MSc Physics
Master Thesis
Astroparticle physics at LHC
Dark matter search in ATLAS
by
Michaël Muusse
6171885
September 2015
60 EC
September 2013 – September 2015
Supervisor/Examiner:
Examiner:
Dr. D. Berge
Dr. M. Vreeswijk ... .Astroparticle physics at the LHC
Astroparticle physics at the LHC
Dark matter search in ATLAS
Master thesis in Physics
Michael Muusse
July 2015
Track: Grappa, University of Amsterdam
Supervisor: Dr. D. Berge
2
ndsupervisor: Dr. D. Salek
Second Reviewer: Dr. M. Vreeswijk
Abstract
Astroparticle physics at the LHC – Dark matter search in ATLAS
In astronomy and cosmology the hypothesized existence of dark matter and its properties are inferred indirectly from observed gravitational effects. An explanation for the missing mass problem in the study of motion of galaxies and clusters, the shape of galactic rotation curves, observations on weak gravitational lens effects near clusters and the correspondence between the cosmic microwave background anisotropies and the large-scale structure of the Universe, could be given by a form of non electromagnetic interacting, invisible matter, hence called dark matter.
An undetected heavy elementary relic particle that interacts only trough the gravitational and weak forces is a leading candidate for dark matter. Furthermore elementary particles with Weakly Interacting Massive Particle (WIMP) properties arise often in theories beyond the standard model. Since WIMPs are ought to be produced in particle accelerator experiments, important evidence could be found in the analysis of high energy collisions in the LHC.
In this thesis simulations of the pair production of WIMPs in the upcoming 2015 LHC collisions with energy of 14 TeV are analysed and the sensitivity of a signature of large missing energy accompanied by a jets is studied. A ROOT program was written which optimises the selection criteria for signature events for all different scenarios of WIMP and mediator particle properties. Different models with a contact interaction represented by the D5 operator or an interaction trough a generic Z’ intermediate state with mediator masses in the range between Mz’ = 100 GeV and Mz’ = 15 TeV and with a dark matter mass of Mx = 50
GeV and Mx = 400 GeV were studied and show harder selection cuts don’t always improve
signal to background ratios but a tuned asymmetric selection is preferred. Considering all 48 studied models in 62.9% of the cases significance was enhanced by an average of 3.0% with respect to the application of any of the standard used selection methods, while there was no situation in which significance was decreased. This confirms the viewpoint of multiple jets being created opposed to the invisible WIMPs and shows the importance of using tuned selection criteria to enhance significance.
Table of contents
Front cover 1 ... i
Front cover 2 ... iii
Abstract ... v
Table of contents ... vii
Preface ... xi
Introduction ... 1
Chapter 1. Dark matter in astrophysics ... 5
1.1 Introduction ... 5
1.2.1 Evidence for dark matter - Motion of galaxies and clusters ... 6
1.2.2 Evidence for dark matter - Gravitational lensing ... 8
1.2.3 Evidence for dark matter - The CMB... 9
1.2.4 Evidence for dark matter - Bullet Cluster ... 12
1.3 Dark matter candidates ... 13
1.4 Alternatives for dark matter ... 16
Chapter 2. Dark matter in Big Bang cosmology ... 17
2.1 Introduction ... 17
2.2 Spacetime & special relativity ... 17
2.3 Spacetime and gravity, general relativity ... 19
2.4 The cosmological principle... 22
2.5 The expanding Universe ... 23
2.6 The Big Bang model... 24
2.7.1 Problems with Big Bang theory: The horizon problem ... 29
2.7.2 Problems with Big Bang theory: The flatness problem ... 29
2.7.3 Inflation theory ... 30
2.8.1 The thermal history of the Universe ... 31
2.8.2 The Planck era ... 31
2.8.3 Grand Unification ... 31
2.8.4 Inflation era ... 32
2.8.5 Baryogenesis... 32
2.8.6 Electroweak symmetry breaking ... 33
2.8.7 The hadron epoch ... 33
2.8.8 The lepton epoch ... 34
2.8.9 Nucleosynthesis ... 34
2.8.10 Matter radiation equality ... 34
2.8.12 Structure formation... 35
2.8.14 Reionization... 35
2.8.15 Formation of galaxies ... 36
2.8.16 Formation of the Solar System ... 37
2.8.17 Today... 37
2.9 ΛCDM ... 37
Chapter 3. Dark matter in particle physics ... 39
3.1 Introduction ... 39
3.2.1 Symmetries and conservation laws ... 39
3.2.2 External symmetries of spacetime ... 39
3.2.3 Internal symmetries of spacetime ... 40
3.3.1 The standard model of particle physics ... 42
3.3.2 Additional global symmetries in the Standard Model ... 44
3.3.3 The Higgs mechanism ... 44
3.4.1 Problems with the Standard Model ... 45
3.4.2 Extensions of the Standard Model ... 50
3.5.1 WIMP detection ... 53
3.5.2 Direct detection of WIMPs ... 53
3.5.3 Indirect detection of WIMPs ... 54
3.5.4 Accelerator searches ... 55
3.6.1 Dark matter search in lepton colliders ... 56
3.6.2 Dark matter search in proton colliders ... 57
Chapter 4. Dark matter at the LHC ... 59
4.1 Introduction ... 59
4.2.1 LHC ... 59
4.2.2 Event rate and luminosity in LHC ... 61
4.2.3 LHC detectors ... 64
4.3.1 The ATLAS Detector ... 65
4.3.2 ATLAS coordinate system ... 66
4.3.3 The Inner detector ... 67
4.3.4 Calorimetry ... 68
4.3.5 The muon spectrometer ... 68
4.3.6 The magnet system... 69
4.3.7 Computing grid ... 70
4.4.1 Event reconstruction and identification ... 71
4.4.2 Particle identification ... 74
4.5.1 Transverse momentum and missing energy ... 76
4.5.2 Smallest angle ... 77
Chapter 5. Methods ... 79
5.1 General objectives and assumptions ... 79
5.2 Interactions in Effective Field Theory ... 81
5.3 Interactions trough a Z’ mediator ... 84
5.4 Background events ... 87
5.5 Simulation methods... 91
5.6 Simulations of background... 93
5.7 WIMP production simulations ... 94
5.8 Data samples ... 95
5.9 General selection criteria ... 97
5.10 Data computation ... 99
5.11 Data analysis and display ... 101
5.12 Schematic overview of the analysis ... 103
Chapter 6. Results ... 107
6.1 Results analysis A ... 107 6.2 Results analysis B ... 109 6.3 Results analysis C ... 116 6.4 Results analysis D ... 131Chapter 7. Conclusion ... 151
7.1 Conclusion analysis A ... 151 7.2 Conclusion analysis B ... 152 7.3 Conclusion analysis C ... 155 7.4 Conclusion analysis D ... 1597.5 General conclusion of the analysis ... 161
7.6 General conclusion ... 162
7.6 Outlook ... 163
Preface
In this thesis I performed an efficiency study of the dark matter detection in the ATLAS experiment combining the fields of particle physics and cosmology. The thesis is written to obtain a master’s degree in Physics at the University of Amsterdam and supervised by Dr. David Berge, researcher at the Grappa institute and David Salek, researcher at ATLAS and Nikhef.
