DEPARTMENT OF PHYSICS BACHELOR OF PHYSICS
Jacinto Paulo da Silva Neto
An Introduction to Large-Scale Structure Formation
Natal-RN
Jacinto Paulo da Silva Neto
An Introduction to Large-Scale Structure Formation
Monografia de Gradua¸c˜ao apresentada ao Departamento de F´ısica Te´orica e Experimental do Centro de Ciˆencias Exatas e da Terra da Universidade Federal do Rio Grande do Norte como requisito parcial para a obten¸c˜ao do grau de bacharel em F´ısica.
Advisor:
Prof. Dr. Farinaldo Queiroz
Federal University of Rio Grande do Norte — UFRN Department of Physics – DF
Natal-RN November 20, 2019.
Silva Neto, Jacinto Paulo da.
An introduction to large-scale structure formation / Jacinto Paulo da Silva Neto. - 2019.
72f.: il.
Monografia (Bacharelado em Física) - Universidade Federal do Rio Grande do Norte, Centro de Ciências Exatas e da Terra, Departamento de Física Teórica e Experimental. Natal, 2019. Orientador: Farinaldo da Silva Queiroz.
1. Física - Monografia. 2. Formação de estruturas em largas escalas Monografia. 3. Evidências de matéria escura
Monografia. 4. Espectro de potência de matéria escura fria -Monografia. I. Queiroz, Farinaldo da Silva. II. Título. RN/UF/CCET CDU 53
Catalogação de Publicação na Fonte. UFRN - Biblioteca Setorial Prof. Ronaldo Xavier de Arruda - CCET
Monografia de Gradua¸c˜ao sob o t´ıtulo An Introduction to Large-Scale Structure Formation apresentada por Jacinto Paulo da Silva Neto e aceita pelo Departamento de F´ısica Te´orica e Experimental do Centro de Ciˆencias Exatas e da Terra da Universidade Federal do Rio Grande do Norte, sendo aprovada por todos os membros da banca examinadora abaixo especificada:
Prof. Dr. Farinaldo S. Queiroz Federal University of Rio Grande do Norte
International Institute of Physics
Prof. Dr. Raimundo Silva
Federal University of Rio Grande do Norte Department of Physics
Dr. Jamerson Rodrigues
Federal University of Rio Grande do Norte Department of Physics
Acknowledgment
I thank my mother, my oldest sister and my stepfather. I have no words to describe how much important they are to me. I keep them close to heart! I also want to thank my girlfriend who encourages me and loves me! She always helps me make right decisions and follow my dreams! Of course, I could not forget my advisor who gives me relevant advice about my studies and career. I also thank my friends who have shared moments inside and outside the academic world talking about physics and everything else.
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No self is of itself alone. It has a long chain of intellectual ancestors. The “I” is chained to ancestry by many factors. . . This is not mere allegory, but an eternal memory.
Resumo
O presente trabalho apresenta uma abordagem simplificada sobre o processo de forma¸c˜ao de estruturas em largas escalas. ´E estudado como as estruturas em largas escalas, tais como gal´axias e aglomerados de gal´axias se formam, evoluem e est˜ao distribu´ıdas no Uni-verso. Tamb´em s˜ao apresentadas evidˆencias para existˆencia de Mat´eria Escura para que possamos fundamentar sua relevˆancia como sendo um dos principais ingredientes desse processo. Finalmente, derivamos uma grandeza chamada Espectro de Potˆencia de Mat´eria que nos permite calcular a distribui¸c˜ao de mat´eria no Universo atual, onde as observa¸c˜oes indicam a existˆencia de uma abundˆancia enorme de Mat´eria Escura Fria e um pouco de Mat´eria Escura Quente.
Palavras-chave: Forma¸c˜ao de estruturas em largas escalas, evidˆencias de Mat´eria Escura, espectro de potˆencia de mat´eria, Mat´eria Escura Fria, Mat´eria Escura Quente.
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Abstract
The present work presents a simplified approach on studies of large-scale structure for-mation. It is studied how large-scale structures such as galaxies and galaxy clusters form, evolve and are distributed in the universe. Evidence for the existence of Dark Matter are also addressed so highlight its relevance as one of the main ingredients of this process. Finally, we give a more quantitative description via the so called Matter Power Spectrum that allows us to calculate the distribution of matter in the current universe where obser-vations indicate the existence of a huge abundance of Cold Dark Matter and a bit of Hot Dark Matter.
Keywords: Large-scale Structure Formation, Evidence of Dark Matter, Matter Power Spectrum, Cold Dark Matter, Hot Dark Matter.
List of Figures
2.1 The rotation curve of a typical galaxy is very similar to that of the NGC 3196. The dots with error bars comes from observations. The upper solid line is considering the dark matter halo, in agreement with observations. The lower solid line is the expected to a galaxy with only baryonic matter. The middle solid line accounts only the contribution of the dark matter halo [1]. . . 4 2.2 Rotation curve data for M31. The purple points are emission line data in
the outer parts from Babcock in 1939 [2]. The black points are from Rubin and Ford in 1970 [3]. The other labels were not mentioned in this work but are more recent then those we treated. The figure was taken from [4]. . . . 5 2.3 An illustration of the CMB angular power spectrum with rough details on
the meaning of the acoustic peaks [5]. . . 6 2.4 A BAO-cartoon produced by the Baryon Oscillation Spectroscopy Survey
(BOOS). The bar traced between two galaxies indicates the scale size of 500 million light-years [6]. . . 7 2.5 An illustration of the concept of BAO, which are imprinted in the CMB
and can still be seen today in galaxy surveys like BOSS. The bar represents the scale distance of 500 million light-years between two galaxies [7]. . . 8 2.6 BBN predictions for light nuclides versus the baryon-to-photon ratio η.
Curve widths: 1σ theoretical uncertainties. Vertical red band: Wilkinson Microwave Anisotropy Probe (WMAP) determination of η. Red circle: predicted values [8, 9]. . . 11 2.7 The solid red line and black dots are the observational CMB power
spec-trum. The green solid line represents the Big Bang model which consider the Universe with abundance splitted into Ωb ' 0.05, ΩDM ' 0.22 and ΩΛ ' 0.74 [8]. . . 13
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2.8 The solid red line and black dots are the observational CMB power spec-trum. The green solid line represents the Big Bang model which con-sider the Universe with abundance splitted into Ωb ' 0.05, ΩDM ' 0 and ΩΛ ' 0.95 [8]. . . 13 2.9 The WMAP all sky map of fluctuations in the temperature of the cosmic
microwave background. The differences in temperature between the hot and cold regions are just a few parts in a hundred thousand as given by equation (2.24) [10]. . . 16 3.1 Data from the Sloan Digital Sky Survey (SDSS). They call it as the SDSS’s
map of the Universe. Each dot is a galaxy and the color is the g-r color of that galaxy [11]. The interesting reader can read more about the g-r color in [12]. . . 18 3.2 The evolution of a sphere of radius R with uniform mass density ρ ≡ ¯ρ. . . 22 4.1 Illustrating how density fluctuations can be a Fourier superposition of
waves. The first panel shows a single wave; the second adds together 13 similar waves running in different directions; The third shows how the reg-ularity of this pattern is destroyed when the waves are given a random shift in phase [10]. . . 32 4.2 Galaxy power spectrum (at small scales) and of the cosmic microwave
back-ground (at large scales). The Lyman-α data measure the linear power spectrum at high k because they can be seen at high redshift, z ≈ 3 [13, 14]. 39 4.3 The correlation function quantify the probability of finding a galaxy in the
small volume dV2 if there is a galaxy in the small volume dV1, a distance r away. Otherwise there is no correlation between the two small volumes, see [15]. . . 39 4.4 The plot shows the suppression of structure growth during the
radiation-dominated phase. Before aeq the fluctuations δ ∝ a2 and after that δ ∝ a. It illustrates the growth of a fluctuation with λ < dH(aeq) which is small enough to enter the Hubble sphere at aenter < aeq [16]. . . 48 4.5 Different transfer functions to different models on structure formation, see
[17]. . . 52 4.6 N-body simulations of structure formation for Hot (left), Warm (middle)
and Cold (right) Dark Matter. Top row: it shows how the Universe would look like at early times (high redshift). Bottom row: it shows how the Universe would look like today (zero redshift), see [18]. . . 53
Contents
Acknowledgment i Resumo iii Abstract iv List of Figures v 1 Introduction 12 Evidence for Dark Matter in the Universe 3
2.1 Galaxy rotation curves . . . 3
2.2 Baryonic Acoustic Oscillations . . . 6
2.3 Big Bang Nucleosynthesis . . . 8
2.4 Cosmic Microwave Background . . . 12
2.5 Large-Scale Structure Formation . . . 14
3 Structure Formation in an Expanding Universe 17 3.1 Inflation Provides the Seeds for Structure Formation . . . 17
3.1.1 What is Inflation? . . . 18
3.1.2 Quantum Fluctuations During Inflation . . . 19
3.2 Small Density Fluctuations in an Expanding Universe . . . 21
3.2.1 Gravitational instability and Jeans length . . . 21
3.2.2 Equation of Motion for Small Density Fluctuations . . . 25
3.2.3 Small Density Fluctuation Solutions . . . 27
4 The Matter Power Spectrum 31 4.1 Properties of the Density Fluctuation Field . . . 31
4.2 Defining Matter Power Spectrum & Correlation Function . . . 35
4.3 Standard Matter Power Spectrum . . . 38
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4.3.2 Primordial Matter Power Spectrum . . . 40 4.3.3 Cold Dark Matter Power Spectrum . . . 41
5 Conclusion 55
Chapter 1
Introduction
If we try to describe how structures form using baryonic matter as the only pressureless component of the Universe we will not be able to explain what is observed today. Fur-thermore, if we want to explain the distribution of galaxies and galaxy clusters in the Universe via the same assumption we will deal with the same problem. In this scenario a nonbaryonic matter (dark matter) arises as a great candidate to solve such Structure Formation problems. However, Cosmology is only another branch of physics where dark matter appears. Particle physicists have studied dark matter as being composed of ele-mentary particles so that they also have tried to solve problems in the standard model via dark matter physics.
