University of Amsterdam
MSc Astronomy and Astrophysics
GRAPPA
Master Thesis
The small scale dark matter power spectrum using the halo
model
by
Nastasha Wijers
10193308
July 2016
60 ECTS
September 2015 – July 2016
Supervisor:
Dr. Shin’ichiro Ando
Examiner:
Dr. Gianfranco Bertone
Anton Pannekoek
Institute
API and GRAPPA
Samenvatting
Voor dit masterproject heb ik de verdeling van materie in ons heelal onderzocht. Uit onderzoek
blijkt namelijk dat ongeveer 80% van de materie in het universum, donkere materie is. Deze bestaat
niet uit de atomen, ionen, en elektronen waaruit de materie die wij kennen bestaat. Het is ook niet
een van de andere deeltjes uit het standaardmodel, dat bijvoorbeeld in deeltjesversnellers gevonden
is. We kunnen het niet zien, maar we kunnen het wel meten: het oefent namelijk zwaartekracht uit
op de materie die we wel kunnen zien.
Waar deze donkere materie wel uit bestaat, is onbekend. Er zijn veel modellen voor wat voor
deeltjes dit kunnen zijn. Die modellen voorspellen ook dat als twee donkere-materiedeeltjes op elkaar
botsen, ze een lichtflits kunnen veroorzaken. De kans dat dit gebeurt is erg klein; als deze kans groot
was, zou de materie niet donker zijn. Aan de andere kant is er heel veel donkere materie in het
universum, dus er is een kans dat al die lichtflitsjes bij elkaar zichtbaar zijn. Er wordt veel naar
gezocht met Fermi-LAT, een gamma-stralingstelescoop. Dit is omdat veel modellen voorspellen dat
de lichtflitsjes uit gammastraling bestaan, en niet uit zichtbaar licht. Tot nu toe is deze straling niet
gevonden. Dat is gebruikt om een maximumkans te vinden dat deze deeltjes botsen.
De reden dat de verdeling van donkere materie uitmaakt voor deze zoektochten, is dat de kans
dat twee deeltjes botsen, afhangt van het kwadraat van hun dichtheid. Dat is omdat als je het
aantal deeltjes ergens verdubbelt, elk deeltje een twee keer zo grote kans heeft op een ander deeltje te
botsen, en er twee keer zo veel deeltjes met deze verdubbelde botsingskans zijn, dus het totaal aantal
botsingen wordt verviervoudigd. Wat dat betekent, is dat als het zelfde aantal deeltjes in klontjes is
verdeeld, er meer botsingen tussen de deeltjes zullen zijn dan als de deeltjes gelijkmatig verdeeld zijn.
Deze verhoging van het aantal botsingen ten opzichte van gelijkmatig verdeelde donkere
materie, en
dus ook de verhoging voor het aantal lichtflitsjes dat we zouden zien, noemen we δ 2 . Dit is het
gemiddelde van het kwadraat van de dichtheidsverhoging.
Ik heb gezocht naar deze verhogingsfactor δ 2 . Dit heb ik gedaan aan de hand van het halomodel.
In dit model zit alle donkere materie in klonten die halo’s heten. Deze halo’s hebben een verdeling
en een structuur. De grootste halo’s hebben een massa van ongeveer een triljoen (1015 ) zonsmassa’s.
Deze bevatten grote clusters van melkwegstelsels. De kleinste halomassa is onbekend, en hangt
onder meer van het deeltjesmodel van de donkere materie af. Dit kan zo groot zijn als ongeveer
een miljoen zonsmassa’s. Sommige modellen voorspellen dat deze kleinste halomassa een biljoenste
(10−12 ) zonsmassa’s is, vergelijkbaar met sommige maantjes in ons zonnestelsel. Ik heb een model
bekeken waarin de kleinste halo’s 30 keer zoveel massa hebben als de Aarde, ofwel een tienduizendste
van de massa van de zon.
Om de structuur van halo’s goed te beschrijven, is het belangrijk hun substructuur mee te nemen.
Dit zijn kleinere klonten binnen de halos, die bijvoorbeeld melkwegstelsels bevatten in clusters, en
de Magelhaense
wolken
binnen de halo van de Melkweg. Hiervoor heb ik termen toegevoegd aan de
berekening van δ 2 , en een programma voor de berekening geschreven.
2 Zonder subhalo’s, en voor een minimummassa van 30 aardmassa’s, vind ik een verhogingsfactor
δ = 64 000. Voor een naı̈ef model met subhalo’s vind ik een
verhogingsfactor
van 89 000, en
voor twee schattingen van realistischere subhalomodellen, vind ik δ 2 = 140 000 en δ 2 = 230 000.
Er zitten vrij grote onderzekerheden in deze modellen, omdat de kleinere (sub)halo’s moeilijk te
onderzoeken zijn. Voor massa’s kleiner dan 10 miljoen zonsmassa’s, zijn (sub)halo’s alleen nog in
simulaties gevonden. Deze waarden die ik vind, zijn wat kleiner dan wat sommige andere onderzoekers
verwachten of gevonden hebben. Ik heb iets andere parameters gebruikt in mijn model, volgens
nieuwere metingen, en een grotere minimummassa voor (sub)halos. Dit verklaart niet noodzakelijk
het hele verschil. Deze waarden kloppen beter met een onderzoek waarin de parameters meer lijken
op wat ik gebruikt heb. Voor een
subhaloparameter heb ik een conservatievere waarde
belangrijke
gekozen, dus deze schatting van δ 2 is aan de lagere kant.
Deze resultaten tonen aan dat het effect van de klonterige verdeling van donkere materie erg
groot is. In het algemeen verhoogt de toevoeging van subhalo’s de voorspelde waarnemingen van
gammastraling van deze botsingen in een gegeven deeltjesmodel. Met ons huidig gebrek een detectie
van deze lichtflitsjes, geeft het een kleiner maximum voor de botsingskans van de deeltjes dan een
model zonder subhalo’s, en legt het dus sterkere beperkingen op aan theorieën over de aard van
donkere materie.
1
Summary
For this MSc project, I have studied the distribution of matter in our universe. I have done this,
because research shows that about 80% of the matter in the universe is dark. It does not consist of
the atoms, ions, and electrons that make up the matter we know. It is also not made up of the other
particles in the standard model, for example, those that have been found in particle accelerators.
We cannot see it, but we can measure it: it gravitationally attracts visible matter.
We do not know what dark matter is made of. There are many models for what kind of particles
it could be made of. Those models predict that if two dark matter particles collide, this could cause
a flash of light. The probability of this happening is small; if it were large, the matter would not
be dark. On the other hand, there is a lot of dark matter in the universe, so there is a chance that
all those flashes of light together can be seen. A lot of searches for this light have been done with
Fermi-LAT, a gamma-ray telescope. This is because many models predict that the flashes of light
are made of gamma rays, not visible light. So far, this radiation has not been found. This has been
used to find maximum collision probabilities for these particles.
The reason the dark matter distribution matters for these searches, is that the probability of
two particles colliding, depends on their density squared. This is because if the number of particles
doubles somewhere, each particle has twice the chance of colliding with another, and there are
twice as many particles with the doubled collision probability, so the total number of collisions is
quadrupled. What this means, is that if the particles have a clumpy distribution, there will be more
collisions than if they are evenly distributed. This increase in the number of collisions, relative to
evenly distributed
dark matter, and therefore the increase in the number of light flashes we would
see, is called δ 2 . This is the average of the squared
density increase.
I have looked for this enhancement factor δ 2 . I have done this using the halo model. In this
model, all dark matter is contained in clumps called halos. These halos have a distribution and a
structure. The largest halos have a mass about a quadrillion (1015 ) times that of the sun. These
contain large clusters of galaxies. The smallest halo mass is unknown. It could be as large as about
a million solar masses. Some models predict this smallest halo mass could be a trillionth (10−12 )
of the mass of the sun, comparable to some moons in our solar system. I have considered a model
where the smallest halo mass is 30 times that of the Earth, or a ten thousandth of that of the sun.
To describe the structure of halos correctly, it is important to include substructure. This consists
of smaller clumps within the halos, that, for example, contain galaxies in clusters, and
the Magellanic
clouds in the Milky Way halo. To do this, I added terms to the calculation of δ 2 , and wrote a
program to compute them.
Without
subhalos, and for a minimum halo mass of 30 Earth masses, I found an enhancement
factor of 89 000, and
factor of δ 2 = 64 000. For a naive subhalo model,
If found an enhancement
for two more realistic subhalo models, I found δ 2 = 140 000 and δ 2 = 230 000. There are fairly
large uncertainties in these models, because the smaller (sub)halos are difficult to research. Below
10 million solar masses, (sub)halos have only been found in simulations. The values I find for the
enhancement factor, are somewhat smaller than other researchers expect or have found. I have used
some different parameters in my model, from newer measurements, and a larger minimum mass
for (sub)halos. This does not necessarily explain the entire difference. These values are in better
agreement with a study using parameters more similar
to mine. For an important subhalo parameter,
I used a more conservative value, so this estimate of δ 2 is on the low side.
These results show that the clumpy distribution of dark matter makes a large difference in predicting gamma-ray observations. Generally, adding subhalos increases the expected observations of
gamma rays from dark matter particle collisions in any particle physics model. With our current
non-detection of these signals, it gives us a smaller maximum probability for dark matter collisions
than a halo model without subhalos. This means that adding subhalos, tightens the constraints on
theories on the nature of dark matter.
2
Abstract
To calculate the strength of gamma-ray emission produced by annihilating dark matter, we need
to take structure into account. This is because an annihilation rate is proportional to the
squared
density. We can parametrise the effect of structure by the so-called flux multiplier, 1 + δ 2 , which
is determined by the average squared overdensity of the dark matter: 1 + δ 2 = ρ2 / hρi2 . This
can be found by integrating the dimensionless power spectrum, which is, in turn, determined by
the Fourier transform of a two-point overdensity correlation function. To model this correlation
function, I have used the halo model. In this model, all (dark) matter is assumed to be contained
in virialised halos. These halos also have substructure: they contain smaller halos called subhalos,
which are formed when a smaller halo merges with a larger one, and is not completely destroyed by
tidal forces.
This substructure is important for the annihilation flux, because it means dark matter is clumpier
than if it were not present. Clumpier dark matter has, on average, a larger squared density, and will
therefore produce a larger annihilation signal. Even if these (sub)structures are unresolved, the total
flux coming from a halo will be larger. In the power spectrum, the effect of the substructure is the
largest on small scales (since subhalos are smaller than their parent halos), where much about the
power spectrum is still uncertain. This is because many model parameters have to be extrapolated
from simulations to smaller scales.
I have found a formalism to include this substructure into the calculation of the power spectrum
using the halo model, and have written a notebook to compute this power spectrum for certain
parameters of the halo model. This can be easily modified to calculate the power spectrum for
similar kinds of models, and can be extended to include more complicated models.
For a toy model of subhalo parameters, I have found the power spectrum, and compared it to the
power spectrum without subhalos. As predicted, subhalos
increased 4the power spectrum on small
scales. For this toy model, I found a flux multiplier 1 + δ 2 = 8.9 · 10
for a minimum halo mass of
10−4 M . For the same minimum mass without subhalos, I found 1+ δ 2 = 6.4·104 . Using the same
scale cut-off as ref. [1], I found a smaller flux multiplier than they did, using a different method. The
subhalo parameters in the toy model provide a conservative estimate for the subhalo contribution to
the annihilation signal, but not necessarily a lower limit. For a generally
conservative model, with
a rough estimate for the subhalo shape based on ref. [2], I find 1 + δ 2 ∼ 1.4 · 105 –2.3 · 105 . This
estimate is also likely on the low side.
3
Contents
1 Introduction
2 The
2.1
2.2
2.3
2.4
7
halo model
Cosmology in a homogeneous, isotropic universe . .
Fourier transformations and notation . . . . . . . . .
The linear power spectrum . . . . . . . . . . . . . .
The halo model formalism . . . . . . . . . . . . . . .
2.4.1 Halo ensemble averages . . . . . . . . . . . .
2.4.2 Further definitions, assumptions, and results
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9
9
10
13
17
18
20
3 Parameters of the halo model
3.1 The halo mass function . . . . . . . . . . . . . . . . .
3.1.1 The Sheth-Tormen mass function . . . . . . . .
3.1.2 Calculating the mass function . . . . . . . . . .
3.1.3 A definition and two parameters . . . . . . . .
3.2 The halo distribution and the linear bias parameter . .
3.3 The halo density profile . . . . . . . . . . . . . . . . .
3.3.1 The NFW profile and its Fourier transform . .
3.3.2 Smoothing out the Fourier transform . . . . . .
3.3.3 Other profiles . . . . . . . . . . . . . . . . . . .
3.4 The concentration function . . . . . . . . . . . . . . .
3.5 Mass definitions and conversions . . . . . . . . . . . .
3.5.1 Converting between mass definitions . . . . . .
3.5.2 Errors in the halo mass function . . . . . . . .
3.6 Subhalo modelling . . . . . . . . . . . . . . . . . . . .
3.6.1 The subhalo mass function . . . . . . . . . . .
3.6.2 The subhalo mass fraction . . . . . . . . . . . .
3.6.3 Points on which to improve the subhalo model
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23
23
23
25
28
29
34
34
36
40
40
41
42
44
47
47
48
49
4 Results
4.1 Sanity checks . . . . . . . . . . . . . . . . . . . . . .
4.1.1 The halo model without subhalos . . . . . . .
4.1.2 Adding the subhalos . . . . . . . . . . . . . .
4.1.3 Checking the flux mulitplier . . . . . . . . . .
4.2 Error estimates for the power spectrum . . . . . . .
4.3 Towards more realistic models . . . . . . . . . . . . .
4.3.1 A more realistic subhalo concentration . . . .
4.3.2 Estimating the effect of concentration scatter
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51
51
52
54
58
60
62
62
65
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5 Discussion
68
5.1 Model uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.2 Comparison to other works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6 Conclusion
72
7 Acknowledgements
72
8 References
73
Appendices
75
4
A Halo model power spectrum derivation
A.1 Justification for bringing the expectation values inside the integrals
2
A.2 Warm-up and preparation: hρi . . . . . . . . . . . . . . . . . . . .
A.3 The main calculations: hρ(z1 )ρ(z2 )i . . . . . . . . . . . . . . . . .
A.3.1 Halo-halo term, i = j . . . . . . . . . . . . . . . . . . . . .
A.3.2 Halo-halo term, i 6= j . . . . . . . . . . . . . . . . . . . . .
A.3.3 Halo-subhalo term, i = j . . . . . . . . . . . . . . . . . . . .
A.3.4 Halo-subhalo term, i 6= j . . . . . . . . . . . . . . . . . . . .
A.3.5 Subhalo-subhalo term, i = j, ai = bi . . . . . . . . . . . . .
A.3.6 Subhalo-subhalo term, i = j, ai 6= bi . . . . . . . . . . . . .
A.3.7 Subhalo-subhalo term, i 6= j . . . . . . . . . . . . . . . . . .
A.3.8 A very long equation . . . . . . . . . . . . . . . . . . . . . .
A.4 Separating smooth and subhalo mass out from the halo mass . . .
A.5 Fourier transforming hρ(z1 )ρ(z2 )i . . . . . . . . . . . . . . . . . . .
A.5.1 A simplifying assumption . . . . . . . . . . . . . . . . . . .
A.5.2 Including a factor that is not a full convolution . . . . . . .
5
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75
75
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1
Introduction
One of the big mysteries in cosmology is the nature of dark matter [3]. We know from various observations
that baryonic matter accounts for only a small portion of the total matter in the universe. The nature
of the rest of the matter is largely unknown. Galaxies and clusters are embedded in halos, made of this
dark matter. There are many theories about this matter; the leading theories propose that dark matter
consists of non-standard-model particles. Observations have so far only been able to rule out parts of
parameter spaces for these models (see e.g. ref. [4]).
One way to investigate dark matter is to look for the results of the rare interactions these particles may
have with standard model particles or with themselves. One way this may happen is if two dark matter
particles annihilate to standard model particles, that then produce photons. A particle physics model
will predict a spectrum for this, given a mass and annihilation cross-section, which can be compared to
observations. This can be difficult in practice, though, since the interaction cross-section of dark matter
is entirely unknown, and there are still large uncertainties in the distribution of dark matter on the sky,
which makes any annihilation signal difficult to distinguish from astrophysical backgrounds that are not
well known. Ref. [5] discusses a possible way to distinguish these signals by their anisotropies across the
sky.
In this thesis, we will investigate the distribution of dark matter on small scales. This is important
for such dark matter searches focussing on annihilation. That is because annihilation is a two-particle
interaction, so its rate depends on the density squared. In particular, the local interaction rate at any
point is proportional to n2χ hσvi: the squared (total) number density of the particles, multiplied by
the velocity-averaged interaction cross-section. The proportionality constant is 12 in the case of selfannihilation, and in the case of two species, the product of their fractional number densities. This is
a general result for interactions between two particles. For dark matter, it is preferable to discuss the
mass density in stead of the number density. This is because we have a reasonable idea of what the
mass densities in galaxies and clusters are, and what the total dark matter density is, but the particle
mass is unknown. We will ignore correlations between dark matter average velocities and densities in
this thesis but these can be computed using the description of the dark matter distribution we will build
up, combined with a model for the particle physics.
Assuming hσvi is position-independent, the dark matter annihilation rate at any point can then be
parametrised as δ 2 + 1 times the annihilation rate if the density in the universe were completely uniform
(δ = 0). This
2
ρ
δ2 =
−1
ρ̄
is the focus of this thesis. Here, ρ is the dark matter density at some point and ρ̄ is the average dark
matter density in the universe. Therefore, δ is the local dark matter overdensity. In the early universe,
this is typically described using linear perturbation theory. Nowadays, on scales smaller than a few
Megaparsecs, the perturbative approach breaks down [6].
Therefore, we will use an analytical model to describe the dark matter distribution today: the halo
model. In this model, dark matter is assumed to be contained in halos: spherical blobs of dark matter
that may host galaxy clusters, galaxies, or dwarf galaxies. The distribution of dark matter can then be
separated into a description of the distribution of these halos and a description of the internal structure
of the halos [6]. For the small scales we focus on here, the internal structure of the halos will be the
most important. In particular, these halos generally contain subhalos [7]. These are smaller halos that
have been absorbed by the main halo, but not (yet) torn apart. We will look at the halo model in more
detail in section 2.4.
For the average annihilation rate, what matters about the dark matter distribution is δ 2 : the
average squared overdensity in the universe. More generally, we can study hδ(z2 )δ(z1 − z2 )i, which is
averaged over z2 . This describes the correlation between overdensities at points separated by z1 . In
practice, we will study the (spatial) Fourier transform of this correlation function, the power spectrum,
as it is generally easier to calculate. It contains the same information, though: an indication of how
much structure there is in the dark matter distribution at a given scale.
In this highly non-linear regime, simulations are also used to study the evolution of dark (and baryonic) matter. It is difficult to study the smallest scales this way alone, though. The largest halos,
containing large galaxy clusters, have masses around 1015 M . Depending on the dark matter model,
6
the smallest have masses of no more than ∼ 106 M , though possibly as small as 10−12 M . This differs
by so many orders of magnitude, that it is so far impossible to simulate this full range of structure in a
statistically representative volume of the universe. Simulations of smaller volumes have also been done,
mainly for Milky-Way sized halos of ∼ 1012 M , but for smaller masses and scales as well [8]. Analytical
modelling is needed to combine and extrapolate these results, and has the significant advantage over
large simulations that it allows relatively easy exploration of parameter space for these models.
Ref. [1] uses a different approach to finding the power spectrum: the authors find it directly from
simulations of a representative volume of the universe (meaning that average quantities in the simulations
should accurately reproduce averages in the observable universe). These simulations resolved structures
to scales of ∼ 102 kpc. However, the minimum size of halos is much smaller. They therefore extrapolate
the power spectrum to smaller scales using a model based on the stable clustering hypothesis.
In the halo model, on the smaller scales we will focus on here, it becomes important to take into
account the fact that dark matter halos have substructure. In galaxy clusters, for example, the galaxies
are contained in subhalos of the cluster halo, and dwarf galaxies around the Milky Way are contained in
dark matter subhalos of the Milky Way halo. Substructure in halos has been studied before, but usually
focussing on a single halo. This has been done by e.g. refs. [9, 10, 8]. The reason for this is that many
dark matter searches focus on a single likely source of annihilation signals, such as dwarf galaxies or the
Galactic Centre. Ref. [7] describes how to include substructure in the power spectrum of a single halo.
Ref. [11] has included subhalos in the power spectrum calculation.
In this thesis, we will calculate and examine the dark matter power spectrum, focussing on small
scales, using the halo model. We will discuss the formalism in section 2.4 and appendix A, the model
parameters in section 3, and the results in section 4. We set up a framework for the calculation of this
power spectrum, and examine the effect that including subhalos has on expected annihilation signals.
Finally, we calculate the flux multiplier for four different models, and find that the flux multiplier δ 2 +1
depends strongly on which model is used.
7
2
The halo model
We will begin by reviewing the halo model in some detail. First we will review some basic cosmology.
Then we will look into the general idea of the halo model, with some discussion of its scope and limitations.
Next, we will discuss the formalism of the halo model under certain assumptions. Finally, we will discuss
some of the parameters in the halo model and what they mean.
2.1
Cosmology in a homogeneous, isotropic universe
To understand the evolution of matter overdensities, we must first look at the evolution of the average
matter density. We will therefore start with a general discussion of some of the cosmology we will need.
This section is intended as a reminder and as an introduction of notation. If you are new to the topic,
ref. [3] gives a more detailed and didactic discussion of cosmology in general. Refs. [12, 13] provide an
introduction to general relativity, and discusses homogeneous and isotropic universes in his chapter 8.
First, we will give an overview of a cosmological timeline, based on ref. [13]. Then we will discuss some
equations and notation.
Generally, the universe is expanding. This means that in the past, the universe was much denser and
hotter, meaning different energy scales determined what physical processes were happening. In the very,
very early universe (a tiny fraction of a second after it began), there was a period of rapid expansion
called inflation. This was driven by a quantum field, which has an uncertain density at any point. During
this expansion, the quantum fluctuations in this density were blown up to macroscopic scales. These
fluctuations form the basis for inhomogeneities in the universe today. Thus, the progenitors for today’s
inhomogeneities were generated stochastically by a quantum field.
About 300 seconds after the big bang, big bang nucleosynthesis occurred. Earlier, the number of
protons and neutrons had been kept in equilibrium through interactions with neutrinos and electrons.
When temperatures and densities had dropped too low, the neutrino interaction rates became negligible,
and the only remaining interaction was neutron decay to protons (and neutrinos). At these temperatures,
any forming nuclei were quickly destroyed by collisions. After some of the neutrons had decayed, the
temperature dropped far enough for forming nuclei to remain intact, before dropping too far for fusion
to occur at all. The abundances of the elements produced in this fusion, depend sensitively on the
baryon density. Therefore, measuring primordial element abundances gives us a good measurement of
the baryon density in the universe.
About 240 000 years later, recombination occurred. When the temperature became low enough,
ambient photons were no longer energetic enough to ionise neutral hydrogen atoms, and the hydrogen
nuclei and electrons combined to form neutral hydrogen. This is much less opaque than a plasma, and
the universe became more transparent. On average, photons from the early universe last scattered off
baryons 350 000 years after the big bang. These photons from the early universe make up the cosmic
microwave background (CMB). This is the earliest time we can see back to. Collaborations like those
for the Planck satellite have measured the CMB and its anisotropies [14]. These depend on various
cosmological parameters that we will discuss, and provide evidence that the initial density perturbations
are as predicted by inflation.
On large scales, galaxy surveys and CMB measurements show that our universe is homogeneous
(symmetric under spatial translations) and isotropic (symmetric under spatial rotations) [13]. In general
relativity, such a universe is described by the FLRW (Friedmann-Lemaı̂tre-Robertson-Walker) metric:
dr2
2
2
2
2
2
+ r dΩ .
ds = dt − a (t)
1 − κr2
Here, r is a radial coordinate and dΩ2 = dθ2 +sin2 θdφ2 , with θ and φ being spherical angular coordinates.
The time coordinate is t. The parameter κ indicates the curvature of the spatial surfaces. We have
used units where c = 1. For this choice of spatial slices, the metric is invariant under spatial shifts
and rotations. The universe is, however, expanding in the spatial directions, and the metric is timedependent. This expansion is parametrised by the Hubble parameter H = ȧ/a, where the dot denotes a
derivative with respect to t. The Hubble parameter today is called H0 . The scale factor a is normalised
such that a = 1 today.
8
An important consequence of the expansion of the universe is cosmological redshift: light waves
expand along with the universe, so that the ratio of emitted to received wavelengths is
a(tem )
1
λem
=
=
,
λrec
a(trec )
1+z
where z is the redshift. Today, z is equal to 0, whereas it was larger in the past. The larger the
redshift, the earlier the time. We will usually describe time using the redshift z. This redshift can be
measured observationally by comparing observed atomic emission and absorption lines to those measured
on Earth. Since the wavelengths of these lines are determined by atomic physics, they should be the
same at emission at any place or time in the universe. Comparing the observed wavelength of these lines
to the known emission (absorption) wavelength then allows us to calculate the redshift at which they
were emitted (absorbed). (Proper motions of sources relative to us mean that there is also a Doppler
shift involved, but at large distances, this effect is relatively small.) This means that the redshift z is a
measure of cosmological time/distance that we can observe.
The Friedmann equation describes the relation between the matter content of the universe and its
expansion:
H 2 (z) = H02 Ωr,0 (1 + z)4 + Ωm,0 (1 + z)3 + Ωk,0 (1 + z) + ΩΛ,0 .
Here, Ωi,0 = ρi,0 /ρcrit,0 , where ρi,0 is the current density of a component of the universe. The critical
density is ρcrit = 3H 2 /(8πG). If the universe has exactly this density, it is flat. (That is, the spatial slices
of the FLRW metric are flat. The space-time is not flat because of the expansion of the spatial slices.)
The subscript m refers to matter (baryonic + dark). More precisely, it describes all non-relativistic
particles. Λ indicates a cosmological constant, which is equivalent to a precisely uniform field, of which
the density does not change with expansion. (Note that its fractional density ΩΛ will change with
time.) The subscript r denotes radiation, which means all ultrarelativistic particles, including photons.
Ωk = −κ/H02 is a measure of curvature. In the Friedmann equation, this can be described as if it were a
matter component. It is also equal to Ωk = 1 − Ωr − Ωm − ΩΛ . The Ω parameters show how important a
component is for the expansion of the universe at a particular time. Today, Ωr is negligible compared to
Ωm and ΩΛ . The curvature Ωk is compatible with zero. Here, these Ω parameters are given a subscript
zero to explicitly indicate that they refer to the ratio of a component density to the critical density today.
In the more general definition, for a component i,
Ωi = ρ i
8πG
.
3H 2
The scaling of the different density components with 1 + z = a−1 is determined by their equations of
state: for a component with density ρi and pressure pi , related by pi = wi ρi , ρi ∝ a−3(1+wi ) . This is
why all the non-relativistic (w = 0) and ultrarelativistic (w = 13 ) particle species in the universe are
considered one component in the Friedmann equation.
We will use the ref. [14] results for cosmological parameters. These parameters are described in table 1.
The densities Ω are as previously described; the subscript b refers to ‘baryons’, which, in cosmology,
refers to atomic nuclei, electrons, and atoms. The parameter Ωc = Ωm − Ωb . In this model, Ωk =0.
The parameter h is a dimensionless characterisation of H0 , defined through H0 = h · 100 km s−1 Mpc−1 .
The parameters ns and keq will be explained in section 2.3; they are important in calculating the linear
power spectrum. The parameter σ8 will be explained in section 3.1; it determines the amplitude of the
power spectrum.
2.2
Fourier transformations and notation
As mentioned in the introduction, we will be working mostly in Fourier space. Here, I will briefly outline
the conventions we will use. What we are after is the average correlation between overdensities on various
scales:
ρ(z1 )
ρ(z2 )
hδ(z1 )δ(z2 )i =
−1
−1
.
ρ̄
ρ̄
We will use angle brackets hi to denote an average over the (observable) universe, or, since the initial overdensities are thought to have been generated stochastically from inflation, over an ensemble of universes
generated from the same cosmological parameters and statistical distribution of initial perturbations. We
9
Parameter
Ωc h2
Ωb h2
ns
h keq Mpc−1
σ8
Value
0.1186
0.02226
0.9677
0.678
0.01027
0.815
±0.0020
±0.00023
±0.0060
±0.009
±0.00014
±0.009
Table 1: The values of cosmological parameters used in this thesis, from ref. [14]. We include the 68%
confidence level values to illustrate the accuracy of these parameters, but do not include them in our
analysis.
define ρ̄ ≡ hρm i: the average density of matter in the universe. Since we ignore baryons in our analysis,
this is equal to the average dark matter density.
We will follow ref. [6] and use the Fourier convention
Z
A(x) =
d3 k
exp(ik · x)A(k).
(2π)3
Generally, we will use the same name for a function and its Fourier transform. This is in order to
simplify notation, as we will already be working with functions describing one or two different halos
and subhalos in a single equation. The vector x, and, later, vectors y and z, refer to coordinates on
the spatial slices of the FLRW metric, in a frame where the universe is isotropic and homogenous on
large scales. These are comoving coordinates: proper motions aside, particles keep the same comoving
coordinates as the universe expands. This means that their Fourier counterparts k are also comoving.
This allows for straightforward comparisons between scales of structures today, and the perturbations
in the early universe they originate from. Today, at a = 1, comoving coordinate distances correspond
to distances measured over our spatial slice. Note that for bound structures, such as dark matter halos,
their physical sizes do not change due to cosmological expansion, and their comoving sizes therefore do.
Earlier, we discussed that the universe is isotropic and homogeneous on large scales. We expect from
inflation that the initial perturbations were generated by a homogenous and isotropic quantum field. That
means that we expect the initial and present perturbations to be statistically isotropic and homogeneous.
This means that in an ensemble of observable universes, generated from the same inflationary quantum
field, and with the same cosmological parameters, any quantity averaged over the ensemble is symmetric
under spatial translations and rotations. More practically, it means that our statistical descriptions of
the dark matter distribution must be invariant under spatial translations and rotations. We will also
describe our halos and subhalos as spherically symmetrical. This means we will use a lot of functions
A(|x|), that only depend on the distances between points. For such functions defined in R3 ,
Z
A(k) =
d3 x exp(−ik · x)A(|x|)
R3
∞
Z
Z
π
Z
=
0
0
∞
2π
r2 dr sin(θ)dθdφA(r) exp(−ir|k| cos(θ))
0
θ=π
exp(−i|k|r cos(θ)) = 2π
r drA(r)
i|k|r
0
θ=0
Z ∞
exp(i|k|r)
−
exp(−i|k|r)
= 2π
r2 drA(r)
i|k|r
0
Z ∞
sin(|k|r)
=
4πr2 drA(r)
.
|k|r
0
Z
2
In the second line, we define spherical coordinates for x, such that k lies in the θ = 0 direction. For any
k, we can do this. We then see that the Fourier transform of a spherically symmetric function is also
spherically symmetric, and that the Fourier transform only involves one integral.
10
We will now apply this to the density correlations we want to consider: hδ(z1 )δ(z2 )i. We define ρ̄ to
be the average density over a spatial slice of the universe. Then
ρ(z1 ) − hρ(z1 )i ρ(z2 ) − hρ(z2 )i
hδ(z1 )δ(z2 )i =
hρ(z1 )i
hρ(z2 )i
1
= 2 hρ(z1 )ρ(z2 )i − ρ̄(hρ(z1 )i + hρ(z2 )i) + ρ̄2
ρ̄
hρ(z1 )ρ(z2 )i
−1
=
ρ̄2
from statistical homogeneity: hρ(z)i ≡ ρ̄, for all z. Furthermore, we note that statistical homogeneity
means than for any z1 , z2 , z3 ,
hδ(z1 )δ(z2 )i = hδ(z1 − z3 )δ(z2 − z3 i) = hδ(z1 − z2 )δ(0)i .
