Ph.D. thesis

Ph.D. thesis
Optimal Weights and Multiple Tracers
in Large-Scale Structure
Dissertation
zur
Erlangung der naturwissenschaftlichen Doktorwürde
(Dr. sc. nat.)
vorgelegt der
Mathematisch-naturwissenschaftlichen Fakultät
der
Universität Zürich
von
Nico Hamaus
aus
Deutschland
Promotionskomitee
Prof. Dr. Uroš Seljak (Vorsitz)
Prof. Dr. Ben Moore
Prof. Dr. Lucio Mayer
Zürich, 2012
3
Nico Hamaus
Institute for Theoretical Physics
University of Zurich
Winterthurerstrasse 190
CH-8057 Zürich
Switzerland
[email protected]
c Nico Hamaus 2012
All rights reserved. No part of this document may be reproduced without the written
permission of the publisher.
4
Contents
Preface
7
Acknowledgments
9
Abstract
11
Zusammenfassung
13
1. Introduction
1.1. The Homogeneous Cosmos . . . . . . . . .
1.1.1. Expansion . . . . . . . . . . . . . .
1.1.2. Geometry . . . . . . . . . . . . . .
1.1.3. Inventory . . . . . . . . . . . . . .
1.1.3.1. Dust . . . . . . . . . . . .
1.1.3.2. Radiation . . . . . . . . .
1.1.3.3. Cosmological Constant . .
1.1.3.4. Scalar Field . . . . . . . .
1.1.4. Observational Status . . . . . . . .
1.2. The Inhomogeneous Cosmos . . . . . . . .
1.2.1. Random Fields . . . . . . . . . . .
1.2.2. Initial Perturbations . . . . . . . .
1.2.2.1. Inflation . . . . . . . . . .
1.2.2.2. Quantum Fluctuations . .
1.2.2.3. Primordial Potential . . .
1.2.3. Evolution of Perturbations . . . . .
1.2.3.1. Hot Phase . . . . . . . . .
1.2.3.2. Cold Phase . . . . . . . .
1.2.3.3. Nonlinear Regime . . . . .
1.2.4. Formation of Large-Scale Structure
1.2.4.1. Halo Mass Function . . .
1.2.4.2. Halo Bias . . . . . . . . .
1.2.4.3. Halo Model . . . . . . . .
1.2.4.4. Stochasticity . . . . . . .
1.2.4.5. Galaxies . . . . . . . . . .
1.2.4.6. Redshift-Space Distortions
1.3. Scientific Tools . . . . . . . . . . . . . . .
1.3.1. Statistical Techniques . . . . . . . .
1.3.1.1. Parameter Estimation . .
1.3.1.2. Error Forecast . . . . . .
1.3.2. Simulations . . . . . . . . . . . . .
1.3.2.1. Particles . . . . . . . . . .
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17
18
18
19
20
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21
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22
22
24
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28
28
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30
31
32
32
34
34
35
35
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37
38
38
38
6
1.3.2.2. Halos . . . . . . . . . . . .
1.3.3. Numerical Analysis . . . . . . . . . .
1.3.3.1. Discrete Fourier Transform
1.3.3.2. Mesh Interpolation . . . . .
1.3.3.3. Estimators . . . . . . . . .
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39
40
40
40
41
2. Paper I
43
3. Paper II
49
4. Paper III
69
5. Paper IV
97
6. Outlook
115
References
117
Preface
This doctoral thesis contains a selection of scientific publications that originated
from research projects I was involved in during my PhD studies at the Institute for
Theoretical Physics (ITP) at the University of Zurich. It is complemented with a
general introduction to the field of cosmology, which describes the basic language
that is used throughout the professional literature and presents the tools that have
been applied to reach the final conclusions. In the end, a brief outlook on possible
extensions of the presented research is given.
The general motivation of this thesis is the attempt to model the large-scale
structure of the universe with the help of analytical and numerical methods, and
the potential application of these models to real observations in order to gain new
information on the underlying cosmological framework. Each of the included journal
publications deals with different aspects of this framework, the details of which are
separately being summarized in each chapter. Additionally, the individual contribution of each author is briefly mentioned. The scientific articles themselves have
largely been left unmodified from their published versions so as to make a direct
connection to the literature. Each paper comes with a separate bibliography for
references used within the article, they should not be confused with the references
cited in the remaining part of the thesis, which are listed in the very end of this
document.
Zurich, June 2012
Nico Hamaus
8
Acknowledgments
During the course of my PhD, I appreciated the support of a number of people
that enabled me to complete this thesis. First of all, I am grateful to my advisor
Uroš Seljak, who gave me the opportunity to join this exciting field of fundamental
research and guided me through numerous projects with his expertise. Many thanks
to Vincent Desjacques and Robert Smith for assistance with the numerical analysis of
their simulations and their expert contributions to my research projects. I also wanna
thank the group members Tobias Baldauf, Lucas Lombriser, Zvonimir Vlah, Jaiyul
Yoo, Andreu Font and Michael Busha for endless discussions and support. I am
grateful to Ben Moore and Lucio Mayer for co-advising my thesis, Regina SchmidRichmond, Suzanne Wilde and Esther Meier for their assistance in the everyday life
at the institute, and all the members of the ITP for discussions, support and making
my time in Zurich so worthwhile.
Moreover, I appreciated the hospitality of Lawrence Berkeley National Laboratory, the UC Berkeley Physics Department and the Institute for the Early Universe
at Ewha University Seoul, while parts of the research presented in this thesis was
conducted, special thanks to Jonathan Blazek and Teppei Okumura for showing
me around. Finally, I wanna thank my family and my girlfriend for supporting me
throughout my studies and their appreciation for my endeavors.
10
Abstract
The past few decades designate a golden age for the science of cosmology. Groundbreaking discoveries and innovative ideas have emerged to form the standard model
of cosmology as we know it today. However, major ingredients of this model, such as
dark energy and inflation, are still being disputed and more data is needed to find
answers to the open questions. Large-scale structures carry a wealth of information
about the cosmological origin and evolution of the universe. They can be explored
with the help of redshift surveys that map out the three-dimensional distribution
of galaxies within a large fraction of the observable universe. The surveys aim at
reconstructing the large-scale distribution of all matter to maximize the statistical
significance of the inferred cosmological conclusions. However, there is strong evidence for the fact that galaxies are only the tracers of a predominant distribution
of dark matter, limiting the attainable accuracy on this reconstruction.
On the basis of numerical N -body simulations, this thesis presents means to
optimize the information content encoded in the statistics of biased tracers of the
dark matter density field and investigates their usefulness in cosmological applications. Particular attention is dedicated to an investigation of the stochastic relation
between the density fields of halos and dark matter. It is found that the degree of
stochasticity decreases as a function of halo mass and that halos of different mass
show correlations in their stochastic clustering component. A detailed analysis reveals an optimal weighting scheme to minimize the stochasticity of halos with respect
to the underlying dark matter density field.
As a matter of fact, the clustering of dark matter is itself characterized by random
quantum fluctuations generated in the early universe, so it can only be described
in a statistical manner up to an uncertainty referred to as cosmic variance. Here,
a technique to circumvent cosmic variance is applied by considering multiple tracers of the density field and correlating them against each other. By means of two
specific applications, it is demonstrated how in combination with optimal weighting
schemes, this approach can improve upon the conventional inference of cosmological
parameters from large-scale structure data sets. One such application deals with the
signatures of primordial non-Gaussianity that can originate in the inflationary phase
of the early universe and might still be visible in the clustering statistics of galaxies
today. The other one addresses the growth rate of structure that can be constrained
from redshift-space distortions in the clustering pattern of galaxies. The magnitude
of the growth rate is of particular interest in cosmology, as it is very sensitive to the
underlying theory of gravity.
12
Zusammenfassung
Die vergangenen Jahrzehnte markieren ein goldenes Zeitalter der Kosmologie. Bahnbrechende Entdeckungen und innovative Ideen haben das heutige Standardmodell der Kosmologie hervorgebracht. Zentrale Bestandteile dieses Modells, wie zum
Beispiel dunkle Energie und Inflation, sind jedoch immer noch umstritten und mehr
Daten werden benötigt, um Antworten auf die noch ausstehenden Fragen zu finden.
Großräumige Strukturen führen eine Fülle an Informationen über den Ursprung und
die Entwicklung des Universums mit sich. Man kann sie anhand von Himmelsdurchmusterungen untersuchen, wobei eine dreidimensionale Kartierung eines Großteils
aller beobachtbaren Galaxien entsteht. Das Ziel ist, die großräumige Verteilung
sämtlicher Materie zu rekonstruieren und so die statistische Aussagekraft daraus
gewonnener kosmologischer Schlußfolgerungen zu maximieren. Es gibt allerdings
starke Hinweise dafür, dass die Verteilung der Galaxien von einer dominierenden
Komponente dunkler Materie bestimmt wird, eine Tatsache welche die erzielbare
Genauigkeit einer solchen Rekonstruktion erschwert.
Auf der Grundlage von N -Teilchen Simulationen werden in dieser Doktorarbeit
Methoden entwickelt, um den Informationsgehalt der statistischen Verteilung sogenannter Tracer zu optimieren, jener Objekte, die den Anhäufungen der dunklen
Materie in gewisser Weise nachfolgen. Der Nutzen dieser Methoden hinsichtlich kosmologischer Anwendungen wird zudem analysiert. Besondere Aufmerksamkeit gilt
der Untersuchung der stochastischen Beziehung zwischen den Dichteverteilungen von
Halos und dunkler Materie. Es stellt sich heraus, dass das Ausmaß an Zufälligkeit
in dieser Beziehung als Funktion der Halomasse abnimmt, und dass Halos verschiedener Massen Korrelationen in ihrer stochastischen Komponente aufweisen.
Anhand detaillierter Analysen wird ein optimales Gewichtungsschema ermittelt,
mit Hilfe dessen diese stochastische Komponente bezüglich der zugrunde liegenden
Verteilung dunkler Materie minimiert werden kann.
Tatsächlich ist die Verteilung der dunklen Materie selbst von zufälligen Quantenfluktuationen des frühen Universums bestimmt und kann daher nur auf statistische
Weise beschrieben werden, die damit verknüpfte Ungewissheit wird als kosmische
Varianz bezeichnet. In dieser Arbeit kommt eine Methode zur Anwendung, bei der
mehrere Tracer der dunklen Materie berücksichtigt und gegeneinander korreliert
werden. Am Beispiel zweier konkreter Anwendungen wird verdeutlicht, wie dies,
kombiniert mit optimalen Gewichtungsschemata, die konventionellen Methoden zur
Bestimmung kosmologischer Parameter verbessert. Erstere beschäftigt sich mit den
Anzeichen primordialer nicht-Gaußianität, welche in der inflationären Phase des
frühen Universums hervorgerufen werden kann und möglicherweise heute noch in
der Statistik von Galaxienansammlungen sichtbar ist. Die zweite Anwendung befasst sich mit der Wachstumsrate von Strukturen im Universum, welche anhand von
Rotverschiebungsverzeichnungen in den Häufungsmustern von Galaxien ermittelt
werden kann. Der Wert dieser Wachstumsrate ist von besonders hohem Interesse für
die Kosmologie, da er stark von der zugrunde liegenden Gravitationstheorie abhängt.
14
“Nature shows us only the tail of the lion. But I do not doubt that
the lion belongs to it even though he cannot at once reveal himself
because of his enormous size.”
– Albert Einstein
16
1
I NTRODUCTION

Chapter 1: Introduction
1.1. The Homogeneous Cosmos
An important ingredient of contemporary cosmology is the Copernican Principle,
also known as the Cosmological Principle. It states:
There is no privileged observer in the universe.
Although this statement degrades humanity’s former vantage point from the center
of the world to any insignificant location in the universe, it also implies a promising
perspective: observations made from a single location are representative for the entire
cosmos. Thus, astronomical observations enable us to reveal the underlying laws of
the universe without having to travel far away from our home planet.
The Copernican Principle is based on the assumption of homogeneity and
isotropy, meaning the properties of the universe do neither change with location, nor
direction, respectively. Clearly, the immediate world around us looks very distinct
from being homogeneous or isotropic, but on very large scales beyond ∼ 100Mpc
most of the astronomical data at hand are in favor of this assumption.
In this section we will focus on those large scales and discuss the properties of
a homogeneous and isotropic universe in the framework of General Relativity. This
will lead us to the so-called Standard Model of Cosmology and its ingredients.
1.1.1. Expansion
Edwin Hubble’s discovery of a connection between the distance and the color of
galaxies [21] initiated the field of observational cosmology as we know it today. In
particular, he observed a relative shift ∆λ of the wavelength λ towards the red in
the spectra of galaxies and found their redshift z ≡ ∆λ/λ to be proportional to their
distance r. Thus far this result does not imply any cosmological consequences unless
interpreted as a Doppler shift originating from the motion of the galaxies away from
the observer. In this case the redshift can be expressed as the ratio of the galaxy’s
velocity to the speed of light, z ' v/c, and the Hubble law becomes
v = Hr ,
(1.1)
where H is the so-called Hubble constant. It can be shown that the functional form
of Hubble’s law is the only one that obeys homogeneity and isotropy. In conjunction
with the Copernican Principle it yields a seemingly paradoxical conclusion, namely
that every observer in the cosmos witnesses all galaxies around him to recede. However, this paradox can be resolved, if we allow the universe itself to have a dynamical
character. In this interpretation, it is empty space that is expanding and thus increasing the distance bewteen the galaxies, not the motion of the individual galaxies
themselves. The idea was first formulated by Georges Lemaı̂tre even before Hubble’s
discovery [27].
Lemaı̂tre’s groundbreaking interpretation has another inevitable implication: if
the universe is expanding today, it must have been smaller in the past. Assuming
the expansion to be monotonic, this leads to the conclusion that the universe must
have had a beginning and started out from a point-like object, a so-called singularity
of spacetime. This evolution is often referred to as the Big Bang scenario and works
remarkably well in describing the expansion history of the universe.
1.1. The Homogeneous Cosmos

1.1.2. Geometry
General Relativity (GR) relates the distribution of mass and energy to the geometry
of spacetime, which is beautifully embodied in Einstein’s field equations
1
Rµν − gµν R = 8πTµν ,
2
(1.2)
with Greek indices representing the four spacetime coordinates xµ = (t, x, y, z) and
gµν being the metric of spacetime. From here on we will use Planck units, setting
c = ~ = G = kB = 1. The Ricci tensor
Rµν ≡
∂Γαµν
∂Γαµα
−
+ Γαβα Γβµν + Γαβν Γβµα
∂xα
∂xν
(1.3)
and its contraction R = g µν Rµν characterizes the curvature of spacetime. It is determined through Christoffel symbols
1 µν ∂gαν ∂gβν
∂gαβ
µ
Γαβ = g
+
−
,
(1.4)
2
∂xβ
∂xα
∂xν
which in turn are defined via the metric gµν . The energy-momentum tensor Tµν describes any form of mass-energy. To ensure its conservation, the covariant divergence
of the energy-momentum tensor must vanish,
∇ν T µν ≡
∂T µν
+ Γµαν T αν + Γναν T µα = 0 .
∂xν
(1.5)
The same must hold for the left-hand side of Eq. (1.2) and is indeed satisfied as a
consequence of the so-called Bianchi identities [5].
In order to properly describe the evolution of the universe one needs to find a
solution to Eq. (1.2) that obeys homogeneity and isotropy. It can be expressed as a
spacetime interval
dr2
2
2
2
2
2
2
µ
ν
2
2
+ r dϑ + r sin ϑdφ
,
(1.6)
ds ≡ gµν dx dx = dt − a (t)
1 − Kr2
and was first derived by Alexander Friedman [16], followed by Lemaı̂tre [28], Robertson [41] and Walker [52]. It is composed of the temporal part dt and the spatial part
in the brackets, which is multiplied by the scale factor a(t). As the name suggests,
a(t) describes the scale of the universe evolving with time. The geometry of space
is expressed in spherical coordinates, with radius r, polar angle ϑ and azimuthal
angle φ. It can take on three different types of curvature depending on the value
of K. K = 0 describes flat space (no curvature), K > 0 a three-dimensional sphere
(positive curvature, closed universe) and K < 0 a hyperbolic geometry (negative
curvature, open universe). Because the magnitude of K can be absorbed in a redefinition of the scale factor a(t), only the sign of K actually matters.
According to Eq. (1.6), physical distances in the universe are proportional to the
scale factor a(t), so the Hubble law in Eq. (1.1) can be rewritten as
ȧ = Ha ,
(1.7)

Chapter 1: Introduction
where the dot represents a total time derivative. Moreover, the redshift can be
expressed as
∆λ
a(tre ) − a(tem )
z=
=
= a−1 − 1 ,
(1.8)
λ
a(tem )
so it is simply a consequence of the expansion of spacetime and determined by the
ratio of the scale factor at the time of reception tre and the time of emission tem of
light. Since the light is observed today, tre = t0 , and we fix a(t0 ) ≡ a0 = 1.
Equation (1.6) is a general solution to Eq. (1.2) obeying homogeneity and
isotropy without specification of any particular form of energy density Tµν . However, we know that the universe is filled with nonrelativistic matter in the form
of stars or galaxies, as well as relativistic matter in the form of radiation. On large
scales, its distribution is fairly homogeneous and isotropic, so it can be approximated
by a perfect fluid with energy-momentum tensor
Tµν = (ρ + p) uµ uν − pgµν ,
(1.9)
of energy density ρ and pressure p moving with four-velocity uµ . The relation between energy density and pressure is determined by the equation of state,
p = wρ ,
(1.10)
which for the most common forms of mass-energy can be expressed with is a constant
number w. Plugging this form of energy-momentum together with the metric from
Eq. (1.6) into Einstein’s field equations yields two relations for the scale factor of
the universe and its matter content. They are known as the Friedman equations,
ä
ȧ2
4π
(ρ + 3p) a ,
3
8π 2
=
ρa − K .
3
=−
(1.11)
(1.12)
The first Friedman equation relates the second time-derivative of the scale factor
with the energy density and the pressure of the matter in the universe. Assuming
cold (non-relativistic) matter with negligible pressure to be dominant, we expect a
deceleration of the expansion, ä < 0. The second Friedman equation reveals another
important fact: for a given expansion rate ȧ the energy-density of matter determines
the geometry of the universe (its curvature K). Hence, Eq. (1.12) defines a critical
energy-density
3H 2
,
(1.13)
ρc ≡
8π
which yields a flat geometry of spacetime (K = 0). Larger or smaller values of the
energy-density implicate a closed or an open geometry, respectively.
1.1.3. Inventory
According to the Friedman equations, astronomical observations of the local matter
content may reveal geometrical properties of the cosmos as a whole. Vice versa,
constraints on the universe’s curvature determine its total energy-density budget.
1.1. The Homogeneous Cosmos

Because there are different forms of matter, it is useful to define a density parameter
ρ
8πρ
=
,
(1.14)
ρc
3H 2
to distinguish the relative contributions to the energy density of various different
components. In terms of today’s density parameter Ωi,0 of component i, Eq. (1.12)
can be recast into
2 X
H
=
Ωi,0 aγi + ΩK,0 a−2 ,
(1.15)
H0
i
P
where γi is the scaling index of component i and ΩK ≡ −K/H 2 a2 = 1 − i Ωi
the relative curvature density. An index of zero denotes quantities evaluated today
(z = 0). In the following, some of the most common such components are discussed
briefly.
Ω≡
1.1.3.1. Dust
Non-relativistic matter is often referred to as dust, because it has negligible pressure.
Solving the Friedman equations of the flat universe (K = 0) for ρ and a(t) assuming
w = 0 in Eq. (1.10) yields
ρm = ρm,0 a
−3
,
a(t) =
2/3
3
H0 t
2
(1.16)
and the age of the universe is t0 = 3H2 0 . An example for a dust-like fluid is cold
dark matter, since it mainly interacts gravitationally. On large scales, baryons also
behave dust-like, as their electromagnetic interactions can be neglected.
1.1.3.2. Radiation
Ultra-relativistic matter has a nonzero pressure with equation of state parameter
w = 1/3. In a flat radiation-dominated universe the expansion is characterized by
ρr = ρr,0 a−4
,
a(t) = (2H0 t)1/2
(1.17)
Hence, compared to the dust-dominated universe, the expansion is slower, yielding
an age of merely t0 = 2H1 0 . An example for ultra-relativistic matter is electromagnetic
radiation or neutrinos.
1.1.3.3. Cosmological Constant
The Einstein field equations allow the inclusion of an additional term Λgµν on the
left-hand side of Eq. (1.2), with Λ being a so-called cosmological constant. When
absorbed into Tµν on the right-hand side, it can be interpreted as a form of vacuum
energy density with equation of state parameter w = −1. Then, the Friedman
equations for K = 0 yield
ρΛ ≡ ρΛ,0 =
Λ
3H02
,
a(t) = exp(H0 t)
(1.18)

Chapter 1: Introduction
Note that both energy density and Hubble parameter H ≡ ȧ/a remain constant in
time, so there is no evolution. This kind of solution was first studied by Willem de
Sitter [12], it has the remarkable property that it describes the same spacetime for
any type of curvature.
1.1.3.4. Scalar Field
A more exotic form of matter is a classical scalar field. Unlike baryons, a scalar
field only has a single degree of freedom, which is given by its field value ϕ. If it is
homogeneous, its energy density and pressure can be expressed as
1
ρϕ = ϕ̇2 + V (ϕ) ,
2
1
pϕ = ϕ̇2 − V (ϕ)
2
(1.19)
Thus, the potential V (ϕ) determines its equation of state parameter w and in general
ρϕ = ρϕ,0 a
−3(1+w)
,
2
3(1+w)
3(1 + w)
a(t) =
H0 t
2
(1.20)
for w > −1. If ϕ̇ = 0, we have w = −1 and the properties of a cosmological constant
are restored. In principle, w can change as a function of time, the scalar field is then
sometimes dubbed quintessence.
1.1.4. Observational Status
Astronomical observations of the homogeneous universe aim at constraining the
free parameters of the cosmological standard model presented above. These are the
various energy-density components Ωi , the effective equation of state parameter w
from all components, the relative curvature density ΩK and the expansion rate H0 .
A measurement of distance and redshift of objects in the universe following the
large-scale Hubble flow allows to constrain the cosmic expansion history and thus
the parameters that go into it. Although it is generally hard to infer astronomical
distances on such large scales, a special class of objects denoted as standard candles
allow a fairly accurate determination of their absolute luminosity.
The most famous example are supernovae of type Ia that have revealed a latetime acceleration of the universe [37] (ä > 0), suggesting the energy density today to
be dominated by a cosmological constant with ΩΛ,0 ' 0.7 and w ' −1. Furthermore,
the data favor a flat universe with Ωm,0 ' 0.3 and ΩK consistent with zero. The expansion rate today is constrained to be somewhere close to H0 ' 70 km s−1 Mpc−1 ,
which yields the age of the universe to be roughly 14 billion years. Due to degeneracies with other cosmological parameters, the value of H0 is often left unspecified
with the dimensionless Hubble parameter
h≡
H0
.
100 km s−1 Mpc−1
(1.21)
In order to lift degeneracies in the parameter space of the standard model, independent probes of cosmology are necessary. The theory of nucleosynthesis and observations of the abundance of light elements in the cosmos is very sensitive to the cosmic
1.1. The Homogeneous Cosmos

baryon- and photon-content Ωb and Ωγ of the universe, but does neither have strong
implications on its dark components, nor on its geometry. The same is true for the
theory of recombination and observations of the spectrum of the cosmic microwave
background (CMB) radiation, first discovered by Penzias and Wilson [36].
However, a sufficiently accurate spatial resolution of the CMB reveals its
anisotropy [50] and provides a wealth of new information on cosmology. Similarly,
galaxy redshift surveys aiming at reconstructing the fluctuations of the density field
of large-scale structure (LSS) achieve a similar goal [11], with the advantage of covering three dimensions instead of only two in the case of CMB observations. In
order to exploit the information that is available with these new probes, the theory
of perturbations in a slightly inhomogeneous universe is presented in the following
section.

Chapter 1: Introduction
1.2. The Inhomogeneous Cosmos
The universe we observe in our vicinity (on small scales) is far from homogeneous
and isotropic. Conglomerates of galaxies form gravitationally bound systems such
as clusters, filaments and walls surrounding large regions of empty space, known
as voids. This so-called cosmic web of structure could not have formed in a perfectly homogeneous universe, initial perturbations must have caused gravitational
instabilities leading to its present appearance.
The CMB provides the earliest “snapshot” of perturbations present in the universe and thus the closest connection to what their initial properties must have been.
In this section we will briefly sketch the evolution of those perturbations from their
creation in the very early universe until today and discuss their properties.
1.2.1. Random Fields
The structure of a slightly inhomogeneous universe can be described by a scalar
function f (x), representing the energy density, the temperature, or the gravitational
potential, for example. For practical reasons it is often useful to consider the Fourier
coefficients fk of that function. In a finite region of volume V , they are defined via
1 X
fk exp(ik · x) ,
f (x) = √
V k
where k is denoted wavevector. The inversion of Eq. (1.22) is given by
Z
1
f (x) exp(−ik · x) d3 x .
fk = √
V V
(1.22)
(1.23)
In general, the Fourier coefficients are complex numbers fk = ak + ibk with real and
imaginary parts ak and bk , respectively. Because f (x) is real, the complex conjugate
fk∗ must be equal to f−k according to Eq. (1.23), so we have a−k = ak and b−k = −bk .
Suppose the field f (x) is a realization of a homogeneous and isotropic Gaussian
random process. Then, the joint probability distribution function for the real and
imaginary parts of its Fourier coefficients will be a bivariate Gaussian [32]
2
2
ak
1
b
p(ak , bk ) =
exp − 2 exp − k2 ,
(1.24)
2
πσk
σk
σk
with variance σk2 only depending on the magnitude k ≡ |k| of the wavevector (the
wavenumber), not its direction. This means, when two Fourier coefficients of f (x)
are correlated by taking the ensemble average hfk fk0 i, this only yields a contribution
from the variance of ak and bk and their covariance vanishes,
K
hfk fk0 i = hak ak0 i − hbk bk0 i + i (hak bk0 i + hak0 bk i) = σk2 δk,−k
0 ,
(1.25)
K
where δk,−k
0 is the Kronecker symbol. Hence, real and imaginary parts are statistically independent, just as the coefficients at different wavevectors k 6= k0 . The limit
1.2. The Inhomogeneous Cosmos

V → ∞ of Eqs. (1.22) and (1.23) defines the Fourier transform f (k) via the integrals
Z
f (x) = f (k) exp(ik · x) d3 k ,
(1.26)
Z
1
f (k) =
f (x) exp(−ik · x) d3 x ,
(1.27)
(2π)3
and the ensemble average of Eq. (1.25) defines the power spectrum Pf (k),
hf (k)f (k0 )i ≡ Pf (k)δ D (k + k0 ) ,
(1.28)
where δ D (k + k0 ) is the Dirac delta function. The power spectrum describes the
variance of fluctuations in the function f in an infinitesimally thin shell of radius k.
In the case of a purely Gaussian process, it completely characterizes the properties
of the random field, i.e. all higher-order correlations can be expressed in terms of the
power spectrum. Since it has the dimension of a volume, it is sometimes convenient
to consider the dimensionless power spectrum
∆f (k) ≡
k3
Pf (k) .
2π 2
(1.29)
Taken in configuration space, the ensemble average from Eq. (1.28) defines the twopoint correlation function of f ,
hf (x)f (x0 )i ≡ ξf (x0 − x) .
(1.30)
Due to statistical homogeneity and isotropy, ξ only depends on the distance r ≡
|x0 − x| between any two locations. With Eqs. (1.26) and (1.28) it follows that the
correlation function and the power spectrum are Fourier pairs,
Z
ξf (r) = Pf (k) exp(ik · r) d3 k .
(1.31)
1.2.2. Initial Perturbations
1.2.2.1. Inflation
It is by now widely accepted that the very early universe must have undergone a
phase of inflation, closely described by an exponentially expanding spacetime as
proposed by de Sitter. The main motivation for inflation comes from its solutions to
the horizon- and the flatness problems, as pointed out by Alan Guth [18]. The former
arises, because with a decelerating expansion many different regions in the observed
universe would have never been in causal contact with each other in the past, posing
the question of why then we observe such a high degree of isotropy, for example in
the CMB. The latter comes about the question of why the observed curvature of
the universe is so small, respectively consistent with zero. In a decelerating universe
the initial expansion rate must have been extremely fine-tuned in order to produce
the observed flatness.

Chapter 1: Introduction
One way to realize accelerated expansion is to assume a scalar field ϕ that is evolving
sufficiently slowly to achieve a negative pressure, see Eq. (1.19). The evolution of a
scalar field is dictated by the Klein-Gordon equation
√
1
∂
∂V
µν ∂ϕ
√
−gg
+
=0,
(1.32)
µ
ν
−g ∂x
∂x
∂ϕ
which follows from Eqs. (1.5) and (1.19) with g ≡ det gµν . From it, the so-called slowroll conditions for the potential V (ϕ) necessary for inflation can be derived [32],
2
2
∂ V /∂ϕ2 ∂V /∂ϕ
1.
1 , (1.33)
V
V
In this regime the scalar field can resemble a cosmological constant with w ' −1
and yield an exponentially growing scale factor. However, our existence tells us that
inflation has to end at some point, so de-Sitter spacetime is only an approximation
to the inflationary phase. It comes to an end when the scalar field decays into the
minimum of its potential and evolves faster due to oscillations. Then its equation of
state changes, because the potential term in Eq. (1.19) can become comparable to
the kinetic term ϕ̇2 and the universe may enter a decelerated phase with w ' 0.
1.2.2.2. Quantum Fluctuations
Considering a mildly inhomogeneous scalar field, we can separate ϕ into a homogeneous part ϕ0 (t) and a linear perturbation δϕ(x, t). These perturbations are generated by quantum fluctuations of the scalar field in the very early universe. From
Heisenberg’s uncertainty principle for a massless scalar field it follows that the minimal amplitude of δϕ(R) on a given scale R is inversely proportional to that scale,
δϕ(R) ∼ 1/R .
(1.34)
In Fourier space, with Eq. (1.22) the typical amplitude of quantum fluctuations of
comoving wavenumber k ∼ a/R becomes
|δϕ(k)| ∼ k −1/2 a .
(1.35)
Because the creation- and annihilation operators for the field quanta permute for
wavemodes with different values of k, these wavemodes are independent of each
other with random relative phases. Thus, according to the central limit theorem,
a superposition of many wavemodes yields a Gaussian random field with variance
|δϕ(k)|2 .
1.2.2.3. Primordial Potential
The fluctuations of the scalar field induce perturbations in the metric gµν from
Eq. (1.6), resulting in the so-called perturbed Friedman metric. In conformalNewtonian gauge, it can be expressed as
ds2 = a2 (1 + 2Φ)dη 2 − (1 − 2Ψ)δij dxi dxj ,
(1.36)
1.2. The Inhomogeneous Cosmos

with the scalar perturbations Φ and Ψ and conformal time η defined as dη ≡ dt/a.
Here, roman indices denote only spatial coordinates and δij is the spatial unit metric. Plugged into Eq. (1.32), this relates the perturbation δϕ and its evolution to
the generalized potentials Φ and Ψ, which are identical for a perfect fluid without
anisotropic stress. With initial amplitude given in Eq. (1.35), the solution for the
perturbation in Fourier space reads [32]
( −1
a exp(±ikη) ,
for k Ha
(1.37)
δϕ(k) ∼ k −1/2 ∂V /∂ϕ −1 V a ∂V /∂ϕ
, for k Ha
V
k=Ha
On subhorizon scales (k Ha), the scalar field perturbation oscillates and its
amplitude decays inversely with the scale factor. However, during the inflationary
phase ȧ = Ha increases with time and a perturbation with given k eventually leaves
the horizon. On superhorizon scales (k Ha) the perturbation freezes out to a
roughly constant value. Only when inflation ends and the slow-roll conditions from
Eq. (1.33) are violated, ∂VV/∂ϕ increases to unity.
Finally, the universe enters the decelerated phase (ȧ < 0) and wavemodes of a
given scale k reenter the horizon. Without an inflationary phase, wavemodes from
originally small scales with large amplitude would decay immediately and could not
be preserved until today with sufficiently high amplitude (e.g. as observed in the
CMB). This is another strong argument for the theory of inflation.
From the solution of Eq. (1.32) we also obtain the potential Φ, which on superhorizon scales becomes
1
∂V /∂ϕ
Φ(k) = − δϕ(k)
.
(1.38)
2
V
At the end of inflation we have ∂V /∂ϕ ' V and the dimensionless power spectrum
of the potential becomes
2
V
∆Φ (k) ' H
.
(1.39)
∂V /∂ϕ k=Ha
Because H is nearly constant during inflation, ∆Φ has very weak scale dependence.
In fact, it can be shown that it is logarithmic [33] and within a narrow range of scales
is well approximated by a power law with amplitude A0 and spectral index nS ,
∆Φ ' A0 k nS −1 .
(1.40)
Inflation predicts a slight deviation from a flat spectrum with nS = 1, the so-called
spectral tilt [33]
3
d ln ∆Φ
nS − 1 ≡
'−
d ln k
8π
∂V /∂ϕ
V
2
+
1 ∂ 2 V /∂ϕ2
.
4π
V
(1.41)
For a massive scalar field with V ∝ ϕ2 , the result is nS ' 0.96. Thus, the considered inflationary scenario makes strong predictions on the properties of the initial
fluctuations in our universe: they can be described by a Gaussian random field of
adiabatic perturbations with a nearly scale-invariant spectrum. So far, this is in very

Chapter 1: Introduction
good agreement with observations of the CMB [25]. However, if any of these properties can be ruled out by measurement, the theory of single field slow-roll inflation
likely has to be extended.
One such possible extension is the introduction of additional inflationary fields
and thus entropy perturbations. Couplings between the Fourier modes of different
fields can break the Gaussian character of fluctuations and produce so-called primordial non-Gaussianity (PNG). This fact makes the detection of PNG a smoking
gun for inflation.
1.2.3. Evolution of Perturbations
The fluctuations of the scalar field produced during the inflationary phase of the
universe are not directly observable. We can only observe their relics that have
evolved after reentering the horizon in the decelerated phase. The CMB provides
the earliest picture of the angular energy density distribution in the universe, when
the perturbations are still relatively small and evolve linearly. However, by the time
of recombination the initial perturbations have translated into fluctuations in the
energy density of various different components, such as dark matter, baryons and
radiation.
1.2.3.1. Hot Phase
In order to describe their properties at the time of recombination, it is necessary to
solve the Boltzmann equations for an imperfect fluid with relativistic and nonrelativistic components. Although this cannot be done analytically, numerical methods
yield a high accuracy [46]. The solution is usually expressed in the form of a transfer
function T (k) and the linear growth factor D(t) that multiplies the initial fluctuation
field f (k, ti ) at the time ti when it enters the horizon,
f (k, t) = T (k)D(t)f (k, ti ) .
(1.42)
For adiabatic perturbations, the asymptotics of T (k) are qualitatively easy to understand. Like in the case of the scalar field, fluctuations in the energy density of
both relativistic and nonrelativistic matter are frozen (constant) on superhorizon
scales. A wavemode entering the horizon during radiation domination grows more
slowly compared to a mode entering during dust domination (see below). Therefore,
T (k) is suppressed for wavenumbers kdeq 1, where deq is the comoving horizon
at dust-radiation equality. At kdeq 1 the transfer function approaches a constant
value, which is normalized to unity for convenience.
Additionally, the competition between gravity and pressure in the hot plasma of
baryons and radiation causes standing waves in their energy density, so-called acoustic oscillations. They are only produced within the horizon before dust-radiation
equality, so for wavenumbers with kdeq > 1, and can be directly observed in the radiation component via the CMB [25]. The structure and location of these oscillations
is very sensitive to cosmological parameters, such as ΩK , Ωm and Ωb .
1.2. The Inhomogeneous Cosmos

1.2.3.2. Cold Phase
After recombination, photons propagate freely and the acoustic oscillation signature
in the radiation component is washed out. Thanks to gravitational interaction, the
oscillations in the baryonic component are carried over to the dominant dark matter
distribution and thus survive until the present day. These so-called baryon acoustic
oscillations (BAO) have been detected recently in galaxy redshift surveys [14] and
serve as a standard ruler for cosmology.
In order to relate the observed density fluctuations in the present universe to
their initial conditions after recombination, one has to understand their gravitational evolution. The dark matter density ρm , respectively the fluctuation around
its mean ρ̄m ,
ρm (x, t) − ρ̄m (t)
,
(1.43)
δ(x, t) ≡
ρ̄m (t)
is related to the gravitational potential by the Poisson equation
∆Φ = 4π ρ̄m δ ,
(1.44)
which is only valid on subhorizon scales in the Newtonian regime. In Fourier space
with Φ given by Eq. (1.40), this yields
√
A0 nS −1
2
|δ(k)| = k
k 2 .
(1.45)
4π ρ̄m
Taking into account the effects of baryons and radiation described by the transfer
function T (k), the dark matter power spectrum at recombination can be written as
Pδ (k, trec ) =
A0 2
D (trec )T 2 (k)k nS .
8ρ̄2m
(1.46)
Because it only interacts gravitationally, the dark matter density field can be described as a perfect fluid in an expanding background. As long as δ 1, it obeys
the linearized Continuity equation
∂δ ∇ · v
+
=0,
∂t
a
(1.47)
and Euler equation
∂v
∇Φ
+ Hv +
=0,
(1.48)
∂t
a
where v is the peculiar velocity field and derivatives are taken with respect to Lagrangian coordinates moving with the Hubble flow. Here, pressure has been neglected, which is a fair approximation on large scales where |v| 1. Together with
Eq. (1.44), this yields a closed form differential equation for the dark matter density
perturbation,
∂ 2δ
∂δ
+ 2H
= 4π ρ̄m δ .
(1.49)
2
∂t
∂t
Its solution can be expressed as
δ = D+ (t)δ (+) (x, ti ) + D− (t)δ (−) (x, ti ) ,
(1.50)

Chapter 1: Introduction
with the linear growth factors D+ and D− of the growing and decaying modes,
respectively, and the corresponding initial density field configurations δ (+) and δ (−) .
With Eq. (1.47), we can solve for the divergence of the peculiar velocity field
∇ · v = −aH f+ δ (+) (x, ti ) + f− δ (−) (x, ti ) ,
(1.51)
where f ≡ d ln D/d ln a is denoted as logarithmic growth rate. In the following we
can neglect the decaying mode and refer to the growing mode with a single growth
factor D and growth rate f .
The growth of perturbations depends on the expansion rate of the universe,
which in turn is dictated by its constituents. With H taken from Eq. (1.15) and
plugged into Eq. (1.49), one can in principle obtain the growth factor and rate for
an arbitrary cosmology. While a general analytic solution does not exist, expressions
for the following limiting cases can be derived [32]:
(i) Dust domination (Ωm Ωr , ΩΛ , Ωk ): D(t) ∝ a(t)
(ii) Radiation domination (Ωr Ωm , ΩΛ , Ωk ): D(t) ∝ ln a(t)
(iii) Cosmological constant domination (ΩΛ Ωm , Ωr , Ωk ): D(t) ∝ const.
Hence, gravity can enhance density perturbations substantially only in the dustdominated regime, where they grow linearly with the scale factor and the growth
rate is close to unity. As soon as a cosmological constant term becomes important
in the energy budget of the universe, the growth of perturbations is stalled and
the growth rate drops below unity. In a flat universe only filled with dust and
cosmological constant, it can be neatly parametrized by
f ' Ωγm ,
(1.52)
with γ ' 0.6 [35] denoted growth index. This parametrization turns out to be more
general and valid for various different models of dark energy with γ still being consistent with the above value [29]. Even modified gravity theories obey Eq. (1.52),
but in general predict a growth index distinct from 0.6. This indicates that measurements of peculiar velocities allow tests of general relativity on cosmological scales,
which makes them very attractive.
1.2.3.3. Nonlinear Regime
When perturbations in the density field reach values of δ ∼ 1, higher-order terms
such as δ 2 become important and the above linear treatment breaks down. It is very
useful to develop a perturbative approach for the evolution of the density field, as it
allows to extend analytical predictions into the semilinear regime where δ approaches
unity [3]. However, regions in the universe with δ ≥ 1 evolve nonlinearly and become
non-perturbative.
A simple example is the collapse of a spherical overdensity δsc of pressureless
matter in a flat dust-dominated universe. The overdense region can itself be treated
1.2. The Inhomogeneous Cosmos

as a universe with Ωm = 1 + δsc > 1, implying ΩK < 0 and thus a positive curvature.
In this case, the solution of Eq. (1.12) can be written in parametric form as
a(η) =
ata
(1 − cos η)
2
,
t=
tta
(η − sin η) ,
π
(1.53)
where ata ∝ Ωm /(Ωm − 1) is the maximal scale factor reached at time tta , after
which the overdense region collapses (turn-around ). At this time the overdensity
reaches its minimal value, which, in order to collapse, must be at least δsc (tta ) + 1 =
9π 2 /16 ' 5.55. After turn-around, the overdensity detaches from the Hubble flow
and increases towards a singularity at t = 2tta . In reality, this singularity is never
realized due to deviations from exact spherical symmetry. Instead, by this time
the dark matter particles making up the overdense region will form a virialized
stable object, denoted dark matter halo. Application of the virial theorem yields
δsc (2tta ) + 1 = 18π 2 ' 178.
The nonlinear character of Eqs. (1.53) becomes evident when we consider their
Taylor expansion only up to linear order. Then, the overdensity at turn-around
becomes δsc,lin (tta ) ' 1.06 and δsc,lin (2tta ) ' 1.69. Hence, when linear theory is
applied to evolve the density field, any regions exceeding the critical value δc ≡ 1.69
will collapse to form virialized structures.
1.2.4. Formation of Large-Scale Structure
Evidently, the formation of structure in the universe is a fairly complex and nonlinear
process. The typical amplitude of initial dark matter fluctuations as in Eq. (1.46)
increases with k, suggesting nonlinear processes to happen earlier on smaller scales.
This phenomenon is referred to as hierarchical clustering and allows the application
of smoothing, i.e., the removal of fluctuations below some threshold scale R. In
Fourier space, the smoothed density field can be expressed as
δR (k) = W (k, R)δ(k) ,
(1.54)
where W is a window function of smoothing scale R. Usually, a tophat window in
configuration space with Fourier transform
W (k, R) = 3
sin(kR) − kR cos(kR)
,
(kR)3
(1.55)
is used, but depending on the application, a Gaussian window or a tophat window in Fourier space are common, too. The variance of the smoothed field is then
determined by applying Eq. (1.31) at r = 0,
Z
2
2
σ (R) ≡ hδR (x)i = Pδ (k) |W (k, R)|2 d3 k ,
(1.56)
it increases with decreasing smoothing scale R. Thus, the procedure of smoothing
allows to control the degree of nonlinearity in the smoothed field δR by erasing
nonlinear structure below that scale.

