Galaxy formation and the structure of the Universe Marcel van Daalen

Galaxy formation and the structure of the Universe Marcel van Daalen
Galaxy formation and the
structure of the Universe
Marcel van Daalen
Galaxy formation and the
structure of the Universe
Proefschrift
ter verkrijging van
de graad van Doctor aan de Universiteit Leiden,
op gezag van de Rector Magnificus prof. mr. C. J. J. M. Stolker,
volgens besluit van het College voor Promoties
te verdedigen op dinsdag 9 december 2014
klokke 16:15 uur
door
Marcel Pieter van Daalen
geboren te ’s-Gravenhage
in 1986
Promotiecommissie
Promotores:
Prof. dr. J. Schaye
Prof. dr. S. D. M. White (MPA, Garching)
Overige leden:
Prof. dr. M. Franx
Prof. dr. K. H. Kuijken
Prof. dr. H. J. A. Röttgering
Dr. H. Hoekstra
Dr. A. R. Zentner (University of Pittsburgh, USA)
“Space is big. You just won’t believe how vastly, hugely,
mind-bogglingly big it is. I mean, you may think it’s a long way down
the road to the chemist’s, but that’s just peanuts to space.”
– Douglas Adams,
The Hitchhiker’s Guide to the Galaxy
The front over the cover shows an image of the Coma cluster
over a silhouette of Leiden, while the back shows an image of the
AGN−WMAP7−L100N512 simulation from the OWLS project over
a silhouette of Munich.
Coma cluster image by Jim Misti, all other cover images and design by the
author.
Table of contents
1
2
3
Introduction
1.1 Large-scale properties of the Universe .
1.2 Testing the standard cosmological model
1.2.1 Linear structure formation . . .
1.2.2 Non-linear evolution . . . . . . .
1.2.3 The role of galaxy formation . .
1.3 Numerical simulations . . . . . . . . . .
1.4 This thesis . . . . . . . . . . . . . . . . .
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1
2
3
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8
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11
13
The Effects of Galaxy Formation on the Matter Power Spectrum: A Challenge for Precision Cosmology
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 The reference simulation . . . . . . . . . . . . . . . . . . . .
2.2.2 Other models . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Power spectrum calculation . . . . . . . . . . . . . . . . . .
2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Comparison of a dark matter only simulation to models . .
2.3.2 The relative effects of different baryonic processes . . . . .
2.3.3 Contributions of dark matter, gas and stars . . . . . . . . .
2.3.4 The back-reaction of baryons on the dark matter . . . . . .
2.3.5 A closer look at the effects of AGN feedback . . . . . . . .
2.4 Comparison with previous work . . . . . . . . . . . . . . . . . . . .
2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.A Convergence tests . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.A.1 Box size . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.A.2 Numerical resolution . . . . . . . . . . . . . . . . . . . . . .
2.B Tabulated power spectra . . . . . . . . . . . . . . . . . . . . . . . .
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16
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21
23
25
26
28
33
36
36
39
43
45
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46
49
The Impact of Baryonic Processes on the Two-Point Correlation Functions of Galaxies, Subhaloes and Matter
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Calculating correlation functions . . . . . . . . . . . . . . .
3.2.3 Linking haloes between different simulations . . . . . . . . .
3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Clustering of galaxies . . . . . . . . . . . . . . . . . . . . .
3.3.2 Clustering of subhaloes . . . . . . . . . . . . . . . . . . . .
3.3.3 Accounting for the change in mass . . . . . . . . . . . . . .
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through
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clustering
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Table of contents
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3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.A Convergence tests . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.B Linked fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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73
77
The Effects of Halo Alignment and Shape on the Clustering
of Galaxies
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Simulation and SAM . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Calculation of the galaxy correlation function . . . . . . . .
4.2.3 Testing the importance of alignment and ellipticity . . . . .
4.2.4 Testing the dependence of galaxy bias on halo shape . . . .
4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Alignment and ellipticity . . . . . . . . . . . . . . . . . . .
4.3.2 Shape-dependent galaxy bias . . . . . . . . . . . . . . . . .
4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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91
94
The Contributions of Matter Inside and Outside of Haloes
to the Matter Power Spectrum
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.2 Power spectrum calculation . . . . . . . . . . . . . . . . . .
5.2.3 Halo particle selection . . . . . . . . . . . . . . . . . . . . .
5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1 Fractional mass in haloes . . . . . . . . . . . . . . . . . . .
5.3.2 Halo contribution to the power spectrum . . . . . . . . . .
5.4 Summary & conclusions . . . . . . . . . . . . . . . . . . . . . . . .
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The Galaxy Correlation Function as a Constraint on Galaxy
Formation Physics
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.1 Estimating the correlation function . . . . . . . . . . . . . .
6.2.2 The SAM and MCMC . . . . . . . . . . . . . . . . . . . . .
6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.1 Comparison with observations . . . . . . . . . . . . . . . . .
6.3.2 Change in parameters . . . . . . . . . . . . . . . . . . . . .
6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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References
139
Table of contents
Nederlandse samenvatting
De vorming van sterrenstelsels en
Structuurvorming . . . . . .
Feedback . . . . . . . . . .
Numerieke simulaties . . . .
Clustering . . . . . . . . . .
In dit proefschrift . . . . . . . . .
de structuur van het
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Universum
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Publications
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Curriculum vitae
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Acknowledgements
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ix
1
Introduction
Our understanding of both the large-scale properties of our Universe and the
processes through which galaxies form and evolve has greatly improved over the
last few decades, thanks in part to new observational probes and more refined
numerical simulations. While the precision with which we measure the cosmic
background radiation, the distribution of matter and the properties of galaxies
continues to increase, we are improving our simulations to include more physical
processes and to resolve ever smaller scales. We are learning just how deeply
cosmology and galaxy formation are intertwined, and the need to model them
simultaneously in order to advance both fields is growing rapidly. In this thesis,
we investigate how galaxy formation can alter the structure of the Universe on a
large range of scales, and how measuring the structure of the Universe can in turn
help us to constrain models of galaxy formation.
Introduction
1.1 Large-scale properties of the Universe
About 13.8 billion years ago, the Universe came into existence in an event we call
the Big Bang. From that moment on, it has been continually expanding. As a
consequence of the Universe being both isotropic and homogeneous on large scales,
the rate of its expansion at any particular time can be related to the current one
through a simple function of only four parameters.1 These are the present-day
matter content of the Universe, Ωm,0 ; its radiation content, Ωγ,0 ; its curvature,
Ωk,0 ; and the contribution of the cosmological constant, ΩΛ,0 , which we presently
refer to mainly as dark energy. Since Ωk,0 is defined such that the sum of these
parameters is by definition equal to unity, this means that only three are independent. Determining the values of these parameters with ever-growing precision is
one of the main aims of cosmology, as together with the Hubble constant, H0 (the
present rate of expansion), they fully determine the evolution of the Universe as a
whole.
Currently, the strongest constraints on these numbers come from observations
of the oldest light in the Universe: the cosmic microwave background, or CMB, a
relic from the Big Bang. The CMB was last scattered when the Universe was only
about 380, 000 years old, at a time when the expansion had cooled the Universe
down sufficiently for protons and electrons to combine and form neutral hydrogen
and helium (an event called “recombination”), allowing light to travel freely for
the first time. It shows us the Universe at the earliest time we could possibly
observe its light, therefore informing us about the initial conditions, from which
any successful model should be able to explain the properties of the Universe as
we see it today.
The CMB is incredibly smooth, indicating a very high level of homogeneity –
the relative variations in the density of baryonic matter (or, “normal”, visible matter) we see in it are of the order of 10−5 (see Figure 1.1). In order for these fluctuations to grow into the galaxies we see today, most matter in the Universe needs
to be (cold) dark matter2 , which is indeed what different observations indicate.
Through very precise CMB measurements using e.g. the Wilkinson Microwave
Anisotropy Probe (WMAP) and Planck satellites, the dark matter fraction and
a host of other cosmological parameters (including all parameters mentioned thus
far) can be determined with ever greater accuracy (e.g. Hinshaw et al., 2013; Planck
Collaboration et al., 2013). These measurements indicate, for example, that the
1 In
more detail, the evolution of the Universe also depends on the equations of state (the ratios of
pressure and density) that characterise these parameters. The equation of state of the radiation
content depends on the number of relativistic particle species, while that of the cosmological
constant depends on the nature of dark energy, both of which are not completely certain yet.
2 If dark matter is “warm” or “hot”, this means that it consists of particles with velocities that are
sufficiently high at the time of decoupling to stream out of density fluctuations, thus preventing
their growth. This process, called free streaming, sets a minimum scale above which fluctuations
can form and depends mainly on the masses of the particles, and their cross sections. While
we already know that not all dark matter can be hot, warm dark matter is not yet completely
ruled out, although its particle mass is strongly constrained (e.g. Viel et al., 2013).
2
1.2 Testing the standard cosmological model through clustering
Figure 1.1: The Cosmic Microwave Background (CMB) as captured by the Planck satellite.
This is a snapshot of the Universe only 380, 000 years after the Big Bang. It is homogeneous and
isotropic on large scales, but very small fluctuations exist nonetheless. The colour scale shows
relative differences of order 10−5 .
Universe is “flat” (i.e. space is not curved but Euclidean), and although everything
we observe directly is baryonic matter, the Universe is in fact dominated by dark
matter and dark energy. We refer to the model that contains all these ingredients
as the ΛCDM-model, currently the standard model of Big Bang cosmology.
1.2 Testing the standard cosmological model through
clustering
As we mentioned, in order for any model of the Universe to be truly successful, it
has to be able to explain all that we see on large scales today. This includes, for
example, the current (accelerating) rate of expansion, but also the different galaxy
populations we observe and the distribution of matter. The latter is the main
focus of this thesis: the clustering of matter, i.e. the structure of the Universe.
The matter distribution is completely determined by the initial conditions of the
Universe; therefore, it is in principle possible to derive all cosmological parameters by examining how matter is organised in the present-day Universe, provided
one understands how structure forms and evolves. In what follows, we present a
simplified view of the formation of structure, starting from the very first density
fluctuations in our Universe.
3
Introduction
Figure 1.2: Illustration of the density field as a field of fluctuations. As a one-dimensional
analogue, we show how seven independent harmonic waves with different amplitudes, wavelengths
and phases (indicated by dotted lines) together add up to the fluctuation field indicated by the
solid curve. Notice that because of contributions of large waves, high-δ fluctuations are often
found close together (i.e. they cluster).
1.2.1
Linear structure formation
Let us consider some part of the Universe with a mean density ρ̄. At3 any threedimensional position x we can calculate a local density ρ(x), which may differ from
the mean. We can now define the density contrast field, or density fluctuations
field, as:
ρ(x) − ρ̄
δ(x) =
.
(1.1)
ρ̄
If δ is positive at some position x, this means that there is a local overdensity.
Under the influence of gravity this overdensity will grow4 , attracting more and
more matter and thereby forming structure.
In order to understand what happens as these overdensities grow, let us first
consider the simplest picture of structure formation: the linear one. Linear structure formation applies when the density fluctuations are very small, i.e. δ " 1.
This is indeed valid for the early Universe, which was extremely homogeneous and
3 To
be precise, a density can never truly be defined at a singular location: one needs to assume
some smoothing scale.
4 For baryonic matter, overdensities may be stable against collapse due to pressure forces. Dark
matter – which dominates the matter content of the Universe – does not feel pressure, however,
and is able to form structure more freely. We will briefly return to this point later in this
chapter.
4
1.2.1 Linear structure formation
therefore contained only small fluctuations.5 As density fluctuations influence each
other through gravity, they do not evolve independently. However, if we consider
each density fluctuation as a superposition of plane waves, then in the linear regime
these waves do evolve independently. Additionally, this view allows us to consider
the growth of structure in a statistical sense, at given scales instead of at given
locations, which is far more meaningful in a largely isotropic and homogeneous
Universe.
An illustration of this wave picture is shown in Figure 1.2. A one-dimensional
density fluctuation is shown as a solid black line. Each such fluctuation can be
uniquely decomposed into harmonic waves with different amplitudes, wavelengths
and phases; in this case, into the seven waves shown as dotted lines. Notice that,
mainly due to the contributions of the longer waves, high-δ fluctuations tend to
cluster – i.e. they are likely to be found close together. As we will see later, this
has large consequences for how matter is organised today.
The relation between the spatial fluctuations δ and the density waves δk can
be expressed through a discrete Fourier transform:
!
δk e−ik·x ,
(1.2)
δ(x) =
k
where k is the wave vector, related to the wavelength by λ = 2π/|k|. We can
now quantify the amount of structure on any Fourier scale k = |k| by squaring
the amplitudes of these density waves, averaging over all waves with the same
wavenumber k to obtain a statistic called the matter power spectrum:
#
"
(1.3)
P (k) ≡ |δk |2 k .
Inflationary theory6 predicts that the primordial power spectrum should scale as
a power law:
(1.4)
P (k) ∝ k ns ,
with a spectral index ns that is very close to unity, meaning that to good approximation the fluctuations in the gravitational potential were scale invariant.7
How the linear density fluctuations evolved from primordial times is best described in terms of the scale factor of the Universe, a(t), which is a dimensionless
length scale that gives a measure of the size of the Universe when it has an age
t. By definition, a(t0 ) = 1, where t0 is the current age of the Universe. Since the
Universe is continually expanding, a was smaller in the past, and infinitesimally
5 Note
that a completely homogeneous, uniform density field cannot exist: at the very least,
microscopic variations in density must exist due to quantum mechanics. Incidentally, such
quantum fluctuations are expected to be the seeds of the variations we see in the CMB –
stretched out to macroscopic scales by a process called inflation very shortly after the Big Bang
– and consequently of all structure existing today.
6 Recently, the first direct evidence for inflation was found by the BICEP-2 team in the form of
a gravitational wave signal in the CMB, see BICEP2 Collaboration et al. (2014).
7 The scale-invariant power spectrum is also called the Harrison-Zel’dovich power spectrum. Current CMB measurements by the Planck satellite indicate ns ≈ 0.96.
5
Introduction
small at the moment after the Big Bang. It is related to the redshift z through
a = 1/(1 + z); at z = 1, the distance between two points in the Universe was thus
twice as small as it is now. For reference, the redshift of the CMB/recombination
is z ≈ 1100.
As we mentioned at the beginning of this chapter, the evolution of the Universe
is determined by its constituents, most importantly matter, radiation and dark
energy.8 Consequently, the growth of fluctuations at any time depends on which of
these constituents dominates. In our simplified picture, a linear density fluctuation
in some spherical volume is expected to grow as:9
δ∝
1
,
ρ̄a2
(1.5)
where ρ̄ is the mean (energy) density of the dominant component of the Universe
at some time t. The amount of matter in the Universe is (to a very high degree)
constant, meaning that its density just scales inversely with the volume of the
Universe:
ρ̄m ∝ a−3 .
(1.6)
Therefore, when matter dominates the Universe, linear fluctuations grow as δ ∝ a.
The energy density of radiation, on the other hand, does not only scale inversely
with the volume, but by an additional factor a since its energy is not conserved
due to photons being redshifted during expansion. Hence:
ρ̄γ ∝ a−4 ,
(1.7)
meaning that during radiation domination linear fluctuations may grow as δ ∝ a2 .
Finally, for dark energy, which is a property of space itself and therefore has a
constant density, we have:
ρ̄Λ ∝ a0 ,
(1.8)
meaning that when dark energy dominates, density fluctuations cannot grow at
all: they are damped by the expansion of the Universe.
However, this damping is not exclusive to the Λ-dominated era. This is related to the existence of the horizon, the maximum distance between causally
connected regions. If two regions are farther apart than this, i.e. farther than light
(and gravity) could have travelled within the age of the Universe, then they could
not have been in causal contact. Fluctuations on scales smaller than the horizon
can be damped if the Universe expands faster than they collapse, which is the
case during the era of radiation domination. In short, this limits the growth of
linear fluctuations to at most logarithmic growth – much slower than the otherwise
power-law growth10 . A summary of the growth rates for both sub- and superhorizon fluctuations is shown in the table at the top of the next page. Here λ is the
wavelength of a fluctuation and rH is the horizon scale.
Since the densities of each of the constituents of the Universe scales differently
with the scale factor of the Universe, it is clear that each dominates in some era.
8 As
6
observations show the Universe to be almost completely flat, geometrically speaking, these
1.2.1 Linear structure formation
λ < rH
λ > rH
γ-dom.
m-dom.
Λ-dom.
damped
δ∝a
damped
δ∝a
2
δ∝a
damped
Radiation, which scales most steeply with a, must have dominated when the Universe was very young (i.e. a was very small), followed by matter, followed by dark
energy. Indeed, radiation dominated the content of the Universe up to a redshift
of z ≈ 3600 (corresponding to an age of the Universe of approximately 50, 000
years), and dark energy has been dominating since z ≈ 0.4 (for approximately the
last 4.2 billion years), meaning that the Universe was matter-dominated during
the majority of its existence, allowing new structure to form.
Combining this insight with the table shown above, we conclude that a special
scale exists, namely the scale of the horizon between the radiation and matter
dominated eras, rH,eq . This scale depends on several cosmological constants, but
roughly11 rH,eq ∼ 102 h−1 Mpc. Fluctuations larger than this scale were able to
grow before the matter-dominated era, while smaller fluctuations were damped.
Afterwards, linear fluctuations on all scales could grow at the same rate.
It can be shown that the damping of subhorizon fluctuations depends on their
size: smaller fluctuations were damped more strongly by the expansion of the Universe. This means that the theoretical linear power spectrum, that started out as
P (k) ∝ k ns , changed shape on scales λ < rH,eq during radiation domination. Consequently, after the radiation-dominated era the power spectrum roughly looked
as follows:
$ n −4
k s
for λ < λeq
P (k, t) ∝
(1.9)
k ns
for λ > λeq .
We show the detailed power spectrum of linear fluctuations in Figure 1.3. The
exact shape and features of this power spectrum depend on all the cosmological
parameters of the standard ΛCDM model, most of which we have already mentioned: besides Ωm,0 , ΩΛ,0 , Ωγ,0 , Ωk,0 , H0 and ns , these are Ωb,0 (the baryonic
matter content of the Universe) and σ8 (the normalisation of the power spectrum).
As all of these influence the power spectrum independently, every one of these can
be determined just by measuring the linear power spectrum to very high precision.
This is essentially what we try to do when observing the CMB, which makes it
the single most powerful observable for understanding our Universe as a whole.
Note that up until shortly before recombination, baryonic subhorizon fluctuare the only constituents of consequence.
approximation is derived by considering the gravitational evolution of a linear fluctuation
in an expanding Universe with mean density ρ̄, which is not trivial.
10 Which, in turn, is much slower than the exponential growth in a non-expanding Universe.
11 The unit shown here for r
H,eq is the typical unit of distance used in cosmology. “Mpc” is
shorthand for “megaparsec”, i.e. one million parsecs (a bit over three million light years), while
“h” is the dimensionless Hubble constant, defined as h ≡ H0 /[100 (km/s)/Mpc] ≈ 0.7.
9 This
7
Introduction
Figure 1.3: The theoretical matter power spectrum. The wavelength of fluctuations decreases
towards the right. A dotted line shows the corresponding primordial power spectrum for ns ≈ 1.
At roughly the horizon scale at matter-radiation equality, rH,eq ∼ 102 h−1 Mpc, the power
spectrum turns over and the power law index asymptotes to ns − 4 ≈ −3. The dashed line shows
a correction for non-linear growth at later times, which only affects scales of a few tens of Mpc
or less.
ations were unable to grow, even though the matter-dominated era had already
begun. This is because photons were capable of dragging baryons along, damping
their fluctuations.12 Therefore, up until the time of the CMB only fluctuations
made up of cold dark matter (which is the dominant form of matter in our Universe) were able to grow. Afterwards, when the baryons could collapse, they followed the dark matter perturbations that were already present. The distribution
of dark matter therefore dictated where stars and galaxies would form.
1.2.2
Non-linear evolution
As we mentioned before, a successful theory need not (and, within reason, cannot)
predict the exact distribution of matter around us in absolute terms. Rather, it
12 Related
to this are the baryon acoustic oscillations (BAO). Gravity and pressure forces (the
latter mainly caused by the photons) counteracted one another, causing oscillations in the
baryonic fluctuations. These can be seen as small wiggles in the matter power spectrum
around 100 h−1 Mpc (see Figure 1.3) and are still imprinted on structure today.
8
1.2.3 The role of galaxy formation
predicts how matter is organised in a statistical sense – for example by predicting
its power spectrum. Up until now, we only considered what would happen to
linear density fluctuations, i.e. fluctuations that are very small (δ " 1). However,
if we want to predict the clustering of matter not only in the very early Universe,
but also today, we need to consider what happens to fluctuations that grow large
enough to actually collapse, for which the simplified picture sketched above no
longer applies. Without a theory that accurately predicts the amount of nonlinear structure as a function of scale in the Universe today, we cannot test our
model against observations.
Many useful insights can be gained from taking a perturbative analytical approach to non-linear structure formation. For example, it grants us expressions for
the time it takes for a halo to form, the relative density at which it forms, and its
final size and mass.13 However, since non-linear collapse is such a complex process
– even when only considering dark matter – clustering predictions nowadays are
mainly made by fitting to the results of simulations. These predictions are then
compared to clustering measurements from observations in order to learn more
about the underlying cosmology.
The general picture of non-linear structure formation looks as follows. As fluctuations grow, they will generally not be spherically symmetric; consequently, they
will collapse first along one direction, forming sheets of matter (also called pancakes, see e.g. Zel’dovich, 1970). It is around this time that fluctuations will enter
the non-linear regime, meaning the approximations used in the previous subsection
are no longer valid. These sheets will collapse along a second direction, forming
filaments, which finally collapse to make what we call haloes, forming a cosmic
web of filaments with very massive haloes at the nodes (hosting galaxy groups
and clusters) and smaller ones throughout. The dark matter collapse then stops
as these haloes virialise, meaning that they attain a quasi-static dynamic equilibrium between the internal gravitational forces and the random motions of their
particles. Further growth then proceeds through the merging of haloes, especially
in clustered environments where the probability of two haloes encountering one
another is large.
This non-linear evolution has important consequences for the clustering of matter, as shown by the dashed line in Figure 1.3. It is the haloes that are of most
interest to us, as these are the regions where galaxies form.
1.2.3
The role of galaxy formation
Haloes are the highest-density regions of dark matter, which constitute the potential wells into which gas flows. Contrary to dark matter, gas feels pressure and
can radiate its energy away as photons, allowing it to cool to the centres of haloes
and form stars and galaxies, which merge and grow and evolve.
13 A
formalism generally referred to as (extended) Press-Schechter theory, see e.g. Press &
Schechter (1974), Bond et al. (1991) and Sheth, Mo & Tormen (2001).
9
Introduction
When we look out into the night sky, we do not see the majority of matter,
which is in the shape of filaments and haloes. Instead, we see only the very
peaks of the matter distribution, as this is where the galaxies reside (see bottom
row of Figure 1.4). Galaxies are therefore biased tracers of the cosmic density
field14 , which means that we must understand the complicated physical processes
through which galaxies formed and evolved in order to be able to use them to
derive cosmological information. The better we understand the galaxy bias, the
better we can constrain the large-scale properties of our Universe by measuring
how galaxies cluster.
Galaxies are not just important to the structure of the Universe because they
are biased tracers; they influence the clustering of (dark) matter as well, which
we can measure through the gravitational effect of matter on light (called lensing). Through gas cooling, baryons can attain much higher densities than dark
matter, and form more structure on galactic scales than dark matter alone could.
The dark matter haloes respond to the formation of galaxies in their centres by
contracting somewhat, thereby changing the amount of structure on small scales
(e.g. Blumenthal et al., 1986). This needs to be taken into consideration when one
tries to predict the clustering of matter based on the relatively simple dark matter
only picture of the Universe. Even though dark matter is dominant and baryons
trace it initially, they act differently on small scales.
However, galaxy formation is not only more complex, but also more violent
than the formation of dark matter haloes. For example, when stars die they may
explode as supernovae, potentially heating up large amounts of gas, which prevents this gas from forming stars. Together, supernovae in a galaxy may drive
galactic fountains of gas, ejecting the gas out of the galaxy. For small enough
galaxies (occupying low-mass haloes), supernovae may even destroy the galaxy
altogether. Very massive galaxies may host an active galactic nucleus (AGN) in
their centre, heating mass amounts of gas and ejecting it far out of the galaxy.
Because of these feedback processes, the pressure of the gas is increased and it
will resist forming structure, meaning that the clustering of matter is lower than
what would be expected from the simple dark matter picture. If enough gas is
driven out of the galaxy, the dark matter haloes may respond in a way opposite
to that we would naively expect: expanding on super-galactic scales (e.g. Velliscig
et al., 2014; also see Chapters 2 and 3 of this thesis). This can even occur without
feedback, due to pressure smoothing of virialised gas on large scales. Many other
physical processes involved in the formation and evolution of galaxies may also
influence clustering predictions. Currently, clustering measurements are becoming
so precise that the need to understand the physics of galaxy formation to comparable accuracy is rapidly increasing.15 Without it, we cannot test our theoretical
14 Galaxies
are often seen as biased tracers of haloes, which are in turn biased tracers of the entire
matter distribution.
15 Even if we do not fully understand the physics of galaxy formation, we may still be able to selfcalibrate our models or marginalise over parameters that describe the effects of e.g. feedback
on halo profiles (see e.g. Zentner, Rudd & Hu, 2008; Yang et al., 2013; Zentner et al., 2013).
10
1.3 Numerical simulations
models of cosmology against observations in a meaningful way.
1.3 Numerical simulations
Because of the complexity and immense range of scales involved in the formation of
galaxies, numerical simulations are the only way to test our models at the precision
of current and upcoming observations.
Different approaches to cosmological simulations are available. For example,
one could run a simulation in which one assumes all matter acts like dark matter
(N-body or collisionless simulations, see Chapter 5), making it possible to simulate
a given region at a far higher resolution than otherwise possible, and base the
formation and evolution of the baryonic component of the Universe on these,
generally assuming the dark matter is not affected by the baryons. By solving
sets of coupled equations for the evolution of baryons and galaxies and using
observations as constraints for the parameters involved, one can very quickly obtain
predictions for other quantities on a large range of scales. This is the approach
taken by semi-analytic models of galaxy formation (see Baugh, 2006, for a review
on the methodology; also, see Chapters 4 and 6 of this thesis).
Another way is to include the baryons in the simulation directly along with
the dark matter, solving the gravitational and other physical equations involved simultaneously (hydrodynamical simulations, see Tormen, 1996, for an introduction
and Springel, 2010, for a review on the method employed here; also, see Chapters
2 and 3 of this thesis). Because there are more complicated equations to solve,
because there are more variables to track and because there are higher densities
involved (decreasing the time steps), such simulations are often run at much lower
resolution than pure N-body simulations in order to keep the computational time
and memory consumption down. However, the trade-off is that fewer approximations have to be made and that the effects of the baryons on the dark matter are
modelled explicitly. Since not all the relevant scales can be resolved (yet), the
physics of e.g. star formation and supernova feedback have to be modelled in a
comparable way to semi-analytics, which brings some uncertainties with it. It is
here that much may be gained in coming years, as these physical recipes in both
semi-analytical and hydrodynamical simulations are constantly being improved,
leading to more realistic representations of our Universe.
The matter distribution in a hydrodynamical simulation from the OWLS project
(Schaye et al., 2010) referred to often in this thesis, called AGN−L100N512 (see
Chapter 2), is shown in Figure 1.4. The region shown is 100 h−1 Mpc on a side
and 10 h−1 Mpc thick. The distributions of cold dark matter, gas and stars are
shown in separate columns, and from top to bottom cosmic time increases. For
z = 127, the fluctuations are still linear, and no stars have formed yet. At z = 6,
However, this requires us to model the way baryons affect clustering in some way, and may
come at the cost of decreasing the statistical significance of the measurements.
11
Introduction
Figure 1.4: This figure illustrates the growth of structure from a linear to a highly nonlinear state, for a comoving (meaning that we scale out the expansion of the Universe) volume
100 h−1 Mpc on a side. The slice shown here, showing the projected mass density, is 10 h−1 Mpc
thick. Each column shows the evolution of a different component: from left to right, these are
cold dark matter, gas, and stars. Each row shows the volume at a different cosmic time. At
the starting redshift of the simulation, z = 127 (only 12 million years after the Big Bang), no
significant structure has formed yet, and density fluctuations are still very small. At z = 6
(almost a billion years after the Big Bang), the dark matter is clearly collapsing and starting to
form a cosmic web. Gas still traces the dark matter on the scales visible here. Galaxies, visible
as small clumps of stars, have started forming at the points of highest density relatively recently.
Finally, at z = 0 (the present, 13.8 billion years after the Big Bang), all the structure we see
today has formed. Note that the gas no longer perfectly traces the dark matter, but is distributed
somewhat more smoothly. This is mainly caused by energetic feedback processes associated with
galaxy formation, heating the gas. The galaxies themselves, seen in the right-most panel, are
clearly biased tracers of the overall mass distribution, having formed where the dark matter
densities are highest.
12
1.4 This thesis
the cosmic web has started to take shape and galaxies have formed in the more
massive haloes. The gas still traces the dark matter almost perfectly on large
scales. By z = 0 (the present time), the cosmic web is more pronounced and all
the galaxies we see today have formed, which trace the large-scale structure of the
cold dark matter. The gas has been heated by gravitational accretion shocks and
by feedback from both supernovae and AGN, and is distributed somewhat more
smoothly than the dark matter.
1.4 This thesis
In the near future it will be possible to measure the distribution of galaxies and
matter to unprecedented precision. To get the most out of these observations and
to avoid unwanted biases, our theoretical models will have to match the accuracy
of real-life measurements. The fields of cosmology and galaxy formation are now
more tightly tied together than ever before: we need to understand the processes
involved in galaxy formation to interpret the clustering of matter and tie our observations to a set of cosmological parameters. Additionally, small-scale clustering
measurements – which are less sensitive to cosmology – may help us to constrain
our galaxy formation models. We explore all these topics in this thesis.
In Chapter 2 we investigate the effects of galaxy formation on the clustering
of matter through the use of the OWLS suite of simulations (Schaye et al., 2010; Le
Brun et al., 2014), in which different physical processes were varied one at a time.
We compare the results of hydrodynamical simulations to those of dark matter
only models, which are generally used to interpret weak lensing measurements
of the matter distribution, and show that feedback from galaxy formation can
have much larger effects on the matter power spectrum than previous studies have
shown. We also investigate how the clustering of dark matter changes when such
processes are included.
Since the clustering of galaxies and the galaxy-galaxy lensing signal may be
similarly affected, we also examine the two-point galaxy correlation function and
the galaxy-matter cross-correlation in these simulations, in Chapter 3. We will
show that efficient feedback can change the predictions by ∼ 10%, and although
this shift is mainly due to the masses of both galaxies and haloes being systematically lowered, significant residual effects remain after correcting for the change in
mass.
Next, we explore the validity and consequences of several assumptions that are
typically used in models based on the halo model and halo occupation distribution.
Specifically, in Chapter 4 we investigate if and how the shapes and alignments of
haloes are reflected in the clustering of galaxies, using the Guo et al. (2011) semianalytical model run on the Millennium Simulation (Springel et al., 2005). We also
ask the question whether it is possible to measure this form of “assembly bias” from
galaxy surveys, without knowing the distribution of dark matter. In Chapter
5 we test the postulate of halo models that all matter resides in haloes, using
13
Introduction
collisionless simulations from the OWLS project. We calculate the clustering of
matter in our simulations and compare it to the clustering of matter within haloes
above a certain mass, exploring also the effects of using different halo definitions.
Finally, in Chapter 6, we present a fast and accurate clustering estimator for
use in semi-analytical models of galaxy formation. Using this halo model based
estimator, it is possible to predict the projected galaxy correlation function to
an accuracy of ∼ 10% using only a very small subsample of haloes, meaning it
can be used efficiently while exploring the parameter space of a model. By using
clustering data as a constraint in addition to the usual one-point functions (such as
the stellar mass or luminosity functions), degeneracies can be removed, improving
both the match of the model to multiple data sets at once and our understanding
of galaxy formation. We apply our model to the semi-analytical model of Guo
et al. (2013), and show how the best-fit parameters change to bring the model into
agreement with the newly-added constraints.
14
2
The effects of galaxy formation on the
matter power spectrum: A challenge for
precision cosmology
Upcoming weak lensing surveys, such as LSST, EUCLID, and WFIRST, aim to
measure the matter power spectrum with unprecedented accuracy. In order to fully
exploit these observations, models are needed that, given a set of cosmological parameters, can predict the non-linear matter power spectrum at the level of 1% or
better for scales corresponding to comoving wave numbers 0.1 ! k ! 10 h Mpc−1 .
We have employed the large suite of simulations from the OWLS project to investigate the effects of various baryonic processes on the matter power spectrum.
In addition, we have examined the distribution of power over different mass components, the back-reaction of the baryons on the CDM, and the evolution of the
dominant effects on the matter power spectrum. We find that single baryonic
processes are capable of changing the power spectrum by up to several tens of
per cent. Our simulation that includes AGN feedback, which we consider to be
our most realistic simulation as, unlike those used in previous studies, it has been
shown to solve the overcooling problem and to reproduce optical and X-ray observations of groups of galaxies, predicts a decrease in power relative to a dark
matter only simulation ranging, at z = 0, from 1% at k ≈ 0.3 h Mpc−1 to 10% at
k ≈ 1 h Mpc−1 and to 30% at k ≈ 10 h Mpc−1 . This contradicts the naive view
that baryons raise the power through cooling, which is the dominant effect only
for k " 70 h Mpc−1 . Therefore, baryons, and particularly AGN feedback, cannot
be ignored in theoretical power spectra for k " 0.3 h Mpc−1 . It will thus be necessary to improve our understanding of feedback processes in galaxy formation, or
at least to constrain them through auxiliary observations, before we can fulfil the
goals of upcoming weak lensing surveys.
Marcel P. van Daalen, Joop Schaye,
C. M. Booth and Claudio Dalla Vecchia
Monthly Notices of the Royal Astronomical Society
Volume 415, Issue 4, pp. 3649-3665 (2011)
Galaxy formation and the matter power spectrum
2.1 Introduction
One of the aims of cosmology is to find the initial conditions for structure formation
in the Universe. These can be characterised by a single set of cosmological parameters, which directly influence the formation, growth and clustering of structure,
and hence the distribution of matter as we observe it today.
A powerful measure of the statistical distribution of matter (and a sufficient
one for the case of Gaussian fluctuations), is the matter power spectrum, P (k),
where k is the comoving wave number corresponding to a comoving spatial scale
λ = 2π/k. Given a sufficiently accurate model for the formation of structure,
we can infer the initial, linear power spectrum from the observed, non-linear one.
Moreover, as the rate of growth of structure depends on the expansion history,
such a model also allows us to convert observations of the evolution of the power
spectrum into measurements of other cosmological parameters such as the equation
of state of the dark energy.
Some of the most accurate measurements of the matter power spectrum come
from studies of weak, gravitational lensing (e.g. Massey et al., 2007; Fu et al., 2008;
Schrabback et al., 2010), galaxy clustering (e.g. Cole et al., 2005; Reid et al., 2010)
and the Lyα forest (e.g. Viel, Haehnelt & Springel, 2004; McDonald et al., 2006).
Up to a few years ago, the statistical errors were sufficiently large that one could use
analytical predictions (always assuming, amongst other things, that the Universe
contains only dark matter), such as those by Peacock & Dodds (1996), Ma et al.
(1999) and Smith et al. (2003a). The latter used ideas from the “halo model” (e.g.
Peacock & Smith, 2000; Seljak, 2000; Cooray & Sheth, 2002) to improve upon the
accuracy of simpler analytical predictions. In recent years the further improvement
of this model has become increasingly dependent on the results from N -body
simulations, such as the derived concentration-mass relation for dark matter haloes
(e.g. Neto et al., 2007; Duffy et al., 2008; Hilbert et al., 2009). If baryonic effects
were negligible, then these methods would allow the matter power spectrum to be
predicted with an accuracy of ∼ 1% for wave numbers k ! 1 h Mpc−1 (Heitmann
et al., 2010). However, we will show here that baryonic effects are larger than this
on the scales relevant for many observations.
Upcoming weak lensing surveys aim to measure the matter power spectrum
on scales of 0.1 < k < 10 h Mpc−1 . In order to reach the level of precision their
instruments are capable of, surveys such as LSST,1 EUCLID,2 and WFIRST3 need
to be calibrated using theoretical models that retain 1% accuracies on these scales
(Huterer & Takada, 2005; Laureijs, 2009).4 This is, however, not as straightforward
1 http://www.lsst.org/lsst
2 http://www.euclid-imaging.net/
3 http://wfirst.gsfc.nasa.gov/
4 Since
cosmological parameters are inferred from cosmic shear using a complicated weighting of
the power spectrum over a range of scales and redshifts, the relation between the accuracy with
which these parameters can be determined and the uncertainty in the models depends on the
survey and is different for different parameters. Semboloni et al. (2011) present a more detailed
study of the consequences of our findings for weak lensing surveys.
16
2.1 Introduction
as increasing the resolution of existing N -body simulations: many authors have
demonstrated that on these scales baryonic matter, which is not accounted for in
currently employed theoretical models, introduces deviations of up to 10% (White,
2004; Zhan & Knox, 2004; Jing et al., 2006; Rudd, Zentner & Kravtsov, 2008;
Guillet, Teyssier & Colombi, 2010a; Casarini et al., 2011a).
Recent hydrodynamic simulations include many of the physical processes associated with baryons, such as radiative cooling, star formation and supernova (SN)
feedback. However, the processes which cannot be resolved in simulations are generally also not entirely understood, and different prescriptions exist that aim to
model the same physics. Because of this, different authors may find significantly
different results even when including the same baryonic processes. Furthermore,
it is not a priori clear which physical effects are capable of changing the matter
power spectrum at the 1% level and should therefore be included. These modelling uncertainties may thus prevent upcoming surveys from further constraining
the cosmological parameters of our Universe.
Here we employ a large suite of state-of-the-art cosmological, hydrodynamical
simulations from the OWLS project (Schaye et al., 2010) to systematically study
the effects of various baryonic processes on the matter power spectrum over a wide
range of scales, k ∼ 0.1 − 500 h Mpc−1 . These processes include metal-line cooling,
different prescriptions for SN feedback, and feedback from active galactic nuclei
(AGN). We will see that all of our results are heavily influenced by the inclusion
of AGN feedback, which was not considered by earlier studies and which has been
shown to solve the overcooling problem that has long plagued hydrodynamical
simulations and to lead to an excellent match to both the optical and X-ray properties of groups of galaxies (McCarthy et al., 2010, 2011). Outflows driven by AGN
strongly increase the scale out to which baryons modify the power spectrum. We
also investigate how the power is distributed over different components (i.e. CDM,
gas and stars) and examine the back-reaction of the baryons on the dark matter.
In a follow-up paper (Semboloni et al., 2011), we quantify the implications for
current and proposed weak lensing surveys and we show how the uncertainty due
to baryonic physics can be reduced by making use of additional observations of
groups and clusters.
This chapter is organised as follows. In §2.2 we discuss the simulations and the
power spectrum estimator employed. In our main results section, §2.3, we compare our dark matter only simulation to analytical estimates (§2.3.1), we compare
power spectra of simulations with different baryonic processes (§2.3.2), and we investigate how the power is distributed over different physical components (§2.3.3).