The first equality expresses invariance under spatial translations; the second expresses this relation for
the choice z3 = z2 . From rotational invariance, we also know that hδ(z1 − z2 )δ(0)i only depends on the
magnitude |z1 − z2 |. Therefore, we can describe this correlation between overdensities using a power
spectrum P (k) that only depends on k = |k|. Then we can write:
Z
Z
3
hδ(k1 )δ(k2 )i = d z1 d3 z2 hδ(z1 )δ(z2 )i exp(−i(k1 · z1 + k2 · z2 ))
Z
= d3 (z1 − z2 )
Z
d3 z2 hδ((z1 − z2 ) + z2 )δ(z2 )i exp(−i(k1 · (z1 − z2 ) + (k2 + k1 ) · z2 ))
Z
= d3 (z1 − z2 ) hδ(z1 − z2 )δ(0)i exp(−ik1 · (z1 − z2 ))
Z
d3 z2 exp(−i(k2 + k1 ) · z2 )
Z
= (2π)3 δD (k1 + k2 ) d3 (z1 − z2 ) hδ(z1 − z2 )δ(0)i exp(−i(k1 · (z1 − z2 )).
Here, δD is the Dirac delta function. We then follow ref. [6] in defining
hδ(k1 )δ(k2 )i = (2π)3 δD (k1 + k2 )P (|k1 |),
where P (k) is also equal to the Fourier transform of hδ(z)δ(0)i. This is the power spectrum we are after.
Finally, we define the dimensionless power spectrum
∆2 (k) ≡
k3
P (k).
2π 2
Plotting this will
show the features of the power spectrum clearly. It also relates back to the flux
multiplier 1 + δ 2 :
Z
2
δ = d log(k)∆2 (k).
Here, as in the rest of this thesis, log means the natural logarithm, and not log10 .
As the above derivation shows, we can find the power spectrum by tracking the evolution of overdensities in momentum space from their generation. This is what is used to find the linear power spectrum,
which we will discuss in the next section. In the halo model, we usually start with a description in
position space, which we will have to transform to momentum space.
11
Figure 1: The relative errors from interpolating the linear power spectrum calculated using the CAMB
transfer functions. Here, PC is the power spectrum calculated at http://hmf.icrar.org/hmf_finder/
form/create/, and PC,2 uses only every other point from this calculated set. The functions are compared
at the half of the points not used to find PC,2 .
2.3
The linear power spectrum
We have reviewed in section 2.1 how a perfectly homogeneous, isotropic universe evolves. However, our
universe is only homogeneous and isotropic when the density field is smoothed over very large scales.
In the early universe, however, for example around the time the cosmic microwave background was
produced, the universe was close to homogeneous and isotropic. Assuming some initial perturbations,
we can use linear perturbation theory to describe the universe up to those times, given cosmological
parameters such as those we just discussed.
We can extend linear perturbation theory to today as well. This will not accurately describe the
matter distribution on the scales we are interested in, but it will be the starting point for parts of our
description.
In this thesis, we will use the linear power spectrum calculated using http://hmf.icrar.org/hmf_
finder/form/create/ [15]. This website is aimed at calculating the halo mass function (which we will
discuss in section 3.1), but calculates the linear power spectrum to do so. In section 3.1, we will see why
that is necessary. This power spectrum is calculated at discrete points. To get a value for the power
spectrum at all scales k, we will use a cubic spline interpolation in log-log space. Specifically, we use
SciPy’s scipy.interpolate.InterpolatedUnivariateSpline.
To check the accuracy of this interpolation, we use the same spline interpolation on half the points,
and compare the interpolated values for the other half of the points to their calculated values. This
is shown in figure 1. This shows that the interpolation should be accurate to at least 1%, and should
mostly be accurate to a few promille. The errors seem to be largest where the power spectrum oscillates
most.
The linear power spectrum P (k) at z = 0 is generally described as a primordial power spectrum,
multiplied by a squared transfer function T (k):
Plin (k) = 2π 2 δh2
k ns
T (k)2 .
H 3+ns
0
c
For each k, T describes how overdensities at this scale have evolved since inflation. The δh is a normalisation factor; the power law index ns is set by inflation. In the linear power spectrum calculation by
12
Figure 2: The log-log derivative of the linear power spectrum using different calculations. Above a
threshold of about k = 3 Mpc−1 , the CAMB-calculated transfer function is extrapolated with a power
law. Above this k, the log-log derivative becomes constant. The small-scale approximation is calculated
using Tlarge , which is valid only for k 0.01 Mpc−1 . The BBKS transfer function approaches the smallscale approximation for large k, and matches the CAMB-calculated linear power spectrum at smaller
k.
ref. [15], the transfer function is calculated using the Code for Anisotropies in the Microwave Background
(CAMB) [16]. This code takes the effects of baryonic matter (i.e. matter coupling to photons ) and the
known neutrino species on the transfer function into account [17].
We will not go into the details of the CAMB calculations here. What we need to take into account,
however, is that this code by ref. [15] only calculates the transfer function from the CAMB code up to
5 hMpc−1 . For larger k (small scales), it extrapolates the transfer function using a power law. However,
we are mainly interested in parts of the power spectrum at these smaller scales.
For these small scales, ref. [3] describes an approximation:
Plin (k) =
2π 2 δh2
k ns
Tlarge (k)2 ,
H 3+ns
0
c
Tlarge (k) = 12
keq
k
2
log
k
8keq
.
This transfer function for large k, Tlarge , is calculated by considering how perturbations in the dark
matter (i.e. matter that interacts only through gravity) density behave in different eras. In the cosmology
described above, different density components dominate at different times. At the earliest times, radiation
was dominant. Since its density is proportional to a−4 , the matter component with density proportional
to a−3 overtook it. The constant-density cosmological constant is now starting to dominate. In these
different eras, perturbations grew at different rates. For a given scale k, what then matters is when
particles separated by that scale came into causal contact; the perturbations on a given scale can only
evolve if particles can interact over that scale. The scale keq gives a measure of this: overdensities of this
scale came into causal contact for the first time since inflation, when the matter and radiation densities
in the universe were equal. The approximation given here holds for k keq ; these overdensity modes
were in causal contact with themselves well before matter-radiation equality. According to ref. [14],
keq ≈ 0.01 Mpc−1 .
Figure 2 shows the log-log derivatives of the linear power spectra calculated using this approximation
and using the CAMB calculation. We use derivatives, because we have not yet chosen some normalisations. As this figure shows, a difficulty with this transfer function is that, unfortunately, it does not
13
quite match the CAMB-calculated linear power spectrum in its range of validity (below about 3 Mpc−1 ).
These derivatives clearly show where the power-law approximation from the CAMB calculation starts.
The BBKS (Bardeen, Bond, Kaiser, and Szalay) approximation will help. This was derived by ref. [18],
assuming dark matter is dominant over baryonic matter. This approximation for the transfer function,
TBBKS , as given by ref. [3], is:
TBBKS (x) =
−0.25
log(1 + 0.171x) ,
1 + 0.284x + (1.18x)2 + (0.399x)3 + (0.490x)4
0.171x
where x = k/keq and all matter is assumed to be dark. This is a fit to the transfer function over wider
range of scales k, and asymptotes to the transfer function Tlarge at large k. The derivative of the linear
power spectrum computed using TBBKS is also displayed in figure 2.
What we will therefore do, is use the CAMB-calculated linear power spectrum for small k and the
BBKS approximation for large k. We will ‘stitch together’ these two power spectrum calculations in
the region where they seem to agree. First, for the BBKS approximation, we set the normalisation
of the overall power spectrum such that it agrees with the CAMB calculation at 1 Mpc−1 . We then
transition linearly from using the CAMB-calculated linear power spectrum Plin, C to using the linear
power spectrum Plin, B from the BBKS approximation to the transfer function. We make this transition
between 1 Mpc−1 and 3 Mpc−1 using the weight function
wlin (k) =
k − 1 Mpc−1
.
3 Mpc−1 − 1 Mpc−1
We will then use the following formula for our linear power spectrum:


if k < 1 Mpc−1
Plin, C (k)
Plin (k) = wlin (k)Plin, B (k) + (1 − wlin (k)) Plin, C
if 1 Mpc−1 ≤ k ≤ 3 Mpc−1


Plin, B (k)
if k > 3 Mpc−1 .
The different linear power spectra discussed here are shown in figure 3. This figure displays the
dimensionless power spectrum
k 3 P (k)
∆2 (k) =
.
2π 2
It shows that the transition between the two different descriptions of the linear power spectrum in the
used model is reasonably smooth. The used function displays the baryon acoustic oscillations (the very
small wiggles around 0.1 Mpc−1 ) from the CAMB calculation, and has the expected behaviour on small
scales. The baryon acoustic oscillations are the result of density oscillations in the early universe. Note
that in this log-log plot, the used linear power spectrum and the small-scale approximation become
parallel at large k. This suggests that 1 Mpc−1 is not an appropriate place to normalise the small-scale
approximation: it may simply not be accurate at that scale. Avoiding these normalisation issues is why
we chose to compare derivates in figure 2 to find a reasonable description for the linear power spectrum
at all scales.
At late times (well after matter-radiation equality zeq and recombination), all overdensity modes grow
at the same rate. We can then parametrise the linear power spectrum at general late time Plin (k, z) as
Plin (k, z) = Plin (k, z = 0)G2 (z),
where G is a growth function: δ(k, z) = G(z)δ(k, 0) [19]. This approximation only works in the linear
regime, where δ 1. A good approximation to G(z), accurate to a few percent according to ref. [20], is:
G(z) =
5 Ωm (z)
1
4/7
2 1 + z Ωm (z) − ΩΛ (z) + 1 + 1 Ωm (z) 1 +
2
1
70 ΩΛ
.
This is necessary to generalise the work we will do for z = 0 to more general redshifts.
14
Figure 3: Different models for the dimensionless linear power spectrum as a function of comoving scale k.
The used linear power spectrum is equal to the CAMB-calculated spectrum at k < 1 Mpc−1 and equal to
the BBKS approximation at k > 3 Mpc−1 . The CAMB-calculated power spectrum and the small-scale
approximation are normalised such that they are equal to the CAMB power spectrum at k = 1 Mpc−1 .
The CAMB-calculated power spectrum is Plin, C , the BBKS approximation to the linear power spectrum
is Plin, B , and the small-scale approximation to the power spectrum was calculated using the transfer
function Tlarge .
15
2.4
The halo model formalism
Now we have discussed the linear power spectrum, we can begin to work on the actual, non-linear, power
spectrum P (k) that we observe today. Specifically, we are interested in the dark matter power spectrum.
In this non-linear regime, this can differ significantly from the baryon power spectrum. Galaxy rotation
curves, for example, show that the dark matter halos surrounding galaxies extend much further out than
their luminous matter [13].
Broadly speaking, observations and simulations show that the largest structures in the universe are
sheets, made up of filaments, that connect at nodes [6]. These nodes typically host galaxy clusters,
while galaxies tend to lie along the filaments and in the sheets. Simulations show dark matter halos
surrounding these galaxies and clusters, making up the bulk of the matter in these structures.
In the halo model, dark matter is assumed to be contained in spherical halos in virial equilibrium
[6]. The dark matter power spectrum can then be described by considering the distribution of the halos
(in these sheets, filaments, and nodes), and the internal structure of the halos. On the largest scales,
the distribution of the halos determines the power spectrum. This distribution follows the linear power
spectrum fairly closely. On smaller scales, the internal structure of the halos is most important. The
transition between these regimes occurs around 1–10 Mpc at z = 0.
The internal structure of a halo is modelled as a smooth matter distribution by ref. [6]. However, as
ref. [7] discusses, halos generally contain substructure as well. These subhalos also have internal structure
and a distribution in the halo.
To relate the power spectrum to the parameters of the halo model, we will therefore split the density
into terms for each subhalo and halo in the patch of universe we are considering. We will assume that
these halos can be described fully by their positions and masses. Adding in extra parameters will be
fairly straightforward. We will express δ(z1 )δ(z2 ) in terms of these halo and subhalo densities, and will
then consider its expectation value. This will show what parameters we need to describe the power
spectrum using the halo model, and how to combine them.
The description of the halo parameters will be a statistical one: we expect the initial perturbations
from inflation to have been generated stochastically. Therefore, when we measure cosmological parameters, that only tells us about the statistics of the initial perturbations that generated the structure we
see today.
In the formal derivation of the halo model formalism, we will follow the outline given by ref. [21].
First, consider a member of an ensemble of universes, or a realisation of a statistical distribution of
universes. In this, we consider a compact volume (e.g., an observable universe). This bit of universe
contains dark matter halos, which we can number.
A given halo i is described by a central coordinate xi , a mass Mi , and a density function ρh (x−xi | Mi ).
A subhalo ai of halo i is described by a central coordinate yai , a mass mai , and a density function (not
including the underlying halo density) ρs (x − yai , yai − xi | mai , Mi ), which may also depends on the
position of the subhalo in the parent halo yai − xi , and the parent halo mass Mi . We will use the indices
i, j, k, ... for halos and ai , bi , ci , ... for subhalos in halo i.
The density at a point z is then given by the sum of all halo and subhalo densities at that point:
X
X
ρ(z) =
ρh (z − xi | Mi ) +
ρs (z − yai , yai − xi | mai , Mi )
ai
i
Z
=
d3 x1
"
Z
dM1
X
δD (x1 − xi )δD (M1 − Mi ) ρh (z − x1 | M1 )
i
Z
+
3
d y1
#
Z
dm1
X
δD (y1 − yai )δD (m1 − mai )ρs (z − y1 , y1 − x1 | m1 , M1 ) .
ai
Here, the subscripts 1 and 2 of the integration variables are unrelated to the indices i, ai of the halos
and subhalos. This way of writing things seems more complicated, but it will prove convenient to have
the delta functions be only the parts of the equation that are stochastic.
16
What we are looking for is hδ(z1 )δ(z2 )i. This is
hδ(z1 )δ(z2 )i =
hρ(z1 )ρ(z2 )i
− 1,
ρ̄2
as shown in section 2.2. The brackets denote an average over an ensemble of universes, or, equivalently,
a set of statistically independent subsets of a universe.
To calculate this, note that the order of integrating and taking expectations can be switched (see
appendix A.1). We will assume that the halo and subhalo density profiles ρh and ρs are determined only
by the given parameters, i.e. they do not depend on the positions of other halos or subhalos. This ignores
tidal effects and halo mergers, which may have some effect on the power spectrum. Ref. [22] finds that
at z = 0, . 5% of halos with masses 1010.9 –1013 M are forming from mergers of halos of similar masses.
Roughly speaking, then, the error in the power spectrum from ignoring these mergers should be . 5%.
This is smaller than other model errors we will find in section 3. Similarly, subhalo mergers are ignored
here.
In keeping with this, we will also assume that the distribution of subhalos in a halo depends only on
the parameters of that halo and the subhalos. We will also assume that the subhalos are correlated with
each other only through the general subhalo distribution around the halo centre: if the subhalos tend to
clump together, that is, in effect, a level of structure between the halo and subhalo. We will not consider
substructure below the level of subhalos, but this sub-substructure does exist [23].
Mathematically, this means that any expectation values only ‘act on’ the Dirac δD -functions. We
can use this to define the functions that will describe the halo and subhalo shapes and distributions. We
begin with the quantities that only involve halos. For single halos, we define
*
+
X
dN
.
δD (x − xi )δD (M − Mi ) =
dM
i
dN
Here, N is the number density of halos. The halo mass function is dM
: the number density of halos of
a given mass. Indeed, integrating both sides over halo mass and space gives the number of halos in the
given volume and mass range. The halo mass function should not depend on position x, from statistical
homogeneity.
For the largest scales, what will matter is the distribution of the halos. This is encoded by a halo-halo
correlation function ξhh . This is defined through the equation
*
+
XX
δD (x1 − xi )δD (M1 − Mi )δD (x2 − xj )δD (M2 − Mj )
i
j,j6=i
dN dN
=
(1 + ξhh (x1 − x2 | M1 , M2 )).
dM1 dM2
In other words, ξhh describes how much more likely it is to find a halo of some mass M2 at position x2 , if
we know there is a halo of mass M1 at position x1 . In principle, ξhh may be negative, if, for example, x1
and x2 are close enough that the halos would overlap. We will not, however, model this halo exclusion.
In section 3.2, we will discuss what this function looks like. On large scales, so for small k, we will find
that ξhh (k | M1 , M2 ) ≈ b1 (M1 )b1 (M2 )Plin (k), where b1 is called the linear bias parameter.
2.4.1
Halo ensemble averages
For subhalos, matters are a bit more complicated, since each halo has its own distribution of subhalos.
For subhalos, we need to consider
*
+
X
X
δD (x − xi )δD (M − Mi )
δD (y − yai )δD (m − mai ) .
i
ai
Since this is a product of delta functions, it is only meaningful when integrated. Integrating over halo
masses M and positions x and subhalo masses and positions m, y gives the number of subhalos in the
given subhalo mass range with centres in the subhalo position range, which are enclosed in halos with
17
centres in the halo position range and masses in the halo mass range. Integrating over all halo masses
and centre positions gives
*
+
XX
δD (y − yai )δD (m − mai ) .
i
ai
This is similar to the halo mass function discussed above: it is the expected number of subhalos at mass
m and position y, or more precisely, it gives these expected numbers within some position and mass
range when integrated over m and y. This means that the expression above contains a subhalo mass
function analogous to the halo mass function, as well as a halo mass function.
P However, the expression cannot easily be split into halo and subhalo parts: the subhalo distribution
ai δD (y − yai )δD (m − mai ) depends on the halo index i: trying to separate out the halo and subhalo
sums into a product of the halo expectation and a conditional subhalo expectation would not work
with the sums. To get something workable, we use two kinds of expectation values: universe ensemble
averages as described above, denoted by h...i, and halo ensemble averages h...ihalo . This halo ensemble
average averages halo and subhalo properties in any given halo over all halos of the given mass, position,
and other halo properties like concentration (see section 3.3). In other words, it only averages over the
subhalo properties of each halo. Basically, it turns an explicit dependence on which halo a subhalo is in
(the index i) into a dependence on its properties (mass, etc.). On average, we expect this to be a good
description. In using these halo ensemble averages, we do ignore some differences between halos. Using
the halo ensemble average is, in practice, equivalent to assuming that the halo properties fully specify
the statistical properties of the subhalos.
The subhalo mass function will have to depend on the properties of the parent halo: given the range
of halo masses (at least ∼ 10 orders of magnitude), even for a modest subhalo mass fraction, the total
subhalo mass in a large halo will be greater than the total mass of a much smaller halo. Furthermore,
the subhalos should be contained within some boundary of the halo: the subhalo mass function will
contain a spatial dependence on the position relative to the halo centre. Ref. [24] finds from their
simulations, that the mass distribution of subhalos does not depend on position. We will therefore
sub
split the subhalo sescription into a subhalo mass function dn
dm that depends only on the halo mass,
a dimensionless subhalo position distribution usub , and a mass-position correlation function ξres, sh to
absorb any remaining correlation:
**
X
+
δD (x − xi )δD (M − Mi )
X
*
i
*
=
halo
*
X
=
δD (y − yai )δD (m − mai )
ai
i
X
i
δD (x − xi )δD (M − Mi )
+
+
X
+
δD (y − yai )δD (m − mai )
ai
halo
dnsub
(Mi )usub (y − xi | Mi )(1 + ξres, sh (y − xi , m, Mi ))
δD (x − xi )δD (M − Mi )
dm
+
dN dnsub
=
(M )usub (y − x | M )(1 + ξres, sh (y − x, m, M )).
dM dm
sub
There is some ambiguity in the normalisations of dn
dm , usub , and ξres, sh . Following e.g. ref. [23], we
dN dnsub
sub
will normalise dn
dm such that integrating dM dm (M ) over all halo and subhalo masses and positions,
gives the expected number of subhalos in that volume, and usub such that the integral over all subhalo
positions is 1 for each M . For a correlation function equal to zero, these normalisations are consistent
with the equation above. The correlation function ξres, sh should then ‘average out’ to zero in more
complicated cases.
Finally, we will need the subhalo mass fraction, fsub (M ), which simply describes what fraction of
a halo’s mass M is contained in subhalos. The rest of the halo mass, sometimes called the smooth
halo mass, follows the density profile ρh . Since the halo density profile ρh will be computed using the
total halo mass M , we note that the actual density of a halo i, with mass Mi and centre xi , at z, is
(1 − fsub (Mi ))ρh (z − xi | Mi ).
18
Figure 4: A sketch of origin of the different terms in the dark matter power spectrum. The black circles
represent halos, the light gray circles are subhalos. The relative sizes of the halos and subhalos and the
distances between them are not to scale.
2.4.2
Further definitions, assumptions, and results
We have just outlined the basic parameters of the halo model: the halo and subhalo mass functions,
density profiles, and distributions. All these parameters will be used to describe the dark matter power
spectrum. In section 3, we will describe the different models we use for these parameters.
For the subhalo distribution on larger scales, we note that using the halo ensemble average means
that subhalo masses and positions will only correlate with other halos and their subhalos through the
halo parameters. In other words, ξhh will describe the distribution of subhalos on large scales as well.
In this section, we will not describe the details of how to derive the power spectrum from this model.
We will simply present the results and describe what they mean. The (rather technical) calculation can
be found in appendix A. The basic idea of this calculation, however, is to split up the correlation between
the densities at two points into correlations between various halo and subhalo densities at those points.
Figure 4 illustrates this split. It shows the seven different terms in the power spectrum P (k).
When calculating the correlation between densities at two points, we consider the density at each
point to be the sum of the densities of all halos and subhalos that contribute at that point. For two
points, we then have three kinds of density combinations: halo-halo, halo-subhalo, and subhalo-subhalo.
Furthermore, we can consider whether those halos and subhalos are the same or not, or whether they
are contained in the same halo or not. That gives us the seven combinations in figure 4. We expect to
be able to group these combinations roughly into three sets. On the largest scales, the terms that will
matter most will be the correlations between (subhalos of) different halos. These should follow ξhh and
be roughly similar, since the subhalo properties are part of the internal structure of halos, which should
not matter on the largest scales.
On smaller scales, the halo structure should be most important. This does not only mean the ‘halo
structure’ term in figure 4. The correlation between subhalos in the same halo and the distribution of
subhalos in the halo are on the same scale as the (smooth) halo structure. However, ref. [25] shows that
the subhalos are not distributed in their parent halo just like dark matter particles in the smooth halo.
Subhalos are formed when smaller halos fall into a larger halo. Tidal forces strip material from these
subhalos, sometimes destroying them. This means that subhalos will be less likely to be found close to
the halo centre than dark matter particles in the smooth halo, because subhalos that come close to the
halo centre will not survive as long as their counterparts further out in the halo.
On even smaller scales, the internal structure of the subhalos will become important. Since most
subhalos are much smaller than their parent halos, their internal structure will contribute to the power
spectrum P (k) on smaller scales.
This split and grouping of terms will be an important ‘sanity check’ for the calculated halo model:
we can check whether the terms that should look alike, do look alike, and which terms are important on
which scales.
To find the power spectrum, we further assume that requiring a subhalo of a certain mass to be
19
contained in the halo, does not significantly affect the subhalo mass function for the other subhalos.
This approximation is discussed in more detail in appendices A.3.6 and A.4. Using halo and subhalo
expectation values, and the model parameters we described, we then find that
ρ̄2 P (k) =
Z
dN
ρh (k | M1 )ρh (−k | M1 )(1 − fsub (M1 ))2
dM1
dM1
Z
dN dN
+ dM1 dM2
ρh (k | M1 )ξhh (k | M1 , M2 )ρh (−k | M2 )
dM1 dM2
(1 − fsub (M1 ))(1 − fsub (M2 ))
Z
dN dnsub
+ 2 dM1 dm1
(M1 )ρs (k | m1 , M1 )usub (k | M1 )ρh (−k | M1 )(1 − fsub (M1 ))
dM1 dm1
Z
dN dN dnsub
(M1 )
+ 2 dM1 dM2 dm1
dM1 dM2 dm1
ρs (k | m1 , M1 )usub (k | M1 )ξhh (k | M1 , M2 )ρh (−k | M2 )(1 − fsub (M2 ))
Z
dN dnsub
+ dM1 dm1
(M1 )ρs (k | m1 , M1 )ρs (−k | m1 , M1 )
dM1 dm1
Z
dnsub
dN dnsub
(M1 )
(M1 )
+ dM1 dm1 dm2
dM1 dm1
dm2
ρs (k | m1 , M1 )usub (k | M1 )usub (−k | M1 )ρs (−k | m2 , M1 )
Z
dN dN dnsub
dnsub
+ dM1 dM2 dm2 dm1
(M1 )
(M2 )
dM1 dM2 dm1
dm2
ρs (k | m1 , M1 )usub (k | M1 )ξhh (k | M1 , M2 )usub (−k | M2 )ρs (−k | m2 , M2 ).
This expression suggests that we will have to integrate over 4 parameters to find some terms. However,
with the expression for ξhh that we found, we find that no more than 2 nested integrations are needed
in these terms:
ρ̄2 P (k) =
Z
dN
dM1
ρh (k | M1 )ρh (−k | M1 )(1 − fsub (M1 ))2
dM1
2
Z
dN
(1 − fsub (M1 ))ρh (k | M1 )b1 (M1 ) Plin (k)
+
dM1
dM1
Z
dN dnsub
+ 2 dM1 dm1
(M1 )ρs (k | m1 , M1 )usub (k | M1 )ρh (−k | M1 )(1 − fsub (M1 ))
dM1 dm1
Z
dN dnsub
(M1 )b1 (M1 )ρs (k | m1 , M1 )usub (k | M1 )
+ 2 dM1 dm1
dM1 dm1
Z
dN
dM2
b1 (M2 )ρh (−k | M2 )(1 − fsub (M2 ))Plin (k)
dM2
Z
dN dnsub
+ dM1 dm1
(M1 )ρs (k | m1 , M1 )ρs (−k | m1 , M1 )
dM1 dm1
Z
2
Z
dN
dnsub
+ dM1
dm1
(M1 )ρs (k | m1 , M1 )usub (k | M1 )
dM1
dm1
Z
2
dN dnsub
+
dM1 dm1
(M1 )ρs (k | m1 , M1 )usub (k | M1 )b1 (M1 ) Plin (k).
dM1 dm1
Here, the first two terms describe correlations between points in smooth halos, the second two terms
are for one point in a smooth halo and the other in a subhalo, and the last three terms are for both
points in subhalos. Comparing this to figure 4, the first term describes the smooth halo structure, and
the second term the smooth halo distribution. If subhalos are left out of the description, these two terms,
with fsub = 0, describe the entire power spectrum. The third term describes the correlation between
a subhalo and its parent halo, and the fourth describes the correlation between a subhalo and another
20
(non-parent) halo. The fifth term describes subhalo structure, the sixth term describes the correlation
between subhalos in the same halo, and the last term describes the correlation between subhalos of
different halos. Terms 2, 4, and 7 therefore describe the large-scale structure, and terms 1, 3, and 6
describe the structure of the whole halo.
Figure 5: The final calculated dimensionless power spectrum ∆2 (k). This plot is for the toy model we
will use, where subhalos have the same shape as halos of the same mass. The terms are grouped by
colour and line style, and the total power spectrum is shown.
We will discuss these terms and how they relate to each other in a lot of detail in section 4. However,
to illustrate these different terms and their groupings, we show a sneak preview of the power spectrum
in figure 5. Here, the large-scale structure terms are shown in red, the halo structure terms in blue, and
the subhalo structure term in green. The line style indicates what levels of structure are related. The
dash-dotted lines are for correlations between points in the smooth halos, the dashed lines have one point
in the smooth halo and one in a subhalo, and the solid lines (except the black one) have both points in
subhalos.
As expected, the large-scale structure terms are dominant on large scales, the halo structure terms
are dominant on intermediate scales, and the subhalo structure becomes important, alongside the halo
structure, on the smallest scales. The scales of the different features make sense: the halo structure
becomes dominant around the scale of the largest halos, k ∼ 1 Mpc−1 . The slope of the power spectrum decreases around a scale where the halo mass function changes its behaviour, k ∼ 10 Mpc−1 (see
section 3.1). The drop-off of the power spectrum at the smallest scales, k ∼ 108 Mpc−1 , happens at the
sizes of the smallest (sub)halos.
21
3
Parameters of the halo model
Now we have discussed the halo model in general, we will discuss the different ingredients needed to
find the power spectrum: the density profiles and distributions of the halos and subhalos. For the halos,
many of these parameters come from analytical models, fit to simulations. Other parameters have been
fit to simulations with a more general idea of what is reasonable.
We include a quick preview of the (fiducial) model we will use. We will explain the meaning of
these terms in the subsections that follow, but give an overview for future reference. Our standard mass
definition for halos is that their average density must be 200ρcrit , but we will need to convert between
mass definitions to include the results of authors using different definitions. For main/field/parent halos
(halos that are not subhalos of larger halos), we will use the Sheth-Tormen mass function [26], a very
slightly modified NFW density profile [27] (see section 3.3), and the mass-concentration relation from
ref. [8]. For subhalos, we will use the same density profile and mass-concentration relation, and we will
assume their distribution in their parent halo follows the parent halo NFW profile. For the subhalo
mass function, we use the ∝ m−1.9 relation from ref. [23], with 10% of the total halo mass contained in
subhalos with masses 10−5 –10−2 times the halo mass. We use a minimum (sub)halo mass of 10−4 M .
3.1
The halo mass function
The first parameter we will discuss, is the halo mass function. We will be using the Sheth-Tormen mass
function [26]. First we will discuss the mass function itself, then we will discuss its calculation. Finally,
we will remark on some parameters and definitions related to the mass function.
3.1.1
The Sheth-Tormen mass function
The Sheth-Tormen mass function is a type of mass function which can be related to the idea of excursion
set theory [28]. The idea behind it, is that the initial density perturbations are enough to predict which
regions will eventually collapse into halos. A model for the non-linear collapse of an isolated overdense
region (e.g. an overdense sphere) is traced back to early times, where its evolution is still well described
by linear perturbation theory. This means we can compare the early overdensity of this collapsing region
to overdensities in linear perturbation theory; in practice, linear perturbation theory, extrapolated to
z = 0, is used for this. Since the late-time linear structure growth is scale-independent, this is equivalent
to comparing the collapsing regions to earlier linear perturbations.
The formalism for this type of mass function was first introduced by ref. [29]. To calculate the mass
function at some time, they consider the maximum radius that an eventually collapsed halo has, before
it virialises. They find the mass function at a later time as a function of the average density and mean
square overdensity at an earlier time, and of the ratio between the maximum and virialised radii of the
(eventually) collapsed object. Later, ref. [28] derived the same mass function as ref. [29], by considering
early mean squared overdensities, smoothed over some scale. They find the mass function by considering
an object of some size (corresponding to the smoothing scale) to be collapsed, if the linear overdensity
in that region exceeds some critical overdensity for collapse. For the critical overdensity, they use the
value 1.686, which is the critical overdensity for an isolated overdense sphere in an Ωm = 1 universe.
Ref. [6] describes this collapse of a spherical region. In an Ωm = 1 universe, a spherically symmetric
region would have an overdensity of 1.686 according to linear theory, when it collapses according to
the analytical calculation of its evolution. Its collapse time and final radius are estimated using the
virial theorem: assuming that all particle motions are radial, all energy in the system is potential when
the physical size of the region is at its maximum. Using energy conservation, and the virial equation
−2 hKi = hU i, the radius of a virialised halo is then found to be half the maximum physical radius of
the object, and its average overdensity at virialisation is 18π 2 ≈ 178. The kinetic energy is K and the
potential energy is U . These quantities are averaged over large time scales. This is where the assumption
that halos are virialised, is important.
To calculate the halo mass function, we will therefore first consider the smoothed density perturbations. We can calculate the smoothed overdensity over a radius R through
Z
δ0,R (x) = d3 yδ0 (x − y)W (|y|, R) ,
22
where W is the smoothing function. The subscript 0 indicates that these are the overdensities predicted
by linear perturbation theory at z = 0, and not the actual overdensities at that time. Since hδ0,R i = 0,
what we need to consider is the average squared overdensity. Taking the average in Fourier space, this
convolution integral becomes a multiplication. Since
Z
d3 k
Plin (|k|) exp(ik · (z1 − z2 ))
hδ0 (z1 )δ0 (z2 )i =
(2π)3
and the smoothing function W does not depend on the points z1 , z2 under consideration, we can find
the mean squared smoothed density on scale R through [28]:
Z
k 3 Plin (k)
2
σ (R) = d log(k)
|W (kR)|2 .