Chapter 1: Introduction
1.2.4.1. Halo Mass Function
Restricting to a range of scales where the probability distribution function of fluctuations is still well described by a Gaussian
δR2
1
exp − 2
p(δR ) = p
,
(1.57)
2σ (R)
2πσ 2 (R)
this allows to estimate the number of collapsed structures in a smoothed field of
given R as follows [38]. Regions in the universe whose variance exceeds the critical
overdensity for collapse, σ(R) > δc , form virialized objects of mass M ∝ R3 . The
probability for such a region is obtained by integrating Eq. (1.57),
Z ∞
1
ν
p(δR ) dδR = erfc √
,
(1.58)
F (ν) ≡
2
2
δc
yielding the complementary error function, which depends on the so-called peak
height ν ≡ δc /σ(R). Thus, the number of collapsed objects between masses M and
M + dM is n(M )dM = ρ̄Mm |dF (ν)| and with Eq. (1.58) we obtain
2
1 ρ̄m dν
ν
n(M ) = √
exp −
.
(1.59)
2
2π M dM
The typical mass M? of halos collapsing at present is defined via ν(M? ) = 1, respectively σ(M? ) = δc . It distinguishes two regimes of the mass function: a power law at
M M? and an exponential drop-off at M M? .
Thanks to its simplistic assumptions, this approach yields a qualitatively good
description of halo formation, but fails to accurately reproduce the halo mass functions studied in simulations. One caveat is the so-called cloud-in-cloud problem,
referring to the fact that a spherical region below the threshold of collapse may
still virialize when located within a larger region above threshold. Taking this into
account amounts to an additional factor of 2 in Eq. (1.59) [6]. Furthermore, the
treatment of halos forming from only spherical overdensities can be extended to the
more general case of ellipsoidal collapse. This results in a collapse threshold that is
no longer constant, but a function of the shape of the overdense region and leads to
a more accurate halo mass function as compared with simulations [47].
1.2.4.2. Halo Bias
To some degree, the large-scale distribution of dark matter halos in the universe
can itself be treated as a continuous field, just as the distribution of dark matter
particles as in Eq. (1.43). In this case the overdensity of halos is defined as the
relative fluctuation in the number of halos around its mean,
δh (x) ≡
n(x) − n̄
.
n̄
(1.60)
The spatial variation of n(x) is caused by dark matter density fluctuations δl on large
scales, which have not turned nonlinear yet. They can be obtained by smoothing
1.2. The Inhomogeneous Cosmos

over a sufficiently large scale by application of Eq. (1.54). Regions with δl > 0 will
then contain more collapsed objects than slightly underdense regions with δl < 0,
because nonlinear small-scale perturbations δs reach the collapse threshold δc earlier
there. In this manner (referred to as the peak-background split formalism [8]), the
initial halo density field can be related to the long-wavelength perturbations of the
dark matter as
n(x) ' n̄ (1 + bL δl ) ,
(1.61)
where bL is a proportionality factor referred to as linear Lagrangian bias. With an
explicit expression for the mass function, it can be calculated as [31]
bL = n̄−1
ν2 − 1
∂n
'
,
∂δl
δc
(1.62)
where in the second step Eq. (1.59) was approximated to linear order. A better
model for the mass function also yields a more accurate formula for bL , for example
in the framework of ellipsoidal collapse [48]. Plugged into Eq. (1.60), this reveals
that dark matter and halo overdensities are proportional on large scales, δh = bL δ,
and the proportionality constant bL increases with halo mass. Because a region with
positive long-wavelength perturbation δl expands more slowly than the homogeneous
background, its final volume becomes smaller, respectively the overdensity of halos
larger and vice versa. With the help of Eqs. (1.53), the final halo density field
becomes [31]
δh = (bL + 1) δ ≡ bδ ,
(1.63)
where b is the so-called linear Eulerian bias, according to Eq. (1.62) it is an increasing
function of halo mass and redshift. Objects forming at the present epoch with typical
mass M? have b(M? ) = 1. Because the linear Eulerian bias is independent of scale,
Equation (1.63) yields a very simple relation between the clustering statistics of
halos and dark matter, both in configuration- and in Fourier space. For instance, to
linear order their two-point correlation functions and power spectra are related as
ξδh (r) = b2 ξδ (r) ,
Pδh (k) = b2 Pδ (k) .
(1.64)
When higher orders are considered in the derivation, this relation becomes more
complex. More generally, Eq. (1.63) can be written as a Taylor series in the dark
matter density [17],
∞
X
δh (x) =
bi δ i (x) ,
(1.65)
i=0
with bias parameters bi for each order i, which can be calculated analytically expanding Eq. (1.62) to corresponding order [9]. This ansatz is referred to as the local
bias model, as the halo overdensity is a function of the dark matter overdensity at
the same location in space. Note that higher powers of δ in Eq. (1.65) correspond to
convolutions of δ in Fourier space, meaning that scales of different size are no longer
independent due to mode coupling.

Chapter 1: Introduction
1.2.4.3. Halo Model
An alternative ansatz to describe the clustering statistics of halos and the dark
matter beyond linear order is provided by the halo model. The idea is to split the
power spectrum into two parts, a 2-halo- and a 1-halo term [43],
P (k) = P 2h (k) + P 1h (k) .
(1.66)
The former describes the clustering of different halos within large-scale structure,
while the latter only considers the clustering of matter within a single halo, which
can still yield contributions on large scales due to mode coupling. For the dark
matter power spectrum the two contributions are given by [9]
Z ∞
Z ∞
M
M0
2h
n(M ) U (k|M )
n(M 0 ) U (k|M 0 )b(M )b(M 0 )Pδlin (k) dM dM 0 ,
Pδ =
ρ̄m
ρ̄m
0
0
(1.67)
2
Z ∞
M
Pδ1h =
n(M )
|U (k|M )|2 dM ,
(1.68)
ρ̄m
0
where U (k|M ) is the normalized Fourier space density profile of a dark matter halo
with mass M and Pδlin the linear-order dark matter power spectrum as given in
Eq. (1.46). For the halo power spectrum the term M/ρ̄m is exchanged by 1/n̄ and
the integration limits have to be adjusted to the mass range that is considered. The
halo model provides a simple phenomenological picture for the clustering statistics
of large-scale structure and is able to reproduce simulation results fairly well even
in the nonlinear regime [49].
1.2.4.4. Stochasticity
The assumption that the distribution of halos can be treated as a continuous field
is only accurate in the high number density limit. However, in a given volume V
the number of virialized objects above a certain mass is finite, because only regions
above threshold will collapse, whereas regions below threshold remain devoid of
halos. Therefore, halos sample the dark matter density field in a discrete fashion,
resulting in a certain degree of scatter in the relation between their overdensities.
For this reason halos are considered being not only biased, but also stochastic tracers
of the dark matter, and Eq. (1.65) must be complemented with an additional noise
field [13],
∞
X
δh (x) =
bi δ i (x) + (x) .
(1.69)
i=0
Since is simply defined as a residual of the Taylor series in δ, it is uncorrelated
with the latter, i.e. h(x)δ(x0 )i = 0 ∀ x, x0 . Assuming to arise from a uniform
Poisson point process [35], it is also spatially uncorrelated with itself, which means
h(x)(x0 )i = 0 ∀ x 6= x0 . However, the field has a finite variance due to its
Poissonian nature for the distribution of N points in a finite volume V ,
2
Var(N )
N̄
1
N − N̄
2
i=
= 2 =
.
(1.70)
h (x)i = h
2
N̄
N̄
N̄
N̄
1.2. The Inhomogeneous Cosmos

This variance is commonly referred to as Poisson shot noise and results in an extra
contribution in the two-point clustering statistics of halos as in Eq. (1.64),
ξδh (r) = b2 ξδ (r) + N̄ −1 δ D (r) ,
Pδh (k) = b2 Pδ (k) + n̄−1 ,
(1.71)
where n̄ = N̄ /V is the average number density of halos in the considered volume.
Thus, Poisson shot noise leads to a scale-independent power with amplitude n̄−1 in
Fourier space, which is why it is also referred to as white noise.
Note that the 1 -halo term of the halo model in Eq. (1.68) already incorporates
this behavior on large scales, where U (k|M ) → 1. However, for the dark matter
power spectrum this is unphysical due to local mass and momentum conservation
of the dark matter particles. Suppose a uniform distribution of particles is split into
a large number of small cells, such that the particles in each cell will collapse onto a
point within the cell’s center of mass. Then, a Taylor expansion of the exponential
in Eq. (1.23) reveals the lowest non-vanishing order in the Fourier mode of the field
to be δk ∝ k 2 , so Pδ (k) ∝ k 4 on large scales [35, 53].
1.2.4.5. Galaxies
The structure of the universe that we observe today consists of galaxies made from
baryonic matter, a fact that has been completely neglected so far. Nevertheless,
due to the overabundance of dark matter it seems reasonable to assume baryons to
behave as test particles in the gravitational potential of the dark matter and galaxies
to form deep inside its potential wells, the dark matter halos [54].
As opposed to halos, the formation of galaxies is subject to electromagnetic interactions among baryons. This causes new physical phenomena to appear, such as
star formation, radiation pressure and gas cooling. Thus, a self-consistent analytical
treatment of structure formation is much harder to achieve and it is more convenient to resort to semi-analytic models for galaxy formation. One such approach is
known as halo occupation distribution (HOD) modeling. The idea is to populate dark
matter halos with galaxies in a probabilistic fashion, with probability distributions
constrained both numerically and observationally.
One generally distinguishes between two such distributions, one for central galaxies that reside close to the center of mass of their parent halo, and satellite galaxies
orbiting the latter. While there is typically one central galaxy in each halo above a
given mass threshold, the number of satellites scales with the host-halo mass [58].
Therefore, it is reasonable to describe the clustering of galaxies by means of the halo
model, with satellites playing the role of bound dark matter particles in the halo
and central galaxies being a proxy for the halo’s center of mass.
1.2.4.6. Redshift-Space Distortions
A complication in the description of galaxy clustering arises observationally, due
to the fact that radial distances of galaxies are inferred via their redshift z and
the Hubble law, Eq. (1.1). However, besides their motion within the Hubble flow,
galaxies acquire peculiar velocities v from the mutual gravitational interaction of

Chapter 1: Introduction
their dark matter halos. Neglecting those results in a redshift-space coordinate s that
differs from their real space coordinate r by
s=r+
r̂ · v
r̂ ,
H
(1.72)
where r̂ ≡ r/|r| is the unit vector along the line of sight. The volume element in the
mapping between the two coordinate frames changes as
2 d r̂ · v
r̂ · v
3
1+
d3 r ,
(1.73)
d s= 1+
Hr
dr H
(s)
but the number of objects is invariant, so δg d3 r = δg d3 s must hold for the overdensities of galaxies in real- and redshift-space, respectively. To linear order, a planewave perturbation with wavevector k k v in the distant-observer approximation, i.e.
kr 1, yields [23]
(1.74)
δg(s) (k, µ) = b + f µ2 δ(k) ,
where µ ≡ k̂ · r̂, galaxies are assumed to be linearly biased with respect to the
dark matter overdensity field δ and Eq. (1.51) has been used to relate the velocity
perturbation v with the latter. Therefore, the galaxy redshift-space power spectrum
becomes anisotropic and can be written as
2
Pδg(s) (k, µ) = 1 + βµ2 b2 Pδ (k) ,
(1.75)
with the observable redshift-space distortion (RSD) parameter β ≡ f /b. Equations (1.74) and (1.75) are only accurate on linear scales, where peculiar velocities are generated from large-scale flows towards overdense regions (Kaiser effect).
On nonlinear scales, random motions are generated in the process of virialization
which washes out power along the line of sight, the so-called Finger-of-God effect.
It is often modeled phenomenologically by an additional Gaussian damping factor
exp(−k 2 µ2 σv2 /2) in Eq. (1.74), with σv being the velocity dispersion [34].
1.3. Scientific Tools

1.3. Scientific Tools
The increasing wealth and complexity of contemporary cosmological data sets requires high logistical demands for their manipulation and analysis. Computers have
become inevitable for performing computationally intensive tasks, either for reducing
the relevant scientific output from observations to a manageable size, or to reproduce complex experiments in the most realistic way. This section is devoted to the
scientific tools that are commonly being utilized in the scientific community for the
analysis of large-scale structure.
1.3.1. Statistical Techniques
Every observation of nature is affected by uncertainty to some degree, no theory
can ever be proven wright, it can only be rejected or updated by a new one. The
selection process to find the most accurate description of the observation thus has
to be probabilistic, irrespective of the universe’s own stochastic nature.
1.3.1.1. Parameter Estimation
In general, a given set of observed data d is described by a model with parameters θ.
The task is to find the posterior probability distribution p(θ|d) of θ given the data
d and to extract the most likely value of the parameters including their uncertainties [19]. The model usually provides the opposite quantity p(d|θ), which is also
referred to as the likelihood L of the data, given the model. The two probability
distributions are related via Bayes’ Theorem [2],
p(θ|d) =
p(d|θ)p(θ)
,
p(d)
(1.76)
with so-called prior p(θ) and evidence p(d). For a given model, the evidence is
just a normalization constant and can be ignored. If the prior is assumed to be
flat, i.e. every parameter value equally likely a priori, then the posterior probability
distribution is proportional to the likelihood, p(θ|d) ∝ L (d, θ).
In this case, a Taylor expansion of the natural logarithm of the likelihood around
the true parameter values θ0 yields a multivariate Gaussian close to its peak,
1
|
(1.77)
L (d, θ) ' L (d, θ0 ) exp − (θ − θ0 ) H (θ − θ0 ) ,
2
2
where H is the Hessian of the log-likelihood, in index notation Hij ≡ − ∂∂θiln∂θLj . The
diagonal elements
of this matrix contain the conditional errors of the parameters
√
as σθi = 1/ Hii , but in general it also has non-vanishing off-diagonals, meaning
different parameters are correlated with each other. Therefore, the marginal errors
of the parameters are calculated via inversion of the Hessian,
q
σθi = (H−1 )ii .
(1.78)

Chapter 1: Introduction
1.3.1.2. Error Forecast
With a particular model at hand one can choose fiducial values for its parameters,
calculate their likelihood and according to Eq. (1.78) their uncertainties. So before
even collecting any data, with a reasonable guess for what the model parameters
should roughly be, it is possible to forecast the precision they can be determined
with. It can be shown that every unbiased estimator of the parameter θi satisfies
the Cramér-Rao bound [39]
q
σθi ≥
(F−1 )ii ,
(1.79)
with the so-called Fisher information matrix Fij [15] defined as the expectation
value of the Hessian,
∂ 2 ln L
Fij ≡ h−
i.
(1.80)
∂θi ∂θj
Suppose the likelihood of N measurements is a multivariate Gaussian in the data,
1
1
|
−1
√
L (θ, d) =
exp − (d − µ) C (d − µ) ,
(1.81)
2
(2π)N/2 det C
with mean µ ≡ hdi and covariance matrix C ≡ h(d − µ) (d − µ)| i depending on
the parameters θ. Plugged into Eq. (1.80) with a bit of algebra yields [19]
∂C −1 ∂C −1
1
−1
C
C + C Mij ,
(1.82)
Fij = Tr
2
∂θi
∂θj
|
|
∂µ ∂µ
∂µ ∂µ
+ ∂θ
. The Fisher matrix obtained from independent exwhere Mij ≡ ∂θ
i ∂θj
j ∂θi
periments can simply be added, a convenient property reflecting the informational
nature of this quantity.
1.3.2. Simulations
Numerical simulations have become an inevitable tool for studying the nonlinear
regime of structure formation. Thanks to the rapid increase in computer performance and the development of efficient algorithms, substantial progress in the field
of numerical cosmology has been achieved in the last two decades.
1.3.2.1. Particles
The idea is to solve nonlinear fluid equations with a discretized set of fluid elements,
the particles. From the classical point of view this is very natural, as baryons and
dark matter are described as particles on the fundamental level, as well. However, the
actual number of physical particles in the observable universe exceeds the computationally manageable one by many orders of magnitude, which limits the accuracy
of these so-called N -body simulations.
To set up the initial conditions, particles have to be distributed such as to satisfy
the background cosmology and the initial power spectrum of density fluctuations,
as in Eq. (1.46). The initial amplitude A0 of these fluctuations is constrained from
observations of the CMB, but traditionally it is also common to use σ8 , the linear
1.3. Scientific Tools

extrapolation of fluctuations to z = 0, smoothed according to Eq. (1.54) with R =
8h−1 Mpc, to normalize the initial power spectrum. A realization of the latter can be
generated by slightly displacing particles on a uniform lattice using the Zel’dovich
approximation [57], assuming linear growth as in Eqs. (1.50) and (1.51). Extensions
to this technique have been developed, utilizing perturbation theory to improve
accuracy beyond linear growth [42].
For collisionless particles, such as the dark matter, it is sufficient to evolve them
only gravitationally. On sub-horizon scales this amounts to calculating the gravitational forces between each particle and all its neighbors according to Newton’s
laws and updating their positions and velocities iteratively within an expanding coordinate system [4]. However, the efficiency of this so-called particle-particle (PP)
method quickly degrades as the number Np of particles increases: the amount of
computing steps scales as O(Np2 ). The high number of force calculations can be reduced by expanding the potential of extended regions in multipoles hierarchically
and truncating the series at sufficient order, resulting in a scaling of O(Np ln Np ) of
this so-called tree method.
Further improvements, especially in regard of memory consumption, are achieved
with the particle-mesh (PM) method. Here, particles are interpolated on a mesh
with Nc3 cells and the Poisson Eq. (1.44) is solved in k-space using a fast Fourier
transform (FFT) algorithm. In the next step, forces are interpolated back on the
particles, which are moved accordingly, and the whole procedure is reiterated. The
scaling of this algorithm is O(Np , Nc3 ln Nc ) and thus allows the highest number of
particles of all methods, with the cost of lower spatial resolution due to the mesh
interpolation.
A widely used state-of-the-art N -body code is is gadget-2, which uses a combination of the PM method and a tree algorithm to more accurately sample large
and small scales, respectively [51]. In addition to collisionless particles, it offers the
possibility to evolve gas particles that exhibit heat and pressure, a feature referred
to as smoothed particle hydrodynamics (SPH).
1.3.2.2. Halos
When the dark matter particles in a simulation are evolved deeply into the nonlinear
regime, they form a cosmic web of collapsed structures. The classification of these
structures into halos, filaments and walls is not unique and many identification
schemes have appeared in the literature. Two of the most common ones for halos
are the so-called spherical overdensity (SO) finder and the friends-of-friends (FoF)
finder. The former defines halos according to the spherical collapse model, localizing
spherical regions with overdensities above some critical threshold [26], while the
latter groups particles together that are separated by less than a specified linking
length [10]. Typically, it is chosen to be 20% of the mean inter-particle distance,
yielding a corresponding critical overdensity of δc,FoF ∼ 1/0.23 = 125.
Although neither of these methods guarantees all identified halo particles to be
gravitationally bound to a virialized structure, they yield reasonable agreement in
identifying halos in the high resolution limit. The SO method tends to miss aspherical
objects that are easily identified by the FoF finder. Conversely, the FoF finder may

Chapter 1: Introduction
link adjacent halos with distinct spherical overdensities to a single object. A number
of alternatives for defining halos have been proposed to overcome these ambiguities,
but a consensus on an optimal identification scheme has not been achieved so far [24].
1.3.3. Numerical Analysis
Most of the mathematical operations in the description of large-scale structure assume continuous and differentiable functions of time and space, such as the density
fluctuation δ(x, t) or the peculiar velocity field v(x, t). In numerical simulations however, everything is discretized, so the mathematical operations utilized so far have
to be modified accordingly.
1.3.3.1. Discrete Fourier Transform
Consider a cubic box of edge length L and volume L3 , divided into Nc3 equally sized
cubical cells. The discrete analog of Eqs. (1.26) and (1.27) can be formulated as
f (xn ) =
f (kn ) =
1 X
f (kn0 ) exp(ikn0 · xn ) ,
L3/2 n0
L3/2 X
f (xn0 ) exp(−ikn · xn0 ) ,
Nc3 n0
(1.83)
(1.84)
where xn = n NLc and kn = n 2π
with ni ∈ [1, Nc ] ∀i ∈ [1, 2, 3]. The resolution in
L
configuration space is limited by the cell size L/Nc , which corresponds to the Nyquist
wavenumber kN ≡ πNc /L in Fourier space.
1.3.3.2. Mesh Interpolation
The overdensity field of Np point particles can formally be written as a superposition
of Dirac delta functions
Np
1 X D
δ(x) =
δ (x − xi ) − 1 ,
Np i=0
(1.85)
where at the location xi of particle i, δ is infinite. In order to manipulate δ(x) using
mathematical operations involving derivatives, it has to be interpolated on a mesh.
This corresponds to a smoothing procedure as in Eq. (1.54) with a suitable window
function. A common choice is the so-called cloud-in-cell (CIC) window [20]
(
n|
1 − |x−x
, if |x − xn | ≤ L/Nc
L/Nc
WCIC (x − xn ) =
(1.86)
0
, otherwise
which is being convolved with the point distribution from Eq. (1.85) over the whole
volume V = L3 in configuration space,
δ(xn ) =
Z
V
Np
1 X
δ(x)WCIC (x − xn ) d x =
WCIC (xi − xn ) .
Np i=1
3
(1.87)
1.3. Scientific Tools

In Fourier space, the CIC window function of Eq. (1.86) reads
WCIC (kn ) =
3
Y
sin2 (πni /Nc )
i=1
(πni /Nc )2
,
(1.88)
so the structure introduced in the particle density field caused by the mesh interpolation can be easily corrected for by simply dividing out the window function in
Fourier space as suggested by Eq. (1.54). Due to discreteness this is not exact, but
an accurate correction can be achieved iteratively [22].
1.3.3.3. Estimators
With the interpolated overdensity field δ(kn ) in Fourier space, it is straight-forward
to compute an estimator P̂ for its power spectrum via shell averaging,
P̂δ (k) =
1 X
δ(kn )δ ∗ (kn ) ,
Nk ∆k
(1.89)
where ∆k denotes a shell in Fourier space containing all wavevectors kn with lengths
4πk2 ∆k
|kn |−∆k ≤ |kn | < |kn |+∆k. This amounts to a number of Nk ' (2π/L)
3 independent
P
1
Fourier modes in a bin of average wavenumber k = Nk ∆k |kn |.
Likewise, the estimator for the two-point correlation function of δ(xn ) can be determined as
1 X 1 X
ξˆδ (r) =
δ(xn0 )δ(xn0 + rn ) ,
(1.90)
Nr ∆r Nc3 n0
where all cells of mutual distance rn within the interval |rn | − ∆r ≤ |rn | < |rn | + ∆r
3
are summed over. Here, the normalization isP
given by the number of cells Nr ' 4πrL2 ∆r
in the bin of average separation r = N1r ∆r |rn |. The Fourier-pair relationship
between the power spectrum and the two-point correlation function from Eq. (1.31)
is only restored for their estimators in the limit ∆k, ∆r → 0 and Nc → ∞.

