ABSTRACT
This research project aims at exploring the validity of the recently published work by Di Valentine et al. (2019) which used the Planck Legacy 2018 (PL18) data that suggested a possibility of a closed universe in which the amplitude of Cosmic Microwave Background radiations is enhanced and prefers a positive curvature at 99% cadence level. The study by Di Valentino et al. (2019) is based on the observations of the ancient light called Cosmic Microwave Background (CMB). In the report, the amplitude of the CMB is larger compared to that of the standard ΛCDM model and the data deviates by 3.4 standard deviations. This research work investigates this amplitude abnormality, derive equations governing dynamics of a closed Universe within Einstein General Relativity, and develop relevant theory behind possible crisis with regard to the proposed evidence of a closed universe by considering the Friedmann-Robertson-Walker (FRW) metric which assumes a homogeneous and an isotropic universe. We analyze the implications of a closed universe in cosmology. This research work begins by deriving the first and second Friedmann equations using the Einstein Field Equations (EFE). Then the continuity equation is derived by considering a perfect fluid. Three coupled differential equations for Hubble parameter, scale factor and density as functions of time are obtained and transformed to two coupled differential equations of Hubble parameter and density parameter as functions of scale factor. The two equations are solved simultaneously using Python – Spyder package called Odeint and plotted graphs of evolution of Hubble parameter and density parameter for Einstein de Sitter (EdS) model, the standard ΛCDM model and compared to that of closed universe. From the graphs obtained, the Hubble parameter decreases with increase in the scale factor. The value of Hubble parameter in EdS at decoupling is greater than that of ΛCDM and closed models but their values converge today. The density parameter for a closed universe is greater than one compared to that Einstein de Sitter model and ΛCDM values which is one. This implies that the closed cosmos has enough matter to cause a deceleration in its expansion. The deceleration implies that at some time in future the expansion will stop and big crunch will occur. If indeed the universe is closed, then the current cosmology is in a crisis. Since the Planck spectra from Planck‟s Legacy 2018 prefers a closed universe, however, the anomalies might have risen from undetected systematics and/or statistical fluctuations, this study recommends that more observations to be carried out to ascertain whether there is a possible paradigm shift in cosmology and new physics is required.
TABLE OF CONTENTS
DECLARATION i
DEDICATION ii
ACKNOWLEDGEMENT iii
ABSTRACT iv
LIST OF TABLES vii
LIST OF FIGURES viii
LIST OF ABBREVIATIONS/ACRONYMS AND SYMBOLS
CHAPTER ONE: INTRODUCTION
1.1 Research Background 1
1.1.1 Geometry of our Universe 1
1.1.2 The Cosmic Microwave Background 3
1.1.3 Standard cosmological model 4
1.2 Theory: History of cosmology 4
1.3 Statement of the Problem 8
1.4 Objectives 8
1.4.1 Main Objective 8
1.4.2 Specific Objectives 8
1.5 Justification and Significance of the Study 9
CHAPTER TWO: LITERATURE REVIEW
CHAPTER THREE: THEORETICAL FRAMEWORK 23
3.1 Einstein Field Equations 23
3.2 Friedman–Lemaitre–Robertson–Walker (FLRW) Metric 23
3.3 Density and Density parameter 26
3.4 Cosmic Microwave Background (CMB) Radiation 27
CHAPTER FOUR: RESEARCH METHODOLOGY
4.1.1 Source of Data 29
4.1.2 Planck, WMAP 29
4.1.3 Data type and Description 29
4.1.4 Data Analysis Tools 30
4.2 The Governing Equations 30
4.2.1 Einstein Field Equations 30
4.2.2 Energy-Momentum Conservation 44
4.2.3 Matter density And Density Parameter 47
4.2.4 Dynamical Equations 51
CHAPTER FIVE: RESULTS AND DISCUSSION
CHAPTER SIX: CONCLUSION AND RECOMMENDATION
References 62
LIST OF TABLES
Table 2. 1: Tensions between PL18 and BAO and CMB Lensing. 18
LIST OF FIGURES
Figure 2. 1: Preference for a closed universe (Di Valentino E, 2019) 14
Figure 2. 2: Degeneracy between curvature and lensing (Di Valentino E, 2019) 16
Figure 2. 3: Curvature and parameters shift (Di Valentino E, 2019) 16
Figure 2. 4: Tension with CMB lensing (Di Valentino E, 2019) 18
Figure 2. 5: Tension with cosmic shear measurements (Di Valentino E, 2019) 19
Figure 2. 6: Tension with combined data (Di Valentino E, 2019) 20
Figure 2. 7: Tensions in combined data (Di Valentino E, 2019) 20
