Simulations of Small Mass Structures in the Local Universe to Constrain the Nature of Dark Matter

Simulations of Small Mass Structures in the Local Universe to Constrain the Nature of Dark Matter
ABSTRACT
Title of dissertation:
SIMULATIONS OF SMALL MASS
STRUCTURES IN THE LOCAL
UNIVERSE TO CONSTRAIN THE
NATURE OF DARK MATTER
Emil Polisensky, Doctor of Philosophy, 2014
Dissertation directed by:
Professor Massimo Ricotti
Department of Astronomy
I use N-body simulations of the Milky Way and its satellite population of dwarf
galaxies to probe the small-scale power spectrum and the properties of the unknown
dark matter particle. The number of dark matter satellites decreases with decreasing
mass of the dark matter particle. Assuming that the number of dark matter satellites
exceeds or equals the number of observed satellites of the Milky Way, I derive a
lower limit on the dark matter particle mass of mW DM > 2.1 keV for a thermal dark
matter particle, with 95% confidence. The recent discovery of many new dark matter
dominated satellites of the Milky Way in the Sloan Digital Sky Survey allows me to
set a limit comparable to constraints from the complementary methods of Lyman-α
forest modeling and X-ray observations of the unresolved cosmic X-ray background
and of halos from dwarf galaxy to cluster scales.
I also investigate the claim that the largest subhalos in high resolution dissipationless cold dark matter (CDM) simulations of the Milky Way are dynamically
inconsistent with observations of its most luminous satellites. I quantify the ef-
fects of the adopted cosmological parameters on the satellite densities and show the
tension between observations and simulations adopting parameters consistent with
WMAP9 is greatly diminished. I explore warm dark matter (WDM) cosmologies
for 1–4 keV thermal relics. In 1 keV cosmologies subhalos have circular velocities at
kpc scales 60% lower than their CDM counterparts, but are reduced by only 10% in
4 keV cosmologies. Recent reports of a detected X-ray line in emission from galaxy
clusters has been argued as evidence of sterile neutrinos with properties similar to
a 2 keV thermal relic. If confirmed, my simulations show they would naturally reconcile the densities of the brightest satellites and be consistent with the abundance
of ultra-faint dwarfs.
I conclude by using N-body simulations of a large set of dark matter halos
in different CDM and WDM cosmologies to demonstrate that the spherically averaged density profile of dark matter halos has a shape that depends on the power
spectrum of initial conditions. Virialization isotropizes the velocity dispersion in
the inner regions of the halo but does not erase the memory of the initial conditions in phase space. I confirm that the slope of the inner density profile in CDM
cosmologies depends on the halo mass with more massive halos exhibiting steeper
profiles. My simulations support analytic models of halo structure that include
angular momentum and argue against a universal form for the density profile.
SIMULATIONS OF SMALL MASS STRUCTURES IN THE
LOCAL UNIVERSE TO CONSTRAIN THE NATURE OF DARK
MATTER
by
Emil Joseph Polisensky
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2014
Advisory Committee:
Professor Massimo Ricotti, Chair/Advisor
Professor Kevork Abazajian
Professor Michael Boylan-Kolchin
Professor Jordan Goodman
Professor Cole Miller
Preface
“Who can argue against the glories of art, music, and literature that existed
long before science did? And what can science offer us to compare with such beauty?
For one thing, it is possible to point out that the vision of the Universe made
apparent by the careful labor of four centuries of modern scientists far outweighs in
beauty and majesty (for those who would take the trouble to look) all the creations
of all human artists put together, or all the imaginings of mythologists, for that
matter.”
–Isaac Asimov, Best Foot Backward
The material presented in this thesis has been published in three refereed journal
articles:
E. Polisensky, M. Ricotti, 2011, “Constraints on the dark matter particle mass
from the number of Milky Way satellites,” PhRvD, 83, 4
E. Polisensky, M. Ricotti, 2014, “Massive Milky Way satellites in cold and warm
dark matter: dependence on cosmology,” MNRAS, 437, 2922
E. Polisensky, M. Ricotti, 2014, “Fingerprints of the initial conditions on the density profiles of cold and warm dark matter haloes,” MNRAS, (to be submitted)
I have given four talks on this material:
ii
E. Polisensky, M. Ricotti, 2014, “Too Big To Fail: A Sensitive Test of Cosmological
Parameters and Dark Matter Properties,” 223rd Meeting of the American
Astronomical Society, 408.03, Washington, DC, January 2014
E. Polisensky, M. Ricotti, K. Keating, K. Holley-Bockelmann, G. Langston, “Constraints on Particle Mass and the Origins of HI Clouds with Dark Matter
Simulations,” Near Field Cosmology as a Probe of Dark Matter, Early Universe and Gravity, Annapolis, MD, 29 November - 1 December 2011
E. Polisensky, M. Ricotti, “Constraints on Particle Mass and the Origins of HI
Clouds with Dark Matter Simulations”, GUN meeting, College Park, MD, 20
April 2012
E. Polisensky, M. Ricotti, “Constraints on Warm Dark Matter from the Local
Group of Galaxies,” Nuclear Particle Astrophysics and Cosmology Lunch,
Physics Dept. UMD, College Park, MD, 21 April 2011
The material in Chapter 3 was chosen for a featured article in the 2011 yearly review
of research and development projects at the Naval Research Laboratory. Fewer than
10% of submitted abstracts and chosen for inclusion in the review, and only five of
those for featured articles.
E. Polisensky, M. Ricotti, “Constraining the Very Small with the Very Large:
Particle Physics and the Milky Way,” 103, 2011 NRL Review
iii
The images in Figures 3.3, 4.2, and 4.3 were created with software I developed myself
for visualizing N-body simulations.
iv
Dedication
To the next generation, may you be better fitted to this world than the current.
v
Acknowledgments
This thesis is the culmination of a long journey that would not have been
possible without the support of many people.
I’d like to thank my advisor, Professor Massimo Ricotti for teaching me about
and giving me a chance to do research in the field of cosmology. I also thank the
members of my committee for their service and suffering through the reading of a
document of this length.
I owe an immeasurable debt to my supervisors, Dr. Namir Kassim and Dr.
Kurt Weiler, at the Naval Research Laboratory for allowing me to participate in the
Edison Memorial Graduate Training Program and attend school while employed.
Without their continued support, and incredible patience, I never would’ve made it.
I also owe a great many thanks to Dr. William Erickson for setting me on the
right path and getting this journey started. And even further back, many thanks to
Dr. Paul Krehbiel for taking a young undergrad under his wing. If I had a second
lifetime I’d be a lightning researcher.
Additional thanks to the high performance computing support staff at the Air
Force Research Lab, U.S. Army Engineer Research and Development Center, and
the University of Maryland. Without their assistance I wouldn’t have made it very
far beyond saying hello to the world.
To my wife for all her love and support and sharing the waking nightmare of
being a graduate student while holding a full-time job. To my family and friends
for sharing a sense of humor and helping the days go by. To my coworkers at NRL
vi
who make it an enjoyable place to work and the wonderful secretarial staff for help
dealing with the never ending harassment of paperwork.
vii
Table of Contents
List of Tables
x
List of Figures
xi
1 Science Motivation and Structure of Thesis
2 Foundations of Structure Formation
2.1 Cosmological Principle . . . . . . . . .
2.2 Friedmann Equations . . . . . . . . . .
2.3 Jeans Instability . . . . . . . . . . . . .
2.4 The Power Spectrum . . . . . . . . . .
2.5 Abundance of Dark Matter Halos . . .
2.6 Beyond Linear Theory - Spherical Halo
2.6.1 The Virial Theorem . . . . . . .
2.6.2 The NFW Profile . . . . . . . .
2.7 Beyond Linear Theory - Simulations .
1
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Collapse
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3 Particle Mass Constraints from Subhalo Abundances
3.1 Overview . . . . . . . . . . . . . . . . . . . . . .
3.2 Simulations . . . . . . . . . . . . . . . . . . . .
3.2.1 Identification of Satellites . . . . . . . .
3.3 Results . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Satellite Distribution Functions . . . . .
3.3.2 Convergence Study . . . . . . . . . . . .
3.3.3 Comparison to Observations . . . . . . .
3.4 Discussion . . . . . . . . . . . . . . . . . . . . .
3.5 Summary . . . . . . . . . . . . . . . . . . . . .
4 Dependence of Satellite Densities on Cosmology
4.1 Overview . . . . . . . . . . . . . . . . . . .
4.2 Simulations . . . . . . . . . . . . . . . . .
4.3 Results . . . . . . . . . . . . . . . . . . . .
4.3.1 Cold Dark Matter . . . . . . . . . .
viii
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4.3.1.1 Velocity profiles
Warm Dark Matter . . .
4.3.2.1 Velocity profiles
Discussion . . . . . . . . . . . .
4.3.2
4.4
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5 The Universal Density Profile That Wasn’t
5.1 Overview . . . . . . . . . . . . . . . . .
5.2 Numerical Simulations . . . . . . . . .
5.2.1 Cosmological Models . . . . . .
5.2.2 Software . . . . . . . . . . . . .
5.2.3 Simulations . . . . . . . . . . .
5.3 Results I - Non-universality Of Profiles
5.3.1 Density Structure . . . . . . . .
5.3.2 Internal Kinematics . . . . . . .
5.3.3 Convergence Tests . . . . . . .
5.4 Results II - Testing Cosmic Variance .
5.5 Origin of the Core . . . . . . . . . . .
5.6 Discussion . . . . . . . . . . . . . . . .
5.7 Summary . . . . . . . . . . . . . . . .
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6 Conclusion
150
6.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
A Testing for bias in subhalo abundances from BBKS
ix
154
List of Tables
3.1
3.2
3.3
Properties of Milky Way halos in Chapter 2 . . . . . . . . . . . . . . 38
Properties of Observed Milky Way Satellites . . . . . . . . . . . . . . 54
Dark matter particle mass constraints . . . . . . . . . . . . . . . . . . 63
4.1
4.2
4.3
Cosmological Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 81
Properties of Milky Way halos in Chapter 3 . . . . . . . . . . . . . . 82
Test results for the Set B halo . . . . . . . . . . . . . . . . . . . . . . 90
5.1
5.2
5.3
Properties of simulations . . . . . . . . . . . . . . . . . . . . . . . . . 115
Properties of Halo A in high resolution simulations . . . . . . . . . . 116
Properties of high resolution simulated halos A-G . . . . . . . . . . . 137
x
List of Figures
2.1
Cold, Warm, and Hot Dark Matter Power Spectra . . . . . . . . . . . 17
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
3.12
3.13
Numerical artifacts in WDM simulations . . . . . . . .
Power Spectra used in Chapter 2 . . . . . . . . . . . .
Portraits of the Set B Milky Way halo . . . . . . . . .
Density profiles of Milky Way halos . . . . . . . . . . .
Cumulative mass functions of subhalos . . . . . . . . .
Cumulative velocity functions for subhalos within R100
Cumulative velocity functions for subhalos within R50 .
Cumulative velocity functions in WDM . . . . . . . . .
Convergence tests with velocity functions . . . . . . . .
Number of set A simulated satellites in SDSS footprint
Number of set B simulated satellites in SDSS footprint
Binned number of satellites . . . . . . . . . . . . . . .
Number of satellites within 50 kpc . . . . . . . . . . . .
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27
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4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
Comparison of Power Spectra . . .
Portraits of CDM halos . . . . . . .
Portraits of CDM and WDM halos
Mass growth histories . . . . . . . .
Satellite densities . . . . . . . . . .
Distribution functions . . . . . . .
Circular velocity profiles . . . . . .
WDM satellites . . . . . . . . . . .
WDM circular velocity plots . . . .
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5.1
5.2
5.3
5.4
5.5
5.6
5.7
WDM transfer functions used in Chapter
Mass growth of Halo A . . . . . . . . . .
Density profiles of Halo A . . . . . . . .
Mass profiles of Halo A . . . . . . . . . .
Axial ratios of Halo A . . . . . . . . . .
Comparison of mass profiles of Halo A .
Comparison of density slopes of Halo A .
4
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109
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5.8
5.9
5.10
5.11
5.12
5.13
5.14
5.15
5.16
5.17
5.18
5.19
Velocity dispersion profiles of Halo A . . . . . . . . . . . . . . . . . .
Phase space density profiles of Halo A . . . . . . . . . . . . . . . . .
Velocity anisotropy profiles of Halo A . . . . . . . . . . . . . . . . . .
Results of convergence tests . . . . . . . . . . . . . . . . . . . . . . .
Mass growth of 15 largest halos . . . . . . . . . . . . . . . . . . . . .
Mass profiles of 15 largest halos . . . . . . . . . . . . . . . . . . . . .
Mass profiles of seven halos simulated at high resolution . . . . . . .
Core growth of Halo A . . . . . . . . . . . . . . . . . . . . . . . . . .
Stability of phase space density profiles of Halo A . . . . . . . . . . .
Portraits of core particles of Halo A . . . . . . . . . . . . . . . . . . .
Portraits of core particles in seven halos simulated at high resolution
Circular velocity profiles of Halo A . . . . . . . . . . . . . . . . . . .
125
127
129
131
133
134
136
139
141
142
142
145
A.1 Comparison of CDM power spectra . . . . . . . . . . . . . . . . . . . 155
A.2 Subhalo velocity function comparison . . . . . . . . . . . . . . . . . . 156
xii
Chapter 1: Science Motivation and Structure of Thesis
Evidence that the majority of gravitating mass in the universe is composed
of an unseen form was first presented by Zwicky in the 1930s (Zwicky 1933, 1937).
By measuring the line of sight velocities of galaxies in clusters the dynamical mass
to light ratio was determined M/L > 100, and is more than an order of magnitude
greater than typical of the older stellar populations found in elliptical galaxies,
M/L ∼ 10.
The first imaging X-ray observatories discovered most baryonic matter in
clusters is in the form of hot gas unbound to member galaxies. This gas emits
bremsstrahlung radiation at X-ray energies. If the cluster is relaxed and the gas
in hydrostatic equilibrium it can be used as a probe of the gravitational potential
(Fabricant et al. 1980; Böhringer 1995). Cluster masses have been determined from
the gas density and temperature, determined from the X-ray emissivity and spectral
cutoff, and confirms the high mass to light ratios in these systems.
Further evidence comes from the flat rotation curves observed in galaxies (Rubin et al. 1978). The circular velocity v at radius R depends on the enclosed mass
M, v 2 = GM/R. For the rotation curve to be flat the mass must increase linearly
with distance but the distribution of light from stars and gas drops off much more
1
rapidly. The mass to light ratio increases rapidly in outer parts of galaxies indicating
most of the matter in galaxies is dark.
Secondary evidence comes from simulations of interacting galaxies that cannot
produce the tidal tails observed unless galaxies have extended halos of dark matter
(Dubinski et al. 1996). Simulations have also shown dark matter halos are needed
to stabilize the disks of spiral galaxies against bar formation (Ostriker and Peebles
1973).
Several lines of evidence point to a non-baryonic composition for dark matter.
The rate of neutron capture in the early era of universal nucleosynthesis is dependent
on the density of baryons and the resulting abundance of light elements (Deuterium,
Helium, Lithium, and Beryllium) are also sensitive to the amount of baryonic matter.
Measurements of the primordial abundances of these elements show baryonic matter
can only account for about 17% of the mass in the universe (Cooke et al. 2014).
Non-baryonic dark matter is also required to form galaxies below cluster scales.
The Cosmic Microwave Background (CMB) is the primordial radiation at redshift
z ∼ 1000 when the baryonic plasma recombined and decoupled from the radiation, allowing the photons to propagate freely. The CMB is observed to be highly
homogenous with temperature variations of only 10−5 . Prior to this epoch the baryonic matter was coupled to the radiation field by Compton scattering and would
have density fluctuations of the same magnitude as the temperature fluctuations.
Adiabatic perturbations grow proportional to (1 + z) in the matter dominated era.
Density variations on the order of one are seen at the current time on 8 Mpc scales.
To produce overdensities of one today the perturbations must be on the order of
2
10−3 at z ∼ 1000, much greater than the observed temperature fluctuations. Nonbaryonic dark matter perturbations are required to provide the seeds for galaxy
formation since they dynamically decouple from the radiation earlier and grow for
a longer period of time.
An alternative explanation for the observed dynamics of galaxies and clusters
is a modification to the theory of gravity at the small accelerations observed on
large scales (Milgrom 1983). However, observations of the merging cluster 1E 0657558 show that due to the merger the dominant baryonic component, the collisional
X-ray emitting gas, has been displaced from the collisionless stellar component.
Gravitational lensing was used to map the gravitational potential and show the
source of gravity traces the auxiliary galaxies, providing empirical evidence that
dark matter exists and is collisionless in nature (Clowe et al. 2006).
Cosmology has entered an age of precision where studies of the CMB, supernovae, and cluster surveys have determined the cosmological parameters to <
∼ 10%.
The universe is geometrically flat and composed of approximately 4% baryonic matter, 23% dark matter, and 73% dark energy (Hinshaw et al. 2012). Additionally,
the recently reported detection of polarization in the B-mode power spectrum of
the CMB is consistent with gravitational waves from an early era of inflation (BICEP2 Collaboration et al. 2014). Many nonbaryonic particles are predicted to exist
in proposed extensions to the standard model of particle physics. Many of these
models (but not all) predict the dark matter to be composed of weakly interacting massive particles (WIMPs) with masses ∼ 100 GeV. These particles decouple
from the other particle species in the early universe with non-relativistic velocities
3
and represent a class of models called cold dark matter (CDM). Recently, excessive
gamma rays with energies 1 − 5 GeV have been observed from the Galactic Center
(GC) region by the Fermi Gamma-ray Space Telescope and claimed as evidence for
WIMPs of mass ∼ 30 GeV annihilating to quarks (Goodenough and Hooper 2009;
Hooper and Goodenough 2011; Daylan et al. 2014). Gamma ray spectral lines have
been observed at 110 and 130 GeV toward the GC and in observations of clusters
(Su and Finkbeiner 2012; Weniger 2012; Finkbeiner et al. 2013; Hektor et al. 2013)
and have also been claimed as evidence of WIMP annihilations, although of WIMPs
with a different mass. Before either of these signals can be confidently attributed
to dark matter other astrophysical explanations have to be ruled out (Abazajian
and Kaplinghat 2012; Abazajian et al. 2014; Finkbeiner et al. 2013). Despite very
sensitive searches, direct detection experiments have been unsuccessful in discovering the dark matter particle (LUX Collaboration et al. 2013). All particle theories
for dark matter are based on extrapolations beyond the range of energies explored
experimentally. In the absence of direct detection cosmological studies offer the
important possibility of constraining theories of elementary particles at ultra-high
energies.
Cold dark matter models have been extremely successful at describing the large
scale features of matter distribution in the universe but face potential problems on
sub-Mpc scales. CDM predicts numbers of satellite galaxies for Milky Way-sized
halos about an order of magnitude in excess of the number observed (Klypin et al.
1999; Moore et al. 1999a). Also, there is a dynamical discrepancy between highresolution CDM simulations and observations of the stellar velocities in the most
4
luminous satellites (Boylan-Kolchin et al. 2011, 2012a). Additional issues include
the number of galaxies in voids and observed density cores in low surface brightness
and dwarf galaxies (van den Bosch and Swaters 2001; Swaters et al. 2003; Weldrake
et al. 2003; Donato et al. 2004; Gentile et al. 2005; Simon et al. 2005; Gentile et al.
2007; Salucci et al. 2007; Kuzio de Naray et al. 2010).
One solution to the issues with CDM is the power spectrum of density fluctuations may be truncated which can arise if the dark matter particles are “warm”
with masses ∼ 1 keV. Warm dark matter (WDM) particles have relativistic velocities in the early universe and only become nonrelativistic when less than a Galactic
mass (∼ 1012 M⊙ ) is within the horizon. Streaming motions while the particles are
still relativistic can erase density fluctuations on sub-Galactic scales and reduce the
number of satellites in Milky Way-sized galaxies as well as the number of galaxies
in voids. Halo formation on dwarf scales is also delayed in WDM resulting in lower
densities for the brightest satellites in agreement with the observations.
WDM models are also useful for exploring the role of substructure and mergers in determining the structural and dynamical profiles of relaxed halos. It is
generally accepted that the density profiles of halos are universal in form with all
information about the initial conditions and assembly history erased in the process
of virialization. However, some analytic models and simulation studies have found
a dependence of the inner profile on the power spectrum with a flattening of the
inner profile with decreasing halo mass.
A frequently studied class of WDM particles are thermal relics. These particles
couple to the relativistic cosmic plasma in the early universe and achieve thermal
5
equilibrium prior to the time of their decoupling. A candidate for a thermal relic
WDM particle is the gravitino, the superpartner of the graviton in supersymmetry
theories. The lightest stable particle (LSP) in supersymmetry theories is a natural
dark matter candidate. If the scale where supersymmetry is spontaneously broken
6
is <
∼ 10 GeV, as predicted by theories where supersymmetry breaking is mediated
by gauge interactions, then the gravitino is likely to be the lightest stable particle
and can have a mass reaching into the keV regime (Gorbunov et al. 2008). Galaxy
formation in gravitino cosmologies was first investigated by Blumenthal et al. (1982).
In general WDM particles may have decoupled before achieving thermal equilibrium or may already be decoupled from the cosmic plasma at the time of their
production. These WDM particle models are called nonthermal relics. An example
of a nonthermal WDM relic is the sterile neutrino (see Kusenko 2009 and references
therein), a theoretical particle added to standard electroweak theory. Sterile neutrinos have been proposed (Gninenko 2010; Gninenko and Gorbunov 2010; Karagiorgi
et al. 2009; Sorel et al. 2004; Melchiorri et al. 2009; Maltoni and Schwetz 2007;
Päs et al. 2005; Akhmedov and Schwetz 2010) as an explanation for the anomalous
excess of oscillations observed between muon and electron neutrinos and antineutrinos (Athanassopoulos et al. 1995, 1996, 1998a,b; Aguilar-Arevalo et al. 2007, 2009,
2010). There have been recent reports of the detection of a X-ray emission line in
the spectrum of galaxy clusters consistent with a decay line from sterile neutrinos of
mass ms = 7.1±0.7 keV (Bulbul et al. 2014; Boyarsky et al. 2014). Abazajian (2014)
calculated the transfer function for one of the sterile neutrino production mechanisms and showed it approximates that of a thermal particle of mass ∼ 2 keV. This
6
detection is provisional but if confirmed has important implications for the observations of small scale structure in the local universe, as my work in this thesis will
show.
There are also ways other than WDM to reduce small scale power. Brokenscale invariance inflation models (Kamionkowski and Liddle 2000) have a cutoff
length below which power is suppressed. Particle theories where the LSP dark
matter particle arises from the decay of the next lightest supersymmetric particle
(NLSP) can also suppress small scale power if the NLSP is charged and coupled to
the photon-baryon plasma (Sigurdson and Kamionkowski 2004) or if the NLSP decay
imparts a large velocity to the LSP (Kaplinghat 2005). Further possibilities include
composite dark matter models where stable charged heavy leptons and quarks bind
to helium nuclei by Coulomb attraction and can play the role of dark matter with
suppression of small scale density fluctuations (Khlopov 2005, 2006; Belotsky et al.
2006a,b; Khlopov and Kouvaris 2008a,b; Khlopov 2008). Wilkinson et al. (2013)
showed CDM with a non-zero elastic scattering cross section with photons can reduce
power at small scales. The method used in this work could potentially be applied
to constrain these models as well, however, I do not examine the consequences my
work has on these theories.
I review in Chapter 2 the foundations of structure formation in an expanding
universe and the dependence of that structure on the nature of the dark matter.
I show in Chapter 3 how the faint population of dwarf satellites discovered in the
Sloan Digital Sky Survey (SDSS) allow an improved lower limit to be set on the dark
matter particle mass. I study the effects of WDM on the densities of the largest
7
satellites in Chapter 4 and quantify the dependence of the densities on the adopted
cosmological parameters. I conclude in Chapter 5 by using WDM simulations to
examine the claim that the virialization process in gravitationally collapsed dark
matter halos erases all information about the initial conditions from which they
form.
8
Chapter 2: Foundations of Structure Formation
2.1 Cosmological Principle
Most cosmological models rely on the Cosmological Principle that on sufficiently large scales the universe is homogeneous and isotropic. Since gravity is the
dominant force on large scales every cosmological model requires a theory of gravity.
Modern cosmology uses general relativity which is a geometric theory that makes
gravity a property of space-time. The geometry of space-time is given by the metric tensor and related to the mass-energy content given by the energy-momentum
tensor. In colloquial terms, mass-energy affects the curvature of space-time while
the curvature of space-time affects how mass-energy moves. The interval ds2 between two events in space-time depends on the metric. For a universe in which
the Cosmological Principle applies space-time can be taken to be a continuous fluid
with geometric properties described by the Robertson-Walker metric in comoving
spherical polar coordinates:
dr 2
2
2
2
2
ds = c dt − a (t)
,
dθ
+
sin
θdφ
+
r
1 − Kr 2
2
2
2
2
"
#
(2.1)
where a(t) is the time variable scale factor and related to the observable redshift
a/a0 = (1 + z)−1 , where a0 is the scale factor at present. K is the curvature
9
parameter and is −1, 0, and 1, for open, flat, and closed universes, respectively. It
is useful to characterize the expansion rate with the Hubble parameter, H ≡ ȧ/a,
where the dot represents a derivative with respect to proper time. In a flat universe
the proper distance is simply the comoving distance scaled by the scale factor and
it is convenient to set the present value of the scale factor a0 = 1. The Hubble
parameter at the current epoch then measures the universal expansion rate at the
current time and is ≈ 70 km s−1 Mpc−1 . It is conventional to express the Hubble
parameter at the current time in terms of a dimensionless parameter h: H0 = 100h
km s−1 Mpc−1 , with h = 0.7.
2.2 Friedmann Equations
For universes described by the Robertson-Walker metric, Einstein’s field equations of general relativity can be written in the form of the Friedmann Equations:
4πGa
P
Λc2 a
ä = −
ρ+3 2 +
3
c
3
2 2
8πG 2 Λc a
ȧ2 + Kc2 =
ρa +
3
3
(2.2)
(2.3)
for a perfect fluid of inertial mass density ρ and pressure P with a cosmological
constant Λ. It is useful to define the critical density, ρc = 3H 2/8πG, and the density
parameter for each fluid component ΩX = ρX /ρc . The total density of mass-energy
Ω is the sum of the individual components, radiation Ωr , matter Ωm , curvature ΩK ,
and cosmological constant ΩΛ . The second equation can be written in terms of the
10
Hubble parameter:
2
H =
H02
"
Ω0r
a
a0
−4
+ Ω0m
a
a0
−3
+ Ω0K
a
a0
−2
#
+ Ω0Λ ,
(2.4)
where “0” indicates values at the current time. The curvature term in a flat universe
is zero.
In this form the Friedmann equation can readily be solved for the evolution of the scale factor with time when one component is dominant. In a matter
√
dominated universe a = a0 (3/2 H0 Ω0m t )2/3 . In a radiation dominated universe
√
a = a0 (2H0 Ω0r t )1/2 . A cosmological constant dominated universe grows exponentially, a = a0 eH0
√
Ω0Λ t
.
The dependence of the matter and radiation densities on the scale factor can
be determined from thermodynamics. For a system with volume V and pressure P
expanding adiabatically the change in internal energy E is equal to the work done
by the system, dE = −P dV . This equation can be solved for ρ(a) if the pressure
is known since V ∝ a3 and E = ρc2 V . For non-relativistic matter, P = 0, and
ρm = ρ0m a−3 . For radiation and relativistic matter, P = ρc2 /3, and ρr = ρ0r a−4 .
The radiation dominates the density of the universe at early times when the scale
factor is small, but it dilutes faster than matter resulting in an epoch of equality
between the matter and radiation densities after which matter dominates. This is
an important epoch and is given by 1 + zeq = 2.6 × 104 Ω0m h2 .
Different regions of the universe can only communicate by causal processes
when they are within each other’s particle horizon. The particle horizon is given by:
RH (t) = a(t)
11
Z
0
t
c dt′
.
a(t′ )
(2.5)
When the universe is radiation dominated the particle horizon is RH = 2ct ∝
a2 , while in the matter dominated regime RH = 3ct ∝ a3/2 . These results have
important consequences for the growth of density perturbations.
2.3 Jeans Instability
The condition for a self-gravitating region to be unstable to gravitational collapse was first analyzed by Jeans (1902) in the context of star formation. Jeans
found there is a minimum mass for collapse determined by the condition that the
free fall time is less than the sound crossing time. His analysis also applies to cosmological density perturbations with the exception that the expanding background
slows the rate of collapse from exponential to a power law.
The equations of motion for a self-gravitating fluid in a smooth background
with velocity and pressure fields ~v and P , and gravitational potential φ are:
∂ρ
+ ∇ · ρ ~v = 0
∂t
∂~v
1
+ (~v · ∇) ~v + ∇P + ∇φ = 0
∂t
ρ
(2.6)
(2.7)
∇2 φ − 4πGρ = 0.
(2.8)
Small perturbations are then applied to the fields: φ = φ0 + δφ, ~v = v~0 + δ~v ,
ρ = ρ0 + δρ, P = P0 + δP . The equations of motion are expanded to first order
in small quantities and the solutions for the unperturbed field subtracted. The
density contrast is defined as δ ≡ δρ/ρ, and the adiabatic sound speed is given by
~
vs2 = δP/δρ. Wave solutions are sought for δ of the form δ = δ0 ei(k·~r−ωt) , which give
12
the wave equation in an expanding background:
δ̈ + 2
ȧ
δ̇ = δ(4πGρ0 − k 2 vs2 )
a
(2.9)
The scale separating gravitational collapse from internal pressure supported
stability is the same as for a static medium with ȧ = 0. This gives the dispersion
relation for wave solutions:
ω 2 = vs2 k 2 − 4πGρ0
(2.10)
The scale separating collapse from stability is given by ω = 0 and is called the Jeans
scale. The Jeans scale is expressed in terms of wavelength, λ = 2π/k:
λJ = vs
π
Gρ0
!1/2
.
(2.11)
Gravitational collapse occurs when λ > λJ , otherwise the perturbations oscillate in
density as sound waves with angular frequency ω.
There are two solutions for the growth rate of perturbations above the Jeans
scale, a growing mode and a decaying mode. In the matter dominated era the
decaying mode δ ∝ t−1 and quickly becomes unimportant. The growing mode
solution is δ ∝ t2/3 . Since a = (3/2 H0 t)2/3 , the amplitude of the density contrast
grows linearly with the scale factor in the matter dominated era, δ ∝ a.
In the radiation dominated era the Jeans analysis must be conducted for a
√
relativistic fluid with P = ρc2 /3 and vs = c/ 3. In this case the growing mode
solution is δ ∝ t. Since a = (2H0 t)1/2 , the amplitude of the density contrast grows
with the square of the scale factor in the radiation dominated era, δ ∝ a2 . The sound
speed for radiation is a constant making the Jeans mass proportional to ρ−1/2 . Since
13
ρr ∝ a−4 and a ∝ t1/2 , the Jeans scale is a linear function of time in the radiation
era and grows at the same rate as the horizon scale.
In an expanding universe the Jeans length is time dependent and perturbations
can switch between stability and growth depending on when they enter the horizon
and which component is dominating the inertial mass of the universe. This has
important consequences for the development of perturbations. It must be noted the
Jeans analysis is valid only for collisional particles such as baryons. For dark matter
the collisionless Boltzmann equation must be solved. If the phase space distribution
of particles is Maxwellian the result is similar to Equation 2.11 but with the sound
speed replaced with the particle thermal velocity dispersion. Dark matter particles
can free-stream out of overdense regions and damp perturbations on the smallest
scales, as discussed in the next section.
2.4 The Power Spectrum
Models of the inflation epoch predict the field of density perturbations, δ(~x),
at the end of inflation will be adiabatic and Gaussian random with statistically
independent wave modes and random phases. A Gaussian random field is completely
described by its power spectrum.
The Fourier transform of the density perturbation field is given by:
1
δ(~x) =
(2π)3
Z
~
δ~k e−ik·~x d3 k.
(2.12)
Applying Parseval’s Theorem that the integral of the square of a function is equal
14
to the integral of the square of its Fourier transform:
Z
1
δ (~x)d x =
(2π)3
2
3
Z
|δ~k |2 d3 k.
(2.13)
The left hand term is the mean square amplitude of density fluctuations, hδ 2 i. On
the right hand side, |δ~k |2 , is the power spectrum of the perturbation field written
P (k). Thus,
1
hδ i =
(2π)3
2
Z
P (k)d3 k.
(2.14)
Inflation models give a primordial power spectrum at the end of inflation
P (k) ∝ k ns with ns ≈ 1. The power spectrum is altered from its primordial form
by the changes in the horizon scale and Jeans scale during the radiation era.
In the radiation dominated era both the horizon scale and the Jeans scale are
of the same magnitude and scale at the same rate. Density perturbations outside
the horizon grow linearly with time until they enter the horizon. When they enter
the horizon they become smaller than the Jeans scale and growth stops. Baryonic
perturbations are coupled to the radiation via Compton scattering and oscillate with
the radiation as sound waves. Dark matter perturbations do not oscillate but their
growth is stalled because the radiation dominates the inertial mass. This is called
the Meszaros effect (Meszaros 1974). At the epoch of equality the dynamics of the
expansion and the inertial mass become dominated by the matter. Dark matter
perturbations within the horizon resume their growth while perturbations just entering the horizon continue to grow uninterrupted. The baryonic plasma, however,
remains coupled to the radiation where the Jeans scale reaches a peak value at
equality and remains nearly constant until recombination. Baryonic perturbations
15
less than the horizon scale at equality continue to oscillate until the epoch of recombination and experience Silk damping due to photon diffusion. After recombination
the baryonic Jeans scale drops and perturbations are regenerated as they fall into
the potential wells of the perturbations in the dark matter. The Meszaros effect
causes the power spectrum of density fluctuations to peak at the horizon scale at
equality and turnover at smaller scales. This can be seen in the power spectrum
plotted in Figure 2.1.
