoSurvey has just a few modules. The core modules are constants
(which, in general, defines physical units), cosmology, and structure. Additional important
modules are the survey modules, which are easy to write. These modules, their details, and
the ways they relate are described in greater detail in the following paragraphs, and in the
76

appendix.
The cosmology module defines a user-defined type named cosmoparams, that collects in
one place all of the physical parameters of the joint cosmology-cluster-survey model. The
cosmology module defines a global variable of type cosmoparams and named theta_G. This
makes the model parameters in theta_G available to all functions that use the cosmology
module, which in general are all other modules and programs in the CosmoSurvey suite. This
choice was made for several reasons. First, it is convenient to have these model parameters
available globally, to avoid the cumbersome ritual of passing so many values into all of
the other functions that use them, of which there are many. Second, it turns out it is
necessary to have these model parameters available globally, since the Netlib and CUBA
numerical integration libraries prohibit the ability to communicate parameters into their
integrands directly. Third, Fortran 95 custom or user-defined types are a convenient way
to aggregate related data. The cosmology module also provides convenience functions for
returning instances of cosmoparams, the model parameter priors, and String names for these
model parameters. Finally and most important, the cosmology module provides functions
for computing global geometry, expansion history, comoving distances, comoving volumes,
matter and Dark Energy density parameters, and various commonly-used distance measures.
The structure module provides functions for computing the growth of linear perturbations, halo concentration, the M-T and L-M mass-observable relations for galaxy clusters,
for describing intrinsic scatter in these relations, and for calculating the luminosity and flux
of distant sources.
The survey module ties together the constants, cosmology, and structure modules.
It mainly provides functions for convolving the halo concentration, the comoving volume,
the mass-observable scatter, and the survey flux limit, to calculate integrated galaxy cluster
77

number counts in redshift-luminosity observable bins.
Finally, the cosmosurvey program brings these modules together, provides a run-time
context for the survey and its dependent modules, arranges for the specification and collection of command-line arguments, and generates the program output.

4.2.1.5

Trade-offs

It is important to acknowledge several trade-offs that were accepted in the design and writing
of CosmoSurvey. First, as a compiled language, Fortran is not as convenient or as flexible
as dynamic programming environments, like IDL or Python. This was accepted for the sake
of computational efficiency and for the convenience of running outside of a run-time, which
introduces other forms of complexity. Second, as noted above Fortran is out of fashion.
Fortran 95 and Fortran 2003 often are ignored, and Fortran in general is not as modern
a language, like even C is by comparison. This was accepted because of the many modern programming features Fortran 95 provides, because of the reported numerical efficiency
edge over C, and because though older, Fortran actually is easier to read and easier to program in than C/C++ is, so long as one avoids low-level system access, string and character
manipulation, and object-oriented programming. Third, CosmoSurvey uses PLplot. As a
library compiled into one’s executables, PLplot is not as flexible as data-oriented plotting
tools, like GNUplot or Excel, and is not as convenient as exploratory dynamic programming
environments, like IDL or R. This was accepted because those tools still can be used with
cosmosurvey output, and it is often convenient to build plotting and visualization functions
directly into your executables. In addition, PLplot is powerful, produces high-quality output in a dizzying array of formats, is well-documented, and is free. Fifth, while it may be
possible to build CosmoSurvey in a Windows environment, the package and especially its
78

build system is not designed for and has never been tested in this setting. This was accepted
because high-quality Unix-like programming, development, and analysis environments are
now widely available, and are nearly ubiquitous in astrophysics. Finally and probably most
important the cosmological, galaxy cluster, and survey models are without a doubt quite
simple. This was a deliberate choice, and was accepted because it is relatively simple to
adapt and augment them with additional model components, though it does require programming to do so. These simple component models are described in detail in the next
section.

4.2.1.6

Cosmological Model

We begin by describing in broad outline the cosmological model of the Universe in CosmoSurvey. For this we choose the concordance model for its simplicity, for its accessibility to
semi-analytical computational methods on commodity hardware, and for its success in explaining the observed properties of the Universe. Also called the ΛCDM model—emphasizing
the importance of Dark Energy and Cold Dark Matter—it rests on a small set of assumptions
[Longair(2008), Voit(2005)]
• The Universe is homogeneous and isotropic on large scales.
• It comprises an expanding, pressureless, curl-free fluid that obeys Weyl’s Postulate, so
that the fluid particles move along world lines that only meet at a common point in
the past.
• Its geometry and dynamics are governed by its contents and obey the principles of
General Relativity.
• It contains radiation, baryonic matter, Cold Dark Matter, and Dark Energy.
79

• At an early epoch its matter component received an imprint of Gaussian-random “primordial” density fluctuations that had a scale-free Fourier-space power spectrum.
A Lagrangian spherical coordinate system in which the radial component r moves along
with a fluid free of gradients and vorticity is the natural choice for an expanding, homogeneous Universe. This so-called comoving coordinate system has the virtue of separating out
of proper distances ρ a non-local, time-dependent scale factor a(t)

a(t) =

ρ(t)
r

(4.1)

that is taken to be unity at the current epoch. The scale factor can be put on an empirical
basis using Hubble’s Law
ρ = Hρ
˙