We have collected enough evidence for dark matter in the past decades. In this work we plain to correlate the existence of dark matter with large-scale structure Formation. We start reviewing the general aspects of such evidence for dark matter including galaxy rotation curves, Baryonic Acoustic Oscillations (BAO), Big Bang Nucleosynthesis (BBN), Cosmic Microwave Background (CMB) and large-scale structure Formation.
Furthermore we will describe the basic concepts behind structure formation. We begin discussing inflation which provides the seeds for structure formation via quantum fluctu-ations that take place in the early Universe. Later on we will give a detailed description of the small density fluctuations and how they evolve in an expanding Universe. We will describe the solutions behind the equations of motion and their implications.
Thereafter, we present a statistical description of the small density fluctuations intro-ducing a density fluctuation field which obeys the equation of motion already obtained. The physics of the early Universe allow us to set the properties of the density fluctuation field which obeys a Gaussian distribution, as well as its statistical properties that can be described using only either the power spectrum or the correlation function. Then, we will define mathematically the matter power spectrum and its relation with the correlation function. Finally, we will be able to derive the current matter power spectrum and discuss
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the possibility of introduction of a new sort of dark matter in order to improve the fit to data on small-scales.
Chapter 2
Evidence for Dark Matter in the
Universe
From a historical perspective, it is hard to determine exactly when we started using the term dark to refer to some kind of exotic matter. People has been used terms as “dark” and “black” to name something which is not well known. For instance, almost one hundred years after Sir. Isaac Newton published his treatise Philosophiæ Naturalis Principia Mathematica, concerning astronomical body dynamics, John Michell realized that since the gravitational force is universal then it could affect light so that could be astronomical bodies so massive that even light could not escape from its gravitational attraction. Pierre Simon Laplace seems to be the first one to name it as black hole. More historical information can be found in [4]. Currently, two major puzzles science are related to some kind of unknown dark component of the Universe: dark energy and dark matter. In this chapter we will assume the reader is unfamiliar with dark matter physics thus we will review its evidence. The Dark Matter problem is at the interface of Cosmology, Astrophysics and Particle Physics. There are thousands of scientists around the globe trying to detect signals of dark matter particles via direct or indirect detection methods. These methods are not the goal of our work thus we refer to for further details [19]. Throughout we will adopt the Cosmological Standard Model, known as the ΛCDM model, where the dark energy composes ∼68% of the total density, dark matter ∼27% and baryonic matter ∼5%.
2.1
Galaxy rotation curves
Galaxy rotation curves refer to the measurement of the velocity of the stars bounded in a given galaxy. In figure (2.1) we show this radial velocity versus their respective distance away from the galactic center. Studying galaxy rotation curves one can infer how massive
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Figure 2.1: The rotation curve of a typical galaxy is very similar to that of the NGC 3196. The dots with error bars comes from observations. The upper solid line is considering the dark matter halo, in agreement with observations. The lower solid line is the expected to a galaxy with only baryonic matter. The middle solid line accounts only the contribution of the dark matter halo [1].
galaxies are and how that mass is distributed. These rotational velocities are taken from the measurement of the Doppler effect of the spectral lines at different positions along the luminous disk of the galaxy as explained in more detail in [20].
Yet in the late 30’s some “anomalies” were observed in the rotation curves of the NGC 3115 and Andromeda galaxies [4, 21, 2]. It was reported that the ratio of luminosity and mass was not consistent with visible matter only. At the time, it was argued that might exist some kind of faint objects like dwarf galaxies not observed yet to account for the discrepancy.
Studying the Andromeda Galaxy in the 60’s, a team headed by Vera Rubin used a new method that allowed to probe the velocity curve on the tail of the spiral galaxies with unprecedented accuracy. They found that the stellar rotational velocity remains either constant or flat for objects far from the galactic center,see figure (2.1).
It is not an expected result since from Newton’s gravitational law one knows that the velocities of objects orbiting at distances far from the center are smaller than those near the center. Therefore there are two possible explanations: (i) Newton’s gravity does not work to explain galaxy dynamics and there should be a new gravitational force; (ii) there is a dark matter surrounding the galaxy.
Focusing on the latter, this mass should not emit light. We will use the Newton’s shell theorem to derive mathematically what those words mean.
Figure 2.2: Rotation curve data for M31. The purple points are emission line data in the outer parts from Babcock in 1939 [2]. The black points are from Rubin and Ford in 1970 [3]. The other labels were not mentioned in this work but are more recent then those we treated. The figure was taken from [4].
has a stable orbit is
mv2 r = G
M m
r2 . (2.1)
If M = const., therefore the velocity will depend on r via
v ∝ r−1/2, (2.2)
which is not in agreement with observations in figures (2.1) and (2.2). Instead, if M = M (r) ∝ r, therefore, requiring it has a spherical symmetry distribution
ρ ∝ r−2. (2.3)
and
v ∝ const., (2.4)
a power law which is in agreement with observations. The important aspect of galaxy rotation curves is the fact that the mass extracted from the luminous object is incapable of producing such large velocities at sufficiently large distances, begging then for the presence of a non-visible, thus dark matter component in galaxies. Interestingly it has been shown that about 90% of a galaxy mass is in the form of dark matter.
It is important to emphasize that even today the astronomers did not observe any kind of invisible baryonic matter (baryonic dark matter) within galaxies which could explain the galaxy rotation curves. Although the Modified Newtonian Dynamics (MOND) models can also explain galaxy rotation curves they cannot solve other problems where dark matter is required such as clusters of galaxies, Cosmic Background Radiation and
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Figure 2.3: An illustration of the CMB angular power spectrum with rough details on the meaning of the acoustic peaks [5].
Baryonic Acoustic Oscillations. We shall see in next subsections how a kind of weakly interacting dark matter is important to explain the observations.
