thesis_mark5

Topics in Microlensing and Dark Energy
By
Mark Yashar
B.A. (San Francisco State University) 1994
M.S. (San Francisco Sate University) 1999
DISSERTATION
Submitted in partial satisfaction of the requirements for the degree of
Doctor of Philosophy
in
Physics
in the
OFFICE OF GRADUATE STUDIES
of the
UNIVERSITY OF CALIFORNIA
DAVIS
Approved:
Committee in Charge
2008
i

Mark Yashar
December 2008
Physics
Topics in Microlensing and Dark Energy
Abstract
In this dissertation we describe two separate research projects. The first project
involves the utilization and development of reddening models, color magnitude diagrams
(CMDs), and microlensing population models of the Large Magellanic Cloud (LMC) to
constrain the locations of micro-lensing source stars and micro-lensing objects in the Large
Magellanic Cloud and the Milky Way (MW) halo using data of 13 microlensing source stars
obtained by the MACHO (massive compact halo objects) collaboration with the Hubble
Space Telescope. This analysis suggests that the source stars are located in the LMC disk
and the lenses are located in the MW halo. For the second project, we report on the
results of a Markov Chain Monte Carlo (MCMC) analysis of an inverse power law (IPL)
quintessence model using the Dark Energy Task Force (DETF) simulated data models as a
representation of future dark energy experiments. Simulated data sets were generated for
a Lambda cold dark matter (ΛCDM) background cosmology as well as a case where the
dark energy is provided by a specific IPL fiducial model. The results are presented in the
form of error contours generated by these two background cosmologies which are then used
to consider the effects of future dark energy projects on IPL scalar field models and are
able to demonstrate the power of DETF Stage 4 data sets in the context of the IPL model.
We find that the respective increase in constraining power with higher quality data sets
produced by our analysis gives results that are broadly consistent with the DETF results
ii

for the w0 − wa parameterization of dark energy. Finally, using our simulated data sets
constructed around a fiducial IPL model, we find that for a universe containing dark energy
described by such a scalar field, a cosmological constant can be excluded by Stage 4 data
at the 3σ level.
iii

To my parents,
M. and F. Yashar.
iv

Acknowledgements
First of all, I would like to thank my adviser Professor Andreas Albrecht for all of
his help, guidance, time and generosity throughout this process. I would also like to send
my sincere condolences to him and his family for the tragedy that they have been through.
I also thank my coauthors and collaborators, Brandon Bozek, Michael Barnard,
and Augusta Abrahamse, for their help, time, generosity, and patience. Significant portions
of Chapter 3 have been drawn from Yashar, M et al., “Exploring Parameter Constraints
on Quintessential Dark Energy: the Inverse Power Law Model”, (2008), in preparation. I
acknowledge the contribution of my co-authors on this work: B.Bozek, A. Abrahamse, A.
Albrecht, M Barnard.
I also acknowledge David Ring for useful discussions, technical assistance, and for
finding an error in our code. I also thank Tony Tyson and his group for the use of their
computer cluster, and, in particular, Perry Gee and Hu Zhan for their expert advice and
computing support. I also thank Gary Bernstein for providing us with Fisher matrices
suitable for adapting the DETF weak lensing data models to our methods.
Additional thanks go to Professors Lloyd Knox, Lori Lubin, Warren Pickett, John
Rundle, and Andrew Waldron for serving on my oral qualifying exam committee and again
thanks to Professors Knox and Lubin for being on my thesis committee and reading this
manuscript.
I would also like to thank Kem Cook, Sergei Nikolaev, and Mark Huber for the
opportunities, support, and patience they provided me at LLNL.
I also express graditude to the following current and former members of the UCD
Physics department staff: Kristi Case, Michael Hannon, Kari Kilpatrick, Laura Peterson,
Lynn Rabena, Georgie Tolle, Robyn Tornay, Bill Tuck, Daniel Wang, Onelia Yan, and
v

Phillip Young.
I am deeply thankful and appreciative of my parents and sister for being there and
being supportive, patient, and generous throughout.
Finally, I thank Deepa, Alvin, and Oskar for their friendship, support, advice,
help, generosity, and for checking up on me from time to time throughout this challenging
process.
vi

Contents
Abstract ii
Contents vii
List of Tables viii
List of Figures ix
1 Introduction 1
2 Constraining the Locations of Microlenses Towards the LMC 6
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Microlensing Searches for Halo Dark Matter . . . . . . . . . . . . . . . . . . 14
2.3 The Location of the Microlenses . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4 Information from the Microlensing Source Star CMD . . . . . . . . . . . . . 31
2.5 Problems, Challenges, and Suggestions for Future Work . . . . . . . . . . . 41
2.6 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3 Exploring Parameter Constraints on Quintessential Dark Energy: The
Inverse Power Law Model 57
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.2 Tracking Quintessence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.2.1 Tracking Solutions and behaviors . . . . . . . . . . . . . . . . . . . . 63
3.2.2 The Inverse Power Law Potential . . . . . . . . . . . . . . . . . . . . 65
3.2.3 The non-tracking case . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.2.4 The transition from tracking to acceleration . . . . . . . . . . . . . . 75
3.2.5 Current constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.3 Parameterization of the Inverse Power Law Model . . . . . . . . . . . . . . 78
3.4 MCMC Results and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.4.1 General approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.4.2 Cosmological Constant Fiducial Model . . . . . . . . . . . . . . . . . 83
3.4.3 Inverse Power Law Fiducial Model . . . . . . . . . . . . . . . . . . . 89
3.4.4 Non-Tracking Parameter Space Regions . . . . . . . . . . . . . . . . 96
3.5 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4 Summary and Conclusions 105
vii

List of Tables
2.1 Expected non-dark matter contributions to the optical depth for standard
models of the MW and LMC (left column). The right column gives the
expected number of microlensing events for the given population [1]. When
adding up the microlensing contribution from each of these populations, one
obtains a total optical depth value τ ∼ 2.4 × 10−8 that is still about 5 times
smaller than the measured value obtained by the MACHO collaboration. . . 29
3.1 Fiducial Parameter Values (energy densities in units of h2) for ΛCDM model. 84
3.2 Fiducial Parameter Values (energy densities in units of h2) for Inverse Power
Law model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
viii

List of Figures
2.1 The observed rotation curve of the nearby dwarf spiral galaxy M33 extends
considerably beyond its superimposed optical image. (From [2]). . . . . . . 11
2.2 Geometry of a microlensing event. The path of a ray of light from a source
object (S) is deflected by the presence of a massive lens (L), creating two
images of the source at S’ and S”. The source lies a distance Ds and the
lens a distance DL from the observer (O). Rs is the impact parameter, the
smallest distance between the observer-source line of sight and the lens. R is
the smallest distance between the observer-image line of sight and the lens. 17
2.3 The lensing mass (M), the small circular source (S), and the two images (I1
and I2), are shown. In the presence of mass M the source is seen only at I1
and I2 and not at S. The Einstein ring is shown as a dashed circle. The
radius of the circle is typically ∼1 milliarcsecond for microlensing by MW
stars. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4 The variation of the magnification due to a point lens (from [3]) is shown
in stellar magnitudes as a function of time. t0, the characteristic time scale
for a microlensing event, is defined as the time the background source takes
to move a distance equal to the Einstein ring radius, RE. The six light
curves correspond to the six values of the dimensionless impact parameter:
p = 0.1, 0.3, 0.5, 0.7, 0.9, 1.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.5 This table lists the LMC model parameter values for the disk, bar, and halo
models used by [1] to obtain their estimate of the self-lensing optical depth
of the LMC. (From [1]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.6 Four possible LMC microlensing geometries that have been developed and
discussed to try to explain the microlensing events towards the LMC. They
correspond to four competing explanations for the microlensing signal to-
wards the LMC. The Milky Way is depicted as the large spiral galaxy in the
upper right hand corner of each panel, and the LMC is depicted as the irreg-
ular galaxy in the lower left hand corner. The arrow in each panel indicates a
line of sight from the location of the Earth in the Milky Way towards random
positions in the LMC. The white dots indicate the position of source stars
and the green dots indicate the position of lensing objects. The figures are
not drawn to scale. (Adapted from [4]). . . . . . . . . . . . . . . . . . . . . 33
ix

2.7 The observed MACHO microlensing source stars (large red) stars, over-
plotted on two model source star populations (small black dots). The left
panel represents a source star population drawn entirely from the LMC
disk+bar (fBKG = 0.0). The right panel represents a model in which all
of the stars belong to a background population (fBKG = 1.0) with ∆µ = 0.3
and ∆E(V − I) = 0.18. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.8 The 2-D KS test probability P that the observed distribution of source stars
was drawn from a source star population in which a fraction of the source
stars are located behind the LMC. The distance moduli of the background
stars with ∆µ = 0.0, 0.30, 0.45 are shown as red circles, green triangles and
blue squares respectively. The error bars indicate the scatter about the mean
value for 20 simulations of each model. . . . . . . . . . . . . . . . . . . . . . 40
2.9 Left Panel: Observed composite HST color magnitude diagram of the 13
LMC fields surrounding each of the microlensing events. Right Panel: An
example of a composite best-fit StarFISH generated model CMD for the 13
LMC fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.10 The observed MACHO microlensing source stars (large red stars), over-
plotted on two model source star populations (small black dots) generated
with a synthetic CMD algorithm The left panel represents a source star pop-
ulation drawn entirely from the LMC disk+bar (fBKG = 0.0, ∆µ = 0.0).
The right panel represents a model in which all of the stars belong to a
background population (fBKG = 1.0) with ∆µ = 0.3 and with the ’1 cloud’
Poisson reddening model applied. . . . . . . . . . . . . . . . . . . . . . . . . 53
was drawn from a model source star population with the ’one cloud’ Poisson
reddening model applied and in which a fraction of the source stars are
located behind the LMC. The distance moduli of the background stars with
∆µ = 0.0, 0.30, 0.45 are shown as red circles, green triangles and blue squares
respectively. The error bars indicate the scatter about the mean value for 20
simulations of each model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.12 The observed MACHO microlensing source stars (large red stars), over-
plotted on two model source star populations (small black dots) generated
with a synthetic CMD algorithm The left panel represents a source star pop-
ulation drawn entirely from the LMC disk+bar (fBKG = 0.0, ∆µ = 0.0).
The right panel represents a model in which all of the stars belong to a
background population (fBKG = 1.0) with ∆µ = 0.3 and with the ’2 cloud’
Poisson reddening model applied. . . . . . . . . . . . . . . . . . . . . . . . . 55
was drawn from a model source star population with the ’two cloud’ Poisson
reddening model applied and in which a fraction of the source stars are
located behind the LMC. The distance moduli of the background stars with
∆µ = 0.0, 0.30, 0.45 are shown as red circles, green triangles and blue squares
respectively. The error bars indicate the scatter about the mean value for 20
simulations of each model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
x

