THE STUDY OF THE INFLUENCE OF INFLATIONARY POTENTIALS AFFECTING THE POWER-SPECTRUM CHIRAPHAN CHINCHO

Transcription

1 THE STUDY OF THE INFLUENCE OF INFLATIONARY POTENTIALS AFFECTING THE POWER-SPECTRUM CHIRAPHAN CHINCHO Submitted in partial fulfillment of the requirements for the award of the degree of Bachelor of Science in Physics B.S. (Physics) Department of Physics, Faculty of Science Naresuan University March, 2012

2 This independent study entitled The study of the influence of inflationary potentials affecting the power-spectrum submitted in partial fulfillment of the requirements for Bachelor of Science Degree in Physics is hereby approved.... (Dr. Teeraparb Chantavat, D.Phil.) Advisor... (Assistant Professor Dr. Pornrad Srisawad, Ph.D.) Advisor... (Associate Professor Charan Phromsuwan, M.Sc.(Physics)) Member... (Arjan Suphornphun Chootin, M.Sc.(Physics)) Member... (Assistant Professor Dr. Thiranee Khumlumlert, Ph.D.) Head Department of Physics March 2013

3 ACKNOWLEDGEMENT I would like to express my sincere thanks to Asst. Prof. Dr. Pornrad Srisawad and Dr. Teeraparb Chantavat who were supervisor for their suggestion through improve the project until it is absolute succeed. In addition, I would like to thanks to Dr. Burin Gumjupai for his teaching that make me understand in introduction cosmology. I would like to thanks to Dr. Nattapong Yongram and Mr. Napatsakon Sarapat for his teaching that make me understand in Mathermatica Program. Furthermore, I would like to thanks to Assoc. Prof. Charan Phromsuwan and Arjan Suphornphun Chootin who were committee of Independent study test that give instruction to correct and check until the project is complete. Finally I acknowledge my family and my friends for all their support the period of this project. Chiraphan Chicho

4 Title Candidate Advisor The study of the influence of inflationary potentials affecting the power-spectrum Miss Chiraphan Chincho Asst. Prof. Dr. Pornrad Srisawad Dr. Teeraparb Chantavat Degree Bachelor of Science Programme in Physics Academic Year 2012 ABSTRACT This project study effect of inflationary potentials to power-spectrum which we consider potential 3 kinds as follows: Massive scalar field, Self-interacting scalar field and Hill-top potential. We calculate inflaton field that can make slow-roll approximation valid under various potentials. Then calculate the spectrum index to compare with the observed value. Finally we analyze effect of spectrum index to power-spectrum.

5 LIST OF CONTENT Chapter Page Acknowledgment Abstract List of content List of tables List of figures I INTRODUCTION... 1 Background Objectives Procedure Application Frameworks Tools II STANDARD COSMOLOGY. 3 Cosmology The Hubble s law... 3 The Cosmological Principle... 3 Comoving Coordinates... 4 Friedmann Equation Fluid Equation Geometry of the Universe The Density Parameter... 6 The Cosmological Constant... 7 The Problem with the Big Bang Theory... 8

6 LIST OF CONTENT (CONT.) Chapter Page The Problem with Inflation Slow-roll approximation.. 12 III POWER SPECTRUM IV CALCULATION V ANALYSIS VI CONCLUSION. 27 REFERENCES APPENDIX BIOGRAPHY... 39

7 LIST OF TABLE Table Page 1 Random sampling of and to find the spectrum index of hill-top potential Deviation between observable index and calculated values A list of spectral indices List of the deviation between observable index and calculated values. 27

8 LIST OF FIGURES Figures Page 1 Graph of relation between and. In case and Graph of spectrum index at.where Graph of spectrum index at. Where The graph of the Gaussian distribution with confidence level at different multiple of the standard deviation. 25