2.2
Baryonic Acoustic Oscillations
The Baryonic Acoustic Oscillations (BAO) are sound waves produced in the early Uni-verse when the baryonic matter (electrons and nucleons) were tightly coupled to photons forming a “baryon-photon fluid”. At high temperature the photon prevented electron and proton to combine to form hydrogen. Therefore, the free electrons strongly scattered the photons of the Cosmic Microwave Background. Indeed, Thomson scattering between the photons and electrons and coulomb interactions between the electrons and baryons were sufficiently rapid that the photon-baryon system behaves as a single tightly coupled fluid. The BAO happens before photons decoupling to form the Cosmic Microwave Back-ground (CMB) and the recombination epoch that took place almost at the same redshift z ∼ 1100, however they are different processes. Such sound waves are formed due to fluc-tuations in the baryonic matter. Those flucfluc-tuations give rise to the pattern on large-scale structure, that is, the distribution of galaxies and galaxy clusters we observe today. It is possible to infer BAO from the CMB angular power spectrum, see (2.3).
Since the photons had a higher density than the baryons in the baryon-photon fluid, the sound waves could travel at high speed cs ∼ 170, 000 km/sec. Those sound waves travelled 400, 000 years before recombination at a large fraction of the speed of light so that it covered large distances at that epoch, about a radius of about 400, 000 light-years. After recombination it remained to expand up to now getting a current size of 500
Figure 2.4: A BAO-cartoon produced by the Baryon Oscillation Spectroscopy Survey (BOOS). The bar traced between two galaxies indicates the scale size of 500 million light-years [6].
million light-years. Therefore, we expected to see the galaxy pairs distributed with such a distance separation today. That is, the average distance between two galaxies should be 500 million light-years today. This is exactly what we observe!
Up to now we have given no explanation to how those waves really form. The standard mechanism is based on the existence of a weakly interacting dark matter which started to cluster even before the baryonic matter. Therefore, at the BAO epoch there were already dark matter gravitational potential wells since it does not interact with radiation and baryons via any fundamental known force except gravity. These potential wells allowed the baryonic matter falls into them, forming baryonic clumps, but thereafter, the radiation which were abundant washed the baryons out because of its pressure gradient.
The dark matter do not feel the radiation so that it could not to be also washed out. Thus, the potential wells remained. This process continued taking place until the epoch when the photons decoupled from the baryon-photon fluid because it had no energy enough to scatter the electron from the nucleons and then free-stream while the baryons are left sitting in a spherical shell around the initial clump of dark matter. We say the expansion rate of the Universe becomes larger than the scattering reaction rate. The results of this mechanism are in great agreement with the observations, see figures (2.4) and (2.5).
The BAO’s measurements is done with great accuracy so that they are used as a standard ruler, that is, a trustworthy way to set distances in the Universe. It improves the way we make distance measurements in the Universe. As we could see, dark matter is an important ingredient in this process which happened at small scales, where MOND had no effect. Thus MOND does not accommodate BAO. In the next section it will become
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Figure 2.5: An illustration of the concept of BAO, which are imprinted in the CMB and can still be seen today in galaxy surveys like BOSS. The bar represents the scale distance of 500 million light-years between two galaxies [7].
even more interesting.
2.3
Big Bang Nucleosynthesis
This section aims to provide a rough idea of how the Big Bang Nucleosynthesis (BBN) constitutes an evidence for dark matter highlighting in particular the necessity for a non-baryonic origin. Big Bang Nuclesynthesis represents an area of research which goes beyond the scope of this work. Here we are only interested in the basic ideas that allow us to understand the importance of the results.
The BBN took place at early time t ≈ 0.01 to 180 sec, i.e. at a redshift z ∼ 109, when the Universe was radiation-dominated. At that epoch the Universe was in a thermal bath with temperature T ∼ 0.1 − 10 MeV [22]. The BBN provides the explanation to the production and abundance of the light elements (D, 3He, 4He, 7Li) in the Universe. The heavier elements are formed when stars start to burn hydrogen atoms. The BBN’s results can be obtained through only one free parameter, the baryon-to-photon ratio
η ≡ nB/nγ, (2.5)
where nB = NB/V is the number density of baryons and nγ = Nγ/V is number density of photons. BBN is one of the pillars of the ΛCMD model as its theoretical predictions were confirmed by many different observations.
weak interaction are quick so that the reactions below can occur
n + e+ ↔ p + ¯νe (2.6)
n + νe↔ p + e− (2.7)
n ↔ p + e−+ ¯νe. (2.8)
Since these reactions are in equilibrium the number of protons is equal to the number of neutrons, that is, Nn/Np = 1. These reactions will remain in equilibrium while the reaction rate Γν is larger than the expansion rate H. When Γν = H, neutrinos decouple from the thermal bath at temperature T ≈ 0.8 MeV [8] and then free-stream. Therefore, protons cannot to convert into neutrons anymore. Defining the particle number density as
N = mT 2π
3/2
e−m/T, (2.9)
this leads to a fixed value to the neutrons-protons ratio Nn
Np
' e−∆m/T ' 1
6, (2.10)
where ∆m = 1.2 MeV is the mass difference between neutrons and protons. However, free neutron can decay into photon increasing the number of protons related to that of neutrons. That is,
Nn Np
' 1
7. (2.11)
Now the number of proton are seven times larger than that of neutrons. At this point the synthesis of light elements start to run.
The first element to form was deuterium D via reaction
n + p → D + γ, (2.12)
which has a binding energy of ED = 2.22 MeV.
As CMB photons have a Plackian distribution, even if the average temperature of the photon is below 2.22 MeV, that will be still a large number of photons capable of distroying Deuterium. The baryon-to-photon number density, baryonmeter η = nb/nγ = 6 × 10−10 [8], thus we have much more photons than baryons indeed. At the BBN epoch they were in thermal bath with baryons at high energy. They are the most perfect natural black-body spectrum ever seen. The ones which drive the dissociation of D are those that belong to high energy tail of the energy distribution.
As the Universe evolves and expands the temperature goes down. When the average temperature of the photon of the CMB is T ≈ 0.1 MeV, photons in thermal bath do not have enough energy to prevent the deuterium to be formed. Then, the deuterium abundance quickly rises [9, 8]. From D other light elements can be formed via strong
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interactions (nuclear interaction). For instance, the most stable and abundant element, the 4He, can be formed via
D + D → 4He. (2.13)
Since 4He is the most stable BBN element, once the hydrogen H is formed before, it is plausible to assume that all neutron become 4He. This gives N
4He = 2Nn, thus the total
mass fraction transformed into4He is Yp = 2Nn Np+ Nn = 2(Nn/Np) 1 + Nn/Np ≈ 0.22. (2.14)
Therefore, approximately 22% of the total baryon abundance Ωb is converted into 4He. The H comprises 75% of Ωb so that adding up 4He and H we find that they amount to 97% of Ωb [8]. The other elements contribute poorly as one can see [8]
YD ≈ 10−5 (2.15)
Y3He ≈ 10−5 (2.16)
Y7Li ≈ 10−10. (2.17)
Since the 4He is the most abundant element and its governs Ω
b, then if its abundance is changed by a small amount, the abundance of the other elements will significantly change to keep Ωb unchanged.
In order to determine η we need to obtain the abundances of the other elements. However, BBN alone does not account for the entire abundance of the light elements, some productions arise from Stars, Nebulae and other astrophysical objects. To account those contributions we need to understand at least the concept of metalicity which is associated to the abundance of different chemical elements relative to H and4He [8, 9]. It is known that the4He abundance is inversely proportional to the metalicity of astrophysical objects, and by studying the relation between metalicity and Helium abundance it was concluded that
Y4He = Yp = 0.238 ± 0.002 (statistical) ± 0.005 (systematical). (2.18)
The contribution of other elements can be observed and computed in similar way. For instance, as deuterium D is not produced in any astrophysical object its total abundance remained unchanged up to now. The outcomes gives [8]
D H = (2.9 ± 0.3) × 10 −5 (2.19) 9.0 × 10−11< 7Li H < 2.8 × 10 −10 , (2.20)
which is in good agreement with BBN prediction in figure (2.6). Finally, in the end of the day, combining the BBN predictions and astrophysical observations, the baryon-to-photon
Figure 2.6: BBN predictions for light nuclides versus the baryon-to-photon ratio η. Curve widths: 1σ theoretical uncertainties. Vertical red band: Wilkinson Microwave Anisotropy Probe (WMAP) determi-nation of η. Red circle: predicted values [8, 9].
