thesis cpenzo

thesis cpenzo
Dissertation in Astronomy
submitted to the
Combined Faculties of the Natural Sciences and Mathematics
of the Ruperto-Carola-University of Heidelberg, Germany
for the degree of
Doctor of Natural Sciences
put forward by
Camilla Penzo, M.Sc.
born in Ponte Dell’Olio, PC, Italy
oral examination: 7 July 2015
2
left=3cm,bottom=0.1cm
Galaxy and Structure Formation
in Dynamical and Coupled Dark Energy
Camilla Penzo
Max-Planck-Institut für Astronomie
Referees: Dr. Andrea V. Macciò
Prof. Dr. Björn M. Schäfer
4
Galaxy and Structure Formation
in Dynamical and Coupled Dark Energy
Abstract:
In this thesis I study the effects of different Dark Energy models on galaxy
formation via numerical simulations. I investigate systems around and below Milky-Way masses and describe the effects of dark energy at galactic
and sub-galactic scales. Firstly, I analyze high-resolution hydrodynamical
simulations of three disc galaxies in dynamical dark energy models. While
overall stellar feedback remains the driving mechanisms in shaping galaxies,
the effect of the dark energy parametrization plays a larger role than previously thought. Secondly, I broaden the galaxy sample by simulating a 80
Mpc/h side cube of our universe using the same dynamical dark energy models. I show that resolution is a crucial ingredient so that baryonic feedback
mechanisms can enhance differences between cosmological models. Thirdly,
I investigate the effects of dynamical dark energy on dwarf mass scales. I find
that there is more variation from object to object (due to the stochasticity
of star formation at these scales) than between the same object in different
cosmological models, which makes it hard for observations to disentangle
different dark energy scenarios. In the second part of this thesis I investigate the effects of coupled dark energy models on galactic and sub-galactic
scales via dark matter only high-resolution simulations. I find that coupled
models decrease concentrations of (Milky-Way-like) parent haloes and also
reduce the number of subhaloes orbiting around them. This improves the
agreement with observations and, hence, makes these cosmologies attractive
alternatives to a cosmological constant.
5
Galaxienentstehung und Strukturbildung in dynamischen und
gekoppelten Dunkle Energiemodellen
Abstract:
In meiner Arbeit untersuche ich die Effekte verschiedener Modelle für die
Dunkle Energie auf die Galaxienentstehung mit Hilfe numerischer Simulationen. Ich untersuche Systeme mit Massen ähnlich oder kleiner der der Milchstrağe und beschreibe die Auswirkungen der Dunklen Energie auf galaktischen und subgalaktischen Skalen. Zunächst untersuche ich hochauflösende
hydrodynamische Simulationen von drei Scheibengalaxien in dynamischen
Dunklen Energiemodellen. Während die thermische und kinetische Rückkopplung von Sternen bei der Gestaltung der Galaxien weiterhin eine dominante Rolle spielt, wirkt sich die Parametrisierung der Dunklen Energie
stärker aus als bisher angenommen. Als nächstes erweitere ich die Galaxienprobe durch Simulation der gleichen dynamischen Dunklen Energiemodelle in
einem Würfel mit 80 Mpc/h Seitenlänge. Dabei zeige ich, dass die Auflösung
ein entscheidender Faktor für die baryonischen Rückkopplung ist und dass
sie die Unterschiede zwischen kosmologischen Modellen sichtbarer machen
kann. Drittens habe ich die Auswirkungen von dynamischer Dunkler Energie
auf Zwerggalaxien untersucht. In diesen Skalen erzeugt die stochastische
Sternentstehung eine solch groğe Varianz von einem Objekt zum nächsten,
dass die Unterschiede zwischen verschiedenen kosmologischen Modellen nicht
sichtbar sind. In Beobachtungen ist es also schwer, zwischen verschiedenen
Dunklen Energieszenarien zu unterscheiden. Im zweiten Teil dieser Arbeit
benutze ich hochauflösende Simulationen mit reiner Dunkler Materie, um den
Einfluss von gekoppelter Dunkler Energie auf galaktischen und subgalaktischen Skalen zu untersuchen. Die gekoppelten Modelle verringern die Konzentration von - milchstraßenähnlichen - Halos und reduzieren die Anzahl der
Satellitenhalos. Dadurch verbessert sich die Übereinstimmung mit Beobachtungen, was diese Kosmologien zu attraktiven Alternativen gegenüber jenen
mit einer kosmologischen Konstante macht.
6
Contents
Introduction
13
1 Cosmological Models
1.1 Cosmological Assumptions and Equations . .
1.2 An Expanding universe and
the Cosmological Constant . . . . . . . . . . .
1.3 The ΛCDM Model . . . . . . . . . . . . . . .
1.3.1 Baryons . . . . . . . . . . . . . . . . .
1.3.2 Radiation . . . . . . . . . . . . . . . .
1.3.3 Dark Matter . . . . . . . . . . . . . .
1.3.4 Problems of a Cosmological Constant .
1.4 Dark Energy . . . . . . . . . . . . . . . . . .
1.4.1 Dynamical Dark Energy . . . . . . . .
1.4.2 Coupled Dark Energy . . . . . . . . .
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2 Galaxy Formation and Evolution
2.1 Linear Structure Formation . . . . . . . . . . . . . . . . . . .
2.1.1 Remarks on Linear Evolution and Dark Energy . . . .
2.2 Non-Linear Structure Formation . . . . . . . . . . . . . . . .
2.2.1 Spherical Collapse . . . . . . . . . . . . . . . . . . . .
2.2.2 Press-Schechter Mass Function . . . . . . . . . . . . .
2.2.3 The Zel’dovich Approximation . . . . . . . . . . . . .
2.3 N-Body Simulations of the Large-Scale Structure of the universe
2.3.1 Particle-Particle Method . . . . . . . . . . . . . . . . .
2.3.2 Particle-Mesh Method . . . . . . . . . . . . . . . . . .
2.3.3 Tree Algorithms . . . . . . . . . . . . . . . . . . . . .
2.3.4 Halo mass Function . . . . . . . . . . . . . . . . . . .
2.3.5 Radial Density Profiles . . . . . . . . . . . . . . . . . .
2.4 Hydrodynamical Processes in Galaxy Formation . . . . . . . .
2.4.1 Accreting Gas and Cooling Process . . . . . . . . . . .
2.4.2 Star Formation and Feedback Mechanisms . . . . . . .
2.5 Hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.1 SPH Method . . . . . . . . . . . . . . . . . . . . . . .
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40
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3 Numerical Methods
3.1 Generating Initial Conditions
3.1.1 grafic2-de . . . . . .
3.2 N-body and SPH Solvers . . .
3.2.1 gasoline-de . . . . .
3.2.2 gadget2-cde . . . .
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43
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4 Galaxy Formation in Dynamical Dark Energy
51
4.1 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2 Disc Galaxies simulations in Dynamical Dark Energy . . . . . 53
4.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2.2 Results and Discussion . . . . . . . . . . . . . . . . . . 55
4.2.3 Conclusions on Disc Galaxies simulations in Dynamical Dark Energy . . . . . . . . . . . . . . . . . . . . . 67
4.3 Hydrodynamical Cosmological Volumes in Dynamical Dark
Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 68
4.3.2 Numerical and Cosmological Settings . . . . . . . . . . 69
4.3.3 Results and Discussion . . . . . . . . . . . . . . . . . . 70
4.3.4 Conclusions on Hydrodynamical Cosmological Volumes
in Dynamical Dark Energy . . . . . . . . . . . . . . . 77
4.4 Dwarf Galaxies Simulations in Dynamical Dark Energy and
ΛCDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 78
4.4.2 Results and Discussion . . . . . . . . . . . . . . . . . . 80
4.4.3 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.4.4 ΛCDM Follow-Up . . . . . . . . . . . . . . . . . . . . 83
4.4.5 Conclusions on Dwarf Galaxies Simulations in Dynamical Dark Energy and ΛCDM . . . . . . . . . . . . . . 86
5 Halo and Subhalo properties in Coupled Dark
5.1 Details on the Cosmological Models . . . . . . .
5.2 Numerical Set-Up . . . . . . . . . . . . . . . . .
5.3 Results on Milky-Way size Halo simulations . .
5.3.1 Host Halos Properties . . . . . . . . . .
5.3.2 Subhalos . . . . . . . . . . . . . . . . . .
5.4 Zooming-in on a dwarf halo . . . . . . . . . . .
5.5 Conclusions . . . . . . . . . . . . . . . . . . . .
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89
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105
108
6 Summary and Conclusions
111
Bibliography
119
8
List of Figures
1.1
CMB temperature map from Planck Collaboration et al. (2015). 19
2.1
Examples of power spectra in dynamical dark energy models
and ΛCDM. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Dark matter large-scale structure from Springel et al. (2006).
2.2
3.1
3.2
3.3
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
4.14
4.15
Example of mesh refinement. . . . . . . . . . . . . . . . . . .
Fiducial galaxy and its rotation curves when varying feedback
recipes from Stinson et al. (2013b). . . . . . . . . . . . . . . .
Dark matter large scale structure in ΛCDM and two coupled
dark energy models from Baldi (2012b). . . . . . . . . . . . .
Our choices for dynamical dark energy models plotted over
the 2σ contours from WMAP7. . . . . . . . . . . . . . . . . .
Evolution of H(a) and D+ (a). . . . . . . . . . . . . . . . . . .
Dark matter density profiles of galα in all five cosmological
models in a dark matter only and hydrodynamical simulation.
Galaxy sample in the stellar-halo mass plane. . . . . . . . . .
Evolutions of the stellar-halo mass relation. . . . . . . . . . .
Evolution of the dark matter and gas mass. . . . . . . . . . .
Star formation histories. . . . . . . . . . . . . . . . . . . . . .
Ratios of SFR/SFRΛ . . . . . . . . . . . . . . . . . . . . . . .
Rotation curves. . . . . . . . . . . . . . . . . . . . . . . . . .
Comparison rotation curves in dark matter only and hydrodynamical simulations. . . . . . . . . . . . . . . . . . . . . . .
Evolution of the mean metallicity in “bulge”, “disc” and “halo”.
Evolution of cool gas in “bulge”, “disc” and “halo”. . . . . . . .
Feedback–Dark Energy degeneracy: evolution of stellar-halo
mass relation. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ratios of the halo mass functions with ΛCDM for the LR
volume. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ratios of the halo mass functions with ΛCDM for the HR
volume. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
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35
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4.16 Ratios of the stellar mass functions with ΛCDM for the LR
volume. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.17 Ratios of the stellar mass functions with ΛCDM for the HR
volume. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.18 Ratio of the mean stellar mass to ΛCDM as function of halo
mass for the LR volume. . . . . . . . . . . . . . . . . . . . . .
4.19 Ratio of the mean stellar mass to ΛCDM as function of halo
mass for the HR volume. . . . . . . . . . . . . . . . . . . . . .
4.20 Ratio of the mean stellar mass to ΛCDM as function of redshift for the single-galaxy simulations. . . . . . . . . . . . . .
4.21 Evolution of the stellar-halo mass relation for galβ and stellar
mass functions for the HR volume. . . . . . . . . . . . . . . .
4.22 Evolutions of w ≡ p/ρ for our choices of dynamical dark energy models and for the early dark energy models used in
Fontanot et al. (2012). . . . . . . . . . . . . . . . . . . . . . .
4.23 Stellar-halo mass relation for our dwarf galaxies in dynamical
dark energy models. . . . . . . . . . . . . . . . . . . . . . . .
4.24 SFRs for our dwarf sample. . . . . . . . . . . . . . . . . . . .
4.25 Stellar-halo mass relation for our 22 dwarf galaxies in ΛCDM
cosmology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.26 Star Formation Rates (SFR) of the 15 dwarf galaxies that
succeeded in forming stars in our ΛCDM sample. . . . . . . .
4.27 SFRs and density profiles for four dwarf galaxies from our
sample with comparable masses. . . . . . . . . . . . . . . . .
4.28 Dark matter mass evolution within 2 and 4 kpc from the center for three dwarf galaxies from our sample with comparable
masses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.29 Dark matter density and gas temperature maps for one of the
two dwarf galaxies in the sub-sample experiencing a major
merger. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1
72
73
74
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75
76
77
81
82
84
85
86
87
87
D+ and mass functions of the chosen coupled dark energy
models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.2 Density maps for all simulated halos. . . . . . . . . . . . . . . 94
5.3 Parent halos density profiles. . . . . . . . . . . . . . . . . . . 95
5.4 Parent halos rotation curves. . . . . . . . . . . . . . . . . . . 97
5.5 Evolution of the NFW scale radius. . . . . . . . . . . . . . . . 97
5.6 Parent halos accretion histories. . . . . . . . . . . . . . . . . . 98
5.7 Fit to the accretion history of haloβ. . . . . . . . . . . . . . . 99
5.8 Cumulative number of subhalos as function of their mass. . . 100
5.9 Cumulative number of subhalos as function of distance from
the parent halo. . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.10 Differential number of subhalos as function of distance from
the parent halo. . . . . . . . . . . . . . . . . . . . . . . . . . . 102
10
5.11
5.12
5.13
5.14
5.15
Differential number of subhalos within 3R200
Rotation curves of subhalos. . . . . . . . . .
Density maps of dwarf halos. . . . . . . . .
Profiles and rotation curves for dwarf halos.
Accretion histories for dwarf halos. . . . . .
11
of the parent halo.103
. . . . . . . . . . 106
. . . . . . . . . . 107
. . . . . . . . . . 107
. . . . . . . . . . 108
12
Introduction
Since the first Type Ia Supernova data were published (Riess et al., 1998;
Perlmutter et al., 1999), it has been clear that our universe is expanding with
a positive acceleration. On one hand, this led to a general acceptance of a
Cosmological Constant as the repulsive force driving the expansion of our
universe, on the other, it triggered the search for alternative theories leading
to Modified Gravity and to Dark Energy.
Nowadays numerous cosmological models that depart from the standard
paradigm have been proposed, many of those triggered by the long-discussed
problems of coincidence and fine-tuning (Wetterich, 1988). Whether or not
these problems are illdefined, some alternative theories can be as viable as
the standard paradigm, and they need to be tested against observational
constraints, at and beyond linear level.
Given the wide spectrum of observational data that is available, our most
powerful approach to constrain a given cosmological model is to broaden the
scales at which we test the model with data. Comparisons to observations on
cosmologically large scales need to be complementary to those on the highly
non-linear end of the power spectrum, down to galactic and sub-galactic
scales. At such small scales models that depart from the standard ΛCDM
paradigm are, at the time being, scarcely investigated.
To make predictions at highly non-linear scales we need to employ high
resolution numerical simulations. Additionally, the effects of baryonic feedback processes are known to strongly affect galactic and sub-galactic scales
and, for this, they ought to be taken into account. Nevertheless, modeling
hydrodynamical processes is a very challenging task and understanding their
role in shaping differences among cosmologies is crucial to make use of the
very high resolution data coming from the next generation of galaxy surveys.
My work concentrates on exploring structure formation at galactic and
sub-galactic scales in two of the most popular alternatives to the ΛCDM
model, dynamical dark energy and coupled dark energy, and is organized
as follows. In Chapter 1 I outline the basic features of the standard cosmological model to then describe the models behind dynamical dark energy
and coupled dark energy. In Chapter 2 I describe the basis of the theory of
structure formation at linear and non-linear level. In Chapter 3 I explain the
numerical methods that will be used throughout this thesis. In Chapter 4
13
I show our results on galaxy formation in dynamical dark energy. We start
with single-object hydrodynamical simulations of disc galaxies, to then extend the project to hydrodynamical simulations of a cosmological volume of
the universe; I end the Chapter by describing our work in exploring smaller
scales with hydrodynamical simulations of dwarf galaxies. In Chapter 5 I
outline our study on halo formation and evolution in coupled dark energy.
We performed single-object high dark matter only simulations of Milky Way
haloes and a dwarf halo, and we study the effects on the coupling between
dark matter and dark energy on halo concentration and number of subhaloes.
Finally, in Chapter 6 I summarize our work and draw conclusions.
14
Chapter 1
Cosmological Models
1.1
Cosmological Assumptions and Equations
Einstein’s theory of gravitation and its cosmological implications (Einstein,
1917) represent the guiding principles for modern theoretical cosmology. In
the theory of General Relativity the geometry of the space-time is included in
the metric tensor gµν , while the matter content of the universe is expressed
by the energy-momentum tensor Tµν and they are related by Einstein’s field
equations
1
8πG
Gµν ≡ Rµν − gµν R = 4 Tµν ,
(1.1)
2
c
where G is Newton’s gravitational constant, c is the speed of light in vacuum,
Gµν is the Einstein tensor, R and Rµν are the Ricci scalar and Ricci tensor
and are defined as follows:
R = g µν Rµν ,
Rµν =
where
∂Γλµλ
∂Γλµν
−
+ Γλµν Γδλδ − Γδµλ Γλνδ ,
∂xν
∂xλ
1
Γλµν = g λτ (gτ µ,ν + gτ ν,µ − gµµ,τ ) ,
2
(1.2)
(1.3)
(1.4)
are the Christoffel symbols (see e.g. Landau and Lifshitz 1976; Riotto 2010).
There exists no general solutions to Eq. (1.1), thus Einstein and others
realized that, in order to apply the field equations to the description of the
universe, simplifying assumptions were needed. Firstly, space was assumed
to be spatially homogeneous and isotropic on sufficiently large scales (Cosmological Principle), which meant that our universe should look the same
at each point and in all directions. From only geometric considerations, the
most generic space-time metric for a homogeneous and isotropic universe
15
takes the following form and is referred to as Friedman-Lemaître-RobertsonWalker metric (Friedmann, 1922; Lemaître, 1927; Robertson, 1935):
ds2 = −c2 dt2 + a(t)2
dr2
2
2
2
2
+
r
(dθ
+
sin
θ
dφ
)
,
1 − kr2
(1.5)
where t is physical time, r, θ and φ are the co-moving spherical coordinates,
a(t) is the scale factor which describes the expansion or contraction of the
universe and k represents the spacial curvature, which is k = 0 in the case
of a flat universe, k = ±1 for a positive or negative curvature.
Secondly, the various matter-energy components are assumed to be well
described by a continuous fluid, to which spacial comoving coordinates are
assigned. When assuming a perfect fluid, the energy-momentum tensor becomes
Tµν = (p + ρc2 )uµ uν − pgµν ,
(1.6)
where p is the pressure, ρc2 is the energy density and uµ is the fluid fourvelocity.
Inserting the FLRW metric of Eq. (1.5) into Einstein’s field equations,
Eq. (1.1), gives non-null identities only in the cases of the time-time and
space-space components, which respectively are:
ä
4πG p
= −
ρ+3 2 ,
a
3
c
p
2
2
aä + 2ȧ + 2kc = 4πG ρ − 2 a2 ,
c
which lead to
ȧ2 + kc2 =
(1.7)
(1.8)
8πG 2
ρa .
3
(1.9)
Eq. (1.7) and Eq. (1.9) are commonly known as Friedmann equations.
1.2
An Expanding universe and
the Cosmological Constant
The theory of General Relativity was given form at the beginning of the 1900s
and at that time the scientific community thought to be certain about living
in a static universe. Einstein realized that only a fluid with negative density
or negative pressure would allow a static solution, in fact, from Eq. (1.9),
ä > 0 if w < 1/3. Such fluid did not seem a reasonable assumption to make,
so the simplest way out of the problem was to introduce a constant term in
the field equations, the Cosmological Constant Λ. Eq. (1.1) then became
1
8πG
Rµν − gµν R − Λgµν = 4 Tµν .
2
c
16
(1.10)
Cosmological Models
Around the same years, observational astronomy was making huge steps
forward. Larger telescopes were built and this allowed to increase the precision on spectra measurements. In 1929, Hubble discovered a number of
objects that not only were distant and not part of the Milky Way, but were
also moving away from us. Furthermore, he discovered that their receding speed was proportional to their distance (Hubble, 1929), which is today
known as Hubble Law :
v = H0 d ,
(1.11)
where v and d are the receding velocity and the distance of the object and
H0 is the Hubble constant.
Consequently to the discovery of the expansion of the universe, there
was no more need to force the theory to predict a static solution, and the
motivation behind the cosmological constant dropped. On the contrary of
what Einstein believed, the interest around the cosmological constant did not
drop as well. In fact Λ was reintroduced to justify the accelerated expansion
of our universe and has been one of the most active research topics since
then.
1.3
The ΛCDM Model
Nowadays the most widely accepted description for our universe is given
by the Lambda Cold Dark Matter (ΛCDM) model. It describes a flat universe composed of baryonic and dark matter, radiation and a cosmological
constant, and it is in astonishing agreement with observational constraints
(Tegmark et al., 2006; Jones et al., 2009; Alam et al., 2015).
The evolution of the i-th component is often described by the density
parameter Ωi ≡ ρi /ρcrit , where ρcrit = 3H2 /(8πG) is the critical density and
H ≡ ȧ/a the Hubble function. As a consequence, Eq. (1.9) can be rewritten
as:
H(a)2 = H02 [Ωb (a) + ΩDM (a) + Ωrad (a) + ΩΛ (a)] ,
(1.12)
where the various Ω’s stand for baryons, dark matter, radiation and cosmological constant contributions and a flat universe has been assumed (see e.g.
Coles and Lucchin 2002 for details). The behavior of the components of the
universe as functions of the scale factor determine the evolutionary history
of the universe and depend on their specific equations of state, i.e. p = wρc2 ,
with 0 ≤ w ≤ 1.
Dust: The case w = 0 represents dust, or pressureless matter, i.e. the typical thermal pressure of particles is much less than their rest mass, which is
a good approximation for a non-relativistic fluid. Both baryonic and dark
matter belong to this case.
Radiative fluid: a fluid composed of non-degenerate, relativistic particles
in thermal equilibrium has w = 1/3. Photons obey to this equation of state.
17
Cosmological Constant: By definition, it has a constant density ρΛ , with
equation of state w = −1
The adiabatic expansion of the universe, can be written as
p ȧ
ρ̇ + 3 ρ + 2
= 0,
c a
(1.13)
which is known as continuity equation. Using the equations of state for
respectively for baryonic and dark matter, radiation and cosmological constant, w = 0, 1/3, −1, we obtain the scaling of density with the expansion
factor:
ρb a3 = ρb0 a30 ,
ρDM a3 = ρDM0 a30 ,
ρrad a4 = ρrad0 a40 ,
ρΛ = const .
(1.14)
With today’s scale factor a0 = 1, we can then rewrite Eq. (1.12) as
H(a)2 = H02 Ωb0 a−3 + ΩDM0 a−3 + Ωrad0 a−4 + ΩΛ .
(1.15)
From Eq. (1.15), it is clear that the expansion rate of the universe is determined by the energy density and the equation of state of its constituents
and that, in order to determine the expansion history, we must measure
the values of the cosmological parameters today and the Hubble constant.
The most recent values for these quantities are (Planck Collaboration et al.,
2015):
H0 = 67.8 km/s/Mpc ,
(1.16)
Ωb0 = 0.0490 ,
(1.17)
(1.18)
ΩDM0 = 0.2685 ,
−5
Ωrad0 = 8.5 × 10
ΩΛ = 0.6824 .
1.3.1
,
(1.19)
(1.20)
Baryons
One of the puzzling questions that astronomers were trying to answer in the
first decades of the XX century was why our universe is mainly composed
of only hydrogen and helium and what produced the remaining heavier elements. In the 1940s, astronomers were debating whether heavier elements
could come directly from the synthesized primordial hydrogen. Gamow
(1946) and Alpher et al. (1948) suggested that the universe in the past was
much hotter and denser, but they showed that the amount of heavier elements present today could not come from what was then referred to as Big
Bang Nucleosynthesis (BBN) alone. Today we know that they indeed come
from star cores and supernovae explosions. From BBN we can have accurate
estimates of the amount of baryons present in the universe, which today
accounts for about 5% of the total energy content.
18
Cosmological Models
1.3.2
Radiation
After having accepted that there must have been an epoch where the universe was extremely hot and dense, Alpher and Herman (1948) realized that
the expansion of the universe had to be accounted for. Furthermore, they
realized that in such conditions the universe had to be dominated by radiation which was in thermal equilibrium with matter. While the universe
was expanding it was also cooling down, which led to a decrease in the temperature until protons and electrons could recombine to form atoms. After
what we refer to as “recombination”, photons stopped interacting with atoms,
maintained their temperature and freely to propagated through the universe.
Thus, Alpher and collaborators predicted the existence of an isotropic thermal radiation background that propagated till today and they estimated
its temperature to be around a few degrees Kelvin. The discovery of the
relic radiation was firstly made by Penzias and Wilson (Penzias and Wilson,
1965) and then confirmed by Roll and Wilkinson (1966). They had found
a constant, isotropic, unpolarized excess in temperature with a black body
spectrum, what was then called Cosmic Microwave Background (CMB) radiation, which convinced the community of the validity of the Big Bang Theory.
Today we know that there also exists a relic radiation of neutrinos, which
are nearly as numerous as CMB photons and were highly relativistic in the
early universe. Finally, also stars emit radiation, but compared to photons
and neutrinos, radiation from stars is completely negligible. Fig. 1.1 shows
the latest CMB temperature map from Planck Collaboration et al. (2015).
Figure 1.1: CMB temperature anisotropies from Planck Collaboration et al.
(2015).
19
1.3.3
Dark Matter
For many years it was believed that the total matter in galaxies was nothing
more that what can be seen, until Zwicky realized that there must have been
some “missing” mass in Coma Cluster. Visible matter was rotating way faster
that it ever could have been if the total mass accounted for was only stars and
gas. He estimated that there had to be 400 times more mass than what we
could see (Zwicky, 1933). Rubin et al. (1980) showed that NGC3115 rotation
curve (i.e. rotational velocity as a function of √
the distance from the galactic
center) did not have the expected Keplerian d fall, but instead appeared
to be constant to large radii. The only explanation was that the luminous
mass was not all the mass that there was to account for, but instead in the
outskirts there had to be some dark mass. Today we know that dark matter
seems to interact only gravitationally and, despite the fact that it accounts
for more than 20% of the energy content of the universe, no dark matter
particle has yet been detected.
1.3.4
Problems of a Cosmological Constant
Even long before the introduction of the cosmological constant, particle
physicists realized that vacuum energy should be an underlying background
energy that must exist throughout space. Once Λ was re-introduced to explain the late-time accelerated expansion of the universe, the most straightforward explanation was to associate the cosmological constant with the
vacuum energy density. By summing up the zero-point energy contributions, the estimate for the vacuum energy density is (see e.g. Amendola and
Tsujikawa, 2010):
ρvac = 1074 GeV4 ,
(1.21)
where natural units are used (h̄ = c = 1). On the other hand, the observed
value for the cosmological constant energy density is:
ρΛ = 10−47 GeV4 .
(1.22)
On top of this, Amendola and Tsujikawa point out that if there were such
a vacuum energy density in the past, this would have triggered an eternal
state of cosmic acceleration, which in turn would of course not agree with the
existence of a radiation dominated nor with a matter dominated era, both
being big successes of the ΛCDM paradigm. The inconsistency between the
theoretical ρvac and the observationally measured ρΛ gave rise to fine-tuning
objections on the cosmological constant.
As showed in Eq. (1.14), matter density is proportional to the cubic of the
scale factor, while the energy density of the cosmological constant is constant
by definition. After the universe was coming out of its matter dominated
era, the contribution of the cosmological constant started taking over. This
happened around redshift z ∼ 1, which compared to the entire history of
20
Cosmological Models
the universe, could appear to be suspiciously close to today. The fact that
we happen to live right about when the epoch of cosmic acceleration started
has been referred to as the coincidence problem.
Wether these two discussed issues are truly a problem for the introduction
of a cosmological constant in the model or not, in the last decades they have
surely motivated cosmologists to come up with alternative theories to explain
the late-time accelerated expansion.