2π 2
This is the inverse Fourier transform of the average squared overdensity in momentum space.
There are different options for the smoothing function W . Ref. [28] names a gaussian filter (with
standard deviation R), a momentum space tophat (1 for k < 1/R, 0 outside this sphere), and a position
space tophat:
(
3
R−3 if |x| ≤ R
W (x, R) = 4π
0
if |x| > R,
3
[sin x − x cos x] , x = |k|R.
x3
For the Sheth-Tormen mass function, the position spaceptophat is used [30]. The cosmological parameter
σ8 (table 1) also refers to this value: it is equal to σ 2 (8h−1 Mpc) [27]. Setting this parameter is
equivalent to setting the normalisation of the linear power spectrum.
Since we want the halo mass function, we will have to associate the smoothing scale R to a mass M .
To do this, we remember that in the early universe, overdensities were small. For some initial overdensity
δi , we can then reasonably approximate
W (k, R) =
M=
4π 3
4π 3
R ρ̄(1 + δi ) ≈
R ρ̄,
3
3
where the radius R is the initial comoving size of the (future) halo. Therefore, we define
1/3 !
3M
2
2
σ (M ) = σ R =
4π ρ̄
to measure the linear overdensity associated with a non-linearly collapsed structure of mass M . Since
the universe is homogeneous on large scales, σ 2 (M ) → 0 as M → ∞. It should decrease monotonically
with M , since, on average, smoothing the density fluctuations over larger areas should reduce deviations
from the average density.
Ref. [29] first used an overdensity at early times to find the non-linear z = 0 mass function. The form
of this mass function dN/dM , is
νf (ν)d log(ν) =
M 2 (M, z) dN
d log(M ),
ρ̄
dM
where ν(M, z) = δc2 (z)σ(M )−2 . This is the ratio of the (squared) critical linear overdensity for collapse
to a typical squared overdensity for mass M . Thus, ν measures how likely it is for structures of mass
M to have collapsed at redshift z. Since σ 2 decreases as M increases, ν increases as M increases. Some
sources use ν 0 (M, z) = δc (z)σ(M )−1 , and get an equivalent definition of the mass function.
The formula for δc includes the growth function G(z), thus allowing a calculation of the mass function
at general z. In an Ω = 1 universe, for spherical collapse, δc (0) = 1.686. Ref. [31] discusses the critical
density for spherical collapse in a flat (Ωk = 0) universe. Using the calculation described in its appendix,
I calculated the critical density and collapse redshift for various linear perturbations δ0 , then interpolated
them using the familiar cubic spline, to find δc (z). At large redshift, when Ωm (z) → 1, δc (z) approaches
the Ωm (z) ≡ 1 value of 1.686. We will use the value δc (0) = 1.676, calculated using Ωm,0 = 0.306.
23
Comparing this to the value found using a root-finder, the relative error in δc from numerically calculating
it, is . 10−5 , meaning that the relative error in ν from using this value is . 2 · 10−5 . For this value of
the critical density, we find a typical halo mass M∗ (z = 0) = 6.7 · 1012 M . The value M∗ (z) is defined
as the mass M , such that ν(M, z) = 0.
Ref. [26] introduced the Sheth-Tormen mass function
qν 1/2
exp(−qν/2),
2π
√ −1
where the normalisation A(p) = [1 + 2−p Γ(1/2 − p)/ π] . We use p = 0.3, and q = 0.75, following
ref. [6]. This form for the mass function is a generalisation of the Press-Schechter mass function [29],
which is of the same form, with p = 1/2 and q = 1. Figure 6a shows the Sheth-Tormen mass function. It
is monotonically decreasing in mass M : the smallest halos are the
most prevalent. However, as we will
show in section 4.1.3, the contribution of a halo of mass M to δ 2 is very roughly proportional to the
mass M . (There are other factors with a weaker mass-dependence.) In a logarithmic interval, which we
dN
dN
dN
dM = M 2 dM
d log M . This function M 2 dM
, shown
will typically consider, what matters is then M dM
13
15
in figure 6b, peaks roughly around M∗ , at ∼ 3 · 10 M . It decreases sharply above ∼ 10 M, but
drops off very weakly towards lower masses. That means that to calculate the flux multiplier δ 2 and
the associated power spectrum, we will need to take a large range of masses into account.
Ref. [26] uses q = 0.707; this factor is mainly determined by the number of massive halos in the
simulations the authors used [32]. It was found by fitting the modified Press-Schechter mass function
to mass functions measured from simulations for different cosmological parameters, with ν in the range
∼ 0.4–20. This corresponds roughly to a mass range of ∼ 4 · 1011 –5 · 1015 M using our model. Ref. [6]
(which uses q = 0.75) notes that the Sheth-Tormen mass function describes a fitting formula to these
simulations ‘very well’, and states that this fitting formula is accurate to ∼ 20% in a σ(M ) range
corresponding to 5 · 1010 –6 · 1015 M using our model.
Using q = 0.75 predicts about 1% more halos than q = 0.707 at small masses ( M∗ ); the halo mass
functions are equal at M∗ , and increase quite a lot at large masses: q = 0.75 predicts 14% less halos
at 1015 M , 46% less at 1016 M , and about 97% less at 1017 M . As discussed above, however, above
1015 M , q = 0.75 halos are also exceedingly rare, and do not matter much for the power spectrum.
Ref. [32] considered the critical density for collapse of elliptical overdensities as a function of the ellipse
parameters. The authors assume typical parameters for these shapes, and argue that the Sheth-Tormen
mass function with p = 0.3 and q = 0.707 matches the mass function found from excursion set theory,
using the elliptical collapse critical density. They estimate that their formulas are accurate to ∼ 10%.
They tested their models using Ωm = 1, white-noise initial condition simulations using 106 particles, and
probed the largest two orders of magnitude in halo mass. They note other initial power spectra produce
similar results.
Comparing the levels of accuracy described for this mass function to the largest difference between
the mass functions for the two values of q, we will simply use q = 0.75. We will consider the difference
between these mass functions at larger masses to be part of the model uncertainty in the mass function.
νf (ν) = A(p) 1 + (qν)−p
3.1.2
Calculating the mass function
Now that we have all the formulas, we want to actually calculate the mass halo function
we need to find
dN
dM .
Therefore,
M 2 dN
d log(ν)
= νf (ν)
ρ̄ dM
d log(M )
M dσ 2 (M )
= −νf (ν) 2
.
σ (M ) dM
This is positive, because dσ 2 (M )/dM < 0. To calculate the halo mass function, we will need νf (ν), and
σ 2 and its derivative. Calculating σ 2 involves an integral. We will have to evaluate it many times in our
calculations. It is not just required in the mass function, but also in the linear bias function (section 3.2).
Furthermore, we need its derivative. We will therefore calculate σ 2 at a number of fixed points R, and
use the same cubic spline interpolation as for the linear power spectrum to interpolate between them.
This will allow a faster calculation of σ 2 , and will allow us to estimate its derivative.
24
2
dN
, in units of Megaparsecs. This func(b) M178
dM178
tion drops off exponentially above M ∼ 1015 M , and
decreases slowly towards small masses. This estimates
the relative importance
of various mass scales in the
calculation of δ 2 .
dN
, in
(a) The Sheth-Tormen halo mass function dM
178
units of solar masses and Megaparsecs. This function
drops off exponentially above M ∼ 1015 M and resembles a power law at small masses.
Figure
2 6: The Sheth-Tormen mass function and the estimated contributions of different halo masses to
δ .
For this calculation, the smoothing function W provided a challenge. A Taylor expansion of sin and
cos shows that 3x−3 (sin x − x cos x) ≈ 1 − (1/10)x2 at small x. However, this requires terms ∝ x in
sin x and x cos x to cancel. For small enough x, this difference will be dominated by the truncation
error: in floating point calculations, computers only store a finite number of digits. Truncation error
is the error induced by ‘chopping off’ the digits too far to the right of the decimal (or rather, binary)
point. At small x, sin x − x cos x is therefore small, but essentially random, and gets magnified by
the x−3 factor. To avoid this problem, we use the Taylor expansion for the smoothing function when
x < exp(−5.835158611477012). This is the (numerically calculated) point where the Taylor expansion is
equal to the smoothing function. The smoothing function only differs from 1 by 8.5 · 10−7 at that point,
so we do not use any linear transition such as for the linear power spectrum here.
The scales we consider for the interpolation are the scales we will require. We use a maximum
mass of 1017 M , which corresponds to a radius of 85 Mpc. Our fiducial minimum mass is 10−4 M ,
corresponding to 8.5 · 10−6 Mpc. A mass of 10−9 M corresponds to 1.8 · 10−7 Mpc, and 10−12 M
corresponds to 1.8 · 10−8 Mpc.
To interpolate values for σ 2 , we use points from R = 10−12 Mpc to 103 Mpc, evenly spaced in log
space, at every integer power of 10. We use extra points at the ends of the interval, and interpolate σ 2
as a function of log10 (R/Mpc). Figure 7a shows the errors induced by the spline interpolation halfway
between the calculated points and at random points (uniform in log space). Below R ∼ 10 Mpc, the
relative errors are smaller than 10−4 , and roughly decreasing in R. At R = 562 Mpc, the relative error is
∼ 31, and the spline is clearly not a good approximation. Above ∼ 102 Mpc, the relative errors become
larger than 0.1.
This shows that in the mass range we consider, the errors induced by using a spline function to
calculate σ 2 (M ) are smaller than 1%, and mostly smaller than 10−4 . Note that halos with masses above
1015 M , corresponding to R = 18 Mpc, are very rare, as shown in figure 6a , so for integrals involving
the halo mass function, it is reasonable to assume the errors in σ 2 are . 10−4 .
Since we will also need dσ 2 /dM , we need to estimate the error in dσ 2 /dR as well. We note that
Z
k 3 Plin (k) dW 2 (kR)
dσ 2 (R)
= d log(k)
,
dR
2π 2
dR
25
(a) The relative error in calculating σ 2 from a spline in
2
log space. The values of σspl
were calculated using the
2
spline; the values of σ were found by direct calculation.
(b) The relative error in calculating dσ 2 /dR from the
σ 2 spline. The value ∆rel (dσ 2 (R)/dR) is the derivative
of σ 2 calculated using the spline, divided by the value
found from differentiating the smoothing function inside the integral for σ 2 , minus 1.
Figure 7: Relative errors from using a spline to find σ 2 and dσ 2 /dR. The black points are the ones used
to find the spline. The blue points are halfway between the points used in the calculation in log space;
the red points are randomly chosen, according to a uniform distribution of log10 (R/Mpc).
and
dW (x)
−9
1 2
= 4 sin x − x cos x − x sin x
dx
x
3
1
≈ − x for x 1.
5
We transition from using the analytical formula for dW (x)/dx to the Taylor approximation at x =
exp(−5.835158611477012), just as for W itself. Figure 7b shows that the relative errors in the derivative
are similar to those in σ 2 ; they are larger at small R, but generally smaller than 10−4 , except at
R & 10 Mpc.
To estimate the effect of the error in σ 2 on the halo mass function, we need to consider the derivative
dN
of νf (ν): the relative error in dM
due to errors in σ 2 , ignoring effects on its derivative, is
∆rel
dN
dM
d log(νf (ν))
= −∆rel (σ (M )) 1 +
d log(ν)
2
,
where ∆rel is the relative error. We then find
d log(νf (ν))
1
p
= (1 − qν) −
d log(ν)
2
1 + (qν)p
for the Sheth-Tormen mass function. This is decreasing in ν (for sufficiently large ν), so decreasing with
mass. Figure 8 shows this error multiplier Fν as a function of mass. It shows that for masses M < M∗ ,
the relative errors in σ 2 and in the halo mass function are roughly the same. At large masses, however,
the error multiplier becomes large, where the relative error in σ 2 is large as well. For 1017 M , the
relative error in the halo mass function then becomes ∼ 1. For 1016 M , it is much less, however, at
∼ 10−2 , and it is ∼ 10−3 for 1015 M . Again considering the rarity of the largest halos, we assume that
when integrating over the mass function, the large errors at the largest masses have little impact, and
that the relative error in the halo mass function from using an interpolation for σ 2 (R), is ∼ 10−3 .
Since errors in σ 2 and its derivative may easily be correlated, we simply add the relative errors from
these two calculations to estimate the relative error in the halo mass function from using interpolation
to calculate σ. For the maximum errors, σ 2 clearly dominates the error. Since the calculation of the
halo mass function seems to require no subtraction beyond what we have discussed in the smoothing
26
Figure 8: Fν (M ) = d log(νf (ν))/d log(ν) as a function of mass at z = 0. For ν = 1, so for M∗ , Fν =
−0.45. At large masses, it decreases quickly: Fν (1015 M ) = −3.3, Fν (1016 M ) = −12, Fν (1017 M ) =
−60.
function W and its derivative, we do not need to consider truncation errors (normally ∼ 10−15 ) further
dN
here. This means that, so far, the relative numerical error in the calculation of dM
is ∼ 10−3 .
A final factor in the numerical error is from Plin . Earlier, in section 2.3, we estimated errors from
interpolating the linear power spectrum were . 1%. Considering that calculating σ 2 (R) and its derivative
means integrating a integrand ∝ Plin , the error induced by this in σ 2 and its derivative is . 1% as well.
In section 3.5, we will examine a further source of error, but its contribution will be negligible. This 1%
error then dominates the approximation error in calculating the halo mass function.
This numerical error is dwarfed by the ∼ 20% accuracy estimate for νf (ν), meaning that in further
sections, when we consider the error possibly arising from errors in the halo mass function, we can
effectively ignore the ∼ 1% approximation error estimates, and consider only the model uncertainties.
3.1.3
A definition and two parameters
A technical point to consider, is that we need to define the mass of a halo. We will consider this matter
in more detail in sections 3.3 and 3.5. For now, we will simply give the definition used by ref. [26] for
the Sheth-Tormen mass function. The mass of a halo is defined to be the mass enclosed in a sphere, of
which the centre coincides with the halo centre, and of which the radius is chosen such that the average
density inside the sphere is equal to 178ρ̄ [30]. In an Ωm = 1 universe, this is the average density of a
spherical overdensity that has just virialised. (Ref. [6] shows a derivation of this.) We will call this mass
M178 , to distinguish it from differently defined masses we will encounter later. Similarly, we will call the
dN
, to make clear which mass definition this is valid for.
mass function dM
178
Finally, we consider minimum and maximum masses for the halos. I have mentioned minimum and
maximum masses, to determine at which masses the halo mass function must be accurate. The maximum
mass is required because it is numerically impossible to integrate to M = ∞. A change of variables could
solve this problem, but considering the mass function shown in figure 6, a simple cut-off at large mass
should work. We choose Mmax = 1017 M . The minimum mass is a physical parameter of the halo
model. It is the only parameter than we cannot reasonably assume to be independent of the particle
physics model for dark matter.
The minimum mass is determined by the Jeans mass or free-streaming scale of dark matter: if an
overdense region is too small, it will not be able to collapse due to the velocity dispersion of the dark
matter, or perturbations on those scales will have been smoothed out by free streaming [5]. Dark matter
masses predicted from particle physics models are roughly 10−4 –10−12 M , e.g. ref. [2, 33]. Ref. [5] points
out that if dark matter annihilation signals come from interactions of annihilation products with ambient
baryonic matter, the Jeans mass for baryons, ∼ 106 M or greater, may be the relevant minimum mass.
Even if the photons come more directly from annihilations, tidal disruption by these baryon condensations
may have destroyed the smaller dark matter halos. Ref. [2] points out that this destruction may be more
effective for subhalos. We will use Mmin = 10−4 M in our calculations.
27
3.2
The halo distribution and the linear bias parameter
In this section, we will relate the distribution of halos to the distribution of mass. Its end result will
be a relation ξhh (k | M1 , M2 ) = b1 (M1 , z)b2 (M2 , z)Plin (k), where the linear bias parameter b1 describes
the relation between mass and halo number overdensities on large scales. Ref. [26] not only proposed a
mass function; the authors also showed how to get the linear bias parameter from a mass function. The
derivation is also shown in ref. [6].
We model the distribution of halos as following linear perturbation theory. This is not as entirely
straightforward as it seems: the linear power spectrum we have found, describes correlations between
overdensities. That is not the same as the correlation between the centres of dark matter halos. On
large scales, however, ref. [26] discusses a way to find an approximation to the halo-halo correlation ξhh ,
given a mass function of the type described above, and the linear power spectrum.
For the first part, we will follow ref. [6]. We start by defining the halo overdensity:
δh =
N (M1 , z1 | M, V, z0 )
− 1.
n(M1 , z1 )V
Here, we are looking for halos of mass M1 at redshift z1 . Unlike in the rest of the thesis, N denotes the
average number (and not a number density) of halos in a cell of volume V , that has mass M at z0 . Here,
n denotes the average comoving number density of halos (and not subhalos) of mass M1 . We already
know what n looks like: it is the halo mass function discussed in section 3.1. We remember that the
overdensity is defined as M/V ≡ ρ̄(1 + δ), thus relating δ and δh .
Here, the volume V is comoving, and the cell is defined by the mass it contains. Since the mass
inside it is constant, at early times, its volume will have been M/ρ̄ = V (1 + δ): the comoving volume of an overdense clump of mass will shrink with time. We can then write N (M1 , z1 | M, V, z0 ) =
n(M1 , z1 | M, V, z0 )V (1 + δ). This relation uses the early time comoving volume V (1 + δ), and not just
V , because the halo distribution is set at early times by the density perturbations present then. The
number density in an average cell of volume V is determined by n(M1 , z1 ); because it has the average
density, its comoving volume is constant.
Ref. [28] showed that n(M1 , z1 | M, V, z0 ) can be related to a halo mass function of the form discussed
in section 3.1, by replacing δsc by δsc (z1 ) − δ0 (δ, z0 ), where δ0 is the overdensity predicted by linear
theory from an overdensity that has actually evolved to δ at redshift z0 . Furthermore, we need to replace
σ 2 (M1 ) by σ 2 (M1 ) − σ 2 (M ). When considering scales much larger than halo sizes, the cell mass M
will be much larger than any halo mass M1 . On the largest scales, overdensities are very small, and
σ 2 (M1 ) − σ 2 (M ) ≈ σ 2 (M1 ).
This means that
M12 n(M1 , z1 | M, V, z0 ) dM1
dν 0
= ν 0 f (ν 0 ) 0 ,
ρ̄
M1
ν
ν0 =
[δsc (z1 ) − δ0 (δ, z0 )]2
.
σ 2 (M1 ) − σ 2 (M )
This can be understood intuitively as follows. The formalism for finding the ‘normal’ halo mass function
considers how likely a region is to collapse, given the ratio of the required overdensity to a typical
overdensity on that scale. When the scale M is much larger than M1 , we may consider the small-scale
perturbations on scale M1 to be independent of the large-scale ones. (Indeed, in linear theory, the
different Fourier modes evolve independently.) The fluctuations in δ0 on small scales will then be of the
same amplitude in a large, dense cell as in an average one, but their value will be offset by δ0 (δ, z0 )
(where δ is the overdensity of the cell). Effectively, then, the dependence of the halo mass function on
the critical density will be the same as in an average cell, but with a lower critical density. As for σ 2 :
by considering a cell of mass M , and volume V , we have effectively fixed the fluctuation on large scales.
The density fluctuations left to consider are then σ 2 (M1 ) − σ 2 (M ); effectively, we only integrate the
linear density perturbations between the scale of the halos we want, and the largest scale available (the
cell scale).
We resume the derivation from ref. [6]. As mentioned, we can effectively ignore the effect on σ 2 , if the
cell size M is large enough. The linear overdensities of large cells are also typically small: δ0 (δ, z0 ) 1,
28
and δ 1 on large scales. Then
[δsc (z1 ) − δ0 (δ, z0 )]2
σ 2 (M1 ) − σ 2 (M )
[δsc (z1 ) − δ0 (δ, z0 )]2
≈
σ 2 (M1 )
2
δ (z1 )
δsc (z1 )
≈ 2sc
− 2δ0 (δ, z0 ) 2
σ (M1 )
σ (M1 )
dν(M1 , z1 )
= ν(M1 , z1 ) − δ0 (δ, z0 )
,
dδsc
ν0 =
to first order in δ0 , for large cells. Since δ0 is small, inserting this in the mass function gives a small
effect, again. Since using ν(M1 , z1 ) just gives the average number density of such halos:
dν(M1 , z1 ) ∂n(M1 , z1 )
dδsc
∂ν
dn(M1 , z1 )
= n(M1 , z1 ) − δ0 (δ, z0 )
.
dδsc
n(M1 , z1 | M, V, z0 ) ≈ n(M1 , z1 ) − δ0 (δ, z0 )
Going back to the equation we started with, this gives us
δh (M1 , z1 ) ≈ δ − (1 + δ)δ0 (δ, z0 )
d log(n(M1 , z1 ))
,
dδsc
which holds well for small overdensities and large cells. Unsurprisingly, δ = δ0 + O(δ02 ). On these large
scales, we can then use δ ≈ δ0 , since the universe is close to homogeneous on large scales, so on large
scales, overdensities are small. Then, to lowest order in δ:
d log(n(M1 , z1 ))
δh (M1 , z1 , δ) ≈ δ 1 −
≡ b1 (M1 , z1 )δ,
dδsc
where b1 is the linear bias parameter we wanted to find. We then note that
hδh (M1 , z1 , δ(x))δh (M2 , z1 , δ(y))i = b1 (M1 , z1 )b1 (M2 , z1 ) hδ(x)δ(y)i .
Fourier transforming the right-hand side of this equation, on scales where the density perturbations are
still well-described as linear, then gives b1 (M1 , z1 )b1 (M2 , z1 )Plin (k). This only holds if x and y are far
apart.
To relate this to the halo-halo correlation, we remind ourselves of its definition:
dN dN
(1 + ξhh (x1 − x2 | M1 , M2 ))
dM1 dM2
*
+
XX
=
δD (x1 − xi )δD (M1 − Mi )δD (x2 − xj )δD (M2 − Mj ) .
i
j,j6=i
P
dN
We further note that dM
= h i δD (x − xi )δD (M − Mi )i. On large scales, the requirement i 6= j is not
necessary: if x1 and y1 are far apart (in fact, strictly speaking, if they are not equal), the δD -function
requirements for xi and xj cannot be satisfied for i = j. (Note that this does not enforce halo exclusion:
for halos not to overlap, ξhh would have to be negative if the distance between x1 and x2 were too small.)
29
Therefore, on large scales (|x1 − x2 | large),
ξhh (x1 − x2 | M1 , M2 )
DP P
E
δ
(x
−
x
)δ
(M
−
M
)δ
(x
−
x
)δ
(M
−
M
)
1
i D
1
i D
2
j D
2
j
i
j D
E DP
E −1
= DP
i δD (x1 − xi )δD (M1 − Mi )
j δD (x2 − xj )δD (M2 − Mj )
−1 *
X
dN dN
dN
dN dN
=
−
δD (x1 − xi )δD (M1 − Mi )
dM1 dM2
dM1 dM2
dM
2
i
−
X
δD (x2 − xj )δD (M2 − Mj )
j
dN
dM1
+
+
XX
i
δD (x1 − xi )δD (M1 − Mi )δD (x2 − xj )δD (M2 − Mj )
j
!
−1 * X
dN dN
dN
=
δD (x1 − xi )δD (M1 − Mi ) −
dM1 dM2
dM1
i

+
X
dN 

δD (x2 − xj )δD (M2 − Mj ) −
dM
2
j
!
*
−1 X
dN
dN
δD (x1 − xi )δD (M1 − Mi ) −
=
dM1
dM1
i
+

−1 X
dN
dN


δD (x2 − xj )δD (M2 − Mj ) −
dM2
dM2
j
= hδh (M1 , z, δ(x1 ))δh (M2 , z, δ(x2 ))i .
dN dN
For the second equality, we have written dM
in three different ways, using its definition and the
1 dM2
linearity of averaging. We find that ξhh is the halo overdensity correlation we have been looking for. We
therefore find that, on large scales,
ξhh (k | M1 , M2 ) = b1 (M1 , z)b2 (M2 , z)Plin (k).
This leaves us with two more tasks: first, we need to find an expression for the linear bias parameter
b1 (M1 , z1 ) = 1−d log(n(M1 , z1 ))/dδsc for the Sheth-Tormen mass function. Secondly, we need to examine
the normalisation of the linear bias factor. This is because in the basic excursion set theory approach,
the halo mass function extends over all masses, whereas we have a minimum (sub)halo mass (and an
unimportant maximum mass).
To find d log(n(M1 , z1 ))/dδsc , we note that we have already found d log(n(M1 , z1 ))/d log(ν) for the
numerical error estimates in section 3.1.2. We can use this to find
d log(n(M1 , z1 ))
b1 (M1 , z) = 1 −
dδsc
d log(νf (ν)) d log(ν)
=1−
d log(ν)
dδsc
1
p
2
(1 − qν) −
=1−
p
2
1 + (qν)
δsc
qν − 1
2p/δsc
=1+
+
δsc
1 + (qν)p
for the Sheth-Tormen mass function, where p and q are the parameters as described in section 3.1.1.
Seemingly, we can now simply plug this into the formula we found for P (k) . However, as mentioned
before, there is a normalisation issue. The Sheth-Tormen mass function is normalised, such that
Z ∞
Z ∞
M dN
dν
=
dM = 1.
νf (ν)
ν
ρ̄ dM
0
0
30
This means that all mass is contained in halos, when the halo mass can take any (positive real) value.
This makes physical sense: in the extreme limit, a single dark matter particle would count as a halo if
it did not belong to any larger ones. Therefore, all mass would be contained in halos. When we do have
a minimum halo mass, we would expect the matter predicted to be in smaller halos to simply not have
collapsed. Therefore, in reality, not all mass is modelled as being part of a halo in these calculations.
However, for uncollapsed matter, we expect the overdensities
to be very small, and therefore, we may
reasonably not include them in the calculation of δ 2 .
The linear perturbations in the early universe should, however, describe all matter: Plin is based on
observed anisotropies in the cosmic microwave background, and linearly extrapolated using cosmological
parameters measured from this. To ‘fix’ our large-scale structure description, we will therefore compare
the halo model prediction of large-scale structure to the one we just described.
According to the halo model description, on very large scales, the halo distribution is the dominant
component in the power spectrum. Including subhalos, there are three terms in our power spectrum
model that describe this. They are the ones involving ξhh :
Z
dN dN
µh (k | M1 )ξhh (k | M1 , M2 )µh (−k | M2 )
Pls (k) = dM1 dM2
dM1 dM2
(1 − fsub (M1 ))(1 − fsub (M2 ))
Z
dN dN dnsub
+ 2 dM1 dM2 dm1
(M1 )
dM1 dM2 dm1
µs (k | m1 , M1 )usub (k | M1 )ξhh (k | M1 , M2 )µh (−k | M2 )(1 − fsub (M2 ))
Z
dnsub
dN dN dnsub
+ dM1 dM2 dm2 dm1
(M1 )
(M2 )
dM1 dM2 dm1
dm2
µs (k | m1 , M1 )usub (k | M1 )ξhh (k | M1 , M2 )usub (−k | M2 )µs (−k | m2 , M2 ).
Here, Pls is the part of the power spectrum that describes the halo distribution, and dominates at large
scales. Since the linear bias parameter does not depend on k, we can compare the descriptions on any
scale we want. We will find it useful to consider the k → 0 limit here. We have not yet discussed the halo
shape, but on the largest scales, we can assume it is roughly a real-space delta function. The Fourier
transform of this is 1. Indeed, we will see in section 3.3 that this is the large-scale limit of the density
profile we will use. The same applies to the subhalo distribution in the halos, and the subhalo density
31
profile. The normalisations remain the same. Then we get
Z
dN dN M1
M2
Pls (k) ≈ dM1 dM2
ξhh (k | M1 , M2 )
dM1 dM2 ρ̄
ρ̄
(1 − fsub (M1 ))(1 − fsub (M2 ))
Z
dN dN dnsub
(M1 )
+ 2 dM1 dM2 dm1
dM1 dM2 dm1
m1
M2
ξhh (k | M1 , M2 )
(1 − fsub (M2 ))
ρ̄
ρ̄
Z
dnsub
dN dN dnsub
(M1 )
(M2 )
+ dM1 dM2 dm2 dm1
dM1 dM2 dm1
dm2
m1
m2
ξhh (k | M1 , M2 )
ρ̄
ρ̄
Z
dN dN M1
M2
= dM1 dM2
ξhh (k | M1 , M2 )
dM1 dM2 ρ̄
ρ̄
(1 − fsub (M1 ))(1 − fsub (M2 ))
Z
dN dN
+ 2 dM1 dM2
dM1 dM2
M1
M2
fsub (M1 )ξhh (k | M1 , M2 )
(1 − fsub (M2 ))
ρ̄
ρ̄
Z
dN dN
+ dM1 dM2
dM1 dM2
M1
M2
fsub (M1 )ξhh (k | M1 , M2 )
fsub (M2 )
ρ̄
ρ̄
Z
M2
dN dN M1
ξhh (k | M1 , M2 )
= dM1 dM2
dM1 dM2 ρ̄
ρ̄
as k → 0. This is indeed what is to be expected on the largest scales: only the mass and distribution
of the halos matters, not their internal (sub)structure. However, we will not simply replace these terms
in the expression for the power spectrum by Plin : the linear power spectrum does not drop off at the
minimum mass for structures. (This is not because this cut-off does not physically exist at early times.
It is just not included in our model.) This cut-off is expected, and will occur in the power spectrum if
we include the halo density profiles in the description. Furthermore, comparing the power spectrum we
will find to the linear power spectrum will give us a way to check our model.
In stead, we now substitute our expression for ξhh on large scales, and find
!2
Z Mmax
M dN
Pls (k) = Plin (k)
dM
b1 (M, z)
as k → 0.
ρ̄ dM
Mmin
Since we expect Pls (k) → Plin (k) as k → 0, and the integral does not depend on k, we can fix the
normalisation
Z Mmax
M dN
dM
b1 (M, z) = 1.
ρ̄ dM
Mmin
This consistency relation is given by ref. [6], along with others for higher-order approximations for δh (δ).
The authors do not specify the integral bounds, however. We can infer this normalisation from the
definition of the linear bias parameter. Note that the mass overdensity contribution of halos of mass M
dN
is δh (M, z, δ) dM
M/ρ̄. We then find that
Z
Mmax
Mmin
M dN
dM
b1 (M, z) =
ρ̄ dM
Z
Mmax
dM
Mmin
Z Mmax
1
=
δ
=1
32
M dN δh (M, z, δ)
ρ̄ dM
δ
dM
Mmin
M dN
δh (M, z, δ)
ρ̄ dM
We do not need to consider higher-order contributions to δh , since this holds in the limit δ → 0 (large
scales, small overdensities), where higher-order terms approach 0, and the integral we are interested in,
is independent of δ. This means that, even if not all the mass is in halos, if the halos are the only part of
the matter contributing to the overdensity, then we should choose the normalisation of b1 to match the
integral constraint above. Therefore, to calculate the power spectrum, we will adjust the normalisation
of b1 to match this constraint. Essentially, this corrects for the fact that the mass overdensity is with
respect to the halo and uncollapsed mass, whereas the halo overdensity is only with respect to halos and
their mass.
We then find that the approximation Pls (k) = Plin (k) on large scales is consistent with our model for
the halo distribution. Unlike for the other components, we will not consider the numerical errors in this
model component. We will simply compare the large-scale terms to the linear power spectrum on large
scales; on smaller scales, their contribution to the total power spectrum will be very small, and errors in
the calculation of these terms will not matter.
3.3
The halo density profile
Now that we have described the masses and distributions of halos, we will consider the halo shape. This
means finding a function that describes the halo density as a function of position relative to its centre.
We will use the Navarro-Frenk-White (NFW) profile for this. We will first discuss this profile, and then
a minor modifiication we make to it. Finally, we will discuss some uncertainties in the modelling of the
halo shape.