In this section we also examine the back-reaction of galaxy formation on the dark
matter (§2.3.4) and we consider the evolution of the most dominant effects on
the power spectrum (§2.3.5). We compare to the results found by other authors
in §2.4 and provide a summary in §2.5. Finally, we test the convergence of our
results in Appendix A and provide tables of the power spectra of all simulations
in Appendix B.
We note that all distances quoted in this chapter are comoving and all power
17
Galaxy formation and the matter power spectrum
spectra are obtained at redshift zero, unless stated otherwise.
2.2 Simulations
The OWLS project (Schaye et al., 2010), where OWLS is an acronym for OverWhelmingly Large Simulations, is a suite of large, cosmological, hydrodynamical
simulations. The code used is a heavily extended version of gadget iii, a Lagrangian code which was last described in Springel (2005a). It uses a TreePM
algorithm to efficiently calculate the gravitational forces (where PM stands for
Particle Mesh and the “Tree” describes the structure in which the particles are organised for this calculation, see for example Barnes & Hut, 1986; Xu, 1995; Bagla,
2002) and Smoothed Particle Hydrodynamics (SPH) to follow and evolve the gas
particles (see Rosswog, 2009, for a review).
There are two main sets of simulations, which have periodic boxes of size L = 25
and 100 h−1 comoving Mpc on a side, and are run down to redshifts z = 2 and
0, respectively. Most simulations use 5123 collisionless cold dark matter (CDM)
particles and an equal number of baryonic (collisional gas or collisionless star)
particles. We will refer to the particle number used in a simulation with the
1/3
parameter N = Npart (= 512 for the high-resolution simulations). In this work we
will focus on z = 0 and hence on the simulations using a 100 h−1 Mpc box. The
particle masses are 4.06 × 108 h−1 M( [L/(100 h−1 Mpc)]3 [N/512]−3 for the dark
matter and 8.66 × 107 h−1 M( [L/(100 h−1 Mpc)]3 [N/512]−3 for the baryons. The
gravitational forces are softened on a comoving scale of 1/25 of the initial mean
interparticle spacing, L/N , but the softening length is limited to a maximum
physical scale of 2 h−1 kpc[L/(100 h−1 Mpc)] which is reached at z = 2.91. The
SPH calculations use 48 neighbours.
For the initial conditions, a theoretical matter power spectrum – which of
course depends on the chosen set of cosmological parameters – is generated using
cmbfast (Seljak & Zaldarriaga, 1996, version 4.1). Prior to imposing the linear
input spectrum, the particles are set up in an initially glass-like state, as described
in White (1994). The particles are then evolved to redshift z = 127 using the
Zel’dovich (1970) approximation.
On small scales, the physics of galaxy formation is unresolved, and subgrid
models are needed to include baryonic effects like radiative cooling, star formation
and supernova feedback. Although each OWLS run is a state-of-the-art cosmological simulation in itself, the real power of the OWLS project lies in the fact that
it is composed of more than 50 simulations that all incorporate different sets of
physical processes, parameter values, or subgrid recipes. In this way the effects
of turning off or tweaking a single process can be studied in detail, making it especially well-suited to investigate which processes can, by themselves, change the
power at k ∼ 1 − 10 h Mpc−1 by > 1%. In this chapter we briefly describe the
subgrid physics included in the reference simulation, as well as the differences with
respect to simulations we compare to in §2.3.2. For a more detailed treatment of
18
2.2.1 The reference simulation
the simulations and the different physics models included, we refer to Schaye et al.
(2010).
2.2.1
The reference simulation
As the intention of the OWLS project is to investigate the effects of altering or
adding a single physical process, it is convenient to have a single simulation that
acts as the basis for all other simulations. Such a “default” simulation should
of course include many of the physical processes that we know to be important
already, as ideally we would only want to vary one thing at a time. We call
this simulation the reference simulation, or REF for short. Note that this is not
intended to be the “best” simulation, but simply a model to build on. In fact, it
has for example been shown that AGN feedback, which was not included in the
REF model and which we briefly discuss in the next section, is required to match
observations of groups and clusters of galaxies (McCarthy et al., 2010, 2011).
We assume cosmological parameter values derived from the Wilkinson Microwave Anisotropy Probe (WMAP) 3-year results (Spergel et al., 2007): {Ωm ,
Ωb , ΩΛ , σ8 , ns , h} = {0.238, 0.0418, 0.762, 0.74, 0.951, 0.73}. Except for σ8 , all
of these are consistent with the WMAP 7-year data (Komatsu et al., 2011). This
specific parameter describes the root mean square fluctuation in spheres with a
radius of 8 h−1 Mpc linearly extrapolated to z = 0 and effectively normalises the
matter power spectrum. Measurements in the last few years have systematically
increased the value of σ8 , which may influence the validity of our results. To check
the effects of using “wrong” values for this and other cosmological parameters, we
have re-run our two most important simulations – one with only dark matter and
one in which AGN feedback is added to the reference model – using the WMAP7
cosmology. We briefly discuss these at the end of section §2.2.2. As we shall see
in §2.3.5, this change in cosmology does not affect our conclusions.
The reference simulation includes both radiative cooling and heating, which are
modelled using the prescription of Wiersma, Schaye & Smith (2009). Net radiative
cooling rates are computed on an element-by-element basis in the presence of the
cosmic microwave background and the Haardt & Madau (2001) model for the UV
and X-ray background radiation from quasars and galaxies, taking into account the
contributions of eleven different elements pre-computed using the publicly available
photo-ionization package CLOUDY, last described by Ferland et al. (1998). The
effects of hydrogen ionization are modelled by switching on the Haardt & Madau
(2001) model at z = 9.
Cosmological simulations do not yet come close to resolving the process of star
formation, and so a subgrid recipe has to be included for this as well. In our
simulations, gas particles can be converted into star particles once their hydrogen
number densities exceed the threshold for thermo-gravitational instability (n∗H =
0.1 cm−3 ; Schaye, 2004). Cold gas particles with higher densities follow an imposed
equation of state, P ∝ ργeff . Here γeff = 4/3, for which Schaye & Dalla Vecchia
(2008) showed that both the Jeans mass and the ratio of the Jeans length to the
19
Galaxy formation and the matter power spectrum
Simulation
Description
AGN
Includes AGN (in addition to SN feedback)
AGN−WMAP7
Same as AGN, but with a WMAP7 cosmology
DBLIMFV1618
Top-heavy IMF at high pressure, extra SN energy in wind velocity
DMONLY
No baryons, cold dark matter only
DMONLY−WMAP7
Same as DMONLY, but with a WMAP7 cosmology
MILL
Millennium simulation cosmology (i.e. WMAP1), η = 4 (twice the SN energy of REF )
NOSN
No SN energy feedback
NOSN−NOZCOOL
No SN energy feedback and cooling assumes primordial abundances
NOZCOOL
Cooling assumes primordial abundances
WDENS
Wind mass loading and velocity depend on gas density (SN energy as REF )
WML1V848
Wind mass loading η = 1, velocity vw = 848 km s−1 (SN energy as REF )
WML4
Wind mass loading η = 4 (twice the SN energy of REF )
Table 2.1: The different variations on the reference simulation that are compared in this chapter.
Unless noted otherwise, all simulations use a set of cosmological parameters derived from the
WMAP3 results and use identical initial conditions.
SPH kernel are independent of the density, thus preventing spurious fragmentation
due to a lack of numerical resolution. Using their pressure-dependent prescription
for star formation, the observed Kennicutt-Schmidt relation, a surface density
scaling law for the star formation rate that can be written as Σ̇∗ ∝ Σng (Kennicutt,
1998), is reproduced by construction, independent of the imposed equation of state.
The reference simulation assumes a Chabrier (2003) stellar Initial Mass Function (IMF) with low and high mass cut-offs at 0.1 and 100 M(, respectively. The
release of hydrogen, helium and heavier elements by these stars to the surrounding
gas is tracked as well: gas can be ejected through Type II SNe and stellar winds
for massive stars, and Type Ia SNe and Asymptotic Giant Branch (AGB) stars for
intermediate mass stars. This implementation of stellar evolution and chemical
enrichment is discussed in Wiersma et al. (2009).
Finally, the reference simulation includes a prescription for supernova feedback,
discussed in Dalla Vecchia & Schaye (2008). Supernovae are capable of depositing a
significant amount of energy in the surrounding gas, driving large-scale winds that
may eject large amounts of gas, dramatically suppressing the formation of stars.
In the model used here, the energy from SNe is injected into the gas kinetically.
After a delay time of 30 Myr, a new star particle j will “kick” a neighbouring
%Nngb
SPH particle i with a probability ηmj / i=1
mi in a random direction, giving
it an extra velocity vw . The reference simulation uses the values η = 2 for the
initial wind mass loading and vw = 600 km s−1 for the initial wind velocity, which
corresponds to 40% of the available kinetic energy for our IMF.
20
2.2.2 Other models
2.2.2
Other models
The OWLS project includes many variations on the reference simulation. We will
now briefly discuss the simulations that we compare to in §2.3.2. The different
models are listed in Table 2.1. For more details and other models we again refer
to Schaye et al. (2010).
The simulation DMONLY includes only dark matter, hence the only active
physical process is gravity. This model is useful, as many (semi-)analytical models
for the matter power spectrum assume that baryons are unimportant on large
scales.
The NOSN simulation excludes supernova feedback, and the simulation NOZCOOL assumes primordial abundances when computing cooling rates. The simulation NOSN−NOZCOOL excludes both SN feedback and metal-line cooling. Naturally, none of the three simulations can be considered realistic as we know that
the omitted processes exist, but they are valuable tools to investigate on what
scales and in what measure these processes affect the total matter power spectrum. In fact, the same may be said for the other models we consider as all,
except for AGN, suffer from the overcooling problem and hence apparently miss
an important process that does occur in nature (be it AGN feedback or something
else).
Supernova feedback models suffer from large uncertainties due to the limited
resolution of the simulations and a lack of observational constraints. Though the
product of the initial wind mass loading and the initial wind velocity squared,
2
ηvw
, determines the energy injected into the winds per unit stellar mass and is
therefore limited from above by the energy available from the SNe, the individual
parameters are poorly constrained and can thus be varied. One variation on the
reference model that uses the same SN energy per unit stellar mass as REF is
WML1V848, in which the wind mass loading
√ is reduced by a factor of 2 while the
wind velocity is increased by a factor of 2. Another such variation is WDENS,
in which the wind parameters scale with the density of the gas from which the
1/6
star particle formed: the wind velocity as vw ∝ nH , and the wind mass loading
−1/3
−2
∝ nH . Both parameters are equal to their fiducial values for stars
as η ∝ vw
formed at the density threshold for star formation. For the polytropic EoS that we
impose onto the ISM, the wind velocity in this model scales with the local effective
sound speed, as might be the case for thermally driven winds.
We also compare to models where the SN energy is varied. One scenario in
which the SN energy may be higher than that in the reference model is when,
under certain circumstances, the IMF becomes top-heavy, meaning that relatively
more high-mass stars are produced. It is expected that the IMF is top-heavy at
high redshift and low metallicity (e.g. Larson, 1998), and both observations and
theory suggest that it may be top-heavy in extreme environments like the galactic
centre and starburst galaxies (e.g. Baugh et al., 2005; Bartko et al., 2010). In
the simulation DBLIMFV1618, the latter effect is modelled by a switch from the
Chabrier IMF to one that follows φ(m) ∝ m−1 once the gas reaches a certain
21
Galaxy formation and the matter power spectrum
pressure threshold, which is set so that ∼ 10% of the stellar mass forms with a
top-heavy IMF. In this case, the emissivity in ionizing photons goes up by a factor
7.3, and it is assumed that the SN energy scales up by the same factor. In the
model √
we consider here, this extra energy is used to raise the wind velocity by a
factor 7.3.
The final model that we consider that only differs from REF in terms of its
wind parameters is WML4, in which the SN energy per unit stellar mass is doubled
by simply increasing the wind mass loading by a factor of two. The same is done in
the simulation MILL. However, the most important feature of the latter is that it
uses the same values for the cosmological parameters as the Millennium simulation
(Springel, Di Matteo & Hernquist, 2005). These are derived from first-year WMAP
data and are given by: {Ωm , Ωb , ΩΛ , σ8 , ns , h} = {0.25, 0.045, 0.75, 0.9, 1.0, 0.73}.
The last and, for our purposes, most important physics variation we consider
here adds a phenomenon that has proved to be increasingly necessary to reconcile
theory and observations, from the scales of individual galaxies to clusters: Active
Galactic Nuclei, or AGN. They are caused by the emission of large amounts of
energy from the accreting supermassive black holes that reside at the centres of
galaxies, in the form of radiation that may couple to the gas and relativistic jets
caused by the magnetic field of the infalling material, which can heat and displace
gas out to very large distances. AGN have been invoked to explain, for example,
the low star formation rates of high-mass galaxies and the suppression of cooling
flows in clusters. Moreover, Levine & Gnedin (2006) have used a toy model to
demonstrate that AGN feedback may provide sufficient energy to have a large
effect on the matter power spectrum.
We model the growth of supermassive black holes and the associated feedback
processes using the prescription detailed in Booth & Schaye (2009), which is an
extension of that by Springel et al. (2005). During the simulation, a black hole
seed particle with mass mseed = 9 × 104 h−1 M( (i.e. 10−3 mbaryon ) is placed at the
centre of every dark matter halo whose mass exceeds mhalo,min = 4 × 1010 h−1 M(
(corresponding to 102 dark matter particles). These particles then accumulate
mass from the surrounding gas at an (Eddington-limited) rate based on BondiHoyle-Lyttleton accretion (Bondi & Hoyle, 1944; Hoyle & Lyttleton, 1939), but
scaled up by a factor α to account for the lack of a cold gas phase and the finite
numerical resolution. However, for densities below our star formation threshold
we do not expect a cold phase to be present and we therefore set α equal to unity.
To ensure a smooth transition, α is made to depend on the density of the gas:
&
1
if nH < n∗H
' (β
α=
(2.1)
nH
otherwise.
n∗
H
Here the density threshold n∗H is the critical value required for the formation of
a cold interstellar gas phase (n∗H = 0.1 cm−3 ; see §2.2.1). Models of this type are
called ‘constant-β models’, and the fiducial value β = 2 is used throughout this
chapter.
22
2.2.3 Power spectrum calculation
The black holes inject 1.5 per cent of the rest mass energy of the accreted
gas into the surrounding matter in the form of heat. This feedback efficiency
determines the normalisation, but not the slope, of the relations between black
hole mass and galaxy properties. Booth & Schaye (2009) and Booth & Schaye
(2011) demonstrate that this efficiency reproduces the observed relations between
BH mass and both stellar mass and stellar velocity dispersion, as well as their
evolution. McCarthy et al. (2010) have shown that the AGN simulation, but not
the reference model, provides excellent agreement with both optical and X-ray
observables of groups of galaxies at redshift zero. In particular, it reproduces the
temperature, entropy, and metallicity profiles of the gas, the stellar masses, star
formation rates, and age distributions of the central galaxies, and the relations
between X-ray luminosity and both temperature and mass. We therefore consider
simulation AGN to be more realistic than our other models. As we shall see in
§2.3, the inclusion of AGN feedback greatly affects the power spectrum on a large
range of scales.
Finally, we have re-run two simulations, DMONLY and AGN, with cosmological parameters derived from the WMAP 7-year results (Komatsu et al., 2011):
{Ωm , Ωb , ΩΛ , σ8 , ns , h} = {0.272, 0.0455, 0.728, 0.81, 0.967, 0.704}. These versions
are called DMONLY−WMAP7 and AGN−WMAP7, respectively. We consider the
latter to be our most realistic and up-to-date model. Note that the linear input
power spectra used for the initial conditions of these simulations have not been
generated by cmbfast, but by the more up-to-date f90 package camb (Lewis &
Challinor, 2002, version January 2010).
2.2.3
Power spectrum calculation
The distribution of matter in the Universe can be described by a continuous density
function, ρ(x), where the vector x specifies the position relative to some arbitrary
origin. Given this density field, we consider fluctuations, δ(x), defined as:
δ(x) ≡
ρ(x) − ρ̄
,
ρ̄
(2.2)
where ρ̄ is the global mean density. We can relate this density contrast to wave
modes δ̂k via a discrete Fourier transform:
!
(2.3)
δ̂k e−ik·x .
δ(x) =
k
The density field can thus be seen as made up of waves with certain amplitudes
and phases, with wave vectors k. We now define the power spectrum, P (k), as:
*
)
(2.4)
P (k) ≡ V |δ̂k |2 ,
k
where V is the volume under consideration. The power spectrum is therefore
obtained by collecting the amplitudes-squared of all wave modes with the same
23
Galaxy formation and the matter power spectrum
wave number k = |k|, and averaging them. This makes it clear that the power
spectrum is a statistical tool, whose accuracy increases when more waves of the
same length are available (i.e. when the scale 2π/k is small compared to the size of
the box). We will present our results using what is often called the dimensionless
matter power spectrum, which is defined as:
∆2 (k) =
k3
P (k).
2π 2
(2.5)
The dimensionless power spectrum scales with the mass variance, σ 2 (M ), where
+ 2π ,3
ρ̄. Note that using ∆2 (k) instead of P (k) does not affect the relative
M = 4π
3
k
differences between power spectra.
The code we have chosen to use to obtain accurate power spectrum estimations
from our simulations is the publicly available f90 package called powmes (Colombi
et al., 2009). The advantages of powmes stem from the use of the Fourier-Taylor
transform, which allows analytical control of the biases introduced, and the use of
foldings of the particle distribution, which allow the dynamic range to be extended
to arbitrarily high wave numbers while keeping the statistical errors bounded. For
a full description of these methods we refer to Colombi et al. (2009). We have
compared the performance of powmes with respect to power spectrum estimators
using simple NGP, CIC and TSC interpolation schemes, and found that powmes
is capable of obtaining far more accurate power spectra over a larger spectral range
within the same computation time.
We have expanded powmes with the possibility to consider only one group
of particles at a time, in order to see which parts of the power spectrum are
dominated by the contribution of, for example, cold dark matter (see §2.3.3).
Finally, we performed extensive timing tests using different grids, foldings, CPUs
and particle numbers which, combined with the performance results from Colombi
et al. (2009), resulted in the fiducial values G = 256 and F = 7 for the number of
grid points on a side and the number of foldings, respectively.
2.2.3.1
Discreteness and other numerical limitations
In Appendix A we demonstrate that the simulations are sufficiently converged
with respect to increases in the numerical resolution to predict the power spectrum
with better than 1% accuracy for k ! 10 h Mpc−1 . This range is greatly expanded
in both directions if we only consider the relative differences in power between
simulations.
Besides numerical resolution, the predicted power spectra may be affected by
sample variance, which is generally called cosmic variance in cosmology. This
is caused by the finite volume of the box and by the fact that each simulation
provides only a single realisation of the underlying statistical distribution. Note
that finite volume effects are different for the simulations than for observational
surveys, because the mean density in the simulation boxes is always equal to the
cosmic mean. In Appendix A we show that finite volume effects may cause us to
24
2.3 Results
underestimate the effects of baryons on scales of several tens of Mpc, i.e. close to
the size of the box. While the fact that we only use a single realisation of the
initial conditions prevents us from obtaining highly accurate absolute values for
the power spectrum on scales close to the box size, it does not prevent us from
investigating the relative changes in power caused by baryon physics.
Finally, we are limited in our determination of the power spectrum by the
discreteness of the density field. Because we use particles to represent a continuous
field, there will always be non-zero power present at all scales, called white noise or
shot noise. If we assume the particle distribution to be a local Poisson realisation
of a stationary random field, an assumption used in any calculation of the power
spectrum and one that is expected to be valid for an evolved distribution,5 this
white noise component can be calculated (see, for example, Peebles, 1980, 1993;
Colombi et al., 2009). Subtracting the shot noise from the initial estimate of the
power spectrum will make the latter somewhat more accurate, but one should
still expect the uncertainty on the estimate of the power spectrum to increase
dramatically when the intrinsic power spectrum falls far below the shot noise
level. The contribution of shot noise to P (k) is independent of k. Hence, if we
use ∆2 (k) as the measure of the power spectrum, then the shot noise level will
scale as k 3 . In the following section, the scale at which the shot noise of each
simulation is equal to (the white noise corrected) ∆2 (k) is denoted by a circle,
while it is shown explicitly in Appendix A. Note that the theoretical shot noise
level has been subtracted for all power spectra shown in this chapter.
2.3 Results
In this section we present the power spectra obtained from our simulations. In
§2.3.1 we compare the power spectrum of our dark matter only simulation to
predictions from the literature. We investigate the effects of adding or modifying
prescriptions for baryonic processes in §2.3.2. We examine how well CDM, gas, and
stars trace each other and consider the contributions of these different components
to the total power in the reference simulation in §2.3.3, and we examine the backreaction of baryons on the CDM for the two most important simulations, REF
and AGN, in §2.3.4. Finally, in §2.3.5, we take a closer look at model AGN,
which we consider to be our most realistic simulation because it reproduces the
optical and X-ray observations of groups of galaxies (McCarthy et al., 2010). We
investigate the effect of using the WMAP7 rather than the WMAP3 cosmology,
compare to widely used model power spectra, and consider the evolution of the
effect of baryons on the matter power spectrum.
5 The
discreteness noise can initially be much smaller if the particles are arranged on a grid or
in a “glass-like” fashion. Particles in low-density regions may retain memory of their initial
distribution, reducing the noise below the level expected for a Poisson distribution.
25
Galaxy formation and the matter power spectrum
2.3.1
Comparison of a dark matter only simulation to
models
In this section we compare the power spectrum of our DMONLY simulation to
those predicted by the widely used models of Peacock & Dodds (1996, hereafter
PD96) and Smith et al. (2003a, hereafter HALOFIT).
The PD96 model is an extension of what is known as the HKLM model (Hamilton et al., 1991), which first introduced a universal analytical formula to map the
linear correlation function into a non-linear one, the coefficients of which were estimated using N-body simulations. Both of these models assume spherical collapse
of fluctuations that have reached a certain overdensity, followed by stable clustering (which states that the mean physical separation of particles is constant on
sufficiently small scales). Peacock & Dodds (1994), followed by PD96, expanded
on the groundwork laid by HKLM by presenting a version of the method that
worked with power spectra instead and allowed for Ω *= 1, a non-zero cosmological
constant and large negative spectral indices.
However, numerical simulations have shown that the assumption of stable clustering is not always valid. The more recent HALOFIT model by Smith et al.
(2003a) aimed to improve on PD96 by taking this into account. This method is
based on concepts from the “halo model”, in which the density field is viewed as a
distribution of isolated haloes (e.g. Peacock & Smith, 2000; Seljak, 2000; Cooray
& Sheth, 2002). It is then assumed that the power spectrum can be split into
two parts: a large-scale quasi-linear term that is due to the clustering of separate
haloes (the 2-halo term), and a small-scale term caused by the correlation of subhaloes within the same parent halo (the 1-halo term). Their resulting analytical
formulae were fit to power spectra obtained from N-body simulations.
To create power spectra that conform with these models and the cosmological
parameters used in our simulations, we have utilised the publicly available package
iCosmo, described in Refregier et al. (2011). We chose to generate the linear power
spectra using the Eisenstein & Hu (1999, EH) transfer function. We have also
tried using the Bardeen et al. (1986, BBKS) transfer function to generate initial
conditions for the PD96 model, as this is the one originally used by the authors,
which introduced only minor differences with respect to the results shown here
(1 − 10% lower power for k " 10 h Mpc−1 ).
In Figure 2.1 we compare these models to the simulation that, like the theoretical models for non-linear growth, includes only dark matter (DMONLY ). For
reference, the dashed curve shows the linear input power spectrum of the simulations. The bottom panel shows the ratio of the analytical predictions to our
results. Note that we have omitted the first wave mode (at λ = 100 h−1 Mpc)
in all of our figures because we cannot sample the power spectrum on the scale
of the simulation box. We see that the dark matter power spectrum follows the
analytical predictions pretty well on large scales (except on the scale of the simulation box), and that HALOFIT provides a better match than the PD96 model,
as expected. However, on scales below a few Mpc the theoretical models start to
26
2.3.1 Comparison of a dark matter only simulation to models
Figure 2.1: Comparison of the matter power spectrum of DMONLY−L100N512 with analytical
fits by Peacock & Dodds (1996, PD96) and Smith et al. (2003a, HALOFIT) at redshift zero. The
small circle, drawn in this and all following plots showing ∆2 (k), indicates the scale below which
the (subtracted) shot noise in the simulation becomes significant, and the dashed purple curve
shows the linear input power spectrum of the simulations. The bottom panel shows the ratios of
the power spectra from theoretical models and the simulation. There is good agreement down
to scales of a few Mpc, especially for the more recent HALOFIT model, but on smaller scales
DMONLY predicts up to twice as much power as HALOFIT. For λ < 102 h−1 kpc the power in
the DMONLY simulation drops due to a lack of resolution.
severely underestimate the amount of structure formed in the simulation, and the
difference between HALOFIT and the DMONLY simulation reaches a factor of
2 on scales of 1 − 3 × 10−1 h−1 Mpc. The rapid decline of the DMONLY power
spectrum for k " 100 h Mpc−1 (λ < 102 h−1 kpc) is mostly due to the underproduction of low-mass haloes due to the finite resolution (see Appendix A). While we
will always show the power spectrum up to k ≈ 500 h Mpc−1 , we are mainly interested in the scales relevant for upcoming surveys, k ! 10 h Mpc−1 . As discussed
in Appendix A, for k + 10 h Mpc−1 numerical convergence may become an issue.
Note that the power spectrum of the simulation remains reasonably well-behaved
far below the theoretical shot noise level (i.e. well to the right of the small circle),
indicating that the subtraction of this noise component is fairly accurate.
Newer implementations of the halo model exist, based on fits to more recent
N-body simulations. These models improve on the HALOFIT model by including
a variable concentration-mass relation (such as those derived by Neto et al., 2007;
27
Galaxy formation and the matter power spectrum
Figure 2.2: A comparison of the total matter power spectra of DMONLY−L100N512 (black),
REF−L100N512 (green) and AGN−L100N512 (red), at redshift z = 0. The bottom panel
shows the absolute value of the relative difference of the latter two with respect to DMONLY ;
solid (dashed) curves indicate that the power is higher (lower) than for DMONLY. The dotted,
horizontal line shows the 1% level. Note that the first wave mode has been omitted as it holds
no information. While pressure forces smooth the baryonic density field on intermediate scales,
cooling allows the baryons to increase the total power on small scales. The addition of AGN
feedback, which is required to match observations of groups, has an enormous effect, reducing
the power by " 10% for k " 1 h Mpc−1 .
Duffy et al., 2008) and have been shown to reproduce the power spectra from
simulations with higher accuracy (e.g. Hilbert et al., 2009). Since no suitable
codes employing these models were available, we do not compare to their results
here. However, as Hilbert et al. (2009) have shown that using the halo model
with the concentration-mass relation of Neto et al. (2007) increases the power at
intermediate scales, we suspect that such models would provide a better match to
the power spectrum of DMONLY.
2.3.2
The relative effects of different baryonic processes
In this section we present our main results, demonstrating how single baryonic
processes, or implementations thereof, can influence the matter power spectrum.
While we will focus mainly on the range of scales relevant to upcoming weak
lensing surveys, 0.1 < k < 10 h Mpc−1 , we will also discuss the differences at the
28
2.3.2 The relative effects of different baryonic processes
much smaller scales that our simulations allow us to probe. We note again that all
power spectra are taken from simulations with L = 100 h−1 Mpc and N = 512 at
redshift zero, and that, unless stated otherwise, all simulations are evolved from
the same initial conditions.
We start by comparing the power spectra of DMONLY, the reference simulation
(REF ) and AGN, in Figure 2.2. The panel at the bottom of most plots in this
section shows the absolute value of the relative difference between power spectra.
The dotted, horizontal line shows the 1% level: any differences above this level
will thus affect the statistics of surveys that aim to measure the power spectrum
to this accuracy.
It is immediately clear from the comparison between DMONLY and REF that
the contribution of the baryons is significant, decreasing the power by more than
1% for k ≈ 0.8 − 5 h Mpc−1 . This is because gas pressure smooths the density
field relative to that expected from dark matter alone. On scales smaller than
1 h−1 Mpc (k " 6 h Mpc−1 ), the power in the reference simulation quickly rises far
above that of the dark matter only simulation, because radiative cooling enables
gas to cluster on smaller scales than the dark matter. These results confirm the
findings of previous studies, at least qualitatively (e.g. Jing et al., 2006; Rudd,
Zentner & Kravtsov, 2008; Guillet, Teyssier & Colombi, 2010a; Casarini et al.,
2011a).
However, when AGN feedback is included, the results change drastically. In
this case, the reduction in power relative to DMONLY already reaches 1% for
k ≈ 0.3 h Mpc−1 (λ ≈ 20 h−1 Mpc) and exceeds 10% for 2 ! k ! 50 h Mpc−1 . We
thus see that AGN feedback even suppresses the total matter power spectrum on
very large scales. The enormous effect of AGN feedback is due to the removal of
gas from (groups of) galaxies. That large amounts of gas are indeed being moved
to large radii in this simulation has been shown by, for example, Duffy et al. (2010,
e.g. Figures 1 & 2) and McCarthy et al. (2011, e.g. Figure 3). Because the AGN
reside in massive and thus strongly clustered objects, the power is suppressed out
to scales that exceed the scale on which individual objects move the gas.
Figure 2.3 shows the difference in the power spectra predicted by a variety of
simulations relative to that predicted by the reference simulation. The models are
listed in Table 2.1 and were described in §2.2.2. The top panel shows the effect of
turning off SN feedback and/or metal-line cooling. Since SN feedback heats and
ejects gas, we expect it to decrease the small-scale power. Indeed, the power in
NOSN is > 1% higher than in the reference simulation for k > 4 h Mpc−1 and
the difference reaches 10% at k ≈ 10 h Mpc−1 . The absence of SN feedback also
increases the star formation rate, making stars the dominant contributor to the
total matter power spectrum out to larger scales (not shown).
Turning off metal-line cooling reduces the power on small scales because less
gas is able to cool down and accrete onto galaxies. Indeed, model NOZCOOL
predicts 10 − 50% less power for k " 30 h Mpc−1 . However, the absence of metalline cooling increases the power by several percent for λ ∼ 1 h−1 Mpc because
the lower cooling rates force more gas to remain at large distances from the halo
29
Galaxy formation and the matter power spectrum
Figure 2.3: Comparisons of z = 0 power spectra predicted by simulations incorporating different
physical processes to that predicted by the reference simulation. The panels are similar to the
bottom panel of Figure 2.2, but now show differences relative to REF. The thin black curve that
is repeated in all panels shows the relative difference with DMONLY. Colours indicate different
simulations, while different line styles indicate whether the power is reduced or increased relative
to the reference simulation.
Top: A simulation without SN feedback (blue), one without metal-line cooling (green) and one
that excludes both effects (red). SN feedback decreases the power on all scales. Metal-line cooling
decreases the power for λ > 0.4 h−1 Mpc but increases the power on smaller scales. The effects
of removing both SN feedback and metal-line cooling are > 10% for k > 20 h Mpc−1 and > 1%
for k > 2 h Mpc−1 .
Middle: Different SN wind models which all use the same amount of SN energy per unit stellar
mass (see text). The effects of varying the implementation of SN feedback, while keeping the SN
energy that is injected per unit stellar mass the same, are > 10% for k > 10 h Mpc−1 and > 1%
for k > 1 h Mpc−1 .
Bottom: Models with different feedback energies and processes, see text for details. Including
a top-heavy IMF at high pressure (DBLIMFV1618 ) or AGN feedback (AGN ) greatly reduces
the power. The reduction caused by the latter is > 10% for k > 2 h Mpc−1 and > 1% for
k > 0.4 h Mpc−1 .
30
2.3.2 The relative effects of different baryonic processes
centres.
Even though the effects of SN feedback and metal-line cooling are somewhat
opposite in nature, as the former increases the energy of the gas while the latter
allows the gas to radiate more of its thermal energy away, removing both processes
in the simulation NOSN−NOZCOOL still introduces differences of about 1 − 10%
for k " 2 h Mpc−1 relative to the reference simulation. It is therefore vital to take
both SN feedback and metal-line cooling into account if one wants to predict the
matter power spectrum with an accuracy better than 10%.
We compare models that use different prescriptions for SN feedback, but the
same amount of SN energy per unit stellar mass as REF, in the middle panel of
Figure 2.3. In WML1V848 the SN energy
√ is distributed over half as much gas,
but the initial wind velocity is a factor 2 higher, resulting in more effective SN
feedback in all but the lowest mass galaxies. The differences with respect to the
reference model extend to even larger scales than when SN feedback is removed
entirely: the power is reduced by > 1% for k " 1 h Mpc−1 and by " 10% for k "
10 h Mpc−1 . In model WDENS the initial wind velocity increases with the local
sound speed in the ISM, but the mass loading is adjusted so as to keep the amount
of SN energy per unit stellar mass equal to that in REF. This implementation
results in an even stronger decrease in power on scales < 10 h−1 Mpc. In both
these models, the reduction in power is caused by the increased effectiveness of
SN feedback in driving outflows of gas. We stress that because of our lack of
understanding of the effects of SN feedback, there is a priori no reason to assume
that the model used in the reference simulation is a better approximation to reality
than the models we compare to here.
In fact, it is possible that the SN energy per unit stellar mass is different from
the value assumed in the REF model, or that it varies with environment. Model
DBLIMFV1618, which we compare with REF in the bottom panel of Figure 2.3,
uses a top-heavy IMF in high-pressure environments. Such an IMF yields more
SNe per unit stellar mass which decreases the power by > 1% for k > 0.7 h Mpc−1
and by > 10% for k > 4 h Mpc−1 . Clearly, it will be necessary to understand
any environmental dependence of the IMF in order to predict the matter power
spectrum to 1% accuracy on the scales relevant for upcoming surveys.
On the other hand, doubling the wind mass loading, while keeping the wind
velocity fixed to the value used in REF, as is done in WML4, has a far more
modest effect. This is because the wind velocities are too low to significantly
disturb the high-pressure ISM of massive galaxies. The differences with respect to
the reference model are limited to ! 1% for k ! 10 h Mpc−1 .
The bottom panel of Figure 2.3 also compares the reference simulation to model
AGN, which differs from REF by the inclusion of a phenomenon that has been
shown to play a role in many contexts and that strongly improves the agreement
with observations of groups of galaxies (McCarthy et al., 2010). Like SN feedback,
AGN feedback decreases the power by heating and ejecting gas, but the effect is
more dramatic than that of the standard SN feedback model, both in scope and
magnitude. With respect to the reference model, the power is decreased by " 30%
31
Galaxy formation and the matter power spectrum
Figure 2.4: Difference of the z = 0 matter power spectrum in a simulation using a WMAP1
cosmology (MILL) relative to that of the REF model, which assumes the WMAP3 cosmology,
after rescaling the former to match the latter on the scale of the simulation box (λ = 100 h−1 Mpc,
not shown). WML4 is shown for reference as this simulation uses the same baryonic physics as
MILL. For k " 3 h Mpc−1 , the effect of AGN feedback is at least as strong as that of this
unrealistically large change in cosmology.
for k > 10 h Mpc−1 and by " 5% for k > 1 h Mpc−1 . The reduction in power only
falls below 1% for k < 0.4 h Mpc−1 (λ " 10 h−1 Mpc). Note that the effect of AGN
feedback is strikingly similar to, albeit stronger than, that of the stellar feedback
model that uses a top-heavy IMF in high-pressure environments.
It is clear that many different baryonic processes, and even slightly different
implementations thereof, are capable of introducing significant differences in the
matter power spectrum on scales relevant for observational cosmology. To put the
effects of baryons into perspective, we compare to a simulation with a very different
cosmology, MILL, in Figure 2.4. The difference between the cosmology derived
from the first-year WMAP data used in MILL and the one derived from the 3-year
WMAP data used in the other simulations is large; in fact, the difference is much
larger than the error bars of the most recent data allow. For reference, we note
that the currently favoured cosmology (Komatsu et al., 2011) lies in between those
given by WMAP1 and WMAP3. To account for the difference in normalisation of
the MILL power spectrum, which is caused mainly by its higher Ωm and σ8 values,
we have rescaled it to have the same power at the box size as REF. Still, the effect
on the power spectrum exceeds 10% for k " 0.2 h Mpc−1 . A quick comparison
with WML4, which uses the exact same baryon physics as MILL and twice the
SN wind mass loading used in REF, shows that the effect of the change in mass
loading is relatively small, as we had already shown in Figure 2.3. However, we
32
2.3.3 Contributions of dark matter, gas and stars
see that for k " 3 h Mpc−1 , the effect of AGN feedback is at least as strong as that
of this unrealistically large change in cosmology. We thus conclude that baryonic
effects are not only significant at the ∼ 1% level, but can even be larger than a
“very wrong” choice of cosmology.
Almost all theoretical models used in the literature consider only CDM, assuming that the baryons follow the dark matter perfectly for k ! 1 h Mpc−1 . We have
shown (see Fig. 2.2) that the fact that baryons experience gas pressure reduces
the power on large scales, while their ability to radiate away their thermal energy
increases the power on small scales. If we ignore AGN feedback, as has been done
in all previous work, we find that the power is reduced by at least a few percent
for 0.8 < k < 5 h Mpc−1 and that the power is increased for k > 7 h Mpc−1 ,
with the difference reaching approximately 6% at k = 10 h Mpc−1 for the reference model. However, the single process of AGN feedback, which improves the
agreement with observations of groups of galaxies, reduces the power by " 10%
over the whole range 1 ! k ! 10 h Mpc−1 and the reduction only drops below
1% for k < 0.3 h Mpc−1 . Highly efficient SN feedback, as may for example result
from a top-heavy IMF in starbursts, would have nearly as large an effect. One can
therefore not expect to constrain the primordial power spectrum more accurately
until such processes are better understood and included in theoretical models.
2.3.3
Contributions of dark matter, gas and stars
Generally, power spectra are calculated using all matter inside the computational
volume. This total matter power spectrum is what is measurable using e.g. gravitational lensing surveys. However, as we have a larger freedom of measurement
using simulations, we can also consider the power in different components, for example to see which parts of the power spectrum are dominated by baryonic matter
or how baryons change the distribution of cold dark matter.
On sufficiently large scales the baryons will trace the dark matter. Hence, when
averaged over these scales, the baryonic and CDM densities are given by
ρcdm
=
ρbar
=
Ωm − Ωb
ρtot ,
Ωm
Ωb
ρtot .
Ωm
(2.6)
We can now use these expressions to estimate the relative contributions of correlations between particle types to the total matter power spectrum. Using Ptot (k) ∝
,|ρ̂tot (k)|2 - ∝ ,|ρ̂cdm|2 -+,ρ̂cdm ρ̂∗bar -+,ρ̂∗cdm ρ̂bar -+,|ρ̂bar |2 -, we find, for sufficiently
33
Galaxy formation and the matter power spectrum
small k:
Pcc
=
Pcb + Pbc
=
Pbb
=
(Ωm − Ωb )2
Ptot ≈ 0.68Ptot ,
Ω2m
2Ωb (Ωm − Ωb )
Ptot ≈ 0.29Ptot ,
Ω2m
Ω2b
Ptot ≈ 0.03Ptot .
Ω2m
(2.7)
Hence, on large scales we expect the power due to the auto-correlation of CDM to
dominate the total matter power spectrum, with a significant contribution from
the cross terms Pcb and Pbc .