1.4
Dark Energy
A cosmological constant does provide an explanation for the late-time accelerated expansion, but it is far from being the only possible explanation.
Also motivated by the fundamental problems in assuming the existence of a
cosmological constant, in the last decades cosmologists came up with alternative theories. In this work we will only summarize two of the many models
available in literature, dynamical dark energy and coupled dark energy. For
clarity, we will assume c = 1.
1.4.1
Dynamical Dark Energy
From Eq. (1.9), it can be concluded that in order to give rise to the cosmic
acceleration a fluid with equation of state that satisfies wDE < −1/3 (assuming a positive density) needs to be introduced. By integrating the continuity
equation (1.13), one obtains the general evolution of the dark energy density
as function of redshift:
Z z
3[1 + wDE (z̃)]
ρDE (z) = ρDE (0) exp
dz̃ ,
(1.23)
1 + z̃
0
where we have made use of H = ȧ/a and z = 1/a − 1. Eq. (1.23) gives
in turn the evolution for the density parameter ΩDE (z) which then enters
in the Friedmann equation (1.15). Unlike the constant w = −1 equation
of state for the cosmological constant, in the case of dynamical dark energy
(or quintessence) the equation of state dynamically changes with time. To
do so, a scalar field φ with a self-interacting potential V (φ) is introduced.
The scalar field only interacts through gravity, thus the standard Lagrangian
becomes
1
1
L=
R − g µν ∂µ φ∂ν φ − V (φ) + Lm+r ,
(1.24)
16πG
2
where R is the Ricci scalar and Lm+r is the lagrangian for the matter and
radiation components. The energy-momentum tensor for the scalar field is
√
√
2 δ[− 21 −gg µν ∂µ φ∂ν φ − −gV (φ)]
(φ)
(1.25)
Tµν = − √
−g
δg µν
1 αβ
= ∂µ φ∂ν φ − gµν
g ∂α φ∂β φ − V (φ) ,
(1.26)
2
21
where g ≡ det (g µν ). Using a FLRW metric, we can find pressure and energy
density of the scalar field to obtain its equation of state parameter
wφ ≡
1 2
φ̇ − V (φ)
pφ
= 21
.
2
ρφ
2 φ̇ + V (φ)
We then can rewrite Friedmann equations as
8πG 1 2
2
H =
φ̇ + V (φ) + ρm + ρrad ,
3
2
8πG 2
φ̇ + ρm + ρrad + pm + prad .
Ḣ = −
2
(1.27)
(1.28)
(1.29)
Furthermore, the variation of the action with respect to the scalar field holds
an extra equation compared to the ΛCDM case
0
φ̈ + 3H φ̇ + V = 0 ,
(1.30)
0
where V ≡ dV /dφ. The scalar field needs to satisfy the condition wφ <
−1/3. Thus, from Eq. (1.27) we obtain that φ̇2 < V (φ), which means that
the potential needs to be flat enough so that the scalar field can slowly
00
move along the potential. Additionally, under the condition H 2 V ,
the kinetic term becomes negligible compared to the potential and the dark
energy behavior approaches the false vacuum (i.e. w = −1).
The behavior of such field strongly depends on the form of the potential.
In the following we list some of the most popular choices that can be found
in literature:
a. V (φ) =
M 4+α
φn ,
with α > 0 (Ratra and Peebles, 1988)
b. V (φ) = M 4 e−λφ , with λ > 0 (Ferreira and Joyce, 1998; Ratra and
Peebles, 1988)
c. V (φ) =
M 4+α 4πGφ2
,
φn e
with α > 0 (Brax and Martin, 1999)
d. V (φ) = V0 + M 4−α φα , with α > 0 (Linde, 1987)
In this work we will only focus on a potential of the (c) form, called SUGRA
(SUper GRAvity) potential. And more details on the specifics used will be
given in Section 4.1.
The interesting feature of adopting a scalar field with a potential to
describe dark energy is that they can exhibit tracking solutions. These are
trajectories in the phase space that “attract” other trajectories and ensure
that, independently of the chosen initial conditions for the field, the behavior
is going to converge towards wDE ' −1 today. With this feature, a dynamical
scalar field responsible dark energy has been claimed to improve the finetuning behind the introduction of a cosmological constant.
22
Cosmological Models
Parametrization
The model for quintessence that we have briefly outlined is completely selfconsistent, one can start from the Lagrangian for a specific field and work out
the guiding equations for its linear evolution. Leaving aside the theoretical
model, being of course the only physical motivated route towards a complete
theory, what affects structure formation is the evolution of the linear background, given that the field for dark energy does not cluster or interacts with
any component other than gravity. For this reason, often parametrizations
of the background quantities are used (e.g. w ≡ p/ρ), not accounting for the
physically complete model that may stand behind. In this work we will use
the so-called CPL parametrization (Chevallier and Polarski 2001 and Linder
2003),
w(a) ≡ w0 + (1 − a)wa ,
(1.31)
with a scale factor and w0 and wa constant values.
To be noted that the parametrization becomes necessary when trying
to use the observed luminosity distance dL (z) to infer H(z), and therefore
obtain w(z) (Amendola and Tsujikawa, 2010). In the case of dark energy, the
reconstruction of the scalar field potential is also possible. The luminosity
distance is defined as
1
2
L
dL ≡
,
(1.32)
4πF
where L is the source absolute luminosity and F is its observed flux. Furthermore,
Z z
c(1 + z)
dz̃
dL =
sinh H0
.
(1.33)
H0
0 H(z̃)
Given that we have measurements of the luminosity distance only at discrete
redshifts values. The direct differentiation of dL with respect to the redshift is
not possible unless a parametrization is used. Different authors have chosen
to parametrize either dL (z), H(z) or wDE (z).
1.4.2
Coupled Dark Energy
The coincidence problem is one of the puzzling points of the ΛCDM model
(see Section 1.3.4). On the other hand, by introducing an interaction between dark matter and dark energy (Amendola, 2000), the fact that their
contributions are of the same order round about today would not come as
a surprise. Indeed, the background evolution of constant coupling models is
characterized by a regime during matter domination where the two interacting fluids share a constant ratio of the total energy content of the universe.
This regime if fueled by the energy transfer from the dark matter particles
to the dark energy scalar field. The fact that dark matter and dark energy
density evolutions are strongly coupled in turn alleviates the coincidence
problem (Mangano et al., 2003; Matarrese et al., 2003).
23
The Lagrangian is the following (Amendola, 2000),
L=
1
1
R − ∂ µ ∂µ φ − V (φ) − m(φ)ψ̄ψ + Lkin [ψ] + Lbar+rad
16πG
2
(1.34)
where the mass m(φ) of the dark matter field ψ is a function of the dark energy scalar field φ, which has a potential V (φ), Lkin [ψ] accounts for possible
terms describing the dark matter Lagrangian and Lbar+rad groups the terms
describing baryons and radiation. The dark energy model is as described in
the previous Section, i.e. a dynamical scalar field φ rolling down its potential
V (φ), thus its energy density and pressure are
1
ρφ = g µν ∂µ φ∂ν φ + V (φ) ,
2
1 µν
pφ = g ∂µ φ∂ν φ − V (φ) .
2
(1.35)
(1.36)
The interaction appears as a source term in the conservation equations for
the single species
∇µ T µν(φ) = +Q(φ)T (DM) ∇ν φ ,
∇µ T µν(DM) = −Q(φ)T (DM) ∇ν φ ,
∇µ T µν(bar) = 0 ,
∇µ T µν(rad) = 0 .
(1.37)
(1.38)
(1.39)
(1.40)
where T µν is the energy-stress tensor, Q is a function of φ and represents the
coupling strength and T is the trace of T µν . In this way, the conservation
of the total stress-energy tensor is not violated. When assuming a FLRW
metric, the continuity equations become
0
φ̈ + 3H φ̇ + V = +Q(φ)ρDM ,
(1.41)
ρ̇DM + 3HρDM = −Q(φ)ρDM φ̇ ,
(1.42)
ρ̇bar + 3Hρbar = 0 ,
(1.43)
ρ̇rad + 4Hρrad = 0 .
(1.44)
where dots are time derivative and the prime is a derivative with respect to
φ. Using the notation from Amendola (2000), a coupling function βc can be
defined
r
3
βc (φ) ≡
MPl Q(φ) ,
(1.45)
2
√
where MPl = 1/ 8πG is the reduced Planck mass. Additionally, by integrating Eq. (1.42) and using Eq. (1.45), we find
!
s
Z φ
2
βc (φ̃)dφ̃ ,
(1.46)
ρDM (a) = ρDM0 a−3 exp −
2
3MPl
φ0
24
Cosmological Models
where ρDM0 and φ0 are today’s values and a0 is taken to be unity. Assuming
that particles are neither created or destroyed, their number density must
be conserved. Thus, the mass of coupled particles must vary in time by
following the variation of the scalar field
!
s
Z φ
2
βc (φ̃)dφ̃ ,
(1.47)
mDM (a) = mDM0 exp −
2
3MPl
φ0
where mDM0 is the dark matter particle mass today. In this thesis, the choice
of the self-interacting potential will be limited to an exponential potential of
the form
V (φ) ∝ e−αφ
(1.48)
with α = 0.1 (Amendola, 2000; Macciò et al., 2004; Baldi et al., 2010).
25
Chapter 2
Galaxy Formation and
Evolution
How the universe transitioned from a nearly homogeneous state to the formation of galaxies is referred to as structure formation. In the previous
Chapter we saw that baryonic matter only accounts for about 15% of the
total amount of matter. As a first approximation, baryonic matter can be
neglected to focus on the collapse of large-scale structures. Then, we focus on
baryonic processes involved in the formation of galaxies to describe smaller
scale effects of structure formation.
One of the landmarks of the standard model for the structure formation is undoubtedly Jean’s gravitational instability theory (Jeans, 1902). He
realized that even when starting with an homogeneous and isotropic fluid,
perturbations in density and velocity can form and these may evolve with
time. The collapse of density perturbations is the result of an interplay
between the smoothing effect of pressure and the collapse under the fluid
self-gravity. When the effect of pressure is much smaller than the effect
of gravity, the density perturbation will continue to grow by accreting surrounding mass. Eventually, this will become a gravitationally bound object.
At the time Jeans demonstrated his criterion, the expansion of the universe
was not known, but nonetheless we still use his adapted theory today.
2.1
Linear Structure Formation
As long as we measure the density distribution on sufficiently large scales,
the assumptions of homogeneity and isotropy are correct. On the other hand,
when looking at smaller and smaller scales our universe is extremely clumpy
due to the clustering of matter. In the standard Big Bang model the universe started by being extremely hot, small and dense, and while expanding,
it cooled down. The theory of Inflation was introduced to justify the fact
that far apart patches of sky have the same CMB temperature, while they
27
should not have been in causal contact at recombination. The assumption
is that only 10−36 seconds after the Big Bang a scalar field (inflaton) drove
an extremely fast accelerated expansion that was able to stretch space, so
that regions that were firstly close to each other ended up being not anymore
in causal contact when inflation ended, 10−32 seconds after the Big Bang.
Not only inflation stretched and flattened space, but it also laid the seeds
for structure formation. In fact, quantum fluctuations in the hot and dense
plasma were now enlarged and spread in space and seeded density perturbations in the homogeneous fluid. Measurements of density perturbations
suggest that they are Gaussian distributed and, as predicted from the theory
of Inflation (e.g. Mukhanov 2005), they have almost the same power at all
scales. This translates into having a primordial power spectrum of the form
Pp (k) ∝ k n
(2.1)
with n = 0.965 from the latest measurements (Planck Collaboration et al.,
2015). Note that n ∼ 1 was a prediction of Inflation, and it would have been
exactly n = 1 in the asymptotic case of inflation lasting forever. We can
quantify density perturbations by introducing the parameter δ, defined by
ρ(~x, t) = ρ̄(t)[1 + δ(~x, t)]
(2.2)
where ρ̄ is the mean density.
The statistical description of density perturbations turns out to be more
conveniently done in Fourier space. The Fourier expansion of the density
perturbation is
Z
d3 k
~
ρ(~x, t) =
δk (~k, t) eik·~x
(2.3)
3
(2π)
where ~k is the wave-number. We define linear regime the case in which
δ(~x, t) 1, which is satisfied for redshifts z . 100. In this regime, the
Fourier modes δk (~k, t) can be treated as they evolved independently from
one an other. The evolution of each mode in Fourier space is governed by
the following differential equation
2 2
cs k
δ̈k + 2H δ̇k +
− 4πGρ̄ δk = 0
(2.4)
a2
where H ≡ ȧ/a is the Hubble function and cs the sound speed. The behavior
of perturbations can be described using the Jeans length,
r
2πa
π
λJ =
= cs
.
(2.5)
kJ
Gρ̄
Perturbations on scales λ λJ are sound waves (i.e. the pressure contribution is much larger than gravity contribution), while on scales λ λJ gravity
28
Galaxy Formation and Evolution
dominates. In a flat, matter-dominated universe the growing solution for the
density perturbations is
δ = const1 t2/3 + const2 t−1
(2.6)
where the dependance on time of the scale factor a ∝ t2/3 in a matterdominated universe has been used. Note how inefficient structure formation
is once the expansion of the universe is taken into account, given that in the
static case the growing solution describes an exponentially fast perturbation
growth. In Eq. (2.4), the interplay between pressure and gravity is clearly
shown. Furthermore, given that dark matter does not interact with radiation, dark matter perturbations could start growing undisturbed even before
recombination. This process helped to speed up structure formation; in fact,
by the time baryons were decoupled and free to collapse, they could fall
into dark matter potential wells that already had already started to form.
On the other hand, radiation and relativistic particles can also dissipate
perturbations in baryonic density. The sum of all these effects shapes the
Gaussian power spectrum coming from Inflation into the power spectrum
at a given scale factor P (k, a), these effects can be included in a function
depending only on the wave number and on the scale factor, the transfer
function T (k, a).
P (k, a) = Pp (k) T (k, a)2 ,
(2.7)
where Pp (k) is the primordial power spectrum at the end of inflation. The
numerical implementation of all processes shaping transfer functions can be
found in the scientific community. For this work, we will use an extended
version of the code CAMB (Lewis and Bridle, 2002), modified to be suitable
for dynamical dark energy cosmologies. The scenario that finally emerges is
what is referred to as "bottom-up", since small linear scale structures form
first and with time merge into more massive clustering. As we pointed out
already, when density perturbations are small enough (δ(~k) 1), we can
treat structure formation with a linear approximation where each Fourier
mode evolves independently from the others. Thus, the power spectrum
scales as (see e.g. Coles and Lucchin, 2002):
P (k, a) = P (k, a1 )
2 (k, a)
2
D+
2 D+ (a)
=
P
(k)
T
(k,
a)
p
2 (k, a )
2 (a )
D+
D+
1
1
(2.8)
2 (k, a) = D 2 (a) when adopting the linear approximation (i.e. scales
where D+
+
2 (a) is actually the growing mode of the
larger than the Jeans length) and D+
density perturbations δ.
2.1.1
Remarks on Linear Evolution and Dark Energy
Dynamical Dark Energy
By changing the evolution of the equation of state parameter for dark energy
w ≡ p/ρ, see Eq. (1.27) and (1.31), the Hubble function in Eq. (1.28) will vary
29
44
Linear & post–linear fluctuation growth
Figure 2.1: Power spectra for ΛCDM and dynamical dark energy models
Figure 2.8:
Powerandspectrum
at z = 40 for ΛCDM and DE models, all power
(SUGRA
RP) at z=40.
spectrum are normalized to have the same σ8 at z = 0.
compared to the ΛCDM case. In turn, the evolution of density perturbations
driven by Eq. (2.4) will be altered. Dark energy is not directly affecting the
On theclustering
contrary,
found a clear
imprint
the nature
of universe.
DE in theThe
evolution
butwe
contributes
to changing
theof
expansion
of the
consequence
is that
structures
in dynamical
darkmass
energyscale
models
of the number
density
of clusters
above
a suitable
M.will
In start
fig. 2.9, we
forming at different times compared to ΛCDM, and,14given
a
dynamical
dark
give the number
N ofthe
thechoice
cluster
with M > will
6.9 determine
· 10 h−1whether
M" expected
energy model,
of normalization
structurein a box
−1 be anticipated or delayed. Figure 2.1 shows power spectra for
of side L formation
= 100 hwill
Mpc, as a function of redshift. The numbers at z > 0 are
RP and SUGRA dark energy models (Macciò, 2004) for z = 40, chosen to
normalized
to an identical number of clusters at z = 0, for all models (N(z = 0) =
have the same σ8 at z = 0. Structures in these dark energy models are more
to thethe
ΛCDM
It is possible
to construct
dynamical
0.13). Inevolved
figure compared
2.9 we give
ratiocase.
between
the number
of clusters
expected
dark energy models so that, at a given redshift, structure will be less evolved
in various models and the number expected in an open CDM model with the
compared to ΛCDM. We will further comment on dynamical dark energy
same value
of Ωmformation
. The mass
here
is selected
so to correspond
a cluster
structure
times
in Section
4.1, addressing
the specific to
models
used whose
in this work.
virial radius is an Abell radius in a standard CDM model. A similar plot, for
a slightly smaller mass, was given by Bahcall, Fan & Cen (1997), for standard
CDM, ΛCDM
andDark
OCDM
only (for a recent analysis of the constraints that
Coupled
Energy
cluster number counts set to the cosmological model, see, e.g., Holder, Haiman
While in the case of dynamical dark energy the effects on perturbation
& Mohr, growth
2001). are only due to changes in the background expansion, in the case
of coupled dark energy the change in the background is summed to the ef-
The main
results
of this analysis
(Mainini
and Bonometto
2003a, see
fects of
the interaction
between the
two darkMacciò
fluids. Consequently,
Eq. (2.4)
also Mainini et al 2003b) is that the cluster evolution depends on the nature of
30
DE in a significant way. Models with RP
potentials approach the evolution of
open CDM models as the energy scale Λ increases. On the contrary, the evolution
of SUGRA models is intermediate between open CDM and ΛCDM models.
Galaxy Formation and Evolution
becomes
3
δ̈c + (2H − βc φ̇)δ̇c − H 2 (1 + 2βc2 )Ωc δc + Ωb δb = 0 ,
2
(2.9)
where Ωc and Ωb are respectively the density parameters Ωi ≡ ρi /ρcrit for cold
dark matter and baryons, and ρcrit = 8πG/3H02 (Amendola, 2000, 2004a;
Macciò et al., 2004; Pettorino and Baccigalupi, 2008). Two extra terms
appear in Eq. (2.9) compared to the ΛCDM case, a friction term −βc φ̇δ̇c
and the factor (1 + 2βc2 ) responsible for the enhancement of the gravitational
force acting on cold dark matter particles, which is known as “fifth force”. In
other words, an effective gravitational constant can be defined,
Geff ≡ G[1 + 2βc (φ)2 ] ,
(2.10)
with G Newtonian gravitational constant. As pointed out in Baldi 2011a, in
the linear regime both these extra terms produce an acceleration of growth
of cold dark matter density perturbations. On the other hand, when considering the non-linear effects, the friction term is responsible for lowering the
concentration of dark matter haloes. This aspect will be further explored in
Section 5.3.1.
The appearance of extra terms becomes clear when calculating the acceleration felt by the i-th dark matter particle in a coupled dark energy
cosmology for the limit of a light scalar field (Baldi et al., 2010):
v¯˙i = βc (φ)φ̇ v̄i + Geff
X mj r̄ij
j6=i
|r̄ij |3
,
(2.11)
where r̄ij is the distance between the i-th and j-th particles and v̄i and mj
are the velocity and the mass of the i-th and j-th particles. In this work,
the coupling function is assumed to be in an exponential form (Amendola,
2004a; Baldi, 2011a)
βc (φ) ≡ β0 eβ1 φ ,
(2.12)
where β0 is today’s value. In this work we will study both the case of a
constant coupling β1 = 0 and the case of a growing coupling β1 > 0. Our
specific choices will be outlined in Section 5.1.
Concluding, the main effects of introducing a coupling between dark matter and dark energy are: (i) dark matter particles masses change with time,
while baryonic masses stay constant, (ii) the gravitational strength felt by
dark matter particles is stronger than in Newtonian dynamics, (iii) an extra
friction term appears, which has the consequence of increasing gravitational
collapse on the linear level, but we will show that on the non-linear level will
be helpful in lowering halo concentrations and decreasing the number and
concentrations of subhaloes (see Chapter 5).
31
2.2
Non-Linear Structure Formation
The success of linear theory is undoubtedly significant, but once density perturbations become δ ∼ 1, the linear approximation inevitably breaks down.
In fact, at few kpc away from the center of a luminous galaxy, the density
is around 105 times larger than the critical density ρc . Thus, to properly
investigate galaxy formation, we need to be able to follow the behavior of
highly non-linear density fluctuations.
2.2.1
Spherical Collapse
In a three-dimensional distribution of density perturbations, we choose one
of them and describe its collapse (Binney and Tremaine, 1987). We assume
that the background model is described by a flat FLRW metric and that the
universe is dark matter dominated. Most structures indeed form between
radiation-matter equality (zrm ' 3100) and matter-dark energy equality
(zmΛ ' 0.5). Additionally, the density perturbation δi 1 at some initial
time ti is assumed to be spherically symmetric with radius ri .
Matter around the perturbation feels the gravitational attraction of the
interior mass M, which can be calculated given the dependance of matter
density on the scale factor in Eq. (1.14). The total mass in the sphere M is
obtained and the evolution of the perturbation radius r(t) can be calculated
using Newton’s law
d2 r(t)
GM
=−
.
(2.13)
2
dt
r(t)2
By solving the second order acceleration differential equation, the turnaround radius can be obtained, radius at which expansion stops and the
collapse starts. The overdensity at turn-around is δt = 4.55.
In reality density perturbations are not spherically symmetric nor isolated and the collapse process is much more complex. Dark matter overdensities finally will collapse into an ellipsoid configuration referred to as
halo. Through collapse, mixing and relaxation processes, haloes reach their
virial equilibrium at about the same time that the toy model collapses into
a singular density at r = 0. Using the virial theorem, one finds that the
halo density is about 200 times the background density. This density ratio
derived from the spherical collapse model is widely used in cosmology for
characterizing dark matter haloes and their formation and we will make use
of it in Chapter 4 and 5 to define haloes virial radii.
Thus, overdensities that will turn into haloes can be identified by a
threshold set by a critical overdensity δc . In fact, they can be traced to
regions where the density at one point in time ti exceeded δc . By using the
turn-around time, one can obtain
δc (ti , t) = 1.686(ti /t)2/3 ,
32
(2.14)
Galaxy Formation and Evolution
where t is the collapse time.
2.2.2
Press-Schechter Mass Function
In the same way that over-dense perturbations continue to become more and
more massive by attracting dark matter mass, under-dense regions (voids)
become emptier and emptier. Voids grow in size with time and merge into
bigger voids, forcing matter into the high density walls around them (sheets).
While the collapse continues, the chance of sheets crossing each other increases, and, when this happens, a high density filament is produced. Mass
tends to move along the filaments towards their intersections (nodes). These
are the high density regions where galaxy clusters will form.
Structure formation is hierarchical, meaning that smaller objects form
first and merge into bigger systems. Galaxies form in the centre of virialized
haloes and are heavily influenced by halo merging. Mergers between haloes
of comparable mass are referred to as major mergers, while those between
different mass haloes are said to be minor mergers. As a result of the specific
merger history, a parent halo may contain smaller haloes (subhaloes) orbiting
around it.
In this picture, at any given redshifts, haloes with a wide range of masses
are present. A very powerful information is their distribution as a function
of mass, i.e. the number density of haloes at given redshift within the mass
range M and M + dM , the mass function.
Press and Schechter (1974) suggested that the probability of finding the
density contrast at or above the linear density contrast for spherical collapse,
δc , is equal to the fraction of the cosmic volume filled with haloes of mass M .
They derived the comoving number density of haloes with masses between
M and M + dM , (see e.g. Bartelmann 2012)
1 ρm δc d ln σR
δc2
dM
N (M, a)dM = √
exp −
2
2
π D+ (a) dM
2 σR D+ (a) M
(2.15)
with σR (a) = σR D+ (a) being the Gaussian variance of the density contrast
distribution at a given scale factor with characteristic scale R = [3M/(4πρm )]1/3 ,
which is defined as
Z
1
2
σR (a) =
d3 k P (k, a) W 2 (kR)
(2.16)
(2π)3
with P (k, a) power spectrum for density perturbations and W 2 (kR) a filter
function on the scale R. Thus, the mass function at a given scale factor
is defined by (i) the primordial power spectrum normalization and (ii) the
transfer function T (k, a), see Eq. (2.1) and (2.7). The normalization length
scale is commonly chosen to be 8 Mpc/h and the variance at this scale is
referred to as σ8 .
33
Despite its simple starting assumptions, the Press-Schechter mass function describes remarkably well the mass distribution of dark matter haloes
in cosmological simulations and its use can be extended to dynamical dark
energy models by evaluating the corresponding transfer functions. For what
concerns coupled dark energy models, Eq. (2.13) is no longer valid, thus a revised spherical collapse and consequent Press-Schechter function would need
to be calculated.
2.2.3
The Zel’dovich Approximation
A kinematical treatment to follow the evolution of density perturbations
further into the non-linear regime was put forward in Zel’dovich (1970).
The basic idea underlying the Zel’dovich approximation is assuming that
particles will move forward in the direction of their initial displacement. In
co-moving coordinates Zel’dovich’s equation is
~x(t) = ~xini − D+ (t)
~ 2 Φini
∇
4πGρini
(2.17)
where Φini and ρini are the gravitational potential and the density at the initial displacement ~xini . This approximation is accurate until particle trajectories cross and their mapping is no longer unique (shell-crossing). Eq. (2.17)
is widely used to determine particle displacements in initial conditions for
numerical simulations. In Section 3.1 we further explain its application.
2.3
N-Body Simulations of the Large-Scale Structure of the universe
Non-linear analytic methods are extremely useful to bring insights in the
physical processes involved in structure formation, but often are not accurate enough to compare their predictions to observations due to the highly
complex behavior of fluctuations once they enter the fully non-linear regime.
Thus, numerical simulations must enter into the picture. Typically, a volume of the universe is represented by a density distribution discretized by
point-like particles that interact with each other gravitationally. The volume
is characterized by periodic boundary conditions to mimic large scale structures. Dark matter only (or N-body) simulations of the large scale structures
(e.g. Springel et al. 2006) are in excellent agreement with observations (e.g.
Alam et al. 2015), which implies that on large scales baryonic matter traces
the dark matter distribution. Figure 2.2 shows the large-scale dark matter
distribution at different redshifts from the Millenium simulation (Springel
et al., 2006).
