1

Cosmological Models based on Higher Dimensional Theories of Gravity

Koh Zhi Xun A0086665N

Department of Physics National University of Singapore

A thesis submitted for the degree of B.Sc (Hons) in Physics 2014/15

Dr Cindy Ng Shao Chin Thesis Supervisor Department of Physics National University of Singapore

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Declaration

I hereby declare that this thesis is my original work and it has been

written by me in its entirety. I have duly acknowledged all the

sources of information which have been used in this thesis.

This thesis has also not been submitted for any degree in any

university previously.

Koh Zhi Xun

6th April 2015

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Abstract

Since the observations on distant Type Ia supernova in 1998, which

affirmed the discovery that the expansion of the Universe is

accelerating, a cosmological constant have been added to current

cosmological models, to explain this accelerating expansion. This

cosmological constant, known as the dark energy, has never been

found and its nature remains unknown.

This thesis explores the realm of adding an extra spatial dimension to

Einstein’s theory of gravitation. This extra dimension changes the

expression of the widely-accepted Friedmann model, by adding a

quadratic density term into the model.

Using this new-found expression, this thesis discusses the amount of

consistency this proposed model has with observational data. The

cosmological constraints used in this thesis are the age constraints,

constraint on the position of the First Doppler peak , the Type Ia

Supernova Test (SNe) and the combined Cosmic Microwave

Background and Baryonic Acoustic Oscillation Test (CMB/BAO).

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Acknowledgements

Firstly, I would like to thank Dr. Cindy for the successful completion of

this thesis. Without her wise counsel, this thesis would not have been

possible. Knowing my weakness in not being able to comprehend

abstract mathematical equations in general relativity and the various

models in Brane cosmology, she took the liberty of sourcing for

research papers which present these difficult concepts in a manner

which I could understand and appreciate, without sacrificing on the

material aspects of this project. This led to my crucial understanding

of this project and most importantly, being able to communicate to

my peers on the implications of my work.

Secondly, I would like to thank all the NUS physics lecturers who have

been instrumental in building up my foundations in physics

throughout my undergraduate years. The journey have not been

smooth-sailing, but I would not have it any other way, for it is these

challenges that will constantly shape and refine your understanding of

the physical universe, broadening your intellectual horizons and

nourishing your hungry minds.

Thirdly, I would like to thank all my friends whom I have made

throughout my four years in NUS for making my undergraduate life a

memorable one. My life would have been very different without you

guys. I am sure we will cross our paths again someday.

Fourthly, I would like to thank my family, especially to my father and

my late mother, for their unwavering commitment and support to me

and my brother in our education. The least I could do is to have a

successful graduation.

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Here’s a special paragraph dedicated to our founding Prime Minister,

the late Mr Lee Kuan Yew. I want to take this opportunity to thank

you for your dedication and commitment in building Singapore to

what it is today. “Your legacy is not cold stone, but a living nation. And

we could no more forget you than we could forget the sky” (Li, 2015)

I give praise to God for your watchful guidance every step of the way.

Koh Zhi Xun

Singapore

6th April 2015

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Contents List of Figures ........................................................................................................................................ 8

Chapter 1 .............................................................................................................................................. 10

Introduction ......................................................................................................................................... 10

1.1 Basic Assumptions .............................................................................................................. 10

1.2 The Coordinate System ....................................................................................................... 10

1.3 Geometry of the Universe ................................................................................................... 11

1.4 Baryon mass fraction in clusters of galaxies ..................................................................... 13

1.5 Introduction to Dark Matter ............................................................................................... 13

1.6 Brief comments on Dark Energy ........................................................................................ 14

Chapter 2 .............................................................................................................................................. 16

A Review of Cardassian Expansion .................................................................................................... 16

2.1 Review of Cardassian Expansion (Freese & Lewis, 2002) ..................................................... 16

2.1.1 Introduction to Redshift .................................................................................................... 18

2.1.2 Deriving the new critical density for the Cardassian model ........................................... 19

2.1.3 Age of the Universe ............................................................................................................ 21

2.2 Our Proposed Model ................................................................................................................. 22

Chapter 3 .............................................................................................................................................. 23

Brane Cosmology ................................................................................................................................. 23

3.1 Brief Introduction to Brane Cosmology ................................................................................... 23

3.2 Review of Brane Cosmology (Binetruy P. , Deffayet, Ellwanger, & Langlois, 2000) ............ 24

3.2.1 The energy-momentum tensor ......................................................................................... 27

3.2.2 Explaining the Junction Conditions................................................................................... 28

3.2.3 Reviewing Equation (51) ................................................................................................... 29

Chapter 4 .............................................................................................................................................. 31

Proposed Models and their Age Constraints ..................................................................................... 31

4.1 Our Proposed Model 1 .............................................................................................................. 31

4.1.1 Setting C=0 .......................................................................................................................... 33

4.2 Our Proposed Model 2 .............................................................................................................. 34

4.3 Evaluation of Results ................................................................................................................ 36

4.3.1 Explanation of Evaluation .................................................................................................. 37

4.4 Findings from other similar models ........................................................................................ 39

4.4.1 Similar Model 1a ................................................................................................................. 39

Chapter 5 .............................................................................................................................................. 42

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Constraints from the First Doppler Peak ........................................................................................... 42

5.1 The First Doppler Peak ............................................................................................................. 42

5.2 Finding the First Doppler Peak ................................................................................................ 46

5.3 The First Doppler Peak for Proposed Model 1 ........................................................................ 49

5.3.1 The First Doppler Peak for Proposed Model 2 ................................................................. 51

5.4 Evaluation of Doppler Peak results ......................................................................................... 52

Chapter 6 .............................................................................................................................................. 56

Further Cosmological Constraints ..................................................................................................... 56

6.1 Further Cosmological Constraints ........................................................................................... 56

6.2 Constraints from CMB (Feng & Li, 2009) ................................................................................ 56

6.2.1 CMB Constraints for Proposed Model 1 ........................................................................... 58

6.3 Constraints from BAO (Feng & Li, 2009) ................................................................................. 59

6.3.1 BAO Constraints for Proposed Model 1 ............................................................................ 60

6.4 Combining both 𝝌𝑪𝑴𝑩𝟐 and 𝝌𝑩𝑨𝑶𝟐 ..................................................................................... 61

6.5 Similar Analysis for Proposed Model 2 ................................................................................... 63

6.5.1 CMB Constraints for Proposed Model 2 ........................................................................... 63

6.5.2 BAO Constraints for Proposed Model 2 ............................................................................ 64

6.6 Supernova as Standard Candles (Cheng, 2005) ...................................................................... 68

6.6.1 Investigating the Changing expansion rate (Feng & Li, 2009) ....................................... 68

6.6.2 SN Constraints for Proposed Model 1 ............................................................................... 71

6.6.3 SN Constraints for Proposed Model 2 ............................................................................... 74

Chapter 7 .............................................................................................................................................. 77

Summary and Future Work ................................................................................................................ 77

7.1 Summary .................................................................................................................................... 77

7.2 Future Work .............................................................................................................................. 78

Appendix A: Mathematica code for solving Einstein’s equations at Figure 8 ................................. 79

Appendix B: SALT Light Curve Fit Parameters and Distances. Constitution Set. ........................... 83

Appendix C: Calculation of 𝝌𝑪𝑴𝑩𝟐, 𝝌𝑩𝑨𝑶𝟐 𝒂𝒏𝒅 𝝌𝑺𝑵𝟐 for Proposed Model 2. .......................... 92

Bibliography ...................................................................................................................................... 100

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List of Figures

Figure 1: Diagram showing the coordinate system used to describe the Universe. ............................ 10

Figure 2:Diagram showing the geometry of the closed, flat and open universes. ............................... 11

Figure 3: Diagram showing the percentage composition of the Universe. .......................................... 12

Figure 4: Diagram showing evidences of dark matter nearby a galaxy. Source: (Cheng, 2005)........... 13

Figure 5: Diagram showing the contour plots for the range of values for z_eq and n. Source: (Freese

& Lewis, 2002) ....................................................................................................................................... 20

Figure 6: Table showing the ages of the Universe using the Cardassian Model for the range of values

for z_eq and n. Source: (Freese & Lewis, 2002) .................................................................................... 22

Figure 7: Figure illustrating a 4-dimensional brane embedded in a five dimensional space. Source: . 23

Figure 8: Equations showing the solved Einstein’s tensors. Source: (Binetruy P. , Deffayet, Ellwanger,

& Langlois, 2000) ................................................................................................................................... 26

Figure 9: Diagram showing the brane (at y=0) embedded in the five-dimensional metric. ............... 28

Figure 10: Table showing the different redshift values that resulted in different values of B and the

age of the Universe. .............................................................................................................................. 32

Figure 11: Table showing the different redshift values that resulted in different values of A, B and the

age of the Universe. .............................................................................................................................. 33

Figure 12: Table showing the different parameters of A, B, C and the age of the Universe. ............... 34

Figure 13: Table showing the different parameters of A, B, C and the age of the Universe. ............... 35

Figure 14: Graph plot showing the time evolution of the scale factor a(t). ......................................... 38

Figure 15: Table showing the different values of B-term for the different graph plots. ...................... 38

Figure 16: A MatLab plot of the contour lines for Equation (37), extended on the horizontal axis to

n=3. ....................................................................................................................................................... 40

Figure 17: This table shows the different values of F(n) and the age of the Universe as z_eq varies. . 40

Figure 18: The figure shows a baryonic shell of matter approximately 150Mpc in radius. This radius is

called the sound horizon. ...................................................................................................................... 43

Figure 19: The figure illustrates the relationship between angular separation 𝛼1 and the sound

horizon 𝜆1. ............................................................................................................................................ 43

Figure 20: The figure shows the graph of the power spectrum. The first peak lies at l=200. .............. 45

Figure 21: This table shows the values for x and the ratio of x against the FRW model. ..................... 47

Figure 22: This table shows the values for 𝒔 ∗and the ratio of 𝒔 ∗against the FRW model. .................. 48

Figure 23: This table shows the ratio to which the Cardassian model deviates from the location of the

First Doppler peak ................................................................................................................................. 48

Figure 24: This table shows the values for x and the ratio of x against the FRW model. ..................... 49

Figure 25: This table shows the values for s* and the ratio of s* against the FRW model. ................... 50

Figure 26: This table shows the values for x and the ratio of x against the FRW model. ..................... 51

Figure 27: This table shows the values for s* and the ratio of s* against the FRW model. ................... 52

Figure 28: This table shows the values for 𝑯𝒛𝒓𝒕𝒛𝒓 for the different parameters of B and C .............. 53

Figure 29: This table shows the values for 𝑯𝒛𝒓𝒕𝒛𝒓 for the different parameters of A, B and C ......... 54

Figure 30: A Figure showing the evolution of scale factor a(t) during the early formation of the

Universe. ............................................................................................................................................... 55

Figure 31: Table showing the gradient values at the early formation of the Universe ........................ 55

Figure 32: CMB data showing the value of Parameter R. Source: Invalid source specified. ............... 57

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Figure 33: Table showing the value of Parameter R for different values of B, C and 𝜒𝐶𝑀𝐵2 ............ 58

Figure 34: Table extendng the different values of B, C for 𝜒𝐶𝑀𝐵2 = 0 ............................................. 58

Figure 35: Plot showing the value of 𝜒𝐶𝑀𝐵2 for different values of 𝑧𝑒𝑞 ........................................... 59

Figure 36: SDSS data showing the value of Parameter A. Source: (Eisenstein, et al., 2005) ................ 59

Figure 37: Table showing the value of Parameter A for different values of B and C ............................ 60

Figure 38: Plot showing the value of 𝜒𝐵𝐴𝑂2 for different values of 𝑧𝑒𝑞 ............................................ 61

Figure 39: Table showing the value 𝜒𝐶𝑀𝐵2 + 𝜒𝐵𝐴𝑂2 for different 𝑧𝑒𝑞............................................. 61

Figure 40: Plot showing the value of 𝜒𝐶𝑀𝐵2 + 𝜒𝐵𝐴𝑂2 for different values of 𝑧𝑒𝑞 ........................... 62

Figure 41: Table showing the different chi square_CMB values for different parameters of A,B and C

.............................................................................................................................................................. 63

Figure 42: Plot showing the values of chi square_CMB for different parameters of B ........................ 64

Figure 43: Table showing the value of Parameter A for different values of A, B and C ....................... 64

Figure 44: Plot showing the values of chi square_BAO for different parameters of B ......................... 65

Figure 45: Table shows the combined CMB and BAO analysis of Model 2 .......................................... 66

Figure 46: Plot shows the combined CMB and BAO analysis of Model 2 against B ............................. 66

Figure 47: Figure shows a plot of Luminosity Distance ∆𝑟 against Redshift 𝑧. An accelerating Universe

would have the Hubble curve bends upward, as a supernova at a given redshift would be further out

in distance than anticipated. ................................................................................................................ 69

Figure 48: Table showing the value of 𝜒𝑆𝑁2 for different values of 𝑧𝑒𝑞, 𝐵 𝑎𝑛𝑑 𝐶 ............................ 71

Figure 49: Plot showing how 𝝌𝑺𝑵𝟐 varies with 𝒛𝒆𝒒 ........................................................................... 72

Figure 50: Table showing the combined 𝝌𝑩𝑨𝑶𝟐 + 𝝌𝑪𝑴𝑩𝟐 + 𝝌𝑺𝑵 𝟐 and 𝒛𝒆𝒒 .............................. 72

Figure 51: Plot showing the combined 𝝌𝑩𝑨𝑶𝟐 + 𝝌𝑪𝑴𝑩𝟐 + 𝝌𝑺𝑵 𝟐against 𝒛𝒆𝒒 ............................ 73

Figure 52: The different values of 𝝌𝑺𝑵𝟐 for different set of parameters A, B and C .......................... 74

Figure 53: The different values of 𝝌𝑺𝑵𝟐 for different set of parameters A, B and C for 𝒛𝒆𝒒 = 𝟎...... 74

Figure 54: Summarized table of the minimum values of 𝝌𝑺𝑵𝟐 for different 𝒛𝒆𝒒 .............................. 75

Figure 55: Summarized table of the combined 𝝌𝑴𝑰𝑵𝟐 values for different 𝒛𝒆𝒒 ............................... 75

Figure 56: Comparison figure for Proposed Model 1 ........................................................................... 75

Figure 57: Different Cardassian Models for Future Work..................................................................... 78

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Chapter 1

Introduction

1.1 Basic Assumptions

The study of the Universe begins with the simplifying assumption that on a large enough

distance scale of over 100Mpc, the Universe is homogenous and isotropic (Cheng, 2005). In

layman’s terms, isotropic means the Universe looks the same in every direction and

homogeneous means the Universe has the same properties at every place. This simplifying

feature is summarized in the cosmological principle, where it states,

At any cosmic time, the universe is homogeneous and isotropic.

The evidence for the Universe to be homogeneous and isotropic came in the form of

extremely uniform Cosmic Microwave Background (CMB) radiation. (Cheng, 2005)

1.2 The Coordinate System

The description of the Universe begins by setting up a coordinate system. This coordinate

system would map every galaxy and matter in the Universe, and all this matter would be

fixed on their respective grids on the coordinates. When the matter moves, the grid would

move along with these matter, this is called co-moving coordinate. Hence, if galaxy A is

mapped onto coordinate grid A, it would be fixed onto coordinate grid A even if it moves, as

the coordinate grid A would move along with it. Another way to understand this concept is

to visualize two points on an expanding balloon, where the two points are fixed but it is the

coordinate system that expands that brings the two points further apart.