(4.2)

which describes the proportionality of proper distances to expansion velocity ρ = a r and
˙
˙
defines the Hubble parameter H.
H=

a
˙
a

(4.3)

The value H0 of the Hubble parameter at the current epoch is an important cosmological
parameter, sought by nearly a century of astronomers. It is currently estimated to be 72
km/s/Mpc.
Because the Universe is homogeneous it also possesses a non-local curvature κ and a
corresponding global radius of curvature R ∝ κ2 . A compact way to summarize these
geometric and kinematic properties is by way of the Robertson-Walker metric ds

ds2 = c2 dt2 − a2 (t)

dx2
+ x2 dφ2
1 − κ x2 /R2
80

(4.4)

where dφ is a transverse angle and where we make the transformation

x = { R sin(r/R), κ > 0, R sinh(r/R), κ < 0.

(4.5)

The Robertson-Walker metric is the starting point for developing useful relations between
time intervals, proper distances, and coordinate distances in this coordinate system. For
instance, as a light signal propagates it accumulates coordinate distance dr

dr(t) =

c dt
a(t)

(4.6)

whose integral from an observer at the origin O to an event at an earlier epoch t defines the
distance measure D
D=

t
to

dr(t′ )

(4.7)

In addition to these effects, light rays also are deflected by spatial curvature, and this deflection distorts observed angles. The angle subtended by a coordinate distance dr at coordinate
distance r becomes
dφ =

dr
x

(4.8)

where x is defined in equation 4.5. Equation 4.8 defines the angular-diameter distance
measure
DA = a x

81

(4.9)

Equation 4.8 allows us to specify a comoving volume element dV (t) at distance ρ(t) in solid
angle dΩ and comoving coordinate interval dr.

dV (t) = x dθ x dφ dr
= x2 dΩ

c dt
a(t)

(4.10)

The time coordinate t in equations 4.6 and 4.10 can be replaced with the observable
redshift z, which measures the stretch a photon wavetrain experiences in an expanding
Universe. Expressed as a ratio of observed to emitted wavelengths, this stretch is merely the
ratio of the scale factor between the observed and emitted times.

λo
a(to )
=
λe
a(te )

(4.11)

Since observations are always made at the current epoch when a(to ) = 1, the scale factor at
the emitted time can be relabeled simply a, so that a = λe /λo . The observed redshift is the
fractional difference between a photon’s emitted wavelength λe and its observed wavelength
λo .
z=

λo − λe
λe

(4.12)

Therefore, redshift z measures scale factor a by way of the important relation

a=

1
1+z

82

(4.13)

with intervals of z and a related by

da = −

1
dz = − a2 dz
(1 + z)2

(4.14)

Connecting the time interval dt to the Hubble parameter H and scale factor a through
equation 4.3,
dt =

da
aH

(4.15)

using equation 4.14, and remembering that x is itself a function of r and t, equations 4.6
and 4.10 become
dr(z) =

c
dz
H(z)

dV (z) = x2 dΩ

c
dz
H(z)

(4.16)
(4.17)

The geometry of a homogeneous and isotropic universe, characterized by its curvature
κ and corresponding radius of curvature R, is a function of its mass-energy content and
is governed by General Relativity. This is described with the Friedmann-Lemaitre model,
which uses the dynamical equation

a
¨
4
= − πGρ
a
3

(4.18)

where ρ is the mean energy density owing to matter, energy, and pressure. Note that in
this context the symbol ρ as a density should not be confused with the same symbol used to
signify proper distances, as in the discussion leading to equation 4.17.
Of course, the density ρ is itself a function of the scale factor a, which can be factored
out along with a scale-independent density ρ0 , so that ρ = ρ0 f (a). Multiplying equation

83

4.18 by a and using this substitution we obtain

a=−
¨

4πGρ0
.
a2

(4.19)

Equation 4.19 can be integrated once using energy conservation

ρ = −3
˙

a
˙
ρ+p ,
a

(4.20)

so long as we have an equation of state to relate the pressure p to the density ρ. If the
pressure is proportional to the density with p = w ρ, then we obtain

8
a2 = πGρ0 a−(1+3w) + const.
˙
3

(4.21)

The constant of integration in equation 4.21 relates to the global curvature of the Universe.
Notwithstanding the radiation-dominated era at high redshift, and before Dark Energy dominates, the Universe is well-represented by a pressureless dust model with w = 0. If we imagine
a flat Universe with zero curvature everywhere, from equation 4.21 we can define a critical
density ρc which would bring it about,
2
3H0
.
ρc =
8πG

(4.22)

This is a useful unit for measuring density and when factored out allows us to define a
dimensionless density parameter Ω0 ,
ρ
Ω0 = 0 .
ρc

84

(4.23)

In fact, we can define a family of density parameters for any given component x with density
ρx —such as radiation or Dark Energy—in a similar way, so that

Ωx =

ρx
.
ρc

(4.24)

Equation 4.21 then becomes for a flat Universe, when incorporating the equation of state
parameter w, the Hubble constant H0 , the density parameter Ω0 , and switching from scale
factor a to its observable analog the redshift z

H 2 (z) =

a 2
˙
2
2
= H0 E 2 (z) = H0 ΩM (1 + z)3 + ΩΛ .
a

(4.25)

This is another important relation, and E(z) is one of the critical computations in a semianalytical cosmology model.