Chapter 1: Introduction
2
PAPER I

Chapter 2: Paper I
Summary The following letter describes a curious feature of dark matter halos in N -body simulations: As opposed to the Poisson assumption, the shot noise of
halos is found to be dependent on the mass of the considered halos, with values ranging from below to above the expected inverse halo number density 1/n̄. Moreover,
weighting each halo in the simulation by its mass (number of dark matter particles)
yields a considerable reduction to the shot noise measured in the unweighted halo
density field. The paper presents the results for various halo samples of different
mass range and number density and finds even higher suppression in shot noise
feasible with slightly modified weights. As a physical explanation it is argued that
local mass and momentum conservation of the halos is responsible for the observed
effects. Although strictly obeyed only by the dark matter particles themselves, mass
and momentum conservation may, to some degree, also apply to central galaxies in
halos. It is argued that a proper implementation of the weighting scheme in current
and future galaxy surveys may have the potential to significantly improve on parameter constraints in cosmology. A number of possible applications are mentioned,
such as measurements of baryon acoustic oscillations, the dark energy equation of
state, primordial non-Gaussianity and redshift-space distortions.
The input for this article was contributed by a collaboration of three authors.
Uroš Seljak initiated a project to study the stochasticity of halos in N -body simulations and proposed the presented estimator for the shot noise. He also suggested the
use of weights to observe how this estimator is affected. The numerical analysis was
carried out by myself on the basis of N -body simulations and halo catalogs provided
by Vincent Desjacques. The main part of the manuscript was written by Uroš Seljak,
I provided the figures. It got published in August 2009 in Physical Review Letters
(vol. 103, ID 091303, arXiv:0904.2963).
How to Suppress the Shot Noise in Galaxy Surveys
Uroš Seljak,1, 2, 3 Nico Hamaus,1 and Vincent Desjacques1
2
1
Institute for Theoretical Physics, University of Zurich, 8057 Zurich, Switzerland
Physics Department, Astronomy Department and Lawrence Berkeley National Laboratory,
University of California, Berkeley, California 94720, USA
3
Ewha University, Seoul 120-750, S. Korea
(Dated: August 26, 2009)
Galaxy surveys are one of the most powerful means to extract cosmological information and
for a given volume the attainable precision is determined by the galaxy shot noise σn2 relative to
the power spectrum P . It is generally assumed that shot noise is white and given by the inverse
of the number density n̄. In this Letter we argue one may considerably improve upon this due
to mass and momentum conservation. We explore this idea with N-body simulations by weighting
central halo galaxies by halo mass and find that the resulting shot noise can be reduced dramatically
relative to expectations, with a 10-30 suppression at n̄ = 4 × 10−3 (h/Mpc)3 . These results open up
new opportunities to extract cosmological information in galaxy surveys and may have important
consequences for the planning of future redshift surveys.
Galaxy clustering has been one of the leading methods to measure the clustering of dark matter in the past
and with upcoming redshift surveys such as SDSS-III and
JDEM/EUCLID this will continue to be the case in the
future. Galaxies are easily observed and by measuring
their redshift one can determine their three-dimensional
distribution. This is currently the only large scale structure method that provides three-dimensional information. On large scales galaxies trace the dark matter up to
a constant of proportionality called bias b, so the galaxy
power spectrum can be directly related to the dark matter power spectrum shape, which contains a wealth of
information such as the scale dependence of primordial
fluctuations, signatures of massive neutrinos and matter density, etc. In recent years the baryonic acoustic
oscillations (BAO) feature in the power spectrum has
been emphasized, which can be used as a standard ruler
and in combination with cosmic microwave background
anisotropies can provide a redshift distance test [1].
For the power spectrum measurement there are two
sources of error: one is the sampling (sometimes called
cosmic) variance, the fact that each mode is a Gaussian
random realization and all the cosmological information
lies in its variance, which cannot be well determined on
the largest scales because the number of modes is finite.
Second source of noise is the shot noise due to the discrete sampling of galaxies, σn2 , which under the standard
assumptions of Poisson sampling equals the inverse of the
number density n̄. The total error on the power spectrum
P is σP /P = (2/N )1/2 (1+σn2 /P ), where N is the number
of modes measured and scales linearly with the volume of
the survey. While the above expression suggests there is
not much benefit in reducing the shot noise to σn2 /P ≪ 1
since sampling variance error remains, recent work suggests there are potential gains in that limit, since we may
be able to reduce the damping of the BAO better [2].
Recently a new multitracer method has been developed
where by comparing two differently biased tracers of the
same structure one can extract cosmological information
in a way that the sampling variance error cancels out [3].
There are several applications of this method, such as
measuring the primordial non-Gaussianity [3], redshiftspace distortion parameter β [4] or relation between the
Hubble parameter and the angular diameter distance [4].
In all these applications one can achieve significant gains
in the error of the extracted cosmological parameters if
σn2 /P ≪ 1. Thus in all of these applications the galaxy
shot noise relative to the power spectrum is the key quantity that controls the ultimate level of cosmological precision one can achieve with galaxy surveys.
The relation between the galaxy and the dark matter
clustering can be understood with the halo model [5–7],
where all of the dark matter is divided into collapsed
halos of varying mass. There are two contributions to
the dark matter clustering: first is the correlation between two separate halos, which is assumed to be proportional to the linear theory spectrum times the product of the two halo biases, while the second contribution
is the one halo term which includes the clustering contributions from the individual halo itself. One obtains the
dark matter power spectrum prediction by adding up the
contributions from all the halos. Since galaxies are assumed to form inside the halos one can write analogous
expressions for galaxy clustering power spectra once one
specifies the occupation distribution of galaxies as a function of halo mass.
One consequence of the halo model is that the one halo
term is dominated by the most massive halos and reduces
to white noise k 0 for very small wave-mode amplitude
k ≪ R−1 , where R is the size of the largest halos. For
galaxies this is believed to be a valid description of the
shot noise amplitude in the low k limit. It distinguishes
ii
For the dark matter, the nonlinear evolution of structure requires local mass and momentum conservation and
as a result the low k limit of nonlinear contribution is predicted to scale as k 4 and not k 0 [9]. This is indeed seen in
simulations [10], making the prediction of the halo model
invalid. While this is often seen as a deficiency of the halo
model, here we take it as an opportunity: if the dark matter has no white noise tail in the k → 0 limit then in the
context of the halo model where all the dark matter is
in the halos and the halo size becomes irrelevant in the
k ≪ R−1 limit it should be possible to achieve the same
effect with galaxies, if one can enforce local mass and
momentum conservation. The most natural possibility is
to weight the galaxies by the halo mass.
The purpose of this Letter is to explore this idea
with numerical simulations. We employ a suite of
large N-body simulations using the Gadget II code,
which include four simulations with 10243 particles in
a (1.6h−1 Gpc)3 box and one simulation with 15363 particles in a (1.3h−1 Gpc)3 box. The fiducial cosmological model has a scale invariant spectrum with amplitude
σ8 = 0.81, matter density Ωm = 0.28 and Hubble parameter H0 = 70km/s/Mpc. We ran a Friends of Friends
halo finder and kept all the halos with more than 20 particles, with the lowest halo mass of 6 × 1012 h−1 M⊙ and
1012 h−1 M⊙ , respectively.
If a tracer has an overdensity δh with a bias bh , then
the relation to the dark matter overdensity δm in Fourier
space can be written as δh = bh δm
+ n, where n is shot
noise with a power spectrum n2 = σn2 and we assume
it is uncorrelated with the signal, i.e. hδm ni = 0 (the
operations should be taken separately on real and imaginary components
of the Fourier modes). Thus we define
σn2 = (δh − bh δm )2 and bias
(Phh /Pmm )1/2 =
2 is
bh =
2
Phm /Pmm , where
2 Phh = δh − σn , Phm = hδm δh2i
. This is equivalent to choosing σn
and Pmm = δm
such that the cross correlation coefficient is unity, r ≡
Phm /(Phh Pmm )1/2 = 1. Thus our definition of the shot
noise includes all sources of stochasticity between the
halos and the dark matter, so it is the most conservative. This can be done as a function of k and so allows for a possibility that noise is not white. We do
not assume a constant bias, although we find that for
k ≪ 0.1h/Mpc this is generally true. Another way to
define the
shot noise is through
fluc the power2 spectrum
+ (σn2 )2 ). We
tuations, (δh2 − Phh − σn2 )2 = (2/N )(Phh
find this definition in general has larger variance, but is
on average in agreement with the definition above, which
σn2
104
103
uniform (bin)
uniform (threshold)
mass
f(mass)
n = 1×10-4 (h/Mpc)3
n = 3×10-4 (h/Mpc)3
n = 7×10-4 (h/Mpc)3
n = 4×10-3 (h/Mpc)3
103
σn2
between the galaxy and the halo number density, but for
a typical survey the fraction of halos with more than one
galaxy in it is small, 5-30% [8], and here we will ignore
this distinction and assume for simplicity there is only
one galaxy in each halo at its center.
102
0.01
0.10
k [h/Mpc]
0.01
0.10
k [h/Mpc]
FIG. 1. Shot noise power spectrum σn2 measured in simulations for uniform weighting of halos in a mass bin and
mass threshold, mass weighting and f (M ) = M/[1 +
(M/1014 h−1 M⊙ )0.5 ] weighting, for several different abundances, corresponding at z = 0 to mass thresholds of
4 × 1013 h−1 M⊙ , 1.4 × 1013 h−1 M⊙ /h, 6 × 1012 h−1 M⊙ and
1012 h−1 M⊙ /h, from the lowest to the highest abundance, respectively. Straight lines (same color or line style) are the
expected shot noise σe2 for each of the weightings (equal for
the mass bin and mass threshold with uniform weighting).
we will use in the following.
We begin by first investigating the shot noise when
each halo has equal weight. The simplest case is that
of a mass threshold, where all of the halos above a certain minimum cutoff are populated. Second possibility
is that of a bin in halo mass, for which we remove the
top 10% of the most massive halos in a simulation and
take the remaining ones to match a given abundance.
As shown in Fig. 1 this leads to a larger shot noise
than the mass threshold case, neither of which in general agrees with the prediction σn2 = n̄−1 . The latter
can dramatically underestimate the shot noise at higher
abundances, by a factor of 3 for our highest number density of n̄ = 4 × 10−3 (h/Mpc)3 . The standard error analysis assumes σn2 = n̄−1 and may be overly optimistic:
shot noise should be a free parameter determined from
the data itself.
Next we investigate the shot noise for nonuniform halo
dependent weighting wi for the same mass threshold sample. WePcompare
P the simulations to the expectation
σe2 = V i wi2 /( i wi )2 , where V is the volume and
the sum is over all the halos. At a given number density this expression is minimized for uniform weighting
(where it equals n̄−1 ), so nonuniform weighting generally increases the expected shot noise. As argued above
wi = Mi , where Mi is the halo mass, is the natural implementation of the idea to enforce mass and momen-
iii
100
σn 2 / P
10-1
10-2
10-3
n = 1×10-4 (h/Mpc)3
bub = 1.48
but = 2.00
bM = 2.61
bf(M)= 2.37
n = 3×10-4 (h/Mpc)3
bub = 1.31
but = 1.55
bM = 2.22
bf(M)= 1.96
n = 4×10-3 (h/Mpc)3
σn 2 / P
10-1
10-2
-4
10-3 n = 7×10 (h/Mpc)
3
0.01
bub = 0.97
but = 1.02
bM = 1.76
bf(M)= 1.50
bub = 1.22
but = 1.31
bM = 2.02
bf(M)= 1.75
0.10
0.01
0.10
k [h/Mpc]
k [h/Mpc]
FIG. 2. Same as Fig. 1, but for σn2 /P . Also shown are the
bias values b for the different cases, which affect σn2 /P , since
P = b2 Pmm , where Pmm is the matter power spectrum.
σn2
104
103
10
2
σM = 0
σM = 0.1
σM = 0.5
σM = 1.0
M weighting
n = 3×10-4 (h/Mpc)3
f(M) weighting
n = 3×10-4 (h/Mpc)3
M weighting
n = 4×10-3 (h/Mpc)3
f(M) weighting
n = 4×10-3 (h/Mpc)3
103
σn2
tum conservation for the halos. The results are shown in
Fig. 1. We see that the predicted and measured shot noise
amplitudes differ significantly and the difference reaches
a factor of 10-30 at the highest abundance in our simulations, n̄ = 4 × 10−3 (h/Mpc)3 . This demonstrates that
this is not a simple Poisson sampling of the field and that
mass and momentum conservation work to suppress the
shot noise relative to expectations.
Other weightings may also improve the results relative
to naive expectations and may work even better for specific applications. For example, weighting by f (M ) =
M/[1 + (M/1014 h−1 M⊙ )0.5 ], shown in Fig. 1, improves
upon the mass weighting. This weighting equals the halo
mass weighting over the mass range of M < 1014 h−1 M⊙ ,
while giving a lower weight to the higher mass halos relative to the mass weighting. Weighting by the halo mass
gives a very large weight to the most massive halos and
this non-uniform weighting leads to a significant increase
in the naive shot noise prediction σe2 relative to the number density of halos. Therefore, if the conservation of
mass and momentum is not perfect for the most massive
halos the residual shot noise may still be large, which
may explain why downweighting high mass halos may
work better. On the other hand, simply eliminating the
halos above 1014 h−1 M⊙ while preserving mass weighting below that mass completely erased any advantages.
We also tried weighting by the halo bias b, which was
argued to minimize σ 2 /P [11], and found no improvements relative to uniform weighting, as expected since it
is close to uniform weighting for most of the halos and
therefore does not implement the mass and momentum
conservation efficiently. It is possible that one may be
able to further improve the signal to noise by optimizing
the weights, but the optimization will depend on the specific application one has in mind (e.g. non-Gaussianity,
redshift-space distortions, BAO etc.) and is beyond the
scope of this Letter.
For actual applications we want to minimize σn2 /P .
Figure 2 shows the results for the same cases as in Fig. 1.
We see there are significant improvements in σn2 /P relative to the uniform weighting and that mass and modified mass give comparable results, with improvements
in excess of 10 possible relative to the uniform weighting. While these results are all at z = 0 where we have
the highest density of halos, we also computed them at
higher redshifts. At z = 0.5 and n̄ = 3 × 10−4 (h/Mpc)3 ,
target density for SDSS-III, we find a factor of 3-10 improvement at the BAO scale in mass weighting relative
to the uniform, comparable to the z = 0 case at the same
number density. This means that the achievable error on
cosmological parameters from BAO can be improved significantly for the same number of objects measured. Alternatively, a significantly lower number of objects may
be needed to achieve the same precision and one can re-
102
101
0.01
0.10
k [h/Mpc]
0.01
0.10
k [h/Mpc]
FIG. 3. Effects of log-normal scatter σ in halo mass observable on the shot noise σn2 for mass and f (M ) = M/[1 +
(M/1014 h−1 M⊙ )0.5 ] weights, for n̄ = 3 × 10−4 (h/Mpc)3 and
n̄ = 4 × 10−3 (h/Mpc)3 . Scatter hardly affects the bias, so the
relative effects of scatter are the same for σn2 /P and we do
not show them here.
duce the target number density by nearly a factor of 3.
Note that SDSS-III plans to oversample the galaxies at
the BAO scale and to use reconstruction to reduce the
damping of BAO, which can be done better if the shot
noise is lower. It is also possible that imposing the local
mass and momentum conservation will minimize systematic shifts in the BAO position relative to the dark matter
that may otherwise be problematic [12], but we leave this
investigation for the future.
So far we ignored the real world complications such as
the imprecise knowledge of the halo mass. To investi-
iv
gate this we add a log-normal scatter with rms variance
σ to each halo mass and recompute the analysis. Figure 3 shows the results for mass and modified mass f (M )
weighting: for the latter we see that a scatter of 50% in
mass increases σn2 /P by about 50% for lower abundance
and a factor of 2 for higher abundance. Since this is a
realistic scatter for optically selected clusters [13] there
is thus realistic possibility that we can apply such analysis to the real data and achieve these gains. In practical
applications one would try to identify the best halo mass
tracer as a function of halo mass, for example, central
galaxy luminosity in the galactic halos and richness or
total luminosity for the cluster halos. In order to minimize the scatter one must understand the relation between the galaxy observables and the underlying halos,
so progress in galaxy formation studies will be needed to
maximize the gains. We find that for the mass weighting, scatter has a larger effect, such that for σ = 0.5 the
degradation in σn2 /P is a factor of 2-3. Once the scatter
becomes too large there is no longer any local mass and
momentum conservation and we find that for σ = 1 the
shot noise is worse than for uniform weighting. Another
potential complication is the effect of redshift-space distortions, since the observed radial distance is a sum of
the true radial distance and peculiar velocity (divided
by the Hubble parameter). We find a modest (50%) increase in σn2 /P , where P in redshift space is the spherically averaged (i.e. monopole) power spectrum. Since
redshift space contains much more information than just
the monopole it is possible that one may be able to use
the additional information to reduce this degradation and
we leave this for a future investigation.
These results are particularly relevant for the multitracer methods where the data are analyzed in terms of
ratios of different tracers and for which the sampling variance error cancels, such as those recently proposed for
non-Gaussianity [3], redshift-space distortions and Hubble versus angular distance relation [4]. For these there
is no lower limit on the achievable error, as long as σn2 /P
decreases and the method proposed here could lead to
a significant reduction of errors relative to previous expectations. We see from Fig. 2 that for mass weighting at n̄ = 4 × 10−3 (h/Mpc)3 , σn2 /P ∼ 10−3 on large
scales, so this could give a signal to noise of 30 for a single mode, compared to 0.7 for the single tracer method,
equivalent to 3 orders of magnitude reduction in volume
needed to reach the same precision. Note that this is
not unreachable, since the existing SDSS survey achieves
n̄ ∼ 10−2 (h/Mpc)3 for the redshift survey of the main
sample.
Equally impressive improvements may be possible for
future redshift surveys such as JDEM/EUCLID or BigBOSS, which are expected to operate at redshifts up to
z ∼ 2. Their target number density could be as high
as n̄ ∼ 10−3 (h/Mpc)3 or higher, and the method proposed here could lead to a dramatic reduction of errors or,
equivalently, to a several-fold reduction in the number of
measured redshifts required to reach the target precision,
with potentially important implications for the design of
these missions. The weights can be further optimized for
specific applications, specially for the multitracer methods that cancel out the sampling variance error. This
approach holds the promise to become the most accurate
method to extract both the primordial non-Gaussianity
and the dark energy equation of state and its full promise
should be explored further with more realistic simulations. In parallel we should develop our understanding
of galaxy formation better to relate the galaxy observables to the underlying halo mass with as little scatter as
possible.
We thank V. Springel for making the gadget ii code
available to us and P. McDonald, D. Eisenstein and R.
Smith for useful comments. This work is supported by
the Packard Foundation, the Swiss National Foundation
under contract 200021-116696/1 and WCU grant R322008-000-10130-0.
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3
PAPER II

