University of Amsterdam - Institute of Physics (IoP)

94 Pages·2016·2.62 MB·English

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: Examiner:
Dr. Shin’ichiro Ando 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
blijktnamelijkdatongeveer80%vandematerieinhetuniversum,donkerematerieis. Dezebestaat
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
deeltjesditkunnenzijn. Diemodellenvoorspellenookdatalstweedonkere-materiedeeltjesopelkaar
botsen,zeeenlichtflitskunnenveroorzaken. Dekansdatditgebeurtisergklein;alsdezekansgroot
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
delichtflitsjesuitgammastralingbestaan,ennietuitzichtbaarlicht. Totnutoeisdezestralingniet
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
aantaldeeltjesergensverdubbelt,elkdeeltjeeentweekeerzogrotekansheeftopeenanderdeeltjete
botsen,enertweekeerzoveeldeeltjesmetdezeverdubbeldebotsingskanszijn,dushettotaalaantal
botsingenwordtverviervoudigd. Watdatbetekent,isdatalshetzelfdeaantaldeeltjesinklontjesis
verdeeld,ermeerbotsingentussendedeeltjeszullenzijndanalsdedeeltjesgelijkmatigverdeeldzijn.
Dezeverhogingvanhetaantalbotsingentenopzichtevangelijkmatigverdeeldedonkerematerie,en
dus ook de verhoging voor het aantal lichtflitsjes dat we zouden zien, noemen we (cid:10)δ2(cid:11). Dit is het
gemiddelde van het kwadraat van de dichtheidsverhoging.
Ikhebgezochtnaardezeverhogingsfactor(cid:10)δ2(cid:11). Dithebikgedaanaandehandvanhethalomodel.
In dit model zit alle donkere materie in klonten die halo’s heten. Deze halo’s hebben een verdeling
eneenstructuur. Degrootstehalo’shebbeneenmassavanongeveereentriljoen(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
bekekenwaarindekleinstehalo’s30keerzoveelmassahebbenalsdeAarde,ofweleentienduizendste
van de massa van de zon.
Omdestructuurvanhalo’sgoedtebeschrijven,ishetbelangrijkhunsubstructuurmeetenemen.
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 (cid:10)δ2(cid:11), en een programma voor de berekening geschreven.
Zonder subhalo’s, en voor een minimummassa van 30 aardmassa’s, vind ik een verhogingsfactor
(cid:10)δ2(cid:11) = 64000. Voor een na¨ıef model met subhalo’s vind ik een verhogingsfactor van 89000, en
voor twee schattingen van realistischere subhalomodellen, vind ik (cid:10)δ2(cid:11)=140000 en (cid:10)δ2(cid:11)=230000.
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
simulatiesgevonden. Dezewaardendieikvind,zijnwatkleinerdanwatsommigeandereonderzoekers
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 belangrijke subhaloparameter heb ik een conservatievere waarde
gekozen, dus deze schatting van (cid:10)δ2(cid:11) 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
gammastralingvandezebotsingenineengegevendeeltjesmodel. Metonshuidiggebrekeendetectie
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 theorie¨en over de aard van
donkere materie.
1
Summary
ForthisMScproject,Ihavestudiedthedistributionofmatterinouruniverse. Ihavedonethis,
because research shows that about 80% of the matter in the universe is dark. It does not consist of
theatoms,ions,andelectronsthatmakeupthematterweknow. Itisalsonotmadeupoftheother
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.
Wedonotknowwhatdarkmatterismadeof. Therearemanymodelsforwhatkindofparticles
itcouldbemadeof. Thosemodelspredictthatiftwodarkmatterparticlescollide,thiscouldcause
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. Whatthismeans,isthatiftheparticleshaveaclumpydistribution,therewillbemore
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 (cid:10)δ2(cid:11). This is the average of the squared density increase.
I have looked for this enhancement factor (cid:10)δ2(cid:11). 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.
Todescribethestructureofhaloscorrectly,itisimportanttoincludesubstructure. Thisconsists
ofsmallerclumpswithinthehalos,that,forexample,containgalaxiesinclusters,andtheMagellanic
clouds in the Milky Way halo. To do this, I added terms to the calculation of (cid:10)δ2(cid:11), 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 (cid:10)δ2(cid:11) = 64000. For a naive subhalo model, If found an enhancement factor of 89000, and
for two more realistic subhalo models, I found (cid:10)δ2(cid:11)=140000 and (cid:10)δ2(cid:11)=230000. 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
enhancementfactor,aresomewhatsmallerthanotherresearchersexpectorhavefound. Ihaveused
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
agreementwithastudyusingparametersmoresimilartomine. Foranimportantsubhaloparameter,
I used a more conservative value, so this estimate of (cid:10)δ2(cid:11) is on the low side.