A large quantity of information can be derived from N-body simulations,
such as dynamics of mergers between galaxies (Toomre and Toomre, 1972),
34
Dark matter
Dark matter
Galaxies
z = 8.55
Galaxy Formation
NATURE|Vol 440|27 April 2006
T = 0.6 Gyr
Dark matter
z = 8.55
5.72
Galaxies
TT==0.6
1.0 Gyr
zz == 5.72
1.39
150 h–1 Mpc
1.0 Gyr
Gyr
TT == 4.7
=0
z =z1.39
T = 0.6 Gyr
T = 0.6 Gyr
z = 8.55
5.72
z = 8.55
Figure 4 | Time evolution of the cosmic largeGalaxies
z = 8.55
5.72
z =zz1.39
==00
z=0
scale structure in dark matter and galaxies,
obtained from cosmological simulations of the
ΛCDM model. The panels on the left show the
projected dark matter distribution in slices
of thickness 15 h–1 Mpc, extracted at redshifts
z = 8.55, z = 5.72, z = 1.39 and z = 0 from the
Millennium N-body simulation of structure
formation5. These epochs correspond to times of
and Evolution
600 million, 1 billion, 4.7 billion and 13.6 billion
INSIGHT
REVIEW
years after the Big Bang,
respectively. The
colour
hue from blue to red encodes the local velocity
TT==0.6
Gyr
1.0
TT=
=0.6
1.0 Gyr
T = 0.6 Gyr
dispersion in the dark matter, and the
brightness
of each pixel is a logarithmic measure of the
Figure 4 | Time evolution of the cosmic large5.72
zz ==show
5.72
1.39
projected density. The panels on the right
1.39
zzz===8.55
5.72
scale structure in dark matter and galaxies,
the predicted distribution of galaxies in the same
obtained from cosmological simulations of the
region at the corresponding times obtained by
ΛCDM model. The panels on the left show the
applying semi-analytic techniques to simulate
projected dark matter distribution in slices
galaxy formation–1in the Millennium simulation5.
of thickness 15 h Mpc, extracted at redshifts
Each galaxy is weighted by its stellar mass, and
z = 8.55, z = 5.72, z = 1.39 and z = 0 from the
the colour scale of the images is proportional to
Millennium N-body simulation of structure
the logarithm
of the projected total stellar mass.
formation5. These epochs correspond to times of
The dark matter evolves from a smooth, nearly
600 million, 1 billion, 4.7 billion and 13.6 billion
uniform distribution into a highly clustered state,
years after the Big Bang, respectively. The colour
quite unlike the galaxies, which are strongly
hue from blue to red encodes the local velocity
clustered from the start.
1.0Gyr
Gyr
= 4.7
1.0 Gyr
4.7
Gyr
TTTT====0.6
1.0
dispersion in the dark matter, and theTT =brightness
of each pixel is a logarithmic measure of the
z =zshow
1.39
=0
=0
projected density. The panels on the right
1.39
zzz===z5.72
1.39
the predicted distribution of galaxies in the same
region at the corresponding times obtained by
applying semi-analytic techniques to simulate
galaxy formation in the Millennium simulation5.
Each galaxy is weighted by its stellar mass, and
the colour scale of the images is proportional to
the logarithm of the projected total stellar mass.
The dark matter evolves from a smooth, nearly
uniform distribution into a highly clustered state,
quite unlike the galaxies, which are strongly
150 h–1
Mpc the start.
clustered
from
4.7 Gyr
TTT===13.6
TT==13.6
4.7Gyr
Gyr
1.0
4.7
larger than the value required by cosmology. Postulating instead a con-
Interestingly, it has als
for accelerated expans
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ticle physics scales is a profound mystery.
©2006 Nature Publishing Group
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1141
on the i-th particle:
F~i = −G
X
i6=j
mi mj (~xj − ~xi )
,
+ |xi − xj |2 )3/2
(2
(2.18)
where m and ~x are mass and position of i-th and j-th particles. As a consequence, the distribution of matter can only be trusted at scales & and is often quoted as the spacial resolution of the simulation. The calculation
tends to be very slow since the method requires N (N − 1)/2 evaluations of
Eq. (2.18) for each time-step.
2.3.2
Particle-Mesh Method
In order to improve the efficiency of the PP-method, Bertschinger and Gelb
(1991) suggested the particle-mesh scheme. Forces are calculated on a regular
grid to which mass points are assigned and the potential is calculated using
the Fast Fourier Transform (FFT) method. The big advantage is that the
computational cost of the method becomes proportional to N log N . The
downside of this technique is the poor resolution on small physical scales
due to the finite size of the grid. To solve this issue, hybrid PP-PM methods
have been developed (Brandt and Lubrecht, 1990). The PM part computes
the long range forces which are slowly varying and the PP part computes
the short range forces between close particles.
2.3.3
Tree Algorithms
The technique was firstly proposed by Appel (1985) and its basic idea consists of grouping particles so that distant particle clumps can be treated as
one single massive particle, lowering the total number of calculations. The
grouping is performed with a hierarchical subdivision procedure, which consists in dividing cells that contain more than one particle in sub-cells which
in turn will be subdivided until only one particle is found in the cell. In
this way the fine grid is created to compute small-range forces. With this
optimization the computational cost can be reduced to N log N .
2.3.4
Halo mass Function
From the Press-Schechter theory briefly delineated in Section 2.2.2, one can
obtain a prediction from the number density of parent haloes as a function of
mass which can be compared with N-body simulations. Jenkins et al. (2001)
show that the simulated mass function is almost independent of epoch, of
cosmological parameters and of the initial power spectrum when expressed
in appropriate variables, exactly as predicted by the Press-Schechter model.
On the other hand, the model predicts a mass function shape that differs
from their numerical results, overestimating the abundance of less massive
36
Galaxy Formation and Evolution
haloes and underestimating that of massive systems. Sheth et al. (2001)
generalize the spherical-collapse model to the case of collapsing ellipsoidal
bodies and showed that their prediction is in much better agreement with
numerical simulations.
2.3.5
Radial Density Profiles
In numerical simulations dark matter haloes seem to follow remarkably well
a universal mass density profile that can be described by two power laws,
ρ(r) =
(r/rs
)α
ρc
,
(1 + r/rs )β−α
(2.19)
with β ' 3 and 1 . α . 1.5. Eq. (2.19) with the choice of α = 1 and β = 3 is
known as the Navarro Frank and White (NFW) profile (Navarro et al., 1995),
with ρc critical density and rs scale radius. Haloes in collisionless simulations
seem to settle to this profile regardless the cosmological parameters and the
power spectrum (Navarro et al., 1996). This is also the case for dark matter
haloes in dynamical dark energy (Klypin et al., 2003) and coupled dark
energy (Baldi et al., 2010). Though, the origin of the universality of the
profile is not yet well understood. Navarro et al. (1995) and Moore et al.
(1999, 2004) show that simulations of halo virialization tend to produce
profiles with α ' 1 and it can be shown that haloes that arises from merging
many small haloes will acquire the density profile of their progenitors Binney
and Tremaine (1987). Thus, the universality of the NFW profile may start
from the virialization of the first haloes and survive the consecutive mergers.
2.4
Hydrodynamical Processes in Galaxy Formation
Over the last decades, the astronomical community has come to the conclusion that galaxies are only the visible part of much more massive systems
composed mainly of dark matter; furthermore these dark matter haloes are
embedded in a uniform distribution of dark energy. The way dark matter and
dark energy behave and their properties build up the cosmological model,
which in turn provides the starting conditions and the evolutionary guidelines to the formation of structures. Despite our incomplete understanding
of the dark sector, we are nonetheless able to build up a coherent picture
for the evolution of galaxies by modeling baryonic processes into numerical
simulations to finally test predictions from simulations with observations. In
fact, all the observational evidence that we have gathered on the universe
comes from emissions from baryonic matter. For this reason, in the following
Section we will summarize the behavior of baryons while the formation of
structures took place.
37
2.4.1
Accreting Gas and Cooling Process
As described in Section 1.3.1, prior to decoupling (z ' 1100) baryons were
coupled to photons and their density perturbations were being wiped out by
the radiation pressure contribution. On the other hand, being dark matter
decoupled from photons at all times, its perturbations were growing undisturbedly and after decoupling baryons fell into the already existing dark
matter potential wells. Given that the ratio of baryonic to dark matter mass
is quite small, the gravitational consequences of the collapse due to baryons
from the decoupling to z ' 30 was not significant (e.g. Binney and Tremaine,
1987), while on the contrary, from z ' 30 onwards, the role of baryons was
crucial in the collapse process due to their ability of radiating away energy
gained from the gravitational sinking. The atomic gas was heated by collisions between the newly formed hydrogen and helium atoms and emitted
photons and free electrons. Consequently, hydrogen molecules started to
be formed thanks to the presence of free electrons. By radiating photons
from the excited atoms and molecules, the gas was able to loose its acquired
gravitational energy and sank into the halo center. Thus, baryon density
in the inner regions increased dramatically until the gas became optically
thick to photons. Photons were now confined and started heating the gas.
Increasing the temperature allowed the central regions to reach a dynamical
equilibrium and star formation could begin. Around z ' 20 the first stars
were formed which in turn re-emitted ionizing photons in the universe, and
by z ' 6 the whole universe was once more filled with ionized gas (reionization epoch) with temperature T ' 104 K. Only haloes that at reionization
were massive enough were able to form stars. In fact, the thermal velocity
for hydrogen at a temperature of 104 K is about 10 km/s, which had to be
less than the halo escape velocity in order for the gas to stay in the halo and
form stars. Furthermore, in small haloes gas could not cool below the UV
background temperature. As a consequence, only haloes more massive than
around 108 M were able to start forming stars. This is a crucial point for
the evolution of dwarf galaxies (see Section 4.4).
The cooling time for gas is
tcool ∝
T
,
Ė
(2.20)
where T is the temperature and Ė the rate per particle at which gas radiates.
Ė is proportional to the gas density, which implies that the cooling time is
shorter in the dense inner halo regions. The cooling time has to be compared
to the free fall time of a system with density ρ, namely the time it takes the
system to collapse,
1
tff ∝ √
.
(2.21)
Gρ
Rees and Ostriker (1977) studied the gas cooling process at the epoch when
38
Galaxy Formation and Evolution
the gas has stopped expanding with the background universe but has not
yet fragmented into stars; they defined two distinct cases. (i) tcool > tff , gas
cooling is a slow process and the system has the time to react and adjust, so
that the hydrostatic equilibrium is restored. (ii) tcool < tff , the gas cooling is
too fast for the system to react accordingly and it falls out of hydrostatical
equilibrium; as a consequence, star formation is extremely efficient. Case
(ii) applies for galaxies whose halo mass is .1012 M .
2.4.2
Star Formation and Feedback Mechanisms
When gas cools, it can reach the center and form stars. Thus, a zero-th order
approximation sets the star formation rate density being proportional to the
gas density through the inverse of the free-fall time
ρgas
dρstar
∝
.
dt
tff
(2.22)
Taking the tff from Eq. (2.21) with ρ = ρgas , one obtains
dρstar
3/2
∝ ρgas
.
dt
(2.23)
Observationally, the proportionality between star formation rate surface density and gas surface density ΣSFR ∝ (Σgas )n was firstly found by Schmidt
(1959). Schmidt used measurements of the solar neighborhood and found
n ' 2. More recently Kennicutt (1998) used observations from around a
hundred nearby galaxies finding n ' 1.4.
On the other hand, only a small fraction of the cold molecular gas
(T ' 20K) is transformed into stars, while its majority is blown away into the
inter stellar medium (Krumholz and Tan, 2007). This re-injection of energy
in the inter stellar medium is referred to as stellar feedback and acts towards
suppressing star formation. Larson (1974) showed that due to supernova
explosions the interstellar gas is heated and driven out of the galaxy in a
galactic wind. The gas loss is significant, especially in smaller galaxies. Supernovae can be held responsible for two different types of feedback: firstly,
as mentioned by Larson, the gas is given momentum from the explosion and
can be expelled from the galaxy (kinetic feedback). Secondly, supernovae
can heat the surrounding gas to temperatures T'106 K which can produce
velocities that are greater than the galaxy escape velocity (Dekel and Silk,
1986). The pressure of the surrounding heated gas can drive gas bulk motions which can drive out of the galaxy much more gas in forms of galactic
outflows (thermal feedback).
Furthermore, a second class of feedback mechanisms is associated with
the presence of an active back hole in the center of the galaxy (active galactic
nucleus) and it is as well believed to be able to suppress star formation in
galaxies (Springel et al., 2005; Di Matteo et al., 2005; Croton et al., 2006).
39
2.5
Hydrodynamics
Since gas can cool and be heated, treating its hydrodynamics is a much more
complicated task than the gravity only case. While gravity is a long range
force, baryonic processes mostly act on scales . 1 Mpc, which makes their
implementation crucial to understand the formation of galaxies. Currently
available hydrodynamical codes can be grouped in three types: Eulerian
grid-based hydrodynamics, Lagrangian Smoothed Particles Hydrodynamics
(SPH) and the more recent moving mesh hydrodynamics. Differences lie
in the choices of discretization and reference systems. Eulerian grid-based
methods (Stone and Norman, 1992; Cen, 1992; Teyssier, 2002) build a space
grid and solve the equations of motion in the reference frame of a static
observer. Lagrangian SPH codes (Lucy, 1977; Gingold and Monaghan, 1977;
Monaghan, 1992; Wadsley et al., 2004; Springel, 2005) describe the density
field with a finite number of point masses and solve the equations of motion in
the reference frame of each fluid element. Moving mesh hydrodynamics codes
have a unstructured space grid that moves along with the fluid elements.
In the limit of an inviscid and non-conducting gas, the following set of
equations describes the hydrodynamics of the system
ρ
Dρ
~ · ~u ,
= −ρ∇
Dt
(2.24)
D~u
~ − ρ∇Φ
~ ,
= −∇p
Dt
(2.25)
D
~ · ~u − Λ(, ρ) ,
= −p∇
Dt
(2.26)
ρ
~ ρ, p,
where ~u ≡< ~v > is the mean fluid velocity at (~x, t), D/Dt ≡ ∂/∂t+~u · ∇,
Φ and are density, pressure, gravitational potential and internal energy per
unit mass and Λ(, ρ) is the cooling function, which denotes radiative cooling
and other photon emission processes. With the addition of the equation of
state = 1/(γ − 1)p/ρ, the system completely describes the dynamics of a
perfect gas.
In this work we study the formation of galactic structures in dynamical
and coupled dark energy. To do so, we will use two SPH codes, gasoline
(Wadsley et al., 2004) and gadget2 (Springel, 2005). In the next Section
we will briefly summarize details regarding SPH codes.
2.5.1
SPH Method
Analogously to N-body collisionless simulations, SPH codes discretize a continuous fluid with a finite number of particles. A position ~ri , velocity ~vi ,
density ρi and internal energy i are assigned to each particle. All hydrodynamic quantities are then replaced by their smoothed estimates, i.e. local
40
Galaxy Formation and Evolution
averages from kernel interpolations. Given a scalar field f , its smoothed field
hf i is
Z
hf (~r)i = d3 u f (~u) W (~r − ~u, h) ,
(2.27)
where h is the smoothing length and W (~r − ~u, h) smoothing kernel, which
satisfies
lim hf (~r)i = f (~r) .
h→0
(2.28)
An often used choice for the kernel can be found in Hernquist and Katz
(1989). When the number of particles is large enough, the continuous form
for hf (~r)i can be discretized by
hf (~r)i =
N
X
mj
j=1
ρj
f (~rj ) W (~r − ~rj , h) .
(2.29)
Springel (2010) shows that this approximation is accurate if the number of
neighbors used for the smoothing is N ≈ 33. Thus, ignoring diffusive and
self-gravity terms, the system of equations can be rewritten as
ρi =
N
X
j=1
mj W (~ri − ~rj , h) ,
(2.30)
#
"
N
X
pj ~
D~ui
pi
=−
mj 2 + 2 ∇i W (~ri − ~rj , h) ,
Dt
ρi
ρj
j=1
(2.31)
N
Di
pi X
~ i W (~ri − ~rj , h) .
= 2
mj (~ui − ~uj ) · ∇
Dt
ρi j=1
(2.32)
A second possibility to evaluate the hydrodynamical equations is to use an
entropy conserving scheme. Based on Springel and Hernquist (2002), in
Gadget-2 implementation Eq. (2.32) is exchanged with
i = si
ργ−1
,
γ−1
(2.33)
where si is the particle entropy.
When shocks are generated, gas can no longer be treated as an inviscid
fluid. Since SPH has no explicit treatment of physical viscosity, an artificial
viscosity term needs to be introduced to account for dissipation in shocks.
This term is responsible for smearing out shocks and for introducing dissipation in regions with strong velocity divergence, and appears in Eq. (2.31)
41
and (2.32) with the following extra terms
N
X
D~ui ~ i W (~ri − ~rj , h) ,
=
−
mj Πij ∇
Dt visc
(2.34)
N
Di 1X
~ i W (~ri − ~rj , h) ,
=
mj Πij (~ui − ~uj ) · ∇
Dt visc 2
(2.35)
j=1
j=1
where Πij is the viscous stress tensor. Furthermore, the change in entropy
is given by
N
Dsi 1γ−1X
~ i W (~ri − ~rj , h) .
mj Πij (~ui − ~uj ) · ∇
=
Dt visc 2 ργ−1
i
(2.36)
j=1
To conclude, a recipe for galaxy formation is not an easy task. Limits
arise both from our incomplete knowledge of the physics involved, and from
the computational limitations of present day machines. In fact, in order to
implement at best the baryonic processes involved in galaxy formation, subgrid models need to be adopted. The various implementations can change
significantly depending on the code used, but are nonetheless a key ingredient
to model the formation of galaxies. We will describe more in depth the subgrid model adopted in gasoline, and used in this work, in Section 3.2.1.
42
Chapter 3
Numerical Methods
3.1
Generating Initial Conditions
After recombination all species are free to collapse and by redshift z ∼ 100
(typical starting redshift for numerical simulations) density fluctuations are
present in all components. Depending on the details on the inflation theory,
initial perturbations might have various distributions. To this day, measurements of the CMB temperature anisotropies hint that initial perturbations
were most probably Gaussian, which means that they can be fully described
by a power spectrum. Thus, given amplitude and spectrum, which depend
on the cosmological model, initial density fluctuations can be generated.
Doroshkevich et al. (1980) were the first to use the Zeldovich approximation
to set initial conditions for numerical simulations. Here we show the guiding
equations (Klypin, 2000)
X
~ q Φ(k, ~q) ,
~x = ~q − α D+ (t)
∇
(3.1)
k
2
p~ = −α a Ḋ+ (t)
X
k
~ q Φ(k, ~q) ,
∇
(3.2)
where ~x and ~q are the comoving and Lagrangian coordinates, Φ the velocity
potential, obtained from
X
Φ(k, ~q) =
ak cos(k ~q) + bk sin(k ~q) .
(3.3)
k
ak and bk are random numbers that depend on the value of the power spectrum at a given wave number, as in
ak =
p
random(0, 1)
P (k)
,
k2
bk =
p
random(0, 1)
P (k)
.
k2
(3.4)
Thus, the power spectrum P (k) and the value of the parameter α in Eq. (3.1)
define the normalization of the density fluctuations. The box size Lbox and
43
the number of particles N of the simulations set the grid for the initial
conditions. The phase space is thus divided in cubes of size 2π/Lbox . A
realization of the spectrum of perturbations ak and bk is obtained when
choosing two random numbers and the particle displacements and momenta
are found using Eq. (3.1).
3.1.1
grafic2-de
The process of generating initial conditions described in the previous Section
obtains the same resolution for the full simulation volume L3box . In many
cases it can be convenient to set initial conditions such that the highest
resolution is achieved only in a portion of the full volume, in order to limit the
computational time. This is the case of single-galaxy simulations, in which
we are interested in having the highest possible resolution for the particles
belonging to the halo and its surroundings, but we are not concerned with
the remaining portion of the cosmological volume except for its gravitational
contribution which does not vary significantly if the resolution is set to be
lower. As a consequence, particles with different masses will compose the
cosmological volume; specifically, the least massive particles will build up
the halo and the most massive the outskirts of the volume.
The production of such multi-mass initial conditions is achieved in two
steps. Firstly, we simulate a low resolution dark matter only box, where all
particles have the same mass. We select the halo we want to investigate and
we define a volume around it that will have the highest resolution (often
called Lagrangian volume); in this work, in most cases we choose a sphere
of three times the halo’s virial radius. Using the particles identification
numbers (IDs), all particles belonging to the sphere are traced back to the
redshift at which we want the high resolution simulation to start (z = 99 in
this work). The initial conditions generator enlarges the Lagrangian volume
to the minimum closed volume, i.e. volume where no sub-volumes with low
resolution particles are present. High-frequency harmonics are then added in
the high resolution volume and the phase space is divided in smaller boxes,
e.g. 2π/Lbox · N/2. Figure 3.1 shows an example of mass refinement in 2D
real space. Three central blocks of particles were chosen for the highest mass
resolution. Each block produces 162 of the smallest particles (16 is referred to
as refinement factor RF ). Adjacent blocks have one step lower resolution and
produce 82 particles each (RF = 8). The procedure is repeated recursively
producing fewer more massive particles at each level (Klypin, 2000).
For this work we have modified the multi-mass initial condition generator code grafic2 (Bertschinger, 2001). The original code assumes a ΛCDM
cosmology, and I and L. Casarini adapted it into grafic2-de, a code which
requires the background cosmology as an input. More in detail, the input needs to provide the evolution as function of the expansion factor for
(i) density parameters for dark matter, baryons and dark energy (a flat
44
aprile 2015
Numerical Methods
–8–
Figure 3.1: An example of mesh with four refinement levels.
Fig. 1.— Example of construction of mass refinement in real space (left) and in phase space (righ
real space (left panel) three central blocks of particles were marked for highest mass resolution. Each
universe isproduces
assumed),
growth particles.
factor D+Adjacent
and its blocks
logarithmic
derivative
162(ii)
of smallest
get one
step lower resolution and produce 82 pa
fΩ ≡ dlnDeach.
/dlna
and
furthermore
(iii)
transfer
functions
for
dark
matter
+
The procedure is repeated recursively producing fewer more
massive particles at each level. In
and baryons.
The
codes
performs
interpolation
the background
quan- harmonics used for the low reso
space
(right
panel)
small an
points
in the left of
bottom
corner represent
tities and simulation.
is able to produce
initial
cosmological
For the multi-mass
high resolution
runconditions
with box for
ratios
1:(1/8):(1/16) the phase space is sampled
simulationscoarsely,
for a wide
of non-ΛCDM
I performed
a series markers) represents a small c
butrange
high frequencies
are cosmologies.
included. Each
harmonic (different
of tests to the
ensure
that
the
two
codes
are
equivalent,
given
the
same
ΛCDM
phase space indicated by squares. In this case the matching
of the harmonics is not perfect: th
cosmology.overlapping
Furthermore,
grafic2-de
reproduces
expected
results
for
cos-A and B are missed in the simul
blocks and gaps. In any case, the waves inside domains
mologies previously studied in literature (we used Ratra Peebles dark energy
models for testing). grafic2-de was used to generate initial conditions for
overlapping
blocks
and there
arework.
gaps. We can get much better results, if we assume different
all numerical
simulations
presented
in this
of the sizes of the boxes. For example, if instead of box ratios 1 : (1/8) : (1/16), we chose
1 : (3/32) : (5/96), the coverage of the phase space is almost perfect as shown in Figure: 2.
3.2
3.2.1
N-body and SPH Solvers
gasoline-de
4. Codes
gasoline (Wadsley et al., 2004) is a SPH tree code based on the gravity
algorithms of the N-body code pkdgrav (Stadel, 2001). gasoline solves
There are many different numerical techniques to follow the evolution of a system of
the equations of hydrodynamics and includes radiative cooling. The gas coolparticles.
earlier
reviews
see Hockney and
& Eastwood
(1981);
ing used (Shen
et al.,For
2010)
includes,
photoionization
heating from
the Sellwood (1987), and Bertsch
(1998).(unpublished)
Most of the methods
forbackground
cosmological
applications
take some ideas from three techn
Haart & Madau
ultraviolet
radiation,
Compton
9 K.Particle-Particle code, and the TREE code
Particle and
Mesh
(PM)metal
code,cooling
directfrom
summation
cooling, hydrogen
helium
10 to 10or
Gas particles
begin
formation
when they are cool (T≤ 1.5 (AP
× 1034M)
K) code (Couchman 1991) is a c
example,
the star
Adaptive
Particle-Particle/Particle-Mesh
3
and dense nation
(n ≥ 9.3cm
).
A
sub-set
of
these
particles
is
then
stochastically
of the PM code and the Particle-Particle code. The Adaptive-Refinement-Tree code (
selected to actually form stars, so that the Kennicutt Schmidt (Schmidt,
(Kravtsov et al. 1997; Kravtsov 1999) is an extension of the PM code with the organizat
1959; Kennicutt, 1998) law is reproduced. The star formation equation used
meshes in the form of a tree. All methods have their advantages and disadvantages.
45
is
Mgas
∆M∗
= c∗
,
∆t
tdyn
(3.5)
where ∆M∗ is the mass of star particle formed, ∆t is the time step between star formation events (8 × 105 yr in all the simulations described in
Section 4.2), c∗ is a constant formation efficiency factor and tdyn is the gas
dynamical time (Stinson et al., 2013b).
Supernova (SN) feedback follows the ‘blastwave model’ (Stinson et al.,
2006). At the end of their lives stars more massive than 8 M are assumed to
explode as Type II SNe and their lifetimes are based on Raiteri et al. (1996).
The energy emitted from explosions is transferred to the surrounding dense
gas which would quickly radiate it away. For this reason, cooling is disabled
inside the blast region for a period of time following McKee and Ostriker
(1977).
For SN II, metals produced in stars are released as the main sequence
progenitors die and distributed to the same gas within the blast radius as is
the SN energy ejected from SN II (Shen et al., 2010). Iron and oxygen are
produced in SN II according to the analytic fits used in Raiteri et al. (1996).
In summary, each stellar generation of 104 M produces, in a lifetime of a
1M star, about 49 SN II and 9 SN Ia events. These eject about 600 and 10
M of metal enriched material respectively, which translate into about 7M
of iron and 48M of oxygen that will impregnate the interstellar medium.
Feedback from SN Ia also follows the model in Raiteri et al. (1996), but
in this case radiative cooling was not disabled. Each SN Ia produces 0.6M
of iron and 0.13M of oxygen (Thielemann et al., 1986) and the metals are
ejected into the nearest gas particle for SN Ia.
Finally, the latest addition to the sub-grid hydrodynamical model is early
stellar feedback (Stinson et al., 2013b), which mimics the radiation energy
from young massive stars, and is introduced immediately after stars form.
This is motivated by radiation pressure driving winds out of massive star
clusters. On the other hand, given the resolution, typically a molecular cloud
is described by maximum a few gas particles, which implies that stellar winds
cannot be resolved, thus thermal pressure is used instead. Thermal feedback
provides pressure support and increases the gas temperature, which in turn
decrease star formation.
With the introduction of the early stellar feedback, Stinson et al. (2013b)
drastically improved the over-cooling problem affecting previous galaxy simulations. In short, simulated disc galaxies were producing too many stars
compared to observations and these stars concentrated in the galactic center
produced peaked rotation curves, which are not in agreement with observationally measured flat rotation curves. By accounting for the energy emitted
in the inter stellar medium by massive stars before they explode as SNe,
Stinson et al. simulated realistic disc galaxies (i.e. star formation histories
matching observational constraints, flat rotation curves and exponential sur46
MAGICC
parameter study
133
Table 1. Simulation data.
Name
Colour
c∗ a
! esf b
Ctd c
IMFd
M∗ (M" )e
MV f
Fiducial
MUGS
Low diffusion
High diffusion
No ESF
120 per cent SN energy
High ESF
Red
Black
Green
Yellow
Magenta
Cyan
Blue
0.1
0.05
0.1
0.1
0.1
0.1
0.1
0.1
0
0.1
0.175
0
0
0.125
0.05
0.05g
0.01g
0.05g
0.05
0.05
0.05
C
K
C
C
C
C
C
2.3 × 1010
8.3 × 1010
3.0 × 1010
2.5 × 1010
3.4 × 1010
1.8 × 1010
1.1 × 1010
−20.6
−21.4
−21.1
−21.4
−19.5
−20.0
−19.9
a Star-forming
efficiency.
stellar feedback efficiency.
diffusion coefficient.
d Initial mass function: C = Chabrier (2003); K = Kroupa et al. (1993).
e M is the total stellar mass.
∗
f V-band magnitude.
g All particles diffused thermal energy including those with cooling shut off.