In hot and warm dark matter models the dark matter particles decouple from
the other particle species with relativistic velocities allowing them to stream out of
overdense regions as they enter the horizon in the radiation dominated era. These
streaming motions damp perturbations below the horizon scale at the time the
particles become non-relativistic resulting in a truncation in the power spectrum.
The scale of the truncation is related to the mass of the dark matter particle with
lighter particles decoupling earlier and able to stream longer. Although they are nonrelativistic when they decouple, CDM particles have a small streaming scale due to
their non-zero thermal velocities. The damping scale of CDM perturbations is about
a Jupiter mass on a scale approximately that of the Solar System (to the distance
of the Oort Cloud; Green et al. 2004; Loeb and Zaldarriaga 2005). Figure 2.1 shows
the effects of streaming on the power spectrum for cold, warm, and hot dark matter
(HDM) cosmologies. HDM models damp perturbations up through cluster scales
and are ruled out observationally.
The changes in the linear power spectrum from its primordial form caused by
the physics of the early universe are parameterized by a transfer function, T (k) ∈
16
Figure 2.1: Power spectra for cold, warm, and hot dark matter cosmologies. The dashed line shows the ideal cold dark matter condition of zero
thermal velocities.
17
[0, 1]. At any time t the power spectrum is given by:
P (k, t) = T 2 (k)
D 2 (t)
Pi (k),
D 2 (t0 )
(2.15)
where Pi is the primordial power spectrum at the end of the inflation era and D is
the linear growth factor at time t normalized by its value at present. The transfer
function can be calculated numerically with software such as CAMB (Lewis and
Bridle 2002), CMBFAST (Seljak and Zaldarriaga 1996), and LINGER (Bertschinger
2001); or by using fitting functions that are accurate to a few percent (Eisenstein
and Hu 1998). In practice the transfer function in Equation 2.15 is calculated for
the ideal CDM case with zero thermal velocities. In WDM models the ideal CDM
power spectrum is used but weighted by a second transfer function to account for
the effects of streaming.
Normalization of the power spectrum is determined observationally, either
from observations of the CMB or by the standard deviation of perturbations on 8
Mpc scales at the present epoch, σ8 .
The variance of perturbations on a comoving spatial scale R at time t is:
σ 2 (R) = h|δR2 |i =
1 D 2 (t)
2π 2 D 2 (t0 )
Z
∞
0
P (k) W (k) k 2 dk,
(2.16)
where W (k) is a window function in Fourier space and can be chosen to be a top
hat. On most scales the power spectrum can be approximated as a power law in k
over the filtered scales, P (k) ∝ k n :
2
σ (R) ∝
Z
0
1/R
2
P (k)k dk ∝
18
Z
1/R
0
k n+2 dk ∝ R−(n+3)
(2.17)
Since R3 ∝ M, the variance can be given in terms of mass scales:
σ 2 ∝ R−(n+3) ∝ M −(n+3)/3
(2.18)
In CDM cosmologies n > −3 and smaller scales have larger variance resulting in
bottom-up galaxy formation with smaller halos collapsing first. In WDM the power
spectrum below the truncation scale has n < −3, smaller scales have smaller variance
leading to top-down galaxy formation where the first structures to form are just
above the streaming scale. In the next section it is shown how the variance can be
used to calculate the mass function of collapsed halos at any epoch.
2.5 Abundance of Dark Matter Halos
A method for estimating the number density of dark matter halos and the
fraction of matter in halos of a given mass at any time was pioneered by Press and
Schechter (1974). An outline of their method is given below.
The probability, P, that different regions with the same mass M will have
perturbation amplitudes between δ and δ + dδ is a Gaussian distribution with zero
mean and variance σ 2 (M):
P(σ)dδ = √
1
2
2
e−δ /2σ dδ
2
2πσ
(2.19)
The fraction of collapsed perturbations, Ωc , at scale M is found by integrating the
probability distribution above the collapse criterion δc :
Ωc =
Z
∞
δc
2
erf(x) = √
π
"
1
δc
P(δ)dδ =
1 − erf √
2
2σ(M)
Z
0
x
2
e−t dt
19
!#
,
(2.20)
(2.21)
The criterion for collapse in linear theory is given by δc = 1.69.
Equation 2.20 needs to be multiplied by a factor of two because it does not
account for underdense regions that collapse when they are embedded within a larger
volume above the collapse criterion. This was shown with excursion set formalism
by Bond et al. (1991).
The collapsed fraction per halo mass is found by differentiating:
s
2 δc dσ −δc2 /2σ2
dΩc
=−
e
dM
π σ 2 dM
(2.22)
The number density of halos per mass is found from Mdn = ρ̄dΩc :
s
2 δc dσ −δc2 /2σ2
dn
= ρ̄
e
dM
π Mσ 2 dM
(2.23)
The number density is characterized by an exponential cutoff at high masses and
dn/dM ∝ M −2 at small masses in CDM. In WDM, streaming motions erase perturbations below the streaming scale and reduce the abundances of small mass halos.
This fact is used in Chapter 3 where the Milky Way satellites are used to set limits
on the streaming scale.
2.6 Beyond Linear Theory - Spherical Halo Collapse
An overdense sphere can be treated like a closed universe with K = 1 in the
Robertson-Walker metric and provides a simple model for the nonlinear evolution of
a density perturbation. The Friedmann equations have parametric solutions leading
to the well-known result that closed universes have oscillatory behavior, expanding
to a maximum then contracting in a Big Crunch to a single point.
20
The equations of motion for the sphere’s radius is the same as for the scale
factor of a closed universe. For a matter dominated universe the equations for the
proper radius of the sphere and time are:
r = A(1 − cos θ)
A=
Ω0
2(Ω0 − 1)
t = B(θ − sin θ)
B=
Ω0
2H0 (Ω0 − 1)3/2
(2.24)
(2.25)
These equations can be solved for the nonlinear evolution of the density contrast
and compared to the extrapolations of linear theory. Expanding the equations to
fifth order in θ gives the linear approximation:
δ≃
3 6t
20 B
2/3
(2.26)
The turnaround point where the sphere stops expanding and begins collapsing
is at θ = π. The density enhancement within the sphere is δ = 9π 2 /16 ≃ 5.55.
Extrapolation of linear theory predicts δlin = (3/20)(6π)2/3 ≃ 1.06.
The sphere collapses to a point at θ = 2π with density contrast δ = (6π)2 /2 ≃
178. Extrapolating linear theory to this time gives δlin = (3/20)(12π)2/3 ≃ 1.69.
This is the collapse criterion used in the Press-Schechter formalism of the last section.
In practice real structures will not collapse to a singularity because the assumptions of no internal pressure and no shell crossing will be violated. Dynamical
processes will result in an extended, gravitationally bound halo in an equilibrium
satisfying the virial theorem, described in the next section. The radius enclosing a
density contrast of 178 is called the virial radius of the halo and the enclosed mass
is called the halo virial mass.
21
2.6.1 The Virial Theorem
For a system of N point particles of mass mi and Cartesian coordinates (xi ,
yi , zi ) interacting only through gravity the moment of inertia, I, for the system is
given by:
I=
N
X
mi (x2i + yi2 + zi2 )
(2.27)
i=1
Differentiating twice with respect to time gives:
N
N
X
X
mi 2
1¨
2
2
mi (xi ẍi + yi ÿi + zi z̈i )
ẋi + ẏi + żi +
I=2
2
i=1
i=1 2
(2.28)
The first summation on the right is a summation over the kinetic energy of each
particle and gives the total kinetic energy of the system, K. In the second summation
on the right, the components of the force vector can be recognized (mi ẍi , mi ÿi , mi z̈i ).
The forces are generated as the gradient of the gravitational potential, U:
mi ẍi = −
∂U
,
∂xi
mi ÿi = −
∂U
,
∂yi
mi z̈i = −
∂U
∂zi
(2.29)
The second term can then be written:
N
X
i=1
mi (xi ẍi + yi ÿi + zi z̈i ) = −
N
X
i=1
∂U
∂U
∂U
xi
+ yi
+ zi
∂xi
∂yi
∂zi
!
=U
(2.30)
The last part of this expression uses the fact that the potential is inversely proportional to distance, making it a homogeneous function of order n = −1. Applying
Euler’s theorem for homogeneous functions, xi ∂f /∂xi = nf , shows that the second
term is simply the total gravitational potential energy of the system.
For a system near equilibrium, I¨ = 0, and Equation 2.28 becomes:
2K + U = 0
22
(2.31)
This is the important virial theorem that for a system of self-gravitating particles in
equilibrium the absolute value of the potential energy is equal to twice the kinetic
energy.
Dark matter halos in simulations are seldom in complete equilibrium and it is
useful to define a “virial ratio” as:
2K
−1
|U|
(2.32)
as a metric of the halo relaxation. This is used in Chapter 5.
2.6.2 The NFW Profile
The seminal work of Navarro et al. (1997, 1996) found that the density structure of relaxed dark matter halos are well represented by what has become known
as the NFW profile:
ρ(r) =
ρs
,
(r/rs )(1 + r/rs )2
(2.33)
where ρ(r) is the density in a spherical shell at distance r from the halo center.
By scaling the free parameters rs and ρs , which define a characteristic length and
density, the NFW profile can fit dark matter halos from dwarf galaxy to cluster
scales.
Defining a halo concentration as the ratio of the virial radius to rs , the concentration was found to correlate with mass such that smaller mass halos are more
concentrated. This was understood as a consequence of the earlier formation epoch
of small mass halos in the bottom-up structure formation of CDM. Since small halos
collapse earlier their inner regions reflect the higher universal density of matter at
23
earlier times. Reducing the power spectrum, either by changing the cosmological
parameters or by introducing a truncation, delays halo formation to later epochs
and reduces halo concentrations.
I investigate how dark matter models affect the concentration of the largest
Milky Way satellites in Chapter 4. I explore in Chapter 5 the effects of erasing small
scale perturbations on the mass and dynamical profiles of collapsed halos in detail.
2.7 Beyond Linear Theory - Simulations
Linear theory breaks down when δ ∼ 1. Phases of the Fourier modes become
non-Gaussian and cross-talk between modes affects the power spectrum of perturbations. In the non-linear regime N-body simulations must be employed. N-body
simulations use high performance computing techniques to numerically integrate
the equations of motion for particles started from small initial perturbations in the
linear regime.
Initial conditions for starting simulations are generated using the Zeldovich
approximation (Zel’dovich 1970) relating the density perturbation field to the positions and velocities of a distribution of particles. The initial time ti is chosen by
the resolution of the simulation so that the smallest scales are in the linear regime.
The initial conditions employed in my simulations are generated from a uniform
grid of point masses. Random unit vectors are generated for each particle and
multiplied by
q
P (k, t0 )D(ti )/D(t0 ) to give δk . The velocity field in Fourier space
is determined from δk and inverse Fourier transformed to get the spatial velocity
24
field. Displacements from the uniform grid for each particle are then determined
from ∆x = ~v (xi , ti )ti . Initial conditions for both CDM and WDM are generated by
adopting the appropriate power spectrum P (k, t0).
25
Chapter 3: Particle Mass Constraints from Subhalo Abundances
3.1 Overview
The cold dark matter paradigm has been extremely successful at describing the
large scale features of matter distribution in the Universe but has problems on small
scales. Below the Mpc scale CDM predicts numbers of satellite galaxies for Milky
Way-sized halos about an order of magnitude in excess of the number observed.
This is the ‘missing satellites’ problem (Klypin et al. 1999; Moore et al. 1999a). One
proposed solution is that, due to feedback mechanisms, some dark matter satellites
do not form stars and remain nonluminous dark halos (Efstathiou 1992; Thoul and
Weinberg 1996; Bullock et al. 2001b; Ricotti and Ostriker 2004; Ricotti et al. 2005).
Another solution is the power spectrum of density fluctuations may be truncated
which may arise if the dark matter is warm (particle mass ∼ 1 keV) instead of cold
(particle mass ∼ 1 GeV). WDM particles have relativistic velocities in the early
Universe and only become nonrelativistic when about a Galactic mass (∼ 1012 M⊙ )
is within the horizon. Streaming motions while the particles are still relativistic
can erase density fluctuations on sub-Galactic scales and reduce the number of
satellites. WDM models have been studied by a number of authors (Colı́n et al.
2000; Avila-Reese et al. 2001; Bode et al. 2001; Knebe et al. 2002, 2003; Zentner
26
and Bullock 2003; Maccio’ and Fontanot 2009) in relation to the missing satellites
problem and other issues with CDM such as the apparent density cores in spiral
and dwarf galaxies (van den Bosch and Swaters 2001; Swaters et al. 2003; Weldrake
et al. 2003; Donato et al. 2004; Gentile et al. 2005; Simon et al. 2005; Gentile et al.
2007; Salucci et al. 2007; Kuzio de Naray et al. 2010).
N-body simulations of WDM cosmologies are complicated by the formation of
artificial halos produced by the discrete sampling of the gravitational potential with
a finite number of particles (see Melott 2007 for a review). Matter perturbations
collapse and form filaments with nonphysical halos separated by a distance equal
to the mean particle spacing (see Fig. 3.1) (Wang and White 2007; Melott 2007).
These halos are numerical artifacts. These halos may survive disruption as they
accrete from filaments onto Milky Way-sized halos and may contaminate the satellite
abundances and distributions in WDM simulations.
Figure 3.1: Nonphysical halos formed along a filament and accreting
onto a larger halo at z = 1 in a WDM simulation (mW DM = 1 keV).
These halos are numerical artifacts.
In the past decade, 16 new dwarf spheroidal galaxies have been discovered in
27
the Sloan Digital Sky Survey (Castander 1998; see Table 3 and references therein).
After correcting for completeness the estimated number of Milky Way (MW) satellites is > 60 (see Sec. 3.3.3). These new dwarfs have low luminosities, low surface
brightnesses, and most appear to be dark matter dominated. Since the number of
dark matter halos must be greater than or equal to the number of observed satellites, the new data from the SDSS may provide improved limits on the mass of the
dark matter particle independent of complementary techniques.
Motivated by the recent increase in the number of observed Milky Way satellites, I have performed new simulations of the growth of Milky Way-like galaxies
in CDM and WDM cosmologies for a variety of WDM particle masses. My goal
is to constrain the dark matter particle mass by comparing the number of satellite
halos in the simulated Milky Ways to the observed number of luminous satellites
for the actual Milky Way. Maccio’ and Fontanot (2009) combined N-body simulations with semianalytic models of galaxy formation to compare the simulated and
observed Milky Way satellite luminosity functions for CDM and WDM cosmologies.
I do not make any assumptions on how dark matter halos are populated with luminous galaxies in this work. I simply impose that the number of observed satellites is
less than or equal to the number of dark matter halos for a range of Galactocentric
radii. This guarantees a robust lower limit on the dark matter particle mass.
28
3.2 Simulations
All my simulations were conducted with the N-body cosmological simulation
code GADGET2 (Springel 2005) assuming gravitational physics only. Values for
cosmological parameters were adopted from the third year release of the WMAP
mission (Spergel et al. 2007), (Ωm , ΩΛ , h, σ8 , ns ) = (0.238, 0.762, 0.73, 0.751,
0.951) to facilitate comparison with the Via Lactea II (VL2) simulation (Diemand
et al. 2008). For each simulation set a single realization of the density field was
produced in the same periodic, comoving volume but with the power spectrum of
fluctuations varied appropriately for CDM and WDM cosmologies. Initial conditions
were generated on a cubic lattice using the GRAFIC2 software package (Bertschinger
2001). The power spectra for CDM and WDM are given by
2
PCDM (k) ∝ k ns TCDM
,
2
PW DM (k) = PCDM TW
DM ,
(3.1)
(3.2)
respectively, with the normalization of PCDM determined by σ8 . For the set A
and set B simulations (described below) the transfer function for CDM adiabatic
fluctuations given by Bardeen et al. (1986) (BBKS) was used:
TCDM (k) =
i−0.25
ln(1 + 2.34q) h
1 + 3.89q + (16.1q)2 + (5.46q)3 + (6.71q)4
, (3.3)
2.34q
where q = k/(Ωm h2 ). A potential problem with the BBKS transfer function is that
it underestimates power on large scales. In Appendix A I investigate the effect that
this choice for the CDM transfer function may have on the number of Milky Way
29
satellites. I run one of the simulations adopting the transfer functions from Eisenstein and Hu (1998) and find that this does not affect the results on the number of
satellites. Additional CDM simulations (set C ) were run using the transfer function
calculated by the LINGER program in the GRAFIC2 package (Ωb = 0.04 was used
for calculating the effects of baryons on the matter transfer function). LINGER integrates the linearized equations of general relativity, the Boltzmann equation, and
the fluid equations governing the evolution of scalar metric perturbations, photons,
neutrinos, baryons, and CDM. The mass and circular velocity functions of satellites
are consistent across both transfer functions.
Assuming the WDM to be a thermal particle, a particle like the gravitino
that was in thermal equilibrium with the other particle species at the time of its
decoupling, the transfer function valid for thermal particles given by Bode et al.
(2001) was used:
TW DM (k) = [1 + (αk/h)ν ]−µ ,
(3.4)
where ν = 2.4, µ = 4.167 and
mW DM
α = 0.0516
1 keV
−1.15 Ωm
0.238
0.15
h
0.73
!1.3 gX
1.5
−0.29
.
(3.5)
The parameter gX is the number of degrees of freedom for the WDM particle,
conventionally set to the value for a light neutrino species: gX = 1.5. The parameter
k is the spatial wavenumber in Mpc−1 and mW DM is the mass of the WDM particle
in keV.
If the dark matter is composed of non-thermal particles like the sterile neutrino
the situation is more complicated. There are several mechanisms by which sterile
30
neutrinos can be produced. In the standard mechanism proposed by Dodelson and
Widrow (DW; Dodelson and Widrow 1994), sterile neutrinos are produced when
oscillations convert some of the more familiar active neutrinos into the sterile variety.
The amount produced depends on the sterile neutrino mass and the mixing angle
but such details are not considered here and when analyzing the results for sterile
neutrinos it is simply assumed they compose the entirety of the dark matter. The
transfer function for DW sterile neutrinos with mass ms is given by (Abazajian
2006):
Ts (k) = [1 + (αk/h)ν ]−µ ,
(3.6)
where ν = 2.25, µ = 3.08, and
ms
α = 0.1959
1 keV
−0.858 Ωm
0.238
−0.136
h
0.73
!0.692
.
(3.7)
Viel et al. (2005) give a scaling relationship between the mass of a thermal particle
and the mass of the DW sterile neutrino for which the transfer functions are nearly
identical:
mW DM
ms = 4.379 keV
1 keV
4/3 Ωm
0.238
−1/3
h
0.73
!−2/3
.
(3.8)
Other sterile neutrino production mechanisms include that proposed by Shi & Fuller
(SF; Shi and Fuller 1999) who showed the DW mechanism is altered in the presence
of a universal lepton asymmetry where production can be enhanced by resonance
effects. Sterile neutrinos can also be produced from decays of gauge-singlet Higgs
bosons at the electroweak scale (Kusenko 2006). The momentum distribution of the
sterile neutrinos depends on the production mechanism. In the absence of transfer
function calculations the expressions in Kusenko (2009) for the free streaming length
31
and average momentum are used to derive approximate scaling factors for the SF
and Higgs produced sterile neutrinos: mDW /mSF = 1.5, mDW /mHiggs = 4.5.
In my simulations I assume the dark matter is thermal and scale the results
to the standard sterile neutrino mass using Eq. (3.8). The initial conditions include
particle velocities due to the gravitational potential using the Zeldovich approximation but I do not add random thermal velocities appropriate for WDM to the
simulation particles. Bode et al. (2001) argue that for warm particle masses greater
than 1 keV thermal motions are unimportant for halos on scales of a kiloparsec and
above. Regardless, it is expected thermal motions, if anything, would reduce the
number of small mass halos and by not including thermal motions the mass limits
derived from my simulations will be more conservative.
Simulations were conducted for CDM and WDM cosmologies with particle
masses of mW DM = 1, 2, 3, 4, and 5 keV (ms = 4.4, 11.0, 18.9, 27.8, 37.4 keV).
Figure 3.2 shows the power spectra for these cosmologies along with the spectrum for
an 11 keV standard sterile neutrino using Eq. (3.8). Two separate sets of simulations
were run, both consisting of a comoving cubic box 90 Mpc on a side. Set A consisted
of 2043 particles giving a ‘coarse’ particle mass of 3.0×109 M⊙ and a force resolution
of 8.8 kpc. All force resolutions were fixed in comoving coordinates. The HOP halo
finding software (Eisenstein and Hut 1998) was used at z = 0 to identify Milky
Way-sized halos with masses 1 − 2 × 1012 M⊙ . Halos were examined visually, one was
chosen that was at least several Mpc away from clusters and other large structures
so as to be relatively isolated. Its particles were identified in the initial conditions
and a cubic refinement level, 6.2 Mpc on a side, was placed on the region. For
32
the refinement region in the low resolution simulations 11, 239, 424 (2243 ) particles
were used with mass and force resolutions of 7.3 × 105 M⊙ and 550 pc, respectively.
Higher resolution simulations were run for CDM and WDM particle masses of 1,
2, and 4 keV with 89, 915, 392 (4483 ) particles in the refinement region and mass
and force resolutions of 9.2 × 104 M⊙ and 275 pc, respectively. The simulated Milky
Way halo had a neighbor halo with mass 0.23MM W at a distance of 700 kpc in the
low resolution simulations. The real Milky Way has a massive neighbor in M31, the
Andromeda galaxy (MM 31 ∼ 1 − 3MM W ), at a distance ∼ 700 kpc. Being nonlinear
and chaotic systems, small perturbations to the trajectories of dark matter halos
can be amplified exponentially and in the higher resolution simulation this satellite
is merging with the Milky Way at z = 0. Such a merger may disrupt the equilibrium
of the halo and make it nonrepresentative of the actual Milky Way. The difference
between the high and the low resolution simulations is significant and complicates
the comparison between the resolutions; however, this merger is not a violation of the
selection method used for the set C halos described below and excellent agreement
is found across all simulation sets.
The need to explore the scatter between the subhalo distributions of different
realizations of Milky Way-type halos, in addition to the complications arising with
the high and low resolution simulations of set A, prompted a second set of simulations to be conducted. Set B consisted of 4083 particles giving a coarse particle mass
of 3.8 × 108 M⊙ and a force resolution of 4.4 kpc. HOP was again used to identify
halos with masses 0.8 − 2.2 × 1012 M⊙ . For each halo the nearest neighboring halo
with mass > 0.8 × 1012 M⊙ was also found. A halo whose nearest massive neighbor
33
Figure 3.2: Power spectra for the simulations. The dotted line is
the power spectrum for an 11 keV standard sterile neutrino Abazajian
(2006). The neutrino spectrum is approximately the same as a 2 keV
thermal particle, validating the scaling relation of Viel et al. (2005).
The vertical dashed lines indicate the lattice cell size in the high and low
resolution refinement levels.
34
was at least 5 Mpc away was selected and visually verified that the halo was indeed
isolated. A rectangular refinement level 6.1 × 7.0 × 7.9 Mpc was placed over this
halo’s particles in the initial conditions. Low and high resolutions were conducted
with the same mass and force resolutions as set A. The low resolution simulations
used 16, 515, 072 (∼ 2553) particles in the refinement level while high resolution used
132, 120, 576 (∼ 5103 ) particles in the refinement level. Figure 3.3 shows portraits
of the Milky Way and the surrounding environment in the set B high resolution
simulations.
A third set of CDM only, low resolution simulations was run to further explore
the scatter between the subhalo distributions of different realizations of Milky Waytype halos and to explore the possibility of a bias introduced by the use of the BBKS
transfer function. Set C consisted of 4083 particles but the CDM transfer function
was generated from the LINGER software in GRAFIC2 (Bertschinger 2001) after
correcting a bug where the power spectrum for baryons was used for dark matter when calculating the transfer function. AMIGA’s Halo Finder (AHF) software
(Knollmann and Knebe 2009) was used to find MW-sized halos with no equal sized
neighbor within two virial radii (defined below). Nine halos were selected for refinement at low resolution from a variety of environments, low density with few large
halos to high density with many large halos. The rectangular refinement regions
had lengths 7.5 − 15.8 Mpc and 31, 752, 192 − 69, 009, 408 (3163 − 4103 ) particles.
35
Figure 3.3: Portraits of the set B simulated Milky Way halo at z = 0
in the high resolution simulations. From top to bottom: CDM, 4 keV,
2 keV, 1 keV. The images at left are 4.5 Mpc × 2 Mpc centered on the
Milky Way. Structures within 300 kpc of the MW center are shown at
right.
36
Table 3.1 summarizes the properties calculated by AHF for all simulated Milky
Way halos at z = 0. R∆ is defined as the radius enclosing an overdensity ∆ times
the critical value. The mass and number of particles inside R∆ are M∆ and N∆ ,
2
respectively; v∆ is the circular velocity v∆
≡ GM∆ /R∆ at R∆ , and vmax is the
maximum circular velocity of the halo. The value ∆ = 178Ω0.4
m = 100 is used (Eke
et al. 1996; which is also very close to the value using the definition from Bryan
and Norman 1998) for the virial radius of the MWs and subhalos within R100 are
considered when comparing to other published work. The mass, radius, and velocity
at ∆ = 50 are also used in the literature and these values are also listed in Table 3.1.
Figure 3.4 shows the density profiles of the A and B Milky Way halos calculated
by breaking the halos into spherical shells. Small differences between the high and
low resolution set A halos caused by the merging neighbor are apparent but generally
the profiles are very similar across all simulations of each set. An inner flattening
of the halos in the WDM simulations is not seen because thermal motions were not
added to the simulation particles. If the gamma-ray excess observed by Fermi is due
to annihilating ∼ 30 GeV WIMPs, Daylan et al. (2014) showed the signal within
5◦ of the Galactic Center is consistent with a typical NFW-like density profile with
inner slope ∼ −1.2. In this case, Figure 3.4 shows CDM particles could have a
truncated power spectrum at dwarf-scales and be consistent with the Fermi data.
37
Table 3.1: Properties of simulated Milky Way halos.
Simulation
M100
[1012 M⊙ ]
R100
[kpc]
M50
[1012 M⊙ ]
R50
[kpc]
v50
[km/s]
vmax
[km/s]
N100
N50
Set A
CDM lo
5 keV lo
4 keV lo
3 keV lo
2 keV lo
1 keV lo
1.4867
1.4964
1.5060
1.5141
1.5100
1.4983
288.25
288.86
289.45
290.00
289.74
289.00
1.6786
1.6825
1.6833
1.6850
1.6702
1.6615
378.47
378.75
378.82
378.95
377.84
377.18
138.11
138.22
138.24
138.29
137.88
137.64
183.02
183.38
183.87
182.98
181.59
180.04
2, 026, 414
2, 039, 597
2, 052, 643
2, 063, 747
2, 058, 104
2, 042, 264
2, 287, 923
2, 293, 239
2, 294, 398
2, 296, 714
2, 276, 518
2, 264, 672
CDM hi
4 keV hi
2 keV hi
1 keV hi
1.8403
1.8261
1.8326
1.8373
309.49
308.70
309.06
309.33
2.0331
2.0383
2.0266
2.0244
403.43
403.77
402.99
402.85
147.22
147.34
147.06
147.01
191.94
189.69
183.82
179.39
20, 067, 182
19, 911, 999
19, 982, 705
20, 033, 935
22, 169, 072
22, 225, 367
22, 098, 268
22, 073, 940
Set B
CDM lo
5 keV lo
4 keV lo
3 keV lo
2 keV lo
1 keV lo
1.9005
1.8862
1.8863
1.8800
1.8479
1.8263
312.84
312.04
312.04
311.70
309.92
308.70
2.1325
2.1254
2.1212
2.1185
2.0936
2.0752
409.89
409.44
409.16
409.00
407.38
406.19
149.58
149.41
149.32
149.25
148.67
148.23
195.87
195.76
195.84
195.75
195.24
192.33
2, 590, 475
2, 570, 982
2, 570, 992
2, 562, 445
2, 518, 690
2, 489, 258
2, 906, 549
2, 896, 920
2, 891, 165
2, 887, 566
2, 853, 610
2, 828, 485
CDM hi
4 keV hi
2 keV hi
1 keV hi
1.7533
1.7426
1.7288
1.6230
304.53
303.92
303.11
296.80
1.9948
1.9781
1.9640
1.8655
400.88
399.75
398.81
392.01
146.29
145.88
145.53
143.06
194.01
188.52
185.18
179.59
19, 117, 720
19, 001, 776
18, 850, 480
17, 697, 389
21, 751, 717
21, 569, 680
21, 415, 983
20, 341, 369
163.93
164.35
151.45
160.79
154.77
143.88
163.31
154.29
143.77
214.42
213.86
203.23
199.95
193.06
187.76
187.56
194.43
201.23
3, 351, 495
3, 204, 746
2, 705, 093
3, 078, 658
2, 382, 645
2, 174, 733
2, 549, 352
2, 706, 610
2, 300, 887
3, 761, 164
3, 855, 526
3, 016, 787
3, 610, 116
2, 589, 405
2, 567, 693
3, 782, 898
3, 189, 609
2, 581, 006
Set C
CDM
CDM
CDM
CDM
CDM
CDM
CDM
CDM
CDM
lo
lo
lo
lo
lo
lo
lo
lo
lo
1
2
3
4
5
6
7
8
9
2.4814
2.3512
1.9846
2.2587
1.7665
1.6004
1.8704
1.9858
1.6881
342.19
336.10
317.63
331.63
305.53
295.64
311.41
317.70
300.95
2.8071
2.8287
2.2133
2.6486
1.9226
1.8977
2.7754
2.3401
1.8936
449.22
450.37
415.01
440.60
345.21
394.26
447.52
422.78
393.97
38
Figure 3.4: Density profile of Milky Way halos in the set A and set B
CDM and WDM simulations. Thick lines are the high resolution simulations. Set B simulations are at top, the set A and the WDM cosmologies
in each set have been offset downward for clarity. The profiles are plotted starting from the convergence radius of Power et al. (2003) for both
resolutions (vertical lines).
39
3.2.1 Identification of Satellites
The AHF halo finding software (Knollmann and Knebe 2009) was used to find
the gravitationally bound dark matter halos in my N-body simulations. Unbound
particles were iteratively removed and gravitationally bound halos with ten or more
particles were selected.
AHF calculates properties of the halos it finds such as the total mass and the
maximum circular velocity. For this study the maximum circular velocity is a better
characteristic of a subhalo than the mass because quantifying the outer boundary
of a subhalo embedded in a larger halo is somewhat arbitrary and can introduce
systematic errors. The maximum circular velocity however typically occurs at a
radius well inside the subhalo outskirts.
3.3 Results
3.3.1 Satellite Distribution Functions
I first compare my CDM simulations to other CDM simulations in the literature. Figure 3.5 shows the cumulative mass functions, N(> Msub ), for subhalos
within R50 for the set A and B MWs. Poisson statistic error bars have been added
to the high resolution simulations and fit by N ∝ M −β . The values of β (0.9 and
0.95) agree with other published work that find values of 0.7 − 1.0 (Moore et al.
1999a; Ghigna et al. 2000; Helmi et al. 2002; Gao et al. 2004; De Lucia et al. 2004;
van den Bosch et al. 2005; Diemand et al. 2007; Giocoli et al. 2008; Springel et al.
40
2008). At both high and low resolution the simulated mass functions turn away
from the fit at masses below about 200 times the mass resolution of the simulation
in agreement with the Via Lactea simulation (Diemand et al. 2007). Also plotted
are the mass functions for the set C simulations. The subhalo abundances of the
set A and B halos are within the halo to halo scatter and are consistent with the
set C simulations.
The cumulative maximum circular velocity functions for subhalos within R100
are plotted in Figure 3.6. The maximum velocities of the subhalos have been normalized by the maximum circular velocity of their host MW. The shaded region
shows the minimum and maximum (lighter) and ±1σ (darker) from the mean of the
68 halos with masses 1.5 − 3 × 1012 M⊙ in the simulation of Ishiyama et al. (2009). I
use the fit to the density profile of the Via Lactea II halo (Diemand et al. 2008) to
estimate its R100 (298 kpc) and from the published subhalo catalog I calculate and
plot the Via Lactea II velocity function as the dashed line. The solid straight line
is the fitting formula from the Bolshoi simulation (Klypin et al. 2011),
1/2
(3.9)
x ≡ vmax /vmax,host ,
(3.10)
N(> x) = 1.7 × 10−3 vmax,host x−3 ,
applied to the high resolution halos which provides an excellent fit (the difference
between the fit for the set A and B vmax,host is less than the thickness of the line).
Via Lactea II used the same cosmological parameters as the simulations conducted
here and their subhalo abundance is in good agreement. My simulations are consistent with the Ishiyama et al. simulation but are systematically on the low end of
41
Figure 3.5: Cumulative mass functions for subhalos within R50 for the
CDM set A (top) and set B (bottom) simulations. Subhalo masses have
been normalized by the M50 mass of the host. Poisson error bars have
been added to the high resolution simulations and fit with a straight line.
Both high and low resolution (dotted lines) turn away from the straight
fit below about 200 times the mass resolutions of the simulations (short
vertical lines). Mass functions for the set C halos have also been added
(thin lines) and show the A and B abundances are within the halo to
halo scatter.