The four panels of Figure 2.5 show power spectra for the REF−L100N512 (left)
and AGN−L100N512 (right) simulation at z = 0, both for the total matter (solid
black) and for individual components (coloured curves). For reference, we also
show the power spectrum for DMONLY−L100N512 (dashed black). The top row
shows the power spectra of δi ≡ (ρi − ρ̄i )/ρ̄i . This definition ensures that the power
spectra of all components i converge on large scales, which allows us to examine
how well different components trace each other. The bottom row, on the other
hand, shows the power spectra of δi# ≡ (ρi − ρ̄tot )/ρ̄tot , which allows us to estimate
the contributions of different components to the total matter power spectrum.
Looking at the top-left panel, we see that, as expected, the baryonic components trace the dark matter well at the largest scales. However, significant
differences exist for λ ! 10 h−1 Mpc. Observe that, at scales of several hundred
kpc and smaller, the difference between REF and the dark matter only simulation
is larger than that between the latter and the analytical models we compared to
earlier (see Fig. 2.1). In fact, the difference between the cold dark matter component of the reference simulation and DMONLY is also larger than that between
the latter and the analytic models. This is due to the back-reaction of the baryons
on the dark matter, which we will discuss in §2.3.4.
Next, we turn to the bottom-left panel of Figure 2.5 which shows that cold
dark matter dominates the power spectrum on large scales, as expected, although
the contribution from the CDM-baryon cross power spectrum (not shown) is important as well. The contribution of baryons is significant for λ ! 102 h−1 kpc
and dominates below 60 h−1 kpc. The strong small-scale baryonic clustering is the
direct consequence of gas cooling and galaxy formation. Taking a look at how
the baryonic component is itself built up, we see that gas dominates the baryonic
power spectrum on large scales, but that stars take over for λ < 1 h−1 Mpc. The
gas power spectrum flattens for λ ! 1 h−1 Mpc, which corresponds to the virial
radii of groups of galaxies, but steepens again for λ ! 0.1 h−1 Mpc, i.e. galaxy
scales. The reason for the decrease in slope around 1 h−1 Mpc is threefold. First,
the pressure of the hot gas smooths its distribution on the scales of groups and clusters of galaxies. Second, as the gas collapses it fragments and forms stars. Third,
due to stellar feedback the gas is distributed out to large distances, reducing the
power.
34
2.3.3 Contributions of dark matter, gas and stars
Figure 2.5: Decomposing the z = 0 total power spectra (black) into the contributions from cold
dark matter (blue), gas (green) and stars/black holes (red). The left and right columns show
results for REF−L100N512 and AGN−L100N512. In the top row the density contrast of each
component i is defined relative to its own mean density, i.e. δi ≡ (ρi − ρ̄i )/ρ̄i . This guarantees
that all power spectra converge on large scales, thus enabling a straightforward comparison of
their shapes. In the bottom row the density contrast of each component is defined relative to the
total mean density, i.e. δi ≡ (ρi − ρ̄tot )/ρ̄tot , which allows one to compare their contributions to
the total power. The power spectrum of the gas flattens or even decreases for λ ! 1 h−1 Mpc
as a result of pressure smoothing, but its ability to cool allows it to increase again on galaxy
scales (λ ! 102 h−1 kpc). The power spectrum of the stellar component, which is a product of
the collapse of cooling gas, increases most rapidly towards smaller scales. While stars dominate
the total power for λ % 102 h−1 kpc in REF, dark matter dominates on all scales when AGN
feedback is included.
The inclusion of AGN feedback greatly impacts the matter power spectrum on
a wide range of scales. Comparing the top panels of Figure 2.5, we see that AGN
feedback strongly decreases the power in the gas and stellar components relative
to that of the dark matter for λ ! 1 h−1 Mpc. A comparison of the bottom panels
reveals that the contribution of stars to the total power is reduced the most, with
the reduction factor increasing from an order of magnitude on the largest scales
to more than two orders of magnitude on the smallest scales. This clearly shows
that AGN feedback suppresses star formation, as required to solve the overcooling
problem. For the gas component the change is also dramatic. While ∆2gas (k) = 1
for λ ∼ 3 h−1 Mpc in REF, this level of gas power is only reached at 100 h−1 kpc for
AGN. The suppression of baryonic structure by AGN feedback makes dark matter
35
Galaxy formation and the matter power spectrum
the dominant component of the power spectrum on all scales shown, although it
is important to note that the dark matter distribution is also significantly affected
by the AGN, as we shall see next.
2.3.4
The back-reaction of baryons on the dark matter
Even though dark matter is unable to cool through the emission of radiation,
its distribution can still be altered by the inclusion of baryons due to changes in
the gravitational potential. We examine this back-reaction of the baryons on the
dark matter for the reference and AGN simulations in the left and right panels of
Figure 2.6, respectively. In order to make a direct comparison, we have rescaled
the density of the dark matter component of the simulations that include baryons
by multiplying it by the factor Ωm /(Ωm − Ωb ). The blue curve shows the relative
differences between the power spectrum of the rescaled CDM component and that
of DMONLY.
On scales k " 2 h Mpc−1 , corresponding to spatial scales λ ! 3 h−1 Mpc, the
power in CDM structures in the reference simulation is increased by > 1% with
respect to DMONLY. The difference continues to rise towards higher k, reaching
10% around k = 10 h Mpc−1 . Because the baryons can cool, they are able to
collapse to very high densities, and in the process they steepen the potential wells
of virialized dark matter haloes, causing these to contract. The effect is larger
closer to the centres of these haloes, i.e. on smaller scales.
The back-reaction is quite different when AGN feedback is included.6 The dark
matter haloes still contract on small scales, albeit by a smaller amount, but the
power in the dark matter component of the AGN simulation is decreased for scales
> 200 h−1 kpc, corresponding to the sizes of haloes of L∗ galaxies. The reduction in
the power of the CDM component in model AGN relative to DMONLY increases
from roughly 1% at k = 3 h Mpc−1 to almost 10% around k = 10 h Mpc−1 . AGNdriven outflows redistribute gas to larger scales, which reduces the baryon fractions
in haloes and results in shallower potential wells. This is consistent with the results
of Duffy et al. (2010), who used the same simulation to show that AGN feedback
decreases the concentrations of dark matter haloes of groups and clusters. Note,
however, that because AGN can drive gas beyond the virial radii of their host
haloes, their effect on the power spectrum cannot be fully captured by a simple
rescaling of the halo concentrations.
2.3.5
A closer look at the effects of AGN feedback
In this section we examine our most realistic model for the baryonic physics, AGN,
more closely.
6 The
small difference in power between the CDM component of AGN and DMONLY near the
size of the box is most likely caused by errors in the power spectrum estimation.
36
2.3.5 A closer look at the effects of AGN feedback
Figure 2.6: The back-reaction of baryons on the CDM. The blue curves show the relative
difference between the power spectrum of the CDM component, after scaling the CDM density
by the factor Ωm /(Ωm − Ωb ), and that of a dark matter only simulation for either the REF (top
panel) or AGN (bottom panel) model. For comparison, the relative differences between the total
matter power spectra of the baryonic simulations and DMONLY is shown by the black curves.
Baryons increase the small-scale power in the CDM component. However, when AGN feedback
is included, the power in the CDM component drops 1 − 10% below that of the DMONLY
simulations for 0.2 ! λ ! 2 h−1 Mpc.
37
Galaxy formation and the matter power spectrum
Figure 2.7: The dependence of the effect of AGN on cosmology. The curves show the relative
differences between the z = 0 matter power spectra for models AGN and DMONLY for our fiducial WMAP3 cosmology (green) and for the WMAP7 cosmology (red). Changing the cosmology
has little impact on the relative effect of the baryonic processes.
2.3.5.1
Dependence on cosmology
Figure 2.7 shows how the relative difference between the z = 0 power spectra of
models AGN−WMAP7 and DMONLY−WMAP7, both of which use the WMAP7
cosmology, compares to that between the same physical models in the WMAP3
cosmology (the latter case was already shown in Figure 2.2). Even though the
power spectra are themselves strongly influenced by, for example, the much higher
value of σ8 in the WMAP7 cosmology, the relative change in power due to baryons
is nearly identical, at least so long as AGN feedback is included. This is good
news for observational cosmology. It means that, once the large current scatter in
implementations of subgrid physics has converged, it may be possible to separate
the baryonic effects from the cosmological ones when modelling the matter power
spectrum. It also means that we can assume that our results of the previous
sections, which were based on the WMAP3 version of the AGN simulation, apply
also to model AGN−WMAP7.
2.3.5.2
Evolution
Next, we use the AGN−WMAP7 simulation, which we consider to be our most
realistic model, to investigate the dependence of the effect of baryon physics on
redshift. Figure 2.8 shows the relative difference between the power spectra of
DMONLY−WMAP7 and AGN−WMAP7 at redshifts 3, 2, 1, 0.5 and zero. We see
from this plot that on large scales, λ " 1 h−1 Mpc, the reduction in power due
38
2.4 Comparison with previous work
Figure 2.8: Evolution of the relative difference between the matter power spectra of
DMONLY−WMAP7 and AGN−WMAP7. From red to blue, redshift decreases from 3 to zero.
The erratic behaviour of the z = 2 and z = 3 power spectra at the very smallest scales shown
is due to a lack of resolution. For λ " 1 h−1 Mpc the reduction in power due to baryons evolves
only weakly for z ! 1, but the transition from a decrease to an increase in power keeps moving
to smaller scales.
to the gas does not evolve much for z ! 1, although the differences between the
different redshifts remain large compared with the precision of upcoming surveys.
The weak evolution below z = 2 is consistent with McCarthy et al. (2011), who
found that the expulsion of gas due to AGN feedback takes place primarily at
2 ! z ! 4. On scales below 1 h−1 Mpc, on the other hand, the effects of baryonic
processes on the power spectrum keep increasing with time, with the transition
point between a decrease and an increase in power steadily moving towards smaller
scales. This is probably because the ejection of low-entropy halo gas at high
redshift (z " 2) results in an increase of the entropy, and thus a reduction of the
cooling rates, of hot halo gas at low redshift (McCarthy et al., 2011).
2.4 Comparison with previous work
Our predictions for the effect of baryons on the matter power spectrum agree qualitatively with those of other authors, provided we restrict ourselves to including
the same baryonic feedback processes as were considered in those studies. However, previous simulations did not include AGN feedback and hence suffered from
overcooling.7 As we have demonstrated, AGN feedback (or very efficient stellar
7 The
toy model of Levine & Gnedin (2006), which we briefly describe later in this section, did
demonstrate, based purely on energetic grounds, that AGN feedback has the potential to have
39
Galaxy formation and the matter power spectrum
feedback) has a dramatic effect on the matter power spectrum over a large range
of scales. In this section we will consider both the qualitative and quantitative
differences with respect to previous work, and examine how these may have come
about.
Jing et al. (2006) used gadget ii (Springel, 2005a) to run a simulation with
a 100 h−1 Mpc box and 5123 gas and DM particles. Their simulation included
radiative cooling and star formation, and used the Springel & Hernquist (2003)
sub-grid model for the multiphase ISM and for galactic winds driven by star formation. Metal-line cooling and AGN feedback were not considered. They found
that the power at k = 1 h Mpc−1 is reduced by ∼ 1% relative to a dark matter
only simulation at z = 0, which matches our results for the reference simulation
very well. Furthermore, in agreement with our reference model, they find that the
inclusion of baryons increases the power by ∼ 10% at k = 10 h Mpc−1 . However,
they find that the transition from a relative decrease to a relative increase in power
occurs at k ≈ 2 h Mpc−1 , while we find that it lies at k ≈ 6 h Mpc−1 .
As the simulation of Jing et al. (2006) excludes metal-line cooling, we expect
their results to be in better agreement with our own results for NOZCOOL. The
main difference with respect to the reference simulation turns out to be the position
of the transition point from a relative decrease to a relative increase in power, which
shifts to k ≈ 2 − 3 h Mpc−1 when metal-line cooling is turned off. Hence, using the
simulation NOZCOOL, we reproduce both the qualitative and quantitative results
of Jing et al. (2006), even though baryonic processes such as SN feedback are not
implemented in the same way.
Rudd, Zentner & Kravtsov (2008) used the art code (Kravtsov, 1999) expanded with the Eulerian hydrodynamics solver described in Kravtsov, Klypin
& Hoffman (2002). They used a 60 h−1 Mpc box with 2563 particles, and included radiative cooling and heating, metal-line cooling, star formation, thermal
SN feedback (which is described in Kravtsov, Nagai & Vikhlinin, 2005) and chemical enrichment. AGN feedback was not considered. The effect of the baryons on
the matter power spectrum they found is far more dramatic than that found by
Jing et al. (2006) and ourselves: a decrease in power of up to ∼ 10% relative to a
dark matter only simulation for k < 1 h Mpc−1 , and a relative increase in power
at k " 1 h Mpc−1 which already reaches ∼ 50% at k ≈ 5 h Mpc−1 . The reason for
these large differences is unclear.
Guillet, Teyssier & Colombi (2010a) used the MareNostrum simulation, which
was run using the adaptive mesh refinement code ramses (Teyssier, 2002), to
investigate the effects of baryons on both the variance and the skewness of the mass
distribution. They used a 50 h−1 Mpc box with 10243 dark matter particles and
included metal-dependent gas cooling, UV heating, star formation, SN feedback
(using the kinetic feedback prescription of Dubois & Teyssier, 2008) and metal
enrichment. AGN feedback was not considered. Unfortunately, they were not able
to run their simulations down to z = 0, but quote results at redshift 2 instead. In
a large effect on the matter power spectrum.
40
2.4 Comparison with previous work
order to better compare to their results, we have examined the power spectra of
REF−L100N512 and DMONLY−L100N512 at z = 2. In our reference simulation
the scale on which baryons significantly reduce the power increases with time (note
that AGN shows the opposite behaviour, see Fig. 2.8): in REF the 1% level is
first reached at k ≈ 2 h Mpc−1 for z = 2 and at k ≈ 0.8 h Mpc−1 for z = 0.
Meanwhile, the effect on the power on scales k " 10 h Mpc−1 hardly changes, and
the transition scale from a decrease to an increase in power relative to DMONLY
remains fixed at k ≈ 7 h Mpc−1 . Guillet, Teyssier & Colombi (2010a), on the
other hand, do not detect a systematic decrease in power due to baryons at any
scale. They find that the power is increased by 1% relative to a dark matter only
simulation at k ≈ 3 h Mpc−1 , reaching 40% at k ≈ 10 h Mpc−1 . For our reference
model we instead find a 2% decrease for k ≈ 3 h Mpc−1 and only a 6% increase at
k ≈ 10 h Mpc−1 . It is hard to say why these results lie so far apart, and especially
why the baryons in their simulation do not reduce the power on large scales due
to pressure effects.
We also compare to the recent study by Casarini et al. (2011a), who use the
SPH code gasoline (Wadsley, Stadel & Quinn, 2004) to perform their simulations. They use two different volumes: a box of 64 h−1 Mpc on a side, and a much
larger 256 h−1 Mpc box, both with only 2563 dark matter and an equal number of
gas particles. Note that the mass resolution of their L = 64 h−1 Mpc run is comparable to that of our fiducial run, while the resolution of their L = 256 h−1 Mpc run
is much poorer. They include radiative cooling, a UV background, star formation
and SN feedback. For the latter they use the prescription of Stinson et al. (2006),
in which Type II SNe are modelled using an analytical treatment of blastwaves
combined with manually turning off radiative cooling. Metal-line cooling and
AGN feedback were not considered. Using their 64 h−1 Mpc box, Casarini et al.
(2011a) find an ∼ 1% decrease in power at k ≈ 1 − 2 h Mpc−1 and an increase
in power at smaller scales, which reaches 20% at k ≈ 10 h Mpc−1 . These results
are in reasonable agreement with both Jing et al. (2006) and our model NOZCOOL. However, when using their 256 h−1 Mpc box, they – like Guillet, Teyssier
& Colombi (2010a) – find no decrease in power due to baryons at any scale, but
instead a steady increase in power that reaches 1% at k ≈ 1 − 3 h Mpc−1 and 40%
at k ≈ 10 h Mpc−1 .
Finally, we discuss the work by Levine & Gnedin (2006), who used a toy model,
rather than a hydrodynamic simulation, to evaluate the potential effect of AGN
feedback on the matter power spectrum. In their models only the evolution of dark
matter was followed explicitly. The gas was assumed to trace the dark matter at
all scales and galaxy formation and the associated physical processes were not
included. Their standard simulation volume is 64 h−1 Mpc on a side, and the
simulation was run with resolutions of 1, 0.5 and 0.25 h−1 Mpc. We note that even
their highest resolution is more than two orders of magnitude below the spatial
resolution in our standard simulations. The gas was assumed to have a constant
temperature of 1.5 × 104 K at all redshifts. A quasar luminosity function was
used to determine the number of AGN at a given redshift and luminosity, which
41
Galaxy formation and the matter power spectrum
were then each placed at a random location, although biased towards high-density
regions. Of the AGN’s bolometric luminosity, a fraction +k = 1% was used to
drive spherically symmetric outflows. Within these outflow regions the baryon
fraction was assumed to be zero. After computing the power spectrum, they
found a large discrepancy between simulations with different resolutions: when
using a resolution of 1 h−1 Mpc, they found a reduction of roughly 10% in power
for 0.3 ! k ! 3 h Mpc−1 at z = 0, relative to a simulation which did not include
AGN, while their higher-resolution runs produced instead an increase in power at
all scales, of up to 20%. We found a decrease in power of 1% at k ≈ 0.3 h Mpc−1 ,
reaching > 10% for 2 ! k ! 50 h Mpc−1 , which does not agree with their results,
even in terms of the sign of the effect. Nevertheless, we do confirm the conclusion
of Levine & Gnedin (2006) that AGN feedback can greatly affect the matter power
spectrum on a wide range of scales.
Even though our current understanding of galaxy formation still allows for
significant deviations between studies, some qualitative results are the same: in
the absence of AGN feedback, baryons will affect the matter power spectrum
significantly on scales k ∼ 1 − 10 h Mpc−1 . Furthermore, all studies agree that the
increase in power due to baryons is of the order of 10% at k = 10 h Mpc−1 . Jing
et al. (2006), our reference and NOZCOOL models, and Casarini et al. (2011a) for
their high-resolution simulation all predict a relative decrease in power of ∼ 1% at
k ≈ 1 h Mpc−1 . Rudd, Zentner & Kravtsov (2008) also find a decrease in power
due to baryons, but in their case the effect is far stronger than that of any other
study, and is seen at much larger scales (k ! 1 h Mpc−1 ).
However, like our reference simulation, all these simulations suffer from the
well-known overcooling problem. As was demonstrated by McCarthy et al. (2010),
the AGN simulation does not. We have shown that the inclusion of AGN has
a tremendous effect on the matter power spectrum for λ ! 10 h−1 Mpc, both
when compared to a simulation that includes only dark matter and when compared to simulations that include baryons and galaxy formation but not AGN
feedback. Therefore, contrary to what, for example, Guillet, Teyssier & Colombi
(2010a) claim, simulations that suffer from overcooling cannot be considered extreme models for which the effects of baryons on the total matter power spectrum
are maximised. Instead, they are prone to under estimate the effects on large
scales. Indeed, model AGN predicts a relative decrease in power of ∼ 1% already
at k = 0.4 h Mpc−1 . The decrease in power reaches several tens of percent on
scales k ∼ 1 − 10 h Mpc−1 , while simulations that suffer from overcooling instead
predict a strong increase in at least part of this range. Based on our results and
on the comparison to other studies, we argue that the inclusion of AGN in cosmological simulations is at present even more important than the improvement or
convergence of existing prescriptions for other baryonic effects.
Motivated by the results of Rudd, Zentner & Kravtsov (2008), Zentner, Rudd
& Hu (2008) have proposed a method to account for the effects of galaxy formation
on the matter power spectrum. This method assumes that the effects of baryons
can be captured by a change in the halo concentration-mass relation. However,
42
2.5 Conclusions
it is unlikely that such an approach can truly model the effects of baryons on the
power spectrum. Since AGN-driven outflows significantly affect scales much larger
than the sizes of individual haloes, the assumption made by Zentner, Rudd & Hu
(2008) will certainly not be valid when AGN feedback is included.
2.5 Conclusions
Upcoming weak lensing surveys, such as LSST, EUCLID, and WFIRST aim to
measure the matter power spectrum with unprecedented accuracy. In order to fully
exploit these observations, theoretical models are needed that can predict the nonlinear matter power spectrum at the level of 1% or better on scales corresponding
to 0.1 ! k ! 10 h Mpc−1 . Here, we have employed a large suite of simulations
from the OWLS project, as well as the highly accurate power spectrum estimator
powmes, to investigate the effects of various baryonic processes on the matter
power spectrum. These tools have also enabled us to examine the distribution of
power over different mass components, the back-reaction of baryons on the CDM,
and the evolution of the dominant effects on the matter power spectrum.
Our most important finding is that the feedback processes that are required to
solve the overcooling problem (i.e. the overproduction of stars), have a dramatic
effect on the matter power spectrum. Such efficient feedback, most likely in the
form of outflows driven by AGN, were not present in the simulations used in
previous studies of the effects of baryons on the matter power spectrum (Jing
et al., 2006; Rudd, Zentner & Kravtsov, 2008; Guillet, Teyssier & Colombi, 2010a;
Casarini et al., 2011a). Although it was generally assumed that overcooling would
make the simulations conservative, in the sense that they would overestimate the
baryonic effects, we demonstrated that the opposite is true. The efficient outflows
that are required to reproduce optical and X-ray observations of groups of galaxies,
redistribute the gas on large scales, thereby reducing the total power by " 10%
on scales k " 1 h Mpc−1 .
We emphasise that the model from which we draw this conclusion, the simulation that includes AGN feedback, is not extreme. On the contrary, we consider
it our most realistic model. McCarthy et al. (2010, 2011) showed that it provides
excellent agreement with both optical and X-ray observables of groups of galaxies
at redshift zero. In particular, it reproduces the temperature, entropy, and metallicity profiles of the gas, as well as the stellar masses, star formation rates, and age
distributions of the central galaxies, and the relations between X-ray luminosity
and both temperature and mass.
We showed that metal-line cooling, star formation, and feedback from SNe all
modify the matter power spectrum by > 1% on the scales relevant for upcoming
surveys. In the absence of AGN feedback, the simulations with baryons have ∼ 1%
less power relative to a dark matter only simulation on scales 0.8 ! k ! 6 h Mpc−1
(a consequence of gas pressure) and > 10% more power for k " 10 h Mpc−1 (a
consequence of gas cooling). However, as we noted above, AGN feedback can
43
Galaxy formation and the matter power spectrum
decrease the power for 1 ! k ! 10 h Mpc−1 by up to several tens of percent.
Furthermore, some implementations of stellar feedback, e.g. the strong SN feedback
resulting from a top-heavy stellar initial mass function in starbursts, can create
differences of the same scope and magnitude by redistributing gas out to very large
scales. The effects from such baryonic processes on the matter power spectrum can
even exceed those of a very large change in cosmology (e.g. WMAP3 to WMAP1).
Indeed, differences > 1% persist even up to scales as large as those corresponding
to k ≈ 0.3 h Mpc−1 .
In the absence of AGN feedback, the back-reaction of baryons on the dark
matter increases the power in the CDM component by 1% at k ≈ 2 h Mpc−1
and the effect becomes larger towards smaller scales. However, when AGN are
included they redistribute sufficiently large quantities of gas out to large radii to
lower the power in the dark matter component by 1−10% for 3 ! k ! 30 h Mpc−1 .
This is consistent with Duffy et al. (2010), who used the same simulation to show
that AGN feedback decreases the concentrations of dark matter haloes of groups of
galaxies. We stress, however, that the back-reaction of AGN feedback on the CDM
will not be straightforward to implement in dark matter only models. While it may
be possible to roughly model the effect of baryons in simulations without efficient
feedback by raising the concentration parameters of the dark matter haloes (e.g.
Zentner, Rudd & Hu, 2008), feedback from AGN redistributes the gas on scales
that exceed those of their host haloes.
The difference between dark matter only simulations and simulations that do
include baryons is nearly the same for the WMAP3 and WMAP7 cosmologies, at
least when AGN are included. This suggests that the relative effect of the baryons
is roughly independent of cosmology, which will simplify future studies aiming to
disentangle the two.
For our most realistic simulation, which assumes the WMAP7 cosmology and
includes AGN feedback, the difference in power relative to the corresponding dark
matter only simulation does not evolve much for z ! 1 on large scales (k <
10 h Mpc−1 ). This is consistent with McCarthy et al. (2011), who showed that
the expulsion of gas through AGN feedback occurs mostly at z ∼ 2 − 4, in the
progenitors of today’s groups and clusters of galaxies.
We demonstrated that our conclusions are robust with respect to changes in
the size of the simulation box and changes in the resolution (see Appendix A), with
any additional modelling uncertainties only making it less likely that the matter
power spectrum can be predicted with 1% accuracy any time soon. Looking at
the large differences that still exist between the results of different authors, it is
clear that much work remains to be done in understanding processes such as gas
cooling and outflows.
In a follow-up paper (Semboloni et al., 2011), we study the implications of our
findings for weak lensing surveys in more detail. In this work we also demonstrate
that the use of optical and X-ray observations of groups of galaxies can significantly
reduce the uncertainties in the predictions of the matter power spectrum. While
this provides a strong incentive for obtaining better and more observations of
44
2.A Convergence tests
groups of galaxies, it is important to note that such auxiliary data will never
completely remove the uncertainty inherent to cosmological probes of the matter
distribution on scales that are potentially affected by baryonic physics. This is
because one can never be sure that all the relevant effects are constrained by the
secondary observations. For example, it may be that other models for the baryonic
physics exist that also reproduce optical and X-ray observations of groups, but
nevertheless predict different power spectra. It will therefore be crucial to consider
a wide variety of observations, with optical and X-ray as well as Sunyaev-Zel’dovich
observations holding particular promise, and a large range of models.
While the strong baryonic effects that we find imply that the cosmological
constraints provided by upcoming weak lensing surveys will be model-dependent,
it also means that such surveys will provide constraints on the physics of galaxy
formation on scales that are difficult to obtain by other means.
Tabulated values of power spectra for redshifts z = 0 − 6 are available for all
the simulations shown in this chapter at http://www.strw.leidenuniv.nl/VD11/
(see Appendix B).
Acknowledgements
We thank all members of the OWLS team for their contributions to the project.
We are also grateful to Henk Hoekstra, Elisabetta Semboloni and Simon White for
discussions. We thank the Horizon Project for the use of their code powmes. Furthermore, in our comparison with analytical models we have utilized code from the
Cosmology Initiative iCosmo. The simulations presented here were run on Stella,
the LOFAR Blue Gene/L system in Groningen, on the Cosmology Machine at the
Institute for Computational Cosmology in Durham as part of the Virgo Consortium research programme, and on Darwin in Cambridge. This work was sponsored
by National Computing Facilities Foundation (NCF) for the use of supercomputer
facilities, with financial support from the Netherlands Organization for Scientific
Research (NWO). This work was supported by an NWO VIDI grant and by the
Marie Curie Initial Training Network CosmoComp (PITN-GA-2009-238356).
2.A Convergence tests
Here we investigate the effects of changing the box size or resolution of the reference
simulation on its power spectrum.
2.A.1
Box size
In Figure 2.9 we vary the size of the box at constant resolution. The difference
between the power spectrum of the 100 and the 50 h−1 Mpc box is smaller than
the difference between the latter and the 25 h−1 Mpc box for all k, and the power
spectrum of the largest box is nearly converged for k " 20 h Mpc−1 .
45
Galaxy formation and the matter power spectrum
However, there are differences of up to a factor of a few at larger scales. For
reference, we also show the input power spectrum linearly evolved to z = 0 and
the HALOFIT model of the non-linear power spectrum by Smith et al. (2003a)
(see §2.3.1). The first wave mode corresponds to the size of the simulation box,
which means that the power measured on this scale is meaningless; hence, we have
omitted this point in all of our figures. The second and third wave modes closely
follow the linear power spectrum. Note that the curves have very similar shapes
on large scales, with the larger boxes shifted to larger scales. This is a consequence
of employing the same seed for the random number generator used to create the
initial conditions. Perturbations that should go non-linear (λ ! 10 h−1 Mpc) are
unable to collapse if their wavelength is close to the size of the box, which in
turn suppresses the power on smaller scales. One might therefore worry that even
the 100 h−1 Mpc box is not large enough to obtain accurate power spectra for k "
1 h Mpc−1 . Lacking larger simulations to check this, we compare to the HALOFIT
model for the non-linear power spectrum, which shows where the transition from
the linear power spectrum should take place at redshift zero. The power spectrum
for the 100 h−1 Mpc box follows this model very well on large scales, suggesting
that a simulation of this size is very close to converged.
Note that finite volume effects only prevent us from obtaining highly accurate
absolute power spectra, and only for the largest scales, while our results are based
on the relative comparisons between models that used identical initial conditions.
Since the 100 h−1 Mpc box extends up to the largest non-linear scales, and since
all simulations start from the exact same realisation of the linear power spectrum
at z = 127, we do not expect our results to be affected by the finite volume of the
simulations.
2.A.2
Numerical resolution
In Figure 2.10 we investigate the effects of changing the resolution for the reference
simulation by varying the number of particles while keeping the box size fixed. The
power spectrum of REF−L100N128 is quite noisy for k " 100 h Mpc−1 because of
its much higher Poisson noise level. Testing for convergence on these scales is only
possible thanks to the accurate shot noise subtraction. Surprisingly, the relative
difference in power between REF−L100N128 and REF−L100N512 is smaller than
the difference between the latter and REF−L100N256. When increasing the resolution beyond that of REF−L100N256, the power begins to decrease. To examine
if this trend continues, we compare the power spectrum of REF−L050N256, which
has the same resolution as REF−L100N512, to that of REF−L050N512 in the
panel on the right. We see that the power on the smallest scales (k " 10 h Mpc−1 )
converges only slowly, but that the trend of decreasing power with increasing resolution continues. This may indicate that, as lower mass haloes become resolved,
the overall effects of supernova feedback become stronger.
We can verify this by isolating the effects due to baryon physics from those
due to a more straightforward dependence on resolution. To this end, we examine
46
2.A.2 Numerical resolution
Figure 2.9: Test of convergence of the z = 0 matter power spectrum in the reference model
with respect to the size of the simulated volume, where the box size and particle number are
varied in such a way as to keep the resolution constant. Also shown are the linear input power
spectrum and the analytical non-linear power spectrum by Smith et al. (2003a). The red, dotted
line in the top panel shows the (subtracted) theoretical shot noise level. The bottom panel shows
the ratio of REF−L100N512 with respect to the other simulations.
what the effect is of increasing the particle number of the DMONLY simulation
with a 100 h−1 Mpc box in Figure 2.11. The behaviour here is quite different: as N
grows, more low-mass haloes are resolved and the power on small scales increases.
As we observe a reversed trend in Figure 2.10, we conclude that the increased
baryonic effects that accompany a higher particle number are more important for
the power spectrum than the straightforward dependence on resolution.
The difference between REF−L050N256 and REF−L050N512 is ∼ 0.1% at
k = 1 h Mpc−1 and ∼ 2% at k = 10 h Mpc−1 . We conclude that simulation
REF−L100N512 is sufficiently converged for the scales of interest for this study,
k ! 10 h Mpc−1 . Note that, since we are only interested in the relative differences
between simulations with equal resolution, the uncertainty will in practice be much
smaller. With increased resolution we expect feedback processes to become more
effective, meaning that we may have underpredicted the differences between models
with different feedback processes in low-mass haloes on small scales.
Similar tests were performed by Colombi et al. (2009) for the convergence of
powmes, which keeps the statistical error bounded through its use of foldings. Its
value depends on the quantity C(k), which is defined as the number of independent
47
Galaxy formation and the matter power spectrum
Figure 2.10: As in Figure 2.9, but now the numerical resolution is varied while keeping the box size constant. The left and right panels show power
spectra for box sizes of 100 and 50 h−1 Mpc, respectively. On small scales, k > 10 h Mpc−1 , convergence is slow.
48
2.B Tabulated power spectra
Figure 2.11: Same as the left-hand panel of Figure 2.10, but now for DMONLY instead of
REF. Here the behaviour is as expected: as the number of particles goes up, more low-mass
haloes form and the power on small scales increases. A comparison with Figure 2.10 shows that
increasing the resolution leads to stronger baryonic effects which may reverse the sign of the
trend with resolution.
wave modes at a given wave number k; to be more precise, we approximately have
∆P/P ∝ C(k)−1/2 (Colombi et al., 2009). For our fiducial grid with 2563 grid
cells, one can expect the statistical error to remain below |∆P |/P ≈ 1.2% as long
as errors due to shot noise do not dominate. We have checked that this is indeed
the case. Note that this means that we can confidently measure 1% differences
between simulations using our fiducial values, as we are interested in systematic
offsets covering at least a small range of scales in k-space, rather than random
deviations.
2.B Tabulated power spectra
Table 2.2 shows the power spectrum values for our most current and realistic
simulation to date, AGN−WMAP7−L100N512, for a subset of scales at z = 0.
Our fiducial powmes values of 2563 grid points and 7 foldings were used, and
shot noise has been subtracted. The full table, with power spectrum values at
all scales shown in this chapter and redshifts up to z = 6, as well as tabulated data for all other simulations presented in this chapter, are available at
49
Galaxy formation and the matter power spectrum
z
k [h/Mpc]
P (k) [h−3 Mpc3 ]
∆2 (k)
0.000
0.000
0.000
0.000
0.000
0.12566371
0.18849556
0.25132741
0.31415927
0.37699112
4364.4776
1853.4484
1524.3814
1112.5603
847.62970
0.43876514
0.62886024
1.2259802
1.7476056
2.3007519
Table 2.2: Power spectrum values for AGN−WMAP7−L100N512 for a subset of scales at z = 0
(full table available online).
http://www.strw.leidenuniv.nl/VD11/.
50
3
The impact of baryonic processes on the
two-point correlation functions of galaxies,
subhaloes and matter
The observed clustering of galaxies and the cross-correlation of galaxies and mass
provide important constraints on both cosmology and models of galaxy formation.
Even though the dissipation and feedback processes associated with galaxy formation are thought to affect the distribution of matter, essentially all models used to
predict clustering data are based on collisionless simulations. Here, we use large
hydrodynamical simulations to investigate how galaxy formation affects the autocorrelation functions of galaxies and subhaloes, as well as their cross-correlation
with matter. We show that the changes due to the inclusion of baryons are not
limited to small scales and are even present in samples selected by subhalo mass.
Samples selected by subhalo mass cluster ∼ 10% more strongly in a baryonic run
on scales r " 1 h−1 Mpc, and this difference increases for smaller separations.
While the inclusion of baryons boosts the clustering at fixed subhalo mass on all
scales, the sign of the effect on the cross-correlation of subhaloes with matter can
vary with radius. We show that the large-scale effects are due to the change in
subhalo mass caused by the strong feedback associated with galaxy formation and
may therefore not affect samples selected by number density. However, on scales
r ! rvir significant differences remain after accounting for the change in subhalo
mass. We conclude that predictions for galaxy-galaxy and galaxy-mass clustering
from models based on collisionless simulations will have errors greater than 10%
on sub-Mpc scales, unless the simulation results are modified to correctly account
for the effects of baryons on the distributions of mass and satellites.
Marcel P. van Daalen, Joop Schaye, Ian G. McCarthy,
C. M. Booth and Claudio Dalla Vecchia
Monthly Notices of the Royal Astronomical Society
Volume 440, Issue 4, pp. 2997-3010 (2014)
Baryons and the two-point correlation function
3.1 Introduction
Many cosmological probes are used in order to derive the values of the parameters
describing our Universe, often relying on some aspect of large-scale structure. By
combining different probes, degeneracies can be broken and the constraints on the
numbers that characterise our Universe can be improved. However, observations
alone are not enough: strong theoretical backing is needed to interpret the data
and to avoid, or at least to reduce, unexpected biases.
Modelling our Universe as a dark matter only ΛCDM universe was a reasonable
approximation for the interpretation of past data sets. However, over the last few
years it has become clear that for many probes this is no longer the case in the
era of precision cosmology: ignoring processes associated with baryons and galaxy
formation may lead to serious biases when interpreting data. The existence of
baryons and the many physical processes associated with them have been shown
to significantly impact, for example, the mass profiles (e.g. Gnedin et al., 2004;
Duffy et al., 2010; Abadi et al., 2010; Governato et al., 2012; Martizzi et al.,
2012; Velliscig et al., 2014) and shapes of haloes (e.g. Kazantzidis et al., 2004;
Tissera et al., 2010; Bryan et al., 2013), the clustering of matter (e.g. White, 2004;
Zhan & Knox, 2004; Jing et al., 2006; Rudd, Zentner & Kravtsov, 2008; Guillet,
Teyssier & Colombi, 2010b; Casarini et al., 2011b; van Daalen et al., 2011) and,
subsequently, weak lensing measurements (e.g. Semboloni et al., 2011; Semboloni,
Hoekstra & Schaye, 2013; Yang et al., 2013; Zentner et al., 2013), the strong
lensing properties of clusters (e.g. Mead et al., 2010; Killedar et al., 2012), and the
halo mass function (e.g. Stanek, Rudd & Evrard, 2009; Cui et al., 2012; Sawala
et al., 2013; Balaguera-Antolínez & Porciani, 2013; Martizzi et al., 2014; Velliscig
et al., 2014). To complicate matters further, different authors studying the same
aspects of galaxy formation often find different and sometimes even contradictory
results, depending not only on which physical processes are modelled but also on
the choice of numerical code, and particularly on the implementation of subgrid
recipes for feedback from star formation and Active Galactic Nuclei (hereafter
AGN) (e.g. Scannapieco et al., 2012). Until a consensus can be reached, it is
therefore important to determine the range of values that observables can take
depending on whether certain baryonic processes are included in a model, and the
way in which they are implemented.
In this chapter, we aim to quantify the effects of baryons and galaxy formation
on the two-point real-space correlation function. Specifically, we will investigate
how the redshift zero galaxy and subhalo correlation functions and the galaxymatter cross-correlation, which is observable through galaxy-galaxy lensing, are
changed if baryonic processes are allowed to influence the distribution of matter
to varying degrees, i.e. using different feedback models. To this end, we will use the
reference and AGN models from the OverWhelmingly Large Simulations project
(OWLS, Schaye et al., 2010). These were also employed by van Daalen et al.
(2011, see Chapter 2) and we have since repeated them using larger volumes,
more particles and a more up-to-date cosmology. The AGN model is particularly
52
3.1 Introduction
relevant, as it has been shown to reproduce many relevant X-ray and optical
observations of groups and clusters (McCarthy et al., 2010, 2011; Stott et al.,
2012).
Any changes in the clustering of objects brought about by galaxy formation can
enter into the correlation function in two ways. The first and most well-established
effect is due to a change in the mass of the objects. For example, assuming that
higher-mass haloes are more strongly clustered, if supernova feedback systematically lowers the stellar content of haloes, then a model which includes this process
is expected to show increased clustering at fixed stellar mass relative to one that
does not.1 Likewise, the clustering of haloes at fixed halo mass is also expected
to show increased clustering when efficient feedback is included, due to the total
mass of the halo being lowered. Secondly, the positions of galaxies and haloes may
shift due to changes in the physics: if the mass within a certain radius around an
object changes, then the gravitational force acting on those scales will change as
well, affecting the dynamics of nearby galaxies and haloes. Moreover, tidal stripping, and hence also dynamical friction, will affect satellites differently if baryonic
processes change the density profiles of either the satellites or the host haloes. We
will consider both types of effects here; most importantly, we will disentangle the
two and show what effects remain after we account for the change in halo mass, as
could be done approximately by selecting samples with constant number density.