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ratio can be calculated resulting in [23]
η = 2.74 × 10−8Ωbh2 ' (6 ± 0.5) × 10−10 (2.21) which translates into
0.018 < Ωbh2 < 0.023, (2.22) where h is the normalisation factor of the Hubble parameter. We refer the reader to [8, 24] for discussions concerning the relation between Ωbh2 and η.
In summary, the baryonic matter of the Universe is a small part of its total density. In other words, there is something else in the Universe, once Ωb < Ωt, with Ωt = 1. Therefore, if the Universe is consisted of some sort of dark matter, with ΩDM ' 0.23, it cannot be baryonic since ΩDM ' 5Ωb. Next subsections will address the importance of this nonbaryonic matter to explain CMB observations and Large-Scale Structure Formation (LSSF) observations.
2.4
Cosmic Microwave Background
We will introduce the key aspects of the CMB and focus on the findings relevant to our reasoning. For more details we refer to [25, 26, 27]. Before the origin of the CMB, the baryonic matter was formed of free nuclei and electrons that were often scattered by photons and consequently their mean free path was very short. Radiation and baryons were tightly together forming an unique fluid as we discussed above. As the Universe expanded and cooled photons were eventually unable to rip electrons off nuclei. Quanti-tatively speaking the rate expansion H became larger than the scattering rate Γγ−e, i.e. Γγ−e < H. This period is known as CMB decoupling. When it happened photons de-coupled from the plasma and started free-streaming, that is, their mean free path became very large allowing them to travel across the Universe and reach us today. The spherical surface which delimits the period of CMB photons decoupling is called the last scattering surface. This surface was formed at a redshift z ∼ 1100. Those photons reach us today from all directions leading to a spherical symmetry, then it makes the radiation field nearly uniform and isotropic, supporting the assumption of homogeneity and isotropy which are the foundation of the cosmological principle.
Observing astrophysical bodies at large distances means looking at the past. Thus, CMB allows us to probe the early Universe [8]. CMB is a relic from the early universe which carries information imprinted on it from much earlier times [25]. For instance, if any cosmological model wants to modify inflation (which we shall discuss it soon), it needs to account a nearly isotropic CMB spectrum.
Figure 2.7: The solid red line and black dots are the observational CMB power spectrum. The green solid line represents the Big Bang model which consider the Universe with abundance splitted into Ωb' 0.05,
ΩDM ' 0.22 and ΩΛ' 0.74 [8].
Figure 2.8: The solid red line and black dots are the observational CMB power spectrum. The green solid line represents the Big Bang model which consider the Universe with abundance splitted into Ωb' 0.05,
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The CMB radiation field exhibits the most perfect black-body spectrum at tempera-ture T0 = 2.725. The temperature of CMB photons when it was created can be estimated via
T0 = T (z)
1 + z, (2.23)
which gives T ' 3000 K. At that time neither galaxies nor galaxy clusters had been formed. To grasp the importance of the CMB, remember that the Hubble telescope and the Sloan Digital Sky Surveys (SDSS ) probe redshift z ' 5 and z ' 3, respectively [8].
The CMB angular power spectrum is theoretically obtained by assuming a spherical symmetry so that they use the spherical harmonics to describe it. Many statistical con-siderations are made in order to have a radiation field which is described by the angular power spectrum or the two-points correlation function (correlation function). Those con-cepts will be introduced and discussed in more detail in chapter 4. Basically, two points (or angles) are randomly drawn in the sky and respective photons temperature are mea-sured. This procedure is made until the most possible amount of points (or angles) are covered. Then, the correlation function takes each pairs of points (or angles) and com-pute their temperature correlation. Photons coming from different paths of the sky do not have precisely the same temperature, there are fluctuations. As we shall see, the power spectrum is the Fourier transform of the correlation function and vice-versa. Therefore, they provide the same information.
CMB surveys as Planck and WMAP (Wilkinson Microwave Anisotropy Probe) probe the sky in order to measure accurately the size of such fluctuations. It has been noticed that the CMB angular power spectrum strongly depends on the abundance of dark and baryonic matter in the Universe. Therefore, it is possible to extract the dark matter abundance via CMB power spectrum [8].
Comparing the CMB data given by Planck with the results predicted by Big Bang model we can observe the need for dark matter. The abundances must be Ωb ' 0.05 for baryons, ΩDM ' 0.22 for dark matter and ΩΛ ' 0.74 for dark energy; this is shown in figure (2.7). Otherwise, if we assume, for example, ΩDM ' 0, ΩΛ ' 0.95 and Ωb ' 0.05 the values do not corresponds to the data as shown in figure (2.8) [8]. The interested reader can find more details about CMB data and analyses in [28]. In a nutshell, CMB needs dark matter.
2.5
Large-Scale Structure Formation
Since the goal of this work is on Large-Scale Structure Formation (LSSF), we shall review the current standard understanding of the evolution and distribution of structures in the Universe. In this subsection we will briefly discuss how dark matter plays a role in the
formation of galaxies and other large-scale structures as we observe them today.
As discussed above CMB gives information about the Universe when it was very young and we observe a nearly isotropic CMB spectrum we can deduce that the Universe was homogeneous at that time. Although we observe galaxies and cluster of galaxies. Therefore, the fact that the Universe is not perfectly homogeneous and isotropic must be the reason behind the formation of structures.
The current explanation is that those structures arise from small density fluctuations in matter when the Universe was younger than at the CMB epoch. Such a small density fluctuations could grow via gravitational instability as long as the Universe was evolving. Those gravitational collapse allows small clumps to form so that each clump has a grav-itational potential well. In this way, those small clumps generate small inhomogeneity which should be observed in CMB spectrum. Therefore, should exist a relation between inhomogeneity in matter distribution and CMB nearly isotropic spectrum.
Since they are really small, they were hard to be detected. Finally, in 1992, the Cosmic Background Explorer (COBE) satellite observed the expected small fluctuations in the CMB temperature of the order of
δρ ρ ∝
δT T ∼ 10
−5. (2.24)
This observation provides information about the distribution of matter at the time of CMB. Therefore, any model which aims to explain how large-scale structures form and evolve needs to be in agreement with CMB.
Assuming the standard cosmology, when we consider the Universe consisted of only one kind of matter, the baryonic matter, we cannot explain the distribution of matter in the Universe today because more time should be necessary to galaxies and then other structures to form. However, when we put dark matter at play it predicts the distribution of large-scales structures as we observe today!
We will show later that fluctuations in baryonic matter could not start to grow earlier because baryons were coupled to photons for far too long. The radiation pressure did fight against gravitational collapse in baryonic matter so that just after photon decoupling, fluctuations in baryonic matter could then collapse and start to grow. However, as dark matter only interacts weakly with photons it could have started to form clumps of matter and gravitational potential wells much before the photon decoupling. Baryonic matter, therefore, could fall into those potential wells. CMB photons traped in this potential well formed by dark matter appear hotter (red) or cooler (blue) in CMB spectrum if they were falling or escaping from the dark matter gravitational potential wells, respectively; see figure (2.9).
16
Figure 2.9: The WMAP all sky map of fluctuations in the temperature of the cosmic microwave back-ground. The differences in temperature between the hot and cold regions are just a few parts in a hundred thousand as given by equation (2.24) [10].
order to fully grasp the importance of dark matter and its role in the formation of sctruc-turs we will explaining, in the next chapet. without deep details address how inflation provides small density fluctuations. Then, we explain the Jeans scale and gravitational collapse mechanism. Thereafter, we derive the equation of motion which describes how those small density fluctuations evolve in an expanding Universe. Finally, we analyze so-lutions for that equation and we discovery that fluctuations of baryonic matter will only grow in a matter-dominated Universe. This is a finding we will come back to many times throughout this work.