4.2.1.7

Cluster Model

We assumed that the cosmological model of an expanding Universe is homogeneous and
isotropic on large scales, because observations lead us to that conclusion. But observations
closer to home also tell us that the Universe is decidedly not homogeneous on smaller scales,
and this fact is captured by another of our assumptions. In the concordance model, we
posit that the Universe at an early time received an imprint of density fluctuations in its
Dark Matter component, and adopt a model in which these “primordial fluctuations” grow
in contrast to the uniform background density under the influence of their own self-gravity,
eventually collapsing into gravitationally-bound structures.
The natural framework for that discussion and the subject of structure formation begins

85

with the density contrast
ρ(x) − ρ
.
ρ

δ(x) =

(4.26)

We can relate the density contrast field ρ(x) to an analogous mass perturbation field δM/M(x)
if we adopt some appropriate smoothing kernel WR (r) with a characteristic length scale R.
δM
(x) =
M R

δ(x)WR (|x − r|)d3 r

(4.27)

Of course, we are not interested in the density contrast or mass perturbation fields at any
particular location x, but rather in their statistical properties. A useful starting point that
paves the way to observable comparisons is to begin with the field’s Fourier components

δk (k) =

δ(x)eik·x

(4.28)

and their isotropic power spectrum

P (k) = |δk |2 .

(4.29)

Building up our statistical picture, we can imagine sampling the perturbation field to obtain
its variance on the length scale of the smoothing kernel WR , using its Fourier transform Wk

σ2

R

=

1
(2π)3

P (k)|Wk |2 d3 k.

(4.30)

In all these relations, it is useful to consider the mass that would be contained within the
smoothing kernel given the mean background density. In that case, equations like 4.30
pertain not just to a given length scale but also to a corresponding mass scale.
86

In our post-radiation era matter-dominated dust model, the amplitude of matter density fluctuations grows linearly with the scale factor a until they reach a density contrast
δρ/ρ ≈ 1 at which point the growth rapidly becomes non-linear. The fluctuations become
gravitationally bound collapsed structures that have “detached” from the global expansion
of the Universe. In our model they are associated with the Dark Matter halos that trapped
ordinary baryonic matter and are the sites of galaxy and cluster formation. In any case,
the full treatment of the growth of perturbations through the radiation-dominated era and
considering Dark Energy is a complex topic which will not be explored in detail here. Fortunately, numerically-integrated growth functions G(z) ∝ δρ/ρ exist. While it is easy in
CosmoSurvey to use different growth functions, we adopt that of Linder [Linder(2005)] because it opens the door to modeling the effects that departures from General Relativity
might have on structure growth
∞ ΩM (z)γ − 1
1
exp
.
G(z) =
1+z
1+z
z

(4.31)

which appears in figure 4.1 The level of departure from General Relativity is controlled by
the γ parameter, with a value of 0.55 corresponding to no departure.
As the density fluctuations grow in contrast and approach δρ/ρ ≈ 1, the growth of their
corresponding mass fluctuations becomes non-linear and we assume they collapse into bound
Dark Matter halos. We need a prescription for modeling this, and an elegantly simple one
can be found in the Press-Schechter formalism. Assuming the initial perturbation field to
have a Gaussian sample variance σ, then the probability that any particular fluctuation has
√
a contrast above some threshold δc is simply the error function erf c[δc / 2σ]. Keep in mind
that the variance σ will itself be be a function of the scale of the perturbation, according

87

0.0

0.2

0.4

G(z)

0.6

0.8

1.0

Growth Factor

0.5

1.0

1.5

2.0

z
Figure 4.1: Growth factor G(z) for the growth of density fluctuations.

88

2.5

to the power spectrum of initial perturbations. By “scale”, that is the length-scale of the
perturbation, the related wavenumber, or more usefully, the enclosed mass M. It will also be
a function of the redshift z via the growth factor G(z), with σ(M, z) ∝ G(z). Then with the
number density of mass perturbations of mass M simply ΩM ρc /M, the density of collapsed
halos will be
n(M, z) =

ΩM ρc
σc
erf c √
M
2σ(M, z)

(4.32)

and the differential mass-function is then
dn
2 ΩM ρc δc
δ2
=−
exp − c2 .
d ln σ
π M σ
2σ

(4.33)

The Press-Schechter approach is an invaluable way to frame the issue of modeling the
collapse of Dark Matter halos, but these days more accurate results are obtained via numerical simulations. CosmoSurvey adopts a mass function produced from simulations in just
such a way [Jenkins et al.(2001)].