3.1 Semilog plots of the evolution of the density parameters Ωi for radiation
(solid line), matter (dotted line), and dark energy (dashed line) for an IPL
quintessence model with α = 0.05, φI = 10−30, and V0 0.38. Present
values of some density parameters, Ωi,0 and the Hubble parameter, H0, are
also included in the figure. The evolution of the equation of state w as a
function of scale factor a is also depicted. The a scale is logarithmic here in
order to show behavior on all time scales. . . . . . . . . . . . . . . . . . . . 68
3.2 IPL potentials (top panel) and w(z) evolution (lower panel) for different α
values (dashed-dotted: α = 0.05, dashed: α = 0.01 and solid: α = 0.1).
For all curves V0 = 0.38 and φI = 10−30. Smaller values of α lead to flatter
potentials and smaller V (φ). . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.3 An Illustration of how the evolution and tracking behavior of w as a function
of scale factor a is affected by different values of α, V0, and φI. The a scale
is logarithmic here in order to show behavior on all time scales. . . . . . . . 72
3.4 Examples of how the evolution and tracking behavior of w as a function of
scale factor a is affected by different values of φI for given values of V0 and α.
For all curves, V0 = 0.38 and α = 0.1. These examples illustrate how different
values of φI lead to the same values of the equation of state parameter today.
The a scale is logarithmic here in order to show behavior on all time scales. 73
3.5 This figure depicts the evolution and tracking behavior of w as a function
of scale factor a for different values of α for given values of V0 = 0.38 and
φI = 10−30. As long as φI << MP , α will determine w0 and the amplitude of
the w(a) curves. In addition, the smaller α is, the later the tracker is reached
for a given φI. The a scale is logarithmic here in order to show behavior on
all time scales. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.6 V0−α 1σ(68.27%), 2σ(95.44%), and 3σ(99.73%) confidence regions for DETF
“optimistic” combined ΛCDM data models. . . . . . . . . . . . . . . . . . 85
3.7 V0 − log10(φI) 1σ(68.27%), 2σ(95.44%) and 3σ(99.73%) confidence regions
for DETF “optimistic” combined ΛCDM data models. . . . . . . . . . . . 87
3.8 log10(φI)−α 1σ(68.27%), 2σ(95.44%) and 3σ(99.73%) confidence regions for
DETF “optimistic” combined ΛCDM data models. . . . . . . . . . . . . . 88
3.9 The potential of the IPL fiducial model (α = 0.14, φI = 10−15, V0 = 0.31)
(top panel,dashed curve). The corresponding equation of state evolution w(z)
for a potentially observable range of redshift values is shown in the bottom
panel. The solid curve overlaying the potential in the top panel shows the
evolution of the IPL fiducial model scalar field for the range of z values (from
z = 5 to the present time) depicted for w(z) in the bottom panel. . . . . . . 91
3.10 V0 − α 1σ (68.27%), 2σ (95.44%) and 3σ (99.73%) likelihood contours for
DETF optimistic combined data sets generated from a selected IPL back-
ground cosmological model. . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.11 V0 − log(φI ) 1σ (68.27%), 2σ (95.44%) and 3σ (99.73%) likelihood contours
for DETF optimistic combined data sets generated from a selected IPL back-
3.12 log(φI ) − α 1σ (68.27%), 2σ (95.44%) and 3σ (99.73%) likelihood contours
xi

3.13 V0 − α 1σ (68.27%), 2σ (95.44%) and 3σ (99.73%) likelihood contours for
DETF optimistic combined data sets generated from a selected IPL back-
ground cosmological model for the case of a cut-off of log10(φI ) = −3 placed
on the MCMC algorithm. This effectively gives an enlarged and more de-
tailed view of non-tracking and “thawing”-like regions of the parameter space. 98
3.14 log(φI ) − α 1σ (68.27%), 2σ (95.44%) and 3σ (99.73%) likelihood contours
ground cosmological model for the case of a cut-off of log10(φI ) = −3 placed
on the MCMC algorithm. This effectively gives an enlarged and more de-
tailed view of non-tracking and “thawing”-like regions of the parameter space. 100
xii

Chapter 1
Introduction
Astronomers and cosmologists have known for well over a decade that non-baryonic
dark matter makes up the majority of the mass of the universe. It is named “dark” because
it is not luminous. Its presence is known due to its gravitational effects on visible objects.
Indeed, the gravitational attraction of this dark matter is believed to drive the develop-
ment and formation of structure in the universe and influence the evolution of the universe
gravitationally. For example, galaxies in clusters are known to move at speeds that are too
high to be accounted for by the visible galaxies and associated visible gas alone [5; 6; 7; 8].
Moreover, the temperatures of gases in these galaxy clusters have been measured to be too
high for the gas to remain bound to the cluster without the existence of some additional,
unseen, mass. Furthermore, X-ray data have indicated the presence of very extended dark
matter halos much beyond the radius of the luminous galaxies [9; 10]. The total postulated
dark matter mass is often measured to be about 40 times greater than the luminous (visible)
portion of the galaxies alone [11; 12; 13; 14].
Recent observations of the apparent magnitude-redshift relation of type Ia Super-
nova (SNe Ia) [15], as well as the analysis of fluctuations of the cosmic microwave back-
1

CHAPTER 1. INTRODUCTION 2
ground (CMB) [16], combined with measurements of baryon acoustic oscillations (BAO)
[17], weak lensing observations (WL) [18], observations of galaxy clusters [19; 20; 21], and
measurements of light element abundances [22] have now placed us in an era of “precision
cosmology” in which the basic components of the universe have become fairly clear. Taken
together, all of these results nicely complement each other and point to a “Cosmological
Concordance Model” in which the universe is nearly spatially flat, consists of about 25%
non-baryonic dark matter, which is not visible to us, while 4-5% is composed of normal
baryonic matter, both visible and invisible, and the remaining 70% is believed to be com-
prised of a mysterious dark energy which pervades space with negative pressure and causes
the expansion of the universe to accelerate. From the observations gathered so far, this
acceleration appears to be most consistent with the effect of a cosmological constant, but
it may also be caused by a slowly evolving dynamical scalar field giving rise to the negative
pressure, such as quintessence, or it could also point to a fundamental modification to gen-
eral relativity (modified gravity). In this dissertation we describe research projects involving
both baryonic dark matter (in the form of massive compact halo objects (MACHOs)) and
dark energy.
The nature of the dark matter continues to be one of the most important and
still unsolved problems in astronomy, and its identification will be an essential step in our
understanding of nature. Current cosmological models assume that there exists a non-
baryonic cold dark matter (CDM) component consisting of non-relativistic particles that
interact with the baryons (including the luminous matter) only through gravitational forces
[23; 24]. Also, CDM simulations have predicted the existence of CDM halos encompassing
galaxies with constant outer rotation curves, in concurrence with observations [25; 24].
Some of the mass in these halos is also believed to consist of baryonic objects that are too

faint to detect.
Chapter 2 of this dissertation focuses on research performed with Dr. Kem Cook
on a method to constrain the locations of microlensing objects (which could be comprised of
baryonic dark matter objects such as brown dwarfs, white dwarfs, neutron stars and black
holes, collectively referred to as “MAssive Compact Halo Objects” (MACHOs)) towards
the Large Magellanic Cloud (LMC). This project has included the use and development of
reddening models, color magnitude diagrams (CMDs), and microlensing population models
of the LMC to constrain the locations of microlensing source stars and microlensing objects
(MACHOs) in the LMC and the Milky Way halo using data of 13 microlensing source
stars obtained by the MACHO collaboration with the Hubble Space Telescope (HST). We
attempted to distinguish between source stars drawn from the average population of the
LMC and source stars drawn from a population behind the LMC by examining the HST
CMD of microlensing source stars and comparing it to the average LMC population. We
carried out a 2-dimensional Kolmogorov-Smirnoff (KS) test to quantify the probability that
the observed microlensing source stars are drawn from a specific model population. The 13
event KS-test analysis results rule out a model in which the source stars all belong to this
background population at a confidence level of 99%. The results of the KS test analysis,
taken together with external constraints, also suggested that the most likely explanation is
that the lens population comes mainly from the MW halo and the source stars are located in
the LMC disk and/or bar. The strength of this analysis was severely limited by the number
of microlensing events used, but other ongoing microlensing surveys and projects could
provide a sufficient sample of microlensing events over the next few years. The technique
outlined here could prove a powerful method for locating source stars and lenses with the
use of these future data sets. Publication of this work is pending additional improvements

in modeling.
It is generally believed that dark energy, like most dark matter, is also non-baryonic
in origin but that dark energy is distinguished from dark matter by the fact that dark energy
has a large negative pressure which causes the expansion of the universe to accelerate,
whereas the dark matter, whose existence can be inferred on galactic scales by galactic
rotation curves, clusters on sub-Megaparsec scales [2]. Due to a great deal of confusion in the
theoretical domain, the field of cosmic acceleration (i.e., dark energy) is highly data driven at
this stage, and there are a number of exciting new and proposed observational programs that
could have a great impact on the field. Chapter 3 focuses on an MCMC analysis of a dark
energy quintessence model (known as the Inverse Power Law (IPL) or Ratra-Peebles model
[26; 27]) that includes the utilization of Dark Energy Task Force (DETF) data models that
simulate current and future data sets from such new and proposed observational programs
[28]. Following the approach taken by the DETF, we generated data models for future
SNe Ia, BAO, weak gravitational lensing, and CMB observations as a representation of
future dark energy experiments. We generated simulated data sets for a ΛCDM background
cosmology as well as a case where the dark energy is provided by a specific IPL model.
Following the approach taken by [29; 30; 31] we then used an MCMC algorithm to map
the likelihood around each fiducial model via a Markov chain of points in parameter space,
starting with the fiducial model and moving to a succession of random points in space
using a Metropolis-Hastings stepping algorithm. From the associated likelihood contours,
we have found that the respective increase in constraining power with higher quality data
sets produced by our analysis gives results that are broadly consistent with the DETF
for the dark energy parameterization that they used. We also demonstrated, consistent
with the findings of [29; 30; 31], that for a universe containing dark energy described by

a particular ﬁducial IPL model, a cosmological constant can be excluded by high quality
Stage 4 experiments by well over 3 σ. Chapter 3 is drawn from the paper Yashar, M.,
Bozek, B., Albrecht, A., Abrahamse, A., Barnard, M., “Exploring Parameter Constraints
on Quintessential Dark Energy: the Inverse Power Law”, submitted to Phys. Rev. D,
(2008), and is also related to the companion papers [29; 30; 31].