Chapter 3: Paper II
Summary The next article is a more detailed follow-up study on the results
found in the previous paper. The stochasticity of halos is being studied in a more
general way by defining the so-called shot noise matrix. Here, all the halos in a
large sample of N -body simulations are split up into equal number-density mass
bins and the shot noise is calculated from all auto- and cross power spectra between the bins. It is found that low mass halos exhibit more-, and high mass halos
less stochasticity than expected from the Poisson sampling model. Moreover, nonvanishing off-diagonal elements in the shot noise matrix are detected, implying both
correlations and anti-correlations in the shot noise of halos of different mass. An
eigendecomposition of the shot noise matrix reveals two eigenmodes that are distinct
from Poisson sampling, a super- and a sub-Poissonian one. The latter provides an
optimal weighting function to suppress the shot noise in the halo density field. This
function is found to scale linearly with mass at high masses, and saturate towards a
constant value at low masses, determined by the mass resolution of the simulation.
It is demonstrated how an application of this weighting scheme can significantly
reduce the shot noise of halos, even more than the previously studied weights. In
realistic observations the halo masses may be poorly constrained, the influence of
scatter in the mass-observable relation is therefore examined. In consideration of
the halo model, the properties of the shot noise matrix can be described remarkably
well, providing an analytical expression for the optimal weighting function. Finally,
it is shown how the scale dependence of the halo bias emerges in this model.
The paper received contributions from five authors. The numerical analysis and
the writing of the manuscript was conducted by myself, in close consultation with
Uroš Seljak. Vincent Desjacques and Robert Smith provided the simulation data
and were involved in many discussions on technical issues of the analysis. The halo
model approach was put forward by Robert Smith, with additional contributions
from Tobias Baldauf. The article was published in August 2010 in Physical Review D
(vol. 82, ID 043515, arXiv:1004.5377).
Minimizing the Stochasticity of Halos in Large-Scale Structure Surveys
Nico Hamaus,1, ∗ Uroš Seljak,1, 2, 3, † Vincent Desjacques,1 Robert E. Smith,1 and Tobias Baldauf1
2
1
Institute for Theoretical Physics, University of Zurich, 8057 Zurich, Switzerland
Physics Department, Astronomy Department and Lawrence Berkeley National Laboratory,
University of California, Berkeley, California 94720, USA
3
Ewha University, Seoul 120-750, S. Korea
(Dated: August 13, 2010)
In recent work (Seljak, Hamaus and Desjacques 2009) it was found that weighting central halo
galaxies by halo mass can significantly suppress their stochasticity relative to the dark matter, well
below the Poisson model expectation. This is useful for constraining relations between galaxies
and the dark matter, such as the galaxy bias, especially in situations where sampling variance
errors can be eliminated. In this paper we extend this study with the goal of finding the optimal
mass-dependent halo weighting. We use N -body simulations to perform a general analysis of halo
stochasticity and its dependence on halo mass. We investigate the stochasticity matrix, defined as
Cij ≡ h(δi − bi δm )(δj − bj δm )i, where δm is the dark matter overdensity in Fourier space, δi the
halo overdensity of the i-th halo mass bin, and bi the corresponding halo bias. In contrast to the
Poisson model predictions we detect nonvanishing correlations between different mass bins. We
also find the diagonal terms to be sub-Poissonian for the highest-mass halos. The diagonalization
of this matrix results in one large and one low eigenvalue, with the remaining eigenvalues close to
the Poisson prediction 1/n̄, where n̄ is the mean halo number density. The eigenmode with the
lowest eigenvalue contains most of the information and the corresponding eigenvector provides an
optimal weighting function to minimize the stochasticity between halos and dark matter. We find
this optimal weighting function to match linear mass weighting at high masses, while at the lowmass end the weights approach a constant whose value depends on the low-mass cut in the halo
mass function. This weighting further suppresses the stochasticity as compared to the previously
explored mass weighting. Finally, we employ the halo model to derive the stochasticity matrix and
the scale-dependent bias from an analytical perspective. It is remarkably successful in reproducing
our numerical results and predicts that the stochasticity between halos and the dark matter can be
reduced further when going to halo masses lower than we can resolve in current simulations.
I.
INTRODUCTION
The large-scale structure (LSS) of the Universe carries
a wealth of information about the physics that governs
cosmological evolution. By measuring LSS we can attempt to answer such fundamental questions as what the
Universe is made of, what the initial conditions for the
structure in the Universe were, and what its future will
be. Traditionally, the easiest way to observe it is by measuring galaxy positions and redshifts, which provides the
3D spatial distribution of LSS via so-called redshift surveys (e.g., [1, 2]).
However, dark matter dominates the evolution and
relation to fundamental cosmological parameters, while
galaxies are only biased, stochastic tracers of this underlying density field. On large scales, this bias is expected
to be a constant offset in clustering amplitude relative
to the dark matter, which can be removed to reconstruct
the dark matter power spectrum [3]. Nevertheless, this
reconstruction is hampered due to a certain degree of
randomness in the distribution of galaxies, which is based
∗
†
[email protected]
[email protected]
on the nonlinear and stochastic relation between galaxies and the dark matter. In the simplest model one describes this stochasticity with the Poisson model of shot
noise. Shot noise constitutes a source of error in the
power spectrum [4] and therefore limits the accuracy of
cosmological constraints. The Poisson model predicts it
to be determined by the inverse of the galaxy number
density, assuming galaxies to be random and pointlike
tracers. However, galaxies are born inside dark matter
halos and for these extended, gravitationally interacting
objects, the shot noise model is harder to describe. It
is thus desirable to develop estimators that are least affected by this source of stochasticity.
In Fourier space the stochasticity of galaxies is usually
described by the cross-correlation coefficient
Pgm
rgm ≡ q
,
P̂gg Pmm
(1)
where P̂gg is the measured galaxy autopower spectrum,
Pmm the dark matter autopower spectrum, and Pgm the
cross-power spectrum of the two components. The crosscorrelation coefficient rgm can be related to the shot noise
power σ 2 , which is commonly defined via the decomposition P̂gg = Pgg + σ 2 , with Pgg = b2 Pmm and the bias
ii
defined as b = Pgm /Pmm . This yields
2
1 − rgm
σ2
=
.
2
Pgg
rgm
(2)
Thus, the lower the shot noise, the smaller the stochasticity, i.e., the deviation of the cross-correlation coefficient
from unity. Minimizing this stochasticity is important if
one attempts to determine the relation between galaxies
and the underlying dark matter. One example of such an
application is correlating the weak lensing signal, which
traces dark matter, to properly radially weighted galaxies [5]: an accurate determination of the galaxy bias can
be combined with a 3-dimensional galaxy redshift survey to greatly reduce the statistical errors relative to the
corresponding 2-dimensional weak lensing survey.
The ultimate precision on how accurate the galaxy bias
can be estimated from such methods is determined by the
cross-correlation coefficient and previous work has shown
that it can deviate significantly from unity for uniformly
weighted galaxies or halos [4]. However, it was demonstrated recently that weighting halos by mass considerably reduces the stochasticity between halos and the dark
matter [6]. The purpose of this paper is to explore this
more systematically and to develop an optimal weighting
method that achieves the smallest possible stochasticity.
Our definition of the shot noise above is relevant for
the methods that attempt to cancel sampling variance (or
cosmic variance) [7, 8] and this will be our primary motivation in this paper. Alternatively, the shot noise is often
associated with its contribution to the error in the power
spectrum determination, this error usually being decomposed into sampling variance and shot noise. Sampling
variance refers to the fact that in a given volume V the
number Nk of observable Fourier modes of a given wave
vector amplitude is finite. In the case of a Gaussian random field the relative error in the measured galaxy power
spectrum P̂gg due to the sum of the two errors is given
√
by σP̂gg /P̂gg = 1/ Nk (each complex Fourier mode has
two independent realizations and we only count modes
with positive wave vector components). Using the above
decomposition of the measured power P̂gg into intrinsic
power Pgg and shot noise sigma2 , one finds
σPgg
σ2
1
1+
.
(3)
= √
Pgg
Pgg
Nk
This definition is ambiguous, since it leaves the decomposition of the measured power into shot noise and shot
noise subtracted power unspecified. Most of the analyses so far have simply assumed the Poisson model, where
the shot noise is given by the inverse of the number density of galaxies, σ 2 = 1/n̄. A second possibility is to
define
p the shot noise such that the galaxy bias estimator Pgg /Pmm becomes as scale independent as possible.
The third way is to define it via the stochasticity between
halos and the dark matter, i.e., the cross-correlation coefficient rgm as in Eq. (2). We choose the third definition,
but will comment on the relations to the other two methods as well.
It is important to emphasize here that the first two definitions are not directly related to the applications where
the sampling variance error can be eliminated, since they
do not include correlations between tracers (where the
dark matter itself can also be seen as a tracer). While
in this paper we focus on minimizing the error on the
bias estimation using the sampling variance canceling
method, there are other applications where correlating
dark matter and galaxies, or two differently biased galaxy
samples, allows us to reduce the sampling variance error
[7–9]. In such cases the stochasticity, or the shot noise to
power ratio as defined in Eq. (2), is the dominant source
of error and methods capable of reducing it offer the potential to further advance the precision of cosmological
tests. Indeed, since the error on the power spectrum as in
Eq. (3) contains two contributions, in the past there was
not much interest in investigating the situation where the
shot noise is much smaller than sampling variance. It is
the situations where the sampling variance error vanishes
that are most relevant for our study.
In this paper we will focus on the relation between halos and the underlying dark matter, using two-point correlations in Fourier space (i.e. the power spectrum) as a
statistical estimator. A further step to connect halos to
observations of galaxies can be accomplished by specification of a halo occupation distribution for galaxies [10],
but we do not investigate this in any detail. Alternatively, one can think of the halos as a sample of central
halo galaxies from which satellites have been removed.
II.
SHOT NOISE MATRIX
The term shot noise is usually related to the fact that
the sampling of a continuous field with a finite number of
objects yields a spurious contribution of power to its autopower spectrum. In the Poisson model the contribution
to the autopower spectrum is 1/n̄, where n̄ = N/V is the
mean number density of objects sampling the continuous
field, whereas the cross-power spectrum of two distinct
samples of objects is not affected (see, e.g., [11, 12]).
However, in cosmology one studies galaxies residing in
dark matter halos, which are not a random subsample of
the dark matter particles. The Poisson model does not
account for that fact.
In recent work it has been argued that there are other
nonlinear terms that appear like white noise terms in the
power spectrum of halos and so a more general approach
is needed to determine the shot noise [13]. In order to account for this fact we define the shot noise more generally
as the two-point correlation matrix
Cij ≡ h(δi − bi δm )(δj − bj δm )i .
(4)
iii
Here the subscripts i and j refer to specific subsamples
of the halo density field with overdensities δi and δj and
corresponding scale-independent bias bi and bj , respectively. The dark matter density fluctuation is denoted by
δm and the angled brackets denote an ensemble average.
We work in Fourier space and the δ’s are the complex
Fourier components of the density field,
1
δ(k) = √
V
Z
δ(x)e−ik·x d3 x .
(5)
However, we handle the complex density modes δ as real
quantities, since their real and imaginary parts are uncorrelated and one can treat them as two independent
modes. Further, we assume the overdensity of a particular halo sample i to be composed of two terms [4]:
δi = b i δm + ǫ i ,
(6)
where ǫi is a random variable of zero mean assumed to be
uncorrelated with the signal, i.e., hǫi δm i = 0. It follows
that the bias parameter bi can be obtained from cross
correlation with the dark matter,
bi =
hδi δm i
,
2 i
hδm
(7)
and the shot noise matrix can be written as Cij = hǫi ǫj i.
With these definitions the cross-correlation coefficient between any given halo bin i and the dark matter,
hδi δm i
,
rim ≡ p 2
2 i
hδi ihδm
(8)
becomes unity when we subtract the shot noise component Cii from hδi2 i. We thus recover the shot noise
definition from Eq. (2) and define Pii ≡ hδi2 i − Cii to
be the halo autopower spectrum (shot noise subtracted),
Pim = hδi δm i the halo-matter cross-power spectrum, and
2
Pmm = hδm
i the matter autopower spectrum (we assume
the shot noise of the matter auto-, as well as the halomatter cross-power spectrum to vanish). Note that these
relations are still self-consistent if we allow the bias bi to
be scale dependent. Here we will, however, explore the
simpler case assuming scale-independent bias, which is a
good approximation on large scales.
In the Poisson model the shot noise matrix Cij is diagonal, but this is not necessarily the case in our definition. The objective of this paper is to study all the
components of this matrix using N -body simulations. In
particular we divide the halos into bins of different mass,
but equal number density. A diagonalization of the shot
noise matrix will then provide its eigenvalues and eigenvectors, which contain important information about the
stochastic properties of the halo density field.
III.
SIMULATIONS
We use the zHorizon simulations [12], 30 realizations
of numerical N -body simulations with 7503 particles of
mass 5.55 × 1011h−1 M⊙ and a box-size of 1.5h−1 Gpc (total effective volume of Vtot = 101.25h−3Gpc3 ) to accurately sample the density field of cold dark matter. The
simulations were performed at the University of Zurich
supercomputers zbox2 and zbox3 with the gadget ii
code [14]. We chose the cosmological parameters to be
close to the outcome of the WMAP5 data release [15],
namely Ωm = 0.25, ΩΛ = 0.75, Ωb = 0.04, σ8 = 0.8,
ns = 1.0 and h = 0.7. The transfer function was computed with the cmbfast code [16] and the initial conditions were set up at redshift z = 50 with the 2lpt initial
conditions generator [17, 18].
We applied the friends-of-friends (FoF) algorithm bfof by V. Springel with a linking length of 20% of
the mean interparticle distance and a minimum of 30
particles per halo to generate halo catalogs. The resulting catalogs contain about 1.3 × 106 halos (n̄ ≃
3.7 × 10−4 h3 Mpc−3 ) with masses between Mmin ≃ 1.1 ×
1013 h−1 M⊙ and Mmax ≃ 3.1 × 1015 h−1 M⊙ . In order
to investigate the influence of the mass resolution on
our results, we employ another set of 5 N -body simulations [19] of box-size 1.6h−1 Gpc with 10243 particles of
mass 3.0 × 1011 h−1 M⊙ , resolving halos down to Mmin ≃
5.9×1012h−1 M⊙ (n̄ ≃ 7.0×10−4h3 Mpc−3 ). All other parameters of this simulation are similar to the one above,
namely Ωm = 0.279, ΩΛ = 0.721, Ωb = 0.046, σ8 = 0.81,
ns = 0.96, h = 0.7. One further realization with these
parameters was generated with an even higher mass resolution, namely 15363 particles of mass 4.7 × 1010h−1 M⊙
in a box of 1.3h−1 Gpc, resolving halos down to Mmin ≃
9.4 × 1011 h−1 M⊙ (n̄ ≃ 4.0 × 10−3 h3 Mpc−3 ).
The density fields of dark matter and halos in configuration space were computed via interpolation of the
particles onto a cubical mesh with 5123 grid points using
a cloud-in-cell mesh assignment algorithm [20]. We then
applied fast Fourier transforms to compute the modes
of the fields in k-space. All our results are presented at
z = 0 and we do not explore the redshift dependence,
because at higher redshifts the halo number density is
lower and we wish to explore the stochastic properties of
halos in the high density limit.
IV.
A.
ANALYSIS
Estimators for the binned halo density field
The shot noise matrix from Eq. (4) is calculated by
plugging in the Fourier modes provided by our simulations and averaging over a range of wave numbers. The
bias is determined via the ratio in Eq. (7), we thus neglect
any shot noise contribution in this expression. In Eq. (4)
iv
Pii (k) [h-3Mpc3]
105
104
3.0
2.5
bi (k)
we use the scale-independent bias, which is obtained by
averaging over the range k ≤ 0.024 hMpc−1 , corresponding to our first four k-bins. This range of wave numbers
is least affected by scale dependence, as apparent from
the middle panel of Fig. 1.
For the division into subsamples we bin the full halo
catalog into bins of different mass, keeping the number
density of each bin constant. This is done by sorting
the halos according to their mass and then dividing this
sorted array into subarrays with an equal number of halos. We use 10 bins for most of the plots presented here,
since more bins make them increasingly hard to read.
For some plots we also show the results with 30 and 100
bins to provide a more accurate sampling of halo masses.
A convergence of the results can only be reached with
infinitely many bins, which is numerically impossible to
accomplish. However, using linear mass weighting of the
halos within each bin makes the results converge faster,
as will be justified later. We apply this technique to our
100 halo mass bins, as shown in some of the following
plots.
2.0
1.5
Power spectrum, bias and cross-correlation coefficient
We start by looking at the autopower spectrum of the
halos in each mass bin as shown in the top panel of Fig. 1,
using our lower resolution simulation with average halo
number density of n̄ ≃ 3.7 × 10−4 h3 Mpc−3 . The halo
autopower spectra have been subtracted by Cii , the diagonal elements of the shot noise matrix from Eq. (4),
depicted below in Fig. 2. The halo subsamples increasingly gain power with higher mass due to their enhanced
bias, which is plotted in the middle panel of Fig. 1.
This plot shows the bias obtained from Eq. (7) as a
function of k. The scale-independent bias is drawn as
straight dotted lines for comparison. On large scales,
roughly below k ≃ 0.015 hMpc−1 , sampling variance
makes the curves appear more noisy, while on smaller
−1
scales, k >
∼ 0.04 hMpc , possibly nonlinear evolution
of the density field or higher-order bias corrections set
in causing the halo bias to pick up a scale dependence
[21]. This scale dependence is most pronounced for the
highest-mass bin.
The degree of halo stochasticity can also be assessed
in the cross-correlation coefficient between halos and the
dark matter, as depicted in the bottom panel of Fig. 1.
We see that the more massive halos are a less stochastic
tracer of the dark matter. Note that subtracting our definition of the shot noise from the autocorrelation of halos
makes the cross-correlation coefficient become unity. It
has the nice property that the bias determined from halo
auto-correlation and from halo-matter cross-correlation
is identical by definition.
1.0
1.0
0.8
rim (k)
1.
0.6
0.4
0.01
k [hMpc-1]
0.10
FIG. 1. TOP: Autopower spectra for 10 consecutive halo mass
bins (solid colored lines) and the dark matter (dotted black
line). MIDDLE: Bias of the 10 halo bins determined from the
cross power with the dark matter, the dotted lines show the
scale-independent bias. BOTTOM: Cross-correlation coefficients of the 10 halo bins with the dark matter (solid colored
lines) without shot noise subtraction. When the shot noise
Cii is subtracted from hδi2 i, by definition the cross-correlation
coefficient becomes unity (dashed lines). For reference, the
value r = 1 is plotted (dotted black line). The error bars on
all three plots were computed from the ensemble of the 30
independent realizations of our simulations. They show the
standard deviation on the mean of each quantity shown.
v
We now turn to the calculation of the shot noise matrix
for the 10 halo mass bins. Figure 2 shows each element of
Cij plotted against the wave number. As expected, the
diagonal components of the shot noise (solid lines) are
dominant. They all show essentially no scale dependence
and match the usual expectation of 1/n̄i very well (where
n̄i is the mean number density of halos in bin i), except
for the highest-mass bin (solid, black line), which is suppressed by about a factor of 2. The conventional expression for the shot noise breaks down for the highest-mass
halos. This sub-Poissonian behavior of the shot noise at
high masses has been found in simulations before [22, 23].
Moreover we find both negative and positive elements
in the off-diagonal parts of the shot noise matrix. In
the case of N halo bins, in total there are N (N + 1)/2
independent elements, since Cij is a symmetric matrix.
These are composed of N diagonal and N (N − 1)/2 offdiagonal elements. Hence, in the case of N = 10, there
are 45 off-diagonal elements and we find 33 of them to
be positive (dashed lines), while 12 are negative (dotted
lines). While all off-diagonal elements are white noise
like, i.e., scale independent, most of the negative components have a higher magnitude than the positive ones.
The former correspond to the cross correlations of any
given halo mass bin with the highest-mass halos.
This finding is rather surprising, because shot noise
cross correlations are usually being neglected. Since there
appear to be negative off-diagonal components in the shot
noise matrix and their magnitude exceeds the positive
ones, one might expect that a suitable linear combination
of the halo bins can reduce the total shot noise, as found
in [6]. In the subsequent section we will show that this
expectation is indeed fulfilled.
B.
4•104
Shot noise matrix
Eigensystem of the shot noise matrix
In order to find the principal components of the shot
noise matrix we have to diagonalize it by determining its
eigenvalues λ(l) and eigenvectors V (l) , defined via
X
(l)
(l)
(9)
Cij Vj = λ(l) Vi .
3•104
Cij (k) [h-3Mpc3]
2.
2•104
1•104
0
-1•104
0.01
k [hMpc-1]
0.10
FIG. 2. Elements of the shot noise matrix as defined in Eq. (4)
with 10 halo mass bins. Most of the diagonal components
(solid lines with stars) agree with Poisson white noise, i.e.,
Cii = 1/n̄i (dotted black line on top), except the highestmass bin which is clearly suppressed (solid black line). There
are both positive (dashed lines with circles, scaled in red)
and negative (dotted lines with triangles, scaled in blue) offdiagonal components.
a function of k. The eigenvalues are computed separately
for every k-bin and then ordered by their magnitude. As
apparent from the figure, we find two eigenvalues to differ
significantly from all the others. One of them is enhanced
by roughly a factor of 1.5 and one is suppressed by a
factor of about 2.5 compared to the other ones that lie
close to the value 1/n̄i . The spread of the curves at low k
is likely due to the low number of modes available there,
making the eigenvalue determination inaccurate.
This result reveals the fact that one of the eigenvectors, which represents a particular linear combination of
the halo mass bins, yields a very low shot noise level.
This shot noise level is determined by the lowest eigenvalue of the shot noise matrix, which we will denote as
λ− . Increasing the number of halo bins we find an even
stronger suppression of λ− compared to the expectation
of 1/n̄i (see Sec. IV C).
j
The superscript (l) is used to enumerate the eigenvalues
and eigenvectors, while the subscripts i and j refer to the
components of the vectors and matrices. We use routines
from [24] to do the calculations.
1.
Eigenvalues
The left panel of Fig. 3 shows the eigenvalues λ(l) of
the shot noise matrix from Fig. 2 for the 10 halo bins as
The other eigenvalue that differs from the value 1/n̄i
represents the highest shot noise level. We designate this
eigenvalue λ+ . Since it does not carry much information
(see below) we do not investigate it further in this paper
beyond noticing that it is likely to be connected to the
second-order bias. We note that had we investigated the
halo covariance matrix hδi δj i, we would not have been
able to reveal the lowest eigenvalue as cleanly, because it
would have been swamped by sampling variance. Indeed,
previous work focused its attention mostly on the largest
eigenvalues of hδi δj i [25].
vi
5•104
1.0
0.5
3•104
V(l)
i
λ(l) (k) [h-3Mpc3]
4•104
2•10
0.0
4
-0.5
1•104
-1.0
0
0.01
-1
0.10
k [hMpc ]
-1
1014
M [h MO• ]
FIG. 3. The 10 eigenvalues (left) and eigenvectors (right) of the shot noise matrix from Fig. 2 in corresponding colors. The
black dotted line shows the value 1/n̄i . The eigenvectors are averaged over the entire k-range.
2.
Eigenvectors
(l)
Every eigenvector Vi is a function of the wave number, just like the eigenvalues. However, as can be seen
from Figs. 2 and 3, over a reasonable range of wave numbers this dependence can be ignored and we average the
eigenvectors over the entire k-range.
P We also divide each
eigenvector by its length |Vi | = ( i Vi2 )1/2 to normalize
it.
The right panel of Fig. 3 displays the 10 eigenvectors
corresponding to the 10 eigenvalues in the left panel.
Each component of an eigenvector corresponds to one
halo mass bin. Since we have equal number densities per
bin, the mass range per bin gets wider with increasing
mass due to the rapid decline of the halo mass function.
Every data point in the figure is plotted at the respective average mass of each halo bin. Only one eigenvector shows exclusively positive components, while at least
one negative component can be found in the remaining
eigenvectors. It is this eigenvector that corresponds to
the lowest eigenvalue λ− and we will denote it as Vi− .
Its components continuously increase with mass. The
eigenvector Vi+ corresponding to the largest eigenvalue
λ+ also shows a monotonic behavior, but its components
decrease with mass and turn negative at the high-mass
end (similarly to the second-order halo bias with an opposite sign).
One can think of the eigenvectors as weighting functions for the halos, since each component acts as a weight
for the associated halo mass bin. Hence, the weighted
halo density field δw (x) in configuration space can be
written as a weighted sum over the halo mass bins, normalized by the sum of the weights,
P (l)
V δi (x)
(l)
δw
(x) = iPi (l)
.
(10)
i Vi
We specifically want to investigate the eigenvector Vi− ,
since it yields the lowest eigenvalue of the shot noise matrix, λ− . In Fig. 4 we plot the components of Vi− in a
log-log plot to investigate this eigenvector in more detail and compare to the results with 30 and 100 mass
bins. The components of Vi− increase linearly with mass
above M ≃ 1014 h−1 M⊙ , while at lower masses the slope
tends to become shallower. We compare this eigenvector
to two different smooth weighting functions for the halo
density field. The first weighting function simply takes
the halo mass as a weight for each halo, w(M ) = M , we
will denote it as linear mass weighting. However, as apparent from the dotted lines in Fig. 4, it only matches the
components of Vi− at high mass. In order to account for
the saturation effect at low masses, we consider a second
weighting function that mimics this behavior,
w(M ) = M + M0 .
(11)
The free parameter M0 determines the shape of this
weighting function, it specifies the mass threshold where
the saturation sets in. For M ≪ M0 , Eq. (11) approaches
uniform weighting, whereas in the limit M ≫ M0 it
matches linear mass weighting. We call this weighting
scheme modified mass weighting, it is shown as a dashed
curve in Fig. 4 and obviously provides a much better fit
to Vi− than linear mass weighting. The fit is shown for
each case of our mass binning. The best-fit value for
M0 increases with the number of bins and in the case of
100 mass-weighted bins becomes M0 ≃ 1.7×1013h−1 M⊙ .
Note that for visibility reasons this eigenvector is shifted
downwards by a factor of 2 in the plot.
Similar weighting schemes have already been applied
to the halo density field in [6], where a significant reduction of the stochasticity between halos and the dark
matter could be achieved. In particular, a trial weighting function also denoted as modified mass weighting was
vii
Vi –
1.00
C.
0.10
M0 = 7.0⋅1012 h-1MO•
M0 = 1.2⋅1013 h-1MO•
M0 = 1.5⋅1013 h-1MO•
M0 = 1.7⋅1013 h-1MO•
0.01
1013
1014
M [h-1MO• ]
FIG. 4. The normalized eigenvector Vi− , corresponding to
the lowest shot noise eigenvalue λ− , computed for 10, 30, 100
uniformly weighted bins and 100 mass-weighted bins (from
top to bottom). The latter was shifted downwards by a factor
of 2 for visibility. The dotted (blue) and the dashed (red) lines
represent linear and modified mass weighting, respectively.
The best-fit values for M0 are given in the bottom right for
the respective cases.
shown to improve on linear mass weighting. However, in
that paper the functional form was found empirically and
was not demonstrated to be optimal. In this work we
show why modified mass weighting as defined in Eq. (11)
is the optimal weighting to suppress the stochasticity in
halos: in the limit of many halo mass bins it converges to
the components of Vi− , the eigenvector of the shot noise
matrix with the lowest eigenvalue. What remains to be
shown is what determines the value of M0 : we will argue it depends on the lower boundary of the halo mass
function considered.
These results also justify why applying linear mass
weighting to the halos within each bin leads to a better
convergence of the eigenvector towards a smooth weighting with infinitely many bins: linear mass weighting already reduces the stochasticity of each bin as compared
to the uniformly weighted case (see [6]). The resulting
eigenvector is then determined more accurately, corresponding to an effectively higher sampling with more
bins. This mainly has an effect on the highest-mass
bins, since they have the broadest range in mass. At low
masses the bins are increasingly narrow, so uniform and
linear mass weighting become increasingly similar within
a given bin.
Other attempts to find an optimal weighting scheme
for the halo (galaxy) density field found the halo bias to
yield the best constraining power on dark matter statistics and cosmological parameters when used as weighting function [25–27]. We have tried using only b(M ) as
a weighting function, but found less suppression in halo
stochasticity as compared to modified mass weighting.
Signal-to-noise
While we have demonstrated that it is possible to suppress the stochasticity of a given halo density field using
a single linear combination of halo bins, it remains to be
shown how the information content in this single eigenmode compares to the complete information content. We
cannot answer this question in general, since it depends
not only on the property of the shot noise matrix, but
also on derivatives of the halo density field with respect
to the cosmological parameters one wants to estimate.
Those two ingredients depend on halo mass and determine the Fisher information content of the halo density
field. We will not explore the general case here and instead focus on the simple case where the information content is expressed via the ratio of the autopower spectrum
to the shot noise of a particular tracer (signal-to-noise
ratio per mode). Its inverse appears in Eqs. (2) and (3).
We compute it for the weighted halo density field,
2
S
Pw
b2
rwm
≡ 2 = w2 Pmm =
.
2
N
σw
σw
1 − rwm
(12)
Pw denotes the autopower spectrum of the weighted halo
2
density field, σw
its shot noise, bw the corresponding
weighted bias and rwm the cross-correlation coefficient
between the fields δw and δm as defined in Eq. (8). The
weighted bias can be computed from the halo bins via
P
Vb
hδw δm i
Pi i i .
(13)
bw =
=
2
hδm i
i Vi
It is clear from this expression that in order to maximize
the signal, the eigenvector components should all be of
equal sign, since otherwise different halo bins cancel each
2
others signal. Using Eq. (4) to express σw
in terms of the
weighted density field δw and Eq. (10) for the definition
of δw (we omit the superscripts for clarity), we have
P
P
2
Vi δi
Vi bi
2
σw
≡ h(δw − bw δm )2 i = h Pi
− Pi
δm i =
i Vi
i Vi
P
P 2
V
V
C
V
i,j i j ij
= λ Pi i 2 ,
(14)
= P
V
V
i
j
V
(
i,j
i i)
P
where i Vi2 = 1 in case the eigenvectors are normalized.
Hence, the signal-to-noise ratio becomes
P
P
2
2
( i Vi bi )
S
b2w ( i Vi )
P 2 Pmm =
P 2 Pmm .
=
(15)
N
λ i Vi
λ
i Vi
Note that this is only the signal-to-noise ratio for one particular weighting of the halo density field, corresponding
to one eigenmode of the shot noise matrix. The complete information content of the halos is calculated by
summing up all N contributions,
!
N
(l) 2
X
bw
S
Pmm .
(16)
=
(l)
N
σw
l=1
viii
1.00
σr2 = P
1
(l)
2
l (1/σw )
.
(17)
Alternatively, the signal-to-noise ratio of the halo density field can be derived from a χ2 distribution. Since the
modes of the halo bins are assumed to be independent,
normally distributed variables, the expression
X
−1
χ2 ≡
(δi − bi δm ) Cij
(δj − bj δm )
(18)
i,j,k
−1
Cij
2
follows a χ distribution. Here,
refers to the ij component of the inverse shot noise matrix and the
index k connotes a summation over all Fourier modes.
The derivative of the χ2 distribution with respect to
the inferred dark matter density field δm must vanish,
∂χ2 /∂δm = 0. This yields
P
−1
i,j,k Cij bi δj
δm = P
.
(19)
−1
i,j,k Cij bi bj
P −1
bi conducts a weighting of the
Here, the vector i Cij
halo mass bins again. The difference to the weighting
with one particular eigenvector of Cij is that this vector
contains the complete information of all eigenmodes and
thus provides an optimal estimator for the dark matter
density field. However, as can be seen in Fig. 5, it has
a very similar shape as Vi− and modified mass weighting
provides an equally thorough fit to this vector. The only
difference is a slight increase in the best-fit value for the
parameter M0 .
The second derivative of the χ2 distribution leads to
the signal-to-noise ratio of the halo density field,
X
S
S ∂ 2 χ2
−1
Cij
bi bj Pmm ,
≡
=
2
N
2Nk ∂δm
i,j
(20)
This expression is equivalent to the Fisher information
on the dark matter density mode δm . Here, the reduced
shot noise is simply computed as
σr2 = P
(l)
1
−1 .
i,j Cij
(21)
We first show the Vi -weighted bias for the 10 halo
bins in the left panel of Fig. 6. The highest bias, bw ≃ 2,
is achieved by weighting with Vi− , as expected, since its
components are all positive and give the largest weight to
the highest halo masses. All the other eigenvectors produce lower values of the bias, distributed around unity.
Σi C-1
ij bi
However, since we find one very low eigenvalue of the shot
noise, most of the signal will be contained in the halos
weighted by Vi− . Adding up the denominators of Eq. (16)
and taking the inverse yields the total noise contribution
of the halos. We call it the reduced shot noise,
0.10
M0 = 2.4⋅1013 h-1MO•
0.01
1013
1014
M [h-1MO• ]
P −1
FIG. 5. The normalized vector
i Cij bi that provides an
optimal estimator for the dark matter, computed for 100 uniformly weighted mass bins. Since this vector is very similar
to Vi− , modified mass weighting (dashed red line) still yields
a reasonable fit with a slightly higher value of M0 (bottom
right).
Note, however, that the weighted bias alone is not sufficient to describe the complete information content of
the weighted halo density field. It is given by the signalto-noise ratio in Eq. (15), which contains the weighted
bias, the eigenvalue and the sums over the eigenvector
components.
The total signal-to-noise ratio of the 10 eigenmodes is
shown in the right panel of Fig. 6. This panel also shows
the sum of all signal-to-noise ratios of each eigenmode,
i.e. Eq. (16), and the signal-to-noise ratio as defined in
Eq. (20) as a cross-check. Clearly, the weighting with
Vi− dominates the signal-to-noise ratio. The eigenvector
corresponding to the largest eigenvalue yields the second largest contribution, which may appear surprising
since its effective bias is around unity and its eigenvalue
is by far the largest. However, the sum of this eigenvector’s components is large, which makes its signal-to-noise
ratio dominant in comparison to the other eigenmodes.
Still, it is suppressed by roughly 1 order of magnitude
compared to the weighting with Vi− and can be safely
ignored. This fact
be cross-checked when we comP can
−1
bi appearing in Eq. (19) to Vi− .
pare the vector i Cij
We found no mentionable discrepancy between the two.
Thus, the main conclusion from this analysis is that the
lowest eigenvalue contains most of the information and
the other eigenmodes can be neglected.
So far we explored the signal-to-noise ratio of only 10
eigenmodes, a relatively sparse mass binning of the halo
density field. Do these results converge with increasing
the number of halo mass bins? It is interesting to plot the
inverse signal-to-noise ratio, since it appears in Eq. (3)
and thus determines the relative error on the power spectrum. We display it in the left panel of Fig. 7, where the
2.5
102
2.0
100
S/N
b(l)
w (k)
ix
1.5
10-2
10-4
1.0
10-6
0.5
0.01
k [hMpc-1]
0.10
0.01
k [hMpc-1]
0.10
FIG. 6. Weighted bias (left) and signal-to-noise ratios (right) of the 10 V (l) -weighted halo density fields. Colors correspond
to the eigenvalues and eigenvectors of Fig. 3. The sum of all 10 signal-to-noise ratios is plotted as a dotted (blue) curve. The
dashed (red) curve shows the signal-to-noise ratio as defined in Eq. (20).
results for different numbers of halo bins are presented.
Increasing the number of halo bins improves the signalto-noise ratio. In the limit of a large number of bins
this should be equivalent to applying the smooth weighting function we found from the eigenvector Vi− (modified
mass weighting) to each halo individually. We thus compute the inverse signal-to-noise ratio from the smoothly
weighted halo density field, defined as
P
w(Mi )δi (x)
δw (x) = iP
.
(22)
i w(Mi )
In practice, for every halo of mass M in the simulation we
assign a weight w(M ) according to the smooth weighting
function of Eq. (11). Summing over all such weighted
halo overdensities and normalizing yields the smoothly
weighted halo density field δw .
Its inverse signal-to-noise ratio results in the lowest
(solid black line) curve in the left panel of Fig. 7. We
also compare to the case when we assume all off-diagonal
elements of the shot noise matrix to vanish (long dashed
blue line). Clearly, a lot of information is lost when doing
so and any improvements compared to uniform weighting (dotted black line) are canceled: we find roughly a
factor of 5 improvement in the best case compared to
uniform weighting. We optimized the value for M0 by
iteration to reach a minimal shot noise level and find
M0 ≃ 3.4 × 1013 h−1 M⊙ , larger than our best-fit values
for the highest resolution eigenvector from Fig. 4 and the
vector from Fig. 5. This is expected, since we only tested
up to 100 halo mass bins and have not fully converged
yet (M0 increases with the number of bins). However,
the results from 100 mass-weighted halo bins closely approach the results obtained from smoothly weighting the
halo density field (solid black lines in Fig. 7).
Looking at the reduced shot noise, we see a similar be-
havior. The right panel of Fig. 7 shows the same improvements when accounting for the off-diagonal elements of
the shot noise matrix and increasing the number of bins.
The shot noise of the halo density field can drastically be
reduced using the appropriate weighting, on average by a
factor of 4 in this case. Since the bias increases with our
weighting, the improvement in the inverse signal-to-noise
ratio is more striking, though.
This is well in agreement with the results in [6], where
we applied linear and a different kind of modified mass
weighting to halo density fields with different abundances. The modified mass-weighting function we applied there was rather a trial function that happened to
suppress the shot noise better than linear mass weighting, and we did not derive it via any formal procedure
like we do here. In order to directly compare to these
older results, we apply modified mass weighting to one of
the simulations presented in that paper. In particular, we
use the simulation with 10243 dark matter particles and
a mean halo number density of n̄ ≃ 7.0 × 10−4 h3 Mpc−3 ,
resolving halos down to Mmin ≃ 5.9 × 1012 h−1 M⊙ . This
yields the inverse signal-to-noise ratio and the reduced
shot noise presented in Fig. 8. We also show the results
from the binned halo density field, as before. Note that
the strong decline of the curves corresponding to 100 bins
is likely due to noise at low k. It is the same effect present
already in our first simulation, however it is pronounced
here, since the number of modes per k-bin is lowered by
a factor of 6. Compared to the best case shown in Fig. 2
of [6], we managed to further reduce the inverse signalto-noise ratio by an additional factor of 2.
The overall improvement compared to uniform weighting is even more striking, about a factor of 10 on average
in signal-to-noise and roughly a factor of 4-5 in shot noise.
Hence, owing to the higher mass resolution of this simula-
x
3000
2500
N/S
σr2(k) [h-3Mpc3]
10-1
2000
1500
1000
500
10-2
0
0.01
k [hMpc-1]
0.10
0.01
k [hMpc-1]
0.10
FIG. 7. Inverse signal-to-noise ratio (left) and reduced shot noise (right) of the halo density field, sliced into 10 (dot-dot-dashed
green line), 30 (dashed red line), 100 uniformly weighted (dotted orange line) and 100 mass-weighted (dot-dashed yellow line)
mass bins. The upper curves (long dashed blue line) show the results when neglecting the off-diagonal elements of the shot
noise matrix. They agree well with uniform weighting (dotted black line, left panel) and the value 1/n̄ (dotted black line, right
panel), respectively. The lowest curves (solid black line) display the results obtained from weighting the halo density field with
the smooth function w(M ) = M + M0 .
2000
10-1
N/S
σr2(k) [h-3Mpc3]
1500
10
-2
1000
500
10-3
0
0.01
k [hMpc-1]
0.10
0.01
k [hMpc-1]
0.10
FIG. 8. Same as Fig. 7, computed from our higher resolution simulation with 10243 particles and a mean halo number density
of n̄ ≃ 7.0 × 10−4 h3 Mpc−3 , resolving halos down to Mmin ≃ 5.9 × 1012 h−1 M⊙ .
tion, we include many more low-mass halos and therefore
roughly double the signal. Via iteration we find the value
M0 ≃ 1.7 × 1013 h−1 M⊙ to yield the lowest shot noise
level. Comparing to the value we found in the previous
simulation, M0 ≃ 3.4 × 1013 h−1 M⊙ , it is roughly half
as large. In our lower resolution simulation the lowest
halo mass we can resolve is Mmin ≃ 1.1 × 1013 h−1 M⊙ ,
while in the higher resolution simulation it is Mmin ≃
5.9 × 1012 h−1 M⊙ .
The decline of M0 with the increase in the resolved
halo mass fraction is expected, since one needs to account for the unresolved halos in the simulations. The
relation between M0 and Mmin should be monotonic: in
the limit of perfect mass resolution we would expect all
the dark matter to be in halos of a certain mass. Weighting all these halos by their mass should then recover the
statistics of the dark matter density field without shot
noise, at least on large scales. Within the tested domain
of our simulations we find the relation M0 ≃ 3Mmin to
be a good approximation in order to determine the appropriate choice for M0 given Mmin.
xi
3000
2500
N/S
σr2(k) [h-3Mpc3]
10
-1
10-2
2000
1500
1000
500
0
0.01
k [hMpc-1]
0.10
0.01
0.10
k [hMpc-1]
FIG. 9. Inverse signal-to-noise ratio (left) and reduced shot noise (right) of halos with a log-normal mass scatter of σln M =
0, 0.1, 0.3, 0.5, 0.8 → 0.4 and 1.0 (bottom to top), weighted by w(M ) = M + M0 . The dotted (black) lines show the results
from uniform weighting.
Mass uncertainty
Up to now we have always been assuming to precisely
know the mass of each halo (up to the sampling variance
of the halo finder). However, in realistic observations
the halo mass can only be determined with a limited
accuracy. Commonly, the uncertainty is expressed as a
log-normal scatter in halo mass. We can mimic this uncertainty by adding a Gaussian random variable G with
zero mean and unit variance, scaled by σln M , to the exponent of the mass,
2
M̃ = M exp(σln M G − σln
M /2) .
our model with linearly decreasing σln M from 0.8 to
0.4 and the high value of σln M = 1.0 yields inverse
signal-to-noise ratios and shot noise levels that are below common expectations. The optimal values for M0
increase with higher mass scatter. We find M0 ≃ 3.5 ×
1013 , 5.1 × 1013 , 9.0 × 1013 , 4.4 × 1014 h−1 M⊙ for the
cases of σln M = 0.1, 0.3, 0.5, 1.0.
Figure 10 shows the resulting eigenvector Vi− when
applying a log-normal scatter with σln M = 0.5, 0.8 →
0.4 and 1.0 to the halo masses. In this case we only
present it with uniformly weighted bins, since due to the
(23)
This yields the noisier mass M̃ , which then follows a
log-normal distribution. The value σln M is the log2
normal scatter and the term σln
M /2 is subtracted to
maintain the same mean. For optical tracers of clusters σln M is about 0.5 [28] and is expected to be much
lower for SZ or X-ray proxies, such as YX [29]. At the
lower mass end the log-normal scatter is, however, poorly
constrained. For simplicity, we will consider a constant
log-normal scatter for all halos and apply the values of
σln M = 0.1, 0.3, 0.5, 1.0. As a more complicated model
we vary the scatter linearly with mass, with σln M = 0.8
at M = 1012 h−1 M⊙ and σln M = 0.4 at M = 1015 h−1 M⊙
(abbreviated as σln M = 0.8 → 0.4).
We again apply modified mass weighting to construct
the smoothly weighted density field as in Eq. (22) and
compute the inverse signal-to-noise ratio as well as the
reduced shot noise from it. For each case we adjust
the value for M0 in the weighting function separately
by iteration. The results are depicted in Fig. 9. When
using modified mass weighting, a 50% log-normal scatter in halo mass still yields about half the shot noise
level of what is expected from uniform weighting. Even
1.00
0.10
Vi –
D.
0.01
1012
M0 = 3.1⋅1013 h-1MO•
M0 = 4.2⋅1013 h-1MO•
M0 = 4.8⋅1013 h-1MO•
1013
1014
M [h-1MO• ]
1015
FIG. 10. The normalized eigenvector Vi− computed for 100
uniformly weighted bins with a log-normal scatter of σln M =
0.5, 0.8 → 0.4 and 1.0 added to the halo masses (top to bottom). For visibility, the lower two eigenvectors are shifted
downwards by a factor of 2 and 5. The dotted (blue) and the
dashed (red) lines represent linear and modified mass weighting, respectively.
xii
scatter, mass weighting the bins does not improve on
the results. Clearly, the saturation effect at low masses
is more pronounced the stronger the scatter, resulting
in an increase of the value for M0 . Also the halo mass
range becomes wider. However, the smooth function for
modified mass weighting still fits the data well. The only
impact that mass scatter has on the eigenvector Vi− is to
raise its saturation tail and thus the value of M0 . This is
even the case for our model of linearly varying σln M with
mass: modified mass weighting still provides a reasonable
fit to the simulation data.
Pij2H (k) =
1
n̄i n̄j
2H
Pim
(k) =
RR
HALO MODEL APPROACH
In order to interpret our results, let us consider the halo
model [21, 30–36]. The basic assumption of this model
is that the power spectra of either dark matter or halos
can be written as the sum of a one-halo term P 1H and a
two-halo term P 2H . The former describes the clustering
of substructure within one single halo, whereas the latter
represents the clustering among different halos. Moreover, it is assumed that all the dark matter is confined
within virialized halos. The two terms can be expressed
dn
analytically via the halo mass function dM
(M ), the normalized Fourier transform of the halo profile u(k, M ), the
analytic bias b(M ), the linear power spectrum Plin (k),
and the mean density of dark matter ρ̄m . Considering
all the possibilities of auto- and cross-power spectra between halos in distinct mass bins i and j and the dark
matter, the halo model for uniform weighting yields:
dn
dn
′
′
′
′
dM (M ) dM (M )b(M )b(M )Θ(M, Mi )Θ(M , Mj )Plin (k)dM dM
1
n̄i ρ̄m
2H
Pmm
(k)
V.
=
RR
= bi bj Plin (k)
dn
dn
′
′
′
′
dM (M ) dM (M )b(M )b(M )M Θ(M, Mi )Plin (k)dM dM = bi Plin (k)
RR
1
dn
dn
′
′
′
′
ρ̄2m
dM (M ) dM (M )b(M )b(M )M M Plin (k)dM dM = Plin (k)
Pij1H (k) =
1
n̄i n̄j
1H
Pim
(k) =
R
dn
dM (M )Θ(M, Mi )Θ(M, Mj )dM
1
n̄i ρ̄m
1H
Pmm
(k) =
R
1
ρ̄2m
dn
dM (M )M Θ(M, Mi )dM
R
dn
2
dM (M )M dM
Here we assume the large-scale limit for the halo profile,
K
i.e., u(k → 0, M ) = 1. Moreover, δij
denotes the Kronecker symbol and Θ a product of two Heaviside step
functions ϑ:
Θ(M, Mi ) ≡ ϑ(M − Mi )ϑ(Mi+1 − M ) .
(30)
Since the integrals all go from 0 to ∞, this function selects
the considered halo bin i with mass range Mi < M <
Mi+1 . The corresponding average halo mass of that bin
is simply denoted as Mi , whereas for an average over all
halos we omit the index.
In the simple approach adopted here the halo model
predicts a white noise term not only for the autopower
spectrum of halos, but also for the halo-matter crossand the matter autopower spectra. However, simulations
have shown that the low-k behavior of the dark matter
one-halo term is incorrect: subtracting off the component correlated with the linear power spectrum [which
is approximately Plin (k) at large scales] from the simulated one indeed yields a k 4 -scaling instead of a constant white noise in the residual power (mode-coupling
power) [34, 37]. This k 4 -tail is a consequence of local
mass and momentum conservation of the dark matter on
≡
=
hnM 2 i
ρ̄2m
=
1 K
n̄i δij
Mi
ρ̄m
.
(24)
(25)
(26)
(27)
(28)
(29)
small scales and the same conservation laws should also
apply to halo-matter correlations. We defer further discussions of this point to a future publication, where we
show that a proper implementation of mass conservation
in the k = 0 limit still yields similar results to those
presented here.
The shot noise matrix as defined in Eq. (4) can be
written as
2
Cij = hδi δj i − bi hδj δm i − bj hδi δm i + bi bj hδm
i.
(31)
Plugging in the sum of the corresponding one- and twohalo terms for each of the angled brackets, we see that
the two-halo terms cancel each other and we are left with
Cij =
K
δij
Mj
Mi
hnM 2 i
− bi
− bj
+ bi bj
.
n̄i
ρ̄m
ρ̄m
ρ̄2m
(32)
Our lower resolution simulation determines the dark
matter one-halo term to be hnM 2 i/ρ̄2m ≃ 428h−3Mpc3 .
Note that this value is by almost 2 orders of magnitude
smaller than the first term in Eq. (32). However, for
highly biased halo bins it can become important in the
off-diagonal terms of Cij . The same applies to the one-
xiii
Mi ≡ Mi − bi
hnM 2 i
.
2ρ̄m
(33)
Now, Eq. (32) can be written more succinctly:
Cij =
K
δij
Mj
Mi
− bi
− bj
.
n̄i
ρ̄m
ρ̄m
(34)
It is straightforward to work out the eigenvalues and
eigenvectors of this matrix. For d > 2 mass bins, there
are d − 2 degenerate eigenvalues with the value λ = 1/n̄i .
The two remaining eigenvalues with corresponding eigenvectors are
sX
X
1
1
1 X
±
b2i , (35)
M2i
Mi bi ±
λ =
−
n̄i
ρ̄m i
ρ̄m
i
i
bi
Mi
∓ pP 2 .
Vi± = pP
2
i Mi
i bi
(36)
They are shown in Fig. 12 for the case of 10 halo bins. It
is remarkable how well the halo model reproduces the distribution of eigenvalues we found in our numerical analysis. The mass dependence of the eigenvectors also shows
a good agreement. This canp
bePseen when we renormalize
2
Vi± by multiplication with
i Mi in Eq. (36), we get
Vi± = Mi ∓ bi M̃0± ,
(37)
with
M̃0±
pP
M2
hnM 2 i
≡ pPi 2i ±
.
2ρ̄m
i bi
(38)
In other words, Vi± is nothing else than a superposition of mass and bias weighting. The relative weight
4•104
3•104
Cij (k) [h-3Mpc3]
halo term of the halo-matter cross-power spectrum, because it scales with the mean halo mass of each bin. For
example, it yields M1 /ρ̄m ≃ 164h−3Mpc3 for the lowest, and M10 /ρ̄m ≃ 2394h−3Mpc3 for the highest of our
10 mass bins. In Fig. 11 we compare each matrix element of Eq. (32) to the numerically determined shot
noise matrix (from Fig. 2). The model yields a good
agreement with the data, especially the observed subPoissonian shot noise power of the highest-mass halos, as
well as the negative off-diagonal components are nicely
reproduced. The off-diagonal elements with low power
are more affected by scatter and therefore show stronger
deviations from the theory.
For the comparison of our model to the numerical data
it is, however, more convenient to look at the eigenvectors and eigenvalues of the shot noise matrix, since they
describe the complete information on halo stochasticity
in a more concise manner. Let us redefine the halo mass
as
2•104
1•104
0
-1•104
0.01
k [hMpc-1]
0.10
FIG. 11. Elements of the shot noise matrix as described by
the halo model in Eq. (32), compared to the simulation results
taken from Fig. 2 (symbols). The diagonal components (solid
lines) monotonously decrease from the lowest (yellow) to the
highest-mass bin (black), in good agreement with the numerical data. The halo model also reproduces both the positive
(dashed lines, scaled in red) and the negative (dotted lines,
scaled in blue) off-diagonal elements of the shot noise matrix
fairly well.
between the two is determined by M̃0± . Equation (37)
has a very similar form as the modified mass-weighting
fitting function from Eq. (11). Evaluating Eq. (38) using
bi and Mi from the simulation with 10 mass bins yields
M̃0− ≃ 1.2 × 1013 h−1 M⊙ . At the high-mass end, i.e.,
M ≃ 1015 h−1 M⊙ , the second term in Eq. (37) is negligible compared to the first one, since bi <
∼ 10 in this
regime. However, at lower masses the two terms become
closer in magnitude and finally the second term dominates at the low-mass end, i.e., M ≃ 1013 h−1 M⊙ . In
this regime the bias is a slowly varying function of mass
and thus well approximated by a constant. Hence, the
analytical form of Vi− predicted by the halo model agrees
well with the functional form for modified mass weighting
that we found earlier.
In order to check our model more quantitatively, we
compare its predictions directly to our numerical results
in Fig. 12. Here we focus on the nontrivial eigenvalues λ±
and eigenvectors Vi± , since only they contain information
on the halo statistics. The agreement between simulation
and theory is remarkable, only for the eigenvalue λ+ we
find a stronger discrepancy, but since it shows a slight
scale dependence it probably involves more detailed modeling. We did the same comparison for the case of 30 and
100 mass bins and find the agreement in the eigenvectors
to become even better. The offset in λ+ however does
not vanish with an increasing number of bins.
One might argue that the way we estimate the bias
from the simulation in Eq. (7) is not correct in this approach, since the halo model predicts a nonzero white
xiv
5•104
1.0
0.5
3•104
V±i
λ± (k) [h-3Mpc3]
4•104
0.0
4
2•10
-0.5
1•104
-1.0
0
0.01
0.10
k [hMpc-1]
M [h-1MO• ]
1014
FIG. 12. The eigenvalues λ± (left) and eigenvectors Vi± (right) from Fig. 3 compared to the predictions of the halo model
(dashed red line). The dotted line in the left panel shows the value 1/n̄i .
noise term for both the halo-matter cross, as well as the
dark matter autopower spectrum. We repeated the same
analysis with a shot noise corrected halo bias defined as
bi =
Mi
ρ̄m
hnM 2 i
ρ̄2m
hδi δm i −
2 i−
hδm
.
(39)
However, we find essentially no differences in the shot
noise matrix and its eigenvalues and eigenvectors. As
can be seen in Eq. (31), this is because small changes
in the bias are compensated by terms of opposite sign.
For the same reason it does not matter much whether we
use the scale-dependent or scale-independent bias in our
analysis.
Another way to compare our model to the simulations
is to look at the estimators for the halo bias itself,
s
hδi δm i
hδi2 i
and
.
(40)
2
2 i
hδm i
hδm
Since the halo model yields white noise terms for all three
correlators appearing in these estimators, this can partly
account for their scale dependence. We get
i
bi Plin (k) + M
hδi δm i
ρ̄m
=
2i ,
2
hnM
hδm i
Plin (k) + ρ̄2
(41)
b2i Plin (k) + n̄1i
hδi2 i
=
2i ,
2 i
hδm
Plin (k) + hnM
ρ̄2
(42)
m
m
where we take bi as an average over large scales from
Eq. (39) or compute it from the halo mass function via
the peak-background split formalism [3]. The simulation
results for both estimators are shown in Fig. 13 for the
case of 10 mass bins. The halo model reproduces the numerical results very well up to scales of k ≃ 0.1 hMpc−1 .
The deviation on smaller scales is expected, since higherorder bias effects, the nonlinear evolution of the density
field [34, 38], and the detailed shape of the halo profile
begin to matter [33].
The figure also shows the result of the two estimators when accounting for all of the halos in the simulation. In the case of uniform weighting (dashed lines)
they both agree on large scales, but show a different
scale dependence towards higher k-modes. With modified mass weighting (dotted lines) however, both estimators agree even up to smaller scales, a consequence of
the small stochasticity in this estimator. Note that for a
weighted field we need to account for the weights in the
averaged quantities, so in Eqs. (41) and (42) we have to
exchange Mi by the weighted mean halo mass Mw , bi by
the weighted bias bw , and 1/n̄i by 1/n̄ × hw2 i/hwi2 , with
Z
dn
n̄ =
(M ) dM ,
(43)
dM
Z
dn
1
(M )w(M ) dM ,
(44)
hwi =
n̄
dM
Z
1
dn
hw2 i =
(M )w2 (M ) dM ,
(45)
n̄
dM
Z
dn
1
(M )w(M )M dM ,
(46)
Mw =
n̄
dM
Z
dn
1
(M )w(M )b(M ) dM ,
(47)
bw =
n̄
dM
where we integrate over all resolved halo masses.
Last but not least we can utilize the halo model to determine the reduced shot noise as a function of mass resolution. For this we need analytic expressions for the halo
dn
(M ) and the halo bias b(M ) to commass function dM
pute the eigenvalues and eigenvectors of the shot noise
matrix. We use the functional forms of Sheth-Tormen [3]
with the parameters given in [39]. For infinitesimal bins,
4.0
4.0
3.5
3.5
(<δi2>/<δm2>)1/2
<δiδm>/<δm2>
xv
3.0
2.5
2.0
3.0
2.5
2.0
1.5
1.5
1.0
1.0
0.01
0.01
0.10
k [hMpc-1]
k [hMpc-1]
0.10
FIG. 13. Scale-dependent bias estimators from halo-matter cross correlation (left panel) and halo-auto correlation (right panel)
as predicted by the halo model. The simulation results from 10 mass bins are shown as crosses with error bars (in color); the
solid lines show the halo model results. The black dots with error bars show the results for only one mass bin (all halos) for
both uniform and modified mass weighting, with the halo model prediction overplotted in dashed and dotted, respectively.
106
Eqs. (35) and (36) can be rewritten as
with
1 p
1
1
hMbi ±
−
hM2 ihb2 i ,
dn ρ̄m
ρ̄m
b
M
∓p
,
V ± (M ) = p
hM2 i
hb2 i
Z
1
n̄
Z
1
2
hM i =
n̄
Z
1
hb2 i =
n̄
hMbi =
(49)
dn
(M )Mb dM ,
dM
dn
(M )M2 dM ,
dM
dn
(M )b2 dM .
dM
(50)
(51)
We neglect all eigenmodes except λ± and V ± for this
calculation, since they have the largest contribution in
signal-to-noise. This yields
σr2 (Mmin ) =
1
2
σw
+
+
1
2
σw
−
−1
.
104
103
102
101
100
1011
1012
1013
Mmin [h-1MO• ]
1014
(52)
The integrals run from Mmin to ∞ and we can compute
the reduced shot noise σr2 of the weighted halo density
field for various values of Mmin using Eqs. (14) and (17)
in their infinitesimal form:
R dn
1
2
2
n̄
dM (M )V (M ) dM
σw (Mmin ) = λ
(53)
R
2 .
1
dn
n̄
dM (M )V (M ) dM
105
(48)
σ2(Mmin) [h-3Mpc3]
λ± =
(54)
The result can then be compared to the expected shot
noise from uniform weighting, which, according to the
FIG. 14. Stochasticity between halos and the dark matter as
a function of mass resolution as predicted by the halo model
for the cases of uniform (σu2 , solid red line) and modified
mass weighting (σr2 , dashed green line). The dotted (blue)
line shows the Poisson prediction 1/n̄. The results from our
highest resolution simulation are overplotted as red diamonds
(uniform weighting) and green circles (modified mass weighting) for five different low-mass cuts. The black crosses show
the corresponding values of 1/n̄ taken from the simulation.
halo model, is given by
σu2 (Mmin ) =
1
hnM 2 i
M
+ b2 2
− 2b
,
n̄
ρ̄m
ρ̄m
(55)
where n̄, M and b depend on Mmin and can be computed
from Eqs. (43), (46) and (47) using uniform weights, i.e.
w(M ) = 1. The functions σu2 (Mmin ) and σr2 (Mmin) are
depicted in Fig. 14. Apparently, at low resolution (high
xvi
Mmin), the improvement due to modified mass weight12 −1
ing is quite modest. However, for Mmin <
∼ 10 h M⊙
2
the function σu (Mmin ) approaches a constant, while
σr2 (Mmin ) still decreases linearly with Mmin . This linear
trend leads to a suppression of stochasticity by almost
2 orders of magnitude if one can resolve halos down to
Mmin = 1010 h−1 M⊙ .
In order to cross-check these results we computed the
k-averaged shot noise (shown as filled symbols) for various low-mass cuts from our highest resolution simulation consisting of 15363 particles resolving halos down
to Mmin ≃ 9.4 × 1011 h−1 M⊙ . Taking into account
all halos in this simulation we obtain a minimal shot
noise level when applying modified mass weighting with
M0 ≃ 3.1 × 1012 h−1 M⊙ , which again satisfies the anticipated relation M0 ≃ 3Mmin .
Overall, the agreement between the simulations and
the halo model is reasonable, but not perfect. At low
Mmin the halo model underestimates the shot noise of the
uniformly weighted halo density field and the shot noise
suppression due to modified mass weighting relative to
uniform weighting is even larger than predicted by the
model. At high Mmin the shot noise in the simulation
is not perfectly scale independent anymore and since we
are taking the average over the whole k-range the result
becomes more inaccurate.
VI.
CONCLUSIONS
In a previous paper [6] it was shown that weighting
dark matter halos by their mass can lead to a suppression of stochasticity between halos and the dark matter
relative to naive expectations. In this work we investigated the shot noise matrix, defined as the two-point correlator Cij ≡ h(δi − bi δm )(δj − bj δm )i in Fourier space,
split into equal number density mass bins. The eigensystem of this matrix reveals two nontrivial eigenvalues, one
of them being enhanced, the other suppressed compared
to the Poisson model expectation. It is the latter that
leads to a reduced stochasticity. The optimal estimator
of the dark matter and the eigenvector corresponding to
the lowest eigenvalue are very similar and the latter dominates the signal-to-noise ratio of the halo density field.
We fit both vectors by a smooth function of mass which
we denote modified mass weighting. It is proportional to
halo mass at the high-mass end and approaches a constant towards lower masses which is determined by the
minimum halo mass resolved in the simulations. This
constant is roughly 3 times the minimum halo mass over
the range of masses we explored.
Applying this function to weight the halo density field
results in a field that is more correlated with the dark
matter with a suppressed shot noise component, improving upon previous results [6] by a factor of 2 in signal-tonoise. We investigate the effect of uncertainty in halo
mass, finding that it does not change our fundamental conclusions, even if it weakens the strength of the
method: a realistic amount of log-normal scatter in mass
at the level of 0.5 increases the shot noise by a about a
factor of 2. Our results can directly be applied to methods that attempt to eliminate sampling variance by investigating the relation between galaxies and the dark
matter both tracing the same LSS. In this case the error is determined by the stochasticity between the two
and reducing it can improve the ultimate reach of these
methods [4, 5].
Considering the halo model as a theoretical approach
to describe the shot noise matrix, we find analytical expressions for its eigenvalues and eigenvectors. In particular, the two nontrivial eigenvectors can be written as
a linear combination of halo bias and halo mass, which
yields a considerable agreement with our simulation results. Furthermore, the two estimators
of the scalep
2
2 i, are well
dependent bias, hδh δm i/hδm
i and hδh2 i/hδm
reproduced. However, our model suffers from the lack of
mass and momentum conservation: its implementation,
together with higher-order perturbation theory and halo
exclusion, further improves the agreement and will be
presented elsewhere.
The halo model suggests the stochasticity between
modified mass-weighted halos and the dark matter to
decrease linearly with mass resolution below M ≃
1012 h−1 M⊙ , yielding a suppression by almost 2 orders of
magnitude at Mmin ≃ 1010 h−1 M⊙ as compared to uniformly weighted halos. While we focused on the question
of how well halos can reconstruct the dark matter, our
analysis is also applicable to the study of stochasticity
between halos themselves. Indeed, reducing the stochasticity between different halo tracers by optimal weighting
techniques, while at the same time canceling sampling
variance, should be possible even if the dark matter field
is not measured. This will be addressed in more detail in
a future work.
Specific applications are the best way to test the efficiency of our method. There is probably not much advantage in applying it to the standard power spectrum
determination, where the sampling variance error dominates the error budget in the limit of small stochasticity,
while in the opposite limit of rare halos, when the shot
noise power is comparable to the intrinsic halo power, we
do not see much gain (as demonstrated by the fact that
all the points in Fig. 14 overlap for the highest Mmin ,
which corresponds to halos with the lowest number density). More promising applications are those where the
sampling variance error is eliminated and the error budget is dominated by stochasticity, or the ratio of shot
noise power to the halo power.
In this paper we have focused on the bias determination from galaxy and dark matter correlations [5] as a
specific application, but other applications are possible,
such as constraining fNL from non-Gaussianity [27, 40]
xvii
and the redshift-space parameter β from redshift-space
distortions [8, 41], to name a few. Upcoming surveys like
SDSS-III [42], JDEM/EUCLID [43, 44] or BigBOSS [45]
and LSST [46] will increase the available number of galaxies significantly, providing both 3D galaxy maps and 2D
to 3D dark matter maps (via weak lensing techniques,
enhanced by lensing tomography).
Our results suggest that correlating modified halo
mass-weighted galaxies against the dark matter has the
potential to lead to dramatic improvements in the precision of cosmological parameter estimation. We will
explore more explicit demonstrations of the above mentioned applications in future work.
We thank Patrick McDonald and Martin White for
useful discussions, V. Springel for making public his gadget ii code and for providing his b-fof halo finder,
and Roman Scoccimarro for making public his 2lpt initial conditions code. RES acknowledges support from a
Marie Curie Reintegration Grant and the Swiss National
Foundation. This work is supported by the Packard
Foundation, the Swiss National Foundation under Contract No. 200021-116696/1, and WCU Grant No. R322009-000-10130-0.
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4
PAPER III