42
their distribution. This is likely due to the different cosmology used in the Ishiyama
et al. simulation (discussed below).
Figure 3.7 also plots the cumulative velocity functions but includes all subhalos
within R50 and the subhalo velocities have been normalized by the circular velocity
at R50 of their host MW. The Ishiyama et al. halos are again plotted as in Figure 3.6
as well as Via Lactea II. The solid straight line is the result from the Aquarius
simulations (Springel et al. 2008). Again there is good agreement between my
simulations and Via Lactea II but the simulations of Ishiyama et al. and Aquarius
are offset. To first order, the abundance of halos of any size depends on the power
spectrum of density perturbations which depends on the normalization, σ8 , and the
tilt of the power spectrum, ns . Larger values of either parameter increases the power
on small scales and leads to a larger number of satellites for a given mass and vmax of
the host. The values (σ8 = 0.9, ns = 1) were used in the Aquarius simulations and
(0.8, 1) were used by Ishiyama et al. Both are significantly greater than the values
adopted here (0.74, 0.951), and this is the likely cause of the abundance offset.
I adopted a WMAP3 cosmology to facilitate comparison to the Via Lactea II
simulation. The WMAP3 values of ns , σ8 , and Ωm are 1.0, 2.9, and 2.5 standard
deviations below the WMAP7 values (Jarosik et al. 2010). The Bolshoi simulation
used parameters in agreement with WMAP7 and constraints from other cosmology
projects. A comparison of the subhalo abundances of 4960 Bolshoi halos with circular velocities and masses comparable to the Via Lactea II halo indicated Bolshoi
has more subhalos by about 10%. Although Via Lactea II is just one halo and
may not be representative of the average for a WMAP3 cosmology, this agrees with
43
Figure 3.6: Cumulative velocity functions for subhalos within R100 in
the low resolution set C (thin lines) and high resolution set A and set B
(thick lines) MW halos. Subhalo circular velocities have been normalized
to the maximum circular velocity of the host halo. The dashed line is
the subhalo velocity function of Via Lactea II, the straight solid line is
the fitting formula from the Bolshoi simulation applied to the A and B
halos. The shaded regions show the minimum and maximum and ±1σ
from the mean of the 68 MW-sized halos of Ishiyama et al. (2009)
44
Figure 3.7: Cumulative velocity functions for subhalos within R50 in
the low resolution set C (thin lines) and high resolution set A and set
B (thick lines) simulations. Subhalo circular velocities have been normalized to the circular velocity of the host halo at a radius enclosing
an overdensity of ∆ = 50. The dashed line is the velocity function for
Via Lactea II and the straight solid line is the average abundance from
the Aquarius simulations (Springel et al. 2008). The shaded region is
the minimum/maximum range and ±1σ about the mean for halos from
the simulations of Ishiyama et al. (2009).
45
expectations from the 10% smaller value of σ8 used by Via Lactea II. I used the
same value of σ8 as Via Lactea II but the Bolshoi fitting formula applied to my high
resolution simulations in Figure 3.6 provides an excellent fit with no indication of an
offset. This could be because, as shown in the Appendix, the BBKS power spectrum
used in my high resolution simulations has about 10% more power at sub-Galactic
scales. Below I argue that an intrinsic scatter in subhalo abundance of 30% (1σ)
is reasonable to adopt and conclude this can account for variations in the adopted
cosmology without the need for a separate correction.
From my simulations and those of Ishiyama et al. in Figures 3.6 and 3.7 it is
clear that the subhalo abundances of halos of similar sizes have a scatter. For a given
cosmology the scatter in abundance includes an intrinsic scatter and a statistical
scatter from the number of subhalos. The Aquarius simulation suite included 6
MW-sized halos simulated at very high resolution. For subhalos within R50 , at
high values of the abundances where the statistical scatter is small, the 1σ intrinsic
scatter was determined to be 10%. In Figure 3.7 the scatter in the Ishiyama et
al. abundances decreases for increasing N and appears to be converging to the
10% found in Aquarius. However the variation in the Ishiyama et al. abundances in
Figure 3.6 is clearly converging to a larger value. As argued in Ishiyama et al. (2009),
the smaller scatter in Figure 3.7 can be explained by the inclusion of subhalos at
distances up to R50 which are outside the virial radius and, hence, their evolution
has not been affected by the structure of the host halo. Using v50 to normalize the
subhalo velocities can also reduce scatter since, unlike vmax , it is less dependent on
the central concentration. I will be interested in the number of subhalos in the inner
46
regions of the host MW which are expected to be sensitive to the host concentration
so I adopt the higher value for the intrinsic variation in the number of subhalo from
Figure 3.6 which is estimated to be about 30% (1σ) after subtracting the Poissonian
statistical scatter expected from the number of subhalos.
In Figure 3.8 the cumulative circular velocity functions for subhalos within R100
for the high resolution set A and set B CDM and WDM simulations are plotted.
The set A abundances have been increased 7% to normalize the CDM abundances to
those of the set B simulation and illustrate that the relative suppression of subhalo
abundances for each WDM simulation compared to CDM is the same across both
simulation sets. The straight line is the Bolshoi fitting function applied to the set B
CDM halo. The vertical lines in Figure 3.8 show where vmax = 6 and 8 km/s. Below
8 km/s the high resolution CDM simulations begin to fall away from the Bolshoi line
due to the resolution limits of the simulations. For vmax > 8 km/s my simulations are
reasonably complete within R100 of each Milky Way although numerical destruction
of a small fraction of satellites in the inner Milky Way would not be apparent in
Figure 3.8, especially for the CDM and 4 keV cosmologies. Before comparing the
simulations to observations the convergence distance of the simulations needs to be
determined.
3.3.2 Convergence Study
Satellites orbiting in the halo of a larger galaxy are destroyed by tidal stripping
and heating through encounters with other satellites. Satellites in simulations are
47
Figure 3.8: Cumulative velocity functions for satellites in the high resolution set A and set B CDM and WDM simulations. The set A abundances (thin lines) have been increased by 7% to normalize the CDM
abundances to those of the set B and show that the relative suppression
of halos in WDM cosmologies compared to CDM is similar across both
simulations. The straight gray line is the Bolshoi fitting formula applied
to the set B halo.
48
also destroyed artificially by numerical effects that become dominant for poorly
resolved halos in the inner halo region. There will therefore be a radius inside of
which the simulations will not converge to a realistic representation of the actual
Milky Way.
To determine the convergence of the simulations and have an idea of the variance of the results, simulations at lower and higher resolution of two different realizations of a Milky Way-sized galaxy were performed. Convergence studies were
conducted following the argument elucidated below, in combination with results of
published high resolution simulations found in the literature. Using the work of
Moore et al. (1999a); Klypin et al. (1999); De Lucia et al. (2004); Ishiyama et al.
(2009), it is assumed that the shape of the cumulative satellite velocity function for
host halos of different masses is nearly constant and the total number of satellites
scales linearly with the host mass. If the simulations are convergent, the cumulative
circular velocity function for satellites, N(R), within a given Galactocentric radius,
R, should be proportional to the enclosed mass, M(R), and a function of R that
represents the fraction of satellites that survive destruction from physical effects:
N(R) ≡ f (R)M(R),
(3.11)
where f (R) ∝ Rα . The normalization of f (R) can be set using values of N(R) and
M(R) at a distance R0 :
R0
N(R)
R
α
M0
M(R)
!
= N0 = const.
(3.12)
The velocity functions normalized in this way will be constant with radius where the
simulations are convergent. Where numerical effects destroy satellites the velocity
49
functions will normalize to a lower value. The parameter α is expected to be constant
because there is no characteristic scale for the destruction rate in dark matter only
simulations. Hence, α can be determined at large radii where convergence is certain.
Figure 3.9 shows the normalized velocity functions for the simulations. The
normalization constants M0 , R0 have been chosen at 200 kpc and the value of α
(0.55) was adjusted by hand until a good fit was achieved for the set B velocity
functions above 200 kpc in the high resolution CDM cosmology at circular velocities
> 6 km/s (vertical lines). This α also provides a good fit for the WDM cosmologies
and for the set A simulations, although the 1 and 2 keV velocity functions have
a wider scatter due to the smaller numbers of satellites in these simulations. The
mW DM = 4 keV simulation is convergent for vmax > 6 km/s to distances > 100 kpc.
At 75 kpc the effects of numerical resolution are apparent. The same value of α has
been used in the normalization of the low resolution sets and appears to provide
a good fit for the velocity functions > 200 − 250 kpc. The effects of numerical
resolution on the destruction of satellites are apparent at larger distances in these
simulations: < 200 kpc for CDM and < 150 kpc for WDM.
The 1 and 2 keV WDM velocity functions in Figures 3.8 and 3.9 show a flattening when going from high to low velocities until about 6 km/s, below which the
number of subhalos increases greatly. This is a generic feature of WDM simulations
(Bode et al. 2001; Barkana et al. 2001) and is usually explained as top-down fragmentation of matter filaments. Given that WDM simulations are known to form
nonphysical halos along filaments (Melott 2007; Wang and White 2007), it is likely
the low velocity upturn in the velocity function of subhalos is actually caused by
50
Figure 3.9: (left) Velocity functions for set B high (top) and low (bottom) resolution simulations normalized with Eq. (3.12). Solid lines are
R = 400, 300, 250, and 200 kpc (thick), dotted line is R = 150 kpc,
dashed line is R = 100 kpc, dot-dashed line is R = 75 kpc. The WDM
cosmologies have been shifted down vertically for clarity. The value
α = 0.55 was set by the high resolution simulation and provides good
normalization for the low resolution as well but the effects of incompleteness become apparent at much larger radii (150 − 200 kpc compared to
75 − 100 kpc for high resolution). (right) Same as the left panel, but for
the set A high (top) and low (bottom) resolution simulations. The value
α = 0.55 also provides good normalization for this set of halos.
51
these numerical artifacts accreting onto the MW. Since these nonphysical halos form
with separations typical of the mean particle distances in the initial conditions, the
number of halos should increase with the mass resolution of the simulation. The
low resolution simulations do not show clear evidence of upturns in the velocity
functions, simulations with resolution higher than the high resolution sets would
be required to confirm this effect. Regardless, only satellites with velocities greater
than 6 km/s in the high resolution simulations will be considered when deriving
constraints on the dark matter particle mass.
3.3.3 Comparison to Observations
Before the Sloan Digital Sky Survey there were only 12 classically known
satellite galaxies to the Milky Way. Sixteen new satellites have been discovered in
the SDSS. All known Milky Way satellites are listed in Table 3.2 where the given
satellite distances are used as their Galactocentric distances. Before comparing the
observed satellites to the simulations it is important to recognize the limitations
of the SDSS that affect the observed satellite abundances. The primary limitation
is the sky coverage of the survey, 28.3% for Data Release 7 (11663 deg2 ). Second,
being a magnitude limited survey, the SDSS has a luminosity bias. The detection
efficiency of dwarfs in the SDSS is a function of dwarf size, luminosity, distance,
and Galactic latitude as shown by Walsh et al. (2009). An approximate expression
is given in Tollerud et al. (2008) (using the work of Koposov et al. (2008)) for the
distance which galaxies of luminosity > L are completely detected: d ≈ 66 kpc
52
(L/1000 L⊙ )1/2 . Galaxies with L > 104 L⊙ should be approximately complete to
200 kpc, with L > 2300 L⊙ to 100 kpc. The distance range 100 − 200 kpc is thus
suited for comparisons because the simulations are convergent and the observations
are nearly, but not quite, complete. For the subsequent analysis only satellites with
distances < 200 kpc are used.
I account for the partial sky coverage of the SDSS by correcting the simulated
satellite abundances to the survey area with a series of random trials, described
in more detail later in this section. My observed data set consists of the SDSS
discovered dwarfs combined with the classic Milky Way satellites within the SDSS
footprint; Ursa Minor, Draco, Sextans, and Leo I and II. A conservative luminosity
correction for the SDSS dwarfs is considered using the formulas in Walsh et al.
(2009). This adds only two satellites at distances 150 − 200 kpc and does not affect
the conclusions, therefore I do not consider luminosity corrections when comparing
the observations and simulations. It is important to note the formulas in Walsh et al.
(2009) assume the size-luminosity distribution of known dwarfs is representative of
all satellites. There may be a population of dwarfs with surface brightnesses below
the detection limit of the SDSS (Ricotti and Gnedin 2005b; Ricotti et al. 2008b;
Ricotti 2010; Bovill and Ricotti 2009b).
53
Table 3.2: Summary of known Milky Way satellites.
Name
dist
[kpc]
σstar
[km/s]
MV
References
Classical (pre-SDSS)
Sagittar
LMC
SMC
Ursa Minor
Draco
Sculptor
Sextans
Carina
Fornax
Leo II
Leo I
Phoenix
24 ± 2 11.4 ± 0.7
49 ± 2
...
58 ± 2
...
66 ± 3
9.3 ± 1.8
79 ± 4
9.5 ± 1.6
79 ± 4
6.6 ± 0.7
86 ± 4
6.6 ± 0.7
94 ± 5
6.8 ± 1.6
138 ± 8 10.5 ± 1.5
205 ± 12 6.7 ± 1.1
270 ± 30 8.8 ± 0.9
405 ± 15
...
-13.4
-18.4
-17.0
-8.9
-8.8
-11.1
-9.5
-9.3
-13.2
-9.6
-11.9
-10.1
a
a
a
a
a
a
a
a
a
a
a
a
-1.5
-3.8
-2.5
-2.5
-3.7
-3.1
-5.8
...
-5.6
-6.0
-4.8
-5.8
-5.2
-5.0
-7.9
-7.1
b
c, d
e
f, c
h, d
i
c
j, k
d
h, d
h, d
h, d
l, m
n
o, d
p, d
SDSS discovered
Segue I
Ursa Major II
Segue II
Willman I
Coma Berenics
Bootes II
Bootes I
Pisces I
Ursa Major I
Hercules
Canes Venatici II
Leo IV
Leo V
Pisces II
Canes Venatici I
Leo T
23 ± 2
30 ± 5
∼ 35
38 ± 7
44 ± 4
60 ± 10
62 ± 3
80 ± 14
106+9
−8
140+13
−12
150+15
−14
160+15
−14
175 ± 9
∼ 180
220+25
−16
∼ 420
4.3 ± 1.2
6.7 ± 1.4
3.4 ± 2.0
4.3+2.3
−1.3
4.6 ± 0.8
...
6.5+2.0
−1.4
...
7.6 ± 1.0
5.1 ± 0.9
4.6 ± 1.0
3.3 ± 1.7
2.4 ± 1.8
...
7.6 ± 0.4
7.5 ± 1.6
References: (a) Mateo (1998), (b) Geha et al. (2009), (c) Martin et al. (2007),
(d) Simon and Geha (2007), (e) Belokurov et al. (2009), (f) Willman et al.
(2005), (g) Martin et al. (2007), (h) Belokurov et al. (2007), (i) Walsh et al.
(2007), (j) Watkins et al. (2009), (k) Kollmeier et al. (2009), (l) de Jong et al.
(2009), (m) Belokurov et al. (2008), (n) Belokurov et al. (2010), (o) Zucker
et al. (2006), (p) Irwin et al. (2007)
54
Willman 1 is an exceptional case in that it may not be a dark matter dominated dwarf galaxy but a globular cluster undergoing tidal disruption. Its stellar
velocity dispersion implies a large mass to light ratio like other dwarf spheroidals
and it has a size and luminosity intermediate between MW dwarfs and globular
clusters (Willman et al. 2005), but unresolved binaries and tidal heating may contaminate the velocity dispersion and lead to an overestimated mass. Although it has
a large metallicity spread unlike the stellar population of a globular cluster (Martin
et al. 2007), follow-up spectroscopy (Siegel et al. 2008) suggests there may be contamination by foreground stars and when these are excluded the metallicity spread
can be consistent with a metal-poor globular cluster. When deriving constraints on
the dark matter particle mass I will consider both including and excluding Willman 1
as a Milky Way satellite.
When comparing observations and simulations I apply cuts to the simulated
subhalos and consider only those with velocities above 6 and 8 km/s. As discussed
in the previous section this is to avoid potential contamination from numerical effects in the WDM simulations. That these velocity cuts are a reasonable estimate
of the minimum vmax of the dark matter halos the observed galaxies are presumably embedded in can be shown as follows. Ricotti and Gnedin (2005b) found in
simulations that the maximum circular velocities of satellites are at least twice the
velocity dispersion of the stellar component. Assuming the stellar velocity dispersions of the observed dwarfs are
√
3 times the line-of-sight velocity dispersions (σstar
in Table 3.2), then all dwarfs with measured velocity dispersions have maximum
circular velocities greater than 8 km/s. Assuming dwarfs without measured velocity
55
dispersions are similar to the other known dwarfs it is conservative to conclude all
dwarfs reside in dark matter halos with vmax greater than 8 km/s. An alternative
approach is the work of Wolf et al. (2010) relating the circular velocity at half light
radius to the velocity dispersion: vc (r1/2 ) =
√
3σstar . All the observed dwarfs except
Leo V have circular velocities at half light radius about 6 km/s or greater. Since the
maximum circular velocity must be greater than or equal to the half light circular
velocity, it is also reasonable to consider that all observed dwarfs reside in halos
with vmax > 6 km/s. It should be stressed that these cuts reflect the need to reduce
the numerical effects of the nonphysical halos in WDM simulations that dominate
the high resolution simulations at subhalo velocities below 6 km/s rather than an
assumption on the relationship between luminous satellites and dark matter halos.
To correct the simulations to the partial sky coverage of the SDSS I first
calculate the coordinates of all simulated satellites in a spherical coordinate system
centered on the Milky Way halo. I run a series of 10,000 trials with a field of view
(FOV) center randomly chosen on the sky and calculate the number of subhalos
within a solid angle corresponding to the total sky coverage of the SDSS (11663
deg2 ). I apply circular velocity and distance cuts to the subhalos and calculate the
median number of subhalos in the FOV and the 1σ and 2σ ranges from the trials.
Figures 3.10 and 3.11 show probability distributions of the number of subhalos in
the sky footprints in each cosmology of the set A and set B simulations for subhalos
with vmax > 6 km/s and distances 100 − 200 kpc. The missing satellites problem is
dramatically illustrated by the CDM simulations which have ∼ 40 − 80 satellites in
this distance range yet only 7 are observed in the SDSS. Satellite abundances are
56
reduced in the WDM simulations with the 2 keV cosmologies in 2σ agreement with
the observations.
I further compare my simulations to the observations in Figure 3.12 where I
plot the number of satellites within 200 kpc in bins of 50 kpc for the observed data
set and both sets of simulation data with vmax > 6 km/s. The observed number
of satellites are plotted with wide black bars and upward arrows indicating these
are only lower limits due to the surface brightness limits of the SDSS; it is possible
there are more dwarfs yet to be discovered. Willman 1 has not been included as a
MW satellite in this figure. The median number of satellites in each simulation set
are plotted as short lines enclosed in shaded dark and light rectangles that give the
1σ and 2σ ranges, respectively. The 6 km/s cut to the simulation data assures the
high resolution simulations are convergent to at least r = 100 kpc. Focusing on the
100−200 kpc bins it is clear the 1 keV simulations have far too few satellites to match
the observations. The 2 keV set B simulation is consistent with the observations in
all bins but set A is only consistent in the 100 − 150 kpc bin. The 4 keV simulations
can be consistent with the observations although they may require galaxy formation
to be suppressed in some of the dark matter halos. Strong conclusions cannot be
drawn from this plot because it is not clear how variance in the abundances and
numerical destruction in the inner bins for the simulated satellites may affect the
results.
The number of satellites in the simulations can be corrected for the effects of
numerical destruction using the convergence equation, Eq. (3.12). The mass and
number of satellites inside R50 were used for the normalization and the number
57
Figure 3.10: Number of subhalos in the set A simulations within a field
of view (FOV) equivalent to the sky coverage of the SDSS. 10,000 trials
were run for each cosmology with the FOV center chosen randomly.
The probability distributions for the number of subhalos are plotted by
the hatched histograms. The long vertical lines shows the median of
the distributions while dark and light shaded bans show the 1σ and 2σ
ranges, respectively. The simulated satellites have been cut by vmax >
6 km/s and by distance from 100 − 200 kpc. The number of satellites
observed in the SDSS in this distance range is given at the top of each
plot and plotted with a short vertical line where appropriate.
58
Figure 3.11: Same as Figure 3.10 but for the set B simulations.
59
Figure 3.12: The number of satellites in 50 kpc distance bins from
the Milky Way center. The satellites in simulations have been cut by
vmax > 6 km/s. The median number of satellites within the equivalent
sky footprint of the SDSS and 1σ and 2σ ranges are plotted as the colored rectangles in each bin for both the set A and set B simulations. The
wide black bars with arrows are the observed satellites within the footprint of the SDSS but do not include Willman 1. The observations are
incomplete at distances greater than about 50 − 100 kpc (depending on
the luminosity and surface brightness of the dwarfs), while simulations
have not converged for less than about 100 kpc. The 1 keV cosmologies
are inconsistent with the observed satellite abundances.
60
of satellites in 50 kpc bins for the simulations were calculated. The 0 − 50 kpc
bin is most important for constraining the dark matter particle mass because the
observations are most complete in this bin.
Plotted in Figure 3.13 are the differences in the number of observed and simulated satellites within 50 kpc of the MW center for the set B simulations as a
function of the dark matter particle mass with interpolation between the simulated
data. The variance in simulated subhalo abundances are calculated for a 30% intrinsic rms scatter plus a Poissonian variance in the number of subhalos and corrected
to the partial sky coverage of the SDSS assuming an isotropic distribution on the
sky. The dark and light shaded regions in the plot show the 1σ and 2σ ranges,
respectively. The number of satellites in simulation must be at least equal to the
number of observed satellites, therefore where this quantity equals zero defines a
lower limit on the dark matter particle mass. The arrowed lines indicate the lower
limits at 1σ and 2σ for this case of the set B simulations with a 6 km/s cut to the
subhalos and excluding Willman 1 from the observed set.
The same analysis was repeated using a vmax > 8 km/s cut to the simulation
data. The effects when Willman 1 was included in the observed data set were also
considered for both the 6 km/s and 8 km/s analysis. The results are presented in
Table 3.3. The set B halo is slightly more abundant in subhalos but both the set
A and B simulations give the same results to within about 10%. Rather than take
the average of the two simulations, the more conservative of the two constraints is
adopted.
In the most conservative case, where Willman 1 is not a dark matter dominated
61
Figure 3.13: The number of satellites within 50 kpc observed in the SDSS
sky footprint, excluding Willman 1, minus the number of satellites in
simulation with 1σ and 2σ limits (dark and light shaded regions). The
number of satellites from 0-50 kpc was calculated from the convergence
equation in the set B high resolution simulation for vmax > 6 km/s and
corrected to the partial sky coverage of the observations assuming an
isotropic distribution on the sky. Where the difference in the number of
observed and simulated satellites equals zero sets a lower limit on the
dark matter particle mass and is given by the arrows.
62
Table 3.3: Dark matter particle mass constraints (in keV) from the high resolution
set A and B MW halos. Constraints for simulated subhalo vmax cuts of 8 and 6 km/s
and including or excluding Willman 1 from the observed data set are given.
vmax > 8 km/s
vmax > 6 km/s
Will 1?
Included
Excluded
Included
Excluded
MW
A
A
A
A
B
> 2.3
> 2.9
> 2.1
> 2.6
2σ
1σ
> 3.2
> 4.4
B
B
B
> 2.9 > 2.7 > 2.5 > 2.7 > 2.4
> 4.0 > 3.6 > 3.3 > 3.4 > 3.0
dwarf galaxy and all observed satellites correspond to dark matter halos with vmax >
6 km/s, a formal limit of mW DM > 2.1 keV can be adopted with 95% confidence.
3.4 Discussion
I found that a model with mW DM ∼ 4 keV produces the best fit to observations
at < 50 kpc, i.e. this model has a number of dark matter satellites equal to the
number of observed luminous satellites. However, due to the large uncertainties in
the number of observed satellites due to partial sky coverage and on the number
of simulated satellites due to Poisson and intrinsic scatter, that partially reflects
observational uncertainties on the mass and vmax of the Milky Way, I find much
weaker lower limits on mW DM than 4 keV. In the future however, the lower limit
on mW DM will improve as observations of MW satellites become more complete.
The scatter of the simulation can also be reduced using constrained simulations of
the Local Group (also including the effect of baryons) in combination with more
accurate determination of the mass, rotation curve, and concentration of the Milky
Way.
63
Considering the various uncertainties in the number of observed and simulated satellites, I found a conservative lower limit of mW DM > 2.1 keV (2σ) on the
dark matter particle mass. I also found the 1 keV WDM simulations have too few
satellites to match the Milky Way observations. This agrees with the semianalytic
modeling and Milky Way satellite luminosity functions in WDM cosmologies work of
Maccio’ and Fontanot (2009); however, I only apply a cut to the simulated halos to
avoid numerical effects and do not make assumptions on how the dark matter halos
are populated by luminous galaxies. Lovell et al. (2013) performed a study similar
to mine, simulating one of the Aquarius Milky Ways in several WDM cosmologies,
and favor a similar but slightly warmer limit of 1.6 keV. Horiuchi et al. (2013) have
examined the abundances of dwarf spheroidal galaxies around M31. The advantage
in this approach is our external viewpoint gives a volume-complete census of dwarfs
around M31 without needing a sky coverage correction as for the MW, although
the census in only complete to higher luminosities. These authors derived a similar
limit of mW DM > 1.8 keV. Schultz et al. (2014) studied using high redshift galaxy
counts as a means to constrain WDM and conclude masses < 1.3 keV are inconsistent with galaxy counts in the Hubble Ultra Deep Field at > 2.2σ. They also
find these models are inconsistent with optical depths to the CMB due to Thomson
scattering observed by the Planck observatory, but with weaker confidence.
My result can also be compared to limits on the particle mass from the Lymanα forest in high redshift quasars. Lyman-α absorption by neutral hydrogen along
the line of sight to distant quasars over redshifts 2–6 probes the matter power
spectrum in the mildly nonlinear regime on scales 1–80 Mpc/h. Viel et al. (Viel
64
et al. 2005, 2006, 2008) have numerically modeled the Lyman-α forest flux power
spectra for varied cosmological parameters and compared to observed quasar forests
to obtain lower limits on the dark matter particle mass. Their 2006 work (Viel
et al. 2006) used low resolution spectra for 3035 quasars (2.2 < z < 4.2) from the
SDSS (McDonald et al. 2006) and found a 2σ lower limit of 2 keV for a thermal
WDM particle. This limit agrees with my results that a 2.1 keV particle is the
lower limit that can reproduce the observed number of Milky Way satellites and
approximately agrees with the Lyman-α work of Seljak et al. (2006) who find a 2σ
limit > 2.5 keV for a thermal particle. Viel et al. (2008) use high resolution spectra
for 55 quasars (2.0 < z < 6.4) from the Keck HIRES spectrograph in addition to
the SDSS quasars. With the new data they report a lower limit of 4 keV (2σ). A
caveat arises in Viel et al. (2009), who show the flux power spectrum from the SDSS
data prefer larger values of the intergalactic medium (IGM) temperature at mean
density than expected from photoionization. The flux power spectrum temperature
is also higher than that derived from an analysis of the flux probability distribution
function of 18 high resolution spectra from the Very Large Telescope and also higher
than constraints from the widths of thermally broadened absorption lines (Ricotti
et al. 2000; Schaye et al. 2000). This could be explained by an unaccounted for
systematic error in the SDSS flux power spectrum data which may also affect the
derived dark matter particle mass limits. The most recent 2σ lower limit from Ly-α
is 3.3 keV (Viel et al. 2013).
Using the scaling relation for sterile neutrinos I find a lower limit ms >
11.8 keV with 95% confidence for a DW produced sterile neutrino particle. Scaling
65
to the other production mechanisms gives ms > 7.9 keV for the SF mechanism and
ms > 2.6 keV for Higgs decay sterile neutrinos; however, it must be noted this is
not based on transfer function calculations for the SF and Higgs mechanisms but
assumes a simple scaling for the average momentum for the different production
mechanisms (Kusenko 2009). The image fluxes in gravitationally lensed quasars
have been shown to require sterile neutrino masses greater than a few keV (Miranda
and Macciò 2007). The Lyman-α forest observations discussed above in the context
of a thermal particle also set limits on the sterile neutrino mass. The 2006 work
of Viel et al. sets ms > 11 keV and is similar to the Seljak et al. (2006) limit
ms > 14 keV. The 2008 work of Viel et al. sets the highest limit of ms > 28 keV
but is subject to the caveats mentioned above. The limit for the most recent 2013
work is ms > 21 keV (Viel et al. 2013).
Sterile neutrinos are expected to radiatively decay to a lighter mass neutrino
and a X-ray photon with energy Eγ = ms /2. X-ray observations of the diffuse X-ray
background (Boyarsky et al. 2006b) and dark matter halos in clusters (Abazajian
and Koushiappas 2006; Boyarsky et al. 2006a; Riemer-Sorensen et al. 2007; Boyarsky
et al. 2008), M31 (Watson et al. 2006, 2012), dwarf spheroidal galaxies (Boyarsky
et al. 2007; Riemer-Sørensen and Hansen 2009; Boyarsky et al. 2009; Loewenstein
et al. 2009), and the halo of the Milky Way (Riemer-Sørensen et al. 2006; Abazajian
et al. 2007; Boyarsky et al. 2007) have all been used to set constraints on the
sterile neutrino mass. Observations of the diffuse X-ray background have set ms <
9.3 keV (Boyarsky et al. 2006b), while the Virgo and Coma clusters have been
used to set ms < 6.3 keV (Abazajian and Koushiappas 2006) which also agrees
66
with limits from the Bullet cluster, 1E 0657-56, ms < 6.3 keV (Boyarsky et al.
2008) and is close to results from the Milky Way halo ms < 5.7 keV (Abazajian
et al. 2007). Tighter constraints have been determined from the dwarf spheroidal
Ursa Minor ms < 2.5 keV (Loewenstein et al. 2009) and from M31 observations
ms < 2.2 keV (Watson et al. 2012).
These upper limits are well below the lower limits derived in this work and
from Lyman-α observations and seem to rule out the DW and SF production mechanisms. However, all of these mass limits, including the constraints set in this work,
are model dependent and make certain assumptions. In general X-ray constraints
depend on the sterile neutrino mass, the mixing angle with active neutrinos θ, and
the cosmic matter density of sterile neutrinos Ωs . There are also assumptions about
the initial conditions, that there were no sterile neutrinos in the early Universe at
temperatures > 1 GeV, there was no entropy dilution after creation, and no coupling
to other particles. There are also uncertainties with the calculation of production
rates because these occur at temperatures where the plasma is neither well described
by hadronic nor quark models (Asaka et al. 2006; Boyarsky et al. 2006b). Depending on the assumptions made and the adopted production model the relationship
between ms , θ, and Ωs changes so that robust constraints cannot be placed on any
one model parameter. However, Horiuchi et al. (2013) show the combined limits
in parameter space rule out that all of the dark matter is DW produced sterile
neutrinos although a mix of resonant and non-resonant production mechanisms is
allowed. This possibility corresponds to a mixed dark matter cosmology with the
non-resonant produced neutrinos being the warm component and the resonant the
67
cold component. There have also been reports of a detection of a dark matter X-ray
emission line in the spectrum of galaxy clusters consistent with ms = 7.1 ± 0.7 keV
(Bulbul et al. 2014; Boyarsky et al. 2014). This detection is provisional but if confirmed the limits derived in this work imply the sterile neutrinos are not produced
by the DW mechanism or do not constitute the entirety of the dark matter. Abazajian (2014) calculated the transfer function for sterile neutrinos produced by the SF
mechanism consistent with the X-ray line and showed it matches that of a thermal
particle of mass 2.02 keV. As I have shown, this would be consistent with MW
satellite abundances and solve the missing satellites problem.
3.5 Summary
I conducted N-body simulations of the formation of MW-sized dark matter
halos in CDM and WDM cosmologies. Such simulations are complicated by the
formation of artifical small mass halos due to the discreteness of the initial conditions
but with sufficient resolution they are only important at small scales and can be
avoided with an appropriate circular velocity cut.
I studied the number of satellite halos as a function of distance from the MW.
The 4 keV and 2 keV WDM simulations can adequately reproduce the observed
number of satellites at distances up to hundreds of kiloparsecs while the 1 keV simulation is severely deficient. The high resolution simulations followed the formation
of two MW-sized halos. Numerical simulations of MW-sized halos show significant
variance in the number of satellites, an effect that can be easily quantified using
68
published studies and was incorporated in my results. I calculated the number of
satellites in the inner 50 kpc, corrected for the effects of numerical destruction, and
accounted for the variance by conservatively adopting a 30% (1σ) intrinsic scatter in
the number of satellites in addition to a scatter from Poisson statistics. I corrected
the number of satellites in simulation to the survey area of the SDSS and derived a
very conservative lower limit on the dark matter particle mass of > 2.1 keV (95%
C.L.). This agrees with the earlier Lyman-α forest modeling work of Viel et al.
(2006) that mW DM > 2 keV but is below their latest limit of mW DM > 3.3 keV
(Viel et al. 2013). However, the two methods are independent and almost certainly
are subject to different systematic errors, if any exist.
My lower limit of 2.1 keV for a thermal dark matter particle scales to lower
limits of 11.8, 7.9, 2.6 keV (95% C.L.) for DW, SF, and Higgs decay produced
sterile neutrinos. Sterile neutrinos, if they exist, are expected to decay into X-rays
and active neutrinos. Observations of the unresolved cosmic X-ray background and
X-ray observations of dark matter halos on scales from dwarf galaxies to clusters
set upper limits below my lower limit and the limits of Lyman-α forest modeling.