Figure 6. 1: A graph of h as a function of the scale factor 56
Figure 6. 2: A graph of a density parameter against scale factor 56
Figure 6. 3: A graph of h as a function of the scale factor for ΛCDM model 57
Figure 6. 4: A graph of density parameter for ΛCDM against the scale factor 57
Figure 6. 5: A graph of h against scale factor (a) in closed model 58
Figure 6. 6: A graph of density parameter with respect to scale factor for a closed universe 58
Figure 6. 7: Comparing the graphs of dimensionless Hubble parameter in EdS, ΛCDM and closed models.59 Figure 6. 8: Comparing the graphs of density parameter as a function of the scale factor in EdS, ΛCDM and closed models 59
LIST OF ABBREVIATIONS/ACRONYMS AND SYMBOLS
FLRW Friedman-Lemaitre-Robertson-Walker
CP Copernican Principle
BAO Baryon acoustic oscillation
SKA Square Kilometer Array
CMB Cosmic Microwave Background Radiation
OHD Observational Hubble parameter data
SDSS Sloan Digital Sky Survey
Mpc mega parsecs
H Planck‟s constant
Kpc kilo parsecs
WMAP Wilkinson Microwave Anisotropy Probe
LAMBDA Legacy Archive for Microwave Background Data Analysis
NASA National Aeronautics and Space Administration
FWHM Full Width Half Maximum
EFE Einstein Field Equations
DM Dark matter
PL Plank‟s Legacy
EoS (w) Equation of sate
CHAPTER ONE
INTRODUCTION
1.1 Research Background
Cosmology is a discipline in astronomy that studies the universe as a whole with an assumption that at the largest scales the universe obeys the homogeneity and isotropy. It aims at understanding, the origin, structure, composition, evolution, and fate of the Universe. Homogeneity means that the universe is the same place to place and isotropy means it looks the same in all directions. This assumption of the universe being homogenous and isotropic is important because observations made from any single point can be used to represent the universe as a whole and in turn this information can therefore be legitimately used in testing cosmological models. This theoretical assumption was made by Albert Einstein in his earliest work in the twentieth century and was meant to simplify the mathematical analysis (Amandola, 2021)
1.1.1 Geometry of our Universe
The geometry of the universe simply means its curvature which is denoted by k and its shape. The curvature can be positive, negative or zero. There are many shapes but only three basics ones are considered which are flat, open and closed shapes of the universe. In the year 1925, Edwin Hubble discovered that our universe is expanding. Hubble came up with evidence showing that the farther a galaxy is the faster it moves away from us and this is now known as Hubble law. The Hubble law simply means the rate of expansion of space and it applies to any system that expands and or contracts in a uniform and isotropic manner (Piattela, 2018). The equation (1) below describes the Hubble law.
ν = 
(1) where, ν is the velocity it moves away from us, r the distance, and H0 is the Hubble constant. The H0 value as determined by recent measurements is, H0 = 67.6 Km/s/Mpc. This value means that for a Mpc away, a source moves away at a speed of 67.6 km/s faster.
Towards the end of 20th century, observations made on the radiation emitted from type Ia supernovae confirmed that the universe is expanding and discovered that this expansion is accelerating. This discovery about the accelerated expansion of the universe posed a great challenge in physics. There was need for cosmological models that could explain this anomalous because our knowledge on gravity is that it should attract matter, and that we should expect the expansion to decelerate (Shu W, 2015). One of the solutions settled on was via the spacetime geometry structure in which length, time and mass are said to be related. It was assumed that there could be some form of new energy which is acting as anti-gravity called dark energy. Other observations from different sources and of different nature at different distances have indicated that there is dark component of matter called dark matter (DM). We can use this dark energy, and dark matter together with normal matter to obtain the universe‟s density parameter. The value of this parameter is derived by finding the ratio of the average total matter and energy density to the critical density. The critical density can be explained as density in which the universe would halt its expansion and that is only after an infinite time. See the equation (2) below.
(2)
Where, ρ and ρc are actual density and critical density of the universe respectively.