A B

Δ𝑥

𝐷𝐴𝐵 = 𝑎(𝑡)∆𝑥𝐴𝐵

𝑣𝐴𝐵 = (𝑡)∆𝑥𝐴𝐵

𝑣𝐴𝐵

𝐷𝐴𝐵=

𝑎= 𝐻

𝑣 = 𝐻𝐷

Figure 1: Diagram

showing the coordinate

system used to describe

the Universe.

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𝐷𝐴𝐵 is the proper actual distance between points A and B. 𝑎(𝑡) is the scale factor and it

depends on time. This scale factor 𝑎(𝑡) will determine how much expansion or contraction

between the two points A and B. ∆𝑥𝐴𝐵 is the coordinate grid separation between points A

and B and these points are observed to be one grid separated from each other from

Figure 1. Differentiating 𝐷𝐴𝐵 with respect to time gives 𝑣𝐴𝐵, the velocity at which the grids

are moving apart from each other. The ratio of 𝑣𝐴𝐵

𝐷𝐴𝐵 gives the Hubble constant 𝐻.

The derivative of the scale factor (𝑡) , would tell us the whether the Universe is expanding

or contracting. The second time derivative of the scale factor, (𝑡), would tell us whether

the expansion or contraction is accelerating or decelerating.

1.3 Geometry of the Universe

The assumption that the Universe is homogeneous and isotropic has resulted in only three

basic cosmological models: the ‘closed’, ‘flat’ and ‘open’ universes. (Schutz, 1982)

Current observations using the Wilkinson Microwave Anisotropy Probe (WMAP) show that

the geometry of the Universe is flat. According to the Friedmann Model of Cosmology, the

density parameter is defined to be Ω =𝜌

𝜌𝑐. The parameter ρ refers to the actual observed

matter density and 𝜌𝑐 refers to the critical density needed for the Friedmann Universe to be

flat.

Figure 2: Diagram

showing the geometry

of the closed, flat and

open universes.

Source: Wikipedia

Figure 2:Diagram showing the geometry of the closed, flat and open universes.

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If the current observed density of the Universe 𝜌 is equal to 𝜌𝑐, which also means Ω = 1,

the Universe would have enough density to be flat, and would be consistent with the WMAP

observations.

However, observations of the average density of ordinary matter in the Universe is just 0.2

atoms per cubic meter, well short of the critical density of 5 atoms per cubic meter for a flat

Friedmann Universe (Rees, 2000). This shortfall of ordinary matter density directly

contradicts the WMAP results that the Universe is flat. Hence, Dark matter and Dark energy

need to be introduced to make Ω = 1.

Hence, current estimates of ordinary matter is around 4%, dark matter is around 22% and

dark energy is around 74%. Together they make up 100%, which gives Ω = 1 and we have

an explanation for a flat universe.

Hence, for a flat universe,

Ω = Ω𝑀+𝐷𝑀 + Ω𝛬

Where Ω𝑀+𝐷𝑀 is the proportion of matter and dark matter and Ω𝛬 is the proportion of dark

energy in the Universe.

Figure 3: Diagram

showing the

percentage

composition of the

Universe.

Source: Wikipedia

Figure 3: Diagram showing the percentage composition of the Universe.

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1.4 Baryon mass fraction in clusters of galaxies

Cluster masses are reliably measured from consistent results from the velocities of the

galaxies, and the gravitational deflection of light from background galaxies etc. and current

measurements shows

Ω𝑀+𝐷𝑀~0.3.

(1)

1.5 Introduction to Dark Matter

The most direct evidence of dark matter’s existence comes from measured “rotation

curves” in galaxies.

Figure 4: Diagram showing evidences of dark matter nearby a galaxy. Source: (Cheng, 2005)

According to Newton’s law of gravitation, the velocity curve with respect to distance would

fall off like r-1/2 after a distance R, however, actual observations shows that the velocity

remained a constant after a distance R. Distance 0 to R is where most of the

luminous/ordinary matter is concentrated.

Figure 4: Diagram showing evidences of dark matter nearby a galaxy. Source: (Cheng, 2005)

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𝑣 = √𝑀𝐺

𝑟

Hence to account for the discrepancies between the observed and theoretical curves, it is

postulated that there must be matter beyond the distance R, such that M is changed to

M(r), total mass as a function of distance r. 𝑀(𝑟) is also postulate to be proportional to 𝑟, so

that a constant velocity 𝑣 can be obtained.

𝑣 = √𝑀(𝑟)𝐺

𝑟 𝑤ℎ𝑒𝑟𝑒 𝑀(𝑟)𝛼 𝑟

This matter is hence referred to as dark matter as they do not emit, reflect or absorb light.

From observations, it is further inferred that this dark matter form a spherical halo

surrounding the luminous portion of the galaxy, as shown in Fig 7.5 of Figure 3.

This dark matter does not collapse by their own mass as unlike luminous matter which loses

energy to radiation, collisions, electromagnetic interactions; this dark matter is weakly

interacting and they have no charge. As such, they do not interact and lose energy with

other matter; hence dark matter doesn’t collapse due to gravitational pull.

1.6 Brief comments on Dark Energy

Dark energy has not been directly observed, and it has been postulated to cause the

accelerating expansion of the universe. Dark energy is a characteristic of empty space. The

value of the density is fixed, and it does not change as the volume of space increases.

𝜌0 = 𝛬

3

8𝜋𝐺

(2)

𝐸 = 𝜌0𝑉

(3)

𝑑𝐸 = 𝜌0𝑑𝑉 + 𝑑𝜌0𝑉 = −𝑤𝜌0𝑑𝑉

(4)

𝑑𝜌0 = 0

(5)

𝑤 = −1 (6)

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Introducing the dark energy term (a cosmological constant) would cause an exponential

growth in the scale factor.

(

𝑎)2 = 𝛬

(7)

𝑎 = 𝑐𝑒√𝛬𝑡

(8)

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Chapter 2

A Review of Cardassian Expansion

2.1 Review of Cardassian Expansion (Freese & Lewis, 2002)

There is a paper by Freese and Lewis, from the University of Michigan that presents

interesting results.

They are interesting because:

a) You do not need to introduce dark energy to explain that the universe is flat because

the critical density for a flat universe is much smaller in this Cardassian model as

compared to the Friedmann model.

b) Their model does not need dark energy to show an accelerating universe, you only

need matter, dark matter and radiation.

c) This Cardassian model can be shown to be compatible with observations.

For a flat universe, the standard equation in the Friedmann model is

𝐻2 = 𝐴𝜌; 𝐴 =

8𝜋𝐺

3

(9)

For the Cardassian model, they modified the equation to be

𝐻2 = 𝐴𝜌 + 𝐵𝜌𝑛

(10)

The density of ordinary matter can be expressed in terms of the scale factor R:

𝜌 = 𝜌0(

𝑅0

𝑅)3

(11)

17

Given (10), if the term 𝐵𝜌𝑛 dominates, the scale factor 𝑅(𝑡) would be,

(𝑅(𝑡)

𝑅(𝑡))2 = 𝐻2 = 𝐵𝜌𝑛

(12)

𝜌 = 𝜌0(

𝑅0

𝑅)3

(13)

𝑅(𝑡) ∝ 𝑡

23𝑛

(14)

It should be noted that 𝑎(𝑡) =𝑅(𝑡)

𝑅0.

If 𝑛 < 2/3, for example 𝑛 = 1/3, 1/6, then 𝑅(𝑡) would take the form

𝑅(𝑡) ∝ 𝑡2, 𝑡4 (15)

This means that the expansion of the Universe is accelerating given that the second

derivative (𝑡) > 0.

Derivation of (11) is as follows:

𝑃 = 𝑤𝜌 (16)

𝑤 = 0 𝑓𝑜𝑟 𝑚𝑎𝑡𝑡𝑒𝑟 (17)

𝐸 = 𝜌𝑉 (18)

𝑑𝐸 = −𝑃 𝑑𝑉 (19)

𝜌 𝑑𝑉 + 𝑉 𝑑𝜌 = −𝑃 𝑑𝑉 (20)

𝑉 𝑑𝜌 = −(1 + 𝑤)𝜌 𝑑𝑉 (21)

𝑑𝜌

𝜌= −(1 + 𝑤)

𝑑𝑉

𝑉

(22)

𝜌 ∝

1

𝑉1+𝑤

(23)

𝜌 𝛼

1

𝑅3

(24)

𝜌 = 𝜌0 (

𝑅0

𝑅)

3

= 𝜌0(1 + 𝑧)3 (25)

𝑤ℎ𝑒𝑟𝑒 𝑧 𝑖𝑠 𝑡ℎ𝑒 𝑟𝑒𝑑𝑠ℎ𝑖𝑓𝑡

18

2.1.1 Introduction to Redshift

When a light source emits a light wave, it emits a certain wavelength 𝜆𝑒𝑚𝑖𝑡𝑡𝑒𝑑. As this light

wave travel through time, this wavelength increases as the Universe expands and the

wavelength detected 𝜆𝑑𝑒𝑡𝑒𝑐𝑡𝑒𝑑 would be longer. The difference is measured and this is the

concept of redshift, denoted by 𝑧.

𝜆𝑑𝑒𝑡𝑒𝑐𝑡𝑒𝑑

𝜆𝑒𝑚𝑖𝑡𝑡𝑒𝑑− 1 = 𝑧

(26)

𝑎(𝑡) − 𝑡ℎ𝑒 𝑠𝑐𝑎𝑙𝑒 𝑓𝑎𝑐𝑡𝑜𝑟 𝑣𝑎𝑟𝑖𝑒𝑠 𝑤𝑖𝑡ℎ 𝑡𝑖𝑚𝑒

This increase in the wavelength is caused by the expansion of the Universe, which is caused

by the increase of the scale factor 𝑎(𝑡), hence the scale factor can be defined in terms of

redshift.

𝑎(𝑡𝑟𝑒𝑐𝑒𝑖𝑣𝑒𝑑)

𝑎(𝑡𝑒𝑚𝑖𝑡𝑡𝑒𝑑)− 1 = 𝑧

(27)

𝑤ℎ𝑒𝑟𝑒 𝑎(𝑡) =

𝑅(𝑡)

𝑅0

(28)

In this Cardassian model, it is assumed that the new term of the equation is initially

negligible and only comes to dominate recently, at a redshift indicated by supernovae

observations.

The second term starts to dominate at a redshift 𝑧𝑒𝑞when 𝐴𝜌(𝑧𝑒𝑞) = 𝐵𝜌𝑛(𝑧𝑒𝑞). Using (25),

we then get

𝐵

𝐴= 𝜌0

1−𝑛(1 + 𝑧𝑒𝑞)3(1−𝑛) (29)

𝐻02 = 𝐴𝜌0 + 𝐵𝜌0

𝑛 (30)

𝐴 =

𝐻02

𝜌0− 𝐵𝜌0

𝑛−1 (31)

𝐵 =

𝐻02(1 + 𝑧𝑒𝑞)3(1−𝑛)

𝜌0𝑛[1 + (1 + 𝑧𝑒𝑞)

3(1−𝑛)]

(32)

19

2.1.2 Deriving the new critical density for the Cardassian

model

Using (29) and (30), we can get the following relation,

𝐻2 = 𝐴 [𝜌 + 𝜌01−𝑛(1 + 𝑧𝑒𝑞)

3(1−𝑛)𝜌𝑛]

(33)

Setting 𝐴 =8𝜋

3𝑚𝑝𝑙2 , we have the following

𝐻0

2 =8𝜋

3𝑚𝑝𝑙2 𝜌0 [1 + (1 + 𝑧𝑒𝑞)

3(1−𝑛)]

(34)

The new critical density for a flat Cardassian Model is the following

𝜌𝑐 =

3𝐻02𝑚𝑝𝑙

2

8𝜋 [1 + (1 + 𝑧𝑒𝑞)3(1−𝑛)

]

(35)

𝜌𝑐 = 𝜌𝑐,𝑜𝑙𝑑 × 𝐹(𝑛) (36)

𝐹(𝑛) = [1 + (1 + 𝑧𝑒𝑞)3(1−𝑛)

]−1 (37)

The factor 𝐹(𝑛) in (35) shows that it is the factor that reduces the critical density for a flat

Friedmann Universe. This means that for a flat Cardassian Universe, you do not need a

critical density as high as that required by the Friedmann Model.

In section 1.4 where we discussed on the findings of the baryon mass fraction, we derived

the result of Ω𝑀+𝐷𝑀~0.3. This means that the current proportion of matter and dark matter

observed in our universe is 0.3 out of 1, where 1 is the density parameter needed for a flat

Friedmann Universe. This paper focuses on setting 𝐹(𝑛) = 0.3, so as to constraint the

Cardassian Model to fit the current observation. With this constraint in mind, this paper

focuses on all values of 𝑧𝑒𝑞 and 𝑛 that would fit the constraint of 𝐹(𝑛) = 0.3.

20

Equation (37) is then plotted on Figure 5, to show all values of 𝑧𝑒𝑞 and 𝑛 that would fit the

constraint of 𝐹(𝑛) = 0.3.

Figure 5: Diagram showing the contour plots for the range of values for z_eq and n. Source: (Freese & Lewis, 2002)

21

2.1.3 Age of the Universe

Only taking the values of 𝑧𝑒𝑞 and 𝑛 which make 𝐹(𝑛) = 0.3, we can solve for the age of the

Universe from the Cardassian Model. Using the following relations,

=

𝑑𝑎

𝑑𝑡

(38)

𝑑𝑡 =

𝑑𝑎

(39)

∫ 𝑑𝑡 = ∫

𝑑𝑎

= ∫

𝑑𝑎

𝑑𝑧

𝑑𝑧

(40)

𝑎(𝑡) =

1

1 + 𝑧

(41)

𝑑𝑎

𝑑𝑧= −

1

(1 + 𝑧)2

(42)

𝑎= 𝐻

(43)

∫ 𝑑𝑡 = ∫

𝑑𝑎

= ∫

𝑑𝑎

𝑑𝑧

𝑑𝑧

= ∫ −

1

(1 + 𝑧)2

𝑑𝑧

𝑎𝐻

(44)

∫ −

1

1 + 𝑧

𝑑𝑧

𝐻

(45)

𝐻(𝑧) = 𝐻0𝐸(𝑧) (46)

Where

𝐸(𝑧) = (Ω𝑀0(1 + 𝑧)3 + Ω𝑅0(1 + 𝑧)4 + Ω𝐾0(1 + 𝑧)2 + Ω𝛬0𝑓(𝑧))1/2 (47)

Given that Ω𝐾0 and Ω𝛬0 equals to zero and Ω𝑅0 is negligible, we only consider the Ω𝑀0 term.

Using (45) and (46), the equation to calculate the age of the Universe is given by

𝑡(𝑧) = 𝐻0

−1 ∫𝑑𝑧

(1 + 𝑧)𝐸(𝑧)

∞

𝑧

(48)

22

From (47) and (30), we get

Ω𝑀0 =

𝐴𝜌0 + 𝐵𝜌0𝑛

𝐻02

(49)

Using (48) and all values of 𝑧𝑒𝑞 and 𝑛 that would fit the constraint of 𝐹(𝑛) = 0.3, we get the

following results.

Recent analyses of the ages of old stars indicate the expansion time is in the range 11 – 17

Gyr, which translates to the range of 0.72 – 1.17 for the dimensionless age parameter H0t0.

The results of the values obtained above are still within the range of analysis. (Freese &

Lewis, 2002)

2.2 Our Proposed Model

The proposed cosmological model for this project is given by the following expression

𝐻2 = 𝐴𝜌 + 𝐵𝜌2 + 𝐶 (50)

where 𝐴, 𝐵 𝑎𝑛𝑑 𝐶 are constants.

This proposed model is motivated by brane cosmology, which considers a fifth dimension in

addition to our ordinary four-dimensional spacetime.