These results show that the clumpy distribution of dark matter makes a large difference in pre-
dicting 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
Tocalculatethestrengthofgamma-rayemissionproducedbyannihilatingdarkmatter,weneed
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+(cid:10)δ2(cid:11), which
is determined by the average squared overdensity of the dark matter: 1+(cid:10)δ2(cid:11) = (cid:10)ρ2(cid:11)/(cid:104)ρ(cid:105)2. 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.
Thissubstructureisimportantfortheannihilationflux,becauseitmeansdarkmatterisclumpier
thanifitwerenotpresent. Clumpierdarkmatterhas,onaverage,alargersquareddensity,andwill
thereforeproducealargerannihilationsignal. Evenifthese(sub)structuresareunresolved,thetotal
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.
Ihavefoundaformalismtoincludethissubstructureintothecalculationofthepowerspectrum
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.
Foratoymodelofsubhaloparameters,Ihavefoundthepowerspectrum,andcomparedittothe
power spectrum without subhalos. As predicted, subhalos increased the power spectrum on small
scales. Forthistoymodel, Ifoundafluxmultiplier1+(cid:10)δ2(cid:11)=8.9·104 foraminimumhalomassof
10−4M . Forthesameminimummasswithoutsubhalos,Ifound1+(cid:10)δ2(cid:11)=6.4·104. Usingthesame
(cid:12)
scalecut-offasref.[1],Ifoundasmallerfluxmultiplierthantheydid,usingadifferentmethod. The
subhaloparametersinthetoymodelprovideaconservativeestimateforthesubhalocontributionto
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+(cid:10)δ2(cid:11) ∼ 1.4·105–2.3·105. This
estimate is also likely on the low side.
3
Contents
1 Introduction 7
2 The halo model 9
2.1 Cosmology in a homogeneous, isotropic universe . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Fourier transformations and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 The linear power spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 The halo model formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4.1 Halo ensemble averages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4.2 Further definitions, assumptions, and results . . . . . . . . . . . . . . . . . . . . . 20
3 Parameters of the halo model 23
3.1 The halo mass function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.1.1 The Sheth-Tormen mass function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.1.2 Calculating the mass function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.1.3 A definition and two parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2 The halo distribution and the linear bias parameter. . . . . . . . . . . . . . . . . . . . . . 29
3.3 The halo density profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3.1 The NFW profile and its Fourier transform . . . . . . . . . . . . . . . . . . . . . . 34
3.3.2 Smoothing out the Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3.3 Other profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.4 The concentration function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.5 Mass definitions and conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.5.1 Converting between mass definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.5.2 Errors in the halo mass function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.6 Subhalo modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.6.1 The subhalo mass function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.6.2 The subhalo mass fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.6.3 Points on which to improve the subhalo model . . . . . . . . . . . . . . . . . . . . 49
4 Results 51
4.1 Sanity checks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.1.1 The halo model without subhalos . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.1.2 Adding the subhalos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.1.3 Checking the flux mulitplier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.2 Error estimates for the power spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.3 Towards more realistic models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.3.1 A more realistic subhalo concentration . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.3.2 Estimating the effect of concentration scatter . . . . . . . . . . . . . . . . . . . . . 65
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 75
A.1 Justification for bringing the expectation values inside the integrals . . . . . . . . . . . . . 75
A.2 Warm-up and preparation: (cid:104)ρ(cid:105)2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
A.3 The main calculations: (cid:104)ρ(z )ρ(z )(cid:105) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
1 2
A.3.1 Halo-halo term, i=j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
A.3.2 Halo-halo term, i(cid:54)=j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
A.3.3 Halo-subhalo term, i=j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
A.3.4 Halo-subhalo term, i(cid:54)=j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
A.3.5 Subhalo-subhalo term, i=j, a =b . . . . . . . . . . . . . . . . . . . . . . . . . . 82
i i
A.3.6 Subhalo-subhalo term, i=j, a (cid:54)=b . . . . . . . . . . . . . . . . . . . . . . . . . . 83
i i
A.3.7 Subhalo-subhalo term, i(cid:54)=j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
A.3.8 A very long equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
A.4 Separating smooth and subhalo mass out from the halo mass . . . . . . . . . . . . . . . . 86
A.5 Fourier transforming (cid:104)ρ(z )ρ(z )(cid:105) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
1 2
A.5.1 A simplifying assumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
A.5.2 Including a factor that is not a full convolution . . . . . . . . . . . . . . . . . . . . 92
5
1 Introduction
Oneofthebigmysteriesincosmologyisthenatureofdarkmatter[3]. Weknowfromvariousobservations
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]).