I approached the topic by studying the vast literature about dark matter in particle physics and cosmology and by taking part of courses in (particle) cosmology and quantum field theory. Furthermore with this thesis I learned to become familiar with the ROOT Analysis Framework, and develop and apply analytical methods to perform efficiency study and I compared and inspected the results found in simulations to write the conclusions. By writing this thesis I also hope to present an accessible overview of the dark matter problem in physics.
Finalizing the project and thesis took almost two years and had a serious impact on my personal life. Therefore I would like to thank first and especially my girlfriend Mariska Dijkstra for supporting me so much the past years. Furthermore I would like to thank dr. D. Berge and dr. D. Salek for giving the opportunity to investigate and complete this project, and for their supervision of this thesis. Many thanks to my employer and colleagues of the Utrechts Stedelijk Gymnasium and to my friends that supported me during the last years. Finally, thanks to my family for their constant support and encouragement.
Introduction
Dark matter is a presumed form of invisible massive matter which makes up 83.9% of the physical matter density in the Universe. The leading candidate for dark matter is an undetected heavy elementary relic particle which interact only trough gravitation and the weak force. Such a Weakly Interacting Massive Particle (WIMP) is often predicted in theories beyond the standard model and is ought to be produced in particle collider experiments with sufficient available energy. Important evidence could be found in the analysis of high energy collisions in the Large Hadron Collider (LHC), the World’s biggest and most powerful particle accelerator.
This research project aims to analyse simulations of WIMP pair production in the upcoming 14 TeV LHC collisions, and develop a ROOT program which optimises the selection criteria for signature events in ATLAS studying scenarios with different WIMP and mediator particle properties.
The topic is approached by giving a short historical and introductional overview of the physics behind dark matter in particle physics and cosmology. Present conceptions and theories are mostly not attained in a straightforward manner, but can have different concepts as foundation since science is not static but influenced by local traditions, cultural and social tendencies and their context.
In chapter 1 the observational evidence for dark matter in astrophysics is described. Already in the 19th century it was noticed from the study of the motion of celestial bodies that a form of non-luminous mass exerting gravitation should be present. Evidence for the existence of so called dark matter was prolonged in the 20th century by observations inferred from its gravitational effects on the movement of galaxies and clusters, gravitational lenses and could explain structure formation in the Universe and the observed anisotropies in the cosmic microwave background (CMB). The leading candidate for dark matter are cold dark matter particles that were produced by falling out of thermal equilibrium with the hot early Universe, such as the WIMPs studied in this thesis. Observations show that the energy density of all matter in the Universe consists of 83.9% dark matter and only for 16.1% of baryonic (ordinary) matter.
In chapter 2 the foundations and concepts of the present standard model of cosmology in which general relativity is a fundamental theory are explained. In Big Bang cosmology the evolution of the Universe is described by its energy density content and curvature, starting from a hot dense state that inflated rapidly to the accelerated expansion today. Remarkable in the present cosmological model of the Universe (ΛCDM) 71.4% of the critical energy density of the Universe consists of dark energy whereas 28.6% of the energy density is determined by dark matter and ordinary baryonic matter.
Dark energy is the unknown form of energy responsible for the accelerated expansion of the Universe. Extrapolating the expansion of the Universe back in time will give the description of the thermal history of the Universe trough its different era’s and provide more insights in the role of dark matter in the evolution of the Universe.
The discovery of new elementary particles was always the realm of particle physics and in this thesis we study the possibility to produce dark matter WIMPs in the high energy particle collider LHC. Any produced dark matter particles would escape the detector unseen leaving an energy imbalance as signature. This idea and its advantages and drawbacks will be discussed in chapter 3, also in comparison with most dark matter research that aims to detect dark matter directly interacting with normal matter (direct detection) or indirect via self annihilation (indirect detection). Furthermore we describe how particles with WIMP like properties often arise in theories beyond the Standard Model. The Standard Model combines the quantum field theories describing the interaction of electromagnetism and the weak and strong nuclear force and classifies all fundamental particles known.
In this thesis I will focus on the possibility WIMPs might be produced and detected by the ATLAS detector when the LHC high energy particle accelerator has restarted in June 2015 running at a higher energy of 14 TeV. Large missing energy accompanied with jet production is the signature of an event in which WIMPS could be produced. In chapter 4 we describe the signature of such event considering the characteristics and limitations of the ATLAS experiment.
For this thesis simulations of WIMP pair production and background processes are studied, and an optimalisation of the selection criteria placing constraints on physical variables is performed by a code written in ROOT using different scenarios of WIMP and mediator particle properties. In chapter 5 the assumptions and constraints for this research are presented with the method used to study the simulations. In chapter 6 the results for optimising signal to background efficiencies are shown and in chapter 7 the conclusion is discussed.
Chapter 1. Dark matter in astrophysics
1.1 Introduction
Cosmology, literally meaning in Greek “study of“ (λογία - logia) the “world“ (κόσμος -
kosmos), describes the physical laws of the origin, evolution, structure, dynamics and
ultimate fate of the Universe. Big Bang cosmology describes 13.8 billion years of cosmic evolution and expansion, since the birth of the Universe from a singularity untill the present and future. Astronomical observations and new discoveries about the laws of physics in the early Universe could give us answers to fundamental questions unavoidable to mankind as “How was matter created?“, ”What is the age of the Universe?“ and “How do the conditions of our era compare to the history and future of the Universe?”.
As expansion cools the Universe down, we try to understand processes at earlier times by extrapolating our theories to higher energies, describing the Universe when it was hot and dense. At early times all matter interacted energetic and frequently and was bound by a unified force. As this primordial plasma cooled while the Universe expanded, elements were formed in a process called nucleosynthesis, and photons were able to stream freely trough the Universe. These initial free photons can be seen today as the cosmic microwave background radiation (CMB) and this “baby photo” of the Universe contains small temperature fluctuations indicating tiny variations in the primordial density of matter, probably originating from quantum fluctuations stretched to cosmic sizes during the early inflationary epoch of rapid expansion. These primordial fluctuations are density variations in the early Universe which are considered the beginning of all structure in the Universe, which under the influence of gravitation let to the formation of stars, galaxies and clusters.