Chapter 3
Structure Formation in an
Expanding Universe
Large-scale Structure Formation is a branch of Cosmology which studies how structures evolve and are formed in an expanding Universe. Although we assume the cosmological principle to describe the Universe as a whole, we observe some irregularities in mass density distribution. It happens because the Universe is homogeneous and isotropic only at large-scales, around ∼ 100 Mpc (see figure 3.1). On smaller scales such as ∼ 20 Mpc, which characterise a distribution of galaxies in space, we already start to see such an irregularities.
As we said above, COBE (Cosmic Background Explorer) and WMAP telescopes have taught us that there were small density fluctuations in the matter distribution even at the time of CMB. The main idea concerns that those small density fluctuations grew as the Universe expanded and formed the building blocks of the observed structures.
In order to understand how these fluctuations arose we need to introduce the inflation mechanism. We will adopt the standard adiabatic fluctuation (also called curvature fluctu-ation). Then, we go through the concept of gravitational instability and the Jeans length which are the important ingredients in our reasoning. Finally, we derive the equation of motion which describes the evolution of the small density fluctuations in an expanding Universe via a Newtonian approach.
3.1
Inflation Provides the Seeds for Structure
For-mation
Inflation is an important episode of the Universe’s history which is needed to address some problems in cosmology such as the flatness problem and the horizon problem. The interested reader can obtain more information about inflation in reference [29, 30, 31, 32,
18
Figure 3.1: Data from the Sloan Digital Sky Survey (SDSS). They call it as the SDSS’s map of the Universe. Each dot is a galaxy and the color is the g-r color of that galaxy [11]. The interesting reader can read more about the g-r color in [12].
33, 34, 35].
3.1.1
What is Inflation?
It was an exponential expansion of the Universe in which the proper distance to the Hubble sphere (the horizon) remained constant and it was coincident with the event horizon. Then, every physical entity beyond the Hubble scale is not casually connected with those which are inside it! Inflation took place at the GUT time when the Universe was very young. As the Universe entered the inflationary phase it moved from the micro-scale to macro-micro-scale size. It is like that the Universe had the size of a rice grain and in a short time became a watermelon.
Inflation is described by a scalar field which governed the energy density of the Universe during such period. This scalar field is called inflaton and its potential has a property of mimicking the cosmological constant behaviour, then when its potential dominates the Universe the inflaton was in a false quantum vacuum state and as the field rolled down to its true quantum vacuum state it caused the Universe to expand exponentially. After this period the Universe the energy density of any relic species will be virtually zero, but after the inflation decay, the universe is reheated, and the matter and radiation component arise.
3.1.2
Quantum Fluctuations During Inflation
Assuming adiabatic fluctuations, any scalar quantity χ is described by an unique fluctu-ation in expansion with respect to the background [30]
Hδt = Hδχ ˙
χ . (3.1)
For the energy density and the pressure we obtain δρ ˙ ρ = δp ˙ p , (3.2)
From equation (3.1) we can understand why it also is called curvature fluctuation: a given time displacement δt causes the same relative change δχ/ ˙χ for all quantities. The fluctuations are democratically shared by all components of the universe [30]. There are also isocurvature fluctuations but it is different approach to inflation which we are not interested here. The interested reader can read more about it in [30, 29].
The exponential expansion provided excitation in the quantum aspects of the inflaton and stretches its fluctuations from microscopic to macroscopic scales. During the process of expansion some fluctuations became larger than the Hubble sphere, that is, they were outside the horizon. However, as inflation was finishing fluctuations which had a size close to the horizon size could renter the Hubble sphere. After inflation the Universe enters a reheating phase where many physical processes are evolved. Then, the Universe becomes radiation-dominated providing a growing rate of the fluctuations which were inside the Hubble sphere.
In order to comprehend how the structures on large-scale formed and evolved we need to use linear perturbation theory to describe the process which the inflaton field underwent during the (quasi) de Sitter phase (exponential expansion, H = cte). The perturbations is associated to quantum fluctuations in the field so that the inflaton field as a whole is both quantum and classical.
The Lagrangian density of a scalar field φ(x, t) is given by L = 1
2∂µφ∂
µφ − V (φ) (3.3)
where µ = 0, 1, 2, 3 and V (φ) is the potential of the field. The form of the potential will change according the inflation model we consider.
Since the potential energy of the inflaton V (φ) dominates the energy density of the Universe during inflation we can set that a perturbation in φ causes the same perturbation in the energy content of the Universe. Then, we can write
20
where Tµν is the energy-moment tensor of the inflation
Tµν = ∂µφ∂νφ − gµνL. (3.5)
Using the same logic we can think that a perturbation in the energy content causes a perturbation in the metric, once the Einstein’s equation tell us that there exists an equivalence between energy and geometry. Then, we obtain
δTµν → 8πGδTµν = δRµν− 1
2δ gµνR → δgµν, (3.6) As the small perturbations modify the metric it implies that there will be some changes in the dynamics of the field, that is, the Kelin-gordon equation will also be perturbed
δgµν → δ − ∂µ∂µφ + ∂V ∂φ = 0. (3.7)
Therefore, we conclude that during inflation the inflaton field was tightly coupled with the metric and then they must be studied together
δφ → δgµν. (3.8)
We will not go into the consequences of it because it outside the scope. Although one can imagine that one of the consequences is the production of primordial gravitational waves. In order to understand the physics behind we will study the dynamics of the inflaton field. The equation of states for ρ and p of the inflaton field φ read
ρ = ( ˙φ) 2 2 + (∇φ)2 2a2 + V (φ) and P = ( ˙φ)2 2 − (∇φ)2 2a2 − V (φ), (3.9) which leads to [36, 30] ¨ φ + 3H ˙φ − ∇ 2φ a2 + ∂V (φ) ∂φ = 0. (3.10)
As we said before, the inflaton field has two parts in this approach: the classical and the quantum. Equation equation (3.11) shows that the quantum part is the perturbed one. Regarding the energy-momentum tensor it is the small density fluctuation which we want to describe the evolution while the Universe expands during the Sitter phase.
φ(x, t) = φ0(t) + δφ(x, t) (3.11)
The classical equation of motion to the classical part will be as the follow where we are neglecting the spatial derivatives because the energy-momentum tensor has a form of a perfect fluid ¨ φ0 + 3H ˙φ0+ ∂V (φ0) ∂φ0 = 0. (3.12)
Now we can perform the total field in equation (3.11) back into the equation (3.10) in order to split and to get only the equation for the density fluctuation. The result is
δ ¨φ + 3Hδ ˙φ + ∂ 2V
∂φ2δφ = 0. (3.13)
When we differentiate equation (3.12), we obtain ... φ0+ 3H ¨φ0+ ∂2V ∂φ2 0 ˙ φ0 = 0. (3.14)
Equations (3.13) and (3.14) will have the same solution when we consider k a2H (scale λ outside the Hubble radius). That is, φ0 and δφ can solve the same equation, see [30]. Therefore, those solutions must be related to each other by a constant of proportionality which depends on space only [30]
δφ = −δt(x) ˙φ0. (3.15)
Put it into the equation (3.11) we obtain
φ(x, t) = φ0 x, t − δt(x)
(3.16)
Conclusion
From the equation above we conclude that the inflaton field does depend on the spatial coordinates. Furthermore, we see that it is not entirely homogeneous since it has fluctua-tions present. These fluctuafluctua-tions and inhomogeneities will arise in the metric and change the energy density distribution [36].