As we will see, not all shifts in position average out, nor can the modification of
the halo profiles be ignored.
Quantifying the significance of the various ways in which clustering measurements may deviate from those in a dark matter only universe is vital for the
improvement of current models employed in clustering studies. Typically these
are based on the distribution of dark matter alone, be they semi-analytical models
(see Baugh, 2006, for a review), a combination of halo occupation distribution
(HOD) and halo models (e.g. Jing, Mo & Börner, 1998; Berlind & Weinberg, 2002;
Cooray & Sheth, 2002; Yang, Mo & van den Bosch, 2003; Kravtsov et al., 2004;
Tinker et al., 2005; Wechsler et al., 2006; van den Bosch et al., 2013) or subhalo
abundance matching (SHAM) models (e.g. Vale & Ostriker, 2004; Shankar et al.,
2006; Conroy, Wechsler & Kravtsov, 2006; Moster et al., 2010; Guo et al., 2010;
Behroozi, Conroy & Wechsler, 2010; Simha & Cole, 2013). It is therefore important
to investigate which ingredients may currently be missing from such efforts.
The effects of galaxy formation on subhalo-subhalo clustering were previously
considered by Weinberg et al. (2008) and Simha et al. (2012). Weinberg et al.
(2008) compared the clustering of objects at fixed number density in a dark matter
only simulation with a baryonic simulation including weak supernova feedback but
no feedback from AGN, and with identical initial conditions. They found that
1 Situations
in which feedback would have the reverse effect are possible in principle. For example,
if the stellar mass - halo mass relation were flat where AGN feedback is important and had a
large scatter, then the stellar mass of some galaxies inhabiting such haloes could be lower than
that of galaxies in lower-mass haloes. As a result, the most massive galaxies would reside in
intermediate mass haloes. However, such a scenario is not supported by our simulations.
53
Baryons and the two-point correlation function
subhaloes cluster more strongly on small scales in the baryonic simulation due to
the increased survival rate of baryonic satellites during infall. While we find a
similar increase in the autocorrelation of subhaloes on small scales (r ! rvir ) –
with a corresponding decrease in clustering on slightly larger scales – we point out
that such results may be biased, due to the difficulties of detecting infalling dark
matter satellites (e.g. Muldrew, Pearce & Power, 2011, , see our Appendix 3.B).
Simha et al. (2012) extended the work of Weinberg et al. (2008) in several
ways, among which are the addition of more effective stellar feedback and the use
of the mass of the subhalo at infall, rather than the current mass, when assigning
galaxy properties to the subhaloes. They find that the addition of effective feedback causes the discrepancies between clustering in hydrodynamical simulations
and results from subhalo abundance matching to increase. They demonstrate that
the two-point correlation function of baryonic subhaloes can be recovered to better
than 15% on scales r > 2 h−1 Mpc when winds are included, but that the discrepancy at smaller scales in these simulations can be up to a factor of a few. The
galaxy correlation function is reproduced much better if the stellar mass threshold
is raised; however, as these simulations do not contain any form of feedback that
is effective at high stellar masses, we would expect the further addition of a process like AGN feedback to exacerbate the discrepancy between subhalo abundance
matching results and hydrodynamical simulations for massive galaxies.
This chapter is organized as follows. We will briefly introduce our simulations
and explain how we calculate the relevant quantities in §3.2. Here we will also
discuss how we identify the same halo in different simulations, an essential step in
order to separate the change in halo mass from other effects. We present our results
in §3.3 and summarise our findings in §3.4. Finally, we show the convergence with
resolution and box size in Appendix 3.A and consider the fraction of subhaloes
successfully linked between simulation in Appendix 3.B.
3.2 Method
3.2.1
Simulations
We consider three models from the OWLS project (Schaye et al., 2010): DMONLY,
REF and AGN. All of these simulations were run with a modified version of
gadget iii, the smoothed-particle hydrodynamics (SPH) code last described in
Springel (2005b). We will discuss the models employed briefly below.
In order to study relatively low-mass objects while also simulating a volume
that is sufficiently large to obtain a statistical sample of high-mass objects, we combine the results of simulations with different box sizes. For each model, we ran
simulations in periodic boxes of comoving side lengths L = 200 and 400 h−1 Mpc,
both with N 3 = 10243 CDM particles and – with the exception of DMONLY –
an equal number of baryonic particles. The gravitational forces are softened on a
comoving scale of 1/25 of the initial mean inter-particle spacing, L/N , but the softening length is limited to a maximum physical scale of 1 h−1 kpc[L/(100 h−1 Mpc)].
54
3.2.2 Calculating correlation functions
The particle masses in the baryonic L200 (L400 ) simulations are 4.68×108 h−1 M(
(3.75 × 109 h−1 M( ) for dark matter and 9.41 × 107 h−1 M( (7.53 × 108 h−1 M( )
for the baryons. We will use the higher-resolution L200 simulations to study the
clustering of galaxies with stellar mass M∗ < 1011 h−1 M( and subhaloes with
total mass Msh < 1013 h−1 M( , while taking advantage of the larger volume of
the L400 simulations to study higher masses. When considering cross-correlations
with the matter distribution, resolution is more important than volume, and we
use the L200 simulations at all masses. We discuss our choice of mass limits in
Appendix 3.A, where we also show resolution tests. All the simulations we employ
in this chapter were run with a set of cosmological parameters derived from the
Wilkinson Microwave Anisotropy Probe (WMAP) 7-year results (Komatsu et al.,
2011), given by {Ωm , Ωb , ΩΛ , σ8 , ns , h} = {0.272, 0.0455, 0.728, 0.81, 0.967,
0.704}. It is important to note that all simulations with identical box sizes were
run with identical initial conditions, which allows us to compare the effects of
baryons and galaxy formation for the exact same objects.
The DMONLY simulation, as its name suggests, contains only dark matter.
This provides us with a useful baseline model for testing the impact of baryon
physics.
The REF simulation is the reference OWLS model. It includes sub-grid recipes
for star formation (Schaye & Dalla Vecchia, 2008), radiative (metal-line) cooling
and heating (Wiersma, Schaye & Smith, 2009), stellar evolution, mass loss from
massive stars and chemical enrichment (Wiersma et al., 2009) and a kinetic prescription for supernova feedback (Dalla Vecchia & Schaye, 2008). The reference
simulation is not intended to be the most realistic, but instead includes only those
physical processes most typically found in simulations of galaxy formation.
The third and final simulation we consider here, AGN, adds feedback from
accreting supermassive black holes to the reference simulation. AGN feedback
was modelled following the prescription of Booth & Schaye (2009), which built
on the model of Springel, Di Matteo & Hernquist (2005). We believe AGN to
be our most realistic model, as it is the only model that solves the well-known
overcooling problem (e.g. Balogh et al., 2001) and that reproduces the observed
properties of groups (McCarthy et al., 2010, 2011; Stott et al., 2012). Specifically,
this model has been shown to reproduce the gas density, temperature, entropy,
and metallicity profiles inferred from X-ray observations, as well as the stellar
masses, star formation rates, and stellar age distributions inferred from optical
observations of low-redshift groups of galaxies. van Daalen et al. (2011) used
this model to show that AGN feedback has a dramatic effect on the clustering of
matter; here we wish to investigate whether the effect on the clustering of galaxies
and subhaloes is equally important.
3.2.2
Calculating correlation functions
The correlation function, ξ(r), returns the excess probability, relative to a random
distribution, of finding two objects at a given separation r. It is therefore a measure
55
Baryons and the two-point correlation function
of the clustering of these objects as a function of scale. As our simulations contain
only a moderate number of resolved objects (i.e. galaxies and (sub)haloes), we do
not need to resort to approximations that are common in the calculation of twopoint clustering statistics. Instead, we can use a parallelised brute force approach
in which we obtain the (cross-)correlation function through simple pair counts,
using the relation:
DDXY (r)
ξXY (r) =
− 1.
(3.1)
RRXY (r)
Here X and Y denote two (not necessarily distinct) sets of objects (e.g. galaxies
and particles or galaxies and galaxies), DDXY (r) is the number of unique pairs
consisting of an object from set X and an object from set Y separated by a distance
r, and RRXY (r) is the expected number of pairs at this separation if the positions
of the objects in these sets were random. As our simulations are carried out with
periodic boundary conditions, more complicated expressions involving cross terms
of the form DRXY (r) (e.g. Landy & Szalay, 1993) are not necessary, nor do we
need to actually create random fields; instead, we can simply compute the term
in the denominator analytically.
The basic functions that we will consider in this chapter are the galaxy autocorrelation function, ξgg , the galaxy-mass cross-correlation function, ξgm , the subhalo
autocorrelation function, ξss , and the subhalo-mass cross-correlation function, ξsm .
We divide galaxies and subhaloes into different bins according to their stellar and
subhalo dark matter mass, respectively. When cross-correlating with matter, we
weight particles by their mass. To keep the computation time manageable, we use
only 25% of all particles for the lowest mass bin of the simulations with (2×)10243
particles, randomly selected. In all other mass bins, we cross-correlate with the
full particle distribution. We have verified that this does not influence our results
in any way. Throughout this chapter we will focus on the three-dimensional correlation function. We will only show the correlation functions in radial bins where
the number of pairs exceeds 10, to prevent our results from being dominated by
spurious clumping. We take the position of our objects to be the position of their
most-bound particle, and assign each galaxy a mass equal to the total mass in
stars in its subhalo. Finally, we confine our analysis to scales r ! 20 h−1 Mpc, corresponding to at most 1/10th of box size, in order to avoid the effects of missing
large-scale modes.
3.2.3
Linking haloes between different simulations
As discussed previously, there are two main ways in which the two-point correlation
function may be affected by baryonic processes: through changes in the masses
of objects, and through shifts in their positions. To disentangle the two effects,
we make use of the fact that all OWLS models were run from identical initial
conditions, allowing us to identify the same objects in different simulations. In
this way we can assign each object in simulation B the mass that the same object
56
3.3 Results
possesses in simulation A, thereby isolating the effect of changes in the positions
of objects on the clustering signal.
Haloes are identified in our simulations using the Friends-of-Friends algorithm
(run on the dark matter particles, with linking length 0.2) combined with a spherical overdensity finder, as implemented in the subfind algorithm (Springel et al.,
2001; Dolag et al., 2009). For every (sub)halo in simulation A we flag the Nmb
most-bound dark matter particles, meaning the particles with the highest absolute
binding energy. Next, we locate these particles in the other simulations, using the
unique number associated with every particle. If we find a (sub)halo in simulation B that contains at least 50% of these flagged particles, a first link is made.
The link is confirmed if, by repeating the process starting from simulation B, the
previous (sub)halo in simulation A is found.
Here we use Nmb = 50, but we have verified that our results are insensitive
to this choice (see Velliscig et al., 2014). For haloes with less than Nmb dark
matter particles, all dark matter particles are used. The fraction of haloes linked
quickly increases as a function of mass, reaching essentially unity for sufficiently
well-resolved haloes. For all subhaloes employed in this work, the linked fraction of DMONLY subhaloes typically exceeds 99%, the exception being the lowest
mass bin where the linked fraction is around 98%. However, at small separations the linked fraction can be much smaller. This is explored in more detail in
Appendix 3.B.
3.3 Results
In this section we will explore the effects of baryon physics on the two-point correlation function at redshift zero. We will first consider the galaxy-galaxy and galaxymatter correlation functions as these are the most directly observable. Since stellar
masses are strongly model-dependent, we will switch from galaxies to subhaloes
in §3.3.2, which allows us to examine how clustering statistics derived from dark
matter only simulations will differ from those including baryons. Finally, in §3.3.3,
we will take the change in the mass of subhaloes out of the equation, and consider
the change in the correlation function for the exact same objects as a function of
the model used.
3.3.1
Clustering of galaxies
3.3.1.1
Autocorrelation
In Figure 3.1 we plot the galaxy autocorrelation functions, ξgg (r), for models REF
and AGN in three different bins of stellar mass, as indicated in the legend. The
bottom panel shows the relative difference in the clustering strength of galaxies in
these models. Since the clustering of haloes increases with mass, and since AGN
feedback reduces the stellar content of massive haloes, one would expect galaxies
in the AGN simulation to be more strongly clustered at fixed (high) stellar mass.
57
Baryons and the two-point correlation function
Figure 3.1: The galaxy autocorrelation function for the REF and AGN simulations (top),
as well as the fractional difference between the two (bottom). Different colours correspond to
different stellar masses, as indicated in the legend. The legend also shows the number of galaxies
in each bin for each simulation (REF,AGN ). At any mass, galaxies in AGN are more highly
clustered than those in REF on large scales, an effect that increases sharply above 1012 h−1 M& ,
where AGN feedback is most important. Note that these effects may be underestimated for the
two highest mass bins for reasons discussed in §3.3.1.3. The relative decrease in clustering for
the AGN simulation on small scales is mostly a numerical effect (see text).
As higher-mass galaxies are expected to host more powerful AGN, this effect is
expected to increase with mass. This is indeed what we observe in Figure 3.1:
as long as we consider sufficiently large scales, galaxies in the AGN simulation
show increased clustering relative to those in REF, and the relative difference
between clustering strengths in the two simulations tends to increase with mass.
For galaxies with stellar masses M∗ < 1010 h−1 M( we expect the effect to be
minor, since in such low-mass objects feedback is controlled by stellar rather than
AGN feedback in these models (e.g. Haas et al., 2013).
Also indicated in the legend are the number of galaxies in each mass bin
for each simulation, the first number corresponding to REF and the second to
AGN. Because AGN feedback systematically lowers the stellar content of massive haloes, and since the number density of haloes decreases with mass, the
AGN simulation suffers from somewhat worse statistics at high stellar masses
than the REF simulation. However, this effect is only seen in the highest mass
bin, M∗ > 1012 h−1 M( , and even in this mass range we can still draw robust
conclusions for scales r > 2 h−1 Mpc.
Note that any two subhaloes must have a finite minimum distance between
58
3.3.1 Clustering of galaxies
them in order to, on the one hand, be recognised as separate objects, and on the
other, not be tidally destroyed. As we identify galaxies by the subhaloes they
occupy, this causes a slight turnover in the galaxy correlation functions on small
scales. Since this minimum distance increases with the size and therefore mass of
the subhaloes hosting the galaxies, at fixed stellar mass this turnover is seen at
larger scales in the model AGN than in REF. This in turn causes the galaxies in
AGN to appear less clustered on small scales.
3.3.1.2
Cross-correlation with matter
Figure 3.2 shows the galaxy-matter cross-correlation functions for these simulations, which are relevant for galaxy-galaxy lensing. Due to the high number of
particles relative to the number of galaxies, the statistics are significantly improved relative to Figure 3.1, and we can see clearly that including AGN feedback
greatly increases the clustering of matter and galaxies at fixed stellar mass.2 . The
relative increase of clustering with mass is more strongly scale-dependent than
for the galaxy-galaxy case. The relative difference in clustering strength between
AGN and REF is largest around 1 h−1 Mpc for the most massive galaxies, where
galaxies at fixed stellar mass are nearly twice as strongly clustered with matter
when AGN are included. At larger scales, AGN always shows ∼ 50% stronger
clustering than REF for M∗ > 1012 h−1 M( . Even for galaxies in the stellar mass
range 1011 < M∗ /[M( /h] < 1012 we see an increase in clustering of up to 150%
around 70 h−1 kpc, and an offset of ∼ 20% at all larger scales.
Interestingly, the relative difference in the galaxy-matter cross-correlation functions between AGN and REF increases towards smaller scales before suddenly
dropping, causing galaxies to become less strongly clustered with the matter distribution in the AGN simulation on the very smallest scales probed here. This
behaviour is caused by two competing effects, a point we will return to when discussing the subhalo-matter cross-correlation function in the next sections. On the
one hand, the lowering of the stellar mass by AGN feedback tends to increase
clustering at fixed stellar mass, and more so towards smaller scales, as galaxies of
the same stellar mass now inhabit denser environments. On the other hand, as
shown in e.g. Velliscig et al. (2014), a large amount of gas – and even dark matter
– is removed from the galaxy, and sometimes from the halo entirely, decreasing
the density peaks in the matter distribution (see e.g. van Daalen et al., 2011).
As we can see from Figure 3.2, the latter effect dominates on sub-galaxy scales
(r ! 10 h−1 kpc).
2 Note
that the number of objects in the two most massive bins, shown in the legend, is lower
than for the autocorrelation function. This is because we now use the higher-resolution L200
for all mass bins, whereas we previously used L400 for the two highest mass bins to obtain
better statistics (see §3.2.1).
59
Baryons and the two-point correlation function
Figure 3.2: As in Figure 3.1, but now showing the galaxy-matter cross-correlation function
for the REF and AGN simulations. Except for sub-galactic scales, AGN feedback tends to
increase the clustering of galaxies with matter at fixed stellar mass. Both the overall magnitude
of the effect and the length scales over which it occurs increase with stellar mass, and for M∗ >
1012 h−1 M& the increase in clustering with the matter distribution reaches values as high as
180%.
3.3.1.3
Caveats
We note that the effect of AGN feedback may be underestimated for massive
galaxies due to two effects. The first only applies to the two highest mass bins
and only to results based on the L400 runs (i.e. the autocorrelation functions):
the implementation of AGN feedback in these simulations is somewhat resolution
dependent, and as a consequence its effect is weaker in the 400 h−1 Mpc box than
in the 200 h−1 Mpc simulation. This is because the seed black holes can only
be injected into resolved haloes, which corresponds to a minimum mass, that
is 8 times higher in the L400 simulation than in the L200 simulation (i.e. the
difference in mass resolution). The result is that AGN feedback in the 400 h−1 Mpc
box, used in the two highest mass bins in Figures 3.1 and 3.2, may be too weak
for galaxies occupying haloes with masses M ! 1013 h−1 M( . In fact, while the
effect of resolution is small for galaxies with masses M∗ > 1012 h−1 M( , for 1011 <
M∗ /[M( /h] < 1012 the effect is significant: when using the higher-resolution L200
simulation in this mass bin, we find an increase in galaxy-galaxy clustering relative
to REF of ∼ 50% for r " 2 h−1 Mpc.
The second effect is due to the way stellar mass is estimated in observations,
where the use of an aperture excludes intracluster light. For the more massive
60
3.3.2 Clustering of subhaloes
galaxies in our sample, which host the most powerful AGN, this aperture size
is typically significantly smaller than the size of the region containing the stars.
However, simulated galaxies are assigned a stellar mass equal to the total mass
in stars in its subhalo. The stellar mass of our most massive galaxies is therefore
significantly higher than would be estimated observationally. Hence, the strong
effects of AGN feedback that we find will be relevant for lower observed stellar
masses than suggested by our plots.
Regardless, even without taking these effects into account, it is clear that
AGN feedback plays an important role in the clustering of galaxies and matter,
and should not be ignored in theoretical models that aim to predict ξgm (r) to
∼ 10% accuracy or better, even when only considering relatively low stellar masses
(M∗ = 1010 − 1011 h−1 M( ).
At this point it is important to note that although our model AGN reproduces
the stellar masses of group-sized haloes relatively well (McCarthy et al., 2010,
2011), predicted stellar masses are generally strongly model-dependent, as well as
cosmology-dependent. Abundance matching studies, on the other hand, reproduce
the stellar mass-halo mass relation by construction (e.g. Moster et al., 2010). Since
clustering models typically employ the results from such studies, which in turn
rely on dark matter only simulations, it is useful to consider the clustering of
the subhaloes that host the galaxies and to select objects by their total subhalo
mass, instead of by their stellar mass. This also allows us to consider the effect
of galaxy formation relative to a dark matter only scenario. For the remainder of
this chapter, we will therefore focus on the clustering of subhaloes.
3.3.2
Clustering of subhaloes
3.3.2.1
Autocorrelation
The top panel of Figure 3.3 shows the subhalo autocorrelation function, ξss (r), for
three different simulations: DMONLY, REF and AGN. Different colours indicate
different subsamples, selected by the total mass of the subhaloes, Msh,tot , though
we note that the results would have been very similar had we selected by dark
matter mass. The correlation functions are displayed in the top panel, while in
the middle panel and bottom panels the baryonic simulations are compared to
DMONLY. From the top panel we can already see that subhalo clustering in the
dark matter only simulation behaves quite differently from that in the baryonic
models, especially on small scales (r ! 1 h−1 Mpc). Vertical dotted lines indicate
the median virial radii3 of subhaloes in each mass bin, which are similar to the
scale at which the subhalo correlation functions for DMONLY turn over.
3 We
computed a characteristic size, rvir , for each subhalo by taking its total mass, Msh,tot , and
treating it as the mass within a region with a mean overdensity of ∆ = 200 relative to ρcrit
(i.e. rvir ≈ r200c ). For reference, for a typical dark matter halo r500c ∼ 0.65 − 0.75 r200c , where
r500c corresponds to the radius out to which the dominant baryonic component (hot gas) of
groups and clusters is typically measured (e.g. Vikhlinin et al., 2006).
61
Baryons and the two-point correlation function
Figure 3.3: The subhalo autocorrelation function, ξss (r), for DMONLY (solid), REF (dashed)
and AGN (dot-dashed lines), and the fractional differences between them. Different colours are
used for different total subhalo masses, and the number of objects in each bin is indicated in
the legend (DMONLY, REF, AGN ). Top: The correlation functions for the three simulations.
Vertical dotted lines indicate the median rvir of the subhaloes. Middle: The fractional difference
of subhalo clustering in REF relative to DMONLY. The curves are greyed out for radii where
they may be biased due to subhalo non-detections (see Appendix 3.B). Bottom: The fractional
difference of subhalo clustering in AGN relative to DMONLY. Both baryonic simulations show
increased clustering, and this effect is stronger on smaller scales. Note that the range on the
y-axis is much smaller here than in Figure 3.1.
62
3.3.2 Clustering of subhaloes
At the high-mass end, all three simulations show very similar behaviour. Looking at the middle and bottom panels, where we compare the autocorrelation of
subhaloes in REF and AGN respectively to that in DMONLY, we see that all subhaloes in the baryonic simulations are typically ∼ 10% more strongly clustered on
large scales than their dark matter only counterparts. As we will demonstrate in
§3.3.3, this difference is due to the reduction of subhalo mass caused by baryonic
processes. For the larger subhaloes, 1013 < Msh,tot /[M( /h] < 1014 , this offset
is somewhat larger when AGN feedback is included, because supernova feedback
alone cannot change the subhalo mass by as much as it can for lower halo masses
(e.g. Sawala et al., 2013; Velliscig et al., 2014). The offset in clustering strength
relative to DMONLY of the lowest-mass subhaloes is also slightly increased by
the addition of AGN: while the masses of these subhaloes may seem to be somewhat low to be significantly affected by AGN feedback, we should keep in mind
that satellite subhaloes may have lost part of their mass through tidal stripping.
Moreover, these would correspond to subhaloes of a higher mass in a DMONLY
simulation, as a significant fraction of the mass has been expelled. Additionally,
low-mass subhaloes do not need to host AGN themselves to be affected by them:
satellites in groups and clusters are sensitive to changes in the host halo profile
and possibly increased stripping caused by the powerful AGN in the more massive
galaxies in their environment.
The differences between the baryonic and dark matter only simulations increase
rapidly for r < 2rvir , at least for Msh,tot < 1014 h−1 M( . As we can see most easily
in the top panel, subhaloes in the REF simulation are significantly more clustered
on small scales than those in the AGN simulation, which seems to contradict the
results of the previous section. This is because subhaloes in the REF simulation
are more compact at fixed mass than those in the AGN simulation, due to the
additional form of feedback in the latter which removes more material from the
centre and lowers the concentration in the inner parts of the subhaloes. However,
the haloes in the AGN simulation are still more compact than those in DMONLY
(see e.g. Velliscig et al., 2014). The increased concentration of subhaloes in baryonic simulations allows them to be identified as separate objects down to smaller
scales, and also to withstand the effects of tidal stripping longer than their dark
matter only counterparts. Both these effects tend to increase the clustering on
small scales. This relative increase in the number density of subhaloes close to
the centres of haloes in baryonic simulations was seen before by e.g. Macciò et al.
(2006), Libeskind et al. (2010), Romano-Díaz et al. (2010) and Schewtschenko &
Macciò (2011) (although Romano-Díaz et al. 2010 note that without strong feedback, the effect may be reversed). On the other hand, baryonic subhaloes are
generally less massive when they are centrals, and those that become satellites
typically fall in later due to the smaller virial radius of the main halo compared to
a pure dark matter run, which means that they should experience less dynamical
friction on scales where tidal stripping is not yet important. This is indeed what
Schewtschenko & Macciò (2011) find, although this effect cannot be seen for the
mass-selected sample shown in Figure 3.3 due to the much larger effect of the
63
Baryons and the two-point correlation function
change in mass.
We explore the clustering behaviour of baryonic satellites in more detail in
§3.3.3.1. For now, we note that if our ability to detect baryonic subhaloes down to
smaller radii than pure dark matter ones were the dominant cause of an increased
number density of subhaloes at small separations in REF and AGN, this would
introduce a bias towards observing a stronger clustering signal in baryonic models
on scales r ! 2rvir .4 We discuss this possible source of error in Appendix 3.B, and
based on the results reported there we have chosen to show the relative differences
in clustering as grey dot-dot-dot-dashed curves in Figure 3.3 for subhalo masses
and scales that may be significantly affected by this bias.
Comparing Figures 3.1 and 3.3, we see that the single act of adding AGN feedback affects the clustering of galaxies and subhaloes very differently. For galaxies,
a strong increase in clustering is found for the highest-mass galaxies, and on large
scales, since the same subhaloes host galaxies with a much lower stellar mass when
AGN feedback is added. Low-mass galaxies are, however, not strongly affected by
AGN feedback. For subhaloes, on the other hand, we find that the largest effects
are found on small scales, and especially at the lowest masses: we find a strong
decrease in clustering for r ! rvir when adding AGN feedback to the reference
model, regardless of halo mass, and far less change on large scales. These two
main differences have two different causes. The large-scale differences between
the effect of AGN feedback on galaxies and on subhaloes is that while AGN are
powerful enough to quench star formation and to remove a lot of gas from galaxies, thus lowering the stellar mass, they are not powerful enough to significantly
change the halo mass. However, as is shown in detail by Velliscig et al. (2014),
and as we will also see in the next section, they do have a significant effect on the
density profiles of subhaloes, and through this on their distribution. At fixed mass,
the subhaloes in REF are more compact and more massive than those in AGN,
causing both the satellite survival rate and the dynamical friction experienced by
satellites to increase, which in turn causes the small-scale differences in clustering
we just discussed.
3.3.2.2
Cross-correlation with matter
We consider the subhalo-mass cross-correlation function in Figure 3.4. From the
top and middle panels we observe, as was the case for galaxy-galaxy clustering, that
on the smallest scales and at fixed total mass, subhaloes cluster far more strongly
with matter in the baryonic simulations than in the dark matter only simulations.
Additionally, there is a constant 5% offset in favour of baryonic simulations on
the largest scales, for all halo masses. The baryonic bias increases as we move
from large scales towards the virial radius, but, interestingly, the strength of the
effect decreases below scales approximately corresponding to rvir before picking
up again at the smallest scales shown. This decrease below rvir even causes the
lowest-mass DMONLY subhaloes to be more strongly clustered than their REF
4 We
64
thank Raul Angulo for pointing out this potential problem.
3.3.2 Clustering of subhaloes
Figure 3.4: As Figure 3.3, but now for the subhalo-mass cross-correlation function, ξsm (r).
Subhaloes are generally more strongly clustered with matter in the baryonic simulations than in
DMONLY. The largest differences are found for REF, for which ξsm (r) can be up to 40% higher
on intermediate scales for the lowest-mass subhaloes, and much higher still for any subhalo mass
if sufficiently small scales are considered. There is also a constant 5% difference in favour of the
baryonic simulations on large scales, regardless of subhalo mass. While the AGN model seems
to increase clustering at fixed subhalo mass less than REF, it does show a stronger decrease
in clustering up to scales r ∼ 102 h−1 kpc. Note that in both cases the clustering differences
between the models are strongly non-monotonic, which is caused by the interplay between the
change in the total subhalo mass and the change in the subhalo mass profiles.
65
Baryons and the two-point correlation function
counterparts around r = 20 h−1 kpc. For AGN, this happens even for the highestmass subhaloes, and over a larger range of scales.
As we will show in the next section, the strongly non-monotonic behaviour of
the relative difference in ξsm between the baryonic simulations and DMONLY is
caused by two counteracting effects. On the one hand, the lowered halo masses in
the baryonic simulations tend to increase clustering at fixed mass on all scales. On
the other hand, while the dissipation associated with galaxy formation causes the
inner halo profile to steepen, increasing clustering on small scales, the associated
feedback causes the outer layers of the halo to expand, decreasing clustering on
intermediate scales. This effect is stronger when AGN feedback is included. Note
that we observe similar behaviour for the relative differences between the galaxymatter cross-correlation functions for REF and AGN.
Furthermore, by comparing the bottom two panels, we can see that for low halo
masses (Msh,tot < 1012 h−1 M( ), for which AGN feedback is not very important,
the small-scale clustering of haloes in REF and AGN is nearly identical, while subhaloes and matter cluster much more weakly on a range of scales around rvir in
AGN. On the other hand, for higher-mass haloes (Msh,tot > 1012 h−1 M( ), significant differences can be seen down from the smallest scales out to r ∼ 1 h−1 Mpc.
This again confirms the strong effect that AGN feedback has on the mass distribution: the higher the mass of the halo, the more important feedback from
supermassive black holes is in removing material from the centre. This in turn
flattens the mass profiles of the haloes and smooths out the density peaks, decreasing the small-scale lensing signal relative to REF.
As we have already pointed out several times, the most important cause of
the increase in clustering due to galaxy formation with strong feedback is the
lowering of the mass of objects. However, secondary effects, such as the resulting
changes in the dynamics and density profiles of haloes, are also expected to be
significant. To disentangle these types of effects, we will use our linking scheme to
match subhaloes between different simulations, allowing us to see if any significant
difference in the clustering remains once the change in mass has been accounted
for.
3.3.3
Accounting for the change in mass
As we are mainly interested in how galaxy formation changes the clustering of
objects with respect to a dark matter only scenario, we use the linking algorithm
described in §3.2.3 to link subhaloes in REF and AGN to those in DMONLY,
and assign all objects the mass of their DMONLY counterpart. Note that this
means that there are in fact two different DMONLY versions of each correlation
function: one derived using all subhaloes for which a counterpart was found in
REF, and one derived using all subhaloes for which a counterpart was found in
AGN. In practice, however, the linked halo samples are nearly identical, and the
resulting correlation functions for DMONLY are virtually indistinguishable. We
therefore show only one of these in the top panels of Figures 3.5 and 3.6, although
66
3.3.3 Accounting for the change in mass
Figure 3.5: As Figure 3.3, but now only showing the autocorrelation functions for subhaloes
linked between a baryonic simulation and DMONLY, and selected based on their mass in the
latter. Relative to Figure 3.3, this procedure removes the effects of changes in the subhalo
masses. As the numbers in the legend imply, almost the exact same haloes are linked with dark
matter only haloes in both cases. The bottom two panels immediately show that in all cases
no differences " 5% in ξss remain on scales r ' rvir , indicating that the differences we saw in
Figure 3.3 on these scales were due to the masses of the objects changing. For smaller scales,
and especially for low-mass subhaloes, the change in dynamics of the objects in the baryonic
simulations can have significant effects, which can primarily be seen as a decrease in clustering
on scales r ! 2rvir . Shaded areas indicate the regions allowed by 1σ bootstrap errors, which show
that the relative small-scale decrease of clustering of low-mass baryonic subhaloes is significant.
67
Baryons and the two-point correlation function
both are used to determine the differences with respect to REF and AGN.
3.3.3.1
Autocorrelation of linked subhaloes
We first consider Figure 3.5, where we show the impact of galaxy formation on the
clustering of subhaloes once the change in mass has been accounted for. Comparing first the sample sizes (numbers in the legend) to those in Figure 3.3, we see
that nearly all DMONLY subhaloes have a match in each of the baryonic simulations.5 Note that the first number in the legend now indicates the sample size of
subhaloes linked between DMONLY and REF, while the second gives the number
of subhaloes linked between DMONLY and AGN.
We have now also performed 500 bootstrap resamplings for each pair of simulations, and show the 1σ errors derived from these as shaded areas in the figure.
As we are now using the exact same (linked) sample of subhaloes for any pair of
simulations, we are able to avoid overestimating the errors due to the false assumption that the halo samples of the simulations are independent. Similar errors
are expected for Figure 3.3.
Comparing the bottom two panels of Figure 3.5 to those of Figure 3.3, we
immediately see that essentially nothing of the ∼ 10% difference in the clustering
amplitude on large scales remains, confirming that this was solely due to galaxy
formation changing the masses of these subhaloes. By accounting for the change
in the masses of objects due to the effects of baryon physics, one will therefore
automatically obtain the correct autocorrelation function at all halo masses, on
scales r + rvir .
However, on smaller scales the changes in the dynamics of subhaloes in the
baryonic runs become important. This is especially the case for low-mass objects,
which are often satellites. As we discussed in §3.3.2.1, Schewtschenko & Macciò
(2011) have shown that, initially, satellites in dark matter only simulations move
in closer to the centre of the main halo in the same amount of time, which is due
in part to the decrease in the virial radius of the main halo when baryons are
included (also found for baryonic haloes in our simulations, see Velliscig et al.,
2014), and in part to the increased dynamical friction experienced by the more
massive dark matter satellites. However, as the satellites undergo tidal stripping,
baryonic subhaloes are able to retain more of their mass due to their increased
concentrations, which causes the situation to reverse on small scales, increasing
the number density of baryonic subhaloes relative to pure dark matter ones. This
was also found by e.g. Macciò et al. (2006), Libeskind et al. (2010) and RomanoDíaz et al. (2010). However, at the same time one expects to see an increase in the
number density – and consequently, the clustering – of baryonic satellite subhaloes
at small scales due to the ability to trace baryonic subhaloes longer during infall.
This resolution effect could lead to a bias at small separations.
5 As
we now select subhaloes by the mass of their DMONLY counterpart, the number of subhaloes
can only be directly compared to those of DMONLY in Figure 3.3, not to the number of baryonic
subhaloes in Figure 3.3.
68
3.3.3 Accounting for the change in mass
To account for this potential bias, we consider the fraction of subhaloes in
DMONLY for which a link could be found in REF in Appendix 3.B. There
we show that the fraction of linked subhaloes decreases strongly on small scales
for low-mass subhaloes. Higher-resolution simulations are needed to investigate
whether the increased survival rate of baryonic subhaloes, and the resulting increase in clustering seen in Figures 3.3 and 3.5 on scales r ! rvir , is physical or
not. We have therefore greyed out the curves in these figures on scales where this
bias may play a significant role.
However, even after accounting for this potential bias, interesting differences
in clustering remain on scales r ! 2rvir , as Figure 3.5 shows. Especially in the
AGN simulation, subhaloes tend to be ∼ 10% less clustered at r ∼ rvir . A
very small increase in clustering (∼ 1%) can be seen on slightly larger scales,
r ∼ 3 − 4rvir . Both these differences could be explained by the combination of the
greater dynamical friction initially experienced by dark matter only subhaloes,
together with the delayed infall of baryonic subhaloes. We plan to investigate
these effects further in a follow-up paper where we consider the differences in the
satellite profiles due to galaxy formation.
Note that small changes in the simulation code (such as changing the level
of optimisation when compiling the simulation code) can shift the positions of
satellite galaxies and subhaloes by small amounts, even if we start from identical
initial conditions.6 However, as almost all these shifts are random, they average
out for two-point statistics. Shifts due to dynamical friction and similar effects
acting on satellites are the exceptions, as these tend to systematically move satellite
subhaloes closer to their respective centrals.
3.3.3.2
Cross-correlation with matter
Finally, we consider what remains of the baryonic effects on the subhalo-matter
cross-correlation function after accounting for the change in the masses of subhaloes. Here, too, we show 1σ errors in all panels, now derived from 10000 bootstrap resamplings. In many cases, the errors are smaller than the widths of the
lines.
Comparing the bottom panels of Figure 3.6 to those of Figure 3.4, we see
that while the large-scale offset is now completely removed, we are left with a
non-negligible effect on scales r ! 1 h−1 Mpc for all subhalo masses. This again
shows the strong effect that feedback can have on the mass distribution: both
supernova and AGN feedback move matter to large scales, decreasing ξgm (r). We
see that, especially when AGN feedback is included, this can significantly affect
clustering out to several times the virial radius, which matches the findings of van
Daalen et al. (2011) and Velliscig et al. (2014). Note that this also confirms that
the findings of van Daalen et al. (2011), namely that AGN feedback decreases
the matter power spectrum at the 1 − 10% level out to extremely large scales
6 The
rms shift in position for subhaloes between DMONLY and AGN is about 0.04 rvir . Similar
values are found for shifts between subhaloes in DMONLY and REF.
69
Baryons and the two-point correlation function
Figure 3.6: As Figure 3.4, but now only showing the cross-correlation functions between matter
and subhaloes that have been linked between a baryonic simulation and DMONLY, and that have
been selected based on their mass in the latter. Relative to Figure 3.4, this procedure removes
the effects of changes in the subhalo masses, leaving only the effect on the mass profiles and the
changes in the positions of the subhaloes. As can be seen from the bottom panel, the change
of the mass profile tends to increase the clustering on the very smallest scales (where baryons
cool to), but decreases it on intermediate scales (where baryons are evacuated). The latter effect
is stronger when AGN feedback is included, and significant over a larger range of scales, for all
masses. Shaded areas indicate the regions allowed by 1σ bootstrap errors, which are typically
much smaller than the widths of the lines.
70
3.4 Summary
(r ∼ 10 h−1 Mpc), are caused by the effect (in Fourier space) of a systematic
change in the profile of haloes, rather than by AGN somehow having a significant
effect the mass distribution out to more than 10 times the virial radius of the
haloes they occupy.
There are strong similarities between the relative differences that remain for
ξsm and the relative differences of halo profiles shown in Velliscig et al. (2014) for
the same models, leaving no doubts as to the origin of the signal we see here.
The strength of the baryonic effect decreases with increasing mass, but is still
highly significant at the mass scales of groups and clusters, although it does not
extend beyond the virial radius for the highest-mass subhaloes. The lowest-mass
subhaloes we consider here experience a maximum decrease in the cross-correlation
with matter of 30%, relative to a dark matter only scenario, and even the most
massive subhaloes are 10% less strongly clustered with the matter distribution
around r = 100 h−1 kpc when AGN are included. On the smallest scales, the
increased clustering due to the cooling of baryons still dominates. Note also that
the small-scale differences that we found in Figure 3.4 between REF and AGN
remain.
These results show us that assigning subhaloes in a dark matter only simulation
the masses they would have had if galaxy formation and efficient feedback had
been included, allows one to obtain the correct clustering predictions on scales
r + 1 h−1 Mpc. However, on smaller scales one cannot correctly predict the crosscorrelation with matter, and hence the galaxy-galaxy lensing signal, to better than
∼ 10% accuracy without taking into account the change in the mass distribution.
3.4 Summary
In this work we investigated how the galaxy and subhalo two-point autocorrelation functions and the cross-correlations with the matter, a measure of the galaxygalaxy lensing signal, are modified by processes associated with galaxy formation.