Chapter 2
Constraining the Locations of
Microlenses Towards the LMC
2.1 Introduction
As pointed out in the introductory chapter, measurements over the last several
years from the Wilkinson Microwave Anisotropy Probe (WMAP) of the CMB [32; 16], com-
bined with supernovae [33; 34; 15] and other cosmological observations (as well as estimates
from Big Bang Nucleosynthesis) have fairly clearly revealed to us the basic composition of
the universe. These observations indicate that the present universe is flat, in agreement
with predictions from inflationary theory, with the ratio of the energy density of the uni-
verse to the critical density needed for a flat universe being Ω0 1 with a Hubble Constant
of H0 = 100h km sec−1 Mpc−1 with h = 0.72 ± 0.08 according to measurements from the
Hubble Space Telescope Key Project [35], and with the critical density today defined as
ρcrit =
3H2
0
8πG
= 1.9 × 10−29
h2
g cm−3
= 2.8 × 1011
h2
M Mpc−3
, (2.1.1)
6

2.1. Introduction 7
where G is the gravitational constant and M is the mass of the sun. These observations also
indicate that about 26% of the universe is made of non-baryonic dark matter (i.e., density
parameter Ωm,0 ≡
ρm,0
ρc
0.26 ), 4% consists of normal baryonic matter, Ωb,0 0.04 (visible
baryonic matter such as stars and gas and normal baryonic matter not detectable to us at
this time such as white dwarfs, neutron stars, brown dwarfs, and extra-solar planets), and
the remaining 70% of the universe is believed to consist of a mysterious ”dark energy”
which pervades space with negative pressure and causes the expansion of the universe to
accelerate (ΩDE 0.70) [36].
The very small visible (baryonic) matter content fraction of the universe (ΩV,0
0.007) [12; 13; 14; 37; 38] in the form of stars, neutral atomic gas (mostly HI), and molecular
gas (mostly H2) in galaxies estimated from HI 21 cm and CO surveys of galaxies and
using methods which include calculations of mass-to-light ratios of galaxies and clusters of
galaxies [12; 14; 11; 2], is less than 10% of the lower limit predicted by standard primordial
nucleosynthesis and less than 1% of the critical density of the universe [12]. Furthermore,
galaxy rotation curves [39; 2], X-ray measurements relating to hot diﬀuse ionized gas in
galaxy clusters and around groups of galaxies [9; 13] and, observed features of high redshift
Lyman-α forest absorption lines of neutral hydrogen in the spectra of background quasars
[40], along with CMB observations [16], point to a total baryon content fraction of the
universe of Ωb,0 0.04 [38] when taken together with the observations of stars and cooler
gas. All of these ﬁndings taken together with the combination of measurements of the
primeval abundance of light elements (especially deuterium) with big bang nucleosynthesis
theory and calculations [22; 41] indicate that only a small fraction (4 − 5%) of the total
matter content of the universe is in baryonic form while about 95% of the mass in galaxies
and clusters of galaxies is made of an unknown (non-baryonic) dark matter component [36].

2.1. Introduction 8
And, moreover, since Ωb,0 > ΩV,0 at least some of the baryonic mass which does exist has
not been observationally accounted for due to the absence of detection of electromagnetic
radiation (i.e., “dark baryons”) [12; 42]. The orbital speeds of objects around the centers
of galaxies are determined by the mass interior to the orbital radius of the object, M(r),
according to Kepler’s third law and Newtonian mechanics and are given by
v(r) =
GM(r)
r
, (2.1.2)
In this way, the “rotation curve” v(r) at a given radial distance from the galactic center
can be determined. However, observations of many spiral galaxies indicate that most if
not all such galaxies (including the Milky Way galaxy) contain stars and gas at large radii
from the centers of these galaxies that have orbital speeds that, instead of declining at the
expected rate v ∝ r−1/2 for M constant, are greater than they would be if the stars and
gas were the only matter present. Hence, these galaxies actually possess velocity curves
which ﬂatten out to v constant, implying M(r) ∝ r (see Fig. 2.1). These observations
suggest that the mass of galaxies continues to grow even without the presence of a luminous
component to account for the increase, pointing to the presence of a possible dark matter
halo within which the visible stellar disk is embedded (e.g., [43; 11]). From observational
techniques such as measurements of Doppler shifts of the 21 cm radio emission line from
neutral hydrogen gas (HI) from the inner portions of galactic disks, radial velocities have
been obtained and ultimately rotation curves have been compiled for well over 1000 spiral
galaxies [44; 39; 2]. The results indicate a mass-to-luminosity ratio of M
LB
= (10 − 20) M
LB
in spiral and elliptical galaxies, where the luminosity LB refers to the total luminosities
of the galaxies in the blue band, while this ratio can increase to M
LB
(200 − 600) M
LB
in

2.1. Introduction 9
dwarf and low surface brightness galaxies, with M
LB
being highly dependent on the radius in
the galaxies and increasing with radius to scales beyond the luminous portions of galaxies
[45; 2]. This increasing mass-to-luminosity ratio with galaxy radius reflects the well-known
inference of the existence of galactic dark halos [45].
On larger scales and higher redshifts, another piece of evidence for dark matter
comes from the detection of hot gas in galaxy clusters. X-ray images (such as those taken
of the Coma Cluster by the ROSAT satellite [10]) indicate that the temperatures of gases
in these galaxy clusters are too high to remain bound to the cluster without the existence of
some additional, unseen mass. For the specific example of the Coma Cluster, the presence of
a vast reservoir of dark matter has been confirmed by the fact that the hot, X-ray emitting
intracluster gas is still in place; if there were no dark matter to anchor the gas gravitationally,
the hot gas would have expanded beyond the cluster on time scales much shorter than the
Hubble time [11]. Further evidence for dark matter comes from the velocity dispersions of
galaxies within clusters. The use of the virial theorem tells us that the velocity dispersions
and thus the speeds of galaxies in clusters are too high to be accounted for by the luminous
matter alone, such as is the case with the motion of satellite galaxies in the Local Group.
Finally, gravitational lensing provides another piece of evidence for the existence of dark
matter via the deflection of light caused by massive objects (as discussed in 3.2 below).
The extent to which, for example, the light from a distant background galaxy or quasar is
bent depends on the amount of intervening mass along the line of sight. More specifically,
a cluster of galaxies can act as a lens to a background galaxy such that, in the case of
strong gravitational lensing, the image of the observed background galaxy will be distorted
into arc shapes and multiple images are formed (as in the case, for example, of the Galaxy
Cluster Abell 2218, [8]). The cluster mass can be estimated by the degree to which it lenses

2.1. Introduction 10
the background galaxy. Masses calculated in this way are in general agreement with masses
found by applying the virial theorem to the motions of galaxies in clusters or applying the
hydrostatic equilibrium equation to hot X-ray emitting intra-cluster gas [11].
From our understanding of the content of the universe, we arrive at the following
conclusions. Although the addition of the dark energy term, ΩDE, greatly reduces the
amount of matter needed for a flat universe, it is still the case that most matter in the
universe has not been observationally accounted for (i.e., has not been detected via the
emission of electromagnetic radiation) (Ωm,0 >> ΩV,0). Second, most of the dark matter
in the universe is non-baryonic (Ωm,0 >> Ωb,0), consistent, for example, with predictions of
primordial nucleosynthesis and measurements of anisotropies in the CMB. However, finally,
some of the baryonic mass has also not been observationally accounted for through the
detection of electromagnetic emission at any wavelength (Ωb,0 > ΩV,0) and is often referred
to as “baryonic dark matter” or “dark baryons” (e.g., [46; 42; 41]). The large values of both
Ωb,0 and Ωm,0 compared to ΩV,0 and Ωm,0 compared to Ωb,0, as well as the corresponding
indication that the amount of baryonic dark matter is much smaller than the total amount
of dark matter are examples of the ”dark matter problem” (as discussed, for example, in
[46]), i.e., the universe contains much more matter than can be accounted for in visible
objects.
Evidence for dark matter has been claimed in many different contexts and gives
rise to a number of different and separate dark matter problems without one single solution
[46; 47]. For example, dark matter my exist locally in the Galactic disk, it may exist
in the halos of our own and other galaxies, and/or it may be associated with clusters of
galaxies. Large-scale structure observations, Big Bang nucleosynthesis calculations and
constrains, CDM simulations, and contemporary theories of galaxy formation, demand that

Figure 2.1. The observed rotation curve of the nearby dwarf spiral galaxy M33 extends
considerably beyond its superimposed optical image. (From [2]).

dark matter in galaxy clusters and possibly in individual galaxy halos as well exist mainly in
non-baryonic form [23; 25; 41; 41; 11; 43]. Moreover, we would expect non-baryonic matter
(e.g., WIMPs, as explained below) to naturally fall into galactic potential wells (given that
non-baryonic matter is clearly required to account for observations on galaxy cluster scales),
possibly playing a crucial role in the process of galaxy formation itself. Thus, the natural
place to find non-baryonic matter could be in galactic halos. On the other hand, however, it
is also possible for baryonic matter to make up a significant portion of the mass of individual
galactic dark halos. Mass to light ratios greater than one occur on nearly all scales, but in
general increase with scale, reaching a maximum value on scales of rich clusters (hundreds
of kpc) before flattening out and remaining approximately constant on larger scales [45; 48].
Since the solution to the dark matter problem(s) need not necessarily be the same on all
scales, and since dark baryons can contribute to only a small fraction of the total dark
matter in the universe, one popular scenario is that dark baryons are more likely to solve
or alleviate the dark matter problem on small scales than on large scales (such as galaxy
clusters) [49; 47]. Ignoring here the possible dark matter problem in the MW disk, the next
larger scale for which to consider dark matter baryons is the MW halo.
In considering the baryonic contribution to dark matter in galaxy halos, the crucial
issue is how far the halo extends. We can express the mass-to-light ratio of a component
of the universe as a ratio of the average density of the component to the critical density,
Ωi, where, for example, i can be a galaxy halo [46; 50]. The ratio Ωi is given in terms
of the average optical luminostiy density of the universe (measured in the B-band), £ =
2.4h × 108LB Mpc−3 [46; 50], and critical density ρcrit = 1.9 × 10−29h2g cm−3 = 2.8 ×
1011h2M Mpc−3, as [46]
Ωi = (M/L)i
£
ρc
. (2.1.3)