Chapter 4: Paper III
Summary One promising application of the previous findings is the search for
primordial non-Gaussianity (PNG) in the clustering of galaxies. It has been shown
that certain types of PNG induce a scale-dependent correction to the halo bias,
which grows towards larger scales. When multiple tracers of the dark matter density
field are considered, such as halos of different mass, the relative bias between the
tracers is not as severely affected by cosmic variance as the power spectrum of each
tracer. In this case, the accuracy is limited by the shot noise of the tracers only.
In the following paper, optimal weighting techniques are applied to improve the
measurement of a scale-dependent component of the halo bias arising in simulations
with non-Gaussian initial conditions. On the basis of a Fisher matrix analysis, it
is shown that the signatures of PNG are more likely to be found when multiple
tracers are considered, due to the cancellation of cosmic variance. If, in addition,
the dark matter density field itself is available, it is shown how the previously derived
optimal weights can help improve upon current constraints on PNG by more than
an order of magnitude. Utilizing the halo model to extrapolate our numerical results
to higher mass resolution suggests that the same gains can be expected even without
observations of the dark matter density field, by considering a wide enough mass
range of halos split into multiple tracers.
Three authors were involved in this study: Uroš Seljak proposed the topic of
PNG as an application for the multitracer analysis, as put forward in an earlier
publication of his, and suggested to include the weighting techniques developed
above. Vincent Desjacques provided N -body simulations with both Gaussian and
non-Gaussian initial conditions and contributed to some parts of the manuscript,
such as the introduction and details about the simulation. I carried out the numerical
analysis and prepared the main body of the paper, always in close consultation with
Uroš Seljak. The manuscript got accepted in October 2011 in Physical Review D
(vol. 84, ID 083509, arXiv:1104.2321).
Optimal Constraints on Local Primordial Non-Gaussianity from the Two-Point
Statistics of Large-Scale Structure
Nico Hamaus,1, ∗ Uroš Seljak,1, 2, 3 and Vincent Desjacques1
1
2
Institute for Theoretical Physics, University of Zurich, 8057 Zurich, Switzerland
Physics Department, Astronomy Department and Lawrence Berkeley National Laboratory,
University of California, Berkeley, California 94720, USA
3
Ewha University, Seoul 120-750, S. Korea
(Dated: October 10, 2011)
One of the main signatures of primordial non-Gaussianity of the local type is a scale-dependent
correction to the bias of large-scale structure tracers such as galaxies or clusters, whose amplitude
depends on the bias of the tracers itself. The dominant source of noise in the power spectrum
of the tracers is caused by sampling variance on large scales (where the non-Gaussian signal is
strongest) and shot noise arising from their discrete nature. Recent work has argued that one can
avoid sampling variance by comparing multiple tracers of different bias, and suppress shot noise by
optimally weighting halos of different mass. Here we combine these ideas and investigate how well
the signatures of non-Gaussian fluctuations in the primordial potential can be extracted from the
two-point correlations of halos and dark matter. On the basis of large N -body simulations with
local non-Gaussian initial conditions and their halo catalogs we perform a Fisher matrix analysis of
the two-point statistics. Compared to the standard analysis, optimal weighting and multiple-tracer
techniques applied to halos can yield up to 1 order of magnitude improvements in fNL -constraints,
even if the underlying dark matter density field is not known. In this case one needs to resolve all
halos down to 1010 h−1 M⊙ at z = 0, while with the dark matter this is already achieved at a mass
threshold of 1012 h−1 M⊙ . We compare our numerical results to the halo model and find satisfactory
agreement. Forecasting the optimal fNL -constraints that can be achieved with our methods when
applied to existing and future survey data, we find that a survey of 50h−3 Gpc3 volume resolving
all halos down to 1011 h−1 M⊙ at z = 1 will be able to obtain σfNL ∼ 1 (68% cl), a factor of ∼ 20
improvement over the current limits. Decreasing the minimum mass of resolved halos, increasing
the survey volume or obtaining the dark matter maps can further improve these limits, potentially
reaching the level of σfNL sim0.1. This precision opens up the possibility to distinguish different
types of primordial non-Gaussianity and to probe inflationary physics of the very early Universe.
I.
INTRODUCTION
A detection of primordial non-Gaussianity has the potential to test today’s standard inflationary paradigm
and its alternatives for the physics of the early Universe.
Measurements of the CMB bispectrum furnish a direct
probe of the nature of the initial conditions (see, e.g.,
[1–5] and references therein), but are limited by the twodimensional nature of the CMB and its damping on small
scales. However, the non-Gaussian signatures imprinted
in the initial fluctuations of the potential gravitationally evolve into the large-scale structure (LSS) of the
Universe, which can be observed in all three dimensions
and whose statistical properties can be constrained with
galaxy clustering data (for recent reviews, see [6, 7]).
One of the cleanest probes is the galaxy (or, more
generally, any tracer of LSS including clusters, etc.)
two-point correlation function (in configuration space)
or power spectrum (in Fourier space), which develops a characteristic scale dependence on large scales in
the presence of primordial non-Gaussianity of the local
∗
[email protected]
type [8]. The power spectrum picks up an additional
term proportional to fNL (bG − 1), where bG is the Gaussian bias of the tracer and fNL is a parameter describing
the strength of the non-Gaussian signal. However, the
precision to which we can constrain fNL is limited by
sampling variance on large scales: each Fourier mode is
an independent realization of a (nearly) Gaussian random field, so the ability to determine its rms-amplitude
from a finite number of modes is limited. Recent work
has demonstrated that it is possible to circumvent sampling variance by comparing two different tracers of the
same underlying density field [9–12]. The idea is to take
the ratio of power spectra from two tracers to (at least
partly) cancel out the random fluctuations, leaving just
the signature of primordial non-Gaussianity itself.
Another important limitation arises from the fact that
galaxies are discrete tracers of the underlying dark matter
distribution. Therefore, with a finite number of observable objects, the measurement of their power spectrum
is affected by shot noise. Assuming galaxies are sampled
from a Poisson process, this adds a constant contribution
to their power spectrum, it is given by the inverse tracer
number density 1/n̄. This is particularly important for
massive tracers such as clusters, since their number density is very low. Yet they are strongly biased and there-
ii
fore very sensitive to a potential non-Gaussian signal.
Recent work has demonstrated the Poisson shot noise
model to be inadequate [13–15]. In particular, [13, 14]
have shown that a mass-dependent weighting can considerably suppress the stochasticity between halos and the
dark matter and thus reduce the shot noise contribution.
In view of constraining primordial non-Gaussianity from
LSS, this can be a very helpful tool to further reduce the
error on fNL .
Both of these methods (sampling variance cancellation
and shot noise suppression) have so far been discussed
separately in the literature. In this paper we combine
the two to derive optimal constraints on fNL that can
be achieved from two-point correlations of LSS. We show
that dramatic improvements are feasible, but we do not
imply that two-point correlations achieve optimal constraints in general: further gains may be possible when
considering higher-order correlations, starting with the
bispectrum analysis [16] (three-point correlations).
This paper is organized as follows: Sec. II briefly reviews the impact of local primordial non-Gaussianity on
the halo bias, and the calculation of the Fisher information content on fNL from two-point statistics in Fourier
space is presented in Sec. III. In Sec. IV we apply our
weighting and multitracer methods to dark matter halos
extracted from a series of large cosmological N -body simulations and demonstrate how we can improve the fNL constraints. These results are confronted with the halo
model predictions in Sec. V before we finally summarize
our findings in Sec. VI.
II.
NON-GAUSSIAN HALO BIAS
Primordial non-Gaussianity of the local type is usually
characterized by expanding Bardeen’s gauge-invariant
potential Φ about the fiducial Gaussian case. Up to second order, it can be parametrized by the mapping [17–20]
Φ(x) = ΦG (x) +
fNL Φ2G (x)
,
(1)
where ΦG (x) is an isotropic Gaussian random field and
fNL a dimensionless phenomenological parameter. Ignoring smoothing (we will consider scales much larger than
the Lagrangian size of a halo), the linear density perturbation δ0 is related to Φ through the Poisson equation in
Fourier space,
δ0 (k, z) =
2 k 2 T (k)D(z)c2
Φ(k) ,
3
Ωm H02
(2)
where T (k) is the matter transfer function and D(z) is
the linear growth rate normalized to 1 + z. Applying the
peak-background split argument to the Gaussian piece of
Bardeen’s potential, one finds a scale-dependent correction to the linear halo bias [8, 21, 22]:
b(k, fNL ) = bG + fNL (bG − 1)u(k, z) ,
(3)
where bG is the scale-independent linear bias parameter
of the corresponding Gaussian field (fNL = 0) and
u(k, z) ≡
3δc Ωm H02
2
k T (k)D(z)c2
.
(4)
Here, δc ≃ 1.686 is the linear critical overdensity for
spherical collapse. Corrections to Eq. (3) beyond linear
theory have already been worked out and agree reasonably well with numerical simulations [23–26]. Also, the
dependence of the halo bias on merger history and halo
formation time affects the amplitude of the non-Gaussian
corrections in Eq. (3) [22, 27–29], which we will neglect
here.
III.
FISHER INFORMATION FROM THE
TWO-POINT STATISTICS OF LSS
It is believed that all discrete tracers of LSS, such as
galaxies and clusters, reside within dark matter halos,
collapsed nonlinear structures that satisfy the conditions
for galaxy formation. The analysis of the full complexity
of LSS is therefore reduced to the information content
in dark matter halos. In this section we introduce our
model for the halo covariance matrix and utilize it to
compute the Fisher information content on fNL from the
two-point statistics of halos and dark matter in Fourier
space. We separately consider two cases: first halos only
and second halos combined with dark matter. While the
observation of halos is relatively easy with present-day
galaxy redshift surveys, observing the underlying dark
matter is hard, but not impossible: weak-lensing tomography is the leading candidate to achieve that.
A.
Covariance of Halos
1.
Definitions
We write the halo overdensity in Fourier space as a
vector whose elements correspond to N successive bins
⊺
δ h ≡ (δh1 , δh2 , . . . , δhN ) .
(5)
In this paper we will only consider a binning in halo mass,
but the following equations remain valid for any quantity that the halo density field depends on (e.g., galaxyluminosity, etc.). The covariance matrix of halos is defined as
Ch ≡ hδ h δ ⊺h i ,
(6)
i.e., the outer product of the vector of halo fields averaged
within a k-shell in Fourier space. Assuming the halos to
be locally biased and stochastic tracers of the dark matter
density field δ, we can write
δ h = bδ + ǫ ,
(7)
iii
and we define
b≡
hδ h δi
hδ 2 i
(8)
as the effective bias, which is generally scale-dependent
and non-Gaussian. ǫ is a residual noise-field with zero
mean and we assume it to be uncorrelated with the dark
matter, i.e., hǫδi = 0 [30].
In each mass bin, the effective bias b shows a distinct
dependence on fNL . In what follows, we will assume that
b is linear in fNL , as suggested by Eq. (3):
b(k, fNL ) = bG + fNL b′ (k) .
(9)
′
Here, bG is the Gaussian effective bias and b ≡ ∂b/∂fNL .
Finally, we write P ≡ hδ 2 i for the nonlinear dark matter
power spectrum and assume ∂P/∂fNL = 0. This is a
good approximation on large scales [31–33]. Thus, the
model from Eq. (7) yields the following halo covariance
matrix:
Ch = bb⊺ P + E ,
(10)
where the shot noise matrix E was defined as
E ≡ hǫǫ⊺ i .
(11)
In principle, E can contain other components than pure
Poisson noise, for instance higher-order terms from the
bias expansion [34–36]. Here and henceforth, we will define E as the residual from the effective bias term bb⊺ P in
Ch , and allow it to depend on fNL . Thus, with Eqs. (8)
and (10) the shot noise matrix can be written as
E = hδ h δ ⊺h i −
hδ h δihδ ⊺h δi
.
hδ 2 i
(12)
This agrees precisely with the definition given in [14]
for the Gaussian case, however it also takes into account the possibility of a scale-dependent effective bias in
non-Gaussian scenarios, such that the effective bias term
bb⊺ P always cancels in this expression [37].
Reference [10] already investigated the Fisher information content on primordial non-Gaussianity for the idealized case of a purely Poissonian shot noise component in
the halo covariance matrix. In [15], the halo covariance
was suggested to be of a similar simple form, albeit with a
modified definition of halo bias and a diagonal shot noise
matrix. In this work we will consider the more general
model of Eq. (10) without assuming anything about E.
Instead we will investigate the shot noise matrix with the
help of N -body simulations.
The Gaussian case has already been studied in [14].
Simulations revealed a very simple eigenstructure of the
shot noise matrix: for N > 2 mass bins of equal number
density n̄ it exhibits a (N − 2)-dimensional degenerate
(N −2)
subspace with eigenvalue λP
= 1/n̄, which is the
expected result from Poisson sampling. Of the two remaining eigenvalues λ± , one is enhanced (λ+ ) and one
suppressed (λ− ) with respect to the value 1/n̄. The shot
noise matrix can thus be written as
E = n̄−1 I + (λ+ − n̄−1 )V+ V+⊺ + (λ− − n̄−1 )V− V−⊺ , (13)
where I is the N × N identity matrix and V± are the
normalized eigenvectors corresponding to λ± . Its inverse
takes a very similar form
⊺
⊺
−1
E −1 = n̄I + (λ−1
+ − n̄)V+ V+ + (λ− − n̄)V− V− . (14)
The halo model [38] can be applied to predict the functional form of λ± and V± (see [14] and Sec. V). This
approach is however not expected to be exact, as it does
not ensure mass- and momentum conservation of the dark
matter density field and leads to white-noise-like contributions in both the halo-matter cross and the matter
auto power spectra which are not observed in simulations [39]. Yet, the halo model is able to reproduce the
eigenstructure of E fairly well [14] and we will use it for
making predictions beyond our N -body resolution limit.
In the Gaussian case one can also relate the dominant
eigenmode V+ with corresponding eigenvalue λ+ to the
second-order term arising in a local bias-expansion model
[34, 35], where the coefficients bi are determined analytically from the peak-background split formalism given a
halo mass function [40, 41]. In non-Gaussian scenarios
this can be extended to a multivariate expansion in dark
matter density δ and primordial potential Φ including
bias coefficients for both fields [16, 23]. For the calculation of E we will however restrict ourselves to the Gaussian case and later compare with the numerical results
of non-Gaussian initial conditions to see the effects of
fNL on E and its eigenvalues. The suppressed eigenmode
V− with eigenvalue λ− can also be explained by a haloexclusion correction to the Poisson-sampling model for
halos, as studied in [33].
In what follows, we will truncate the local bias expansion at second order. Therefore, we shall assume the
following model for the halo overdensity in configuration
space
δ h (x) = b1 δ(x) + b2 δ 2 (x) + nP (x) + nc (x) .
(15)
Here, nP is the usual Poisson noise and nc a correction to
account for deviations from the Poisson-sampling model.
In Fourier space, this yields
δ h (k) = b1 δ(k) + b2 (δ∗δ) (k) + nP (k) + nc (k) , (16)
where the asterisk-symbol denotes a convolution. The
Poisson noise nP arises from a discrete sampling of the
field δ h with a finite number of halos, it is uncorrelated
with the underlying dark matter density, hnP δi = 0,
and its power spectrum is hnP n⊺P i = 1/n̄ (Poisson white
iv
noise). We further assume hnP n⊺c i = hnc δi = 0, which
leads to
h(δ∗δ) δi
,
hδ 2 i
(17)
Ch = b1 b⊺1 hδ 2 i + (b1 b⊺2 + b2 b⊺1 ) h(δ∗δ) δi
+ b2 b⊺2 h(δ∗δ)2 i + hnP n⊺P i + hnc n⊺c i ,
E = n̄−1 I + b2 b⊺2 h(δ∗δ)2 i −
2
h(δ∗δ) δi
hδ 2 i
(18)
(20)
where we define
h(δ∗δ) δi2
.
hδ 2 i
(21)
In [36] this term is absorbed into an effective shot noise
power, since it behaves like white noise on large scales and
arises from the peaks and troughs in the dark matter density field being nonlinearly biased by the b2 -term [42]. We
evaluated Eδ2 along with the expressions that appear in
Eq. (21) with the help of our dark matter N -body simulations for Gaussian and non-Gaussian initial conditions
(for details about the simulations, see Sec. IV).
The results are depicted in Fig. 1. Eδ2 obviously shows
a slight dependence on fNL , but it remains white-noiselike even in the non-Gaussian cases. The fNL -dependence
of this term has not been discussed in the literature yet,
but it can have a significant impact on the power spectrum of high-mass halos which have a large b2 -term; see
Eq. (20). A discussion of the numerical results for halos, specifically the fNL -dependence of λ+ , is conducted
later in this paper. It is also worth noticing the fNL 2
dependence of h(δ∗δ) i and h(δ∗δ) δi. The properties of
the squared dark matter field δ 2 (x) are similar to the
ones of halos, namely, the k −2 -correction of the effective
bias in Fourier space, which in this case is defined as
bδ2 ≡ h(δ∗δ) δi/hδ 2 i and appears in Eq. (17).
The last term in Eq. (19) corresponds to the suppressed
eigenmode of the shot noise matrix. Both its eigenvector
and eigenvalue can be described reasonably well by the
halo model [14]. The argument of [33] based on halo
exclusion yields a similar result while providing a more
intuitive explanation for the occurrence of such a term.
2.
εδ 2
<(δ∗δ)δ>
+ hnc n⊺c i .
λ+ = b⊺2 b2 Eδ2 + n̄−1 ,
2
105
104
(19)
Hence, we can identify the normalized vector b2 /|b2 | with
the eigenvector V+ of Eq. (13) with corresponding eigenvalue
Eδ2 ≡ h(δ∗δ) i −
<(δ∗δ)2>
P [h-3Mpc3]
b = b1 + b2
106
Likelihood and Fisher information
In order to find the best unbiased estimator for fNL ,
we have to maximize the likelihood function. Although
<δ2>
103
0.01
k [hMpc-1]
0.10
FIG. 1. Shot noise Eδ2 of the squared dark matter density field
δ 2 as defined in Eq. (21) with both Gaussian (solid green) and
non-Gaussian initial conditions with fNL = +100 (solid red)
and fNL = −100 (solid yellow) from N -body simulations at
z = 0. Clearly, Eδ2 is close to white-noise like in all three
cases. The auto power spectrum h(δ∗δ)2 i of δ 2 in Fourier
space (dashed), its cross power spectrum h(δ∗δ) δi with the
ordinary dark matter field δ (dot-dashed), as well as the ordinary dark matter power spectrum hδ 2 i (dotted) are overplotted for the corresponding values of fNL . The squared dark
matter field δ 2 can be interpreted as a biased tracer of δ and
therefore shows the characteristic fNL -dependence of biased
fields (like halos) on large scales.
we are dealing with non-Gaussian statistics of the density
field, deviations from the Gaussian case are usually small
in practical applications (e.g., [22, 43, 44]), so we will
consider a multivariate Gaussian likelihood
1
1 ⊺ −1
√
L =
exp
−
δ
C
δ
.
(22)
2 h h h
(2π)N/2 det Ch
Maximizing this likelihood function is equivalent to minimizing the following chi-square,
χ2 = δ ⊺h C−1
h δ h + ln (1 + α) + ln(det E) ,
(23)
where we dropped the irrelevant constant N ln(2π) and
used
det Ch = det (bb⊺ P + E) = (1 + α) det E ,
(24)
with α ≡ b⊺ E −1 bP . For a single mass bin, Eq. (23)
simplifies to
χ2 =
δh2
+ ln b2 P + E .
b2 P + E
(25)
The Fisher information matrix [45] for the parameters
θi and θj and the random variable δ h with covariance Ch ,
v
as derived from a multivariate Gaussian likelihood [46,
47], reads
1
∂Ch −1 ∂Ch −1
Fij ≡ Tr
Ch
Ch
.
(26)
2
∂θi
∂θj
With the above assumptions, the derivative of the halo
covariance matrix with respect to the parameter fNL is
∂Ch
= bb′⊺ + b′ b⊺ P + E ′ ,
∂fNL
(27)
with E ′ ≡ ∂E/∂fNL . The inverse of the covariance matrix can be obtained by applying the Sherman-Morrison
formula [48, 49]
−1
C−1
−
h =E
E
−1
⊺
−1
bb E P
,
1+α
(28)
where again α ≡ b⊺ E −1 bP . On inserting the two previous relations into Eq. (26), we eventually obtain the full
expression for FfNL fNL in terms of b, b′ , E, E ′ and P (see
Appendix A for the derivation of Eq. (A11)). Neglecting
the fNL -dependence of E, i.e., setting E ′ ≡ 0, the Fisher
information on fNL becomes
αγ + β 2 + α αγ − β 2
FfNL fNL =
,
(29)
(1 + α)2
with α ≡ b⊺ E −1 bP , β ≡ b⊺ E −1 b′ P and γ ≡ b′⊺ E −1 b′ P .
For a single mass bin, Eq. (A11) simplifies to Eq. (A12),
2
′
bb P + E ′ /2
.
(30)
FfNL fNL = 2
b2 P + E
This implies that even in the limit of a very well-sampled
halo density field (n̄ → ∞) with negligible shot noise
power E (and neglecting E ′ ) the Fisher information content on fNL that can be extracted per mode from a single
2
halo mass bin is limited to the value 2 (b′ /b) . This is due
to the fact that we can only constrain fNL from a change
in the halo bias relative to the Gaussian expectation, not
from a measurement of the effective bias itself. The latter can only be measured directly if one knows the dark
matter distribution, as will be shown in the subsequent
paragraph. However, the situation changes for several
halo mass bins (multiple tracers as in [9]). In this case,
the Fisher information content from Eqs. (A11) and (29)
2
can exceed the value 2 (b′ /b) (see Sec. IV and V).
B.
Covariance of Halos and Dark Matter
1.
Definitions
We will now assume that we possess knowledge about
the dark matter distribution in addition to the halo density field. In practice one may be able to achieve this by
combining galaxy redshift surveys with lensing tomography [50], but the prospects are somewhat uncertain. We
will simply add the dark matter overdensity mode δ to
the halo overdensity vector δ h , defining a new vector
⊺
δ ≡ (δ, δh1 , δh2 , . . . , δhN ) .
(31)
In analogy with the previous section, we define the covariance matrix as C ≡ hδδ ⊺ i and write
2
hδ i hδ ⊺h δi
P b⊺ P
=
.
(32)
C=
bP Ch
hδ h δi hδ h δ ⊺h i
2.
Likelihood and Fisher information
Upon inserting the new covariance matrix into the
Gaussian likelihood as defined in Eq. (22), we find the
chi-square to be
χ2 = δ ⊺ C−1 δ + ln (det E) ,
(33)
where we used
2
= P det E , (34)
det C = det Ch det P − b⊺ C−1
h bP
and we still assume P to be independent of fNL and therefore drop the term ln (P ) in Eq. (33). In terms of the halo
and dark matter overdensities, the chi-square can also be
expressed as
⊺
χ2 = (δ h − bδ) E −1 (δ h − bδ) + ln (det E) ,
(35)
which is equivalent to the definition in [14] (where the
last term was neglected). The corresponding expression
for a single halo mass bin reads
χ2 =
(δh − bδ)2
+ ln (E) .
E
(36)
For the derivative of C with respect to fNL we get
∂C
0
b′⊺ P
=
.
(37)
′
′⊺
′ ⊺
′
bP
bb P + b b P + E
∂fNL
Performing a block inversion, we readily obtain the inverse covariance matrix,
(1 + α)P −1 −b⊺ E −1
C−1 =
.
(38)
−E −1 b
E −1
As shown in Appendix B, the Fisher information content
on fNL now becomes
FfNL fNL = γ + τ ,
(39)
with γ ≡ b′⊺ E −1 b′ P and τ ≡ 12 Tr E ′ E −1 E ′ E −1 . For a
single halo mass bin this further simplifies to
2
b′2 P
1 E′
.
(40)
FfNL fNL =
+
E
2 E
vi
It is worth noting that, in contrast to Eq. (30), the Fisher
information from one halo mass bin with knowledge of
the dark matter becomes infinite in the limit of vanishing E. In this limit the effective bias can indeed be determined exactly, allowing an exact measurement of fNL [9].
IV.
APPLICATION TO N-BODY SIMULATIONS
We employ numerical N -body simulations with both
Gaussian and non-Gaussian initial conditions to find signatures of primordial non-Gaussianity in the two-point
statistics of the final density fields in Fourier space. More
precisely, we consider an ensemble of 12 realizations of
box-size 1.6h−1 Gpc (this yields a total effective volume of
Veff ≃ 50h−3 Gpc3 ). Each realization is seeded with both
Gaussian (fNL = 0) and non-Gaussian (fNL = ±100) initial conditions of the local type [31], and evolves 10243
particles of mass 3.0 × 1011 h−1 M⊙ . The cosmological
parameters are Ωm = 0.279, ΩΛ = 0.721, Ωb = 0.046,
σ8 = 0.81, ns = 0.96, and h = 0.7, consistent with the
wmap5 [51] best-fit constraint. Additionally, we consider one realization with each fNL = 0, ±50 of box-size
1.3h−1 Gpc with 15363 particles of mass 4.7×1010 h−1 M⊙
to assess a higher-resolution regime. The simulations
were performed on the supercomputer zbox3 at the University of Zürich with the gadget ii code [52]. The initial conditions were laid down at redshift z = 100 by perturbing a uniform mesh of particles with the Zel’dovich
approximation.
To generate halo catalogs, we employ a friends-offriends (FOF) algorithm [53] with a linking length equal
to 20% of the mean interparticle distance. For comparison, we also generate halo catalogs using the ahf halo
finder developed by [54], which is based on the spherical
overdensity (SO) method [55]. In this case, we assume an
overdensity threshold ∆c (z) being a decreasing function
of redshift, as dictated by the solution to the spherical
collapse of a tophat perturbation in a ΛCDM Universe
[56]. In both cases, we require a minimum of 20 particles per halo, which corresponds to a minimum halo mass
Mmin ≃ 5.9 × 1012 h−1 M⊙ for the simulations with 10243
particles. For Gaussian initial conditions the resulting total number density of halos is n̄ ≃ 7.0×10−4 h3 Mpc−3 and
4.2×10−4h3 Mpc−3 for the FOF and SO catalogs, respectively. Note that the FOF mass estimate is on average
20% higher than the SO mass estimate. For our 15363particles simulation we obtain Mmin ≃ 9.4 × 1011 h−1 M⊙
and n̄ ≃ 4.0 × 10−3h3 Mpc−3 resulting from the FOF halo
finder.
The binning of the halo density field into N consecutive
mass bins is done by sorting all halos by increasing mass
and dividing this ordered array into N bins with an equal
number of halos. The halos of each bin i ∈ [1 . . . N ] are
selected separately to construct the halo density field δhi .
The density fields of dark matter and halos are first com-
puted in configuration space via interpolation of the particles onto a cubical mesh with 5123 grid points using a
cloud-in-cell mesh assignment algorithm [57]. We then
perform a fast fourier transform to compute the modes
of the fields in k-space.
For each of our Gaussian and non-Gaussian realizations, we match the total number of halos to the one realization with the least amount of them by discarding halos
from the low-mass end. This abundance matching technique ensures that we eliminate any possible signature of
primordial non-Gaussianity induced by the unobservable
fNL -dependence of the halo mass function. It guarantees
a constant value 1/n̄ of the Poisson noise for both Gaussian and non-Gaussian realizations. A dependence of the
Poisson noise on fNL would complicate the interpretation of the Fisher information content. Note also that,
in order to calculate the derivative of a function F with
respect to fNL , we apply the linear approximation
∂F
F (fNL = +100) − F (fNL = −100)
≃
,
∂fNL
2 × 100
(41)
which exploits the statistics of all our non-Gaussian runs.
All the error bars quoted in this paper are computed from
the variance amongst our 12 realizations.
A.
Effective bias and shot noise
At the two-point level and in Fourier space, the clustering of halos as described by Eq. (10) is determined
by two basic components: effective bias and shot noise.
Since the impact of primordial non-Gaussianity on the
nonlinear dark matter power spectrum P is negligible on
large scales (see Fig. 1), the dependence of both b and
E on fNL must be known if one wishes to constrain the
latter. In the following sections, we will examine this
dependence in our series of N -body simulations.
1.
Effective bias
In the top left panel of Fig. 2, the effective bias b in
the fiducial Gaussian case (fNL = 0) is shown for 30
consecutive FOF halo mass bins as a function of wave
number. In the large-scale limit k → 0, the measurements are consistent with being scale-independent, as indicated by the dotted lines which show the average of
b(k, fNL = 0) over all modes with k ≤ 0.032hMpc−1 ,
denoted bG . At larger wave numbers, the deviations
can be attributed to higher-order bias terms, which are
most important at high mass. Relative to the low-k averaged, scale-independent Gaussian bias bG , these corrections tend to suppress the effective bias at low mass,
whereas they increase it at the very high-mass end (see
Eq. (17)). The right panel of Fig. 2 shows the large-scale
average bG as a function of halo mass, as determined
vii
b(fNL=0)
3.5
3.0
3.0
2.5
2.5
2.0
1.5
bG
1.0
∂b/∂fNL
0.015
2.0
0.010
1.5
0.005
0.000
1.0
0.01
k [hMpc-1]
0.10
1013
M [h-1MO• ]
1014
FIG. 2. LEFT: Gaussian effective bias (top) and its derivative with respect to fNL (bottom) for the case of 30 mass bins. The
scale-independent part bG is plotted in dotted lines for each bin; it was obtained by averaging all modes with k ≤ 0.032hMpc −1 .
RIGHT: Large-scale averaged Gaussian effective bias bG from the left panel (dotted lines) plotted against mean halo mass.
The solid line depicts the linear-order bias derived from the peak-background split formalism. All error bars are obtained from
the variance of our 12 boxes to their mean. Results are shown for FOF halos at z = 0.
from 30 halo mass bins, each with a number density of
n̄ ≃ 2.3×10−5h3 Mpc−3 . The solid line is the linear-order
bias as derived from the peak-background split formalism [40, 41]. We find a good agreement with our N -body
data, only at masses below ∼ 8 × 1012 h−1 M⊙ deviations
appear for halos with less than ∼ 30 particles [58].
The bottom left panel of Fig. 2 depicts the derivative
of b with respect to fNL for each of the 30 mass bins.
The behavior is well described by the linear theory prediction of Eq. (3), leading to a k −2 -dependence on large
scales which is more pronounced for more massive halos (for quantitative comparisons with simulations, see
[31, 32, 59]). Thus, the amplitude of this effect gradually diminishes towards smaller scales and even disappears around k ∼ 0.1hMpc−1 . Note that [31] argued for
an additional non-Gaussian bias correction which follows
from the fNL -dependence of the mass function. This kindependent contribution should in principle be included
in Eq. (3). However, as can be seen in the lower left plot,
it is negligible in our approach (i.e., all curves approach
zero at high k) owing to the matching of halo abundances
between our Gaussian and non-Gaussian realizations.
2.
Shot noise matrix
The shot noise matrix E has been studied using simulations with Gaussian initial conditions in [14]. Figure 3
displays the eigenstructure of this matrix for fNL = 0
(solid curves) and fNL = ±100 (dashed and dotted
curves). The left panel depicts all the eigenvalues (top)
and their derivatives with respect to fNL (bottom), while
the right panel shows the two important eigenvectors V+
and V− (top) along with their derivatives (bottom). The
eigenstructure of E is accurately described by Eq. (13),
even in the non-Gaussian case. Namely, we still find one
enhanced eigenvalue λ+ and one suppressed eigenvalue
(N −2)
are degenλ− . The remaining N − 2 eigenvalues λP
erate with the value 1/n̄, the Poisson noise expectation.
This means that our Gaussian bias-expansion model from
Eq. (16) still works to describe E in the weakly nonGaussian regime.
Note however that, owing to sampling variance, the decomposition into eigenmodes becomes increasingly noisy
towards larger scales. This leads to an artificial breaking
of the eigenvalue degeneracy which manifests itself as a
scatter around the mean value 1/n̄. This scatter is the
major contribution of sampling variance in the halo covariance matrix Ch . Although we can eliminate most of
(N −2)
it by setting λP
≡ 1/n̄, a residual degree of sampling
variance will remain in λ+ and λ− , as well as in b and P .
As is apparent from the left panel in Fig. 3, the dominant eigenvalue λ+ exhibits a small, but noticeable fNL dependence similar to that of Eδ2 in Fig. 1, which is about
2% in this case. Its derivative, ∂λ+ /∂fNL , clearly dominates the derivative of all other eigenvalues (which are all
consistent with zero due to matched abundances). Only
the derivative of the suppressed eigenvalue λ− shows a
similar fNL -dependence of ∼ 2%, albeit at a much lower
absolute amplitude. To check the convergence of our results, we repeated the analysis with 100 and 200 bins
and found both derivatives of λ+ and λ− to increase,
supporting an fNL -dependence of these eigenvalues.
By contrast, the eigenvectors V+ and V− shown in
viii
1.0
8•104
0.5
6•104
V±
λ [h-3Mpc3]
1•105
0.0
4•104
4
-0.5
5•10-5
20
∂V±/∂fNL
-3
3
∂λ/∂fNL [h Mpc ]
2•10
0
30
10
0
0
-10
-5•10-5
-20
-1•10-4
0.01
k [hMpc-1]
1013
0.10
M [h-1MO• ]
1014
FIG. 3. Eigenvalues (left panel) and eigenvectors (right panel) of the shot noise matrix E for fNL = 0 (solid), +100 (dashed) and
−100 (dotted) in the case of 30 mass bins. Their derivatives with respect to fNL are plotted underneath. For clarity, only the
two eigenvectors V± along with their derivatives are shown in the right panel. The straight dotted line in the upper left panel
depicts the value 1/n̄ and the red (dot-dashed) curve in the top right panel shows b2 (M ) computed from the peak-background
split formalism, scaled to the value of V+ at M ≃ 3 × 1013 h−1 M⊙ . Results are shown for FOF halos at z = 0.
the right panel of Fig. 3 exhibit very little dependence
on fNL (the different lines are all on top of each other).
The derivatives of V+ and V− with respect to fNL shown
in the lower panel reveal a very weak sensitivity to fNL
which is less than 0.5% for most of the mass bins (for
the most massive bin it reaches up to 1%). We repeated
the same analysis with 100 and 200 mass bins and found
that the relative differences between the measurements in
Gaussian and non-Gaussian simulations further decrease.
We thus conclude that the eigenvectors V+ and V− can
be assumed independent of fNL to a very high accuracy.
Our findings demonstrate that the two-point statistics of halos are sensitive to primordial non-Gaussianity
beyond the linear-order effect of Eq. (3) derived in
[8, 21, 22]. However, the corrections are tiny if one considers a single bin containing many halos of very different
mass (see [37]) due to mutual cancellations from b2 -terms
of opposite sign. Only two specific eigenmodes of the shot
noise matrix (corresponding to two different weightings of
the halo density field) inherit a significant dependence on
fNL . This is most prominently the case for the eigenmode
corresponding to the highest eigenvalue λ+ . Its eigenvector, V+ , is shown to be closely related to the second-order
bias b2 in Eq. (19). As can be seen in the upper right
panel of Fig. 3, V+ measured from the simulations, and
the function b2 (M ) calculated from the peak-background
split formalism [40, 41], agree closely ( note that b2 (M )
has been rescaled to match the normalized vector V+ ).
In the continuous limit this implies that weighting the
halo density field with b2 (M ) selects the eigenmode with
eigenvalue λ+ given in Eq. (20). Since λ+ depends on fNL
through the quantity Eδ2 defined in Eq. (21), the result-
ing weighted field will show the same fNL -dependence.
However, this fNL -dependence cannot immediately be exploited to constrain primordial non-Gaussianity, because
the Fourier modes of Eδ2 are heavily correlated due to
the convolution of δ with itself in Eq. (21), and thus do
not contribute to the Fisher information independently.
The bottom line is that for increasingly massive halo bins
with large b2 , the term Eδ2 makes an important contribution to the halo power spectrum and shows a significant
dependence on fNL . It is important to take into account
this dependence when attempting to extract the best-fit
value of fNL from high-mass clusters, so as to avoid a
possible measurement bias. Although it provides some
additional information on fNL , we will ignore it in the
following and quote only lower limits on the Fisher information content.
B.
Constraints from Halos and Dark Matter
Let us first assume the underlying dark matter density
field δ is available in addition to the galaxy distribution.
Although this can in principle be achieved with weaklensing surveys using tomography, the spatial resolution
will not be comparable to that of galaxy surveys. To
mimic the observed galaxy distribution we will assume
that each dark matter halo (identified in the numerical
simulations) hosts exactly one galaxy. A further refinement in the description of galaxies can be accomplished
with the specification of a halo occupation distribution
for galaxies [15, 60], but we will not pursue this here.
Instead, we can think of the halo catalogs as a sample
ix
of central halo galaxies from which satellites have been
removed. We also neglect the effects of baryons on the
evolution of structure formation, which are shown to be
marginally influenced by primordial non-Gaussianity at
late times [61].
1.
Single tracer: uniform weighting
In the simplest scenario we only consider one single halo mass bin. In this case, all the observed halos
(galaxies) of a survey are correlated with the underlying dark matter density field in Fourier space to determine their scale-dependent effective bias, which can then
be compared to theoretical predictions. In practice, this
translates into fitting our theoretical model for the scaledependent effective bias, Eq. (3), to the Fourier modes of
the density fields and extracting the best fitting value of
fNL together with its uncertainty. For a single halo mass
bin, we can employ Eq. (36) and sum over all the Fourier
modes.
In the Gaussian simulations, we measure the scaleindependent effective bias bG via the estimator hδh δi/hδ 2 i
and the shot noise E via h(δh − bG δ)2 i, and average over
all modes with k ≤ 0.032hMpc−1 . In practice, bG and
E are not directly observable, but a theoretical prediction based on the peak-background split [40, 41] and the
halo model [14] provides a reasonable approximation to
the measured bG and E, respectively, (see Sec. V). Note
that for bins covering a wide range of halo masses, the
fNL -dependence of the shot noise is negligible [37] and it
is well approximated by its Gaussian expectation.
Figure 4 shows the best fits of Eq. (3) to the simulations with fNL = 0, ±100 using all the halos of our
FOF (left panel) and SO catalogs (right panel). In order
to highlight the relative influence of fNL on the effective bias, we normalize the measurements by the largescale Gaussian average bG and subtract unity. The resulting best-fit values of fNL along with their one-sigma
errors are quoted in the lower right for each case of initial
conditions. The 68%-confidence region is determined by
the condition ∆χ2 (fNL ) = 1. Note that we include only
Fourier modes up to k ≃ 0.032hMpc−1 in the fit, as linear
theory begins to break down at higher wave numbers.
Obviously, the best-fit values for fNL measured from
the FOF halo catalogs are about 20% below the input
values. A suppression of the non-Gaussian correction to
the bias of FOF halos has already been reported by [32,
59]. These authors showed that the replacement δc → qδc
with q = 0.75 in Eq. (3) yields a good agreement with
their simulation data. In our framework, including this
“q-factor” is equivalent to exchanging fNL → fNL /q and
σfNL → σfNL /q, owing to the linear scaling of Eq. (3)
with δc . Repeating the chi-square minimization with q =
0.75 yields best-fit values that are consistent with our
input values, namely fNL = +107.0 ± 8.3, +1.8 ± 8.7 and
−104.0 ± 8.5. In fact, the closest match to the input
fNL -values is obtained for a slightly larger q of ≃ 0.8.
Note that [59] attributed this suppression to ellipsoidal
collapse. However, this conclusion seems rather unlikely
since ellipsoidal collapse increases the collapse threshold
or, equivalently, implies q > 1 [62]. A more sensible explanation arises from the fact that a linking length of 0.2
times the mean interparticle distance can select regions
with an overdensity as low as ∆ ∼ 1/0.23 = 125 (with
respect to the mean background density ρ̄m ), which is
much less than the virial overdensity ∆c (z = 0) ≃ 340
associated with a linear overdensity δc (see [56, 63, 64]).
Therefore, we may reasonably expect that, on average,
FOF halos with this linking length trace linear overdensities of height less than δc .
In the case of SO halos, however, we observe the opposite trend. As is apparent in the right panel of Fig. 4, the
model from Eq. (3) overestimates the amplitude of primordial non-Gaussianity by roughly 40%. This is somewhat surprising since the overdensity threshold ∆c ≃ 340
used to identify the SO halos at z = 0 is precisely the
virial overdensity predicted by the spherical collapse of a
linear perturbation of height δc . As we will see shortly,
however, an optimal weighting of halos can remove this
overshoot and therefore noticeably improve the agreement between model and simulations.
2.
Single tracer: optimal weighting
As demonstrated in [14], the shot noise matrix E exhibits nonzero off-diagonal elements from correlations between halos of different mass. Thus, in order to extract
the full information on halo statistics, it is necessary to
include these correlations into our analysis. For this purpose, we must employ the more general chi-square of
Eq. (35). The halo density field is split up into N consecutive mass bins in order to construct the vector δ h ,
and the full shot noise matrix E must be considered.
However, this approach can be simplified, since we
know that E exhibits one particularly low eigenvalue λ− .
Because the Fisher information content on fNL from
Eq. (39) is proportional to the inverse of E (this is true
at least for the dominant part γ), it is governed by the
eigenmode corresponding to this eigenvalue. In [14] it has
been shown that this eigenmode dominates the clustering
signal-to-noise ratio. In the continuous limit (infinitely
many bins), it can be projected out by performing an
appropriate weighting of the halo density field. The corresponding weighting function, denoted as modified mass
weighting with functional form
w(M ) = M + M0 ,
(42)
was found to minimize the stochasticity of halos with
respect to the dark matter. Here, M is the individual
halo mass and M0 a constant whose value depends on
x
uniform FOF-halos
bG = 1.31, ε = 1939.9 h-3Mpc3
0.5
b(k,fNL)/bG -1
b(k,fNL)/bG -1
0.5
uniform SO-halos
0.0
fNL = +79.9
fNL = +3.1
fNL = -78.2
-0.5
0.01
k [hMpc-1]
± 6.6
± 6.3
± 6.3
bG = 1.32, ε = 2871.5 h-3Mpc3
0.0
fNL = +137.6 ± 7.8
fNL = +2.1
± 7.0
fNL = -143.8 ± 7.0
-0.5
0.10
0.01
k [hMpc-1]
0.10
FIG. 4. Relative scale dependence of the effective bias from all FOF (left panel) and SO halos (right panel) resolved in our
N-body simulations (Mmin ≃ 5.9 × 1012 h−1 M⊙ ), which are seeded with non-Gaussian initial conditions of the local type with
fNL = +100, 0, −100 (solid lines and data points from top to bottom). The solid lines show the best fit to the linear theory
model of Eq. (3), taking into account all the modes to the left of the arrow. The corresponding best-fit values are quoted in the
bottom right of each panel. The dotted lines show the model evaluated at the input values fNL = +100, 0, −100. The results
assume knowledge of the dark matter density field and an effective volume of Veff ≃ 50h−3 Gpc3 at z = 0.
weighted FOF-halos
bG = 1.73, ε = 307.2 h-3Mpc3