3.3.1
The NFW profile and its Fourier transform
We will use the NFW profile to describe the density of the halos. This density is given by ref. [27] as
ρ(r|m) =
ρs
,
(r/rs )(1 + r/rs )2
where ρs is a scale density and rs is a scale radius; r is the distance to the halo center. This scale radius
marks the transition from the shallower ∝ r−1 inner slope of the profile to the steeper ∝ r−3 outer slope.
The density diverges to infinity at r = 0, but the mass enclosed in a sphere of radius rvir = crs converges:
c
3
m = 4πρs rs ln(1 + c) −
.
1+c
This equation does show that the enclosed mass diverges to infinity as r → ∞. Therefore, it is necessary
to cut off the halos at some radius rvir in order to assign some mass to them. Note that this cut-off is
something we impose. In reality, the density should drop off smoothly. We will examine the definition of
this mass and radius in some detail soon. The parameter c is called the concentration. For a fixed rvir
and mass, a larger concentration means a higher density at the center of the halo, and a smaller density
at the edges. We will examine the concentration function in more detail in section 3.4. Roughly, it varies
from a minimum of 5 for large halos, to ∼ 50 for halos of 10−4 M ; for smaller masses, the concentration
is even larger.
We will normalise this density profile for the mass, defining u(r | m) ≡ m−1 ρ(r | m). The Fourier
transform of this density profile is [6]
sin cx
u(x | m) = fu (c) sin(x)[Si((1 + c)x) − Si(x)] −
+ cos(x)[Ci((1 + c)x) − Ci(x)] ,
(1 + c)x
−1
Rx
R∞
−1
.
where x = krs , Si(x) = 0 t−1 sin t dt, Ci(x) = − x t−1 cos t dt, and fu (c) = ln(1 + c) − c(1 + c)
Figure 9 shows this profile in position space and momentum space for different concentrations. In
momentum space, this profile approaches 1 at krs 1; it drops off around krs = 1. The oscillations at
large krs will prove to be a problem. There are two relevant scales in this profile: the scale radius rs
and the cut-off radius rvir . The scale radius is a physical radius, corresponding to scale where the slope
of the density profile changes.
The virial radius has some physical meaning as well. The subscript ‘vir’ stands for ‘virial’; the dark
matter inside this radius is thought to be virialised, whereas the matter outside this radius is still falling
33
Figure 9: The cut-off NFW profile in position space and momentum space. All plotted quantities are
in units of the scale radius rs . In position space, r2 u(r | m) is plotted, since the density itself has a
singularity at r = 0. The Fourier transform of the normalised density profile is dimensionless. At large
krs , (starting at krs ∼ 107 ), the calculated Fourier transform of the NFW profile becomes dominated
by numerical noise. There, u(k, m) is about 10−15 , which is about the relative precision of the floating
point numbers used.
onto the halo. Ref. [34] showed that in an Einstein-de Sitter (i.e. Ωm = 1, ΩΛ = Ωk = 0) universe, the
boundary between these regions lies at about the radius r178 , which encloses a sphere of average density
178ρcrit . The authors did point out that an overdensity of e.g. 300 in stead of 178 also seems to work
fairly well.
This average overdensity of 18π 2 ≈ 178 corresponds to the predictions of a spherical collapse model
[34]. Ref. [6] shows a derivation of this prediction. The authors start with a slightly overdense spherical ‘tophat’ region in an Einstein-de Sitter universe. This sphere has a uniform (initially very small)
overdensity that cuts off sharply at its edge. There is then an analytical solution to the evolution of
such an overdensity, assuming that the mass enclosed by each spherical shell is constant. They find the
maximum (physical) radius of this shell, then assume that its corresponding radius at virialisation is
half that maximum radius. This relation comes from the virial theorem for Newtonian gravity (which
holds to good approximation in bound systems in cosmology), and the assumption that all motion is
this collapse is radial. They conclude that at virialisation, in an Einstein-de Sitter universe, the average
density of the object is 18π 2 ρcrit .
This relation holds at virialisation, but since infall of dark matter onto halos is ongoing, this should
hold today for the portion of the halo that is just virialised. This virialisation is not actually predicted
by the analytical description; this predicts collapse into a single point. That calculation assumes perfect
spherical symmetry and a constant enclosed mass for spherical shells, which does not occur in reality.
(Inner shells will have moved through each other and themselves and have started re-expanding as outer
shells fall in.)
Note that in an Einstein-de Sitter universe, the critical density ρcrit is equal to the average matter
density ρ̄. This is not true in our universe, where Ωm = 0.3. This means that there are two ways to
extrapolate this overdensity definition of the virial radius from an Einstein-de Sitter universe to our
universe. We can define the virial radius to enclose an average density of 18π 2 ρ̄ or of 18π 2 ρcrit , and
different authors choose different definitions. Furthermore, this 18π 2 is sometimes rounded off to 178,
and sometimes to 200.
For our models, which portion of a halo is virialised and which part is infalling matter does not matter
directly; what we want is the density of the matter as a function of position. However, this spherical
collapse and virialisation underlies the mass function (see section 3.1), so it does matter indirectly.
More practically, the virial radius defines a cut-off for the halo, which defines the mass an NFW halo.
Therefore, a definition of the virial radius is equivalent to a definition of the halo mass. We will find
different quantities we need are defined using different definitions of the halo mass. Using the NFW
profile, we can convert between these definitions. In section 3.5, we will explore this in more detail.
34
3.3.2
Smoothing out the Fourier transform
This sharp cut-off of the halo density will be something of a problem. To find the power spectrum, we
will integrate a product of functions, including the Fourier transform of the NFW profile, over mass.
Specifically, we will integrate over log(M/M ). The scale radius rs is dependent on the mass directly
and through the concentration parameter. Roughly, rs ∝ m1/3 . This means that numerically integrating
over log(M/M ) means that we sample u(x | c) roughly evenly in log space in x = krs . At large x,
however, the sharp cut-off causes oscillations in u. As we will shortly show, these oscillations have a
constant period of 2π/c in x. At large x, which will be for some halo masses at most k we consider, an
even sample of x in log space will mean that for large x, the phase of the oscillation is essentially random.
The effect of these oscillations will then roughly be to add noise to the function we want to integrate.
This will limit the precision to which we can calculate these integrals with a reasonable sampling of the
function.
Therefore, we will shortly examine these oscillations more closely, and show that their period corresponds to the virial radius and not the scale radius. This means that, for the halo density profile,
these oscillations are likely not physical. The sharp cut-off we impose to define a halo mass, does not
correspond to a sharp cut-off in the actual density profiles of halos. We will therefore seek to ‘smooth
out’ these oscillations. We will check whether or not this profile is reasonable by comparing its Fourier
transform back into position space to the NFW profile.
To investigate these oscillations, we consider the behaviour of u(x | c) at large x. To do this, we will
find approximations to the Si and Ci functions for large arguments. First, we note that the way these
functions appear in u is through
Z x(1+c)
sin t
dt
Si(x(1 + c)) − Si(x) =
t
x
and
Z
x(1+c)
Ci(x(1 + c)) − Ci(x) =
x
cos t
dt.
t
This means that for a fixed concentration, at large x, the sin(t) and cos(t) functions will oscillate many
times in the integration domain, whereas 1/t changes relatively slowly. This points us towards a useful
approximation strategy. For a function f , we will use f [1] to denote its integral function (and not, as is
often meant, its first derivative). Similarly, f [n] will mean the result of integrating a function n times.
Using integration by parts, we remark that
b Z b
Z b
f [n] (t)
f [n+1] (t) f [n+1] (t)
n! n+1 dt = n! n+1 −
−(n + 1)! n+2 dt.
t
t
t
a
a
a
Of course, we want to use this for f = sin or f = cos, of which the integral functions are easily found.
We then have, for any positive integer N :
x(1+c) Z
N
x(1+c)
X
sin[n] (t) sin[N ] (t)
Si(x(1 + c)) − Si(x) =
(n − 1)!
+
(N )! N +1 dt.
n
t
t
x
n=1
x
A similar equation holds for the Ci term. We note that the sum will not converge for general x and
c, since (n − 1)! grows more rapidly with n than tn . For large x, however, the approximation given for
large, but not too large N should be good, since the integral is bounded by
Z
Z
x(1+c)
sin[N +1] (t) x(1+c)
sin[N +1] (t) dt ≤
dt
N!
N! N +1
x
tN +1
t
x
Z x(1+c)
1
≤
N ! N +1 dt
t
x
x(1+c)
1 = − (N − 1)! N t x
(N − 1)!
1
=
1
−
.
xN
(1 + c)N
35
The same bounds apply when replacing sin by cos. Now, we consider the ‘trigonometric part’ of u(x | c).
−1
We define Gu (x | c) = u(x | c)(fu (c))−1 + (x(1 + c)) sin(cx), and consider what it looks like in this
approximation. We consider the terms of this function in this approximation Gu,n (x | c), which come
from taking the terms n from the Si and Ci approximations above.
For n ≡ 0 mod 4,
sin x(1 + c)
cos x(1 + c)
(n − 1)!
2
2
− sin (x) + sin(x)
+ − cos (x) + cos(x)
Gu,n (x | c) =
xn
(1 + c)n
(1 + c)n
(n − 1)!
cos cx
=
−1 +
.
xn
(1 + c)n
This follows from trigonometric identities. For n ≡ 2 mod 4, the term looks the same, but with the
opposite (overall) sign.
For n ≡ 1 mod 4,
cos x(1 + c)
sin x(1 + c)
(n − 1)!
sin(x)
cos(x)
−
sin(x)
−
cos(x)
sin(x)
+
cos(x)
Gu,n (x | c) =
xn
(1 + c)n
(1 + c)n
(n − 1)! sin cx
=
.
xn (1 + c)n
Again, for n ≡ 3 mod 4, the term has the same form, but with the opposite (overall) sign. The n = 1
−1
term will cancel against the −((1 + c)x) sin(cx) term in u(x | c). This means that the n = 2 term will
be the lowest-order term in the large x expansion of u(x | c). Expanding up to n = 4 then gives
1
6
cos cx
2 sin cx
cos cx
u(x | c) ≈ uL,o (x | c) = fu (c) 2 1 −
− 4 1−
+ 3
.
x
(1 + c)2
x (1 + c)3
x
(1 + c)4
This explains why the Fourier transform of the NFW appears to become numerically noisy when
u(x | c) ∼ 10−15 . The floating point numbers used, store the equivalent of about 15 digits in base 10.
Since the ∝ x−2 behaviour of u(x | c) arises from a cancellation of ∝ x−1 terms, the cancellation between
these two will become dominated by floating point round-off errors when u(x | c) ∼ 10−15 .
Figure 10 shows this approximation uL,o and the relative differences between this approximation and
u(x | c). From the maths we have done, we can see a few important things. First, we have confirmed
that the oscillations in u(x | c) at large k have a period of 2π/rvir . This means that these oscillations
do indeed correspond to the artificially sharp cut-off of the halo, and not to the physical transition
between inner and outer profile slopes. Therefore, trying to smooth out these oscillations seems to be
justified. Furthermore, we see that the amplitude of the oscillations decreases with concentration. This
makes sense, since a larger concentration means that the density at the sharp cut-off is smaller, and
the discontinuity in the density is therefore also smaller. The error bound on the approximation is the
smallest for small c. This is likely to do with the smaller integration interval this implies. Note that the
bound we derived on this approximation error is not very stringent.
To smooth out the oscillations, the difficulty will be greatest at the smallest concentrations: there,
the oscillations begin closer to x = 1, where the smoothed out behaviour of u(x | c) is not yet the ∝ x−2
decline of large x. For x ∼ 1, our error bound shows that including more terms probably will not solve
this issue. Therefore, it is clear that the oscillations begin at smaller x than for which we can reasonably
expect the approximation to be reasonable. Our goal, however, is not to remove these oscillations entirely;
it is to remove them at large x, where their period becomes small in log space.
Figures 10a and 10c show that the approximation uL,o to u is good to about a factor of 10−3 or better
beyond x ∼ 101.5 . The difference is systematic, but small. At larger x, the error seems to become larger
again. These errors seem fairly random, and are likely numerical errors. Our main interest, however, is
not in approximating u at large x, but in smoothing it out. Therefore, we define
6
1
uL (x | c) = fu (c) 2 − 4 .
x
x
This is just uL,o with the oscillating parts removed. Figures 10b and 10d show this approximation and
its relation to u. The amplitude of the oscillations in the relative difference matches the expectations
from the uL,o approximation, i.e. (1 + c)−2 .
36
(a) The Fourier transform of the cut-off NFW profile
u(x | m) (solid lines) and its small-scale approximation
uL,o (x | m) (dashed lines). Below the sharp downward
peak, the values of the approximation are negative.
(b) The Fourier transform of the cut-off NFW profile u(x | m) (solid lines) and its oscillation-free smallscale approximation uL (x | m) (dashed lines). Below the
sharp downward peak, the values of the approximation
are negative.
(c) The relative error in the approximation uL,o (x | m).
(d) The relative difference between the smoothed and
analytical NFW profiles in momentum space. The
range of x is the same as in figure 10c.
Figure 10: The approximation to the NFW profile at large krs for different concentrations, including
oscillations (uL,o ) and excluding the oscillations (uL ). As before, x = krs . In figure 10d, the oscillations
at large x are poorly resolved. However, their amplitude is clear.
37
(a) The position-space cut-off NFW profile for different
concentrations as a function of radius (solid lines) and
the numerical inverse Fourier transform of the exact
Fourier transform of the cut-off NFW profile u(k | m)
for the same concentrations (dashed lines).
(b) The position-space cut-off NFW profile for different concentrations as a function of radius (solid lines)
and the numerical inverse Fourier transform of the
smoothed Fourier transform of the cut-off NFW profile
uNFW (k | m) for the same concentrations (dashed lines).
(c) The estimated relative error in the numerical inverse
Fourier transform integral ∆ρ/ρ. This is plotted for the
inverse Fourier transform of the exact Fourier transform
of the cut-off NFW profile.
(d) The relative difference between the numerical inverse Fourier transform of the analytical Fourier transform of the NFW profile ρan and the numerical inverse
Fourier transform of the smoothed Fourier transform of
that profile ρsm .
Figure 11: Comparisons between the position space cut-off NFW profile, and the numerical inverse
Fourier transform of the analytical Fourier transform u(k | m) of the NFW profile and its smoothed
version uNFW (k | m). All mass-normalised densities and radii are in units of the scale radius rs . The
behaviour of the c = 10 and c = 20 profiles is similar to that of the c = 5 and c = 50 profiles, respectively.
These spectra and errors were left out for legibility.
Based on this, we choose to smooth the oscillations in u at large x by transitioning from using the
analytical function for the cut-off NFW profile u below x = 101.6 and the smoothed approximation uL
above x = 102.0 . Between those values, we will use a weighted average of the two functions, just as will
did in section 2.3 for the linear power spectrum. That is, we define


if x < 40.0
u(x | m)
uNFW (x | m) = wNFW (x)uL (x | m) + (1 − wNFW (x)) u(x | m)
if 40.0 ≤ x ≤ 100.0


uL (x | m)
if x > 100.0,
where once again, x = krs . The weight function wNFW = (x − 40.0)/60.0.
Finally, we want to compare uNFW to the cut-off NFW profile in position space. Figure 11 shows
the results of this. The numerical inverse Fourier transform of the exact Fourier transform of the NFW
profile is shown to indicate the precision of the numerical inverse Fourier transform. At concentration
c = 50, the noisy behaviour at large radii seems to set in at smaller densities than for c = 5. This is
likely because the inverse Fourier transformation integrals were done using twice as many points for the
larger concentration. This was in order to better calculate this density profile until its cut-off at a larger
38
radius r/rs .
Generally, figures 11a and 11b show that in position space, the smoothed Fourier transform of the
NFW profile follows the exact cut-off profile well, until the calculations become ‘noisy’ at large radii.
Figure 11d shows the estimated relative errors in calculating the inverse Fourier transform numerically.
These do become large where the numerical inverse transforms become noisy. These relative errors
agree well with the difference between the exact cut-off NFW profile and the numerical inverse Fourier
transform of the exact Fourier transform of this profile, where the exact profile is non-zero. This means
that the integration error estimates are reasonable in this case. Figure 11d shows the relative difference
between the inverse Fourier transformed analytical Fourier transform of the NFW profile and smoothed
Fourier transform of the NFW profile. The relative differences are similar to the numerical errors in
the inverse Fourier transform. This is also true at radii r/rs < 1, for which the difference cannot be
seen in these plots. Therefore, smoothing the Fourier transformed NFW profile at large krs seems to be
reasonable. We further note that these smoothed profiles do not get ‘noisy’ at at uNFW ∼ 10−15 , as the
analytical calculation of the density profile in momentum space does.
3.3.3
Other profiles
In the literature, other profiles than NFW are also used. The outer slope of the density profile, ∝ r−3 ,
is fairly certain [24]. However, there is uncertainty about the inner slope, which in the NFW profile is
∝ r−1 . Ref. [35] found that small halos, ∼ 10−6 M , are ‘cuspier’: their inner slopes seem to be steeper
then ∝ r−1 . Ref. [2] finds that including this effect, increases the subhalo annihilation rate in halos by
∼ 30%, for a minimum halo mass of 10−6 M .
In the Via Lactea simulations, ref. [24] finds inner slopes ∼ r−1.2 . These simulations do not include
any processes associated with baryons. Ref. [23] names some possible effects that baryons may have: gas
cooling and the dynamics of stars and galactic structures may increase or decrease the central density of
a halo, depending on which processes dominate. Ref. [36] points out that observed inner slopes of halos
hosting low-mass galaxies tend to be > −1, or even ∼ 0. The authors find from simulations that include
baryonic effects, that depending on that halo mass, the inner slope may be larger or smaller than −1.
Finally, ref. [37] finds that halos are generally not spherical; they tend to be prolate.
Here, we use the NFW profile, because it generally is a good fit to simulations [27, 6]. Spherical
symmetry keeps the number of integrations tractable, and the NFW profile has a conveniently analytical
Fourier transform.
3.4
The concentration function
To describe the halo density profile, we will also need to describe the concentration c of halos. This
concentration is largely determined by the masses of halos. We will describe this relation using the
Sánchez-Conde Prada (SCP) concentration function [8]. The mass definition used here is M200 , meaning
that the radius of the spherical halo is chosen, such that the average density of the halo is 200ρcrit .
The SCP concentration function is based on measured properties of simulated halos over a large
range of masses [8]. The authors used data from many different simulations. They had much data in
the 1010 –1015 h−1 M mass range, some data at ∼ 5 · 108 h−1 M and in the 106 –109 h−1 M range, and
some with masses 2 · 10−6 –4 · 10−4 M . Their figure 1 shows the concentration-mass relation they find,
and the simulations it is based on. Some of the z = 0 concentrations they find, have been extrapolated
from higher redshifts. They use different cosmological parameters in their model than we do, but show
that simulations using different cosmological parameters all fit their model to within about 1σ.
The SCP concentration-mass relation has an upturn at the largest masses, above ∼ 1015 M . The
authors propose that this is due to these largest halos not yet having fully collapsed and virialised, and
their outer parts therefore still being more extended. To find this relation, they found c as a function of
σ(M ). This is described in ref. [38], where the approach comes from. Broadly speaking, the motivation
for this parametrisation, is that σ(M ) determines at what redshift a halo forms, and characterises its
merger history, and this formation history determines its concentration.
Ref. [8] parametrises this model at z = 0 by the relation
5
X
i
M200
,
c200 (M200 , z = 0), =
Ci · log
h−1 M
i=0
39
Figure 12: The halo concentration as a function of mass according to ref. [8], using an average halo
density of 200ρcrit to define the halo mass.
where the subscript 200 indicates the mass/cut-off radius definition for which this description is valid.
The coefficients Ci are given by [37.5153, −1.5093, 1.636 · 10−2 , 3.66 · 10−4 , −2.89237 · 10−5 , 5.32 · 10−7 ].
Figure 12 shows this relation. They state that this parameterisation is accurate to 1% in the mass range
10−6 h−1 M < M200 < 1015 h−1 M . As before, we do not worry too much about possible errors at the
high-mass end. Our fiducial minimum mass is within the range where the parametrisation is accurate,
but we note that this parametrisation may be problematic at smaller minimum masses.
Figure 12 shows that in this model, c200 (M200 ) peaks at ∼ 10−9 M . This mass is outside the range
of the fitted data, and ref. [8] does not show or address this feature. Since maxima are a general feature
of polynomial fits, we conclude that certainly by this mass, the SCP concentration-mass relation is no
longer a good description. It should therefore not be used for the lower minimum halo masses considered
in the literature.
This function describes the mean halo concentration. Ref. [8] points out that a typical concentration
scatter is about 0.14 dex (so a factor of ∼ 1.4) at 1σ. A typical distribution for describing halo concentration scatter is a lognormal distribution, i.e. a normal distribution of log(c) [6]. Ref. [2] finds similar
concentration scatter distributions for subhalos, with the 1σ scatter being a factor of 1.2–1.5.
Ref. [6] investigated the effect of including halo concentration scatter on the power spectrum. They
find the effect depends on the wave number k, increasing towards smaller scales. For the smallest scales
they consider, 102 h/Mpc, the power spectrum increases by 5–25%, depending on the size of the scatter.
We will not include this scatter in our calculations, despite the fact that at smaller scales this may mean
a very real underestimation of the power spectrum.
We do not anticipate numerical errors as much of a problem here; there do not seem to be any major
cancellations involved in this calculation, as there are with the NFW profile. Estimating the impact
of inaccuracies in c200 on the final power spectrum will be difficult, since it is a parameter inside the
halo density profile,
of which the dependence on c200 is fairly complicated. We will use a more direct
calculation of δ 2 in section 4.1.3 to estimate the impact of model errors in c200 on the power spectrum.
3.5
Mass definitions and conversions
We have now discussed the parameters for modelling the main (not sub) halos in our model. One issue
here is the definition of the halo mass. The definitions we will use, all rely on fixing the average density
of halos, ρh . When the halo centre is found, the radius is then defined as the radius enclosing a region
of this average density, and the mass as the mass enclosed in this radius. This mass can be extracted
from simulations for comparison with theory. Converting between definitions using different ρh requires
assuming a density profile for the halos. This is done in position space, and we will simply use the
cut-off NFW profile from section 3.3 for this. The definitions we have seen are ρh = 200ρcrit for the
concentration-mass relation, and ρh = 178ρ̄ for the halo mass function, and the associated linear bias
parameter. For the subhalo mass function, we will need the mass, defined by ρh = 200ρ̄. The mass we
40
will integrate over to find P (k) will be ρh = 200ρcrit .
Therefore, we will need to convert the integration variable mass to the masses used to define these
functions. Furthermore, we will need the derivatives of these mass conversions, since the (sub)halo mass
functions are derivatives with respect to mass. Besides the mass, the concentration also depends on
the chosen mass definition: since c = rvir /rs , the definition of the virial radius directly impacts the
concentration of the halo. We will discuss how to convert between these mass definitions, and estimate
the errors these conversions and the approximations we use for them will cause.
3.5.1
Converting between mass definitions
To make these mass conversions, we will have to convert between two characterisations of halos: the first
uses the variables (c, M, ρh ), the last of which is typically implicit. These three variables are used in the
calculation of the power spectrum, but depend on the definition of the halo mass and virial radius. The
second set is (ρs , rs , rvir ). These are the variables that define the NFW profile. For the same halo, we
assume that only rvir depends on the mass definition. If the profiles are not exactly NFW (and generally,
they are, for example, to some degree elliptical [37]), the best-fitting rs , ρs may actually depend on
the choice of mass definition, so any mass conversions we apply, add some model uncertainty to our
calculations. We will label quantities defined for different average halo densities by A and B for now.
When converting between these sets of variables, the mass and virial radius are the easiest to relate:
rvir,A =
3MA
4πρh,A
1/3
.
Using the definition of the concentration, the scale radius then follows:
1/3
rs =
3MA
4πρh,A
4πρs rs3
ln(1 + cA ) −
c−1
A .
For ρs , we use an equation from ref. [6]:
MA =
cA
1 + cA
for a given halo. This then gives us
ρs =
−1
c3 ρh,A
c
log(1 + c) −
.
3
1+c
For completeness, we also give the inverse relations. We have used these to check the algebra in the
mass conversions, by converting back and forth from the two sets of variables, and comparing ρs and rs
from the initial and converted masses and concentrations. Here, the simplest one is the definition
cA =
rvir,A
.
rs
The mass is then also easily obtained:
MA = 4πρs rs3 ln(1 + rvir,A rs −1 ) −
rvir,A rs −1
.
1 + rvir,A rs −1
Finally, we can also reconstruct the average density:
−3
ρh,A = 3ρs rs3 rvir,A
ln(1 + rvir,A rs −1 ) −
rvir,A rs −1
.
1 + rvir,A rs −1
These equations, alone, are not enough to convert between mass definitions: they only work if both sets
of variables are using the same mass definition; specifically, we cannot simply swap rvir,A with rvir,B
without first calculating MB . We can, however, make good use of the equations for rs and ρs : since they
41
Figure 13: The mass ratios M200 /M178 = F200/178 and M200 /M200m = F200/200m as a function of
concentration c200 .
do not depend on the mass definition, we can set expressions for these variables equal to each other. For
rs , this yields
1/3
MA ρh,B
cA
=
.
cB
MB ρh,A
For ρs , using the definition of c, gives
ln(1 + cA ) −
MA
=
MB
ln(1 + cB ) −
cA
1+cA
cB
1+cB
.
This will allow us to convert between definitions, but it will not be possible analytically. We will find
an equation for MA /MB ≡ FA/B , which depends on the concentration cA , and the ratio of densities
ρh,A /ρh,B ≡ PA/B . Combing the above equations, we find that
FA/B
cA
log(1 + cA ) − 1+c
A
=
.
1/3
1/3 −1
PA/B FA/B
cA
−1
log 1 + PA/B FA/B
cA −
1/3
−1
1+ PA/B FA/B
cA
We can solve this equation numerically for different cA , and for the two values of PA/B we will need.
We will then interpolate the mass ratio for other concentrations using cubic splines. To make these
interpolations, we will use concentrations between 0 and 300. This is a wider range than we expect
for average halo concentrations, but should give better accuracy at the ends of the interval in which
concentrations actually lie. It will also make it easier to include concentration scatter, and larger subhalo
concentrations, if we want to expand the model. The spacing of the points varies, being smallest at the
low-concentration end, where the mass ratio changes fastest. Figure 13 shows the mass ratio as a function
of concentration for the two ρh ratios. Note that PA/B depends on redshift, because the ratio ρ̄/ρcrit
does, so when making these calculations for many different redshifts, this approach could take some
time. For this, it may be useful to use an approximation to this relation for general PA/B and c, given
by ref. [39].
From this, we get an approximation FA/B (cA , PA/B ). We will need this to convert from the M200
mass we use in the density profile and concentration function, to the M178 mass we need in the halo
mass function and linear bias parameter, and from the m200 subhalo mass to the m200m subhalo mass
needed for the subhalo mass function. (Here, the mass m200m is defined as the mass contained in a
radius enclosing a region of average density 200ρ̄.)
In the halo mass function, we do not just need to insert a converted mass. Since it is a derivative
42
Figure 14: The relative error ∆rel (FA/B ) in the mass ratio from the two mass definitions, from comparing the interpolated mass ratio to the directly solved mass ratio. The mass ratios are M200 /M178
and M200 /M200m . The errors are calculated at concentrations halfway between those used for the interpolation, and random concentrations, uniformly distributed in log space. The legend does not obscure
any plotted points. Because the interpolations for the two mass conversions were based on the same
concentrations, the halfway points for the M200 /M200m conversion often overlap those for M200 /M178 .
with respect to mass, we find that:
ρ̄
M178 (M200 ) dσ 2 (M178 (M200 ))
dN
=−
ν(M178 (M200 ))f (ν(M178 (M200 ))) 2
dM200
M200
σ (M178 (M200 ))
dM200
dM178
ρ̄ M178 (M200 ) dN
(M178 (M200 ))
,
=
M200
ρ̄
dM178
dM200
using the chain rule for the mass derivative. In other words, considering the definition of the mass
function in terms of νf (ν) and σ 2 , which were fitted to simulations using their definition from M178 , we
can get the halo mass function for M200 by ‘plugging in’ M178 as obtained from M200 by conversion, and
multiplying by dM178 /dM200 . To put it another way
fM (M178 ) =
M178 dN
,
ρ̄ dM178
dM178
M200 dN
M200
fM (M178 (M200 ))
=
.
ρ̄ dM200
M178 (M200 )
dM200
Therefore, we need to find a form for dM178 /dM200 . We find that
dM178
d
M200
=
dM200
dM200 F200/178 (c200 (M200 ))
=
dF200/178 (c200 (M200 ) dc200 (M200 )
M200
1
− 2
.
F200/178 (c200 (M200 )) F200/178 (c200 (M200 ))
dc200
dM200
Since we use an analytical fit for c200 (M200 ), its derivative is easily calculated analytically. To find
F200/178 (c200 ), we interpolate values, so its derivative can be estimated as the derivative of the interpolation. For the mass conversion between m200 and m200m for the subhalo mass function, similar equations
apply; only the labels differ.
The linear bias parameter was also derived using the M178 mass definition. The derivation of section 3.2 shows that this depends on a derivative with respect to δsc , but depends on the mass only
through its dependence on ν. We can therefore simply use b1 (M200 ) = b1 (M178 (M200 )): we convert the
mass to the appropriate definition, and simply use that mass to find the linear bias parameter.
3.5.2
Errors in the halo mass function
Estimating the errors induced by these conversions will prove to be somewhat complicated. For the
functions F200/178 (c200 ) and F200/200m (c200 ), this will be easiest. To find the points to interpolate, we
43
Figure 15: An estimate of the relative errors ∆rel in dF200/178 /dc200 and dF200/200m /dc200 . They are
plotted as a function of log10 (c200 ), but are not derivatives with respect to log10 (c200 ). The errors are
estimated by comparing the derivative of the used interpolation, to the derivative of an interpolation
between the same concentrations, and the concentrations halfway between those. The lines for the two
mass conversions largely overlap.
use the sage function sage.numerical.optimize.find root. To check the interpolation, we simply solve
the equation for F200/178 and F200/200m for values of c200 not used in the spline. The relative error in
the used mass M178 is then simply equal to the relative error in F200/178 (but with the opposite sign),
and similarly for m200m . We show these relative errors in figure 14. This shows that for concentrations
c200 > 2, the relative errors in these mass conversions are smaller than ∼ 10−5 for both mass conversions.
The trends in these errors may be somewhat odd, because the points between which the interpolation
was made, were chosen by hand, and are somewhat irregularly spaced.
The difficulty is in finding the relative error in the halo mass function from this. For the subhalo
mass function, we postpone this discussion until section 3.6, when we actually discuss this mass function.
Certain steps will be very similar, however, so some estimates we make here, will apply to the suhalo
mass function as well. To estimate these relative errors, we will find the derivatives of the mass function
with respect to mass numerically.
Error estimates for the derivatives dM178 /dM200 and dm200m /dm200 will be more difficult. Unlike
with dσ 2 /dR, we cannot easily compare the interpolation to a direct calculation of dF200/178 (c200 )/dc200 ,
and the error is not easily estimated from just the error in F200/178 . To get a rough idea, we will simply
compare the derivative of this function to the derivative obtained using twice as many points to find the
cubic spline. Figure 15 shows the estimated errors from this. The most difficult component is, however,
dc200 (M200 )/dM200 . We have an analytical fit for c200 (M200 ), but this is not a fit we have calculated.
Ref. [8] gives an accuracy for the analytical fit, but not for its derivative. Without the data this was fit
to, the relative error in the derivative simply cannot be reliably estimated.
To estimate the errors in dM178 /dM200 and dm200m /dm200 from approximations we make, we will
consider its two terms separately. For the first term error, figure 14 gives a good estimate. As remarked
before, the relative errors in this term are . 10−5 for concentrations c200 > 2. In the second term, the
−2
relative error in F200/178
(c200 (M200 )) is then . 2 · 10−5 . Figure 15 shows that the relative errors in the
derivative dF200/178 (c200 )/dc200 are larger: they are . 10−3 for 2 < c200 < 200. Therefore, the relative
errors in these derivatives are the dominant source of error we can check in the second term.