Numerical Methods
b Early
c Thermal
MAGICC
Figure 2. Face-on and edge-on images of the fiducial galaxy at z = 0. The
images are 50 kpc on a side and were created using the Monte Carlo radiative
transfer code SUNRISE. The image brightness and contrast are scaled using
a sinh as described in Lupton et al. (2004).
Downloaded from http://mnras.oxfordjournals.org/ at MPI Astronomy on April 2, 2015
Figure 3. Stellar mass plotted as a function of halo mass. The abundance
matching fit from Moster et al. (2012) and the cosmic baryon fraction lines
are shown as reference. All the MUGS formed too many stars along a line
where half of the baryons have turned into stars and half are in the hot
gas halo. The simulations of g1536 are shown as coloured stars and lines.
The legend provides a brief description of which simulation each of these
colours represents, while a more extensive description can be found in Table
1. The fiducial simulation is coloured red and lies closest to the Moster et al.
(2012) relationship. The yellow star representing the simulation where all
gas particles diffuse thermal energy also lies close to the relationship line,
but it is rapidly forming stars and will quickly overshoot the relationship.
The blue point uses the same physics as the fiducial run, but increases the
! esf by 25 per cent and forms far fewer stars. The magenta line shows a
simulation run without ESF. While it ends up close to the fiducial run at z =
0, it follows a dramatically different evolutionary path.
is turned off for particles with their cooling disabled. This change
significantly decreases the mass of stars formed, though the stellar
mass still lies above the Moster et al.c (2012) relationship. This
motivated us to add ESF.
As a check to see whether the addition of more energy or the
timing of that energy addition had a stronger effect, we ran a series
of simulations without the 1051 erg per SN energy constraint. The
best result from that series is represented by the cyan line and star. It
shows that a 20 per 10
cent increase of the SN energy lowered the stellar
mass to the Moster et al. (2012) relationship. Its evolution more
closely follows the z = 0 Moster et al. (2012) relationship than the
simulation using 1051 erg. Again, we refer the reader to Section 3.3
Figure 6. Plot of v =
√
GM(r)/r as a function of r for all the models
Figure
3.2: Right panel, face-on
and
the
fiducial
of g1536
and edge-on
g5664 at z =images
0 modelledofwith
varying
amountsgalaxy
of stellar
is shown in Section 3.3 to be similar to how the observed stellar to
halo mass ratio evolves. A small change in the amount of feedback
feedback.
at
= change
0 from
Stinson
et al. (2013b). Left panel, Rotation curves of the
causes az
significant
in the number of
stars that form. When
! increases from 0.1 to 0.125 (represented in blue), the stellar
mass decreasesgalaxies
by a factor of 2.
same
when feedback 2recipes
varied (Early Stellar Feedback and
× 10 Mare
The magenta star and line represent the simulation in which
# , which means that the rotation curves do not reach
only the SN feedback was increased from the MUGS value of 4 ×
feedback
from
SNs).
their peak until well beyond the disc scale length.
In this simulation,
the thermal diffusion
10 erg per SN to 10 erg.
esf
50
51
Even though the rotation curves slowly rise in the plotted region,
the peak of the rotation curve, 180 km s−1 , is still higher than the
−1
110 km stheir
. In other
words,
the rotation
turn
value
of vwe
vir , show
face density profiles). In Figure
3.2
fiducial
galaxy
(leftcurves
panel)
over and decline outside the plotted region.
venerdì 3 aprile 2015
and the change in rotation curves
when curve
changing
recipes.
The rotation
for the feedback
fiducial galaxy
is separated into the
contributions
from its
constituent
particle
types in as
Fig.well
7. Stars
To be noted that in the last
years other
groups
have
succeeded
in
dominate the potential in the central regions of the galaxy. The
reproducing realistic disc galaxies (see Robertson et al. 2006, Governato et al.
curves represent the mass included from just that component, so it
2007, Agertz et al. 2011, Guedes
et represent
al. 2011,
et al.those
2012,
Scannapieco
does not
the Brook
speed at which
particles
are orbiting.
Each component
orbit close
to theon
totalgalaxy
velocity.formation
et al. 2012, Marinacci et al. 2013),
but all will
of these
works
The rotation velocity information enables another comparison
assumed a ΛCDM model.
between the amount of stars that have formed and the mass of the
Hydrodynamical simulations
that
will be presented
Section
4.21977).
extend
galaxy,
the Tully–Fisher
relationshipin
(Tully
& Fisher
The
Tully–Fisher
relationship
compares
the
luminosity
of
a
galaxy
with
the work done in Stinson et al. (2013b) to the case of dynamical dark energy.
circular velocity. It is possible to argue for the choice of many
The implementation of the its
sub-grid
feedback model is thus the same as
different radii at which to measure the circular velocity. In Stinson
in Stinson et al. (2013b), butet al.
the
darktheenergy
extension
(2010),
galaxies were
reported aswas
fittingimplemented
the Tully–Fisher
(Casarini et al., 2011b). Given that the dark energy field does not interact
with any component other than through gravity, for gasoline-de it was
sufficient to modify the background evolution in which matter perturbations
grow. Thus, the implementation consisted of allowing the code to accept
the evolution of the Hubble function H ≡ ȧ/a as an external input, so that
its value at each time step could be calculated with an interpolation of the
input values.
3.2.2
p
gadget2-cde
The coupled dark energy model described in Section 1.4.2 and 2.1.1 was
implemented in the N-body code gadget2 (Baldi et al., 2010), which we
here briefly summarize and refer to as gadget2-cde. As pointed out in
√ constant coupling models is
the model section, the background
Figure 7. evolution
Plot of vc = of
GM(r)/r as a function of r for the simulated
galaxy at z = 0. Each component, dark matter (solid red), gas (green,
dotted) and stars (light blue, dashed), is plotted separately to show the
47 of each component.
matter distributions
Figure 8. Peak circular velocity as a func
pared to the Reyes et al. (2011) observa
with the standard colour scheme used th
simulation (red) lies directly in the centre
tions without ESF lie significantly above
relationship because the circular ve
where Rd is the scale radius, which w
peak. Here, we instead choose to c
Reyes et al. (2011), who use an arc
peak velocities found using that fit
peak velocity from our simulations in
The fiducial simulation along with
ESF lies in the middle of the observed
the high-diffusion simulation, which
due to its significant late star forma
velocity. The simulations without ESF
in the observed relationship due to th
No ESF and MUGS lie well above the
SN energy lies only slightly above th
3.3 Star formation history
The star formation history in Fig. 9 sh
a significant impact on when stars fo
fairly standard star formation history
in the literature (Scannapieco et al. 20
buildup of total halo mass as shown in
without ESF experience a fairly bro
after 2 Gyr. This is the time period w
mergers are rapidly building the hal
is increased to 120 per cent, star for
below 3 M# yr−1 . Introducing the E
burst from the simulations shown h
gas from collapsing into the central
Following the early peaks of star form
exponentially declines with intermitt
The star formation rate has been c
137 Myr wide in Fig. 9. Fig. 1 show
culated using 25 Myr bins, which sh
formation that is characteristic of all
MUGS. There are minor episodic bu
laid on to the underlying shape of the
minor bursts correspond to the dyna
proximately the time it takes for gas
characterized by a regime during matter domination where the two interacting fluids share a constant ratio of the total energy content of the universe.
Thus, differently from the ΛCDM case, a non-negligible early dark energy
fraction is present during structure formation, which in turn implies a different background evolution compared to the one in a ΛCDM model. As
in the previous sections, an interpolation of the Hubble parameter H allows
the code to model the correct expansion history. Furthermore, the same
procedure is implemented also for the coupling function for cold dark matter
βc (φ), the correction term for dark matter masses ∆mDM and the parameter
for the kinetic energy density of the scalar field Ωkin (φ). In fact, as showed
in Eq. (1.47), dark matter particle masses change with time, thus they need
to be corrected at each time step. Eq. (2.11) shows the acceleration felt by a
dark matter particle. The extra term depending on the velocity of the particle is not present in ΛCDM and was implemented in terms of the velocity
variable p~ ≡ a(t)2 ~x˙ in the code, with ~x comoving coordinates


X
G̃ij mj ~xij 
1
φ̇
p~˙i = βc (φ) a p~i +
,
(3.6)
a
M
|~xij |3
i6=j
where G̃ij is the effective gravitational constant between the i-th and j-th
particle, which is given in Eq. (2.10) and implemented in the code as
G̃ij = GN (1 + 2βi βj ) ,
(3.7)
with i and j denoting the particle species (i.e. only DM-DM interaction will
give an enhanced gravitational constant) and GN is the Newtonian gravitational constant. Finally, also the multipole expansion of the tree algorithm
had to be modified. In standard gadget2 the gravity calculation is either
computed on the whole node or the node is once more divided in other eight
smaller nodes merely using the opening criterium, which sets the accuracy of
approximating a distribution of particles with a more massive particle positioned in the distribution center of mass. In coupled dark energy models not
all particles exert the same gravitational pull, thus one massive particle per
each particle species in the respective centers of mass has to be accounted
for.
gadget-cde has been used to investigate the formation of large scale
structures and cluster of galaxies in follow up papers, Baldi (2011a,b); Baldi
and Pettorino (2011); Baldi (2012a,b). In Figure 3.3 shows the dark matter density distribution in a slice with size 1000×250 Mpc/h and thickness 30 Mpc/h from the CoDECS simulations (Baldi, 2012b). EXP003 and
SUGRA003 are two coupled dark energy models chosen in his work. Given
that all three cosmological models have the same σ8 at z = 0, there are
still differences in the central densities of clusters. Despite the deep study
carried out by Baldi et al., no high resolution multi-mass simulations have
48
Numerical Methods
8
M. Baldi
Figure 3. The CDM density distribution in a slice with size 1000 × 250 Mpc/h and thickness 30 Mpc/h as extracted from the L-CoDECS simulations of
a few selected models. The middle slice shows the case of the standard ΛCDM cosmology, while the top slice is taken from the EXP003 simulation and the
bottom slice from the bouncing cDE model SUGRA003. While the latter model shows basically no difference with respect to ΛCDM at z = 0, due to the
very similar value of σ8 for the two models, clear differences in the overall density contrast and in the distribution of individual halos can be identified by eye
for the EXP003 cosmology. (A higher resolution version of this figure is available online through the CoDECS website, see the Appendix)
Figure 3.3: Dark matter density distribution in ΛCDM and two coupled dark
energy models taken from Baldi (2012b).
tered in the position of the most massive halo identified at z = 0
in the ΛCDM simulation of the H-CoDECS suite, for three different redshifts z = 0, 1, and 2. In this case, since the figure shows
the redshift evolution of the three models, it is possible to identify
the differences that characterize the growth of structures in these
selected scenarios. Also this comparison shows the same features
already described above for the comparison between ΛCDM and
EXP003. By comparing the two cosmologies at z = 0, in fact, it
clearly appears how the latter has a more evolved structure of the
cosmic web around the central massive halo and shows a larger
density contrast between the highly nonlinear overdense regions
and the voids. It also appears much more clearly than before, from
these plots, the lack of small structures in the EXP003 universe
with respect to ΛCDM, due to the faster merging processes occurring in the cDE cosmology. Despite this, the relative abundance of
substructures as a function of their fractional mass with respect to
their host main halo is not significantly affected by the coupling,
as will appear from the Subhalo Mass Function displayed in Fig. 8
below.
The higher amplitude of density perturbations in the EXP003
model is visible5
alsowe
at higher
redshifts, especially
at z = study
1, where of
been performed before this work. In Chapter
present
the first
the difference in the structure of the cosmic web around the central
coupled dark energy on galactic and sub-galactic
scales.
halo between EXP003 and
ΛCDM is particularly evident, as well as
the different merging history of the two models. Particularly interesting is in this case also the comparison between ΛCDM and the
bouncing cDE model SUGRA003. At z = 0, as already discussed
above, the two models show much weaker differences with respect
to the case of the EXP003 cosmology, due to the comparable amplitude of their linear density perturbations. However, at high redshifts (e.g. at z = 2) the SUGRA003 model appears clearly more
evolved as compared to ΛCDM: the main filamentary structure and
the high density peaks appear more luminous than the corresponding ΛCDM case, and the main halos are more extended due to the
merging of the surrounding satellite structures. This shows the main
characteristic feature of the bouncing cDE scenario: a larger amplitude of density perturbations (and a correspondingly higher number
of massive clusters) with respect to ΛCDM at high redshifts, followed by a reconnection of the two models that show very little
c 2011 RAS, MNRAS 000, 1–18
�
49
50
Chapter 4
Galaxy Formation in
Dynamical Dark Energy
In this Chapter we investigate the effects of dynamical dark energy on the
formation of galaxies via hydrodynamical numerical simulations. The cosmological model is outlined in Section 1.4.1 and 2.1.1 and more specifics will
be given in the following sections.
Section 4.2 reports the results on high-resolution single-object simulations of three disc galaxies from Penzo et al. (2014). Here we study the role
of baryons in differentiating dynamical dark energy models. In this work I
produced initial conditions, ran the simulations and carried out the analysis
with the supervision of A. V. Macciò. In Section 4.3 we performed hydrodynamical simulations of a large volume of the universe (Lbox = 114 Mpc).
The aim is to test the effects of viable dark energy models across a wide
range of galaxy masses, environments and merger histories. This Section is
part of a work in preparation that is being carried out in collaboration with
L. Casarini and A. Macciò, where L. Casarini performed the simulations and
I worked on the analysis with the supervision of A. V. Macciò. To conclude,
in Section 4.4 we investigate the effects of dynamical dark energy on satellite dwarf galaxies to then focus on the star formation stochasticity of these
small mass systems in a ΛCDM scenario (A. V. Macciò, C. Penzo et al. in
preparation).
4.1
Models
We choose four dynamical Dark Energy (dDE) models, each of which is
consistent with WMAP7 data (Komatsu et al. 2011) at two sigma level,
waCDM0, waCDM1, waCDM2 and SUCDM. All the models have at z = 0:
Ωb0 = 0.0458, ΩDM0 = 0.229, H0 = 70.2 km−1 s−1 Mpc−1 , σ8 = 0.816, ns =
0.968, where these parameters are density parameters for baryons and dark
matter, Hubble constant, root mean square of the fluctuation amplitudes and
51
Figure 4.1: Two-sigma contours from WMAP7 in the wa -w0 plane from
WMAP7. Each cosmological model is represented by a star.
waCDM0 (green)
waCDM1 (yellow)
waCDM2 (red)
w0
-0.8
-1.18
-0.67
wa
-0.755
0.89
-2.28
Table 4.1: Parameters of the waCDM cosmological models
primeval spectra index. waCDM0, waCDM1 and waCDM2, are based on a
linear CPL parametrization of the equation-of-state parameter w showed in
Eq. (1.31). In Table 4.1 we present the values we chose for w0 and wa in
each of the three cases. In Figure 4.1 the two-sigma contours from WMAP7
in the wa -w0 plane are shown, and each cosmological model is represented
by a star. waCDM0 is a model very close to LCDM as shown in Casarini
et al. (2009a), while waCDM1 and waCDM2, already studied in Casarini
et al. (2011a), are the most distant. Furthermore, we have included a fourth
cosmological model, SUCDM, in which dark energy is described by a scalar
field with a SUGRA self-interacting potential (see Section 1.4.1), where we
chose α = 2.9 and M = 10 GeV in agreement with Alimi et al. (2010). It
is important to note that all of these models are viable models according to
WMAP7 data. The blue star representing the SUCDM model is here shown
only for comparison, but clearly its position on this plot holds only at z = 0,
since its equation-of-state parameter w(a) cannot be described by the CPL
parametrization.
In order to show how the background evolution changes between cosmological models, Figure 4.2 shows both expansion velocity (left panel) and
52
Galaxy Formation in Dynamical Dark Energy
Figure 4.2: The left panel shows universe expansion velocities in units of the
Hubble constant as a functions of the scale factor. The right panel shows
the ratio of the growth factor at a given scale factor to the growth factor at
a = 1 divided by the same ratio for ΛCDM.
growth factor (right panel) as functions of the scale factor for each cosmology.
Note that we chose to compare different cosmological models by normalizing
mercoledì
25 marzoto
2015 the same σ
them
8 today. With this choice, a model with a faster cosmic
expansion will have to start producing structure earlier than a model with
slower expansion. This means that statistically, the SUCDM model (blue
lines) will show collapsed structures at an earlier epoch than the waCDM2
model (red lines), in order to compensate for the faster expansion of the universe. The earlier structure formation also leads to earlier accretion of the
substructures onto the parent halo. In turn, we expect that earlier accretion
will lead to earlier star formation in the simulated galaxies.
4.2
4.2.1
Disc Galaxies simulations in Dynamical Dark
Energy
Introduction
In their pioneering dark-matter-only simulations with a dark energy equation of state evolving with time w(z), Klypin et al. (2003) found that the
differences between the cosmological models were not significant at z=0 both
in the non-linear matter power spectrum and in the halo mass function, although differences between models became significant at higher redshifts
with a higher number of clusters for the dark energy models compared to
ΛCDM.
Subsequently, multiple groups investigated the properties of dark matter
structures in DE cosmologies; see, for instance, Dolag et al. (2004), Bartelmann et al. (2005), Francis et al. (2009), Grossi and Springel (2009). They
studied halo concentrations, velocity dispersions, abundance relations and
linear density contrast at collapse time in dark energy and early dark energy
53
models. Generally, significant differences between ΛCDM and dark energy
cosmologies were found mostly at high redshifts (when the same value for
the mean density amplitude σ8 at z=0 was assumed).
Several studies compared the inner structure of haloes simulated in ΛCDM
and dark energy cosmologies in collisionless simulations. In all cases, a
Navarro-Frenk-White (Navarro et al., 1997) density profile well described
the matter distribution. Furthermore, they found higher central concentrations in haloes simulated in a dark energy cosmology compared to those in
ΛCDM, due to earlier formation times of haloes (Klypin et al. 2003; Linder
and Jenkins 2003; Kuhlen et al. 2005).
It is important to note that, while these studies all considered dark energy
cosmologies that featured an earlier collapse time than ΛCDM, it is also
possible for dark energy cosmologies to form structure at a later stage than
ΛCDM. This happens when the equation-of-state parameter, w(a), “crosses
over the cosmological constant boundary from below”. In other words, w can
evolve from w < −1 at high redshift to w > −1 at z = 0. Such models have
less collapsed structure at high redshift than ΛCDM (see Pace et al. 2012).
While it is useful, and surely computationally much cheaper, to study
collisionless simulations of dark energy cosmologies, we can only directly observe baryons. Even though they account for ∼ 15 of the mass density of
dark matter in the universe, baryons can have a strong impact on the formation of small scale structures (White, 1976; Gallagher et al., 1984; Zhan and
Knox, 2004; Puchwein et al., 2005; Jing et al., 2006; Maio et al., 2006; Rudd
et al., 2008; Casarini et al., 2011c; De Boni et al., 2011; van Daalen et al.,
2011; Casarini et al., 2012; Fedeli et al., 2012). So far, simulations including
dark energy have focused on massive galaxy clusters since cosmology has the
largest effect on the formation of the largest structures.
In the last decade different groups have been studying galaxy formation
and evolution by performing high resolution hydrodynamical simulations in
a cosmological context. Only recently they have succeeded in simulating
realistic disc galaxies, e.g. star formation history matching with observational constrains, flat rotation curves, exponential surface density profiles
(see Robertson et al. 2006, Governato et al. 2007, Agertz et al. 2011, Guedes
et al. 2011, Brook et al. 2012, Scannapieco et al. 2012, Stinson et al. 2013b,
Marinacci et al. 2013). In all of these high resolution simulations a ΛCDM
cosmology has always been assumed. Recently an attempt to study galaxy
formation in different cosmological models has been presented in Fontanot
et al. (2012, 2013), where N-body simulations where combined with a Semi
Analytical Model (SAM) for galaxy formation. SAMs use merger trees from
dark matter only simulations and model the evolution of the baryonic component with approximate, yet physically motivated, analytical prescriptions.
While Fontanot et al. (2012, 2013) were able to address the effect of cosmology on global properties of galaxies (e.g. the cosmic star formation), due to
SAM’s limitations, they were not able to study the effects of Dark Energy
54
Galaxy Formation in Dynamical Dark Energy
parametrization on the internal structure of simulated galaxies.
In this chapter we show the first detailed study of the effect of dark
energy on galactic scale using high resolution hydrodynamical simulations
for three different disc galaxies. Our study is an extension of the MaGICC
project (Making Galaxies In a Cosmological Context) and we dubbed it
DarkMaGICC. The MaGICC project uses gasoline2 (see Section 3.2.1)
and has been quite successful in reproducing several properties of observed
disc galaxies, including star formation rates and stellar masses (Brook et al.
2012; Stinson et al. 2013b), metals production and distribution (Stinson et al.
2012; Brook et al. 2013b), flat rotation curves and cored profiles (Macciò et al.
2012; Di Cintio et al. 2014) and disc properties as observed in the Milky-Way
(Brook et al. 2013a; Stinson et al. 2013a).
We adopted the same set of numerical parameters describing the baryonic
physics as in Stinson et al. (2013b), and perform high resolution hydrodynamical simulations with different dark energy backgrounds, to study the
impact of dark energy on disc galaxy properties.
4.2.2
Results and Discussion
By using grafic2-de and gasoline-de (see Section 3.1.1 and 3.2.1), we
simulated three disc galaxies, galα, galβ, galγ, each of them in all five cosmological models. In Table 4.2 we summarize the main proprieties of the
chosen three disc galaxies in ΛCDM, SUCDM and waCDM2 cosmologies.
For all three galaxies the softening for gas and dark matter particles are
respectively 0.45 and 1 kpc. Note that, for all galaxies, the SUCDM equivalents are always the most massive and, on the other hand, the waCDM2
are always the least massive. Using hydrodynamical and dark matter only
simulations, in the next sections we present how galα, galβ and galγ evolved
and their z = 0 properties. These include dark matter distribution, gas,
star and total halo masses, star formation histories, rotation curves, and the
chemical enrichment of the galaxies.
Stellar and Halo Mass
Figure 4.3 shows dark matter profiles for galα in simulations with and without baryons for all different cosmological models; galβ and galγ show similar
results.
The five radial density profiles from the dark matter only simulations (left panel) are almost indistinguishable. This confirms previous
findings from N-body simulations, which showed that dark matter only simulations on galactic scales weakly depend on the dark energy model. The
right panel of Figure 4.3 shows the radial density profiles of dark matter
in hydrodynamical simulations. In contrast to the dark matter only simulations, the density profiles start differentiating with respect to the dark
energy model used.
55
galα
ΛCDM
SUCDM
waCDM2
galβ
ΛCDM
SUCDM
waCDM2
galγ
ΛCDM
SUCDM
waCDM2
Mvir
[M ]
MDM
[M ]
Mgas
[M ]
M∗
[M ]
7.74×1011
8.23×1011
7.33×1011
6.76×1011
7.12×1011
6.46×1011
5.34×1010
5.52×1010
5.68×1010
4.52×1010
5.62×1010
3.07×1010
5.65×1011
5.70×1011
5.45×1011
4.91×1011
4.97×1011
4.82×1011
3.48×1010
2.84×1010
3.47×1010
3.87×1010
4.50×1010
2.91×1010
3.44×1011
3.65×1011
3.35×1011
3.09×1011
3.24×1011
3.02×1011
2.76×1010
2.82×1010
2.69×1010
6.68×109
1.33×1010
5.77×109
Table 4.2: Physical properties of the three disc galaxies in ΛCDM , SUCDM
and waCDM2. We show virial mass (total mass), dark matter mass, gaseous
mass and stellar mass, all calculated within one virial radius.
DM-only simulation
107
hydro simulation
ρ/ρcr
106
105
104
LCDM
waCDM0
waCDM1
waCDM2
SUCDM
103
102
100
101
r [kpc]
102
100
101
r [kpc]
102
Figure 4.3: Dark matter density profiles of galα simulated in all five cosmological models, respectively in a dark matter only (left panel) and in a
hydrodynamical simulation (right panel). We plot the density in units of
critical density, as a function of the distance from the center of mass of the
galaxy.
56
Galaxy Formation in Dynamical Dark Energy
Figure 4.4: Values for stellar mass and halo mass at z = 0. Each star
represents a galaxy in a given cosmological model. The solid black line
shows the prediction from observational constraints and the shaded area its
2σ errors.
Figure 4.4 sets galα, galβ and galγ in the abundance matching plot at
z = 0. The abundance matching technique relates the galaxy luminosity
function at each epoch of interest to the dark matter halo and subhalo mass
function from N-body simulations. Galaxies are ranked by luminosity and
haloes by mass and matched one-to-one, so that lower luminosity galaxies
are associated with haloes of lower mass, and galaxies above a given luminosity threshold are assigned to haloes above a given mass threshold with the
same abundance or number density. Throughout this work we will compare
our simulations with the abundance matching predictions from Moster et al.
(2013), represented in Figure 4.4 by the black line. The shaded area represents the errors on the prediction. The abundance matching prediction does
not vary from ΛCDM to the other cosmologies since it is calculated using
the value for σ8 at z = 0, and all our cosmological models share the same
value. While statistical conclusions are not possible because of the limited
sample, the three galaxies do show the same trend as a function of cosmology. By simply varying the cosmological model, the change among the three
galaxies is of only a few percent in the dark matter mass, while the stellar
mass almost doubles. Galaxies simulated in the waCDM2 cosmology (red
symbols) always make the least stars at z = 0, while the galaxies formed in
the SUCDM cosmology (blue symbols) always make the most stars. As expected, galaxies formed in a ΛCDM cosmology always lie in the middle. The
hierarchy is in agreement with the behaviors of the cosmological background
evolutions of these cosmological models (see Figure 4.2 and Section 4.1),
since we expect more structures to be formed in a cosmological model that
57
Figure 4.5: Evolutions of the stellar-halo mass relation as a function of scale
factor for galα, galβ and galγ.
begins forming structures earlier.
Evolution of the stellar-halo mass relation
Figure 4.5 shows how the ratio of stellar mass and total mass evolve with
scale factor a = 1/(z + 1). Each panel relates to a specific galaxy and the
different colors describe each galaxy run in a different cosmology. Again, the
black solid line represents the expected evolution for a ΛCDM model using
the abundance matching technique. The predicted evolutions do change with
the change in cosmology, but they all do not distance themselves significantly
from the ΛCDM prediction. Hence, out of clarity, we have only plotted the
ΛCDM predicted behavior from abundance matching. As in the z = 0 case,
the stella-halo mass trends for the galaxies simulated in different cosmologies
are in agreement with the evolution of their cosmological backgrounds. In
the SUGRA cosmology (blue lines), we expect higher density perturbations
to compensate for the faster expansion of the universe. These higher density
perturbations trigger a more efficient star formation . This is in agreement
with previous works on SUCDM cosmology (e.g. Zhan and Knox 2004;
Puchwein et al. 2005; Jing et al. 2006; Maio et al. 2006; Rudd et al. 2008;
Casarini et al. 2011c; De Boni et al. 2011; van Daalen et al. 2011; Casarini
et al. 2012; Fedeli et al. 2012).
On the contrary, the waCDM2 galaxy (red lines) always makes less stars
throughout its evolution. The cosmological models waCDM0 and waCDM1
are not far apart from the ΛCDM model, in the wa −w0 plane, thus we would
expect galaxies that live in those models not to differ greatly from galaxies
that live in the ΛCDM universe. This expectation is nicely reproduced for
all three haloes.
As shown in Figure 4.5, it is noticeable how both galα and galβ undergo a
significant merger around a = 0.8 which raises their star formation efficiency
and increases their dark matter mass. The merger is visible also from the
dark matter mass of the haloes as a function of the scale factor. Figure 4.6
58
Galaxy Formation in Dynamical Dark Energy
Figure 4.6: Evolution of the dark matter mass (solid lines) and gas mass
(dashed lines) for galα, galβ and galγ in all different cosmological models.