0.5
b(k,fNL)/bG -1
b(k,fNL)/bG -1
0.5
weighted SO-halos
0.0
fNL = +80.0
fNL = -1.3
fNL = -85.4
-0.5
0.01
k [hMpc-1]
± 1.0
± 1.1
± 1.1
0.10
bG = 1.83, ε = 863.9 h-3Mpc3
0.0
fNL = +100.0 ± 1.5
fNL = -1.6
± 1.6
fNL = -108.1 ± 1.6
-0.5
0.01
k [hMpc-1]
0.10
FIG. 5. Same as Fig. 4, but for weighted halos that have minimum stochasticity relative to the dark matter. Note that the
one-sigma errors on fNL are reduced by a factor of ∼ 5 compared to uniform weighting. In the case of SO halos the input values
for fNL are well recovered by the best-fit, while FOF halos still show a suppression of ∼ 20% (q ≃ 0.8) in the best-fit fNL .
the resolution of the simulation. It is approximately 3
times the minimum resolved halo mass Mmin, so in this
case M0 ≃ 1.8 × 1013 h−1 M⊙ . The weighted halo density
field is computed as
P
w⊺δh
i w(Mi )δhi
(43)
δw = P
≡ ⊺ ,
w 11
i w(Mi )
where we have combined the weights of the individual
mass bins into a vector w in the last expression. Because the chi-square in Eq. (35) is dominated by only one
eigenmode, it simplifies to the form of Eq. (36) with the
halo field δh being replaced by the weighted halo field δw .
Note also that bG and E have to be replaced by the corresponding weighted quantities (see [14]).
The results are shown in Fig. 5 for both FOF and SO
halos. We observe a remarkable reduction in the error
on fNL by a factor of ∼ 4 − 6 (depending on the halo
finder) when replacing the uniform sample used in Fig. 4
by the optimally weighted one. While for the FOF halos
the predicted amplitude of the non-Gaussian correction
xi
to the halo bias still shows the 20% suppression (again,
this can be taken into account by introducing a q-factor
into our fit), for the SO halos the best-fit values of fNL
now agree much better with the input values, i.e., q ≃ 1.
Therefore, the large discrepancy seen in Fig. 4 presumably arises from noise in the SO mass assignment at low
mass. To ascertain whether this is the case, we repeat
the analysis, increasing the threshold for the minimum
number of particles per halo and discarding all halos below that threshold. If this threshold reaches 40 particles
per halo, we find the best-fit fNL to be much closer to the
input values, namely fNL = +102.6 ± 4.6, +2.1 ± 4.5 and
−103.5 ± 4.4. This suggests that most of the discrepancy seen in the right panel of Fig. 4 is due to poorly
resolved halos of mass M . 2Mmin [58]. However, modified mass weighting removes this discrepancy since halos
at low mass are given less weight.
Our findings are consistent with the ones of [31], where
the non-Gaussian bias of SO halos has been measured
also at higher redshifts and mass thresholds, and the
results of [65], where the fractional deviation from the
Gaussian mass function for both FOF and SO halos was
presented (see their Fig. 5). The remarkable improvement in the constraints on fNL follows from the fact that
the stochasticity (shot noise) of the optimally weighted
halo density field is strongly suppressed with respect to
the dark matter [14]. This means that the fluctuations
of the halo and the dark matter overdensity fields are
more tightly correlated and the variance of the estimator hδh δi/hδ 2 i for the effective bias is minimized. Also,
cosmic variance fluctuations inherent in both δ and δh
are canceled in this ratio (see Appendix C). Since the
scale dependence of this estimator is a direct probe of primordial non-Gaussianity, the error on fNL is significantly
reduced. At the same time, modified mass weighting increases the magnitude of bG . We will show below that
the constraints on fNL are indeed optimized with this
approach.
Finally, we can test our assumption about the likelihood function as defined in Eq. (22) being of a Gaussian form and thus yielding the correct Fisher information. Non-Gaussian corrections could arise from correlated k-modes in the covariance matrix (as present in
the eigenmode λ+ of the shot noise matrix), preventing
the Fisher information from being a single integral over
k. The error σb on the effective bias in Figs. 4 and 5
is determined from the variance amongst our sample of
12 realizations and thus provides an independent way of
testing the value for σfNL : from Eq. (3) we can determine σfNL = σb /(bG − 1)u(k, z) and compare it to the
value obtained from the chi-square fit with Eq. (35). Applying the two methods, we find no significant differences
in σfNL , so at least up to the second moment of the likelihood function, the assumption of it being Gaussian seems
reasonable for the considered values of fNL .
3.
Multiple tracers
Let us now estimate the minimal error on fNL achievable with a given galaxy survey for the general case, dividing halos into multiple mass bins. The Fisher information is given by Eq. (39) or (29), depending on whether
the dark matter density field is known or not, and the
minimal error on fNL is determined via integration over
all observed modes in the volume V ,
Z kmax
V
=
σf−2
(44)
FfNL fNL (k) k 2 dk .
NL
2π 2 kmin
The largest modes with wave number kmin = 2π/Lbox ≃
0.0039hMpc−1 available from our N -body simulations
are smaller than the largest modes in a survey of
50h−3 Gpc3 volume (kmin ≃ 0.0017hMpc−1 ), since
we only obtain an effective volume by considering 12
smaller simulation boxes. Because the signal from fNL
is strongest at low k, our results slightly underestimate
the total Fisher information. However, we can roughly
estimate that on larger scales (kmin < 0.0039hMpc−1 ),
FfNL fNL (k) ∼ u2 (k)P (k) ∼ k −4 k ns [see Eqs. (4), (29)
and (39), as well as Figs. 6 and 8], and thus σfNL ∼
−1/2
ln (kmax /kmin )
assuming ns ≃ 1, a relatively weak
dependence on kmin . In our case this amounts to an
overestimation of σfNL by roughly 20%.
Note that we only consider the fNL -fNL -element of
the Fisher matrix. In principle we would have to consider various other parameters of our cosmology
and
then marginalize over them, i.e., compute F −1 fNL fNL
[66]. However, any degeneracy with cosmological parameters is largely eliminated when multiple tracers are considered, since the underlying dark matter density field
mostly cancels out in this approach [9]. A mathematical
demonstration of this fact is presented in Appendix C.
Recent studies have developed a gauge-invariant description of the observable large-scale power spectrum
consistent with general relativity [67–73]. In particular,
it has been noted that the general relativistic corrections
to the usually adopted Newtonian treatment leave a signature in the galaxy power spectrum that is very similar
to the one caused by primordial non-Gaussianity of the
local type [74–76]. However, in a multitracer analysis
the two effects can be distinguished sufficiently well, so
that the ability to detect primordial non-Gaussianity is
little compromised in the presence of general relativistic
corrections [77].
In order to make the most conservative estimates we
will discard all the terms featuring E ′ in the Fisher matrix, since it is not obvious how much information on
fNL can actually be extracted from the shot noise matrix. E is indeed close to a pure white-noise quantity and
we find its Fourier modes to be highly correlated. Therefore, in order to extract residual information on fNL , one
would have to decorrelate those modes through an in-
xii
10-2
10-4
σfNL
FfNLfNL
10
10-6
10-8
10-10
1
0.01
k [hMpc-1]
0.10
0.01
kmax [hMpc-1]
0.10
FIG. 6. Fisher information (left panel) and one-sigma error on fNL (right panel, kmin = 0.0039hMpc −1 , Veff ≃ 50h−3 Gpc3 )
from simulations of FOF halos and dark matter at z = 0. The lines show the results for 1 (solid black), 3 (dotted blue), 10
(dashed green) and 30 (dot-dashed red) uniform halo mass bins, as well as for 1 weighted bin (long-dashed yellow).
version of the correlation matrix among k-bins (see [78]).
However, in light of the limited volume of our simulations
this can be a fairly noisy procedure, especially when the
halo distribution is additionally split into narrow mass
bins. Hence, for the Fisher information content on fNL
assuming knowledge of both halos and dark matter, we
will retain only the first term in Eq. (39) and provide a
lower limit:
FfNL fNL ≥ γ ≡ b′⊺ E −1 b′ P .
(45)
To calculate FfNL fNL , we measure the functions b(k),
b′ (k), E(k) and P (k) from our N -body simulations (see
Figs. 1, 2 and 3). In order to mitigate sampling variance
in the multibin case, we then use Eq. (13) to recalculate
the shot noise matrix. Namely, we set all the eigenvalues
(N −2)
λP
equal to the average value 1/n̄, and measure λ+ ,
λ− , as well as V+ and V− directly from the numerical
eigendecomposition of E.
Figure 6 depicts FfNL fNL (k) and σfNL (kmax ) with fixed
kmin = 0.0039hMpc−1 for the cases of 1, 3, 10 and 30 halo
mass bins. Clearly, the finer the sampling into mass bins,
the higher the information content on fNL . The weighted
halo density field with minimal stochasticity relative to
the dark matter (corresponding to a continuous sampling
of infinitely many bins) yields more than a factor of 6 reduction in σfNL when compared to a single mass bin of
uniformly weighted halos. This improvement agrees reasonably well with that seen in Figs. 4 and 5, although
the estimates for σfNL are slightly larger than those we
obtained from the fitting procedure. This may be expected, since we only obtain an upper limit on σfNL from
Eqs. (44) and (45).
The inflection around k ∼ 0.1hMpc−1 in FfNL fNL
and σfNL marks a breakdown of the linear model from
Eq. (9). We should not trust our results too much at
high wave number, where higher-order contributions to
the non-Gaussian effective bias may become important.
It should also be noted that the inflection disappears for
the weighted field, suggesting numerical issues to be less
problematic in that case.
Further improvements can be achieved when going to
lower halo masses (see Sec. V): the error on fNL is
proportional to the shot noise of the halo density field
(Eq. (40)), which itself is a function of the minimum
halo mass Mmin . References [14, 15] numerically investigated the extent to which the shot noise depends on
Mmin and proposed a method based on the halo model
for extrapolating it to lower mass. It predicts the shot
noise of the weighted halo density field to decrease linearly with Mmin , anticipating about 2 orders of magnitude further reduction in E when resolving halos down
to Mmin ≃ 1010 h−1 M⊙ . In terms of fNL -constraints
this is however somewhat mitigated by the fact that
the Gaussian bias also decreases with Mmin , so the nonGaussian correction to the effective bias in Eq. (3) gets
smaller. Furthermore, [14] studied the effect of adding
random noise to the halo mass (to mimic scatter between
halo mass and the observables such as galaxy luminosity), while [15] explored the redshift dependence of the
optimally weighted halo density field and extended the
method to halo occupation distributions for galaxies.
C.
Constraints from Halos
The scenario described above is optimistic in the sense
that it assumes the dark matter density field is available.
In the following section we will show that it is possible to
xiii
uniform FOF-halos
bG = 1.31, ε = 1939.9 h-3Mpc3
0.5
b^(k,fNL)/bG -1
b^(k,fNL)/bG -1
0.5
weighted FOF-halos
0.0
fNL = +90.2
fNL = +11.0
fNL = -68.9
-0.5
0.01
k [hMpc-1]
± 22.7
± 19.6
± 15.1
0.10
bG = 1.73, ε = 307.2 h-3Mpc3
0.0
fNL = +80.8
fNL = +1.1
fNL = -87.5
-0.5
0.01
k [hMpc-1]
± 11.9
± 8.6
± 5.5
0.10
FIG. 7. Relative scale dependence of the effective bias b̂ estimated from all uniform (left panel) and weighted FOF halos (right
panel) resolved in our N-body simulations (Mmin ≃ 5.9 × 1012 h−1 M⊙ ), which are seeded with non-Gaussian initial conditions of
the local type with fNL = +100, 0, −100 (solid lines and data points from top to bottom). The solid lines show the best fit to the
linear theory model of Eq. (3), taking into account all the modes to the left of the arrow. The corresponding best-fit values are
quoted in the bottom right of each panel. The dotted lines show the model evaluated at the input values fNL = +100, 0, −100.
The results assume no knowledge of the dark matter density field and an effective volume of Veff ≃ 50h−3 Gpc3 at z = 0.
considerably improve the constraints on fNL even without this assumption. This is perhaps not surprising in
light of the results in [14, 15], where it was argued that
halos can be used to reconstruct the dark matter to arbitrary precision, as long as they are resolved down to the
required low-mass threshold.
1.
Single tracer
Considering a single halo mass bin, we must again sum
over all the Fourier modes in Eq. (25) and minimize this
chi-square with respect to fNL . Although we pretend to
have no knowledge of the dark matter distribution, we
determine bG and E from our simulations. In realistic
applications, however, these quantities will have to be
accurately modeled. In addition, we use the linear power
spectrum P0 (k) instead of the simulated nonlinear dark
matter power spectrum P (k) in Eq. (25).
Since, in this case, we cannot determine the scaledependent effective bias directly from the estimator
hδh δi/hδ 2 i, we define the new estimator
s
hδh2 i − E
b̂ ≡
,
(46)
P0
which solely depends on the two-point statistics of halos.
In Fig. 7 we plot this estimator together with the best-fit
solutions for the scale-dependent effective bias obtained
from the chi-square fit of Eq. (25). The left panel depicts
the results obtained for uniform FOF halos. Compared
to the previous case with dark matter, we observe the
constraints on fNL to be weaker by a factor of ∼ 3. The
main reason for this difference is the fact that sampling
variance inherent in δh is not canceled out by subtracting δ, as is done in Eq. (36). This can also be seen in the
estimator b̂, where a division of the smooth linear power
spectrum P0 does not cancel the cosmic variance inherent in hδh2 i. Hence, b̂ shows significantly stronger fluctuations than b = hδh δi/hδ 2 i, which demonstrates how well
the basic idea of sampling variance cancellation works.
Exchanging the uniform halo field δh with the weighted
one, δw , the constraints on fNL improve by about a factor
of 2 − 3, as can be seen in the right panel of Fig. 7.
However, this improvement is mainly due to the larger
value of bG of the weighted sample, since the relative
scatter among the data points remains unchanged. This
is expected, because we do not consider a second tracer
(e.g., the dark matter) in this case, and therefore do not
cancel cosmic variance.
Comparing the uncertainty on fNL obtained from
Eq. (23) with the one determined via the variance of b̂
amongst our 12 realizations, we can check once more the
assumption of a Gaussian likelihood as given in Eq. (22).
Again, we find both methods to yield consistent values
for σfNL , suggesting any non-Gaussian corrections to the
likelihood function to be negligible at this order.
xiv
10-2
10-4
σfNL
FfNLfNL
10
10-6
10-8
10-10
1
0.01
k [hMpc-1]
0.10
0.01
kmax [hMpc-1]
0.10
FIG. 8. Fisher information (left panel) and one-sigma error on fNL (right panel, kmin = 0.0039hMpc −1 , Veff ≃ 50h−3 Gpc3 )
from simulations of FOF halos only (z = 0). The lines show the results for 1 (solid black), 3 (dotted blue), 10 (dashed green)
and 30 (dot-dashed red) uniform halo mass bins, as well as for 1 weighted bin (long-dashed yellow).
2.
Multiple tracers
If we want to exploit the gains from sampling variance
cancellation in the case where the dark matter density
field is not available, we have to perform a multitracer
analysis of halos (see Appendix C), which is the focus
of this section. We now consider Eq. (23) for the chisquare fit. In order to calculate the Fisher information,
we use Eq. (29) and thus neglect any possible contribution emerging from the fNL -dependence of the shot noise
matrix E.
Numerical results for the Fisher information content
and the one-sigma error on fNL are shown in Fig. 8 for
1, 3, 10 and 30 uniform FOF halo mass bins, as well
as for 1 weighted bin. Clearly, the cases of 10 and 30
uniform bins outperform a single bin of the weighted field
in terms of Fisher information. This suggests that further
improvements compared to the single weighted halo field
can be achieved when all the correlations of sufficiently
many halo mass bins are taken into account.
In principle, we want to split the halo density field into
as many mass bins as possible and extrapolate the results
to the limit of infinitely many bins (continuous limit).
Note that in the high-sampling limit of n̄ → ∞, FfNL fNL
from Eq. (30) is limited to 2 (b′ /b)2 , whereas the same
quantity for several mass bins, Eq. (29), may surpass this
bound (see Sec. V). In Sec. IV B we showed that a single
optimally weighted halo sample combined with the dark
matter reaches the continuous limit in FfNL fNL , which
corresponds to a splitting into infinitely many bins in
the multitracer approach. It is unclear, whether a similar
goal can be achieved from halos alone, e.g., by considering
two differently weighted tracers that would preserve all
of the information on fNL , because we do not know the
continuous limit of the Fisher information in that case.
We will therefore turn to theoretical predictions by the
halo model in the following paragraph.
V.
HALO MODEL PREDICTIONS
A useful theoretical framework for the description of
dark matter and halo clustering is given by the halo
model (see, e.g., [38]). Despite its limitations [39], the
halo model achieves remarkable agreement with the results from N -body simulations [14, 38]. In particular, it
provides an analytical expression for the shot noise matrix in the fiducial Gaussian case, given by
E = n̄−1 I − bM⊺ − Mb⊺ ,
(47)
where M ≡ M /ρ̄m − bhnM 2 i/2ρ̄2m and M is a vector
containing the mean halo mass of each bin (see [14] for
the derivation). The Poisson model is recovered when we
set M = 0. Here, b can be determined by integrating the
peak-background split bias b(M ) over the Sheth-Tormen
halo mass function dn/dM [40] in each mass bin. The
expression hnM 2 i/ρ̄2m originates from the dark matter
one-halo term, so it does not depend on halo mass and
from our suite of simulations we determine its Gaussian
value to be ≃ 418h−3Mpc3 at z = 0 and ≃ 45h−3 Mpc3
at z = 1. In the case of one single mass bin, Eq. (47)
reduces to E = n̄−1 − 2bM/ρ̄m + b2 hnM 2 i/ρ̄2m , while if
we project out the lowest eigenmode V− and normalize,
we obtain the weighted shot noise
Ew ≡
V−⊺ EV−
V−⊺ V−
=
λ
2 .
−
2
V−⊺ 11
V−⊺ 11
(48)
xv
108
10
8
ε [h-3Mpc3]
106
bG
6
4
104
102
2
0
1010
10
11
10
12
13
10
10
Mmin[h-1MO•]
14
10
15
10
16
100
1010
1011
1012
1013
1014
Mmin[h-1MO•]
1015
1016
FIG. 9. Halo model predictions for the mean scale-independent Gaussian bias (left panel) and shot noise (right panel) as a
function of minimum halo mass from uniform- (solid lines) and weighted halos (dashed lines) from a single mass bin at z = 0
(blue) and z = 1 (red). N -body simulation results are overplotted, respectively, as squares and circles for different low-mass
cuts. The dotted line in the left panel depicts bG = 1, the ones in the right panel show the Poisson-model shot noise n̄−1
tot .
The eigenvalues λ± with eigenvectors V± can be found
from Eq. (47),
√
λ± = n̄−1 − M⊺ b ± M⊺ M b⊺ b ,
(49)
.√
.√
−1
⊺
⊺
M M ∓b
b b ,
V± = N±
M
(50)
where
N± ≡
r
2 ∓ 2M⊺ b
.√
M⊺ M b⊺ b
(51)
is a normalization constant to guarantee V±⊺ V± = 1. It is
easily verified that V±⊺ V∓ = 0, i.e., they are orthogonal.
In the continuous limit of infinitely many bins (N → ∞)
we can replace V± by the smooth function
.p
.p
hM2 i ∓ b
hb2 i ,
(52)
V± = N± −1 M
and obtain
hV 2 i
p
−
Ew = n̄−1
hM2 ihb2 i
,
tot − hMbi −
hV− i2
hV− bi
bw =
,
hV− i
(53)
(54)
where bw is the weighted effective bias and we exchanged
the vector products by integrals over the mass function:
Z Mmax
N
dn
⊺
x y −→
(M )x(M )y(M ) dM ≡ N hxyi ,
n̄tot Mmin dM
(55)
Z Mmax
dn
(M ) dM ≡ N n̄ .
(56)
n̄tot =
dM
Mmin
Figure 9 depicts the halo model prediction for the
scale-independent Gaussian bias bG and shot noise E as a
function of minimum halo mass Mmin at z = 0 and z = 1
for both the uniform and the weighted case of a single
mass bin. Simulation results are overplotted as symbols
for a few Mmin [we approximate the weighting function
V− (M ) by w(M ) from Eq. (42) in the simulations]. Obviously, modified mass weighting increases bG , especially
when going to lower halo masses. It is also worth noticing
that in contrast to the uniform case, bG is always greater
than unity when weighted by w(M ) (at least in the considered mass range). Going to higher redshift further
increases bG at any given Mmin .
For the shot noise we observe the opposite behavior: modified mass weighting leads to a suppression
of E, which is increasingly pronounced towards lower halo
masses. Moreover, it is always below the Poisson-model
prediction of n̄−1
tot . Our N -body simulation results generally confirm this trend (at least down to our resolution limit), although the halo model slightly underestimates the suppression of shot noise between uniform and
weighted halos at lower Mmin. At higher redshifts, this
suppression becomes smaller at given Mmin , but the magnitude of Ew at z = 1 approaches the one at z = 0 towards
low Mmin and is still small compared to the Poissonmodel prediction of n̄−1
tot .
A.
Single tracer
With predictions for bG and E at hand, we can directly compute the expected Fisher information content
on fNL from a single halo mass bin. If the dark mat-
xvi
103
z=0
σfNL
102
101
n→∞
one uniform bin
one weighted bin
multiple bins
blue (open): only halos
red (filled): halos & matter
100
10-1
1010
1011
1012
1013
Mmin[h-1MO•]
1014
1015
1016
FIG. 10. Halo model predictions for the one-sigma error on fNL (inferred from an effective volume of Veff ≃ 50h−3 Gpc3 ,
taking into account all modes with 0.0039hMpc −1 ≤ k ≤ 0.032hMpc−1 at z = 0) as a function of minimum halo mass from
uniform- (solid lines) and weighted halos (dashed lines) from a single mass bin. The N -body simulation results are overplotted,
respectively, as squares and circles for different low-mass cuts. Results that assume knowledge of halos and the dark matter are
plotted in red (filled symbols), those that only consider halos are depicted in blue (open symbols). The dotted lines (triangles)
show the results from splitting the halo catalog into multiple mass bins and taking into account the full halo covariance matrix
2
in calculating FfNL fNL . The high-sampling limit for one mass bin (n̄ → ∞, FfNL fNL = 2 (b′ /b) ) is overplotted for the uniform(thin solid line) and the weighted case ( thin dashed line). Arrows show the effect of adding a log-normal scatter of σln M = 0.5
to all halo masses, they are omitted in all cases where the scatter has negligible impact.
ter density field is known we apply Eq. (40), if it is
not we use Eq. (30). In order to obtain most conservative results, we neglect terms featuring derivatives of the
shot noise with respect to fNL . We then apply Eq. (44)
with kmin = 0.0039hMpc−1 , kmax = 0.032hMpc−1 and
V ≃ 50h−3 Gpc3 to compute the one-sigma error forecast
for fNL .
Results are shown in Fig. 10 for z = 0. When
the dark matter density field is available (red lines and
filled symbols), weighting the halos (red dashed lines
and filled circles) is always superior to the conventional
uniform case (red solid lines and filled squares), especially when going to lower halo masses. In particular,
σfNL substantially decreases with decreasing Mmin in the
weighted case, while for uniform halos it shows a spike
at Mmin ≃ 1.4 × 1012 h−1 M⊙ . This happens when bG becomes unity and the non-Gaussian correction to the halo
bias in Eq. (3) vanishes, leaving no signature of fNL in
the effective bias. Since in the weighted case bG > 1 for
all considered Mmin, this spike does not appear, although
we notice that the error on fNL begins to increase below
Mmin ∼ 1011 h−1 M⊙ .
The simulation results are overplotted as symbols for
a few values of Mmin , the agreement with the halo
model predictions is remarkable. Note that the first two
data-points at Mmin = 9.4 × 1011 h−1 M⊙ and Mmin =
2.35×1012 h−1 M⊙ resulting from our high-resolution simulation were scaled to the effective volume of our 12 lowresolution boxes. The simulations yield a minimum error
of σfNL ≃ 0.8 at Mmin ≃ 1012 h−1 M⊙ in the optimally
weighted case with the dark matter available. This value
is even lower than what is anticipated by the halo model
(σfNL ≃ 1).
The results without the dark matter are shown as blue
lines and open symbols. σfNL exhibits a minimum at
Mmin ≃ 1014 h−1 M⊙ with σfNL ∼ 10 for both uniform
and weighted halos. Thus, weighting the halos does
not decrease the lowest possible error on fNL from the
uniform case, as expected. This suggests that only the
highest-mass halos (clusters at z = 0) need to be consid-
xvii
103
z=1
σfNL
102
101
n→∞
one uniform bin
one weighted bin
multiple bins
blue (open): only halos
red (filled): halos & matter
100
10-1
1010
1011
1012
1013
Mmin[h-1MO•]
1014
1015
1016
FIG. 11. Same as Fig. 10 at z = 1.
ered to optimally constrain fNL from a single-bin survey
without observations of the dark matter.
2
In the limit of n̄ → ∞, FfNL fNL → 2 (b′ /b) . Then,
according to Eq. (3) for high Mmin, b′ → bG u, and hence
FfNL fNL → 2u2 becomes independent of Mmin . The corresponding σfNL in the limit n̄ → ∞ is plotted in Fig. 10
for both uniform- (thin blue solid line) and weighted halos (thin blue dashed line) and it indeed approaches a
constant value at high Mmin . It is about a factor of 2
below the minimum in σfNL without setting n̄ → ∞.
The results for redshift z = 1 are presented in Fig. 11.
In comparison to Fig. 10 one can observe that all the
curves are shifted towards the lower left of the plot, i.e.,
the constraints on fNL improve with increasing redshift.
This is mainly due to the increase of the Gaussian effective bias bG with z, as evident from the left panel
of Fig. 9. For example, the location of the spikes in
σfNL (Mmin ) requires bG = 1. At z = 1 this condition
is fulfilled at lower Mmin (≃ 5 × 1010 h−1 M⊙ ) than at
z = 0, thus shifting the spikes to the left. Further, since
the Fisher information from Eqs. (30) and (40) increases
with bG , σfNL (Mmin ) decreases, especially at low Mmin .
In the case of optimally weighted halos with knowledge
of the dark matter, our simulations suggest σfNL ≃ 0.6
when reaching Mmin ≃ 1012 h−1 M⊙ at z = 1, in good
agreement with the halo model. It even forecasts σfNL ≃
0.2 when including halos down to Mmin ≃ 1010 h−1 M⊙ .
B.
Multiple tracers
The more general strategy for constraining fNL from
a galaxy survey is to consider all auto- and crosscorrelations between tracers of different mass, namely,
the halo covariance matrix Ch . If the dark matter density field is known, one can add the correlations with this
field and determine C. The Fisher information on fNL is
then given by Eq. (29) and Eq. (39), respectively. Again,
the halo model can be applied to make predictions on the
Fisher information content. In Appendix D, the analytical expressions for α, β and γ are derived for arbitrarily
many mass bins and the continuous limit of infinite bins.
The dotted lines in Fig. 10 show the halo model predictions at z = 0 in this continuous limit of infinitely many
mass bins. When the dark matter is available (red dotted
line), σfNL coincides with the results from the optimally
weighted one-bin case (dashed red lines). This confirms
our claim that with the dark matter density field at hand,
modified mass weighting is the optimal choice for constraining fNL and yields the maximal Fisher information
content. Only below Mmin ∼ 1012 h−1 M⊙ the optimally
weighted halo field becomes slightly inferior to the case
of infinite bins.
From multiple bins of halos without the dark matter
(blue dotted line) we observe a different behavior. While
at high Mmin the error on fNL still matches the results
σfNL
xviii
102
102
101
101
100
100
10-1
-1
z=2
σfNL
10-2
z=3
-2
102
101
101
100
100
10-1
-1
z=4
10-2
108
z=5
-2
109
1010
1011 1012
Mmin[h-1MO•]
1013
1014
109
1010
1011 1012
Mmin[h-1MO•]
1013
1014
1015
FIG. 12. Same as Figs. 10 and 11 at higher redshifts, as indicated in the bottom right of each panel. Here, only the halo model
predictions are shown.
from one mass bin, either uniform (blue solid line) or
weighted (blue dashed line), below Mmin ∼ 1014 h−1 M⊙
it departs towards lower values and finally reaches the
same continuous limit as in the case where the dark matter is available at Mmin ∼ 1010 h−1 M⊙ . Thus, galaxies
in principle suffice to yield optimal constraints on fNL ,
however, one has to go to very low halo mass.
Our simulation results for multiple bins (triangles in
Fig. 10) support this conclusion. Although we can only
consider a limited number of mass bins in the numerical
analysis (we used N = 30 for our 12 low-res boxes and
N = 10 for our high-res box), the continuous limit of the
halo model can be approached closely. However, note
that residuals of sampling variance in the numerical determination of E, as described in Sec. IV B and shown in
Fig. 3, can result in an overestimation of FfNL fNL . This
is especially the case when the number of mass bins N is
high, resulting in a low halo number density per bin n̄.
Hence, we chose N such that the influence of sampling
variance on our results is negligible, and yet clear improvements compared to the single-tracer case are established.
One concern in practical applications is scatter in the
halo mass estimation. Although X-ray cluster-mass proxies show very tight correlations with halo mass with a
log-normal scatter of σln M . 0.1 [79, 80], optical massestimators are more likely to have σln M ≃ 0.5 [81]. We
applied a log-normal mass scatter of σln M = 0.5 to all of
our halo masses and repeated the numerical analysis for
all the cases (symbols) shown in Fig. 10. The arrows in
that figure show the effect of adding the scatter, pointing
to the new (higher) value of σfNL . We find the effect of
the applied mass scatter to be negligible in most of the
considered cases (arrows omitted). Only in the case of
xix
one weighted halo bin with knowledge of the dark matter (red filled circles) we observe a moderate weakening
in fNL -constraints, especially towards lower Mmin . This
is expected, since we make most heavy use of the halo
masses when applying modified mass weighting. Yet, the
improvement compared to the uniform one-bin case remains substantial, so the method is still beneficial in the
presence of mass scatter.
At higher redshifts, we observe the same characteristics
as in the single-tracer case: the σfNL -curves are shifted
towards the lower left of the plot in Fig. 11, due to the
increase in the effective bias with z. Moreover, the impact
of mass scatter on σfNL becomes less severe at higher
redshifts, as evident from the smaller arrows in Fig. 11
as compared to Fig. 10. High-redshift data are therefore
more promising for constraining fNL . This is good news,
since the relatively large effective volume assumed in the
current analysis (Veff ≃ 50h−3 Gpc3 ) can only be reached
in practical applications when going to z ∼ 1 or higher.
On the other hand, the convergence of the constraints
obtained with and without the dark matter is pushed
to even lower halo masses at higher redshifts. This can
be seen in Fig. 12, where we show the halo model predictions for even higher redshifts, going up to z = 5.
With a mass threshold of Mmin = 1010 h−1 M⊙ , the optimal constraints on fNL from only halos start to saturate
above z ≃ 2, where σfNL ≃ 0.5. This is however not
the case when the dark matter is available: the error
on fNL decreases monotonically up to z = 5 reaching
σfNL ≃ 0.06, although for practical purposes it will be
difficult to achieve this limit. Yet, reaching σfNL ∼ 1 at
z = 1 and Mmin ∼ 1011 h−1 M⊙ with a survey volume of
about 50h−3 Gpc3 seems realistic.
VI.
CONCLUSIONS
The aim of this work is to assess the amount of information on primordial non-Gaussianity that can be extracted from the two-point statistics of halo- and dark
matter large-scale structure in light of shot noise suppression and sampling variance cancellation techniques
that have been suggested in the literature. For this purpose we developed a theoretical framework for calculating the Fisher information content on fNL that relies on
minimal assumptions for the covariance matrix of halos
in Fourier space. The main ingredients of this model
are the effective bias and the shot noise matrix, both of
which we measure from N -body simulations and compare
to analytic predictions. Our results can be summarized
as follows:
(i) On large scales the effective bias agrees well with
linear theory predictions from the literature, while
towards smaller scales, it shows deviations that can
be explained by the local bias-expansion model.
The shot noise matrix exhibits two nontrivial eigenvalues λ+ and λ− , both of which show a considerable dependence on fNL . We further show that
the eigenvector V+ is closely related to the secondorder bias and that the corresponding eigenvalue
λ+ depends on the shot noise of the squared dark
matter density field Eδ2 , which itself depends on
fNL weakly. This property can become important
when constraining fNL from very high-mass halos
(clusters). However, since the Fourier modes of Eδ2
are highly correlated, it is questionable how much
information on primordial non-Gaussianity can be
gained from the fNL -dependence of the shot noise
matrix. We demonstrate, though, that for the considered values of fNL the assumption of a Gaussian
form of the likelihood function is sufficient to determine the correct Fisher information.
(ii) With the help of N -body simulations we demonstrate how the parameter fNL can be constrained
and its error reduced relative to traditional methods by applying optimal weighting- and multipletracer techniques to the halos. For our specific simulation setup with Mmin ∼ 1012 h−1 M⊙ , we reach
almost 1 order of magnitude improvements in fNL constraints at z = 0, even if the dark matter density
field is not available. The absolute constraints on
fNL depend on the effective volume and the minimal halo mass that is resolved in the simulations,
or observed in the data, and are expected to improve further when higher redshifts or lower-mass
halos are considered.
(iii) We confirm the existence of a suppression factor
(denoted q-factor in the literature) in the amplitude
of the linear theory correction to the non-Gaussian
halo bias. We argue that this only holds for halos generated with a friends-of-friends finding algorithm and depends on the specified linking length
between halo particles. For a linking length of 20%
of the mean interparticle distance, our simulations
yield q ≃ 0.8. For halos generated with a spherical
overdensity finder, we demonstrate that the bestfit values of fNL measured from the simulations are
fairly consistent with the input values, i.e., q ≃ 1.
(iv) We calculate the Fisher information content from
the two-point statistics of halos and dark matter
in Fourier space, both analytically and numerically,
and express the results in terms of an effective bias,
a shot noise matrix and the dark matter power
spectrum. In the case of a single mass bin and assuming knowledge of the dark matter density field,
the Fisher information is inversely proportional to
the shot noise and, therefore, not bounded from
above if the shot noise vanishes. However, when
only the halo distribution is available, the Fisher
xx
information remains finite even in the limit of zero
shot noise. In this case, the amount of information
on fNL can only be increased by dividing the halos
into multiple mass bins (multiple tracers).
(v) Utilizing the halo model we calculate σfNL and find
a remarkable agreement with our simulation results. We show that in the continuous limit of infinite mass bins, optimal constraints on fNL can in
principle be achieved even in the case where dark
matter observations are not available. With an effective survey volume of ≃ 50h−3 Gpc3 out to scales
of kmin ≃ 0.004hMpc−1 this means σfNL ∼ 1 when
halos down to Mmin ∼ 1011 h−1 M⊙ are observed
at z = 0. In comparison to this, a single-tracer
method yields fNL -constraints that are weaker by
about 1 order of magnitude. Further improvements
are expected at higher redshifts and lower Mmin ,
potentially reaching the level of σfNL . 0.1.
(vi) In realistic applications, additional sources of noise,
such as a scatter in halo mass will have to be considered. We test the impact of adding a log-normal
scatter of σln M = 0.5 to our halo masses and find
our results to be relatively unaffected. Assuming
the dark matter to be available to correlate against
halos is even more uncertain. Weak-lensing tomography can only measure the dark matter over a
broad radial projection and more work is needed
to see how far this approach can be pushed. Moreover, one would also need to include weak-lensing
ellipticity noise into the analysis, which we have not
done here.
We conclude that the shot noise suppression method
(modified mass weighting) as presented in [14] when the
dark matter density field is available, and the sampling
variance cancellation technique (multiple tracers) as proposed in [9] when it is not, have the potential to significantly improve the constraints on primordial nonGaussianity from current and future large-scale structure data. In [16] it was found (their Fig. 15) that
while the power spectrum analysis of a single tracer
with Mmin ∼ 1014 h−1 M⊙ (close to our optimal mass
for a single tracer without the dark matter) predicts
σfNL ∼ 10 for Veff ≃ 50h−3 Gpc3 , in good agreement
with our results, the bispectrum analysis improves this
to σfNL ∼ 5. Our results suggest that the multitracer
analysis of the halo power spectrum can improve upon
a single-tracer bispectrum analysis, potentially reaching
significantly smaller errors on fNL . In principle the multitracer approach can also be applied to the halo bispectrum, but it is not clear how much one can benefit
from it, since the dominant terms in the bispectrum do
not feature any additional scale dependence that changes
with tracer-mass.
In this paper we only focused on primordial nonGaussianity of the local type and the two-point correlation analysis. Yet, our techniques can be applied to some,
but not all, other models of primordial non-Gaussianity,
which have only recently been studied in simulations [82–
87]. Theoretical calculations of the non-Gaussian halo
bias generally suggest different degrees of scale dependence and amplitudes depending on the model [88–92].
Our methods may help to test those various classes of
primordial non-Gaussianity and thus provide a tool to
probe the physics of the very early Universe.
ACKNOWLEDGMENTS
We thank Pat McDonald, Tobias Baldauf and Jaiyul
Yoo for fruitful discussions, V. Springel for making public his N-body code gadget ii, and A. Knebe for making public his SO halo finder AHF. This work is supported by the Packard Foundation, the Swiss National
Foundation under contract 200021-116696/1 and WCU
grant R32-10130. VD acknowledges additional support
from FK UZH 57184001. NH thanks the hospitality of
Lawrence Berkeley National Laboratory (LBNL) at UC
Berkeley and the Institute for the Early Universe (IEU)
at Ewha University Seoul, where parts of this work were
completed.
Appendix A: FISHER INFORMATION ON PRIMORDIAL NON-GAUSSIANITY FROM THE
COVARIANCE OF HALOS
Plugging Eq. (27) into Eq. (26) and using the cyclicity of the trace yields
FfNL fNL =
1
Tr
2
∂Ch −1 ∂Ch −1
C
C
∂fNL h ∂fNL h
=
i
1 h ′ ⊺ −1
′ −1 2
P
+
E
C
=
Tr b b Ch P + bb′⊺ C−1
h
h
2
1 ′ −1 ′ −1 ′⊺ −1 ′ 2
⊺ −1 ′ ⊺ −1 ′ 2
⊺ −1 ′ −1 ′
. (A1)
= b⊺ C−1
h bb Ch b P + b Ch b b Ch b P + 2b Ch E Ch b P + Tr E Ch E Ch
2
xxi
Applying Eq. (28) yields
α2
α
=
,
1+α
1+α
βα
β
′
b⊺ C−1
=
,
h b P = β−
1+α
1+α
β2
γ + αγ − β 2
′
b′⊺ C−1
=
,
h bP = γ−
1+α
1+α
αβµ
αν + βµ
′ −1 ′
+
,
b⊺ C−1
h E Ch b P = ν −
1+α
(1 + α)2
1 ′ −1 ′ −1 ρ
µ2 /2
,
Tr E Ch E Ch
=τ+
+
2
1 + α (1 + α)2
b⊺ C−1
h bP = α −
(A2)
(A3)
(A4)
(A5)
(A6)
where
Finally, we get
α ≡ b⊺ E −1 bP , β ≡ b⊺ E −1 b′ P , γ ≡ b′⊺ E −1 b′ P ,
′
′
µ ≡ −b⊺ E −1 bP , ν ≡ −b⊺ E −1 b′ P ,
′
′
ρ ≡ b⊺ E −1 E E −1 bP ,
1
τ ≡ Tr E ′ E −1 E ′ E −1 .
2
FfNL fNL =
(1 + α) (αγ + 2ν − ρ) + (1 − α) β 2 + (µ/2 − 2β) µ
(1 + α)
2
(A7)
(A8)
(A9)
(A10)
+τ .
(A11)
For a single mass bin we have αγ = β 2 , αν = βµ, αρ = µ2 , γρ = ν 2 and ατ = ρ/2. In this case, Eq. (A11) becomes
!2
p
2
′
β + τ /2
bb P + E ′ /2
FfNL fNL = 2
=2
.
(A12)
1+α
b2 P + E
Appendix B: FISHER INFORMATION ON PRIMORDIAL NON-GAUSSIANITY FROM THE
COVARIANCE OF HALOS AND DARK MATTER
Now we need to work out Eq. (26) by plugging in Eq. (37) and (38). Let us first note that
∂C −1
−β
b′⊺ E −1 P
.
C =
b′ − βb − E ′ E −1 b bb′⊺ E −1 P + E ′ E −1
∂fNL
(B1)
This yields
1 2
Tr β + b′⊺ E −1 b′ P − βb′⊺ E −1 bP − b′⊺ E −1 E ′ E −1 bP + b′ b′⊺ E −1 P − βbb′⊺ E −1 P
2
−E ′ E −1 bb′⊺ E −1 P + bb′⊺ E −1 bb′⊺ E −1 P 2 + bb′⊺ E −1 E ′ E −1 P + E ′ E −1 bb′⊺ E −1 P + E ′ E −1 E ′ E −1 =
1 2
=
β + γ − β 2 − ν + γ − β 2 − ν + β 2 + ν + ν + 2τ = γ + τ . (B2)
2
FfNL fNL =
Appendix C: CANCELLATION OF DARK MATTER DENSITY AND COSMIC VARIANCE
In the case where the dark matter density field is known, one can immediately see from the first term in the chisquare of Eq. (35), that with the model δ h = bδ + ǫ from Eq. (7), the underlying density field δ is completely canceled
(including its sampling variance) and the residual is
hχ2 i = hǫ⊺ E −1 ǫi = Tr E −1 hǫǫ⊺ i = N ,
(C1)
xxii
where N is the number of halo bins. If we only consider halos, the first term from Eq. (23) reads
⊺ −1
χ2 = δ ⊺h C−1
δh −
h δh = δh E
2
P
δ ⊺h E −1 b ,
1+α
(C2)
where we used Eq. (28) in the second equality. Plugging in the model δ h = bδ + ǫ, we get
2
P
b⊺ E −1 bδ + b⊺ E −1 ǫ =
χ2 = b⊺ E −1 bδ 2 + 2b⊺ E −1 ǫδ + ǫ⊺ E −1 ǫ −
1+α
2
2
α2
P
δ
α
α−
δ + ǫ⊺ E −1 ǫ −
b⊺ E −1 ǫ . (C3)
+ 2b⊺ E −1 ǫ 1 −
1+α P
1+α
1+α
A large fraction of the first two terms in the last expression obviously cancel when α ≫ 1. The quantity α, also
denoted as signal-to-noise ratio in [14], monotonically increases with the number of halo bins N . In [14] it was shown
to reach O(102 ) in the continuous limit. Even higher values can be reached when the mass resolution of the simulation
is increased [15]. Hence, in the limit α ≫ 1 the residual of the chi-square in Eq. (C3) becomes
hχ2 i =
2
hδ 2 i
hδi
P
b⊺ E −1 hǫǫ⊺ iE −1 b
+ 2b⊺ E −1 ǫ
+ hǫ⊺ E −1 ǫi − h b⊺ E −1 ǫ i = 1 + N −
=N ,
P
α
α
b⊺ E −1 b
(C4)
the same as in Eq. (C1) with knowledge of the dark matter. Note that in the case of one single halo bin as in Eq. (25),
a cancellation neither of the underlying dark matter field, nor of the sampling variance is possible.
Appendix D: HALO MODEL PREDICTION FOR ALPHA, BETA AND GAMMA
In the halo model the shot noise matrix is given by Eq. (47). In order to invert E, we write E = A − Mb⊺ with
A ≡ n̄−1 I − bM⊺ and apply the Sherman-Morrison formula:
E −1 = A−1 +
A−1 Mb⊺ A−1
.
1 − b⊺ A−1 M
(D1)
bM⊺ n̄
.
n̄−1 − M⊺ b
(D2)
Likewise, we apply the Sherman-Morrison formula to invert A:
A−1 = n̄I +
With
b⊺ A−1 b
P ,
1 − b⊺ A−1 M
b⊺ A−1 b′
β ≡ b⊺ E −1 b′ P =
P ,
1 − b⊺ A−1 M
b′⊺ A−1 b′ 1 − b⊺ A−1 M + b′⊺ A−1 Mb⊺ A−1 b′
γ ≡ b′⊺ E −1 b′ P =
P ,
1 − b⊺ A−1 M
α ≡ b⊺ E −1 bP =
and
b⊺ b
,
n̄−1 − M⊺ b
⊺
⊺
M b n̄−1 − M b + b⊺ bM⊺ M
n̄ ,
b⊺ A−1 M =
n̄−1 − M⊺ b
b⊺ b′ n̄−1 − M⊺ b + b⊺ bM⊺ b′
b⊺ A−1 b′ =
n̄ ,
n̄−1 − M⊺ b
M⊺ b′ n̄−1 − M⊺ b + b⊺ b′ M⊺ M
n̄ ,
b′⊺ A−1 M =
n̄−1 − M⊺ b
′⊺ ′
⊺
′
′
b b n̄−1 − M⊺ b + b b M⊺ b
n̄ ,
b′⊺ A−1 b′ =
n̄−1 − M⊺ b
b⊺ A−1 b =
(D3)
(D4)
(D5)
(D6)
(D7)
(D8)
(D9)
(D10)
xxiii
after some algebra we get
b⊺ b −1
n̄ P ,
λ+ λ−
b⊺ b′ n̄−1 − M⊺ b + b⊺ bM⊺ b′
β=
P ,
λ+ λ−
2
2
b⊺ b M⊺ b′ + M⊺ M b⊺ b′ + 2b⊺ b′ M⊺ b′ n̄−1 − M⊺ b
n̄P ,
γ = b′⊺ b′ n̄P +
λ+ λ−
α=
(D11)
(D12)
(D13)
2
with λ+ λ− = n̄−1 − M⊺ b − b⊺ bM⊺ M. According to Eq. (3) we can write b′ = (b − 11) u. Moreover, in the
continuous limit we can exchange the vector products by integrals as in Eq. (55). This finally yields
α=
hb2 i
n̄−1
2
tot P ,
2 ihM2 i
n̄−1
−
hMbi
−
hb
tot
2
hb2 i − hbi n̄−1
tot − hMbi + hb i (hMbi − hMi)
β=
uP ,
2
n̄−1
− hb2 ihM2 i
tot − hMbi
γ = hb2 i − 2hbi + 1 n̄tot u2 P
+
2
hb2 i (hMbi − hMi)2 + hM2 i hb2 i − hbi + 2 hb2 i − hbi (hMbi − hMi) n̄−1
tot − hMbi
n̄tot u2 P .
2
−1
2
2
n̄tot − hMbi − hb ihM i
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
Chapter 4: Paper III
5
PAPER IV