Ω ρc
dn
= −AJ M exp (−|BJ − ln σ|ǫJ )
d ln σ
M

(4.34)

where AJ , BJ , ǫJ are parameters fit from numerical simulations with AJ = 0.301, BJ = 0.64,
and ǫJ = 3.82.
Again, in these relations σ is the variance in density fluctuations on a given length scale,
which can be mapped to a given mass. Simple application of the chain rule translates
equations 4.33 and 4.34 into true mass functions. The number density of clusters of mass
M is then
Ω (z)ρc
dn
= M
AJ exp (−|BJ − ln σ|)
d ln M
M
89

(4.35)

Mass functions like this appear in figure 4.2, where the exponential cutoff for the largest
mass galaxy clusters is apparent.
The cosmological and cluster model provide the basic framework for the CosmoSurvey
model. The final remaining ingredient for translating that into a model that could, in
principle, be compared to observations, is a treatment of the survey itself. This necessarily
also requires delving into the the observable properties of galaxy clusters, and how they
relate to the only property we have considered so far, which is their mass.

4.2.1.8

Survey Model

As introduced above, the survey model necessarily must consider the observable properties of
galaxy clusters. Two observables that are featured prominently in the CosmoSurvey model
are the cluster luminosity in a broad X-ray band and the temperature of the hot, tenuous
gas that emits the X-rays.
For it turns out that in certain respects galaxy clusters are only incidentally about galaxies. To start, the bulk of a cluster’s mass is in its collapsed Dark Matter halo whose gravity
binds the cluster together. In addition, the remaining baryonic matter comprises mostly a
dilute, hot, ionized hydrogen gas that copiously emits X-rays in the 1-10 keV energy range
via the thermal bremsstrahlung process. The X-ray luminosity in some well-defined suitably
chosen range then offers one important observable. What is more, we can adopt a thermal
model for the X-ray emitting intra-cluster medium (ICM) and infer temperatures for the
components of that model. Chapter 3 of this work uses simulated galaxy clusters to examine
models with multiple thermal components, but a common and useful starting point is simply
to assume the entire ICM is in thermal equilibrium at one temperature. That is the approach
CosmoSurvey uses, so that we can talk about “the temperature” as another observable in
90

10

-3

Mass Function

-6

10

-7

10
10

-10

10

-9

10

-8

dn/dlnM (M)

10

-5

10

-4

z = 0.0
z = 0.5
z = 1.0

13

10

14

15

10

10
M

Figure 4.2: Differential mass function for the number density of collapsed Dark Matter halos,
at three different redshifts

91

addition to luminosity.
Next, we need to link the X-ray luminosity and temperature to the theoretical model
we have built up so far, which as equations 4.32, 4.33, and 4.34 show has only considered a
cluster’s mass. The simplest non-trivial way we could relate mass M, X-ray luminosity L,
and X-ray temperature T is in power-law form, and this turns out to be a good choice on
both theoretical and observational grounds. Structure in general arises from initial density
perturbations whose spectrum is assumed to be scale-free, and its growth is driven by gravitational processes that have no way to impart scale. If we assume that the processes that
heat the ICM also to be scale free (significant line emission, non-linear thermal and dynamical instabilities, and cluster energy feedback will violate this assumption), then a scale-free
power law is the only choice for relating galaxy cluster mass, X-ray luminosity, and X-ray
temperature.
L
=
L0

M
M0

β

T
=
T0

M
M0

α

(4.36)

(4.37)

Where L0 , T0 , and M0 are some fiducial values for luminosity, temperature, and mass, and
β and α are the power-law slopes for these relationships. Together, they comprise additional
parameters for our model, made more vivid when equations 4.36 and 4.37 are cast in a more
convenient logarithmic form. Defining λ = ln L, τ = ln T , and µ = ln M, then equation 4.36
becomes
λ = βµ + ln L0 − β ln M0 .

92

(4.38)

Similarly, equation 4.37 becomes

τ = αµ + ln T0 − α ln M0 .

(4.39)

This formulation for the luminosity-mass (L-M) and temperature-mass (T -M) relationships makes it easy to ratchet up by one notch the complexity and richness of what is
otherwise still a very simple model. In particular, we can consider the phenomenon of scatter in these mass-observable relationships. Scaling arguments aside, we expect clusters not
to adhere rigidly to the mass-observable relationships, and this expectation is supported by
observations. There is scatter in the L-M and T -M relationships. Scale-imposing processes
in the ICM, in galaxy cluster feedback, and in galaxy formation break the symmetry of selfsimilarity. X-ray luminous active galactic nuclei (AGN) may “contaminate” X-ray surveys
for galaxy clusters and introduce confusion. Further complicating matters, scatter itself may
evolve with redshift z. Still, given a model for scatter with new parameters, and enough
of the right kind of data, perhaps we could fit for the scatter parameters. Whether that is
practical or even possible for a given scatter model is the question CosmoSurvey originally
was intended to help answer. CosmoSurvey adopts a simple model of log-normal scatter in
the L-M and T -M relationships, constant in mass but evolving with redshift z. Defining σλ
and στ to be the standard deviations in log luminosity λ and log temperature τ , we adopt
a simple power-law redshift evolution, with

σλ = σλ0 (1 + z)φ ,

(4.40)

στ = στ 0 (1 + z)ψ .