3.2
Small Density Fluctuations in an Expanding
Uni-verse
3.2.1
Gravitational instability and Jeans length
The density of any component can be written as a function of position and time [33]. Therefore, the spatial average density is given by
¯ ρ(t) ≡ 1 V Z V ρ(r, t)d3r. (3.17)
For such a component we may assume the existence of some deviations in its distribution, which can be treated as perturbations or even as fluctuations. To turn it readable, the dimensionless density fluctuation is written in this way
δ(r, t) ≡ ρ(r, t) − ¯ρ(t) ¯
22
Figure 3.2: The evolution of a sphere of radius R with uniform mass density ρ ≡ ¯ρ.
There are two possibilities for the level of fluctuations. If δ > 0 we get an over-dense region. Mathematically speaking there is no upper limit but physically might there exist since the Universe’s energy could be finite. If δ < 0, that region is said to be under-dense and as we will see exists a minimum at δ = −1, ρ = 0.
Model Building
We assume these fluctuations are small. It means that |δ| 1 and therefore it is possible to apply linear perturbation theory to study their evolution under the influence of gravity. We start supposing a approximately static, homogeneous and matter-only Universe with uniform mass density ¯ρ (see figure 3.2). If we add a small uniform amount of mass within a sphere of radius R then its density will becomes ρ = ¯ρ(1 + δ) with δ 1. As assumed that this small density excess added is uniform then we can compute its contribution to the gravitational acceleration at the sphere’s surface
¨ R = −G(∆M ) R2 , ∆M = 4π 3 R 3ρδ¯ (3.19) ¨ R = −G R2 4πR3ρδ¯ 3 = −4π 3 GR ¯ρδ ¨ R(t) R(t) = − 4π 3 G ¯ρδ(t). (3.20)
Note that δ > 0 then ¨R < 0, the sphere collapses inward. Now we have two unknown variables δ(t) and R(t). We ought to assume the existence of such a fluctuation which causes the sphere collapses inward to build the model for gravitational instability. In order to describe how δ(t) evolves we need to find another equation within this system to determine R and then extract δ(t). Therefore, take δ > 0 but with δ 1. From the conservation of mass we obtain that the mass of the sphere does not changes during the
collapse so that we can use it to construct the another equation. Given that before the collapse we have [33] M0 = 4π 3 ρR¯ 3 0, and R0 = 3M0 4π ¯ρ 1/3 = cte (3.21)
and during it we have
M = 4π
3 ρ1 + δ(t)R¯
3(t), (3.22)
using mass conservation M0 = M , we obtain
R(t) = [1 + δ(t)]−1/3R0.
Taking δ 1 then we expand the above equation via Taylor series, neglecting second order terms in δ, we obtain an equation for the radius like
R(t) ≈ R0 1 −δ(t) 3 . (3.23)
Finally, we differentiating the equation (3.23) to get ¨
R R0
≈ −δ¨
3. (3.24)
Now we have two equations and two variables to determine. Combining the equa-tions (3.20) and (3.24) we find an equation which describes how these small density fluctuations evolve as the sphere gravitationally collapses
¨
δ = 4πG ¯ρδ, (3.25)
δ(t) = A1et/τdyn + A2e−t/τdyn, (3.26) where τdyn is the dynamical time for the sphere collapses and it is defined as
τdyn = c2 4πG ¯ρ 1/2 (3.27) We note that the dynamical time does not depends on R but only on ¯ρ. As the time passes only the growing exponential term of the above equation will be important. From the equation (3.26) we see that gravity tends to make small density fluctuations grow exponentially with time for a static and pressureless medium. Next paragraph we will introduce the concept of Jeans length which arises from the generalization of the assump-tion about the medium. We will consider that δ > 1 but the sphere could not collapse immediately because the pressure of the medium.
24
Jeans length
As we studied before for a static and pressureless medium there will be collapse only when the fluctuations becomes positive and it happens when the region is over-dense. We also studied that in such a case the small fluctuations will grow exponentially. Note, however, that when the medium does not fit those conditions we will have another behavior. For instance, the air in the Earth’s atmosphere yields a dynamical time for collapse of τdyn ≈ 9 hours but it does not collapse anytime [33]. What is happening? The pressure is holding the gravity collapse.
In order to get a more general way to understand gravitational instability we need to assume that such a Universe is not pressureless. Since a sphere of gas is collapsing inward by its own gravity if the dynamical time is larger than the time for the pressure gradient to be build up then a hydrostatic equilibrium will be achieved. Otherwise, if the dynamical time is smaller than the pressure gradient time then the sphere collapses and the hydrostatic equilibrium is broken.
Taking an over-dense sphere (δ > 0) with initial radius R where the pressure is present (p 6= 0), when it tries to collapse it will be countered balanced by a steepening of the pressure gradient within the fluctuation. Anyway this ”response” is not instantaneous because any change in pressure travels at the sound speed. As the particles change their positions they perturb the others around and generated sound waves. Then, the time it takes for the pressure gradient do build up in such a region is given by
tpre ∼ R cs
. (3.28)
Without pressure (p = 0) it would collapse on a timescale given by the dynamical time written as τdyn ∼ 1 (G ¯ρ)1/2 ∼ c2 G ¯ρ 1/2 . (3.29)
For a medium with equation of state given by w = kT /µc2, the speed of sound is described by cs= c dp d ¯ρ 1/2 = c√w. (3.30)
Having discussed the basic ideas about the relationship gravity and pressure in a over-dense region, we need to define a reference size associated to the collapse time of an over-dense region and the sound speed in this region. This reference size is so-called Jeans length, it is written as
λJ = cs πc2
G ¯ρ 1/2
= 2πcsτdyn. (3.31)
From that definition we can immediately analyze the conditions which cause the grav-itational collapse. Take, for instance, the case when the dynamical time is larger than
pressure gradient time, i.e, tpre < τdyn R cs < λJ cs → R < λJ, (3.32)
that is, the hydrostatic equilibrium will be attained and there is no collapse. In other words, for a density fluctuation to be stabilized by pressure against collapse this region must be smaller than the Jeans length. Over-dense regions smaller than Jeans length produce stable sound waves due to oscillations in density [33].
Taking the case when the dynamical time is smaller than the time for the pressure gradient to build up, i.e, tpre > τdyn we get
R cs
> λJ cs
→ R > λJ, (3.33)
that is, the hydrostatic equilibrium is broken and the sphere collapses inward. Over-dense regions larger than the Jeans length collapse under their own gravity.
To develop a description for density fluctuations on cosmological scales we consider the Universe being flat (k = 0, Ωtotal = 1) and that there are density fluctuations with amplitude |δ| 1 in the mean density ¯ρ. Since the characteristic time for expansion is
H−1 = 3c2 8πG ¯ρ 1/2 (3.34) and the dynamical time is given by the equation (3.27), rearranging the terms
H−1 = 3 2
1/2
τdyn (3.35)
and the Jeans length in an expanding flat universe becomes
λJ = 2πcsτdyn= 2π 2 3 1/2 cs H. (3.36)
As we can see from the above equations we can associate the dynamical time, the time for an over-dense region collapse, to the Hubble time, the timescale for expansion. Now we can go beyond on our study about the cosmological implications of such a concepts. It means we are more close to get a nice description about how the small density fluctuations grew generating the anisotropies in CMB and large-scale structures we observe today.
3.2.2
Equation of Motion for Small Density Fluctuations
Spherical Newtonian Model
The goal of this section is to build a model to describe the evolution of small density fluctuations in an expanding universe. A Newton’s analysis seems to be sufficient to
26
understand some main ideas and concepts [33]. Consider a Universe that is expanding and it is filled up by a pressureless matter with mass density ¯ρ(t) that evolves with the scale factor via the relation ¯ρ ∝ a(t)−3.