Chapter 5: Paper IV
Summary Another application for the methods developed above is the analysis
of redshift-space distortions (RSD) in spectroscopic galaxy surveys. It has been
demonstrated that the use of multiple tracers allows to measure the RSD parameter
β without cosmic variance limitation. The accuracy to determine β is then only
limited by the shot noise of the tracers, which can be reduced through optimal
weighting. On the basis of N -body simulations, the following article investigates the
combination of optimal weights and multiple tracers to improve constraints on the
RSD parameter. By means of a Fisher analysis it is shown that the best constraints
can be achieved when the two principal components of the clustering signal-to-noise
ratio of halos are considered only. These two principal components correspond to the
two non-Poisson eigenmodes of the shot noise matrix found in paper II. Although
one of them exhibits a super-Poissonian shot noise level, a combination of the two
eigenmodes is beneficial due to their large relative bias. The inclusion of threedimensional dark matter maps into the analysis breaks the degeneracy between halo
bias, normalization of the power spectrum and growth rate. Using optimal weights,
it is demonstrated that the gains in the accuracy on the growth rate can potentially
be even much higher.
The contributions of the three authors of this paper are as follows: Uroš Seljak
motivated the RSD analysis as a possible application of the methods described above.
I conducted the numerical analysis using N -body simulations provided by Vincent
Desjacques, and composed the manuscript in consultation with Uroš Seljak. It has
been submitted to Physical Review D in July 2012 and is currently under review.
Optimal Weighting in Galaxy Surveys:
Application to Redshift-Space Distortions
Nico Hamaus,1, ∗ Uroš Seljak,1, 2, 3 and Vincent Desjacques4
1
2
Institute for Theoretical Physics, University of Zurich, 8057 Zurich, Switzerland
Physics Department, Astronomy Department and Lawrence Berkeley National Laboratory,
University of California, Berkeley, California 94720, USA
3
Ewha University, Seoul 120-750, S. Korea
4
Département de Physique Théorique & Center for Astroparticle Physics,
Université de Genève, 24 Quai Ernest Ansermet, 1211 Genève 4, Switzerland
(Dated: July 4, 2012)
Using multiple tracers of large-scale structure allows to evade the limitations imposed by sampling
variance for some parameters of interest in cosmology. We demonstrate the optimal way of carrying
out a multitracer analysis in a galaxy redshift survey by considering the principal components of the
shot noise matrix from two-point clustering statistics. We show how to construct two tracers that
maximize the benefits of sampling variance and shot noise cancellation using optimal weights. On
the basis of high-resolution N -body simulations of dark matter halos we apply this technique to the
analysis of redshift-space distortions and demonstrate how constraints on the growth rate of structure
formation can be substantially improved. The primary limitations are nonlinear effects, which cause
significant biases in the method already at scales of k < 0.1hMpc−1 , suggesting the need to develop
nonlinear models of redshift-space distortions in order to extract the maximum information from
future redshift surveys. Nonetheless we find gains of a factor of a few in constraints on the growth
rate achievable when merely the linear regime of a galaxy survey like EUCLID is considered.
I.
INTRODUCTION
One of the deepest mysteries of contemporary cosmology is the nature of the observed accelerated expansion
of the Universe. So far, its evolution can be described
remarkably well by Einstein’s theory of gravitation including a nonzero cosmological constant Λ. However, in
order to fit the astronomical observations (e.g., [1]), Λ
must be many orders of magnitude smaller than what our
standard model of particle physics would expect. This
hierarchy problem inspired various departures from the
cosmological standard model, such as modifications of
Einstein’s field equations or the introduction of exotic
forms of matter.
A particularly sensitive probe of cosmology is the largescale structure (LSS) of the Universe. Galaxy redshift
surveys map out large fractions of its observable volume
and thereby reconstruct a three-dimensional map of density fluctuations whose statistical properties directly relate to fundamental cosmological parameters [2–4]. Unfortunately, this reconstruction is hampered by the fact
that galaxies are biased and stochastic tracers of the dominating dark matter density field. Even if the density
field could be inferred perfectly well, the finite number
Nk of independent Fourier modes in the survey sets a
fundamental lower limit on the achievable uncertainty,
which is known as sampling variance (or cosmic variance, in case the survey size is the whole observable Universe). For example, in a measurement of the dark matter
∗
[email protected]
power spectrum
p P , the sampling variance limit is given
by σP /P ≥
2/Nk , an uncertainty floor that propagates into all the parameters one wants to infer from P .
This limit decreases towards smaller scales as more and
more Fourier modes can be sampled, but at the same
time linear theory starts to break down and higher-order
perturbation theory has to be adopted to model P (see,
e.g., [5–8]). Further complication arises in relating the
observed galaxy power spectrum to the latter, as galaxy
bias becomes nonlinear and nonlocal [9, 10].
An alternative approach to accurately probe cosmological parameters is to consider multiple tracers of the
density field within the well-understood linear regime.
The relative clustering amplitude between multiple tracers can be inferred without sampling variance limitation
because the underlying density fluctuations cancel out in
taking ratios [11, 12]. Therefore, any cosmological information that remains in the relative clustering amplitude
between different tracers can potentially be inferred with
a much higher accuracy.
In this paper we focus on a particular contribution to
the clustering amplitude of galaxies coming from redshiftspace distortions (RSD). These are caused by peculiar
velocities along the line of sight, causing their clustering
statistics to become anisotropic. First treated as a contamination, this effect has been realized to be a powerful
probe of cosmology, as an understanding of RSD allows
to infer the growth rate of structure formation, which is
directly tied to the expansion history of the Universe as
well as the theory of gravity (e.g., [13–23]).
After recapping the fundamentals of RSD and introducing a general formalism for the multitracer analysis
ii
FORMALISM
growth rate can provide viable tests on the theory of
gravitation.
Galaxies only form in specific, discrete locations of the
density field; they are referred to as biased and stochastic
tracers of the dark matter. On linear scales, this relation
can be described locally by
Galaxies in redshift space
δg(r) (r) = bg δ(r) + ǫ ,
in Sec. II, we present our results on the RSD analysis
from N -body simulations in Sec. III. Finally we draw
our conclusions in Sec. IV.
II.
A.
In galaxy surveys, radial distances are inferred via the
individual redshift of objects, assuming they follow the
Hubble flow. Due to gravitational attraction, however,
galaxies (respectively, their host halos) build up peculiar velocities v which contribute to their redshift via the
Doppler effect. Hence, the real-space and redshift-space
locations r and s of a galaxy are related as
s = r+
v · r̂
r̂ ,
H(z)
(1)
where r̂ is the unit vector along the line of sight and H(z)
is the Hubble constant as a function of redshift z. On
large scales, gravity causes coherent infall of test particles
into the potential wells of the dark matter. Thus, galaxies are moving towards overdense regions in the Universe,
resulting in an enhancement of their inferred overdensity
along the line of sight. According to linear perturbation
theory, the redshift-space and real-space galaxy overdensities are related as
δg(s) (k, µ) = δg(r) (k) + f µ2 δ(k) ,
(2)
where µ the cosine of the angle between any wave vector
k in the survey and the line of sight, δ the dark matter
overdensity field and f the growth rate of structure. This
well-known result [24] further makes use of the planeparallel approximation, assuming the separation of any
galaxy pair to be much smaller than their distance to
the observer. On nonlinear scales, random motions are
generated in the process of virialization, which causes
a damping in the clustering amplitude of galaxies along
the line of sight. This so-called Finger-of-God effect is often modeled phenomenologically by an additional Gaussian damping factor in Eq. (2) [25], but more elaborate
schemes have been developed (e.g., [26–34]). We will neglect nonlinear corrections in this paper and focus on the
large linear scales.
The growth rate f is the logarithmic derivative of the
growth factor D with respect to the scale factor a. In
linear theory, it can be expressed as
f ≡ d ln D/d ln a ≃ Ωγm ,
(3)
with the matter density parameter Ωm and the growth
index γ [35]. In Einstein gravity the value of γ is about
0.55, but can take on distinctly different values in modified gravity scenarios [36]. Therefore, constraints on the
(4)
where the factor bg is the linear galaxy bias and ǫ a
random variable denoted as shot noise, describing the
stochastic nature of this relation. Both bg and ǫ depend
on redshift, as well as various properties of the type of
galaxies one is considering (e.g., luminosity, color, hosthalo mass).
Together with Eq. (2) this yields a model for the overdensity field of galaxies in redshift space,
δg(s) (k, µ) = bg + f µ2 δ(k) + ǫ .
(5)
Since the phenomena of galaxy biasing and RSD are to
multiply the density field δ by some factor, we can simply define a more general effective bias parameter b that
contains both contributions,
b ≡ b g + f µ2 .
(6)
In the following, we will drop the superscript that distinguishes between real-space and redshift-space quantities
for clarity. If not explicitly mentioned otherwise, all symbols should be understood as given in redshift space.
B.
Multiple tracers
In order to exploit the gains of sampling variance and
shot noise cancellation, we need to consider multiple tracers of the dark matter density field [11, 12]. One way to
achieve this is splitting some galaxy catalog into bins of a
certain observable property of the galaxies (like luminosity, color, host-halo mass, etc.). However, the following
framework is not limited to galaxies and may be adopted
for other tracers of the dark matter density field as well.
1.
Covariance matrix
We start by writing the density fields of N tracers as a
vector δ g ≡ (δg1 , δg2 , . . . , δgN ). The outer product of this
vector, once ensemble averaged within a k-shell in Fourier
space, yields the covariance matrix C ≡ hδ g δ †g i, where
the † symbol denotes the Hermitian conjugate (transpose
and complex conjugation). Plugging in the model for
galaxy overdensities in redshift space from Eq. (5), it
reads
C = bb† P + E ,
(7)
iii
with the effective bias vector b = bg + f µ2 I, the dark
matter power spectrum P ≡ hδδ ∗ i and the shot noise matrix E ≡ hǫǫ† i (by definition hǫδ ∗ i = 0). The covariance
matrix contains all auto-power and cross-power spectra
of the considered tracers, so in total N (N − 1)/2 independent elements per k-shell. However, since all tracers
follow the same dark matter density distribution, these
elements are correlated.
Let us compare the analysis for a single tracer with
the one for two tracers. From a single tracer we can only
observe an estimator of its auto-power spectrum
2
(8)
C(k, µ) = 1 + βµ2 b2g P (k) + E ,
cancel out completely and the three ratios will be different and scale dependent. This gives rise to residual
sampling variance inherent to a measurement of α and
β, which in general can be much smaller than in the
single-tracer case. In turn, a better measurement of β
allows a more precise estimate on f 2 P [see Eq. (10)].
However, the accuracy on f 2 P is still p
limited by sam2
2 P /f P ≥
pling variance, yielding
σ
2/Nk , respecf
p
tively σf σ8 /f σ8 ≥ 1/2Nk [12].
If both δ g and δ are known, we can simply add the
dark matter overdensity mode δ to the overdensities of
the tracers and write δ ≡ (δ, δ g ). In this case, the effectivebias can be obtained directly by taking the ratio
hδ g δ ∗ i hδδ ∗ i. The covariance matrix then becomes
! hδδ ∗ i hδδ †g i
P
b† P
, (15)
C=
=
bP bb† P + E
hδ g δ ∗ i hδ g δ †g i
P ≡ σ82 P0 ,
with N (N + 1)/2 independent elements. Now the degeneracy between bg , σ8 and f is lifted because P is known
separately. Considering the cross-correlation coefficient
of a tracer δg with the dark matter δ, we find
!−1
∗ 2
E
hδ
δ
i
g
= 1+
r2 =
, (16)
2
hδg δg∗ ihδδ ∗ i
(bg + f µ2 ) P
where we define β ≡ f /bg . In this parametrization it
is obvious to see that bg is degenerate with the power
spectrum P ; respectively, its normalization σ8 defined
via
(9)
where P0 describes the shape of the power spectrum and
is assumed to be known, at least up to linear order. From
Eq. (8), only the combination b2g P can be determined at
µ = 0 (usually, the shot noise is assumed to be Poissonian, meaning it is scale independent and given by the
inverse number density of galaxies n̄−1 ). The same is
true for the growth rate f , we can only determine the
product
C(k, µ) − E − [C(k, 0) − E] 1 + 2βµ2
2
f P =
, (10)
µ4
but this does not apply for β, which can be extracted
directly from observations of C where µ 6= 0 (e.g., [37]).
Note, however, that in this case the achievable error on
β is limited by the sampling variance inherent to P .
In case two distinct tracers of the density field with
biases bg and αbg are observed, where α is their relative
galaxy bias, we obtain the three following power spectra:
2
(11)
C11 (k, µ) = 1 + βµ2 b2g P (k) + E11 ,
2
(12)
C22 (k, µ) = α + βµ2 b2g P (k) + E22 ,
2
2
2
α + βµ bg P (k) + E12 . (13)
C12 (k, µ) = 1 + βµ
The degree to how well they are correlated is quantified
2
by the cross-correlation coefficient r2 ≡ C12
/C11 C22 , so
in the idealistic case of no shot noise (E = 0), r = 1 and
the ratios
C12 /C11 = C22 /C12 =
p
α + βµ2
C22 /C11 =
1 + βµ2
(14)
all yield the same expression, which is independent of
P . Hence, a combination of observations at different
µ yields α and β without sampling variance [12]. Unfortunately, in realistic surveys E 6= 0, so P will not
so the deviation of r from unity is only controlled by the
shot noise of the tracer. Again, if E = 0, the dark matter
power spectrum disappears completely, so that bg and f
can be determined without sampling variance.
2.
Fisher information
In order to determine more quantitatively how much
information on cosmology is buried in the clustering
statistics of biased tracers and the dark matter, we have
to compute the Fisher information matrix [38], a derivation of which is presented in the following: we start with
a multivariate Gaussian likelihood of the data vector δ g
1 † −1
1
√
(17)
exp − δ g C δ g ,
L =
2
(2π)N/2 det C
which is a reasonable assumption on large scales, where
δ ≪ 1. The Fisher information matrix for the parameters θi and θj is obtained by ensemble averaging over the
Hessian of the log-likelihood [39, 40],
2
∂ ln L
∂C −1 ∂C −1
1
Fij ≡ −
C
C
. (18)
= Tr
∂θi ∂θj
2
∂θi
∂θj
According to the model from Eq. (7), the derivative of the
halo covariance matrix with respect to the parameters is
∂C †
(19)
= bbi + bi b† P + bb† Pi ,
∂θi
iv
where bi ≡ ∂b/∂θi , Pi ≡ ∂P/∂θi and we assume
∂E/∂θi = 0. Utilizing the Sherman-Morrison formula
[41, 42], the inverse of the covariance matrix becomes
C−1 = E −1 −
−1
†
−1
E bb E P
,
1 + b† E −1 bP
(20)
provided the shot noise matrix is not singular (e.g., with
vanishing E also C becomes singular by construction).
The full Fisher matrix for this case is calculated in Appendix A. The result is
Σi Σj
Σi Σj
+ Σij −
Fij = Σ−1 Σij +
Σ
Σ
−2
Σj Pi
Pi Pj
Σi Pj
, (21)
1 + Σ−1
+
+
+
2
Σ P
Σ P
2P
where
Σij ≡ b†i E −1 bj P ,
(22)
and Σi as well as Σ are defined accordingly by simply
omitting the corresponding indices (derivatives). We can
identify the first two terms in the square brackets as a
single-tracer and a multitracer term, respectively. The
single-tracer term is suppressed by a factor Σ−1 as compared to the multitracer term. By definition, the latter vanishes for the case of only one tracer, since then
ΣΣij = Σi Σj and Eq. (21) simplifies to
−2
bi bj
E
bi Pj
bj Pi
Pi Pj
Fij = 2 2 +
1+ 2
.
+
+
b
b P
b P
2P 2
b P
(23)
While the Fisher information for multiple tracers can in
principle become infinite in the limit of no shot noise,
Eq. (23) reaches a finite limit when E → 0.
We consider two sets of parameters separately: the
ones that influence only the effective bias,
θ(b) ≡ (bg , f )
(24)
and the ones that go into the matter power spectrum,
θ (P ) ≡ (σ8 , ns , h, ΩΛ , Ωm , Ωb , Ωk ) .
(25)
The elements of the shot noise matrix are usually not
considered as quantities of interest, but as nuisance parameters that can be marginalized over. From Eq. (21)
it is evident that there are degeneracies between θ (b) and
θ(P ) due to mixed terms of the form Σi Pj . However,
if Σ is sufficiently large, those terms are suppressed in
comparison to the term Σij .
In case the dark matter density field is known in addition to the galaxies, the derivative of C from Eq. (15)
becomes
∂C
Pi
b†i P + b† Pi
. (26)
=
∂θi
bi P + bPi bb†i P + bi b† P + bb† Pi
Furthermore, a block inversion of C yields
(1 + Σ)P −1 −b† E −1
−1
C =
,
−E −1 b
E −1
(27)
and the Fisher information becomes (see Appendix A)
Fij = Σij +
Pi Pj
.
2P 2
(28)
For a single tracer this further simplifies to
Fij =
bi bj P
Pi Pj
.
+
E
2P 2
(29)
This expression may increase indefinitely for sufficiently
small E. In the limit E → 0, the effective bias can be determined exactly, allowing an exact measurement of the
parameters θ(b) [12, 43]. On the other hand, constraints
on the parameters θ (P ) are always limited by the variance of the power spectrum Var(P ) = 2P 2 . Here, there
are no mixed terms depending both on the effective bias
and the power spectrum, so parameter degeneracies between θ(b) and θ(P ) are absent. We note that in the limit
of small Σi /Σ, Eq. (21) reduces to Eq. (28).
3.
Reparametrization
As mentioned above, the growth rate f cannot be determined from a galaxy redshift survey alone, since it is
degenerate with the power spectrum P , respectively its
normalization σ8 . This degeneracy can only be broken
with knowledge of the dark matter density field, or other
prior constraints. For this reason it is sometimes convenient to reparametrize by the mapping
θ̃
(b)
ref
= θ(b) bref
g = (bg , f ) bg ≡ (α, β) ,
σ̃8 =
σ8 bref
g
,
(30)
(31)
where bref
g is an arbitrary reference galaxy bias so that
α is the relative galaxy bias of all considered tracers to
this reference. This mapping leaves the covariance matrix
from Eq. (7) unchanged. We can conveniently choose the
lowest bias of all tracers as the reference, bref
g = bg1 , such
that α = (1, bg2 /bg1 , . . . , bgN /bg1 ) is always larger than
unity and β = f /bg1 . Another popular parametrization is
θ̃
(b)
= θ(b) σ8ref = bg σ8ref , f σ8ref
σ̃8 =
σ8 /σ8ref
,
,
(32)
(33)
with σ8ref being some arbitrary reference normalization
of the power spectrum, which we choose to be identical
with our simulation input value of σ8 = 0.81. In this
paper we will quote constraints on both β and f σ8 when
considering a galaxy redshift survey only, and on f when
the latter is combined with dark matter observations.
v
III.
1.0
ANALYSIS
0.8
A.
w- = M + M 0
w+ = M - M1 + M22/M
Numerical setup
0.6
B.
Optimal tracer selection
In order to fully exploit the benefits of the multitracer
approach, the question remains on how to ideally construct different tracers from a given galaxy catalog. In
this section we will derive the answer to that question
and test it on the basis of N -body simulations.
1.
Principal components of the signal-to-noise ratio
A quantity that plays a crucial role for the cosmological information content contained in the two-point clustering statistics of LSS is the clustering signal-to-noise
ratio Σ ≡ b† E −1 bP . For dark matter halos, it has been
0.4
w±
Our numerical analysis is based on a high-resolution
N -body simulation performed at the University of Zürich
supercomputer zBox3 with the gadget-2 code [44]. It
contains 15363 particles of mass 4.7×1010h−1 M⊙ in a box
of 1.3h−1 Gpc a side. We chose our fiducial cosmology to
match the WMAP5 best fit with σ8 = 0.81, ns = 0.96,
h = 0.7, ΩΛ = 0.721, Ωm = 0.279, Ωb = 0.046, Ωk = 0
[45]. We further employ a friends-of-friends algorithm
[46] with a linking length of 20% of the mean interparticle
distance to generate halo catalogs at different redshifts.
With a minimum of 20 dark matter particles per halo we
resolve halo masses down to Mmin ≃ 9.4 × 1011 h−1 M⊙ ,
resulting in a mean halo number density of n̄ ≃ 4.0 ×
10−3 h3 Mpc−3 at z = 0 and n̄ ≃ 3.3 × 10−3 h3 Mpc−3 at
z = 1.
In order to transform the real-space halo catalog into
redshift space, we apply Eq. (1) using the velocities
of the halos along the three independent directions of
the box (x, y and z axis). Thus, three independent
redshift-space catalogs can be constructed from a single real-space catalog, yielding a total effective volume of
Veff ≃ 6.6h−3 Gpc3 . Density fields are created by cloudin-cell interpolation [47] onto a mesh of 10243 grid points,
and the Fourier modes are obtained using a FFT algorithm.
We utilize the idl algorithm mpfit [48] to fit our models to the numerical data and find the best-fit parameters
including their uncertainties. It is based on the minpack
distribution by [49] and uses the Levenberg-Marquardt
technique to find the minimum of a multidimensional
nonlinear least-squares problem. Parameter uncertainties are calculated via the Jacobian of the chi square,
which is determined numerically using finite-difference
derivatives.
0.2
0.0
-0.2
-0.4
1012
1013
M [h-1MO•]
1014
FIG. 1. Three different eigenvectors of the shot noise matrix
obtained from a halo catalog that has been split into 30 mass
bins of equal number density. The two non-Poisson eigenvectors (upper red and lower blue stars) are overplotted with
their best-fit weighting functions w± as solid lines with functional form given in the top left of the panel in corresponding
colors. One representative Poisson eigenvector is shown as
green circles; when used as a weighting function it yields a
very low clustering signal-to-noise ratio Σ, due to its oscillatory behavior.
demonstrated on the basis of numerical N -body simulations that Σ is dominated by only two principal components corresponding to the two nontrivial (non-Poisson)
eigenvectors of the shot noise matrix E (see Fig. 6 in [50]).
We denote these components with a plus and a minus
subscript, according to their super- and sub-Poissonian
shot noise levels, respectively. In this manner we can
expand the clustering signal-to-noise ratio as
Σ ≡ b† E −1 bP =
N
2
X
(V ⊺ b)
i
i=0
λi
P ≃
b2−
b2
P + +P ,
E−
E+
(34)
where V ⊺i and λi are the N eigenvectors and eigenvalues
of the shot noise matrix E, and b± and E± are the effective
bias and shot noise of the two principal components of
the tracer density field,
P
w± (Mi )δg (Mi )
δ± ≡ i P
,
(35)
i w± (Mi )
where the summation runs over all individual objects in
the volume. The w± (M ) are weighting functions corresponding to the two non-Poisson eigenvectors of the shot
noise matrix, as depicted in Fig. 1. Equation (34) states
that a splitting of the tracer density field into N mass
bins yields, in the limit N → ∞, about the same clustering signal-to-noise ratio as simply considering the two
weighted fields δ+ and δ− . Therefore, these are the most
promising candidates to carry out a multitracer analysis
with.
vi
105
In [50] the functional form of w− (M ) was found to be
well described by
(36)
where M0 is about three times the minimum halo mass
resolved in the simulation, M0 ≃ 3Mmin. For the second
weighting function we find
w+ (M ) = M − M1 + M22 /M
(37)
to be a good fit to the second eigenvector of the shot noise
matrix, as shown in Fig. 1. For the constants M1 and M2
we find M1 ≃ 1 × 1014 h−1 M⊙ , M2 ≃ 3 × 1013 h−1 M⊙ at
z = 0 and M1 ≃ 1 × 1014 h−1 M⊙ , M2 ≃ 1 × 1015 h−1 M⊙
at z = 1.
The physical origin of the first principal component
is related to halo exclusion effects [51]. The sampling of
the density field with halos is less stochastic than Poisson
sampling (sampling with points) due their finite extension. Since the exclusion volume of halos is proportional
to their mass, this effect is strongest at the high mass end
[50]. The second principal component can be interpreted
as a loop correction to the galaxy bias [52], coming from
the second-order term in a local bias expansion model
[53]. Due to its nonlinear character, it adds a superPoissonian shot noise contribution to the two-point clustering statistics of halos, originating from the squared
density field [54, 55]. However, through mode coupling
it also yields a second-order clustering signal that originates from the bispectrum (three-point function) and
adds valuable information coming from smaller scales.
Figure
2 displays
the three
matrix
elements
covariance
∗
∗
∗
in
and C−− = δ− δ−
, C+− = δ+ δ−
C++ = δ+ δ+
real space (µ = 0) at redshift z = 0, extracted from our
simulation. The three power spectra are obviously highly
correlated and closely follow the shape of the estimated
dark matter power spectrum up to k ≃ 0.1hMpc−1 . By
construction, the shot noises of the two fields are not correlated, i.e., hǫ− ǫ+ i = 0. Taking the ratios as in Eq. (14)
yields three possible estimators for the relative galaxy
bias α ≡ bg− /bg+ ≃ 1.6. In the following two subsections,
we will provide evidence for the claim that the fields δ+
and δ− are indeed the optimal choice for a multitracer
analysis.
2.
Sampling variance cancellation
The idea of utilizing multiple tracers is to cancel sampling variance from the underlying density field δ. To
quantify the magnitude of cancellation between any two
tracers δg1 and δg2 , we define the following statistic:
2
σSV
≡
h|b2 δg1 (k, µ) − b1 δg2 (k, µ)|2 i
2
2
h|b2 δg1 (k, µ)| i + h|b1 δg2 (k, µ)| i
.
(38)
C(k) [h-3Mpc3]
w− (M ) = M + M0 ,
<δ-δ*->
<δ+δ*->
<δ+δ*+>
104
103
102
0.01
0.1
k [hMpc-1]
1
FIG. 2. Real-space auto-power and cross-power spectra of the
two fields δ+ and δ− (as indicated), obtained through weighting the halo catalog by the two non-Poisson eigenvectors of
the shot noise matrix w+ and w− . The long-dashed black line
shows the dark matter power spectrum from the simulation
and the dotted line its linear theory prediction. The three different estimators for the relative bias α between δ+ and δ− , as
defined in Eq. (14), are depicted in dot-dashedp(C+− /C++ ),
dot-dot-dot-dashed (C−− /C+− ) and dashed ( C−− /C++ ).
For visibility, they were shifted upwards by a factor of 103 .
Here, b1 and b2 is the effective bias as defined in Eq. (6).
If the two tracers are completely uncorrelated (r = 0),
there is no cancellation and σSV = 1. Yet, if they are
perfectly correlated (r = 1), σSV = 0. Because the real
and imaginary parts of any given Fourier mode are uncorrelated, we can swap them for one of the tracers to
mimic the case of no correlation.
We consider the following selection criteria for two
tracers from our halo catalog:
20/80: lightest 20% vs heaviest 80% of all halos,
50/50: lightest 50% vs heaviest 50% of all halos,
80/20: lightest 80% vs heaviest 20% of all halos,
u/w− : all uniformly weighted vs all w− -weighted halos,
w+ /w− : all w+ -weighted vs all w− -weighted halos.
The first three are simply obtained via cutting the halo
catalog in two at different mass thresholds Mcut to
yield the indicated abundances in each bin. The fourth
selection utilizes the whole uniform halo catalog (not
weighted) and its w− -weighted form, while the last one
uses both weighting functions w± to construct two tracers from one and the same halo catalog. Further details about the tracers with these selection criteria can
be found in Table I.
Figure 3 shows σSV for the different tracer pairs; on
large scales they all exhibit significant cancellation of
vii
TABLE I. Details of the tracer selection.
Selection
20/80
50/50
80/20
u/w−
w+ /w−
Mcut
[h−1 M⊙ ]
M̄1
[h−1 M⊙ ]
M̄2
[h−1 M⊙ ]
1.20 × 1012
2.00 × 1012
5.00 × 1012
-
1.04 × 1012
1.33 × 1012
1.97 × 1012
6.30 × 1012
2.29 × 1012
7.72 × 1012
1.13 × 1013
2.22 × 1013
6.76 × 1013
6.76 × 1013
n̄1
n̄2
E11
E22
E12
[h3 Mpc−3 ] [h3 Mpc−3 ] [h−3 Mpc3 ] [h−3 Mpc3 ] [h−3 Mpc3 ]
8.50 × 10−4
2.00 × 10−3
3.15 × 10−3
4.00 × 10−3
4.00 × 10−3
3.15 × 10−3
2.00 × 10−3
8.50 × 10−4
4.00 × 10−3
4.00 × 10−3
1.2
1.0
σSV
0.8
0.6
0.4
0.2
0.0
0.01
0.1
k [hMpc-1]
1
FIG. 3. Sampling variance statistic σSV for five different
tracer selections at z = 0 as described in the text, 20/80
(dot-dot-dot-dashed, blue), 50/50 (dot-dashed, green), 80/20
(dashed, red), u/w− (solid, orange) and w+ /w− (long-dashed,
yellow). Dotted lines show the corresponding results for uncorrelated modes by swapping the real and imaginary parts
of one of the tracer’s Fourier modes.
sampling variance as compared to the reference case with
switched real and imaginary parts (dotted lines). Towards smaller scales this effect is deteriorated due to the
onset of nonlinearities and velocity dispersion (Fingerof-God effects) [26]. As evident from the plot, a combination of the uniform with the w− -weighted halos yields
the highest cancellation of sampling variance. However,
the combination of w+ -weighted and w− -weighted halos
shows a comparable suppression, as opposed to cutting
the halo catalog in two, which yields less cancellation,
especially for a low mass cut. If a mass cut is imposed
to construct two tracers, the highest sampling variance
cancellation is achieved when the same abundance of objects in each of the resulting catalogs is chosen (50/50),
which is in agreement with the findings of [56].
3.
Fisher information
Cancellation of sampling variance alone is not a sufficient indicator on how well cosmological parameters can
1432
911
854
812
804
964
1243
1770
51
51
448
547
606
53
7
bg1
0.899
0.896
0.901
0.969
0.902
bg2
Σmax
0.988 74
1.042 73
1.217 73
1.457 2067
1.457 2098
be constrained in a multitracer analysis. This is because
one is looking for relative changes in the clustering signal
from multiple tracers, and not for the absolute clustering
amplitude in each tracer. In this paragraph we will show
that it is desirable to have tracers with a high relative
galaxy bias ratio α.
Quantitatively, the achievable accuracy on a given cosmological parameter θ is determined by the inversion of
the Fisher matrix of Eq. (21). For the sake of simplicity,
let us consider only the f -f element,
ΣΣf f + Σ2f + Σ ΣΣf f − Σ2f
Ff f =
.
(39)
1 + Σ2
In the high signal-to-noise regime, Σ ≫ 1, and expressing Σ, Σf , as well as Σf f in terms of the two principal
components as in Eq. (34), we get
2
2
b2
b
(b− −b+ )2
2 E−2 + E+2 + (b−E−+bE++ )
E E
−
+
4
µ + b2 − +b2 µ4 P .
Ff f ≃
b2
b2+ 2
−
+
−
+
E− + E+
E−
E+
(40)
As a second approximation, we can make use of the fact
that b2− /E− ≫ b2+ /E+ and E− ≪ E+ (see Table I), which
yields
2 4
2µ4
b+
µ P
+
1
−
=
2
b−
b−
E+
2 4
1 + βµ2
µ P
2µ4
+ 1−
.
=
(bg− + f µ2 )2
α + βµ2
E+
Ff f ≃
(41)
The single-tracer term (as derived in [57]) is dominated
by the first principal component of Σ, while the multitracer term depends on the bias ratio α = bg− /bg+ of
both principal components and the shot noise E+ of the
second principle component. In order to maximize Ff f ,
it is thus desirable to have a large α and a low E+ at the
same time. In the special case of uniform Poisson shot
noise, Eq. (39) reproduces the expression derived in [58].
If the dark matter density field is known separately,
this Fisher matrix element becomes
Ff f = Σf f ≃
µ4 P
,
E−
(42)
viii
103
Fff (k,µ=1)
10
20/80
50/50
80/20
u/w-
w+/w-
2
101
100
10-1
0.01
0.1
k [hMpc-1]
1
0.01
0.1
k [hMpc-1]
1
0.01
0.1
k [hMpc-1]
1
0.01
0.1
k [hMpc-1]
1
0.01
0.1
k [hMpc-1]
1
FIG. 4. Fisher matrix element Ff f as a function of k at µ = 1 and z = 0 for various tracer selections (indicated in the top
right of each panel). Results are shown for a single-tracer analysis with the first (solid, blue) and the second tracer (dotted,
green) used separately, both of the tracers combined in a multitracer analysis (dashed, red), and each of the tracers combined
with idealistic dark matter observations (dot-dashed orange and long-dashed yellow, respectively).
thus independent of the effective bias and limited only by
the low shot noise level of the first principal component.
Figure 4 depicts Ff f (k, µ = 1) for all of our five tracer
selections. For each case we further distinguish between
the following scenarios:
• single-tracer analysis with each of the two tracers
taken separately,
• multiple-tracer analysis with both tracers combined,
• combined analysis of each tracer with the dark matter density field.
Clearly, in a single-tracer analysis the tracer with the
lowest galaxy bias yields the highest information on the
growth rate f . This is evident from Eq. (23), where bg
enters in the denominator of the first term and the shot
noise E is negligible if E/b2g P ≪ 1. Since bg1 is very
similar in all five cases, the Fisher information from a
single tracer cannot be increased much by any particular
choice of tracer.
On the contrary, a combination of two tracers can cancel out sampling variance and therefore considerably increase the available information on f if the tracers are
selected appropriately. As evident from the dashed red
lines in Fig. 4, the multitracer term in Eq. (41) gains importance over the single-tracer term as moving from the
left to the right panel. Again, the two tracers obtained by
a low mass cut yield the worst results, owing to the fact
that the shot noise of both tracers is super-Poissonian in
this selection. It is more optimal to impose a high mass
cut in order to benefit from the sub-Poissonian shot noise
level of the heaviest halos [50]. However, the highest
Fisher information content is obtained when correlating
the two orthogonally weighted fields δ+ and δ− . The
main reason for this is the large relative galaxy bias between the two fields and their relatively low shot noise
level [see Table I and Eq. (41)].
We have also explored the possibility of splitting the
halo catalog into more than two mass bins, considering up
to N = 10 tracers. We find that the Fisher information
from multiple tracers increases with the number of bins N
and approaches the result obtained with the two weighted
fields δ+ and δ− in the limit of high N .
Finally, adding the information from the dark matter
density field to each one of the tracers increases the information content on f . In this case, sampling variance
inherent in the density field δ is known and can thus
be removed from the halo fields directly. According to
Eq. (42), the Fisher information is inversely proportional
to the shot noise of the tracer, so the lowest stochasticity
weight w− yields the best results.
C.
Multitracer fit
So far we have investigated the Fisher information content on the growth rate using multiple tracers of the LSS.
The question of how to actually constrain parameters of
interest from a data set in the optimal way will be answered in this section. For this task we want to maximize
the likelihood function from Eq. (17), which is equivalent
to minimizing its negative logarithm, the chi square
χ2 ≡
X1
k
2
δ †g (k, µ)C−1 δ g (k, µ) +
1
ln (det C) .
2
(43)
Here, the covariance matrix C is given by the clustering
model of Eq. (7) and we sum over all individual Fourier
modes δ g (k, µ) from our halo catalog. When we add the
dark matter density field as an observable, we use the
model from Eq. (15) and δ = (δ, δ g ) as our data vector.
Figure 5 presents the fitting results for the RSD parameter β and the product of growth rate f with the
normalization of the power spectrum σ8 from our halo
catalogs at redshift z = 0. While β has been obtained
from a single-parameter fit, we have marginalized over
ix
single uniform tracer
2.0
combined weighted tracers
β
1.5
1.0
0.5
0.0
fσ8
1.0
0.5
0.0
0.01
0.1
k [hMpc-1]
1
0.01
0.1
k [hMpc-1]
1
FIG. 5. Fit for the redshift-space distortion parameter β (top) and the product f σ8 (bottom) from the two-point clustering
statistics of halos in an N -body simulation with effective volume Veff ≃ 6.6h−3 Gpc3 and halo-mass resolution Mmin ≃ 9.4 ×
1011 h−1 M⊙ at z = 0. LEFT: Conventional single-tracer analysis utilizing all halos (not weighted) from the same catalog.
RIGHT: Multitracer analysis with the two fields δ+ and δ− , obtained through weighting the halo catalog with its principal
components w+ and w− . The fitting results are shown in logarithmic bins of k (points with 1-σ error bars), as well as cumulative
as a function of k = kmax with fixed kmin = 0.0048hMpc −1 (red solid lines with shaded region). Dotted lines show the linear
0.55
theory prediction with f = Ωm
and σ8 = 0.81, dashed lines the cumulative sampling variance limit.
the galaxy bias of each tracer as a free parameter in the
fit for f σ8 . In the left column, the standard single-tracer
analysis utilizing all objects in the halo catalog is performed. The best fits along with their 1σ-error bars
are shown both in k bins (points with error bars), as
well as cumulative as a function of k = kmax with fixed
kmin = 0.0048hMpc−1 (solid lines with shaded region).
The constraints on β and f σ8 are clearly affected by
sampling variance, as evident from the large scatter of the
points at low k. When the number of available Fourier
modes grows towards higher k, this scatter becomes
smaller; however, beyond a scale of k ≃ 0.2hMpc−1 , linear theory breaks down and the fits depart from their
scale-independent linear value assuming f = Ω0.55
(dotm
ted line). The cumulative sampling variance limit for the
determination of f σ8 is shown as a dashed line. Clearly,
the single-tracer fit yields a substantially larger uncertainty compared to this limit.
When combining the two w+ /w− -weighted tracers in
a multitracer analysis as shown in the right column of
Fig. 5, the scatter of the fit is significantly suppressed. On
the largest scales, the errors are reduced by up to a factor
of 4 and the constraints on f σ8 reach the sampling variance limit closely. However, the fit seems to deviate from
the linear theory prediction already at k ≃ 0.04hMpc−1 .
This is likely due to the high bias of the w− -weighted
tracer: as shown in [59], more highly biased halos show a
stronger scale dependence in redshift space, invalidating
the Kaiser formula on even larger scales. This could be
corrected for by nonlinear RSD models, which is beyond
the scope of this paper. On the other hand, the apparently more linear behavior of the single-tracer analysis
may likely be coincidental at k > 0.04hMpc−1 . This is
supported by results shown in the left panel of Fig. 6.
Here, we combine the uniform halo catalog (without
weighting) with the dark matter field from our simulation to fit for f directly. Obviously, sampling variance
has decreased even further, but deviations from linear
theory already kick in at a scale of k ≃ 0.04hMpc−1 .
Most impressive constraints on the growth rate are obtained when we combine the w− -weighted halos with the
dark matter density field, as depicted in the right panel
of Fig. 6. In this case, sampling variance has almost
canceled out completely and the error bars on f have
x
0.8
uniform tracer & matter
weighted tracer & matter
f
0.6
0.4
0.2
0.0
0.01
0.1
k [hMpc-1]
1
0.01
0.1
k [hMpc-1]
1
FIG. 6. Fit for the growth rate f from the two-point clustering statistics of halos and the dark matter combined. LEFT: All
uniform halos (not weighted) and the dark matter. RIGHT: All halos, weighted with the lowest stochasticity weight w− , and
the dark matter. The meaning of lines and symbols is the same as in Fig. 5.
diminished by up to a factor of 142 (at the peak of the
power spectrum), when compared to the standard singletracer analysis. Moreover, deviations from linear theory
are very small up to scales of k ∼ 0.2hMpc−1 , making
this kind of experiment the most promising one out of
the four considered scenarios.
Methods to combine galaxy clustering and weak lensing data have been studied extensively in the recent
literature (e.g., [60–66]), suggesting high improvements
for precision cosmology. Unfortunately, obtaining the
dark matter density field in 3D from observations is a
highly nontrivial problem and is subject of active research. Weak lensing tomography is the technique aiming
to achieve this goal (e.g., [67–71]), but the resolution in
the radial direction is not expected to be high because
of the relatively broad lensing kernels along the line of
sight. Moreover, we assume an ideal reconstruction of
the dark matter density field without considering additional sources of error involved in the lensing measurement, such as shape noise and intrinsic alignment, for
example.
Without knowledge about δ, in principle we would have
to marginalize over all the parameters θ(P ) of the dark
matter power spectrum as well. This implies calculating
the transfer function in each iteration of the fitting procedure, which goes beyond the scope of this paper. In the
high signal-to-noise regime of the multitracer analysis,
the degeneracy between the parameters θ(P ) and θ(b) is
expected to be rather weak (except the fundamental degeneracy between f and σ8 ), as the mixed terms between
Σi and Pj in Eq. (21) are suppressed by Σ. Moreover, in
Appendix C of [52] it has been shown that P (k) cancels
out to a high degree in the chi square of Eq. (43). Of
course, in case the dark matter density field is available,
we do not have to worry about those issues, since bg and
P are directly observable.
In order to quantify the gains in accuracy, we compare the size of the error bars on β and f σ8 from the
multitracer to the single-tracer analysis in Fig. 7 (solid
and dashed blue lines, respectively). Obviously, sampling
variance mostly cancels on the largest scales, yielding improvements in accuracy of up to a factor of 4. Beyond
scales of k ≃ 0.1hMpc−1 , the improvement is deteriorated due to the onset of nonlinear clustering and mode
coupling. Deviations from linear theory increase towards
smaller scales, making the fit of this model to the data increasingly biased (see Figs. 5 and 6). Therefore, the drop
of the curves at k > 0.3hMpc−1 is likely an artifact of the
fitting procedure using an incorrect model and should not
be trusted. At redshift z = 1 this turnover is moved to
smaller scales, the overall improvement in the error ratio
is, however, slightly deteriorated. In order to access the
cosmological information content buried in the semilinear
regime of galaxy clustering in redshift space, one cannot
avoid having to invoke more elaborate models involving
perturbative methods, such as the ones proposed in [72].
More gains can be achieved when galaxies and ideal
dark matter observations are combined, the improvement in the accuracy on β compared to the ordinary
single-tracer analysis amounts to about a factor of 10
in this case (solid green line). If, additionally, galaxies
are weighted optimally, it increases by another factor of
10, two orders of magnitude better than what a singletracer analysis can achieve. In this case the improvement
even persists down to smaller scales of k ≃ 0.3hMpc−1 .
However, the effect of optimal weighting is diminished
towards higher redshifts.
Unfortunately, the halo masses we used to construct
our weighted density fields are not directly observable in
reality. Yet, they correlate with many observables, such
as X-ray luminosity, galaxy richness, weak lensing shear,
velocity dispersion or the thermal Sunyaev-Zel’dovich
(SZ) effect (e.g., [73]). Scaling relations between these
xi
σβmulti / σβsingle
100
10-1
10-2
z=0
z=1
-3
10
0.01
0.1
k [hMpc-1]
1
0.01
0.1
k [hMpc-1]
1
FIG. 7. Ratios of the binned one-sigma error bars (from Fig. 5) on β (solid blue) and f σ8 (dashed blue) when comparing the
optimal multitracer to the single-tracer analysis at z = 0 (left panel) and z = 1 (right panel). Additionally, the improvement in
constraints on β from a combined clustering analysis of halos and dark matter as compared to the single-tracer case is shown
in solid green (no weighting of halos) and in solid red (optimally weighted halos). Adding a log-normal scatter of σln M = 0.1
(dot-dashed) and σln M = 0.5 (dotted) to the halo masses results in a degradation of the constraints from the weighted fields.
observables and halo mass can be calibrated with numerical simulations to obtain unbiased mass proxies with
minimal scatter (e.g., [74, 75]). While optical methods
show a rather large scatter of σln M ≃ 0.45 [76], X-ray or
SZ observations yield tighter relations with σln M reaching below 0.1 [77].
We artificially add a constant log-normal scatter to
the halo masses of our catalog in order to mimic the observational uncertainties in the mass determination for
a rather pessimistic scenario (σln M = 0.5) and a more
optimistic scenario (σln M = 0.1). The results are depicted in Fig. 7 as dotted and dot-dashed lines, respectively. While the constraints from the multitracer analysis on β are only marginally affected for σln M = 0.1 and
degrade by roughly 20–30% for σln M = 0.5, the combined analysis using optimally weighted halos and the
dark matter is more severely deteriorated by mass scatter. Here, even the optimistic scenario increases the uncertainty on the growth rate by roughly 50%, while the
benefits from optimal weighting are completely lost when
going to σln M = 0.5.
Clearly, the high level of precision that can be obtained
with this kind of experiment demands precise mass estimates. Fortunately, a whole industry of existing and
planned experiments devoted to cluster cosmology will
provide those high quality data (e.g., [78–83]).
D.
Halo model predictions
In this section we want to investigate how our results
depend on the resolution of the simulation; respectively,
the minimum resolved halo mass. In a real experiment,
this corresponds to the depth of the galaxy survey with
a corresponding luminosity threshold. Because N -body
simulations with a given volume become increasingly expensive with higher resolution, we will turn to theoretical
predictions henceforth.
The clustering properties of dark matter and halos can
be neatly described by the halo model (see, e.g., [84]).
The basic idea is to separately describe the clustering
within a given halo (one-halo term) and the clustering
amongst different halos (two-halo term). In [50] the halo
model is utilized to derive an analytical expression for
the shot noise matrix. The result can be written as
E = n̄−1 I − bg M† − Mb†g ,
(44)
where n̄ is the number density of halos per bin, I the
identity matrix, M ≡ M /ρ̄m − bg hnM 2 i/2ρ̄2m and M
a vector containing the mean halo mass of each bin. In
Appendix B we utilize this expression to derive the clustering signal-to-noise ratio Σ (as well as Σi and Σij ) from
the halo model. With Eq. (39) we can then determine
the Fisher information on the growth rate and compare
the single-tracer analysis to the multitracer analysis.
Figure 8 displays the same ratio as Fig. 7 for the uncertainty on β, but now as a function of minimum halo
mass Mmin at a fixed scale of k ≃ 0.016hMpc−1 (peak
xii
IV.
CONCLUSIONS
In this paper we investigated the benefits of using
weights in a multitracer analysis of LSS, with a particular focus on constraining the growth rate of structure
formation. On the basis of earlier results on the clustering properties of dark matter halos and their stochasticity
[50], we argue that the gains from a multitracer analysis
in the sense of [12] can be achieved by considering only
the two principal components of the clustering signalto-noise ratio Σ (or, equivalently, the two non-Poisson
eigenvectors of the shot noise matrix E).
We present their explicit functional forms in terms of
weights, showing that the first one coincides with the
weighting function explored in previous work [50], giving
rise to low stochasticity and high bias. For the second
one the weights are also mass dependent, but have a zero
crossing, such that the overall bias is low. This yields
a high relative galaxy bias α between the two tracers,
maximizing the Fisher information content on the cosmological parameters [12]. All of the other eigenvectors
oscillate around zero and add very little information. The
advantage of reducing the information to two eigenvectors is that all of the objects in a given catalog can be
used to construct the two principal components, while
in the conventional multitracer analysis the catalog has
100
σβmulti / σβsingle
of the power spectrum). The gains from the multitracer
method kick in at Mmin ≃ 1014 h−1 M⊙ , where Σ ≃ 1, and
increase towards lower Mmin due to the growing signalto-noise ratio.
A combination of galaxy and dark matter observations may increase these gains further, especially when
the galaxies are weighted optimally. In this case there
is no saturation towards lower Mmin and the improvement compared to the single-tracer analysis continues to
grow. In contrast, the uniform galaxy overdensity field
(not weighted) combined with dark matter already shows
up a saturation at Mmin ≃ 1010 h−1 M⊙ , so no more information on the growth rate can be gained when even
more lighter halos are included in this kind of analysis. Also note that the halo model predictions underestimate the improvements obtained when adding dark matter clustering information, as our N -body results with
Mmin ≃ 9.4×1011h−1 M⊙ yield higher improvements (see
Fig. 7).
In the shot noise dominated regime above a minimum
mass of Mmin ∼ 1014 h−1 M⊙ the error ratio decreases
again towards higher Mmin because the Fisher information on f from a single tracer here roughly scales as E −2 ,
while for a tracer combined with dark matter as E −1 [see
Eqs. (23) and (29), respectively]. We refer the reader to
Figs. 10 and 11 of [52], where a similar plot is shown for
the individual error bars on the non-Gaussianity parameter fNL .
10-1
10-2
10-3
1010
1011
1012
1013
1014
Mmin [h-1MO•]
1015
1016
FIG. 8. Halo model prediction for the improvement on σβ
from the single-tracer analysis to the optimal multitracer
analysis (blue) at k ≃ 0.016hMpc−1 as a function of the lower
halo-mass threshold Mmin at z = 0 (solid) and z = 1 (dashed).
The results for the combined analysis of uniform halos (not
weighted) and dark matter (green), as well as the combination of optimally weighted halos with the dark matter (red)
are also depicted.
to be split into many lower number density subsamples
with higher shot noise [56, 58].
On the basis of numerical N -body simulations of dark
matter halos, we demonstrate that the constraints on β
and f σ8 can be improved by up to a factor of 4 relative to a single-tracer method, but most of the improvement comes from large scales (low k), while for higher k
the gains are smaller and vanish above k ∼ 0.1hMpc−1 ,
where nonlinear effects introduce additional stochasticity
between the two tracers. This technique is fairly insensitive to the observational uncertainty on the halo masses,
as even a 50% log-normal scatter does not degrade the
improvements significantly. Halo model considerations
suggest even higher gains of the method with increasing
mass resolution.
One potential concern for our method is the possibility
that galaxies might be bad tracers of their host-halo centers [85] and therefore exhibit less pronounced principal
components in the clustering signal-to-noise ratio that
are distinct from Poisson sampling. However, there are
strong indications that certain types of galaxies do show
strong correlations in both position and mass with their
host halo, e.g. luminous red galaxies (LRGs) [86]. Techniques to distinguish satellite galaxies from central galaxies have been developed and the satellite fraction can be
used as an estimator of the host-halo mass [87]. Mock
LRG catalogs obtained from a halo occupation distribution (HOD) model suggest some reduction in stochasticity is possible even without explicit knowledge of halo
masses [88]. Therefore, an achievement of the presented
gains seems feasible in light of upcoming spectroscopic
xiii
galaxy surveys such as EUCLID [3], which will attain
galaxy number densities, host-halo mass ranges and a
survey volume comparable to the simulations used in this
paper [89, 90].
Whether these gains translate into a useful constraint
on the final cosmological parameters depends on our ability to model nonlinear RSD effects. We find nonlinear
effects are important for β already at k > 0.03hMpc−1 ,
although they appear to be important for f σ8 only at
k > 0.1hMpc−1 . In the most pessimistic case where
the RSD model cannot be trusted for k > 0.03hMpc−1 ,
the multitracer method provides major gains relative to
the single-tracer case, but neither method provides very
strong constraints overall because of the limited number
of available Fourier modes. In the case where we can
use all the modes up to k ∼ 0.1hMpc−1 , the overall errors are considerably smaller and the multitracer method
provides less of an advantage. It is clear that a better
modeling of the nonlinear effects in RSD is needed to understand the ultimate reach of RSD in both single-tracer
and multitracer methods.
In a more idealistic scenario, we also consider the joint
analysis of halos and the dark matter density field, which
in principle is achievable via a combination of spectroscopic redshift surveys and weak lensing tomography.
Here, utilizing optimal weights can yield up to two orders
of magnitude improvements in constraining β as compared to a single-tracer analysis, but the method is more
prone to uncertainties in the halo mass estimates. It is
unlikely that this gain can be achieved in practice, since
it is very difficult to measure dark matter clustering in
the radial direction directly.
A further technique to construct differently biased
tracers of the density field makes use of nonlinear transformations [91]. Although it is difficult to describe the
effects of a nonlinear transformation on both signal and
noise in galaxy clustering data, combined with optimal
weights this may provide another tool for the multitracer
analysis.
In this paper we have focused on the information that
can be extracted from RSD, in particular β and f σ8 , but
our method is not limited to constraints on the growth
rate, but may be applied to the analysis of primordial
non-Gaussianity [52], general relativistic corrections in
large-scale clustering [92], the Alcock-Paczyński test [12]
or any other quantity that influences the effective bias of
tracers of the density field. It is possible that a better
model of nonlinear RSD may yield a more efficient multitracer method, where the gains relative to the singletracer analysis described here on large scales can be extended to smaller scales. We leave these directions for
the future.
ACKNOWLEDGMENTS
We thank Jaiyul Yoo, Jonathan Blazek, Tobias Baldauf and Zvonimir Vlah for fruitful discussions and
Volker Springel for making public his N -body code
gadget-2. This work is supported by the Packard Foundation, the Swiss National Foundation under Contract
No. 200021-116696/1 and WCU Grant No. R32-10130.
V.D. acknowledges support by the Swiss National Science Foundation. N.H. appreciated the hospitality of
Lawrence Berkeley National Lab and the Institute for
the Early Universe at Ewha University Seoul while parts
of this work were completed.
xiv
Appendix A: FISHER MATRIX FOR MULTIPLE BIASED TRACERS
With Eqs. (19) and (20) plugged into Eq. (18), we have
Pj
Pi
1
C−1 bb†j + bj b† + bb†
C−1 P 2 =
Fij = Tr bb†i + bi b† + bb†
2
P
P
1
Pj
=
b† C−1 bb†i C−1 bj + b† C−1 bb†j C−1 bi + b† C−1 bi b† C−1 bj + b†i C−1 bb†j C−1 b + b† C−1 bb† C−1 bi +
2
P
P
P
P
P
P
i
i
i j
j
+ b† C−1 bb† C−1 bj
+ b† C−1 bb†j C−1 b + b† C−1 bb† C−1 b 2 P 2 =
b† C−1 bb†i C−1 b
P
P
P
P
2
Σi Σj
Σi Σj
Σ
Pi Pj
Σj Σ Pi
Σ
Σi Σ Pj
Σij −
+
=
=
+
+
+
2
2
2
1+Σ
1+Σ
1+Σ
2P 2
(1 + Σ)
(1 + Σ) P
(1 + Σ) P
−2
Σi Σj
Σi Σj
Σi Pj
Σj Pi
Pi Pj
= Σ−1 Σij +
+ Σij −
+
. (A1)
1 + Σ−1
+
+
Σ
Σ
Σ P
Σ P
2P 2
With additional knowledge about the dark matter density field we need to work out Eq. (18) by plugging in Eqs. (26)
and (27). Let us first note that
∂C −1
Pi /P − Σi
b†i E −1 P
C =
,
∂θi
bi + b (Pi /P − Σi ) bb†i E −1 P
so
∂C −1 ∂C −1
C
C =
∂θi
∂θj
(Pi /P − Σi ) b†j E −1 P + Σi b†j E −1 P
† −1
bi bj E P + bb†j E −1 P (Pi /P − Σi ) + bΣi b†j E −1 P
(Pi /P − Σi ) (Pj /P − Σj ) + Σij + Σi (Pj /P − Σj )
[bi + b (Pi /P − Σi )] (Pj /P − Σj ) + bΣij + bΣi (Pj /P − Σj )
!
This yields
1
Fij = Tr
2
∂C −1 ∂C −1
C
C
∂θi
∂θj
= Σij +
Pi Pj
.
2P 2
(A2)
Appendix B: HALO MODEL PREDICTION FOR THE CLUSTERING SIGNAL-TO-NOISE RATIO
In the halo model the shot noise matrix is given by Eq. (44). In order to invert E, we write E = A − Mb†g with
A ≡ n̄−1 I − bg M† and apply the Sherman-Morrison formula twice:
E −1 = A−1 +
A−1 Mb†g A−1
1 − b†g A−1 M
With
Σij ≡ b†i E −1 bj P =
,
A−1 = n̄I +
bg M† n̄
.
n̄−1 − M† bg
b†i A−1 bj 1 − b†g A−1 M + b†i A−1 Mb†g A−1 bj
and
b†i A−1 bj =
1 − b†g A−1 M
b†i bj n̄−1 − M† bg + b†i bg M† bj
n̄ ,
n̄−1 − M† bg
2
n̄−1 − M† bg − b†g bg M† M
n̄ ,
1 − b†g A−1 M =
n̄−1 − M† bg
(B1)
P ,
(B2)
(B3)
(B4)
xv
and similar terms combining b, bi , bg and M, after some algebra we get
b†i Mb†g bg M† bj + b†i bg M† bj + b†i Mb†g bj n̄−1 − M† bg + b†i bg M† Mb†g bj
Σij = b†i bj n̄P +
n̄P .
2
n̄−1 − M† bg − b†g bg M† M
(B5)
In the continuous limit (N → ∞), we can exchange the vector products by integrals over the mass function and set
n̄tot = n̄N . This finally yields
2
hb2g ihMbi ihMbj i + (hbi bg ihMbj i + hMbi ihbj bg i) n̄−1
tot − hMbg i + hbi bg ihbj bg ihM i
n̄tot P .
Σij = hbi bj in̄tot P +
2
n̄−1
− hb2g ihM2 i
tot − hMbg i
(B6)
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6
O UTLOOK