(4.41)

93

The CosmoSurvey cluster model also treats covariance in luminosity and temperature scatter,
with a covariance parameter ρ, and a covariance matrix C





C=

σλ σλ
ρστ σλ



ρσλ στ 

στ στ




(4.42)

With this Gaussian scatter the likelihood that a cluster of log mass µ will have log luminosity
λ and log temperature τ is straightforward. Its joint probability density function is merely
the multivariate normal distribution. With two random variables λ and τ whose expectation
values λ0 and τ0 at fixed µ are given by the L-M and T -M relationships we can define the
vector X as





X=




λ − λ0 

τ − τ0

.


(4.43)

The joint conditional probability P (λ, τ |µ) that a cluster of log mass µ will have log luminosity λ and log temperature τ is

P (λ, τ |µ) =

1
1
exp − XT C−1 X
2π |C|
2

(4.44)

where |C| is the determinant of the scatter covariance matrix C and C−1 is its inverse.
While CosmoSurvey easily could be adapted to support other survey types, out-of-the box
it models flux-limited observations of galaxy cluster number counts in luminosity and redshift
space. The product of equations 4.35 and 4.44 can be integrated over a log luminosity interval
∆λ = [λ1 , λ2 ], a redshift interval ∆z = [z1, z2], and over all log mass µ and log temperature
τ , multiplied by a solid angle ∆Ω, and restricted by a suitable flux limit calculation. The

94

integrand is
dN
dn
=
P (λ, τ |µ)
dµ dτ dλ dz dΩ
d ln M

(4.45)

The result is the expected number of observable clusters in that luminosity-redshift bin, Ni

Ni =

z=z2
z=z1

λ=λ2
λ=λ1

τ =∞

z=∞

dN
dµ′ dτ ′ dλ′ dz ′ ∆Ω.
τ =−∞ z=−∞ dµ dτ dλ dz dΩ

(4.46)

CosmoSurvey adopts a hard flux limit by applying a redshift-dependent cutoff to the integration in log luminosity λ, given a flux f

λc = ln

2
4πDL f
L0

(4.47)

where DL is the luminosity distance measure DL . This flux limit calculation is depicted in
figure 4.3
With these labors and using an adaptive multi-dimensional integration package, CosmoSurvey obtains a matrix of observables N comprising counts Nij . These are the expected
galaxy cluster number counts in log luminosity and redshift bins. Partly as a computational
convenience, but also to emphasize their role as independent observables, CosmoSurvey unrolls the matrix into a vector of observables θ = (θ1 , θ2 , . . . , θi , . . .) when generating its final
output. The matrix of observed number counts is plotted in figure 4.4.

4.3

Summary

CosmoSurvey is a program and a framework for quickly and easily modeling flux-limited
galaxy cluster surveys on commodity hardware. Written in modern Fortran 95, it is fast,

95

10

42

10

43

L (z)

10

44

10

45

Flux Limit

0.5

1.0

1.5

2.0

2.5

z
Figure 4.3: Flux limit calculation used to cut off the integration of galaxy cluster number
counts over a luminosity bin.

96

Cluster Number Counts

104

dN

103
102
101
100
10-1
0.5

1.0

1.
z 5 2
.0

45

LObs

2
10
m)
/c
(ergs/s

Figure 4.4: CosmoSurvey simulated cluster survey counts in observable bins, obtained using
the cosmoplot program.

97

efficient, and easily adapted. It and programs written with its library run on the commandline, using a simple interface. It also emits estimated number counts of observed galaxy
clusters in luminosity and redshift bins, straight to standard output, which is convenient
for applications in a scripting environment. CosmoSurvey adopts the standard ΛCDM cosmological model, and its model for structure formation is rooted in the Press-Schechter formalism. It has found application in the evaluation of Monte Carlo Markov Chain (MCMC)
engines, and future extensions may find application in modeling the effects of scatter in the
mass-observable relations for clusters, redshift evolution in that scatter, and in the effect of
point-source contamination of cluster surveys from X-ray luminous Active Galactic Nuclei
(AGN).