The evidence of the Big Bang comes from the observations of the expansion of the Universe, the CMB and light element abundances which are in accordance with Big Bang nucleosynthesis theory. From this evidence and other observational data it is calculated that the majority of the Universe today consists of different forms of invisible energy and matter called dark energy and dark matter. What the nature of dark energy and dark matter exactly is remains still a mystery today.
Dark energy is an unknown form of energy present everywhere in space and is required to exist in order to describe the observed accelerated expansion of the Universe. It is currently believed 71.4 % of the critical energy density of the Universe consists of dark energy whereas 28.6 % consists of matter (dark matter and normal baryonic matter). [3] [4]
Dark matter is an invisible form of matter which presence has been inferred from its gravitational effects. Dark matter was first imposed to explain the motion of stars, but also observations on the movement of our galaxy, galaxy rotation curves (movement of stars in galaxies) and the velocity dispersion of galaxies in clusters, require a form of invisible matter to be present. Dark matter could also give an explanation for cosmic structure formation and its effects on the anisotropies observed in the cosmic microwave background. Observations show that of all matter 83.9% is dark matter and 16.1% baryonic (ordinary) matter. From the total critical energy density of the Universe 24.0% is composed of non baryonic dark matter and only 4.6% of baryonic matter, see fig 1.1.
Ordinary matter is in cosmology named baryonic matter since in atoms around 99% of the mass is in the nucleus in the form of protons or neutrons which consist of 3 quarks (called a baryon) while the mass contribution from electrons is small. From the 4.6% baryonic matter an estimated 0.4% baryonic matter can be found in luminous matter like stars or hot gas while 4.2% is in non-luminous baryonic matter, see fig. 1.1. [2] [4]
Fig. 1.1. Estimated distribution of dark energy, dark matter and baryonic matter in the Universe. [5]
1.2.1 Evidence for dark matter - Motion of galaxies and clusters
Different astronomical bodies emit or absorb light in different ways with different efficiencies and this is used to study the content of the Universe in astrophysics. The effectiveness of emissivity is reflected in the mass to light (M/L) ratio of an object in a fixed photometric system and the mass can be determined by the amount of light emission. A different way to determine the mass of an astronomical object is by its gravitational effect on the motion of other bodies nearby the studied object. For most objects the calculated mass from gravitational effects exceeds the calculated mass by its M/L ratio, and the apparent non-luminous mass exerting a gravitational force is what is commonly referred to as dark matter. A large amount of observational evidence of dark matter comes from the study of motion of galaxies and clusters. The presence of dark matter was first noticed by Bessel in 1844 when he showed that the positional measurements of Sirius and Procyon implied each star was moving in orbit with another invisible object [6]. Jeans and Oort analysed in 1922 the vertical motions of stars near the plane of our Galaxy and showed that the density of all stars near the galactic plane was insufficient to justify the vertical motions. Therefore they assumed the presence of two invisible stars next to each bright star. [7] [8]
In 1933 Zwicky measured the redshift of galaxies in the Coma cluster and estimated the clusters total mass by observing the motion of galaxies near its edge. He found the calculated mass for the cluster was about 400 times larger than expected based on the number of galaxies in the cluster and its total brightness. Therefore Zwicky concluded an invisible form of matter or “missing mass“ exists, massive enough to keep the galaxies in the cluster gravitationally bounded [9]. Babcock showed in 1939 that the galaxy rotation curves of the Andromeda galaxy, which show the star velocity of rotation versus the distance from the galactic center, could not be explained by the amount of visible matter. [10]
More precise estimates in line with modern observations were done in 1959 by Kahn & Woltjer who noticed that while the light from most galaxies is redshifted due to the expansion of the Universe, the Andromeda galaxy has a blueshift and thus moves towards our Galaxy [11]. They concluded the Andromeda galaxy and our Galaxy form a two body system orbiting around eachother and are currently approaching towards eachother. Calculation of the total mass of the double system showed its mass to be about ten times higher then presumed. Using the same method with modern data Bell & Einasto estimated in 1982 the total mass of the Local Group (the galaxy group that includes our Galaxy and the Andromeda Galaxy) to be in agreement with present measurements of the Andromeda Galaxy and our Galaxy including their dark matter halos [12]. Rubin & Ford calculated in 1980 the rotation curve of spiral galaxies with greater precision and showed that about 6 times as much mass from dark matter is present than there is from visible stars. All these measurements indicate the presence of a form of invisible matter with very large M/L ratio and large radius and mass in galaxies and clusters [13]. Dark matter is assumed to be centrally concentrated and have a roughly spherical symmetric distribution called a dark matter halo, see fig. 1.2.
1.2.2 Evidence for dark matter - Gravitational lensing
A gravitational lens is formed when clusters, galaxies or stars are so massive that their excerted gravity bends and focuses light from distant objects that lie behind. Gravitational lenses were hypothised by Einstein and Zwicky but not discovered until 1979 [15]. Gravitational lenses can be categorised into 3 classes. [16]
A strong lens shows an image with visible distortions such as Einstein rings, arcs and multiple images, see fig 1.3. The mass of the cluster creating this distortion can be calculated by the size of the distortions as the path light is bended near of the cluster. This mass corresponds to dynamical estimates of the mass of clusters based on the movement of its galaxies.
A weak lens distorts the image of background objects only slightly. Statistical analyses on numerous objects can search for shear deformation and shape distortions in the image of the adjacent background galaxies and determine the mass of the object causing the distortion. [17]
The correspondence of the masses of objects measured by using the gravitational lens techniques with other independent dark matter measurements as in the discovery of the Bullet Cluster (Chapter 1.2.4) has convinced most physicists that dark matter exists as a major component of the Universe. [18]
Microlenses have the property that no distorted image is visible, but the flux received from a background object changes with time. Microlenses can lead to the discovery of dark objects in our Galaxy and determine whether dark matter in our Galaxy is made of baryonic normal matter.
Fig. 1.3. The strong lensing effect caused by the gravity of the luminous red galaxy (LRG 3-757) distorted the image from a more distant blue galaxy such that the light bending resulted in an Einstein Ring. Image from Hubble Space Telescope’s Wide Field Camera 3. [19]
1.2.3 Evidence for dark matter – The CMB
The existence of the cosmic microwave background radiation (CMB) was predicted by Dicke in 1965 in the same year that Penzias & Wilson discovered a signal by accident what turned out to be the CMB radiation. The small fluctuations (order 10-5) in the CMB were first detected by the COBE satellite (1992) but the experiment was not as accurate as the observations from the Wilkinson Microwave Anisotropy Probe (WMAP) spacecraft (2009) or the Planck satellite (2013). [20] For comparison the maps of the CMB made by the three satellite experiments is shown in fig. 1.4.