Chapter 6: Outlook
In addition to the presented ones, further applications for multitracer analyses utilizing optimal weights are conceivable. One example is the so-called Alcock-Paczyński
test [1], where the conversion factor between radial and angular coordinates in the
universe is determined. Because this factor depends on the cosmic expansion history, it is particularly sensitive to the dark energy at late times. The use of multiple
tracers can break degeneracies between various parameters in the model, which,
combined with optimal weighting techniques, can potentially be constrained with
unprecedented accuracy [30].
Another example deals with general relativistic corrections to the common Newtonian treatment of galaxy clustering [55]. These corrections become important only
on very large scales, comparable to the size of the horizon. In a single-tracer analysis
their detection is hampered due to cosmic variance, i.e. the limited number of independent Fourier modes available in the observable universe. In a recent study it was
demonstrated that the use of multiple tracers and optimal weights can potentially
reveal those corrections in the galaxy clustering pattern, opening up the unique possibility to test general relativity on the largest cosmic scales [56]. An extension of
this study to modified gravity scenarios is feasible.
The methods developed in this thesis bear promising implications for the attainable precision in cosmological analyses of large-scale structure. However, their
performance has only been tested on the basis of dark matter simulations so far. A
next step could be the inclusion of baryons using full-fledged hydrodynamic simulations, in order to close the gap between the formation of dark matter halos and
galaxies. Unfortunately, due to the enormous range of scales important for galaxy
formation, a self-consistent treatment is numerically not achievable yet, and semianalytic models have to be adopted. HOD-models, for example, are already quite
successful in reproducing the observed clustering properties of galaxies [40]. Within
this framework, it has been shown that for certain types of galaxies, optimal weighting techniques are still very efficient [7]. The actual benefit of these methods still
has to be tested on observational data, taking into account all real-world complications of galaxy surveys, this presents a true challenge for future investigations in
this direction.
A further challenge is posed on the theory side. In order to exploit the gains of
using optimal weights and multiple tracers beyond the linear regime, it is necessary
to consider more elaborate models of large-scale structure. While perturbative approaches for the description of dark matter clustering are already quite successful [3],
complications arising due to nonlinear galaxy biasing and redshift-space distortions
increasingly impede the practicability of smaller and smaller scales that are available
in galaxy surveys. In order for the unused data to be of any value for cosmological
inference, new approaches have to be developed towards this direction (e.g., [45]).
Conversely, one can think of the inverse approach of nonlinearly transforming the
data in order to describe it with a more simple model [44]. Although it is difficult to
describe the effects of nonlinear transformations on both signal and noise in galaxy
clustering data, combined with optimal weights this may provide another tool for
the multitracer analysis. In general, this framework may even be extended to tracers
of large-scale structure other than galaxies.
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