To combine these relative errors into a relative error for the derivative dM178 /dM200 , we consider
the relative sizes of these terms. First, we consider the part of this relation that depends on the halo
masses. For our mass-concentration relation, |M200 dc200 /dM200 | ≤ 1.5 for the masses at which we trust
this relation, and ≤ 2.5 for all masses 10−12 M < M200 < 1017 M . We then focus on the parts that
depend only on the concentration. Figure 16 shows the relative sizes of the two terms in dM178 /dM200
and dM200m /dM200 . The contribution of the second term without the mass-dependent part is ≤ 10%
for concentrations c200 > 2.5. For concentrations c200 > 4.5, which are the concentrations that we
find for our concentration-mass relation, its contribution is ≤ 5%. Including the factor ∼ 2 from the
44
Figure 16: The relative sizes of the two terms in dMB /dMA . The terms are T1,A/B = FA/B −1 and
T2,A/B = FA/B −2 dFA/B (cA )/dcA . The parts of dMB /dMA that depend on the mass-concentration
relation are not included. This ratio is plotted for the two mass conversions we use; the lines for these
two largely overlap.
mass-dependent part of the second term, in our model, the second term will contribute no more than
∼ 10% of dM178 /dM200 and dM200m /dM200 . (Note that in a more realistic subhalo mass-concentration
relation, subhalo concentrations would be larger than halo concentrations, so the contribution would be
smaller.) The relative error contributed by the second term to dM178 /dM200 and dM200m /dM200 is then
. 0.1 · 10−3 = 10−4 , and from the first term, . 10−5 . Then, we can approximate the relative errors in
both dM178 /dM200 and dM200m /dM200 as . 10−4 , for the mass-concentration relation we use. Errors
in this mass-concentration relation and its derivative are not included.
dN
induced by these errors in the mass conversion.
Finally, we want to estimate the relative error in dM
200
The final ingredient we need is then
d
M178 dN
d
log
log (fM (M178 ))
≡
d log(M178 )
ρ̄ dM178
d log(M178 )
log (fM ((1 + ∆)M178 )) − log (fM ((1 − ∆)M178 ))
≈ M178
2∆M178
log (fM ((1 + ∆)M178 )/ (fM ((1 − ∆)M178 )))
.
=
2∆
This equation is exactly true in the limit ∆ → 0. To calculate this quantity numerically, we must choose
some value of ∆ that is small, but not too small: for very small ∆, we will simply be measuring the
truncation error. In figure 17, this derivative is plotted for ∆ = 0.001. Using ∆ = 0.01 and ∆ = 0.0001
gave the same result. This logarithmic derivative is −2.3 for 1015 M , and drops very quickly at large
masses, reaching −8.1 at 1016 M and −46 at 1017 M . As before, we will base our error estimates on
halo masses below 1015 M , where 2.3 is an upper limit on (the absolute value of) this derivative.
We then find that the total maximum relative error from mass conversions on dMdN
is a combination
M
200
dN
of 10−5 for the mass ratio itself, 3 · 10−5 for the mass ratio error in the calculation of M178 /ρ̄ dM
, and
178
−4
−4
10 in the derivative dM178 /dM200 . This gives a total relative error of . 1.4 · 10 in the halo mass
dN
function dM
from numerical errors and approximations in the mass conversion. Combing this with
200
the estimated relative error of ∼ 10−2 from numerical errors and approximations in the calculation of
dN
dM178 (section 3.1.2), means that this previous error is dominant, and that the total relative accuracy of
dN
the halo mass function dM
is ∼ 10−2 .
200
We now have our model for halos and their distribution, as well as estimates for the numerical and
approximation errors in these quantities. Next, we will move on to the modelling of subhalos.
45
dN
.
Figure 17: The derivative d log (fM (M178 )) d log(M178 ), where fM (M178 ) = M178 /ρ̄ dM
178
3.6
Subhalo modelling
For the subhalos, we will mainly be using a toy model. We model the subhalo distribution in the parent
halo, usub (k | M ), as being the NFW profile (section 3.3) of the parent halo, with the same concentration.
The distribution of subhalo centres then follows the distribution of the mass in the main halo. Similarly,
we model subhalos as having the same NFW density profile as (parent) halos of that same mass, with the
same mass-concentration relation. Ref. [8] uses their main/field/parent halo c(m) relation to predict the
enhancement of halo annihilation rates from substructure. The only thing we cannot reasonably copy
from our model for halos, is the subhalo mass function. We will discuss how we model this mass function,
as well as name some better models for the model parameters we just discussed. We will also find an
approximation to the subhalo mass fraction, as this will simplify many of our calculations. Appendix A.4
shows where and why this is done.
3.6.1
The subhalo mass function
For the subhalo mass function, we will use a model from ref. [23]. This is based on simulations of MilkyWay scale (∼ 1012 M ) halos in the ‘Via Lactea’ I and II simulations. In these simulations, subhalos are
resolved down to masses of ∼ 106 M . Ref. [23] fits power laws to its measured subhalo mass functions.
For the cumulative mass function (nsub (< m)), the authors find a best fit slope of α = −2.0, but for the
differential mass function, they find best fits of α = −1.8–−1.9. We will use a slope of α = −1.9, but
note that different authors have found that choosing a slope of −2.0 or −1.9 makes a large difference in
the enhancement (boost) of annihilation signals from single (parent) halos. Ref. [23] finds an difference
in enhancement of a factor ∼ 2–8 for parent halos of 106 –1012 M . Ref. [2] finds similar differences
in enhancement, up to a factor of ∼ 10 for halos of 1015 M . These are both for minimum masses of
10−6 M . Ref. [9] finds enhancement differences of a factor ∼ 2–7 for parent halos of 106 –1015 M , and
a minimum mass of 10−6 M ; the authors used tidally stripped subhalos in their model. This choice of
slope will therefore probably impact the power spectrum we find significantly.
Following ref. [23], we normalise the subhalo mass function, such that 10% of the halo mass is
contained in subhalos with masses m in the range 10−5 < m/M < 10−2 , where M is the parent halo
mass. This is not the full range of subhalo masses in the halo: larger and smaller subhalos may exist. It
is simply the range we use to fix the normalisation of the mass function. This normalisation does not take
the (sub)halo minimum mass into account. For the smallest halos, we find the subhalo mass function as
if subhalos in the full mass range 10−5 < m/M < 10−2 existed, then cut it off at our minimum mass of
10−4 M . This means that the subhalo mass fraction in the smallest halos will be below 10%.
We also follow ref. [23] in cutting off the subhalo mass function at a maximum mass of 0.1M , where
M is the parent halo mass. For clear notation, we will call this maximum mass fraction fmax . The
authors assume that dynamical friction would cause these largest subhalos to fall to the halo centre and
be tidally destroyed. We note that with this subhalo mass function, there would be few halos with these
larger masses, even if the cut-off were larger. The minimum mass for subhalos is assumed to be the same
46
as for halos. However, as discussed in section 3.1.3, the minimum mass for subhalos, may, in reality, be
larger than that for main halos.
Our subhalo mass function then looks like
α
A(α)
m200m
dnsub
.
(M200m ) =
dm200m
M200m M200m
Here, m200m is the subhalo mass, M200m is the halo mass, and A(α) is a normalisation factor. This
normalisation factor can be found by
(
(α+2)·10%
if α 6= −2
−2 )α+2 −(10−5 )α+2
A(α) = (10 10%
if α = −2.
log(10−2 /10−5 )
Here, 10% should be read as 0.1; it is written this way to make clear that this is the fraction of mass in
subhalos in the chosen mass ratio range, and not the maximum mass fraction in a single subhalo.
Ref. [23] uses the masses M200m and m200m in its mass definitions: the mass and virial radius are
defined such that the mean density of the halo is 200ρ̄. (Our fiducial definition uses an average density
of 200ρcrit .) We will then have to convert between these mass definitions as explained in section 3.5.
This conversion can make up to a ∼ 20% difference in the subhalo mass function, which is largest for
large mass ratios m200m /M200m .
For the subhalo mass function, finding the error induced by errors in the halo and subhalo masses is
fortunately easy:
dnsub
(M200m ))
d log( dm
200m
d log(m200m )
= α ∼ −2,
dnsub
d log( dm
(M200m ))
200m
d log(M200m )
= −α − 1 ∼ 1,
so the . 10−5 relative errors in finding M200m from M200 (which works the same way for subhalo masses;
see section 3.5.2), cause . 3 · 10−5 relative errors in the subhalo mass function. The derivative correction
dm200m /dm200 brings in a relative error of . 10−4 , bringing the relative error from mass conversions to
. 2 · 10−4 for the subhalo mass function. We do not expect other sources of error in the calculation of
this function, although there may of course be errors in the model and its parameters. We are, after all,
extrapolating the subhalo mass function from simulations of ∼ 1012 M halos to our full halo mass range,
and to subhalos with masses ∼ 10 order of magnitude below what was resolved in these simulations.
3.6.2
The subhalo mass fraction
Now that we have the subhalo mass function, we can also calculate the subhalo mass fraction for a given
parent halo. Calculating this for some parent halo masses and interpolating using a cubic spline, will
mean that we do not have to integrate the subhalo mass function for power spectrum terms involving
only the smooth halo mass. The calculation is simple, but not analytical because of the mass conversions:
Z max(Mmin ,0.1·M200 )
1
dnsub
dm200
(M200 )m200 .
fsub (M200 ) =
M200 Mmin
dm200
This means that in our model, the smallest halos with M < 10−3 M contain no subhalos at all.
Figure 18a shows this subhalo mass fraction as a function of host halo mass.
Without the mass conversions, the subhalo mass fraction would be strictly increasing with mass,
although the increase would be fairly weak. However, the derivative correction dm200m /dm200 peaks with
the concentration, at ∼ 1015 M (but it is always positive). The ratios M200 /M200m and m200 /m200m
similarly reach a minimum around this same mass. These effects then conspire to cause the subhalo
mass fraction to decrease at the largest masses, even though the total subhalo mass is increasing.
As before, we will check the accuracy of this cubic spline interpolation by comparing the interpolated
values at some points to their directly calculated values. As before, we use points halfway between the
points the interpolation is based on, in log space, and random points that are uniformly distributed in
log space. Since we will, in practice, use 1 − fsub (M200 ) to calculate the power spectrum, this is what we
examine the error in. Figure 18b shows these estimated errors at points halfway (in log space) between
those used for the spline, and at random points. It shows that the relative errors are generally . 3 · 10−4 ,
and . 2 · 10−5 if we disregard the rare, highest-mass halos.
47
(b) The relative error ∆rel in 1 − fsub (M200 ) from using
an interpolation to find the subhalo mass fraction instead of calculating it directly. This error is evaluated
at points halfway between those used in the interpolation in log space (blue) and a random points, uniformly
distributed in log space (red).
(a) The subhalo mass fraction as a function of parent
halo mass M200 .
Figure 18: The subhalo mass fraction and its estimated errors.
3.6.3
Points on which to improve the subhalo model
As mentioned before, most of the parameters of our subhalo model are mainly chosen for an easy
comparison to the halo model without subhalos. This is very useful, since it will allow us to check
whether the outcome of these rather complicated calculations makes sense. Once we have a working setup, however, we will want to use more realistic models, to get a better estimate of the power spectrum
and the flux multiplier.
First, we note that the distribution of the subhalos does not necessarily follow the distribution of
the smooth halo mass precisely: closer to the halo centre, the tidal forces on subhalos are stronger, so
subhalos will not survive there as long as towards the halo edges. Ref. [25] analysed the Aquarius and
Via Lactea II simulations, with halo masses ∼ 1012 M and mass resolutions of 104.5 and 105 M . For
one simulation, the authors use a non-NFW (Einasto) profile for the main halo. For the subhalos, they
find that the mass distribution profile is more ‘cored’ than that of the main halo: the inner slope of the
profile approaches 0, and is significantly less steep than the total halo density profile toward the centre.
They find different profiles for the two simulations, as well as different subhalo mass function slopes
(α = −1.9 and α = −2.0).
Therefore, it seems clear that an NFW profile is not a very good description. This will mainly matter
for the terms in the power spectrum that measure the overdensity correlation between a halo and its
own subhalos, and between two subhalos of the same halo.
The main difference between our toy model and the results of simulations and analytical models,
however, is in the subhalo shape. Ref. [2] points out that, although subhalos are difficult to resolve in
simulations, we know something about them: subhalo density profiles are modified by tidal stripping.
This stripping causes infalling halos to lose their outer parts, so that subhalo densities drop steeply near
their edges, and their profiles are truncated at smaller radii than halos of the same (remaining) mass.
However, it is expected that an infalling halo’s central regions are largely unaffected by tidal stripping.
However, these regions are not resolved well in current simulations. Ref. [9] indicates that this results in
larger average squared overdensities in subhalos than in main halos of the same mass. This means that
a correct modelling of the subhalo shape, will mean a larger annihilation flux than a toy model assuming
halos and subhalos of the same mass have the same shape. Refs. [23] and [2] find that the subhalo
concentration depends on the distance to the halo centre. This is because tidal forces are stronger near
the centre, so tidal stripping will cause more mass loss there. Furthermore, ref. [37] points out that
subhalos are generally not spherical, but triaxial; subhalo inner regions are usually slightly less spherical
than outer parts.
48
Ref. [2] fits a model for subhalo concentration as a function of subhalo mass and distance to the halo
centre to ∼ 50 halos of ∼ 1012 M , with mass resolutions of 103.6 M , 104.3 M and 105 M . The authors
define concentrations using ρh = 200ρcrit , so we would not have to perform another mass conversion.
The formalism for including a position-dependent subhalo shape in discussed in appendix A.5.2. For the
smallest, innermost subhalos, they find concentrations up to almost 3 times as large as main halos of the
same mass. Typically, they find concentrations of subhalos are ∼ 1.5–2 times as large as those of samemass main halos. Their figures show that this ratio is indeed roughly valid for the SCP concentration
function that they use as a reference.
Ref. [2] finds that including their model for the subhalo concentration function, increases the annihilation signal from subhalos in parent halos by a factor of 2–3 relative to previous subhalo concentration
functions. Including tidal stripping effects reduces these boosts by ∼ 30%. Here, this means truncating
the subhalo density profiles at their tidally stripped radii, instead of the larger virial radius from the
mass definition. Including larger concentrations found for the smallest ∼ 10−6 M halos in different
simulations, increases the boosts by ∼ 30%.
Ref. [9] approached the same issue analytically. They model subhalos as the remnants of halos
with an NFW profile, following the halo mass-concentration relation, following simulation results. The
authors start with the subhalo’s mass and (tidally stripped) size. Then they use analytical models for
the infall times of halos that will become subhalos, and for their mass-loss rates. Given the original
mass of the now-sub-halo, they find its original NFW profile, given its current size and mass, and then
use a distribution function for the infalling halos to find a distribution for the subhalo masses and sizes.
They do not show the mass-concentration relation they find for subhalos, but they find flux multipliers
for single halos. The subhalo contributions to these are ∼ 2–5 times larger than those using the main
halo mass-concentration relation for subhalos, depending on the halo mass, subhalo mass function, and
minimum halo mass. This seems to be roughly consistent with the results of ref. [2].
Finally, we note that subhalos may also have substructure themselves: sub-subhalos. Ref. [9] find
that for any host halo mass, these never contribute more than ∼ 10% to the total annihilation flux. Tidal
stripping tends to remove most of the subhalos from an infalling halo, since subhalos are distributed more
towards the edges of the halo than smooth halo mass. Compared to other model uncertainties, this is a
fairly small effect, and we will further ignore it.
49
4
Results
Now that we have described our model, we will consider its results. First, we will analyse our sanity check
model for self-consistency and expected behaviour. Then, we will consider a somewhat more realistic
subhalo model, and look into what the effects of these changes in the model are. Throughout this section,
the model parameters are as described in section 3, unless stated otherwise. The halo masses we discuss
will be the masses M200 , as defined in section 3.5.
4.1
Sanity checks
In this section, we will analyse our toy model for the (non-linear) power spectrum with and without
subhalos. The main purpose of this, is to test the framework we have developed for finding this power
spectrum. We will compare these results to some in the literature. Since this is a toy model, and we have
often used a different parameter than other authors in some component of our model, these comparisons
will mostly be qualitative. However, as a test for our framework, such ‘sanity checks’ are useful.
This analysis will also be useful to gauge or predict the impact of certain parameter changes. Due to
time constraints, we will not be able to calculate the power spectrum for all the different mass functions,
mass-concentration relations, etc. found in the literature. However, we can sometimes get an idea of the
effects of changing these parameters.
We will often want to discuss different terms in the power spectrum, or different groups of terms. We
used descriptive terms in section 2.4.2, but to be entirely clear, we will introduce some notation:
Z
dN 2
µ (k | M1 )(1 − fsub (M1 ))2 ,
P1h (k) = dM1
dM1 h
Z
2
dN
P2h (k) =
dM1
(1 − fsub (M1 ))µh (k | M1 )b1 (M1 ) Plin (k),
dM1
Z
dN dnsub
Phs (k) = 2 dM1 dm1
(M1 )µs (k | m1 )usub (k | M1 )µh (k | M1 )(1 − fsub (M1 ))
dM1 dm1
Z
dN dnsub
Phos (k) = 2 dM1 dm1
(M1 )b1 (M1 )µs (k | m1 )usub (k | M1 )
dM1 dm1
Z
dN
dM2
b1 (M2 )µh (k | M2 )(1 − fsub (M2 ))Plin (k)
dM2
Z
dN dnsub
P1s (k) = dM1 dm1
(M1 )µ2s (k | m1 )
dM1 dm1
Z
2
Z
dN
dnsub
P2s (k) = dM1
dm1
(M1 )µs (k | m1 )usub (k | M1 )
dM1
dm1
2
Z
dN dnsub
(M1 )µs (k | m1 )usub (k | M1 )b1 (M1 ) Plin (k).
Psos (k) =
dM1 dm1
dM1 dm1
We have used µs/h = ρs/h /ρ̄. Here, we have written the terms of our equation from section 2.4.2 in a
way that makes use of the spherical symmetry of the halo and subhalo density profiles we use. The total
power spectrum is the sum of all these terms.
As a reminder, P1h describes the structure of the smooth component of the halo, Phs describes the
correlation between overdensities in a subhalo and the smooth component of its parent halo, and P2s
describes the correlation between overdensities in different subhalos of the same halo. In a similar vein,
P2h describes overdensity correlations between the smooth components of different halos, Phos describes
this correlation for the smooth component of one halo and a subhalo of another, and Psos describes the
correlation between overdensities of subhalos of different halos. Finally, P1s describes the structure of
subhalos.
We also introduce notation for the halo distribution and halo structure groups of terms:
Pdist = P2h + Phos + Psos ,
Pstruc = P1h + Phs + P2s .
The group of terms Pdist describes the large-scale structure, dominated by the correlation between
(parent) halos, and the group Pstruc describes the structure of the halos, dominated by the density
50
−1
Figure 19: The scales rs−1 (blue) and rvir
(red) associated with a halo of mass M , using the SCP
concentration function (section 3.4). The units are Mpc and M .
profile that, in our toy model, describes the smooth halo mass distribution, as well as the subhalo
distribution.
Furthermore, when we do not include subhalos in our model, the power spectrum is the sum of only
P1h and P2h , with the subhalo mass fraction fsub = 0. We will use the same notation for these terms
with and without subhalos, except for an added subscript ‘ns’ when we want to compare the two kinds
of models. The total power spectrum is Ptot .
4.1.1
The halo model without subhalos
First, we will examine the halo model without subhalos. This model involves fewer terms and parameters,
and can be compared to the halo model of ref. [6]. Since we do not have the authors’ data or programs, this
will be a visual comparison of plots. Quantitative comparisons will therefore be difficult, but qualitative
comparisons will be possible and useful.
A first parameter we check, is the maximum halo mass. I calculated P1h for the maximum mass of
1017 M we normally use, and for 1016 M . The effect of this change on P1h,ns is very small: changing
it from 1017 M to 1016 M , changes this term by less than 0.5%, and the differences appear to be
random errors. (Indeed, they fall within the estimated margin of error we will find for this term.) This
demonstrates that this choice of maximum halo mass was sound: it should simply be a finite bound for
integrations, not a physical parameter.
Now, we move on to investigating the terms of the power spectrum. It will prove to be useful to
associate a length scale k −1 to the main property of a halo: its mass. Figure 19 is included as a quick
reference for converting between a scale k and the mass of a halo M . As shown in section 3.3, the Fourier
transform of the mass-normalised NFW profile approaches 1 at large scales k rs−1 , and decays ∝ k −2
at small scales k rs−1 . Therefore, for a halo of mass M , the scale k ∼ rs−1 (M ) is the relevant physical
scale. When comparing a scale k in the power spectrum to a halo mass M , this mass is therefore not
necessarily the main contributing mass at that scale. It is, however, a rough upper limit to the halo
masses contributing to the power spectrum at that scale.
Figure 20 shows the halo model without subhalos. There are some clear features we can check here.
We will examine the scale at which the halo structure contribution becomes dominant, the largest scale
at which the derivative of this term changes quickly, and the cut-off in the power spectrum at large k.
We will also consider the asymptotic behaviour of the terms.
First, we will focus on the scales of the features. Figure 20 shows that the halo structure term becomes
relevant, then dominant, as we move from large to small scales (increasing k). The scales where this
happens, correspond to the sizes rs and rvir associated with the most massive halos in our model. This
makes sense: at the sizes of the largest halos, the halo structure should become relevant. In general, the
halo distribution term should be starting to shift towards anticorrelation of halos around these scales,
since halo exclusion becomes relevant here. However, we have not modelled this, and simply check that
51
(a) The dimensionless power spectrum without subhalos. Shown are the total power spectrum Ptot , with its
two terms, P1h and P2h , and the linear power spectrum
Plin .
(b) The fractional contributions P1h /Ptot and P2h /Ptot
to the total power spectrum. The ‘blocky’ appearance
is because the power spectra were calculated at a limited number of scales k.
Figure 20: The (non-linear) power spectrum, using the halo model without subhalos, and the relative
contributions of its two terms.
the 2h term in the power spectrum follows the linear power spectrum until scales small enough for the
halo structure term to become dominant.
To check this modelling of the halo structure term, we compare figures 20b and 21. This shows that
when the two terms in the subhalo-free power spectrum are equal, the halo distribution term is still
very close to the linear power spectrum. When the contribution of the halo structure term to the total
power spectrum reaches ∼ 90%, P2h ∼ 0.9Plin ; when P2h /Ptot ≈ 0.95, P2h ∼ 0.8Plin . All in all, we
therefore estimate that the error in the the total power spectrum from approximations we make in the
halo distribution term, is no more than ∼ 1%.
Next, we move on to smaller scales. Around k ∼ 10–100 Mpc−1 , the slope of the power spectrum
changes rather rapidly. These scales correspond to scale radii rs of halos with masses broadly around
M∗ (z = 0) = 6.7 · 1012 M , which is roughly where M 2 dN/dM peaks. The sharpest change in the power
spectrum seems to be at about the scale of M∗ . Specifically, this range in k corresponds to a mass
range of 1011.0 –1013.4 M . These are roughly the masses of the halos that stop contributing to the power
spectrum in this scale range. Since the power spectrum starts growing much less quickly around these
scales, the contributions of halos of these masses must have been important on larger scales. The reason
that ∆2 (k) increases even though less halos contribute to the power spectrum at smaller scales, is the
k 3 weighting in ∆2 (k) = k 3 (2π 2 )−1 P (k).
As we will show in section 4.1.3, the annihilation signal from a single halo is proportional to its mass.
Therefore, M dN/dM dM should roughly measure the importance of any halo mass range dM around M
in the power spectrum. However, the range of masses ‘dropping out’ of the power spectrum calculation
in some interval in d log k, depends on the mass contribution d log M . (As figure 19 shows, the relation
between rs and M is roughly a power law.) Since M dN/dM dM = M 2 dN/dM d log M , this means that
in figure 20a, M 2 dN/dM is what determines how sharply the power spectrum changes with log k.
Finally, the smallest-scale feature in the power spectrum is the drop-off at ∼ 108 Mpc−1 . This
corresponds to the scale radius rs of the smallest halos we include: those of mass 10−4 M . Indeed,
below this scale, there are no halos to contribute to the power spectrum, and a drop-off is to be expected.
Now, we will discuss the asymptotes of the terms in the power spectrum. We have already discussed
the halo distribution term on large scales, and we do not expect it to be accurate at small scales; there
we simply expect it to be negligible, and it is. For the halo structure term, the asymptotes are based on
mathematical considerations, more than on physical ones. Physically, we simply expect some drop-off
on small scales, and dominance of the halo distribution term on large scales. Mathematically, however,
we see that the only parts of the halo structure term that depend on k, are the density profile and the
factor k 3 that converts P tot ∆2 . Then, on the largest scales, where uN F W ≈ 1, the halo structure term
52
Figure 21: The ratio of the large-scale structure term in the non-linear power spectrum to the linear
power spectrum, for the halo model without subhalos.
should be proportional to k 3 . On the smallest scales, where k rs for all halo masses, uN F W ∝ k −2 ,
and ∆1h ∝ k −1 . Comparing the power spectrum to such power laws, shows that the halo structure term
indeed has the asymptotes we expect.
Therefore, the features and behaviour of the power spectrum without subhalos seem to make sense.
Comparing this power spectrum to the one in ref. [6] (figure 11), shows differences and similarities. The
linear power spectrum in this reference is already larger than the one we have found. (The authors use
different cosmological parameters than we do.) Correspondingly, the non-linear power spectrum found in
this model is larger. However, the shapes of the terms, and on what scales they dominate, are the same,
and the scales where the linear power spectrum and the halo distribution term diverge, are also very
similar. We then suspect that the power spectrum we have found for the halo model without subhalos
is reasonable, and that our framework for calculating these terms works.
4.1.2
Adding the subhalos
Now we have checked the power spectrum we calculated for halo model without subhalos, and have an
understanding of the features in these terms, we add the subhalos. As mentioned in section 2.4, we
expect the terms in Pdist together to follow P2h,ns , at least on large scales, since both should follow the
linear power spectrum there. Similarly, Pstruc should roughly follow P1h,ns . At first glance, the subhalo
structure term P1s should not necessarily follow any of the terms in the subhalo-free power spectrum.
However, we have modelled subhalos as copies of halos of the same mass. Therefore, the subhalo structure
term should be like the halo structure term P1h , but with a different mass function.
Figure 22 shows this power spectrum with subhalos, and its various terms. Figure 22a shows that
adding subhalos increases the total power spectrum, particularly on small scales. The grey line showing
Ptot,ns follows Ptot on large scales, and Pstruc on small scales. This small-scale behaviour comes from
the fact that, as figures 20b and 22d show, Ptot,ns ≈ P1h,ns on small scales, and P1h,ns and Pstruc are
similar. On large scales, subhalos are not very important, so adding them does not change the power
spectrum much, there.
Figure 22b shows that the different groups of terms are dominant where that would be expected:
the halo distribution determines the power spectrum on large scales. On intermediate scales, the halo
structure is most important, and on the smallest scales, the halo and subhalo structure both matter.
The halo structure never becomes entirely dominant here: it increases in importance around the scales
of the largest halos, but the largest subhalos are only about an order of magnitude smaller than that.
The halo distribution term is very similar to P2h,ns . This similarity on large scales means that, like
P2h,ns , this term follows the linear power spectrum on the scales where we expect it to. The simlarity
over a larger range of scales is also as expected: for the distribution of halos, the structure and position
of the subhalos should not matter much. Figure 22f shows that the difference between P2h,ns and Pdist
is largest at the scale of relatively small halos.
53
(a) The dimensionless power spectrum. Shown are the
total power spectrum Ptot , with its three (groups of)
terms Pdist , Pstruc , and P1s . The power spectrum without subhalos Ptot,ns is included for reference.
(b) The fractional contributions Pdist /Ptot , Pstruc /Ptot ,
and P1s /Ptot to the total power spectrum. The ratio
Ptot,ns /Ptot is also shown.
(c) The terms P1h , Phs , and P2s (blue) in Pstruc (black).
The term P1h,ns (gray) of the power spectrum without
subhalos is shown for comparison.
(d) The fractional contributions P1h /Pstruc , Phs /Pstruc ,
and P2s /Pstruc to the halo structure term. The ratio
P1h,ns /Pstruc is also shown.
(e) The terms P2h , Phos , and Psos (red) in Pdist (black).
The term P2h,ns is obscured by Pdist . The linear power
spectrum is shown for comparison.
(f) The fractional contributions P2h /Pdist , Phos /Pdist ,
and Psos /Pdist to the halo distribution term. The ratio
P2h,ns /Pdist is also shown.
Figure 22: The dimensionless (non-linear) power spectrum ∆2 , using the halo model with toy model
subhalos, its terms, and their relative sizes.
54
Compared to P1h,ns and P2h,ns , figures 22d and 22f show that Pstruc and Pdist tend to be somewhat
smaller. This makes sense physically and mathematically. Physically, in our toy model, the subhalo
centres are distributed the same way as the smooth halo mass. However, the subhalos are not point
masses. They have extended density profiles, so their effect on the halo structure and distribution in
this toy model, is simply that they ‘smooth out’ the density profiles of the halos. Smoothing decreases
density peaks, and therefore reduces the squared density found in the power spectrum. Since subhalos
are typically much smaller than their parent halos, the effect is not very large. On the largest scales,
the effect is small because the halo structure does not matter much. On the smallest scales, the halos
contain very few subhalos, so this effect becomes small, again.
Mathematically, in the halo structure and distribution terms with subhalos, we are effectively replacing factors M uNFW (k | M )fsub (M ) in the terms without substructure, by factors of
Z
dnsub
muNFW (k | m).
usub (k | M ) dm
dm
If uNFW (k | m) = 1, this means Pdist = P2h,ns and Pstruct = P1h,ns . Since uNFW (k | m) approaches 1
when k is much smaller than rs−1 (m), the terms are equal on scales much larger than subhalo sizes.
At larger k, however, uNFW (k | m) < 1 and the sums of terms involving subhalos are smaller. Since
most subhalos are much smaller than their parent halos, the difference is not very large: the density
profile of the parent halo will already be contributing fairly little to the power spectrum at scales small
enough for the subhalo density profile to depart from 1 significantly. On the smallest scales, the subhalo
mass fractions are smallest, so the contributions from the affected terms become very small. Then,
Pstruct /P1h,ns becomes closer to 1 again, and so does Pdist /P2h,ns .
Comparing the component ratios for the structure and distribution terms in figures 22d and 22f,
shows that they develop similarly for both. This is because scales in these groups of terms relate to the
halos that the subhalos are contained in. On the smallest scales, the halos contain little or no subhalos,
and the terms relating subhalos to other subhalos or their parent halos will be relative small. In larger
halos, the subhalo mass fraction is larger, and the subhalos become important components of the halos.
In both groups of terms, the ratio of the different terms is therefore determined by a weighted average
of the subhalo mass fractions of the halos that are still relevant on that scale. The weighting is different for
each term, so the term ratios are not exactly the same for the two groups of terms. On the largest scales,
all halos contribute to the power spectrum (uNFW (k | M ) ≈ 1 for all masses). At those large scales, the
ratios become constant. On the smallest scales, the density profiles for all masses are dropping off ∝ k −2 ,
but the terms involving subhalos have these profiles describing their density profiles and distributions.
Therefore, the relative contributions of the terms involving subhalos (other than the subhalo structure
term) approach zero at small scales. Since the subhalo mass fraction is small for the smallest halos, the
relative contributions of the terms involving subhalos are already close to zero there.
The subhalo mass fraction peaks at 0.28, for halo masses ∼ 1015 M . It is over 0.25 for a fairly
large range of large masses. Therefore, we may reasonably expect that on the largest scales, the relevant
weighted average of the subhalo mass fraction is ∼ 0.25. This predicts a relative contribution from
the terms involving two subhalos of ∼ 0.06, and of the halo and subhalo terms of ∼ 0.38. For the
halo distribution terms, at the largest scale calculated, these relative contributions are 0.06 and 0.37,
respectively, which matches these predictions very well. For the halo structure terms, these relative
contributions are, respectively, 0.07 and 0.39 on this largest scale.