9 Chapter 1 Introduction 1.1 Background Inflation [1, 2] is a physical phenomenon which is useful to provide the initial conditions of the Universe and it can solve various cosmological problems such as the flatness problem and the horizon problem etc. These problems can be explained by various scalar field potentials that affect the power-spectrum which can be observed in cosmology. 1.2 Objectives To study the effect of various inflationary potentials which affect the power spectrum [2] under the slow-roll approximation in order to find the spectrum index of the power spectrum. 1.3 Procedure Studying background of inflation and conditions of inflation. Calculate the inflaton field [2] that can make the slow-roll approximation valid under various potentials. Take the inflaton field to calculate the amplitude of density perturbations [3] that is a function of wave number. Calculate the spectrum index from the amplitude of density perturbations in order to compare with the observed value. Calculate the spectral index to compare with the observed value. 1.4 Application To compare spectrum indices that was calculated from the potentials of inflation with the observed values. To check the theory and to limit scalar-field that can make the theory is true.

10 1.5 Frameworks Studying various inflaton potentials and calculate the spectral index in order to compare with value which received from observation in cosmology. 1.6 Tools Mathematica Program

11 Chapter2 Standard Cosmology 2.1 Cosmology Cosmology is a study of the Universe which is considered a big structure which galaxies may be approximated as very small particles. Cosmology studies the evolution of the Universe since the beginning of time as well as its future. If we consider the Universe at the beginning, scientists believe that there was a great explosion called Big Bang which happened when the Universe had high density and high temperature. When it had exploded, matter dispersed all over the Universe. As the time moved on, the matter particles started to form, the origin of every objects in the Universe which had become planets, galaxies etc. and are orbiting around each other. In 1920, Edwin Powell Hubble, an American astronomer, observed and discovered that the spectrum of galaxies had shifted towards the red spectrum which indicated movements of galaxies away from us. Edwin Hubble concluded that the Universe was expanding. 2.2 The Hubble s law In 1929 Hubble discovered that there was an expansion of the Universe from the moving out of other galaxies. Where the further the galaxies are, the larger their outward velocity is. Hubble, then, deduced the law of expansion called the Hubble s law as follows: (2.1) where is the radial velocity of the galaxy. is the distance of galaxy towards us. is the Hubble s constant. From the equation above, we see that the radial velocity of a galaxy depends only with distance of galaxy. That means, if there are galaxies or any objects that are much further. The objects will be quickly moving away from us with increasing radial velocity. 2.3 The Cosmological Principle The cosmological principle explains that there is no position in the Universe that is more special than other positions in the Universe since all positions in the Universe are equivalent. That means the Universe must have properties that are called homogeneity and isotropy.

12 From the cosmological principle the Universe have two properties as follows: Homogeneity describes each point in the Universe as statistically the same. Isotropy describes the Universe is same all direction that can be observed. 2.4 Comoving Coordinates The comoving coordinate system is a set of coordinates that expands together with the Universe. It is different from the physical coordinates which are the real distance. This can be defined as follows: (2.2) where is the physical coordinates. is the comoving coordinates and is the scale factor, which is a function of time provided that the Universe is homogeneous and isotropic. From equation (2.2), we obtain (2.3) where. is the Hubble constant, which is also a function of time and dot indicates 2.5 Friedmann Equation The Friedmann equation comes from the conservation of energy in mechanics which is used to explain the expansion of the Universe. If we consider the Universe as a uniform expanding sphere, which have the mass, and a particle mass, which is on surface of the sphere. The distance, which is the distance between the center of the sphere to the particle, we obtain the total energy of the particle as follow: (2.4) (2.5) where is the total energy, is the kinetic energy, is the gravitational potential energy of the particle. is the average density of the sphere and is the universal gravitational constant. We define the curvature parameter as (2.6) where is the curvature parameter, is the speed of light in vacumm and is the comoving coordinates. From the total energy, we can consider

13 In the case where so, the universe expands continuously and never-endingly which is called an open universe,. In the case where so, the universe expands for a certain period of time and then it shrinks back which is called a closed universe,. In the case where so, the universe expands in a similar manner as the open universe. However, an expansion rate of the universe will approach zero as the size of the universe approaches infinity,. From equation (2.5) and (2.6), the Friedmann equation can be written as and ( ) (2.7) (2.8) From the Friedmann equation we see that on right-hand side of the equation, the first term is the term of matter and the second term is term of energy. That means, the expansion rate of the Universe depends on matter and energy of the Universe. 2.6 Fluid Equation We use the first law of the thermodynamics to consider the fluid in the scale of the Universe. We can consider such a fluid as a perfect fluid where there is no heat conduction and viscosity. So the Universe undergoes an adiabatic process which is reversible. The entropy is constant,. We have the Fluid equation as ( ) (2.9) From equation (2.9) we see that in the second term is related to the kinetic energy rate of the expansion of the Universe. is density rate in the Universe. is the pressure and is scale factor rate.