Taking a spherical region with radius R that is larger than the Jeans length λJ and smaller than the Hubble radius cH−1, its mass density after added some amount will be
ρ(t) = ¯ρ[1 + δ(t)], (3.37)
where |δ| << 1 to keep the oscillations at first order like before. However, considering all terms it is possible to perform the Newton’s analysis to get an equation that describes the evolution of the small density fluctuations in terms of the scale factor once there exists a relation between the mass density of some component and the scale factor. The total gravitational acceleration on the sphere’s surface is then given by
¨ R = −GM R2 = − G R2 4π 3 ρR 3 = −4π 3 G ¯ρR − 4π 3 G( ¯ρδ)R (3.38) where M = 4π 3 ρR 3 = 4π 3 ρ(t)[1 + δ(t)]R(t)¯ 3 . (3.39)
The equation of motion for a particle at the surface of this sphere is then ¨ R = − 4π 3 G ¯ρ + 4π 3 G( ¯ρδ) R ¨ R R = − 4π 3 G ¯ρ − 4π 3 G( ¯ρδ) (3.40)
and from the conservation of mass (3.39), using the relations between the mass density and the scale factor, the equation for the evolution of the radius R is
R(t) ∝ ¯ρ(t)−1/3[1 + δ(t)]−1/3
∝ a(t)[1 + δ(t)]−1/3. (3.41)
The equation above inform us the relation between the radius of a spherical region and the scale factor of the Universe given some small density fluctuations within that region. That is, when that region is slightly over-dense the growth of the radius R will be slightly less rapid than the scale factor and the opposite will happen when the region is slightly under-dense. It means that denser the region the lesser is its expansion [33].
Taking the second derivative of the equation (3.41) for |δ| 1 ¨ R R = ¨ a a − 1 3 ¨ δ − 2 3 ˙a a˙δ, (3.42)
and combining the equations (3.42) and (3.40) ¨ a a − 1 3 ¨ δ − 2 3 ˙a a˙δ = − 4π 3 G ¯ρ − 4π 3 G( ¯ρδ), (3.43)
we obtain a coupled equation of motion involving the scale factor and the small density fluctuation. A simplifying case limit occurs when δ = 0, which leads to
¨ a a = −
4π
3 G ¯ρ, (3.44)
which is the known accelerating equation of Friedmann for a homogeneous, isotropic and pressuless matter-dominated Universe. Therefore, putting it into the equation (3.43)
1 3 ¨ δ + 2 3 ˙a a˙δ = 4π 3 G( ¯ρδ), (3.45) ¨ δ + 2H ˙δ = 4πG ¯ρδ, (3.46)
we obtain a linear equation of motion which describes the evolution of small density fluctuations δ, where the term 2H ˙δ is so-called the Hubble friction that slow the grow of the fluctuations. If the universe is static, H = 0, then the equation (3.46) becomes
¨
δ = 4πG ¯ρδ, (3.47)
which is the same as that equation (3.25).
3.2.3
Small Density Fluctuation Solutions
This section brings applications of the equation (3.46) in order to show that small density fluctuations can only grow in amplitude inside the Hubble sphere when the Universe is dominated by pressureless matter.
Λ-dominated Universe
If the Universe is dominated by some kind of material with equation of state’s parameter w ≈ −1/3, then ¯ρ 1 and H = HΛ. The equation of motion will turn into
¨
δ + 2HΛ˙δ ≈ 0, (3.48)
thus
δ(t) ≈ C1+ C2e−2HΛt. (3.49) This solution tell us that the fluctuations in matter density reach a constant fractional amplitude when the Universe is dominated by dark energy [33].
28
Radiation-dominated Universe
Since the Universe is dominated by radiation the energy density parameter of pressureless matter is ¯ρ 1 and the Hubble parameter is written as H = 1/2t, therefore the equation of motion which describes the evolution of a small density fluctuation in matter density becomes ¨ δ + ˙δ t ≈ 0 (3.50) with solution δ(t) ≈ B1 + B2ln(t). (3.51)
Therefore, at the radiation epoch the small density fluctuations grow in a logarithmic rate [33].
Matter-dominated Universe
We consider the Universe being flat (k = 0) and dominated by matter so that Ωm = 1. We have ¨ δ + 4 3t˙δ − 2 3t2δ = 0. (3.52)
A nice guess to solve this equation is a power-law of the form Dtn. Putting it into the above equation it gives us the possible solutions which are n = −1 and n = 2/3
n(n − 1)Dtn−1+ 4 3tnDt n−1− 2 3t2Dt n= 0 n(n − 1) + 4 3n − 2 3 = 0, (3.53)
thus the general solution is
δ(t) ≈ D1t2/3+ D2t−1. (3.54) As the Universe expands and the time is passing the second term (decaying mode) will becomes smaller compared to the first one (growing mode) so that the growing mode becomes dominant. Finally, when only the growing mode survives the density fluctuations will grow at the rate (while |δ| 1)
δ ∝ t2/3∝ a(t) (3.55)
δ ∝ 1
1 + z. (3.56)
Therefore, this gives the solution for the evolution of the small density fluctuations inside the Hubble sphere. This results is of great importance in order to explain the evolution of structure in the expanding Universe [33].
Mixture-dominated Universe
In order to give a more general perspective consider the Universe dominated by radiation and matter at some epoch (e.g., at matter-radiation equality). Therefore, the total density is ρ = ¯ρm(1 + δ) + ¯ρrad so that there are no fluctuations in the radiation density. We define a new parameter which relates both matter density and radiation density,
η ≡ ρ¯m ¯ ρrad
= η0a (3.57)
since ρm ∝ a−3 and ρrad ∝ a−4. Recall that H ≡ ˙a/a, then ˙ η a = η0 ˙a a ˙ η = ηH. (3.58)
Now, since δ(η) we can use equation (3.58) to obtain a new way to define the small density fluctuations
˙δ = δ0
ηH (3.59)
¨
δ = δ00(ηH)2+ δ0η0¨a, (3.60) where δ0 = ∂δ/∂η and δ00 = ∂2δ/∂η2. To derive ¨a in the second equation we use the Friedmann acceleration equation
¨ a a = − 4πG 3 (ρ + 3p) (3.61) = −4πG 3 (ρm+ 2ρrad) (3.62) = −4πG 3 ρrad(η + 2) (3.63)
since wm ≈ 0 and wrad = 1/3. Then, placing this result in equation (3.60), it gives ¨ δ = δ00(ηH)2− δ 0 2 η + 2 η + 1ηH 2. (3.64)
Finally, replacing in the equations (3.59) and (3.64) into the small density fluctuations equation of motion, we obtain
δ00+ 2 + 3η 2η(1 + η)δ
0− 3
2η(1 + η)δ = 0. (3.65)
The solution of this equation gives the growing mode to be δ ∝ η + 2
3. (3.66)
Thus, initially the Universe dominated by radiation and the fluctuations in matter cannot grow δ ≈ const., but since the radiation density decays faster than the matter density, at
30
the end of the day the Universe becomes matter-dominated, i.e. η = ρm/ρrad > 1, so that the growing mode becomes δ ∝ a. Summarising,
δ ∝ (
const., η < 1 (radiation dominates)
a, η > 1 (matter dominates). (3.67)
Note that it is in well agreement with the above results since the logarithmic rate varies very slowly [37].
Comments
Note that we have not distinguished which kind of pressureless matter composes this mixture. However, we know that the pressureless components of the Universe are dark matter and baryonic matter. From many observations and evidences, we have that dark matter must be the most abundant of them. Since dark matter does not feel radiation as baryonic matter does, then the growing mode in dark matter density fluctuations could start even before matter dominates.
The results obtained in this section will be useful to derive the matter power spec-trum which describes the distribution of galaxies and galaxy clusters we observe today. Furthermore, the Jeans scale will allow us to estimate when the fluctuations started to collapse inward to form gravitational bounded objects. Therefore, this chapter will help us to understand the current distribution of structures in the Universe, even on small-scales. In the next chapter we shall study the evolution of those density fluctuations using a statistical approach.
Chapter 4
The Matter Power Spectrum
In this chapter the small density fluctuations become the density fluctuation field. In the first section we shall develop the concept of density fluctuation field and state its main features. Thereafter, we define the matter power spectrum and the correlation function which will allow us to describe the distribution of galaxies and galaxy clusters. Going through pages, we introduce the need for Dark Matter as the most abundant kind of pressureless material in the Universe. Furthermore, we shall derive the Standard Cold Dark Matter model for Structure Formation which gives the best fit to data on large-scales. Finally, we briefly present an alternative Hot Dark Matter model in order to fit the data on small-scales.