These limits are derived under many assumptions and, in general, the constraints
apply to a parameter space of ms , θ, and Ωs .
My constraint is a conservative lower limit since I only correct the satellite
abundances in simulation to the number of SDSS dwarfs by accounting for the
partial sky coverage of the survey. An analysis that takes into account the surface
brightness limits of the observational data may allow tighter constraints; however,
the analysis would be somewhat model dependent. I have also not included the
69
effects on subhalos of baryonic structures in the inner MW halo such as a disk.
The presence of a disk could lead to greater subhalo destruction due to increased
dynamical friction and tidal heating. By increasing the subhalo destruction rate in
the inner halo, disks would increase the lower bounds on the dark matter particle
mass. The assumption of no disk is a conservative one and an analysis that includes
a disk may allow tighter constraints.
I have demonstrated how N-body simulations of the MW and its satellites can
set limits on the dark matter particle mass comparable to, and independent of, complementary methods such as modeling the Lyman-α forest. These limits are helped
greatly by the discovery of many new MW satellites in the SDSS. There may still be
a population of low luminosity, low surface brightness dwarf galaxies undetectable
by the SDSS (Ricotti and Gnedin 2005b; Ricotti et al. 2008b; Ricotti 2010; Bovill
and Ricotti 2009b). Future surveys with instruments like the Large Synoptic Survey Telescope have the potential to discover many more MW satellites and further
improve constraints on the mass of the dark matter particle. Better constraints will
result from the smaller uncertainty in the number of observed satellites achieved by
improving the sky coverage and reducing luminosity corrections. In addition, the
existence of a yet unknown population of even fainter satellites is not unlikely.
70
Chapter 4: Dependence of Satellite Densities on Cosmology
4.1 Overview
The satellite galaxies of the Milky Way, being the closest extragalactic objects
and indeed within the virial radius of the Milky Way’s extended halo of dark matter,
are uniquely suited for testing theories of galaxy formation and evolution and the
nature of dark matter. The MW satellites known before the SDSS numbered too
few to account for predictions from N-body simulations in ΛCDM cosmologies that
were otherwise successful in describing the abundances of galaxies in clusters and
the large scale features of the matter distribution (Klypin et al. 1999; Moore et al.
1999a). The discovery of a population of fainter satellites in the SDSS and more
sophisticated simulations that account for supernova feedback and the heating of
the IGM during reionization have alleviated this problem by predicting a strong
suppression of galaxy formation in low mass halos (Bullock et al. 2000; Ricotti et al.
2002b,a, 2008a).
Recent work focusing on the brightest MW satellites has highlighted dynamical discrepancies with high-resolution CDM simulations. Boylan-Kolchin et al.
(2011, 2012a) compared the most luminous satellites to subhalos in the Aquarius
simulation suite of six Milky Way-sized halos. Abundance matching models set a
71
one-to-one correspondence between luminosity and dynamical mass and place the
brightest satellites in the largest subhalos. However, the observed stellar velocities
cannot be reconciled with the velocity profiles of the largest dark matter subhalos
in simulation. The most massive satellites, either at the present epoch, the epoch
of reionization, or over the complete infall history, are too dense to be dynamically consistent with the Milky Way satellites. Observations of the stellar velocity
dispersions in the bright satellites are consistent with dark matter halos with maximum circular velocities < 25 km s−1 while the Aquarius Milky Ways have about 10
subhalos each with vmax > 25 km s−1 that are also not Magellanic Cloud analogs.
Several solutions to this problem have been proposed. Galaxy formation may be
stochastic on dwarf spheroidal scales and the bright satellites do not reside in the
largest subhalos (Boylan-Kolchin et al. 2011; Katz and Ricotti 2012). This requires
abandoning the monotonic relation between galaxy luminosity and halo mass that
is well-established for brighter galaxies.
Interestingly, models in which some of the ultra-faint dwarfs are fossils of
the first galaxies (Ricotti and Gnedin 2005a; Bovill and Ricotti 2009a) show some
tension with observations only at the bright end of the satellite luminosity function
(Bovill and Ricotti 2011a,b). Simulations that produce a large population of ultrafaint dwarfs also produce an overabundance of bright dwarf satellites especially in
the outer parts of the Milky Way. However, this tension is eased by the expected
stripping of the extended primordial stellar population around bright satellites.
The number of satellites of all size are known to be proportional to the mass
of the host halo (Klypin et al. 1999). Wang et al. (2012) argue the low velocities
72
of the MW satellites may be an indication the MW is less massive than typically
thought. They show there is only a 5% probability for a galaxy of mass 2 × 1012 M⊙
to have 3 satellites or less with maximum circular velocities > 30 km s−1 but 40%
for a galaxy of mass 1012 M⊙ . A low mass for the Milky Way of 8 × 1011 M⊙ is also
favored in the work of Vera-Ciro et al. (2013). Direct measures of the MW mass
typically focus on stellar tracers of the inner halo or radial velocity measurements
of the MW satellites and give a range of virial mass 0.8 − 2.5 × 1012 M⊙ , the reader
is referred to the references in Boylan-Kolchin et al. (2012b) where observations of
the spatial motion of Leo I are used to constrain the mass of the Milky Way to
> 1012 M⊙ at 95% confidence.
Sawala et al. (2012) show the simulations can be reconciled with the observations by including baryonic physics in the simulations. Inclusion of baryonic physics
removes gas from halos through supernova expulsion of the interstellar medium,
prevention of gas accretion through reionization heating of the IGM, and ram pressure stripping from satellites. Removal of baryons from the dark matter halos also
reduces the potential well resulting in less accretion of both gas and dark matter.
They show dark matter only simulations overpredict the subhalo abundance by 30%
at a mass scale of 1010 M⊙ with an increasing number of subhalos with no gas or
stars below this scale.
The influence of baryons was also studied by di Cintio et al. (2011). They
found that while satellites with low baryon fractions have lower concentrations than
their dark matter only counterparts, satellites with high baryon fractions have higher
central densities due to adiabatic contraction. Satellites with high baryon fractions
73
also tend to have the largest maximum circular velocities. However, their recent
work (Di Cintio et al. 2013) finds the subhalo density profiles are better described
by Einasto profiles than Navarro, Frenk, and White profiles (Navarro et al. 1997) and
that this reconciles the observations with simulated satellites of similar luminosities.
Vera-Ciro et al. (2013) also find agreement with Einasto profiles. However, while the
initial work of Boylan-Kolchin et al. (2011) assumes NFW profiles their later work
(Boylan-Kolchin et al. 2012a) uses the subhalo circular velocity profiles directly with
no assumed form.
Another possibility is a change in the nature of the dark matter from standard
CDM assumptions of collisionless particles with low intrinsic thermal velocities.
Vogelsberger, Zavala, and Loeb (2012) simulated one of the Aquarius Milky Way
halos in self-interacting dark matter models. The ability of the dark matter particles
to self-scatter leads to the formation of subhalos with constant density cores. The
lower density decreases the inner circular velocity profiles bringing the simulations
into agreement with the observations.
A truncation in the dark matter power spectrum was investigated as a solution to the paucity of satellites by reducing the abundance of halos at subgalactic
scales. One method for producing a truncated power spectrum is if the dark matter
particles decoupled with relativistic velocities early in the radiation dominated era
and thereby able to stream out of overdense regions before becoming nonrelativistic
at a time before the horizon had reached Galactic scales. The scale of the power
spectrum truncation in WDM is related to the mass of the dark matter particle with
lighter particles decoupling earlier and able to stream longer.
74
Dwarf-scale halos in WDM cosmologies form later and have lower concentrations than halos in CDM, offering a potential solution to the dynamical discrepancies. Lovell et al. (2012) simulated one of the Aquarius halos in a 1 keV thermal
relic WDM cosmology and showed the subhalos have central densities and velocity
profiles in agreement with the bright MW satellites. In Lovell et al. (2013) their
work was extended to particle masses 1.4-2.3 keV. Recently, one Milky Way-like halo
was simulated in WDM at 2, 3, and 4 keV (Schneider et al. 2013). In this work I
investigate the subhalo dynamics in four Milky Way-sized halos in 1, 2, 3, and 4 keV
cosmologies.
Another area potentially affecting the subhalo densities are the adopted cosmological parameters. The Via Lactea II simulation (Diemand et al. 2007, 2008),
which adopted parameters from the 3rd year release of the Wilkinson Microwave
Anisotropy Probe, was found to give similar results as the six Aquarius halos adopting WMAP1 parameters. However, reason to suspect the adopted cosmology is
important comes from Macciò et al. (2008) who explored the dependence of halo
concentration on the adopted cosmological model for field galaxies. They fit NFW
density profiles to the halos in their simulations:
ρ(r) =
ρs
,
(r/rs )(1 + r/rs )2
(4.1)
and determined the concentrations, c200 = R200 /rs , where R200 is the radius enclosing
a density 200 times the critical density, ρcrit . They found the average concentration
of dwarf-scale field halos varies by a factor of 1.55 between WMAP1 and WMAP3.
In this work I also examine the dependence of the CDM subhalo populations on the
75
adopted cosmological parameters.
4.2 Simulations
All simulations were conducted with the N-body cosmological simulation code
GADGET2 (Springel 2005) with gravitational physics only and initial conditions
generated with the GRAFIC2 software package (Bertschinger 2001). I use the high
resolution simulations presented in Chapter 3 and Polisensky and Ricotti (2011)
where two Milky Way-sized halos were simulated in a cubic box with comoving
side length of 90 Mpc, mass resolution of 9.2 × 104 M⊙ , and a 275 pc gravitational
softening length. I refer to these halos as the set A and set B simulations. I also ran a
high resolution simulation of halo C8 from Chapter 2 with a 138 pc softening length
and refer to this as the set C simulations. Finally, an additional set D simulation
was run of another Milky Way-sized halo in a 67 Mpc comoving box with a mass
resolution 8.2 × 104 M⊙ and gravitational softening length 196 pc.
Table 4.1 lists sets of cosmological parameters from measurements of the cosmic microwave background by WMAP and the Planck mission (Spergel et al. 2003,
2007; Komatsu et al. 2009; Larson et al. 2010; Jarosik et al. 2010; Komatsu et al.
2011; Hinshaw et al. 2012; Planck Collaboration et al. 2013). “Bolshoi” are the
parameters from the Bolshoi simulation (Klypin et al. 2011) which were chosen to
be within 1σ of WMAP5, WMAP7, and consistent with the results of supernovae,
and X-ray cluster surveys. These parameters are within 1σ of WMAP9 except the
value of ns which is within 1.7σ. They are also within 1.2σ of Planck1 with the
76
exceptions of Ωm and ΩΛ which are 2.2σ below Planck1. The WMAP1 parameters
are 2.4 − 4.1σ away from Planck1 while σ8 and ns are 3.4σ and 2.2σ above WMAP9,
respectively. In contrast, the value of σ8 in WMAP3 is 3.5σ below WMAP9 and
Planck1.
Figure 4.1 shows the linear power spectra for the parameters listed in Table 4.1
normalized by the Bolshoi power spectrum. On the scale of the dwarfs (k ∼ 10
Mpc−1 ) the power varies greatly across cosmologies with WMAP1 and WMAP3
representing the extremes of high and low power. The Bolshoi parameters, however,
represent a conservative estimate of the power on dwarf scales while being consistent
with the latest CMB measurements from WMAP and Planck.
To investigate the dependence of satellite densities on cosmology I ran CDM
simulations for each of the four sets adopting WMAP1, WMAP3, and Bolshoi parameters with the CDM transfer function from Eisenstein and Hu (1998). The box
size and softening lengths were scaled in each simulation to keep the mass resolution
constant. A series of low resolution tests of the set B halo were also run, these are
described in the next section.
For my investigation of warm dark matter I used the warm dark matter transfer
function given by Bode, Ostriker, and Turok (2001) valid for particles in thermal
equilibrium at the time of their decoupling, such as the gravitino. I adopted Bolshoi
parameters and ran simulations for particle masses of 1, 2, 3, and 4 keV for each
halo. Figure 4.2 and Figure 4.3 present portraits of the Milky Way halos in the
CDM and WDM simulations.
Version 1.0 of the AMIGA’s Halo Finder (AHF) software (Knollmann and
77
Figure 4.1: Power spectra for CDM cosmologies normalized by the Bolshoi power spectrum.
78
Figure 4.2: Milky Way halos in CMD simulations. From left to right:
WMAP3, Bolshoi, WMAP1. From top to bottom: MW A–D.
79
Figure 4.3: Milky Way halos in CMD and WDM simulations adopting
Bolshoi parameters. From left to right: MW A–D. From top to bottom:
CDM–1 keV.
80
Table 4.1: Cosmological parameters.
Name
WMAP1
WMAP3
WMAP5
WMAP7
WMAP9
Planck1
Bolshoi
Ωm
ΩΛ
Ωb
h
σ8
ns
0.25
0.238
0.258
0.267
0.282
0.317
0.27
0.75
0.762
0.742
0.733
0.718
0.683
0.73
0.045
0.040
0.0441
0.0449
0.0461
0.0486
0.0469
0.73
0.73
0.72
0.71
0.70
0.67
0.70
0.90
0.74
0.796
0.801
0.817
0.834
0.82
1.0
0.951
0.963
0.963
0.964
0.962
0.95
Knebe 2009) was used to identify the Milky Way halos and their gravitationally
bound subhalos after iteratively removing unbound particles. Table 4.2 summarizes
the properties calculated by AHF for the simulated Milky Ways at z = 0. I write
R100 to mean the radius enclosing an overdensity 100 times ρcrit . The mass and
number of particles inside R100 are M100 and N100 , respectively; vmax = max(vcirc )
2
is the maximum circular velocity of the halo occurring at a radius Rmax , and vcirc
=
GM(< r)/r. Also given is the NFW c200 concentration for each halo determined
from:
vmax
v200
2
= 0.2162 c200 /f (c200 ),
(4.2)
where f (c) = ln(1 + c) − c/(1 + c).
The SUBFIND program (Springel et al. 2001) was also run on the set B
WMAP3 data and excellent agreement was found with the results from AHF.
81
Table 4.2: Properties of simulations and Milky Way halos at z = 0.
Cosmology
CDM
CDM
CDM
4 keV
3 keV
2 keV
1 keV
WMAP1
Bolshoi
WMAP3
Bolshoi
Bolshoi
Bolshoi
Bolshoi
mres
[M⊙ ]
M100
[1012 M⊙ ]
R100
[kpc]
vmax
[km s−1 ]
Rmax
[kpc]
N100
c200
9.17 × 104
9.17 × 104
9.17 × 104
9.17 × 104
9.17 × 104
9.17 × 104
9.17 × 104
2.1119
1.9803
1.8410
1.9644
1.9724
2.0061
2.0197
324.233
326.357
309.740
325.486
325.929
327.771
328.514
214.78
198.97
192.28
198.11
197.05
197.54
199.24
39.849
55.243
41.027
50.414
54.900
39.871
58.943
23, 028, 026
21, 560, 499
20, 074, 556
21, 387, 017
21, 474, 003
21, 874, 542
22, 022, 816
9.68
8.38
7.77
8.25
7.99
7.96
8.04
210.02
194.90
194.62
194.19
193.65
194.53
195.23
67.068
82.086
79.767
74.900
77.500
79.500
84.286
22, 760, 127
21, 012, 806
19, 125, 479
20, 928, 496
20, 962, 535
20, 981, 724
20, 503, 730
9.01
7.81
8.29
7.69
7.64
7.90
8.06
231.42
215.81
203.42
215.03
215.25
214.40
210.94
44.932
58.943
56.164
56.900
57.100
61.529
64.857
26, 240, 319 11.13
25, 211, 233 9.05
21, 645, 271 8.72
25, 152, 203 8.95
25, 153, 016 9.01
25, 070, 237 8.88
24, 563, 114 8.61
190.95
176.26
164.27
176.62
176.38
175.73
171.97
67.027
69.057
50.164
75.414
74.143
75.729
79.514
22, 135, 114
19, 429, 510
15, 323, 846
19, 412, 993
19, 345, 715
18, 947, 343
18, 276, 956
7.29
6.80
6.56
6.77
6.78
6.81
6.39
68.795
73.630
66.233
71.209
78.429
77.943
98.288
82.740
3, 168, 819
3, 192, 628
3, 237, 093
3, 198, 000
2, 987, 609
2, 834, 081
2, 405, 721
2, 640, 759
9.56
9.92
10.63
10.02
9.29
8.91
7.45
7.92
Set A
Set B
CDM
CDM
CDM
4 keV
3 keV
2 keV
1 keV
WMAP1
Bolshoi
WMAP3
Bolshoi
Bolshoi
Bolshoi
Bolshoi
9.17 × 104
9.17 × 104
9.17 × 104
9.17 × 104
9.17 × 104
9.17 × 104
9.17 × 104
2.0873
1.9271
1.7540
1.9193
1.9224
1.9242
1.8804
322.973
323.414
304.781
322.971
323.157
323.257
320.771
Set C
CDM
CDM
CDM
4 keV
3 keV
2 keV
1 keV
WMAP1
Bolshoi
WMAP3
Bolshoi
Bolshoi
Bolshoi
Bolshoi
9.17 × 104
9.17 × 104
9.17 × 104
9.17 × 104
9.17 × 104
9.17 × 104
9.17 × 104
WMAP1
Bolshoi
WMAP3
Bolshoi
Bolshoi
Bolshoi
Bolshoi
4
2.4195
2.3259
1.9887
2.3195
2.3194
2.3113
2.2607
339.274
344.343
317.808
344.029
344.014
343.614
341.086
Set D
CDM
CDM
CDM
4 keV
3 keV
2 keV
1 keV
8.21 × 10
8.21 × 104
8.21 × 104
8.21 × 104
8.21 × 104
8.21 × 104
8.21 × 104
1.8164
1.5944
1.2575
1.5930
1.5875
1.5548
1.4998
308.342
303.614
272.781
303.526
303.171
301.086
297.486
Set B Low Resolution Tests
CDM
CDM sm
CDM sm hi zi
CDM
CDM
CDM
CDM
CDM hi zi
WMAP1
WMAP1
WMAP1
Planck1
WMAP9
Bolshoi
WMAP3
WMAP3
7.34 × 105
5.92 × 105
5.92 × 105
7.34 × 105
7.34 × 105
7.34 × 105
7.34 × 105
7.34 × 105
2.3249
1.8899
1.9162
2.3463
2.1919
2.0793
1.7650
1.9375
334.795
312.452
313.890
355.582
337.600
331.714
305.411
315.055
82
221.67
208.53
212.12
215.70
210.15
205.26
191.01
198.24
Snapshots of the particle information were saved every 0.05 change in the
universal scale factor, a = (1 + z)−1 , for simulations adopting Bolshoi and WMAP1
parameters. Figure 4.4 shows the mass growth of each MW halo and the VL2 halo
as a function of a. The masses are normalized to the halo mass at a = 1. The
MergerTree tool in AHF was used to construct merger trees for all identified halos.
This allows determination of vinf all for each subhalo, the maximum value of vmax
over a halo’s formation and accretion history: vinf all = max(vmax (z)). I follow
the work of Boylan-Kolchin et al. (2011) and consider subhalos within 300 kpc of
the Milky Way centers. I similarly identify subhalos with vmax > 40 km s−1 and
vinf all > 60 km s−1 as hosts of Magellanic Cloud analogs.
I compare the simulated subhalos to the MW dwarf spheroidal satellites with
luminosities LV > 105 L⊙ . Walker et al. (2009) and Wolf et al. (2010) show lineof-sight velocity measurements provide good constraints on the dynamical masses
of dispersion-supported galaxies like the MW dwarf spheroidals. The Magellanic
Clouds are excluded from the observation sample as they are irregular type galaxies.
The Sagitarius dwarf is also excluded because it is undergoing disruption and far
from equilibrium. The observed sample consists of nine galaxies: Canes Venatici I,
Carina, Draco, Fornax, Leo I, Leo II, Sculptor, Sextans, and Ursa Minor.
83
Figure 4.4: Mass growth histories of simulated Milky Way halos as a
function of scale factor, a.
84
Figure 4.5: Plots of vmax and Rmax for subhalos in the high resolution
CDM simulations for each set of cosmological parameters. The shaded
area shows the 2σ constraints for the bright Milky Way dwarfs from
Boylan-Kolchin et al. (2011) assuming NFW profiles. The sloped red
line shows the mean of the Aquarius subhalos. Magellanic Cloud analogs
in the Bolshoi and WMAP1 simulations are plotted in blue.
85
4.3 Results
4.3.1 Cold Dark Matter
Figure 4.5 is a plot of vmax and Rmax for subhalos in the high resolution CDM
simulations. Boylan-Kolchin et al. (2011) investigated what values of vmax and Rmax
of NFW halos (Navarro et al. 1997) are consistent with the half-light dynamical mass
constraints of the bright MW dwarf spheroidals from Wolf et al. (2010). Their 2σ
confidence region is plotted as the shaded regions in Figure 4.5.
It is easy to see there are many subhalos that lie in the range consistent with
the MW dwarfs, but there are some with vmax > 20 km s−1 that do not. These are
the subhalos highlighted by Boylan-Kolchin et al. (2011) that are massive but have
central densities too high to host any of the MW dwarfs. However the WMAP3 and
Bolshoi simulations have only 1-3 subhalos per parent halo outside the shaded zone
of Milky Way satellites compared to 4-8 subhalos for the WMAP1 simulations. This
is due to Rmax being shifted to higher values from WMAP1 for the same values of
vmax .
Springel et al. (2008) show that the logarithms of vmax and Rmax for the
Aquarius subhalos have a linear relationship. I estimate the equation of their fitting
line:
log Rmax = 1.41 log(vmax /14.72 km s−1 ),
(4.3)
and plot this as the red line. I assumed a constant slope and performed least-squares
fits to my subhalos in each cosmology and plot these as the black lines. The red
86
arrowed lines show the shift in Rmax for each of the simulation sets compared to
Aquarius. My simulations adopting WMAP1 parameters are in good agreement
with the Aquarius simulations, differing by only a factor of 1.07, but in Bolshoi and
WMAP3 my subhalos are offset to higher values of Rmax by factors of 1.45 and 1.50,
respectively.
I compared the fit for each simulation set separately to the corresponding fit in
the WMAP1 cosmology. I found the average scale in Rmax from WMAP1 to Bolshoi
is a factor of 1.35 and a factor of 1.40 for WMAP3, with a 1σ scatter of ±0.10 for
each.
To determine if factors other than the cosmology may be affecting the subhalo
densities I ran a series of tests on the set B halo with the mass resolution decreased a
factor of 8 but the softening length kept the same as the high resolution simulations.
I ran a test adopting WMAP3 parameters starting from the same initial redshift
as the high resolution simulation (zi = 48) and another test starting from a high
redshift (zi = 115), comparable to the starting redshift of Aquarius (zi = 127). I also
ran tests adopting the WMAP1 parameters. The Milky Way halo mass was about
30% greater in this simulation so I ran tests with the box size and mass resolution
decreased to give a halo mass similar to the WMAP3 tests. I ran small box tests
starting from the same low and high redshifts.
I examined applying velocity cuts of vmax > 14 − 20 km s−1 to the subhalos.
At smaller velocities the Rmax values for some subhalos were inside the convergence
radius satisfying the criterion of Power et al. (2003) and therefore affected by the
resolution of the simulations. In Figure 4.6 I normalize the values of Rmax for all
87
Figure 4.6: Distribution functions of Rmax normalized to the Aquarius
values for CDM subhalos with vmax > 18 km s−1 in the WMAP1 and
WMAP3 simulations of the set B halo. Simulations adopting WMAP3
parameters are plotted in red while WMAP1 simulations are plotted
in blue. The offset between simulations is consistent with a cosmology
dependence and not on mass resolution, starting redshift, or mass of the
host Milky Way halo. Solid gray area is the distribution for Via Lactea-II
subhalos.
88
subhalos with vmax > 18 km s−1 to the Aquarius value of Rmax from Equation 4.3
and present binned distributions for these subhalos and those of the Via Lactea-II
(VL2) simulation (zi = 104). I find consistent distributions between the low and high
resolution simulations showing the mass resolution and softening length are sufficient
to sample subhalos with vmax > 18 km s−1 . I also find weak to no dependence on
the starting redshift as the simulations started from zi = 115 have distributions
consistent with the corresponding simulations started from zi = 48. However, I do
see a strong dependence on the cosmology as the WMAP3 simulations are offset
to higher Rmax compared to WMAP1. The offset is only weakly dependent on the
mass of the Milky Way host as the WMAP1 simulations in the large and small boxes
have nearly identical distributions.
Additional low resolution tests were run of the set B halo adopting WMAP9,
Bolshoi, and Planck1 parameters. These simulations also show offsets from WMAP1
but less than the WMAP3 tests (final column in Table 4.3), as expected for the
greater small scale power in these cosmologies. These tests show the subhalo concentrations are largely determined by their formation time. As the small scale power
increases formation occurs earlier and the subhalos are more concentrated at z = 0.
This is supported by examining the high redshift data for these simulations. Table 4.3 gives the number of halos with masses > 2 × 108 M⊙ and the average mass
of the 12 largest halos in the high resolution volume at z = 9 in the test simulations
of the set B halo with mass resolution 7.34 × 105 M⊙ . In the high resolution volume
at z = 9 there are more than six times as many halos with masses > 2 × 108 M⊙ in
the WMAP1 simulation than in WMAP3. Furthermore, the 12 most massive halos
89
are an average of four times as massive in WMAP1 than WMAP3. This is evidence
dwarf-scale halos are collapsing earlier and have more time to grow in a WMAP1
cosmology.
Table 4.3: Comparison of the low resolution CDM tests of the set B halo with a
common mass resolution. See text for an explanation of quantities in the columns.
Name
WMAP1
Planck1
WMAP9
Bolshoi
WMAP3
Nz=9
> 2 × 108M⊙
378
239
193
149
57
< Mtop12 >
[109 M⊙ ]
2.939
1.982
1.612
1.375
0.777
Rmax
Rmax,W M AP 1
1.0
1.06
1.16
1.20
1.57
The distribution of VL2 subhalos is also plotted in Figure 4.6. The VL2
simulation used WMAP3 cosmology but its subhalos have concentrations consistent
with Aquarius. I hypothesize this is because the VL2 halo has a higher redshift
of formation than the mean for a WMAP3 cosmology. Figure 4.4 shows my halos
generally have accreted less of their final mass at a < 0.5 than the VL2 halo. For
example, at a = 0.25 the VL2 halo has 23% of its final mass while my halos have only
5 − 18% of their final masses. Further evidence comes from the halo concentration
which is known to correlate with formation epoch. I determined M200 and R200
(1.417 × 1012 M⊙ , 225.28 kpc) from the fit to the VL2 density profile (Diemand
et al. 2008) and calculate c200 from Eqn 4.2. The concentration of VL2 is 10.7, in
contrast with the 6.6−8.7 concentrations of my WMAP3 halos. VL2 is a 2.4σ outlier
in the WMAP3 simulations of Macciò et al. (2008) where the average concentration
of relaxed 1012 M⊙ h−1 halos is 5.9.
90
4.3.1.1 Velocity profiles
A direct comparison of the subhalo circular velocity profiles to the half-light
circular velocities of the observed dwarfs is desirable but is complicated by two effects. The circular velocity is a cumulative quantity and its profile is affected by the
softening length to greater distances than the density profile (Zolotov et al. 2012)
making reliable inward extrapolation difficult. Additionally, the hosts of the bright
dwarfs are expected to be the largest subhalos over the complete infall history of
the subhalo population or the largest at the epoch of reionization. Many of these
subhalos will experience tidally stripped mass loss thereby reducing their Rmax sufficiently to become affected by the softening length. The largest subhalos at present
(z = 0) are generally subhalos just beginning to infall as indicated by their large
spatial extent (Anderhalden et al. 2013). They are the least affected by stripping
and therefore have the most reliable circular velocities. Excluding Magellanic Cloud
analogs from the simulations, 5-6 of the 10 subhalos with greatest vmax at z = 0
are among the top 10 with greatest vinf all while 2-4 are among the top 10 with
greatest vmax at z = 9. Thus while the largest subhalos at z = 0 are not expected
to completely match the observed dwarf population they are useful for illustrating
the effects of cosmology on the too big to fail problem.
Plotted in Figure 4.7 are the NFW circular velocity profiles with Rmax and
vmax values of the 10 largest subhalos in each CDM simulation adopting WMAP1
and Bolshoi cosmologies. The data points with error bars show the circular velocities
at half light radii from Wolf et al. (2010) for the sample of bright Milky Way dwarfs.
91
While there is some halo-to-halo scatter the reduced densities and shift of the profiles
to larger radii in the Bolshoi cosmology is dramatically clear.
4.3.2 Warm Dark Matter
The results in the previous section show the discrepancy between the largest
subhalos in CDM simulations and observations of bright Milky Way dwarfs may
largely be due to the adopted cosmological parameters of the Aquarius simulation
and that adopting parameters in agreement with the most recent WMAP release
would greatly alleviate this problem. However I also saw that even a WMAP3
simulation like VL2 can have massive satellites dynamically inconsistent with the
bright dwarfs implying a dependence on the formation history of the Milky Way
and its satellites. In this section I investigate the effects warm dark matter has on
the massive subhalos.
Figure 4.8 is a plot of vmax and Rmax for subhalos in each simulation set for
each WDM cosmology. Again, it is clear there are many subhalos that lie in the
area consistent with the MW dwarfs but there are some with vmax > 20 km s−1 that
do not, however the number of outliers decreases as the particle mass decreases. An
average of 2 subhalos per simulation are outside the allowed region decreasing to
1.5 per simulation in 3 keV, < 1 in 2 keV, and 0 in 1 keV. An average of 2 subhalo
outliers per Bolshoi CDM simulation were found demonstrating the minimal effect
a 4 keV cosmology has on the densities.
The effects of WDM are a reduction in the total number of subhalos as well
92
Figure 4.7: NFW circular velocity profiles for the 10 subhalos with
largest vmax at z = 0 in each CDM simulation adopting WMAP1 cosmology (top row ); and Bolshoi cosmology (bottom row ) after filtering
Magellanic Cloud analogs. Subhalos denser than any observed dwarf
(points with error bars) are plotted in bold. Subhalos that are neither
among the 10 with largest vinf all or 10 largest vmax at z = 9 are not
expected to host a bright dwarf and are plotted with dotted lines. Note
that NFW profiles for the 10 subhalos with largest vmax over their infall
history select a few subhalos with lower values of vmax and Rmax than
shown here, further alleviating the discrepancy with observations.
93
Figure 4.8: Plots of vmax and Rmax for subhalos in the high resolution WDM simulations adopting Bolshoi cosmological parameters. The
shaded area shows the 2σ constraints for the bright Milky Way dwarfs
assuming NFW profiles. Magellanic Cloud analogs are colored purple.
as their circular velocities and an increase in their Rmax . I estimate the increase in
Rmax by fitting equations of the form of Eqn 4.3 to the WDM subhalo data and
comparing to the fits for the corresponding CDM simulation. I find, for constant
values of vmax , Rmax values are increased an average of 7% in 4 keV, 15% in 3 keV,
30% in 2 keV, and 46% in 1 keV; however, the small number of subhalos in 1 keV
makes it difficult to achieve a reliable estimate for this cosmology.
The effects of WDM on the circular velocities can be estimated by comparing
the velocities at several radii in the range 1 − 3 kpc for subhalos in WDM compared
to the corresponding CDM simulation. I find the subhalos in 1 keV WDM have
velocities up to 60% less than their CDM counterparts. This reduction decreases to
20% in 2 keV, 15% in 3 keV, and only 10% in 4 keV.
94
4.3.2.1 Velocity profiles
Figure 4.9 shows the NFW circular velocity profiles of the 10 subhalos with
the largest vmax at z = 0 in the WDM simulations after excluding Magellanic Cloud
analogs.
The subhalo profiles are severely affected in the 1 keV cosmology with both the
velocities and Rmax values showing large changes. The 1 keV simulations struggle
to match the observations in number and density with only set D managing to fit
both.
Comparison to the CDM subhalos plotted in Figure 4.7 shows some scatter
among individual subhalos. For example, a few subhalos in set B have increased
density in WDM. In general, subhalo densities are significantly reduced in cosmologies warmer than 2 keV while at higher particle masses the effects are weak. This is
in agreement with the single-halo simulations in Schneider et al. (2013) and Lovell
et al. (2013).
4.4 Discussion
I found the concentrations and velocity profiles of subhalos in CDM simulations
are dependent on the adopted cosmological parameters. I tested and found little to
no dependence on the starting redshift, the mass resolution, the mass of the parent
halo, and the halo finding software.
A cosmological dependence is also seen in other published work of Milky Waysized galaxies. The simulations of Stoehr et al. (2002) used similar parameters
95
Figure 4.9: NFW circular velocity profiles for the 10 subhalos with
largest vmax at z = 0 in each WDM simulation adopting Bolshoi cosmology. Subhalos denser than any observed dwarf (points with error
bars) are plotted in bold.
96
to Aquarius, (Ωm , ΩΛ , h, σ8 , ns ) = (0.3, 0.7, 0.7, 0.9, 1), and are well fit by
Equation 4.3. di Cintio et al. (2011) saw an offset in their simulations using WMAP3
and WMAP5 parameters. A dependence of substructure central densities on the
cosmological parameters is predicted in the work of Zentner and Bullock (2003)
using the semianalytic model of Bullock et al. (2001a). The central densities are
expected to reflect the mean density of the universe at the time of collapse. Adopting
values for cosmological parameters that moves the formation of small mass halos to
later epochs will result in less concentrated subhalos.