98

Chapter 5
Summary
In this work analyze ways of coping with scatter in the mass-observable scaling relationships
for clusters of galaxies, and present software components we have developed to aid in this
effort.
We began by using a sample of galaxy clusters simulated with cooling and feedback.
With this sample, we investigated three substructure statistics and their correlations with
temperature and mass offsets from mean scaling relations in the M-TX plane. First, we
showed that the substructure statistics w, η, P20 and P30 all correlate significantly with
δ ln TX , though with non-negligible scatter. In all cases this scatter is larger for δ ln TSL
than it is for δ ln TEW . Next, we considered the possibility that M-TX scatter is driven by
low-mass clusters. We tested the degree to which scatter can be reduced by filtering out
these systems. This consisted of performing a cut at 2 keV, for which we saw that it yielded
a modest improvement in mass estimates. To see whether incorporating substructure could
refine these mass estimates, we first showed that w, η, P20 , and P30 correlate significantly
with the difference δ ln M between masses predicted from the mean M(TX ) relation and
the true cluster masses, with non-negligible scatter that again is less for M(TEW ) than it
is for M(TSL ). Then we adopted a full three-parameter model, M-TX -S, which includes
substructure information S estimated using w, η, P20 , and P30 . Scatter about the basic twoparameter M-TEW relation was 0.094. Including substructure as a third parameter reduced
the scatter to 0.072 for centroid variation, 0.084 for axial ratio, 0.081 for P20 , and 0.084
99

for P30 . Scatter about the basic two-parameter M-TSL relation was 0.124, and including
substructure as a third parameter reduced the scatter to 0.085 for centroid variation, 0.112
for axial ratio, 0.110 for P20 , and 0.108 for P30 . As one last test, and to increase our
confidence that our substructure measures are not relying on potentially non-physical core
structure in the simulations, we also repeated the comparison of mass-estimates for TEW ,
with the core regions of the clusters excised. First, removing the core slightly increased the
scatter in M-TX possibly by making the average temperature more sensitive to structure
outside the core. Second, even with the cores removed the improvement in mass-estimates
obtained using substructure information remains. Based on these results, it appears that
centroid variation is the best substructure statistic to use when including a substructure
correction in the M-TEW relation. However, the correlations we have found in this sample
of simulated clusters might not hold in samples of real clusters, because relaxed clusters in
the real universe tend to have cooler cores than our simulated clusters do.
Next, we adopted two methods of quantifying temperature inhomogeneity spectroscopically, the temperature ratio THBR [Mathiesen & Evrard(2001), Cavagnolo et al.(2008)] and
the cool residual REScool . The former is the ratio of a hardband X-ray spectral-fit temperature to a broadband temperature, and becomes greater than 1 for clusters whose ICM
contains cool, over-luminous sub-components. The latter is the excess broadband count rate
relative to the count rate predicted by a model fit to the hard X-ray band. Though our
simulated clusters are typically less massive and have lower temperatures than the Chandra archive clusters in [Cavagnolo et al.(2008)], we find that their temperature ratios THBR
occupy generally the same distribution as the observed clusters.
We also looked for an opportunity to combine THBR and the cool residual with the
mean mass-temperature relation to obtain better mass estimates than are achieved just
100

with the scaling relation alone. We find, however, that while both THBR and the REScool
are correlated with offset from the M-TX relation, these correlations are weak, at least for
this sample. We conclude that these measures of temperature inhomogeneity are not very
effective at reducing scatter in the mass-temperature relation.
In examining temperature inhomogeneity, we note a particular difficulty that arises
when trying to use clusters from hydrodynamic simulations to calibrate scatter-correction
observables based on temperature inhomogeneity, such as THBR . Historically, simulated
clusters have tended to exhibit their own kind of “over-cooling problem”, in which dense
lumps of luminous, cool gas associated with merged sub-halos appear. These cool lumps
are often excised from simulated clusters before their global properties are measured, but
masking them appears to make the remaining temperature structure of the simulated clusters overly homogeneous. This finding suggests that real clusters may have cooler subcomponents, that are more diffuse and less concentrated than in their simulated counterparts. Some physical process, perhaps thermal conduction, turbulent heat transport
[Dennis & Chandran(2005), Parrish et al.(2010), Ruszkowski & Oh(2011)], or a more aggressive form of feedback, prevents cool lumps from forming in real clusters and might
not completely eliminate those temperature inhomogeneities. Newer simulations incorporate treatments of conduction and AGN feedback, as well as more accurate treatments of
mixing, and it will be interesting to revisit these temperature inhomogeneity measures in
simulated clusters when large samples of such simulated clusters become available.
Finally we present the CosmoSurvey software for quickly and efficiently modeling simple
flux-limited galaxy cluster surveys on commodity hardware. The software and its library are
easily adapted, and can find applications in a variety of settings. It has been used as the
basis of the likelihood function in evaluating a Monte Carlo Markov Chain (MCMC) engine,
101

and work is underway to adapt it for evaluating the effects of point-source contamination
of flux-limited X-ray surveys, due to Active Galactic Nuclei (AGN). While limited in scope
and sophistication, we hope that CosmoSurvey also helps illustrate an effective approach
for building small, fast, efficient semi-analytical models using general purpose programming
tools.

102

APPENDICES

103

Appendix A
CosmoSurvey User Guide
CosmoSurvey comprises a small set of simple tools and functions for modeling large-area
survey of galaxy clusters. It is small, simple, and efficient on commodity hardware, and is
easily adapted.