For an easier comparison, the gas mass was increased of a factor of five.
Figure 4.7: Star formation histories for galα, galβ and galγ in all the cosmological models.
shows a clear increase in the dark matter mass due to the accretion of a
nearby satellite galaxy.
Star Formation Histories
Figure 4.7 shows the star formation rate (SFR) as a function of physical time.
Different cosmological models show longer or shorter ages of the universe
because how much physical time elapses as the universe expands depends on
the cosmology. The choice of showing the star formation in standard physical
units is made to give more insight. Figure 4.7 shows how dark energy can
suppress and delay star formation. Interestingly, waCDM2 cosmology (red
lines) delays star formation, both in the case of galα and galβ. In all three
galaxies the waCDM2 cosmology drastically suppresses star formation until
recent times.
As pointed out, both galα and galβ undergo a merger. The merger event
is clearly marked by the presence of a peak in the SFRs between 7 and 10 Gyr
(notice how the peak shifts in time according to the cosmological model).
After the star formation burst due to the merger, both galaxies decrease
59
Figure 4.8: For each galaxy we show the ratio of SFR/SFRΛ (Star Formation
Rate) for all cosmological models as a function of scale factor. The top panel
shows galα, the middle panel galβ and the bottom panel galγ.
their star formation activity due to the decrease in the amount of available
cold gas. This is shown in Figure 4.6, where we plot the evolution of the
dark matter mass and the cool gas mass (T<105 K).
In order to compare the effects of dynamical dark energy on galaxies
with different masses, in Figure 4.8 we show for each galaxy the ratio of
the star formation rate in a given cosmology and the star formation rate in
ΛCDM (SFR/SFRΛ ) for all cosmological models as a function of scale factor
to more easily compare with Fontanot et al. (2012). Fontanot et al. used
semi-analytic models to investigate differences in cosmic star formation in
dynamical dark energy cosmologies. They find a larger difference between
cosmological models at high redshifts. In our case the contribution of cosmology seems not to change significantly with redshifts. The reason for this
apparent inconsistency is that, while we are looking at the evolution of an
individual object, Fontanot et al. looked at the evolution of an entire population of galaxies. Clearly, the effect of dark energy on a single halo has to
be convolved with the evolution of the mass function itself. Convolving the
two components explains the stronger redshift evolution seen in Fontanot
et al.. To this stage, we note that, given our limited sample, the effect of
dark energy seems not to have a trend with the mass of the galaxy (taken
into account cosmic variance).
Rotation Curves
Different star formation histories reflect different matter distributions among
the galaxy, as the rotation curves in Figure 4.9 show. Galaxies with delayed
60
Galaxy Formation in Dynamical Dark Energy
Figure 4.9: Rotation curves for galα, galβ and galγ in all cosmological models.
star formation (waCDM2 cosmology, red lines in Figure 4.7) have flatter
rotation curves than galaxies where star formation started earlier (SUCDM
cosmology, blue lines), see Stinson et al. 2013b. Thus, a galaxy can have
a flat or centrally peaked rotation curve based simply on the background
cosmology in which it forms. Centrally peaked rotation curves have long
been the prime symptom of the overcooling problem in disc galaxy formation
simulations (Scannapieco et al., 2012). In the centers of haloes the gas
density becomes high enough that hot gas starts radiating and consequently
cools. In such environments, the cooling process is unstable because, once the
hot gas has cooled, it no longer pressure supports the surrounding gas, which
then becomes denser and cools. Stars then form in excess and primarily in
the central concentration, and they produce peaked rotation curves. Most
solutions have focused on adding energy from stars or AGN (Scannapieco
et al. 2012). Stinson et al. (2013b) showed one solution based on stellar winds
from massive stars (i.e. “early stellar feedback”). Our results show that also
cosmology can have a considerable effect on flattening rotation curves. This
work shows that simply changing the evolutions of the dark energy equation
of state flattens rotation curves of a considerable and definitely observable
amount (i.e. more than 100 km/s in both galα and galβ). Figure 4.10
compares rotation curves for galα in dark matter only simulations (left panel)
and in SPH simulations (right panel) for each cosmological model. The
change is striking. While in the dark matter only case the cosmological
models are almost indistinguishable, they become clearly distinguishable in
the hydrodynamical simulations and they vary of an observable amount.
Feedback and Cosmology, Metallicity Interplay
We showed that a model whose universe velocity expansion is slower compared to the one of ΛCDM (e.g. waCDM2, red lines) has a lower star formation till much later times, and on the other hand a model whose universe
velocity expansion is faster than ΛCDM (e.g. SUCDM, blue lines) has a
61
Figure 4.10: Rotation curve for galα in all different cosmological models,
respectively in a dark matter only simulation (left panel) and in a hydrodynamical simulation (right panel).
higher star formation at all redshifts (see Figure 4.7). We can trace back
this difference to the fact that all five different cosmological models have the
same σ8 today, because, in order for this to happen, structures in a SUCDM
model (blue lines) have to start forming earlier. This implies that, at the
starting redshift (z = 99 for all simulations), density perturbations that
seeded structure formation had to be slightly bigger in the SUCDM model
(blue lines) compared to the initial density perturbations in the waCDM2
model (red lines). Thus, stars will start forming earlier since more gas is
accreted and cools. These differences in the initial perturbations do not significantly affect properties of structures on galactic scales in dark matter only
simulations. On the other hand, the interplay between cooling, metallicity
and star formation not only helps differentiating between the cosmological
models but also enhances their differences. To highlight the “positive feedback” that star formation has on radiative cooling through metal enrichment,
Figure 4.11 shows the evolution of metallicity as a function of scale factor
for three different regions of galα and galβ, a central 2 kpc sphere (“bulge”),
a disc cylinder with radius 20 kpc and 6 kpc thickness (“disc”), and a sphere
of the size of the Rvir (“halo”). The waCDM2 model (red lines) exhibits
the lowest metallicity in the bulge and disc throughout its evolution, which
reflects its lower star formation rate and hence lower enrichment rate. The
effect of increased metal enrichment is non-linear: the more star formation
enriches gas, the faster the gas cools, and the more stars that subsequently
form.
The halo metallicity of waCDM2 is also lower throughout most of the
galaxy evolution, but becomes higher after a ∼ 0.75, as its mean halo metallicity continues increasing while the metallicity in the other models starts to
decrease or flatten out at that time. Both waCDM2 galaxies start to have
more metallicity in the halo due to Supernova explosions being able to move
62
Galaxy Formation in Dynamical Dark Energy
Figure 4.11: Mean metallicity in solar units for galα and galβ as function of
scale factor in “bulge”, “disc” and “halo”. Different colors represent different
cosmological models.
63
the gas outside from the disc.
Comparing the trends for metallicity (Figure 4.11), cool gas (Figure 4.12)
and star formation rates (Figure 4.7) as functions of scale factor, we find
them in agreement. Because of the lower metallicity, the waCDM2 model
(red lines) ends up having the least amount of gas that has been able to cool
and thus also makes the least amount of stars. Having used up a smaller
amount of cold gas at earlier times increases the amount of cold gas left
for star formation at late times. The presence of cool gas that has yet not
formed stars can be seen in Figure 4.12, where after a = 0.7 the disc of the
waCDM2 galaxy has the most amount of cool gas compared to the galaxies
in the other cosmological models. This is also the case for the galaxy halo,
and this is probably due to the cooled gas moved by supernova explosions.
Feedback–Dark Energy Degeneracy
Along with dark energy having a profound effect on disc galaxy evolution,
galaxy formation strongly depends on how feedback is modeled. The effects
of suppressing star formation, and hence flattening rotation curves, can also
be obtained by varying the feedback sub-grid recipes. As shown in Stinson et al. (2013b), pre-supernova feedback from massive stars significantly
suppresses star formation, which is the same effect seen in the dark energy
model that has the most delayed expansion (waCDM2). For this reason, we
wish to explore whether dark energy or stellar feedback have a greater effect
in suppressing star formation and whether their effect changes with redshift.
We select the waCDM2 cosmology (red lines in previous plots), which
showed the most star formation suppression and delay, and we re-simulate
it with a range of stellar feedback strengths. We vary both the supernova
feedback efficiency and the early stellar feedback separately. First, the early
stellar feedback is turned from 10% down to 0% efficiency with the standard
1051 erg of supernova energy. Then, with no early stellar feedback, the
supernova feedback strength is increased to 120% and 150%.
The left panel of Figure 4.13 shows each of these variations implemented
in the waCDM2 model for galα. The stellar mass evolution shows a strong
dependence on the early stellar feedback parameter. A decrease of 25%
to 7.5% increases the final stellar mass 50% and moves most of the star
formation from late to early times. All the simulations with less than 7.5%
efficiency, but more than 0 early stellar feedback end with nearly the same
final stellar mass. What is somewhat surprising is that the simulation with
no early stellar feedback ends with l ess stellar mass than these intermediate
feedback models. Stinson et al. (2013b) also saw this effect and found that it
was due to the higher star formation efficiency at early times driving stronger
outflows due to the greater supernova feedback. Thus, gas was driven to radii
where it could not be re-accreted, whereas the early stellar feedback does not
drive gas so far away.
64
Galaxy Formation in Dynamical Dark Energy
Figure 4.12: Cool gas in solar masses (T < 105 K) for galα and galβ as
function of scale factor in “bulge”, “disc” and “halo”. Different colors represent
different cosmological models.
65
Figure 4.13: Evolution of stellar-halo mass relation with scale factor. We
are now showing only the case of galα. In the left panel we have changed
the feedback parameters for the waCDM2 run. We have increased the early
stellar feedback parameter (ESF) from zero to the fiducial value (see Stinson
et al. (2013b)) while keeping the SN parameter fixed to the fiducial value
of 1.0 (solid lines). We then have changed the supernova parameter (SN)
while keeping the early stellar feedback fixed to zero (dashed and dash-dotted
cyan lines). The dotted black line is the abundance matching prediction from
Moster et al. 2013 and the shaded area its 2σ contours. In the right panel
we compare the effect of early stellar feedback feedback with the effect of
different cosmology, ΛCDM (red and black lines) and waCDM2 (blue and
red).
66
Galaxy Formation in Dynamical Dark Energy
Supernova feedback unambiguously decreases the amount of stars formed
throughout the galaxy’s evolution, but can easily push the trends out of the
expected behaviors from abundance matching techniques suggesting that
Supernova feedback alone is not enough to reproduce realistic galaxies.
The right panel of Figure 4.13 shows a comparison of the stellar masshalo mass evolution of the waCDM2 model (red lines) with the LCDM model
(black lines) for galα. The models separate most notably at late times. The
corresponding simulations with no early stellar feedback are shown in the
dashed lines. They clearly show that both cosmology and pre-SN feedback
have a strong effect in suppressing star formation, but while pre-SN feedback
has the strongest effect at high redshift, cosmology has the strongest effect
more recently (i.e. after a = 0.7).
The degeneracy between feedback and dark energy parametrizations certainly cannot be disentangled with our limited sample of galaxies, on the
other hand Fontanot et al. (2013) pointed out that a possible way to put
constraints on dynamical dark energy models would come from future missions, when we will be able to have more sensitivity at high redshifts.
4.2.3
Conclusions on Disc Galaxies simulations in Dynamical
Dark Energy
The intention of this work was to investigate for the first time the effect
of dark energy on galactic scales in SPH simulations. We find that the
dark energy modeling has an unexpected significant effect on disc galaxy
formation, on the contrary of what is most commonly believed.
The experiment used a suite of SPH multi-mass cosmological simulations with masses of 3.4 × 1011 M , 6.6 × 1011 M and 7.7 × 1011 M , in
four dynamical dark energy models plus the reference ΛCDM model. The
models all employed the same baryonic physics prescriptions. All dynamical
dark energy models are within the two sigma contours given by WMAP7.
We examined the dark matter distribution, gas, star and total halo masses,
star formation histories, rotation curves and the chemical enrichment of the
galaxies.
Changing the dark energy evolution implies changing the expansion rate
of the universe, which in turn affects the accretion history. We show how the
same galaxy evolved in different dark energy cosmologies does not present
significant differences in dark matter only simulations, while in hydrodynamical simulations the galactic properties change greatly.
At z = 0, the stellar mass inside one virial radius can vary by a factor of
2 depending on cosmology, while the dark matter mass only changes of a few
percent. Thus baryons amplify differences between dark energy models, as
the evolution of the stellar mass - halo mass ratio shows: by simply changing
the dark energy parametrization stellar mass either decreases or increase of
a factor of two throughout the whole galaxy evolution.
67
The reason why baryons amplify the differences among the various dynamical dark energy models lies on the non linear response of the hydrodynamical processes. Once the cosmological model introduces slightly different
density perturbations, feedback processes enhances those differences by producing slightly more (or less) stars. More stars introduce more metals in
the feedback cycle and more metals decrease the cooling time, which in turn
allows gas to cool faster and produce even more stars. Through the highly
non-linear response of baryons, dark energy models that would have been
indistinguishable from ΛCDM on galactic scales show distinctive features in
SPH simulations.
The distinctive features of dynamical dark energy become clear when
looking at the star formation rates. We find that certain dark energy models
are able to both delay and suppress star formation until recent epochs. The
delay in star formation is then in turn responsible for the flattening of rotations curves, where we show a change of about 100 km/s in the two most
massive galaxies we considered.
Finally we compare the effect of dynamical dark energy with the effect
of baryonic feedback. We keep the cosmology fixed (waCDM2) and change
the feedback parametrization. Provided that the Supernova feedback is kept
constant, at late times the effect of dark energy is comparable to the effect
of early stellar feedback (see Section 3.2.1 for details on feedback modeling).
Even the degree at which stellar feedback is able to flatten rotation curves,
is comparable to the effect of dark energy. We noted on the other hand, that
in order to obtain the behavior suggested by abundance matching considerations at high redshifts, early stellar feedback had to be introduced since at
high redshifts it has the most important effect compared to the dark energy
modeling.
Having shown that the dark energy modeling has an important effect on
disc galaxy formation and evolution, we would like to stress on the fact that,
especially in the era of high precision cosmology, the details on dark energy
do matter and certainly need further investigations.
4.3
4.3.1
Hydrodynamical Cosmological Volumes in Dynamical Dark Energy
Introduction
In Section 4.2 we have investigated high resolution single-object simulations
of galaxies. Despite the multi-mass technique being an extremely useful
tool to reach high resolutions and minimize the computational costs, only
some tens of objects can be usually simulated. Recently, Wang et al. (2015)
were able to build a larger ΛCDM set of single-galaxy simulations, which
accounts for 100 objects. When aiming at describing the behavior of a signif68
Galaxy Formation in Dynamical Dark Energy
icant distribution of galaxies (∼ thousands of objects at least), single-object
simulations can no longer be employed.
In this Section we will present hydrodynamical simulations of a large
volume of the universe (Lbox = 114 Mpc) in the dynamical dark energy
models described in Section 4.1. The aim is to test the effects of viable
dark energy models across a wide range of galaxy masses, environments and
merger histories. Our work is an extension of Kannan et al. (2014), where
they investigated the effectiveness of the feedback implementation as part of
the MaGICC project in a ΛCDM cosmology. In our case, we adopted the subgrid model for feedback proposed in Stinson et al. (2013b) and discussed in
Section 3.2.1 and we varied the dark energy equation of state. These are the
first fully hydrodynamical simulations of a significant volume of the universe
in non-ΛCDM cosmologies. This type of simulations treats hydrodynamics
in the entire box and all particles of each type have the same mass. Previous
studies have either been dark matter only (Klypin et al., 2003; Dolag et al.,
2004; Bartelmann et al., 2005; Francis et al., 2009; Grossi and Springel, 2009;
Li et al., 2012) or adiabatic hydrodynamical simulations, i.e. no gas cooling
(Baldi, 2012b; Carlesi et al., 2014).
Nowadays we can relay on large catalogues describing the galactic population in the local universe (Kajisawa et al., 2010; Karim et al., 2011; Santini
et al., 2012). These set observational constraints to statistical properties of
galaxies and can be used to determined whether the implemented sub-grid
modeling is able to accurately describe galaxy formation processes. Most importantly, data constrain the galaxy Stellar Mass Function (SMF), namely
the number density of galaxies as a function of their stellar mass.
In the following Section we show results related to the cosmological volumes in different dynamical dark energy models. We focus on the effects of
resolution compared to the multi-mass simulations described in Section 4.2
and finally we determine whether these cosmological models can be constrained by recent observations. This work has been part of a collaboration
with L. Casarini, who is responsible for running the simulation set. I carried
out the analysis with the supervision of A. V. Macciò.
4.3.2
Numerical and Cosmological Settings
We simulate a cosmological volume (114 Mpc)3 , starting at redshift z = 99.
Initial conditions were created with grafic2-de (see Section 3.1.1) using
the cosmological parameters from Section 4.2, i.e. at z = 0: Ωb0 = 0.0458,
ΩDM0 = 0.229, H0 = 70.2 km−1 s−1 Mpc−1 , σ8 = 0.816, ns = 0.968, where
these are density parameters for baryons and dark matter, Hubble constant,
root mean square of the fluctuation amplitudes and primeval spectra index.
We have two sets of simulations, a higher resolution case (HR) with 5123
particles and a lower resolution case (LR) with 2563 particles. In both cases
we simulated both a hydrodynamical and a dark matter only box (in the
69
hydrodynamical simulations the particle number doubles for both resolutions, accounting for dark matter and gas particles). For the HR simulations, due to the very high computational time (∼ six months), we selected
ΛCDM, waCDM0 and SUCDM cosmologies, while we are running the LR
in all cosmological models. In the HR case, the dark matter particle mass
is 3.5×108 M and the initial gas particle mass and initial star particle mass
are 6.9×107 M and 2.3×107 M . The softening for dark matter particles is
3.7 kpc, while for gas and stellar particles is 2.17 kpc. In the LR case, the
dark matter particle mass is 2.8×109 M and the initial gas particle mass and
initial star particle mass are 5.6×108 M and 1.9×108 M . The softening for
dark matter particles is 7.4 kpc, while for gas and stellar particles is 4.34
kpc.
We used the code gasoline2-de for all hydrodynamical simulations and
pkdgrav for the dark matter only runs. To identify haloes in each snapshots
we used the Spherical Overdensity (SO) algorithm (Macciò et al., 2008).
4.3.3
Results and Discussion
Halo Mass Function
In the ΛCDM LR dark matter only volume (2563 particles) we find about
4 000 halos with more than 100 particles at z = 4 and about 19 000 at z = 0.
While, in the ΛCDM HR dark matter only volume (5123 particles) we find
about 63 000 halos with more than 100 particles at z = 4 and 117 000 at
z = 2 (our lowest redshift).
In Figure 4.14 and 4.15, for the 2563 and 5123 sets respectively, we show
the ratio of halo mass functions to the ΛCDM case, namely the number
density of haloes as a function of their halo mass in the dark matter only
simulations. We choose not to include error bars since we are showing the
mass function ratio. Calculating the error
√ propagation on the ratio (where
we would use the Poisson relative error 1/ N with N number counts) makes
the ratio error bar depend on the ratio value itself, which in turn would not
give direct insight on the number of halos present in our analysis. Instead, we
give details on our Milky-Way-like sample in ΛCDM cosmology for general
guidelines. In the LR ΛCDM run we count about 2 000 halos with masses in
the range 11.8<log(Mhalo )<12.2 at z = 4 and about 8 600 at z = 0. In the
HR ΛCDM run we count again about 2 000 halos with 11.8<log(Mhalo )<12.2
at z = 4 and about 6 400 at z = 2.
Figure 4.14 and 4.15 show that the volume simulated in SUCDM cosmology features the highest number of haloes at higher redshifts. This is
the consequence of the choice of normalization, namely, all simulations have
the same σ8 at z = 0. Thus, having SUCDM a faster universe expansion
compared to the ΛCDM case, it must have an earlier formation of structures to reach the same σ8 at z = 0 than ΛCDM (see the evolutions of D+
70
Galaxy Formation in Dynamical Dark Energy
φ/φΛCDM
1.8
1.4
φ/φΛCDM
z= 3
1.4
1.0
1.0
0.6
11.8
1.8
1.4
12.2
12.6
13.0
0.6
12.0
1.8
z= 2
12.4
12.8
13.2
z= 1
1.4
1.0
1.0
0.6
11.8
1.8
φ/φΛCDM
1.8
z= 4
1.4
12.6
0.6
11.8
13.4
12.6
13.4
log(M [M ¯])
z= 0
1.0
0.6
11.8
12.5
13.5
14.0
LCDM
waCDM0
waCDM1
waCDM2
SUCDM
14.5
log(M [M ¯])
φ/φΛCDM
φ/φΛCDM
Figure 4.14: Ratio of the halo mass functions in different cosmologies to
the halo mass function in the ΛCDM case for the LR dark matter only
simulations (2563 ). We show snapshots at redshifts z = 4, 3, 2, 1, 0 in the
different panels.
1.8
1.4
1.0
0.6
1.8
1.4
1.0
0.6
z= 4
11.0
11.5
12.0
12.5
13.0
z= 2
1.8
1.4
1.0
0.6
z= 3
11.0
11.5
12.0
12.5
log(M [M ¯])
13.0
LCDM
waCDM0
SUCDM
11.0 11.5 12.0 12.5 13.0 13.5
log(M [M ¯])
Figure 4.15: Ratio of the halo mass functions in different cosmologies to
the halo mass function in the ΛCDM case for the HR dark matter only
simulations (5123 ). We show snapshots at redshifts z = 4, 3, 2 in the different
panels.
71
φ/φΛCDM
φ/φΛCDM
2.5
2.0
1.5
1.0
0.5
8
2.5
2.0
1.5
1.0
0.5
8
2.5
2.0
1.5
1.0
0.5
8
2.5
2.0
1.5
1.0
0.5
8
z =4
9
10
z =2
9
10
log(M ∗ [M ¯])
11
z =3
LCDM
waCDM0
waCDM1
waCDM2
SUCDM
9
9
10
z =0.8
10
log(M ∗ [M ¯])
11
11
12
Figure 4.16: Ratio of the stellar mass functions to the stella mass function
in the ΛCDM case for the LR hydrodynamical simulations (2563 ). We show
snapshots at redshifts z = 4, 3, 2, 0.8 in the different panels.
in Figure 4.2). As expected, differences in structure distributions between
cosmologies decrease going towards z = 0. waCDM0 shows a halo mass function remarkably close to ΛCDM at all examined redshifts; this was chosen by
construction when setting the combination of wa and w0 as done in Casarini
et al. (2009b). Results from LR and HR are globally in agreement and differences in the halo mass functions among cosmologies show an increase when
halo mass increases, as previously found in the literature (Klypin et al., 2003;
Dolag et al., 2004; Bartelmann et al., 2005; Francis et al., 2009; Grossi and
Springel, 2009).
Galaxy Stellar Mass Function
In the ΛCDM LR hydrodynamical volume simulation (2563 particles) we find
a total of about 900 galaxies at z = 4 and about 6 300 galaxies at z = 0.8 (our
lowest redshift), while in the HR ΛCDM run we find about 7 000 galaxies at
z = 4 and 25 700 galaxies at z = 2 (our lowest redshift).
In the LR ΛCDM run, we count about 52 galaxies with 9.8<log(M∗ )<10.2
at z = 4 and 884 galaxies at z = 0.8. In the HR ΛCDM run, we find 94
galaxies with 9.8<log(M∗ )<10.2 at z = 4 and 841 at z = 2.
In Figure 4.16 and Figure 4.17 we show the galaxy stellar mass functions for LR and HR cases. As expected, the number of galaxies with a
certain stellar mass is higher in SUCDM cosmology compared to ΛCDM or
waCDM2. Snapshots at redshift z = 4, 3 allow a comparison between the
two resolutions and, also in the hydrodynamical case, these globally agree.
72
φ/φΛCDM
φ/φΛCDM
Galaxy Formation in Dynamical Dark Energy
2.5
2.0
1.5
1.0
0.5
7
2.5
2.0
1.5
1.0
0.5
7
2.5
2.0
1.5
1.0
0.5
7
z =4
8
9
10
z =2
z =3
8
9
log(M ∗ [M ¯])
10
11
LCDM
waCDM0
SUCDM
8
9
10
log(M ∗ [M ¯])
11
Figure 4.17: Ratio of the stellar mass functions to the stella mass function
in the ΛCDM case for the HR hydrodynamical simulations (5123 ). We show
snapshots at redshifts z = 4, 3, 2 in the different panels.
Star Formation Efficiency and Resolution
Differences in the galaxy stellar mass functions are due to a combination
of two effects: an increased number of dark matter haloes and a possible
increase of star formation efficiency at a fixed halo mass. Thus, we are interested in disentangling the two quantities and focusing in the variations
of the star formation efficiency. In the case of single-object high resolution multi-mass simulations, described in Section 4.2, differences in the dark
matter accretion histories were amplified by the behavior of baryons. In
fact, thanks to the highly non-linear nature of the star forming mechanisms,
producing just slightly more stars leads to an increase of the inter stellar
medium metallicity and, as in a vicious cycle, higher metallicity leads to
a higher star formation. In turn, this process should be reproduced in all
galaxies in our volumes and eventually differences in stellar mass between
cosmologies should rise. To asses this, we look at the mean stellar mass of
the sample of galaxies with a certain halo mass. We are interested in checking whether a “typical galaxy”, given a halo mass, makes more or less stars
when cosmology changes. In Figure 4.18 and Figure 4.19 we normalize the
mean stellar mass to the mean stellar mass in the ΛCDM case, respectively
for the LR and HR cases.
In Figure 4.18 we show the behavior of the mean stellar mass as a function
of halo mass for z = 4, 3, 2, 0.8. First of all, we note that the ranking in
cosmological models is what we expected from Section 4.2, with SUCDM
having the largest mean stellar masses and waCDM2 having the littlest.
Secondly, the differences compared to ΛCDM are hardly reaching 20% for
all redshifts. In panels showing z = 2 and z = 0.8 there seem to be larger
73
­
® ­
M ∗ / M ∗ΛCDM
®
­
® ­
M ∗ / M ∗ΛCDM
®
1.4
LCDM
waCDM0
1.3 z =4
waCDM1
waCDM2
1.2
SUCDM
1.1
1.0
0.9
11.4 11.6 11.8 12.0 12.2 12.4
1.4
1.3 z =2
1.2
1.1
1.0
0.9
11.5
12.0
12.5
13.0
13.5
log(Mhalo [M ¯])
1.4
1.3 z =3
1.2
1.1
1.0
0.9
11.4 11.6 11.8 12.0 12.2 12.4 12.6 12.8
1.4
1.3 z =0.8
1.2
1.1
1.0
0.9
11.5 12.0 12.5 13.0 13.5 14.0
log(Mhalo [M ¯])
Figure 4.18: Ratio of the mean stellar mass to the mean stellar mass in the
ΛCDM case as function of halo mass for the LR hydrodynamical simulations
(2563 ). We show snapshots at redshifts z = 4, 3, 2, 0.8 in the different panels.
­
® ­
M ∗ / M ∗ΛCDM
®
­
® ­
M ∗ / M ∗ΛCDM
®
1.4
1.3 z =4
1.2
1.1
1.0
0.9
11.0 11.2 11.4 11.6 11.8 12.0 12.2
1.4
1.3 z =2
1.2
1.1
1.0
0.9
11.0 11.5 12.0 12.5 13.0 13.5
1.4
1.3 z =3
1.2
1.1
1.0
0.9
11.0 11.2 11.4 11.6 11.8 12.0 12.2 12.4 12.6
log(Mhalo [M ¯])
LCDM
waCDM0
SUCDM
log(Mhalo [M ¯])
Figure 4.19: Ratio of the mean stellar mass to the mean stellar mass in the
ΛCDM case as function of halo mass for the HR hydrodynamical simulations
(5123 ). We show snapshots at redshifts z = 4, 3, 2 in the different panels.