Fig. 1.4. Maps of the CMB made by COBE, WMAP and Planck (from left to right). [21]
Using the data from WMAP the fluctuations in the CMB are reconstructed as is shown in fig. 1.5. The cosmic microwave background (CMB) has an average temperature of 2.725 K with small fluctuations within 0.01% and appears to be smooth and isotropic at first sight, as can be seen in the upper left part of fig. 1.5. If the average temperature is subtracted and the remaining fluctuations are increased by a factor 400, the dipole pattern and the emission from our Galaxy which dominate the red color in the picture are clearly visible in the upper left part of fig. 1.5. The maximal temperature fluctuations have a 0.00335 K variation with one hot pole and one cold pole which form the dipole pattern. After the dipole pattern is subtracted and the contrast is enhanced by an extra factor 50, the intrinsic fluctuations in the CMB are visible in the lower left part of fig. 1.5. When finally the emission from our Galaxy is subtracted, the picture of the CMB as can be seen in the lower right part of fig 1.5 is obtained in which the anisotropies are clearly visible. [20]
Fig. 1.5. Contrast enhancement in the WMAP data of the CMB showing in the lower right figure a temperature fluctuation range of 0.0005 K from the coldest (blue) to the hottest (red) parts of the CMB. [22]
According to the present cosmological model the Universe was initially ionised and very hot, and photons, electrons and quarks were tightly coupled. Perturbations of baryons oscillated as sound waves in the electron-baryon plasma and these baryonic acoustic oscillations (BAO) became embedded in the distribution of matter when the plasma condensed as the Universe expanded. The largest possible amplitude of the BAOs is at a wavelength equal to the size of the sound horizon at the moment of decoupling. This wavelength can be seen in the first maximum of the angular power spectrum of the CMB while the following maxima correspond to the overtones. When the Universe was 340.000 years old the photons were able to decouple and stream freely trough the Universe. This “first light” can be detected today as the cosmic microwave background radiation currently cooled due to the expansion of the Universe to an average of 2.725 K. The fluctuations in the plasma are reflected by temperature fluctuations in the CMB, and contain information about the large scale distribution of the Universe at the time of decoupling. This distribution is highly dependent on the amount of dark matter present in the early Universe. [23]
In March 2013 the Planck mission released the map of the cosmic microwave background which showed the fluctuations with larger precision. From this data it was calculated that the Universe is slightly older than thought (13.8 Gy) and show the fluctuations created as early as 10−30 seconds after the Big Bang which are the seeds for the present structure of the cosmic web, clusters and dark matter. [24]
The temperature fluctuations are described by a function written in multipole moments by performing spherical harmonic decomposition by angle, thus depending on the direction on the sky. If the fluctuations have Gaussian distributions they have no preferred direction and a power spectrum (the intensity of each spectral component) can be made. Typically the fluctuations in the CMB are showed in an angular power spectrum where the power per natural logarithmic interval l is plotted against the temperature fluctuation in mK. The reciprocal of l is the angular scale or angular wavelength of the fluctuation. For example l = 10 corresponds to about 10 degrees on the sky, and l = 100 to about 1 degree on the sky. On the largest scales (l < 10) it’s difficult to make statistical statements about the scale of the entire Universe if only a part of the Universe can be observed, and for low multipole moments l this effect (cosmic variance) will lead to large errors. For smaller angular scales (l > 100) the power spectrum shows acoustic peaks. On the smallest angular scales (l > 103) the mean free path length of photons suppressed the growth of structures on small scales and the power spectrum is damped by a photon diffusion damping scale called “Silk damping”. This diffusion process limited the fluctuations to grow to the size of galaxies observed today.
The position of the first peak of the power spectrum depends strongly on the total matter/energy density and is in agreement with the predicted value of 1 in units of the critical cosmological density, indicating the Universe is very close to spatially flat. From the acoustic peaks also the baryon density can be estimated as baryons add mass to the photon-baryon plasma. Acoustic peaks that are even numbered are related with the plasma “rebound”, the odd numbered peaks are related to “compression” of the plasma. A large baryon density will enhance the odd peaks over the even peaks. [23] [25]
Furthermore increasing baryon density decreases the frequency of the BAOs and shifts the position of the peaks to higher multipole moments l. A higher baryon density also results in the power spectrum decreasing at higher multipole moments l. The power spectrum depends on the baryon density in many independent ways and can therefore be calculated precisely. With a resolution of 7 degrees on the sky COBE could only see the largest angle fluctuations but combining the experiments of the 2015 data of Planck with measurements from baryon acoustic oscillations, and the Hubble constant the best fit estimate for the baryon energy density is 4.6% and for the cold dark matter energy density 24.0%. [24]
Fig. 1.6. The angular power spectrum of the CMB measured by the Planck experiment based on data from 2015. On the left the error from cosmic variance is clearly shown. The red line shows the current best fitting cosmological model. [24].