A.1

Quick Start

Getting started with CosmoSurvey is simple once a small set of requirements are met. These
are laid out in detail in the rest of this user guide. This provides just the highlights, providing
a quick path to using the tool for the first time.

A.1.1

Requirements

CosmoSurvey has these requirements.
• A POSIX-compliant operating system such as as Linux, MacOS, or Unix
• A Fortran90 compiler, such as GFortran or the Intel ifort compiler
• The PLPlot plotting library
http://http://plplot.sourceforge.net/
• Version 1.6 of the Cuba numerical integration library
http://www.feynarts.de/cuba/Cuba-1.6.tar.gz
104

A.1.2

Download

CosmoSurvey can be download as a ZIP archive or as a tar.gz archive from the following
location.
http://dventimi.github.com/CosmoSurvey/

A.1.3

Install

Follow these steps to build and install CosmoSurvey.
• Download and install requirements: PLPlot and Cuba.
• Download the CosmoSurvey archive file.
• Unzip the archive file to any convenient temporary location.
• In CosmoSurvey’s top-level directory, run the following commands.
./configure
make
make install

Note that the last command make install probably will need to be run with sufficient
permissions. This can be achieved in a variety of ways, such as logging in as or changing to
a “super-user” account, or by running sudo make install.

A.1.4

Run

Running CosmoSurvey evaluates the model for a given set of cosmological, cluster, and
survey parameters. It then emits to standard output a list of numbers that represent galaxy
105

cluster number counts in observable bins. Parameters may be supplied on the command
line, though they are optional since there are a set of default values. For example, simply
running from the command line
cosmosurvey
Produces this output
48926.125
10116.227
1412.8425
120.24377
5.5203319
...
1.55754946E-03
The number counts emitted serially correspond to number counts in observable bins Nij ,
for i bins in redshift z and j bins in log luminosity ln L. Essentially, it is a matrix of
observables with rows for redshift and columns for luminosity, and unrolled into a flat list in
a column-first order.

A.2

Simulating A Survey

As discussed above, simulating a survey, evaluating it at a given set of parameters θi , and
emitting the observables to standard is the primary function of CosmoSurvey. The relevant
input parameters are as follows, with the default values in parentheses.
-z1

(0.01)

Redshift start
106

-z2

Redshift end

-nz

(10)

Number of redshift bins

-l1

(44.0)

log10(L1) luminosity start, [L1] = ergs/s

-l2

(46.0)

log10(L2) luminosity end, [L2] = ergs/s

-nl

(10)

Number of luminosity bins

-do

(12.0)

Survey solid angle dOmega, [dOmega] = steradians

-fl

(1.25e-13)

Survey flux limit, [fl] = ergs/s/cm^2

-w

(-1.0)

Dark Energy equation of state parameter w

-om

(0.30)

Matter density parameter Omega_M

-ol

(0.70)

Dark Energy density parameter Omega_Lambda

-s8

(0.80)

Power spectrum normalization sigma_8

-g

A.3

(2.50)

(0.55)

Linder growth index gamma

Visualizing A Survey

While the cosmosurvey program itself probably is the most useful insofar as it can be adapted
to other uses, such as integration into a Monte Carlo Markov Chain (MCMC) engine, it
sometimes is helpful to visualize survey observables. Of course the output of cosmosurvey,
being a series of numbers to standard out, can be plotted in any number of plotting tools
(e.g., GNUplot, SuperMongo, and R). However, as a convenience the results of a survey
can be plotted directly, using the cosmoplot program. The cosmoplot program takes the
same set of cosmological, cluster, and survey parameters as the cosmosurvey program does.
However, it also prompts the user for an output device, so that the plot can be displayed
directly to the user or written out to a file in a variety of formats (e.g., SVG, JPEG, GIF,

107

PostScript, and PDF). The output is displayed as a three-dimensional plot, with redshift
bins in z along one horizontal axis, log luminosity bins along the other horizontal axis, and
galaxy cluster number counts along the vertical axis. An example of its output for the same
set of default parameters that cosmosurvey uses appears in figure A.1.

108

Cluster Number Counts

104

dN

103
102
101
100
10-1
0.5

1.0

1.
z 5 2
.0

45

LObs

2
10
s/cm )
(ergs/

Figure A.1: CosmoSurvey simulated cluster survey counts in observable bins, obtained using
the cosmoplot program.

109

A.4

Fortran API

The Fortran API comprises six Fortran95 modules. Along with programs, modules are the
primary unit for code packaging and re-use in Fortran95. The CosmoSurvey packages are
quadpack, quadrature, utils, constants, cosmology, and structure.

A.4.1

quadpack

The quadpack module is a Fortran 90 port of a Fortran 77 library for numerical integration of one-dimensional functions, and is in the public domain. The Fortran 77 version of
the integration sub-programs are available on Netlib, a repository of software for scientific
computing maintained by AT&T, Bell Laboratories, the University of Tennessee, and Oak
Ridge National Laboratory.

A.4.2

quadrature

The quadrature module is an adapter for the one-dimensional integrators in the quadpack
module and the multi-dimensional integrators in the Cuba library.