4.1
Properties of the Density Fluctuation Field
The description of the density fluctuation field after matter-radiation equality has a sta-tistical character. Equation (3.18) defines the density fluctuation field. Another similar and usual way to define is via
δ(r) ≡ δρ ¯
ρ . (4.1)
Notice we have hidden the time-dependence because we are only interested in the spatial description, i.e. in the shape of the density fluctuation field. This treatment provides intuitive thoughts about the amplitude of the density fluctuation field as we will see later. The density fluctuations do not change the cosmological principle, which is applied for large-scale observables in the Universe. Then, while it remains in low-amplitude |δ| 1 the expansion of the Universe remains nearly isotropic. In this way, we can use the Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) metric to set up a comoving coordinate system which will be based on the metric in order to describe the position and motion of an object in the expanding Universe . The FLRW metric [33] can be written as
32
Figure 4.1: Illustrating how density fluctuations can be a Fourier superposition of waves. The first panel shows a single wave; the second adds together 13 similar waves running in different directions; The third shows how the regularity of this pattern is destroyed when the waves are given a random shift in phase [10].
where Sk(r) corresponds to different values depending on the curvature and a(t) is the scale factor that we already know. Moreover, the spatial part depends on the cosmic time only via the scale factor otherwise the expansion would not be so isotropic and homogeneous. Once we choose a point as the origin, the proper distance of any point from that one reads
xphys(t) = a(t) r (4.3)
where r is the comoving distance such that when a = 1 (today) we have the proper distance (physical) equals to the comoving one. While the density perturbation field evolves in a linear regime the comoving coordinates remain nearly constant.
In order to study the statistical properties of the density fluctuation field we will divide the Universe into patches of volume V with periodic conditions. It is usual to assume that the distribution of the matter density in the Universe is a superposition of plane waves that evolve independently, as long as the the linear regime is valid, |δ| 1 [33, 36, 22, 16, 38, 39, 40].
To illustrate the idea we shall make a mental exercise. Imagine a single plane wave moving in some direction. Then, take another single plane wave moving in any other direction and superimpose one over the other. What do you see? Constructive interference pattern at some points and destructive interference pattern at some others. Since we have only two plane waves they maybe become only constructive or only destructive. However, when we take several plane waves (maybe infinity) but propagating in various different directions with random shift in phase, i.e. moving randomly along their direction of propagation, the result is an image similar to that in figure (4.1) which shows step-by-step the building of a superposition of several plane waves each one with its own direction and with random shift phase distribution, see [10].
Looking at the Universe in large scale where the cosmological principle is applicable we note that the distribution of matter is very similar to our assumption. Thus, we can
expand equation (3.18), which is equivalent to equation (4.1), in a Fourier series. Hence, δ(r) = ∞ X nx,ny,nz=−∞ δkeik·r → V (2π)3 Z V δkeik·rd3k, (4.4)
where δk corresponds to the individual Fourier components. A particular Fourier com-ponent is characterised by its amplitude |δk| and its comoving wave number k [36]. We already defined the comoving and physical distance, so we can define the comoving and physical wave number by
kphys = k
a(t), (4.5)
and the comoving and physical wavelength of a fluctuation λ ≡ 2π
k , λphys = a(t)λ. (4.6)
As in the case of the distance we have that for today the comoving wave number equals to the physical wave number and the same is applied for wavelengths. During the linear regime the physical size of a given fluctuation grows with the Hubble flow so that we use the equation (4.6). Since the comoving coordinates do not change as the Universe expands (time passes by) it turns very useful to label a given mode with them because it ensures that the same physical fluctuation is characterised by the same comoving label, see [36].
The periodic conditions on the surfaces of each fluctuation is given by
δ(L, y, z) = δ(0, y, z), (4.7)
which can be applied for other components of the field; and ki =
2πni
L , i = x, y, z. (4.8)
each Fourier component can be written from equation (4.4) as δk = 1 V Z V δ(r)e−ik·r d3r, (4.9)
where for δk=0 = 0, it ensures the mass conservation inside the volume V and for δk∗ = δ−k the field δ(r) is real as it must be once it is an observable [36]. If we know all the Fourier components therefore we can determine the density fluctuation field δ(r). The inverse procedure works as well. In order to extract those components we recall that they are complex quantities and therefore,
δk= |δk|e−iφk. (4.10)
If we assume that the phases φk of each different Fourier component are random and uncorrelated with each other, when V → ∞ the central limit theorem provides that the
34
density fluctuation field δ(r) is a Gaussian field [36, 22]. The fluctuation amplitude may be written
|δ(r)|2 = δ(r)δ∗
(r) =X k
|δk|2. (4.11)
The central limit theorem says that the distribution of a quantity which is obtained by superposition of random contributions which are all drawn from the same probability distribution (with finite variance) turns into a Gaussian in the limit of infinity contribu-tions [36]. The probability distribution for the Gaussian density fluctuation field δ(r) is given by p(δ) = 1 σ√2πexp − δ2 2σ2 ! (4.12) where the standard deviation σ is intimately associated with the variance σ2 of the density field. This Gaussian field is taken to be a homogeneous and isotropic field so that it depends only on |k| = k and |r| = r. Thanks to its properties the average of the field vanishes hδ(r)i = 0 and the variance is defined only from the mean-square
σ2 ≡ δρ ρ
2
≡ hδ2(r)i (4.13)
which characterises the amplitudes of inhomogeneities of the density fluctuation field. Notice a statistical homogeneity means that the statistical properties of the translated field are the same as the original field, implying that
ˆ
Taδ(r) ≡ δ(r − a), (4.14)
that is,
P r[δ(r)] = P r[ ˆTaδ(r)] (4.15)
and the statistical isotropic means the statistical properties of the rotated field remains the same as the original field
ˆ
Rδ(r) ≡ δ(R−1r), (4.16)
that is,
P r[δ(r)] = P r[ ˆRδ(r)] (4.17)
where R is a rotation matrix and P r[...] is a functional which gives the probability of obtaining some field configuration. More details can be found in [41]. We have reviewed how δ(r) is connected to the probability distribution, in what follows we will connect it to the matter power spectrum.
4.2
Defining Matter Power Spectrum & Correlation
Function
The power spectrum can be defined as the mean square amplitude of the fluctuations of a given wave number k = |k|, since we are assuming density fluctuation field obeys a Gaussian distribution, i.e.
P (k) = h|δk|2i , (4.18)
where this quantity is evaluated over all possible directions of the wave number k [36, 40, 33]. Moreover, we must have P (k) → 0 as k → 0 if homogeneity is to be recovered on large scale [40]. Therefore, the power spectrum is the tool that tell us the balance between the contributions of the density fluctuations from different scales [10]. Despite this definition, other authors use the power spectrum being the square of the amplitude of the modes in Fourier space, i.e. in the wave number space [10, 22]
P (k) = |δk|2. (4.19)
There is another way to define the power spectrum which is complementary to the above definitions using useful statistical tools. First, we define the correlation function (or two-point correlation function) as the quantity which describes the probability of finding a given value for the density fluctuation field δ(r) at two different points r1 and r2 of the configuration space
ξ(r1, r2) = hδ(r1)δ(r2))i . (4.20) Since the density fluctuation field is homogeneous and isotropic, then translation and rotation invariance work as well to the correlation function. Applying that conditions developed in equations (4.15) and (4.17) into the correlation function we obtain for trans-lation,
ξ(r1, r2) = ξ(r1− a, r2− a), ∀ a
ξ(r1, r2) = ξ(r1 − r2), (4.21) and for rotation
ξ(r1, r2) = ξ(R−1(r1− r2)) ∀R−1
ξ(r1, r2) = ξ(|r1 − r2|). (4.22) We conclude that ξ depends only on the distance between the two points, i.e, |r| = |r1− r2| = r [41].