The value of this parameter density Ω0 is almost one. There are studies going on aiming at finding on whether the value of Ω0 is greater than 1, less than 1 or exactly 1, which in turn can give the geometry of the universe as follows;
1. This means that the universe is open which tells us that it will continue to expand forever. An open universe‟s shape is likened to that of 3D saddle on which two parallel lines diverge.
2. This means that the universe is closed which tells us that it will eventually stop its expansion and re-collapse. A closed universe‟s shape is likened to that of a 3D sphere in which two initially parallel lines will finally converge.
3. This means that the universe is flat and that it has matter to stop the expansion but won‟t to re-collapse it. The shape of a flat universe is likened that of a flat sheet or Euclidean such that any two initially parallel lines on it will always remain parallel to each other.
(3)
where, Ωρ is matter density, Ωk dark energy density/ curvature density and ΩΛ is cosmological density.
1.1.2 The Cosmic Microwave Background
The cosmic microwave background (CMB) is the electromagnetic radiation remained after the Big bang. This radiation is a powerful tool in investigating the early universe and the information obtained is used in constraining the standard cosmological model parameters. The CMB radiations gives us a picture on how the universe looked like when it was a few hundreds of thousands years of age, a time at which the neutral atoms could form and photons decouple from matter. This CMB radiation was found to have black body spectrum by Cosmic background explorer (COBE) satellite from which it can be concluded that matter and radiation balanced in the early periods. So the distribution of photons should reflect that of matter at the time decoupling took place and if there is an inhomogeneity in matter density it means that fluctuations of CMB temperature occurred.
In the early 1990s, COBE detected anisotropy in the CMB temperature, though the level was very small, it made it simple to predict theoretically anisotropy pattern by applying linear perturbation theory. This anisotropy pattern gives cosmological information which is mostly concentrated at angular scales which is less than a degree on the sky and this corresponds to the perturbations that were inside the horizon before decoupling. It is through these scales that physical processes left CMB imprint in the early Universe.
The CMB power spectrum shape is determined by the cosmological parameters. With perturbations in density, given its initial distribution in the early Universe, the relative peaks‟ height indicates baryonic matter density in the Universe. However, the peaks‟ position depends on the mapping of the sound horizon‟s physical scale into angular dimensions on the sky at decoupling which also depends on the geometry of the Universe. For instance, in an open Universe, at decoupling, the angle of physical scales is small compared to that of a flat Universe. Therefore, the peaks‟ position of CMB power spectrum is a good approximation of the total density of the universe.
Planck‟s Legacy 2018 used the Gravitational lensing to measure the density matter of the Universe. Gravitational lensing can be defined as the process by which radiations from distant astronomical objects is bent by the gravity of massive objects it encounters as it travels towards us. This bending makes the images of background astronomical objects appear slightly distorted and such observations is used to obtain useful cosmological information. The degree at which CMB light has been bent or 'gravitationary lensed' while travelling through the universe over the past 13.8 billion years is what the Planck Telescope uses to measure and be able to gauge the density of the universe. The amount of matter that intervenes CMB photons as they travel towards the earth, gives the extent at which they are deflected so that their direction does not crisply reflects their starting in the early universe (Balbi A, 2004).
1.1.3 Standard cosmological model
The current Standard Cosmological Model is denoted by ɅCDM, where Lambda (Ʌ) is a cosmological constant associated with dark energy and CDM is an abbreviation for cold dark matter which is the sufficient massive dark matter particles of the Universe. This model assumes that the origin of the Universe is from pure energy that underwent the Big Bang and that about 5% of it makes normal matter while 27% makes dark matter and 68% dark energy. This model assumes further that in the large scales the universe is not only homogeneous but also isotropic. This model is based on two theoretical models which are; the Standard Model of Particle Physics (SMPP) also called physics of the very small and General Theory of Relativity (GTR) which is the physics of the very large. However, these two models have their shortcomings. For instance, the SMPP does not give an understanding on how the three generations of leptons and quarks came to exist and even their mass hierarchy, nature of gravity and the nature of dark matter. GTR on the other hand is short of information about Big Bang cosmology, inflation, the asymmetrical of the matter-antimatter in the universe, and the nature of dark energy (Robson B, 2019).