74
Galaxy Formation in Dynamical Dark Energy
galα
galβ
2.0
M ∗/M ∗ΛCDM
2.0
1.5
1.5
1.5
1.0
1.0
1.0
1
2
3
redshift
4
1
2
3
redshift
galγ
2.0
4
1
2
3
redshift
4
Figure 4.20: Ratio of the stellar mass to the stellar mass in the ΛCDM
case as function of redshift for the single-galaxy simulations studied in Section 4.2. The dynamical dark energy models are the same used in the volume
simulations.
differences but we would like to point out that for those masses feedback from
active galactic nuclei is believed to be significant and gasoline does not have
such feedback implemented. Thus we will not further discuss masses greater
than 1012 M . Furthermore, there seem to be an increase in the departures
from ΛCDM from redshift z = 4 onwards of about 5-10% at all masses. This
could be the effect of two distinct causes, (i) star formation at redshift z = 4
is not yet at its peak and thus has not reached its fully potential in fueling
differences between cosmological models, or (ii) the resolution is too low at
this redshift and the number of particles describing haloes is not sufficient
to correctly capture star formation.
Figure 4.19 shows a clearer trend with increasing masses. Differences between SUCDM and ΛCDM reach 20% for a 1012 M halo, while are negligible
till 3×1011 M for z = 4. This could have a double explanation: (i) differences are larger at 1012 M because this is the mass at which the peak for
star formation efficiency is reached (Moster et al., 2010), (ii) we do not have
enough resolution to describe objects less massive than 3×1011 M for z = 4.
At z = 4 differences in the LR case are only of 10%, while in the HR case
they reach 20% for masses around 3×1011 M , suggesting that resolution is
playing an important role in describing the star formation processes.
To better compare Figure 4.18 and Figure 4.19 with results from Section 4.2, in Figure 4.20 we show the ratio of stellar mass to ΛCDM stellar
mass for galα, galβ and galα. Departures from ΛCDM are clearly much
more significant for these three galaxies, especially in the SUCDM case,
where differences at z = 1, 2, 3, 4 range from 50% to 100%. Given the striking difference between the single-galaxy runs and the cosmological volumes
in departures from the ΛCDM reference model, it would be hard not to point
the finger at the role that resolution plays in enhancing differences between
star formations in the various cosmologies. As already pointed out, the nonlinear cycle involving star formation and metallicity is what fuels differences
75
15
Figure 4.21: The left panel shows the evolution of the stellar-halo mass relation as a function of scale factor for galβ for ΛCDM, waCDM2 and SUCDM.
Each cosmology has a high resolution (dashed lines) and low resolution (solid
lines) run. The high resolution is the
same resolution used in Section 4.2,
domenica 12 aprile 2015
while
the low resolution is comparable to the resolution in the HR volume.
mercoledì
15 aprile 2015
The right panel shows the stellar mass functions for ΛCDM, waCDM0 and
SUCDM for the HR volume. In red observational constraints from Santini
et al. (2012) are shown.
in the single-objects runs. When these processes cannot be represented by
a sufficiently high number of particles, their role in enhancing differences in
the accretion history due to cosmology is no longer dominant. To further
support the last statement, we ran galβ with the lowest possible resolution
for a multi-mass simulation, i.e. refinement factor RF = 2, while the simulations showed in Section 4.2 and in Figure 4.20 have RF = 8. This implies
that the gas mass resolution for the HR set is mgas = 6.9 × 107 M , while in
the single-galaxy runs mgas = 4.2 × 105 M . The evolution of the stellar-halo
mass relation as a function of the scale factor is shown in the left panel of
Figure 4.21. The differences between cosmologies are significantly reduced
when the resolution is lowered. Note that the dark matter particle mass in
the low-resolution multi-mass galβ is mDM = 1.4 × 108 M and in the HR
cosmological volume is mDM = 4.0 × 108 M .
Furthermore, the right panel of Figure 4.21 shows the mass functions for
the HR volume runs at z = 3. In red we show the observational constraints
from Santini et al. (2012) for redshift z ' 2.5. Despite the normalization of
the stellar mass functions that can be explained with a non-precise redshift
matching between our snapshot and the data, it is clear that the error bars on
observations are comparable to the difference that we find between SUCDM
and ΛCDM (which are the two most distant models in our choices). Thus, at
our resolution, viable dynamical dark energy models cannot be constrained
using current data on the galaxy mass function.
Fontanot et al. (2012) investigated the effects of early dark energy models
76
Galaxy Formation in Dynamical Dark Energy
0.0
0.5
w(a)
1.0
1.5
LCDM
waCDM0
waCDM1
waCDM2
SUCDM
EDE1
EDE3
EDE5
2.0
2.5
3.00.0
0.2
0.4
0.6
a =1/(1 + z)
0.8
1.0
Figure 4.22: Evolutions of the equation of state parameter w ≡ p/ρ for the
dynamical dark energy models used in this Chapter (solid lines) and for the
Early Dark Energy (EDE) models used in Fontanot et al. (2012) (dashed
lines).
(Wetterich, 1988) using Semi Analytical Models (SAMs). Early dark energy
models exhibit a non-negligible amount of dark energy at early times. In Figure 4.22 we show the evolutions of the equation of state parameter w ≡ p/ρ
for his choices of early dark energy parametrization (EDE1, EDE3, EDE5)
and our choices for dynamical dark energy models. Fontanot et al. concluded
that, for their models, differences due to cosmology were not significant given
the current data precision and were especially evident at high redshifts. In
Section 4.2 we attributed their small differences between cosmological models compared to our findings to the fact that SAMs cannot account for the
large role of non-linearity in star formation processes. The same happens
in hydrodynamical simulations when resolution is not large enough to fully
describe the interplay between star formation and metallicity. Thus, SAMs
or hydrodynamical cosmological volume simulations (at this state) are not
the best tools to set precise constraints to differentiate among dynamical
dark energy models.
4.3.4
Conclusions on Hydrodynamical Cosmological Volumes
in Dynamical Dark Energy
We ran cosmological hydrodynamical simulations of a volume with side
Lbox = 114 Mpc in dynamical dark energy models plus the ΛCDM reference model in two resolutions, Np = 2563 and Np = 5123 . The code used
is gasoline-de for hydrodynamical runs, which features gas cooling, supernova feedback and feedback from young massive stars, while we used
pkdgrav for dark matter only runs. The dynamical dark energy models
77
chosen lie within the 2σ constraints from WMAP7. Our goal was to test
the effects of dynamical dark energy on galaxy formation across a wider
range of galaxy masses, environments and merger histories compared to the
single-object runs shown in Section 4.2 in the same cosmological models.
In the cases of single-galaxy runs we find that baryonic processes enhance
already present differences in dark matter accretion histories across the various dynamical dark energy models, and produce an increase in stellar mass
that can reach a factor of two in SUCDM compared to ΛCDM, while dark
matter mass only increases of a few percents. We ascribe this boost to the
feedback cycle that connects the increase in star formation to an increase
of metallicity, which is in turn connected to a decreasing cooling time. We
expected the same process to be manifest at all mass scales, possibly with a
different degree of efficiency as a function of halo mass.
For all examined redshifts, z = 0.8, 2, 3, 4, we find that in the galaxy
sample contained in our cosmological volumes the increase in the mean stellar
mass given a halo mass is at maximum 20% in SUCDM, our most distant
model from ΛCDM. We attribute the lower differences among cosmologies
compared to the single-galaxy runs to the change in resolution. In fact,
in order for baryonic feedback processes to be well sampled they require a
high enough resolution. For comparison, the mass resolution of the most
refined of the two sets of volumes (Np = 5123 ) is for gas particles (initial
masses) mgas = 6.9 × 107 M , while in the single-galaxy runs mgas = 4.2 ×
105 M . To support the resolution argument, we ran one of the single-galaxy
simulations with a resolution for gas particles mgas = 2.7 × 107 M and
differences between cosmologies became comparable to the differences found
in the hydrodynamical volumes.
To conclude, we would like to stress that in the era of precision cosmology
making detailed numerical predictions is essential. In this Section we showed
that in order to distinguish between ΛCDM and a dynamical dark energy
models we need to employ high enough resolutions to fully describe the
baryonic processes that fuel differences between cosmologies.
4.4
4.4.1
Dwarf Galaxies Simulations in Dynamical Dark
Energy and ΛCDM
Introduction
In Section 4.2 we investigated the effects of dynamical dark energy on disc
galaxies with masses ∼5×1011 M . We found that the effects of dynamical
dark energy on the star formation efficiency were significant and they shaped
galaxy properties. In this Section we aim at extending our study to dwarf
mass scales, precisely to haloes with masses between a few 109−10 M .
Inconsistencies in the ΛCDM paradigm are mostly showing at dwarf
78
Galaxy Formation in Dynamical Dark Energy
galaxy scales. ΛCDM predicts an extremely low efficiency in converting gas
into stars at these halo masses, given that in dark matter only simulations
the number of subhaloes in a Mily-Way-size host halo is much higher than the
number of satellite dwarf galaxies observed (Klypin et al., 1999; Moore et al.,
1999). Furthermore, recent claims (Boylan-Kolchin et al., 2011, 2012) suggested that the central density of these dwarf haloes must be lower than expected from dark matter only simulations (Flores and Primack, 1994; Kuzio
de Naray et al., 2008; de Blok et al., 2008), questioning whether dwarf haloes
can be described by a Navarro, Frenk and White (NFW) profile (Navarro
et al., 1997).
Dekel and Silk (1986) and Bullock et al. (2000) suggested that the low
efficiency in converting gas into stars for dwarf haloes could be attributed to
both feedback from supernovae and from the UV ionizing background. In
fact, supernova explosions not only heat the gas but, for these low masses,
can also easily blow the gas away from the halo; while the existence of a
UV background radiation prevents less massive haloes from cooling gas and
start star formation. Thus, the possibility for a dwarf halo to form stars or to
remain dark until z = 0 is strictly connected to how much its virial mass has
grown by the time reionization happens. As shown in Section 4.2, varying
the background cosmology translates into varying accretion histories, so that
for all redshifts galaxies in SUCDM were always more massive than their
equivalent galaxies in ΛCDM. These variations were only of a few percents
in the case of disc galaxies, but for dwarf galaxies such variations in mass
can become crucial in determining whether star formation will start at all.
Supernova feedback can also be responsible for halo density profile that
is less cuspy when rapid starburst events occur. These can transfer energy to
dark matter particles which in turn will make the system expand (Navarro
et al., 1996; Governato et al., 2010; Pontzen and Governato, 2012; Di Cintio
et al., 2014) and thus lower the density in inner regions. On the other hand,
Trujillo-Gomez et al. (2015) find that in their simulations the formation of a
core in the dwarf density profiles is strictly linked to radiation pressure more
than feedback from supernovae. Recently Oñorbe et al. (2015) simulated
dwarf haloes with Mvir =109.5−10 M and they were able to produce a cored
density profile when the dwarf star formation was mainly happening at later
times. A delayed star formation as cause for lower density in the halo center
was also a proposed argument in Stinson et al. (2013b) and is the reason
for flat rotation curves in the waCDM2 models discussed in Section 4.2 (see
Figure 4.9). The effects on the density profiles are less evident for these more
massive galaxies, but anyhow Figure 4.3 shows the slightly less cuspy profile
of the waCDM2 case. Based on these reasons, we are interested in extending
the dynamical dark energy analysis to dwarf galaxy mass scales.
The dwarf galaxies from Oñorbe et al. (2015) reproduce the observationalinferred relation between stellar mass and dark matter halo mass obtained
via the abundance matching technique, but one could argue that a larger
79
set of dwarf halo simulations is needed to better sample the behavior of star
formation efficiency in dwarf haloes. Boylan-Kolchin et al. (2011) pointed
out that there could be a mass scale under which star formation will become
stochastic, and, more recently, Sawala et al. (2014a,b) suggest that the mass
scale could be Mvir ' 109−9.5 M . To better test these suggestions with a
larger sample of dwarf galaxies, in the second part of this Section we will
describe our study on 22 dwarf galaxies with masses ∼109−10 M within a
ΛCDM scenario.
This is part of the study that will be published in Macciò, Penzo et al.
in preparation. I and A. V. Macciò performed simulations and carried out
the analysis for half of the ΛCDM dwarf galaxy sample each. Furthermore,
I ran and analyzed the set of dwarf galaxy simulations in dynamical dark
energy.
4.4.2
Results and Discussion
We present a set of four dwarf galaxies with virial masses of a few 109 M .
We chose the two most distant cosmological models based on the results
from Section 4.2, waCDM2 and SUCDM. We selected the galaxies from a
volume of side 15 Mpc/h and then re-simulated them at a much higher resolution. Particle masses are mDM =4.1 × 103 M , mgas =8.2 × 102 M (initial),
mstar =2.7×102 M (initial), while the softening length for dark matter particles is 64 pc and for stellar and gas particles is 29 pc. Initial conditions were
generated using grafic2-de and they were evolved using gasoline-de (for
details on the codes, see Sections 3.1.1 and 3.2.1). Our aim is to study dwarf
galaxies as satellites of a Milky-Way size halo, because of this, each galaxy
was evolved from z = 99 to z = 1 assuming that for z ≥ 1 they would evolve
as single objects and for z ≤ 1 the satellites would be accreted and would
start to feel the influence of the parent halo.
Stellar-Halo Mass Relation
Figure 4.23 shows stellar and halo masses for our four dwarfs in ΛCDM,
waCDM2 and SUCDM cosmologies. We show the prediction from abundance matching (Moster et al., 2013), but we should remark that this is an
extrapolation to lower masses of the original relation. The hierarchy of cosmological models is maintained, with the SUCDM dwarfs making more stars
than ΛCDM and waCDM2, but the difference in stellar mass compared to
ΛCDM varies in the four dwarfs and is negligible in two out of four cases.
That differences between cosmological models vary among our four cases is
in agreement with the statement that star formation becomes stochastic at
these mass scales, thus also the enhancement due to baryonic processes of
differences between cosmological models is expected to vary. In the case of
the dwarf with the lowest halo mass, its waCDM2 realization does not make
80
Galaxy Formation in Dynamical Dark Energy
Figure 4.23: Stellar-halo mass relation for our four dwarf galaxies in ΛCDM,
waCDM2 and SUCDM cosmologies. The black line shows an extrapolation
to lower masses of the prediction from abundance matching. Symbolically,
the empty red circle represents the waCDM2 dwarf galaxy that did not make
stars. Different realizations of the same galaxy in the various cosmologies
are grouped by the grey ellipses.
lunedì 27 aprile 2015
81
lunedì 27 aprile 2015
Figure 4.24: Each row shows the Star Formation Rate (SFR) of one of the
three dwarf galaxies in ΛCDM, waCDM2 and SUCDM cosmologies. Virial
mass is written in each panel.
any stars and will remain dark throughout its life. This shows that even
small departures from ΛCDM can affect the number of luminous subhaloes
when their virial mass is around a few 109 M .
Star Formation Histories
Figure 4.24 shows the star formation histories for our four dwarf galaxies.
Each row represents one galaxy and each column a cosmological model. Our
four SUCDM dwarf galaxies show a global increase of star formation compared to ΛCDM and, on the other hand, the three dwarf galaxies in waCDM2
show a decrease compared to their ΛCDM realizations. The trend of delaying star formation in waCDM2 and, on the other hand, anticipating star
formation in SUCDM is no longer clear at these masses as it was for disc
galaxies in Figure 4.7. Even given the limited number of simulations, we
can notice quite different behaviors even within cosmologies, supporting the
suggested stochasticity on these masses (Boylan-Kolchin et al., 2011; Sawala
et al., 2014a,b).
82
Galaxy Formation in Dynamical Dark Energy
4.4.3
Remarks
We conclude our experiment on dynamical dark energy at dwarf mass scales
by stating that the impact that cosmology has on our four dwarf galaxies is
not as significant as in the case of ∼5×1011 M galaxies showed in Section 4.2.
Stellar mass at z = 1 has important differences only in two out of four cases.
Star formation histories are globally enhanced or decreased but no delay
or anticipation shows, as instead was the case for ∼5×1011 M galaxies.
Furthermore, for all our dwarf galaxies, density profiles do not vary between
different dynamical dark energy models (differences are less than few percents
at inner radii). On the other hand, diversity in star formation histories
within a given cosmological model may be important. Indeed, Figure 4.24
shows a significant difference in star formation histories between the two most
massive dwarfs galaxies which have comparable masses; in all cosmologies
we note a significantly higher star formation in the 6.3×109 M galaxy.
4.4.4
ΛCDM Follow-Up
To better investigate what fuels these different star formation histories, in
the next Section we will increase the dwarf sample and focus on the ΛCDM
model. This is part of the study that will be carried out in Macciò, Penzo
et al. in preparation.
Figure 4.25 shows our sample of 22 galaxies; circular symbols indicate
galaxies that remained dark. Resolution and settings are the same as described in the previous Section. The solid line shows the extrapolation of
the abundance matching relation from Moster et al. (2013), suggesting a
quite large scatter (∼ a factor of ten) around the extrapolated relation. To
be noted that 7 out of 22 galaxies never started forming stars. Their halo
masses span from 8×108 M to ∼4×109 M . In our sample we have 8 galaxies in the same halo mass range that on the contrary were active in forming
stars, suggesting that halo mass is not uniquely what determines these different star forming modes.
As suggested in the previous Section, at a given halo mass, we find heterogeneous star formation histories. Figure 4.26 shows star formation histories
for all our dwarf galaxies that formed stars. Virial masses in M are shown
for each panel. We note that generally more massive systems tend to have
more active star formation, but we can find systems with comparable masses
that have very different star formation behaviors.
To better investigate what triggers such variety of star formation modes,
we investigate four galaxies with comparable masses, dwarfα (blue), dwarfβ
(red), dwarfγ (cyan) and dwarfδ (black). Precisely, their masses are 5.0×109 M ,
5.7×109 M , 6.0×109 M and 1.0×1010 M . The left panel of Figure 4.27
shows their star formation rates with a different binning compared to the
previous figures. The right panel shows density profiles normalized to the
83
LCDM
Moster+ (2013)
Stellar Mass [M ¯]
107
106
105
104
109
Halo Mass [M ¯]
1010
Figure 4.25: Stellar-halo mass relation for our 22 dwarf galaxies in ΛCDM
cosmology at z = 1. The black line shows an extrapolation to lower masses of
the abundance matching from Moster et al. (2013). The round solid symbols
represent galaxies that did not make any stars.
radius at which the density profile slope reaches −2 (dashed lines show an
NFW profile). While for dwarfβ the scatter is high for the inner points, we
can confidently say that dwarfδ does show a cored density profile. Oñorbe
et al. (2015) also were able to produce cores in their two dwarf galaxies,
which have virial masses that are comparable with the virial mass of dwarfδ.
Figure 4.28 shows for dwarfα, dwarfβ, dwarfγ the evolution of the amount
of dark matter mass contained in a sphere centered on the potential minimum
of each halo with radius 4 kpc (left panel) and 2 kpc (right panel). Note
that virial radii for these systems are ∼35-45 kpc, thus we are estimating the
change in dark matter mass in their centers, regions where star formation
is expected to happen. The steep increase shown in dwarfβ and dwarfγ is
the result of a major merger, which is what triggers further star formation
episodes in these two systems. As a result, their stellar masses at z = 1 is one
order of magnitude higher than the stellar mass in dwarfα. To justify this
claim, in Figure 4.29 we show density maps (first row) and gas temperature
maps (second row) for a cube of size 20 kpc for dwarfβ. Panels are snapshots
taken at z = 4.97, 4.49, 3.98. The major merger can clearly be followed in
the dark matter row. The gas temperature snapshots show a temperature
increase after the major merger happened. At this high resolution, we can
84
Galaxy Formation in Dynamical Dark Energy
0.007
0.005
0.003
0.001
SFR [M ¯ yr−1 ] SFR [M ¯ yr−1 ] SFR [M ¯ yr−1 ] SFR [M ¯ yr−1 ] SFR [M ¯ yr−1 ]
1.8e9
2.0e9
4.5e9
LCDM
0.007
0.005
0.003
0.001
4.1e9
3.9e9
5.3e9
0.007
0.005
0.003
0.001
3.5e9
4.5e9
4.5e8
0.007
0.005
0.003
0.001
3.0e9
5.0e9
6.0e9
0.007
0.005
0.003
0.001
0
5.7e9
1.0e10
1.3e10
1
2
3
4
time [Gyr]
5
60
1
2
3
4
time [Gyr]
5
60
1
2
3
4
time [Gyr]
5
6
Figure 4.26: Each panel shows the Star Formation Rate (SFR) of one of
the 15 dwarf galaxies that succeeded in forming stars in ΛCDM cosmology.
Virial mass in M is written on each panel.
85
Halo to Halo scatter
Halo to Halo scatter
Figure 4.27: The left panel shows star formation rates of four dwarf galaxies
Variety of SFH at afrom
fixed
halo
mass!
our sample,
dwarfα
(blue), haloes
dwarfβ (red),
dwarfγ accretion!
(cyan) and dwarfδ
Cored
and
cusped
before
(black), with masses 5.0×109 M , 5.7×109 M , 6.0×109 M and 1.0×1010 M .
The right panel shows density profiles for the same galaxies with distance
from the center normalized to the radius at which the density profile slope
is −2.
domenica 19 aprile 2015
fully describe the blast wave that is being produced. The presence of such
strong increase in temperature is due to the strong feedback, which is an
evidence for a significant increase in star formation. As a consequence, we
find that more than 85% of the stellar mass is produced during the merger.
4.4.5
Conclusions on Dwarf Galaxies Simulations in Dynamical Dark Energy and ΛCDM
In this Section we aimed at studying the properties of simulated satellites
of Milky-Way size galaxies. To do so, we built a suite of hydrodynamical
single-object simulations of dwarf galaxies with masses ∼109−10 M . A subsample of four objects were run in waCDM2 and SUCDM dynamical dark
energy models described in Section 4.1. Our goal is to study the properties
of dwarf satellites before the in-fall into their parent halo. For this reason, we
simulated isolated dwarf galaxies from z = 99 to z = 1, with the assumption
that for z > 1 the influence of the parent halo can be neglected. At these mass
scales, galaxies have typical ratios M∗ /MDM ∼ 10−5 and R∗ /RDM . 0.001,
which make them interesting environments to test the effects of baryons on
the dark matter distribution.
Comparing our findings on dwarf galaxies in dynamical dark energy models with Section 4.2, differences in stellar mass and in star formations histories
compared to the ΛCDM reference models are less significant for dwarf scales
than for disc galaxy scales. Since differences between cosmologies are fu86
Galaxy Formation in Dynamical Dark Energy
Figure 4.28: We show the dark matter mass evolution for three dwarf galaxies
with masses 5.0×109 M (blue), 5.7×109 M (red) and 6.0×109 M (cyan) for
a sphere centered on the halo center with radius 4 kpc (left panel) and 2 kpc
(right panel).
lunedì 20 aprile 2015
Figure 4.29: First row shows dark matter density, while the second row the
gas temperature at three different redshifts z = 4.97, 4.49, 3.98.
martedì 28 aprile 2015
87
eled by baryonic feedback mechanisms, we expected larger departures from
ΛCDM close to the peak of star formation efficiency (∼ 1012 M ). We as
well realized that scatter in star formation histories within a given cosmology among different dwarf galaxies was more significant than the scatter
among cosmologies given a certain galaxy. Due to this, we enlarged the
sample and focused only in a ΛCDM cosmology.
In our sample of 22 dwarf galaxies, we find that 7 out of 13 haloes with
masses 8 × 108 . Mhalo /M . 4 × 109 do not form any stars. The reason
for these haloes to remain dark lies in the presence of the UV background
radiation from the reionization epoch. We turned off its contribution in one of
these simulations and the gas in the halo was able to cool and, consequently,
it started star formation.
Our sample follows the extrapolation of the abundance matching prediction for the stellar-halo mass relation at z = 1, but shows a large scatter (∼
factor of 10). We find a variety of star formation histories which supports
the suggestion that star formation at these mass scales might be stochastic. Though, the stochasticity of star formation can be connected to the
stochasticity of halo formation histories. We find that whether a dwarf halo
undergoes major mergers in its recent past or not produces very different
galaxy properties at fixed halo mass. Because of this, we expect dwarf satellites to be heterogeneous objects even before their in-fall due to their very
different formation histories.
88
Chapter 5
Halo and Subhalo properties in
Coupled Dark energy
Despite the highly successful inflationary ΛCDM paradigm, the fundamental problems associated with the introduction of a cosmological constant,
namely fine-tuning and coincidence problems (Weinberg, 1989), have served
as motivations for alternative descriptions of the dark sector.
Given the currently still unknown nature of the dark sector, the possibility of a non-null coupling between dark matter and dark energy has been
considered (see Chapter 1.4.2). Since in these models dark matter and dark
energy density evolutions are strongly coupled, the coincidence problem outlined in Section 1.3.4 is alleviated.
The effects of such interaction might be seen on the Cosmic Microwave
Background (CMB), on supernovae and on the growth of structures, as
pointed out by Matarrese et al. (2003); Amendola (2004b); Amendola et al.
(2004); Koivisto (2005); Guo et al. (2007) and many others. Structure formation has been as well investigated via numerical simulations by Macciò
et al. (2004); Baldi et al. (2010); Li and Barrow (2011); Carlesi et al. (2014)
and their follow up works, where the statistical distribution of structures
has been studied. Both Baldi et al. (2010) and Carlesi et al. (2014) found
that, when introducing a coupling between dark energy and dark matter,
halo concentrations decrease.
This Chapter is based on Penzo et al. (2015), and will describe the analysis of the first high resolution simulations on galactic scales in coupled dark
energy cosmology. The aim is to obtain high enough resolutions to investigate the properties of the dark matter distribution at sub-galactic scales,
mass scales at which the effects of the coupling have not yet been studied.
The subhalos will in turn be the hosts of dwarf galaxies and their properties can be compared with observations of satellite dwarf galaxies of both
Milky Way and Andromeda. In fact, despite ΛCDM predictions on large
scales being in very good agreement with galaxy clustering surveys (Jones
89
et al., 2009; Alam et al., 2015), on galactic scales challenges between ΛCDM
predictions and observations have appeared.
Firstly, the missing satellites problem, i.e. overabundance of substructures in ΛCDM Milky-Way size halo simulations when compared to observations of the Milky Way dwarf galaxies (Klypin et al., 1999; Moore et al.,
1999). On the other hand, as showed in Madau et al. (2008) and Macciò et al.
(2010), accounting for the baryonic physics drastically reduces the number
of visible satellites.
Secondly, the core/cusp problem, namely the inconsistency between the
constant density cores estimated from observations and the cuspy inner density profiles found in ΛCDM simulations. See Flores and Primack (1994);
Moore (1994); Diemand et al. (2005); Gentile et al. (2009); Walker and Peñarrubia (2011); Agnello and Evans (2012); Salucci et al. (2012), but also van
den Bosch and Swaters (2001); Swaters et al. (2003); Simon et al. (2005).
While this inconsistency can be attributed to baryonic feedback processes
(e.g. Governato et al., 2012; Di Cintio et al., 2014; Oñorbe et al., 2015), for
the case of Milky Way satellites the baryonic explanation is not straightforward since these objects can be almost completely dark matter dominated.
Baldi et al. (2010) and Carlesi et al. (2014) showed that for halos with
M & 1013 M the coupling between dark matter and dark energy produces
density profiles that are less cuspy in the inner density regions, which can
help alleviating the core/cusp problem. The aim of the work presented in
this Chapter is to investigate whether this effect persists at lower masses
M . 1012 M .