1.2.4 Evidence for dark matter - Bullet Cluster
The Bullet Cluster is a system of two clusters which are colliding. Most of the baryonic mass in the clusters is contained in the hot gas between the galaxies rather then in the mass of all the stars of the galaxies. The baryonic mass is bound to the cluster by a dark matter halo with an even greater mass. In most objects dark matter and baryonic matter are concentrated around the same center as they experience mutual gravitational attraction. However in the Bullet Cluster the collision has caused dark matter to separate from baryonic matter. X-ray observations by Chandra show that the baryonic matter in the form of hot gas is slowed down by electromagnetic interaction. Studying the weak gravitational lensing effect of the clusters on background galaxies by measurements of the Hubble Space Telescope, the European Southern Observatory's Very Large Telescope and the Magellan optical telescopes, the location of the center of mass of the clusters was found to be inconsistent with the position of the hot gas, as can be seen in fig 1.7. The interpretation is that dark matter is non-collisional and that the dark matter halos of both clusters have collided and passed eachother without slowing down, whereas baryonic matter behaved collisional and has slowed down causing it to separate from the dark matter halos. The Bullet Cluster is seen as direct evidence of the existence of dark matter independent of the exact laws of gravity. From the temperature and density of the X-ray gas, pressure and gravity balance can be estimated determinating the baryonic mass of the clusters at 12-15% of the total mass in agreement with observations on other clusters. [18] [26]
Fig. 1.7. The Bullet Cluster: a colliding pair of galaxy clusters. Most of the baryonic mass is in the X-ray emitting hot gas shown in red and yellow (left). In green the contours from the weak gravitational lensing effect are shown indicating the gravitational effect from the system dominated by dark matter which is centered on the observed distribution of galaxies (right). A separation is visible between baryonic matter and dark matter. [26]
1.3 Dark matter candidates
Besides the Bullet Cluster and the CMB power spectrum there are more observations that show dark matter mostly consists of non baryonic matter. The theory of Big Bang nucleosynthesis suggest baryonic matter accounts for about 4.6% of the critical energy density of the Universe, and accurately predicts the observed abundance of elements as the hydrogen/deuterium ratio which is discussed in more detail in Chapter 2. Non baryonic dark matter accounts therefore for 24.0% of the energy density in order to combine to the 28.6% of the matter density in the Universe. The amount of ordinary luminous matter in the objects observed on the sky like stars and luminous gas and dust clouds is estimated to be 0.4% of the critical density. This means 4.2% of the baryonic mass is in a non-luminous form of baryonic matter called baryonic dark matter. [27][28]
Baryonic dark matter
Baryonic dark matter or missing baryonic matter was first proposed by Zwicky in 1937 and is a form of normal baryonic matter which presence can be inferred from gravitational effects only, as it does not emit light. Baryonic dark matter could consist of non-luminous interstellar gas, cold gas, brown dwarfs (Jupiter’s) or Massive Astrophysical Compact Halo Objects (MACHOs) such as neutron stars, white dwarfs, planet-like objects, black holes, etc. MACHOs can be detected by gravitational microlensing if such a massive object passes near the line of sight between Earth and distant stars. The gravitational field around the MACHO will focus the light of background stars which appear to brighten for a period of time. Microlenses are sensitive to objects with mass m in the range 10-8 M☉< m <10
3
M☉ which is
in the range of the expected mass of MACHOs. The characteristic time profile of this brightening is independent of the wavelength of the light and therefore distinguishable in observations from other situations with brightning variations like variable stars. Signs of microlensing were found in observations of the Magellanic Cloud and galactic bulge however it’s unclear if the objects are a substantial part of the non luminous dark matter present in the Universe. [2] [29] [30]
Non-baryonic dark matter
There exist a large number of non baryonic dark matter candidates but they can be classified in the categories WIMPs, axions and neutrinos. All of these candidates are stable long lived and electric and color neutral particles with halftime t > tUniverse. An important difference
between these candidates is the effect of dark matter on the formation of structures in the Universe. We differentiate between cold dark matter (CDM) particle candidates that became non-relativistic and hot dark matter (HDM) particle candidates that were relativistic at early times in the Universe. The effect on structure formation is that hot dark matter has a large free-streaming length and would have diffused fluctuations in the early Universe therefore preventing the formation of structures in the Universe as all structure gets “washed out”. Hot dark matter models would imply a top-down formation history of structure in the Universe with the big structures (clusters, groups) formed first and smaller structures (galaxies, stars) formed later in history. Instead cold dark matter would act as a compactor of structure pulling matter together by its gravitational attraction enhancing the formation of filamentary superclusters and voids in the Universe. The evolution of clusters and large scale filaments in the presence of cold dark matter has been studied intensively with computer simulations as showed in fig. 1.8 and fig. 1.9. [16] [31]
Fig. 1.8. Computer simulation of an area of space of 140 million light years across presenting the evolution of clusters and large-scale filaments in the Cold Dark Matter model with dark energy from 100 million years after the Big Bang (left) untill the present (right). [32]
Observations suggest that structure formation in the Universe proceeded bottom up, with the smallest structures being created first followed by galaxies and clusters of galaxies. The fact that early formed high redshift galaxies are found in the Hubble Ultra Deep Field (HUDF) and the fact that our galaxy appears to be older than the Local Group, support the bottom up formation history of the Universe. The bottom up cold dark matter model of structure formation in the Universe is favoured by studied which compare simulations of the structure formation with observational data of galaxy surveys as the Sloan Digital Sky Survey (SDSS) and 2dF Galaxy Redshift Survey (2dF). [32]
Neutrinos as a form of dark matter are at first sight an interesting candidate since their cross section is so small they are practically invisible, while in the last decade it is shown neutrinos are not massless particles as predicted in the Standard model of particle physics, but have a small mass of order of ~ eV. However it can be shown that neutrinos are not abundant enough such that their relic density is the dominant component of dark matter. Furthermore neutrinos are relativistic particles with a large free streaming length, and therefore would act as hot dark matter. From the analysis of CMB anisotropies combined with large scale structure data it can be shown the free streaming length of neutrinos is too large in comparison with the galaxy scale perturbations for neutrinos to be a significant component of dark matter. [34][16]
Fig. 1.9. A comparison of the time evolution structure formation (showed from top to bottom) using cold, warm and hot dark matter models (showed from left to right). In the hot dark matter model the smallest structures dissapear. [33]
The suggestion of the presence of cold dark matter in the Universe explains the observed structure formation as it evolved from an initial hot dense state as illustrated in fig. 1.10. However the nature of cold dark matter particles is still unknown. WIMPs are interesting candidates as cold dark matter particles that would dominate dark matter in the Universe. The leading hypothetical particle physics candidate for dark matter are WIMPs relic particles that were produced by falling out of thermal equilibrium (frozen out) with the hot dense plasma of the early Universe and interacted weakly with standard model particles via a force similar in strength to the weak nuclear force. The foundations and concepts of Big Bang cosmology which describes a Universe that was inflated from a hot dense state is described in chapter 2. An overview is given of the thermal history of the Universe and the process of the thermal creation of WIMPs is discussed in more detail.
Fig. 1.10. The history of 13.82 billion years of evolution of our Universe showing the main events that occurred between the initial hot dense phase with tiny fluctuations, to the cosmic large scale structure observed today. [35]
1.4 Alternatives for dark matter
The large amount of unknown forms of matter has initiated attempts to construct theories with modified laws of gravity at large distances, in comparison to Newton’s laws. Milgrom & Bekenstein suggested in 1987 the theory of Modified Newtonian Dynamics (MOND) but other theories exist as well. The aim of such theories is to explain the observational data without presuming a form of invisible matter. The discovery of the Bullet Cluster is an argument against MOND and other theories without dark matter. In such theories the lensing effect would be centered on the baryonic matter which is dominated by the hot X-ray gas (see chapter 1.2.4) and not outside these regions. This is interpreted as most of the mass in the pair of clusters is in a form of non-luminous collisionless matter indicating the presence of non-baryonic dark matter. The discovery of the Bullet Cluster has made theories with modified gravity laws less popular. [36] [37]
Chapter 2. Dark matter in Big Bang cosmology
2.1 Introduction
In this chapter the foundations and concepts of Big Bang cosmology are reviewed. In this theory general relativity, the cosmological principle and the idea of a Universe that was inflated from a hot dense state are used to describe the observable Universe as a whole based on its cosmic inventory. Extrapolating the expansion back in time tells us the thermal history of the Universe as forces arose, particles condensed and the structures we observe today were formed.