If the MW galaxy is a typical galaxy and the mass-to-light ratios of the halos of other
galaxies parameterize in terms of the (uncertain) halo radius Rhalo in a similar fashion as
estimated for the MW galaxy in [51], then [49; 47]
Ωhalo ≈ 0.03h−1 Rhalo
100 kpc
. (2.1.4)
For h ≈ 0.7, this implies Ωhalo Ωb only for Rhalo < 100 kpc. Even if halo radii are
larger than 100 kpc, and studies by [52] suggest that they could be as large as 300 kpc
[47] while galaxy-galaxy lensing measurements suggest that a typical halo radius may be
more on the order of Rhalo > 300 kpc [53], a substantial fraction of the MW halo could
still consist of baryonic matter even if it is unlikely that all of the MW halo dark matter is
baryonic [49; 47]. A crucial question then arises regarding the extent to which dark matter
halos consist or do not consist of baryons and our ability to place direct observational
constraints on this baryonic mass. Popular non-baryonic candidates include an ultra-light
pseudo-Goldstone boson called an axion, weakly interacting massive particles (WIMPs)
such as supersymmetric neutralinos- particles which have been proposed for other reasons
in various particle physics models, whereas the baryonic matter in the dark halos of galaxies
would be made up of ordinary atoms (consisting of protons, neutrons, and electrons) hidden
in non-luminous forms such as stellar remnants (white dwarfs, neutron stars,and black
holes), brown dwarfs, or even planets, all collectively referred to as “massive compact halo
objects” (MACHOs). The main constraints on the amount of baryons and the baryon
density in the universe comes from the theory of Big Bang nucleosynthesis, which requires
Ωb 0.04( h
0.7 )−2 [54] and corresponds to observed abundances of light elements (such
as Helium, Deuterium, and Lithium) as determined, for example, from the absorption of

2.2. Microlensing Searches for Halo Dark Matter 14
quasar light as it passes through primordial gas clouds [22]. Hence, we see that for a Hubble
constant H0 = 70 km sec−1 Mpc−1, nucleosynthesis constraints along with the mass-to-light
ratio calculations mentioned previously (which point to, not including hot x-ray emitting
gas associated with galaxy clusters, ΩV,0 0.007 [12; 13; 46; 47; 37; 38]), again indicates
that a sizable fraction of the baryons in the universe can not be directly observationally
accounted for, even when the ionized x-ray emitting gas associated with galaxy clusters is
taken into account [46; 47; 42; 41]. These dark baryons could be in the form of MACHOs
and/or possibly cold gas clouds consisting of H2.
Hence, if the halo of our galaxy were made of MACHOs, microlensing (as we
explain in Section 2.2 below) would be the ideal tool to search for them, and the failure
to ﬁnd MACHOs with microlensing could strengthen the case for completely or mostly
non-baryonic galactic dark halos. With all of this in mind, several experiments have been
undertaken in the last 15 or so years, including the MACHO project, to carry out microlens-
ing searches of baryonic halo dark matter in the form of MACHOs in the halo of the Milky
Way along lines of sight towards the LMC.
This chapter includes the results of research carried out with Dr. Kem Cook.
Publication of these results is pending additional improvements in modeling and, possibly,
additional data.
2.2 Microlensing Searches for Halo Dark Matter
Here, we focus on the geometry of microlensing (Fig. 2.2) [55] towards the LMC,
a dwarf galaxy in orbit around the MW, which acts as a crowded stellar background of
source stars. We consider the MACHO to act as a point source point lens. If a MACHO in
the halo of our galaxy passes near our line of sight to a source star in the LMC, the gravi-

tational influence of the MACHO will cause the light from the source star to be deflected,
as predicted by Einstein’s theory of general relativity, by a deflection angle (assuming that
the gravitational field is weak, the deflection angle is small, and the region within which
essentially all deflection occurs is small compared to the scale of the universe)
α =
4GML
Rsc2
, (2.2.1)
where c is the speed of light, G is the gravitational constant, ML is the mass of the lens,
and Rs is the impact parameter between the light ray trajectory and the lensing object,
which is the distance of closest approach between the light ray and the lens (or, equiva-
lently, the smallest distance between the lens and the observer-source line of sight), and two
distorted images on opposite sides of the lens, S and S , will be produced (Fig. 2.3)[3].This
equation is only valid for the case in which the impact parameter is much larger than the
Schwarzschild radius of a body of mass ML. In Fig. 2.3, α is the angle between S and
S at the lens plane, i.e., the angle at which the light ray trajectories from S and S inter-
sect each other at the lens plane. The two images will only be separated by an order of
a milliarcsecond, and thus not enough to be resolved as two separate images. The source
star will instead appear as a single distorted and magnified image due to the conservation
of surface brightness, causing a transient increase in the source star flux which reaches the
observer’s telescope. The MACHO acts as a lens to the source star, and the increase in flux
is known as a microlensing event. If the MACHO is exactly along the line of sight between
us and the lensed source star, the image produced is that of a perfect ring around the lens

with a radius known as the Einstein Ring radius, RE, of the lens, which is given by
RE ≡
2
c
GMLDLSDL
DS
, (2.2.2)
where (see Fig. 2.2) ML is the mass of the lens, DLS is the distance from the lens to the
source, DL is the distance between the lens and observer, and DS is the distance between
the source and observer. RE is the (physical) radius of the projection of the Einstein ring on
the lens plane (a plane containing the lens and orthogonal to the line of sight, i.e., measured
at the location of the MACHO) and depends on the relative positions of source, observer
and lens in addition to the lensing object’s mass. The positions of two source images will be
(physically) separated by approximately 2RE, where RE corresponds to the radius of the
dashed circle depicted in Fig. 2.3. The Einstein radius is an important quantity because
lensing will signiﬁcantly modify the source’s appearance if the source lies within about RE
of the observer-lens line. On the other hand, a source lying further than RE from this line
will have the same appearance as it would have if the lensing object were not there [56].
Numerically, the Einstein radius is given by [57]
RE = 9.0 AU
MLDLS
10 kpc M
(1 −
DLS
DS
), (2.2.3)
where 1 AU is the semimajor axis of the Earth’s orbit (1.4960 x 1011 m), 1 kpc =
3.0857 x 1019 m, and M , the mass of the Sun, is 1.99 x 1030 kg. Thus, for example,
if the lens were an object in the Milky Way at a distance of DL = 10 kpc and the
source an object in the LMC at a of DS = 50 kpc, then the Einstein radius will be
RE = 8 ML
M AU.
The angular separation of the two source images on the sky is typically of order

Figure 2.2. Geometry of a microlensing event. The path of a ray of light from a source object
(S) is deﬂected by the presence of a massive lens (L), creating two images of the source at
S’ and S”. The source lies a distance Ds and the lens a distance DL from the observer (O).
Rs is the impact parameter, the smallest distance between the observer-source line of sight
and the lens. R is the smallest distance between the observer-image line of sight and the
lens.
2θE, where θE = RE
DL
is the angular Einstein ring radius (referred to as the “Einstein angle”)
and is given by
θE ≡
2
c
GMLDLS
DSDL
. (2.2.4)
To get an estimate of the size of the Einstein angle for the case of lensing by stars within
the galaxy, the Einstein angle can be parameterized as [54]
θE 0.9 mas
DLS10 kpcML
DLDSM
, (2.2.5)
where “mas” is short for milliarcseconds. So, for example, for a lens mass of ∼ 1M and

Figure 2.3. The lensing mass (M), the small circular source (S), and the two images (I1
and I2), are shown. In the presence of mass M the source is seen only at I1 and I2 and not
at S. The Einstein ring is shown as a dashed circle. The radius of the circle is typically ∼1
milliarcsecond for microlensing by MW stars.
DL = DLS = 10 kpc, θE ∼ 1 mas. Thus, when the lensing object is a star of approximately
solar mass and the source and lens lie within the Milky Way, the Einstein angle, and thus
the separation between the two images, is too small to be measured with present-day optical
telescopes (including even the Hubble Space Telescope, with a resolution of ∼ 0.1 ) [56; 54].
(This explains why this form of gravitational lensing is referred to as microlensing). We can
instead only see the combined light intensity of the two images, rather than two separate
images, due to the finite resolution of optical telescopes. The combined images appear as a
magnification or amplification, A, of the source star given by
A =
u2 + 2
u (u2 + 4)
, (2.2.6)

where u(t) ≡ Rs(t)
RE
is the impact parameter in units of the Einstein ring radius, which
corresponds to the distance of the lens to the line of sight to the source star in units of RE.
Assuming that the lens drifts with constant velocity with respect to and close to our line of
sight to a background source during the duration of the microlensing event, we have
u(t) = p2 + (
t − tmax
t0
)2, (2.2.7)
where p is the dimensionless or normalized impact parameter (the smallest angular distance
between the source and lens measured in units of the Einstein radius), tmax is the time of
closest approach (u(tmax) = p), which also corresponds to the time of maximum magniﬁca-
tion of the lens, and t0 is the characteristic time scale for a microlensing event (also known
as the event duration) and is given by
t0 =
DLSθE
vt
= 0.214 yr (
ML
M
)(
DL
10 kpc
)(1 −
DL
Ds
) (
200km/s
vt
), (2.2.8)
(where vt is relative transverse velocity of the lens with respect to the source [3; 54; 58]
and is the component of the lens velocity relative to the line of sight to the source star in
a direction perpendicular to this same line). This is the time it takes the source object to
move with respect to the lens by one Einstein ring radius.
Since MACHOs in the dark halo of the Milky Way and the stars in the LMC are
in constant relative motion (which may correspond to Milky Way halo lenses moving across
our line of sight to the source stars in the LMC at a transverse velocity of vt ∼ 200 km/s)
the typical signature of a lensing event is a star that becomes brighter as the angular
distance between source star and MACHO decreases, then becomes dimmer as the angular
distance increases again. So, the magniﬁcation, A, as well as the smallest distance between