Here I show how the subhalo densities can be simply related to the power at
their mass scale and therefore dependent on both σ8 and ns . The parameter σ8 sets
the power at a scale of 8 Mpc h−1 corresponding to a mass of about 2.5×1014 M⊙ . If
the masses of the largest satellites are about 1010 M⊙ , the wave number is ksat ∼ 30k8
where k8 is the wave number corresponding to 8 Mpc h−1 . The change in σ between
WMAP3 and WMAP1 values of ns is given by:
ksat (ns,W M AP 3 −ns,W M AP 1 )/2
∼ 0.92.
k8
(4.4)
σ8,W M AP 3
∼ 0.82.
σ8,W M AP 1
(4.5)
The change due to σ8 is:
The total change at the satellites scale is 0.92×0.82 = 0.76. This is also proportional
to the change of the redshift of formation:
(1 + zf )W M AP 3 = 0.76(1 + zf )W M AP 1.
(4.6)
The virial radius is proportional to Rmax at virialization and the circular velocity at
97
the virial radius is proportional to vmax at virilization and:
Rvir ∝ vvir (1 + zf )−1.5 .
(4.7)
Therefore the following scaling between cosmologies is obtained:
Rmax,W M AP 3
= 0.76−1.5 = 1.51.
Rmax,W M AP 1
(4.8)
Repeating this for the scaling between Bolshoi and WMAP1 cosmologies yields
a factor of 1.31. From my simulations I derived average scaling factors of 1.40
and 1.35 for WMAP3 and Bolshoi, respectively, with a scatter of 0.10. This is in
good agreement with the rough calculation that assumes a mass of 1010 M⊙ for the
large satellites and neglects tidal effects that may introduce a cosmology dependent
change of the present values of Rmax and vmax from the values at virialization. An
approximate general scaling relation for Rmax at a fixed vmax can be written:
Rmax ∝ (σ8 5.5ns )−1.5 .
(4.9)
This equation gives a scaling of 1.24 between Planck1 and WMAP1.
I also investigated how the subhalo densities are affected in a range of WDM
cosmologies and quantified the reduction in circular velocity at kpc scales. In the
previous chapter I showed that the abundance of Milky Way satellites, including the
ultra-faint dwarfs discovered in the SDSS, allow a lower limit of 2.1 keV to be placed
on the dark matter particle mass. The work of Lovell et al. (2013) favors a similar
but slightly warmer limit of 1.6 keV. Several authors have used Lyman-α data to
provide independent constraints on WDM with lower limits ranging from 1.7–4 keV
(Boyarsky et al. 2009; Viel et al. 2006; Seljak et al. 2006; Viel et al. 2008, 2013).
98
Under these constraints I expect the circular velocities of the largest satellites in
WDM to be affected by less than 20%, much less than the 60% changes seen in a
1 keV cosmology. I conclude that WDM cosmologies colder than ∼ 2 keV have only
a mild effect on the density of massive Milky Way satellites, that are instead most
sensitive to the redshift of formation of the Milky Way and the power at small scales
given by σ8 and ns . Interestingly, if the ∼ 3.5 keV X-ray emission observed from
galaxy clusters is caused by a sterile neutrino with properties similar to a ∼ 2 keV
thermal relic my work shows it will naturally account for the observed densities of
the bright Milky Way dwarfs.
While my simulations adopting Bolshoi cosmology reduced the number of “too
big to fail” subhalos in 3/4 of the Milky Way realizations from about four or five in
WMAP1 to about one or two, none of my simulated Milky Ways are completely free
of overdense subhalos. Furthermore, the case of the VL2 halo demonstrates that
large variation in average subhalo density is possible even in WMAP3 cosmologies.
Purcell and Zentner (2012) examined 10, 000 realizations of substructure for three
host Milky Way masses from an analytic model. While their technique is only an
approximation to direct simulation they find ∼ 10% of their subhalo populations
have no massive failures in a WMAP7 cosmology. The Milky Way may thus simply
be mildly atypical. Interestingly, Hammer et al. (2007) show the Milky Way is
deficient in stellar mass, disk angular momentum, and average iron abundance of
stars in the Galactic halo at the 1σ level. Only 7% ± 1% of spiral galaxies with
comparable rotation speeds have similar properties. One way of explaining these
discrepancies is to assume the Milky Way had a quiet accretion history without
99
major merger events for the past ∼ 10 Gyr. Figure 4.4 shows VL2 and the set
B and set C halos assemble ∼ 70% of their mass by z = 1.5 and may better
represent the Milky Way than the other halos, according to this model. Opposite to
expectations these halos have the highest number of outliers. However, Purcell and
Zentner (2012) found selecting hosts for quiet accretion histories did not significantly
increase the probability of consistency.
My simulations assumed the dark matter was purely cold or purely warm, but
a mixture of the two is possible. The transfer function of mixed dark matter is
characterized by a step related to the particle mass and a plateau at smaller scales
related to the fraction of the warm component. This could arise if the dark matter
is composed of multiple particle species or a single species containing warm and
cold primordial momentum distribution components caused by separate production
stages, for example. Boyarsky et al. (2009) allowed for mixed cold and warm dark
matter in their analysis of Lyman-α forest data. They find a particle mass of 1.1 keV
is allowed if the WDM fraction is less than 0.4 (95% confidence). Masses below 1
keV are allowed provided the fraction of WDM is less than 0.35. Anderhalden et al.
(2013) examined a subhalo population in several mixed dark matter cosmologies.
They show a range of models that agree with Lyman-α constraints can be ruled out
for failing to produce subhalos with sufficient density to match the observations,
highlighting the usefulness and uniqueness of the Milky Way satellites as a probe of
small-scale cosmology.
100
Chapter 5: The Universal Density Profile That Wasn’t
5.1 Overview
The seminal work of Navarro et al. (1996) found that the density structure of
relaxed dark matter halos are well represented by what has become known as the
NFW profile:
ρ(r) =
ρs
,
(r/rs )(1 + r/rs )2
(5.1)
where ρ(r) is the density in a spherical shell at distance r from the halo center.
By scaling the free parameters rs and ρs , which define a characteristic length and
density, the NFW profile can describe dark matter halos from dwarf galaxy to cluster
scales. Furthermore, it was found the NFW profile was valid for halos regardless not
only of mass but also the power spectrum of initial density fluctuations and values
of cosmological parameters, establishing that density profiles are universal in form
independent of the cosmological context (Navarro et al. 1997). Another universal
property was found in the coarse-grained phase-space density profile, Q ≡ ρ/σ 3 ,
where σ is the velocity dispersion of simulation particles. Taylor and Navarro (2001)
discovered Q has a remarkably simple form of a power-law, Q ∝ r α , with α ∼ −1.9.
It is useful to recast the free parameters of the NFW profile in terms of a halo
101
mass and concentration. For relaxed halos a radius can be defined in which the
material has reached virial equilibrium:
Mvir =
4π
3
∆(z)ρc (z)Rvir
,
3
(5.2)
where Rvir is the virial radius enclosing a density ∆ times the critical density, ρc , at
redshift z. In a matter dominated Einstein-de Sitter cosmology, ∆ = 178. With the
virial radius defined the characteristic radius rs can be recast as the concentration
parameter, cvir ≡ Rvir /rs . Much effort has gone into understanding the relationship
between cvir and Mvir as well as the dependencies on the background cosmology
and the evolution with redshift (Prada et al. 2012). This is necessary for predicting
the properties of luminous galaxies that reside in the dark matter halos and for
using galaxy observations as probes of the CDM paradigm. The concentration was
found to correlate with mass such that smaller mass halos are more concentrated.
This was understood as a consequence of the earlier formation epoch of small mass
halos in the bottom-up structure formation of CDM. Since small halos collapse
earlier their inner regions reflect the higher universal density of matter at earlier
times. Changing the cosmological parameters or the power spectrum changes the
halo formation epoch and affects the concentrations but does not affect the shape of
the universal profile. This interpretation is consistent with simulations of hot and
warm dark matter which found halos with masses below the truncation scale form
later and have lower concentrations than CDM halos of similar size (Avila-Reese
et al. 2001; Bode et al. 2001; Knebe et al. 2002).
Much effort has also gone into understanding the physical processes that pro102
duce the NFW profile. There are two basic approaches to analytically modeling the
density profile: smooth accretion based on spherical infall (Gunn and Gott 1972;
Gott 1975), and hierarchical merging following Press-Schechter formalism (Press and
Schechter 1974; Peebles 1974; Lacey and Cole 1993; Manrique et al. 2003). Both
approaches have been successful at producing the universal profiles. This has been
explained as a result of the process of virialization. If virialization erases all information about the past merging history of the halo then it will not matter if the mass
accretion is modeled as clumpy or smooth. However, a consensus has not emerged on
the dominant processes occurring during virialization or if the virialization process
erases all memory of the initial conditions.
The early stages of halo formation are marked by rapid accretion and mergers
making it natural to consider violent relaxation as the dominant mechanism determining the dark matter profiles in the fluctuating gravitational potential (White
1996). Violent relaxation was originally proposed to explain the structure of elliptical galaxies (Lynden-Bell 1967) where estimates of star-star encounters would not
establish equilibrium in a Hubble time. The relaxation time of a forming halo is
related to the rate of change of the gravitational potential. Austin et al. (2005)
and Barnes et al. (2006) argue the universal nature of Q(r) results from violent
relaxation.
The works of Wechsler et al. (2002); Zhao et al. (2003a,b, 2009) have shown
there are two main eras of halo growth, a fast accretion phase and a slow phase.
The fast growth phase in CDM is dominated by mergers of objects with similar
mass in contrast to the slow growth phase characterized by quiescent accretion and
103
minor mergers. The inner halo is set at the end of the fast era with the slow growth
phase having little impact on the inner structure and gravitational potential well,
leading to an inside-out growth of halos. These studies find violent relaxation is
only important in forming the inner profile with the outer profile determined by
secondary infall during the slow growth phase.
The NFW profile is characterized by a logarithmic slope, γ ≡ d log ρ/d log r,
that rolls from an asymptotic value γ = −3 at large radii to γ = −1 in the inner halo.
The value of the inner slope has been a matter of controversy. The first concerned
the value of the asymptotic slope (Moore et al. 1999b). As the number of particles in
simulations have increased it has become evident the density profiles do not approach
an asymptotic value in the center but continue to roll slowly with radius (Navarro
et al. 2004; Diemand et al. 2004; Graham et al. 2006) and are better described by
Einasto profiles (Einasto 1965). However, this has not changed the conclusion that
all information about the formation history is lost in the virialization process.
The second controversy is a possible dependence of the inner profile on the
halo mass. Many models have been constructed that explain the emergence of the
universal profile as a consequence of repeated mergers (Syer and White 1998; Nusser
and Sheth 1999; Subramanian et al. 2000; Dekel et al. 2003). Although they differ
in the details, the relevant physical processes determining the halo properties are
the tidal stripping of material from accreting subhalos, dynamical friction, and tidal
compression transferring energy from the satellites to the halo particles and the
decaying of satellite orbits to the halo center. These models predict a dependence of
the inner density profile slope on the slope of the power spectrum at the scale of the
104
halo, P (k) ∝ k n . The steeper spectrum characteristic of dwarf-scales is predicted
to produce softer cores than for galactic and cluster-scale halos. Independent of the
merger models, Del Popolo (2010) questions the universality of both the density
and the Q profile and concludes both should depend on mass. His spherical infall
models with angular momentum show a steepening of the inner density profile with
increasing halo mass, although to a lesser extent than the merger models.
The heart of the issue is, do halos in equilibrium retain any memory of the
initial conditions and mass function of accreting satellites they are built from or is
all information lost in the virialization process?
Ricotti (2003) ran CDM simulations of the same realization of the density field
in boxes of varying side length to compare the profiles at different mass scales. He
examined the average profiles when the box structures showed similar clustering and
the most massive halos were composed of the same number of particles. He found a
systematic dependence of the inner slope on halo mass with dwarf-scale halos having
softer cores than galactic and cluster-scale halos in agreement with the predictions
of Subramanian et al. (2000). These results were reinforced in Ricotti et al. (2007).
Jing and Suto (2000) also saw a dependence of inner slope in their simulations of
halos at galactic and cluster scales.
Another way of testing the importance of substructure is by introducing a
truncation in the power spectrum as in hot and warm dark matter cosmologies
where substructure is suppressed below the particle free-streaming scale and halos
form by monolithic collapse. Many investigations using these cosmologies have been
conducted (e.g. Wang and White 2009, Huss et al. 1999a, Moore et al. 1999a,
105
Colı́n et al. 2000, Busha et al. 2007, Bode et al. 2001). These works find halos that
form below the truncation scale have lower concentrations consistent with the later
formation epochs of these halos but the profiles are well described by the NFW form.
This is in contrast to Colı́n et al. (2008) who find the profiles are systematically
different in their WDM simulations of Galaxy-sized halos. They find profiles are
steeper in the inner regions and the densities greater than the best fitting NFW
profile. Williams et al. (2004) and Viñas et al. (2012) have independently modeled
the density profiles of WDM halos and predict a flattening of the inner profile due
to the truncated power spectrum.
In this work I employ N-body simulations of halo formation in CDM and
WDM cosmologies to explore the effects of the power spectrum on halo structure
and dynamics. I use the method of Ricotti (2003) of scaling the simulation volume
to change the mass scale. However, I do not study a statistical sample of halos but
focus on one halo with about a factor of 100 greater mass resolution than in Ricotti
(2003). The goal is not to rigorously test any particular halo model but simply to
look for evidence the halo retains some memory of the initial power spectrum. This
evidence is expected to manifest itself as trends in the halo profiles as the mass
and truncation scales change. I examine if my results are typical of a larger halo
population and investigate the physical origin of my results.
106
5.2 Numerical Simulations
5.2.1 Cosmological Models
WDM particles are relativistic in the early universe and free-stream out of
overdense regions before the adiabatic expansion of the universe reduces the particles to subrelativistic velocities. WDM thus damps density perturbations below
a characteristic scale that depends on the particle mass and acts as a filter on the
power spectrum of density perturbations. The power spectra for WDM cosmologies
is related to that for CDM by
2
PW DM (k) = PCDM TW
DM ,
(5.3)
where TW DM is the WDM transfer function. The transfer function given by Bode
et al. (2001) is used for thermal relic dark matter particles that were coupled to the
relativistic cosmic plasma at early times and achieved thermal equilibrium prior to
the time of their decoupling. The formula of Eisenstein and Hu (1998) is adopted
for the CDM power spectrum.
I define the WDM filtering mass as in Sommer-Larsen and Dolgov (2001),
Mf ≡
4π 4
Ωm ρc kf−3 ,
3
(5.4)
where ρc is the critical density and kf is a characteristic free-streaming wave number
2
defined where TW
DM = 0.5. For consistency with Sommer-Larsen and Dolgov (2001)
I also define the free-streaming, or filtering length as Rf ≡ 0.46kf−1 .
I adopted values for cosmological parameters from the Bolshoi simulation
107
(Klypin et al. 2011), (Ωm , ΩΛ , Ωb , h, σ8 , ns ) = (0.27, 0.73, 0.0469, 0.7, 0.82, 0.95),
which were chosen to be within 1σ of WMAP5, WMAP7, and consistent with the
results of supernovae, and X-ray cluster surveys. These parameters are also within
1.7σ of WMAP9 and 2.2σ of Planck1. I use a variety of WDM models for thermal
relics in the range 0.75 − 15 keV. Figure 5.1 shows the WDM transfer functions for
the cosmological parameters adopted in this work with the filtering masses indicated
by the colored ticks across the top of the plot.
Since my focus is to examine the effects of the power spectrum on halo structure, the initial conditions include particle velocities due to the gravitational potential using the Zeldovich approximation but random thermal velocities appropriate
for WDM have not been added to the simulation particles. For the WDM cosmologies adopted here the effects of thermal velocities are expected to be small; this is
discussed further in Section 5.6.
108
Figure 5.1: WDM transfer functions used in the simulations. The filtering masses, Mf , are marked along the top.
109
5.2.2 Software
The simulations were conducted with the N-body cosmological simulation code
GADGET-2 (Springel 2005) with gravitational physics only and initial conditions
generated with the GRAFIC2 software package (Bertschinger 2001). I produced a
single realization of the density field but varied the power spectrum of fluctuations
appropriate for CDM and WDM cosmologies.
The AMIGA’s Halo Finder (AHF) software (Knollmann and Knebe 2009)
was used to identify gravitationally bound halos and calculate their properties after
iteratively removing unbound particles. The virial mass of a halo is defined in
Equation 5.2. Since the simulations are confined to high redshifts (z > 4) the
universe is matter dominated at all epochs and I adopt the virial condition for an
Einstein-de Sitter cosmology, ∆(z) = 178. The MergerTree tool in AHF was used
to construct merger trees, identify halo progenitors at all times, and for identifying
halos across cosmologies.
AHF calculates the convergence radius according to the criterion of Power et al.
(2003) and is generally about 10 softening lengths, enclosing ∼ 2000 particles at r ∼
0.006Rvir . I tested this by running low resolution simulations and found the profiles
are actually converged to about 5 − 6 softening lengths, enclosing ∼ 200 particles at
r ∼ 0.003Rvir . The convergence radius given by AHF may be overly conservative
for the simulations but this has no impact on the results. When examining the halo
profiles I adopt the convention of plotting r > 6ǫ but I indicate in bold where the
profiles satisfy the criterion of Power et al.
110
5.2.3 Simulations
I started at small scales by simulating a small cubic box with a comoving side
length of 3.3 Mpc from z = 79 to z = 8 with 5123 particles and mass resolution
∼ 104 M⊙ . A halo of mass ∼ 2 × 108 M⊙ that appeared to have an early formation
epoch and relaxed to virial equilibrium at scale factor a = 0.1 was chosen for resimulation using a zoom technique. I refer to this as “Halo A.” A volume of higher
mass resolution was generated in the initial conditions covering the initial volume of
particles that end within three virial radii of Halo A. I ran high resolution simulations with the mass resolution increased a factor of 83 in CDM and multiple WDM
cosmologies in the range 4 − 15 keV. To test the convergence of the results I also
ran low resolution simulations with the mass refinement reduced to a factor of 43 in
CDM and 6 keV WDM. I further tested the dependence of the results on the initial
conditions by running low resolution tests in CDM and 6 keV starting from z = 120.
I ran additional simulations at medium and large mass scales by increasing the
box size to medium and large side lengths of 7.0 and 22.4 Mpc which increased the
mass scale a factor of 10 and 320, respectively. CDM and WDM cosmologies ranging
from 2 − 5 keV were run for the medium mass scale while CDM and 0.75 − 2 keV
WDM were run for the large mass scale. To compare Halo A across mass scales
I define “normalization times” as the epochs when the CDM halos have grown to
encompass the same number of particles within the virial radius, N ∼ 107 , as the
small mass scale at a = 0.1. This occurred at a = 0.116 and a = 0.155 in the
medium and large mass scales, respectively. Figure 5.2 shows the growth of Halo
111
A at the three mass scales. Halo formation is delayed in the WDM cosmologies
but once it begins it grows quickly until it catches up with the CDM halo, after
which it evolves at a similar rate. The circles show the normalization times when
the halo has entered the slow growth phase and is composed of the same number
of particles in CDM at all mass scales. At the normalization times the halo masses
are approximately 2 × 108 M⊙ , 2 × 109 M⊙ , and 6 × 1010 M⊙ for the small, medium,
and large mass scales, respectively.
To explore if the results for Halo A were typical of halos in general I ran
additional sets of low resolution simulations with the refinement volume increased
a factor of 30 over the Halo A simulations. In these simulations the refinement
volume was cubic with a side length 1/4 the box length, consisted of 5123 particles,
and was centered on Halo A. Simulations were run for CDM and 1.1 keV at the
large mass scale and for CDM and 6 keV at the small scale. The 15 largest halos in
the refinement volume were examined in detail. Six halos were chosen for individual
resimulation at high resolution in an analogous way to Halo A, but only at large
mass scale and only for CDM and 2 keV thermal relic cosmologies that correspond
to the 7.1 keV sterile neutrino (Abazajian 2014) recently claimed to be indirectly
detected via an X-ray line at 3.55 keV (Bulbul et al. 2014; Boyarsky et al. 2014).
Table 5.1 gives a summary of the simulations conducted in this work. Listed
in the table are the WDM filtering mass and length, the simulation particle mass
in low and high resolution, and the force softening length ǫ. All lengths are given
in comoving units, in which the softening length was held constant. For the WDM
cosmologies the box side length Lbox is given in units of the filtering length. This
112
Figure 5.2: Mass growth of Halo A in the small, medium, and large box
simulations (top to bottom) in CDM and select WDM cosmologies. The
circles show the CDM halo at the normalization times when the halo has
grown to ∼ 107 particles at the three mass scales.
113
is a convenient way to show in which simulations the effects of the truncated power
spectrum will be similar across mass scales. The 2 keV simulation in the large
box is expected to be similar to the 5 keV medium box and 10 keV small box.
Likewise, the 1.1 keV large box will be similar to the 3 keV medium box and 6
keV small box. The 0.75 keV large box will be similar to the 2 keV medium box
and 4 keV small box. These simulations are color-coded in Figure 5.2. Another
way of characterizing the similarity of these simulations is by the ratio of filtering
mass to the virial mass of Halo A. For the similar cosmologies given above, the
filtering masses are approximately 7%, 40%, and 170% of the halo virial masses at
the normalization times.
Table 5.2 summarizes the properties of Halo A at the normalization times. An
examination of Figure 5.2 shows Halo A has not suffered a recent major merger and
is in the slow growth phase in all cosmologies at the normalization times. However,
a more rigorous examination of the halo relaxation state is desirable. Differences
from a universal profile are seen in unrelaxed halos and halos with large amounts
of substructure (Jing 2000). Additionally, the inner slope of the density profile is
sensitive to the location of the halo center. An artificial flattening of the profile could
be produced by an ambiguously defined center due to a recently arrived subhalo
at the core, for example. I performed a qualitative visual examination that the
halo centers determined by AHF correspond to the density peak of particles and I
examined quantitative measures of the relaxation. Neto et al. (2007); Macciò et al.
(2007, 2008) have studied large samples of halos and identified several metrics for
separating halos by relaxation: xof f , the offset between the halo center and center
114
Table 5.1: Properties of simulations.
Cosmo
Mf
[M⊙ ]
Rf
[kpc]
Lbox
mlow res
[M⊙ ]
mhigh res
[M⊙ ]
ǫ
[pc]
48,080
48,080
48,080
48,080
6,010
6,010
6,010
6,010
55
55
55
55
1,500
1,500
1,500
1,500
1,500
187
187
187
187
187
17
17
17
17
17
150
150
150
150
150
150
150
150
18.7
18.7
18.7
18.7
18.7
18.7
18.7
18.7
8
8
8
8
8
8
8
8
Large Box: 22 Mpc
CDM
2 keV
1.1 keV
0.75 keV
3.34 × 109
2.63 × 1010
9.84 × 1010
22394 kpc
40.8
548 Rf
81.2
276 Rf
126.2
177 Rf
Medium Box: 7 Mpc
CDM
5 keV
4 keV
3 keV
2 keV
1.41 × 108
3.05 × 108
8.24 × 108
3.34 × 109
14.2
18.4
25.6
40.8
7049 kpc
495 Rf
383 Rf
275 Rf
173 Rf
Small Box: 3 Mpc
CDM
15 keV
10 keV
7 keV
6 keV
5 keV
4.5 keV
4 keV
3.19 × 106
1.29 × 107
4.43 × 107
7.54 × 107
1.41 × 108
2.03 × 108
3.05 × 108
4.0
6.4
9.7
11.5
14.2
16.1
18.4
3270 kpc
812 Rf
510 Rf
338 Rf
283 Rf
230 Rf
203 Rf
178 Rf
of mass of particles within Rvir ; the virial ratio 2K/|U| − 1; and the spin parameter
λ′ from Bullock et al. (2001b) that characterizes the halo angular momentum:
λ′ = √
J
,
2Mvir vvir Rvir
(5.5)
where J is the total angular momentum of all particles within Rvir and vvir is the
circular velocity at Rvir , v 2 ≡ GM/R. These metrics are listed in Table 5.2. The
general conditions for a relaxed halo are: xof f < 0.1Rvir , λ′ < 0.1, and 2K/|U|−1 <
0.5. Halo A satisfies these criteria in all simulations.
115
Table 5.2: Properties of Halo A at the normalization times in high resolution simulations.
Cosmo
(1)
a
Mvir
[108 M⊙ ]
(3)
(2)
λ′
[10−2 ]
(4)
xof f
[Rvir ]
(5)
2K
|U |
−1
(6)
Halo A - Small Box
CDM
15 keV
10 keV
7 keV
6 keV
5 keV
4.5 keV
4 keV
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
1.868
1.888
1.887
1.811
1.713
1.507
1.330
1.074
4.21
4.24
4.63
4.88
4.80
4.30
3.93
3.16
0.06
0.05
0.09
0.08
0.07
0.06
0.06
0.06
0.41
0.42
0.42
0.40
0.40
0.39
0.39
0.41
0.06
0.06
0.07
0.07
0.05
0.39
0.41
0.41
0.40
0.40
0.04
0.08
0.07
0.04
0.39
0.40
0.39
0.38
Halo A - Medium Box
CDM
5 keV
4 keV
3 keV
2 keV
0.116
0.116
0.116
0.116
0.116
18.533
18.894
18.533
16.912
10.126
4.09
4.63
4.77
4.76
3.10
Halo A - Large Box
CDM
2 keV
1 keV
0.75 keV
0.155
0.155
0.155
0.155
600.497
604.681
547.367
351.273
116
3.34
4.07
4.44
3.48
5.3 Results I - Non-universality Of Profiles
I begin by examining the effects of the WDM power spectra on the density
structure of Halo A in the three boxes and thus the three mass scales of the halo.
I then examine the kinematics and conclude by checking the convergence. In Section 5.4 I examine a larger halo sample to check if the results of Halo A are typical
for halos in general.
5.3.1 Density Structure
Figure 5.3 shows the spherically averaged density profiles of Halo A at a = 0.1
in all cosmologies of the small mass simulations, M = 2 × 108 M⊙ . The profiles are
plotted with solid lines where they satisfy the convergence criterion of Power et al.
(2003) and the inner profiles are extended to six force softening lengths with dotted
lines. At r > 0.4Rvir the profiles are consistent across cosmologies. In the range
0.1 − 0.4Rvir the WDM densities are below that of CDM and at r < 0.1Rvir the
WDM simulations have greater density than CDM. The location where the WDM
density begins to increase shows a correlation with filtering scale, moving to larger
radii as the cosmology gets warmer and the filtering scale larger.
This feature is more pronounced in the cumulative mass profiles shown in Figure 5.4. The enclosed mass is equivalent in CDM and WDM at > 0.5Rvir indicating
it is the mass in shells at 0.1 − 0.4Rvir that has been displaced to smaller radii in
the WDM simulations.
I next check for differences in the triaxiality of the halos. Figure 5.5 shows
117
Figure 5.3: Density profiles of the small mass simulations of Halo A at
the normalization time, a = 0.1. The densities have been multiplied
by r 2 to reduce the dynamic range. The radial coordinates have been
normalized to the virial radius in CDM and are plotted to 6ǫ. The WDM
profiles have been grouped and are plotted against the CDM profile for
clarity. All profiles are plotted with solid lines where they satisfy the
convergence criterion of Power et al. (2003). The dashed line gives the
asymptotic slope of the NFW profile at small radii. Clear deviations
from a universal shape are seen in the WDM profiles.
118
Figure 5.4: Cumulative mass profiles of the small mass simulations of
Halo A. The mass profiles have been multiplied by r −2 to reduce the dynamic range. WDM simulations have been grouped and plotted against
the CDM profile for clarity. Mass has been displaced from intermediate
radii to the core in the WDM simulations.
119
radial profiles of the axial ratios of Halo A in the small box simulations. The axial
ratios tend to become more spherical in the inner and outer regions at r < 0.02Rvir
and r > 0.2Rvir and less spherical in the intermediate regions as the cosmology
changes from cold to warm.
To compare the mass profiles of Halo A across the three mass scales I plot
in Figure 5.6 the profiles of enclosed number of simulation particles and normalize
the radial coordinates by the CDM virial radius for each mass scale. Interestingly,
variations as a function of halo mass are seen at r < 0.1Rvir in the CDM halos in
contrast to the WDM simulations where the profiles are nearly identical across the
explored range of halo masses. The enclosed mass in the CDM inner halo becomes
greater as the halo mass increases but when small scale structures are erased, as
in the WDM simulations, the profiles are insensitive to the halo mass. Angular
momentum sets the shape of the inner profile in the models of Del Popolo (2009)
where more massive halos are predicted to have less angular momentum resulting
in steeper profiles. It can be seen from Table 5.2 that the spin of the CDM halo
decreases as the mass scale increases, consistent with this idea.
The CDM halo spin parameter is 26% higher at the small mass scale compared
to the large while the WDM halos vary by <
∼ 10% which may be why the WDM
profiles are very similar. However, the WDM spin parameters are higher than the
CDM halos at all scales yet they have steeper profiles than CDM so this is not
the entire answer. The inner profiles are shallower in WDM in agreement with the
models of Williams et al. (2004) and Viñas et al. (2012). It is important to emphasize
that the differences between WDM and CDM density profiles diminish as the halo
120
Figure 5.5: Axial ratio profiles of the small mass simulations of Halo
A. The CDM and 4 keV (Mf = 1.7M) profiles are plotted in bold for
clarity. The halo is more spherical in the inner and outer regions and
less spherical in the intermediate regions in WDM.
121
mass increases due to the steepening of the CDM profile. This observation may
explain why previous works have not clearly identified the prominent features and
trends in the profile shapes found in this work.
In Figure 5.7 the logarithmic slope of the density profiles is compared across
the mass scales. The large mass CDM halo profile is seen to be steeper than the
medium and small mass halos for r < 0.3Rvir and reaches the NFW value of −1 at a
smaller radius (given by the short gray lines). However, the differences are less than
predicted by the model of undigested subhalo cores of Subramanian et al. (2000)
but in agreement with the predictions of Del Popolo (2010) whose models give an
inner slope mass dependence due to angular momentum.
Unlike the CDM profiles the slopes in the WDM cosmologies are nearly identical across mass scales. The WDM profiles are steeper than CDM for r < 0.1Rvir
and achieve −1 at smaller radii, although this scale moves outward as the filtering
scale gets larger. The inner profiles quickly become softer than CDM at r <
∼ 0.01Rvir
but none of the profiles in any cosmology show signs of approaching an asymptotic
value.
5.3.2 Internal Kinematics
Figure 5.8 shows the profiles of σ 3 for the small mass simulations of Halo A,
where σ is the local 3D velocity dispersion. Similar to the density profiles, the
dispersions are greater in the inner WDM halos compared to CDM and show a
correlation with the filtering scale, growing larger and extending to greater radii
122
Figure 5.6: Comparison of the cumulative mass profiles of Halo A at
the normalization times of the three mass scales when the halos have
grown to ∼ 107 particles. The small, medium, and large mass halos
are plotted with the solid, dotted, and dashed lines, respectively. The
profiles are given by the number of enclosed simulation particles and the
radial coordinates have been normalized by the CDM virial radii. The
WDM profiles are plotted against the small and large mass CDM profiles
for comparison.
123
Figure 5.7: Comparison of the slope of the density profiles of Halo A
across the three mass scales at the normalization times. The small,
medium, and large mass simulations are plotted with the solid, dotted,
and dashed lines, respectively. The radial coordinates have been normalized by the CDM halo virial radii. Short gray lines indicate where the
log slope is −1 in all cosmologies. The CDM halo is steeper at the large
mass scale than the small scale. The WDM halos are generally steeper
than the CDM except at r < 0.01Rvir .
124
Figure 5.8: Velocity dispersion profiles of the small mass simulations of
Halo A. WDM simulations have been grouped and plotted against the
CDM profile for clarity.
as the cosmology becomes warmer and the filtering scale increases. This can be
understood as a consequence of the increased mass in the WDM cores. As the mass
in the core grows the dispersion must get larger to stay in virial equilibrium against
the deeper potential well.
To examine the phase space density profiles of Halo A I adopt α = −1.875
and fit the phase space density profiles of each CDM halo to the form, Qf it =
Ar α . Schmidt et al. (2008) have questioned if Q is a true universal profile and find
125
other choices for the velocity dispersion, such as different weights to the radial and
tangential components, and different values of the parameter α can provide equally
good fits. I am interested in the differences between the CDM and WDM haloes
so the concern is merely adopting the same law for all cosmologies, not which law
provides the best fit. An examination of alternative phase space density definitions
did not change the results.
Simply for illustrative purposes, I show Q normalized to a power law fit Qf it .
I calculate the deviation, Q̃ = Qsim /Qf it , using the CDM Qf it for the WDM simulations. Figure 5.9 shows the deviations from power law for Halo A in all cosmologies
for all three mass scales. A maximum is seen in the phase space density deviation
in the inner regions of the WDM halos. Along the top axis of each plot are ticks
marking the location of 0.037Rf in each WDM cosmology. This scaling was empirically determined but marks the location of the peak remarkably well indicating the
deviations scale with the filtering scale.
Interestingly, a drop in the inner profile of the CDM halos is seen that becomes
more pronounced as the halo mass decreases. This also agrees with the models of
Del Popolo (2010) where he argued for a dependence of the Q profile on halo mass.