Figure 6: Table showing the ages of the Universe using the Cardassian Model for the range of values for 𝑧𝑒𝑞

and 𝑛. Source: (Freese & Lewis, 2002)

Figure 6: Table showing the ages of the Universe using the Cardassian Model for the range of values for z_eq and n. Source: (Freese & Lewis, 2002)

23

Chapter 3

Brane Cosmology

3.1 Brief Introduction to Brane Cosmology

In the braneworld picture, the standard model fields are not universal. They are confined to

lie on a 3+1 dimensional hyper-surface, known as the brane, embedded in some higher

dimensional space-time, known as the bulk. Hence, the extra dimensions need not be

constrained by collider experiments to be small and compact. Test of Standard Model

processes can only constrain how far the brane may extend into the bulk. Constraints of the

bulk can be found from gravitational experiments, as gravity is the only force that extends

into the bulk space-time. (Clifton, Ferreira, Padilla, & Skordis, 2012)

In the course of this project, we shall consider only up to five dimensions, and this fifth

dimension corresponds to the bulk space-time. Only matter is confined to the brane and

gravity interactions extends to the bulk. This concept of adding an extra dimension is

motivated by string theory and has many useful consequences, one of which allows us to

explain the vast hierarchy between the electro-weak scale and the Planck scale. However,

these topics are not relevant to our current project and hence we shall not delve further

into these details.

Figure 7: Figure illustrating

a 4-dimensional brane

embedded in a five

dimensional space. Source:

Invalid source specified.

24

3.2 Review of Brane Cosmology (Binetruy P. , Deffayet,

Ellwanger, & Langlois, 2000)

This paper is by Binetruy, Deffayet, Ellwanger and Langlois from France. From this paper,

they obtained the following results,

2

𝑎2=

𝜅2

6𝜌𝐵 +

𝜅4

36𝜌𝑏

2 +𝐶

𝑎4−

𝑘

𝑎2

(51)

To derive (51), they defined a metric of the form:

𝑑𝑠2 = −𝑛2(𝜏, 𝑦)𝑑𝜏2 + 𝑎2(𝜏, 𝑦)𝛾𝑖𝑗𝑑𝑥𝑖𝑑𝑥𝑗 + 𝑏2(𝜏, 𝑦)𝑑𝑦2 (52)

Where 𝛾𝑖𝑗 is a maximally symmetric 3-dimensional metric.

This metric expressed in its complete form is the following

𝑑𝑠2 = −𝑛2(𝜏, 𝑦)𝑑𝜏2 + 𝑎2(𝜏, 𝑦)[

1

1 − 𝑘𝑟2𝑑𝑟2 + 𝑟2(𝑑𝜃2 + 𝑠𝑖𝑛2𝜃 𝑑𝜙2)]

+ 𝑏2(𝜏, 𝑦)𝑑𝑦2

(53)

The five-dimensional Einstein equation takes the form

𝐴𝐵 ≡ 𝐴𝐵 −

1

2𝐴𝐵 = 𝜅2𝐴𝐵

(54)

Where 𝐴𝐵 is the five-dimensional Ricci tensor, = 𝐴𝐵𝐴𝐵 the scalar curvature and the

constant 𝜅 is related to the five-dimensional Newton’s constant 𝐺(5) and the five-

dimensional reduced Planck mass 𝑀(5), by the relations

𝜅2 = 8𝜋𝐺(5) = 𝑀(5)−3 (55)

25

To solve for the Einstein tensor 𝐴𝐵, one needs to calculate the Christoffel symbols and Ricci

tensor for the metric expression shown in (53).

Γ𝜏𝜏𝜏 =

𝑛

Γ𝑟𝑟𝜏 = −

𝑎

(−1+𝑘𝑟2)𝜂2

Γ𝜃𝜃𝜏 =

𝑟2𝑎

𝜂2

Γ𝜙𝜙𝜏 =

𝑟2𝑎 𝑠𝑖𝑛2𝜃

𝜂2

Γ𝑦𝜏𝜏 =

𝜂′

𝜂

Γ𝑦𝑦𝜏 =

𝑏

𝜂2

Γ𝑏𝜏𝑏 =

𝑏

Γ𝑏𝑏𝑏 =

𝑏′

𝑏

Γ𝑟𝜏𝑟 =

𝑎

Γ𝜃𝜃𝑟 = (−1 + 𝑘𝑟2)𝑟

Γ𝜙𝜙𝑟 = (−1 + 𝑘𝑟2)𝑟 𝑠𝑖𝑛2𝜃

Γ𝑦𝑟𝑟 =

𝑎′

𝑎

Γ𝜃𝜏𝜃 =

𝑎

Γ𝜃𝑟𝜃 =

1

𝑟

Γ𝜙𝜙𝜃 = −𝑐𝑜𝑠𝜃 𝑠𝑖𝑛𝜃

Γ𝑦𝜃𝜃 =

𝑎′

𝑎

Γ𝜙𝑟𝜙

=1

𝑟

Γ𝜙𝜃𝜙

= cot 𝜃

Γ𝜙𝑦𝜙

=𝑎′

𝑎

Γ𝜏𝜏𝑦

=𝜂𝜂′

𝑏2

Γ𝑟𝑟𝑦

=𝑎𝑎′

(−1 + 𝑘𝑟2)𝑏2

Γ𝜃𝜃𝑦

= −𝑟2𝑎𝑎′

𝑏2

Γ𝜙𝜙𝑦

= −𝑟2𝑎 𝑠𝑖𝑛2𝜃 𝑎′

𝑏2

Γ𝑦𝜃𝜃 =

𝑎′

𝑎

𝑅𝜏𝜏 = 3 (𝜂𝑎′𝜂′

𝑎𝑏2+

𝑎𝜂−

𝑎) + 𝜂 (

−𝑏′𝜂′ + 𝑏𝜂′′

𝑏3 ) +𝑏2

𝑏3𝜂−

𝑏

𝑅𝑟𝑟 =−𝑎𝜂3𝑎′𝑏′ + 𝑏𝜂2 (𝑎𝑎′𝜂′ + 𝜂(2𝑎′2

+ 𝑎𝑎′′)) − 𝑎𝑏2𝜂 + 𝑏3(𝑎 − 𝜂(22 + 𝑎))

(−1 + 𝑘𝑟2)𝑏3𝜂3

𝑅𝜃𝜃 =𝑟2𝑎𝜂3𝑎′𝑏′ − 𝑟2𝑏𝜂2 (𝑎𝑎′𝜂′ + 𝜂(2𝑎′2

+ 𝑎𝑎′′)) + 𝑟2𝑎𝑏2𝜂 + 𝑏3(𝑘𝑟2𝜂3 − 𝑟2𝑎 + 𝑟2𝜂(22 + 𝑎))

𝑏3𝜂3

𝑅𝜙𝜙

=𝑠𝑖𝑛2𝜃 (𝑟2𝑎𝜂3𝑎′𝑏′ − 𝑟2𝑏𝜂2 (𝑎𝑎′𝜂′ + 𝜂(2𝑎′2

+ 𝑎𝑎′′)) + 𝑟2𝑎𝑏2𝜂 + 𝑏3(𝑘𝑟2𝜂3 − 𝑟2𝑎 + 𝑟2𝜂(22 + 𝑎)))

𝑏3𝜂3

𝑅𝑏𝜏 = 3(𝜂′

𝜂𝑎+

𝑎′

𝑏𝑎−

𝑎′

𝑎)

𝑅𝑏𝑏 = 3 (𝑎′𝑏′

𝑎𝑏−

𝑎′′

𝑎+

𝑏

𝜂2𝑎)

26

Solving for the Einstein’s equations, one gets the following:

Figure 8: Equations showing the solved Einstein’s tensors. Source: (Binetruy P. , Deffayet, Ellwanger, & Langlois, 2000)

The stress-energy momentum tensor can be decomposed into two parts,

𝐵𝐴 = 𝐵,𝑏𝑢𝑙𝑘

𝐴 + 𝑇𝐵,𝑏𝑟𝑎𝑛𝑒𝐴 (56)

Where 𝐵,𝑏𝑢𝑙𝑘𝐴 is the energy momentum tensor of the bulk matter, which will be assumed in

the present work to be of the form

𝐵,𝑏𝑢𝑙𝑘𝐴 = 𝑑𝑖𝑎𝑔(−𝜌𝐵, 𝑃𝐵, 𝑃𝐵, 𝑃𝐵, 𝑃𝑇) (57)

The energy density 𝜌𝐵 and pressures 𝑃𝐵 𝑎𝑛𝑑 𝑃𝑇 are independent of the coordinate y.

The second term 𝑇𝐵,𝑏𝑟𝑎𝑛𝑒𝐴 corresponds to the matter content in the brane (y=0). Since we

consider here only strictly homogeneous and isotropic geometries inside the brane, the

latter can be expressed quite generally in the form

𝑇𝐵,𝑏𝑟𝑎𝑛𝑒

𝐴 =𝛿(𝑦)

𝑏𝑑𝑖𝑎𝑔(−𝜌𝑏 , 𝑝𝑏 , 𝑝𝑏 , 𝑝𝑏 , 0)

(58)

Where the energy density 𝜌𝑏and pressure 𝑝𝑏 are independent of the position inside the

brane, which demonstrate the isotropic and homogeneous assumptions.

27

3.2.1 The energy-momentum tensor

The energy-momentum tensor describes the matter/energy content of our five-dimensional

space-time. This tensor, given in its component form

𝑇𝐴𝐵 𝑤ℎ𝑒𝑟𝑒 𝐴, 𝐵 𝑟𝑎𝑛𝑔𝑒𝑠 𝑓𝑟𝑜𝑚 0,1,2,3 𝑎𝑛𝑑 5.

𝑇𝐴𝐵 is the flux of momentum 𝑃𝐴 across a surface of constant 𝑥𝐵.

Consider the following component, 𝑇00. This component is defined as the flux of 0-

momentum, which is energy, across a surface of constant time.

In this paper, it is assumed that 05 = 0. This means that there is no flow of matter along

the constant surface of the fifth dimension. This implies that 05 vanishes.

It then turns out that the components (0,0) and (5,5) of Einstein’s equations can be

rewritten in the simple form

𝐹′ =

2𝑎′𝑎3

3𝜅20

0 (59)

=

2𝑎3

3𝜅25

5 (60)

Where F is a function of 𝜏 and 𝑦 defined by

𝐹(𝜏, 𝑦) ≡

(𝑎′𝑎)2

𝑏2−

(𝑎)2

𝑛2− 𝑘𝑎2

(61)

Since 00 = −𝜌𝐵, one can integrate (59), which gives

𝐹 +

𝜅2

6𝑎4𝜌𝐵 + 𝐶 = 0

(62)

Using (61) and (62), we get the following

(

𝑛𝑎)2 =

1

6𝜅2𝜌𝐵 + (

𝑎′

𝑏𝑎)2 −

𝑘

𝑎2+

𝐶

𝑎4

(63)

Using the following Junction conditions derived in (Binetruy, Deffayet, & Langlois, 1999)

[𝑎′]

𝑎0𝑏0= −

𝜅2

3𝜌𝑏

(64)

28

[𝑛′]

𝑛0𝑏0=

𝜅2

3(3𝑝𝑏 + 2𝜌𝑏)

(65)

We derived (51) from (63), (64) and (65).

2

𝑎2=

𝜅2

6𝜌𝐵 +

𝜅4

36𝜌𝑏

2 +𝐶

𝑎4−

𝑘

𝑎2

(66)

3.2.2 Explaining the Junction Conditions

The junction conditions at (64) and (65) relate the jumps of the derivative of the metric

across the brane to the stress-energy tensor inside the brane.

Consider Figure 9, which shows the brane located at y=0, within a fifth dimensional metric.

In order to have a well-defined geometry, the metric is required to be continuous across the

brane localized in y=0. This means 𝑔(0−) = 𝑔(0+).

However, its derivatives with respect to y can be discontinuous in 𝑦 = 0. This means the

existence of a Dirac delta function in the second derivatives of the metric with respect to y.

From (52), the expression for the second derivative of 𝑎 is given as

𝑎′′ = 𝑎′′ + [𝑎′]𝛿(𝑦) (67)

Where 𝑎′′ is the non-distributional part of the double derivative of 𝑎 (the standard

derivative), and [𝑎′] is the jump in the first derivative across y=0, defined by

[𝑎′] = 𝑎′(0+) − 𝑎′(0−) (68)

Matching the Dirac delta functions of the Einstein tensor with those of the brane stress-

energy tensor, you can get (64) and (65).

Brane at y=0

0- 0+ Axis y

Figure 9: Diagram showing the brane (at y=0) embedded in the five-dimensional metric.

29

[𝑎′]

𝑎0𝑏0= −

𝜅2

3𝜌𝑏

[𝑛′]

𝑛0𝑏0=

𝜅2

3(3𝑝𝑏 + 2𝜌𝑏)

For example, given the Einstein tensor component 00

Substitute (67) into 00 and equate to

𝑇𝐵,𝑏𝑟𝑎𝑛𝑒𝐴 =

𝛿(𝑦)

𝑏𝑑𝑖𝑎𝑔(−𝜌𝑏 , 𝑝𝑏 , 𝑝𝑏 , 𝑝𝑏 , 0)

One could get (64) and (65).

3.2.3 Reviewing Equation (51)

Equation (51) given as shown

2

𝑎2=

𝜅2

6𝜌𝐵 +

𝜅4

36𝜌𝑏

2 +𝐶

𝑎4−

𝑘

𝑎2

It has a linear bulk density, a brane energy density that enters quadratically and some

cosmological constant. The quadratic term in (51) does not give us the required standard

cosmology.

We can recover standard cosmology by decomposing the energy density in the brane into

two parts,

𝜌𝑏 = 𝜌𝛬 + 𝜌 (69)

where 𝜌𝛬 is a constant that represents an intrinsic tension of the brane and 𝜌 stands for the

ordinary energy density in cosmology. Inserting (69) into (51), one would have

02

𝑎02 =

𝜅2

6𝜌𝐵 +

𝜅4

36𝜌𝛬

2 +𝜅4

18𝜌𝛬𝜌 +

𝜅4

36𝜌2 +

𝐶

𝑎04 −

𝑘

𝑎02

(70)

30

We recover standard cosmology when we set

𝜅2

6𝜌𝐵 +

𝜅4

36𝜌𝛬

2 = 0 𝑎𝑛𝑑 8𝜋𝐺 ≈𝜅4𝜌𝛬

6

(71)

The quadratic term 𝜅4

36𝜌2 in (70) will diminish at late times as matter density 𝜌2 ∝ 𝑅−6, see

(24) for derivation. This is because R (the scale factor) increases through time, resulting in

the dominance of the linear term 𝜅4

18𝜌𝛬𝜌 in (70), hence recovering standard cosmology.

31

Chapter 4

Proposed Models and their Age Constraints

4.1 Our Proposed Model 1

From equation (70), we proposed a model of the general form

𝐻2 = 𝐴𝜌 + 𝐵𝜌2 + 𝐶 ; 𝐴 =

8𝜋𝐺

3

(72)

where B and C are arbitrary constants. A is set at such a value to follow that of the

Friedmann Model.

We use the observations obtained in (1) as the constraint in our proposed model. Using the

results from the baryon mass fraction that

𝜌

𝜌𝑐= 0.3

(73)

where 𝜌𝑐 is the critical mass density for a flat Friedmann universe and 𝜌 is the actual

observed mass density of matter and dark matter.

We set up the following constraints

𝐻02 = 𝐴𝜌0 + 𝐵𝜌0

2 + 𝐶 (74)

where the subscript 0 denotes all the values of the parameters at present day.