Figure 2.4. The variation of the magnification due to a point lens (from [3]) is shown in
stellar magnitudes as a function of time. t0, the characteristic time scale for a microlensing
event, is defined as the time the background source takes to move a distance equal to
the Einstein ring radius, RE. The six light curves correspond to the six values of the
dimensionless impact parameter: p = 0.1, 0.3, 0.5, 0.7, 0.9, 1.1.
the lens and the observer-source line of sight (the impact parameter), RS,, changes with
time. Inserting Eq. (2.2.7) into Eq. (2.2.6) leads to a time dependent amplification of the
luminosity of the source described by a characteristic light curve, plotted in Fig. 2.4 for
six values of impact parameter, with the corresponding time variability expressed in stellar
magnitudes (∆m ≡ 2.5 log A) [3].
From Eq. (2.2.6) we see that the amplification only depends on the impact pa-
rameter in units of the Einstein ring radius, u(t). The amplification reaches its maximum

value Amax at the time tmax as u(t) approaches p. In particular, for p = 1, we have
Amax = 3√
5
1.34 and a change in stellar magnitude of ∆m ≡ 2.5 log A = 0.3191 mag.
In this way, the event duration t0 is defined in observing units to be the amount of time
in which the flux of the source star is at least 1.34 times greater than its baseline value.
A microlensing event is considered to have taken place when the peak magnification in the
lensing induced light curve is Amax ≥ 1.34 (corresponding to u ≤ 1) when the closest ap-
proach between the point mass lens and source is ≤ θE. Then, the corresponding variation
in intensity of the source is easy to detect with reasonably accurate photometry [3; 58].
If the lenses are located in the Galactic halo and the sources are in the LMC,
then the ratio DL
DS
will be close to unity. If lensing induced light curves are sampled with
time intervals (t0) between about an hour and a year, MACHOs with masses in the range
10−6M to 102M are potentially detectable [58]. For example, for a microlensing event
in which the source distance DS to the LMC is 50 kpc, the lens distance DL in the Milky
Way is 10 kpc, and the relative transverse velocity vt of the lens with respect to the source
is 200 km/s, the lensing event duration, according to Eq. (2.2.8), will be about 9.4 days for
a 0.1M object (such as a brown dwarf). All of the real physical information concerning
the lensing population (mass, distance, and velocity) is contained in this single observable
(t0). The degeneracy contained in Eq. (2.2.8) makes it very difficult to determine the
mass, velocity, and (most importantly) location of the lensing population, given that many
different combinations of mass, velocity and distance can produce the observed t0. Hence,
direct measurements of these fundamental parameters are difficult to extract, and this
makes it difficult to measure where and what the microlensing signal is. The lensing event
duration is also important in calculating the microlensing detection efficiency, which will be
discussed later in this chapter. Event duration can be determined, together with with p (or

equivalently Amax) and tmax by fitting the theoretical light curve to the observed source star
luminosities plotted as a function of time. Light curves of lensed source stars also have the
following characteristic signatures: the light curves are expected to be symmetric in time
and the magnification is expected to be achromatic, due to the gravitational origin of the
lensing effect which causes all wavelengths to be affected equally. By contrast, intrinsically
variable stars typically have asymmetric light curves and variability associated with changes
in color, with the consequence that each waveband yields a different light curve [56; 58; 57].
In some instances the light curve shape can deviate from Eq. (2.2.6) or not be achromatic
due to binarity of the lens or source, finite source size, or blending effects (as discussed in,
e.g, [57]).
Given that direct measurements of the fundamental parameters (mass, velocity,
and distance of the lens from the observer) are difficult to extract, the main results of
microlensing surveys are expressed in terms of a quantity referred to as the ’optical depth’,
τ, to gravitational microlensing. Conceptually, the optical depth can be defined as the
percent chance that in an observation of any single star at any given instant a microlensing
event will be in progress. It is also a number equal to the sum of the durations of all
microlensing events divided by the total number of stars observed times the total time of
observation. The optical depth can also be defined as the fraction of sky covered by the
Einstein rings of the lensing population. Numerically, the total optical depth due to all
lenses between the source and the observer can be expressed as
τ =
DS
0
4πGρ
c2
DLDLS
DS
dDL =
π
4E
i
t0,i
(t0,i)
, (2.2.9)
where the first equality gives a theoretical expression for the optical depth and the second

equality gives an expression for the optical depth in terms of observational quantities. In
the first equality, ρ is the total mass density of lenses (i.e.,MACHOs) along the line of sight.
In the second equality (t0,i) is the efficiency for detecting an event of a given duration,
which is the likelihood of detecting a microlensing event for a source star of a particular
magnitude as a function of the lensing duration t0, and E is the total exposure time–the
length of time each star has been monitored [59; 3]. The optical depth probes directly the
MACHO fraction of dark matter, since it depends on the density profile of microlensing
objects along the line of sight to the target source stars, and it is at its maximum value for
the case in which DL = 1
2 DS. If the distribution of the total mass density along a given
line of sight is known, then the experimental estimate of the optical depth along this line of
sight gives us the fraction of total dark matter in the form of MACHOs. Note that since RE
is proportional to
√
M, while for a given ρ the number density of lenses is proportional to
M−1, the optical depth τ depends on the total mass in all of the lenses but is independent
of the masses and velocities of individual lenses. A simple estimate for the optical depth
for ρ = const using the first equality in Eq. (2.2.9) is
τ =
2π
3
GρD2
S
c2
. (2.2.10)
If we assume that the system of lenses is self-gravitating, and if we suppose that the distance
to the source star DS is approximately the size of the whole system (i.e., a galaxy of lenses),
then we can take the density over a spherical volume of masses, each with the same mass
m, so that ρ = m
4πD3
S/3
, and we can then use the virial theorem to obtain
τ ∼
v2
c2
∼
(220km/s)2
c2
∼ 5 × 10−7
, (2.2.11)

for the optical depth of source stars in the LMC, where v is the circular rotation speed in
the Milky Way galaxy in the gravitational ﬁeld of the halo [59; 3]. This is an estimate of the
optical depth assuming that the Milky Way halo wholly consists of compact objects and has
no dependence on the location of the source stars, as long as the source stars are, on average,
roughly twice as far away as the lenses. We use the rotation velocity of the Milky Way here
because we are estimating τ due to the Milky Way’s mass under the assumption that the
dark halo is completely composed of MACHOs. This very small estimate for τ (which
corresponds to about 1 in 2 million) indicates why millions of stars need to be monitored to
search for microlensing signals. Also implied from this very small lensing probability is the
fact that microlensing should essentially never repeat for the same source star [54]. More
accurate estimates of the optical depth are obtained by evaluating the integral in Eq. (2.2.9)
for any mass density distribution along the line of sight. Such calculations for a popular
isothermal sphere model which involves a parameterization adopted for the density of a
spherically symmetric halo (e.g., [60]) give an optical depth for lensing of source stars in the
LMC by MACHOs in the Milky Way of τLMC ∼ 5.1 × 10−7, under the assumptions that
(1) all of the dark matter is in the form of MACHOs, and (2) the most naive halo model
(spherically symmetric, with a small halo core radius of a few kpc) is correct. This very
low value for τLMC means that, at any given time, only about one star in two million will
be magniﬁed by A > 1.34. This estimate of τLMC also assumes that all of the dark matter
is in MACHOs, and hence is a crude upper limit to the optical depth [60; 54].
Using the optical depth and the distribution of event durations one can draw
conclusions about the fraction of halo dark matter which is made up of MACHOs and the
most likely MACHO mass. Various microlensing survey teams including the MACHO and
EROS2 collaborations have chosen the LMC as a source background because this nearby

galaxy provides a background of bright source stars and is located away from the plane of the
Milky Way, which reduces the amount of foreground confusion that occurs in stellar fields
with crowded and complex backgrounds (due to random superpositions of stars of different
luminosities). Moreover, if one wants to observe microlensing from objects in the dark halo
of the Milky Way, the monitored source stars must be far enough away so that there is a
large amount of MW halo material between the observer and the source stars. The MACHO
collaboration observed 11.9 million stars in the LMC and found 13 to 17 events towards the
LMC in 5.7 years of observations, with a most likely mass for the lenses estimated from a
maximum likelihood analysis to be in the range 0.15−0.9M assuming a standard spherical
Galactic halo and derived optical depth of τ = 1.2 ± 0.35 × 10−7 from microlensing events
with event durations of 2 < t0 < 400 days [61; 62]. (The reported number of events depends
on the specific light curve selection criteria employed). [61] explores two possible selection
criteria. (We also note in passing that a subsequent re-analysis of the same MACHO
collaboration dataset constrained the microlensing event candidate sample to 10 events and
obtained a resulting microlensing optical depth of τ = 1.0 ± 0.30 × 10−7 [63]. However, this
result is not believed to effect the qualitative results of [61] that we discuss here, and we
will not discuss this particular result further). This optical depth is too large by a factor
of about 5 to be accounted for by known populations of stars. This excess is attributed to
MACHOs of mass ∼ 0.5M . Furthermore, the MACHO collaboration found a scarcity of
short duration lensing events, suggesting that there is no significant population of brown
dwarfs (with M < 0.08M ) or other low mass MACHOs in the dark halo of the MW galaxy.
The total number of lensing events detected by the MACHO collaboration suggest that as
much as 20% of the halo mass could be in the form of MACHOs. The long time scales of the
observed lensing events suggest typical MACHO masses of M > 0.15M . The estimated

mass range of 0.15−0.9M may indicate that a reasonable baryonic dark matter candidate
in the MW halo could include a population of white dwarf stars. The results reported in
[61] indicate that a MW dark halo consisting of 100% MACHOs is ruled out at the 95%
confidence level except for the most extreme halo model. In addition, these results indicate
that the total mass in MACHOs out to 50 kpc is 98+4
−3 × 1010M [61].
The EROS collaboration monitored over 49 million stars in the LMC for 6.7 years
between 1996 and 2003 [62; 64]. Using a subsample of 7×106 bright stars spread over 84 deg2
of the LMC, they recently reported finding no candidate microlensing events towards the
LMC [65]. From their sample, they are only able to estimate an upper limit of the optical
depth towards the LMC of τ = 0.36 × 10−7 at a 95% confidence level for lensing objects
with M ∼ 0.4M , corresponding to less than 8% of the MW halo mass being comprised
of lensing objects [65]. These results would indicate that MACHOs in the mass range
0.6 × 10−7M < M < 15M are ruled out as the primary occupants of the Milky Way
Halo [65]. The EROS collaboration estimate of optical depth is significantly lower than
that of the MACHO collaboration. This significant discrepancy is thought to be due to the
different data sets and samples of stars used in these groups’ analysis: Generally speaking,
the MACHO group used faint stars in dense fields (1.1×106 stars over 13.4 deg2 in primarily
the central part of the LMC), whereas the EROS experiment used bright giant stars in sparse
and less crowded fields (0.7 × 107 stars) over a larger solid angle [65; 66].
Given that the Andromeda galaxy (M31) is both nearby and similar to the MW,
it has also been considered as a suitable target for microlensing searches for MACHOs.
Microlensing searches towards M31 allow us to explore the MW halo along different lines
of sight. Moreover, M31 also has its own halo that can be studied globally, and its high
inclination is expected to give a strong gradient in the spatial distribution of microlensing