We utilised a set of cosmological, hydrodynamical simulations with models from
the OWLS project, run with more particles and an updated cosmology relative to
previous OWLS simulations, to examine what the combined effects on the autoand cross-correlation functions are of adding baryons and radiative (metal-line)
cooling, star formation, chemical enrichment and supernova feedback to a dark
matter only simulation, as well as the further addition of a prescription of AGN
feedback that reproduces observations of groups and clusters. As nearly all clustering models employed in the literature rely on pure dark matter distributions,
either from N-body simulations or halo model type prescriptions, it is important
to quantify just how important the effects of baryons and galaxy formation are.
Our findings can be summarised as follows:
• The stellar masses of galaxies are strongly decreased by (AGN) feedback at
fixed subhalo mass, which in turn tends to greatly increase the clustering
of galaxies at fixed stellar mass. More importantly for semi-analytical and
71
Baryons and the two-point correlation function
halo models, the masses of subhaloes are also significantly decreased by the
effects of feedback, the result of which is an increase in clustering of ∼ 10%
on scales r + 1 h−1 Mpc, for the full range of subhalo masses considered here
(Msh,tot = 1011 − 1015.5 h−1 M( ). This effect is much stronger on smaller
scales.
• Both the change in subhalo mass and the modified subhalo profiles act to
change the subhalo-matter cross-correlation function by ∼ 5% on large scales,
and significantly more on sub-Mpc scales. The modulation of the signal is
strongly non-monotonic and mass-dependent, with both significant increases
and decreases in clustering on different scales.
We used the identical initial conditions of our simulations to link each baryonic
subhalo with its dark matter only counterpart, allowing us to effectively exclude
the effect of galaxy formation on the change in the masses of these objects. Nearly
all subhaloes are successfully matched in this way.
• While accounting for the change in mass of subhaloes removes essentially
all of the baryonic effects on the autocorrelation of subhaloes on scales r +
rvir , deviations ∼ 10% remain on scales r ! 2rvir , where rvir is the virial
radius of the subhalo. We argued that these deviations are mainly caused
by the differences in the dynamics of satellites, such as the initially greater
dynamical friction experienced by the more massive, recently accreted pure
dark matter satellites, and the increased concentration of baryonic subhaloes.
• Finally, on scales r ! 1 h−1 Mpc strong deviations in the subhalo-matter
cross-correlation function remain after accounting for the change in the
masses of subhaloes. While on galactic scales (! 10 h−1 kpc) the clustering
of subhaloes with matter is always much higher in a baryonic simulation than
in the corresponding dark matter only simulation, the inclusion of baryons
results in a significant decrease of the cross-correlation for r " 10 h−1 kpc.
These effects are stronger for lower-mass subhaloes, reaching up to 30% for
subhaloes with masses 1011 < Msh,tot /[M( /h] < 1012 . When AGN feedback
is included, ξsm decreases by ∼ 10% relative to a dark matter only simulation for r ∼ 102 h−1 kpc, even for subhalo masses Msh,tot > 1014 h−1 M( .
Mass- and radius-dependent rescalings of halo profiles which extend to several times the virial radius would be needed to account for this effect in dark
matter only simulations.
We note that while many of our results rely on a model that includes AGN feedback, other feedback processes may have similar effects on clustering. In principle,
any other mechanism that is also effective at high masses, sufficiently reducing the
stellar masses of massive galaxies, and allows one to reproduce the global properties of groups and clusters, may show similar effects to those shown here for AGN
feedback. For example, a model in which a top-heavy IMF is used in high-pressure
72
3.A Convergence tests
environments, such as the OWLS model DBLIMF, may have the same qualitative
effect on clustering (see e.g. van Daalen et al., 2011).
We stress that while the effects discussed in this chapter will certainly need
to be modelled in order to achieve the accuracy needed to interpret upcoming
cosmological data sets to their full potential, both our knowledge of the relevant
physics involved and the currently achievable resolution in cosmological simulations still allow for significant uncertainty in the clustering measures discussed
here. The same holds for quantities such as the halo or cluster mass function:
much work is yet to be done before we can converge on a realistic prescription of
galaxy formation, with uncertainties small enough to match observations in the
era of precision cosmology. Although approaches based on dark matter only models, such as semi-analytical modelling or halo occupation distributions, are able to
match the observed galaxy mass function, our results imply that their predictions
for galaxy-galaxy and galaxy-mass clustering will have errors greater than 10% on
sub-Mpc scales, unless the simulation results are modified to correctly account for
the effects of baryons on the distributions of mass and satellites.
Acknowledgements
The authors thank Raul Angulo, Marcello Cacciato and Simon White for useful comments and discussions, and Marcello Cacciato also for comments on the
manuscript. We would also like to thank the anonymous referee for suggestions
that improved the paper. The simulations presented here were run on the Cosmology Machine at the Institute for Computational Cosmology in Durham (which is
part of the DiRAC Facility jointly funded by STFC, the Large Facilities Capital
Fund of BIS, and Durham University) as part of the Virgo Consortium research
programme. This work was sponsored by the Dutch National Computing Facilities Foundation (NCF) for the use of supercomputer facilities, with financial
support from the Netherlands Organization for Scientific Research (NWO). We
also gratefully acknowledge support from the European Research Council under
the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC
Grant agreement 278594-GasAroundGalaxies and from the Marie Curie Training
Network CosmoComp (PITN-GA-2009- 238356).
3.A Convergence tests
Here we investigate the effects of changing the box size or resolution of the simulations used in this chapter on the subhalo autocorrelation function, as this is
the main focus of this chapter. We will also briefly discuss the effects on the
subhalo-matter cross-correlation function.
In Figure 3.7 we show the subhalo autocorrelation functions for models DMONLY
and REF. For clarity the correlation functions for the AGN model are not shown,
but the results are very similar. Contrary to what was done for the figures in the
73
Baryons and the two-point correlation function
Figure 3.7: The relative differences in the subhalo autocorrelation functions between models DMONLY and REF, split by subhalo mass as indicated
in the top left of each panel. Contrary to the plots shown in other sections, no minimum number of pairs per bin is imposed. The box sizes and
particle numbers, as well as the subhalo numbers for DMONLY and REF, respectively, are indicated in the legend, and a vertical dotted line indicates
the mean virial radius in each mass bin. At fixed resolution (same line style) very little changes, although the effect of the better statistics offered by
a larger volume are apparent. At fixed box size (same colour) the results are also very similar, except for the lowest mass bin, where the small-scale
clustering is resolution dependent. Note that all simulations show excellent agreement for 1012 < Msh,tot /[M& /h] < 1013 , where neither resolution
nor volume is an issue.
74
Figure 3.8: As Figure 3.7, but now only showing the autocorrelation functions for subhaloes linked between REF and DMONLY, and selected based
on their mass in the latter. The convergence here is very similar to that seen in Figure 3.7.
3.A Convergence tests
75
Baryons and the two-point correlation function
main text, here we do not impose a minimum number of pairs per bin. We vary
both the box size and particle number in a systematic way: for simulations shown
with the same line style (either solid or dashed) we vary the box size at fixed resolution, while for simulations shown with the same colour we vary the resolution
at fixed box size.
We first consider the effect of changing the size of the simulated volume. Looking at the solid and dashed lines separately, we can see that very little changes at
fixed resolution, except that the results clearly benefit from the better statistics
offered by a larger volume. This is noticeable both for the rare high-mass objects,
on any scale, and for low-mass objects on the very smallest scales, where very few
pairs are found.
If we instead consider each colour of Figure 3.7 separately, we see that at fixed
box size the results are also very similar. The exception is the lowest mass bin,
1011 < Msh,tot /[M( /h] < 1012 , where the correlation function is clearly resolution
dependent when baryons are included. This is because these subhaloes contain
only ∼ 102 particles in the low-resolution simulations, which is not quite enough
for convergence, especially when feedback processes are included. We have verified
that the subhalo mass functions of the highest-resolution simulations shown here
are indeed converged using simulations with smaller volumes and higher resolutions (not shown here). The results for the second mass bin on the other hand,
1012 < Msh,tot /[M( /h] < 1013 , are fully consistent between the different resolutions shown here.
We have repeated these same resolution tests for the autocorrelation functions
of linked subhaloes, shown in Figure 3.8. Here, too, we see that our results are
converged for Msh,tot > 1012 h−1 M( .
Based on these tests, we choose to use the higher-resolution L200N1024 simulations for subhaloes with masses 1011 < Msh,tot /[M( /h] < 1013 , and take advantage of the better statistics offered by the L400N1024 simulations for subhalo
masses Msh,tot > 1013 h−1 M( . Similarly, we opt to use the higher-resolution
simulation for the autocorrelation function of galaxies with stellar masses 109 <
M∗ /[M( /h] < 1011 , and the larger-volume simulation for galaxies with M∗ >
1011 h−1 M( .
We also verified that the cross-correlation functions shown in this work are
sufficiently converged (not shown). For the subhalo-matter (and galaxy-matter)
cross-correlation functions, statistics are less of an issue, as the number of particles is the same for the L200 and L400 simulations. In other words, while for the
2
autocorrelation functions the number of pairs scales as Nobj
, the number of pairs
for the cross-correlation functions scales as Nobj Npart , where Npart + Nobj . Resolution is still an issue, however: while simulations including baryons always show
stronger clustering on galaxy scales than DMONLY, the exact scale on which the
transition of a relative increase to a relative decrease in clustering occurs depends
somewhat on the softening length. Additionally, as we discussed briefly in §3.3.1,
the effect of AGN feedback is resolution-dependent in our simulations, due to the
fact that seed black holes can only be inserted in resolved haloes. AGN feedback
76
3.B Linked fractions
may therefore be weaker at the L400 resolution than at the L200 resolution, while
the strength of the feedback in the latter was deemed realistic. We therefore choose
to use the L200 simulations at all masses when considering the cross-correlation
functions ξgm and ξsm , valuing resolution over volume.
3.B Linked fractions
Here we consider the fraction of subhaloes for which a link can be established
between DMONLY and REF as a function of both mass and, in the case of
satellites, radius. Both numerical and physical effects play a role here. First,
at small radii subfind may fail to detect satellite subhaloes even though these
have not been fully disrupted yet, due to the high background density of the main
halo (e.g. Muldrew, Pearce & Power, 2011). As baryonic subhaloes are typically
more concentrated than dark matter only ones, increasing their density contrast,
these can be detected down to smaller radii. Second, baryonic satellites tend to
be survive longer than their dark matter only counterparts, as their increased
concentration also allows them to better withstand the tidal forces of the main
halo (e.g. Macciò et al., 2006). Because of this, our results for linked samples may
be biased at radii where a significant fraction of satellite subhaloes is unlinked,
as we expect to be better able to detect a pair of identical subhaloes when the
baryonic one is located at smaller radii than the dark matter only one, relative to
a situation in which the dark matter only satellite is located at smaller radii than
its baryonic counterpart.
In Figure 3.9 we show the fraction of subhaloes in DMONLY for which a
counterpart is found in REF. Once again we do not show a comparison with AGN
for clarity, but note that very similar results are obtained.
Horizontal lines show the total fraction of DMONLY subhaloes (both centrals
and satellites) that is recovered in REF, while lines with plot symbols show the
fraction of satellites for which a link is found as a function of radius. It is clear that
the linked fraction depends heavily on both box size and resolution for Msh,tot <
1012 h−1 M( , although the effect of the box size is only significant for the lowresolution simulations. For the simulation employed in this mass bin throughout
the main text of the chapter, L200N1024, the total fraction of linked subhaloes
is around 98%. However, the fraction of linked satellites is significantly lower,
especially for radii r ! 2rvir , where the different survival and detection rates of
baryonic subhaloes are expected to play a role.
Comparing this panel to the corresponding panel in Figure 3.8, we see that
the drop in the fraction of matched satellites at small radii corresponds to the
strong increase in clustering found for baryonic subhaloes, indicating that this
may be a biased result. Similar results are found for satellites with masses 1012 <
Msh,tot /[M( /h] < 1013 , although both the total and the satellite linked fractions
are much higher than for 1012 < Msh,tot /[M( /h] < 1013 , for all simulations and
radii. No drop-off in the linked fraction of satellites is observed at higher masses.
77
Baryons and the two-point correlation function
Figure 3.9: The fraction of subhaloes in DMONLY for which a link was found in REF, split by subhalo mass as indicated in the top left of each
panel. Colours and line styles are as in Figure 3.7, and a vertical dotted line once again indicates the mean virial radius in each mass bin. Horizontal
lines show the total fraction of linked subhaloes (both centrals and satellites) at the corresponding box size and resolution, while the lines with plot
symbols show the fraction of satellite subhaloes linked as a function of radius. For r ! 2rvir the fraction of linked satellites typically drops sharply as
subhaloes are destroyed by tidal stripping or become undetectable. Both the matched satellite and total fractions depends strongly on box size and
resolution for subhalo masses Msh,tot < 1012 h−1 M& .
78
3.B Linked fractions
Based on these results, we haven chosen to grey out the relative difference
curves in the Figures showing autocorrelation functions (Figures 3.3 and 3.5) on
radii where the fraction of linked satellites is < 95% of the total matched fraction. Note that this may not completely remove the possible bias on scales where
the satellite contribution dominates the correlation function. Further investigation with higher-resolution simulations is needed to determine whether the upturn
observed at small radii is physical or numerical in origin.
Note that the occasional downturn of the linked fraction at relatively large radii,
r " 2rvir , is due to small-number statistics, as low-mass subhaloes found at these
radii are rarely satellites. As the autocorrelation function of linked subhaloes at
these radii is dominated by central-central pairs, we do not apply a cut at r ≥ 2rvir .
79
4
The effects of halo alignment and shape
on the clustering of galaxies
We investigate the effects of halo shape and its alignment with larger scale structure on the galaxy correlation function. We base our analysis on the galaxy formation models of Guo et al., run on the Millennium Simulations. We quantify
the importance of these effects by randomizing the angular positions of satellite
galaxies within haloes, either coherently or individually, while keeping the distance
to their respective central galaxies fixed. We find that the effect of disrupting the
alignment with larger scale structure is a ∼ 2 per cent decrease in the galaxy correlation function around r ≈ 1.8 h−1 Mpc. We find that sphericalizing the ellipsoidal
distributions of galaxies within haloes decreases the correlation function by up to
20 per cent for r ! 1 h−1 Mpc and increases it slightly at somewhat larger radii.
Similar results apply to power spectra and redshift-space correlation functions.
Models based on the Halo Occupation Distribution, which place galaxies spherically within haloes according to a mean radial profile, will therefore significantly
underestimate the clustering on sub-Mpc scales. In addition, we find that halo assembly bias, in particular the dependence of clustering on halo shape, propagates
to the clustering of galaxies. We predict that this aspect of assembly bias should
be observable through the use of extensive group catalogues.
Marcel P. van Daalen, Raul E. Angulo and Simon D. M. White
Monthly Notices of the Royal Astronomical Society
Volume 424, Issue 4, pp. 2954-2960 (2012)
The effects of halo shape on clustering
4.1 Introduction
Investigating how matter is organized in our Universe is one of the key ways in
which we can test the validity of cosmological models and constrain their parameters. By comparing theoretical predictions to observed measures of structure, such
as the galaxy correlation function or the matter power spectrum, one can reject
some models and fine-tune others. It is, however, important to keep in mind the
limitations of theoretical models, both numerical and analytical, when making this
comparison, as these may limit the applicability of the results.
There are various ways in which one can predict the organization, or “clustering”, of matter and galaxies theoretically. One can use fully hydrodynamical
simulations, in which dark matter, gas and stars are treated explicitly, to follow
the formation and evolution both of dark matter haloes and of the galaxies within
them. For a recent review of the numerical methods behind such simulations, see
Springel (2010). Such models are computationally expensive, limited to small volumes in comparison to recent galaxy surveys, and sensitive to the ad hoc subgrid
recipes required to include critical processes like star formation and feedback.
An alternative, first implemented by Kauffmann et al. (1999) (see also Springel
et al., 2001; Springel, 2005a), is to combine N-body simulations of the growth of
dark matter structures with semi-analytic models of galaxy formation (e.g. White
& Frenk, 1991; Kauffmann, White & Guiderdoni, 1993; Cole et al., 1994; see
Baugh, 2006 for a review). A great advantage of semi-analytic simulations is that
they require comparatively little CPU time even for a large underlying N-body
simulation. This allows them to be run many times and on many haloes, so that
one can explore the physical processes and the associated parameters that are
required to produce galaxy populations in agreement with selected observational
data (such as the galaxy stellar mass, luminosity or correlation functions). Such
semi-analytic simulations do not focus on the properties of individual objects, but
rather on the underlying statistical properties of the entire population. In this
way, the relative importance of different physical processes can be examined as a
function of the time and place where they are occurring. A disadvantage of such
simulations is that they provide only very crude information on the structure of
individual objects.
Yet another alternative is to take the statistical approach one step further.
If one is interested only in the present-day clustering of galaxies, the physical
processes associated with their formation and evolution may not be relevant. One
can then populate the haloes in an N-body simulation with galaxies using a purely
statistical model that depends on current halo properties, for example halo mass.
Galaxy clustering can then be described in terms of the clustering of the haloes.
This approach is known as halo occupation distribution modelling, or simply HOD
modelling (see Cooray & Sheth, 2002, for a review). Typically, central and satellite
galaxies are treated separately, as each halo will contain one and only one of
the former but may contain none or many of the latter (Kauffmann et al., 1999;
Kravtsov et al., 2004; Zheng et al., 2005). The satellite galaxies assigned to a halo
82
4.1 Introduction
are usually assumed to be spherically distributed following a standard profile such
as that of Navarro, Frenk, & White (1997). Attempts at including substructure
or an environmental dependence have also been made (e.g. Giocoli et al., 2010;
Gil-Marín, Jimenez & Verde, 2011). Note that by assuming spherical symmetry
some information is lost. As the simulations of Davis et al. (1985) first showed,
cold dark matter haloes are typically strongly ellipsoidal. If the distribution of
galaxies follows the mass distribution, this would leave an imprint on the galaxy
correlation function on small scales. Furthermore, halo ellipticity may also have
an effect on larger scales. If neighbouring haloes are aligned, as expected from
tidal-torque theory, this will boost the correlation on scales corresponding to the
typical separations between haloes.
The ellipticity and intrinsic alignment of dark matter haloes and their galaxy
populations have been the subject of many earlier studies, e.g. Carter & Metcalfe
(1980); Binggeli (1982); West (1989); Splinter et al. (1997); Jing & Suto (2002)
and Bailin & Steinmetz (2005), and recently Paz et al. (2011) and Smargon et al.
(2012). Most relevant to the current work are the studies by Smith & Watts (2005)
and Zu et al. (2008). The former authors investigated the effects of halo triaxiality
and alignment on the matter power spectrum in the halo model framework. Inspired by the results of simulations, they took a purely analytic approach in which
they re-developed the halo model to account for ellipsoidal halo shapes. Zu et al.
(2008), on the other hand, used the semi-analytical models of De Lucia & Blaizot
(2007) to investigate environmental effects, including that of halo ellipticity, on
the galaxy correlation function in both real and redshift space.
There exists a deeper connection between halo shape and clustering that we also
explore in this chapter. As Bett et al. (2007) and Faltenbacher & White (2010)
have previously shown, at fixed mass the clustering of haloes depends on their
shape. This is probably a reflection of assembly bias (i.e. the dependence of halo
clustering on properties other than mass, Gao & White, 2007). More specifically,
the most spherical haloes in their samples cluster significantly more strongly than
average, and the most aspherical more weakly. The known correlations between
formation time and halo shape (e.g. Allgood et al., 2006; Ragone-Figueroa et al.,
2010), and between formation time and clustering strength (e.g. Gao, Springel &
White, 2005; Wechsler et al., 2006; Wetzel et al., 2007; Jing, Suto & Mo, 2007)
point in this direction but may by themselves not be strong enough to explain the
magnitude of the effect. Here, we investigate whether this shape-dependence is
also seen in the clustering of the galaxies. If so, this would bring us one step closer
to measuring assembly bias directly in observations. We are herein also motivated
by the results from Zhu et al. (2006) and Croton, Gao & White (2007), who showed
that assembly bias in general is indeed expected to propagate to galaxy clustering.
In this chapter, we expand upon previous work by investigating the effects of
alignment and ellipticity on the galaxy correlation function using the Millennium
Simulation (Springel, 2005a) and the semi-analytic models of Guo et al. (2011).
In Section 4.2, we discuss these simulations and our methods for quantifying the
effects of alignment and ellipticity. We also outline our procedure for determining
83
The effects of halo shape on clustering
the shape-dependence of galaxy bias. We show our results in Section 4.3, and
present our conclusions in Section 4.4.
4.2 Methods
4.2.1
Simulation and SAM
We make use of the galaxy catalogues generated by Guo et al. (2011, , hereafter
G11), who implemented galaxy formation models on the Millennium Simulations
(Springel, 2005a; Boylan-Kolchin et al., 2009). The Millennium Simulation (MS) is
a very large cosmological N-body simulation in which 21603 particles were traced
from redshift 127 to the present day in a periodic box of side 500 h−1 Mpc, comoving. The Millennium-II Simulation (MS-II) follows the same number of particles in
a box of side 100 h−1 Mpc and so has 125 times better mass resolution. Both simulations assume a ΛCDM cosmology with parameters based on a combined analysis
of the 2dFGRS (Colless et al., 2001) and the first-year WMAP data (Spergel
et al., 2003). These cosmological parameters, given by {Ωm , Ωb , ΩΛ , σ8 , ns , h} =
{0.25, 0.045, 0.75, 0.9, 1.0, 0.73}, are not consistent with the latest analyses of the
CMB data, for example the seven-year WMAP results (Komatsu et al., 2011). In
particular, the more recent data prefer lower σ8 and higher Ωm values.1 We will
only make relative comparisons between clustering statistics here, and do not expect our results to be significantly influenced by these small parameter differences
(see Guo et al., 2013).
The galaxy formation models of G11 allow galaxies to grow at the potential
minima of the evolving population of haloes and subhaloes in the simulations.
Each Friends-of-Friends (FoF) group contains a central galaxy at the potential
minimum of its main subhalo, and may contain many satellite galaxies at the
centres of surrounding subhaloes. In some cases, due to tidal effects, a satellite
galaxy may be stripped of its dark matter to the point where its subhalo is no
longer identified as a bound substructure, turning the galaxy into an “orphan”.
Such galaxies follow the orbit of the dark matter particle that had the highest
binding energy immediately before subhalo disruption, except that their distance
to the central galaxy is artificially decreased until they merge with it in order
to mimic the effects of dynamical friction. We note that the treatment of the
orbits of orphans is approximate, and for example does not include the expected
circularization of the orbits (e.g. Boylan-Kolchin, Ma & Quataert, 2008). The
models also include treatments of star formation, gas cooling, gas stripping, metal
enrichment, supernova and AGN feedback, and galaxy mergers. For more details
about the SAM, as well as the treatment of different types of galaxies, we refer
to G11. For our purposes, it is enough to note that the predicted clustering of
galaxies is quite a close match to that seen in the Sloan Digital Sky Survey (Guo
et al., 2011).
1 See
84
Angulo & White (2010a) for a method to correct for this.
4.2.2 Calculation of the galaxy correlation function
4.2.2
Calculation of the galaxy correlation function
The galaxy two-point correlation function, ξ(r), measures the clustering of galaxies
as a function of scale. It effectively encodes the excess probability of finding a pair
of galaxies at a given separation r, relative to the expectation for a uniform random
distribution. In what follows, we will be interested in scales 30 h−1 kpc < r <
50 h−1 Mpc, as these are both well-resolved and well-sampled by the simulation.
In order to get accurate results over this full range, we calculate the correlation
function by direct pair counts on small scales (i.e. r ! 4 h−1 Mpc) and use an
approximate but accurate method to calculate it on intermediate and large scales.
A direct calculation of this function scales as the number of galaxies squared
and is thus unfeasible for the large sample analysed here. We therefore speed up
the calculation by mapping galaxies onto a grid, and we correlate the mean density
contrast in each grid cell with that of every other (a method previously employed
by, for example, Barriga & Gaztañaga, 2002, Eriksen et al., 2004 and Sánchez,
Baugh & Angulo, 2008). We improved the performance on intermediate scales
by folding the density field onto itself before its autocorrelation is calculated (see
e.g. Jenkins et al., 1998). We do not go into these methods here, but note that
tests against higher-accuracy calculations show that the error in the ratio of the
correlation functions, which is the relevant quantity for our main results, is less
than 1 per cent on all scales considered. For the correlation functions we calculate
to determine the galaxy bias a direct pair count over the full range of scales is
feasible, as there we only consider relatively small subsets of galaxies (see §4.3.2).
4.2.3
Testing the importance of alignment and ellipticity
During their lifetime, haloes merge and may accrete more subhaloes. The accretion
of mass is not isotropic since matter flows in preferentially along filaments (see e.g.
Tormen, Bouchet & White, 1997; Colberg et al., 1999 or more recently Vera-Ciro
et al., 2011). As a result, the distribution of subhaloes and thus galaxies within a
FoF group is generally not isotropic either, but is instead approximately ellipsoidal,
following the mass and aligning with surrounding large-scale structure (see e.g.
Angulo et al., 2009). To test whether alignment with neighbouring structure has an
effect on clustering statistics, we randomly rotate the haloes around their centres
and see if this systematically alters the galaxy correlation function. More precisely,
we rotate the satellite population of each FoF group bodily around the central
galaxy to a new randomly chosen orientation, and we repeat this process for every
FoF group in the simulation. We stress that this transformation preserves the
numbers, properties, and relative positions of the galaxies in every halo; only the
orientations of the distributions change. We then calculate the galaxy correlation
function for the new distribution, and compare it to the original. If alignment with
large-scale structure is important, one would expect to see the correlation decrease
systematically on scales slightly larger than individual haloes. To estimate the
uncertainty in our results, we have repeated this process 25 times, each time with
85
The effects of halo shape on clustering
a different set of randomly chosen angles.
The effect of halo ellipticity is tested in a similar way. Here, we randomly rotate
the position of each individual satellite galaxy around its central, rather than
rotating all satellites together. In this way, the galaxy distribution within each
halo is sphericalized. Since the distribution of galaxies within haloes is typically
ellipsoidal, this process should increase the average distance between galaxies, thus
decreasing the correlations between galaxies in the same halo. We note that Zu
et al. (2008) investigated the effect of halo ellipticity in the same way.
4.2.4
Testing the dependence of galaxy bias on halo shape
Faltenbacher & White (2010) showed that the clustering of haloes depends on the
shape of the halo, defined as s = c/a, where c and a are eigenvalues of the inertia
tensor (a > b > c). Specifically, they showed that the large-scale bias of haloes
with more spherical shapes is larger than average, while the inverse is true for
the most aspherical haloes. They also found that this difference decreases with
equivalent peak height, ν(M, z) = δc (z)/σ(M, z), where σ(M, z) is the root-meansquare linear overdensity within a sphere which contains the mass M in the mean,
and δc (z) is the linear overdensity threshold for collapse at redshift z. Here, we
are interested in seeing if this shape-dependent clustering, which might reflect the
assembly bias of the haloes, is also recovered from the galaxy distribution.
We use the halo shape data from Faltenbacher & White (2010), who calculated the inertia tensor from the dark matter particles belonging to the mostpronounced subhalo2 of each FoF halo, which on average comprises ∼ 80 per
cent of its mass. To ensure that the shapes were accurately determined, only
haloes with at least 700 particles were considered, corresponding to a minimum
(sub)halo mass M = 6.02 × 1011 h−1 M( . We compare this to the shape measured
from the galaxy distribution in the same way, using all galaxies with stellar masses
M∗ > 109 h−1 M( within a sphere of radius R200 , defined as the radius enclosing
200 times the mean density of the Universe, centred on the central galaxy. Using
such a distance cut makes it easier to compare our results to observations – in
fact, a similar procedure is often followed when determining the richness of real
groups and clusters. Rejecting galaxies outside the virial radius slightly biases us
to measure more spherical shapes, but we have checked that this effect is small
and does not significantly affect our results.
After splitting the galaxies by the shape of their halo (measured either from the
dark matter or from the galaxies themselves) we determine the large-scale galaxy
bias factor bgal for each subsample. Here, too, we follow Faltenbacher & White
(2010), who in turn followed the approach of Gao & White (2007). The bias is
2 The
most-pronounced subhalo may differ from the most massive subhalo only if the FoF groups
hosts two or more subhaloes with roughly the same mass. In this case one of these is arbitrarily
assigned to be the most massive. The most-pronounced subhalo is more consistently defined by
using the halo merger tree. It is also the subhalo hosting the most luminous galaxy in the FoF
group.
86
4.3 Results
Figure 4.1: The effect of halo alignment on the galaxy correlation function. The x-axis shows
the real-space separation r, while the y-axis shows the fractional difference between the correlation function after random bodily rotations are applied, and that of the original, unrotated
sample. All simulated galaxies with M∗ > 109 h−1 M& from the z = 0 catalogue of G11 have
been used here. The bin size is roughly 0.07 dex. Each of the 25 thin, coloured lines represents
a different set of random rotations, and the thick, black line shows the average of these. There
is a clear signal around r ≈ 1.8 h−1 Mpc, where the correlation function is lowered by roughly 2
per cent.
computed as the relative normalization factor that minimizes the mean square of
the difference log(ξgm ) − log(bgal ξmm ) for four bins spaced equally in log r in the
range 6 < r < 20 h−1 Mpc. Here ξmm is the dark matter autocorrelation function
and ξgm is the cross-correlation function of galaxies and dark matter. Note that
unlike Faltenbacher & White (2010) we are only interested in the results at z = 0.
4.3 Results
4.3.1
Alignment and ellipticity
We will first discuss our results for “bodily” rotations, which test the effect of
halo alignment. Figure 4.1 shows the fractional difference between the correlation
functions of the “rotated” and original samples, plotted against the real-space
separation r. We have only used those galaxies from the catalogue generated by
G11 that have a stellar mass M∗ > 109 h−1 M( , as the Millennium Simulation
is not complete below this limit. This provides a sample of 5 200 801 galaxies.
We note that increasing this mass limit by a factor of ten does not influence our
results, either qualitatively or quantitatively. All random rotations are applied
prior to the mass cut in order to avoid problems in cases where a central galaxy
87
The effects of halo shape on clustering
Figure 4.2: The effect of halo ellipticity on the galaxy correlation function. The bin size,
axes and lines are as in Figure 4.1, although the bodily rotations have been replaced by random
independent rotations of satellites around their central galaxies. A peak of 1 − 2 per cent can be
seen around r ≈ 3.5 h−1 Mpc, but the largest effect is seen on small scales, where the correlation
function is systematically lowered by up to ∼ 20 per cent. Note also that the scatter has been
greatly reduced relative to Figure 4.1. This is mainly due to the larger number of random
rotations used when rotating satellites separately.
below the limiting mass has satellites above it. Coloured lines indicate different
sets of rotations, while the thick, black line shows the average of these. There
is clearly a significant dip around r ≈ 2 h−1 Mpc, with a depth of 2 per cent.
This is due to the disruption of the alignment between haloes and surrounding
structure. Note that the scatter is extremely low, due to the large number of
objects (in fact, the uncertainty at large scales is dominated by the errors due
to our approximate calculation of the correlation). Neglecting the orientation
of haloes when populating them with galaxies will therefore have a modest, but
significant, effect on the derived correlation function.
The ellipsoidal shape of the haloes, and the fact that the galaxy distribution
follows this shape, is a more significant factor when modelling the galaxy distribution. Figure 4.2 shows the result of applying independent rotations, which
sphericalize the galaxy distributions within haloes. This substantially suppresses
correlations for r ! 2 h−1 Mpc, with a ∼ 20 per cent effect on the smallest scales
probed here. The ellipsoidal shape of the galaxy distribution within haloes significantly reduces the typical separations of pairs within them. This is compensated
by a 1 − 2 per cent stronger correlation around r ≈ 3.5 h−1 Mpc. We conclude
that models that assume spherical profiles for the distribution of galaxies within
haloes will underestimate the galaxy correlation function by up to ∼ 20 per cent,
depending on the smallest scale considered. Note that the scatter is even smaller
88
Figure 4.3: Same as Figures 4.1 and 4.2, but now for the fractional differences in the galaxy power spectrum versus the wave number k. Left: Result
when testing for alignment. Again a weak but systematic signal of a few per cent can be seen, now between k ≈ 0.1 h Mpc−1 and k ≈ 2 h Mpc−1 .
Right: Result when testing for ellipticity. A monotonic decline in power sets in at k ≈ 0.1 h Mpc−1 , reaching roughly 20 per cent at k = 30 h Mpc−1 ,
matching the result found in Figure 4.2. This again demonstrates the importance of taking the ellipsoidal shape of the galaxy distribution within
haloes into account.
4.3.1 Alignment and ellipticity
89
The effects of halo shape on clustering
than before on all scales, which is due to the increased number of degrees of freedom here. This result is in excellent agreement with that of Zu et al. (2008),
who applied similar methods to all galaxies from De Lucia & Blaizot (2007) with
r-band luminosities Mr < −19.
One might worry that the necessarily artificial treatment of orphan, or “type
2”, galaxies in the galaxy formation models of G11 influences these results. G11
already showed that the inclusion of the orphans is critical if the radial distribution
of galaxies within rich clusters in the Millennium Simulation is to agree both with
observations and with the much higher-resolution MS-II. We have investigated
effects on our analysis by repeating it with these galaxies removed, reducing our
sample size by ∼ 24 per cent and spoiling the relatively good agreement of its
small-scale correlation with observation. This removal significantly amplifies the
signal found for the effects of alignment. This is because the orphan galaxies
are primarily located near halo centres. Once they are removed, galaxies that
do contribute to the alignment signal receive more weight. Orphan removal also
changes the signal found for the effects of ellipticity, modestly boosting it down
to r ≈ 0.1 h−1 Mpc. As a further check that the distribution of orphans in the
simulation is realistic, we have examined the shapes of the galaxy distributions of
massive haloes (specifically, 14 < log10 (MFoF /[ h−1 M( ]) < 14.5) in the MS and
MS-II, again using G11’s galaxy catalogues and considering only galaxies with
stellar masses M∗ > 109 h−1 M( . The better mass resolution of the MS-II results
in far fewer orphans in this mass range, and consequently the positions of galaxies
in MS-II are determined more accurately. Nevertheless, the shapes of the galaxy
distributions agree very well, thus implying that the distribution of orphans in the
MS is consistent with the distribution of similar, but unstripped, galaxies in the
MS-II. We also found that these shapes agree very well with those of the dark
matter haloes themselves (see Bett et al., 2007). A more detailed discussion of the
shapes is beyond the scope of this chapter.
For completeness, we have also compared the galaxy-galaxy power spectra of
the rotated and unrotated samples. The results are shown in Figure 4.3. Three
foldings in total were used to calculate the power spectra over the full range shown,
each with a fold factor of six (i.e. each folding maps the particle distribution to
1/216th of the volume). The power spectra were re-binned logarithmically to
resemble the bins used for the galaxy correlation functions, and to reduce noise.
Shot noise, which dominates the power for k " 10 h Mpc−1 , was subtracted. The
left-hand figure shows the fractional differences that result from applying bodily
rotations. Just as for the correlation function, there is a weak but clear dip of 1 − 2
per cent present which reflects the alignment of haloes with surrounding structure.
As expected, the right-hand figure shows a much stronger decrease in power, up
to 20 per cent on the smallest scales considered. This again demonstrates the
importance of taking the ellipsoidal distribution of galaxies within FoF groups
into account in, for example, models that use that use the full shape of the power
spectrum to extract cosmological parameters.
Our results differ from those found by Smith & Watts (2005). Like ourselves,
90
4.3.2 Shape-dependent galaxy bias
Figure 4.4: A comparison of the halo shape measured from the dark matter and that measured
from the galaxy distribution. Here shape is defined as the ratio of the smallest to the largest
eigenvalue of the inertia tensor. The shape on the horizontal axis is computed using all galaxies
satisfying M∗ > 109 h−1 M& within R200 . For the shape on the vertical axis, only CDM particles
belonging to the most-pronounced substructure are included, and only if the number of particles
is at least 700. Pixels are colour-coded by mean number of galaxies per halo. The “true” shape is
recovered more accurately when the number of galaxies increases; some scatter remains however,
as the galaxy distribution does not perfectly trace the main subhalo.
they find that the scale at which the contribution from alignments to the power
spectrum is maximal is ∼ 0.5 h Mpc−1 , but they show that the relative contribution
of alignments is strongly model-dependent, varying from 10−12 to 10 per cent.
Furthermore, they find that when haloes are assumed to be spherical, the power
is higher than when they are ellipsoidal by up to 5 per cent. Not only is the effect
we find significantly stronger and increasing towards smaller scales up to at least
k = 30 h Mpc−1 , but its sign is opposite. We attribute these differences to the
fact that in both models explored by Smith & Watts (2005), the radially averaged
density profiles are not conserved when transforming the haloes from spherical to
triaxial, making a comparison with our own results difficult.
4.3.2
Shape-dependent galaxy bias
Having established that the shape of the galaxy distribution significantly affects
the small-scale clustering, we now investigate how the shape-dependent assembly
bias affects clustering. We first examine how well the shape measured from the
91
The effects of halo shape on clustering
galaxy distribution corresponds to that measured from the dark matter particles
belonging to the most-pronounced substructure. Note that while the galaxies trace
the shape of the FoF halo very well, this is not necessarily the case for its mostpronounced substructure. Additionally, even if the galaxy distribution traces the
dark matter perfectly, the right shape may still not be recovered if only a small
number of galaxies is available to sample the halo.
The results of this shape comparison are presented in Figure 4.4. Here the
shape measured from the galaxy distribution is shown on the horizontal axis, while
the shape measured from the dark matter is on the vertical axis. Each pixel is
colour-coded by the mean number of satellites, Nsat , satisfying M∗ < 109 h−1 M(
and |rsat −rcen | < R200 . A low value of s = c/a indicates that the halo is (measured
to be) very aspherical, while a perfectly spherical halo would have s = 1. It is
immediately clear that a low number of satellites leads to a severe underestimate
of s. This is expected: together with the central galaxy, any two satellites will
define a plane, ensuring that c = 0. It is only when the halo is sampled by a large
enough ensemble of points that the inertia tensor can be determined accurately.
Figure 4.4 illustrates that one needs Nsat " 30 to get an unbiased and accurate
shape estimate. However, there is always a significant amount of scatter around
the diagonal. This is due to the galaxies tracing the mass, i.e. the shape of the
whole FoF group, and not just that of the most-pronounced substructure.
Next, we split our galaxy sample by the shape measured from the halo dark
matter distribution and calculate the galaxy bias factor as function of equivalent
peak height, bgal (ν), following the method described in §4.2.4. As we can only
use galaxies for which a dark matter shape has been determined – i.e. those in
FoF haloes of which the most massive substructure is comprised of at least 700
particles – our galaxy sample is reduced to 2 953 050 galaxies. The results are
shown in Figure 4.5. Here the upper panel shows the galaxy bias determined for
each sample at a given equivalent peak height. The points show the median value,
horizontal error bars indicate the width of the bin, and vertical error bars show
1σ deviations calculated from 50 bootstrap resamplings of the galaxy catalogues.