0 = 𝐵𝜌02 + 𝐴𝜌0 + 𝐶 − 𝐻0

2 (75)

0.3 =

𝜌0

𝜌𝑐= (

1

𝜌𝑐)

−𝐴 ± √𝐴2 − 4𝐵(𝐶 − 𝐻02)

2𝐵; 𝜌𝑐 =

3𝐻02

8𝜋𝐺

(76)

This constraint gives a linear relation between parameters B and C, and they are

𝐶 = 𝐻02 − 𝐵𝜌0

2 − 𝐴𝜌0 (77)

𝐶 = (−7.080 ∗ 1036)𝐵 + (3.469 ∗ 109) (78)

32

We can find another relation between 𝑧𝑒𝑞 and B, we set 𝐴𝜌(𝑧𝑒𝑞) = 𝐵𝜌2(𝑧𝑒𝑞) for a

particular redshift.

𝐴𝜌 = 𝐵𝜌2 (79)

𝐵

𝐴=

𝜌(𝑧𝑒𝑞)

𝜌2(𝑧𝑒𝑞)=

𝜌0(1 + 𝑧𝑒𝑞)3

𝜌02(1 + 𝑧𝑒𝑞)6

=1

𝜌0(1 + 𝑧𝑒𝑞)3

(80)

𝐵 =

8𝜋𝐺

3

1

𝜌0(1 + 𝑧𝑒𝑞)3

(81)

𝜌0 = (0.3)

3𝐻02

8𝜋𝐺

(82)

Hence, knowing the relation between zeq and B, and between B and C, we can then find the

age of the universe of this new model using all these parameters.

zeq B C (× 109) H0t0

0 2.1E-28 1.982 0.565

0.1 1.57776E-28 2.352 0.610

0.2 1.21528E-28 2.609 0.651

0.3 9.55849E-29 2.792 0.690

0.4 7.65306E-29 2.927 0.725

0.5 6.22222E-29 3.028 0.759

0.6 5.12695E-29 3.106 0.790

0.7 4.27437E-29 3.166 0.819

0.8 3.60082E-29 3.214 0.845

0.9 3.06167E-29 3.252 0.871

1 2.625E-29 3.283 0.894

Figure 10: Table showing the different redshift values that resulted in different values of B and the age of the Universe.

I would restrict the scope of analysis up till 𝑧𝑒𝑞 = 1, even if higher values of 𝑧𝑒𝑞gives

favorable values of 𝐻0𝑡0. This is because a redshift value z of more than 1 means a Universe

in the distant past. The value of 𝑧𝑒𝑞 > 1 would mean that the B-term is equivalent to the A-

term way back in the past and that the value of B-term is negligible compared to the A-term

currently. Hence, values of 𝑧𝑒𝑞 > 1 would serve to diminish the existence of the B-term

which this model is trying to preserve. The absence of the B-term would bring this proposed

model 1 back to the standard Friedmann model of the flat Universe.

33

4.1.1 Setting C=0

We consider the cosmological consequences by setting 𝐶 = 0.

0.3 =

𝜌0

𝜌𝑐= (

1

𝜌𝑐)

−𝐴 ± √𝐴2 − 4𝐵(−𝐻02)

2𝐵

(83)

The above equation relate A to B by the following,

𝐴 = (−2.655 ∗ 1018)𝐵 + (1.863 ∗ 10−9) (84)

The significance is now A is not fixed but varies with zeq.

Making 𝐴𝜌(𝑧𝑒𝑞) = 𝐵𝜌2(𝑧𝑒𝑞), hence relating B to zeq is the following equation:

𝐵 =

𝐻02(1 + 𝑧𝑒𝑞)−3

𝜌02(1 + (1 + 𝑧𝑒𝑞)

−3)

(85)

Varying the values of zeq from 0 to 1, and getting the corresponding values of B and then A,

we can calculate the age of the Universe with the respective values of A and B.

zeq A B H0t0

0 9.315E-10 3.509E-28 0.390

0.1 1.06377E-09 3.0108E-28 0.402

0.2 1.18008E-09 2.5728E-28 0.415

0.3 1.28027E-09 2.1955E-28 0.427

0.4 1.3654E-09 1.8749E-28 0.439

0.5 1.43717E-09 1.6046E-28 0.451

0.6 1.49742E-09 1.3777E-28 0.462

0.7 1.54793E-09 1.1875E-28 0.472

0.8 1.59031E-09 1.0279E-28 0.482

0.9 1.62595E-09 8.9368E-29 0.491

1 1.656E-09 7.805E-29 0.500

Figure 11: Table showing the different redshift values that resulted in different values of A, B and the age of the Universe.

34

4.2 Our Proposed Model 2

Now we consider setting A as a free parameter, to investigate its consistency with

observations.

𝐻2 = 𝐴𝜌 + 𝐵𝜌2 + 𝐶 ; 𝐴 = 𝐹𝑟𝑒𝑒 (86)

Using the relation in (29), we can find a linear relation between 𝐴 𝑎𝑛𝑑 𝐵.

𝑩

𝑨= 𝝆𝟎

𝟏−𝒏(𝟏 + 𝒛𝒆𝒒)𝟑(𝟏−𝒏)

Setting 𝑧𝑒𝑞 = 1; 𝑛 = 2 , we can get a linear relation between A and B

𝐴 = (2.12773 ∗ 1019)𝐵 (87)

To get the parameter C, we used the relation shown in (77) and (82), reproduced below

𝐶 = 𝐻02 − 𝐵𝜌0

2 − 𝐴𝜌0

𝝆𝟎 = (𝟎. 𝟑)𝟑𝑯𝟎

𝟐

𝟖𝝅𝑮

With the Hubble constant at the present value given by 𝐻0 = 70.4 (𝑘𝑚

sec)/𝑀𝑝𝑐.

Figure 12 below show the values of the A, B, C parameters and the age of the Universe

when 𝑧𝑒𝑞 = 1.

zeq 𝐴 (× 10−10) 𝐵 (× 10−29) 𝐶 (× 109) H0t0

1 44.674 21.000 -8.413 0.445

33.564 15.778 -5.088 0.495

25.853 12.153 -2.780 0.544

20.334 9.558 -1.129 0.592

16.281 7.653 8.420 0.639

13.237 6.222 9.951 0.685

10.907 5.127 1.692 0.729

9.093 4.2747 2.235 0.772

7.660 3.601 2.664 0.814

6.513 3.062 3.007 0.855

5.584 2.625 3.285 0.894

Figure 12: Table showing the different parameters of A, B, C and the age of the Universe.

35

Given relation (87), the parameters of B could be arbitrary. Hence I set the parameters of B

to be those from Figure 10, and use that as a basis to find all other parameters. Knowing the

acceptable age range H0t0 to be from 0.72 – 1.17, I extend Figure 12 using decreasing

parameters of B to get values of H0t0 which are within the acceptable age range.

zeq 𝐴 (× 10−10) 𝐵 (× 10−29) 𝐶 (× 109) H0t0

1 4.829 2.27 3.511 0.933

4.191 1.97 3.702 0.971

3.680 1.73 3.855 1.006

3.234 1.52 3.989 1.042

2.851 1.34 4.103 1.078

2.532 1.19 4.199 1.111

2.276 1.07 4.275 1.142

2.042 0.960 4.345 1.174

Figure 13: Table showing the different parameters of A, B, C and the age of the Universe.

The bolded parameters A, B and C would be used for further analysis in cosmological tests

as their H0t0 is within the acceptable range and hence are not rejected.

36

4.3 Evaluation of Results

It should be mentioned that the values of Figure 12 would reduce to the values of Figure 11

if we take the parameter C to be zero and use the parameter B given in Figure 11 to

evaluate relation (29), and also taking into account the change in the 𝑧𝑒𝑞 values.

Furthermore, Figure 12 would reduce to the values in Figure 10 by using relation (29), using

all the parameters of B given in Figure 12 and taking into account the changes in the 𝑧𝑒𝑞

values.

Comparing the parameters in Figure 10 and 11, we observed generally that the age of the

Universe becomes younger as the B-parameter increases. This means that the B term slows

down the expansion rate of the Universe.

The H0t0 values shown in Figure 10 is within the acceptable range of 0.72 – 1.17 when 𝑧𝑒𝑞 is

at or more than 0.4. This result shows that the B term is present up to 𝑧𝑒𝑞 = 0.4.

The H0t0 values shown in Figure 11 are not within the range as discussed in the above

paragraph. Hence, the data do not yield any interesting findings. However, it does lend

support to the proposition that when the B term increases, the age of the Universe

decrease.

37

4.3.1 Explanation of Evaluation

In order to explain how the B-term slows down the expansion rate of the Universe, we need

to find the evolution of the normalized scale factor 𝑎(𝑡) as a function of time.

First, we solve for 𝑎(𝑡) from equation (72), assuming a matter dominated universe, we solve

the following differential equation

(𝑡) =

𝐴

𝑎(𝑡)+

𝐵

𝑎4(𝑡)+ 𝐶𝑎2(𝑡)

(88)

𝑎(𝑡) = (𝛼 + 𝛽exp (3√𝐶(𝑡 − 𝛾) + 𝜔 exp (−3√𝐶(𝑡 − 𝛾)))1/3 (89)

Where 𝛼, 𝛽, 𝜔 𝑎𝑛𝑑 𝛾 are

𝛼 = −

𝐴

2𝐶

𝛽 =1

4𝐶

𝜔 =𝐴2

4𝐶− 𝐵

𝛾 =ln (𝐴 + √4𝐵𝐶)

3√𝐶

(90)

Where 𝐴 =8𝜋𝐺

3. The values of B and C can be found in Figure 10. This scale factor a(t) is

being drawn for Proposed model 1.

38

Plotting for the scale factor 𝑎(𝑡), we have the following figure

Figure 14: Graph plot showing the time evolution of the scale factor a(t).

The scale factor plot from A to F has the following B values

Plot B-term values (× 10−28)

A 0.622

B 0.765

C 0.956

D 1.215

E 1.578

F 2.1

Figure 15: Table showing the different values of B-term for the different graph plots.

The horizontal axis of Figure (14) is the time axis. The point P is the present epoch. As the

value of the B-term increases, the age of the Universe decrease. This age can be observed

from the length of

𝐴𝑃 > 𝐵𝑃 > 𝐶𝑃 > 𝐷𝑃 > 𝐸𝑃 > 𝐹𝑃

Where the greater the length, means the longer the age of the Universe.

39

The vertical axis of Figure (14) shows the value of the scale factor. As the value of the B-term

increases, the value of the scale factor decreases. One can compare this difference at a fixed

point P.

At this fixed point P, the gradient of all curves should be the same to reflect the same value

of the Hubble constant 𝐻0 observed today. We can observed that for a young Universe, the

gradient of the scale factor as it evolved to point P has to become steeper in order to fit the

gradient constraint at point P. From this observation, it can be shown that as B term

increases, the age of the Universe gets younger. A younger Universe would have to slow its

expansion rate in order to meet the gradient constraint at point P. Hence, the B term

actually slows down the expansion rate of the Universe.

4.4 Findings from other similar models

4.4.1 Similar Model 1a During the course of this project, we have also considered the findings resulting from

variants of our proposed models shown above. We have considered the following model,

where we set

𝐻2 = 𝐴𝜌 + 𝐵𝜌2 𝑤𝑖𝑡ℎ 𝐴 =

8𝜋𝐺

3

(91)

This model follows the exact same formulation as in the Cardassian paper (Freese & Lewis,

2002), except that now the free variable 𝑛 has to be fixed at 𝑛 = 2.

We consider Figure 5, where the maximum value of 𝑛 on the horizontal axis is up till 0.6. For

our purposes, we need to extend the horizontal axis to show the behavior of the contour

lines up till or more than 𝑛 = 2. This is shown in Figure 16 below.

40

Figure 16: A MatLab plot of the contour lines for Equation (37), extended on the horizontal axis to n=3.

From Figure 16, we can observe that for the contour of 𝐹(𝑛) = 0.3, it approaches

asymptotic infinity even before 𝑛 = 1. This means that 𝐹(𝑛) ≠ 0.3 for 𝑛 = 2, which shows

that this model is not consistent with the observations of the baryon mass fraction as shown

in (1).

Further calculations of the age of this model yield the following results

zeq F H0t0

0.1 0.571 0.403

0.2 0.633 0.415

0.3 0.687 0.428

0.4 0.733 0.440

0.5 0.771 0.451

0.6 0.804 0.462

0.7 0.831 0.472

0.8 0.854 0.482

0.9 0.873 0.491

1.0 0.889 0.500

Figure 17: This table shows the different values of F(n) and the age of the Universe as z_eq varies.

41

Figure 17 shows that this model has mass fraction 𝐹(𝑛), which is not compatible with the

observation values of 𝐹(𝑛)~0.3. The ages of the Universe calculated are also not within the

observed range of 0.72 – 1.17 for the dimensionless age parameter H0t0. Hence this model

could be discarded as it does not yield any interesting results.

42

Chapter 5

Constraints from the First Doppler Peak

5.1 The First Doppler Peak

The direct evidence that support the theory that the Universe is flat comes from the

observations of the detailed measurements of the temperature anisotropy of the Cosmic

Microwave Background (CMB) radiation (Cheng, 2005).

In the early Universe, the baryons, photons and dark matter are packed together in a highly

dense fluid. This highly-dense fluid contains high-pressure photon radiation, which together

drives the baryonic matter outwards at a speed given by

𝑐𝑠~√𝑝

𝜌~

𝑐

√3 , 𝑤𝑖𝑡ℎ 𝑝 =

1

3𝜌𝑐2

(92)

This outward radiation pressure is opposed by the gravitational clumping of baryons. This

outward radiation pressure acting on baryonic matter last till the photons decoupled 105

years later, which is the recombination epoch.

Since these photons have decoupled, there is no more radiation pressure driving these

baryonic matter outwards, hence these baryonic matter remained in place as the photons

stream away. This baryonic matter that remained is translated through gravitational redshift

into temperature inhomogeneity, showing up as a series of peaks and troughs in the

temperature power spectrum. (Cheng, 2005)

43

From Figure 19, we show the following relationship (Cheng, 2005)

𝛼1 ≅

𝜆1

𝑑(𝑡𝛾)

(93)

𝜆1 = ∫

𝑐𝑠𝑑𝑡

𝑎(𝑡)

𝑡𝛾

0

≈ 𝑐𝑠 ∫𝑑𝑡

𝑎(𝑡)

𝑡𝛾

0

(94)

𝑑(𝑡𝛾) = 𝑐 ∫

𝑑𝑡

𝑎(𝑡)

𝑡0

𝑡𝛾

(95)

Where 𝑑(𝑡𝛾) is the comoving distance a photon would have traveled to reach us from the

surface of last scattering and 𝜆1 is the radius of the sound horizon.

Figure 18: The figure

shows a baryonic shell of

matter approximately

150Mpc in radius. This

radius is called the sound

horizon.

Source: http://mwhite.berkeley.edu/BAO/SantaFe07.pdf

Figure 18: The figure shows a baryonic shell of matter approximately 150Mpc in radius. This radius is called the sound horizon.

𝜆1

𝛼1

𝑑(𝑡𝛾) Figure 19: The figure

illustrates the

relationship between

angular separation 𝛼1

and the sound horizon

𝜆1.

44

Equation (94) shows us that the radius of the sound horizon is equal to the speed given in

(92) multiplied by the time from the beginning of the Universe to the recombination epoch

with the expansion of the Universe taken into account.