2.3. The Location of the Microlenses 27
events that can in principle provide an unmistakable signature for M31 microlensing halo
events [67; 66]. Furthermore, M31 offers a favorable alternative venue for exploring the
halo dark matter problem in spiral galaxies by applying microlensing search techniques
to source stars in M31 itself for lenses in M31 but also lenses in the Milky Way [68].
With these factors in mind, a number of groups have recently carried out microlensing
searches for MACHOs in the halo of M31. Tantalizing preliminary results from two of these
collaborations [68; 67; 66] suggest that at least 20-25% of the mass of Andromeda’s halo is
in the form of MACHOs with an average mass lying in the 0.5 - 1M range, similar to the
MACHO collaboration’s estimates for the MW halo. We must keep in mind, however, that
the final interpretation of observed microlensing events toward both the LMC and M31 as
reported by various collaborations is complicated by the fact that the statistics are low and
that known stellar populations in the LMC and M31 may be contributing to a “self-lensing”
signal (see Section 3.3). Moreover, some of the conclusions drawn from some of the reported
results are preliminary and somewhat contradictory in nature, and the interpretation of the
detection of microlensing events with respect to the halo dark matter issue is still open
to debate [66]. A resolution of these issues will probably require additional data so that,
for example, the distance to a representative sample of LMC microlensing events can be
determined [63].
2.3 The Location of the Microlenses
There has been a great deal of controversy over the location of the microlenses
detected towards the LMC [69; 70; 71] since the detection of the first microlensing event
[72]. Three competing explanations for the microlensing signal are that microlensing towards
the LMC is caused by: (1) MACHOs in the halo of the MW, (2) the ”self-lensing” of stars

in the LMC by other faint stars in the LMC, and (3) a structure behind the LMC providing
source stars for LMC lensing.
Three possible ways for distinguishing between a self-lensing and a halo-lensing sig-
nal, and, thus, resolving the halo-lensing or self-lensing debate, include: (1) given an agreed
upon LMC model, modeling the contribution to the optical depth from known dark-matter
components, subtracting this from the observed signal, and assuming that the remaining
signal comes from the MW dark halo, (2) comparing the spatial distribution of the observed
microlensing events with the predictions of LMC and halo models [59; 1], and (3) determin-
ing the location of the lenses in some more direct fashion: those events whose lenses lie in
the LMC are self-lensing and those that lie in the halo are halo-lensing.
In a standard (thin disk + bar) model of the LMC, modeling of the microlensing
geometry indicates that self-lensing events are expected to make only a small contribution
(∼ 8 − 13%) to the total optical depth towards the LMC [69; 59; 1]. As a speciﬁc example,
an analytical estimate by [69] of the optical depth for “disk-disk” self-lensing for a self-
gravitating thin disk LMC model yields
τ =
2 < v2 >
c2
1
cos2 i
. (2.3.1)
For an observed line-of-sight velocity dispersion for stars in the inner parts of the LMC of
v ∼ 20 km/s and an LMC inclination angle of i ∼ 30 degrees this yields τ ∼ 1×10−8, which
is about 12 times smaller than or about 8% of the measured value (τ = 1.2 ± 0.35 × 10−7)
obtained by the MACHO collaboration. Standard models of the MW and LMC consistent
with current observations and which include more realistic estimates of the average optical
depth for lensing due to the known stellar LMC populations (and which, for example,

include the use of a double exponential density profile model to describe the LMC stellar
disk using the model parameters in the table depicted in Fig. 2.5 from [1]) yield an LMC
self-lensing optical depth of τ ∼ 2.4 × 10−8 [1]. This value for optical depth is about
5 times smaller than the measured value obtained by the MACHO collaboration. As a
specific example, Fig. 2.5 (from Table 3 of [1]) shows the preferred values of the LMC
model parameters used to compute the contribution to the observed optical depth and the
expected number of microlensing events (Nexp) from each (non-dark matter) population in
the MW and LMC, listed in Table 2.1 [1]. The table in Fig. 2.5 also shows the range of LMC
model parameters accepted in the literature and includes velocity dispersion, scale height,
zd, and inclination angle, all parameters that the LMC disk self-lensing optical depth is
particularly sensitive to. For all reasonable combinations of LMC parameters, however, the
LMC disk self-lensing optical depth can not realistically exceed more than about 20% of
the observed optical depth [1]. The results from such modeling and analysis thus indicate
that the experimental value of optical depth obtained by the MACHO group can not be
explained by LMC “disk-disk” self-lensing alone. However, the structure and dynamics of
the LMC are not completely determined, and more adventurous and creative models may
(or may not) result in a significantly higher self-lensing optical depth [73; 74; 75; 76].
Table 2.1. Expected non-dark matter contributions to the optical depth for standard mod-
els of the MW and LMC (left column). The right column gives the expected number of
microlensing events for the given population [1]. When adding up the microlensing contri-
bution from each of these populations, one obtains a total optical depth value τ ∼ 2.4×10−8
that is still about 5 times smaller than the measured value obtained by the MACHO col-
laboration.
Population τ × 10−8 Nexp
MW Thin Disk 0.4 0.4
MW Thick Disk 0.2 0.2
MW Stellar Halo 0.2 0.2
LMC Disk 1.6 1.3
Total 2.4 2.1

Figure 2.5. This table lists the LMC model parameter values for the disk, bar, and halo
models used by [1] to obtain their estimate of the self-lensing optical depth of the LMC.
(From [1]).

2.4. Information from the Microlensing Source Star CMD 31
Fortunately, the problem of confining the location of the lensing population can
be approached in other ways. We will now explore how we can use the source star color-
magnitude diagram to obtain information about the possible locations of the microlensing
source stars.
2.4 Information from the Microlensing Source Star CMD
Here, we compare four different models for LMC microlensing. Each model involves
a different microlensing geometry, in which the source stars and lenses are located in different
populations. In some models, a substantial fraction of the source stars will lie behind the
LMC disk.
The four LMC microlensing models are:
® MW Halo-lensing: For this model the lens is a MACHO in the MW halo and
the source star is located in the LMC disk or bar. All of the source stars lie in the LMC
disk and/or bar such that the source stars are representative of the average color magnitude
diagram (CMD) of the LMC.
® LMC Disk and/or Bar Self-lensing: Both the lens and the source star
are normal stars in the LMC disk and/or bar and, once again, none of the source stars lie
behind the LMC disk and/or bar.
® LMC Spheroid Self-lensing: This is a reference to both LMC halo and
”shroud” self-lensing. The term ”shroud” here is meant to imply an LMC population which
is like a halo in that it is spatially not part of the LMC disk, but unlike a halo in that
it is non-virialized and takes the form of an extended flattened spheroidal component of
tidal debris [74; 75]. In this model, both the lens and the source star can lie either in the
LMC disk or the LMC halo. In spheroid self-lensing there are four event geometries: (1)

background spheroid source and disk lens, (2) disk source and foreground spheroid lens, (3)
disk source and disk lens, and (4) background spheroid source and foreground spheroid lens.
We might naively expect event geometries (1) and (2) to dominate the number of expected
events and so if we were to ignore the contribution to event geometries (3) and (4) we would
conclude that spheroid lensing would imply that about half the source stars lie in back of
or behind the LMC. However, in order to produce the total observed optical depth, the
spheroid needs to be so massive that it is no longer self-consistent to ignore event geometry
(4). Calculations performed in the formalism of [1] suggest instead that microlensing events
with a background spheroid source and a foreground spheroid lens become an important
contributer and increase the expected fraction of background source stars behind the LMC
disk to ∼ 0.65
® Background Lensing: In this model, introduced and pushed forward by
[77; 78; 4], the observed microlensing events are due to ”background” lensing, in which all
of the source stars are located in some background population, displaced at some distance
behind the LMC. Lenses for this population may then be supplied by the disk and bar of
the LMC. Such a background population, however, may be nearly impossible to conﬁrm
or reject observationally, as there are nearly no limits on its size or content (provided, of
course, that it is small enough to ”hide” or be obscured behind the LMC.)
Each of the four LMC microlensing model geometries described above is pictorially
depicted in Fig. 2.6.
[79] have attempted to distinguish between these possibilities by estimating the
source star locations with predictions for various LMC self-lensing and MW-lensing geome-
tries, and, thus, locating the lenses by ﬁrst locating the source stars. We have continued
with this approach and attempted to improve upon it by using synthetic CMDs and a larger

Figure 2.6. Four possible LMC microlensing geometries that have been developed and
discussed to try to explain the microlensing events towards the LMC. They correspond to
four competing explanations for the microlensing signal towards the LMC. The Milky Way
is depicted as the large spiral galaxy in the upper right hand corner of each panel, and the
LMC is depicted as the irregular galaxy in the lower left hand corner. The arrow in each
panel indicates a line of sight from the location of the Earth in the Milky Way towards
random positions in the LMC. The white dots indicate the position of source stars and
the green dots indicate the position of lensing objects. The ﬁgures are not drawn to scale.
(Adapted from [4]).

number of observed source stars (corresponding to 13 microlensing events, compared to the
8 events used by [79]).
The available self-lensing geometries are constrained by knowledge of the size,
content, and structure of the LMC [61]. The most viable and quantitatively plausible
models for a self-lensing population large enough to explain the total observed optical depth
towards the LMC propose a thick three dimensional structure behind the LMC [78; 4]. This
structure allows for a slight variation on self-lensing in which source stars are drawn from this
background population and the lenses are normal stars in the LMC. Since this population
lies behind the LMC, source stars drawn from it will suffer from the internal extinction
of the LMC. Additionally, since this structure may be displaced behind the LMC by some
amount, source stars drawn from it should be slightly fainter and have a slightly larger
distance modulus, which is defined as
µ ≡ m − M = 5 log10(
d
10pc
), (2.4.1)
where m is the apparent magnitude of the source star, M the absolute magnitude, and d is
the distance of the source star from Earth in units of parsecs. If we find that the MACHO
source stars are drawn mostly from this background population, then we can conclude
that the most likely microlens population for this background source star population is the
LMC itself and, therefore, a source star drawn from this population behind the LMC would
imply that microlensing is dominated by LMC disk or bar self-lensing. If, on the other
hand, the source stars are drawn mainly from the LMC itself, then we might conclude that
the microlensing is dominated by MW halo-lensing, since the contribution from LMC disk
or bar self-lensing is known to be small [69; 1].