Moreover, concentrations of the most massive subhalos orbiting around
a ΛCDM Milky-Way size halo seem to be too high to be hosting the brightest dwarf galaxies observed. This translates into a prediction from ΛCDM
numerical simulations for the existence of massive dark matter subhalos that
seem to have failed at forming stars, and is known as the too big to fail problem (Boylan-Kolchin et al., 2011; Lovell et al., 2012; Rashkov et al., 2012;
Tollerud et al., 2012).
Whether these issues bring serious challenges for the ΛCDM model or
whether they can entirely be treated by invoking baryonic physics is currently
under debate. The aim of this study is to investigate the properties of
halos and their sub-structures to determine whether coupled dark energy
cosmologies can alleviate the aforementioned issues.
This work is based on Penzo et al. (2015), a result of the collaboration
with M. Baldi, who performed the numerical simulations. I produced the
initial conditions and carried out the analysis, under the supervision of A.
V. Macciò.
90
Halo and Subhalo Properties in Coupled Dark Energy
5.1
Details on the Cosmological Models
As described in Section 1.4.2 and 2.1.1, the main effects of an interaction
between dark matter and dark energy are: (i) dark matter particles masses
change with time, (ii) the gravitational strength felt by dark matter particles
is stronger than in Newtonian dynamics, (iii) an extra friction term appears,
which accelerates dark matter particles in the direction of their motion. We
plan to investigate how these departures from ΛCDM influence galactic and
sub-galactic scales.
Based on Baldi et al. (2010) and Baldi (2011a), we chose three coupling
scenarios. EXP003 (cyan) and EXP008e3 (blue) are observationally viable
models (see Pettorino et al. 2012 for CMB constraints on the coupling value),
respectively with a constant coupling βc = 0.15 and a coupling βc (a) varying
with the scale factor. We choose the parametrization
βc (a) = β0 eβ1 φ(a)MPl ,
(5.1)
where β0 = 0.4 and β1 = 3. We also explore an extreme constant coupling
case EXP006 (red) with βc = 0.3 (about 6σ outside present observational
limits, Pettorino et al. 2012) to better understand the implications of the
coupling. All the models share the same cosmological parameters and have
at z = 0: Ωb0 = 0.0458, ΩDM0 = 0.229, H0 = 70.2 km s−1 Mpc−1 , σ8
= 0.816, ns = 0.968, where these parameters are density parameters for
baryons and dark matter, Hubble constant, root mean square of the fluctuation amplitudes and primeval spectra index.
5.2
Numerical Set-Up
We first generated two sets of uniform particle distributions, a 80 Mpc/h
box and a 12 Mpc/h box, both with 3503 particles. We produced the initial
conditions with grafic-2 (see Section 3.1.1), which requires background
evolution and transfer functions. In the left panel of Figure 5.1 we show the
evolution of the linear growth factor D+ for all four cosmological models.
The transfer functions for ΛCDM have been produced using camb (Lewis
and Bridle, 2002), while the transfer functions for the coupled dark energy
models are obtained by weighting the ΛCDM transfer functions with the D+
of the coupled model. we created all initial conditions by using the same
random seeds, in order to be able to identify structures among the models.
Initial conditions were then evolved with the code gadget-2 described
in Section 3.2.2. The right panel of Figure 5.1 shows the ratio between the
mass functions for the coupled dark energy cosmologies with the one from
ΛCDM for redshifts z = 2, 1, 0 for the 80 Mpc/h boxes.
We then chose four dark matter halos in the ΛCDM 80 Mpc/h box and
one dwarf halo in the ΛCDM 12 Mpc/h box, and looked for their corresponding realizations in the coupled dark energy simulations. Halos were chosen so
91
3
Figure 1. Linear growth factor evolutions for all cosmologies
Figure 1. toLinear
growth
for all
cosmologies
normalized
today’s
values factor
dividedevolutions
by the ΛCDM
evolution.
normalized to today’s values divided by the ΛCDM evolution.
3
Figure 2. Ratio between the mass functions for the coupled dark
Figure
2. Ratio and
between
the mass
the coupled
energy
cosmologies
the one
from functions
ΛCDM forforredshifts
z = dark
energy
cosmologies
andboxes
the one
from
ΛCDM for redshifts z =
2, 1,
0 for the
cosmological
of size
80 Mpc/h.
2, 1, 0 for the cosmological boxes of size 80 Mpc/h.
Figure 5.1: The left panel shows the linear growth factor evolutions for all
in a coupled dark energy cosmology for the limit of a light
cosmologies
normalized
tothetoday’s
values divided by the ΛCDM evolution.
in a coupled
dark
energy
for
limit of a light
scalar
field (see
Baldi
et al.cosmology
2010 for calculation):
All the models share the same cosmological parameters and
scalar field
(see panel
Baldi et al.
2010 forthe
calculation):
� mratio
The
right
shows
between
the
for
theHcoupled
r̄
Allatthe
models
share
the same
parameters
have
zmass
=
0: Ω functions
= 0.0458,
Ω cosmological
= 0.229,
= 70.2 km and
v¯˙ = β(φ)φ̇v̄ + G[1 + 2β(φ) ] �
.
(5)
m
r̄
|r̄ |j ij
2
have
at
z
=
0:
Ω
=
0.0458,
Ω
=
0.229,parameters
H0 = 70.2 km
s
Mpc
,
σ
=
0.816,
n
=
0.968,
where
these
DM
b
˙
v¯i = β(φ)φ̇v̄
. one(5)from −1
i + G[1 + 2β(φ) ] and the
dark energy
cosmologies
ΛCDM
for
redshifts
z
=
2,
1,
0 Hubfor
3
−1
|r̄ij |
Mpc parameters
, σ8 = 0.816,
ns = 0.968,
these parameters
ares density
for baryons
and where
dark matter,
j�=i
The term
β(φ)φ̇v̄ accelerates
dark matter
particles
inMpc/h.
the
bleare
constant,
mean square
the fluctuation
densityroot
parameters
for of
baryons
and darkamplitudes
matter, Hubthe
cosmological
boxes
of
size
80
direction of their motion and thus lowers halo concentrai
i
j ij
3
2
j�=i
−1
ij
−1
DM0
b0
8
0
s
0
0
i
martedì 28 aprile 2015
The term β(φ)φ̇v̄i accelerates dark matter particles in the
tions.
direction of their motion and thus lowers halo concentraBased on Baldi et al. (2010) and Baldi (2011), we chose
tions.
threeBased
coupling
EXP003
are we
obseron scenarios.
Baldi et al.
(2010) and
and EXP008e3
Baldi (2011),
chose
vationally
viablescenarios.
models (see
Pettorino
al. 2012 for
three coupling
EXP003
andet
EXP008e3
areCMB
obserconstraints
on
the
coupling
value),
respectively
with
a
convationally viable models (see Pettorino et al. 2012 for CMB
stant
coupling
= coupling
0.15 and value),
a coupling
β(φ) varying
constraints
on βthe
respectively
with with
a conredshift
(for more
details
these
models we
refer
to Baldi
stant coupling
β =
0.15 on
and
a coupling
β(φ)
varying
with
2012).
We
also
explore
an
extreme
constant
coupling
case
redshift (for more details on these models we refer to Baldi
EXP006
with
β
=
0.3
(about
6σ
outside
present
observa2012). We also explore an extreme constant coupling case
tional limits, Pettorino et al. 2012) to better understand its
EXP006 with β = 0.3 (about 6σ outside present observaimplications.
tional limits, Pettorino et al. 2012) to better understand its
implications.
andble
primeval
spectra
index.square of the fluctuation amplitudes
constant,
root mean
and primeval spectra index.
3.2 N-body Simulations
3.2
N-body
Simulations
We first
generate
two sets of uniform particle distribu3
tions, a We
80 Mpc/h
box
andtwo
a 12sets
Mpc/h
box, both
with 350
first generate
of uniform
particle
distribuparticles.
The
initial
conditions
were
evolved
with
the
code
tions, a 80 Mpc/h box and a 12 Mpc/h box, both with 3503
gadget-2
2005),
which includes
the coupled
particles.(Springel
The initial
conditions
were evolved
with dark
the code
energy
implementation
in Baldi
et al.
In dark
gadget-2
(Springel introduced
2005), which
includes
the(2010).
coupled
Fig. 2 we show the ratio between the mass functions for the
energy implementation introduced in Baldi et al. (2010). In
coupled dark energy cosmologies with the one from ΛCDM
Fig. 2 we show the ratio between the mass functions for the
for redshifts z = 2, 1, 0 for our 80 Mpc/h boxes.
coupled dark energy cosmologies with the one from ΛCDM
We chose four dark matter halos in the ΛCDM 80
for redshifts z = 2, 1, 0 for our 80 Mpc/h boxes.
Mpc/h box and one dwarf halo in the ΛCDM 12 Mpc/h box,
We chose four dark matter halos in the ΛCDM 80
and looked for their corresponding realizations in the couMpc/h
andsimulations.
one dwarf halo
the ΛCDM
Mpc/h
pled
dark box
energy
Noteinthat
we used12the
same box,
and looked
forall
their
corresponding
the courandom
seed for
initial
conditions torealizations
be able to infollow
dark
energy
thatOur
we halos
used the
thepled
same
halos
in allsimulations.
cosmologicalNote
boxes.
havesame
random
for no
all other
initialhalos
conditions
to be able
to follow
been
chosenseed
so that
with comparable
masses
thefound
samewithin
halos four
in all
boxes.
were
of cosmological
their virial radii.
We Our
then halos
re-ran have
chosen soboxes
that no
other
halos with
comparable
masses
thebeen
cosmological
with
increased
resolution
in a Lawere found
within
of their
virial radii.
grangian
volume
that four
includes
all particles
thatWe
at then
z = re-ran
0
thefound
cosmological
boxesradii
withof increased
resolution
were
in three virial
each selected
halo. in a Lagrangian
includes of
allthree
particles
that at z = 0
Our finalvolume
samplethat
is composed
Milky-Way-size
11
were
found haloβ
in three
radiia of
each
selected
halos
(haloα,
andvirial
haloγ),
6×10
M
(haloδ)
⊙ halo halo.
Our final
is For
composed
of threeonMilky-Way-size
and a dwarf
halosample
(halo�).
more details
the halos
11
properties
at z =haloβ
0 seeand
Table
1. For
the halo
halos (haloα,
haloγ),
a 6×10
M⊙identificahalo (haloδ)
tion
we aused
thehalo
code(halo�).
Amiga Halo
Finderdetails
(Knollmann
and
dwarf
For more
on the&halos
Knebe
2009). The
chosen
be 1/40
of
properties
at zsoftening
= 0 seelengths
Table are
1. For
thetohalo
identificathetion
intra-particle
distance
the low
resolution
simulation &
we used the
code in
Amiga
Halo
Finder (Knollmann
divided
the refinement
factor
RF; RF
15 fortohaloα,
Knebeby2009).
The softening
lengths
are =
chosen
be 1/40 of
haloβ
halo�, RF distance
= 24 forinhaloγ
and resolution
haloδ. Precisely,
the and
intra-particle
the low
simulation
thedivided
softening
lengths
are
0.54
kpc
for
haloα
and
haloβ,
0.34
by the refinement factor RF; RF = 15 for
haloα,
kpchaloβ
for haloγ
and haloδ,
kpc
for halo�.
The partiand halo�,
RF =0.081
24 for
haloγ
and haloδ.
Precisely,
clethe
masses
at
z
=
0
in
the
high
resolution
volumes
are
softening lengths are 0.54 kpc for haloα and5haloβ, 0.34
kpc for haloγ and haloδ, 0.081 kpc for halo�. Theparti3 resolution volumes are
cle masses at z = 0 in the high
that no other halos with comparable masses were found within four of their
virial radii. We then re-ran the cosmological boxes with increased resolution
in a Lagrangian volume that includes all particles that at z = 0 were found
in three virial radii of each selected halo.
3
NUMERICAL METHODS
3.1
Initial Conditions
and Coupled Dark Energy
3 NUMERICAL
METHODS
The final sample is composed of three Milky-Way-size halos (haloα, haloβ
and haloγ), a 6×10 M halo (haloδ) and a dwarf halo (halo). For more
details on the halos properties at z = 0 see Table 5.1. For the halo identification we used the code Amiga Halo Finder (Knollmann and Knebe, 2009).
in Penzo
et al. and
(2014),
we used
grafic-de,
3.1 AsInitial
Conditions
Coupled
Dark
Energy
an extension of the initial condition
11 generator grafic-2
As in Penzo
et that
al. (2014),
we
usedforgrafic-de,
(Bertschinger
2001) such
initial conditions
a generic
an extensionmodel
of the
condition
generator
grafic-2
cosmological
caninitial
be produced
once
the evolution
of
(Bertschinger
2001)
such that
forgrafica generic
the
cosmological
parameters
areinitial
given conditions
as an input.
cosmological
model functions,
can be produced
thedensity
evolution
de
requires transfer
evolutiononce
of the
pa- of
the cosmological
are given
as an
graficrameters
Ωi , linearparameters
growth factor
D+ and
fΩ , input.
the logarithde requires
functions,
evolution
of the
mic
derivativetransfer
of the growth
factor
with respect
to density
the scaleparameters
growth
D+ and
fΩ , to
thegenerate
logarithfactor.
As Ω
the
original
code, factor
grafic-de
is able
i , linear
multi
mass initial
conditions
from awith
cosmological
In
mic derivative
of the
growth factor
respect tobox.
the scale
Fig.
1 we
evolution
the linear growth
D+
factor.
Asshow
the the
original
code,ofgrafic-de
is able factor
to generate
for
all four
models.from
The atransfer
functions
for In
multi
masscosmological
initial conditions
cosmological
box.
ΛCDM
have
been
BridleD+
Fig. 1 we
show
theproduced
evolutionusing
of thecamb
linear(Lewis
growth&factor
2002),
the transfer functions
for the
coupled
dark en-for
for all while
four cosmological
models. The
transfer
functions
ergy
models
produced
by weighting
the ΛCDM
ΛCDM
havehave
beenbeen
produced
using
camb (Lewis
& Bridle
transfer
functions
with
the
D
of
the
coupled
model.
ini-en+
2002), while the transfer functions
for the coupled All
dark
tial
conditions
were
created
using
the
same
random
ergy models have been produced by weighting the seeds,
ΛCDM
in
order to
be able with
to identify
structures
amongmodel.
the models.
transfer
functions
the D+
of the coupled
All initial conditions were created using the same random seeds,
c — RAS, MNRAS 000, 1–12
�
4
in order to be able to identify structures among the
models.
The softening lengths are chosen to be 1/40 of the intra-particle distance
in the low resolution simulation divided by the refinement factor RF (see
Section 3.1.1 for details on refinements); RF = 15 for haloα, haloβ and
halo, RF = 24 for haloγ and haloδ. Precisely, the softening lengths are 0.54
kpc for haloα and haloβ, 0.34 kpc for haloγ and haloδ, 0.081 kpc for halo.
The particle masses at z = 0 in the high resolution volumes are 3.8×10 M
for haloα and haloβ, 9.4×10 M for haloγ and haloδ, 1.3×10 M for halo.
Figure
shows
the projected density maps of the four most massive halos
c — RAS, 5.2
�
MNRAS
000, 1–12
for each cosmological model. We will discuss the dwarf halo in Section 5.4.
For the density maps and throughout the paper we chose to calculate halo
properties using R200 , radius at which the density is 200 times the critical
density ρc , with ρc ≡ 3H 2 /(8πG).
92
Halo and Subhalo Properties in Coupled Dark Energy
haloα
ΛCDM
EXP003
EXP008e3
EXP006
haloβ
ΛCDM
EXP003
EXP008e3
EXP006
haloγ
ΛCDM
EXP003
EXP008e3
EXP006
haloδ
ΛCDM
EXP003
EXP008e3
EXP006
halo
ΛCDM
EXP006
M200
[M ]
R200
[kpc]
c ≡ R200 /rs
N200
2.6×1012
2.5×1012
2.6×1012
2.1×1012
284
281
282
265
11.8
9.1
10.2
4.6
6.8×106
6.6×106
6.7×106
5.5×106
2.5×1012
2.2×1012
2.2×1012
1.7×1012
278
267
268
246
10.7
8.0
8.7
4.3
6.3×106
5.7×106
5.8×106
4.5×106
9.7×1012
9.3×1012
9.5×1012
7.6×1012
204
201
203
188
10.8
8.6
9.6
3.2
1.0×107
9.9×106
1.0×107
8.1×106
6.4×1011
5.6×1011
5.9×1011
5.3×1011
177
170
172
166
13.3
9.7
11.0
4.7
6.8×106
6.0×106
6.3×106
5.6×106
4×109
3×109
33
30
15.3
6.6
3.1×106
2.4×106
Table 5.1: Physical properties of the five halos in all cosmologies, ΛCDM,
EXP003, EXP008e3 and EXP006. We show mass at R200 , R200 , concentrations and number of particles within R200 .
93
Figure 5.2: Projected density maps of our sample at z = 0. From first row
to last we are showing haloα, haloβ, haloγ,haloδ; from first column to last
we are showing ΛCDM, EXP003, EXP008e3 and EXP006 cosmologies.
94
Halo and Subhalo Properties in Coupled Dark Energy
Figure 5.3: Density profiles for haloα, haloβ, haloγ and haloδ at z = 0, each
for ΛCDM (black), EXP003 (cyan), EXP008e3 (blue) and EXP006 (red).
The inner radius is equal to three times the softening length, while the outer
radius is four times R200 of each halo. The vertical dashed lines mark R200
for each halo in each cosmology.
5.3
5.3.1
Results on Milky-Way size Halo simulations
Host Halos Properties
We will first outline the properties of the four most massive halos, haloα,
haloβ, haloγ and haloδ, by showing concentrations, density profiles, rotation
curves, evolution of the scale radii and accretion histories.
Concentrations and Density Profiles
By introducing a coupling between dark matter and dark energy, halo concentrations decrease. This was also shown in Baldi et al. (2010), Li and
Barrow (2011), and Carlesi et al. (2014) for halos with masses M & 1013 M .
We investigate mass scales M . 1012 M . The resolution that we are able
to reach is higher thanks to the multi-mass technique. In Table 5.1 we show
the concentration values for each halo, for which we use the definition
c ≡ R200 /rs ,
95
(5.2)
where R200 is the radius at which the density equals 200 times the critical
density and rs is the scale radius in the Navarro Frenk and White (NFW)
profile (Navarro et al., 1997), see Eq. (2.19). We computed rs via a χ2 minimization procedure using the Levenberg & Marquardt method. In agreement
with literature, we find that halos which lived in a coupled dark energy cosmology have lower concentrations. Figure 5.3 shows the density profiles for
haloα, haloβ, haloγ and haloδ. The behavior of the profiles as a function
of cosmology is maintained for all four halos, with a significant flattening
of the inner part of the profiles only for the extreme coupled cosmology
EXP006, while differences are less evident in the EXP003 and EXP008e3
halos. Interestingly, the EXP006 realization of haloγ (M = 7.6×1011 M )
produces a much flatter halo profile, with slope α=-0.8, which falls out of
NFW parametrization. On the other hand, all other profiles of haloα, haloβ,
haloγ and haloδ in all cosmologies are well described by the NFW profile.
Additionally, in Figure 5.4 we show the rotation curves at z = 0 for
the four halos. For models within the observational constraints the rotation curves are not significantly affected. The only case in which we observe
a considerable flattening is the extreme model EXP006, for all four cases.
This is in agreement with results from disc galaxies simulations in dynamical dark energy in Section 4.2.2, where we find that differences in rotation
curves among models within observational constraints for dynamical dark
energy are not significant in the dark matter only case. On the contrary, in
hydrodynamical simulations we find observable differences in rotation curves
due to the effects of baryons which enhance the variations in the dark matter accretion. We expect the same enhancement also in coupled dark energy
models once hydrodynamics is taken into account. We leave this aspect to
be investigated in a future work.
NFW Scale Radius Evolution
In Section 5.3.1 we have showed that halos that form in a coupled dark energy
cosmology with a high value for the coupling constant have concentrations
that are significantly lower at z = 0. Given that almost all halos are well
described by a NFW density profile, it means that their NFW scale radii rs
are much larger than the scale radii of the corresponding ΛCDM realizations.
In Figure 5.5 we show the behavior of the scale radius rs as a function of
redshift for haloβ in all four cosmologies; the other Milky-Way size halos
have similar behaviors. Compared to the ΛCDM case, halos which live in
coupled dark energy cosmologies show a larger scale radius at all redshifts.
Parent Halos Accretion Histories
In order to investigate the origin of the different concentrations, especially
in the EXP006 cosmology, in this Section we concentrate on the halo forma96
Halo and Subhalo Properties in Coupled Dark Energy
Figure 5.4: Rotation curves for haloα, haloβ, haloγ and haloδ at z = 0, each
for ΛCDM , EXP003, EXP008e3 and EXP006. The inner radius is equal to
three times the softening length, while the outer radius is four times R200
of each halo. The vertical dashed lines mark R200 for each halo in each
cosmology.
Figure 5.5: Scale radius obtained by fitting an NFW density profile using
the Levenberg & Marquart method for haloβ as a function of redshift.
97
Figure 5.6: Evolution of the mass enclosed in R200 normalized to the mass
at a = 1 as a function of the scale factor for haloα, haloβ, haloγ in all four
cosmologies.
tion times. Firstly, in Figure 5.6 we show the accretion histories, namely the
evolution of the mass enclosed in R200 normalized to the value of the mass at
z = 0 as a function of expansion factor. Halos growing in ΛCDM, EXP003
and EXP008e3 cosmologies show similar accretion histories especially for
haloα and haloγ; while halos living in the EXP006 cosmology accrete their
mass earlier on compared to their ΛCDM realizations. Among the three
halos, coupled cosmologies runs show unexpected drops in the accretion histories. These would be unusual in a ΛCDM scenario since halo total masses
do not decrease unless it is a temporary effect of a merger (see for instance
haloα and haloβ around a = 0.3). On the other hand, in the case of coupled cosmologies the friction term in Eq. (2.11) is responsible for injecting
kinetic energy into the system, which may cause some particles to become
gravitationally unbound.
In order to estimate the time of formation for each halo, we followed
the approach described in Wechsler et al. (2002). In their paper they show
how halo accretion histories are well described by an exponential form that
depends on one parameter, the formation epoch ac , which is defined as the
expansion factor at which the logarithmic derivative of the mass evolution
falls below a critical value S. Specifically, the fitting form is given by
M (a)
1
= exp −ac S
−1
M (1)
a
98
(5.3)
Halo and Subhalo Properties in Coupled Dark Energy
Figure 5.7: Accretion histories for haloβ, where we show for each panel
a different cosmological model. The fit to the accretion histories of each
realization (solid yellow line) is obtained from Eq. (5.3).
haloα
haloβ
haloγ
ΛCDM
0.194
0.261
0.225
EXP003
0.181
0.229
0.210
EXP008e3
0.183
0.230
0.215
EXP006
0.142
0.177
0.173
Table 5.2: Values for the formation epochs ac for haloα, haloβ, haloγ in all
four cosmologies.
with S = 2. We used the fitting function of Eq. (5.3) to compute the
formation epochs for each of the three halos for all cosmologies. As an
example, we show in Figure 5.7 the fitting result for haloβ and we summarize
the formation epochs for all halos in Table 5.3.1.
As pointed out in Wechsler et al. (2002), Dutton and Macciò (2014)
and Ludlow et al. (2013), in a ΛCDM cosmology an early formation epoch
leads to higher concentrations. The same happens for dynamical dark energy
cosmologies, e.g. Klypin et al. (2003); Dolag et al. (2004). Interestingly, in
coupled dark energy cosmology this behavior is not preserved. Despite the
fact that a stronger coupling brings to an earlier halo formation epoch, halo
concentrations decrease when the coupling is increased, as clearly visible
from the EXP006 realizations. This phenomenon has also been showed in
Baldi (2011a), where an analysis of the modifications due to the coupling
has been extensively carried out, but the resolution was significantly lower.
99
Figure 5.8: Cumulative number of subhalos with more than 400 particles as
function of their mass for haloα, haloβ, haloγ and haloδ at z = 0 for each
cosmology.
The friction term in Eq. (2.11) is responsible for making the halo expand by
altering its virial equilibrium through the injection of kinetic energy in the
system, which in turns lowers the concentration. This shows how in coupled
cosmologies the lower concentrations are not the result of formation histories
but the effect of the modified dynamics.
5.3.2
Subhalos
In this Section we describe the study on subhalo abundance, their radial
distribution and circular velocities.
Abundance
The lower number of substructures present in EXP006 halos compared to
ΛCDM can be recognized in Figure 5.2. In Figure 5.8 we show the subhalo mass function, where only subhalos that lie within R200 and that have
more than 400 particles are considered. The total number of subhalos in
EXP006 realizations is always from 50 to 75% lower than in the respective
ΛCDM cases, while differences between EXP003 and EXP008e3 and ΛCDM
are much less evident (∼ 10%). Thus, the missing satellites problem (Klypin
et al., 1999; Moore et al., 1999) can be progressively alleviated when increasing the coupling. Note that the differences in the subhalos minimum mass
100
Halo and Subhalo Properties in Coupled Dark Energy
Figure 5.9: Cumulative number of subhalos with more than 400 particles as
function of distance from the parent halo center for haloα, haloβ, haloγ and
haloδ at z = 0 for each cosmology.
among haloα, haloβ and haloγ, haloδ are due to the different resolutions
used (see Section 5.2).
Radial Distribution
Figure 5.9 shows the cumulative distribution of subhalos as a function of
the distance from the parent halo center normalized to the total number of
subhalos within R200 . All halos in all cosmologies show non-significant differences in the cumulative radial distribution. In order to better understand
the distribution of subhalos, Figure 5.10 shows the differential distribution
in a sphere of constant radius for all cosmologies, the radii are 350 kpc for
haloα and haloβ, 250 kpc for haloγ and 200 kpc haloδ. The number of bins
is kept the same for each halo in all cosmologies and the vertical lines show
the virial radii. The distributions show a clear decrease of the number of
subhalos in EXP006 halos compared to their respective ΛCDM cases, while
for EXP003 and EXP008e3 cosmologies differences are not so evident.
As pointed out in Section 5.3.1, coupled cosmologies decrease halo concentrations thanks to the presence of a friction term in the equation for the
evolution of density perturbations, Eq. (2.11), despite the earlier halo formation epochs. We claim that the same effect can be responsible for the lower
number of subhalos compared to ΛCDM. Thanks to the extra friction term
and to subhalo lower concentrations, subhalos that are falling into the par101
Figure 5.10: Differential number of subhalos in R200 with more than 400
particles as function of distance from parent halo center for haloα, haloβ,
haloγ and haloδ at z = 0 for each cosmology. The vertical dashed lines mark
R200 for each halo in each cosmology. For a given halo, the binning is kept
constant for all cosmologies.
102
Halo and Subhalo Properties in Coupled Dark Energy
Figure 5.11: Number of subhalos in 3R200 with more than 400 particles as
function of distance from parent halo center for haloα, haloβ, haloγ and
haloδ at z = 0 for each cosmology. The vertical dashed lines mark R200 and
2R200 for each halo in each cosmology. For a given halo, the binning is kept
constant for all cosmologies.
ent halo potential well are more heavily stripped, thus less halos with more
than 400 particles survive. If this claim is correct, we should be able to find
a difference in the subhalos number distribution when we reach distances
from the parent halo center that are bigger than the radius from which the
gravitational influence of the host halo is felt. In Figure 5.11 we show the
differential radial distribution of the number of subhalos out to about three
times the virial radius of each halo. For the sake of clarity, we choose to
show only the most extreme cases, EXP006, and ΛCDM for all four halos.