We can consider the scale factor 𝑎(𝑡) ∝ 𝑡2/3 for the matter-dominated flat Universe to

evaluate the integrals of (94) and (95). Hence, we obtain the following expressions

𝑎(𝑡) ∝ 𝑡2/3

(96)

𝑑𝑎

𝑑𝑡=

2

3𝑡−1/3

(97)

𝑑𝑡 = 𝑡1/3𝑑𝑎 = 𝑎1/2𝑑𝑎

(98)

∫𝑑𝑡

𝑎(𝑡)

𝑡0

𝑡𝛾

∝ ∫𝑎

12𝑑𝑎

𝑎(𝑡)∝ 𝑎

12 = (1 + 𝑧)−1/2

(99)

Using (99) to evaluate (93), we have

𝛼1 ≈𝜆1

𝑑(𝑡𝛾)=

𝑐𝑠(1 + 𝑧𝛾)−

12

𝑐 [(1 + 𝑧0)−12 − (1 + 𝑧𝛾)

−12]

≈(1 + 𝑧𝛾)

−12

√3≈ 1°

(100)

Where we have used 𝑧0 = 0, 𝑧𝛾 ≅ 1100.

The multipole number 𝑙 having the form

𝑙1 ≈

𝜋

𝛼1≈ 𝜋√3(1 + 𝑧𝛾)

12 ≈ 200

(101)

will have the first Doppler peak of the power spectrum located at 𝑙 = 200 for a flat

Universe.

Observations by the WMAP on the CMB power spectrum confirmed that the first Doppler

peak lies at 𝑙 = 200. This is shown in Figure 20.

45

In section 1.3 on the geometry of the Universe, I have not explained on how the findings on

WMAP allowed us to reach the conclusion that the Universe is flat. This section on the First

Doppler peak has explained this based on the position of the first acoustic peak on the

power spectrum graph.

Figure 20: The figure shows the graph of the power spectrum. The first peak lies at l=200.

Source:

http://mwhite.berkeley.e

du/BAO/SantaFe07.pdf

46

5.2 Finding the First Doppler Peak

First, we shall find the first Doppler peak for the Cardassian Model as shown in (10). For

ease of reference, (10) is reproduced as shown

𝐻2 = 𝐴𝜌 + 𝐵𝜌𝑛

I will adopt the terminology used in the Cardassian Expansion paper (Freese & Lewis, 2002).

The angular separation relation shown in (93) is recast as

𝜃 =

𝑠∗

𝑑,

(102)

where 𝑠∗ is the radius of the sound horizon, d is the co-moving distance a photon would

have traveled to reach us from the surface of last scattering.

The co-moving distance d is expressed as follows

𝑑 = ∫

𝑑𝑡

𝑎𝑅0

𝑡0

𝑡𝑟

(103)

where 𝑎(𝑡) is defined as the normalized scale factor, while 𝑅(𝑡) is the scale factor.

𝑎(𝑡) =

𝑅(𝑡)

𝑅0.

(104)

We define the redshift dependence of H as

𝐻(𝑧) = 𝐻0𝐸(𝑧)

where 𝐸(𝑧) is defined in (47). Using the methods listed in (39) to (41), we have the following

expression for 𝑑.

𝑑 =

1

𝐻0𝑅0∫

𝑑𝑧

𝐸(𝑧)

𝑧𝑟

0

(105)

47

From Equation (94) and using the method listed in (39) to (41), we get the expression for the

radius of the sound horizon 𝑠∗

𝑠∗ ∝ ∫

1

𝐸(𝑧)𝑑𝑧,

∞

𝑧𝑟

(106)

where 𝑧𝑟 is the redshift at recombination.

Calculating for the value of 𝑑 at (105) for the matter dominated FRW cosmology with

Ω𝑚,0 = 1, 𝐸(𝑧) = (1 + 𝑧)3/2, we have

𝑑 =

1.94

𝐻0𝑅0

(107)

For the Cardassian Model at (10), the 𝐸(𝑧) expression is as shown

𝐸(𝑧)2 = 𝐹 × (1 + 𝑧)3 + (1 − 𝐹) × (1 + 𝑧)3𝑛 (108)

Evaluating the value of 𝑑 using (108) and (105), with

𝐹 ≡𝜌𝑐

𝜌𝑐,𝑜𝑙𝑑= 0.3, 𝑓𝑜𝑟 𝑛 = (0.6, 0.3, 0.2, 0.1)

and finding the ratio of these 𝑑 values with (107), we get the following values listed in the

following table.

𝑛 𝑥,𝑥

𝐻0𝑅0 𝑅𝑎𝑡𝑖𝑜 1𝑎:

𝑥

1.94

0.6 2.852 1.47

0.3 3.104 1.60

0.2 3.127 1.61

0.1 3.162 1.63

Figure 21: This table shows the values for x and the ratio of x against the FRW model.

48

The ratio in the third column shows how much deviation the Cardassian model is from the

FRW model. Calculating the sound horizon 𝑠∗ for the FRW model gives

𝑠∗ ∝ ∫

1

(1 + 𝑧)3/2𝑑𝑧 = 0.063214

∞

𝑧𝑟

(109)

Calculating the values of the sound horizon 𝑠∗ for the Cardassian model gives the following

𝑛 𝑠∗ 𝑅𝑎𝑡𝑖𝑜 1𝑏: 𝑠∗

0.063214

0.6 0.110038 1.74072

0.3 0.110046 1.74085

0.2 0.110046 1.74085

0.1 0.110046 1.74085

Consider the multipole number 𝑙 of (101). The multipole expression is given in this

Cardassian paper as

𝑙 =𝜋

𝜃

(110)

where 𝜃 is defined in (102).

Knowing the ratios from both Figure 21 and 22, we can find the deviation of the multipole

number for the first Doppler peak for the Cardassian model.

𝑛 𝑅𝑎𝑡𝑖𝑜 1𝑎

𝑅𝑎𝑡𝑖𝑜 1𝑏

0.6 0.8445

0.3 0.9191

0.2 0.9248

0.1 0.9363

Figure 22: This table shows the values for 𝒔∗and the ratio of 𝒔∗against the FRW model.

Figure 23: This table shows the ratio to which the Cardassian model deviates from the location of the First Doppler peak

49

5.3 The First Doppler Peak for Proposed Model 1

Proposed Model 1 is set out as shown

𝐻2 = 𝐴𝜌 + 𝐵𝜌2 + 𝐶 ; 𝐴 =8𝜋𝐺

3

Calculating for (105) and (106) using the following 𝐸(𝑧)

𝐸(𝑧) = (

𝐴𝜌0(1 + 𝑧)3 + 𝐵𝜌02(1 + 𝑧)6 + 𝐶

𝐻02 )

12

(111)

𝜌0 = 0.3(

3𝐻02

8𝜋𝐺)

(112)

𝐸(𝑧) = (0.3(1 + 𝑧)3 + (1.429 ∗ 1027)𝐵(1 + 𝑧)6 + (2.018 ∗ 10−10)𝐶)1/2 (113)

Hence, knowing (113), we can proceed to calculate (105) and (106), the following results are

shown in the table below.

zeq B C (× 109) 𝑥, 𝑥

𝐻0𝑅0 𝑅𝑎𝑡𝑖𝑜 2𝑎:

𝑥

1.94

0 2.1E-28 1.982 0.730072 0.376326

0.1 1.57776E-28 2.352 0.795025 0.409807

0.2 1.21528E-28 2.609 0.85647 0.441479

0.3 9.55849E-29 2.792 0.9146 0.471443

0.4 7.65306E-29 2.927 0.969562 0.499774

0.5 6.22222E-29 3.028 1.0216 0.526598

0.6 5.12695E-29 3.106 1.07084 0.551979

0.7 4.27437E-29 3.166 1.11756 0.576062

0.8 3.60082E-29 3.214 1.16186 0.598897

0.9 3.06167E-29 3.252 1.20396 0.620598

1 2.625E-29 3.283 1.46857 0.641222

Figure 24: This table shows the values for x and the ratio of x against the FRW model.

50

zeq B C (×109)

𝑠∗ 𝑅𝑎𝑡𝑖𝑜 2𝑎: 𝑠∗

0.0602

0 2.1E-28 1.982 7.530*10^-7 1.251*10^-5

0.1 1.57776E-28 2.352 8.687*10^-7 1.443*10^-5

0.2 1.21528E-28 2.609 9.898*10^-7 1.644*10^-5

0.3 9.55849E-29 2.792 1.116*10^-6 1.854*10^-5

0.4 7.65306E-29 2.927 1.247*10^-6 2.071*10^-5

0.5 6.22222E-29 3.028 1.383*10^-6 2.30*10^-5

0.6 5.12695E-29 3.106 1.524*10^-6 2.532*10^-5

0.7 4.27437E-29 3.166 1.669*10^-6 2.772*10^-5

0.8 3.60082E-29 3.214 1.818*10^-6 3.020*10^-5

0.9 3.06167E-29 3.252 1.972*10^-6 3.274*10^-5

1 2.625E-29 3.283 2.130*10^-6 3.538*10^-5

Figure 25: This table shows the values for s* and the ratio of s* against the FRW model.

51

5.3.1 The First Doppler Peak for Proposed Model 2

Proposed Model 2 is set out as shown

𝐻2 = 𝐴𝜌 + 𝐵𝜌2 + 𝐶 ; 𝐴 = 𝐹𝑟𝑒𝑒

Using (111) and (112), we got the 𝐸(𝑧) expression

𝐸(𝑧) = ((5.36638 ∗ 108)𝐴(1 + 𝑧)3 + (1.42728 ∗ 1027)𝐵(1 + 𝑧)6

+ (2.01769 ∗ 10−10)𝐶)1/2

(114)

Hence, knowing (114), we can proceed to calculate (105) and (106), the following results are

shown in the table below.

𝑧𝑒𝑞 A B C 𝑥, 𝑥

𝐻0𝑅0 𝑅𝑎𝑡𝑖𝑜 2𝑎:

𝑥

1.94

1 1.09067E-09 5.12695E-29 1692343230 0.9915 0.511082

9.093E-10 4.27437E-29 2235102376 1.0563 0.544485

7.66013E-10 3.60082E-29 2663889614 1.1201 0.577371

6.51318E-10 3.06167E-29 3007116752 1.1828 0.609691

5.58424E-10 2.62500E-29 3285104315 1.2444 0.641443

4.82904E-10 2.27E-29 3511100112 1.3045 0.672423

4.19084E-10 1.97E-29 3702082476 1.3651 0.70366

3.68028E-10 1.73E-29 3854868368 1.4220 0.73299

3.23354E-10 1.52E-29 3988556022 1.4805 0.763144

2.85062E-10 1.34E-29 4103145441 1.5385 0.793041

2.53152E-10 1.19E-29 4198636623 1.5947 0.82201

2.27624E-10 1.07E-29 4275029568 1.6456 0.848247

2.04224E-10 9.60E-30 4345056435 1.6994 0.875979

Figure 26: This table shows the values for x and the ratio of x against the FRW model.

52

5.4 Evaluation of Doppler Peak results

We have to consider the 𝑠∗ values in Figure 25 and 27, and try to make sense of why its

order of magnitude is so vastly different from (109). This small value of 𝑠∗ suggests that the

time taken from Big Bang till the recombination epoch is very small as compared to the FRW

model. Hence, this results in the small radius of the sound horizon.

We use equation (48)

𝑡(𝑧) = 𝐻0−1 ∫

𝑑𝑧

(1 + 𝑧)𝐸(𝑧)

∞

𝑧𝑟

to compare the values of the 𝐻𝑧𝑟𝑡𝑧𝑟

of both Proposed Model 1 and 2 with the FLW model.

FLW model:

∫

𝑑𝑧

(1 + 𝑧)5/2

∞

𝑧𝑟

, 𝑧𝑟 = 1100

(115)

𝐻𝑧𝑟𝑡𝑧𝑟

= 0.00001825 (116)

zzeq A B C 𝑠∗ 𝑅𝑎𝑡𝑖𝑜 2𝑎: 𝑠∗

0.0602

1 1.09067E-09 5.12695E-29 1692343230 1.52479E-6 2.53287E-05

9.093E-10 4.27437E-29 2235102376 1.66995E-6 2.774E-05

7.66013E-10 3.60082E-29 2663889614 1.81945E-6 3.02234E-05

6.51318E-10 3.06167E-29 3007116752 1.97316E-6 3.27767E-05

5.58424E-10 2.62500E-29 3285104315 2.13096E-6 3.5398E-05

4.82904E-10 2.27E-29 3511100112 2.29154E-6 3.80654E-05

4.19084E-10 1.97E-29 3702082476 2.45984E-6 4.08611E-05

3.68028E-10 1.73E-29 3854868368 2.62493E-6 4.36035E-05

3.23354E-10 1.52E-29 3988556022 2.80039E-6 4.65181E-05

2.85062E-10 1.34E-29 4103145441 2.98255E-6 4.9544E-05

2.53152E-10 1.19E-29 4198636623 3.16495E-6 5.25739E-05

2.27624E-10 1.07E-29 4275029568 3.33771E-6 5.54437E-5

2.04224E-10 9.60E-30 4345056435 3.52375E-6 5.85341E-5

Figure 27: This table shows the values for s* and the ratio of s* against the FRW model.

53

Proposed Model 1:

∫

𝑑𝑧

(1 + 𝑧)(0.3(1 + 𝑧)3 + (1.429 ∗ 1027)𝐵(1 + 𝑧)6 + (2.018 ∗ 10−10)𝐶)1/2

∞

𝑧𝑟

(117)

We can calculate the 𝐻𝑧𝑟𝑡𝑧𝑟

for our Proposed Model 1 for the different parameters of B and

C.

zeq B C (×109)

𝐻𝑧𝑟𝑡𝑧𝑟

0 2.1E-28 1.982 7.668*10^-7

0.1 1.57776E-28 2.352 8.847*10^-7

0.2 1.21528E-28 2.609 1.008*10^-6

0.3 9.55849E-29 2.792 1.137*10^-6

0.4 7.65306E-29 2.927 1.270*10^-6

0.5 6.22222E-29 3.028 1.409*10^-6

0.6 5.12695E-29 3.106 1.552*10^-6

0.7 4.27437E-29 3.166 1.700*10^-6

0.8 3.60082E-29 3.214 1.852*10^-6

0.9 3.06167E-29 3.252 2.008*10^-6

1 2.625E-29 3.283 2.169*10^-6

Figure 28: This table shows the values for 𝑯𝒛𝒓𝒕𝒛𝒓

for the different parameters of B and C

Proposed Model 2:

∫

𝑑𝑧

(1 + 𝑧)((5.36638 ∗ 108)𝐴(1 + 𝑧)3 + (1.42728 ∗ 1027)𝐵(1 + 𝑧)6 + (2.018 ∗ 10−10)𝐶)1/2

∞

𝑧𝑟

(118)

We can calculate the 𝐻𝑧𝑟𝑡𝑧𝑟

for our Proposed Model 2 for the different parameters of A, B

and C.

54

The values of 𝐻𝑧𝑟𝑡𝑧𝑟

shown in Figure 28 and 29 indeed supports the hypothesis that for

Proposed Model 1 and 2, the time taken from Big Bang till the recombination epoch is much

smaller than that of the FLW model calculated at (115).

Figure 28 and 29 shows further that as the B-term increases, the time from the Big Bang to

the recombination epoch decreases. This means that the expansion rate for larger B-term of

the Universe in the early times is faster than for the smaller B-term.

This faster expansion rate could cool the Universe down, providing the conditions necessary

for recombination to happen early in the formation of the Universe.

The following figure is drawn for the scale factor a(t) of Proposed Model 1. Figure 30 is a

plot of a(t) against time.

zzeq A B C 𝑯𝒛𝒓𝒕𝒛𝒓

1 1.09067E-09 5.12695E-29 1692343230 3.67627E-8

9.093E-10 4.27437E-29 2235102376 4.02625E-8

7.66013E-10 3.60082E-29 2663889614 4.38668E-8

6.51318E-10 3.06167E-29 3007116752 4.75727E-8

5.58424E-10 2.62500E-29 3285104315 5.13774E-8

4.82904E-10 2.27E-29 3511100112 5.5249E-8

4.19084E-10 1.97E-29 3702082476 5.93067E-8

3.68028E-10 1.73E-29 3854868368 6.32869E-8

3.23354E-10 1.52E-29 3988556022 6.75173E-8

2.85062E-10 1.34E-29 4103145441 7.19092E-8

2.53152E-10 1.19E-29 4198636623 7.63069E-8

2.27624E-10 1.07E-29 4275029568 8.04721E-8

2.04224E-10 9.60E-30 4345056435 8.49575E-8

Figure 29: This table shows the values for 𝑯𝒛𝒓𝒕𝒛𝒓

for the different parameters of A, B and C

55

Figure 30: A Figure showing the evolution of scale factor a(t) during the early formation of the Universe.