The black line shows bgal (ν) for the full sample of galaxies, while the coloured lines
show the bias for the different subsamples. In the bottom panel, the fractional
difference between these subsamples and the full sample is shown. It is immediately
clear that there is a strong dependence on shape: at the lowest equivalent peak
heights probed here, the bias of the galaxies in the most spherical (aspherical)
20 per cent of haloes is up to 40 per cent higher (lower) than that of the full
sample. This shows that the shape-dependence of the halo bias found by Bett
et al. (2007) and Faltenbacher & White (2010) is also strongly present in the
clustering of the galaxies. The effect grows weaker with increasing ν. However,
for ν " 2.5 statistical uncertainties begin to dominate, due to the low number
of haloes available at high equivalent peak heights. Note that this is not seen in
the data of Faltenbacher & White (2010) as they combine the data from different
redshifts, while we only consider z = 0.
We then repeat this exercise, but this time we split our galaxy sample by
92
4.3.2 Shape-dependent galaxy bias
Figure 4.5: The dependence of galaxy bias, bgal , on shape, as a function of peak height at
z = 0. The black line in the top panel shows the galaxy bias of the full galaxy sample, while
coloured lines show the bias of subsamples split by halo shape measured from the dark matter
distribution. In the bottom panel the fractional differences of the bias of these subsamples
relative to the full sample are shown. The vertical error bars show 1σ deviations calculated from
50 bootstrap resamplings of the galaxy catalogues. We find that galaxies in the most spherical
(aspherical) haloes are strongly biased (antibiased) relative to the full sample. The difference
can be as much as 40 per cent for ν ≈ 0.7. For ν " 2.5 statistical uncertainties, due to the low
number of high-mass haloes, begin to play an important role.
the shape measured from the distribution of the galaxies themselves. However,
we expect low-mass haloes to host only a few satellite galaxies, leading to very
unreliable estimates of the halo shape (see Figure 4.4). In order to separate the
signal we are looking for – i.e. the shape-dependence of the galaxy bias – from the
unwanted bias introduced by using too few galaxies in the shape measurement, we
now consider bgal as a function of the number of satellites per halo, Nsat , instead of
the equivalent peak height ν. An additional advantage of this approach is that Nsat
is directly observable. We note, however, that we obtain almost identical results
when considering galaxy bias as a function of peak height instead of Nsat . Since we
do not use the dark matter particle data in this case, we are no longer constrained
by needing haloes with at least 700 particles. However, as we can now only consider
haloes with at least one satellite galaxy that satisfies both M∗ > 109 h−1 M( and
|rsat − rcen| < R200 , we are left with a sample of 2 566 441 galaxies. The results are
shown in Figure 4.6. At the lowest value of Nsat , no significant effect can be seen.
But as Nsat grows, increasing the accuracy of the shape determinations, we again
see a clear dependence of bgal on the shape s: galaxies in more spherical haloes
93
The effects of halo shape on clustering
Figure 4.6: As Figure 4.5, but now showing the bias as a function of the number of satellite
galaxies and split by the halo shape measured from galaxies within R200 . At low Nsat no significant shape-dependence of bgal is recovered, due to the extremely unreliable shape determinations
that follow from using only a handful of galaxies to sample the halo. For 10 ! Nsat ! 400, however, we find again that galaxies in more spherical (aspherical) haloes have a significantly higher
(lower) bias than average. At higher Nsat our results are again dominated by poor statistics.
are ∼ 20 per cent more strongly clustered than average. The inverse is true for
galaxies in the most aspherical haloes. When Nsat grows too high our results are
once more dominated by statistical errors, due to the low number of high-mass
haloes (hosting at least several hundreds of satellites) available.
These results show that halo assembly bias in the form of a shape-dependent
clustering strength propagates to the clustering of galaxies, and can therefore in
principle be measured in sufficiently large surveys. In order to carry out such a
task, a large galaxy survey with appropriately defined group catalogues is needed.
4.4 Summary
We have investigated the effects of halo alignment with larger scale structure and of
halo ellipticity on galaxy correlation functions, using the Millennium Simulations
(Springel, 2005a; Boylan-Kolchin et al., 2009) and the galaxy formation models of
Guo et al. (2011). By rotating satellite galaxies in FoF groups around their central
galaxies, either coherently for each halo or independently for each satellite, and
then comparing the correlation function of the resulting galaxy distribution to the
original one, we were able to quantify the importance of taking halo alignment and
94
4.4 Summary
non-sphericity into account. Furthermore, by measuring the shape of the haloes
as traced by the galaxies we were able to investigate the propagation of shapedependent assembly bias to the clustering of galaxies. Only galaxies with stellar
masses M∗ > 109 h−1 M( were considered in our analysis, though we note that
increasing this mass limit by a factor of ten does not influence our results. Our
findings can be summarized as follows:
• The effects on the galaxy correlation function of the alignment of haloes with
larger-scale structure are small. The main effect of disrupting this alignment
is a 2 per cent reduction in correlation amplitude around r ≈ 1.8 h−1 Mpc,
with minor effects of at most 1 per cent at smaller scales.
• The ellipsoidal shapes of the galaxy distributions within individual haloes
have a much stronger influence on galaxy correlations. By sphericalizing
these galaxy distributions (i.e. randomizing the angular positions of satellites while keeping the distance from the central galaxy fixed), the correlation function is raised by up to 2 per cent around r ≈ 3.5 h−1 Mpc, but
greatly reduced for r ! 1.5 h−1 Mpc, by up to ∼ 20 per cent on the smallest scale probed, r = 30 h−1 kpc. This confirms the results of Zu et al.
(2008). The effect on the galaxy power spectrum extends to scales as large
as k = 0.1 h Mpc−1 .
• The assembly bias of haloes, as characterized by the dependence of clustering
on halo shape, is reflected in the clustering of galaxies. The effect is strongest
at low equivalent peak heights: at ν ≈ 0.7, the galaxy bias of galaxies in
the 20 per cent most spherical and most aspherical haloes deviate from the
average by 40 per cent.
• Even if the shape of the halo cannot be measured directly, but is instead
estimated from galaxies within one virial radius of the central galaxy, the
effect of assembly bias is clearly visible.
By using the plane-parallel approximation and ignoring evolution, we have checked
that comparable results are obtained for the effects of alignment and ellipticity on
the redshift-space correlation functions. Models that assume a spherically symmetric profile for the galaxy distribution, such as HOD models, will therefore significantly underestimate galaxy correlations and power spectra on sub-Mpc scales.
Furthermore, we have demonstrated that the shapes of haloes and of the galaxy
distributions within them can be strongly correlated with their clustering. This
effect should be measurable in galaxy redshift surveys. With the help of extensive
group catalogues it should therefore be possible to measure assembly bias directly.
Acknowledgments
The authors thank Andreas Faltenbacher for kindly providing dark matter shape
determinations for haloes in the Millennium Simulation. The Millennium Sim95
The effects of halo shape on clustering
ulation databases used in this chapter and the web application providing online
access to them were constructed as part of the activities of the German Astrophysical Virtual Observatory. This work was supported by the Marie Curie Initial
Training Network CosmoComp (PITN-GA-2009-238356) and by Advanced Grant
246797 "GALFORMOD" from the European Research Council.
96
5
The contributions of matter inside and
outside of haloes to the matter power
spectrum
Halo-based models have been very successful in predicting the clustering of matter. However, the validity of the postulate that the clustering is fully determined
by matter in haloes remains largely untested, and it is not clear a priori whether
non-virialised matter might contribute significantly to the non-linear clustering
signal. Here, we investigate the contribution of haloes to the matter power spectrum as a function of both scale and halo mass by combining a set of cosmological
N-body simulations to calculate the contributions of different spherical overdensity
regions, Friends-of-Friends groups and matter outside haloes to the power spectrum. We find that on scales k < 2 h Mpc−1 , matter inside spherical overdensity
regions of size R200,mean accounts for less than 85% of the power, regardless of the
minimum halo mass. Its relative contribution increases with increasing Fourier
scale, peaking at ∼ 95% around k = 20 h Mpc−1 and on smaller scales remaining
roughly constant. For 2 ! k ! 10 h Mpc−1 , haloes below ∼ 1011 h−1 M( provide
a negligible contribution to the power spectrum, the dominant contribution on
these scales being provided by haloes with masses M200 " 1013.5 h−1 M( , even
though such haloes account for only ∼ 13% of the total mass. When haloes are
taken to be regions of size R200,crit , the amount of power unaccounted for is larger
on all scales. Accounting also for matter inside FoF groups but outside R200,mean
increases the contribution of halo matter on all scales probed here by 5 − 15%.
Matter inside FoF groups with MFoF > 109 h−1 M( accounts for essentially all
power for 3 < k < 100 h Mpc−1 . We therefore expect a halo model based approach to overestimate the contribution of haloes of any mass to the power on
small scales (k " 1 h Mpc−1 ), while ignoring the contribution of matter outside
R200,mean , unless one takes the halo to be a broader non-spherical region similar
to the FoF group.
Marcel P. van Daalen and Joop Schaye
In preparation
Halo matter and the power spectrum
5.1 Introduction
The matter power spectrum, a measure of how matter clusters as a function of
scale, is a key observable and a powerful tool in determining the other cosmological parameters of our Universe. As future weak lensing experiments which will
measure this quantity to unprecedented accuracy, such as DES1 , LSST2 , Euclid3
and WFIRST4 , draw ever closer, the precision with which the theoretical matter
power spectrum is being predicted steadily increases as well. Currently, some of
the largest uncertainties on fully non-linear scales come from our incomplete understanding of galaxy formation (e.g. van Daalen et al., 2011), which causes large
unwanted biases in the cosmological parameters derived from observations. We
expect that we may be able to account for these using independent measurements
of, for example, the large-scale gas distribution, and/or to marginalise over these
uncertainties using a halo model based approach, although for the largest of these
future surveys more effective and less model-dependent mitigation strategies than
currently exist will be needed (e.g. Semboloni et al., 2011; Zentner et al., 2013).
But even assuming that we can somehow account for the effects of galaxy
formation on the distribution of matter, significant challenges remain before we
are able to predict the matter power spectrum to the sub-percent accuracy needed
to fully exploit future measurements (Huterer & Takada, 2005; Hearin, Zentner
& Ma, 2012). These include converging on the “true” simulation parameters in
N-body codes, although these too can be marginalised over (Smith et al., 2014).
However, with each such marginalisation one should expect the constraining power
of observations to be reduced.
Direct simulations are not the only way to obtain theoretical predictions of the
matter power spectrum, however. Other avenues, such as through the analytical
halo model (e.g. Seljak 2000, Ma & Fry 2000; see Cooray & Sheth 2002 for a
review), exist, and are widely used in clustering studies. The halo model is based
on the assumption that all matter is partitioned over dark matter haloes, which
finds its origin in the model proposed by Press & Schechter (1974, , hereafter PS),
later extended by Bond et al. (1991). The PS formalism is based on the ansatz
that the fraction of mass in haloes of mass M (R) is related to the fraction of the
volume that contains matter fluctuations δR > δcrit , where R is the smoothing
scale and δcrit is the critical density assuming spherical collapse. If the initial field
of matter fluctuations is known, a halo mass function can be derived from this
ansatz, which together with a model for the bias b(M ) (the clustering strength of
a halo of mass M relative to the clustering of matter) and a description of halo
density profiles fully determines the clustering of matter.
Much work has been done to improve the predictions of the halo model since
its introduction. More accurate mass functions have been derived based on, for
1 http://www.darkenergysurvey.org/
2 http://www.lsst.org/lsst
3 http://www.euclid-imaging.net/
4 http://wfirst.gsfc.nasa.gov/
98
5.1 Introduction
example, ellipsoidal collapse (Sheth, Mo & Tormen, 2001), fits to N-body simulations (e.g. Jenkins et al. 2001; Warren et al. 2006; Reed et al. 2007; Tinker et al.
2008; Bhattacharya et al. 2011; Angulo et al. 2012; Watson et al. 2013; see Murray,
Power & Robotham 2013 for a comparison of different models) and simulations
taking into account the effects of baryons (e.g. Stanek, Rudd & Evrard, 2009;
Sawala et al., 2013; Martizzi et al., 2014; Cusworth et al., 2014; Khandai et al.,
2014; Cui, Borgani & Murante, 2014; Velliscig et al., 2014). Similarly, much effort
has gone into deriving more accurate (scale-dependent) bias functions (e.g. Sheth
& Tormen, 1999; Seljak & Warren, 2004; Smith, Scoccimarro & Sheth, 2007; Reed
et al., 2009; Grossi et al., 2009; Manera, Sheth & Scoccimarro, 2010; Pillepich,
Porciani & Hahn, 2010; Tinker et al., 2010) and concentration-mass relations for
halo profiles (e.g. Bullock et al., 2001; Eke, Navarro & Steinmetz, 2001; Neto et al.,
2007; Duffy et al., 2008; Macciò, Dutton & van den Bosch, 2008; Prada et al., 2012;
Ludlow et al., 2014). Current halo models may incorporate additional ingredients
like triaxiality, substructure, halo exclusion, primordial non-Gaussianity and baryonic effects (e.g. Sheth & Jain, 2003; Smith & Watts, 2005; Giocoli et al., 2010;
Smith, Desjacques & Marian, 2011; Gil-Marín, Jimenez & Verde, 2011; Fedeli,
2014), and fitting formulae based on the halo model have also been developed
(e.g. Smith et al., 2003b; Takahashi et al., 2012).
However, the validity of the postulate that the clustering of matter is fully
determined by matter in haloes remains relatively untested. Even though matter
is known to occupy non-virialised regions such as filaments, their mass may simply
be made up of very small haloes itself, although recent results indicate that part
of the dark matter accreted onto haloes is genuinely smooth (Angulo & White,
2010b; Fakhouri & Ma, 2010; Genel et al., 2010; Wang et al., 2011). Either way,
it is not clear a priori whether this non-virialised matter contributes significantly
to the non-linear clustering signal.
Here, we examine the contributions of halo and non-halo mass to the matter
power spectrum with the use of a set of N-body simulations. We will first investigate the contribution to the redshift zero matter power spectrum of haloes
that are defined analogous to the typical halo model approach, also examining the
contributions of matter in smaller overdensity regions and outside of haloes. Next,
we expand the haloes to include all matter associated to Friends-of-Friends (FoF)
groups. Finally, we make predictions for the contribution of halo matter to the
power spectrum as a function of both scale and minimum halo mass, which can
serve as a test for halo models aimed at reproducing the clustering of dark matter.
This chapter is organized as follows. In §5.2 we describe our simulations and
the employed power spectrum estimator. We present and discuss our results in
§5.3 and summarise our findings in §5.4.
99
Halo matter and the power spectrum
Name
L400N1024
L200N1024
L050N512
L025N512
Box size
[h−1 Mpc]
Particle
number
400
1024
3
1024
3
200
50
25
512
3
512
3
[h
−1
mdm
M( ]
[h
−1
+max
kpc]
4.50 × 10
9
4.00
5.62 × 10
8
2.00
7.03 × 10
7
1.00
8.79 × 10
6
0.50
Table 5.1: The different simulations employed in this chapter. From left to right, the columns
list their name, box size, particle mass and maximum proper softening length. All simulations
were run with only dark matter particles and a WMAP7 cosmology.
5.2 Method
5.2.1
Simulations
We base our analysis on a set of dark matter only runs that were run with a
modified version of gadget iii, the smoothed-particle hydrodynamics (SPH) code
last described in Springel (2005b). The cosmological parameters are derived from
the Wilkinson Microwave Anisotropy Probe (WMAP) 7-year results (Komatsu
et al., 2011), and given by {Ωm , Ωb , ΩΛ , σ8 , ns , h} = {0.272, 0.0455, 0.728, 0.81,
0.967, 0.704}.
We generate initial conditions assuming the Eisenstein & Hu (1998) transfer
function. Prior to imposing the linear input spectrum, the particles are set up in
an initially glass-like state, as described in White (1994). The particles are then
evolved to redshift z = 127 using the Zel’dovich (1970) approximation.
The relevant parameters of the simulations we employ here are listed in Table 5.1. The simulation volumes range from 25 h−1 Mpc to 400 h−1 Mpc. The
mass resolution improves by a factor of 8 with each step, corresponding to an
improvement of the spatial resolution by a factor of 2, from the largest down to
the smallest volume. The gravitational forces are softened on a comoving scale
of 1/25 of the initial mean inter-particle spacing, L/N , but the softening length
is limited to a maximum physical scale of 2 h−1 kpc[L/(100 h−1 Mpc)] which is
reached at z = 2.91. As we will demonstrate, by combining these simulations,
we can accurately determine the matter power spectrum from linear scales up to
k ∼ 100 h Mpc−1 .
5.2.2
Power spectrum calculation
The matter power spectrum is a measure of the amount of structure that has
formed on a given Fourier scale k, related to a physical scale λ through k = 2π/λ.
It is defined through the Fourier transform of the density contrast, δ̂k . We will
present our results in terms of the dimensionless power spectrum, defined in the
100
5.2.3 Halo particle selection
usual way:
∆2 (k) =
*
k3
k3 V )
2
|
δ̂
P
(k)
=
|
,
k
2π 2
2π 2
k
(5.1)
with V the volume of the simulation under consideration. As all particles have the
same mass, the shot noise is simply equal to < |δ̂k |2 >k,shot = 1/Np , with Np the
number of particles in the simulation. All power spectra presented here have had
shot noise subtracted to obtain more accurate results on small scales.
We calculate the matter power spectrum using the publicly available f90 package powmes (Colombi et al., 2009). The advantages of powmes stem from the
use of the Fourier-Taylor transform, which allows analytical control of the biases
introduced, and the use of foldings of the particle distribution, which allow the
dynamic range to be extended to arbitrarily high wave numbers while keeping
the statistical errors bounded. For a full description of these methods we refer to
Colombi et al. (2009). As in van Daalen et al. (2011), we set the grid parameter
to G = 256 and use a folding parameter F = 7 for the two smallest volumes. To
calculate the power spectrum down to similar scales for the 200 and 400 h−1 Mpc
boxes, we set F equal to 8 and 9, respectively. Our results are insensitive to this
choice of parameters.
Both box size and resolution effects lead to an underestimation of the power
– at least on scales where a sufficient number of modes is available so that the
effects of mode discreteness can be ignored (k " 8π/L) – while all simulations
show excellent agreement on scales where they overlap (see §5.2). In order to cover
the dynamic range from k = 0.01 h Mpc−1 to 100 h Mpc−1 , we therefore combine
the power spectra of different simulations. By always taking the largest value of
∆2 (k) at each k. In the case of the full power spectrum, i.e. the power spectrum
of all matter, we take the combined power spectrum to be the one predicted by
linear theory up to k = 0.12 h Mpc−1 , where the power starts to become nonlinear. While the largest boxes show excellent agreement with the linear power
spectrum on these scales, we wish to avoid box size effects as much as possible. For
k > 0.12 h Mpc−1 – or, in the case of power spectra of subsets, for the smallest kvalue available – we individually average each power spectrum over each of 25 bins
in Fourier space ki and assign the combined power spectrum the largest ∆2 (ki ) of
all simulations derived in this way.
We combine the power spectra of selections of particles (e.g. all particles that
reside in haloes above a certain mass) in a similar way, but without including the
linear theory power spectrum.
5.2.3
Halo particle selection
In the halo model approach, haloes are commonly defined through a spherical
overdensity criterion, usually relative to the mean density of the Universe. In
order to investigate the contribution of such haloes to the matter power spectrum,
we define our haloes consistently.
101
Halo matter and the power spectrum
Overdense regions are identified in our simulations with a spherical overdensity
finder, as implemented in the subfind algorithm (Springel et al., 2001). We define
a halo as a spherical region with an internal mass overdensity of 200 × Ωm ρcrit ,
where ρcrit is the critical density of the Universe. These haloes therefore have a
mass equal to:
4π
3
Ωm ρcrit R200
M200 = M200,mean = 200 ×
,
(5.2)
3
where R200 = R200,mean is the radius of the region. In the remainder of the chapter,
we will define halo particles as any particle with a distance R < R200 from any
halo centre. All other particles are treated as non-halo particles, irrespective of
their possible FoF group membership, or having been identified as part of a bound
subhalo by subfind.
While we focus on halo matter as defined through R200 , we will also briefly
discuss the contribution of halo matter to the power spectrum for other overdensity regions and halo definitions (i.e. R500 , R2500 , R200,crit and Friends-of-Friends)
during the course of the chapter.
5.3 Results
5.3.1
Fractional mass in haloes
We first examine the fraction of the mass that resides in haloes, fh . As in each
simulation there is a lower limit to the masses of haloes that we can reliably
resolve, we compute fh as a function of the minimum mass of the included haloes.
Knowing the minimum resolved masses also allows us to estimate over which halo
mass range we can probe the contribution of halo particles to the power spectrum
in each simulation.
The results for fh are shown in Figure 5.1. Different colours are used for each
of our four different simulations, as indicated in the legend. Vertical dotted lines
denote the masses corresponding to 100 particles. Below these limits the fraction
of mass in haloes flattens off, indicating that such low-mass haloes are unresolved.
A thick dashed line shows the result of combining the mass fractions of all four
simulation for Mmin > 109 h−1 M( , through fh,comb = max(fh,i ). The bottom
panel shows the ratio of fh of each simulation to this combined fraction.
At the massive end, the high-resolution but low-volume L025 and L050 simulations significantly underestimate fh . This is most clearly seen in the bottom
panel: for L025 the mass fraction in haloes becomes significantly underestimated
for halo masses M200 " 1011 h−1 M( , while for the L050 box this happens for
M200 " 1012 h−1 M( . This is consistent with the points at which the halo mass
functions are underestimated for these simulations (not shown). The fluctuations
seen in the bottom panel for L200 for M200 > 1014 h−1 M( are due to the rarity
of such massive haloes, but as the fraction of the mass residing in such haloes is
< 10% this does not impact our conclusions. All simulations in which haloes at a
102
5.3.1 Fractional mass in haloes
Figure 5.1: The cumulative fraction of mass inside haloes, fh , as a function of minimum halo
mass, for different collisionless simulations as indicated in the legend. The resolution limit,
defined as the mass of haloes containing 100 particles, is shown as a vertical dotted line for
each simulation. Below this limit, the fraction of mass in haloes is underestimated. For the
two highest-resolution simulations these fractions are also significantly underestimated at high
masses, as these haloes are under-represented in these small volumes. Between the limits imposed
by resolution and box size effects, the simulations are in excellent agreement, and show that the
fraction of mass in haloes is ∼ 52% for M200 > 109 h−1 M& . A black dashed line shows the
combined result, taking the maximum fraction of mass in haloes between the different simulations
at every mass, while the bottom panel shows the fraction of this combined function predicted by
each simulation. We also show predictions for the Tinker et al. (2008) mass function as a black
dotted and dot-dashed line (see main text).
certain Mmin are both well-resolved and well-represented show excellent agreement
in fh (M > Mmin ).
The fraction of mass in haloes increases with decreasing halo mass. Only ∼ 19%
of matter is found in groups and clusters (Mmin > 1013 h−1 M( ), which increases
to ∼ 30% for Milky Way haloes and up (Mmin > 1012 h−1 M( ). But even at
the lowest resolved mass of roughly 109 h−1 M( , the fraction of mass in haloes is
still barely more than 50%. We therefore expect a significant contribution from
particles in haloes with M < 109 h−1 M( , and possibly from dark matter particles
that do not reside in haloes of any mass, to the matter power spectrum on large
scales, which we calculate in the next section.
For comparison, the top panel of Figure 5.1 also shows predictions of the fraction of mass in haloes above a certain mass based on the Tinker et al. (2008)
M200 halo mass function. Using the normalized halo mass function fit provided
by these authors, we have calculated fh (M > Mmin ) under the standard halo
103
Halo matter and the power spectrum
model assumption that all mass resides in haloes. The results are shown by the
black dotted line. Under this assumption, far more mass is predicted to reside
within resolved haloes than we find for our simulations, at any mass. However,
the Tinker et al. (2008) mass function converges to a mass density of only about
0.72 × Ωm ρcrit , meaning that either the true mass function predicts far more mass
in haloes M200 ! 1011 h−1 M( (roughly the lowest-mass haloes considered by Tinker et al. 2008), or that about 28% of the dark matter mass is genuinely smoothly
distributed at z = 0, not residing in haloes of any mass. To compensate for this
“missing mass”, we have also calculated the mass in haloes above some Mmin predicted by the Tinker et al. (2008) mass function relative to the total mass in the
Universe. This result is shown by the dot-dashed line, and shows much better
agreement with our simulations.
Up to Mmin ≈ 1012 the relative difference between the Tinker et al. (2008)
prediction and our combined result is constant at about 10% before decreasing
at higher masses. One possible reason for this discrepancy is that we only count
matter in regions where haloes overlap once, which is not taken into account when
integrating the mass function. However, we have checked that the mass residing
in overlap regions in our simulation is always ! 1.7%, with the largest overlap
fraction being found for the most massive haloes. The ! 10% differences found for
fh are therefore likely due to the non-universality of the halo mass function at this
level of precision (e.g. Tinker et al., 2008; Murray, Power & Robotham, 2013).
As it seems that fh (M > Mmin ) continues to rise on mass scales unresolved by
our simulations, the total contribution of matter in haloes to the power spectrum
will be underestimated in our simulations. However, as we will see in §5.3.2.1,
this depends on the scale considered. A range in Fourier space exists where the
fraction of power from halo particles is bounded below unity, and the contribution
of haloes with masses M200 ! 1011 h−1 M( is negligible. Additionally, on scales
where this does not hold we can still constrain the contribution from haloes above
a certain mass.
In the remainder of the chapter, we will only consider particles residing in
haloes with M200 > 109 h−1 M( to be halo particles, as this corresponds roughly
to the smallest haloes we can resolve.
5.3.2
Halo contribution to the power spectrum
We first show the full dimensionless matter power spectrum, i.e. using all particles,
in Figure 5.2. Here each simulation is shown by a different colour, and it is
immediately clear that no single one is converged over the full dynamic range
up to k ∼ 100 h Mpc−1 . The linear theory power spectrum, as generated by the
f90 package camb (Lewis, Challinor & Lasenby, 2000, , version January 2010),
is shown as a long dashed purple line. L400 and L200 show good agreement
with the linear power spectrum on scales where non-linear evolution is negligible
(k ! 0.12 h Mpc−1 ) and a sufficient number of modes is available (k > 0.04 and
0.08 h Mpc−1 respectively, roughly corresponding to λ = 0.4 L), while L050 and
104
5.3.2 Halo contribution to the power spectrum
Figure 5.2: The dimensionless power spectrum derived from each simulation, along with the
linear power spectrum (long-dashed purple line) and the combined power spectrum (dashed
black line). While the L025 and L050 simulations significantly underestimate the power on
large scales due to missing modes, their high resolution allows us to accurately extend the power
spectrum of the larger volumes up to k ∼ 100 h Mpc−1 . The erratic behaviour seen for lowresolution simulations at large k is due to shot noise subtraction. The bottom panel shows for
each simulation (as well as for the linear theory prediction) the fraction of power relative to the
combined power spectrum. For k < 20 h Mpc−1 , multiple simulations show the same results,
indicating convergence on these scales.
L025 show severe box size effects due to their lack of large-scale modes. These box
size effects become negligible only for k > 10 and k > 40 h Mpc−1 respectively.
Due to their finite resolution, all simulations underestimate the power on sufficiently small scales. Note that all power spectra shown here have had shot
noise subtracted, which explains the erratic behaviour of the power spectra on
the smallest scales. The underestimation of small-scale power becomes significant
already on scales corresponding to ∼ 100 softening lengths. However, for every
k ! 100 h Mpc−1 , there is at least one simulation for which neither box size nor resolution leads to an underestimation of the power at the " 1% level. We therefore
combine the different power spectra as described in §5.2.2 to obtain the combined
power spectra, shown as a dashed black line.
The bottom panel of Figure 5.2 shows the fraction of power predicted by each
simulation, as well as the fraction predicted by linear theory, relative to the combined power spectrum. By construction, this fraction is bounded to unity on
105
Halo matter and the power spectrum
Figure 5.3: The combined power spectrum for different sets of particles: R < R200 (halo
particles), R < R500 , R < R2500 and R > R200 (non-halo particles). Only haloes with M >
Mmin = 109 h−1 M& were considered in the cuts made, which in total contain about 38% of
all dark matter. The halo particles easily dominate the power on small scales; however, there
is a significant range of non-linear scales (k < 1.4 h Mpc−1 ) where this subset does not provide
the most power. While the non-halo particles account for a larger fraction of the power for
k < 0.6 h Mpc−1 , it is the cross-terms between the halo and non-halo particles (not shown) which
dominate in the mildly non-linear regime. Note that the horizontal range has been shortened
relative to Figure 5.2.
non-linear scales. Note that on scales k ! 20 h Mpc−1 , the fractions of multiple simulations are within a few percent of unity, indicating convergence on these
scales. For smaller scales, however, convergence is uncertain, although based on
the results for larger scales we expect our combined power spectrum to be accurate
to ∼ 1% up to k ∼ 100 h Mpc−1 .
Next, we repeat this procedure for halo and non-halo particles. We also consider particles within the R500 and R2500 overdensity regions, defined analogously
to R200 , which probe the inner parts of haloes. As we cannot reliably resolve
haloes with less than about 100 particles with our highest-resolution simulation,
we only consider the contribution of haloes with masses M > Mmin = 109 h−1 M(
here, treating matter in lower-mass haloes as non-halo particles. The results are
shown in Figure 5.3. Note that for clarity only the combined power spectra are
shown, and that the horizontal range has been shortened with respect to Figure 5.2, only showing the range of scales for which we can reliably determine the
power spectrum.
106
5.3.2 Halo contribution to the power spectrum
The contribution from halo particles strongly dominates the power on small
scales. The halo contribution is in turn dominated by the very inner regions of
haloes, at least on scales smaller than the size of these regions. However, towards
larger scales this contribution diminishes, and for k < 0.4 h Mpc−1 less than half
of the total power is provided by matter in haloes alone. On large scales the
significant fraction of the mass that occupies non-virialised regions becomes more
important, increasing to about 20%, roughly half of the contribution of halo matter
on the same scales. The remaining ∼ 40% of the total matter power on large scales
is therefore contributed by the cross-terms of halo and non-halo matter (not shown
here).
Note that on the scales shown here, only L400 and L200 contribute to the
combined power spectrum of non-halo particles. However, as these two are in
excellent agreement for k " 0.4 h Mpc−1 even though the mass resolution is eight
times worse in L400, we do not have reason to believe that this component would
significantly change on non-linear scales if lower-mass haloes were resolved. On
linear scales, the contribution of halo matter is mostly determined by the fraction
of mass in haloes, which does of course depend on the minimum halo mass resolved.
We will return to this point in §5.3.2.1.
We investigate the contribution of halo matter in more detail in Figure 5.4,
which shows the ratio of the power spectrum of matter within R200 of haloes with
masses M200 > 109 h−1 M( to the power spectrum of all matter. The black dashed
line shows the ratio of the combined power spectra, obtained from the smoothed
power spectra of all four simulations shown here as described in §5.2.2, relative
to the combined total power spectrum (black line in Figure 5.3). The solid lines
show the relative contribution of halo matter in each simulation separately,.
The contribution of halo matter to the total power increases with decreasing
physical scale. On large (linear) scales, the contribution from haloes seems to
converge to ∼ 30%, in good agreement with fh (M > 109 h−1 M( )2 ≈ 0.27. This
is expected, as the contribution of any subset of matter to the power spectrum on
linear scales should scale only with (the square of) the fraction of mass contained
in such a subset. However, as the fraction of power in haloes on large scales is fully
determined by L200 and L400, with both predicting roughly the same fraction as
can be seen in Figure 5.4, while the fraction of mass in haloes M > 109 h−1 M(
is only accurately measured for L025, this correspondence is actually surprising.
On non-linear scales the ratio rapidly increases down to physical scales of λ ∼
2 h−1 Mpc (k ∼ 3 h Mpc−1 ), reaching at most 95%, before slowly levelling off
towards smaller scales. Note that the combined results are fully determined by
L050 around k ≈ 20 h Mpc−1 , where we are unable to show convergence due
to the too-low resolution of L200 and too-small volume of L025. However, the
results of §5.3.2.1 imply that little would change on these scales if higher-resolution
simulations were available.
While L400 and L200 are in good agreement for 0.2 ! k ! 10 h Mpc−1 , on subMpc scales the contribution of halo matter to the total matter power spectrum
starts to show a strong dependence on resolution. On these scales fluctuations
107
Halo matter and the power spectrum
Figure 5.4: The fraction of power within haloes with masses M200 > 109 h−1 M& as a function of scale. A dashed black line again shows the combined power spectrum derived from the
smoothed power spectra of the four simulations employed in this chapter, each of which is shown
as well. The halo contribution rapidly rises down to λ ∼ 2 h−1 Mpc, peaking at ∼ 95% for
k ≈ 20 h Mpc−1 (λ ≈ 300 h−1 kpc) and remaining roughly constant for larger k. The power
spectrum on smaller scales is dominated by increasingly smaller haloes, while the power spectrum on the largest scales depends mainly on the total mass fraction. The grey dashed line shows
the result if R200,crit is used instead of R200 .
within the same halo dominate the power spectrum (i.e. the 1-halo term in halo
model terminology), so naturally the contribution to the power will be underestimated on scales λ ! R200,min , where R200,min is the virial radius of a halo
with the minimum resolved mass, Mmin, in that particular simulation. In practice, the power is significantly underestimated already on larger scales, due to the
gravitational softening employed in the simulation, which leads to an underestimation of the inner density of haloes. This leads to a significant power deficit
on scales λ ! 100 +max (see Table 5.1). Fortunately, the combination of simulations chosen here still allows us to probe the contribution of halo matter up to
kmax ∼ 100 h Mpc−1 .
As the power on sub-Mpc scales is dominated by the 1-halo term, adding lowermass haloes than those resolved here will have a negligible impact on the measured
contribution of halo matter on the scales considered here, as 2π/R200,min > kmax .
Therefore, 5 − 7% of small-scale power is unaccounted for by halo particles, regardless of resolution effects. Instead, it is the cross-term between halo matter
and matter just outside the R200 regions that makes up the deficit.
108
5.3.2 Halo contribution to the power spectrum
Figure 5.5: As Figure 5.4, but now for all mass inside FoF groups with MFoF > 109 h−1 M& .
While the scale dependence is very similar (i.e. a rapid rise down to λ ∼ 1 h−1 Mpc and roughly
constant on smaller scales), the contribution to the power spectrum is higher than for the R200
overdensity regions on any scale. The contribution of halo matter power spectrum is increased
by 5 − 10% on all scales relative to the results of Figure 5.4, and matter in FoF groups accounts
for essentially all power on scales k > 3 h Mpc−1 . This implies that the R200 overdensity regions
do not fully capture the halo.
To demonstrate that this is indeed the case, we calculate the contribution of
matter in FoF groups (with a linking length of 0.2) to the total power spectrum,
with a mass limit of MFoF > 109 h−1 M( . The results are shown in Figure 5.5.
Here we see that, while the scale-dependence of the contribution is similar to
that shown in Figure 5.4 for the R200 regions, the contribution is significantly
larger on all scales, and is essentially 100% for k " 3 h Mpc−1 . This Fourier scale
corresponds well to the virial radius of the largest clusters in the simulation.
On scales k ! 0.3 h Mpc−1 , the contribution of matter in FoF groups is consistently ∼ 25% higher than that of matter in R200 haloes. Correspondingly, the
fraction of mass in FoF groups is also higher than the fraction of mass in R200
haloes at every mass, from 10% higher at M = 109 h−1 M( to 15% higher at
M = 1014 h−1 M( (not shown).
Finally, we also show the results if R200,crit is used instead (with M200,crit >
109 h−1 M( ), as a dashed grey line in Figure 5.4. As such an overdensity criterion
picks out smaller regions than R200 , containing less mass, the contribution of halo
matter to the power spectrum is also smaller, especially on large scales. On subMpc scales, however, the differences are small, with the contribution to the power
109
Halo matter and the power spectrum
spectrum of halo matter peaking at 94%.
We conclude that what region is chosen to represent a halo has a large impact
on the contribution of haloes to the matter power spectrum, in a scale-dependent
way. In what follows, we will continue to define haloes using the mean overdensity
criterion, as this is typically used in the halo model approach.
5.3.2.1
Contribution as a function of mass
To see which haloes contribute most to the matter power spectrum as a function of scale, while simultaneously examining the dependence of our results on
the mass of the lowest resolved halo, we turn to Figure 5.6. Each panel corresponds to a different simulation and each curve to a different minimum halo mass.
The halo contributions are shown relative to the combined power spectrum of
all matter (black line in Figure 5.3). The legend shows the minimum halo mass
log10 (Mmin /[M( /h]). Note that the minimum masses differ for each simulation,
because these are based on the particle masses of the simulations, in such a way
that the first fractional contribution to the power shown includes matter in haloes
of 100 particles or more, corresponding to our imposed resolution limit in mass
(see Figure 5.1). The minimum halo mass increases by half a dex with each step.
Grey regions indicate the approximate scales on which the full matter power spectrum of the simulation is not converged to ∼ 1% with respect to the combined
one. While this gives an indication of which scales to trust, note that the relative
contribution of each halo mass may to be converged for a different range of scales.
Finally, the bottom half of each panel shows the difference between consecutive
lines, i.e. the contribution added by decreasing the minimum halo mass by half a
dex. Here f∆,i ≡ ∆2200,i /∆2all . As we will show shortly, while the relative contributions of haloes of a certain mass shown in the bottom halves of the panels can be
compared between different simulations, the same does not hold for the absolute
contributions, as box size effects play an important role on a large range of scales.
Several things can be learned from this figure. First, we can ask whether
we are resolved with respect to the minimum mass, and on which scales. Lowmass haloes become increasingly important towards larger values of k on sub-Mpc
scales. As we discussed before, we therefore do not expect to be converged on
scales k " 2π/R200,min , limiting us to k ! 100 h Mpc−1 when all simulations are
combined. Interestingly, on scales k ∼ 10 h Mpc−1 , roughly where the contribution
from halo matter plateaus, the L200 simulation (top right panel) is just about
converged with minimum halo mass. This can be seen most clearly by comparing
the relative halo contribution on this scale at approximately fixed minimum halo
mass between different simulations: as for example the L050 simulation (bottom
left panel) shows, well-resolved haloes below ∼ 1011 h−1 M( (about a factor of
two above the halo resolution limit of L200 ) provide a negligible contribution
(< 1%) at k ≈ 10 h Mpc−1 , indicating convergence on this scale. Indeed, both
simulations agree that most of the power at k ∼ 10 h Mpc−1 comes from haloes
with masses M200 " 1013 h−1 M( . These group and cluster-scale haloes remain
110
5.3.2 Halo contribution to the power spectrum
Figure 5.6: Comparison of the contribution of haloes above a certain mass to the matter power
spectrum relative to the total combined power spectrum of all simulations, for L400 (top left),
L200 (top right), L050 (bottom left) and L025 (bottom right). The legend shows the minimum
halo mass log10 (Mmin /[M& /h]). Note that lines of the same colour do not correspond to the
same minimum halo mass in the four panels, as the binning is based on the minimum resolved
halo mass (see text). The grey regions denote where box size or resolution effects are " 1%
for the full power spectrum; the relative contribution may be converged on a different range,
as can be seen by comparing the panels. The bottom half of each panel shows the difference
between consecutive lines, i.e. the contribution added by decreasing the minimum halo mass
by half a dex. Note that scales exist where we are converged with minimum halo mass: for
example, around k = 4 h Mpc−1 haloes with M200 < 1011 h−1 M& contribute negligibly to the
total matter power spectrum. For 1 < k < 10 h Mpc−1 , the power spectrum is dominated by the
contributions of haloes with M200 > 1013 h−1 M& , even though these account for only about
19% of the total mass. On large scales, haloes in the full mass range probed here contribute
significantly to the power, and no convergence is obtained. Note that box size effects strongly
influence the contributions of large haloes measured for the smallest two boxes, especially in the
case of L025.