A physically reasonable parameter for the extra reddening of the background pop-
ulation of source stars in our model is E(B − V ) = 0.13, as inferred from the mean
extinction of the LMC from [80] and corrected for galactic foreground extinction. This is
the lowest mean value determined using young, hot stars (T > 22000K), but is significantly
higher than the [81] value for old, cooler stars (5500K < T < 6500K). The distance to
the background population, corresponding to a shift and increase in the distance modulus
with respect to the LMC of ∆µ ∼ 0.3 according to the model presented by [4], is very
loosely obtained by the requirement that the background (source star) population be at
least transiently gravitationally bound to the LMC, and we further assume that this back-
ground population does not differ intrinsically from the main LMC population. We can,
thus, look for evidence of background source stars by looking for evidence of extra redden-
ing in the source star CMD. We then define fBKG as the fraction of source stars which
have roughly twice the average reddening of the LMC, i.e., they’re on the far side of the
LMC. We consider three different displacement distances, ∆µ ∼ 0.0 (”model 1”), ∆µ ∼ 0.3
(corresponding to ∼ 7.5 kpc behind the LMC according to Eq. (2.4.1) ; ”model 2”) and
∆µ ∼ 0.45 (corresponding to ∼ 11.5 kpc behind the LMC; ”model 3”), where we have no
constraints on the size or content of the background source star population except that it
must be small enough and similar enough to the LMC stellar population to have avoided
direct detection. Such a stellar population could, for example, be a population of stars 7.5
kpc or 11.5 kpc behind the LMC disk that was pulled from the LMC and Small Magellanic
Cloud due to MW tidal forces (i.e., a tidal debris tail or tidal extension behind the main
body of the LMC), or some other extra three-dimensional thick distinct stellar structure
displaced from the two-dimensional thin and cold disk of the LMC but physically associ-
ated with the LMC and sharing its kinematics [82; 4; 83]. The distance to this background

population of ∆µ ∼ 0.3 from [4], for example, is loosely derived by the requirement that
the background stellar population be gravitionally bound to the LMC.
We attempt to distinguish between source stars drawn from the average popula-
tion of the LMC and source stars drawn from a population behind the LMC by examining a
Hubble Space Telescope composite Color Magnitude Diagram (HST CMD) of microlensing
source stars and comparing it to the HST CMD of the average LMC. From the definition of
fBKG, we can also distinguish between MW halo lensing and LMC disk self-lensing. These
CMDs are created from wide field planetary camera (WFPC2) HST photometry of MA-
CHO microlensing source stars and their surrounding fields with care taken to identify the
proper sources from severely blended ground-based MACHO images (see [79] and [61] and
references therein for technical details concerning how these images were obtained, reduced,
processed, and analyzed and how the photometry was carried out). The identification is
achieved by deriving accurate centroids in the ground-based MACHO images using differ-
ence image analysis (DIA) [84] and then transforming the DIA coordinates to the HST
frame. Before we can properly compare the source star CMD to the HST field CMD, and
in order to take into account the fact that not all possible micro-lensing events are detected
in MACHO images, we must first convolve the HST WFPC2 composite field CMD with
the MACHO microlensing detection efficiency as a function of stellar magnitude and mi-
crolensing event duration in order to reproduce the unblended population observed by the
MACHO experiment. (The microlensing detection efficiency is defined as the likelihood of
(the MACHO experiment, in this case) detecting a source star of a given magnitude). We
then have a model CMD representing a population of source stars which produce detectable
microlensing events.
The next step is to compare the microlensing source stars with model populations

with varying fractions of source stars placed behind the LMC. For each model source star
population we begin with the composite HST CMD and then shift some fraction, fBKG, of
the stars in the WFPC2 HST CMD behind the LMC, i.e., we redden and displace a fraction
fBKG of the CMD by amounts
∆V = Av + ∆µ = 0.43 + ∆µ, (2.4.2)
∆(V − I) = E(V − I) = 1.376E(B − V ) = 0.18 mag, (2.4.3)
where the reddening or V-band extinction is Av = 3.315E(B −V ) = 0.43 mag, E(B −V ) =
0.13 is the mean internal extinction of the LMC found by [80] using UBV and UBVI
photometry, and the coefficient in the conversion from E(B −V ) to E(V −I) is drawn from
Table 6 of [85]. Each model CMD now contains a fraction 1.0−fBKG of source stars drawn
from the LMC disk and bar and a fraction fBKG of source stars drawn from a background
population. We thus now have our final “model 1” and “model 2” distribution of source
stars. Fig. 2.7 shows the resulting two (extreme) model source star populations, with the
observed microlensing source stars overplotted as large red stars.
Next, we need a numerical statistic which will give us the probability that the
observed microlensing source stars are drawn from a given model population. Given that
we do not have an analytical model to test with (as in the case of a χ2 test), we choose the
two-dimensional Kolmogorov-Smirnoff (KS) test [86] as in [79] to quantify the probability
that the observed microlensing source stars are drawn from a specific model population.
KS tests were run using 20 Monte Carlo simulations for each of the three models (“model
1”, “model 2”, and “model 3”) for each background fraction fBKG between 0.0 and 1.0 in
increments of 0.1. An average value of P(D), the probability that the observed microlensing

source stars were drawn from a model population with a fraction fBKG behind the LMC,
as a function of distance statistic D [86], was obtained for each model and for each value
of fBKG. Essentially what we have done here is to generate many synthetic data sets from
each of the three models, with each synthetic data set having the same number of stars as
the real data set. We then compute D for each synthetic data set and count what fraction
of time these synthetic D values exceed the distance statistic D values from the real data. It
is this fraction which is then referred to as our ’significance’. This then corresponds to the
use of the distribution of D statistics to compute the 2-dimensional integrated cumulative
probability function P(D) that if two distributions were from the same parent distribution,
a ’worse’ value of D would result [86]. The KS test results, depicted in Fig. 2.8, indicate that
the 2-D KS-test probability P is highest for fBKG ∼ 0.0 − 0.2 with very little dependence
on the value of the displacement ∆µ. Note that small values of P (corresponding to ’worse’
values of D) indicate that there is a small chance that the microlensing source stars come
from a population of stars of a given fBKG. The error bars indicate the scatter about
the mean value of 20 Monte Carlo simulations for each fBKG and for each of the three
models. Also, because the creation of the efficiency-convolved CMD is a weighted random
draw from the HST CMD, the model (CMD) population created in each simulation differs
slightly. This in turn leads to small differences in KS statistics. We also point out here
that the shifting up or down of the model CMDs with respect to the real CMDs (which
correspond to changes in the distance modulus of the model CMDs) do not make very much
of a difference in the KS test results because most of the stars in the CMDs are contained
in the main sequence and red giant branch, which are (already) vertical structures. It is the
horizontal shifts of the model CMDs with respect to the real CMDs which tend to cause
greater or smaller distances between the circles, triangles, and squares in the KS test result

Figure 2.7. The observed MACHO microlensing source stars (large red) stars, over-plotted
on two model source star populations (small black dots). The left panel represents a source
star population drawn entirely from the LMC disk+bar (fBKG = 0.0). The right panel
represents a model in which all of the stars belong to a background population (fBKG = 1.0)
with ∆µ = 0.3 and ∆E(V − I) = 0.18.
plot for different ∆µ.
We rule out a model in which the source stars all belong to some background
population at a confidence level of 99%. We can rule out spheroid self-lensing models
(fBKG ∼ 0.65) at the statistically marginal confidence level of 80-90%. With the 13 event
KS-test analysis alone, we can’t exclude lensing by an LMC stellar shroud with any real
significance. However, no strong observational evidence exists to support the existence of
an LMC shroud, and some observational work limits the total mass of any type of spheroid

Figure 2.8. The 2-D KS test probability P that the observed distribution of source stars
was drawn from a source star population in which a fraction of the source stars are located
behind the LMC. The distance moduli of the background stars with ∆µ = 0.0, 0.30, 0.45 are
shown as red circles, green triangles and blue squares respectively. The error bars indicate
the scatter about the mean value for 20 simulations of each model.
to about 5% of the total mass of the LMC [87; 88] which is far too small to account
for the observed microlensing signal. The allowed region of the KS test result plot is
consistent with the expected location of the source stars in both the MW-lensing and LMC
disk+bar self-lensing geometries. However, as discussed above, detailed modeling of the
LMC disk+bar self-lensing suggests that it contributes at most 13% of the observed optical
depth. We also note that, by deﬁnition, fBKG for LMC disk-disk lensing is much higher than
for MW halo lensing, indicating that the KS test results disfavor LMC disk-disk lensing.

2.5. Problems, Challenges, and Suggestions for Future Work 41
Therefore, the results of the KS test analysis presented here, taken together with external
constraints, suggest that the most likely explanation is that the lens population comes
mainly from the MW halo, with a smaller self-lensing contribution from the disk and bar
of the LMC. The strength of this analysis is severely limited by the number of microlensing
events. Although we are currently able to exclude the most extreme model (fBKG ∼ 1.0)
at a reasonable degree of confidence, more microlensing events will be necessary to more
accurately determine fBKG and eliminate microlensing models such as LMC ”shroud” self-
lensing. Ongoing and future microlensing surveys and projects such as SuperMACHO
[89] may provide a sufficient sample of microlensing events over the next few years. The
technique outlined here could prove a powerful method for locating source stars and lenses
with the use of these future data sets.
2.5 Problems, Challenges, and Suggestions for Future Work
Another significant problem with the analysis outlined here, however, is that in the
creation of the model source star populations, the background fraction was created from the
normal, observed CMD, which already includes the background fraction and all other pop-
ulations. Also, the observed CMDs we were using to create model source star populations
were already reddened with the observed reddening of the LMC and foreground reddening,
and reddening them again with the specified reddening model could result in too large a
spread along the reddening vector. In order to begin to address these problems, we have
also attempted to do a more sophisticated analysis of the MACHO source stars in relation
to the general LMC population by deriving the underlying un-reddened stellar CMD and
then constructing various reddening models involving uniform reddening as well as Poisson
reddening models with a Poisson distribution of ”cloudlets” (to make a detailed study of