This scale-dependence of the term ratios also points to features in figures 22c and 22e. Here, the
terms involving subhalos clearly start to ‘drop off’ on larger scales than the terms that only involve halos.
Much like P1h,ns , P1h ‘cuts off’ fairly sharply at a scale corresponding to the size of the smallest halo.
The scale relevant to the terms involving subhalos (apart from the subhalo structure term), is the scale
of the halo containing the subhalo, and the distance between (sub)halos in the halo distribution terms.
Since the subhalo mass fraction decreases gradually with halo mass, and is zero for the smallest halos,
these terms will not ‘cut off’ as sharply as P1h , and will drop off on larger scales.
We can also consider the asymptotes of the terms here. We have already investigated the halo
distribution terms. Since these terms involve the linear power spectrum, they will not simply asymptote
to power laws, at least on the scales we consider. For the other terms, however, we can consider their
calculation. Since the only position dependencies in our toy model follow NFW profiles, this check is
fairly easy. On large scales, krs 1 for the scale radii rs of all (sub)halo masses, uNFW (k | M ) ≈ 1 for all
halo and subhalo masses M . Then the only dependence of these structure terms (including the subhalo
55
(a) The ratio of the subhalo structure term P1s to the
subhalo-free halo structure term P1h,ns , and the ratio
of the subhalo and halo mass functions. The ratio of
the mass functions is evaluated at as mass m, such that
k = 1/rs (m).
(b) The ratio of the subhalo number density at mass m,
dN/dm, to the halo number density dN/dM at mass
M = m, as a function of the (sub)halo mass.
Figure 23: A comparison between the subhalo term P1s and the halo structure term P1h,ns , and the
mass functions that determine them.
structure term) on k, is the factor k 3 from converting from the power spectrum to the dimensionless
power spectrum. Indeed, on large scales, P1h , Phs , P2s , and P1s are all ∝ k 3 .
On small scales, the behaviour depends on the term. In the smooth halo and subhalo structure terms,
there are only two copies of uNFW . On small scales, these terms are then ∝ k 3 (k −2 )2 = k −1 . In Phs ,
there is also an NFW profile describing the subhalo distribution, and the small-scale behaviour should
be ∝ k −3 . In P2s , a second subhalo distribution is needed, and the P2s (k) ∝ k −5 on the smallest scales.
The terms in figure 22 indeed have this asymptotic behaviour.
The term we have not examined much yet, is the subhalo structure term P1s . As mentioned before,
the main way we can check that this term behaves as expected, is to compare it to the halo structure
term P1h,ns . We need the halo structure without subhalos, because our toy model subhalos have all
their mass in their smooth component, and we want the terms to differ only in their mass functions. We
compare these terms in figure 23. The subhalo mass function here is
Z
dN
dN dnsub
= dM
(M ),
dm
dM dm
not the subhalo mass function dn/dm(M ) of subhalos in a given halo. It counts the total number density
of subhalos at some mass, independent of their parent halos.
In figure 23b, a peak stands out. This peak in the ratio of the mass functions is affected by the
mass conversions, but these mainly change the magnitude of the ratio. Even if the mass conversions are
ignored, the peak is there, and at the same scale. It must therefore be due to the difference in shape
between the halo and subhalo mass functions, before mass conversions are applied.
The halo mass function is never truly a power law, even for small masses. This is most clearly seen
in figure 6b: in log-log space, the function is convex on small scales. Ignoring mass conversions, the
log-log derivative of dN/dM178 is equal to −1.90 at a mass of 108.6 M . At smaller masses, the slope is
steeper, reaching −1.96 at 10−4 M . The subhalo mass function, ignoring mass conversions, is simply
a power law with a slope of −1.90. Therefore, at small masses, there are two competing effects in the
ratio between the halo and subhalo number densities. At small masses, the subhalo mass function in any
halo increases less strongly with decreasing mass than the halo mass function. On the other hand, the
subhalo number density not only increases with decreasing mass because it increases in every halo, but
also because at smaller subhalo masses, the number of halos that can contain those subhalos increases.
This will boost the increase of the subhalo mass function with decreasing mass. These competing effects
56
seem to cause the peak in the figure, with the additional parent halos for subhalos being the dominant
effect at large masses, and the steeper halo mass function having the larger effect on small scales.
In broad terms, the dominance of the effects at large and small scales makes sense. At the smallest
subhalo masses, the effect of having more available parent halos will be relatively small, since the relative
increase in available parent halos is smaller. At these scales, the halo mass function is also steeper, so
the difference in logarithmic slopes is largest there.
To compare this mass ratio to the ratio of P1s to P1h,ns , we examine figure 23a. Bear in mind that
the masses that contribute to a power spectrum at some scale, are mostly all halos smaller than that
scale. We then expect the mass function ratio for masses smaller than a scale to roughly determine the
ratio of the terms at that scale. This is because the mass function determines how much the number of
contributing halos changes at that mass, and the ratio of the terms is more or less a weighted average of
this halo number density ratio at smaller masses. We see that the ratio of the terms P1s /P1h,ns mostly
trails the increase in the ratio of halo mass functions, but its peak is less extreme than that of the
mass function, and occurs at a slightly smaller scale. This is not consistent with the ratio of the power
spectrum terms behaving like a weighted average of the ratio of the mass functions over only larger
scales: then P1s /P1h,ns would follow the mass function ratio from right to left, averaging out over larger
k. However, larger (sub)halos than at scale k still contribute somewhat to P1s /P1h,ns . In that case, the
weighted average is over scales around and below the scale k we are considering. Then, the two curves
should still be similar, and the peak of P1s /P1h,ns less extreme, but which function follows which is no
longer clearly predicted. It therefore seems that the subhalo structure term P1s behaves in a way that
makes sense.
The log-log convexness of the halo mass function on small scales has another effect. Without subhalos,
the halo model predicts a dimensionless power spectrum that is slightly convex between the start of the
dominance of the halo structure term on large scales, and the beginning of its decay at small scales.
Even if a power law or constant concentration function is used for these halos, this convexness remains.
The rise of the subhalo structure term on smaller scales than the halo structure terms, and the overall
concaveness of the subhalo structure term, mean that when subhalos are included, the power spectrum
becomes concave in log-log space.
The behaviour of the terms in broad strokes, and the ratios of the terms in Pstruc and Pdist , are
predictions made before calculating the power spectrum. The rest is mostly analysis after the power
spectra were found. For a sanity check, I consider this to acceptable. The goal, after all, is to check if
features in the calculated terms may be due to errors in the calculation framework. We find that, so far,
our model and calculation method seem sane.
4.1.3
Checking the flux mulitplier
R
A final sanity check on the model we have developed is a check of δ 2 = d ln k (2π 2 )−1 k 3 P (k). If we
ignore subhalos, this term is can be found by simply integrating the squared density over each halo:
Z
Z
2
ρ(x | M )2
dN
d3 x
.
δ = dM
dM
ρ̄2
This is simply the first term in the overdensity correlation in appendix A.3.8. One reason this is fairly
simple, is that for the NFW profile, the inner integral can be calculated analytically:
Z
Z
ρ(x | M )2
4πρ2s rvir 2
1
d3 x
=
r dr 2 4
2
2
ρ̄
ρ̄
r
r
0
1
+
rs
rs
2 3 Z c
4πρs rs
1
=
dx
4
ρ̄2
(1 + x)
0
4πρ2s rs3
1
=
1−
.
3ρ̄2
(1 + c)3
In sections 3.3 and 3.5, we showed how the mass gives the scale radius rs for a given concentration and
an average halo density 200ρcrit . In section 3.5, we also showed that for this definition of the halo mass,
ρs (c) = 200ρcrit
c3
.
3(log(1 + c) − c/(1 + c))
57
Given
2 the halo mass function and concentration-mass relation, we can then calculate the flux multiplier
δ using a single integral. We find that
Z
1 − (1 + c)−3
200ρcrit 3
.
c M
d3 xρNFW (x | M )2 =
9
(log(1 + c) − c/(1 + c))2
This shows the strong dependence of the squared density on the concentration. The average density
3
ρh = 200ρcrit is not as important as it seems: if this is changed, c3 will also change. Since c3 = rvir
rs−3
is proportional to the volume of the halo, this change in c will counteract any changes in ρh (but it will
not cancel them entirely). This equation is the reason why the relative contribution of halos of mass M
to the power spectrum and flux multiplier can be roughly estimated by M dN/dM .
The reason we only consider the halo structure, is that halos should not overlap. Therefore, particles
in different halos should not be able to annihilate each other. Since we have not modelled this halo
exclusion in our halo distribution
terms
(or in our subhalo distribution), the halo distribution term will,
in fact, contribute somewhat to δ 2 in our calculation. This contribution should be small, however.
When subhalos come into the picture, things become a bit more complicated. There are then three
terms to take into account: the halo-halo term we considered above (with a mass fraction correction),
as well as the subhalo structure term and the correlation between a halo and its subhalo. Again, we
ignore the terms relating (sub)halos that should not overlap. If the subhalo concentration is taken to be
independent of the subhalo position in the halo, the subhalo structure term is relatively easy. It looks
the same as the halo-halo
term, except that the concentration-mass relation may be different, and the
R
dN dnsub
mass function is dM dM
dm (M ).
The halo-subhalo term is trickier. It involves real convolution integrals, and the subhalo distribution.
In the model we are checking here, however, the subhalos are distributed according to a NFW profile.
Since subhalos are mostly much smaller then halos, we approximate the density function for a subhalo
with its centre at y0 as ρs (y − y0 | m) = mδD (y − y0 ). Then the halo-subhalo term in this calculation
becomes
Z
dN dnsub
(M1 )µs (z1 − y1 , y1 − x1 | m1 , M1 )
2 d3 x1 dM1 d3 y1 dm1
dM1 dm1
usub (y1 − x1 | M1 )(1 − fsub (M1 ))µh (z1 − x1 | M1 )
Z
dN dnsub
m1
≈ 2 d3 x1 dM1 d3 y1 dm1
δD (z1 − y1 )
(M1 )
dM1 dm1
ρ̄
usub (y1 − x1 | M1 )(1 − fsub (M1 ))µh (z1 − x1 | M1 )
Z
dN M1
fsub (M1 )usub (z1 − x1 | M1 )(1 − fsub (M1 ))µh (z1 − x1 | M1 )
= 2 d3 x1 dM1
dM1 ρ̄
Z
Z
dN
ρh (x1 | M1 )2
= dM1 2fsub (M1 )(1 − fsub (M1 ))
,
d3 x1
dM1
ρ̄2
where µ = ρ/ρ̄, and ρ is the density of the (sub)halo, assuming all the mass is in the smooth distribution
(and no mass is in subhalos). The third equality only holds when the NFW profile describes both the
subhalo distribution and the halo density profile.
Therefore, in this simple model, we can include the subhalo-halo correlation term with the halo-halo
term:
Z
Z
2
dN
ρh (x | M )2
d3 x
δ ≈ dM 1 − fsub (M1 )2
dM
ρ̄2
Z
Z
Z
dN dnsub
ρs (y | m)2
+ dm dM
(M ) dy
dM dm
ρ̄2
2 R
To calculate δ = d ln k (2π 2 )−1 k 3 P (k), we use a cubic spline interpolation of the calculated P (k)
points in log space, using an extrapolation for k > 3.8 · 1011 Mpc−1 . The extrapolation follows the ∝ k −1
power law we expect. We integrate between k = 2.1 · 10−2 Mpc−1 and k = 1017 Mpc−1 .
Table 2 shows the flux multiplier calculated by these two methods, for the toy model power spectrum
with and without subhalos. It also shows the contributions from various terms. For the halo model
without subhalos, the two calculations agree very well.
58
term(s)
direct
Ptot,ns
P1h,ns
Ptot
P1h
P1s
Phs
6.42 · 104
6.42 · 104
9.50 · 104
4.68 · 104
3.28 · 104
1.54 · 104
R
∆2
6.42 · 104
6.42 · 104
8.91 · 104
4.71 · 104
3.28 · 104
8.36 · 103
Table 2: The flux multiplier δ 2 , calculated by different means. The direct calculation means Rthe one
described in this section. The calculation from integrating the power spectrum is indicated by ∆2 . I
have calculated this for the total power spectrum, as well as for individual terms.
function
Plin
dN
dM200
dnsub
dm200
1 − fsub
uNFW
estimated relative error
−2
10
10−2
2 · 10−4
3 · 10−4
10−3
section(s)
2.3
3.1.2, 3.5.2
3.6.2, 3.5.2
3.6
3.3.2
Table 3: Estimated maximum relative errors for different ingredients of the power spectrum. The last
column indicates the section(s) in which this estimate was made. Note that for the parameter c200 , we
use the formula from ref. [8] directly, and do not anticipate any significant errors in its calculation.
2 The integration of the power spectrum with subhalos, agrees less 2well with the direct calculation of
δ . In the integration of the power spectrum, a contribution of 9 · 10 is not accounted for by the terms
included here. The largest contribution not included, is that of P2s , which accounts for more than 95%
of this discrepancy. Considering its contribution as shown in figure 22, this seems consistent with what
we have already seen. This contribution is about 1% of the total, and suggests a model error of this size
from the lack of halo and subhalo exclusion modelling.
The P1h and P1s terms show good agreement between the two calculation methods. (The differences
are smaller than the errors we will estimate in section 4.2.) However, there are clear problems with Phs .
I suspect the problem is in the approximation in the direct calculation. This is mainly because P1h has
such good agreement between the calculation methods, and the ratio of Phs to P1h is in good agreement
with both qualitative and quantitative expectations. These expectations are, however, based on the
behaviour of the subhalo mass fraction. The idea that the subhalo mass fraction should determine the
between P1h,ns and Phs , is exactly the assumption used in approximation in the calculation of
Rdifference
∆2hs (k)d log k.
Therefore, it is not entirely clear where this discrepancy comes
from, and we note that there may be
some issues with Phs . We note that its contribution to 1 + δ 2 is not very large, and will continue to
use this power spectrum term in the next sections.
We have now checked many aspects of the power spectrum: we have checked its asymptotic behaviour,
the ratios of its terms, and whether the effect of adding subhalos was what we expected. We have also
checked the flux multiplier we are after, using a different calculation (albeit with many of the same
ingredients). After these checks, our calculation framework seems to work, and we trust the calculations
to accurately reflect the parameters used in the model. We note some possible issues with Phs .
4.2
Error estimates for the power spectrum
Now we have established that we trust our calculations in broad terms, we look into the errors induced
by approximations we make in the calculation. We will do this term by term, but focus on the effects on
the total power spectrum. Table 3 shows these errors. The error in the calculation of the NFW profile
is estimated from the difference between the approximation we found and the analytical NFW profile in
momentum space. It is estimated for values krs where we use the approximation. The relative difference
between the smoothed NFW profile and the analytical cut-off NFW profile in momentum space may
59
term
P1h,ns
P2h,ns
P1h
P2h
Phs
Phos
P1s
P2s
Psos
integrand
1.2 · 10−2
1.3 · 10−2
1.4 · 10−2
1.2 · 10−2
1.4 · 10−2
M integration
1.0 · 10−2
1.3 · 10−3
1.0 · 10−2
1.5 · 10−3
9 · 10−3
1.4 · 10−3
2.4 · 10−4
8 · 10−3
3 · 10−3
m integration
total
2 · 10−2
2 · 10−2
4 · 10−4
4 · 10−4
7 · 10−3
8 · 10−4
4 · 10−4
2 · 10−2
2 · 10−2
2 · 10−2
Table 4: The estimated relative error from the integrand and integration in various power spectrum terms.
The integration errors are shown for all terms. They are the maximum estimated relative integration
errors found for each term. The integrand errors are only shown for the Pstruc and P1s terms. The
total error estimates for P2h,ns and Pstruct are not based on the integration errors; these estimates are
explained in the text.
be larger (∼ (1 + c)−2 ), but we consider this a potential model error. In the subhalo mass fraction,
we combine the errors in the interpolation of fsub with those from the subhalo mass function needed to
calculate fsub .
To estimate the errors in the power spectrum and its terms from this, we use a different approach for
different terms. For the sum of the terms involving the linear bias parameter, Pdist , we estimated the
error in the power spectrum from the difference between Pdist and Plin . This is a rather rough estimate,
since we expect such deviations on scales where halo size becomes relevant. In section 4.1.1, we found
the relative error from this in the total power spectrum to be ∼ 1%. Adding the error in Plin of . 10−2 ,
means the relative error in the total power spectrum from the terms Pdist should be no more than ∼ 2%.
This total power spectrum error should become negligible
on scales where the halo structure is dominant.
In section 4.1.3, we found that in the calculation of δ 2 , the relative contribution of Pdist was very
small (≤ 10−3 relative contribution). Therefore, in this calculation, these errors will not be a significant
factor. The same holds for P2h,ns . For completeness, we do point out that model errors in the halo
mass function will be cancelled by corresponding errors in the linear bias parameter: this parameter is
constructed to retrieve Plin from the calculation for any halo mass function described by some function
νf (ν). Therefore, at least at scales much larger than any halo sizes, errors in Plin and approximations
in the calculation should be the only sources of error in Pdist , and in P2h,ns .
For the other terms, we will continue to simply add the relative errors in their factors. These are
shown in table 3. We estimate the total integrand error this way, because some of these errors, e.g. those
from mass conversions, affect multiple terms, so these errors are sometimes correlated. The relative error
in each term should be roughly the same as the relative error in the integrand. A larger relative error
could arise from integration if any of the factors had oscillations, but we only have the minor oscillations
in uNFW left. If the relative errors in the integrand were independent at each point the integrand was
evaluated, integration may smooth out the errors, so that the relative error in the terms is smaller than
the analysis of the integrand predicts. However, the errors we consider come from interpolations and
approximations, and may therefore be systematic. These relative errors in the factors were added to find
the integrand error estimates for the Pstruct and P1s terms in table 4.
A further source of error is the integration itself. I have used quadratic Simpson integration. The
basic idea, is that a number of intervals is chosen, and on each interval, the integrand is evaluated at the
ends and the midpoint. The integrand is then approximated by a quadratic polynomial that takes the
same values as the integrand at those three points. On each interval, there is then an analytical expression
for the integral of the approximating polynomial. These approximate integrals on each interval are then
added up, and form an estimate of the total integral. The error estimate we use for this integration
scheme, is simply the difference between the integral we find, and the integral estimated with half the
number of intervals (which are then twice as long).
Five terms in the power spectrum with subhalos require two nested integrations. For the inner
integrals, over subhalo mass m, the relative error estimates were mostly ≤ 10−3 , and always < 10−2 .
The largest errors in these inner integrals were for P1s ; for the other terms, the relative errors in the
60
inner integrals were always < 10−3 . (The inner integrals for all these terms are the same.) In table 4, the
integration errors are the maximum relative errors for any k in the outer integral (over M ) for that term,
and the maximum relative error for all k and M in the inner integral (over m). Oddly, the maximum
outer integral relative error estimate is smallest for P1s , at 2 · 10−4 . This is not an impossible outcome
if the outer integration smooths out errors in the inner integrals for each mass M . However, the effects
may be systematic, and we add the maximum inner and outer integration error estimates for each term
to estimate the total error from integration.
We find that the maximum relative errors from integration are ∼ 10−2 or ∼ 10−3 , depending on the
term. In the term P2s , the relative error for the inner integral is doubled, because in the term itself,
this integral is squared. The error estimates in previous sections, on which the integrand error estimates
are based, have been fairly conservative. The dominant integrand error, is the error in the halo mass
function. This calculation of this factor is also the most complicated, requiring σ 2 and its derivative, as
well as a mass conversion.
We conclude that the relative errors in the Pstruc , P1s , and P1h,ns terms are all . 2%. Similarly, the
relative error in Pdist and P2h,ns is ∼ 1% on the largest scales, and the relative error from these terms
in the total power spectrum is no more than ∼ 2%, including behaviour on the
where halo sizes
scales
start to matter. The errors in the integration of the power spectrum to find δ 2 are small compared
to the errors we estimate in P (k). This means that the total relative error
approximations made
from
in the calculation of the power spectrum P (k) and the flux multiplier 1 + δ 2 should be no larger than
about 2%. We consider these error estimates to be conservative, since they are based on the maximum
errors of the parameters and integrations, not on the typical errors.
4.3
Towards more realistic models
We have now verified that we have a working framework to calculate the power spectrum for our toy
model, and we have a good idea of how accurate the power spectra we find are. Now, we want to use
this framework to calculate power spectra and flux multipliers for more realistic parameters of the halo
model.
For this, we will focus on the subhalo concentration function. This is because the shape of the
subhalos is something we have deliberately modelled in a way that was useful for predictability of the
power spectrum terms, but not very realistic. We will also use some results from section 4.1.3 to help
estimate the effect of including scatter in the concentrations of halos and subhalos.
4.3.1
A more realistic subhalo concentration
Due to time constraints, we will only be examining a simple modification of the subhalo concentration
function. Ref. [2] found a more realistic model for subhalo concentrations, including a dependence on the
distance between the subhalo and the halo centre. We leave a full implementation of this model to future
work, but use the fact that the subhalos they found typically had concentrations 1.5–2 times larger than
main halos of the same masses. We discussed this in more detail in section 3.6.3. We therefore try out
two more realistic subhalo concentration functions:
csub,1.5 (m200 ) = 1.5cmain (m200 ), and csub,2 (m200 ) = 2cmain (m200 ),
where cmain is the SCP concentration-mass relation for main halos, which we also used for subhalos in the
toy model. We do not model the truncation of these subhalos at smaller radii than rvir . To differentiate
between models, we will use subscripts 1.5 and 2 to label power spectra calculated with these subhalo
concentration functions, and will label those obtained from the toy model from the previous sections
with ‘toy’.
Figure 24 shows the effects of changing the subhalo model on the power spectrum and its terms. This
change has a large effect on the P1s term in the power spectrum, but hardly any effect on Pstruc and
Pdist . Since the subhalo structure term comes into play at smaller scales than the halo structure terms,
the change in the total power spectrum occurs mainly at smaller scales. We note that the slope of the
power spectrum, after the rise around 1 Mpc−1 , is somewhat steeper when the subhalo concentrations
are larger. For comparisons to simulations, bear in mind that a change in the main halo concentration
can produce such a steepening as well. Changes in the mass functions could also have this effect.
61
(a) The dimensionless power spectrum for the toy
model (solid), and the models using csub,1.5 (dotdashed) and csub,2 (dashed).
(b) The fractional contributions Pdist /Ptot (red),
Pstruc /Ptot (blue), and P1s /Ptot (green) to the total
power spectrum. The solid lines are for the toy model
(csub given by the SCP concentration-mass relation),
the dot-dashed lines were found using csub,1.5 , and the
dashed lines were found using csub,2 . The Pdist /Ptot
lines overlap.
(c) The ratios Ptot,1.5 /Ptot,toy ,
Ptot,2 /Ptot,toy ,
P1s,1.5 /P1s,toy , and P1s,2 /P1s,toy .
These show,
for terms using concentrations csub,2 (dashed) and
csub,1.5 (dash-dotted), the effect on the subhalo
structure term (green) and the total power spectrum
(black) of changing the concentration with respect to
the toy model.
(d) The ratios Pstruc,1.5 /Ptot,toy , Pstruc,2 /Ptot,toy ,
Pdist,1.5 /P1s,toy , and Pdist,2 /P1s,toy . These show, for
terms using concentrations csub,2 (dashed) and csub,1.5
(dash-dotted), the effect on the halo structure terms
(blue) and the halo distribution terms (red) of changing the concentration with respect to the toy model.
Figure 24: The dimensionless (non-linear) power spectrum ∆2 , using the halo model with subhalos, and
subhalo concentrations 1, 1.5, and 2 times halo concentrations at the same object mass. The (groups of)
terms, and their relative sizes, are also shown.
62
term(s)
Ptot
P1s
P1h
Phs
toy
csub,1.5
4
8.9 · 10
3.3 · 104
4.7 · 104
8.4 · 103
5
1.4 · 10
8.5 · 104
4.7 · 104
8.8 · 103
csub,2
2.3 · 105
1.7 · 105
4.7 · 104
9.2 · 103
Table 5: The average squared overdensity δ 2 for different models of the subhalo concentration. The
‘toy’ column is for the toy model, where the halo and subhalo concentration functions are the same. The
other two models are labeled by the subhalo concentration functions of their models. The contributions of
the most important terms are shown; the terms not shown never contribute more than 103 put together.
Larger concentrations mean smaller scale radii for the same halo mass. Therefore, in the models
with larger subhalo concentrations, the sizes of the minimum mass subhalos are smaller than those of
the minimum mass halos. This means that the subhalo structure term now cuts off at slightly smaller
scales than Pstruc . This effect is visible in figure 24b. On the largest scales, the subhalo structure is
unimportant, and changing the subhalo concentration function has little effect. On smaller scales, larger
subhalo concentrations cause the contribution of P1s to rise faster, but the shape of the rise is the same.
At even smaller scales, the difference in shape between the halo and subhalo mass functions causes
the Pstruct contribution to rise again. When the halo and subhalo concentrations are equal, this is the
end of the story. However, as we discussed, when the subhalo concentrations are larger than the halo
concentrations at the smallest masses, the subhalo structure term cuts off at a smaller scale than the halo
structure terms, and its contribution rises again at the smallest scales. This is a qualitative difference in
the behaviour of the terms.
Figure 24c shows this smaller cut-off effect too, but in relation to the subhalo structure term in the
toy model. The difference between the P1s terms for different subhalo concentrations rises sharply where
the term has cut off at the smallest subhalo size in the toy model, but has not yet cut off at the smaller
minimum size of the more concentrated subhalos. Close inspection shows that the difference rises longer
for the model using csub,2 than the the model using csub,1.5 , but that these rises start at the same scale.
The effect on the total power spectrum is similar: its cut-off moves to smaller scales. It also becomes
less sharp, since the cut-offs for P1s and Pstuc no longer completely coincide.
On larger scales, larger subhalo concentrations cause the increase in the subhalo structure term
more directly: more concentrated halos have more of their mass towards their centres, and are therefore
effectively smaller. This means that the squared densities are larger in these more concentrated subhalos,
so their power spectrum is larger.
We also consider the effect of the changed subhalo concentration on Pstruc and Pdist . This is shown
in figure 24d. The differences in Pstruc are ∼ 1%. The effect of the altered concentrations on Pdist is
even smaller: no more than 0.4%.
At the smallest scales, these terms very slightly decrease when the subhalo concentration increases.
This is because the asymptotes of the terms involving subhalos are steeper than those of the terms that
only involve the smooth halo component. That means that at the smallest scales, a larger subhalo mass
fraction means smaller Pstruc and Pdist . Because of the mass conversions needed to find the subhalo
mass function, the subhalo mass fraction changes if the subhalo concentration does. The subhalo mass
function increases with subhalo concentration, although this effect is small. Therefore, the subhalo mass
fraction is slightly larger in the models using csub,1.5 and csub,2 , than in our toy model, and Pdist and
Pstruc are slightly smaller at the smallest scales when subhalo concentrations are larger.
Above the scales of the smallest halos, Pstruc and Pdist are larger when the concentration is larger.
This is because, as we discussed previously, the effect of subhalos on the halo structure is that they
smooth it out somewhat, and therefore decrease the average squared density. More concentrated subhalos
are essentially smaller, so they smooth the halo density out less than less concentrated subhalos. The
difference between Pstruc,1.5 and Pstruc,2 , and Pstruc in the toy model, follows the same shape as the
difference between Pstruc in the toy model and P1h,ns shown in figure 22d (the gray line). Similarly, the
differences in Pdist between these subhalo concentration models, follows the difference between Pdist and
P2h,ns shown in figure 22f. This is because the effects are essentially the same. The difference is in how
much the halo structure is being smoothed out.
63
Figure 25: Ratios of subhalo structure contributions to δ 2 , estimated from the analytical integral of
the single (sub)halo squared density. The ratios are for (sub)halos of the same mass, but with different
concentrations. The dashed line shows the contribution ratio of csub,2 subhalos to toy model subhalos,
and the dot-dashed line show the contribution ratio of csub,1.5 subhalos to toy model subhalos.
This behaviour of the power spectrum terms is reflected in their contributions to δ 2 . These are
shown in table 5. The contributions of the two halo structure terms change relatively little with subhalo
concentration; the sum of these terms changes by less than 4 · 102 between the toy model and the csub,2
model. The change in subhalo mass fraction shifts some of the contribution of P1h to Phs , and Phs
is also larger, because the more concentrated subhalos smooth out the halo density less than the toy
model subhalos. However, the subhalo structure contribution increases
very much with concentration,
and determines the changes in the total power spectrum and in δ 2 .
The ratios of the subhalo structure contributions to the toy model case are 2.6 for csub,1.5 , and 5.1
for csub,2 . We compare this to predictions using the work we did in section 4.1.3. There, we found that
a (sub)halo of mass m contributes
Z
1 − (1 + c)−3
≡ Fcont (m, c).
ρ2 (x, m, c)d3 x ∝ c3 m
(log(1 + c) − c/(1 + c))2
to δ 2 . The ratio of these contributions for different concentration functions should then be an indicator
of the ratio between the P1s contributions to δ 2 for these functions. Figure 25 shows this ratio as a
function of (sub)halo mass. The ratios of the contributions to δ 2 we found from P1s , seem consistent with
a weighted average of the single subhalo contribution ratios. The ratios of single subhalo contributions
at the smallest subhalo masses give the best indication of the total ratio of the contributions to δ 2 . This
makes sense, since the smallest subhalos are also the most abundant.
We have found that including a more realistic model of subhalo concentrations is very important
for an accurate description of the power spectrum
on small scales, and can make a difference of more
than a factor 2 in the flux multiplier 1 + δ 2 . The flux multipliers found with these simple changes
to the subhalo concentration function therefore provide a very rough estimate of the flux multiplier. In
appendix A.5.2, a formalism for including a position-dependent subhalo shape is explained.
4.3.2
Estimating the effect of concentration scatter
We have not included any scatter in halo or subhalo concentrations in our model. The reason for this,
is that the extra integrations (over concentration) that this would require, would make the calculations
take an impractically long time. However,
we have just established that the effect of a concentration
change on single halo contributions to δ 2 , is a reasonable predictor of the effect on a term in δ 2 . We
therefore look into the effect of concentration scatter on the
of (sub)halos of a fixed mass
contributions
to the halo structure terms and subhalo structure term in δ 2 .
We model this concentration scatter with a lognormal distribution. The probability of finding a halo
64
R
Figure 26: The enhancement of ρ2 dV for
R (sub)halos with different masses and concentration functions.
This enhancement is the average value of ρ2 dV with concentration scatter, divided by its value without
this scatter. The line styles indicates the concentration-mass relation that was used. Solid lines are for
the main halo concentration-mass relation, that was also used for subhalos in the toy model. Dash-dotted
lines use csub,1.5 , and dashed lines use csub,2 . The colour indicates the value of σlog c that was used: red
is for 0.4, green for 0.32, blue for 0.28, and purple for 0.2.
concentration c, is given by [6]:
"
#
(log(c) − log(c̄))2
d log c
exp −
.
p(c | c̄)dc = q
2
2σlog
2
c
2πσlog
c
For a given halo mass, c̄ is the typical concentration, given by the concentration functions we have
2
discussed. The parameter σlog
c determines the typical scatter of concentrations. Ref. [2] finds subhalo
concentration scatter σlog c in the range 0.2–0.4, with about 0.28 being typical. Ref. [8] reports a typical
scatter σlog c = 0.32 for main halos.
We can then find the contribution of individual (sub)halos to δ 2 with and without subhalos, by
comparing Fcont (m, c̄(m)) to
Z
dc Fcont (m, c)p(c | c̄(m)).
In figure 26, we show the ratio of this individual halo contribution with scatter to its value without
scatter. We do this for the four values of σlog c we mentioned, and for the toy model, csub,1.5 , and csub,2
concentration-mass relations. This shows that increasing the concentration scatter, will increase the flux
multiplier. Annihilation rates are most increased for the smallest halos, which are more concentrated to
begin with. We see the same trend for the concentration-mass relations: larger typical concentrations c̄
mean larger enhancements.