1.2 Theory: History of cosmology
Looking into observational Cosmology, the first model to describe the universe was the
„island universe‟ model that was developed by Descartes that was published in The World of 1636 which involved the solar system problem. In the year 1750 Wright published a book with a title An Original Theory of the Universe which involved stars and the solar system in a sphere. In 1755 Kant and 1761 Lambert came up with first pictures of the Universe which were hierarchical. All these information about the Universe did not have observational validation. Afterwards, the distance of the Sun was known, making it a first star with a known distance. Friedrich Bessel et al. (1830s) made the first parallax measurement of stars.
The quantitative estimations about scale and structure of Universe were made by William Herschel in 18th century. His large-scale structure model was based on the counting of stars and it gave an evidence for the „island universe‟. Herschel derived the famous model for the galaxy on an assumption that the absolute luminosities of the stars were the same.
John Michell, a Geology Woodwardian Professor at Queen‟s College, Cambridge, warned William Herschel on his assumption that stars had fixed luminosity. In 1767, John Michell developed the Cavendish experiment which was used to measure the average Earth density. Michellis greatly remembered from his invention of black hole. In 1802, Herschel after measuring the visual binary magnitudes of our Galaxy and stars, in his conclusion he agreed with John Michell‟s warning about the luminosity of stars and finally he lost faith in his model.
Throughout the 19th century there was a great desire to make observations of the Universe using a telescope of a higher aperture. A 72-inch reflector, the largest telescope then was constructed by William Parsons at Birr Castle, Ireland. The telescope was so big that on tracking the astronomical objects, its barrel was moved by ropes so as to accommodate the platform that could move at the Newtonian focus of the telescope during observations. During this century, the problem of pointing of reflecting telescope was solved by Lewis Morris Rutherfurd, Andrew Common, John Draper and George Carver by inventing plate holder which was adjustable that enabled the observer to maintain pointing and high precision.
The advancement in technology is attributed to achievement made by James Keeler; he was able to obtain spiral nebulae images among them was his famous M51 image. The images showed detailed structures of the spiral nebulae in which a large number were fainter at a smaller angular size. He concluded that, if these fainter objects were similar to Nebula M31 of Andromeda, then they farther away from the solar system.
Carnegie discovered helium through astronomy long before it was identified in the laboratory. This is one way to prove that astronomy can provide information about behavior of matter by just making astronomical observations which can be reproduced later in the laboratory. Carnegie facilitated the construction of 100-inch Hooker Telescope, which was the largest in the world with all other features learned from other earlier telescopes. In the year 1918, it was complete and it dominated for about 30 years until 1948 another larger telescope, Palomar 200-inch telescope was commissioned.
Using 100-inch Hooker Telescope, Scheiner (1899) obtained M31 spectrogram and stated that it suggested Sun-like stars cluster. Opik, in 1922 compared mass-to-light ratio of M31 with our Galaxy and obtained an estate distance of M31 to be 440 kpc. The same year, Duncan discovered variable stars in spiral nebulae that in turn led to a discovery by Hubble of variable stars in M31.
A paper by Hubble (1925&1926) provided a description about galaxies in the extragalactic system. The paper classified the galaxies into Hubble types with an estimation number of the different types, their mass-to-luminosity ratio and average densities. Its at this time the mean mass density of the Universe as a whole was derived. By the year 1929, after Hubble collecting approximation distances of about 24 Galaxies with measured velocities he came up with his law that bears his name; the Hubble law.
Theoretical cosmology is attributed to Albert Einstein with his famous static model of the Universe. First, in the year 1825, Lobachevsky and Bolyai violated the Euclid‟s fifth axiom by solving the problem of existence of geometries. Their work led to an introduction of quadratic differential forms by Riemann resulting to the generalized non-Euclidean geometries. After a long rout of searching for a consistent theory of gravity that was relativistic using ideas such as; the influence of gravity on light, the principle of equivalence, and the Riemannian spacetime, Einstein came up with general relativity. As the year 1912 was ending, he wanted to have a non-Euclidean geometry. He consulted his friend Marcel Grossmann, on a general way to transform frames of reference for metrics of the form.
The Grossmann‟s answer was that Einstein should use the Riemannian geometries, though they were nonlinear a fact Einstein took as an advantage because any theory that satisfies relativistic gravity must be nonlinear. In the year 1915 Einstein formulated general relativity in its definitive form. In the following year, Willem de Sitter and Paul Ehrenfest gave an idea that in order to remove the problems of the boundary conditions at infinity, there has to be a spherical 4-dimensional spacetime. In 1917, Einstein realized that general relativity was a theory that can be used to construct consistent model of the Universe. At this time the expansion of the Universe was yet to be discovered.