A useful metric of the particle orbits is the velocity anisotropy parameter given
by:
β(r) = 1 −
σθ2 + σφ2
,
2σr2
(5.6)
where σθ2 and σφ2 are the angular velocity dispersions and σr2 is the radial velocity
dispersion. For purely radial orbits, β = 1, while isotropic particle motions give
126
Figure 5.9: Deviations from power-law behavior in the phase space density profiles of the simulations of Halo A. The simulations are grouped by
mass scale: small, medium, and large from left to right, and by relation
of filtering mass to halo mass with the cosmology growing warmer from
top to bottom. Deviations from power-law are seen in the inner WDM
halos that reach a peak at ∼ 4% of the filtering length (colored ticks
along the top of each plot).
127
β = 0. At the halo outskirts β → 1 where freshly accreted material is still falling
inward while in the core of a relaxed halo β → 0. In practice the anisotropy
parameter is seldom exactly zero in the inner regions since simulated halos are
generally not spherically symmetric.
Figure 5.10 shows the velocity anisotropy profiles of Halo A in all simulations.
There is a radial bias in the CDM particle orbits at r > 0.1Rvir while particles
inside this scale are well isotropized. The WDM profiles are generally similar to
CDM, although in the warmest cosmologies (bottom row) the halos have radial bias
extending deeper into the inner halo than in the other cosmologies. This may not
be due to an incomplete isotropization of the particle velocities but simply to the
increased triaxiality seen at these radii (Figure 5.5). What is clear is the lack of any
feature at the location of the peak deviations in the Q profile making it apparent
the physical processes that created the increased mass in the WDM cores do not
leave an imprint on the isotropy of particle velocities after virialization.
5.3.3 Convergence Tests
Figure 5.11 shows the profiles of Halo A in the high and low resolution simulations of Halo A and the test simulations initiated from a higher redshift. Excellent
agreement is seen across resolutions in both CDM and WDM and from the simulations started from higher redshift. The convergence criterion of Power et al.
(2003) appears to be not only valid but perhaps overly conservative in measuring
the convergence radii of these simulations.
128
Figure 5.10: Velocity anisotropy profiles of the simulations of Halo A.
The simulations are grouped by mass scale: small, medium, and large
from left to right, and by relation of filtering mass to halo mass with the
cosmology growing warmer from top to bottom. No features are seen at
the peaks of the phase space density profile deviations marked by the
colored ticks.
129
Simulations with truncated power spectra are known to produce numerically
artificial small mass halos along the filaments of collapsed density perturbations
(Wang and White 2007) whose size and separation are dependent on the mass resolution. Figure 5.11 also shows the results are not due to these spurious halos since
a dependence on the mass resolution would be expected to reflect on the shape and
location of the features in the WDM profiles.
5.4 Results II - Testing Cosmic Variance
In this section I examine the effects of truncated power spectra on a larger
sample of halos. I examine halos in the simulations with increased refinement volume
around Halo A sampled with low mass resolution. These simulations were run for
the large and small mass scales with a similar ratio of the WDM filtering scale to
halo scale in both volumes. I compare the same halos across cosmologies and also
compare the same CDM halos at different mass scales.
The 15 largest halos are labeled A-O and their mass growth histories are
plotted in Figure 5.12. The dashed gray lines indicate the WDM filtering mass in
the large box. The largest halos after entering the slow growth phase (A-E and H)
have masses greater than the filtering mass. These halos are also where the WDM
mass catches up with the CDM mass while both halos are still in the fast growth
phase, with both halos growing at similar rates thereafter. Halos F, N, and O are
halos where formation begins later in WDM but the halo mass has just caught up
to the CDM by the end of the simulations. In the other halos WDM halo formation
130
Figure 5.11: Comparison of the high and low resolution small mass simulations of Halo A. The high and low resolution CDM simulations are
plotted in black and gray, respectively, the high and low 6 keV simulations in light and dark blue. The dashed lines are the low resolution 6
keV and CDM simulations started from a higher redshift. Profiles are
plotted with solid lines where they satisfy the convergence criterion of
Power et al. (2003) and are extended to 3ǫ with dotted lines. Consistent results are seen across simulations demonstrating the results are not
affected by the mass resolution or starting redshift.
131
was delayed too long to catch up to the CDM halo before the end of the fast growth
phase.
Figure 5.13 shows cumulative mass profiles for halos A-O at the normalization
times when these halos are composed of > 105 particles in CDM and > 3 × 104 in
WDM. This is about the same resolution as the simulations of Ricotti (2003) but the
volume sampled here is smaller and I examine the halo profiles individually rather
than averaging. Halos J, L, and M have virial ratios, 2K/|U| − 1 > 0.8, at these
times due to recent or ongoing merging. I consider the profiles of these halos to be
unreliable and focus the analysis on the other 12.
I first compare the CDM halos at large and small mass scales I note differences
in nine of the 12 halos: A-G, I, and N. The difference is largely in the inner profile,
< 0.1Rvir , with the larger halos being more dense. The exception is halo C where
the small halo is denser over most of the profile.
Comparing the WDM profiles to the CDM, increased mass is seen in the inner
WDM halos at one or both mass scales for six halos: A-E and H. Halo A is seen
to be the most extreme with 2.5 times more mass at the convergence radius at the
small mass scale while the others are less than a factor of 2. Halo F is unique in that
it has nearly identical inner profiles while in the rest the WDM profiles are below
the CDM.
The halos with dense cores in WDM are the ones where the halo mass catches
up to the CDM halo early then evolves at the same rate afterwards. This is consistent
with the view that if the WDM catches the CDM before the end of the fast growing
phase the overdense core forms. Furthermore, halos A-F and H have masses equal
132
Figure 5.12: Growth in mass for the 15 largest halos in the low resolution,
large refinement volume simulations at the small and large mass scales.
CDM simulations are plotted in black and WDM in blue where the ratio
of filtering scale to mass scale was the same for both simulation volumes.
Gray dashed lines indicate the WDM filtering mass at the large mass
scale.
133
Figure 5.13: Cumulative mass profiles for the 15 largest halos in the low
resolution, large refinement volume simulations at a = 0.1 and a = 0.155
at the small (thin lines) and large mass scales (thick lines), respectively.
The CDM simulations are plotted in black and the WDM in blue. For
comparing across mass scales the profiles are given in number of enclosed
simulation particles and the radial coordinates have been normalized by
the halo virial radii.
134
to or greater than the filtering mass before they enter the slow growth phase while
the other halos are well below the filtering scale when their growth begins to slow.
I examine this further by running individual high resolution simulations of
select halos in a similar manner as Halo A. Halos B-E were chosen for resimulation
because they show dense cores in WDM. Halo F was chosen because it shows nearly
identical profiles in CDM and WDM and because halo G was nearby such that one
refinement volume could sample both halos. I ran the simulations at the large mass
scale for CDM and 2 keV to reduce the filtering mass for halo G and guarantee
the WDM halo formation doesn’t start too late. Since these simulations were only
conducted at one mass scale I compare the halos at the end of the simulations,
a = 0.16, instead of the normalization time. The halos are composed of 2.4 − 10.8
million particles at this time and their properties are listed in Table 5.3. Figure 5.14
shows the enclosed mass profiles for halos A-G. The profile of halo F again shows
almost no difference in CDM and WDM. Curiously, halo F also has a significantly
higher spin than the other halos. Although halo C shows a very dense WDM core it
also has a large amount of substructure in CDM and hence may not be fully relaxed.
I judge halo C to be inconclusive and conclude five out of seven halos show increased
central mass in WDM. Note that the difference bewteen CDM and WDM is not as
evident at large mass scales as for the small scales.
135
Figure 5.14: Cumulative mass profiles of all seven halos in the high
resolution, large mass scale simulations at a = 0.16. The mass profiles
have been multiplied by r −2 to reduce the dynamic range and offset
downward in intervals of 0.5 dex for clarity.
136
Table 5.3: Properties of high resolution simulated halos A-G.
Cosmo
(1)
a
(2)
Mvir
[108 M⊙ ]
(3)
λ′
[10−2 ]
(4)
xof f
[Rvir ]
(5)
2K
|U |
−1
(6)
Halos A-G - Large Box
A CDM
A 2 keV
B CDM
B 2 keV
C CDM
C 2 keV
D CDM
D 2 keV
E CDM
E 2 keV
F CDM
F 2 keV
G CDM
G 2 keV
0.16
0.16
0.16
0.16
0.16
0.16
0.16
0.16
0.16
0.16
0.16
0.16
0.16
0.16
648.777
642.201
411.910
400.569
425.037
413.263
329.871
330.351
223.489
237.997
188.720
149.931
147.183
142.662
2.11
2.83
1.32
1.56
0.96
1.34
2.01
1.47
1.00
0.94
7.73
6.53
3.01
3.16
0.09
0.06
0.01
0.01
0.10
0.04
0.06
0.03
0.05
0.05
0.08
0.06
0.05
0.03
0.38
0.38
0.48
0.44
0.49
0.43
0.28
0.28
0.27
0.24
0.55
0.62
0.51
0.54
5.5 Origin of the Core
The simulations in this work have shown an increased mass in the inner 0.1Rvir
is common for halos near the filtering mass in WDM cosmologies. This region will
be referred to as the core although the scale where the halos diverge from their
CDM counterparts is dependent on the filtering scale. In this section I examine the
formation of the core, its stability, and investigate clues to its origin.
I begin by examining the core of Halo A. Here, I define the core radius for each
mass scale as 3.1% of the CDM halo’s virial radius at the normalization times. This
radius was held constant in proper length and the number of particles within the
137
core was calculated for all cosmologies at all times a halo progenitor was identified.
Figure 5.15 shows the growth in the number of core particles in all large mass scale
simulations (similar results are seen for the medium and small mass scales). To
show the average growth in the number of core particles I fit polynomials up to
seventh order to the simulation data and plot these as thick lines in Figure 5.15.
It is clear the core forms quickly in WDM while core growth is more gradual in
CDM. The number of core particles remain approximately constant after formation
in both CDM and WDM. The epoch of WDM core formation occurs shortly after
the halo virial mass catches up to the CDM halo and the accretion rate slows to the
CDM rate. For example, in the 1.1 keV (Mf ∼ 0.4M) large mass scale simulation
the core forms at a ∼ 0.11, while Figure 5.2 shows the accretion rate slows to about
the CDM rate at a = 0.1. I conclude the core forms as the fast growth phase ends
and the halo transitions to the slow accretion phase.
The stability of the core after its formation is examined next. Figure 5.16 shows
the deviations from power law in the phase-space density profiles of the medium mass
scale simulations of Halo A at multiple epochs between a = 0.105 − 0.121. This is
equivalent to a timescale of ∼ 1.4 × 108 yr. The mass within rprop = 0.5 kpc is
≈ 108 M⊙ , this timescale therefore spans > 6 dynamical times. The profiles show
a scatter of ∼ 20% but the scatter is not correlated with time demonstrating the
inner halo is stable over this timescale.
The accretion rate, Ṁ ≡ d log M/d log a, for these epochs is < 6 for the 3 keV
cosmology and < 4 in CDM, 5 keV, and 4 keV. The 2 keV halo enters the slow growth
phase later than the other cosmologies and Q̃ is only plotted at epochs a > 0.115
138
Figure 5.15: Evolution of the number of particles within the core in the
high resolution, large mass scale simulations of Halo A. The simulation
data is plotted as the thin lines and polynomial fits as thick lines. The
core radius was held fixed in proper length at 620 pc. The scale factors
have been normalized by anorm .
139
in Figure 5.16 when Ṁ < 6 for this simulation. Zhao et al. (2003b) found that halo
growth in the fast phase is M ∝ H −4 , and in the slow phase M ∝ H −1 . In the
matter dominated era the Hubble parameter depends on scale factor as H ∝ a−3/2 .
Therefore, the mass growth rate is Ṁ = 6 in the fast phase and Ṁ = 3/2 in the
slow phase. I found the Q profiles are stable after the accretion rate drops <
∼ 6,
which agrees with the core being formed at the end of the fast era. I conclude the
structural and dynamical features in the inner WDM profiles are stable in the slow
growth phase.
Having established when the core forms and shown it is stable after formation,
I proceed to look for clues to the physical processes responsible for its creation.
Figure 5.17 shows the core particles of Halo A in the small mass scale simulations
at a = 0.05, well before core formation. The images are centered on the particles’
center of mass. As the cosmology gets warmer the core particles become more
symmetrically distributed around the center of mass. As noted by Busha et al.
(2007), the filtered power spectra causes what were multiple clumps in CDM to
become one collapsing clump in WDM.
I also examine the core particles of the high resolution, large mass scale simulations of halos B-G. For these halos the core radius is defined as 5.7 kpc in comoving
units at a = 0.16. Figure 5.18 shows the core particles at a = 0.075. Halo F has
minimal differences between CDM and WDM profiles and also appears to have the
least differences between particle distributions. Its growth is least affected by the
power spectrum cutoff. Halo F was also seen to have a much higher spin parameter
than the other halos at the end of the simulation.
140
Figure 5.16: Q̃ profiles in proper radius for the medium mass scale simulations of Halo A for epochs a = 0.105 − 0.121. The scatter in the
profiles is not correlated with epoch.
141
Figure 5.17: Positions at a = 0.05 of core particles in all cosmologies of
the high resolution small mass scale simulations of Halo A. Images are
centered on the center of mass of core particles.
Figure 5.18: Positions at a = 0.075 of particles within a proper core
radius of 914 pc at a = 0.16 in all seven halos of the high resolution,
large mass scale simulations. CDM is shown in the top row, 2 keV WDM
in the bottom row. Images are centered on the center of mass.
142
The evidence points toward angular momentum playing an important part in
determining the structure of the core. The importance of angular momentum for
the shape of the inner profile has been emphasized by a number of studies (Huss
et al. 1999a,b; Hiotelis 2002; Ascasibar et al. 2004; Lu et al. 2006). Purely radial
orbits give a steep inner profile, ρ ∝ r −2.25 (Bertschinger 1985). As the amount of
angular momentum is increased particles remain closer to their maximum orbital
radii resulting in shallower density profiles. Angular momentum is dominated by
the tangential component of the velocity dispersions which are acquired dynamically
in both the CDM and WDM simulations since thermal velocities were not added
to the WDM particles. Interactions with substructure and the global tidal field
produce tangential components to the particle velocities. An alternative possibility
is radial orbit instability (Belokurov et al. 2008). I speculate that during collapse
the particles in CDM acquire more tangential velocity, either through interactions
with other subclumps or violent relaxation occurring within the subclumps, while in
WDM the particles collapse radially before acquiring tangential dispersions. After
collapse, the virialization process isotropizes the particle velocities equally well in
both CDM and WDM as seen from Figure 5.10. The higher accretion rates in the
WDM fast growth phase may also play a role in generating the core as seen in the
models of Lu et al. (2006).
143
5.6 Discussion
I found the inner structure of dark matter halos in cosmologies with truncated
power spectra may deviate from their profiles in CDM with mass moving from the
intermediate regions to the center. In this section I discuss how my work compares
to previous studies.
It is well established that in WDM cosmologies halos below the truncation
scale form later and have lower concentrations than CDM halos of similar size (AvilaReese et al. 2001; Bode et al. 2001; Knebe et al. 2002). The free parameter rs in the
concentration definition is frequently taken to be the radius where the logarithmic
slope is −2. From Figure 5.3 it is clear that rs and thus the concentration parameter
is minimally affected by the rearrangement of mass in the inner WDM halos, as
expected for halos above the filtering mass.
Investigations of Milky Way satellites in 1 − 4 keV WDM cosmologies have
shown the maximum circular velocity decreases and the radius where this occurs
increases for dwarf galaxy-sized halos (Lovell et al. 2012; Anderhalden et al. 2013;
Polisensky and Ricotti 2014). Figure 5.19 shows the circular velocity profiles of the
medium mass scale simulations of Halo A at the normalization time. It is clear the
increased mass at the core has not affected the maximum circular velocity or its
location to be in disagreement with the conclusions of other work.
My WDM halo profiles are similar to the profiles seen by Colı́n et al. (2008).
They simulated five galactic-sized halos in WDM and fit NFW profiles. They found
their halo profiles were steeper and denser in the inner region than NFW. However,
144
Figure 5.19: Circular velocity profiles of the medium mass scale simulations of Halo A. The WDM profiles have been grouped and plotted
against the CDM profile for clarity.
145
only one of their halos had a corresponding CDM simulation and was seen to have
less mass at the center than the CDM halo. This may be because for the transfer
function they used the filtering mass is Mf = 1.0 × 1013 M⊙ while their 5 halos
range 1.8 − 6.2 × 1012 M⊙ , well below the filtering mass. The simulations presented
in Section 5.4 show halos below the filtering mass have lower core densities than in
CDM.
My simulations also explain a feature seen in the HDM cluster simulations of
Wang and White (2009). They stacked the Q(r) profiles of their 20 most massive
halos in both CDM and HDM and see a flattening in the inner 0.05Rvir of the HDM
average profile (Figure 7 in Wang and White) similar to the flattening seen in the
warmest simulations of Halo A.
My results seem to be in conflict with the work of Busha et al. (2007). They
evolved their CDM and WDM simulations far into the future, until the scale factor a = 100. Past the current epoch (a = 1), the cosmological constant quickly
dominates the density of the universe (ΩΛ → 1) leading to exponential expansion.
Halo accretion and structure growth essentially cease at a ∼ 3. Thus, examining
halo properties in the far future guarantees the halos have ample time to relax into
their equilibrium states. Busha et al. find the average density profiles of halos
in both cosmologies are well fit by the NFW form for r > 0.05Rvir (although the
outer slope is steeper due to the inflating universe, as noted by Ricotti 2003). They
further examine the average density profiles for halos in mass ranges above, near,
and below the filtering mass. They find all profiles are well fit by the NFW form
for r > 0.05Rvir with only lower concentrations below the filtering mass. However,
146
there are two things that complicate comparison of their simulations to mine. First
is the difficulty due to the different epochs the halos are examined at. I examine
my halos shortly after the end of the fast accretion phase when the inner profile is
set but the outskirts are still growing while Busha et al. examine their halos well
after all halo growth has stopped and Rvir has reached a maximum. Therefore, the
effects I see in the inner halo will be at radii smaller than the convergence radius in
their halos, r < 0.05Rvir . Also, their box sizes are larger than mine and they used a
much greater filtering mass, 1.2 × 1014 M⊙ . In this work I have shown how even the
density profiles of CDM halos have a dependence on mass with smaller differences
between CDM and WDM profiles for larger halo masses, for a fixed ratio of the
filtering mass to halo mass.
Finally, I comment on the effects of adding thermal velocities appropriate to
the adopted WDM models to the simulation particles. Thermal WDM particles
decouple with a finite fine-grain phase space density that imposes an upper limit
on their density, resulting in soft cores in collapsed halos. The radius of this core
depends on the mass of the WDM particle and the mass of the halo (Hogan and
Dalcanton 2000). For the warmest cosmologies of Halo A the core radius is ∼
4 × 10−4 Rvir which agrees with the core sizes seen in the simulations of Macciò et al.
(2012). The thermal core would be about the size of the adopted softening lengths,
far below the scales where the WDM profiles deviate from the CDM profile.
147
5.7 Summary
I tested the claim that the virialization process erases all information about
the initial conditions and produces universal mass density and phase space density
profiles in gravitationally collapsed dark matter halos. I simulated an isolated halo
with an early formation epoch at three mass scales in CDM and a variety of WDM
cosmologies where the formation of structures below the filtering scale is suppressed.
I examined the halo at epochs z > 5 when the halo was composed of ∼ 107 particles
at each mass scale and the halo was in the slow growth phase. I found the halos
were changed both structurally and dynamically. Mass was rearranged in the WDM
halos with radii < 0.1Rvir gaining mass at the expense of radii 0.1−0.4Rvir . Particle
velocity dispersions also increase in the inner profiles resulting in deviations from
power law behavior in the inner coarse-grain phase space density profiles. However,
velocity anisotropies after virialization are largely similar across cosmologies.
I also found a dependence on mass in the CDM profiles with larger halos exhibiting a steeper density profile as in Ricotti (2003). The spin parameter decreases
with increasing mass in agreement with the models of Del Popolo (2009) that more
massive halos have less angular momentum resulting in steeper profiles. The WDM
halos have similar spins across mass scales and also have similar profiles.
My work shows that the shape of halo profiles cannot be parameterized simply
by a generalized NFW or Einasto profile with a concentration or scale radius dependent on the mass or cosmology. The halo shape is more complex, with logarithmic
slopes that can vary non-monotonically and with features in the profile that reflect
148
the shape of the power spectrum. Thus, in general halos cannot be fitted by a universal density profile. This is actually good news because it may become feasibile
to find fingerprints of the initial power spectrum of perturbations on galactic or
sub-galactic scales in the density profiles of dark matter dominated dwarf galaxies
or clusters.
149
Chapter 6: Conclusion
I have shown how N-body simulations of the MW can be combined with observations of its satellite population to explore the nature of dark matter. Both the
number of satellites and their densities are sensitive to the small scale power spectrum. One way the power spectrum can be reduced is by WDM where streaming
motions in the early universe truncate the power below a scale dependent on the
mass of the dark matter particle. I have shown how a thermal relic mass of ∼ 2 keV
is able to reproduce the total number of satellites, including the ultra-faint dwarfs
discovered in the SDSS, and reconcile the observed stellar dynamics of the brightest
dwarfs. It is intriguing that this is also the value recently shown for sterile neutrinos
that would be consistent with the observed X-ray emission line from galaxy clusters.
For masses > 2 keV the densities of the largest dwarfs are more sensitive to the small
scale power determined by ns and σ8 than the nature of the dark matter.
If, however, the interpretation of the Fermi data is correct that the gamma ray
emissions from the Galactic Center are caused by annihilating CDM particles with
mass ∼ 30 − 100 GeV the MW satellites still provide an excellent method to probe
the power spectrum. This is important because while the Fermi observations can
probe the inner density profile of the MW they do not constrain the small scale power
150
spectrum of CDM. If the dark matter is cold but with a cutoff like a 2 keV thermal
relic the MW mass is > 300Mf and its density profile is indistinguishable from
standard CDM. Determining the power spectrum is important for testing theories
of CDM production and possible interactions with the photon-baryon plasma in the
early universe. Future surveys will ensure that small scale structures in the local
universe remain a powerful tool for probing the nature of dark matter by improving
sky coverage and luminosity limits and reducing uncertainties in the number of MW
satellites.
I have also shown how N-body simulations in CDM and WDM can be used to
demonstrate the dependence of the structural and dynamical profiles of dark matter
halos on the power spectrum of initial conditions. For halos near the filtering mass,
the inner density profile in WDM deviates from CDM with a mass increase up to
a factor of three in the inner halo. The scale of the deviation scales with the free
streaming length indicating some memory of the initial conditions is retained in
the halo core. A dependence on mass is also seen in the cold dark matter profiles
with more massive halos exhibiting steeper profiles. The increased core mass is at
the expense of matter at intermediate scales and supports analytic models of halo
structure formation that include angular momentum and argue against a universal
form for the density profile.
151
6.1 Future Work
Extensive simulation and modeling work over the past two decades have shown
that reionization of the IGM at z ∼ 10 suppresses gas accretion and prevents galaxy
formation in dark matter halos with masses less than 108−9 M⊙ . Ricotti (2009)
proposed, however, that these minihalos may experience a late-phase of gas accretion
due to the increasing concentration of the dark matter halo and a decreasing IGM
temperature due to helium reionization at z < 3. In this scenario the accreted gas
for isolated minihalos in low density regions is expected to have very low metallicity
and is unlikely to form stars. The gravitational potential well of the minihalo core
grows deep enough for the ionized gas density to increase and allow a fraction of the
gas to recombine. Lyman-α emission quickly cools the neutral hydrogen and allows
the gas to condense isothermally. Core gas densities of 1 − 10 neutral hydrogen
atoms per cubic centimeter are predicted. These objects may be observable by their
21 cm emission with existing and future radio telescopes.
If these gas-rich minihalos exist they would be a unique probe of the power
spectrum at scales smaller than the ultra-faint dwarfs of the Milky Way. They would
further constrain the nature of dark matter and would address the question of what
is the minimum mass a galaxy can have, a fundamental unanswered question in
cosmology. They may also offer a unique probe of the thermal and metal enrichment
history of the IGM and of the ionizing background radiation field.
Gas-rich minihalos offer exciting prospects but simulations must first be conducted to test the predictions of the theoretical arguments. The likelihood of mini152
halo formation, their sensitivity to the properties of the IGM, and their detectability
must be quantified. I have started preliminary work on such simulations by identifying candidate minihalos in void regions of a cosmological volume and testing
methods to decrease the computation time without affecting the accuracy in the
refinement region. My preliminary simulations are limited by their low resolution
and lack the hydrodynamics to model the baryonic gas in the dark matter halos.
Ultimately, a mass dynamic range greater than 106 will be required to resolve the
parsec-scale cores of minihalos while also sampling the gravitational forces from the
mass distribution on Mpc scales.
I plan to pursue funding opportunities to continue this research. If funding can
be acquired I will build on my preliminary work by testing the performance of more
efficient N-body codes (GADGET3) and more robust initial condition generators
(MUSIC, Hahn and Abel 2011). I will also add gas particles with detailed heating
and cooling physics and run multiple simulations varying the free parameters of the
radiation background, temperature of the IGM, and metallicity of the accreted gas.
I will quantify the number, flux, and size distribution of minihalos and produce
synthetic maps for 21 cm observations.
153
Chapter A: Testing for bias in subhalo abundances from BBKS
In Chapter 3 the BBKS formula for the CDM transfer function was used when
generating the initial conditions for the high resolution simulations. This formula
assumes a baryon density of zero. Eisenstein and Hu (1998) calculated transfer
functions for CDM cosmologies that include baryon physics.
Plotted in Figure A.1 are the power spectrum from the fitting formula of
Eisenstein & Hu and the spectrum calculated with the LINGER software (using
Ωb = 0.04) normalized by BBKS. With Ωm = 0.238 a Milky Way-sized halo with
mass ∼ 2 × 1012 M⊙ would form from a spherical region with diameter 4.8 Mpc
(k = 0.28 h/Mpc); this is plotted along with the scale of the simulation box (90 Mpc)
as solid vertical lines. Dashed vertical lines show the cell size in the refinement region
of the low and high resolution simulations.
Figure A.1 shows that, for a fixed value of σ8 , BBKS underestimates power on
scales k <
∼ 0.1 but the power spectra are nearly identical for scales <
∼ 14 Mpc with
BBKS slightly power overabundant by ∼ 10%. The set C halos showed subhalo
abundance variations much greater than 10% and the BBKS power overabundance
is much less than the 30% (1σ) intrinsic scatter in subhalo abundance for MW-sized
halos adopted in Chapter 3.
154
Figure A.1: Comparison of CDM power spectra calculated from the fitting formula of Eisenstein and Hu (1998) (EH97) and from the LINGER
software normalized by BBKS. On scales k > 0.1 h/Mpc (< 14 Mpc) the
power spectra are nearly identical. The ‘MW’ vertical line is the diameter of a spherical region with density Ωm ρc enclosing a Milky Way-sized
mass 2 × 1012 M⊙ (∼ 5 Mpc). This scale is well within the range where
the power spectra are nearly equal.
155
Figure A.2: Subhalo velocity function comparison for CDM high resolution set B simulations using fitting formula from BBKS and Eisenstein &
Hu (thick lines) and the LINGER using set C simulations (thin lines).
(top) Subhalos within R100 and velocities normalized by vmax of their
host. The straight sloped line is the fitting formula from the Bolshoi
simulation. (bottom) Subhalos within R50 , normalized by v50 of their
host. The subhalo abundances between the BBKS and EH97 simulations are in good agreement and within the scatter of the set C halos.
156
To check if BBKS might affect the number of satellites, I reran the set B
high resolution CDM simulation using initial conditions generated from the formula
of Eisenstein & Hu. The panels of Figure A.2 compare the velocity functions of
satellites and show good agreement between the simulations and within the scatter
of the set C simulations. Based on this and the agreement between the BBKS and
set C simulations seen in Section 3.3, I conclude that the use of BBKS has not
introduced a systematic error into the results of Chapter 3.
157
Bibliography
K. Abazajian.
Linear cosmological structure limits on warm dark matter.
Phys. Rev. D, 73(6):063513–+, March 2006. doi: 10.1103/PhysRevD.73.063513.
K. Abazajian and S. M. Koushiappas. Constraints on sterile neutrino dark matter.
Phys. Rev. D, 74(2):023527–+, July 2006. doi: 10.1103/PhysRevD.74.023527.
K. N. Abazajian. Resonantly-Produced 7 keV Sterile Neutrino Dark Matter Models
and the Properties of Milky Way Satellites. ArXiv e-prints, March 2014.
K. N. Abazajian and M. Kaplinghat. Detection of a gamma-ray source in the
Galactic Center consistent with extended emission from dark matter annihilation
and concentrated astrophysical emission. Phys. Rev. D, 86(8):083511, October
2012. doi: 10.1103/PhysRevD.86.083511.
K. N. Abazajian, M. Markevitch, S. M. Koushiappas, and R. C. Hickox. Limits on
the radiative decay of sterile neutrino dark matter from the unresolved cosmic
and soft x-ray backgrounds. Phys. Rev. D, 75(6):063511–+, March 2007. doi:
10.1103/PhysRevD.75.063511.
K. N. Abazajian, N. Canac, S. Horiuchi, and M. Kaplinghat. Astrophysical and
Dark Matter Interpretations of Extended Gamma Ray Emission from the Galactic
Center. ArXiv e-prints, February 2014.
A. A. Aguilar-Arevalo, A. O. Bazarko, S. J. Brice, B. C. Brown, L. Bugel, J. Cao,
L. Coney, J. M. Conrad, D. C. Cox, A. Curioni, Z. Djurcic, D. A. Finley, B. T.
Fleming, R. Ford, F. G. Garcia, G. T. Garvey, C. Green, J. A. Green, T. L.
Hart, E. Hawker, R. Imlay, R. A. Johnson, P. Kasper, T. Katori, T. Kobilarcik, I. Kourbanis, S. Koutsoliotas, E. M. Laird, J. M. Link, Y. Liu, Y. Liu,
W. C. Louis, K. B. M. Mahn, W. Marsh, P. S. Martin, G. McGregor, W. Metcalf, P. D. Meyers, F. Mills, G. B. Mills, J. Monroe, C. D. Moore, R. H. Nelson,
P. Nienaber, S. Ouedraogo, R. B. Patterson, D. Perevalov, C. C. Polly, E. Prebys, J. L. Raaf, H. Ray, B. P. Roe, A. D. Russell, V. Sandberg, R. Schirato,
D. Schmitz, M. H. Shaevitz, F. C. Shoemaker, D. Smith, M. Sorel, P. Spentzouris, I. Stancu, R. J. Stefanski, M. Sung, H. A. Tanaka, R. Tayloe, M. Tzanov,
158
R. van de Water, M. O. Wascko, D. H. White, M. J. Wilking, H. J. Yang, G. P.
Zeller, and E. D. Zimmerman. Search for Electron Neutrino Appearance at
the ∆m2 ∼ 1 eV2 Scale. Phys. Rev. Lett., 98(23):231801–+, June 2007. doi:
10.1103/PhysRevLett.98.231801.
A. A. Aguilar-Arevalo, C. E. Anderson, A. O. Bazarko, S. J. Brice, B. C. Brown,
L. Bugel, J. Cao, L. Coney, J. M. Conrad, D. C. Cox, A. Curioni, Z. Djurcic,
D. A. Finley, B. T. Fleming, R. Ford, F. G. Garcia, G. T. Garvey, C. Green, J. A.
Green, T. L. Hart, E. Hawker, R. Imlay, R. A. Johnson, G. Karagiorgi, P. Kasper,
T. Katori, T. Kobilarcik, I. Kourbanis, S. Koutsoliotas, E. M. Laird, S. K. Linden,
J. M. Link, Y. Liu, Y. Liu, W. C. Louis, K. B. M. Mahn, W. Marsh, G. McGregor,
W. Metcalf, P. D. Meyers, F. Mills, G. B. Mills, J. Monroe, C. D. Moore, R. H.
Nelson, V. T. Nguyen, P. Nienaber, J. A. Nowak, S. Ouedraogo, R. B. Patterson,
D. Perevalov, C. C. Polly, E. Prebys, J. L. Raaf, H. Ray, B. P. Roe, A. D. Russell,
V. Sandberg, R. Schirato, D. Schmitz, M. H. Shaevitz, F. C. Shoemaker, D. Smith,
M. Sodeberg, M. Sorel, P. Spentzouris, I. Stancu, R. J. Stefanski, M. Sung, H. A.
Tanaka, R. Tayloe, M. Tzanov, R. van de Water, M. O. Wascko, D. H. White,
M. J. Wilking, H. J. Yang, G. P. Zeller, and E. D. Zimmerman. Unexplained
Excess of Electronlike Events from a 1-GeV Neutrino Beam. Phys. Rev. Lett.,
102(10):101802–+, March 2009. doi: 10.1103/PhysRevLett.102.101802.
A. A. Aguilar-Arevalo, C. E. Anderson, S. J. Brice, B. C. Brown, L. Bugel, J. M.
Conrad, R. Dharmapalan, Z. Djurcic, B. T. Fleming, R. Ford, F. G. Garcia, G. T.
Garvey, J. Mirabal, J. Grange, J. A. Green, R. Imlay, R. A. Johnson, G. Karagiorgi, T. Katori, T. Kobilarcik, S. K. Linden, W. C. Louis, K. B. M. Mahn,
W. Marsh, C. Mauger, W. Metcalf, G. B. Mills, C. D. Moore, J. Mousseau,
R. H. Nelson, V. Nguyen, P. Nienaber, J. A. Nowak, B. Osmanov, Z. Pavlovic,
D. Perevalov, C. C. Polly, H. Ray, B. P. Roe, A. D. Russell, R. Schirato, M. H.
Shaevitz, M. Sorel, J. Spitz, I. Stancu, R. J. Stefanski, R. Tayloe, M. Tzanov,
R. G. van de Water, M. O. Wascko, D. H. White, M. J. Wilking, G. P. Zeller, and
E. D. Zimmerman. Event Excess in the MiniBooNE Search for νµ → νe Oscillations. Phys. Rev. Lett., 105(18):181801–+, October 2010. doi: 10.1103/PhysRevLett.105.181801.