Plot B-term values (×10−29)

Gradient around 𝑎(𝑡) = 0 (× 108)

A 7.653 3.728

B 6.222 3.361

C 5.127 1.525

D 4.274 1.063

E 3.601 0.806

F 3.062 3.742

G 2.625 2.183

Figure 31: Table showing the gradient values at the early formation of the Universe

Consider the gradients of the graph plots A - G of Figure 30 during the early formation of the

Universe, when a(t)~0. As the value of the B-term increases, the gradient of the scale factor

𝑎(𝑡) from A – G becomes steeper. This shows that a higher value of B-term would increase

the expansion rate of the Universe in the early times, causing the Universe to cool down

faster and reach the recombination epoch earlier, consistent with the findings in Figure 28

and 29.

However, there is some anamoly found for Plot F and G. Nevertheless, the values of their

gradients is still within the same range in the table.

G F E D C B A

56

Chapter 6

Further Cosmological Constraints

6.1 Further Cosmological Constraints

In this section, we will impose further cosmological constraints on Proposed Model 1. We

consider constraints from the Cosmic Microwave Background (CMB) (Elgaroy & Multamaki,

2007), Baryonic Acoustic Oscillation (BAO) observations and the Supernova constitution

data sets. (Hicken, et al., 2009).

6.2 Constraints from CMB (Feng & Li, 2009)

We consider the shift parameter R from the CMB data. The parameter R is defined as

𝑅 = √Ω𝑚0

Ω𝑘0𝑆𝑖𝑛𝑛(√|Ω𝑘0| ∫

𝑑𝑧′

𝐸(𝑧′)

𝑧𝑙𝑠

0

)

(119)

Where 𝑧𝑙𝑠 = 1100 is the redshift of the last scattering surface. Furthermore,

𝑆𝑖𝑛𝑛(𝑥)

sin(𝑥) ; Ω𝑘 < 0 (𝑐𝑙𝑜𝑠𝑒𝑑)

𝑥; Ω𝑘 = 0 (𝑓𝑙𝑎𝑡)

sinh(𝑥) ; Ω𝑘 > 0 (𝑜𝑝𝑒𝑛)

(120)

The term Ω𝑘 tells us the correct form of parameter R to use for different shapes of the

Universe as shown in Figure 2.

The parameter R for a flat Universe is

𝑅 = ∫

𝑑𝑧′

𝐸(𝑧′)

𝑧𝑙𝑠

0

(121)

Where Ω𝑚0 = 1.

57

One can see that Equation (121) is similar to (106). Hence the parameter R could be thought

of as the co-moving distance a photon would have traveled to reach us from the surface of

last scattering as shown in Figure 19. The different forms shown in (120) would then tell us

the ways to calculate this co-moving distance for the different geometries of the Universe.

Using the CMB data,

Where it is shown that 𝑅𝑜𝑏𝑠 = 1.710 ± 0.019, we can calculate the chi-square 𝜒𝐶𝑀𝐵2 for our

Proposed Model 1 and 2.

𝜒𝐶𝑀𝐵

2 = (𝑅 − 1.710

0.019)

2

(122)

Figure 32: CMB data showing the value of Parameter R. Source: Invalid source

specified.

58

6.2.1 CMB Constraints for Proposed Model 1

Evaluating equation (121) with 𝐸(𝑧′) given in (113), we obtained the following results

zeq B (𝟏𝟎−𝟐𝟗) C (× 109) 𝑹 𝝌𝑪𝑴𝑩𝟐

0 21.0 1.982 0.730072 2659.997

0.1 15.8 2.352 0.795025 2319.056

0.2 12.2 2.609 0.85647 2018.043

0.3 9.56 2.792 0.9146 1752.524

0.4 7.65 2.927 0.969562 1518.694

0.5 6.22 3.028 1.0216 1312.727

0.6 5.13 3.106 1.07084 1131.65

0.7 4.27 3.166 1.11756 972.258

0.8 3.60 3.214 1.16186 832.2921

0.9 3.06 3.252 1.20396 709.3531

1 2.63 3.283 1.24397 601.6176

Extending the redshift values zeq beyond 1, we can try to get the point where 𝜒𝐶𝑀𝐵2 = 0.

zeq B (𝟏𝟎−𝟐𝟗) C (× 109) 𝑹 𝝌𝑪𝑴𝑩𝟐

1.1 2.27 3.31 1.28 512.188

1.2 1.97 3.33 1.32 421.330

1.3 1.73 3.35 1.35 359.003

1.4 1.52 3.36 1.39 283.657

1.5 1.34 3.37 1.42 232.964

1.6 1.19 3.38 1.45 187.258

1.7 1.07 3.39 1.48 146.537

1.8 0.96 3.40 1.50 122.161

1.9 0.86 3.41 1.53 89.751

2.0 0.78 3.414 1.56 62.327

2.5 0.490 3.434 1.671 4.213

2.7 0.415 3.440 1.711 0.003

2.8 0.383 3.442 1.730 1.108

3.0 0.328 3.446 1.767 9.000

Figure 33: Table showing the value of Parameter R for different values of B, C and 𝜒𝐶𝑀𝐵2

Figure 28: Table showing the value of Parameter R for different values of B

Figure 34: Table extendng the different values of B, C for 𝜒𝐶𝑀𝐵2 = 0

Figure 28: Table showing the value of Parameter R for different values of B

59

Plotting the graph of 𝜒𝐶𝑀𝐵2 against 𝑧𝑒𝑞 in Figure 33 and 34, we have the following plot

6.3 Constraints from BAO (Feng & Li, 2009)

We consider the parameter A from the Sloan Digital Sky Survey (SDSS) data (Eisenstein, et

al., 2005). The parameter A is defined as

𝐴 =

√Ω𝑚0

𝑧1[

𝑧1

𝐸(𝑧1)

1

|Ω𝑘0|𝑆𝑖𝑛𝑛2(√|Ω𝑘0| ∫

𝑑𝑧′

𝐸(𝑧′))

𝑧1

0

]1/3 (123)

Where 𝑧1 = 0.35. For a flat Universe, we would have the following expression of A

𝐴 =

1

𝑧1(

𝑧1

𝐸(𝑧1))1/3(∫

𝑑𝑧′

𝐸(𝑧′)

𝑧1

0

)2/3 (124)

From the SDSS data,

0

500

1000

1500

2000

2500

3000

0 0.5 1 1.5 2 2.5 3

chis

qu

are_

CM

B

z_eq

Data Set 1

Data Set 1

Poly. (Data Set 1)

Figure 35: Plot showing the value of 𝜒𝐶𝑀𝐵2 for different values of 𝑧𝑒𝑞

Figure 36: SDSS data showing the value of Parameter A. Source: (Eisenstein, et al., 2005)

60

From Figure 36, we know the observational value to be 𝐴𝑜𝑏𝑠 = 0.469(0.96

0.98)−0.35 ± 0.017,

hence we can calculate the values of 𝜒𝐵𝐴𝑂2 given by

𝜒𝐵𝐴𝑂2 = (

𝐴 − 0.469(0.960.98)−0.35

0.017)

2

(125)

6.3.1 BAO Constraints for Proposed Model 1

Evaluating (124) and (125) for the different parameters of B and C of our Proposed Model 1,

we have the following

zeq 𝑩 (× 𝟏𝟎−𝟐𝟗) 𝑪 (× 𝟏𝟎𝟗) 𝑨 𝝌𝑩𝑨𝑶𝟐

0 21.00 1.982 0.70658 189.7585

0.1 15.78 2.352 0.739191 246.289

0.2 12.15 2.609 0.765402 297.0601

0.3 9.558 2.792 0.78649 341.3582

0.4 7.653 2.927 0.803411 379.1296

0.5 6.222 3.028 0.817081 411.0905

0.6 5.127 3.106 0.82812 437.8443

0.7 4.274 3.166 0.837159 460.3785

0.8 3.601 3.214 0.844535 479.1837

0.9 3.062 3.252 0.850647 495.0555

1 2.625 3.283 0.855708 508.3911

2.7 0.415 3.440 0.883219 583.988

Figure 37: Table showing the value of Parameter A for different values of B and C

61

Plotting the graph of 𝜒𝐵𝐴𝑂2 against 𝑧𝑒𝑞 in Figure 37, we have the following plot

6.4 Combining both 𝝌𝑪𝑴𝑩𝟐 and 𝝌𝑩𝑨𝑶

𝟐

We can constrain our model by taking into consideration both 𝜒𝐶𝑀𝐵2 and 𝜒𝐵𝐴𝑂

2 . Therefore,

we get the following results.

zeq 𝝌𝑩𝑨𝑶𝟐

+ 𝝌𝑪𝑴𝑩𝟐

0 2849.756

0.1 2565.345

0.2 2315.103

0.3 2093.882

0.4 1897.824

0.5 1723.818

0.6 1569.494

0.7 1432.637

0.8 1311.476

0.9 1204.409

1 1110.009

150

200

250

300

350

400

450

500

550

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

chi s

qu

are

_B

AO

z_eq

Data Set 2

Data Set

Figure 38: Plot showing the value of 𝜒𝐵𝐴𝑂2 for different values of 𝑧𝑒𝑞

Figure 39: Table showing the value 𝜒𝐶𝑀𝐵2 + 𝜒𝐵𝐴𝑂

2 for different 𝑧𝑒𝑞

62

1000

1200

1400

1600

1800

2000

2200

2400

2600

2800

3000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

𝜒_𝐶𝑀𝐵

^2+ 𝜒

_𝐵𝐴𝑂

^2

𝑧_𝑒𝑞

Data Set 3

Data Set

Figure 40: Plot showing the value of 𝜒𝐶𝑀𝐵2 + 𝜒𝐵𝐴𝑂

2 for different values of 𝑧𝑒𝑞

63

6.5 Similar Analysis for Proposed Model 2

I will do a similar analysis of Proposed Model 2, first to calculate 𝜒𝐶𝑀𝐵2 𝑎𝑛𝑑 𝜒𝐵𝐴𝑂

2 for the

various B parameters. The analysis would be restricted to only the bolded parameters as

they are within the acceptable age condition as shown in Figure 12.

6.5.1 CMB Constraints for Proposed Model 2

Evaluating equation (121) with 𝐸(𝑧′) given in (114), we obtained the following results

𝒛𝒆𝒒 𝑨 (× 𝟏𝟎−𝟏𝟎) 𝑩 (× 𝟏𝟎−𝟐𝟗) 𝑪 (× 𝟏𝟎𝟗) 𝑹 𝝌𝑪𝑴𝑩𝟐

1 10.907 5.127 1.692 0.9915 1430.034

9.093 4.274 2.235 1.0563 1183.722

7.660 3.601 2.664 1.1201 963.9391

6.513 3.062 3.007 1.1828 769.9165

5.584 2.625 3.285 1.2444 600.5079

4.829 2.27 3.511 1.3045 455.4855

4.191 1.97 3.702 1.3651 329.518

3.680 1.73 3.855 1.4220 229.7618

3.234 1.52 3.989 1.4805 145.901

2.851 1.34 4.103 1.5385 81.47438

2.532 1.19 4.199 1.5947 36.82573

2.276 1.07 4.275 1.6456 11.48853

2.042 0.96 4.345 1.6994 0.3112

Figure 41: Table showing the different chi square_CMB values for different parameters of A,B and C

64

Plotting a graph of 𝜒𝐶𝑀𝐵2 against the B-parameter, we have the following figure.

Figure 42: Plot showing the values of chi square_CMB for different parameters of B

6.5.2 BAO Constraints for Proposed Model 2

𝒛𝒆𝒒 𝑨 (× 𝟏𝟎−𝟏𝟎) 𝑩 (× 𝟏𝟎−𝟐𝟗) 𝑪 (× 𝟏𝟎𝟗) 𝑨 𝝌𝑩𝑨𝑶𝟐

1 10.907 5.127 1.692 0.770395 307.2705

9.093 4.274 2.235 0.796011 362.3666

7.660 3.601 2.664 0.818578 414.6694

6.513 3.062 3.007 0.838421 463.5695

5.584 2.625 3.285 0.855846 508.7584

4.829 2.27 3.511 0.870992 549.7424

4.191 1.97 3.702 0.884677 588.1406

3.680 1.73 3.855 0.896223 621.5435

3.234 1.52 3.989 0.906765 652.8486

2.851 1.34 4.103 0.916109 681.2378

2.532 1.19 4.199 0.924252 706.4722

2.276 1.07 4.275 0.931105 728.0639

2.042 0.96 4.345 0.937198 747.5347

Figure 43: Table showing the value of Parameter A for different values of A, B and C

0

200

400

600

800

1000

1200

1400

1600

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

chi s

qu

are

_CM

B

B (*10^-29)

Data Set 4

Data Set

65

Plotting a graph of 𝜒𝐵𝐴𝑂2 against the B-parameter, we have the following figure.

Figure 44: Plot showing the values of chi square_BAO for different parameters of B

It is interesting to note that for the CMB analysis, the 𝜒𝐶𝑀𝐵2 values decreases for smaller

values of parameter B and for BAO analysis, the 𝜒𝐵𝐴𝑂2 values decreases for higher values of

parameter B. This is also similarly observed in the CMB and BAO analysis of Proposed model

1 shown in Figure 35 and 38.

In Figure 43, I did not go ahead to find the smallest value of 𝜒𝐵𝐴𝑂2 , as the B-parameter with

the smallest value of 𝜒𝐵𝐴𝑂2 would not be within the acceptable age constraint.

300

350

400

450

500

550

600

650

700

750

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

chi s

qu

are

_BA

O

B (*10^-29)

Data Set 5

Data Set 5

66

Combining both values of 𝜒𝐶𝑀𝐵2 and 𝜒𝐵𝐴𝑂

2 and plotting a graph of 𝜒𝐵𝐴𝑂2 + 𝜒𝐶𝑀𝐵

2 against B

𝒛𝒆𝒒 𝑩 (× 𝟏𝟎−𝟐𝟗) 𝝌𝑪𝑴𝑩𝟐

𝝌𝑩𝑨𝑶𝟐 𝝌𝑩𝑨𝑶

𝟐 + 𝝌𝑪𝑴𝑩𝟐

1 5.127 1430.034 307.2705 1737.305

4.274 1183.722 362.3666 1546.089

3.601 963.9391 414.6694 1378.609

3.062 769.9165 463.5695 1233.486

2.625 600.5079 508.7584 1109.266

2.27 455.4855 549.7424 1005.228

1.97 329.518 588.1406 917.6586

1.73 229.7618 621.5435 851.3053

1.52 145.901 652.8486 798.7496

1.34 81.47438 681.2378 762.7122

1.19 36.82573 706.4722 743.2979

1.07 11.48853 728.0639 739.5524

0.96 0.3112 747.5347 747.8459

Figure 45: Table shows the combined CMB and BAO analysis of Model 2

Figure 46: Plot shows the combined CMB and BAO analysis of Model 2 against B

700

900

1100

1300

1500

1700

1900

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

𝜒_𝐶𝑀𝐵

^2+ 𝜒

_𝐵𝐴𝑂

^2

B (*10^-29)

Data Set 6

Data Set 6

67

Figure 46 shows that the combined values of 𝜒𝐵𝐴𝑂2 + 𝜒𝐶𝑀𝐵

2 decreases as the B-term

decreases. This means that if the B-parameter is smaller, it would be more consistent with

observational data.