The dotted lines represent one and two times R200 for each halo in each cosmology. What we would like to stress, is that there seem to be a decrease in
the number of subhalos in halos living in EXP006 cosmology compared to
their ΛCDM realizations only within the gravitational influence of the parent
halo. Between 1.5 and 2 R200 this behavior inverts and halos living in the
strongly coupled cosmology seem to have a larger or at least a comparable
number of subhalos with respect to their ΛCDM cases. Thus, we ascribe
the presence of a lower subhalos number to a massive stripping effect rather
103
than EXP006 producing intrinsically a lower number of subhalos. We would
like to stress on the fact that lowering the number of subhalos can also be
achieved by warm dark matter cosmologies (e.g. Anderhalden et al. 2013),
but the fundamental difference lies on the fact that those subhalos in warm
dark matter cosmologies were never formed, while in coupled dark energy
cosmologies subhalos do form but they are heavily stripped.
Circular Velocities
Boylan-Kolchin et al. (2011) first showed that N-body simulations of a MilkyWay size halo predict a significant number of subhalos with circular velocities
higher than the circular velocities that we measure for the brightest satellites
of the Milky Way, which is surprising since these massive subhalos should
not fail in producing stars.
The discrepancy between ΛCDM prediction and observations can be alleviated in multiple ways, starting from baryonic processes. Brooks and
Zolotov (2014) suggest that baryonic feedback processes could be responsible for a dark matter redistribution, with the result of decreasing the central
densities of the most massive subhalos. Rashkov et al. (2012) point out
that the possibility of star formation being stochastic below a certain mass
would justify the Milky Way having massive dark satellites; furthermore,
they highlight the fact that the tension between the Via Lactea II simulation and observations is only a factor of two in mass, which suggests that the
uncertainty on the Milky Way virial mass could be a viable way out from the
tension (Vera-Ciro et al., 2013; Kennedy et al., 2014). Purcell and Zentner
(2012) showed that there exists a significant variation in subhalo properties
even when the host halos have the same virial mass.
Last but not least, the discrepancy can be alleviated by appealing to nonΛCDM cosmologies. The cases for warm, mixed (cold and warm) and selfinteracting dark matter are considered respectively in Lovell et al. (2012),
Anderhalden et al. (2012, 2013), Vogelsberger et al. (2012). In all cases
they find that subhalos are less concentrated due to their late formation
time, suggesting that alternative cosmologies can contribute to alleviate the
tension between predictions and observations.
Figure 5.12 shows the rotation curves for the twelve most massive subhalos at the moment of infall. We used the correlation between orbital energy
and subhalo mass loss found in Anderhalden et al. (2013) to determine the
subhalos ranking. Anderhalden et al. define the binding energy of a subhalo
as the sum of the kinetic plus potential energy, divided by the mass of the
subhalo
1 2 GM
r
eorbit ≡ K + U = ~v −
ln 1 +
,
(5.4)
2
r
rs
where ~v is the velocity of the subhalo with respect to the parent halo center
and the parent halo is fitted with an NFW density distribution which gives
104
Halo and Subhalo Properties in Coupled Dark Energy
its scale radius rs . They argue that the ratio vmax (z = 0)/vmax (zinfall ) is
a proxy for the subhalo mass loss since the time of accretion due to tidal
stripping and they find it correlates with the subhalo binding energy. In fact,
the deeper a subhalo orbits within the potential well of the parent halo, the
more tidal stripping it has undergone. This leads to a reduced value of the
maximum circular velocity at z = 0 relative to the value of velocity at infall.
They give the following fitting form:
vmax (z = 0)
= 1 + ξ · eorbit
vmax (zinfall )
(5.5)
where ξ = 2.89×10−6 (km/s)−2 . This formula suggests a straight-forward
way to calculate the infall velocity once the velocity at z = 0 is known.
Each row of Figure 5.12 illustrates the twelve subhalos rotation curves
for a given parent halo in all considered cosmologies, from the top down
we show haloα, haloβ, haloγ, haloδ. The observed values for vcirc (r1/2 )
for the brightest dwarf galaxies orbiting around the Milky Way are shown
in yellow. Data are taken from Anderhalden et al. (2013) and references
therein. Despite haloα and haloβ having comparable masses, the tension
between simulated curves and measured points in the ΛCDM case is more
evident in haloβ, supporting the fact that subhalo properties can vary even
when host halos have the same virial mass (Purcell and Zentner, 2012).
The tension is alleviated in the case of haloγ and even more haloδ, given
their lower masses (Vera-Ciro et al., 2013). Overall, when looking at all
halos in EXP003 and EXP008e3 cosmologies these do not show significant
improvement compared to their ΛCDM realizations in decreasing the inner
densities of subhalos. On the other hand, in the case of EXP006 cosmology,
all four halos show such a dramatic decrease in subhalos rotational velocity
peaks that rotation curves become incompatible with measured values. The
dramatic decrease was to be expected given the choice of a large coupling
parameter for EXP006 cosmology, but nonetheless it is useful to understand
the effects of the coupling.
5.4
Zooming-in on a dwarf halo
To better explore the effects of the coupling at high resolutions, we simulated
a dwarf galaxy halo, halo. We chose an isolated halo (no structures with
comparable mass within four of its virial radii) and, given the results of
Section 5.3.2, we only focused on the two most distant cosmological cases,
ΛCDM and EXP006 cosmology. The virial masses are respectively 4×109 M
and 3×109 M , with a mass resolution of 1.3×103 M . We show in Figure 5.13
the density maps for halo in both cosmologies, ΛCDM on the left panel.
It is visible how the number of substructures decreases in the case with
coupling. In Figure 5.14 we show density profiles and rotation curves for
105
Figure 5.12: Rotation curves of the most massive subhalos at the moment of
infall for each halo in each cosmology. From the top row down we show haloα,
haloβ, haloγ, haloδ, from left to right we show ΛCDM (black), EXP003
(cyan), EXP008e3 (blues), EXP006 (red). We estimate the subhalo mass
ranking at the moment of infall using the correlation between orbital energy
and subhalo mass loss found in Anderhalden et al. (2013). The yellow points
are the observed values for vcirc (r1/2 ) for the brightest dwarf galaxies orbiting
around the Milky Way. Data are taken from Anderhalden et al. (2013) and
references therein. The masses of each parent halo realization is written on
each panel.
106
Halo and Subhalo Properties in Coupled Dark Energy
Figure 5.13: Density maps for halo in ΛCDM (left panel) and in EXP006
cosmology (right panel). The side of each projection is 2×R200 .
Figure 5.14: Density profiles and rotation curves for halo in ΛCDM (black)
and in EXP006 (red) cosmology. The inner minimum radius is three times
the softening length, while the most outer radius is four times R200 respectively. The vertical dashed lines represent R200 for each cosmology.
107
Figure 5.15: Accretion histories for halo in ΛCDM (black) and in EXP006
(red) cosmology. We show the ratio of M200 at a given time and M200 today
as a function of the scale factor.
halo in both cosmologies. The effect of the coupling is very evident in
lowering the concentration and flattening the rotation curve. The values for
the halo concentrations are c = 15.2 and c = 6.5 for ΛCDM and EXP006
cosmology respectively. Although the density profile in the coupled dark
energy case is less concentrated, it is still cuspy, showing that in coupled
cosmologies, as in ΛCDM, we are not able to produce a dark matter only
cored density profile. The inconsistency with observation thus still persists,
given the observational evidence that supports cored density profiles for the
satellites of the Milky Way (Walker and Peñarrubia, 2011; Amorisco and
Evans, 2012; Amorisco et al., 2013). Interesting to note, by constructing
a model in which both warm and cold dark matter are present and only
the cold component is coupled to dark energy, a very high value (βc ∼ 10)
for the coupling constant is favored (Bonometto et al., 2015) and simulated
dark matter only dwarf halos show a cored density profile (Macciò et al.,
2015). Figure 5.15 shows accretion histories, the ratio of M200 to its value
today is plotted as function of scale factor. As in Section 5.3.1, we calculated
the formation epochs as in Wechsler et al. (2002) and obtained ac = 0.191
(ΛCDM), ac = 0.146 (EXP006), confirming the finding that coupled dark
energy models have earlier formation times, also for less massive halos.
5.5
Conclusions
We have outlined the first study in coupled dark energy cosmologies on high
resolution simulations on galactic scales, with the aim to study the effects
108
Halo and Subhalo Properties in Coupled Dark Energy
of the coupling between dark energy and dark matter on these scales, so
far neglected in previous works. We investigated two viable models, one
with constant coupling and one with varying coupling with redshift; we also
chose a third case where the constant coupling value has been pushed beyond
observational constraints to better investigate its effects. We then selected
three Milky-Way size halos, a 6×1011 M halo and a dwarf halo 5×109 M ,
and studied their properties in a ΛCDM reference model and in the coupled
cosmologies, resolving each halo with ∼ 106 particles.
We computed concentrations and formation epochs for all halos and we
find that, despite the earlier formation epochs of the coupled cosmologies
halos, these have lower concentrations. In a ΛCDM or a dynamical dark
energy scenario, earlier formation epochs would imply higher concentrations,
but in the coupled dark energy case the reason for lower concentrations is
not related to formation histories, but rather to the modified dynamics. In
fact, the equation for the linear evolution of density perturbations, Eq. (2.9),
shows the friction term −βc φ̇δ̇c , an extra term compared to the ΛCDM case
that redistributes the dark matter particles and lowers the central densities,
despite the earlier formation times (see Baldi 2011a). We find that this
behavior is reproduced for all mass scales that we have investigated.
In particular, subhalos can also be significantly less concentrated. When
falling towards their host, they are more heavily stripped once they start
feeling the gravitational influence of the host halo. This translates into decreasing the number of subhalos compared to the ΛCDM realization and, additionally, subhalos are themselves less massive and less concentrated. For
these reasons, coupled cosmologies can be helpful in alleviating satellitescales inconsistencies of ΛCDM. On the other hand, we find that, in order
to try to solve these issues with the coupling alone, one needs to use an
extreme value for the coupling constant that is ruled out by observational
constraints. In fact, only in the case with the highest coupling value the
number of subhalos is significantly reduced (up to 75% less subhalos) than
in the respective ΛCDM cases, while for the viable coupling cosmologies the
decrease is much less significant (10% less subhalos). Moreover, we find that
the distribution of the subhalos inside the parent halo virial radius does not
vary significantly among cosmologies.
Less concentrated coupled cosmologies subhalos can in principle be useful
to reconcile the inconsistency between the observed properties of the Milky
Way dwarf galaxies and ΛCDM simulations predictions, but once more a
high enough value for the coupling must be assumed. Interestingly, allowing
the introduction of massive neutrinos does alleviate the constraints on the
coupling (see e.g. La Vacca et al., 2009), leaving coupled dark energy models
dynamics on sub-galactic scales an interesting option.
Overall coupled dark energy models can be as effective as ΛCDM in
reproducing observations on sub-galactic scales and, for specific choices of the
coupling, they can improve the agreement between predicted and observed
109
properties. Hence, coupled models would need to be further investigated,
possibly taking into account the effects of baryons at sub-galactic scales,
which, as already shown in dynamical dark energy models (Section 4.2), are
expected to amplify differences observed in the dark matter only case.
110
Chapter 6
Summary and Conclusions
Since it was realized that an extra term in Einstein’s fields equations was
needed to explain the acceleration of the universe, alternatives to a Cosmological Constant have been proposed and dark energy models largely developed. In this thesis, we focused on two of the most popular choices nowadays
available in literature, Dynamical Dark Energy (Wetterich, 1988) and Coupled Dark Energy (Amendola, 2000).
Dynamical dark energy only affects the background evolution of the universe and various geometry probes can be used to constrain the model parameters (e.g. luminosity distance, angular-diameter distance, baryonic acoustic
oscillations). In turn, the change in background evolution affects the linear
growth factor, with the result that structure formation is sped up or slowed
down compared to ΛCDM (Klypin et al., 2003). Coupled dark energy exhibits similar changes to the background evolution as dynamical dark energy,
but, additionally, an extra force between dark energy and dark matter alters
the equation for the evolution of linear perturbations. As a consequence, the
modified dynamics further influence structure and galaxy formation (Macciò
et al., 2004).
Thus, structure formation and evolution can show significant departures
from a ΛCDM model. For this reason, it is extremely useful to compare
predictions of dark energy models with observations, which can range from
cosmologically large scales to the highly non-linear end of the power spectrum down to galactic and sub-galactic scales. Given that formation of
structures is a strongly non-linear process, numerical simulations need to
be employed to obtain detailed predictions. Furthermore, when investigating galactic scales, baryonic effects play a significant role and should not be
overlooked. Modeling the luminous matter is necessary for a consistent comparison with observations but treating hydrodynamical processes is a very
challenging task. Nonetheless, understanding the role of baryonic matter in
shaping differences among cosmologies is crucial to make use of the very high
resolution data coming from the next generation of galaxy surveys.
111
Dynamical dark energy and coupled dark energy have been extensively
tested both at linear level and on large-scale predictions, but scarcely investigated on galactic and sub-galactic scales. In this thesis I outlined our
work on structure and galaxy formation in both dynamical and coupled dark
energy across the mass range ∼ 109−12 M via dark matter only and hydrodynamical numerical simulations.
Firstly, we developed an initial condition generator suitable for producing
initial conditions for multi-mass cosmological simulations in a large variety
of non-ΛCDM models, grafic2-de. The code is a generalization of the already available grafic2 (Bertschinger, 2001). Details on initial conditions
generator and its extension to non-ΛCDM cosmologies were given in Chapter 3.
In Section 4.1 I described our choices for the linear parametrization of
the dynamical dark energy equation-of-state parameter w ≡ p/ρ. Four models plus ΛCDM were selected and used throughout Chapter 4: waCDM0,
waCDM1, waCDM2 and SUCDM. All dynamical dark energy models lie
within the two-sigma contours given by WMAP7 data (Komatsu et al., 2011),
employ the same baryonic physics prescriptions and share the same σ8 at
z = 0. The code used is gasoline-de for hydrodynamical runs (Wadsley
et al., 2004), which features gas cooling, supernova feedback and feedback
from young massive stars, while pkdgrav (Stadel, 2001) was used for dark
matter only runs. Both codes have implementations that account for the
background expansion of a given dynamical dark energy model (Casarini
et al., 2011b).
In Section 4.2 I show our results on high resolution hydrodynamical
simulations of three single-galaxies with masses 8×1011 M , 6×1011 M and
3×1011 M . The intention of this work1 is to investigate the effects of dark
energy on galactic scales in hydrodynamical simulations. We find that the
same galaxy evolved in different dark energy cosmologies does not present
significant differences in dark matter only simulations, while in hydrodynamical simulations galactic properties vary greatly. By changing the dark
energy parametrization, stellar mass either decreases or increase of a factor
around two throughout the whole galaxy evolution. Baryons amplify differences among dynamical dark energy models due to their non-linear response
in hydrodynamical processes. Cosmological models set the initial differences
in density perturbations, but these are not sufficient to produce significant
changes in galaxies unless feedback processes start enhancing those differences by producing slightly more (or less) stars. More stars introduce more
metals in the feedback cycle and more metals decrease the cooling time,
which in turn allows gas to cool faster and produce even more stars. Through
1
Published in Penzo et al. (2014)
112
Summary and Conclusions
the highly non-linear response of baryons, dark energy models that would
have been indistinguishable from ΛCDM on galactic scales show distinctive
features in hydrodynamical simulations. We find that, by changing the dark
energy model, we are able to delay and suppress (or anticipate and increase)
star formation until recent epochs. The delay in star formation is then in
turn responsible for flattening rotations curves. We find a significant change,
about 100 km/s, in the two most massive galaxies that we considered.
We compare the effect of dynamical dark energy with the effect of baryonic feedback on disc galaxies. We keep the cosmology fixed (waCDM2) and
change the feedback parametrization. Provided that supernova feedback is
kept constant, at late times the effect of dark energy is comparable to the
effect of feedback from young massive stars. Also the degree at which stellar feedback is able to flatten rotation curves is comparable to the effect
of dark energy. On the other hand, at high redshift, the effect of feedback
from young massive stars becomes more significant than the dark energy
modeling. Nonetheless, the importance of the dark energy modeling can be
compared to the importance of the details in the feedback parametrization.
The implications of the dynamical dark energy modeling on the formation
and evolution of our three high-resolution galaxy simulations are noteworthy
and in principle may be reproduced at all mass scales. To check whether this
is the case, in Section 4.3 we extend the exercise to hydrodynamical simulations of a cube of the universe with side 114 Mpc. This is a hydrodynamical
simulation with the same feedback implementation used in Section 4.2, with
the difference that resolution is constant throughout the box and hydrodynamics are treated in the full volume. Due to the high computational cost of
such settings, the resolution had to be much lower than in the single-object
cases. In fact, the high resolution of single-galaxy runs makes it possible
to simulate only a few tens of objects. When interested in a larger sample
(∼ thousands of objects), single-galaxy simulations can no longer be employed. With a hydrodynamically simulated volume we can study the effects
of dynamical dark energy on galaxy formation across a wider range of galaxy
masses, environments and merger histories.
In the galaxy sample contained in our cosmological volumes the increase
in the mean stellar mass given a halo mass is at most 20% in SUCDM, our
most distant model from ΛCDM, while for our three single-galaxy simulations the increase reaches 100%. We attribute the lower differences among
cosmologies compared to the single-galaxy runs to the change in resolution.
In fact, in order for baryonic feedback processes to be well sampled and fuel
the feedback cycle, they require a high enough resolution. For comparison,
the mass resolution of the most refined of the two sets of volumes (Np = 5123 )
is for gas particles (initial masses) mgas = 6.9 × 107 M , while in the singlegalaxy runs mgas = 4.2 × 105 M . To support the resolution argument, we
ran one of the single-galaxy simulations with a resolution for gas particles
mgas = 2.7×107 M and differences between cosmologies became comparable
113
to the differences found in the hydrodynamical volumes.
Our work2 on the cosmological volumes in dynamical dark energy diverted the focus onto resolution and underlined its crucial role in differentiating between dynamical dark energy models. In particular, we argued that
resolution in our volume simulations may not be sufficient to disentangle
between cosmological models at dwarf galaxy mass scales where we find a
variation in the mean stellar mass from ΛCDM of only a few percents. In
the higher resolution volume (Np = 5123 ), a ∼1010 M galaxy is described
with a total of about 100 particles.
To investigate whether low resolution is suppressing potential differences between cosmological models or simply differences are intrinsically not
present at these mass scales, in Section 4.4 we investigated hydrodynamical
simulations of dwarf galaxies in dynamical dark energy. I present a set of
four dwarf galaxies with virial masses of a few×109 M evolved in ΛCDM
and in the two most distant cosmological models, waCDM2 and SUCDM.
In this case a typical 1010 M dwarf galaxy is described with about four million particles within its virial radius and the softening length for stellar and
gas particles is 29 pc. We find that the hierarchy of cosmological models
is maintained in our four dwarf galaxies, with the SUCDM dwarfs making
more stars than ΛCDM and waCDM2, but the difference in stellar mass
compared to ΛCDM varies and is significant only in two out of four cases.
That differences between cosmological models vary among our four cases
is in agreement with the statement that star formation becomes stochastic
at these mass scales (Boylan-Kolchin et al., 2011), thus also the enhancement due to baryonic processes of variations between cosmological models is
expected to vary.
We find that the waCDM2 realization of the least massive dwarf galaxy
does not make any stars and will remain dark throughout its life. This shows
that even small departures from ΛCDM can affect the number of luminous
subhaloes when their virial mass is around a few×109 M . Star formation
histories are globally enhanced or decreased but no delay or anticipation
shows at these mass scales. Furthermore, for all our dwarf galaxies, density profiles do not vary between different dynamical dark energy models
(differences are less than few percents at inner radii).
I conclude our experiment on dynamical dark energy at dwarf mass scales
by stating that cosmology does not have a significant impact on our four
dwarf galaxy cases compared to the impact we find on disc galaxies. This
is in agreement with expecting higher variations among cosmological models
around the peak of star formation efficiency (Mhalo ∼1012 M ) (Moster et al.,
2010).
2
Collaboration with L. Casarini and A. V. Macciò, where L. Casarini was responsible
for running the simulations, while I performed the analysis under the supervision of A. V.
Macciò
114
Summary and Conclusions
Our findings from dwarf galaxies simulations additionally suggested that
scatter in star formation histories within a given cosmological model may
be extensive. To better investigate what fuels these different star formation
histories, in Section 4.4.4, we increase the dwarf sample and focus on the
ΛCDM model3 . Our goal is to study the properties of dwarf satellites before
in-fall into their Milky-Way size parent halo (z ∼ 1).
Our sample is composed of 22 dwarf galaxies in the mass range a few
109−10 M ; 7 out of 13 haloes with masses 8 × 108 . Mhalo /M . 4 × 109
do not form any stars. The reason for these haloes to remain dark lies in the
presence of the UV background radiation from the reionization epoch (Dekel
and Silk, 1986). Generally, the sample follows the extrapolation of the stellar
mass halo mass relation at z = 1 from abundance matching (Moster et al.,
2013), but it suggests a quite large scatter (∼ a factor of ten) around the
extrapolated relation. We find a wide variety of star formation histories,
which supports the suggestion that star formation at these mass scales may
be stochastic.
We study three dwarf galaxies with comparable masses but diverse star
formation histories and, as a consequence, different stellar mass at z = 1. We
connect their heterogeneity with their various formation histories. Indeed,
we find that what discriminates haloes with comparable masses is whether
they underwent recent major mergers or not. At these mass scales, major
mergers are the cause for a massive increase in star formation (∼85% of the
stellar mass is produced during the merger).
In Chapter 5 I describe our work on structure formation on galactic and
sub-galactic scales in dark matter only simulations in coupled dark energy4 .
I start by describing our choices of coupling functions in a scenario in which
dark matter and dark energy interact with each other. This class of models
features interesting departures from the ΛCDM scenario, (i) an enhancement
of the gravitational strength, (ii) a dark matter particle mass that changes
with time as a function of the dark energy scalar field, (iii) a friction term
that introduces a velocity dependance in the acceleration equation for dark
matter particles (Amendola, 2000). We investigate two viable models (Pettorino et al., 2012), one with constant coupling and one with varying coupling
with redshift; we also choose a third case where the constant coupling value
is pushed beyond observational constraints to better investigate its effects.
We run single-object simulations of five dark matter haloes, precisely
three Milky-Way size haloes, a 6×1011 M halo and a dwarf halo 5×109 M .
These cosmologies have been previously studied in large volume simulations
(Baldi et al., 2010; Carlesi et al., 2014), but, before this work, the effects
of the coupling were never investigated at the aforementioned mass scales
3
4
A. V. Macciò, C. Penzo et al. in preparation
Penzo et al. (2015) submitted, arXiv:1504.07243
115
and at such high resolutions (each halo has a few million particles within
the virial radius). Given that inconsistencies with a ΛCDM model appear
mostly at sub-galactic scales (Klypin et al., 1999; Moore et al., 1999; BoylanKolchin et al., 2011), exploring the behavior of coupled dark energy models
at these scales becomes of interest.
We compute concentrations and formation epochs for all haloes and we
find that, despite the earlier formation epochs of the coupled cosmologies
haloes, these have lower concentrations. In a ΛCDM or a dynamical dark
energy scenario, earlier formation epochs would imply higher concentrations,
but in the coupled dark energy case the reason for lower concentrations is
not related to formation histories, but rather to the modified dynamics. In
fact, the friction term redistributes the dark matter particles and lowers the
central densities, despite the earlier formation times (Baldi, 2011a). We find
that this behavior is reproduced for all haloes that we have investigated.
As a consequence of the lower concentration, when falling into the parent
halo subhaloes are more heavily stripped in coupled cosmologies compared
to ΛCDM. The heavy stripping is responsible for decreasing the number of
subhalo orbiting a Milky-Way size halo and is proportional to the strength
of the coupling. In our two models with coupling values within the observational constraints the decrease is of about 10%, while in the extreme model
the decrease reaches 75%. Given that the high number of subhaloes predicted from ΛCDM simulations has been a concern in the last decade, a
cosmological model that tends to alleviate the tension even before appealing to baryonic effects is undoubtedly useful to improve the matching with
observations.
All subhaloes are heavily stripped in coupled cosmologies. Thus, subhaloes that succeeded in forming stars will be less massive than their equivalent subhaloes in ΛCDM. As a consequence, we find that these dwarf haloes
show rotation curves that are flatter and lower than the dwarf haloes that
reside in a ΛCDM Milky-Way parent halo. This can help in matching observational constraints from velocity dispersions of the Milky Way satellites
(Wolf et al., 2010). In the light of the results from Chapter 4, also in this
case the effects of baryons should be taken into account. We plan to explore
this aspect in a future work.
Less concentrated subhalos of coupled cosmologies can in principle be
useful to reconcile the inconsistency between the observed properties of the
Milky Way dwarf galaxies and ΛCDM predictions from simulations, but a
value for the coupling outside of observational constraints must be assumed.
Interestingly, allowing the introduction of massive neutrinos alleviates these
constraints (see e.g. La Vacca et al., 2009) and suggests that dynamics on
sub-galactic scales in coupled dark energy models leave interesting options
open.
To conclude, considering what our study covered, both dynamical dark
116
Summary and Conclusions
energy and coupled dark energy are plausible alternatives to ΛCDM. We
showed that specific parameter choices can even improve the match with
observations. Additionally, I hope we have drawn the reader’s attention
to the important roles played by baryonic processes and by resolution in
differentiating between cosmological models. In the era of high precision
cosmology, small departures from a cosmological constant do matter and
must be further investigated together with baryonic physics.
117
118
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Acknowledgements
I would like to thank my supervisor Andrea Macciò for his advices and guidance that made this work possible. Despite his numerous students, he has
always been patient and understanding and found time to help and answer
all my questions. Thank you for giving me the chance to see what “science” is
really like and also for trying to find solutions where there seemed to be none.
Additionally, I would like to thank the collaborators that have significantly
contributed to this work, Marco Baldi, Luciano Casarini and Greg Stinson.
Each of them have spent time explaining and clarifying and had useful suggestions and comments.
I am thankful to Andrea Macciò and Björn Schäfer for agreeing to referee
my thesis and also to Luca Amendola and Henrik Beuther for being part of
my thesis committee.
An heart-felt thank you to Christian Fendt, the coordinator of the IMPRS
program. He has been there for me in moments of need, helped to find solutions and always gives caring advices.
I would like to thank my master thesis supervisor, Lauro Moscardini. If it
had not been for you, I would not even have applied for a Ph.D. position.
Thank you for making me rediscover my interest for cosmology and for smiling when I would show up with colorful hair and a longboard.
In particular, I would like to add that, despite this field often being a cradle
for heteronormativity, no one has ever made me regret being open and being
myself. I greatly appreciated your open-mindedness, every single bit.
I would also like to thank all friends who helped to make of Heidelberg a
welcoming place where I could spend most of the last three years. It has been
a pleasure to share laughs and down moments, to share the office, lunches
and the numerous falafel.
An enormous thank you to my parents, who have always supported me in
all my decisions and who keep doing it today. Thank you for having been
able to find an impressive combination of being there for me and giving me
space to make my own choices and mistakes. I Roberti are the strongest
pillar I have, one that I can lean against. Un enorme grazie ai miei genitori,
che mi hanno sempre appoggiato in tutte le mie scelte e che continuano a
farlo ancora oggi. Grazie per essere stati in grado di trovare un’eccezionale
combinazione tra il darmi supporto e lasciarmi lo spazio per fare scelte ed
errori. I Roberti sono la colonna portante su cui so di potermi appoggiare.
Finally, I would like to deeply thank Xenia for all she has done for me. I
thank you with all my love for having stood by me during the ups and downs
of these last two years. Thank you for being my better half.
I’m in love with your brother
What’s his name?
I thought I’d come by
to see him again
when you two dance
oh, what a dance
when you two laughed
oh, what a laugh
[...]
Pass This On, The Knife
136
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