111
Halo matter and the power spectrum
the dominant contributors on somewhat larger scales as well, their contribution
peaking around k = 2 − 3 h Mpc−1 before gradually falling off. Note that haloes
with M200 > 1013 h−1 M( account for only about 19% of the total mass (see
Figure 5.1).
On larger scales, convergence is obtained for increasingly larger minimum halo
masses: as the bottom halves of the panels show, around k = 4 h Mpc−1 haloes
with masses M200 < 1011 h−1 M( do not make a significant contribution to the
power spectrum. While the simulations shown here do not predict exactly the
same contribution to the power spectrum for haloes with M200 > 1011 h−1 M( ,
due to the limited box size of L050 and L025, it is clear that the results for the
relevant importance of such haloes are converged.
However, for k < 1 h Mpc−1 the contributions of haloes which are not well
resolved by L400 and L200 once again become important, and even L050 may
no be longer converged with minimum halo mass at the 1% level. This is because
on these scales the matter power spectrum is dominated by the cross-correlation
of matter in different haloes (i.e. the 2-halo term in halo model terminology).
Therefore, while the bottom panels for each simulation do show that the relative
contribution of haloes on small enough scales decreases with decreasing M200 , on
scales k ! 1 h Mpc−1 our results can only provide a minimum contribution of halo
matter to the total matter power spectrum.
As we noted before, the panels show that even when compared at roughly
the same minimum halo mass, different simulations make different predictions for
the contributions of haloes above a certain mass to the matter power spectrum.
When the box size decreases, the contribution on large scales and that of highmass haloes decreases as well. This is expected, as large-scale modes are missed
in the smaller boxes and massive haloes are under-represented. However, the role
of low-mass haloes is simultaneously overestimated in the smaller boxes, and the
total contribution of halo matter at fixed minimum halo mass is therefore higher
than it should be.
To demonstrate explicitly that this is the case, we show in Figure 5.7 again the
results for L200 (top-right panel in Figure 5.6), but with the results for L100 superimposed as dashed lines. The L100 simulation has 5123 particles and therefore the
same resolution as L200, but in an 8× smaller volume. Comparing the two simulations therefore shows the effects of box size at fixed resolution. On large scales and
for high-mass haloes, the contribution of halo matter is underestimated in L100,
relative to L200. Meanwhile, the contribution of low-mass haloes on small scales
tends to be overestimated, even though the resolution is identical. Interestingly,
there are mass and spatial scales where the simulations are in perfect agreement,
such as for a minimum halo mass of 1013.75 h−1 M( where k > 3 h Mpc−1 . Most
important, however, is that the contributions at a certain halo mass shown in the
bottom half of the panel are in perfect agreement over the entire range of scales,
excepting the very highest mass bin (which is under-represented in L100 ) and the
principal modes. This shows that we can still derive the correct contribution of
haloes within a certain mass range, and investigate whether we are converged with
112
5.4 Summary & conclusions
Figure 5.7: As the top-right panel of Figure 5.6, but with the results for L100 added for the
same minimum halo masses as dashed lines, showing the effects of box size at fixed resolution.
Due to the missing large-scale modes in L100, the large-scale contribution is underestimated.
Additionally, high-mass haloes are under-represented and the role of low-mass haloes on small
scales is overestimated. However, the relative contributions of haloes of a certain mass shown in
the bottom half of the figure are in excellent agreement for all but the highest mass bin.
mass on a certain scale, even when box size effects play a role.
5.4 Summary & conclusions
In this work we investigated the contribution of haloes to the matter power spectrum as a function of both scale and halo mass. This was motivated by the assumption typically made in halo-based models that all matter resides in spherical
haloes of size R200 . To do so, we combined a set of cosmological N-body simulations to calculate the contributions of different spherical overdensity regions, FoF
groups and matter outside haloes to the power spectrum.
Our findings can be summarised as follows:
• On scales k < 1 h Mpc−1 , haloes – defined as spherical regions with an enclosed overdensity of 200 times the mean matter density in the Universe –
with masses M200 ! 109.5 h−1 M( , which are not resolved here, may signif113
Halo matter and the power spectrum
icantly contribute to the matter power spectrum. For 2 < k < 60 h Mpc−1 ,
our simulations suggest their contribution to be < 1%.
• For k " 2 h Mpc−1 , the minimum mass of haloes that contribute significantly
to the matter power spectrum decreases towards smaller scales, with more
massive haloes becoming increasingly less important.
• On scales k < 2 h Mpc−1 , matter in haloes accounts for less than 85% of the
power in our simulations. Its relative contribution increases with increasing
Fourier scale, peaking at ∼ 95% around k = 20 h Mpc−1 . On smaller scales,
its contribution is roughly constant. When R200,crit is used to define haloes
instead of the fiducial R200,mean , the contribution of haloes to the power
spectrum decreases significantly on all scales.
• For 2 ! k ! 10 h Mpc−1 , haloes below ∼ 1011 h−1 M( provide a negligible
contribution to the power spectrum. The dominant contribution on these
scales is provided by haloes with masses M200 " 1013.5 h−1 M( , even though
such haloes account for only ∼ 13% of the total mass.
• Matter just outside the R200 overdensity regions, but identified as part of FoF
groups, provides an important contribution to the power spectrum. Taken
together, matter in FoF groups with MFoF > 109 h−1 M( accounts for essentially all power for 3 < k < 100 h Mpc−1 . Switching from R200 to FoF
haloes increases the contribution of halo matter on any scale probed here by
5 − 15%.
As we have demonstrated, the halo model assumption that all matter resides in
(spherical overdensity) haloes may have significant consequences for the predictions
of the matter power spectrum. Specifically, we expect such an approach to overestimate the contribution of haloes to the power on small scales (k " 1 h Mpc−1 ),
mainly because it ignores the contribution of matter just outside R200 to the
power spectrum. While defining haloes to be larger regions similar to FoF groups
mitigates the small-scale power deficits, the fact that such regions are often nonvirialised and typically non-spherical may lead to other problems.
Clearly, the validity of the postulate that the clustering of matter is fully determined by matter in haloes is strongly dependent on the definition of a halo used –
but it is hard to say what the “best” definition to use in this context is. For example, while haloes defined through R200,crit will be more compact and therefore have
a smaller overlap fraction than R200,mean or FoF haloes, their contribution to the
power spectrum will be smaller for the same minimum halo mass. And while FoF
groups seem to contain nearly all mass important for clustering, the fact that they
are not completely virialised, may be non-spherical and have boundaries which not
necessarily correspond to some fixed overdensity (e.g. More et al., 2011) prohibit
their use in traditional halo based models.
Some optimal choice of halo definition may exist, but whatever definition one
uses, it remains difficult to say what the contribution to the matter power spec114
5.4 Summary & conclusions
trum of haloes below the resolution limit is. Convergence with mass is extremely
slow over a large range of scales, and as the results of §5.3.1 show, apparent convergence over a decade in mass is no guarantee, as successive decades in mass may
contribute equally. This makes it extremely difficult to give a definitive answer
on whether mass outside haloes significantly contributes to the power spectrum,
or even if such mass exists: for example, we cannot exclude the possibility that
matter outside R200 but inside FoF groups itself consists purely of very small R200
haloes. Therefore, any claims about the role of haloes or the mass contained in
them needs to be quoted together with a minimum halo mass in order to have a
meaningful interpretation.
Acknowledgements
The authors thank Simon White for useful discussions. The simulations presented
here were run on the Cosmology Machine at the Institute for Computational
Cosmology in Durham (which is part of the DiRAC Facility jointly funded by
STFC, the Large Facilities Capital Fund of BIS, and Durham University) as part
of the Virgo Consortium research programme. We also gratefully acknowledge
support from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreement 278594GasAroundGalaxies.
115
6
The galaxy correlation function as a
constraint on galaxy formation physics
Semi-analytical models of galaxy formation are generally successful in reproducing
the number densities of galaxies as a function of mass. In order to remove possible
degeneracies and improve the model, having additional orthogonal constraints like
clustering data while exploring parameter space would be useful. However, this
is challenging due to the two-point nature of such quantities, which makes using
them as a constraint computationally very expensive, as the model would have to
be run on the full halo catalogue at every step. Here, we present a fast estimator
for the projected galaxy correlation function that produces ∼ 10% accurate results
using only a very small subsample of haloes. As a first application, we incorporate
it in a recent version of the Munich semi-analytical model and find a set of galaxy
formation parameters that simultaneously reproduces the observed z = 0 stellar
mass function and clustering data from SDSS.
Marcel P. van Daalen, Bruno M. B. Henriques,
Raul E. Angulo and Simon D. M. White
In preparation
Constraining galaxy formation through clustering
6.1 Introduction
Galaxy formation is currently an unsolved problem. Because of this, any model of
galaxy formation – be it hydrodynamical, analytical or semi-analytical in nature
– has to rely on some set of observations in order to constrain the parameters of
the physical processes that cannot be derived from first principles, or be simulated
directly.
Hydrodynamical simulations can simulate baryonic processes directly on large
scales while relying on sub-grid recipes to model relevant processes below the resolution limit. As such simulations are relatively expensive computationally, the
values of the parameters in the sub-grid formulations usually have to be informed
by comparing a set of simulations run at lower resolution or in smaller volumes to
some observational quantity, though these numerical settings themselves may impact which parameter values are “right” for it. Still, as the available computational
resources are ever growing, the number of processes which cannot be simulated
directly is slowly decreasing (e.g. Hopkins et al., 2013), and valiant efforts are
currently being made to improve the accuracy of direct cosmological simulations
(e.g. EAGLE, Schaye et al., in preparation).
Semi-analytical models (hereafter SAMs), on the other hand, necessarily include more physical parameters to calibrate, as baryonic processes are not simulated directly on any scale. However, once the high-resolution collisionless simulations that they are based on have been run a single time, they can be repeated
many times with different parameter values at low computational cost. Coupled
with a method to efficiently explore parameter space such as Monte Carlo Markov
Chains (MCMC, for a review on this and similar methods see Trotta, 2008), this
allows one to find the highest-likelihood set of parameters for any given model,
based on a set of observational constraints.
Typically, SAMs use observational data sets of one-point functions, such as stellar mass or luminosity functions, as constraints for their model parameters (e.g.
Kauffmann, White & Guiderdoni 1993; Baugh, Cole & Frenk 1996; Somerville &
Primack 1998; Kauffmann et al. 1999; Cole et al. 2000; Croton et al. 2006; Bower
et al. 2006; Monaco, Fontanot & Taffoni 2007; Somerville et al. 2008; Henriques
et al. 2009; Guo et al. 2011; Henriques et al. 2013, see Baugh 2006 for a review on
the general methodology). The resulting models of galaxy formation can then be
tested against other observables (i.e. observables that are independent of those used
as constraints) and be used to make predictions for these. A delicate balance must
be maintained here: if the model has too many free parameters, prior regions that
are too wide, or if there are too few (independent) observational constraints, degeneracies may occur (i.e. separate regions of high likelihood in parameter space),
while too little freedom or failing to include some relevant physical process may
leave the model unable to match several observables at once.
SAMs generally have trouble matching the small-scale clustering of galaxies
while simultaneously matching other observational constraints such as the luminosity function (e.g. Kauffmann et al. 1999; Springel et al. 2005; Li et al. 2007; Guo
118
6.2 Method
et al. 2011; Kang et al. 2012; but see e.g. Kang 2014). In order to determine the
cause of this discrepancy, and to test whether the models retain enough freedom to
match the observed clustering at all, it would be instructive to use clustering measurements as constraints while exploring parameter space. As galaxy clustering is
determined by how galaxies with different properties populate haloes of different
mass, it directly constrains galaxy formation, in a way that is complementary to,
for example, the luminosity function.
However, this presents a problem: while one-point functions such as the stellar
mass function can be quickly estimated with known uncertainty by running the
model on only a small sample of representative haloes, allowing large regions of
parameter space to be rejected without having to run the model on the full dark
matter simulation, the same cannot be done simply for two-point functions such
as the correlation function. In principle, any observable that relies on spatial
correlations between galaxies can only be calculated by running the model on the
full simulation, which is computationally infeasible when thousands of models need
to be explored. While running the SAM on a small sub-volume may allow one to
measure small-scale correlations to some degree, cosmic variance will be an issue.
Additionally, if one aims to compare to observations, where clustering is viewed
in projection (unless line-of-sight velocities are used), one still has to account for
large-scale correlations, even at small separations.
Here, we present a method to quickly estimate the projected correlation function, w(rp ), to some known uncertainty from a small sample of haloes using a
halo model based approach, and apply it to constrain the recent version of the
Munich semi-analytical model presented in Guo et al. (2013, , hereafter G13). By
measuring the properties of galaxies within individual haloes and making informed
assumptions about the distribution of these haloes, we are able to circumvent the
aforementioned problems, greatly reducing the CPU time needed to predict their
two-point clustering.
This chapter is organised as follows. In Section 6.2, we present our method
for estimating w(rp ) and briefly describe the semi-analytical model we apply it
to. Next, in Section 6.3, we show the results of using clustering as an additional
constraint on parameter space, on top of the often-used z = 0 stellar mass function. Finally, in Section 6.4 we present a summary of our work and discuss future
improvements and applications.
6.2 Method
6.2.1
Estimating the correlation function
Our approach is slightly different to that of most previous works constructing a
correlation function estimator based on the halo model, where the aim is typically
to reproduce observations given some halo occupation distribution (HOD). Here,
our goal is instead to reproduce the results of the semi-analytical model run on
the full dark matter simulation to within some given accuracy, given the galaxy
119
Constraining galaxy formation through clustering
properties for a small sample of haloes. As we will show, we are able to reproduce
the projected correlation function of the full galaxy sample to within about 20%,
using the properties of semi-analytical galaxies occupying less than 0.04% of the
full halo sample (0.14% of the subhalo sample).
6.2.1.1
The backbone of the model
Our starting point is the linear halo model, introduced independently by Seljak
(2000), Ma & Fry (2000) and Peacock & Smith (2000). In what follows, we will
adhere to the terminology of Cooray & Sheth (2002). In the analytical halo model
the power spectrum, P (k), is written as the sum of two terms:
P (k) = P 1h (k) + P 2h (k).
(6.1)
Here P 1h (k) is the 1-halo term, describing the two-point clustering contribution
of points within the same halo, and P 2h (k) is the 2-halo term, describing the
contribution of points within separate haloes. For the clustering of matter, these
are given by:
. /2
M
1h
Pdm (k) =
n(M )
|u(k|M )|2 dM
ρ̄
.
.
/
/
-M1
M2
2h
Pdm (k) =
n(M1 )
u(k|M1 )n(M2 )
u(k|M2 ) ×
ρ̄
ρ̄
Phh (k|M1 , M2 )dM1 dM2 .
(6.2)
Here M = M200mean is the halo mass definition1 we will be using throughout,
n(M ) is the halo mass function, ρ̄ is the mean matter density of the Universe,
u(k|M ) is the normalised Fourier transform of the density profile of a halo of mass
M , and Phh (k|M1 , M2 ) is the halo-halo power contributed by two haloes of masses
M1 and M2 on a Fourier scale k. We can rewrite the latter term assuming a linear
scale-independent bias relation, Phh (k|M1 , M2 ) = b(M1 )b(M2 )Plin (k), where b(M )
is the halo bias and Plin the linear theory matter power spectrum. We then obtain:
0. /
12
M
2h
Pdm
(k) = Plin (k)
n(M )b(M )
(6.3)
u(k|M )dM .
ρ̄
From these expressions, one can easily derive a model for the galaxy power spectrum. For this we assume that the number of galaxies scales with the halo mass
M ; specifically, M ∝ ,Ngal |M - and M 2 ∝ ,Ngal (Ngal − 1)|M -, leading to:
,Ngal (Ngal − 1)|M 1h
Pgal (k) =
n(M )
|ugal (k|M )|p dM
n̄2gal
012
,Ngal |M 2h
Pgal (k) = Plin (k)
n(M )b(M )
ugal (k|M )dM .
(6.4)
n̄gal
1M
200mean is the mass within a spherical region with radius R200mean and internal density
200 × ρ̄ = 200 × Ωm ρcrit .
120
6.2.1 Estimating the correlation function
2
Here the mean number density of galaxies is given by ngal = n(M ) ,Ngal |M - dM .
Note that we have followed Cooray & Sheth (2002) in replacing the normalised
Fourier transform of the halo density profile, u(k|M ), by one describing the distribution of (satellite) galaxies, ugal (k|M ), and subsequently in changing the powerlaw index on this term in the 1-halo term by p. This is often done in the literature
in order to be able to differentiate between contributions from central-satellite and
satellite-satellite terms, with p = 1 for the former and p = 2 for the latter, based
on the value of ,Ngal (Ngal − 1)-. ,Ngal |M - – the most common form of the HOD –
is often separated into contributions from centrals and satellites as well, with the
former (Ncen ) following a roughly lognormal distribution with respect to M , and
the latter (Nsat ) being very well approximated by a (linear) power law (e.g. Guzik
& Seljak, 2002; Kravtsov et al., 2004; Zehavi et al., 2005; Tinker et al., 2005; Zheng
et al., 2005). From this approximate expressions for ,Ngal (Ngal − 1)- in terms of
Ncen and Nsat can be derived as well.
However, as our aim is to reproduce the results of the semi-analytical model,
for which information on the HOD and the galaxy type is much more readily
available than for observations, we can explicitly separate the contributions from
central and satellite galaxies to the galaxy power spectrum without approximation. Keeping in mind that a halo will contain at most one central, meaning that
,Ncen (Ncen − 1)|M - = 0, that ,Ncen Nsat |M - = ,Nsat Ncen |M -, and using that central galaxies reside in the centre of the halo and should therefore not be weighted
by the profile, we derive:
,Ncen Nsat |M 1h
Pgal
(k) = 2 n(M )
[ugal (k|M ) − W (kR)] dM +
n̄2gal
4
,Nsat (Nsat − 1)|M - 3
n(M )
|ugal (k|M )|2 − W (kR)2 dM
2
n̄gal
0,Ncen|M 2h
Pgal
(k) = Plin (k)
n(M )b(M )
dM +
n̄gal
12
,Nsat |M n(M )b(M )
ugal (k|M )dM .
(6.5)
n̄gal
Note that we have followed Valageas & Nishimichi (2011) in adding a counterterm
to the halo profiles in the 1-halo term, which ensures the 1-halo term goes to zero
for k → 0. Here W (kR) is the Fourier transform of a spherical top-hat of radius
R(M ) = [3M/(4π ρ̄)]1/3 , given by:
.
/
sin(kR) cos(kR)
−
W (kR) = 3
.
(6.6)
(kR)3
(kR)2
In our model, we take Plin (k) to be the realised linear input power spectrum
from the dark matter initial conditions. We calculate the halo mass function,
n(M ), directly from the dark matter simulation as well and spline-fit the results.
Furthermore, we use the fit for the M200mean halo bias function provided by Tinker
121
Constraining galaxy formation through clustering
et al. (2010) for b(M ), and compute each of the four HOD terms directly from the
SAM run on our halo subsample, spline-fitting these results as well.
6.2.1.2
The galaxy distribution
The normalised Fourier transform of the galaxy distribution, ugal (k|M ), is often
derived from the dark matter mass profile of the halo. This in turn is usually
assumed to be equal to the Navarro, Frenk & White (1997, , NFW) profile, cut off
at the virial radius rvir = R200mean , with some concentration-mass relation c(M ):
ρNFW (r) =
ρ0
,
(r/rs )(1 + r/rs )2
(6.7)
where rs = rvir /c is the scale radius. The main advantage of using the oneparameter NFW profile is that this leads to an analytic expression for u(k|M ).
However, many authors have shown that the Einasto (1965) profile provides a more
accurate fit to the mean profile of haloes of a given mass, and to the distribution
of dark matter substructure (e.g. Navarro et al., 2004; Merritt et al., 2005, 2006;
Gao et al., 2008; Springel et al., 2008; Stadel et al., 2009; Navarro et al., 2010;
Reed, Koushiappas & Gao, 2011; Dutton & Macciò, 2014). The two-parameter
Einasto density profile is given by:
$
15
0. /α
r
2
ρEin (r) = ρ0 exp −
−1 ,
(6.8)
α
rs
where the shape parameter α allows additional freedom in the slope of the profile.
This function does not have an analytic Fourier transform, and an extra numerical
integration step is therefore needed when replacing the NFW profile by an Einasto
one. The larger degeneracies in fitting a two-parameter model also mean more data
points are needed to obtain a reliable fit. Still, when the computational expense
is acceptable and enough information on the measured profile is available, the
increased accuracy may be worth the cost.
We find that the Einasto profile provides an excellent fit to the distribution
of satellite galaxies in the inner parts of haloes in our simulation. But even the
Einasto profile over-predicts the number of galaxies at large radii, r " 0.7rvir .
Additionally, standard practice is to cut off the profile at the virial radius, while
we find that ∼ 10% of the satellite galaxies in our simulation are found at distances
1 < r/rvir < 3. Note that these galaxies are not necessarily outside the virialised
region, as haloes are typically not spherical objects. We therefore seek a profile
with the same small-scale behaviour as the Einasto profile, while simultaneously
fitting the galaxy distribution out to ∼ 3rvir .
We find that the following functional form, which we refer to here as the
“gamma” profile, is capable of providing an excellent match to the galaxy distribution over the full range of scales we consider, and at any halo mass:
' r (ac−3
6 ' r (c 7
ng (r) = n0
.
(6.9)
exp −
b
b
122
6.2.1 Estimating the correlation function
Figure 6.1: Galaxy number density profiles for all Guo et al. (2011) galaxies with stellar masses
10.27 < log10 (M∗ /M& ) < 10.77, for five different halo mass bins (shown in different colours).
The legend shows the mean logarithmic mass in each of the bins. Solid lines indicate the measured
profiles, while dashed lines show the best-fit gamma profiles (see equation 6.12). The halo mass
bins are dynamically chosen such that each contains roughly the same number of galaxies, and
the fits are performed using 30 radial bins spaced equally in log-space between log10 x = −2.5
and log10 x = 0.5.
This fitting function has three parameters, a, b and c. Note that the role of b is
similar to that of rs in the Einasto profile. Both the Einasto and gamma profiles
are near universal if defined in terms of x ≡ r/rvir . If we rewrite both profiles in
terms of x and integrate them to obtain N (< r), the similarities and differences
between the profiles are most easily appreciated. For the Einasto profile:
' (α 9
8
γ α3 , α2 rxs
'
(α 9 ,
NEin (< r) = Ntot 8
(6.10)
γ α3 , α2 xmax
rs
while for the gamma profile:
3 + ,c 4
γ a, xb
Ng (< r) = Ntot 3 + xmax ,c 4 .
γ a, b
(6.11)
Here γ(a, b) is the lower incomplete gamma function, and we have assumed the
profiles cut off at some xmax . The similarities in the two profiles are clear, and the
main difference is that the two parameters of the gamma function are independent
for the gamma profile, which effectively allows for a steeper profile at large x and
consequently a better match to the galaxy distribution around the virial radius.
In practice, we fit a normalised number density profile ng (r)/ ,Ng - to the galaxy
distribution before numerically Fourier transforming this to obtain ugal (k|M ). For
123
Constraining galaxy formation through clustering
completeness, ng (r)/ ,Ng - is given by:
6 ' x (c 7
' x (ac−3
ng (r)
c
3
+
,
4
.
=
exp
−
c
3 γ a, xmax
,Ng b
b
4πb3 rvir
b
(6.12)
In our model we set xmax = 3, as > 99.9% of satellites in our fiducial model are
found inside this radius. Even for small halo samples, the three parameters of the
fit are independent enough to ensure degeneracies are not a problem. An example
is given in Figure 6.1, where we show the best-fit model for all galaxies with stellar
masses 10.27 < log10 (M∗ /M( ) < 10.77 in the Guo et al. (2011) semi-analytical
model, for five different halo mass bins. The solid lines show the measured number
density profiles, while the dashed lines show the best-fit gamma profiles. The halo
mass bins are dynamically chosen inside the code such that each contains roughly
the same number of galaxies. We use 30 radial bins spaced equally in log-space
between log10 x = −2.5 and log10 x = 0.5, and fit an Akima spline through each of
the three parameters as a function of halo mass to obtain smooth functions that
are stable to outliers.
6.2.1.3
Correction for non-sphericity
As is common, we have assumed a spherical distribution of satellite galaxies around
each central. In reality, haloes and consequently their galaxy populations are
triaxial. van Daalen, Angulo & White (2012) investigated the effect of assuming
a spherical distribution on the two-point correlation function and galaxy power
spectrum, and found that the effects can be quite large, with the true power being
underestimated by 1% around k = 0.2 h Mpc−1 to 10% around k = 25 h Mpc−1 ,
increasing even more towards smaller scales (see the right panel of their Figure 3,
or Figure 4.3 in Chapter 4 of this thesis). We have repeated their analysis and
found that the functional shape of this underestimation of the power appears to
be completely independent of the mass of the galaxies. We therefore fit a function
e(k) through these results and use this to correct our halo model power spectra
for the combined effects of non-sphericity. The final galaxy power spectrum that
comes out of our model for a given set of galaxies is therefore:
1h
2h
Pgal = [Pgal
(k) + Pgal
(k)]/[1 + e(k)],
(6.13)
1h
2h
with Pgal
(k) and Pgal
(k) given by equation (6.5).
6.2.1.4
Converting to the projected correlation function
To obtain the projected correlation function from the galaxy power spectrum,
we numerically perform two standard transformations. First, to obtain the 3D
correlation function:
- ∞
1
sin kr
ξ(r) =
dk,
(6.14)
k 2 P (k)
2π 2 0
kr
124
6.2.1 Estimating the correlation function
Figure 6.2: The FoF halo mass function, showing number of haloes available in the Millennium
Simulation at z = 0 (black) and the number randomly selected as a function of M200mean in each
subsample (red). The subsamples each comprise of less than 0.04% of the total halo sample, or
0.14% of the total subhalo sample. The selection function was built iteratively by demanding
that ∼ 90% of the random samples it generated lead to projected correlation functions that were
within 30% of the full sample prediction. Low-mass haloes were favoured over high-mass haloes
in order to suppress the size of the trees used in the SAM. Even so, the fraction of FoF groups
needed to match the correlation function within some uncertainty at any stellar mass is higher
for more massive haloes.
and, finally, to obtain the projected galaxy correlation function:
- ∞
- ∞ ':
(
rξ(r)
:
w(rp ) = 2
ξ
rp2 + π 2 dπ = 2
dr.
0
rp
r2 − rp2
(6.15)
Here rp and π are the projected and line-of-sight separation, respectively. It is
in this last step that we also convert the units from Mpc/h to Mpc, in order to
directly compare our model w(rp ) to that of observations.
6.2.1.5
Selection function
The selection function we use to create the halo sample the SAM is repeatedly
run on while exploring parameter space was built through use of the following
algorithm.
At each step, the algorithm adds some number of Friends-of-Friends (FoF)
groups to each halo mass bin in turn, and generates a number of random samples
for each of the resulting selection functions. The correlation functions predicted
using these samples are then compared to determine which mass bin would contribute to the largest reduction in the variance with respect to the full model run
125
Constraining galaxy formation through clustering
Figure 6.3: The fractional difference between our model prediction of the projected galaxy
correlation function and a direct calculation, for galaxies in the Guo et al. (2011) semi-analytical
model. Here we use the full galaxy sample as an input to our model. Results are shown for six
different stellar mass bins, indicated by lines of different colours, over the range where SDSS/DR7
data is available for each. The overall agreement is within 20%, with the model tending to overpredict the clustering on sub-Mpc scales. This can be traced to an overestimation of the power
in the 1-halo term by a similar amount around k = 1 h Mpc−1 . For our application, our model
performs well enough, and we leave improvements to future work.
for all six stellar mass bins. If at any step adding more haloes does not reduce
the variance for any halo mass, FoF groups are added to a random bin. This
continues until at least 90% of the random samples the current selection function
generates lead to projected correlation functions that are within 30% of the full
sample prediction.
In order to suppress the size of the merger trees used in the SAM, low-mass
haloes were favoured over high-mass haloes by weighting the number of FoF groups
added to each mass bin by the inverse of the average number of subhaloes hosted by
FoF groups of that mass. Nonetheless, the fraction of haloes selected at high mass
is still higher than at low mass, since more massive haloes potentially contribute
more galaxies to the sample, increasing the accuracy of the estimates made in the
clustering model described above. Additionally, the most massive galaxies probed
here, M∗ > 1011.27 M( , preferentially occupy the most massive haloes.
After building several selection functions in this way, we found that on average
they were well approximated by the combination of a constant value and a power
law (rounded to integer values). This is the near-optimal selection function shown
in Figure 6.2 (red line), which takes the constant value Nh = 200 below M200mean =
1012.2 h−1 M( . The subsamples generated by this selection function each comprise
less than 0.04% of the total FoF halo sample, or 0.14% of the total subhalo sample.
126
Figure 6.4: The fractional difference between the predictions of our model for 100 halo subsamples and the model prediction for the full sample. Each
subsample consists of about 0.14% of the total subhalo sample (see text for details). The colours indicate the same stellar mass bins as in Figure 6.3.
Left: The predictions of each of the 100 separate realisations, showing the scatter around the full sample result. Right: Same as the left panel, but
now showing only the median, 16th and 84th percentiles. Even using a small random sample, our model can quickly estimate the projected correlation
function to ∼ 10% precision.
6.2.1 Estimating the correlation function
127
Constraining galaxy formation through clustering
6.2.1.6
Performance of the model
We compare our model prediction of w(rp ), using the full halo sample, to that
calculated directly for the galaxies in the Guo et al. (2011) model in Figure 6.3.
Here we show the relative difference between the two for six different bins in stellar
mass, indicated as ranges in log10 (M∗ /M( ). We only show the results over the
range where we constrain w(rp ) using observations. The model performs well,
and any deviations from the true correlation function are typically within 20%.
The magnitude of the mismatch tends to increase with stellar mass. The largescale disagreement is caused by the model slightly under-predicting the power
in the transition region between the 1-halo and 2-halo terms, while the smallscale offset is mostly due to the 1-halo term in the power spectrum being slightly
overestimated around k = 1 h Mpc−1 . However, overall the agreement is good,
especially considering our relatively simple treatment of e.g. the halo bias (linear
and scale-independent), and we leave further improvements – such as using a halohalo power spectrum measured from the dark matter only simulation instead of a
biased linear power spectrum – to future work.
The true power of the model lies in its ability to reproduce the clustering prediction for the full sample from only a small subsample of FoF groups. In Figure 6.4
we compare the predictions for 100 random subsamples selected according to the
selection function shown in Figure 6.2 to the model prediction for the full sample.
The dotted lines indicate offsets of 30% for reference, and the colours indicate the
same stellar mass bins as in Figure 6.3. The scatter is around 7 − 8% for the
first four mass bins, increasing to 10% and 16% for the fifth and sixth mass bin
respectively. This shows that the model is capable of reproducing the full sample
estimate from relatively few haloes.
6.2.2
The SAM and MCMC
As our estimator is able to quickly and accurately recover the projected correlation
function from a very small subsample of haloes, this makes it ideally suited for
constraining the parameter space of semi-analytical models using the projected
correlation function. In this work we present a first application, where we constrain
the model of G13, a recent version of the Munich semi-analytical code, using
both the galaxy stellar mass function (SMF) and the projected galaxy correlation
function. For this we utilise the same data sets as presented in G13. As we
will only utilise the Millennium Simulation, and not Millennium II, we only use
constraints above M∗ > 109 h−1 M( .
The G13 model includes 17 parameters which together determine the outcome
of galaxy formation. These are (see Table 6.1): the star formation efficiency (αSF );
the star formation criterion (M̃crit , or equivalently Σcrit ); the star formation efficiency in the burst phase following a merger (αSF,burst ); the slope on the merger
mass ratio determining the stellar mass formed in the burst (βSF,burst ); the AGN
radio mode efficiency (kAGN ); the black hole growth efficiency (fBH ); the typical halo virial velocity of the black hole growth process (VBH ); three parameters
128
6.2.2 The SAM and MCMC
Parameter
Description
αSF
M̃crit
αSF,burst
βSF,burst
kAGN
fBH
VBH
+
Vreheat
β1
η
Veject
β2
γ
y
Rmerger
αfriction
Star formation efficiency
Star formation threshold
Star formation burst mode efficiency
Star formation burst mode slope
Radio feedback efficiency
Black hole growth efficiency
Quasar growth scale
SN mass-loading efficiency
Mass-loading scale
Mass-loading slope
SN ejection efficiency
SN ejection scale
SN ejection slope
Ejecta reincorporation scale factor
Metal yield fraction
Major-merger threshold ratio
Dynamical friction scale factor
Units
–
M( km s−1 Mpc−1
–
–
h−1 M( yr−1
–
km s−1
–
km s−1
–
–
km s−1
–
–
–
–
–
Table 6.1: Parameters varied in the MCMC. The best-fit values (as well as the G13 values for
the WMAP1 cosmology and the prior ranges) are shown in Figure 6.8. For more information we
refer to G13.
governing the reheating and injection of cold disk gas into the hot halo phase by
supernovae, namely the gas reheating efficiency (+), the reheating cut-off velocity
(Vreheat ) and the slope of the reheating dependence on Vvir (β1 ); three parameters
governing the ejection of hot halo gas to an external reservoir, namely the gas
ejection efficiency (η), the ejection cut-off velocity (Veject ) and the slope of the
ejection dependence on Vvir (β2 ); a parameter controlling the gas return time from
the external reservoir to the hot halo (γ); the yield fraction of metals returned
to the gas phase by stars (y); the mass ratio separating major and minor merger
events (Rmerger ); and finally a parameter controlling the dynamical friction time
scale of orphan galaxies, i.e. the time it takes for satellite galaxies of which the
dark matter subhalo is disrupted (or at least no longer detected) to merge with
the central galaxy (αfriction ).
While in the original G13 paper some of these parameters were held fixed, here
we allow all 17 to vary. We start our Monte Carlo Markov Chains (MCMCs) at
the position in parameter space used by Guo et al. (2011), which was arrived at by
using a combination of SMFs, as well as rest-frame B -band and K -band luminosity
functions between z = 0 and z = 3, as observational constraints. We then use the
same techniques as described in G13 to find a new set of best-fit parameters, with
the projected correlation function as an additional constraint.
129
Constraining galaxy formation through clustering
Figure 6.5: The projected galaxy correlation function in six bins of stellar mass. The points
with error bars show the SDSS data in each bin, while the lines show the model results. The
green dotted line shows the results for the original model from G13, in which the parameter
values were set manually. The blue lines show the results of only using the stellar mass function
as a constraint, while the red lines show the results when the model is simultaneously constrained
by the projected correlation function and the stellar mass function. Finally, dashed and dotted
lines are used to indicate whether these are the results for the sample haloes or for all haloes,
respectively. The clustering on small scales of the full model is systematically underestimated
by the sample, which is mostly due to the clustering estimator (see §6.2.1.6). Note that even
though the lowest mass bin is not used as a constraint, the match to observations is markedly
improved with respect to the other models.
As our model is only accurate to within ∼ 10% on small scales, and additionally since the error bars on the SDSS clustering data were derived from Poisson
statistics alone, and so do not include cosmic variance, we artificially increase the
error bars on the data points used during the fitting. Each data point of the observed projected correlation function was assumed to have an uncertainty of 20%.
As noted before, we do not use the clustering data below M∗ = 109.27 M( , nor the
stellar mass function data below M∗ < 109 M( , when constraining the model, as
the haloes hosting these galaxies are not well resolved in the original Millennium
Simulation which we are using as a basis for the SAM. When fitting to the SMF
and clustering data simultaneously, we increase the relative weighting of the fit to
130
6.3 Results
the SMF by a factor of five to compensate for the fact that the clustering data is
measured in five separate bins. This helps avoid sacrificing the excellent fit to the
SMF in favour of matching the correlation function.
Note that while G13 used a WMAP7 cosmology, here we use the original
WMAP1 cosmology to avoid additional complications introduced by scaling to
a different cosmology. In future work the results will be explored for more up-todate cosmologies. Contrary to what is claimed in G13, the change in cosmology
has a negligible impact on the resulting correlation functions, which are far more
sensitive to the SAM’s physical recipes. Besides updating the cosmology, the only
change made from the WMAP1 Guo et al. (2011) model to the newer WMAP7
G13 model is that the type 2 (orphan) satellite galaxy positions are now correctly
updated in the code, meaning that their orbits now decay as intended and can
therefore be disrupted earlier. This change was the main reason for the improved
agreement with clustering data with respect to Guo et al. (2011).
6.3 Results
6.3.1
Comparison with observations
The results of our MCMC chains for the projected correlation function are shown
in Figure 6.5, for six bins in stellar mass, as indicated in the panels. In each
figure, we indicate the original results found by G13, where the galaxy formation
parameters were set by hand, as a green dotted line. The new results are shown
in blue and red; in blue, we show the correlation functions that follow from only
using the stellar mass function as a constraint (“SMF-only”), while in red we show
the results of fitting to the clustering data simultaneously (“SMF+w(rp )”).
The dashed lines show the predictions made based on the sample of haloes
used in the MCMC, as described in §6.2.1. The dotted lines show the true galaxy
correlation function, as calculated directly from the full galaxy catalogue for the
same model parameters. The true values are generally below the ones estimated
from the sample, as expected from the results of §6.2.1.6, and as a consequence the
new results tend to under-predict the amount of clustering on the smallest scales.
Even so, one immediately sees that the SMF+w(rp ) correlation functions (red
lines) generally provide a better fit to the data, bringing the small-scale clustering down considerably in comparison with the original G13 and SMF-only (blue
lines) models, which are very close together. This effect is larger for low stellar
masses, where the clustering discrepancy between the old model and the data was
larger as well. The much improved match to observations indicate that the model
retains enough freedom to match the clustering data. Note that the match to the
projected galaxy correlation function for galaxies in the first mass bin is greatly
improved as well, even though this data is not used to constrain the model. For the
highest-mass galaxies, 11.27 < log10 (M∗ /M( ) < 11.77, all models perform equally
well, while for galaxies with masses above 1010.27 M( the SMF-only correlation
131
Constraining galaxy formation through clustering
Figure 6.6: The stellar mass functions of the models. The green line again refers to the
original G13 model, of which the parameters were set manually. The blue lines show the results
when the MCMC algorithm is used with only the stellar mass function as a constraint, while
the red lines again show the result when clustering constraints are used additionally. When
the SMF is the only constraint, the model clearly has enough freedom to reproduce it to high
precision. However, the match grows somewhat worse at low mass when the model is additionally
constrained by clustering, and is in some places about 2σ away from the combined observational
constraints shown in black. Still, the SMF+w(rp ) model performs better in matching both sets
of constraints simultaneously.
functions perform better on the smallest scales, due to the clustering estimator
overestimating the small-scale clustering.
However, the improved match to the observed clustering data (at least for lowmass galaxies) comes at a price. In Figure 6.6, we show how the models compare to
the SMF data used to constrain the models. The black points with error bars are
derived by combining several observational data sets (see G13). The original G13
model, in which the parameters were set by hand, is again shown as a green dotted
line, which matches the data well. When we use only the SMF as a constraint for
the galaxy formation model, shown in blue, we obtain a marginally better fit to
the data at low mass.