E. Akhmedov and T. Schwetz. MiniBooNE and LSND data: non-standard neutrino
interactions in a (3+1) scheme versus (3+2) oscillations. J. High Energy Physics,
10:115–+, October 2010. doi: 10.1007/JHEP10(2010)115.
D. Anderhalden, A. Schneider, A. V. Macciò, J. Diemand, and G. Bertone. Hints
on the nature of dark matter from the properties of Milky Way satellites. JCAP,
3:014, March 2013. doi: 10.1088/1475-7516/2013/03/014.
T. Asaka, M. Shaposhnikov, and A. Kusenko.
Opening a new window
for warm dark matter.
Phys. Rev. B, 638:401–406, July 2006.
doi:
10.1016/j.physletb.2006.05.067.
159
Y. Ascasibar, G. Yepes, S. Gottlöber, and V. Müller. On the physical origin
of dark matter density profiles. MNRAS, 352:1109–1120, August 2004. doi:
10.1111/j.1365-2966.2004.08005.x.
C. Athanassopoulos, L. B. Auerbach, D. A. Bauer, R. D. Bolton, B. Boyd, R. L.
Burman, D. O. Caldwell, I. Cohen, B. D. Dieterle, J. B. Donahue, A. M. Eisner,
A. Fazely, F. J. Federspiel, G. T. Garvey, M. Gray, and et al. Candidate events in
a search for ν̄µ → ν̄e oscillations. Phys. Rev. Lett., 75:2650–2653, October 1995.
doi: 10.1103/PhysRevLett.75.2650.
C. Athanassopoulos, L. B. Auerbach, R. L. Burman, I. Cohen, D. O. Caldwell, B. D.
Dieterle, J. B. Donahue, A. M. Eisner, A. Fazely, F. J. Federspiel, G. T. Garvey,
M. Gray, R. M. Gunasingha, R. Imlay, K. Johnston, H. J. Kim, W. C. Louis,
R. Majkic, J. Margulies, K. McIlhany, W. Metcalf, G. B. Mills, R. A. Reeder,
V. Sandberg, D. Smith, I. Stancu, W. Strossman, R. Tayloe, G. J. Vandalen,
W. Vernon, N. Wadia, J. Waltz, Y.-X. Wang, D. H. White, D. Works, Y. Xiao,
and S. Yellin. Evidence for νµ → νe Oscillations from the LSND Experiment at
the Los Alamos Meson Physics Facility. Phys. Rev. Lett., 77:3082–3085, October
1996. doi: 10.1103/PhysRevLett.77.3082.
C. Athanassopoulos, L. B. Auerbach, R. L. Burman, D. O. Caldwell, E. D. Church,
I. Cohen, J. B. Donahue, A. Fazely, F. J. Federspiel, G. T. Garvey, R. M. Gunasingha, R. Imlay, K. Johnston, H. J. Kim, W. C. Louis, R. Majkic, K. McIlhany, G. B. Mills, R. A. Reeder, V. Sandberg, D. Smith, I. Stancu, W. Strossman, R. Tayloe, G. J. Vandalen, W. Vernon, N. Wadia, J. Waltz, D. H. White,
D. Works, Y. Xiao, and S. Yellin. Results on νµ → νe Neutrino Oscillations
from the LSND Experiment. Phys. Rev. Lett., 81:1774–1777, August 1998a. doi:
10.1103/PhysRevLett.81.1774.
C. Athanassopoulos, L. B. Auerbach, R. L. Burman, D. O. Caldwell, E. D. Church,
I. Cohen, J. B. Donahue, A. Fazely, F. J. Federspiel, G. T. Garvey, R. M. Gunasingha, R. Imlay, K. Johnston, H. J. Kim, W. C. Louis, R. Majkic, K. McIlhany, W. Metcalf, G. B. Mills, R. A. Reeder, V. Sandberg, D. Smith, I. Stancu,
W. Strossman, R. Tayloe, G. J. Vandalen, W. Vernon, N. Wadia, J. Waltz, D. H.
White, D. Works, Y. Xiao, and S. Yellin. Results on νµ → νe oscillations from
pion decay in flight neutrinos. Phys. Rev. C, 58:2489–2511, October 1998b. doi:
10.1103/PhysRevC.58.2489.
C. G. Austin, L. L. R. Williams, E. I. Barnes, A. Babul, and J. J. Dalcanton.
Semianalytical Dark Matter Halos and the Jeans Equation. ApJ, 634:756–774,
November 2005. doi: 10.1086/497133.
V. Avila-Reese, P. Colı́n, O. Valenzuela, E. D’Onghia, and C. Firmani. Formation
and Structure of Halos in a Warm Dark Matter Cosmology. ApJ, 559:516–530,
October 2001. doi: 10.1086/322411.
J. M. Bardeen, J. R. Bond, N. Kaiser, and A. S. Szalay. The statistics of peaks of
Gaussian random fields. ApJ, 304:15–61, May 1986. doi: 10.1086/164143.
160
R. Barkana, Z. Haiman, and J. P. Ostriker. Constraints on Warm Dark Matter from Cosmological Reionization. ApJ, 558:482–496, September 2001. doi:
10.1086/322393.
E. I. Barnes, L. L. R. Williams, A. Babul, and J. J. Dalcanton. Density Profiles of
Collisionless Equilibria. I. Spherical Isotropic Systems. ApJ, 643:797–803, June
2006. doi: 10.1086/503025.
V. Belokurov, D. B. Zucker, N. W. Evans, J. T. Kleyna, S. Koposov, S. T. Hodgkin,
M. J. Irwin, G. Gilmore, M. I. Wilkinson, M. Fellhauer, D. M. Bramich, P. C.
Hewett, S. Vidrih, J. T. A. De Jong, J. A. Smith, H.-W. Rix, E. F. Bell, R. F. G.
Wyse, H. J. Newberg, P. A. Mayeur, B. Yanny, C. M. Rockosi, O. Y. Gnedin,
D. P. Schneider, T. C. Beers, J. C. Barentine, H. Brewington, J. Brinkmann,
M. Harvanek, S. J. Kleinman, J. Krzesinski, D. Long, A. Nitta, and S. A. Snedden.
Cats and Dogs, Hair and a Hero: A Quintet of New Milky Way Companions. ApJ,
654:897–906, January 2007. doi: 10.1086/509718.
V. Belokurov, M. G. Walker, N. W. Evans, D. C. Faria, G. Gilmore, M. J. Irwin,
S. Koposov, M. Mateo, E. Olszewski, and D. B. Zucker. Leo V: A Companion of
a Companion of the Milky Way Galaxy? ApJ, 686:L83–L86, October 2008. doi:
10.1086/592962.
V. Belokurov, M. G. Walker, N. W. Evans, G. Gilmore, M. J. Irwin, M. Mateo,
L. Mayer, E. Olszewski, J. Bechtold, and T. Pickering. The discovery of Segue 2:
a prototype of the population of satellites of satellites. MNRAS, 397:1748–1755,
August 2009. doi: 10.1111/j.1365-2966.2009.15106.x.
V. Belokurov, M. G. Walker, N. W. Evans, G. Gilmore, M. J. Irwin, D. Just,
S. Koposov, M. Mateo, E. Olszewski, L. Watkins, and L. Wyrzykowski. Big
Fish, Little Fish: Two New Ultra-faint Satellites of the Milky Way. ApJ, 712:
L103–L106, March 2010. doi: 10.1088/2041-8205/712/1/L103.
K. Belotsky, M. Khlopov, and K. Shibaev. Stable Matter of 4th Generation: Hidden
in the Universe and Close to Detection? In A. Studenikin, editor, Particle Physics
at the Year of 250th Anniversary of Moscow University, pages 180–+, 2006a.
K. M. Belotsky, M. Y. Khlopov, and K. I. Shibaev. Composite dark matter and its
charged constituents. Gravitation and Cosmology, 12:93–99, June 2006b.
E. Bertschinger. Self-similar secondary infall and accretion in an Einstein-de Sitter
universe. ApJS, 58:39–65, May 1985. doi: 10.1086/191028.
E. Bertschinger. Multiscale Gaussian Random Fields and Their Application to
Cosmological Simulations. ApJS, 137:1–20, November 2001. doi: 10.1086/322526.
BICEP2 Collaboration, P. A. R Ade, R. W. Aikin, D. Barkats, S. J. Benton, C. A.
Bischoff, J. J. Bock, J. A. Brevik, I. Buder, E. Bullock, C. D. Dowell, L. Duband,
J. P. Filippini, S. Fliescher, S. R. Golwala, M. Halpern, M. Hasselfield, S. R.
161
Hildebrandt, G. C. Hilton, V. V. Hristov, K. D. Irwin, K. S. Karkare, J. P.
Kaufman, B. G. Keating, S. A. Kernasovskiy, J. M. Kovac, C. L. Kuo, E. M.
Leitch, M. Lueker, P. Mason, C. B. Netterfield, H. T. Nguyen, R. O’Brient, R. W.
Ogburn, IV, A. Orlando, C. Pryke, C. D. Reintsema, S. Richter, R. Schwarz, C. D.
Sheehy, Z. K. Staniszewski, R. V. Sudiwala, G. P. Teply, J. E. Tolan, A. D. Turner,
A. G. Vieregg, C. L. Wong, and K. W. Yoon. BICEP2 I: Detection Of B-mode
Polarization at Degree Angular Scales. ArXiv e-prints, March 2014.
G. R. Blumenthal, H. Pagels, and J. R. Primack. Galaxy formation by dissipationless particles heavier than neutrinos. Nature, 299:37, September 1982. doi:
10.1038/299037a0.
P. Bode, J. P. Ostriker, and N. Turok. Halo Formation in Warm Dark Matter
Models. ApJ, 556:93–107, July 2001. doi: 10.1086/321541.
H. Böhringer. Clusters of Galaxies. In H. Böhringer, G. E. Morfill, and J. E.
Trümper, editors, Seventeeth Texas Symposium on Relativistic Astrophysics and
Cosmology, volume 759 of Annals of the New York Academy of Sciences, page 67,
1995. doi: 10.1111/j.1749-6632.1995.tb17517.x.
J. R. Bond, S. Cole, G. Efstathiou, and N. Kaiser. Excursion set mass functions
for hierarchical Gaussian fluctuations. ApJ, 379:440–460, October 1991. doi:
10.1086/170520.
M. S. Bovill and M. Ricotti. Pre-Reionization Fossils, Ultra-Faint Dwarfs, and
the Missing Galactic Satellite Problem. ApJ, 693:1859–1870, March 2009a. doi:
10.1088/0004-637X/693/2/1859.
M. S. Bovill and M. Ricotti. Pre-Reionization Fossils, Ultra-Faint Dwarfs, and
the Missing Galactic Satellite Problem. ApJ, 693:1859–1870, March 2009b. doi:
10.1088/0004-637X/693/2/1859.
M. S. Bovill and M. Ricotti. Where are the Fossils of the First Galaxies? I. Local
Volume Maps and Properties of the Undetected Dwarfs. ApJ, 741:17, November
2011a. doi: 10.1088/0004-637X/741/1/17.
M. S. Bovill and M. Ricotti. Where are the Fossils of the First Galaxies? II. True
Fossils, Ghost Halos, and the Missing Bright Satellites. ApJ, 741:18, November
2011b. doi: 10.1088/0004-637X/741/1/18.
A. Boyarsky, A. Neronov, O. Ruchayskiy, and M. Shaposhnikov. Restrictions on
parameters of sterile neutrino dark matter from observations of galaxy clusters. Phys. Rev. D, 74(10):103506–+, November 2006a. doi: 10.1103/PhysRevD.74.103506.
A. Boyarsky, A. Neronov, O. Ruchayskiy, and M. Shaposhnikov. Constraints on
sterile neutrinos as dark matter candidates from the diffuse X-ray background.
MNRAS, 370:213–218, July 2006b. doi: 10.1111/j.1365-2966.2006.10458.x.
162
A. Boyarsky, J. Nevalainen, and O. Ruchayskiy. Constraints on the parameters of
radiatively decaying dark matter from the dark matter halos of the Milky Way and
Ursa Minor. A&A, 471:51–57, August 2007. doi: 10.1051/0004-6361:20066774.
A. Boyarsky, O. Ruchayskiy, and M. Markevitch. Constraints on Parameters of
Radiatively Decaying Dark Matter from the Galaxy Cluster 1E 0657-56. ApJ,
673:752–757, February 2008. doi: 10.1086/524397.
A. Boyarsky, J. Lesgourgues, O. Ruchayskiy, and M. Viel. Realistic Sterile Neutrino Dark Matter with KeV Mass does not Contradict Cosmological
Bounds. Phys. Rev. Lett., 102(20):201304–+, May 2009. doi: 10.1103/PhysRevLett.102.201304.
A. Boyarsky, O. Ruchayskiy, D. Iakubovskyi, and J. Franse. An unidentified line
in X-ray spectra of the Andromeda galaxy and Perseus galaxy cluster. ArXiv
e-prints, February 2014.
M. Boylan-Kolchin, J. S. Bullock, and M. Kaplinghat. Too big to fail? The puzzling
darkness of massive Milky Way subhaloes. MNRAS, 415:L40–L44, July 2011. doi:
10.1111/j.1745-3933.2011.01074.x.
M. Boylan-Kolchin, J. S. Bullock, and M. Kaplinghat. The Milky Way’s bright
satellites as an apparent failure of ΛCDM. MNRAS, 422:1203–1218, May 2012a.
doi: 10.1111/j.1365-2966.2012.20695.x.
M. Boylan-Kolchin, J. S. Bullock, S. T. Sohn, G. Besla, and R. P. van der Marel.
The Space Motion of Leo I: The Mass of the Milky Way’s Dark Matter Halo.
ArXiv e-prints, October 2012b.
G. L. Bryan and M. L. Norman. Statistical Properties of X-Ray Clusters: Analytic
and Numerical Comparisons. ApJ, 495:80–+, March 1998. doi: 10.1086/305262.
E. Bulbul, M. Markevitch, A. Foster, R. K. Smith, M. Loewenstein, and S. W. Randall. Detection of An Unidentified Emission Line in the Stacked X-ray spectrum
of Galaxy Clusters. ArXiv e-prints, February 2014.
J. S. Bullock, A. V. Kravtsov, and D. H. Weinberg. Reionization and the Abundance
of Galactic Satellites. ApJ, 539:517–521, August 2000. doi: 10.1086/309279.
J. S. Bullock, T. S. Kolatt, Y. Sigad, R. S. Somerville, A. V. Kravtsov, A. A.
Klypin, J. R. Primack, and A. Dekel. Profiles of dark haloes: evolution, scatter
and environment. MNRAS, 321:559–575, March 2001a. doi: 10.1046/j.13658711.2001.04068.x.
J. S. Bullock, A. V. Kravtsov, and D. H. Weinberg. Hierarchical Galaxy Formation
and Substructure in the Galaxy’s Stellar Halo. ApJ, 548:33–46, February 2001b.
doi: 10.1086/318681.
163
M. T. Busha, A. E. Evrard, and F. C. Adams. The Asymptotic Form of Cosmic
Structure: Small-Scale Power and Accretion History. ApJ, 665:1–13, August 2007.
doi: 10.1086/518764.
F. J. Castander. The Sloan Digital Sky Survey. Ap&SS, 263:91–94, June 1998. doi:
10.1023/A:1002196414003.
D. Clowe, M. Bradač, A. H. Gonzalez, M. Markevitch, S. W. Randall, C. Jones, and
D. Zaritsky. A Direct Empirical Proof of the Existence of Dark Matter. ApJ,
648:L109–L113, September 2006. doi: 10.1086/508162.
P. Colı́n, V. Avila-Reese, and O. Valenzuela. Substructure and Halo Density Profiles in a Warm Dark Matter Cosmology. ApJ, 542:622–630, October 2000. doi:
10.1086/317057.
P. Colı́n, O. Valenzuela, and V. Avila-Reese. On the Structure of Dark Matter Halos
at the Damping Scale of the Power Spectrum with and without Relict Velocities.
ApJ, 673:203–214, January 2008. doi: 10.1086/524030.
R. J. Cooke, M. Pettini, R. A. Jorgenson, M. T. Murphy, and C. C. Steidel. Precision
Measures of the Primordial Abundance of Deuterium. ApJ, 781:31, January 2014.
doi: 10.1088/0004-637X/781/1/31.
T. Daylan, D. P. Finkbeiner, D. Hooper, T. Linden, S. K. N. Portillo, N. L. Rodd,
and T. R. Slatyer. The Characterization of the Gamma-Ray Signal from the
Central Milky Way: A Compelling Case for Annihilating Dark Matter. ArXiv
e-prints, February 2014.
J. T. A. de Jong, N. F. Martin, H.-W. Rix, K. W. Smith, S. Jin, and A. V. Maccio’.
The enigmatic pair of dwarf galaxies Leo IV and Leo V: coincidence or common
origin? ArXiv e-prints, December 2009.
G. De Lucia, G. Kauffmann, V. Springel, S. D. M. White, B. Lanzoni, F. Stoehr,
G. Tormen, and N. Yoshida. Substructures in cold dark matter haloes. MNRAS,
348:333–344, February 2004. doi: 10.1111/j.1365-2966.2004.07372.x.
A. Dekel, J. Devor, and G. Hetzroni. Galactic halo cusp-core: tidal compression in mergers. MNRAS, 341:326–342, May 2003. doi: 10.1046/j.13658711.2003.06432.x.
A. Del Popolo. The Cusp/Core Problem and the Secondary Infall Model. ApJ, 698:
2093–2113, June 2009. doi: 10.1088/0004-637X/698/2/2093.
A. Del Popolo. On the universality of density profiles. MNRAS, 408:1808–1817,
November 2010. doi: 10.1111/j.1365-2966.2010.17288.x.
A. di Cintio, A. Knebe, N. I. Libeskind, G. Yepes, S. Gottlöber, and Y. Hoffman. Too
small to succeed? Lighting up massive dark matter subhaloes of the Milky Way.
MNRAS, 417:L74–L78, October 2011. doi: 10.1111/j.1745-3933.2011.01123.x.
164
A. Di Cintio, A. Knebe, N. I. Libeskind, C. Brook, G. Yepes, S. Gottlöber, and
Y. Hoffman. Size matters: the non-universal density profile of subhaloes in SPH
simulations and implications for the Milky Way’s dSphs. MNRAS, March 2013.
doi: 10.1093/mnras/stt240.
J. Diemand, B. Moore, and J. Stadel. Convergence and scatter of cluster density profiles. MNRAS, 353:624–632, September 2004. doi: 10.1111/j.13652966.2004.08094.x.
J. Diemand, M. Kuhlen, and P. Madau. Dark Matter Substructure and GammaRay Annihilation in the Milky Way Halo. ApJ, 657:262–270, March 2007. doi:
10.1086/510736.
J. Diemand, M. Kuhlen, P. Madau, M. Zemp, B. Moore, D. Potter, and J. Stadel.
Clumps and streams in the local dark matter distribution. Nature, 454:735–738,
August 2008. doi: 10.1038/nature07153.
S. Dodelson and L. M. Widrow. Sterile neutrinos as dark matter. Phys. Rev. Lett.,
72:17–20, January 1994. doi: 10.1103/PhysRevLett.72.17.
F. Donato, G. Gentile, and P. Salucci. Cores of dark matter haloes correlate with
stellar scalelengths. MNRAS, 353:L17–L22, September 2004. doi: 10.1111/j.13652966.2004.08220.x.
J. Dubinski, J. C. Mihos, and L. Hernquist. Using Tidal Tails to Probe Dark Matter
Halos. ApJ, 462:576, May 1996. doi: 10.1086/177174.
G. Efstathiou. Suppressing the formation of dwarf galaxies via photoionization.
MNRAS, 256:43P–47P, May 1992.
J. Einasto. On the Construction of a Composite Model for the Galaxy and on the
Determination of the System of Galactic Parameters. Trudy Astrofizicheskogo
Instituta Alma-Ata, 5:87–100, 1965.
D. J. Eisenstein and W. Hu. Baryonic Features in the Matter Transfer Function.
ApJ, 496:605–+, March 1998. doi: 10.1086/305424.
D. J. Eisenstein and P. Hut. HOP: A New Group-Finding Algorithm for N-Body
Simulations. ApJ, 498:137–+, May 1998. doi: 10.1086/305535.
V. R. Eke, S. Cole, and C. S. Frenk. Cluster evolution as a diagnostic for Omega.
MNRAS, 282:263–280, September 1996.
D. Fabricant, M. Lecar, and P. Gorenstein. X-ray measurements of the mass of M87.
ApJ, 241:552–560, October 1980. doi: 10.1086/158369.
D. P. Finkbeiner, M. Su, and C. Weniger. Is the 130 GeV line real? A search for systematics in the Fermi-LAT data. JCAP, 1:029, January 2013. doi: 10.1088/14757516/2013/01/029.
165
L. Gao, S. D. M. White, A. Jenkins, F. Stoehr, and V. Springel. The subhalo
populations of ΛCDM dark haloes. MNRAS, 355:819–834, December 2004. doi:
10.1111/j.1365-2966.2004.08360.x.
M. Geha, B. Willman, J. D. Simon, L. E. Strigari, E. N. Kirby, D. R. Law,
and J. Strader. The Least-Luminous Galaxy: Spectroscopy of the Milky Way
Satellite Segue 1. ApJ, 692:1464–1475, February 2009. doi: 10.1088/0004637X/692/2/1464.
G. Gentile, A. Burkert, P. Salucci, U. Klein, and F. Walter. The Dwarf Galaxy
DDO 47 as a Dark Matter Laboratory: Testing Cusps Hiding in Triaxial Halos.
ApJ, 634:L145–L148, December 2005. doi: 10.1086/498939.
G. Gentile, P. Salucci, U. Klein, and G. L. Granato. NGC 3741: the dark halo
profile from the most extended rotation curve. MNRAS, 375:199–212, February
2007. doi: 10.1111/j.1365-2966.2006.11283.x.
S. Ghigna, B. Moore, F. Governato, G. Lake, T. Quinn, and J. Stadel. Density Profiles and Substructure of Dark Matter Halos: Converging Results at Ultra-High
Numerical Resolution. ApJ, 544:616–628, December 2000. doi: 10.1086/317221.
C. Giocoli, G. Tormen, and F. C. van den Bosch. The population of dark matter
subhaloes: mass functions and average mass-loss rates. MNRAS, 386:2135–2144,
June 2008. doi: 10.1111/j.1365-2966.2008.13182.x.
S. Gninenko. A resolution of puzzles from the LSND, KARMEN, and MiniBooNE
experiments. ArXiv e-prints, September 2010.
S. N. Gninenko and D. S. Gorbunov. MiniBooNE anomaly, the decay Ds+ → µ+ νµ
and heavy sterile neutrino. Phys. Rev. D, 81(7):075013–+, April 2010. doi:
10.1103/PhysRevD.81.075013.
L. Goodenough and D. Hooper. Possible Evidence For Dark Matter Annihilation
In The Inner Milky Way From The Fermi Gamma Ray Space Telescope. ArXiv
e-prints, October 2009.
D. Gorbunov, A. Khmelnitsky, and V. Rubakov. Is gravitino still a warm dark
matter candidate? Journal of High Energy Physics, 12:55–+, December 2008.
doi: 10.1088/1126-6708/2008/12/055.
J. R. Gott, III. On the Formation of Elliptical Galaxies. ApJ, 201:296–310, October
1975. doi: 10.1086/153887.
A. W. Graham, D. Merritt, B. Moore, J. Diemand, and B. Terzić. Empirical Models
for Dark Matter Halos. II. Inner Profile Slopes, Dynamical Profiles, and ρ/σ 3 . AJ,
132:2701–2710, December 2006. doi: 10.1086/508990.
166
A. M. Green, S. Hofmann, and D. J. Schwarz. The power spectrum of SUSY-CDM on
subgalactic scales. MNRAS, 353:L23–L27, September 2004. doi: 10.1111/j.13652966.2004.08232.x.
J. E. Gunn and J. R. Gott, III. On the Infall of Matter Into Clusters of Galaxies and
Some Effects on Their Evolution. ApJ, 176:1, August 1972. doi: 10.1086/151605.
O. Hahn and T. Abel. Multi-scale initial conditions for cosmological simulations.
MNRAS, 415:2101–2121, August 2011. doi: 10.1111/j.1365-2966.2011.18820.x.
F. Hammer, M. Puech, L. Chemin, H. Flores, and M. D. Lehnert. The Milky Way,
an Exceptionally Quiet Galaxy: Implications for the Formation of Spiral Galaxies.
ApJ, 662:322–334, June 2007. doi: 10.1086/516727.
A. Hektor, M. Raidal, and E. Tempel. Evidence for Indirect Detection of Dark
Matter from Galaxy Clusters in Fermi γ-Ray Data. ApJ, 762:L22, January 2013.
doi: 10.1088/2041-8205/762/2/L22.
A. Helmi, S. D. White, and V. Springel. The phase-space structure of a dark-matter
halo: Implications for dark-matter direct detection experiments. Phys. Rev. D,
66(6):063502–+, September 2002. doi: 10.1103/PhysRevD.66.063502.
G. Hinshaw, D. Larson, E. Komatsu, D. N. Spergel, C. L. Bennett, J. Dunkley, M. R.
Nolta, M. Halpern, R. S. Hill, N. Odegard, L. Page, K. M. Smith, J. L. Weiland,
B. Gold, N. Jarosik, A. Kogut, M. Limon, S. S. Meyer, G. S. Tucker, E. Wollack,
and E. L. Wright. Nine-Year Wilkinson Microwave Anisotropy Probe (WMAP)
Observations: Cosmological Parameter Results. ArXiv e-prints, December 2012.
N. Hiotelis. Density profiles in a spherical infall model with non-radial motions.
A&A, 382:84–91, January 2002. doi: 10.1051/0004-6361:20011620.
C. J. Hogan and J. J. Dalcanton. New dark matter physics: Clues from halo
structure. Phys. Rev. D, 62(6):063511, September 2000. doi: 10.1103/PhysRevD.62.063511.
D. Hooper and L. Goodenough. Dark matter annihilation in the Galactic Center as
seen by the Fermi Gamma Ray Space Telescope. Physics Letters B, 697:412–428,
March 2011. doi: 10.1016/j.physletb.2011.02.029.
S. Horiuchi, P. J. Humphrey, J. Onorbe, K. N. Abazajian, M. Kaplinghat, and
S. Garrison-Kimmel. Sterile neutrino dark matter bounds from galaxies of the
Local Group. ArXiv e-prints, November 2013.
A. Huss, B. Jain, and M. Steinmetz. The formation and evolution of clusters of
galaxies in different cosmogonies. MNRAS, 308:1011–1031, October 1999a. doi:
10.1046/j.1365-8711.1999.02757.x.
A. Huss, B. Jain, and M. Steinmetz. How Universal Are the Density Profiles of
Dark Halos? ApJ, 517:64–69, May 1999b. doi: 10.1086/307161.
167
M. J. Irwin, V. Belokurov, N. W. Evans, E. V. Ryan-Weber, J. T. A. de Jong, S. Koposov, D. B. Zucker, S. T. Hodgkin, G. Gilmore, P. Prema, L. Hebb, A. Begum,
M. Fellhauer, P. C. Hewett, R. C. Kennicutt, Jr., M. I. Wilkinson, D. M. Bramich,
S. Vidrih, H.-W. Rix, T. C. Beers, J. C. Barentine, H. Brewington, M. Harvanek,
J. Krzesinski, D. Long, A. Nitta, and S. A. Snedden. Discovery of an Unusual
Dwarf Galaxy in the Outskirts of the Milky Way. ApJ, 656:L13–L16, February
2007. doi: 10.1086/512183.
T. Ishiyama, T. Fukushige, and J. Makino. Variation of the Subhalo Abundance
in Dark Matter Halos. ApJ, 696:2115–2125, May 2009. doi: 10.1088/0004637X/696/2/2115.
N. Jarosik, C. L. Bennett, J. Dunkley, B. Gold, M. R. Greason, M. Halpern, R. S.
Hill, G. Hinshaw, A. Kogut, E. Komatsu, D. Larson, M. Limon, S. S. Meyer,
M. R. Nolta, N. Odegard, L. Page, K. M. Smith, D. N. Spergel, G. S. Tucker,
J. L. Weiland, E. Wollack, and E. L. Wright. Seven-Year Wilkinson Microwave
Anisotropy Probe (WMAP) Observations: Sky Maps, Systematic Errors, and
Basic Results. ArXiv e-prints, January 2010.
J. H. Jeans. The Stability of a Spherical Nebula. Royal Society of London Philosophical Transactions Series A, 199:1–53, 1902. doi: 10.1098/rsta.1902.0012.
Y. P. Jing. The Density Profile of Equilibrium and Nonequilibrium Dark Matter
Halos. ApJ, 535:30–36, May 2000. doi: 10.1086/308809.
Y. P. Jing and Y. Suto. The Density Profiles of the Dark Matter Halo Are Not
Universal. ApJ, 529:L69–L72, February 2000. doi: 10.1086/312463.
M. Kamionkowski and A. R. Liddle. The Dearth of Halo Dwarf Galaxies: Is
There Power on Short Scales? Phys. Rev. Lett., 84:4525–4528, May 2000. doi:
10.1103/PhysRevLett.84.4525.
M. Kaplinghat. Dark matter from early decays. Phys. Rev. D, 72(6):063510–+,
September 2005. doi: 10.1103/PhysRevD.72.063510.
G. Karagiorgi, Z. Djurcic, J. M. Conrad, M. H. Shaevitz, and M. Sorel. Viability of
∆m2 ∼ 1eV 2 sterile neutrino mixing models in light of MiniBooNE electron neutrino and antineutrino data from the Booster and NuMI beamlines. Phys. Rev. D,
80(7):073001–+, October 2009. doi: 10.1103/PhysRevD.80.073001.
H. Katz and M. Ricotti. Two Epochs of Globular Cluster Formation from Deep
Fields Luminosity Functions: Implications for Reionization and the Milky Way
Satellites. ArXiv e-prints, November 2012.
M. Y. Khlopov. Composite dark matter from 4th generation. ArXiv Astrophysics
e-prints, November 2005.
M. Y. Khlopov. New symmetries in microphysics, new stable forms of matter around
us. ArXiv Astrophysics e-prints, July 2006.
168
M. Y. Khlopov. Composite dark matter from stable charged constituents. ArXiv
e-prints, June 2008.
M. Y. Khlopov and C. Kouvaris. Strong interactive massive particles from a strong
coupled theory. Phys. Rev. D, 77(6):065002–+, March 2008a. doi: 10.1103/PhysRevD.77.065002.
M. Y. Khlopov and C. Kouvaris. Composite dark matter from a model with composite Higgs boson. Phys. Rev. D, 78(6):065040–+, September 2008b. doi:
10.1103/PhysRevD.78.065040.
A. Klypin, A. V. Kravtsov, O. Valenzuela, and F. Prada. Where Are the Missing
Galactic Satellites? ApJ, 522:82–92, September 1999. doi: 10.1086/307643.
A. A. Klypin, S. Trujillo-Gomez, and J. Primack. Dark Matter Halos in the Standard
Cosmological Model: Results from the Bolshoi Simulation. ApJ, 740:102, October
2011. doi: 10.1088/0004-637X/740/2/102.
A. Knebe, J. E. G. Devriendt, A. Mahmood, and J. Silk. Merger histories in warm
dark matter structure formation scenarios. MNRAS, 329:813–828, February 2002.
doi: 10.1046/j.1365-8711.2002.05017.x.
A. Knebe, J. E. G. Devriendt, B. K. Gibson, and J. Silk. Top-down fragmentation
of a warm dark matter filament. MNRAS, 345:1285–1290, November 2003. doi:
10.1046/j.1365-2966.2003.07044.x.
S. R. Knollmann and A. Knebe. AHF: Amiga’s Halo Finder. ApJS, 182:608–624,
June 2009. doi: 10.1088/0067-0049/182/2/608.
J. A. Kollmeier, A. Gould, S. Shectman, I. B. Thompson, G. W. Preston, J. D.
Simon, J. D. Crane, Ž. Ivezić, and B. Sesar. Spectroscopic Confirmation of the
Pisces Overdensity. ApJ, 705:L158–L162, November 2009. doi: 10.1088/0004637X/705/2/L158.
E. Komatsu, J. Dunkley, M. R. Nolta, C. L. Bennett, B. Gold, G. Hinshaw,
N. Jarosik, D. Larson, M. Limon, L. Page, D. N. Spergel, M. Halpern, R. S.
Hill, A. Kogut, S. S. Meyer, G. S. Tucker, J. L. Weiland, E. Wollack, and E. L.
Wright. Five-Year Wilkinson Microwave Anisotropy Probe Observations: Cosmological Interpretation. ApJS, 180:330–376, February 2009. doi: 10.1088/00670049/180/2/330.
E. Komatsu, K. M. Smith, J. Dunkley, C. L. Bennett, B. Gold, G. Hinshaw,
N. Jarosik, D. Larson, M. R. Nolta, L. Page, D. N. Spergel, M. Halpern, R. S.
Hill, A. Kogut, M. Limon, S. S. Meyer, N. Odegard, G. S. Tucker, J. L. Weiland, E. Wollack, and E. L. Wright. Seven-year Wilkinson Microwave Anisotropy
Probe (WMAP) Observations: Cosmological Interpretation. ApJS, 192:18, February 2011. doi: 10.1088/0067-0049/192/2/18.