the effect of patchiness on the significance of the KS test) for the populations supplying the
microlensing source stars, and, thus, recreating any populations we are interested in testing.
We then compare these reddened model CMDs to the observed source star CMDs and the
13 microlensing source stars. The null hypothesis that we are testing is that the population
of microlensing sources in the LMC came from the ”normal” CMD. The alternative is that
the population of microlensing sources came from a different distribution (”background”
stars). So, we are testing what model fits the observed data best, and the statistics that we
use to do the testing come from the 2D KS test. It is clear that we have too few sources to
make strong statements like rejecting the null hypothesis. Rather, we would like to show
that the null hypothesis fits the data better than some other hypothesis (like the model in
[4]). We use the synthetic CMD algorithm ”StarFISH”1 [90; 91] to generate non-reddened
model CMDs.
The StarFISH package is a suite of FORTRAN 77 code and routines designed to
determine the best-fit star formation history (SFH) for a stellar population, given multicolor
stellar photometry and a library of theoretical isochrones. The package constructs a library
of synthetic CMD Hess diagrams based on theoretical isochrones and data-derived deter-
minations of interstellar extinction, photometric errors, completeness, distance modulus,
and binarity, and then uses a minimization routine (using the downhill simplex method)
to determine the linear combination of synthetic CMDs that best matches the observed
photometry. This is then compared statistically to the observed photometry, and when the
best-fitting model (according to the best χ2 fit) is found, the amplitude coefficients modu-
lating each synthetic CMD will describe the SFH of the observed stellar population. We are
then able to use the code to construct an artificial stellar population (the synthetic CMD)
1
Available from http://www.noao.edu/staff/jharris/SFH/

that can then be compared to the observed CMD (Fig. 2.9). Essentially, the StarFISH code
is able to populate the isochrones with stars by using the fact that the probability of finding
a star at any point along an isochrone depends on the initial mass function (IMF) and on
the evolutionary timescale of that particular point. To populate the isochrones in a realistic
manner, i.e., to reproduce observational effects, the stars added to the isochrones are prob-
abilistically removed and scattered according to the photometry-dependent completeness
rate and crowding error tables [90].
In order to successfully use the StarFISH code to determine best-fit star formation
histories and to generate synthetic CMDs, we needed to generate crowding tables for each
of our 13 HST LMC fields because the synthetic CMDs must also include the effects of
completeness in addition to the effects of extinction, photometric errors, and binarity. We
discovered that the HST data used in this work was incomplete brightward of apparent
V-band magnitude V = 20. Hence, crowding tables generated earlier did not have sufficient
coverage of brighter magnitudes. Therefore, we needed to carry out artificial star tests
on short exposure HST images with the brighter stars, and then combine these resulting
crowding tables with the long exposure crowding tables. We then input these combined
crowding tables into the StarFISH code. The purpose of carrying out the artificial star
tests was to estimate the photometric errors and the effects of crowding on the observed
stellar luminosity distribution [56] so that they could be properly simulated in the synthetic
CMDs with the StarFISH code. Again, the crowding tables (constructed from the artifi-
cial star tests) were input into the StarFISH code to simulate seeing, incompleteness, and
photometric error in the synthetic CMDs.
After ”de-reddening” the artificial photometry generated by the StarFISH code,
we then take this uniformly de-reddened synthetic CMD and applied uniform reddening

Figure 2.9. Left Panel: Observed composite HST color magnitude diagram of the 13 LMC
fields surrounding each of the microlensing events. Right Panel: An example of a composite
best-fit StarFISH generated model CMD for the 13 LMC fields.

or Poisson reddening models to it, as we describe below. We then convolve the MACHO
microlensing detection efficiency with this synthetic ”re-reddened” CMD given an input
fBKG and other input parameters such as the reddening values for a single “cloudlet” in
the LMC when implementing Poisson reddening models.
In our implementation of Poisson reddening models we have utilized the “one
cloud” or “cloudlet” model of [92] in which the number of clouds N is Poisson distributed
around the value of 3.3. The internal LMC reddening along a given line of sight is then
given as N times the unit reddening of a cloud, which is 0.04 magnitudes in E(B-V) [92].
The mean reddening of this model, E(B − V ) = 3.3 × 0.04 = 0.132, compares very well
with the current adopted value of E(B − V ) = 0.13. Note that the mean of 3.3 cloudlets
per line of sight was derived by taking the mean LMC reddening, E(B − V ) = 0.13 and
dividing it by the mean reddening in a cloudlet, E(B − V )1cloudlet = 0.04, with the latter
value taken from [92]. We must also take into account the fact that light from objects
in the LMC is influenced by both internal LMC extinction as well as galactic foreground
extinction caused by dust inside our own Galaxy.
In what follows, we introduce realistic reddening models which includes the LMC
structure and describe our attempts to implement them with the use of the StarFISH
synthetic CMD algorithm. First, we take x to be the mean for a Poisson distribution of
cloudlets, and we use 2x for the mean of the Poisson distribution for background redden-
ing, since the optical path traverses the entire dust disk of the LMC, as opposed to just
half of it for an average LMC star. The value for x is derived from the best-fit Av value,
Av,starfish, found by the StarFISH code and parabola fitting. Now that we know the mean
number of cloudlets x for the foreground (2x for the background) reddening, we then use
an acceptance-rejection technique to redden the synthetic CMD with the Poisson model

according to the following procedure:
a. Generate a random number (RN) in [0,1] (by definition, RN is a number between 0 and
1, exclusive of the end-point values, so it will always be smaller than 1.0)
b. If RN > fBKG, then this is a star in the LMC, and only (Milky Way) galactic foreground
reddening, Av,galactic = 0.1989 [93], is applied to the un-reddened StarFISH synthetic CMD.
c. If RN < fBKG then we use the ’one cloud’ Poisson reddening model, in which a
random deviate is drawn from a Poisson distribution with a mean of x = (Av,starfish −
Av,galactic)/0.1326 cloudlets and then multiplied by the mean Av value for a single cloudlet,
Av,1cloudlet, and all this is then added to the magnitude of the star in the de-reddened syn-
thetic CMD so that the final magnitude and color of the (re-reddened) star in the synthetic
CMD becomes
Vsynth = V + Av,1cloudletP(x) + Av,galactic, (2.5.1)
(V − I)synth = V − I + E(V − I)1cloudletP(x) + E(V − I)galactic, (2.5.2)
where V and I are the visual and I-band magnitudes of the stars in the de-reddened synthetic
CMD, E(V −I)1cloudlet = 0.055 is the color extinction of one cloudlet (Eq. (2.4.3) and [92]),
E(V − I)galactic = 0.08 is the galactic foreground color reddening, and P(x) is a random
deviate from a Poisson distribution with a mean of x. We believe that this reddening
model is consistent with observations because we ensure that the mean reddening is always
Av,starfish, no matter what we assume for the background fraction. The corresponding
efficiency-convolved (synthetic) CMD plots for this model, with fBKG = 0.0, ∆µ = 0.0
in the left panel and with fBKG = 1.0, ∆µ = 0.3 in the right panel, are depicted in Fig.
2.10. We can visually see that the model (efficiency-convolved) source star population in
the left panel of this plot appears quite similar to the observed efficiency-convolved source

star population in the left panel of Fig. 2.7. In the right panel of Fig. 2.10 the source
stars appear to be shifted signiﬁcantly to the right (red-ward) with respect to the model
source star population in comparison to the corresponding plot in the right panel of Fig.
2.7. We then use this synthetic CMD and the set of 13 microlensing events to invoke the 2-
dimensional KS test and compute the test statistic using the probability of a random chance
occurrence as the measure of likelihood for a particular model (i.e., with particular values of
fBKG and ∆µ, etc.), as described in the previous section. The corresponding KS test results
plot for this reddening model is shown in Fig. 2.11. We see that the KS probability seems
to peak for the ∆µ = 0.30 and 0.45 cases at around fBKG = 0.30 with an overall greater
dependence on the value of displacements (0 kpc, 7.5 kpc, and 11.5 kpc) behind the LMC
for a given fraction fBKG of microlensing source stars behind the LMC. This may be related
to greater horizontal shifts of the diﬀerent model source star populations with respect to
each other when generated with the synthetic CMD algorithm (Fig. 2.10) as compared to
the corresponding case for when real data was used (Fig. 2.7). The KS test plot does not
show the same overall pronounced trend towards lower 2-D KS test probabilities with higher
background fractions for fBKG > 0.3 that we see in Fig. 2.8 for the case in which real data
was used instead of synthetic CMDs. We also note that the KS test probabilities at lower
background fractions fBKG < 0.3 are considerably lower than in Fig. 2.8, and we can not
rule out a model in which the source stars all belong to some background population to
nearly the same extent as when real data and a uniform reddening model was used. As
discussed below, these KS test results are very preliminary and somewhat uncertain due to
some possible problems with the StarFISH-generated model CMDs. For example, the wider
and more smeared out appearances of the red giant branches and red clumps in the model
CMDs with ∆µ > 0 (e.g., Fig. 2.10) could in part lead to KS test results such as those in

Fig. 2.11. Moreover, the overall appearance of the KS test plot in Fig. 2.11, including the
fact that the KS test probabilities never vary by more than about 20% for different values
of fBKG, may point to the possibility that the significance of the KS test will substantially
decrease if the reddening in the LMC is very patchy.
We have also used and implemented a ’two cloud’ Poisson reddening model, which
is a slight variation on the ’one cloud’ model described above:
a. Generate RN in [0,1]
b. If RN > fBKG, then this is also a star in the LMC, and only galactic foreground
reddening should be applied to the un-reddened StarFISH synthetic CMD.
c. If RN < fBKG then we implement the ’two cloud’ Poisson reddening model such that
the final magnitude and color of a (re-reddened) star in the synthetic CMD becomes
Vsynth = V + Av,1cloudletP(2x) + Av,galactic + ∆µ, (2.5.3)
(V − I)synth = V − I + E(V − I)1cloudletP(2x) + E(V − I)galactic, (2.5.4)
where all definitions are the same as in the case of the ’one cloud’ model , but we additionally
note that P(2x) is a draw or random deviate from a Poisson distribution with a mean of
2x = 2(Av,starfish − Av,galactic)/0.1326 cloudlets, and ∆µ is an added in change in distance-
modulus factor for the distance behind the LMC that the background stars will be displaced
to. The corresponding efficiency-convolved CMD plots for this model (with the fBKG = 0.0,
∆µ = 0.0 case in the left panel and the fBKG = 1.0, ∆µ = 0.3 case in the right panel)
are displayed in Fig. 2.12. The corresponding KS test results for this reddening model are
show in Fig. 2.13. These results seem quite similar, though not completely identical, to the
KS test results for the ’one cloud’ reddening model.

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