A.4.3

utils

The utils module is an assortment of “helper” functions that are optional and operate
outside of the core CosmoSurvey functionality.

A.4.3.1

Subprograms

hist3d dist x1 x2 y1 y2 file floor : This is a subroutine that writes out the twodimensional floating-point array dist to file in a format suitable for the 3-D histogram

110

plotting functions in GNUPlot. The parameters x1, x2, y1, and y2 define the range of
the resulting plot data.
isoline iso x1 x2 y1 y2 file i floor : This is a subroutine that generates a GNUPlotcompatible isoline from one row of data in row i of the data in iso. The parameters
x1, x2, y1, and y2 define the range of the resulting plot data. The data are written
out to file. The parameter floor represents the lowest value in the plot.
hist_3d x y z : This is a subroutine that regrids multi-dimensional data in z into bins,
in a format compatible with the PLPlot plotting library. The parameters x and y are
one-dimensional arrays representing the two independent coordinates.
f_nvpair name:
f_split string char :
f_records filename:
f_record_count fu:
json_matrix m unit:

A.4.4

constants

A.4.4.1

Data

111

A.4.5

cosmology

A.4.5.1

Subprograms

f_lcdm :
f_lcdm_priors :
f_lcdm_names :
f_E z :
E(z) =

ΩM (1 + z)3 + ΩΛ (1 + z)3(1+w)

(A.1)

f_dr_dz z argv :
dr
c
=
dz
H(z)

(A.2)

f_r z1 z2 :
z2

dr ′
dz
z1 dz

(A.3)

dV
r 2 (z)
=
dz dΩ
H(z)

(A.4)

r=

f_dV_dOmega_dz z argv :

f_dV z1 z2 dOmega:
V =

z2

dV
dz ′ dΩ
z1 dz dΩ

(A.5)

f_Omega_M z :
ΩM =

ΩM0 (1 + z)3

112

E 2 (z)

(A.6)

f_Omega_lambda z :
ΩΛ =

1 − ΩM0
E 2 (z)

(A.7)

f_D z :
D = r(z)

(A.8)

DL = (1 + z)D(z)

(A.9)

Ω (z)γ−1
dG
= M
dz
1+z

(A.10)

f_D_L z :

A.4.6

structure

f_Integrand_Linder

f_G
G(z) =

z2

dG
z1 dz

(A.11)

f_ln_inv_sigma
σ
ln σ −1 = ax + bx2 − ln g(z) − ln 8
0.3

(A.12)

αeff = a + 2bx

(A.13)

f (ln σ −1 ) = a exp(−| ln σ −1 + b|ǫ )

(A.14)

f_alpha_eff

f_mass_fraction

113

f_dN_dV_dmu
dN
ρc
= ΩM (z) αeff (M) f (ln σ −1 (M, z)
dV dµ
M

(A.15)

τ (µ) = τ0 + α µ + α ln(E(z))

(A.16)

dµ
=α
dτ

(A.17)

λ(µ) = λ0 + µβ + log E(z)2

(A.18)

log E(z)2
µ(λ) = λ − λ0 −
β

(A.19)

f_tau_of_mu

f_dtau_dmu

f_lambda_of_mu

f_mu_of_lambda

f_Cov






Σ=

στ στ
ρ σλ στ



ρ στ σλ 

σλ σλ




(A.20)

f_C_det
|Σ|−1 = Σ11 Σ22 − Σ12 Σ21

(A.21)

L = 4 π DL (z)2 f

(A.22)

f_luminosity

114

Appendix B
CosmoFisher Installation Guide
CosmoFisher comprises a small set of simple tools and functions for computing and working
with Fisher Information matrices for a numerical model.

B.1

CosmoFisher Quick Start

Getting started with CosmoFisher is simple once a small set of requirements are met. These
are laid out in detail in the rest of this user guide. This provides just the highlights, providing
a quick path to using the tool for the first time.

B.1.1

Requirements

CosmoFisher has these requirements.
• A POSIX-compliant operating system such as as Linux, MacOS, or Unix
• The Python language interpreter
• The Numdifftools differentiation library
http://pypi.python.org/pypi/Numdifftools/0.3.1
• The SciPy scientific analysis library
http://www.scipy.org/

115

• The MatPlotLib plotting library http://matplotlib.sourceforge.net/

B.1.2

Download

CosmoFisher can be download as a ZIP archive or as a tar.gz archive from the following
location.
http://dventimi.github.com/CosmoFisher/

B.1.3

Install

Follow these steps to build and install CosmoFisher.
• Download and install requirements: Python, NumDiffTools, SciPy, and MatPlotLib. x
• Download the CosmoFisher archive file.
• Unzip the archive file to any convenient temporary location.
• In CosmoFisher’s top-level directory, run the following commands.
./configure
make
make install

Note that the last command make install probably will need to be run with sufficient
permissions. This can be achieved in a variety of ways, such as logging in as or changing to
a “super-user” account, or by running sudo make install.

116

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