169
S. Koposov, V. Belokurov, N. W. Evans, P. C. Hewett, M. J. Irwin, G. Gilmore,
D. B. Zucker, H.-W. Rix, M. Fellhauer, E. F. Bell, and E. V. Glushkova. The
Luminosity Function of the Milky Way Satellites. ApJ, 686:279–291, October
2008. doi: 10.1086/589911.
A. Kusenko. Sterile Neutrinos, Dark Matter, and Pulsar Velocities in Models
with a Higgs Singlet. Phys. Rev. Lett., 97(24):241301–+, December 2006. doi:
10.1103/PhysRevLett.97.241301.
A. Kusenko. Sterile neutrinos: The dark side of the light fermions. Phys. Reports,
481:1–28, September 2009. doi: 10.1016/j.physrep.2009.07.004.
R. Kuzio de Naray, G. D. Martinez, J. S. Bullock, and M. Kaplinghat. The
Case Against Warm or Self-Interacting Dark Matter as Explanations for Cores
in Low Surface Brightness Galaxies. ApJ, 710:L161–L166, February 2010. doi:
10.1088/2041-8205/710/2/L161.
C. Lacey and S. Cole. Merger rates in hierarchical models of galaxy formation.
MNRAS, 262:627–649, June 1993.
D. Larson, J. Dunkley, G. Hinshaw, E. Komatsu, M. R. Nolta, C. L. Bennett,
B. Gold, M. Halpern, R. S. Hill, N. Jarosik, A. Kogut, M. Limon, S. S. Meyer,
N. Odegard, L. Page, K. M. Smith, D. N. Spergel, G. S. Tucker, J. L. Weiland,
E. Wollack, and E. L. Wright. Seven-Year Wilkinson Microwave Anisotropy Probe
(WMAP) Observations: Power Spectra and WMAP-Derived Parameters. ArXiv
e-prints, January 2010.
Antony Lewis and Sarah Bridle. Cosmological parameters from CMB and other
data: a Monte- Carlo approach. Phys. Rev., D66:103511, 2002.
A. Loeb and M. Zaldarriaga. Small-scale power spectrum of cold dark matter.
Phys. Rev. D, 71(10):103520, May 2005. doi: 10.1103/PhysRevD.71.103520.
M. Loewenstein, A. Kusenko, and P. L. Biermann. New Limits on Sterile Neutrinos
from Suzaku Observations of the Ursa Minor Dwarf Spheroidal Galaxy. ApJ, 700:
426–435, July 2009. doi: 10.1088/0004-637X/700/1/426.
M. R. Lovell, V. Eke, C. S. Frenk, L. Gao, A. Jenkins, T. Theuns, J. Wang, S. D. M.
White, A. Boyarsky, and O. Ruchayskiy. The haloes of bright satellite galaxies
in a warm dark matter universe. MNRAS, 420:2318–2324, March 2012. doi:
10.1111/j.1365-2966.2011.20200.x.
M. R. Lovell, C. S. Frenk, V. R. Eke, A. Jenkins, L. Gao, and T. Theuns. The
properties of warm dark matter haloes. ArXiv e-prints, August 2013.
Y. Lu, H. J. Mo, N. Katz, and M. D. Weinberg. On the origin of cold dark matter
halo density profiles. MNRAS, 368:1931–1940, June 2006. doi: 10.1111/j.13652966.2006.10270.x.
170
LUX Collaboration, D. S. Akerib, H. M. Araujo, X. Bai, A. J. Bailey, J. Balajthy,
S. Bedikian, E. Bernard, A. Bernstein, A. Bolozdynya, A. Bradley, D. Byram,
S. B. Cahn, M. C. Carmona-Benitez, C. Chan, J. J. Chapman, A. A. Chiller,
C. Chiller, K. Clark, T. Coffey, A. Currie, A. Curioni, S. Dazeley, L. de Viveiros,
A. Dobi, J. Dobson, E. M. Dragowsky, E. Druszkiewicz, B. Edwards, C. H. Faham,
S. Fiorucci, C. Flores, R. J. Gaitskell, V. M. Gehman, C. Ghag, K. R. Gibson,
M. G. D. Gilchriese, C. Hall, M. Hanhardt, S. A. Hertel, M. Horn, D. Q. Huang,
M. Ihm, R. G. Jacobsen, L. Kastens, K. Kazkaz, R. Knoche, S. Kyre, R. Lander,
N. A. Larsen, C. Lee, D. S. Leonard, K. T. Lesko, A. Lindote, M. I. Lopes,
A. Lyashenko, D. C. Malling, R. Mannino, D. N. McKinsey, D.-M. Mei, J. Mock,
M. Moongweluwan, J. Morad, M. Morii, A. S. J. Murphy, C. Nehrkorn, H. Nelson,
F. Neves, J. A. Nikkel, R. A. Ott, M. Pangilinan, P. D. Parker, E. K. Pease,
K. Pech, P. Phelps, L. Reichhart, T. Shutt, C. Silva, W. Skulski, C. J. Sofka,
V. N. Solovov, P. Sorensen, T. Stiegler, K. O‘Sullivan, T. J. Sumner, R. Svoboda,
M. Sweany, M. Szydagis, D. Taylor, B. Tennyson, D. R. Tiedt, M. Tripathi,
S. Uvarov, J. R. Verbus, N. Walsh, R. Webb, J. T. White, D. White, M. S.
Witherell, M. Wlasenko, F. L. H. Wolfs, M. Woods, and C. Zhang. First results
from the LUX dark matter experiment at the Sanford Underground Research
Facility. ArXiv e-prints, October 2013.
D. Lynden-Bell. Statistical mechanics of violent relaxation in stellar systems. MNRAS, 136:101, 1967.
A. V. Maccio’ and F. Fontanot. How cold is Dark Matter? Constraints from Milky
Way Satellites. ArXiv e-prints, October 2009.
A. V. Macciò, A. A. Dutton, F. C. van den Bosch, B. Moore, D. Potter, and J. Stadel.
Concentration, spin and shape of dark matter haloes: scatter and the dependence
on mass and environment. MNRAS, 378:55–71, June 2007. doi: 10.1111/j.13652966.2007.11720.x.
A. V. Macciò, A. A. Dutton, and F. C. van den Bosch. Concentration, spin and
shape of dark matter haloes as a function of the cosmological model: WMAP1,
WMAP3 and WMAP5 results. MNRAS, 391:1940–1954, December 2008. doi:
10.1111/j.1365-2966.2008.14029.x.
A. V. Macciò, S. Paduroiu, D. Anderhalden, A. Schneider, and B. Moore. Cores in
warm dark matter haloes: a Catch 22 problem. MNRAS, 424:1105–1112, August
2012. doi: 10.1111/j.1365-2966.2012.21284.x.
M. Maltoni and T. Schwetz. Sterile neutrino oscillations after first MiniBooNE
results. Phys. Rev. D, 76(9):093005–+, November 2007. doi: 10.1103/PhysRevD.76.093005.
A. Manrique, A. Raig, E. Salvador-Solé, T. Sanchis, and J. M. Solanes. On the
Origin of the Inner Structure of Halos. ApJ, 593:26–37, August 2003. doi:
10.1086/376403.
171
N. F. Martin, R. A. Ibata, S. C. Chapman, M. Irwin, and G. F. Lewis. A
Keck/DEIMOS spectroscopic survey of faint Galactic satellites: searching for
the least massive dwarf galaxies. MNRAS, 380:281–300, September 2007. doi:
10.1111/j.1365-2966.2007.12055.x.
M. L. Mateo. Dwarf Galaxies of the Local Group. ARA&A, 36:435–506, 1998. doi:
10.1146/annurev.astro.36.1.435.
P. McDonald, U. Seljak, S. Burles, D. J. Schlegel, D. H. Weinberg, R. Cen, D. Shih,
J. Schaye, D. P. Schneider, N. A. Bahcall, J. W. Briggs, J. Brinkmann, R. J.
Brunner, M. Fukugita, J. E. Gunn, Ž. Ivezić, S. Kent, R. H. Lupton, and D. E.
Vanden Berk. The Lyα Forest Power Spectrum from the Sloan Digital Sky Survey.
ApJS, 163:80–109, March 2006. doi: 10.1086/444361.
A. Melchiorri, O. Mena, S. Palomares-Ruiz, S. Pascoli, A. Slosar, and M. Sorel.
Sterile neutrinos in light of recent cosmological and oscillation data: a multiflavor scheme approach. JCAP, 1:36–+, January 2009. doi: 10.1088/14757516/2009/01/036.
A. L. Melott. Comment on ’Discreteness Effects in Simulations of Hot/Warm Dark
Matter’ by J. Wang & S.D.M. White. ArXiv e-prints, September 2007.
P. Meszaros. The behaviour of point masses in an expanding cosmological substratum. A&A, 37:225–228, December 1974.
M. Milgrom. A modification of the Newtonian dynamics as a possible alternative to
the hidden mass hypothesis. ApJ, 270:365–370, July 1983. doi: 10.1086/161130.
M. Miranda and A. V. Macciò. Constraining warm dark matter using QSO gravitational lensing. MNRAS, 382:1225–1232, December 2007. doi: 10.1111/j.13652966.2007.12440.x.
B. Moore, S. Ghigna, F. Governato, G. Lake, T. Quinn, J. Stadel, and P. Tozzi.
Dark Matter Substructure within Galactic Halos. ApJ, 524:L19–L22, October
1999a. doi: 10.1086/312287.
B. Moore, T. Quinn, F. Governato, J. Stadel, and G. Lake. Cold collapse and the
core catastrophe. MNRAS, 310:1147–1152, December 1999b. doi: 10.1046/j.13658711.1999.03039.x.
J. F. Navarro, C. S. Frenk, and S. D. M. White. The Structure of Cold Dark Matter
Halos. ApJ, 462:563, May 1996. doi: 10.1086/177173.
J. F. Navarro, C. S. Frenk, and S. D. M. White. A Universal Density Profile from
Hierarchical Clustering. ApJ, 490:493, December 1997. doi: 10.1086/304888.
J. F. Navarro, E. Hayashi, C. Power, A. R. Jenkins, C. S. Frenk, S. D. M. White,
V. Springel, J. Stadel, and T. R. Quinn. The inner structure of ΛCDM haloes
- III. Universality and asymptotic slopes. MNRAS, 349:1039–1051, April 2004.
doi: 10.1111/j.1365-2966.2004.07586.x.
172
A. F. Neto, L. Gao, P. Bett, S. Cole, J. F. Navarro, C. S. Frenk, S. D. M. White,
V. Springel, and A. Jenkins. The statistics of Λ CDM halo concentrations. MNRAS, 381:1450–1462, November 2007. doi: 10.1111/j.1365-2966.2007.12381.x.
A. Nusser and R. K. Sheth. Mass growth and density profiles of dark matter haloes in
hierarchical clustering. MNRAS, 303:685–695, March 1999. doi: 10.1046/j.13658711.1999.02197.x.
J. P. Ostriker and P. J. E. Peebles. A Numerical Study of the Stability of Flattened
Galaxies: or, can Cold Galaxies Survive? ApJ, 186:467–480, December 1973. doi:
10.1086/152513.
H. Päs, S. Pakvasa, and T. J. Weiler. Sterile-active neutrino oscillations and shortcuts in the extra dimension. Phys. Rev. D, 72(9):095017–+, November 2005. doi:
10.1103/PhysRevD.72.095017.
P. J. E. Peebles. The Gravitational-Instability Picture and the Nature of the Distribution of Galaxies. ApJ, 189:L51, April 1974. doi: 10.1086/181462.
Planck Collaboration, P. A. R. Ade, N. Aghanim, C. Armitage-Caplan, M. Arnaud,
M. Ashdown, F. Atrio-Barandela, J. Aumont, C. Baccigalupi, A. J. Banday, and
et al. Planck 2013 results. XVI. Cosmological parameters. ArXiv e-prints, March
2013.
E. Polisensky and M. Ricotti. Constraints on the dark matter particle mass from
the number of Milky Way satellites. Phys. Rev. D, 83(4):043506, February 2011.
doi: 10.1103/PhysRevD.83.043506.
E. Polisensky and M. Ricotti. Massive Milky Way satellites in cold and warm dark
matter: dependence on cosmology. MNRAS, 437:2922–2931, January 2014. doi:
10.1093/mnras/stt2105.
C. Power, J. F. Navarro, A. Jenkins, C. S. Frenk, S. D. M. White, V. Springel,
J. Stadel, and T. Quinn. The inner structure of ΛCDM haloes - I. A numerical convergence study. MNRAS, 338:14–34, January 2003. doi: 10.1046/j.13658711.2003.05925.x.
F. Prada, A. A. Klypin, A. J. Cuesta, J. E. Betancort-Rijo, and J. Primack. Halo
concentrations in the standard Λ cold dark matter cosmology. MNRAS, 423:
3018–3030, July 2012. doi: 10.1111/j.1365-2966.2012.21007.x.
W. H. Press and P. Schechter. Formation of Galaxies and Clusters of Galaxies by
Self-Similar Gravitational Condensation. ApJ, 187:425–438, February 1974. doi:
10.1086/152650.
C. W. Purcell and A. R. Zentner. Bailing out the Milky Way: variation in the properties of massive dwarfs among galaxy-sized systems. JCAP, 12:007, December
2012. doi: 10.1088/1475-7516/2012/12/007.
173
M. Ricotti. Dependence of the inner dark matter profile on the halo mass. MNRAS,
344:1237–1249, October 2003. doi: 10.1046/j.1365-8711.2003.06910.x.
M. Ricotti. Late gas accretion on to primordial minihaloes: a model for Leo T, dark
galaxies and extragalactic high-velocity clouds. MNRAS, 392:L45–L49, January
2009. doi: 10.1111/j.1745-3933.2008.00586.x.
M. Ricotti. The First Galaxies and the Likely Discovery of Their Fossils in the Local
Group. Advances in Astronomy, 2010, 2010. doi: 10.1155/2010/271592.
M. Ricotti and N. Y. Gnedin. Formation Histories of Dwarf Galaxies in the Local
Group. ApJ, 629:259–267, August 2005a. doi: 10.1086/431415.
M. Ricotti and N. Y. Gnedin. Formation Histories of Dwarf Galaxies in the Local
Group. ApJ, 629:259–267, August 2005b. doi: 10.1086/431415.
M. Ricotti and J. P. Ostriker. X-ray pre-ionization powered by accretion on the first
black holes - I. A model for the WMAP polarization measurement. MNRAS, 352:
547–562, August 2004. doi: 10.1111/j.1365-2966.2004.07942.x.
M. Ricotti, N. Y. Gnedin, and J. M. Shull. The Evolution of the Effective Equation of State of the Intergalactic Medium. ApJ, 534:41–56, May 2000. doi:
10.1086/308733.
M. Ricotti, N. Y. Gnedin, and J. M. Shull. The Fate of the First Galaxies. II. Effects
of Radiative Feedback. ApJ, 575:49–67, August 2002a. doi: 10.1086/341256.
M. Ricotti, N. Y. Gnedin, and J. M. Shull. The Fate of the First Galaxies. I. Selfconsistent Cosmological Simulations with Radiative Transfer. ApJ, 575:33–48,
August 2002b. doi: 10.1086/341255.
M. Ricotti, J. P. Ostriker, and N. Y. Gnedin. X-ray pre-ionization powered by
accretion on the first black holes - II. Cosmological simulations and observational signatures. MNRAS, 357:207–219, February 2005. doi: 10.1111/j.13652966.2004.08623.x.
M. Ricotti, A. Pontzen, and M. Viel. Is the Concentration of Dark Matter Halos at
Virialization Universal? ApJ, 663:L53–L56, July 2007. doi: 10.1086/520113.
M. Ricotti, N. Y. Gnedin, and J. M. Shull. The Fate of the First Galaxies. III.
Properties of Primordial Dwarf Galaxies and Their Impact on the Intergalactic
Medium. ApJ, 685:21–39, September 2008a. doi: 10.1086/590901.
M. Ricotti, N. Y. Gnedin, and J. M. Shull. The Fate of the First Galaxies. III.
Properties of Primordial Dwarf Galaxies and Their Impact on the Intergalactic
Medium. ApJ, 685:21–39, September 2008b. doi: 10.1086/590901.
S. Riemer-Sørensen and S. H. Hansen. Decaying dark matter in the Draco dwarf
galaxy. A&A, 500:L37–L40, June 2009. doi: 10.1051/0004-6361/200912430.
174
S. Riemer-Sørensen, S. H. Hansen, and K. Pedersen. Sterile Neutrinos in the
Milky Way: Observational Constraints. ApJ, 644:L33–L36, June 2006. doi:
10.1086/505330.
S. Riemer-Sorensen, K. Pedersen, S. H. Hansen, and H. Dahle. Probing the nature
of dark matter with cosmic x rays: Constraints from “dark blobs” and grating
spectra of galaxy clusters. Phys. Rev. D, 76(4):043524–+, August 2007. doi:
10.1103/PhysRevD.76.043524.
V. C. Rubin, N. Thonnard, and W. K. Ford, Jr. Extended rotation curves of highluminosity spiral galaxies. IV - Systematic dynamical properties, SA through SC.
ApJ, 225:L107–L111, November 1978. doi: 10.1086/182804.
P. Salucci, A. Lapi, C. Tonini, G. Gentile, I. Yegorova, and U. Klein. The universal
rotation curve of spiral galaxies - II. The dark matter distribution out to the virial
radius. MNRAS, 378:41–47, June 2007. doi: 10.1111/j.1365-2966.2007.11696.x.
T. Sawala, C. S. Frenk, R. A. Crain, A. Jenkins, J. Schaye, T. Theuns, and J. Zavala.
The abundance of (not just) dark matter haloes. ArXiv e-prints, June 2012.
J. Schaye, T. Theuns, M. Rauch, G. Efstathiou, and W. L. W. Sargent. The thermal
history of the intergalactic medium∗ . MNRAS, 318:817–826, November 2000. doi:
10.1046/j.1365-8711.2000.03815.x.
K. B. Schmidt, S. H. Hansen, and A. V. Macciò. Alas, the Dark Matter Structures
Were Not That Trivial. ApJ, 689:L33–L36, December 2008. doi: 10.1086/595783.
A. Schneider, D. Anderhalden, A. Maccio, and J. Diemand. Warm Dark Matter:
The End is Nigh. ArXiv e-prints, September 2013.
C. Schultz, J. Oñorbe, K. N. Abazajian, and J. S. Bullock. The High-$z$ Universe
Confronts Warm Dark Matter: Galaxy Counts, Reionization and the Nature of
Dark Matter. ArXiv e-prints, January 2014.
U. Seljak, A. Makarov, P. McDonald, and H. Trac. Can Sterile Neutrinos Be
the Dark Matter? Phys. Rev. Lett., 97(19):191303–+, November 2006. doi:
10.1103/PhysRevLett.97.191303.
Uros Seljak and Matias Zaldarriaga. A line of sight approach to cosmic microwave
background anisotropies. Astrophys. J., 469:437–444, 1996.
X. Shi and G. M. Fuller. New Dark Matter Candidate: Nonthermal Sterile
Neutrinos. Phys. Rev. Lett., 82:2832–2835, April 1999. doi: 10.1103/PhysRevLett.82.2832.
M. H. Siegel, M. D. Shetrone, and M. Irwin. Trimming Down the Willman 1 dSph.
AJ, 135:2084–2094, June 2008. doi: 10.1088/0004-6256/135/6/2084.
175
K. Sigurdson and M. Kamionkowski. Charged-Particle Decay and Suppression of
Primordial Power on Small Scales. Phys. Rev. Lett., 92(17):171302–+, April 2004.
doi: 10.1103/PhysRevLett.92.171302.
J. D. Simon and M. Geha. The Kinematics of the Ultra-faint Milky Way Satellites:
Solving the Missing Satellite Problem. ApJ, 670:313–331, November 2007. doi:
10.1086/521816.
J. D. Simon, A. D. Bolatto, A. Leroy, L. Blitz, and E. L. Gates. High-Resolution
Measurements of the Halos of Four Dark Matter-Dominated Galaxies: Deviations from a Universal Density Profile. ApJ, 621:757–776, March 2005. doi:
10.1086/427684.
J. Sommer-Larsen and A. Dolgov. Formation of Disk Galaxies: Warm Dark Matter and the Angular Momentum Problem. ApJ, 551:608–623, April 2001. doi:
10.1086/320211.
M. Sorel, J. M. Conrad, and M. H. Shaevitz. Combined analysis of short-baseline
neutrino experiments in the (3+1) and (3+2) sterile neutrino oscillation hypotheses. Phys. Rev. D, 70(7):073004–+, October 2004. doi: 10.1103/PhysRevD.70.073004.
D. N. Spergel, L. Verde, H. V. Peiris, E. Komatsu, M. R. Nolta, C. L. Bennett,
M. Halpern, G. Hinshaw, N. Jarosik, A. Kogut, M. Limon, S. S. Meyer, L. Page,
G. S. Tucker, J. L. Weiland, E. Wollack, and E. L. Wright. First-Year Wilkinson
Microwave Anisotropy Probe (WMAP) Observations: Determination of Cosmological Parameters. ApJS, 148:175–194, September 2003. doi: 10.1086/377226.
D. N. Spergel, R. Bean, O. Doré, M. R. Nolta, C. L. Bennett, J. Dunkley, G. Hinshaw, N. Jarosik, E. Komatsu, L. Page, H. V. Peiris, L. Verde, M. Halpern, R. S.
Hill, A. Kogut, M. Limon, S. S. Meyer, N. Odegard, G. S. Tucker, J. L. Weiland, E. Wollack, and E. L. Wright. Three-Year Wilkinson Microwave Anisotropy
Probe (WMAP) Observations: Implications for Cosmology. ApJS, 170:377–408,
June 2007. doi: 10.1086/513700.
V. Springel. The cosmological simulation code GADGET-2. MNRAS, 364:1105–
1134, December 2005. doi: 10.1111/j.1365-2966.2005.09655.x.
V. Springel, S. D. M. White, G. Tormen, and G. Kauffmann. Populating a cluster
of galaxies - I. Results at [formmu2]z=0. MNRAS, 328:726–750, December 2001.
doi: 10.1046/j.1365-8711.2001.04912.x.
V. Springel, J. Wang, M. Vogelsberger, A. Ludlow, A. Jenkins, A. Helmi, J. F.
Navarro, C. S. Frenk, and S. D. M. White. The Aquarius Project: the subhaloes
of galactic haloes. MNRAS, 391:1685–1711, December 2008. doi: 10.1111/j.13652966.2008.14066.x.
176
F. Stoehr, S. D. M. White, G. Tormen, and V. Springel. The satellite population of
the Milky Way in a ΛCDM universe. MNRAS, 335:L84–L88, October 2002. doi:
10.1046/j.1365-8711.2002.05891.x.
M. Su and D. P. Finkbeiner. Strong Evidence for Gamma-ray Line Emission from
the Inner Galaxy. ArXiv e-prints, June 2012.
K. Subramanian, R. Cen, and J. P. Ostriker. The Structure of Dark Matter Halos in Hierarchical Clustering Theories. ApJ, 538:528–542, August 2000. doi:
10.1086/309152.
R. A. Swaters, B. F. Madore, F. C. van den Bosch, and M. Balcells. The Central
Mass Distribution in Dwarf and Low Surface Brightness Galaxies. ApJ, 583:
732–751, February 2003. doi: 10.1086/345426.
D. Syer and S. D. M. White. Dark halo mergers and the formation of a universal
profile. MNRAS, 293:337, February 1998. doi: 10.1046/j.1365-8711.1998.01285.x.
J. E. Taylor and J. F. Navarro. The Phase-Space Density Profiles of Cold Dark
Matter Halos. ApJ, 563:483–488, December 2001. doi: 10.1086/324031.
A. A. Thoul and D. H. Weinberg. Hydrodynamic Simulations of Galaxy Formation.
II. Photoionization and the Formation of Low-Mass Galaxies. ApJ, 465:608–+,
July 1996. doi: 10.1086/177446.
E. J. Tollerud, J. S. Bullock, L. E. Strigari, and B. Willman. Hundreds of Milky
Way Satellites? Luminosity Bias in the Satellite Luminosity Function. ApJ, 688:
277–289, November 2008. doi: 10.1086/592102.
F. C. van den Bosch and R. A. Swaters. Dwarf galaxy rotation curves and the
core problem of dark matter haloes. MNRAS, 325:1017–1038, August 2001. doi:
10.1046/j.1365-8711.2001.04456.x.
F. C. van den Bosch, G. Tormen, and C. Giocoli. The mass function and average
mass-loss rate of dark matter subhaloes. MNRAS, 359:1029–1040, May 2005. doi:
10.1111/j.1365-2966.2005.08964.x.
C. A. Vera-Ciro, A. Helmi, E. Starkenburg, and M. A. Breddels. Not too big, not
too small: the dark haloes of the dwarf spheroidals in the Milky Way. MNRAS,
428:1696–1703, January 2013. doi: 10.1093/mnras/sts148.
J. Viñas, E. Salvador-Solé, and A. Manrique. Typical density profile for warm
dark matter haloes. MNRAS, 424:L6–L10, July 2012. doi: 10.1111/j.17453933.2012.01274.x.
M. Viel, J. Lesgourgues, M. G. Haehnelt, S. Matarrese, and A. Riotto. Constraining
warm dark matter candidates including sterile neutrinos and light gravitinos with
WMAP and the Lyman-α forest. Phys. Rev. D, 71(6):063534–+, March 2005.
doi: 10.1103/PhysRevD.71.063534.
177
M. Viel, J. Lesgourgues, M. G. Haehnelt, S. Matarrese, and A. Riotto. Can Sterile
Neutrinos Be Ruled Out as Warm Dark Matter Candidates? Phys. Rev. Lett.,
97(7):071301–+, August 2006. doi: 10.1103/PhysRevLett.97.071301.
M. Viel, G. D. Becker, J. S. Bolton, M. G. Haehnelt, M. Rauch, and W. L. W.
Sargent. How Cold Is Cold Dark Matter? Small-Scales Constraints from the Flux
Power Spectrum of the High-Redshift Lyman-α Forest. Phys. Rev. Lett., 100(4):
041304–+, February 2008. doi: 10.1103/PhysRevLett.100.041304.
M. Viel, J. S. Bolton, and M. G. Haehnelt. Cosmological and astrophysical constraints from the Lyman α forest flux probability distribution function. MNRAS,
399:L39–L43, October 2009. doi: 10.1111/j.1745-3933.2009.00720.x.
M. Viel, G. D. Becker, J. S. Bolton, and M. G. Haehnelt. Warm dark matter as
a solution to the small scale crisis: New constraints from high redshift Lymanα forest data. Phys. Rev. D, 88(4):043502, August 2013. doi: 10.1103/PhysRevD.88.043502.
M. Vogelsberger, J. Zavala, and A. Loeb. Subhaloes in self-interacting galactic
dark matter haloes. MNRAS, 423:3740–3752, July 2012. doi: 10.1111/j.13652966.2012.21182.x.
M. G. Walker, M. Mateo, E. W. Olszewski, J. Peñarrubia, N. Wyn Evans, and
G. Gilmore. A Universal Mass Profile for Dwarf Spheroidal Galaxies? ApJ, 704:
1274–1287, October 2009. doi: 10.1088/0004-637X/704/2/1274.
S. M. Walsh, H. Jerjen, and B. Willman. A Pair of Boötes: A New Milky Way
Satellite. ApJ, 662:L83–L86, June 2007. doi: 10.1086/519684.
S. M. Walsh, B. Willman, and H. Jerjen. The Invisibles: A Detection Algorithm
to Trace the Faintest Milky Way Satellites. AJ, 137:450–469, January 2009. doi:
10.1088/0004-6256/137/1/450.
J. Wang and S. D. M. White. Discreteness effects in simulations of hot/warm
dark matter. MNRAS, 380:93–103, September 2007. doi: 10.1111/j.13652966.2007.12053.x.
J. Wang and S. D. M. White. Are mergers responsible for universal halo properties?
MNRAS, 396:709–717, June 2009. doi: 10.1111/j.1365-2966.2009.14755.x.
J. Wang, C. S. Frenk, J. F. Navarro, L. Gao, and T. Sawala. The missing massive satellites of the Milky Way. MNRAS, 424:2715–2721, August 2012. doi:
10.1111/j.1365-2966.2012.21357.x.
L. L. Watkins, N. W. Evans, V. Belokurov, M. C. Smith, P. C. Hewett, D. M.
Bramich, G. F. Gilmore, M. J. Irwin, S. Vidrih, L. Wyrzykowski, and D. B.
Zucker. Substructure revealed by RRLyraes in SDSS Stripe 82. MNRAS, 398:
1757–1770, October 2009. doi: 10.1111/j.1365-2966.2009.15242.x.
178
C. R. Watson, J. F. Beacom, H. Yüksel, and T. P. Walker. Direct x-ray constraints
on sterile neutrino warm dark matter. Phys. Rev. D, 74(3):033009–+, August
2006. doi: 10.1103/PhysRevD.74.033009.
C. R. Watson, Z. Li, and N. K. Polley. Constraining sterile neutrino warm dark
matter with Chandra observations of the Andromeda galaxy. JCAP, 3:018, March
2012. doi: 10.1088/1475-7516/2012/03/018.
R. H. Wechsler, J. S. Bullock, J. R. Primack, A. V. Kravtsov, and A. Dekel. Concentrations of Dark Halos from Their Assembly Histories. ApJ, 568:52–70, March
2002. doi: 10.1086/338765.
D. T. F. Weldrake, W. J. G. de Blok, and F. Walter. A high-resolution rotation
curve of NGC 6822: a test-case for cold dark matter. MNRAS, 340:12–28, March
2003. doi: 10.1046/j.1365-8711.2003.06170.x.
C. Weniger. A tentative gamma-ray line from Dark Matter annihilation at the
Fermi Large Area Telescope. JCAP, 8:007, August 2012. doi: 10.1088/14757516/2012/08/007.
S. D. M. White. Violent Relaxation in Hierarchical Clustering. In O. Lahav, E. Terlevich, and R. J. Terlevich, editors, Gravitational dynamics, page 121, 1996.
R. J. Wilkinson, J. Lesgourgues, and C. Boehm. Using the CMB angular power
spectrum to study Dark Matter-photon interactions. ArXiv e-prints, September
2013.
L. L. R. Williams, A. Babul, and J. J. Dalcanton. Investigating the Origins of Dark
Matter Halo Density Profiles. ApJ, 604:18–39, March 2004. doi: 10.1086/381722.
B. Willman, M. R. Blanton, A. A. West, J. J. Dalcanton, D. W. Hogg, D. P. Schneider, N. Wherry, B. Yanny, and J. Brinkmann. A New Milky Way Companion:
Unusual Globular Cluster or Extreme Dwarf Satellite? AJ, 129:2692–2700, June
2005. doi: 10.1086/430214.
J. Wolf, G. D. Martinez, J. S. Bullock, M. Kaplinghat, M. Geha, R. R. Muñoz,
J. D. Simon, and F. F. Avedo. Accurate masses for dispersion-supported galaxies.
MNRAS, 406:1220–1237, August 2010. doi: 10.1111/j.1365-2966.2010.16753.x.
Y. B. Zel’dovich. Gravitational instability: An approximate theory for large density
perturbations. A&A, 5:84–89, March 1970.
A. R. Zentner and J. S. Bullock. Halo Substructure and the Power Spectrum. ApJ,
598:49–72, November 2003. doi: 10.1086/378797.
D. H. Zhao, Y. P. Jing, H. J. Mo, and G. Börner. Mass and Redshift Dependence of
Dark Halo Structure. ApJ, 597:L9–L12, November 2003a. doi: 10.1086/379734.
179
D. H. Zhao, H. J. Mo, Y. P. Jing, and G. Börner. The growth and structure of
dark matter haloes. MNRAS, 339:12–24, February 2003b. doi: 10.1046/j.13658711.2003.06135.x.
D. H. Zhao, Y. P. Jing, H. J. Mo, and G. Börner. Accurate Universal Models for the
Mass Accretion Histories and Concentrations of Dark Matter Halos. ApJ, 707:
354–369, December 2009. doi: 10.1088/0004-637X/707/1/354.
A. Zolotov, A. M. Brooks, B. Willman, F. Governato, A. Pontzen, C. Christensen,
A. Dekel, T. Quinn, S. Shen, and J. Wadsley. Baryons Matter: Why Luminous
Satellite Galaxies have Reduced Central Masses. ApJ, 761:71, December 2012.
doi: 10.1088/0004-637X/761/1/71.
D. B. Zucker, V. Belokurov, N. W. Evans, M. I. Wilkinson, M. J. Irwin, T. Sivarani,
S. Hodgkin, D. M. Bramich, J. M. Irwin, G. Gilmore, B. Willman, S. Vidrih,
M. Fellhauer, P. C. Hewett, T. C. Beers, E. F. Bell, E. K. Grebel, D. P. Schneider,
H. J. Newberg, R. F. G. Wyse, C. M. Rockosi, B. Yanny, R. Lupton, J. A. Smith,
J. C. Barentine, H. Brewington, J. Brinkmann, M. Harvanek, S. J. Kleinman,
J. Krzesinski, D. Long, A. Nitta, and S. A. Snedden. A New Milky Way Dwarf
Satellite in Canes Venatici. ApJ, 643:L103–L106, June 2006. doi: 10.1086/505216.
F. Zwicky. Die Rotverschiebung von extragalaktischen Nebeln. Helvetica Physica
Acta, 6:110–127, 1933.
F. Zwicky. On the Masses of Nebulae and of Clusters of Nebulae. ApJ, 86:217,
October 1937. doi: 10.1086/143864.
180
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