In a lognormal halo concentration distribution, log c is normally distributed; c is not. Its distribution
is lopsided, with a relatively large tail towards large concentrations. These higher-density (sub)halos
enhance the average squared density more than the lower-density halos decrease it, because squaring a
quantity enhances large values more than it decreases small ones. Since the scatter σlog c measures the
scatter in c/c̄, typical values of c − c̄ will be larger at a given scatter, when c̄ is larger. Therefore, the
enhancement of the flux multiplier by concentration scatter should indeed be larger, when the typical
concentration is larger to begin with. That is why the enhancement factor from including scatter, roughly
follows the shape of the concentration function, and this factor increases when the concentration function
does.
We note that for the enhancement through the concentration function increase, the enhancements in
individual halos gave a good rough estimate of the overall enhancement of δ 2 . This overall enhancement
should roughly be a mass-function weighted average of the enhancements as a function of halo mass. For
the subhalo structure contribution enhancements, figure 25 shows that the enhancements at m ∼ M
65
were a good estimate for the overall enhancement. We expect the smallest-mass subhalos to be the most
important for this, since they are the most abundant.
We then estimate
that, for the typical scatters of refs. [8] and [2], the enhancement of the Pstruc
contribution to δ 2 is probably ∼ 20–30%. We have rougher estimates
for c̄ for subhalos, but likely
enhancements seem to be ∼ 20–40% for the P1s contribution to δ 2 .
Ref. [6] compares power spectra with and without subhalos in its figure 17. It shows that the
enhancement of the power spectrum by scatter increases with scale, at least up to the minimum scale
k = 102 hMpc−1 shown. The power spectrum is enhanced by ∼ 25% for σlog c = 0.4 at this minimum
scale, and by ∼ 5% for σlog c = 0.2. The authors do use a different mass-concentration relation than we
do. Figure 24c shows that the effect of larger subhalo concentrations, is still growing at k = 102 hMpc−1 .
The same is true, bit to a lesser degree, for the main halo concentration, if its concentration function is
doubled. Therefore, the power spectrum enhancement for a term at
k = 102 hMpc−1 may very well be
smaller than the overall enhancement of the corresponding term in δ 2 .
In conclusion,
we find that realistic modelling of halo and subhalo concentrations is important in
2
finding
δ
.
A
more
realistic mean subhalo mass-concentration relation makes a very large difference
in δ 2 . Including the concentration scatter probably makes a real difference as well, but a smaller one.
66
5
Discussion
We have now found
power spectra, using the framework we developed, and the corresponding flux
multipliers 1 + δ 2 . The calculations of the power spectra are accurate to about 2%, but this does not
include uncertainties in the model. We will discuss some of these model uncertainties and their expected
impacts, and then compare our results to results from the literature.
5.1
Model uncertainties
We have already discussed many model uncertainties in section 3. We will repeat some of these here,
and focus on their implications for the overall calculations.
To find the halo mass function, ref. [26] fitted a function to halos in the mass range ∼ 4 · 1011 –
5 · 1015 M . The authors give an accuracy of ∼ 20% for this function in a somewhat larger mass range.
Therefore, we have extrapolated the mass function over many orders of magnitude in our calculations.
This mass function is, however, physically motivated, from excursion set theory and elliptical collapse
of halos [32]. Therefore, the extrapolation of this mass function to smaller scales is uncertain, but
reasonable. The ∼ 20% errors in the fitted mass range translate fairly directly to ∼ 20% errors in the
Pstruc and P1s terms in the power spectrum, and therefore to similar errors in 1 + δ 2 .
As we discussed before, we expect
the errors in the halo distribution modelling to have little effect on
the total power spectrum and 1+ δ 2 . In the power spectrum, this is because this distribution modelling
is mostly constrained by the fact that Pdist should reproduce the linear power spectrum on large scales.
There is only a very small range of scales where Pdist has started to diverge from this, on the scales of the
largest
halos, but is not yet negligible compared to Pdist and P1s . We did find a contribution of ∼ 1% to
1 + δ 2 from Pdist . This is an error, induced by the fact that we have not modelled the non-overlapping
of halos.
For the halo shape, we have used the NFW profile. There are some uncertainties about the inner slope
of this profile. Ref. [4] finds that the annihilation signal contribution from individual halos, decreases by
∼ 10% if a particular profile with a flat inner slope is used instead of the NFW profile. Ref. [2] notes
that some simulations show that the smallest halos may have steeper inner slopes. For a minimum mass
of 10−6 M , they find that these steeper profiles increase the annihilation rate from subhalos in a parent
halo by ∼ 30%.
Furthermore, ref. [1] discusses how the subhalo shape may be different from the NFW profile at masses
close to the minimum halo mass. This is because for the smallest halos, the properties of the dark matter
particles, such as their velocity dispersion, become important. After all, these particle properties are
what prevent even smaller halos from forming.
We have seen that the one shape parameter we use, the concentration, has the most impact on the
smallest scales. This is in part because it determines the size of the smallest halos, and in part because,
as we have seen, the effect a fractional change in concentration has on the flux multiplier is largest
when the concentration is larger to begin with.
this uncertainty in the shapes of the smallest
Therefore,
halos could have a significant impact on 1 + δ 2 . Since ref. [1] expects the smallest halos to be more
extended, and have a shallower inner slope,
than the NFW profile would predict, including such effects
would decrease the predicted value of 1 + δ 2 . The effect might be amplified, if subhalos of these smaller
masses are also less likely to survive tidal stripping in halos, which would be the case if they are less
concentrated.
We have not modelled any deviations of the halo and subhalo shapes from spherical symmetry. We
have also not modelled the fact that subhalo density profiles cut off at radii smaller than rvir . According
to ref. [2], this would decrease the predicted subhalo annihilation signal in halos by ∼ 30%.
The main halo concentration function we used, was calibrated to a large range of simulated halo
masses [8]. The authors state an accuracy of 1% in all but the highest-mass end of the mass range we
integrate over. Therefore, the uncertainties in this parameter seem relatively small compared to some
of the others. The subhalo concentration functions we used, were simply multiples of this function, and
constitute a rough estimate. The difference between the flux multipliers for the two subhalo concentration
functions we tried, is almost a factor 2.
In our calculation of the power spectrum, we have ignored concentration scatter entirely. From our
own estimates, and those shown
at
large scales by ref. [6], we estimate that this scatter adds about 5–30%
to the Pstruc contribution to δ 2 , and about 5–40% to the subhalo structure contribution. The size of
67
this effect, depends on the size of the concentration scatter, and on the concentration-mass relation for
the typical concentration c̄.
There is also a model uncertainty in the subhalo mass function. Ref. [23] finds a slope for this mass
function of −1.9 or −2.0. We have used a slope of −1.9. Ref. [2] find that changing the subhalo mass
function slope to α = −2.0 changes the subhalo contribution to a single-halo annihilation signal by a
factor of 2–8 for Mmin = 10−6 M , and parent halo masses of 106 –1012 M . Refs. [9] and [2] find similar
differences. The largest difference is for the most massive, least abundant halos. For Mmin = 10−4 M ,
this difference should be smaller, since the relative effect of the steeper slope becomes larger at smaller
masses.
The largest uncertainty, however, is in the minimum mass of the (sub)halos. This parameter is
almost entirely unknown. Ref. [4] states that the smallest observed (sub)halo mass is 107 M , so we
know Mmin cannot be larger than that. Otherwise, particle physics models predict minimum masses in
the 10−4 –10−12 M mass range [2, 33]. Ref. [5] points out that tidal disruptions by baryon condensations
may have disrupted halos with masses below the baryon Jeans mass ∼ 106 M . Since the dimensionless
power spectrum is growing in k until the minimum size cut-off, this minimum mass is very important
for finding the flux multiplier. Ref. [23] finds that the relative enhancement of the halo annihilation rate
from subhalos, is about ten times larger when the minimum mass is changed from 10−12 M to 1 M .
Given a particle physics model, this flux multiplier should be calculated for the specific minimum mass
of that model.
All in all, the model uncertainties are fairly large, especially for the smallest scales. They are certainly
much larger than the 2% errors we estimate from approximations in the calculations. Even when the halo
model parameters are calibrated to simulations, at the smallest halo masses, the simulation predictions
for the dark matter distribution have not been compared to observations; the minimum
dark
observed
matter halo mass is ∼ 107 M [4]. The large difference between the predictions of 1 + δ 2 from csub,1.5
and csub,2 , seems to be a reasonable measure for the general model uncertainty in the halo model.
However, an error in the slope of the subhalo mass function, could have a larger effect.
The use of the actual concentration-mass relation from ref. [2], instead of the simplified versions
we used, could reduce some of the uncertainty in this model. In appendix A.5.2, we outline a way to
include the subhalo position dependence of this concentration-mass relation into the calculation of the
halo model. This is an effect that is not accounted for in the equations of ref. [11], although the authors
mention the possibility of including it.
5.2
Comparison to other works
Now that we have discussed some of the uncertainties in the halo model, we will compare the flux
multipliers and power spectra we found to the results of others.
Ref. [1] found power spectra at small scales as well, but used a different method. The authors use
a stable clustering approach to extrapolate large scale power spectra from the Millennium simulations,
to smaller scales. They predict a flux multiplier from this, and bracket this value by a power-law
extrapolation of the measured power spectrum, using the slope of the power spectrum at the smallest
measured scale they trust, and the assumption that the power spectrum is non-decreasing until the
cut-off. These limits seem consistent with the behaviour of the power spectra with subhalos we found.
Until the small-scale cut-offs, their log-log slopes are decreasing in k, but the power spectra themselves
are increasing. They also find some tighter bounds.
They cut off their power spectra at a scale kmax =
106 hMpc−1 . With this cut-off, they predict 1 + δ 2 = 1.3 · 105 , with the tightest error bounds being
8.4 · 104 –2.4 · 105 .
Our linear power spectrum seems to be similar to the one used in ref. [1]. However, the power
spectrum found in the reference is generally larger than the ones we have found. This already occurs on
scales where the power spectrum in the reference, was measured from simulations. The simulations do
use a larger value of σ8 (normalisation of the linear power spectrum), but ref. [1] states that the errors
from uncertainties in the cosmological parameters, should be no more than ∼ 10%. The shapes of the
power spectra seem to
besimilar, including on small scales.
The values of 1 + δ 2 in ref. [1], are very close to the predictions we find for the different subhalo
concentration models, using Mmin = 10−4 M . For a fair comparison, however, we cut off our
power
spectra at the same kmax , and find values of 2.9 · 104 , 3.7 · 104 , 5.2 · 104 , and 7.4 · 104 for 1 + δ 2 for
the model without subhalos, the toy model subhalo concentration, csub,1.5 , and csub,2 respectively. This
68
difference in flux multipliers reflects the generally smaller power spectrum we found. The looser lower
bound (from assuming the power spectrum is non-decreasing in k) in ref. [1] is 4.6 · 104 , which does allow
the values we found for the more realistic models for subhalo concentrations. The authors found that
the halo model usually predicts larger flux multipliers than their approach. For our calculations, that
does not apply.
Ref. [11] uses the same approach as us to find the small-scale power spectra. The equations the
authors use for the power spectrum terms, agree with what we have found, except that they include
concentration scatter. They use a similar subhalo mass function, with the same slope at small masses,
but a different concentration-mass relation. The minimum mass used, is Mmin = 10−6 M .
These authors find concentration scatter enhancements for host halos of 15–20%. This is consistent
with our rough estimates. Their ratios of halo and subhalo structure terms to P1h,ns and of halo
distribution terms to P2h,ns have the same general behaviour as a function of k as we found, but are not
the same. The authors do use a different concentration-mass relation for subhalos, so their halo mass
fractions may simply be different. As for ref. [1], the power spectra they find, are larger than ours. This
makes sense, since they are calibrated to the same Millennium simulation. Their ratio Ptot /Ptot,ns , so
the enhancement of the power spectrum by inclusion of subhalos, is larger than what we found.
Unlike us, ref. [4] normalised its halo mass function in order to impose that all mass is contained
in halos. Other authors did not mention such normalisation changes, but if they used them, this may
explain (some of) the difference between the power spectra we found, and those in the literature. In
our model, only 59% of the mass is contained in halos. The rest is ignored in the formalism, since it
is assumed to have little to no structure. The reference uses cosmological parameters from an earlier
5
−4
Planck release, similar to ours, and finds a flux
2 multiplier of ∼ 2 · 10 for α = −1.9, and 10 M . This is
similar to the larger value we found for 1 + δ ; including the difference in mass function normalisation,
it is similar to the csub,1.5 estimate.
For a comparison to work focussed on annihilation signals from single
we will consider
(sub)halos,
2 − 1, where δ is the
the substructure boost. For a halo of a given mass, this is defined as δ 2 / δns
overdensity
with subhalos, and δns without them. We will compare this to the same ratio for the total
2
δ values for the universe. We find a boost of 0.39 for the toy model, 1.2 for csub,1.5 , and 2.5 for csub,2 .
Ref. [8] found individual halo boosts for a subhalo mass function slope of −1.9 of about 1–3 , with
the smallest boosts being for the smallest halo mass shown, 106 M . They used Mmin = 10−6 M . This
is for the subhalo model corresponding to our toy model: the SCP concentration-mass relation is used
for halos and subhalos. Since the smallest halos are most abundant, we expect the boost to the isotropic
annihilation signal to be at the low end of this range. Still, the boost we find for this toy model is clearly
smaller.
Ref. [2] found boost factors of 2–10, when using the more realistic model for the subhalo concentrationmass relation that we based our simple models csub,1.5 and csub,2 on. These factors are for an α = −1.9
subhalo mass function, without including tidal stripping, and with Mmin = 10−6 M .
Ref. [9] also uses Mmin = 10−6 M . For the smallest halos these authors show, they find a boost of
0.5–1.1 for halo masses from 106 M using a subhalo mass function slope of α = −1.9, and main halo
concentrations for subhalos. When they include their analytical model for subhalo concentrations, they
find boosts of 2–5. These figures are closer to what I found, with the csub,2 model predicting an overall
boost which is plausibly compatible with that of ref. [9].
We note that the boost factor for main halos due to the inclusion of subhalos, is decreasing with
main halo mass. This makes sense, because larger halos have larger subhalo mass fractions. Since boosts
are not shown for main halo masses below 106 M in the references we discussed, the overall boost may
be lower than these larger halo boost factor ranges suggest, especially since these smallest halos are the
most abundant.
We have found that in comparing power spectra, flux multipliers, and boosts, we find smaller power
spectra than most others. The smaller minimum mass used in these references, may play part in that.
This does not, however, explain the differences with the power spectra on larger scales, such as those
found from the Millennium simulations. Differences in cosmological parameters appear to be too small
to account for this difference, but they may account for part of it. The lack of concentration scatter
in our calculations may also play a part. The boost factors we found, are more similar to some values
found in the literature. However, they still seem to be on the small side. The flux multipliers for our
most realistic models agree well with those of ref. [4], but are small compared to the findings of ref. [1].
For the subhalo mass function, we have chosen to use the more conservative of the two commonly
69
used models. We have also not
included concentration scatter, which increases the power spectrum.
Therefore, our estimate of 1 + δ 2 ∼ 1.4 · 105 –2.3 · 105 for Mmin = 10−4 M , is likely on the conservative
side. It is not, however, a lower limit.
70
6
Conclusion
We have investigated the flux multiplier 1 + δ 2 , which determines the magnitude of a dark matter
annihilation signal, relative to the same signal if dark matter were distributed completely smoothly. To
calculate this, we have developed a model for including the subhalo structure and distribution in the
halo model, which agrees with that found by ref. [11]. I have set up a framework to calculate this power
spectrum for the model parameters we used, which can be modified to use different parameter values.
We found that including subhalos in the power spectrum, has a large effect on this power spectrum,
especially on small scales. The difference between the power spectrum with and without subhalos,
is mainly determined by the term describing the subhalo structure. For the subhalo concentration
parameter, we have used rough estimates, based on a model calibrated to simulations by ref. [2]. An
implementation of the subhalo concentration function from this reference would reduce the uncertainty
in our calculations from making these rough assumptions. Since this halo concentration function depends
on the position of the subhalo in the halo, we would need to include it in the power spectrum as described
in appendix A.5.2. This describes a new addition to the expression for the power spectrum from the
halo model.
We used a minimum (sub)halo mass of 10−4 M , the more conservative of the two commonly used
subhalo mass functions, and rough
estimates for the subhalo concentration function, without scatter.
From this, we estimate that 1 + δ 2 ∼ 1.4 · 105 –2.3 · 105 . This estimate is likely on the low side. However,
it is clearly larger than the estimates that do not include subhalos, or use the same concentration function
for subhalos as for main halos.
7
Acknowledgements
I would like to thank my supervisors, Shin’ichiro Ando and Mattia Fornasa, for their help with this
research, and its presentation. I would also like to thank my parents, for their support during the
writing of this thesis.
71
8
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73
Appendices
A
Halo model power spectrum derivation
In this appendix, we will give the full derivation of the power spectrum in terms of halo model parameters.
The basic assumptions and methods are described in section 2.4. Section A.1 gives some technical
mathematical background. After that, we will consider the power spectrum itself.
A.1
Justification for bringing the expectation values inside the integrals
In the following sections, we will often want to switch the order of integrating over mass and position
and of taking expectation values. Taking the expectation value of ρ2 formally means convolving it with
a probability density function p(X1 , ...., Xn ) and integrating over the stochastic variables X1 , .., Xn .
In this case, the stochastic variables for a compact patch of the universe seem to be the number of
halos N , their locations x1 , ..., xN and masses M1 , ..., MN , the number of subhalos n1 , ...nN in each halo
i, and their positions y1i , ..., yni and their masses m1i , ..., mni . This is not a good description, however,
since the number of stochastic variables is itself variable. A more accurate description would have to
take an infinite set of halo and subhalo mass and position variables, and assign p = 0 to any realisation
for which an infinite number of halo or subhalo masses is nonzero.
This infinite set of stochastic variables is mathematically problematic, since with an infinite set of
variables, the definition of the integral becomes problematic. However, we can set some upper limits on
things. First, we fix the cosmological parameters here, so the average density ρ̄ = Ωm ρcrit is fixed. This
gives the total mass in the patch of universe we consider, hence a maximum halo mass and subhalo mass.
Independent of the clustering model, the dark matter particle mass places a lower limit on the mass
of a bound structure. Combined with the total mass limit, this gives an upper limit on the number of
halos and subhalos in the universe patch of our chosen size, and therefore also on the number of subhalos
in every halo.
These maxima limit the number of stochastic variables to a finite set. This means our integration is
well-defined. The compact patch of universe chosen is the integration domain for the position variables,
and the mass upper limit means the halo and subhalo mass integration domains are bounded. We may
choose our integration domains in each variable to be closed. For the non-stochastic mass and position
variables used to calculate ρ and ρ2 , the same arguments apply. Therefore, our total integration domain
for ρ2 p is compact and we may change the order of integration.
A.2
Warm-up and preparation: hρi2
2
2
We will begin our calculation by finding hρi . This should help subtract the 1 in ρ2 / hρi − 1 from
2 the
individual terms, and will introduce the general approach we will use to calculate each term in ρ . In
section 2.4, we introduced the following parameterisation of the density in the universe:
"
Z
Z
X
3
ρ(z) = d x1 dM1
δD (x1 − xi )δD (M1 − Mi ) ρh (z − x1 | M1 )
i
Z
+
3
d y1
#
Z
dm1
X
δD (y1 − yai )δD (m1 − mai )ρs (z − y1 , y1 − x1 | m1 , M1 ) .
ai
74
In this formalism,
Z
hρ(z)ihalo =
Z
3
dM1 ρh (z − x1 | M1 )
d x1
*
X
+
δD (x1 − xi )δD (M1 − Mi )
i
Z
Z
Z
Z
X
halo
Z
+ d3 x1 dM1 d3 y1 dm1 ρs (z − y1 , y1 − x1 | m1 , M1 )
*
+
XX
δD (x1 − xi )δD (M1 − Mi )δD (y1 − yai )δD (m1 − mai )
ai
i
Z
=
3
halo
d x1
dM1
δD (x1 − xi )δD (M1 − Mi )ρh (z − x1 | M1 )
i
Z
+
3
Z
d x1
Z
dM1
Z
3
d y1
dm1
X
δD (x1 − xi )δD (M1 − Mi )
i
dnsub
(M1 )usub (y1 − x1 | M1 )(1 + ξres, sh (y1 − x1 , m1 , M1 ))
dm1
ρs (z − y1 , y1 − x1 | m1 , M1 ).
Then, taking the universe expectation value,
*
+
Z
Z
X
3
hhρ(z)ihalo i = d x1 dM1
δD (x1 − xi )δD (M1 − M1 ) ρh (z − x1 | Mi )
i
Z
+
d3 x1
Z
Z
dM1
d3 y1
*
Z
dm1
+
X
δD (x1 − xi )δD (M1 − Mi )
i
dnsub
(M1 )usub (y1 − x1 | M1 )(1 + ξres, sh (y1 − x1 , m1 , M1 ))
dm1
ρs (z − y1 , y1 − x1 | m1 , M1 )
Z
Z
dN
3
= d x1 dM1
ρh (z − x1 | M1 )
dM
Z
Z
Z 1 Z
dN dnsub
(M1 )usub (y1 − x1 | M1 )
+ d3 x1 dM1 d3 y1 dm1
dM1 dm1
(1 + ξres, sh (y1 − x1 , m1 , M1 ))ρs (z − y1 , y1 − x1 | m1 , M1 ).
This should be equal to ρ̄, if the two step expectation value works like we hope. Here, the halo and
dN
sub
subhalo mass functions dM
and dn
dm describe the halo and subhalo number density at each mass as
outlined in section 2.4. In this section, we also defined the subhalo distribution usub and the correlation
between subhalo mass and position ξres, sh .
2
We see that hhρihalo i consists of three terms in this formalism. Here, we have assumed that ξres, sh =
0, to simplify the expression shown. This is also the value it takes throughout this thesis. It can easily
be reinserted at any combination of the subhalo mass and spatial distribution functions. We then find
that
Z
Z
dN dN
2
hhρ(z)ihalo i = d3 x1 dM1 d3 x2 dM2
ρh (z − x1 | M1 )ρh (z − x2 | M2 )
dM1 dM2
Z
Z
dN
+ 2 d3 x1 dM1 d3 y1 dm1 d3 x2 dM2
ρh (z − x2 | M2 )
dM2
dN dnsub
(M1 )usub (y1 − x1 | M1 )ρs (z − y1 , y1 − x1 | m1 , M1 )
dM1 dm1
Z
Z
3
3
+ d x1 dM1 d y1 dm1 d3 x2 dM2 d3 y2 dm2
dN dnsub
dN dnsub
(M1 )
(M2 )usub (y1 − x1 | M1 )usub (y2 − x2 | M2 )
dM1 dm1
dM2 dm2
ρs (z − y1 , y1 − x1 | m1 , M1 )ρs (z − y2 , y2 − x2 | m2 , M2 ).
75
We have written this using two mass and position integrals so that it will be easy to compare to the
terms in hρ(z1 )ρ(z2 )i later.
A.3
The main calculations: hρ(z1 )ρ(z2 )i
Now, we will proceed to the main event: the description of hρ(z1 )ρ(z2 )i in terms of halo model parameters.
Here, again, we will assume ξres, sh = 0 to keep things (somewhat) legible.
We begin by examining the general appearance of ρ(z1 )ρ(z2 ):
Z
ρ(z1 )ρ(z2 ) = d3 x1 dM1 d3 x2 dM2
X
X
δD (x1 − xi )δD (M1 − Mi )
δD (x2 − xj )δD (M2 − Mj )
i
j

 ρh (z1 − xi | Mi )ρh (z2 − xj | Mj )
+ ρh (z2 − xj | Mj )
Z
X
d3 y1 dm1 ρs (z1 − y1 , y1 − x1 | m1 , M1 )
δD (y1 − yai )δD (m1 − mai )
ai
+ ρh (z1 − xi | Mi )
Z
X
d3 y2 dm2 ρs (z2 − y2 , y2 − x2 | m2 , M2 )
δD (y2 − yaj )δD (m2 − maj )
aj
Z
+
d3 y2 dm2 d3 y1 dm1
ρs (z2 − y2 , y2 − x2 | m2 , M2 )ρs (z1 − y1 , y1 − x1 | m1 , M1 )

X
δD (y2 − yaj )δD (m2 − maj )
aj
X
δD (y1 − yai )δD (m1 − mai ) .
ai
In the above equation, in each of the middle terms, z1 and z2 play different parts. However, when we
take the expectation value, we expect the terms to be the same. This is because for a general statistically
homogeneous and isotropic term f (z1 )g(z2 ) , with some functions f and g,
hf (z1 )g(z2 )i = hf (0)g(z2 − z1 )i = hf (0)g(|z1 − z2 |)i = hf (0)g(z1 − z2 )i = hf (z2 )g(z1 )i .
This means that for given z1 and z2 , the middle terms will have the same expectation value, and do not
76
need to be considered separately:
** Z
hhρ(z1 )ρ(z2 )ihalo i =
d3 x1 dM1 d3 x2 dM2
X
δD (x1 − xi )δD (M1 − Mi )
i
X
δD (x2 − xj )δD (M2 − Mj )
j

 ρh (z1 − x1 | M1 )ρh (z2 − x2 | M2 )
Z
d3 y1 dm1
X
ρs (z1 − y1 , y1 − x1 | m1 , M1 )
δD (y1 − yai )δD (m1 − mai )
+ 2ρh (z2 − x2 | M2 )
ai
Z
d3 y2 dm2 d3 y1 dm1
+
ρs (z2 − y2 , y2 − x2 | m2 , M2 )ρs (z1 − y1 , y1 − x1 | m1 , M1 )
X
δD (y2 − ybj )δD (m2 − mbj )
bj
#+
X
δD (y1 − yai )δD (m1 − mai )
ai
+
.
halo
This makes clear which combinations of density profiles and delta functions should be considered
here. First, there is the halo-halo term, with two options: the halos i and j are the same or different.
Then, in the halo-subhalo term, the subhalo is ai either of the halo j (i = j) or of a different halo.
Finally, the subhalo-subhalo term has three options: the subhalos could be in different halos, they could
be different subhalos of the same halo, or they could be the same entirely. These are the seven terms
described in figure 4. These distinctions are important because they determine how many ‘copies’ of e.g.
halo mass functions are required for a term. In the following sections, we will consider these seven terms,
and show how to calculate them if the parameters of the halo model are known.
A.3.1
Halo-halo term, i = j
The first term we will consider is the structure of the smooth halo: this is the halo-halo term for i = j.
We will call this term Th=h (z1 , z2 ). Working out ρρ,
Z
XX
d3 x1 dM1 d3 x2 dM2
i
j,j=i
δD (x1 − xi )δD (M1 − Mi )δD (x2 − xj )δD (M2 − Mj )ρh (z1 − xi | Mi )ρh (z2 − xj | Mj )
Z
X
= d3 x1 dM1 d3 x2 dM2
i
δD (x1 − xi )δD (M1 − Mi )δD (x2 − xi )δD (M2 − Mi )ρh (z1 − xi | Mi )ρh (z2 − xi | Mi )
Z
X
= d3 x1 dM1 d3 x2 dM2
i
δD (x1 − xi )δD (M1 − Mi )δD (x2 − x1 )δD (M2 − M1 )ρh (z1 − x1 | M1 )ρh (z2 − x2 | M2 )
Z
X
= d3 x1 dM1
δD (x1 − xi )δD (M1 − Mi )ρh (z1 − x1 | M1 )ρh (z2 − x1 | M1 ).
i
77
Taking expectation values,
**Z
Th=h (z1 , z2 ) =
d3 x1 dM1
X
δD (x1 − xi )δD (M1 − Mi )
i
+
+
ρh (z1 − x1 | M1 )ρh (z2 − x1 | M1 )
halo
*Z
=
d3 x1 dM1
X
δD (x1 − xi )δD (M1 − Mi )
i
+
ρh (z1 − x1 | M1 )ρh (z2 − x1 | M1 )
Z
d3 x1 dM1
=
A.3.2
dN
ρh (z1 − x1 | M1 )ρh (z2 − x1 | M1 ).
dM1
Halo-halo term, i 6= j
Next we consider the correlation between two points in different halos. We will call this term Th6=h . The
term
Z
XX
d3 x1 dM1 d3 x2 dM2
i
j,j6=i
δD (x1 − xi )δD (M1 − Mi )δD (x2 − xj )δD (M2 − Mj )ρh (z1 − xi | Mi )ρh (z2 − xj | Mj )
cannot be simplified further. The halo expectation value does not affect this term. The expectation value
here will involve the halo-halo correlation function ξhh , defined in section 2.4. This function is equal to
zero if the position of the second halo is statistically independent of the position of the first. Then
Z
Th6=h (z1 , z2 ) = d3 x1 dM1 d3 x2 dM2 ρh (z1 − x1 | M1 )ρh (z2 − x2 | M2 )
*
+
XX
δD (x1 − xi )δD (M1 − Mi )δD (x2 − xj )δD (M2 − Mj )
i
Z
=
A.3.3
j,j6=i
dN dN
ρh (z1 − x1 | M1 )ρh (z2 − x2 | M2 )
dM1 dM2
(1 + ξhh (x1 − x2 | M1 , M2 )).
d3 x1 dM1 d3 x2 dM2
Halo-subhalo term, i = j
We now move on to the terms involving subhalos. The first of these terms describes a relation between
subhalos and their parent halos. This term will be called Th=h−s . The term in ρ(z1 )ρ(z2 ) simplifies as
78
follows:
Z
d3 x1 dM1 d3 x2 dM2 d3 y1 dm1 ρh (z2 − x2 | M2 )ρs (z1 − y1 , y1 − x1 | m1 , M1 )
X
X
δD (x1 − xi )δD (M1 − Mi )
δD (x2 − xj )δD (M2 − Mj )
i
j,j=i
X
δD (y1 − yai )δD (m1 − mai )
ai
Z
=
d3 x1 dM1 d3 x2 dM2 d3 y1 dm1 ρh (z2 − x2 | M2 )ρs (z1 − y1 , y1 − x1 | m1 , M1 )
X
δD (x1 − xi )δD (M1 − Mi )δD (x2 − x1 )δD (M2 − M1 )
i
X
δD (y1 − yai )δD (m1 − mai )
ai
Z
=
d3 x1 dM1 d3 y1 dm1 ρh (z2 − x1 | M1 )ρs (z1 − y1 , y1 − x1 | m1 , M1 )
X
X
δD (x1 − xi )δD (M1 − Mi )
δD (y1 − yai )δD (m1 − mai ).
ai
i
Applying hi and hihalo as before, gives:
Z
Th=h−s (z1 , z2 ) = d3 x1 dM1 d3 y1 dm1 ρh (z2 − x1 | M1 )ρs (z1 − y1 , y1 − x1 | m1 , M1 )
dN dnsub
(M1 )usub (y1 − x1 | M1 ).
dM1 dm1
A.3.4
Halo-subhalo term, i 6= j
Our next term describes the correlation between a subhalo and a non-parent halo. We will call his term
Th6=h−s . The delta functions in the term in ρ(z1 )ρ(z2 ) that describes this, do not simplify much. Here,
taking the halo expectation value means that the correlation between a subhalo and a non-parent halo
79
is dictated by the correlation between halos ξhh . This term is
Th6=h−s (z1 , z2 )
Z
= d3 x1 dM1 d3 x2 dM2 d3 y1 dm1 ρh (z2 − x2 | M2 )ρs (z1 − y1 , y1 − x1 | m1 , M1 )
**
X
X
δD (x1 − xi )δD (M1 − Mi )
δD (x2 − xj )δD (M2 − Mj )
i
j,j6=i
+
X
δD (y1 − yai )δD (m1 − mai )
ai
Z
=
+
halo
3
3
3
d x1 dM1 d x2 dM2 d y1 dm1 ρh (z2 − x2 | M2 )ρs (z1 − y1 , y1 − x1 | m1 , M1 )
*
X
i
X
δD (x1 − xi )δD (M1 − Mi )
δD (x2 − xj )δD (M2 − Mj )
j,j6=i
+
dnsub
(Mi )usub (y1 − xi | Mi )
dm1
Z
= d3 x1 dM1 d3 x2 dM2 d3 y1 dm1 ρh (z2 − x2 | M2 )ρs (z1 − y1 , y1 − x1 | m1 , M1 )
*
X
X
δD (x1 − xi )δD (M1 − Mi )
δD (x2 − xj )δD (M2 − Mj )
i
j,j6=i
+
dnsub
(M1 )usub (y1 − x1 | M1 )
dm1
Z
= d3 x1 dM1 d3 x2 dM2 d3 y1 dm1 ρh (z2 − x2 | M2 )ρs (z1 − y1 , y1 − x1 | m1 , M1 )
dN dN
dnsub
(1 + ξhh (x1 − x2 | M1 , M2 ))
(M1 )usub (y1 − x1 | M1 ).
dM1 dM2
dm1
Note that in the third equality, δD (x1 − xi )δD (M1 − Mi ) allows us to substitute 1 for i, which means that
for the fourth equality, we do not need to include the subhalo parameters in the (universe) expectation
value.