Onewaytoinvestigatedarkmatteristolookfortheresultsoftherareinteractionstheseparticlesmay
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 n2 (cid:104)σv(cid:105): the squared (total) number density of the particles, multiplied by
χ
the velocity-averaged interaction cross-section. The proportionality constant is 1 in the case of self-
2
annihilation, 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
thisthesisbutthesecanbecomputedusingthedescriptionofthedarkmatterdistributionwewillbuild
up, combined with a model for the particle physics.
Assuming (cid:104)σv(cid:105) is position-independent, the dark matter annihilation rate at any point can then be
parametrised as δ2+1 times the annihilation rate if the density in the universe were completely uniform
(δ =0). This
(cid:18)ρ (cid:19)2
δ2 = −1
ρ¯
is the focus of this thesis. Here, ρ is the dark matter density at some point and ρ¯ 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 (cid:10)δ2(cid:11): the
average squared overdensity in the universe. More generally, we can study (cid:104)δ(z )δ(z −z )(cid:105), which is
2 1 2
averaged over z . This describes the correlation between overdensities at points separated by z . In
2 1
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 bary-
onic) matter. It is difficult to study the smallest scales this way alone, though. The largest halos,
containing large galaxy clusters, have masses around 1015M . Depending on the dark matter model,
(cid:12)
6
the smallest have masses of no more than ∼106M , though possibly as small as 10−12M . This differs
(cid:12) (cid:12)
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,
mainlyforMilky-Waysizedhalosof∼1012M , butforsmallermassesandscalesaswell[8]. Analytical
(cid:12)
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
simulationsofarepresentativevolumeoftheuniverse(meaningthataveragequantitiesinthesimulations
should accurately reproduce averages in the observable universe). These simulations resolved structures
to scales of ∼102kpc. 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,wecalculatethefluxmultiplierforfourdifferentmodels,andfindthatthefluxmultiplier(cid:10)δ2(cid:11)+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.
Thenwewilllookintothegeneralideaofthehalomodel,withsomediscussionofitsscopeandlimitations.
Next,wewilldiscusstheformalismofthehalomodelundercertainassumptions. Finally,wewilldiscuss
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, theuniverseisexpanding. Thismeansthatinthepast, theuniversewasmuchdenserand
hotter, meaningdifferentenergyscalesdeterminedwhatphysicalprocesseswerehappening. Inthevery,
very early universe (a tiny fraction of a second after it began), there was a period of rapid expansion
calledinflation. Thiswasdrivenbyaquantumfield,whichhasanuncertaindensityatanypoint. 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.
Whentemperaturesanddensitieshaddroppedtoolow,theneutrinointeractionratesbecamenegligible,
andtheonlyremaininginteractionwasneutrondecaytoprotons(andneutrinos). Atthesetemperatures,
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 240000 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 350000 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
cosmologicalparametersthatwewilldiscuss, andprovideevidencethattheinitialdensityperturbations
are as predicted by inflation.
On large scales, galaxy surveys and CMB measurements show that our universe is homogeneous
(symmetricunderspatialtranslations)andisotropic(symmetricunderspatialrotations)[13]. Ingeneral
relativity, such a universe is described by the FLRW (Friedmann-Lemaˆıtre-Robertson-Walker) metric:
(cid:20) dr2 (cid:21)
ds2 =dt2−a2(t) +r2dΩ2 .
1−κr2
Here,risaradialcoordinateanddΩ2 =dθ2+sin2θdφ2,withθandφbeingsphericalangularcoordinates.
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 time-
dependent. This expansion is parametrised by the Hubble parameter H =a˙/a, where the dot denotes a
derivative with respect to t. The Hubble parameter today is called H . The scale factor a is normalised
0
such that a=1 today.
8