2.2 Spacetime & special relativity
In physics the interaction of particles by forces moving in space and time is studied. Space and time can be described by a single space-time continuum and Minkowski space is the manifold combining three spatial dimensions and one time dimension. Incas described space and time already by a single perception called “pacha” but its mathematical analysis was unexplored untill the 19th century by the theory of electromagnetism (Maxwell) and in Lagrangian mechanics (Lagrange and Hamilton).
Untill the end of the 19th century empty space was considered to be filled with a thin, transparent medium called “aether” through which light and electromagnetic waves were thought to propagate. However Michelson and Morley tried to measure with the famous aether drift experiment in 1887 the relative difference in the speed of light using measurements moving towards and away from the Sun as the Earth rotated. While it was expected that as the Earth moved through the ether towards the Sun, the speed of light increased and vice versa, no change in the velocity of light was observed [38]. Based on the conclusions of this experiment Albert Einstein wrote in 1905 a paper that led to the development of the theory of special relativity, describing the relation between space and time based on two postulates; [39]
1. The laws of physics are invariant in all inertial systems 2. The speed of light in vacuum is equal for all observers
The consequences of this theory are relativity of simultaneity, time dilation and length contraction, Lorentz transformations, and mass-energy equivalence described by formula [2.1];
E = mc2 [2.1]
In special relativity it is described how with the increase of speed of an object one needs more and more energy to make it move faster such that the maximum speed for any real particle is approaching the speed of light. While inertial mass measures how hard it is to accelerate an object, it is the true rest mass of the objects that remains constant and is an intrinsic property of a particle and independent of the frame. It is the dynamical law that relates momentum and energy that depends on velocity in special relativity and approaches Newton’s laws in the low velocity limit.
The finite speed of light means that the farther away an object is from the Earth, the longer it takes for its light to reach us. Thus observing a more distant object means the observer is actually looking back farther in time. Observing bodies at various distances allows us to study their evolution at different epochs in time. The largest distance we can look back to is the moment the CMB was created, and the first photons were able to stream freely trough the Universe, see fig. 2.1.
Fig. 2.1. This illustration places the "look-back time" of the Hubble Ultra Deep Field (HUDF) in the context of the history and age of the Universe. [40]
Everywhere in the Universe special relativity holds, and Minkowski visualized a light cone can be drawn, defining the boundary between past and future accessible locations, since no information is allowed to travel faster than the speed of light, see fig. 2.2.
Fig. 2.2 (left). Past and present lightcones. [41]
Fig. 2.3 (right). Lightcones in flat Minkowskian spacetime. [42]
The geometric and causal structure of spacetime is described by using a coordinate grid called a metric that is laid down over all spacetime, which determines the distances between specified points. Everywhere on the metric special relativity holds and light cones can be drawn at every point on the metric, see fig. 2.3.
In general relativity spacetime is not necessary flat and the shape of the Universe is described by its local geometry, which mostly concerns the curvature of spacetime at any arbitrary point of the observable Universe (averaged on a sufficiently large scale) and its global geometry, which defines the topology of the Universe as a whole. Observations from Planck have shown that the cosmological curvature parameter ΩK is 0.000±0.005, consistent
with a flat universe. [24]
The geometry and curvature of space-time can be mathematically described by the Pythagorean theorem, see fig. 2.4. Zero curvature means spacetime is flat and the Pythagorean theorem holds. If spacetime has positive curvature then the ythagorean theorem is correct for triangles with their sum of angles larger then . When spacetime has negative curvature the Pythagorean theorem holds for triangles with their sum of angles smaller then .
Fig. 2.4. Three types of curvature of two-dimensional surfaces as an analogy to the 3-dimensional structure of the Universe. [43]
2.3 Spacetime and gravity, general relativity
While special relativity only holds for non-accelerating inertial reference frames, Einstein developed between 1907 and 1915 the theory of general relativity, applicable to any frame and able to handle gravitational effects and general coordinate-transformations. The first postulate in this theory is that local physics is still governed by the theory of special relativity as is showed in fig. 2.5. The second postulate is the equivalence principle that states an observer is not able to distinguish between gravity and acceleration. General relativity generalizes Newton’s law of universal gravity and special relativity describing gravity as a geometric property of spacetime directly relating the curvature of space-time to the local energy and momentum density present within that spacetime, see fig 2.6. [44]
Fig. 2.5 (left). Local lightcones near strongly curved spacetime near a black hole formed by collapsing matter. [45]
Fig. 2.6 (right). When a small mass is close to a large mass, it will curve towards it since spacetime itself is curved near the large mass. [46]
In 1915 Einstein did a Gedankenexperiment and imagined two elevators, one at Earth and one accelerating in space, see fig. 2.7. A blinded observer inside an elevator would not be able to distinguish between the gravitational force experienced while standing on Earth and the force experienced by an observer in a non-inertial accelerated frame of reference. [83]
Fig. 2.7. Scientist on Earth (Newton, left) and in an elevator (Einstein, right) that can’t distinguish between observing gravity or acceleration. [47]
Since Newton's equation of motion by the act of gravity is given by
Inertial mass x Acceleration = Gravitational mass x Intensity of the gravitational field
Einstein assumed in 1907 that “complete physical equivalence exist in the Universe between
a gravitational field and the corresponding acceleration of the reference system” [48]. From
this idea the weak principle is stated: “Gravitational mass and inertial mass are equal, and
different test particles which have no mutual gravitational interactions at equal spacetime points in a gravitational field, will experience an equal acceleration, independent of their properties including their rest mass”. [48] Therefore the gravitational motion of a small test
body depends only on its initial position in spacetime and velocity, and not on its composition.
The strong equivalence principle is an even stronger statement: “Everywhere in the Universe
a local experiment (with or without mutual gravitational interactions) in a freely falling laboratory is independent of the laboratory’s velocity and its position in spacetime”. [49] The
free falling object must be small such that local tidal forces can be neglected. The strong equivalence principle applies to objects with significant internal gravitation interactions such as stars, planets, black holes etc., and requires the gravitational constant to be equal everywhere in the Universe.