When the projected galaxy correlation function is used as an additional constraint, shown in red, the agreement with the stellar mass function suffers considerably in favour of the clustering predictions. While the agreement for galaxies
with masses M∗ " 1010.5 M( is still comparable to that obtained by G13, the
new model over-predicts the number densities of lower-mass galaxies, although
the results are still within 2σ of the data. Note that the sample results (dashed
lines) agree perfectly with the full catalogue ones (dotted lines) for both SMF-only
132
6.3.1 Comparison with observations
Figure 6.7: Comparison of the galaxy distribution profiles for the SMF-only (solid lines) and
SMF+w(rp ) (dashed lines) best-fit parameters. The different panels show the profiles of galaxies
in the six correlation function mass bins, as indicated in the top right of each panel. As in
Figure 6.1, different colours are used for different halo mass bins, which are set to be the same
for both models to allow for an unbiased comparison. Note that the mass bins do change as a
function of stellar mass in order to make sure each bin in halo mass is roughly equally populated.
For clarity, we show only the fits to the measured profiles (see equation 6.12) here, but stress
that each provides an excellent fit over the full range shown here. Note that the dynamic range
in scales has been extended relative to Figure 6.1 to better appreciate the differences between
the profiles. Mainly because of the reduced dynamical friction time scale in the latter model, the
profiles of galaxies in every mass bin are slightly flatter at any halo mass, reducing the correlation
function on small scales.
133
Constraining galaxy formation through clustering
and SMF+w(rp ), which indicates that the discrepancy observed for the correlation
function is indeed due to the inaccuracy of the estimator at small separations.
The slight mismatch for low-mass galaxies could indicate that the SAM is
missing some physical ingredient needed in order to reproduce observations, but
other viable explanations also exist. For both the clustering and SMF data the
uncertainties may be underestimated; for example, the error bars on the correlation function do not take into account cosmic variance, which could have a quite
significant effect. If the observed correlation functions are biased low because of
this, the clustering in our model may have been brought artificially low, preventing us from matching the SMF simultaneously. Another possible source of errors
could be systematic uncertainties in the observations that lead to samples that
are not volume limited. Additionally, changing the cosmology to one that is more
up-to-date may help. We will explore some of these possibilities in future work.
Note, however, that the SMF+w(rp ) model is in far closer agreement with both
the SMF and the clustering data simultaneously than both the original G13 and
the SMF-only models: while the latter models are in strong disagreement with
the clustering data for low-mass galaxies on small scales, the SMF+w(rp ) model
is generally in agreement with both the low-mass clustering data and the SMF
within 2σ. This shows the merit of using a clustering estimator while exploring
parameter space.
6.3.2
Change in parameters
Even though we vary 17 galaxy formation parameters, by far the largest role in
bringing the clustering predictions in agreement with observations is played by
only two of these: αfriction , which controls the time it takes for satellite galaxies
to merge with the central once their dark matter subhalo has been disrupted, and
γ, which controls the time it takes for ejected gas to re-enter the halo.
The way these parameters influence the clustering and stellar mass function
predictions is as follows. When the clustering data is included as an additional
constraint, the dynamical friction time scale of orphan galaxies decreases by more
than a factor of three with respect to the SMF-only results. This causes galaxies at
small separation scales to merge with their centrals much quicker, flattening the
galaxy distribution profile within the haloes and greatly decreasing the amount
of clustering on small scales, especially for low-mass satellites. This change in
the galaxy distribution profiles from the SMF-only to the SMF+w(rp ) model is
shown in Figure 6.7. The halo mass bins are set to be the same for both models
to allow for an unbiased comparison. Note that the mass bins do change as a
function of stellar mass in order to make sure each bin in halo mass is roughly
equally populated. Although we only show the fits to the measured profiles here,
we stress that each provides an excellent fit to the data, over the full range in scales
shown here. The change in slope of the profiles is relatively small, meaning that
the galaxy distributions are still consistent with SDSS data for rich clusters (see
Figure 14 of Guo et al., 2011). This is because even though the friction time scale
134
Figure 6.8: The preferred parameter values in both models. The best-fit values are shown as dashed vertical lines. The dotted vertical line again
shows the result for the original G13 model, and the grey regions indicate values deemed non-physical, and which are therefore made inaccessible to
the model. For most parameters the best-fit values are consistent between the model where clustering is not used as a constraint and the model where
it is. The largest shifts occur for αSF , M̃crit , y, γ and αfriction . The last two are most important for bringing the clustering data into agreement with
data, while the rest mainly serve to preserve the match to the SMF.
6.3.2 Change in parameters
135
Constraining galaxy formation through clustering
decreases by more than a factor of three when using clustering as an additional
constraint, the number of type 2 galaxies at z = 0 decreases only by a factor 0.87,
as the merging time scale for many of these galaxies is still long compared to the
Hubble time.
Additionally, however, the decrease in the dynamical friction time scale causes
the number density of galaxies above the knee (M∗ > 1010.5 M( ) to decrease as
well. This counter-intuitive change in the SMF comes about because the cold gas
in the merging satellites directly feeds the supermassive black holes in the centres
of the central galaxies, increasing feedback from AGN and thereby the suppression
of star formation.
The γ parameter, on the other hand, increases by more than a factor of five in
SMF+w(rp ) with respect to SMF-only, meaning that the hot gas reincorporation
time scale decreases by the same factor. This raises the number densities of galaxies
at any mass, but most significantly below the knee of the SMF (M∗ < 1010.5 M( ).
The change in γ is the main source of the higher low-mass number densities from
SMF-only to SMF+w(rp ). The upside is that this parameter shift also lowers the
clustering of galaxies, especially for galaxies with masses M∗ > 109.77 M( . While
it may seem counter-intuitive to have the number of galaxies at some mass increase
while their clustering decreases, keep in mind that it is the (normalised) galaxy
distribution within each halo that is driving the clustering prediction, and this
distribution flattens when the aforementioned time scales decrease.
The parameter changes in γ and αfriction alone, with respect to the best-fit
parameters of the SMF-only data, already produce predictions that are very close
to those of the SMF+w(rp ) model. While the decrease in the dynamical friction
and reincorporation time scales each bring the clustering into better agreement
with data separately, a change in both simultaneously is needed as they affect the
SMF in different (adverse) ways.
We show the shift in parameter values in Figure 6.8. We again indicate the
results for all three models: the original G13 model (green dotted lines), the
SMF-only model (blue lines), and the SMF+w(rp ) model (red lines). Histograms
indicate the Bayesian likelihood regions as derived from the full MCMC chains,
while the vertical dashed lines indicate the best-fit values. Both the likelihood
regions and the best-fit values of the SMF-only and SMF+w(rp ) models are generally consistent. The largest exceptions to these are the star formation efficiency
αSF , the cold gas mass star formation threshold M̃crit , the metal yield y, and
the previously mentioned reincorporation scale factor γ and dynamical friction
scale factor αfriction . The latter two cause the main decrease in the clustering
predictions, needed to bring them in agreement with observations. The significant
increase in the star formation efficiency and the decrease in the cold gas mass
threshold for star formation, on the other hand, mainly affect the SMF, compensating for the decrease in high-mass galaxies due to the more active AGN, caused
in turn by the change in αfriction . Finally, the large change in the metal yield is
of little consequence, as this parameter is largely unconstrained by both the SMF
and correlation functions.
136
6.4 Summary
Figure 6.9: The effect of the changes in the supernova parameters ), Vreheat , β1 , η, Veject and
β2 . The mass-loading factor (left panel) goes up slightly when using the correlation function as
an additional constraint, but the change is not significant with respect to the 2σ regions allowed
(also shown). The same goes for the supernova ejection efficiency (right panel).
To show the effect of the changes in the feedback parameters (+, Vreheat , β1 ,
η, Veject and β2 ), we turn to Figure 6.9. In the left-hand panel, we show the
SN mass loading as a function of the maximum virial velocity of the halo, for all
three models. We also indicate the 2σ regions allowed by the parameters for the
SMF+w(rp ) model. It is clear that while the supernova mass loading increases
when the clustering data is used as an additional constraint, the change is not
significant.
The right-hand panel of Figure 6.9 shows how the SN ejection efficiency changes
between the different models. Because the parameter η is significantly higher in the
SMF+w(rp ) model with regards to the others, the high-Vmax horizontal asymptote
of this function is increased, meaning SNe are more effective at ejecting material
for galaxies occupying massive haloes. However, the large 2σ regions again indicate
that the constraints used here are not very sensitive to these changes.
6.4 Summary
We have developed a fast and accurate clustering estimator, capable of predicting
the projected galaxy correlation function for a full galaxy catalogue to within
∼ 10% accuracy using only a very small subsample of haloes (< 0.1% of the total
sample). In this work, we have described our estimator and demonstrated its
effectiveness for use in constraining parameter space for semi-analytical models of
galaxy formation, using the Guo et al. (2013) version of the Munich SAM as a test
case.
Our estimator determines the halo occupation distribution of galaxies in the
subsample and fits a profile to the galaxy distribution within haloes as a function of
halo mass, using these quantities in a halo model based approach to determine the
137
Constraining galaxy formation through clustering
galaxy clustering of the full sample. By being able to quickly predict the two-point
galaxy correlation function for the first time while exploring parameter space, one
can use clustering observations to limit the range allowed to the galaxy formation
parameters of any SAM, adding constraints complementary to those of one-point
functions typically used today, such as the stellar mass or luminosity function. As
we have demonstrated, this may lead to different sets of parameters through which
the resulting model is able to provide a better match to the observed stellar mass
and correlation functions simultaneously.
For the G13 model tested here, the improved match to the correlation function
is achieved mainly by significantly decreasing the time it takes for stripped (orphan) satellites galaxies to merge with their centrals, as well as the time it takes
for gas ejected into the hot halo by feedback processes to be reincorporated. Both
changes cause the galaxy distribution profiles within haloes to flatten, lowering
the clustering on small scales. Other parameter shifts mainly serve to keep the
changes in the SMF caused by the reduced time scales in check.
While the use of the clustering estimator presented here clearly has merit,
some issues remain to be solved. The estimator tends to over-predict clustering on
small scales, leading to final results that tend to fall ∼ 10% below the observational
constraints. Improving the model, for example by adding higher-order terms to
the linear halo bias currently used, or basing the clustering predictions of galaxies
directly on the measured clustering of the haloes in N-body simulations may help.
Additionally, the agreement with the SMF could be improved at low mass. We
will explore these topics in future work.
Acknowledgements
The authors thank Joop Schaye for useful discussions and comments on the manuscript.
The Millennium Simulation databases used in this chapter and the web application providing online access to them were constructed as part of the activities
of the German Astrophysical Virtual Observatory. This work was supported by
the Marie Curie Initial Training Network CosmoComp (PITN-GA-2009-238356)
and by Advanced Grant 246797 "GALFORMOD" from the European Research
Council.
138
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Nederlandse samenvatting
De vorming van sterrenstelsels en de structuur van
het Universum1
Als we naar de nachthemel kijken, zijn de meeste objecten die we met het blote
oog kunnen zien sterren. Al deze sterren staan kosmisch gezien heel dicht bij ons
in de buurt, namelijk binnen ons eigen sterrenstelsel, de Melkweg. Toch is zelfs de
dichtstbijzijnde ster na de Zon, Proxima Centauri, op 4,24 lichtjaar afstand2 , voorlopig nog compleet onbereikbaar. De gehele Melkweg is ongeveer 100.000 lichtjaar
groot en bevat honderden miljarden sterren, en is zelf slechts één van de honderd
miljard sterrenstelsels in het zichtbare Universum. Bijna alle onderwerpen binnen
de sterrenkunde liggen dus ontzaglijk ver buiten ons bereik. Desondanks kunnen
we heel veel te weten komen over sterren, sterrenstelsels en zelfs het Universum
als geheel, dankzij het licht dat we zien en onze kennis van de natuurwetten.
In dit proefschrift is het Universum als geheel onderwerp van onderzoek. Er
wordt gekeken naar hoe de structuur van het Universum (hoe alle materie verdeeld
is binnen het Universum) en de vorming van sterrenstelsels met elkaar in verband
staan.
Structuurvorming
Dankzij het licht dat we zien en onze kennis van de natuurwetten weten we inmiddels dat ons heelal uitdijt en dat die uitdijing momenteel steeds sneller verloopt
door wat we donkere energie3 noemen. Verder weten we dat baryonische materie
slechts een klein deel is van alle materie in het Universum. Baryonische materie is
alle materie die we kunnen zien, zo ook het materiaal waar alles op Aarde van is
gemaakt. We kunnen baryonische materie zien doordat het directe interactie met
licht heeft: het kan licht uitzenden, absorberen en verstrooien. Dit in tegenstelling
tot donkere materie, dat alleen zwaartekracht voelt en voor zover wij weten geen
interactie heeft met licht, waardoor het niet direct waarneembaar is. De meeste
materie in het Universum is donkere materie.
1 Hoewel
we maar bekend zijn met één universum, gaan wij er vanuit dat er meerdere universa
bestaan. Binnen de sterrenkunde verwijzen we naar ons universum als “het Universum”.
2 Om enkele vergelijkingen op menselijke maat te geven: dat is ongeveer 40.000.000.000.000 km
(40 biljoen kilometer), oftewel een miljard rondjes om de Aarde, of 50 miljoen keer naar de
Maan en terug, of 130.000 keer heen en weer naar de Zon. De Voyager 1, gelanceerd in 1977, is
het verst van ons verwijderde mensgemaakte object, en zelfs deze ruimtesonde bevindt zich na
37 jaar reizen pas op de rand van ons Zonnestelsel, op 1/21.000e van de afstand tussen ons en
Proxima Centauri.
3 Donkere energie is een vreemde eigenschap van de lege ruimte die er kort gezegd voor zorgt dat
er meer ruimte komt. Het is alleen verantwoordelijk voor de versnelling van de uitdijing: de
uitdijing zelf is een gevolg van de Oerknal. Donkere energie is overal en in gelijke hoeveelheid,
maar heeft op Aarde en zelfs binnen sterrenstelsels geen effect, omdat zwaartekracht alles bij
elkaar houdt. Sterrenstelsels die ver genoeg uit elkaar staan (en dus nauwelijks elkaars zwaartekracht voelen), lijken echter versneld uit elkaar geduwd te worden door de donkere energie in
de tussenliggende ruimte.
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Figuur 8.1: De evolutie van structuur in het Universum. Hier weergegeven is de dichtheid van
donkere materie in een vierkant stukje Universum waarvan de zijden nu meer dan 300 miljoen
lichtjaar lang zijn. We bewegen bij deze weergave mee met de uitdijing van het Universum. Van
zwart naar wit neemt de dichtheid toe. In het vroege Universum (links, 12 miljoen jaar na de
Oerknal) zijn de dichtheidsverschillen nog erg klein. Deze groeien langzaam onder invloed van
zwaartekracht uit tot een kosmisch web (midden, bijna een miljard jaar na de Oerknal, en rechts,
13,8 miljard jaar na de Oerknal – nu). Waar de dichtheden het hoogst zijn vormen halo’s van
donkere materie, waarin sterrenstelsels vormen. Bij de grootste knooppunten in het web vormen
clusters van sterrenstelsels.
Deze verschillende soorten materie zijn niet gelijk verdeeld over de ruimte, op
sommige plaatsen is de dichtheid hoger dan op andere plaatsen. Omdat zwaartekracht dichtheidsverschillen versterkte, waren deze in het vroege Universum dus
kleiner dan nu. Vanuit minieme dichtheidsverschillen aan het begin van het Universum, 13,8 miljard jaar geleden, is langzaam onder invloed van zwaartekracht
een kosmisch web van donkere materie gevormd (zie Figuur 8.1). De baryonische
materie, die in het begin bijna uitsluitend bestond uit waterstof en helium4 , volgt
de zwaartekracht van donkere materie. Doordat het gas wel interactie heeft met
licht, kan het energie kwijt door licht uit te zenden (“gaskoeling”). Hierdoor kan
baryonische materie nog hogere dichtheden bereiken dan donkere materie, waardoor kernfusie mogelijk wordt en sterren gevormd kunnen worden. Dit gebeurt
daar waar de concentratie van donkere materie (en dus de zwaartekracht) het
hoogst is, in “halo’s” van donkere materie. Hoe meer gas er naar het centrum van
een halo stroomt, hoe meer sterren er kunnen vormen en hoe groter het sterrenstelsel wordt. Onder invloed van zwaartekracht kunnen halo’s (en hun sterrenstelsels) met elkaar versmelten om zo nog groter te worden, een proces dat we
“merging” noemen. Sommige halo’s zijn groot genoeg om meerdere sterrenstelsels
te bevatten. Verzamelingen van sterrenstelsels die dezelfde halo bewonen, noemen
we “groepen” (enkele tot tientallen sterrenstelsels) en “clusters” (honderden tot
duizenden sterrenstelsels, zie Figuur 8.2, links).
4 Alle
zwaardere elementen, inclusief koolstof, zuurstof en ijzer, zijn gemaakt in sterren en verspreid door supernovae en stellaire winden.
148
Feedback
Net zoals de verdeling van materie bepaalt waar en hoe sterrenstelsels vormen,
beïnvloedt de vorming en evolutie van sterrenstelsels andersom ook de verdeling
van materie. De halo’s van donkere materie reageren op de vorming van sterrenstelsels in hun centra door meer samen te trekken. Maar dat niet alleen: de
vorming van sterrenstelsels gaat namelijk gepaard met veel geweld en heeft invloed
op de verdeling van (alle) materie.
Als sterren sterven gebeurt dat meestal stilletjes, maar hele zware sterren (tenminste 8x zo zwaar als de Zon) kunnen exploderen als supernovae. Bij een supernova kan gas weg worden geslingerd (wat supernova-feedback wordt genoemd),
waardoor de verdeling van het gas wordt beïnvloed. Het gas kan hierbij ook verhit
worden, wat de vorming van nieuwe sterren tegengaat: het gas moet deze energie
namelijk eerst kwijtraken voordat de dichtheid ervan hoog genoeg kan worden voor
kernfusie.
Soortgelijke processen gebeuren op veel grotere schaal bij de supermassieve
zwarte gaten in de centra van sterrenstelsels. Gas kan hierbij zelfs tot ver uit het
sterrenstelsels worden gedreven. Ook hierbij wordt het gas verhit, soms tot zulke
hoge temperaturen dat het miljarden jaren kan duren voordat het gas deze energie
kwijt is en weer structuur kan vormen. De donkere materie kan op het uitstoten
van grote hoeveelheden gas reageren door uit te zetten. Een supermassief zwart
gat dat sterke interactie vertoont met het gas eromheen wordt een AGN (Active
Galactic Nucleus) genoemd. Het proces waarbij gas wordt verhit en naar buiten
wordt gedreven, heet AGN-feedback (zie Figuur 8.2, rechts). AGN-feedback en
supernova-feedback komen veel terug in de verschillende hoofdstukken van dit
proefschrift, niet alleen omdat ze de verdeling van materie beïnvloeden maar ook
omdat hun effect groter is dan voorheen werd aangenomen.
Numerieke simulaties
Omdat de sterrenstelsels alleen vormen waar de dichtheden het hoogst zijn, zijn ze
“biased tracers” van de algehele materieverdeling. Dit houdt in dat ze ons een gekleurd beeld geven van waar alle materie zich bevindt. Mede door gedetailleerde
computersimulaties uit te voeren, leren we steeds beter hoe de totale materieverdeling en de verdeling van sterrenstelsels zich tot elkaar verhouden. Met deze
kennis kunnen we door het nauwkeurig bestuderen van hoe sterrenstelsels in het
Universum verdeeld zijn dus steeds meer leren over de structuur van het Universum
als geheel.
Er zijn verschillende manieren om de relatie tussen de vorming van sterrenstelsels en de verdeling van materie te modelleren. Omdat deze allemaal terugkomen
in dit proefschrift, worden ze hieronder kort beschreven.
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Nederlandse samenvatting
Figuur 8.2: Links: Voorbeeld van een cluster van sterrenstelsels, Abell 2744. Veel van de
sterrenstelsels in dit plaatje (niet die op de achtergrond) bewonen dezelfde halo, en draaien om
elkaar heen. De sterrenstelsels zelf, het enige dat we op dit plaatje kunnen zien, bevatten slechts
5 procent van de materie: 20 procent zit in het gas tussen de sterrenstelsels in, dat zo heet is dat
het bijna uitsluitend röntgenstraling uitzendt. De overige 75 procent van de massa is donkere
materie. Rechts: Voorbeeld van AGN-feedback in het sterrenstelsel NGC1275. Dit sterrenstelsel
zit in het midden van de Perseus cluster (Abell 426). Het bolvormige witte licht in het midden
komt van het sterrenstelsel zelf. De enorme paarse wolk eromheen laat de röntgenstraling van
het door AGN-feedback uitgestote en verhitte gas eromheen zien. Dit gas heeft een temperatuur
van tientallen miljoenen graden.
Hydrodynamische simulaties
Ten eerste kan men kosmologische, hydrodynamische simulaties uitvoeren: computerberekeningen waarbij een significant deel van het Universum wordt gesimuleerd,
met zowel donkere materie als gas. Hierbij worden alle relevante natuurkundige
vergelijkingen, zoals die voor zwaartekracht en gaskoeling, doorgerekend vanaf het
vroege Universum tot nu. Hoe krachtiger de computer, hoe meer deeltjes de simulatie kan bevatten, en daarmee hoe fijner de resolutie wordt. Momenteel zijn
we nog niet in staat om in dergelijke kosmologische simulaties processen zoals
stervorming en supernovae direct te simuleren, omdat daar veel hogere resolutie voor nodig is dan zelfs de beste supercomputers van vandaag de dag kunnen
leveren.5 Daarom maken we gebruik van “sub-grid recepten”: formuleringen die
voorschrijven hoe processen op kleinere schaal dan we direct kunnen simuleren
afhangen van eigenschappen op grotere schaal (bijvoorbeeld hoe de hoeveelheid
nieuwe sterren die gevormd wordt per eenheid tijd binnen één gasdeeltje afhangt
van de dichtheid en temperatuur gemeten rond het gasdeeltje). Wat uit de simulatie komt hangt dus voor een groot deel af van wat we er zelf instoppen. Het is
derhalve belangrijk dat de sub-grid recepten gebaseerd zijn op de natuurwetten
en op wat de waarnemingen in het echte Universum ons vertellen. Verder is het
belangrijk dat we een beschrijving hebben van elk proces dat we niet direct kunnen
5 Om
een idee te geven: in de hydrodynamische simulaties die in dit proefschrift worden
beschreven, is elk deeltje ongeveer zo zwaar als 100 miljoen zonnen.
150
simuleren: als wij de simulatie niet vertellen dat er iets zoals een AGN bestaat
en hoe deze zich gedraagt, dan zal de simulatie het verkeerde antwoord geven op
plekken waar AGN belangrijk zijn.
N-body simulaties
Ook kunnen simulaties worden uitgevoerd onder de aanname dat alle materie
donkere materie is (“N-body” simulaties). Zulke simulaties zijn een stuk simpeler en kunnen met een veel hogere resolutie worden uitgevoerd dan hydrodynamische simulatie: immers, de enige vergelijking die doorgerekend hoeft te worden
is zwaartekracht. Omdat donkere materie de dominante vorm van materie is, en
zwaartekracht het dominante proces bij structuurvorming, kunnen we met zulke
simulaties nog steeds veel leren over de verdeling van materie in het echte Universum. We moeten echter in gedachten houden dat de effecten van baryonische
processen op de donkere materie (zoals gaskoeling en feedback) hierbij worden verwaarloosd. Het hangt af van de schaal waar naar gekeken wordt of deze processen
significant zijn.
Semi-analytische modellen
Verder is het mogelijk om te bestuderen hoe sterrenstelsels gevormd zouden zijn in
de donkeremateriehalo’s van N-body simulaties met behulp van semi-analytische
modellen (“SAMs”). Hierbij wordt aangenomen dat de groei en evolutie van sterrenstelsels volledig bepaald wordt door de eigenschappen van de donkere materie.
Een semi-analytisch model kan gezien worden als een collectie in elkaar hakende
sub-grid recepten: niets wordt direct gesimuleerd behalve de donkere materie.
Het voordeel van dergelijke “simulaties” is dat ze erg snel uitgevoerd kunnen worden, waardoor het makkelijk wordt om het effect van verschillende voorschriften
voor de vorming van sterren etcetera te testen. Wel kennen dergelijke simulaties
meer beperkingen dan hydrodynamische simulaties, aangezien de donkerematerieverdeling vast staat.
HOD- en halomodellen
Tot slot is het mogelijk om nog een stap verder af te wijken van het doen van directe
simulaties, door de halo’s voor te stellen als bolvormige objecten met een bepaalde
verdeling in de ruimte en een bepaald dichtheidsprofiel (meestal gebaseerd op de
resultaten van donkeremateriesimulaties). Voorbeelden hiervan zijn halomodellen
en HOD-modellen (“halo occupation distribution”), waarmee de verdeling van respectievelijk materie en sterrenstelsels relatief snel en simpel voorspeld kan worden,
tot op een zekere nauwkeurigheid.
Elk van deze methodes heeft zijn eigen voor- en nadelen. Directe, hydrodynamische simulaties zijn het meest complex en kunnen daardoor de meeste verschillende processen en effecten bevatten. Ze kosten echter ook de meeste computertijd,
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Nederlandse samenvatting
en worden daarom vaak maar eenmaal gedraaid. Modellen die verder afwijken van
directe simulaties zijn minder precies, maar kunnen sneller tot een uitkomst leiden, wat ook als voordeel heeft dat hetzelfde model vele malen opnieuw gedraaid
kan worden met kleine variaties in de vrije parameters (bijvoorbeeld in de kosmologische parameters die ons Universum karakteriseren). Elk van de verschillende
uitkomsten kan dan met de waarnemingen vergeleken worden, om zo te kijken
welke parameters de werkelijkheid het beste beschrijven. Hierbij moet echter wel
rekening worden gehouden met de genomen benaderingen (zoals dat de baryonische
materie en feedback de donkere materie niet beïnvloeden).
Clustering
Als we de werkelijkheid en simulaties met elkaar willen vergelijken, dan hebben
we daar kwantificeerbare grootheden voor nodig: meetbare eigenschappen waar
we een getal aan kunnen verbinden. We kunnen bijvoorbeeld kijken naar de hoeveelheid sterrenstelsels met een bepaalde massa in sterren, of die een bepaalde
hoeveelheid licht uitzenden. In dit proefschrift ligt de nadruk zoals gezegd op de
verdeling van sterrenstelsels en materie in de ruimte, wat we “clustering” noemen.
We kunnen de hoeveelheid clustering op een bepaalde schaal kwantificeren met
behulp van de correlatiefunctie en het “power spectrum”.6
De correlatiefunctie van sterrenstelsels geeft de waarschijnlijkheid dat twee
sterrenstelsels zich op een bepaalde afstand van elkaar bevinden, ten opzichte van
een willekeurige verdeling. Door in waarnemingen en simulaties van duizenden
sterrenstelsels de onderlinge afstanden te bepalen, kunnen we de correlatiefunctie
berekenen door simpelweg te tellen hoe vaak sterrenstelsels op een bepaalde afstand van elkaar staan. Als de correlatiefunctie positief is op een bepaalde schaal
(dus voor een bepaalde onderlinge afstand), betekent dit dat de sterrenstelsels
“graag” op deze afstand van elkaar zitten. De correlatiefunctie neemt sterk toe
naar kleinere schalen, wat betekent dat sterrenstelsels heel vaak dicht bij elkaar te
vinden zijn, wat past bij het beeld van een kosmisch web waarbij de sterrenstelsels
vormen waar de dichtheden het hoogst zijn.
Het power spectrum is iets ingewikkelder, maar kort gezegd gebruiken we het
in dit proefschrift om de clustering van (alle) materie te karakteriseren. In de
simulatie is dit makkelijk te meten, omdat we precies weten waar alle materie
zich bevindt, maar in het echte Universum is dit wat lastiger. We kunnen immers alleen het licht van sterrenstelsels en gas direct waarnemen. Gelukkig zijn
we steeds beter in staat om de verdeling van alle materie, baryonisch en donker,
in kaart te brengen, dankzij waarnemingen van “lensing”: het effect dat licht van
sterrenstelsels een klein beetje wordt afgebogen onder invloed van zwaartekracht.
Door deze afbuigingen heel precies in kaart te brengen kan het zwaartekrachtsveld
(en daarmee de verdeling van materie) gereconstrueerd worden en kan een power
6 De
Nederlandse vertaling hiervan is de “spectrale vermogensdichtheidsfunctie”, maar dit rolt
toch wat minder goed van de tong. Overigens mist de Nederlandse taal helaas een even bondige
doch sterk beschrijvende vertaling voor het woord “clustering”.
152
spectrum gemeten worden.
Door de correlatiefuncties en power spectra van waarnemingen en simulaties te
vergelijken, kunnen we meer leren over ons Universum. We moeten hierbij wel
zorgvuldig zijn met onze simulaties: als we mogelijk belangrijke processen zoals
feedback negeren, gaat de vergelijking met de werkelijkheid niet op en kloppen onze
interpretaties van de waarnemingen misschien niet. In dit proefschrift richten we
ons daarom voornamelijk op de effecten die de vorming van sterrenstelsels kan
hebben op de clustering van materie en sterrenstelsels zelf.
In dit proefschrift
Hieronder volgt een vereenvoudigde samenvatting van de inhoud van dit proefschrift.
In Hoofdstuk 1 geef ik een uitgebreidere introductie van de studie van het
Universum als geheel: kosmologie. Ik beschrijf in meer detail hoe kleine dichtheidsverschillen in het vroege Universum groeien onder invloed van zwaartekracht. Ook
ga ik dieper in op de rol die de vorming van sterrenstelsels speelt in de algehele
verdeling van materie.
In Hoofdstuk 2 onderzoeken we de effecten die verscheidene fysische processen
verwant aan de vorming en evolutie van sterrenstelsels – waaronder supernovaen AGN-feedback – kunnen hebben op de clustering van materie. We vergelijken
daarbij hydrodynamische simulaties met N-body simulaties en we wisselen af welke
fysische processen in overweging worden genomen, zodat we het effect van elk
afzonderlijk proces kunnen testen. We laten hierbij zien dat feedback een veel
grotere invloed op het power spectrum kan hebben dan in voorgaande studies is
aangetoond. We onderzoeken ook hoe de clustering van de donkere materie hierbij
verandert. We concluderen dat het nodig is om processen zoals feedback in acht
te nemen, omdat dit grote invloed kan hebben op hoe goed we in staat zullen zijn
om de nauwkeurige clusteringdata die in de nabije toekomst verkregen zal worden
te interpreteren.
Omdat dergelijke fysische processen ook belangrijk zouden kunnen zijn voor
de clustering van sterrenstelsels, beschouwen we in Hoofdstuk 3 de relevante
correlatiefuncties. Ook hier treden significante veranderingen op als feedback in
overweging wordt genomen. Deze veranderingen komen voornamelijk tot stand
doordat de massa’s van de sterrenstelsels en hun halo’s door feedback afneemt,
maar er zijn daarnaast ook kleinere, complexere effecten die een rol spelen. We
laten zien dat het belangrijkste van deze effecten de herverdeling van materie door
feedback is.
We onderzoeken in dit proefschrift ook de geldigheid van enkele aannames waar
in HOD- en halomodellen gebruik van wordt gemaakt. In Hoofdstuk 4 bekijken
we hoe de aanname dat halo’s van donkere materie bolvormig zijn de voorspelde
clustering van sterrenstelsels kan beïnvloeden. We maken hierbij gebruik van semi153
Nederlandse samenvatting
analytische modellen om het verschil tussen het gebruik van realistische halo’s of
kunstmatige, ronde halo’s te bestuderen. Met de aanname dat halo’s rond zijn, kan
de correlatiefunctie van sterrenstelsels sterk onderschat worden op kleine schaal.
Het is daarom belangrijk om realistische vormen in acht te nemen, al laten we
zien dat de oriëntatie van de halo’s weinig verschil maakt. We laten ook zien dat
het verschil in clustering waar de vorm van de halo’s voor zorgt, meetbaar zou
moeten zijn in het echte Universum. Met gebruik van N-body simulaties toetsen
we vervolgens in Hoofdstuk 5 de aanname van halomodellen dat alle materie
in halo’s zit. We doen dit door de clustering van materie te berekenen en die te
vergelijken met de clustering van alleen de materie die in halo’s zit. We laten zien
dat materie buiten de halo’s ook belangrijk kan zijn voor de clustering, afhankelijk
van de halomassa’s die in beschouwing worden genomen en de gebruikte definitie
van een halo.
Tot slot presenteren we in Hoofdstuk 6 een snelle en nauwkeurige computercode voor het schatten van de correlatiefunctie van sterrenstelsels in semianalytische modellen. De correlatiefunctie voor de gehele simulatie wordt hierbij
geschat gebruik makende van slechts 1 op de 1.000 sterrenstelsels. Hiermee kan
de clustering veel sneller bepaald worden (met een bekende onzekerheid) dan door
deze direct te berekenen. Daardoor wordt het mogelijk om op efficiënte wijze de
parameters van het semi-analytische model te vinden die de clustering in het echte
Universum het beste reproduceren. We demonstreren dit in hetzelfde hoofdstuk
door onze methode toe te passen in een semi-analytisch model.
154
Publications
1. The clustering of baryonic matter. II: halo model and hydrodynamic simulations
Cosimo Fedeli, Elisabetta Semboloni, Marco Velliscig, Marcel P. van Daalen,
Joop Schaye, Henk Hoekstra
2014, accepted by JCAP, arXiv:1406.5013
2. The impact of galaxy formation on the total mass, profiles and abundance of
haloes
Marco Velliscig, Marcel P. van Daalen, Joop Schaye, Ian G. McCarthy,
Marcello Cacciato, Amandine M. C. Le Brun, Claudio Dalla Vecchia
2014, MNRAS, 442, 2641
3. The impact of baryonic processes on the two-point correlation functions of
galaxies, subhaloes and matter
Marcel P. van Daalen, Joop Schaye, Ian G. McCarthy, C. M. Booth, Claudio
Dalla Vecchia
2014, MNRAS, 440, 2997
4. The effects of halo alignment and shape on the clustering of galaxies
Marcel P. van Daalen, Raul E. Angulo, Simon D. M. White
2012, MNRAS, 424, 2954
5. Quantifying the effect of baryon physics on weak lensing tomography
Elisabetta Semboloni, Henk Hoekstra, Joop Schaye, Marcel P. van Daalen,
Ian G. McCarthy
2011, MNRAS, 417, 2020
6. The effects of galaxy formation on the matter power spectrum: A challenge
for precision cosmology
Marcel P. van Daalen, Joop Schaye, C. M. Booth, Claudio Dalla Vecchia
2011, MNRAS, 415, 3649
155
Curriculum vitae
I was born on January 8 1986 in The Hague. Around the age of 5 I realised I
wanted to go into science, not yet knowing how broad a term “science” was and
that it might therefore be useful to think about what kind of science I wanted to
do. All I knew back then is that I wanted to discover and learn things that nobody
knew yet.
After exasperating my parents with endless “how’s” and “why’s” about how
the world worked all through elementary school, my interests began to point more
towards space. During my time at the Alfrink College high school I especially
loved mathematics and physics, and – inspired by the great math teachers I had –
briefly debated studying and then teaching mathematics. I was – and still am – also
fascinated with computers and technology. However, the idea of “unravelling the
Universe” won out, and in 2004 I started studying Astronomy at Leiden University.
It only took one programming class in my first year to get me hooked, and I
haven’t bored of programming since – a good thing, considering my preference for
the theoretical side of astronomy. After a Bachelor’s project with Clovis Hopman
on binary star break-up in the Galactic centre and a minor Master’s project with
Jarle Brinchmann on gamma-ray bursts from Wolf-Rayet stars, I started working
on the effects of galaxy formation on clustering with Joop Schaye. After obtaining
my Master’s degree (cum laude) I continued working on the interface of galaxy
formation and cosmology throughout my PhD project, working with Joop Schaye
and with Simon White at the Max-Planck-Institut für Astrofysik in Garching,
Germany. After having spent the first half of my PhD living in Munich, I moved
back to Leiden, travelling back and forth between the two places during my PhD.
The love for teaching that I had acquired in high school has remained, and
during my studies and doctorate I very much enjoyed tutoring first-year students,
assisting in courses, answering astronomy-related questions from the public as
member of the outreach committee, and participating in the popularisation of
astronomy through outreach at high schools and giving public lectures.
In April I married my wonderful wife Marieke, and in September we will move
across the Atlantic to start my first position as a post-doctoral researcher, at the
University of California in Berkeley, USA. I plan to remain there for three years,
working as a TAC Fellow.
157
Acknowledgements
In the last four years there have been a lot of people that helped me along the
way and/or made my time in Leiden and Munich more enjoyable, and to whom I
would like to express my gratitude.
First of all, I’m grateful for the support of my parents in my choice of career,
even when they didn’t always understand it. I’m also grateful to the teachers who
inspired me, especially Anton Dolle (my math teacher in the final years of high
school) and Walter Kosters (who taught me the joys of C++).
A big thanks goes to all my colleagues at Leiden Observatory and MPA. Thanks
to you, both institutes have great atmospheres that are enjoyable to work in. Clovis
and Jarle, thank you for taking me through my first two research experiences and
writing me many letters of recommendation (even after one of you left astronomy).
Had I been allowed to thank my current supervisors (Joop and Simon) here, I would
have said that I greatly appreciate the time and effort they put in mentoring me,
sharing their expertise and making me a better scientist. Xander, thanks for
keeping an eye on me when it seemed like I might not finish my thesis on time.
Thanks to all members (past and current) of Joop’s awesome OWLS research
group, which I’m happy to have been a part of for the last five years. Your
company and the twice-weekly discussions all contributed greatly to my time at
the Sterrewacht. Craig, Claudio and Rob C., thanks for always making time
for questions from us students. Olivera, I always enjoyed our conversations, and
practising your Dutch. Ali, it was fun having you around, both here and in Munich.
Marco and Marcello, it was great working with you, and I hope we can continue to
do so in the future. Marco, I also very much enjoyed complaining about subfind
with you. Ben, Milan, Monica, Alex, Joki and Marijke, thank you for making the
group more fun as well. Freeke, you were the best office mate anyone could wish
for, and it was great having you around in Munich too; Ann-Marie, you were an
awesome neighbour (twice). See you both in Berkeley!
I would also like to thank the people I worked with in Garching. Chervin and
Rob Y., thanks for all the laughs. Ben and Laura, thanks for being so much fun.
Raul and Bruno, it was awesome working with you, and I learned a lot.
The support staff at both Leiden Observatory and MPA is wonderful, and
they’ve helped me with a lot of issues over the years. Evelijn and Arianne, thanks
for all your help and enjoyable conversations, both at the physics department and
at the Sterrewacht. Also a big thanks to the computer group at Leiden, Erik,
David, Tycho, Niels and Aart, for their support (which I’ve made use of many,
many times). I’m also extremely grateful to Jeanne, Anita and Liesbeth in Leiden,
and Maria, Gabi and Cornelia at MPA, who helped me with many problems and
questions along the way.
Finally, I would like to thank my amazing wife Marieke. Not only for her
help with turning my propositions into actual propositions and making my Dutch
summary more understandable (compounded by my stubbornness), but most of
all for her undying love and support. Even when I moved to a different country.
Without her, this thesis would have lost its meaning.
159
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