Similarly in Figure 40, the plot shows that the combined values of 𝜒𝐵𝐴𝑂2 + 𝜒𝐶𝑀𝐵

2 is smaller

for a larger 𝑧𝑒𝑞, and this larger 𝑧𝑒𝑞 corresponds to a smaller B-parameter.

Hence, both models conclude that the B-parameter should be smaller in order to fit

observational data.

Another interesting observation is to compare Figure 45 with 39 at 𝑧𝑒𝑞 = 1. The values of

𝜒𝐵𝐴𝑂2 + 𝜒𝐶𝑀𝐵

2 is smaller at 739.55 in Figure 45, than compared to 1110.01 in Figure 39. This

shows that proposed model 2 is more promising in getting a better fit with observations.

68

6.6 Supernova as Standard Candles (Cheng, 2005)

In order to obtain observational evidence for any changing expansion rate of the universe

(i.e. to measure the curvature of the Hubble curve, in our case, Figure 14), one would have

to measure great cosmic distances (a distance of over 5 billion light years). Clearly some

very bright light sources are required. This also means that we must measure objects back in

a time interval that is a significant fraction of the age of the universe, hence these objects

must be present at the early cosmic era.

The type Ia supernova (SNe Ia) is the optimum choice to serve as such standard candles for

investigating any changing expansion rate of the universe. These supernova explosions

begin as white dwarfs, which are collapsed old stars sustained by degenerate pressure of

their electrons, with mass comparable to the sun. If the white dwarf has a large companion

star, its powerful gravitational attraction will draw matter from its companion. This happens

till its mass increases until the “Chandrasekhar limit”~ 1.4 𝑚𝑎𝑠𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑢𝑛. As it can no

longer be countered by the electron pressure, the gravitational contraction develops and

the resultant heating of the interior core would trigger the thermonuclear blast that rips it

apart, resulting in a Supernova explosion. These explosions produce a million times more

light than Cepheid Variables (the common standard candles used in cosmology).

6.6.1 Investigating the Changing expansion rate (Feng & Li,

2009)

Because light from distant galaxies was emitted long ago, to measure a star (or a supernova)

farther out in distance is to probe the cosmos further back in time. An accelerating

expansion means that the expansion rate was smaller in the past. Hence, to reach a given

redshift, the star must be located further away than expected (as compared to a

decelerating or an empty universe). Observationally, the star (or a supernova) would be

measured to be dimmer than expected.

69

Therefore, by observing these supernovas and plotting them in Figure 47, one can

determine whether the expansion of the Universe is accelerating. In fact, observations made

by the Supernova Cosmological Project and the High-z Supernova Search Team have

concluded that the expansion of the Universe is accelerating.

The theoretical expression for Luminosity Distance, denoted by 𝐷𝐿𝑡ℎ is given as

𝐷𝐿

𝑡ℎ =1 + 𝑧

√|Ω𝑘0|𝑆𝑖𝑛𝑛[√|Ω𝑘0| ∫

𝑑𝑧′

𝐸(𝑧′; 𝑎1, … , 𝑎𝑛)]

𝑧

0

(126)

Referring to (120), for a flat Universe, we have the following expression

𝐷𝐿

𝑡ℎ(𝑧𝑖; 𝑎1, … , 𝑎𝑛) = (1 + 𝑧𝑖) ∫𝑑𝑧′

𝐸(𝑧′; 𝑎1, … , 𝑎𝑛)

𝑧𝑖

0

(127)

The 𝑆𝑁2 then would be given by the following expression

𝑆𝑁2 (𝑎1, … , 𝑎𝑛) = ∑

[𝜇𝑜𝑏𝑠(𝑧𝑖) − 𝜇(𝑧𝑖)]2

𝜎𝑖2

397

𝑖=1

(128)

Figure 47: Figure shows a plot

of Luminosity Distance ∆𝑟

against Redshift 𝑧. An

accelerating Universe would

have the Hubble curve bends

upward, as a supernova at a

given redshift would be

further out in distance than

anticipated.

70

Where

𝜇(𝑧𝑖) = 5 log10 𝐷𝐿(𝑧𝑖) + 𝜇0

(129)

𝜇0 = −5 log10 ℎ + 42.38; ℎ =

𝐻0

100

(130)

The term 𝜇𝑜𝑏𝑠(𝑧𝑖) and its associated uncertainties 𝜎𝑖 are observational data available in the

following paper, Improved Dark Energy Constraints from ~100 New CfA Supernova Type Ia

Light Curves.

We can analytically marginalized the expression 𝑆𝑁2 above to get the following expression

𝜒𝑆𝑁

2 = −2 ln ( ∫ 𝑒−𝜒2

2 𝑃(𝜇0)𝑑∞

−∞

𝜇0) = 𝑎 −𝑏2

𝑐+ ln(

𝑐

2𝜋)

(131)

𝑎 = ∑[𝜇𝑜𝑏𝑠(𝑧𝑖) − 5 log10 𝐷𝐿

𝑡ℎ(𝑧𝑖; 𝑎1, … , 𝑎𝑛)]2

𝜎𝑖2

397

𝑖=1

(132)

𝑏 = ∑[𝜇𝑜𝑏𝑠(𝑧𝑖) − 5 log10 𝐷𝐿

𝑡ℎ(𝑧𝑖; 𝑎1, … , 𝑎𝑛)]

𝜎𝑖2

397

𝑖=1

(133)

𝑐 = ∑1

𝜎𝑖2

397

𝑖=1

(134)

71

6.6.2 SN Constraints for Proposed Model 1

Using the Supernova constitution data sets. (Hicken, et al., 2009), we can calculate for 𝜒𝑆𝑁2

in equation (131). The figure below shows the various results of 𝜒𝑆𝑁2 for the different

parameters of B and C.

In this SN constraint test, I want to find the minimum value of 𝜒𝑆𝑁2 by extending the values

of 𝑧𝑒𝑞 beyond 1.

zeq B (× 10−29) C (× 109) 𝜒𝑆𝑁2

0 21.000 1.982 1418.19

0.1 15.778 2.352 1140.35

0.2 12.153 2.609 949.84

0.3 9.558 2.792 817.77

0.4 7.653 2.927 725.17

0.5 6.222 3.028 659.51

0.6 5.127 3.106 612.47

0.7 4.274 3.166 578.40

0.8 3.601 3.214 553.46

0.9 3.062 3.252 535.04

1 2.625 3.283 521.30

1.1 2.27 3.31 510.94

1.2 1.97 3.33 503.01

1.3 1.73 3.35 497.05

1.4 1.52 3.36 492.46

1.5 1.34 3.37 488.80

1.6 1.19 3.38 485.95

1.7 1.07 3.39 483.81

1.8 0.96 3.40 481.97

1.9 0.86 3.41 480.42

2.0 0.78 3.414 479.34

2.7 0.415 3.440 475.36

3.0 0.328 3.446 474.70

4.0 0.168 3.457 473.81

5.0 0.0972 3.462 473.56

6.0 0.0612 3.464 473.46

Infinity 0.00 3.469 473.35

Figure 48: Table showing the value of 𝜒𝑆𝑁2 for different values of 𝑧𝑒𝑞 , 𝐵 𝑎𝑛𝑑 𝐶

72

Plotting the values of 𝜒𝑆𝑁2 against 𝑧𝑒𝑞, we have the following figure

Figure 49: Plot showing how 𝝌𝑺𝑵𝟐 varies with 𝒛𝒆𝒒

Using the data shown in Figure 48, I would now combine all these observational values,

𝜒𝐵𝐴𝑂2 + 𝜒𝐶𝑀𝐵

2 + 𝜒𝑆𝑁 2

𝑧𝑒𝑞 𝜒𝐵𝐴𝑂2 + 𝜒𝐶𝑀𝐵

2 + 𝜒𝑆𝑁 2

0 4267.946

0.1 3705.695

0.2 3264.943

0.3 2911.652

0.4 2622.994

0.5 2383.328

0.6 2181.964

0.7 2011.037

0.8 1864.936

0.9 1739.449

1.0 1631.309

Figure 50: Table showing the combined 𝝌𝑩𝑨𝑶𝟐 + 𝝌𝑪𝑴𝑩

𝟐 + 𝝌𝑺𝑵 𝟐 and 𝒛𝒆𝒒

400

600

800

1000

1200

1400

0 1 2 3 4 5 6

chi s

qu

are

_SN

z_eq

Data Set

Data Set

73

Plotting the values of 𝜒𝐵𝐴𝑂2 + 𝜒𝐶𝑀𝐵

2 + 𝜒𝑆𝑁 2 against 𝒛𝒆𝒒

Figure 51: Plot showing the combined 𝝌𝑩𝑨𝑶𝟐 + 𝝌𝑪𝑴𝑩

𝟐 + 𝝌𝑺𝑵 𝟐 against 𝒛𝒆𝒒

One consistent observation that can be made from Figure 48, 49 and 51 is that the

combined 𝝌𝑩𝑨𝑶𝟐 + 𝝌𝑪𝑴𝑩

𝟐 + 𝝌𝑺𝑵 𝟐 values decrease with increasing 𝒛𝒆𝒒 and decreasing values of

the B-parameter. This goes to show that observations do not favor the existence of a B-term

in our proposed model.

1500

2000

2500

3000

3500

4000

4500

0 0.2 0.4 0.6 0.8 1

Co

mb

ine

d c

hi s

qu

are

z_eq

Data Set

Data Set

74

6.6.3 SN Constraints for Proposed Model 2

Calculating the 𝝌𝑺𝑵𝟐 values for the different sets of parameters A, B and C.

𝑧𝑒𝑞 𝐴 (× 10−10) 𝐵 (× 10−29) 𝐶 (× 109) 𝐴𝑔𝑒 𝜒𝑆𝑁2

1 10.907 5.127 1.692 0.729 1416.87

9.093 4.2747 2.235 0.772 1216.15

7.660 3.601 2.664 0.814 1058.59

6.513 3.062 3.007 0.855 934.64

5.584 2.625 3.285 0.894 836.92

4.829 2.27 3.511 0.933 760.30

4.191 1.97 3.702 0.971 698.26

3.680 1.73 3.855 1.006 651.10

3.234 1.52 3.989 1.042 612.14

2.851 1.34 4.103 1.078 580.87

2.532 1.19 4.199 1.111 556.65

2.276 1.07 4.275 1.142 538.82

2.042 0.960 4.345 1.174 523.51

Figure 52: The different values of 𝝌𝑺𝑵𝟐 for different set of parameters A, B and C

Currently, Proposed Model 2 is calculated on the basis that 𝑧𝑒𝑞 = 1. I could extend this

calculation to include smaller values of 𝑧𝑒𝑞to see how small 𝝌𝑺𝑵𝟐 can get.

Calculating similarly for Figure 52, with 𝑧𝑒𝑞 = 0.

Figure 53: The different values of 𝝌𝑺𝑵𝟐 for different set of parameters A, B and C for 𝒛𝒆𝒒 = 𝟎.

𝑧𝑒𝑞 𝐴 (× 10−10) 𝐵 (× 10−29) 𝐶 (× 109) 𝐴𝑔𝑒 𝜒𝑆𝑁2

0 2.542 9.558 3.604 0.719 1156.86

2.035 7.653 3.873 0.766 977.66

1.655 6.222 4.076 0.812 845.88

1.364 5.127 4.231 0.855 748.49

1.137 4.2747 4.351 0.897 676.20

0.958 3.601 4.447 0.938 622.37

0.814 3.062 4.523 0.977 582.20

0.698 2.625 4.585 1.015 552.19

0.604 2.27 4.635 1.05 529.91

0.524 1.97 4.677 1.09 512.93

0.460 1.73 4.711 1.12 500.81

0.404 1.52 4.741 1.15 491.48

75

Similarly, the same figure could replicated for 𝑧𝑒𝑞 = 0.3, 0.5 𝑎𝑛𝑑 0.8. However, I would

leave that in the appendix and would summarized those results in the following figure.

𝑧𝑒𝑞 𝜒𝑆𝑁/𝑚𝑖𝑛2 𝐴𝑔𝑒 𝐵 (× 10−29)

0 491.48 1.150 1.52

0.3 495.88 1.165 1.34

0.5 498.93 1.178 1.19

0.8 514.02 1.171 1.07

1.0 523.51 1.174 0.96

Figure 54: Summarized table of the minimum values of 𝝌𝑺𝑵𝟐 for different 𝒛𝒆𝒒

The table in Figure 54 shows that the value of 𝜒𝑆𝑁2 is the smallest when 𝑧𝑒𝑞 = 0. The B-

parameter value is also the largest for the smallest 𝜒𝑆𝑁2 value.

Combining the values of 𝜒𝐵𝐴𝑂2 + 𝜒𝐶𝑀𝐵

2 + 𝜒𝑆𝑁 2 for each 𝑧𝑒𝑞 and selecting the minimum

𝜒𝑀𝐼𝑁2 = 𝜒𝐵𝐴𝑂

2 + 𝜒𝐶𝑀𝐵2 + 𝜒𝑆𝑁

2 for each 𝑧𝑒𝑞, we obtained the following.

Proposed Model 2 𝑧𝑒𝑞 𝜒𝑀𝐼𝑁

2 𝐴𝑔𝑒 𝐴 (× 10−10) 𝐵 (× 10−29) 𝐶 (× 109)

0 1332.15 1.150 0.404 1.52 4.741

0.3 1320.23 1.165 0.783 1.34 4.653

0.5 1298.94 1.178 1.068 1.19 4.588

0.8 1281.51 1.171 1.660 1.07 4.439

1.0 1271.36 1.174 2.042 0.96 4.345 Figure 55: Summarized table of the combined 𝝌𝑴𝑰𝑵

𝟐 values for different 𝒛𝒆𝒒

Proposed Model 1

𝑧𝑒𝑞 𝐴𝑔𝑒 𝐴 (× 10−10) 𝐵(× 10−29) 𝐶 (× 109) 𝜒𝐵𝐴𝑂2 + 𝜒𝐶𝑀𝐵

2

+ 𝜒𝑆𝑁 2

0 0.565 5.588 21.0 1.982 4267.946

0.1 0.610 15.778 2.352 3705.695

0.2 0.651 12.153 2.609 3264.943

0.3 0.690 9.558 2.792 2911.652

0.4 0.725 7.653 2.927 2622.994

0.5 0.759 6.222 3.028 2383.328

0.6 0.790 5.127 3.106 2181.964

0.7 0.819 4.274 3.166 2011.037

0.8 0.845 3.601 3.214 1864.936

0.9 0.871 3.062 3.252 1739.449

1.0 0.894 2.625 3.283 1631.309

Figure 56: Comparison figure for Proposed Model 1

76

Figure 56 shows that Proposed Model 1 is more consistent with observations for a higher

𝑧𝑒𝑞 value and a decreasing B-parameter. Note that for Proposed Model 1 in Figure 56, each

values of 𝑧𝑒𝑞 leads to a unique set of B and C parameters, and hence a unique value of 𝜒2.

This is unlike in Proposed Model 2 in Figure 55. Each value of 𝑧𝑒𝑞 can have a range of A, B

and C parameters leading to different values of 𝜒2. It is only in Proposed Model 2, that we

can choose the smallest value of 𝜒2called 𝜒𝑀𝐼𝑁2 , for every value of 𝑧𝑒𝑞.

Following from this line of reasoning, one can conclude that for Proposed Model 1, a higher

𝑧𝑒𝑞 would lead to a smaller B-parameter, and also a smaller value of 𝜒2. This means that the

B-parameter cannot be preserved in Proposed Model 1, as higher values of 𝑧𝑒𝑞 which fits

observations indicates that the B-parameter is negligible today, leading Proposed Model 1

to reduce to the Friedmann model. Indeed, Figure 34 and 48 shows that the values of 𝜒2

would drop for higher values of 𝑧𝑒𝑞 and smaller values of the B-parameter.