An immediate consequence of the equivalence principles is that gravity can bend the trajectory of massless particles such as photons (light), see fig. 2.8. To visualize why this is true one imagines a photon crossing the elevator that is accelerating into space. As the photon crosses the elevator, the floor is accelerated upwards and the photon appears to fall downwards. By the equivalence principle the same act must occur when the photon is moving in a gravitational field. [52]
Thus in general relativity gravity can be described as motion caused by the geometric properties of curved spacetime, and the relation is specified by the Einstein field equations, described by Wheeler as “Matter tells spacetime how to curve, and spacetime tells matter
and light how to move” [51]. Furthermore space contracts near mass and dilates away from
it, time dilates near mass and contracts away from it, both effects that have been observed with great precision. The strong equivalence principle suggests that the nature of gravity is entirely geometrical and the metric solely determines the effect of gravity. Einstein proposed three experiments that could be explained by general relativity, accurate measurements on the perihelion precession of Mercury’s orbit, deflection of light by the Sun and gravitational redshift of light, and was able to write down the field equations in 1915. [52] [53] [54]. The first experimental confirmation of general relativity was made in 1919 when Eddington was photographing a total solar eclipse and measurements of stellar positions near the solar limb darkening proved Einstein was right. [55] Einstein furthermore showed that general relativity agrees closely with the observed amount of perihelion shift of Mercury’s orbit and with the gravitational redshift of light measured by the Pound-Rebka experiment in 1959. [56] Another strong prove from modern observations came from the strong and weak lensing effects first measured in respectively 1980 and 1986. Furthermore general relativity predicts a time dilation in a gravitational field and this was confirmed by the Hafele Keating experiment with atomic clocks flying airplanes in 1972. [57]
In 1916 Einstein predicted the existence of gravitational waves, ripples in the curvature of spacetime that can propagate outwards from the source as waves and transport energy as gravitational radiation. These gravitational waves have been indirectly detected in 1974 from a binary pulsar by Hulse [58]. Accurate timing of the pulses showed that the binary pulsars gradually spiral towards eachother demonstrating an energy loss in close agreement with the predicted energy radiated by gravitational waves. [59]
Fig. 2.8. The sun curves spacetime, here represented by the grid. The preceived position of the star A differs from its actual position B, an observation that shows gravity can bend light. [50]
2.4 The cosmological principle
The Aristotelian model placing the Earth at a unique place in the center of the solar system assuming everything is orbiting around it, was the commonly accepted viewpoint untill the Middle Ages. At the end of the 16th century it was the idea of Copernicus that the Earth has no special place in the Universe, and that the Sun was the center of our solar system with the Earth in orbit around the Sun, while Galilei in the beginning of the 17th century showed by observations on the moons of Jupiter and the motion of Venus, the Copernican heliocentric viewpoint to be correct. [60] [61]. This new perspective had its implications on religion, philosophy metaphysics and science, and encountered much resistance from the Catholic Church.
On large scales, larger then galaxies and clusters, the Universe appears to look the same in all directions by any observer, and the average density of matter is about equal at all places and fairly smooth distributed. The Copernican viewpoint can therefore be extended by stating that in the Universe no special directions (isotropy) and no special places (homogeneity) exist. The assumption that on large scales the Universe is homogeneous and isotropic is called the cosmological principle. Of course the Universe on smaller scales looks clumpy and is far from isotropic and homogenous. [62]
Initially the cosmological principle was a postulate, but recent observations confirm this idea. In 1997 the 2dF Galaxy Redshift survey started to collect data followed by the larger Sloan Digital Sky Survey (SDSS) which began in 2000 to determine the large scale structure of the local Universe, see fig. 2.9. The data of the observations done in New Mexico (USA) of around 500 million objects covering 35% of the sky, were released in 2011, and the Universe has been measured to be statistically homogeneous on scales larger than 70 Mpc at the 10% level. [63] In chapter 2.5 we will see that the cosmological principle singles out a unique form of the spacetime geometry.
Fig. 2.9. A slice through the SDSS map of the distribution of galaxies. The Earth is at the center of the image and each point represents a galaxy typically containing about 100 billion stars. The colors of the galaxies indicate the age of their stars with the redder more strongly clustered points showing galaxies that are made of older stars. The outer circle is at a distance of two billion light years from the centre. The region between the wedges was not mapped by the SDSS experiment because in these directions dust in our own Galaxy obscures the view of the more distant Universe [64]
2.5 The expanding Universe
Einstein applied the equations of general relativity to the Universe as a whole, being the first to describe a testable theory of the Universe and initiating the field of relativistic cosmology. The evolution of the Universe is then described by its energy density content and curvature. A model Universe is typically filled with different types of “matter” which are classified by their energy density. Einstein found that a Universe filled with matter and radiation would exhibit an infinite lasting expansion that will be slowed down by gravitation. This can be understood considering expansion leads to a decrease of the energy density, further slowing down expansion, etc. A slightly positive curved Universe would either effect in a recollapse, or a slightly negative curved Universe would expand infinitely. The idea of an expanding Universe originating from a smaller hotter state, was early in the 20th century an unusual idea. But the idea of a Universe that would either expand forever or recollapse was so abnormal that it made Einstein to develop a way to describe a static non-expanding Universe. Realizing empty space is not “nothing” and that it is possible for more space to come into existence in an expanding Universe, he made the prediction that empty space has its own energy. As more space comes into existence this vacuum energy has the property that its energy density remains constant. Adding such a cosmological constant or vacuum energy to the content of the Universe, is uniquely allowed theoretically, and has an exponentially expanding effect on the Universe. Einstein hoped to describe a static non expanding Universe by assuming the cosmological constant would have the exact right value to balance the slowing down expansion of the matter radiation content. However such a Universe could be static non-expanding for a short period of time, but is fundamentally unstable since small deviations would increase rapidly, leading again to fast expansion or collapse. Since the model is fundamentally unstable and a perfectly balanced value for the cosmological constant seems rather unlikely, it’s not applicable to our Universe.
Lemaitre (1927) and Hubble (1929) observed in 1929 that the velocities of faraway galaxies were strongly correlated with their distance, see fig. 2.10. [65]. Nowadays the so called Hubble constant is measured to be 67.74 km·s-1·Mpc-1 giving the speed in km/s of a galaxy 1Mpc (3.09·1019 km) away. Assuming that the Earth is not the center of the Universe the interpretation is that all observable regions of the Universe moving away from all others with a velocity proportional to their distance, proving the Universe is overall expanding. With respect to the past viewpoint of the Universe Einstein therefore concluded the cosmological constant to be his greatest blunder, and dropped it from his equations. Since we know that the distance between galaxies increases today, reversing the argument means that galaxies were closer located in the past. Thus the expansion of the Universe leads to the conclusion the Universe was really denser (and hotter) in the past.