(4) In his theory, Einstein wanted to incorporate the fact that, in the large-scale Universe, distribution of matter should determinate the local inertial frame of reference. Another problem emerged; Newton noted that a static model of the Universe is unstable under gravity. This forced Einstein to introduce another term called the cosmological constant denoted by into the field equation that solved the problem.
(5) In the same year, de Sitter found the solutions of Einstein‟s field equations in the absence of matter ρ= p = 0 meaning that Einstein did not achieve his objectives. de Sitter‟s metric was in the form.
(6)
In 1922, Kornel Lanczos interpreted de Sitter solution by coordinate transformation as follows.
(7) In the same year, Alexander Alexandrovich Friedmann wrote a paper about relativistic cosmology. He noted that for an isotropy world model, the curvature has to be isotropic. He formulated model showing a solution of expanding world with closed spatial geometrie.
̇
(8)
On solving these equations one can get exactly the standard world models of general relativity. In 1927, Georges Lemaître also discovered the solutions of Friedman. Lemaître and Howard P. Robertson in 1928 became aware that the Friedman solutions were actually a discovery that was taken as an evidence for the expansion of the universe. In 1935, Robertson and George Walker independently solved the problem of time and distance in cosmology. For homogeneity and isotropy world, they introduced a metric of the form
(9)
where k is the space curvature at the present epoch, r is a radial distance of comoving coordinate and R(t) is proportional to the distance between any two worldlines changing with cosmic time t and is scale factor, (Longair S, 2004).
Cosmology uses the Friedmann-Lemaitre-Robertson-Walker (FLRW) cosmological model in understanding the evolution of the universe. This model is so successful and for that reason it has become the standard cosmological model which now is used to predict about the universe even at earliest times of 10-43 sec after the Big Bang (Piattela, 2018).
It is until towards the end of the twentieth century when firm empirical data was obtained to confirm the homogeneity and isotropy of the universe exactly the same as the Cosmological Principle had predicted. The temperature of the cosmic microwave background (CMB) radiations which is uniform serves as the best evidence for the isotropy of the observed universe.
1.3 Statement of the Problem
The standard cosmological model ɅCDM predicts the shape of the universe to be flat which agrees with many cosmological observations. The knowledge of the shape of the universe is of great importance as it can be used to predict the evolution and fate of the universe which is in continuous accelerated expansion and it depends on the density parameter. However, the recent cosmological observations, from The Planck Legacy 2018 data indicates that, the Cosmic Microwave Background light‟s amplitude is larger. This can only be explained by the closed universe model. This poses a challenge to the current Standard cosmological Model. Therefore, there is a lot of concern to both observational and theoretical cosmologists that the present model which assumes the shape of the universe to be flat may be incomplete or inaccurate. This concern has shifted our focus to thorough scrutiny through research on whether the current model is incorrect and if so then what will be its implications in cosmology. Although the recent data has suggested possible model of a closed universe, more observations are required to ascertain these claims. In this research we aim to explore the evidence of a closed universe and see if there could be crisis in cosmology.
1.4 Objectives
1.4.1 Main Objective
The main objective of this work is to explore the evidence of the closed universe as suggested by Planck Legacy 2018 data, which shows enhanced amplitude of CMB, and establish a model of a closed universe.
1.4.2 Specific Objectives
Specific objectives of this study are:
1 To derive equations governing dynamics of a closed universe within Einstein Theory of General Relativity considering isotropy and homogeneity.
2 To obtain equations governing the evolution of matter density and matter density contrast of the universe.
3 To derive the equations and develop relevant theory behind possible crisis with regard to the proposed evidence of a closed universe.
4 To study the implications of the closed universe evidence for current cosmology.
1.5 Justification and Significance of the Study
The shape of the universe is key on formation of a standard cosmological model which gives the insight of the dynamics and the future of the universe. The universe is flat an assumption made by the current Standard Cosmological Model ɅCDM. Considering the importance of the model of the shape of the universe in cosmology, the valid way to explain the abnormality in PL18 is to model a closed universe shape. Exploring PL18 is of great significance because: It will help challenge the existing model of a flat universe and shift to a closed universe model; It will help us to predict the future and fate of the universe as the accelerated expansion of the closed universe will halt and Big Crunch occur; It will help to solve the problem of the enhanced amplitude in CMB by PL18.
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