Preliminary observations of the trend shown in Figure 55 seems to suggest that a more

consistent fit with observations will occur at a higher value of 𝑧𝑒𝑞, with an increasing A-

parameter and decreasing B-parameter. For Proposed Model 2, I have restricted the range

of A, B and C parameters to fit the age constraints. The design of Proposed Model 2 which

allowed the variation of the A-parameter has indeed lead to more consistent results with

observations as compared to Proposed Model 1 with a fixed parameter A. Based on the

limited trend available in Figure 55, my preliminary view is that Proposed Model 2 also do

not favor the addition of a B-parameter. This is because the limited trend goes towards

showing that a growing A and a diminishing B would be more consistent with observational

data.

77

Chapter 7

Summary and Future Work

7.1 Summary

This project is motivated by the need to explain the phenomenon of the accelerating

Universe. The Λ Cold Dark Matter (Λ𝐶𝐷𝑀) model explained it using Dark energy, however,

its nature remains unknown and it has not been physically detected. Therefore, this

explanation remains unsatisfactory. Due to the lack of experimental evidence for this Dark

energy, alternative viable cosmological models have been formulated in recent years to

circumvent the need for Dark energy to explain this accelerating phenomenon.

Brane cosmology is one branch of study that explore the realm of higher spatial dimensions

that might exist in our physical Universe. The test for the existence of higher dimensions

need not be exclusively done in particle accelerators, it could also be done on a large scale

using precision cosmology. This study of Brane cosmology also hopes to come up with

models that will be consistent with observational data, such as the accelerating

phenomenon (Langlois, 2002).

This thesis starts with the addition of a quadratic density term into the current Friedman

model. Firstly, using the age and baryonic matter constraints, we can rule out certain

variants of our proposed model. Next, we take our 2 Proposed Models further through the

First Doppler Peak test, the CMB/BAO and the SNe tests, hoping to get consistent results

with observations.

In conclusion, based on the above cosmological tests done on both Proposed Model 1 and 2,

observational data do not favour an addition of a quadratic density term in the cosmological

models. The Λ cold dark matter (Λ𝐶𝐷𝑀) model still remains as the best candidate for the

explanation of an accelerating Universe (Primack, 2004). Section 7.2 will discuss on the

possibility of future work pertaining to these Cardassian models.

78

7.2 Future Work

Future developments in this area include the cosmological testing of other Cardassian

models, like the modified polytropic Cardassian model (MPC), the exponential model (EC)

and their flat and viscous versions (Feng & Li, 2009).

𝐻2 = 𝑔(𝜌)

Where 𝑔(𝜌) could take different forms.

Models 𝑔(𝜌)

Original Cardassian Model 𝜌𝑚 [1 + (

𝜌𝑚

𝜌𝑐𝑎𝑟𝑑)

𝑛−1

]

Exponential Model 𝜌𝑚 exp [(

𝜌𝑚

𝜌𝑐𝑎𝑟𝑑)

−𝑛

]

Modified Polytropic Cardassian Model

𝜌𝑚 [1 + (𝜌𝑚

𝜌𝑐𝑎𝑟𝑑)

𝑞(𝑛−1)

]

1𝑞

Figure 57: Different Cardassian Models for Future Work

79

Appendix A: Mathematica code for solving

Einstein’s equations at Figure 8

80

81

82

83

Appendix B: SALT Light Curve Fit Parameters

and Distances. Constitution Set.

84

85

86

87

88

89

90

91

92

Appendix C: Calculation of 𝝌𝑪𝑴𝑩𝟐 , 𝝌𝑩𝑨𝑶

𝟐 𝒂𝒏𝒅 𝝌𝑺𝑵𝟐

for Proposed Model 2.

𝒛𝒆𝒒 = 𝟎:

zzeq A B C

𝑹 𝝌𝑪𝑴𝑩𝟐

0 2.54223E-10 9.55849E-29 3603816507 0.951791 1592.468

2.03545E-10 7.65306E-29 3873406377 1.02183 1311.85

1.65490E-10 6.22222E-29 4075848855 1.09008 1064.545

1.36359E-10 5.12695E-29 4230813192 1.15662 848.281

1.13684E-10 4.27437E-29 4351440521 1.22154 660.9229

9.57696E-11 3.60082E-29 4446737781 1.28492 500.5346

8.14300E-11 3.06167E-29 4523019449 1.34683 365.353

6.98161E-11 2.62500E-29 4584801728 1.40735 253.7314

6.03743E-11 2.27E-29 4635028923 1.46612 164.7575

5.23953E-11 1.97E-29 4677474440 1.525 94.80609

4.60121E-11 1.73E-29 4711430853 1.5803 46.59859

4.04268E-11 1.52E-29 4741142715 1.6367 14.88335

93

B Age Chi square _SN

9.55849E-29 0.719 1156.86

7.65306E-29 0.766 977.66

6.22222E-29 0.812 845.88

5.12695E-29 0.855 748.49

4.27437E-29 0.897 676.20

3.60082E-29 0.938 622.37

3.06167E-29 0.977 582.20

2.62500E-29 1.015 552.19

2.27E-29 1.05 529.91

1.97E-29 1.09 512.93

1.73E-29 1.12 500.81

1.52E-29 1.15 491.48

A B C α β 𝑨 𝝌𝑩𝑨𝑶𝟐

0 2.54223E-10 9.55849E-29 3603816507 0.899445553 0.911984 0.82028 418.7559

2.03545E-10 7.65306E-29 3873406377 0.914331722 0.926073 0.846738 484.875

1.65490E-10 6.22222E-29 4075848855 0.9267449 0.937534 0.868855 543.8631

1.36359E-10 5.12695E-29 4230813192 0.937102191 0.955513 0.895413 619.1705

1.13684E-10 4.27437E-29 4351440521 0.945758787 0.95459 0.902812 641.0186

9.57696E-11 3.60082E-29 4446737781 0.953013126 0.96094 0.915788 680.2536

8.14300E-11 3.06167E-29 4523019449 0.959112122 0.966211 0.926705 714.1617

6.98161E-11 2.62500E-29 4584801728 0.964258961 0.970612 0.935921 743.4326

6.03743E-11 2.27E-29 4635028923 0.968589763 0.974282 0.94368 768.5275

5.23953E-11 1.97E-29 4677474440 0.972358354 0.977451 0.950433 790.7101

4.60121E-11 1.73E-29 4711430853 0.975448636 0.980032 0.955971 809.1377

4.04268E-11 1.52E-29 4741142715 0.978210025 0.982326 0.960921 825.7886

94

𝒛𝒆𝒒 = 𝟎. 𝟑:

zzeq A B C

𝑹 𝝌𝑪𝑴𝑩𝟐

0.3 4.47190E-10 7.65306E-29 3225312410 0.985101 1455.619

3.63582E-10 6.22222E-29 3548924559 1.0536 1193.521

2.99582E-10 5.12695E-29 3796641061 1.12051 962.5996

2.49763E-10 4.27437E-29 3989468524 1.18588 760.9467

2.10406E-10 3.60082E-29 4141804888 1.24976 586.7614

1.78902E-10 3.06167E-29 4263744092 1.31222 438.3073

1.53386E-10 2.62500E-29 4362505462 1.37333 313.9797

1.32642E-10 2.27E-29 4442795580 1.4327 213.0063

1.15113E-10 1.97E-29 4510646384 1.49219 131.4161

1.01089E-10 1.73E-29 4564927028 1.5481 72.60834

8.88179E-11 1.52E-29 4612422591 1.60512 30.4704

7.83000E-11 1.34E-29 4653133074 1.66195 6.395575

95

B Age Chi square _SN

7.65306E-29 0.73756 1147.69

6.22222E-29 0.78343 979.16

5.12695E-29 0.82763 853.03

4.27437E-29 0.87026 758.27

3.60082E-29 0.9114 686.84

3.06167E-29 0.95115 632.84

2.62500E-29 0.98958 591.95

2.27E-29 1.0265 561.15

1.97E-29 1.0631 537.27

1.73E-29 1.09713 519.92

1.52E-29 1.1315 506.26

1.34E-29 1.1654 495.88

A B C α β 𝑨 𝝌𝑩𝑨𝑶𝟐

0.3 4.47190E-10 7.65306E-29 3225312410 0.898356309 0.908619 0.819287 416.3678

3.63582E-10 6.22222E-29 3548924559 0.91236925 0.922233 0.844921 480.1784

2.99582E-10 5.12695E-29 3796641061 0.92421728 0.933488 0.866485 537.3809

2.49763E-10 4.27437E-29 3989468524 0.934237946 0.94282 0.892677 611.1846

2.10406E-10 3.60082E-29 4141804888 0.942723576 0.95060 0.899914 632.4175

1.78902E-10 3.06167E-29 4263744092 0.949923886 0.95710 0.91282 671.1753

1.53386E-10 2.62500E-29 4362505462 0.956049411 0.96257 0.923745 704.8883

1.32642E-10 2.27E-29 4442795580 0.96124016 0.96715 0.932991 734.0633

1.15113E-10 1.97E-29 4510646384 0.965784901 0.97113 0.940947 759.6406

1.01089E-10 1.73E-29 4564927028 0.96953183 0.97438 0.94767 781.5967

8.88179E-11 1.52E-29 4612422591 0.972895693 0.97728 0.953469 800.7865

7.83000E-11 1.34E-29 4653133074 0.975845234 0.97981 0.958598 817.9538

96

𝒛𝒆𝒒 = 𝟎. 𝟓:

zzeq A B C

𝑹 𝝌𝑪𝑴𝑩𝟐

0.5 6.86962E-10 7.65306E-29 2587519154 0.953523 1585.201

5.58525E-10 6.22222E-29 3030375123 1.02197 1311.317

4.60210E-10 5.12695E-29 3369369600 1.08895 1068.43

3.83680E-10 4.27437E-29 3633249658 1.15449 854.8237

3.23220E-10 3.60082E-29 3841718556 1.21861 668.8757

2.74825E-10 3.06167E-29 4008589621 1.28138 508.9061

2.35628E-10 2.62500E-29 4143742340 1.34284 373.4251

2.03762E-10 2.27E-29 4253617566 1.4026 261.7583

1.76833E-10 1.97E-29 4346469870 1.46251 169.6712

1.55290E-10 1.73E-29 4420751714 1.51884 101.2248

1.36440E-10 1.52E-29 4485748326 1.57632 49.50233

1.20282E-10 1.34E-29 4541459709 1.6336 16.16886

1.06818E-10

1.19E-29

4587885861

1.6888 1.244986

97

B Age Chi square _SN

7.65306E-29 0.71291 1319.92

6.22222E-29 0.75899 1116.18

5.12695E-29 0.80348 962.05

4.27437E-29 0.84646 845.04

3.60082E-29 0.888 755.91

3.06167E-29 0.92818 687.83

2.62500E-29 0.96707 635.72

2.27E-29 1.0044 596.03

1.97E-29 1.04154 564.89

1.73E-29 1.076 541.95

1.52E-29 1.1109 523.63

1.34E-29 1.1453 509.46

1.19E-29

1.17808

498.93

A B C α β 𝑨 𝝌𝑩𝑨𝑶𝟐

0.5 6.86962E-10 7.65306E-29 2587519154 0.884364331 0.893022 0.789757 348.496

5.58525E-10 6.22222E-29 3030375123 0.899618373 0.908402 0.817215 411.4099

4.60210E-10 5.12695E-29 3369369600 0.912657131 0.921239 0.840775 469.5515

3.83680E-10 4.27437E-29 3633249658 0.923794862 0.931979 0.860957 522.4113

3.23220E-10 3.60082E-29 3841718556 0.933311327 0.940993 0.878239 569.9157

2.74825E-10 3.06167E-29 4008589621 0.941450974 0.948584 0.893045 612.2578

2.35628E-10 2.62500E-29 4143742340 0.948425062 0.955 0.905746 649.7879

2.03762E-10 2.27E-29 4253617566 0.954371979 0.96041 0.916588 682.7105

1.76833E-10 1.97E-29 4346469870 0.959608085 0.96512 0.926137 712.3779

1.55290E-10 1.73E-29 4420751714 0.963945925 0.968994 0.934058 737.4669

1.36440E-10 1.52E-29 4485748326 0.967856965 0.97246 0.941202 760.4688

1.20282E-10 1.34E-29 4541459709 0.971299824 0.97549 0.947493 781.016

1.06818E-10

1.19E-29

4587885861

0.974235432 0.97806 0.952861 798.7629

98

𝒛𝒆𝒒 = 𝟎. 𝟖:

zzeq A B C

𝑹 𝝌𝑪𝑴𝑩𝟐

1 9.65141E-10 6.22222E-29 1948776950 0.966224 1532.418

7.95251E-10 5.12695E-29 2478160392 1.03281 1270.322

6.63006E-10 4.27437E-29 2890243016 1.09817 1036.942

5.58530E-10 3.60082E-29 3215793992 1.1623 830.9565

4.74902E-10 3.06167E-29 3476384608 1.22521 651.0287

4.07169E-10 2.62500E-29 3687442948 1.28694 495.7888

3.52104E-10 2.27E-29 3859027235 1.34706 364.8904

3.05571E-10 1.97E-29 4004028041 1.40743 253.5972

2.68344E-10 1.73E-29 4120028685 1.46426 167.2802

2.35770E-10 1.52E-29 4221529250 1.52231 97.5832

2.07850E-10 1.34E-29 4308529733 1.58023 46.6489

1.84583E-10 1.19E-29 4381030136 1.63604 15.15258

1.65970E-10 1.07E-29 4439030459 1.68706 1.457739

99

B Age Chi square _SN

6.22222E-29 0.716 1410.91

5.12695E-29 0.7606 1200.70

4.27437E-29 0.8038 1038.19

3.60082E-29 0.8457 912.21

3.06167E-29 0.8864 814.29

2.62500E-29 0.9258 738.03

2.27E-29 0.9638 678.90

1.97E-29 1.0015 631.64

1.73E-29 1.0367 596.15

1.52E-29 1.0723 567.20

1.34E-29 1.1074 544.28

1.19E-29 1.1409 526.79

1.07E-29 1.1713 514.02

A B C α β 𝑨 𝝌𝑩𝑨𝑶𝟐

1 9.65141E-10 6.22222E-29 1948776950 0.876519883 0.882726 0.773727 314.1796

7.95251E-10 5.12695E-29 2478160392 0.891421205 0.89821 0.800683 372.9066

6.63006E-10 4.27437E-29 2890243016 0.904363739 0.911357 0.824198 428.2422

5.58530E-10 3.60082E-29 3215793992 0.915593241 0.922541 0.844672 479.5386

4.74902E-10 3.06167E-29 3476384608 0.925333348 0.932073 0.862478 526.5087

4.07169E-10 2.62500E-29 3687442948 0.933784892 0.940217 0.87796 569.1324

3.52104E-10 2.27E-29 3859027235 0.941073838 0.947147 0.891335 607.2898

3.05571E-10 1.97E-29 4004028041 0.947557397 0.953237 0.903247 642.3147

2.68344E-10 1.73E-29 4120028685 0.952977741 0.958276 0.913216 672.3823

2.35770E-10 1.52E-29 4221529250 0.957904396 0.962815 0.922285 700.3331

2.07850E-10 1.34E-29 4308529733 0.962273118 0.966807 0.930332 725.6127

1.84583E-10 1.19E-29 4381030136 0.966022948 0.970208 0.937243 747.6789

1.65970E-10 1.07E-29 4439030459 0.969097859 0.97298 0.942913 766.0288

100

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