Fig. 2.10. Original plot of Hubble expansion as published by Hubble in 1929. [a65]
2.6 The Big Bang model
The cosmological principle implies that the Universe and therefore the metric are homogeneous and isotropic on large scales as observed, and physical laws are universal and independent of time leading to constant physical constants. Observations show that the largest possible deviation of the fine structure constant over the age of the Universe is of order 10−5. [81] The cosmological principle singles out the Friedmann–Lemaître–Robertson– Walker metric (FLRW metric) as a solution from Einstein‘s equations. The FLRW metric together with the Friedmann equation describes a homogeneous, isotropic expanding or contracting Universe, and was developed independently by the authors in the 1920s and 1930s. It describes the observable Universe as a whole based on the mathematics of fluid dynamics taking into account the inventory of the Universe as a perfect fluid. [25]
The greatest consequence of the cosmological principle is that it implies that all parts of space are causally connected at some time in the past (although they may no longer be connected today) and leads to the conclusion that the whole Universe appeared at a single moment of time, a Big Bang. We note that due to isotropy there is not a point where the Big Bang occurred since there is no definition of a center point in the Universe.
The metric now contains a scale factor a(t) which describes how the size of the Universe expands or contracts as a function of time. This allows the choice of a coordinate system to be made called comoving coordinates, which can be seen as a grid that expands along with the Universe, see fig. 2.11 and fig. 2.12. Using co-moving coordinates objects which move with the expansion of the Universe conveniently are fixed points on the metric. The coordinate distance called comoving distance remains constant but the physical distance between two comoving objects expands with the scale factor of the Universe. The FLRW metric assumes a uniform distribution of mass and energy and therefore it applies to our Universe only on large scales. Therefore local concentrations of matter such as galaxies are gravitationally bound and do not expand in proportion to the expansion of the Universe, however it is the space between galaxies that is expanding accelerated. The ratio between the physical distance and the comoving distance is the scale factor a(t). The scale factor increases for an expanding Universe, is dependent on time and determines the Hubble constant by formula [2.2] which is the time dependent constant of proportionality between the proper distance and its velocity.
[2.2]
The cosmic inventory is described by the categories (dark)matter, radiation and dark energy. All matter consists of non-relativistic particles, and when a given volume of matter expands by a factor a(t) as V ~ a3 the energy density dilutes as ρmatter ~ a-3. For relativistic particles or
radiation the energy density dilution contains an extra factor a-1 to account for the Doppler effect and therefore ρradiation ~ a-4. The present radiation density comes from photons and
neutrinos and is neglectible compared to matter or dark energy densities. [25] [66]
The Universe today seems to expand accelerating and therefore is presumed to contain a large component that can be described by a cosmological constant called dark energy. The energy density of dark energy Λ is constant and a property of space itself, and doesn‘t dilute for expanding space. The nature of dark energy is not known. [3]
'( )
( )
( )
a t
H t
a t
Fig. 2.11 (left). Expansion of the Universe. The comoving distance between points on the grid remains constant as the Universe expands. The physical distance is proportional to the comoving distance times the scale factor a(t) and hence gets larger as time evolves. [74] Fig. 2.12 (right). Expansion visualized in three dimensions in which the space between the objects expands. [68]
With the Friedmann equation [2.3] the behaviour of the scale factor and thus the Hubble constant is described by matter and radiation energy density ρ, the curvature k of the Universe and the dark energy density Λ.
[2.3]
The curvature is indicative for the rate of expansion and whether or not the expansion rate is increasing or decreasing and therefore implies the future fate of the Universe. When the curvature is zero, the energy density of the Universe is equal to the so called critical density, and the Universe will expand forever but with an ever decreasing rate. In a slightly positive curved Universe gravitational attraction will eventually stop the expansion and recollapse the Universe in a “Big Crunch”, a negative curved Universe will expand forever since no sufficient energy density is present to stop the expansion. Combined observations from the WMAP and Planck missions in 2008 and 2013 show the Universe is flat within 0.4% margin of error and 13.798 billion years old, and using the measured Hubble constant today 67.8 km·s-1·Mpc-1 it is possible to calculate the critical density today, that is equal to the sum of all contributions to the energy density, ρcrit = 1.9 10-29 g·cm-3 or about 5 hydrogen atoms per m3.
[66] [69] [70]
When the dependence of the density of radiation and matter on the scale factor is used in formula [2.3] and the critical density rescales the Friedmann equation with dimensionless density parameters Ω ρ ρ
Ω
ρ ρ Ω
ρ
ρ and Ωk the curvature density
parameter, the ratio of the Hubble constant at time t, H(t) and at present time H0 can be
expressed conveniently by formula [2.4]. [25]
2 4 3 2 2 0 R M K
H
a
a
a
H
[2.4] 2 2 2'
8
( )
3
3
a
G
k
H t
a
a
At present time this ratio is 1 and the scale factor is a(t) = 1. Considering the Universe is measured to be flat at present and at earlier times, curvature is completely neglectable since Ωk < . and the present the sum of density parameters ΩM , ΩR and ΩΛ must add up to 1.
Combined observations from the combined WMAP and Planck mission and baryon acoustic oscillations (BAO’s) show ΩR = 9.4.10-5,, ΩM = 0.308 ,ΩΛ = 0.692, see fig. 2.13. [3] [70]
Fig. 2.13. Dark energy and dark matter content of the Universe. The intersection of the supernova (SNe), cosmic microwave background (CMB) and baryonic acoustic oscillations (BAOs) measurements indicate a topologically flat Universe with an energy density composed 69.2% of dark energy (y-axis) and 30.8% of dark matter plus normal matter (x-axis). [71]
In the early Universe the scale factor was shorter meaning that the wavelength from CMB photons was blue shifted, and energy and temperature were higher. The fact that the scale factor is inversely proportional to temperature implies the initial state of the Universe was a state of extreme high temperature and density. From formula [2.4] it can be seen that an early hot and dense Universe is radiation dominated as the contribution to the energy density from radiation increases most for small scale factors. With expansion the radiation density drops faster than the matter density and the moment in time where radiation and matter energy density were equal is called matter radiation equality and is estimated at 47.000 years after the Big Bang. The Universe remained optically thick to radiation untill the Universe was about 378.000 years old and photons were last scattered composing the CMB. Then a period of matter domination followed in which dark matter determined the growth of the large scale of the Universe. Expansion was first slowed down by respectively radiation and matter, but recently dark energy dominates the energy density of the Universe accelerating its expansion, see fig. 2.14, fig. 2.15 and fig. 2.16. [3] [25] [66]
Fig. 2.14. Examples of Universes with different matter density parameters from which ΩM =
0.30 ,ΩΛ = 0.70 is in close agreement with latest observations. It is showed how the Universe
recently started its accelerating expansion, whereas for models with no dark energy ΩΛ = 0
the Universe would expand with a constant speed ΩM = 0 or end with a Big Crunch ΩM > 0.
[66].
Fig. 2.15. Different epochs in the thermal history of the Universe. The radiation dominated era is followed by a matter dominated era and the present dark energy dominated era. [72]