80
A.3.5
Subhalo-subhalo term, i = j, ai = bi
We now consider the term describing the structure of the subhalos. This term will be called Th−s=s .
There are some simplifications to be made in ρ(z1 )ρ(z2 ) here:
Z
Z
Z
d3 x1 dM1 d3 x2 dM2 d3 y2 dm2 d3 y1 dm1
ρs (z1 − y1 , y1 − x1 | m1 , M1 )ρs (z2 − y2 , y2 − x2 | m2 , M2 )
X
X
δD (x1 − xi )δD (M1 − Mi )
δD (x2 − xj )δD (M2 − Mj )
i
j,j=i
X
δD (y1 − yai )δD (m1 − mai )
ai
X
δD (y2 − ybj )δD (m2 − mbj )
bj ,bj =ai
Z
d3 x1 dM1 d3 x2 dM2
=
Z
d3 y2 dm2
Z
d3 y1 dm1
ρs (z1 − y1 , y1 − x1 | m1 , M1 )ρs (z2 − y2 , y2 − x2 | m2 , M2 )
X
δD (x1 − xi )δD (M1 − Mi )δD (x2 − x1 )δD (M2 − M1 )
i
X
δD (y1 − yai )δD (m1 − mai )δD (y2 − y1 )δD (m2 − m1 )
ai
Z
=
d3 x1 dM1 d3 y1 dm1 ρs (z1 − y1 , y1 − x1 | m1 , M1 )ρs (z2 − y1 , y1 − x1 | m1 , M1 )
X
X
δD (x1 − xi )δD (M1 − Mi )
δD (y1 − yai )δD (m1 − mai ).
i
ai
Taking the halo and universe expectation values then shows that
Z
dN dnsub
(M1 )usub (y1 − x1 | M1 )
Th−s=s (z1 , z2 ) = d3 x1 dM1 d3 y1 dm1
dM1 dm1
ρs (z1 − y1 , y1 − x1 | m1 , M1 )ρs (z2 − y1 , y1 − x1 | m1 , M1 ).
81
A.3.6
Subhalo-subhalo term, i = j, ai 6= bi
Now, we move on to the correlation between subhalos in the same parent halo. We will call this term
Th−s6=s . Again,we begin by simplifying ρ(z1 )ρ(z2 ):
Z
Z
Z
d3 x1 dM1 d3 x2 dM2 d3 y2 dm2 d3 y1 dm1
ρs (z1 − y1 , y1 − x1 | m1 , M1 )ρs (z2 − y2 , y2 − x2 | m2 , M2 )
X
X
δD (x1 − xi )δD (M1 − Mi )
δD (x2 − xj )δD (M2 − Mj )
i
j,j=i
X
δD (y1 − yai )δD (m1 − mai )
ai
X
δD (y2 − ybj )δD (m2 − mbj )
bj ,bj 6=ai
Z
d3 x1 dM1 d3 x2 dM2
=
Z
Z
d3 y2 dm2
d3 y1 dm1
ρs (z1 − y1 , y1 − x1 | m1 , M1 )ρs (z2 − y2 , y2 − x2 | m2 , M2 )
X
δD (x1 − xi )δD (M1 − Mi )δD (x2 − x1 )δD (M2 − M1 )
i
X
X
δD (y1 − yai )δD (m1 − mai )
ai
δD (y2 − ybi )δD (m2 − mbi )
bi ,bi 6=ai
Z
=
d3 x1 dM1 d3 y1 dm1 d3 y2 dm2 ρs (z1 − y1 , y1 − x1 | m1 , M1 )
X
ρs (z2 − y1 , y1 − x1 | m1 , M1 )
δD (x1 − xi )δD (M1 − Mi )
i
X
X
δD (y1 − yai )δD (m1 − mai )
ai
δD (y2 − ybi )δD (m2 − mbi ).
bi ,bi 6=ai
Taking the halo expectation value here comes with a difficulty: the second halo mass and position are
not necessarily independent of the first. This may be remedied by introducing a subhalo correlation term
akin to ξhh for halos. The correlation here, however, is not expected to be due to a level of substructure
between halos and subhalos. Such a correlation would then only account for non-overlapping of subhalos,
and the fact that the masses of the subhalos may depend on each other. We will not model the spatial
dnsub,2
exclusion here, but will include a conditional mass function dm
(m1 , M1 ) that represents the halo mass
2
function for a halo of mass M1 , given that it contains a different subhalo of mass m1 . In other words (or
rather, in mathematical symbols),
dnsub,2
(m1 , M1 )usub (y2 − x1 | M1 )
dm2
*
+
X
the halo i of mass Mi contains
=
δD (y2 − ybj )δD (m2 − mbj ) some subhalo ai of mass m1
bj ,bj 6=ai
DP
E
P
δ
(y
−
y
)δ
(m
−
m
)
δ
(y
−
y
)δ
(m
−
m
)
1
ai D
1
ai
2
bj D
2
bj
ai D
bj ,bj 6=ai D
halo
P
.
=
ai δD (y1 − yai )δD (m1 − mai ) halo
P
Then, multiplying by
ai δD (y1 − yai )δD (m1 − mai ) halo , it follows that
*
+
X
X
δD (y1 − yai )δD (m1 − mai )
δD (y2 − ybj )δD (m2 − mbj )
ai
bj ,bj 6=ai
halo
*
+
dnsub,2
(m1 , M1 )usub (y2 − x1 | M1 )
=
dm2
X
δD (y1 − yai )δD (m1 − mai )
ai
dnsub
dnsub,2
=
(M1 )
(m1 , M1 )usub (y2 − x1 | M1 )usub (y1 − x1 | M1 ).
dm1
dm2
82
halo
Then, it is clear that
Th−s6=s (z1 , z2 )
Z
= d3 x1 dM1 d3 y1 dm1 d3 y2 dm2 ρs (z1 − y1 , y1 − x1 | m1 , M1 )ρs (z2 − y2 , y2 − x1 | m2 , M1 )
dN dnsub
dnsub,2
(M1 )
(m1 , M1 )usub (y2 − x1 | M1 )usub (y1 − x1 | M1 ).
dM1 dm1
dm2
A.3.7
Subhalo-subhalo term, i 6= j
Finally, we consider the correlation between subhalos in different parent halos. We will call this term
Ts−h6=h−s . This ρ(z1 )ρ(z2 ) term cannot be simplified much. It is simply
Z
Z
Z
d3 x1 dM1 d3 x2 dM2 d3 y2 dm2 d3 y1 dm1
ρs (z1 − y1 , y1 − x1 | m1 , M1 )ρs (z2 − y2 , y2 − x2 | m2 , M2 )
X
X
δD (x1 − xi )δD (M1 − Mi )
δD (x2 − xj )δD (M2 − Mj )
i
j,j6=i
X
δD (y1 − yai )δD (m1 − mai )
ai
X
δD (y2 − ybj )δD (m2 − mbj )
bj
Z
d3 x1 dM1 d3 x2 dM2
=
Z
d3 y2 dm2
Z
d3 y1 dm1
ρs (z1 − y1 , y1 − x1 | m1 , M1 )ρs (z2 − y2 , y2 − x2 | m2 , M2 )
X
X
δD (x1 − xi )δD (M1 − Mi )
δD (y1 − yai )δD (m1 − mai )
i
X
ai
δD (x2 − xj )δD (M2 − Mj )
j,j6=i
X
δD (y2 − ybj )δD (m2 − mbj ).
bj
Since the halo expectation value acts on each halo separately, the presence of two subhalos is not an
83
issue here as it was in the previous term. Considering only the δD -function parts:
**
X
X
δD (x1 − xi )δD (M1 − Mi )
δD (y1 − yai )δD (m1 − mai )
ai
i
+
X
δD (x2 − xj )δD (M2 − Mj )
j,j6=i
X
X
=
δD (y2 − ybj )δD (m2 − mbj )
bj
*
δD (x1 − xi )δD (M1 − Mi )
halo
*
X
δD (x2 − xj )δD (M2 − Mj )
*
X
j,j6=i
=
δD (x1 − xi )δD (M1 − Mi )
i
X
j,j6=i
δD (y1 − yai )δD (m1 − mai )
+halo +
δD (y2 − ybj )δD (m2 − mbj )
bj
*
X
+
ai
i
X
+
halo
dnsub
(M1 )usub (y1 − x1 | M1 )
dm1
+
dnsub
(M2 )usub (y2 − x2 | M2 )
δD (x2 − xj )δD (M2 − Mj )
dm2
dnsub
dnsub
=
(M1 )usub (y1 − x1 | M1 )
(M2 )usub (y2 − x2 | M2 )
dm
dm2
* 1
+
X
X
δD (x1 − xi )δD (M1 − Mi )
δD (x2 − xj )δD (M2 − Mj )
i
j,j6=i
dN dN
(1 + ξhh (x1 − x2 | M1 , M2 ))
=
dM1 dM2
dnsub
dnsub
(M1 )usub (y1 − x1 | M1 )
(M2 )usub (y2 − x2 | M2 ).
dm1
dm2
The term we want is then
Z
Ts−h6=h−s =
3
3
d x1 dM1 d x2 dM2
Z
3
d y2 dm2
Z
d3 y1 dm1
ρs (z1 − y1 , y1 − x1 | m1 , M1 )ρs (z2 − y2 , y2 − x2 | m2 , M2 )
dN dN
(1 + ξhh (x1 − x2 | M1 , M2 ))
dM1 dM2
dnsub
dnsub
(M1 )usub (y1 − x1 | M1 )
(M2 )usub (y2 − x2 | M2 ).
dm1
dm2
A.3.8
A very long equation
Now we have all the separate terms, we can put them together. To get hδ(z1 )δ(z2 )i, which we expect
2
to be well approximated by hhδ(z1 )δ(z2 )ihalo i, we need to subtract hρi from these terms, and divide by
84
ρ̄2 . Putting everything together, and denoting ρs/h /ρ̄ = µs/h ,
hδ(z1 )δ(z2 )i =
Z
dN
d3 x1 dM1
µh (z1 − x1 | M1 )µh (z2 − x1 | M1 )
dM1
Z
dN dN
µh (z1 − x1 | M1 )
+ d3 x1 dM1 d3 x2 dM2
dM1 dM2
ξhh (x1 − x2 | M1 , M2 )µh (z2 − x2 | M2 )
Z
dN dnsub
+ 2 d3 x1 dM1 d3 y1 dm1
(M1 )µs (z1 − y1 , y1 − x1 | m1 , M1 )
dM1 dm1
usub (y1 − x1 | M1 )µh (z2 − x1 | M1 )
Z
dN dN dnsub
(M1 )µs (z1 − y1 , y1 − x1 | m1 , M1 )
+ 2 d3 x1 dM1 d3 x2 dM2 d3 y1 dm1
dM1 dM2 dm1
usub (y1 − x1 | M1 )ξhh (x1 − x2 | M1 , M2 )µh (z2 − x2 | M2 )
Z
dN dnsub
+ d3 x1 dM1 d3 y1 dm1
(M1 )usub (y1 − x1 | M1 )
dM1 dm1
µs (z1 − y1 , y1 − x1 | m1 , M1 )µs (z2 − y1 , y1 − x1 | m1 , M1 )
Z
dnsub,2
dN dnsub
(M1 )
(m1 , M1 )
+ d3 x1 dM1 d3 y1 dm1 d3 y2 dm2
dM1 dm1
dm2
µs (z1 − y1 , y1 − x1 | m1 , M1 )usub (y1 − x1 | M1 )usub (y2 − x1 | M1 )
µs (z2 − y2 , y2 − x1 | m2 , M1 )
Z
Z
dN dN dnsub
dnsub
3
3
3
+ d x1 dM1 d x2 dM2 d y2 dm2 d3 y1 dm1
(M1 )
(M2 )
dM1 dM2 dm1
dm2
µs (z1 − y1 , y1 − x1 | m1 , M1 )usub (y1 − x1 | M1 )
Z
ξhh (x1 − x2 | M1 , M2 )usub (y2 − x2 | M2 )µs (z2 − y2 , y2 − x2 | m2 , M2 ).
2
Note that the effect of subtracting hρi is to remove the 1 from 1 + ξhh in the terms that relate different
2
(parent) halos. The difference between hρi and hρ(z1 )ρ(z2 )i is in the structure of the halos and subhalos,
and in the correlations between the positions of the halos and subhalos.
A.4
Separating smooth and subhalo mass out from the halo mass
In these formulas, the halo mass M is the mass in the smooth component. Generally speaking, however,
the mass M in the halo mass function refers to the total (halo + subhalo) mass. Intuitively, this should
M −mtot,sub
µh , where
change the normalisation of the halo mass profile µh to
M
Z
dnsub
mtot,sub (M ) = dm
(M )m.
dm
However, we need to examine the impact this has on the halo expectation value: this would have another
function to act on. The subhalo mass fraction correction is needed in the first four terms in the power
spectrum. In the Th6=h and Th6=h−s , the correction is the simplest. Here, the halo expectation value for
the/each main halo acts only on the subhalo mass of that halo. We note that
M − hmtot,sub ihalo
M − mtot,sub
=
M
M
halo
85
from linearity, and
*
Z
hmtot,sub ihalo =
+
X
dm
δD (m − mi )
ai
Z
=
Z
dm
m
halo
3
d y
*
X
+
δD (m − mi )δD (y − yi )
ai
Z
=
Z
dm
Z
=
dm
m
halo
dnsub
(Mi )usub (y − xi )ξres, sh (y − xi , m, M )
d ym
dm
3
dnsub
(Mi )m
dm
by definition of the subhalo mass function. This means that for these two terms, we may simply correct
the halo density profile(s) by a factor of the subhalo mass fraction.
The situation is more complicated for Th=h and Th=h−s , however, since here, the halo expectation
value relating to ρh applies to more than one halo involved. Before taking any expectation values, we
simply insert a factor of
R
P
M − dmd3 y ai δD (m − mai )δD (y − yai )m
M
in front of each instance of ρh . (Note that we must use different integration variables y and m for each
instance.) We will discuss Th=h−s first. It is clear that the M
M part of the subhalo mass fraction correction
will just give us a copy of the term we had already found. Therefore, we only repeat our earlier work
with the other part of the correction:
Z
d3 x1 dM1 d3 x2 dM2 d3 y1 dm1 ρh (z2 − x2 | M2 )ρs (z1 − y1 , y1 − x1 | m1 , M1 )
X
X
δD (x1 − xi )δD (M1 − Mi )
δD (x2 − xj )δD (M2 − Mj )
i
j,j=i
R
X
δD (y1 − yai )δD (m1 − mai )
dm2 dy2
P
bj
δD (m2 − mbj )δD (y2 − ybj )m2
M2
ai
Z
=
d3 x1 dM1 d3 x2 dM2 d3 y1 dm1 d3 y2 dm2 ρh (z2 − x2 | M2 )ρs (z1 − y1 , y1 − x1 | m1 , M1 )
X
δD (x1 − xi )δD (M1 − Mi )δD (x2 − x1 )δD (M2 − M1 )
i
X
δD (y1 − yai )δD (m1 − mai )
X
ai
δD (y2 − ybj )δD (m2 − mbj )m2 M2−1
bj
Z
=
d3 x1 dM1 d3 y1 dm1 d3 y2 dm2 ρh (z2 − x1 | M1 )ρs (z1 − y1 , y1 − x1 | m1 , M1 )m2 M1−1
X
X
δD (x1 − xi )δD (M1 − Mi )
δD (y1 − yai )δD (m1 − mai )
i
X
ai
δD (y2 − ybi )δD (m2 − mbi ).
bi
To apply the halo expectation value to the δD -functions, we split the term into a bi 6= ai and a bi = ai
86
part, and get results as with Th−s6=s and Th−s=s :
**
X
X
δD (x1 − xi )δD (M1 − Mi )
δD (y1 − yai )δD (m1 − mai )
ai
i
+
X
δD (y2 − ybi )δD (m2 − mbi )
bi 6=ai
=
+
halo
*
X
δD (x1 − xi )δD (M1 − Mi )
i
dnsub
dnsub,2
(M1 )
(m1 , M1 )usub (y2 − x1 | M1 )usub (y1 − x1 | M1 )
dm1
dm2
=
+
dN dnsub
dnsub,2
(M1 )
(m1 , M1 )usub (y2 − x1 | M1 )usub (y1 − x1 | M1 )
dM1 dm1
dm2
and
**
X
δD (x1 − xi )δD (M1 − Mi )
X
δD (y1 − yai )δD (m1 − mai )
ai
i
+
X
δD (y2 − ybi )δD (m2 − mbi )
bi =ai
=
+
halo
**
X
δD (x1 − xi )δD (M1 − Mi )
X
δD (y1 − yai )δD (m1 − mai )
ai
i
+
+
δD (y2 − y1 )δD (m2 − m1 )
halo
dN dnsub
(M1 )usub (y1 − x1 | M1 )δD (y2 − y1 )δD (m2 − m1 )
=
dM1 dm1
We can nowR compare these terms to what we previously found for Th=h−s . For the ai 6= bi part, we
first note that d3 y2 usub (y2 − x1 | M1 ) = 1. Therefore, the additional part in the term, compared to
Th=h−s , is
Z
dnsub,2
(m1 , M1 )m2 .
M1−1 dm2
dm2
This is similar to the subhalo mass fraction we found before, except that in the subhalo mass function,
we
R 3know that there is a subhalo of mass m1 , and we only consider other halos. In the second part,
d y2 δD (y2 − y1 ) = 1, so again, we only need to consider and additional m2 integral. This time,
however, that integral is just a delta function, and only contributes an extra factor m1 M1−1 . This is
simply the the contribution to the halo mass fraction from the one subhalo that was not captured in the
other part of the correction.
dnsub,2
sub
In this thesis, we will use dm
(m1 , M1 ) = dn
dm2 (M1 ). We will ignore this ‘last subhalo’ term of
2
the halo mass fraction; it will generally be very small, since most subhalos are much smaller than their
parent halos, and among subhalos, the largest are the rarest [23]. Formally, however, the correction we
need to apply in Th=h−s is
Z
dnsub,2
−1
1 − M1
m1 + dm2
(m1 , M1 )m2 ,
dm2
using the m1 and M1 integration variables used in the equations for P (k) and Th=h−s (z1 , z2 ). With the
approximations we described, this is the same as the halo mass fraction correction we use for Th6=h and
Th6=h−s .
87
For the final term, Th=h , our approach will be similar. Here, before taking expectation values, we
insert a factor
!
Z
X
3
−1
1 − M1
dm1 d y1
δD (m1 − mai )δD (y1 − yai )m1
ai


M2−1
1 −
Z
3
dm2 d y2
X
δD (m2 − mbj )δD (y2 − ybj )m2 
bj
= 1 − 2M1−1
Z
dm1 d3 y1
X
δD (m1 − mai )δD (y1 − yai )m1
ai
+
M1−1
Z
dm1 d3 y1
X
δD (m1 − mai )δD (y1 − yai )m1
ai
M2−1
Z
dm2 d3 y2
X
δD (m2 − mbj )δD (y2 − ybj )m2 .
bj
The ‘1’ here will simply give an unchanged term. The effect of the second term we be the same as
in Th6=h and Th6=h−s : the halo expectation value will only act on one copy of the subhalo δD -functions,
and the correction will be (twice) the expected subhalo mass fraction. For the final term, we revisit the
calculation of Th=h :
Z
XX
d3 x1 dM1 d3 x2 dM2
i
j,j=i
δD (x1 − xi )δD (M1 − Mi )δD (x2 − xj )δD (M2 − Mj )ρh (z1 − xi | Mi )ρh (z2 − xj | Mj )
Z
X
−1
M1
dm1 d3 y1
δD (m1 − mai )δD (y1 − yai )m1
ai
M2−1
Z
3
dm2 d y2
X
δD (m2 − mbj )δD (y2 − ybj )m2
bj
Z
=
d3 x1 dM1 d3 x2 dM2 dm1 d3 y1 dm2 d3 y2
X
δD (x1 − xi )δD (M1 − Mi )δD (x2 − xi )δD (M2 − Mi )
i
XX
ai
δD (m2 − mbi )δD (y2 − ybi )δD (m1 − mai )δD (y1 − yai )
bi
M1−1 m1 M2−1 m2 ρh (z1 − xi | Mi )ρh (z2 − xi | Mi )
Z
X
= d3 x1 dM1 d3 x2 dM2 dm1 d3 y1 dm2 d3 y2 M1−1 m1 M2−1 m2
i
δD (x1 − xi )δD (M1 − Mi )δD (x2 − x1 )δD (M2 − M1 )ρh (z1 − x1 | M1 )ρh (z2 − x2 | M2 )
XX
δD (m2 − mbi )δD (y2 − ybi )δD (m1 − mai )δD (y1 − yai )
ai
Z
=
bi
d3 x1 dM1
X
δD (x1 − xi )δD (M1 − Mi )ρh (z1 − x1 | M1 )ρh (z2 − x1 | M1 )M1−2
i
Z
dm1 d y1 dm2 d3 y2 m1 m2
XX
δD (m2 − mbi )δD (y2 − ybi )δD (m1 − mai )δD (y1 − yai ).
ai
3
bi
Just like before, in Th=h−s , we find that the halo expectation value acts on two copies of the δD -
88
functions for one subhalo set:
**
X
X
δD (x1 − xi )δD (M1 − Mi )
δD (y1 − yai )δD (m1 − mai )
ai
i
+
X
+
δD (y2 − ybi )δD (m2 − mbi )
bi
halo
dN dnsub
dnsub,2
=
(M1 )
(m1 , M1 )usub (y2 − x1 | M1 )usub (y1 − x1 | M1 )
dM1 dm1
dm2
dN dnsub
+
(M1 )usub (y1 − x1 | M1 )δD (y2 − y1 )δD (m2 − m1 ).
dM1 dm1
As before, the integrals over y1 and y2 just give factors of 1 in each term. The first term here gives a
correction of
Z
Z
dnsub,2
dnsub
−2
M1
(M1 ) dm2 m2
(m1 , M1 )
dm1 m1
dm1
dm2
and the second term gives a correction of
M1−2
Z
dm1
dnsub
(M1 )m21
dm1
to the original Th=h term. Again, we will use approximations here. Generally,
subhalo will
R a single
dnsub,2
(m1 , M1 ).
have mass m1 much smaller than the combined mass of all the other subhalos dm2 m2 dm
2
We will therefore neglect the second term here, and only use the first term. Furthermore, as discussed
dnsub,2
sub
before, we will approximate dm
(m1 , M1 ) = dn
dm2 (M1 ). In that case, the first term is simply the
2
squared total subhalo mass fraction. Adding the two terms together, we see that this is exactly the same
approximation we made in Th=h−s . Combining all the correction terms in Th=h , we then see that they
add up to
2
Z
dnsub
dnsub
−1
1−
dm1
(M1 )m1 + M1
dm1
(M1 )m1
dm1
dm1
2
Z
dnsub
= 1 − M1−1 dm1
(M1 )m1 .
dm1
2M1−1
Z
This concludes our analysis of how to correct the smooth halo mass for the subhalo mass contribution.
We introduce the notation
Z
dnsub
−1
fsub (M ) = M
dm
(M )m
dm
for the subhalo mass fraction. Within reasonable approximations, we can then correct the smooth halo
mass for the subhalo contribution by simply inserting a factor of 1 − fsub (M ) for each halo density
function ρh (... | M ) we encounter in the power spectrum.
A.5
Fourier transforming hρ(z1 )ρ(z2 )i
The expression for hδ(z1 )δ(z2 )i contains a lot of convolution integrals. Those can be more easily expressed
in the Fourier transforms of the expressions. As discussed in section 2.2, we will follow the ref. [6] Fourier
convention
Z
d3 k
A(x) =
A(k) exp(ik · x).
(2π)3
R
Then if A(x) = d3 yB(x − y)C(y), A(k) = B(k)C(k). By induction, using multiple convolutions just
multiplies the Fourier transform by more such factors. As discussed in section 2.2, the power spectrum
P (k) is the Fourier transform of hδ(x)δ(0)i.
This power spectrum is therefore easier to compute than hδ(x)δ(0)i itself: the terms in hδ(z1 )δ(z2 )i
are convolution integrals, except that both the subhalo density profile ρs and the subhalo distribution in
the halo usub depend on the distance from the subhalo centre to the parent halo centre. In refs. [7, 11], the
89
subhalo density profile is assumed to be independent of the position of the subhalo in the halo. However,
ref. [2] has analysed subhalos of simulated Milky Way mass halos, and found that the concentrations of
subhalos depend on their distances from the halo center. The concentration is a parameter describing
the shape of (sub)halos, as described in section 3.3.
A.5.1
A simplifying assumption
As stated above, the power spectrum will simply be a product of Fourier transforms, if we assume that
the subhalo density profile does not depend on the position of the subhalo in the halo. As we will
show, excluding this effect requires two less integrals, so it makes the calculations much more feasible.
Considering this, we will drop the dependence of ρs on y − x and the spatial integrals will become only
convolution integrals; this continues the assumption that ξres, sh = 0.
In this case, we simply leave out the second argument in µs . Furthermore, we note that we may
interchange x and −x freely in these integrals as long as the functions have a reflection symmetry. This
is certainly true for ξhh , from statistical isotropy. For spherical, or even triaxial, halo and subhalo density
profiles and subhalo distributions, this should also hold. Furthermore,
Z
d3 y1 usub (y1 − x1 | M1 ) = 1
for all x1 and M1 by choice of normalisation, so part of the fifth term in hδ(z1 )δ(z2 )i ‘drops out’ if the
dependence of µs on y1 − x1 is ignored.
This reflection symmetry is not quite necessary, however. If F (x) and F (k) are a Fourier transform
pair and G(x) = F (−x), then G(k) = F (−k). Under these assumptions, and those in section A.4,
Fourier transforming our equation for hhδ(z)δ(0)ihalo i and inserting the appropriate subhalo mass fraction
corrections gives:
P (k) =
Z
dN
µh (k | M1 )µh (−k | M1 )(1 − fsub (M1 ))2
dM1
dM1
Z
dN dN
µh (k | M1 )ξhh (k | M1 , M2 )µh (−k | M2 )
+ dM1 dM2
dM1 dM2
(1 − fsub (M1 ))(1 − fsub (M2 ))
Z
dN dnsub
+ 2 dM1 dm1
(M1 )µs (k | m1 , M1 )usub (k | M1 )µh (−k | M1 )(1 − fsub (M1 ))
dM1 dm1
Z
dN dN dnsub
(M1 )
+ 2 dM1 dM2 dm1
dM1 dM2 dm1
µs (k | m1 , M1 )usub (k | M1 )ξhh (k | M1 , M2 )µh (−k | M2 )(1 − fsub (M2 ))
Z
dN dnsub
+ dM1 dm1
(M1 )µs (k | m1 , M1 )µs (−k | m1 , M1 )
dM1 dm1
Z
dN dnsub
dnsub,2
+ dM1 dm1 dm2
(M1 )
(m1 , M1 )
dM1 dm1
dm2
µs (k | m1 , M1 )usub (k | M1 )usub (−k | M1 )µs (−k | m2 , M1 )
Z
dN dN dnsub
dnsub
+ dM1 dM2 dm2 dm1
(M1 )
(M2 )
dM1 dM2 dm1
dm2
µs (k | m1 , M1 )usub (k | M1 )ξhh (k | M1 , M2 )usub (−k | M2 )µs (−k | m2 , M2 ).
Note that we use the same notation for functions and their Fourier transforms for simplicity of notation,
since there are already a lot of different functions flying around.
This Fourier transform greatly reduces the number of integrals to be calculated, and will therefore
make the calculations much more doable, provided that the Fourier transforms of the spatial distributions
functions are not too difficult to calculate. In section 3, we discuss the various functions we use for these
halo model parameters. In section 2.4, we discuss some aspects of these functions, and how they apply
to the power spectrum calculation.
90
A.5.2
Including a factor that is not a full convolution
For functions of two variables, Fourier transforms are done over each argument separately. For convolutions, the same rules that apply to functions of a single variable then apply to each argument. (This is
essentially what is done with Fourier transforms over vectors.) The problem with ρs is not that is has
two arguments. The issue is that the y − x argument(s) appear in two functions, meaning the expression
for the power spectrum is not just a convolution of functions.
However, we can still apply a Fourier transform. We can use the convolution theorem, if we take the
product of the two functions as the function to be convoluted. Analogous to convolutions in position
space, a product in position space is a convolution in momentum space: if F (x) = f (x)g(x),
Z
F (k) =
d3 l
f (k − l)g(l).
(2π)3
Therefore, the formalism is not very hard to adapt to additional position dependences, although it would
require two or four1 extra integrations in some terms. Considering that some of these terms already
require many integrations, this would slow down calculations significantly.
Adding in a position-dependance of the subhalo density profile, we get
P (k) =
Z
dN
dM1
µh (k | M1 )µh (−k | M1 )
dM1
Z
dN dN
+ dM1 dM2
µh (k | M1 )ξhh (k | M1 , M2 )µh (−k | M2 )
dM1 dM2
Z
dN dnsub
(M1 )µh (−k | M1 )
+ 2 dM1 dm1
dM1 dm1
Z
d3 l
µs (k, k − l | m1 , M1 )usub (l | M1 )
(2π)3
Z
dN dN dnsub
+ 2 dM1 dM2 dm1
(M1 )ξhh (k | M1 , M2 )µh (−k | M2 )
dM1 dM2 dm1
Z
d3 l
µs (k, k − l | m1 , M1 )usub (l | M1 )
(2π)3
Z
Z
Z
d3 l2
dN dnsub
d3 l1
+ dM1 dm1
(M1 )
dM1 dm1
(2π)3
(2π)3
µs (k, k − l1 | m1 , M1 )usub (l1 − l2 | M1 )µs (−k, l2 | m1 , M1 )
Z
dN dnsub
dnsub,2
+ dM1 dm1 dm2
(M1 )
(m1 , M1 )
dM1 dm1
dm2
Z
d3 l1
µs (k, k − l1 | m1 , M1 )usub (l1 | M1 )
(2π)3
Z
d3 l2
µs (−k, −k − l2 | m2 , M1 )usub (l2 | M1 )
(2π)3
Z
dN dN dnsub
dnsub
+ dM1 dM2 dm2 dm1
(M1 )
(M2 )ξhh (k | M1 , M2 )
dM1 dM2 dm1
dm2
Z
d3 l1
µs (k, k − l1 | m1 , M1 )usub (l1 | M1 )
(2π)3
Z
d3 l2
usub (−k, −k − l2 | M2 )µs (l2 | m2 , M2 ).
(2π)3
The subhalo mass fractions were left out for legibility. The effect of including a function ξres, sh (y −
x, m, M ) would be similar to including this second position dependence in µs .
1 In practice, the functions we will use are spherically symmetric in each argument. To find F (|k|) will still require two
integrations, however, since |k − l| is determined not just by |l| and |k|, but also by k · l.
91
In practice, finding µ(k, l | m, M ) means Fourier transforming µs (z1 − y1 , y1 − x1 | m, M ). The dependence of the subhalo density profile on position is through the concentration [23]. The dependence of
ρs on the concentration is complicated; it is shown in section 3.3. This means that, in practice, finding
the Fourier transform of ρs in this second position argument will require an additional numerical integral. Using the ref. [2] model for this dependence, and otherwise using the functions from section 3, it
should be possible to do this in advance by interpolating the function in two arguments: in this model,
the concentration is the product of a mass-dependent and a position-dependent part, and the position
dependence is easily inverted. The subhalo density profile then depends fiducially on three parameters,
one of which is the second position dependence.
92