14 2.7 Geometry of the Universe Geometry of the Universe has 3 types as follows: Flat Geometry We consider two parallel lines that are in the Euclidean geometry, we can see that distance between the lines is always fixed. So we can conclude, in this geometry, that the sum of the angles of a triangle is equal to and the perimeter of a circle, whose radius is, is equal to. So the universe has a flat geometry that is known as a flat universe Spherical Geometry We cannot use the Euclidean geometry to describe a spherical geometry (while we use the Riemannian geometry to define a spherical geometry). If we consider two parallel lines that are on surface of a spherical geometry, we can see that the parallel lines will meet which is opposed to the Euclidean geometry. We can find in this geometry that the sum of angles of a triangle is more than and the perimeter of a circle, whose radius is is less than. The universe that has a spherical geometry is also known as a closed universe Hyperbolic Geometry We use the Riemannian geometry to describe a hyperbolic geometry. If we consider two parallel lines that are on the surface of a hyperbolic geometry, we can see that the parallel lines will diverge away from each other. We can find, in this geometry, that the sum of angles of a triangle is equal to less than and the perimeter of a circle, whose radius is is bigger than. The Universe that has a hyperbolic geometry is also known as an open Universe. 2.8 The Density Parameter obtain Consider the Friedmann equation when the Universe is flat,, So we (2.10) (2.11) where is known as the critical density. So from / and m. We get (2.12)

15 which seems a tiny value when compared with the density of water. But if we convert to an astronomical unit, we get (2.13) where is mass of the sun. The observation shows that the present density of the Universe is very close to the critical density. We can rescale the density of the Universe. This new quantity is called the density parameter as follows:. (2.14) From the given definition of the density parameter, we can rewrite the Friedmann equation as (2.15) We set so (2.16) If we define the curvature term in the density parameter as (2.17) We get (2.18) is the density parameter of matter and radiation and is the density parameter of curvature. If we consider the case where,,, this indicates a flat universe.,,, this indicates an open universe.,,, this indicates a closed universe. 2.9 The Cosmological Constant Einstein adds the term of the cosmological constant in Einstein s field equation. Since he believed that the Universe was static. So we rewrite the Friedmann equation as follows: (2.19) where is the cosmological constant. We obtain (2.20)

16 where is the density parameter of the cosmological constant, we define as so we can relate the equation (2.20) with different geometries of the universe. That is, if, indicates an open universe. If, indicates a flat universe., indicates a closed universe Fluid Explanation of the Cosmological Constant If we consider the cosmological constant as a perfect fluid, from the Friedmann equation, we can add the cosmological constant term with the matter term. We have (2.21) where is the energy density of the cosmological constant which we define as. We obtain the fluid equation with the cosmological constant as follows: ( ) (2.22) Because is constant so we have (2.23) where is the pressure of the cosmological constant. We can see that value of pressure is negative which implies the expansion of the universe, although the volume of the Universe is increasing. The energy density is constant The Problem with the Big Bang Theory The Big Bang theory is the standard model of the evolution of the Universe which initially all of matters were crowded to a point. The point has high density and high temperature. Then it exploded largely which is the cause of the expansion of the Universe. This situation is the origin of time and space The flatness problem We start at equation (2.8), we set We define, we have (2.24)

17 When is density parameter of curvature. We consider a conventional Universe where the usual radiation or matter dominated the curvature and the cosmological constant. Let start with radiation dominated case, we have In matter dominated case, we have (2.25) (2.26) So we obtain : Radiation dominated (2.27) : Matter dominated (2.28) And : Radiation dominated (2.29) : Matter dominated (2.30) In the Big Bang theory, equation (2.29) and (2.30) we see that the dissimilarity between and is an increasing function of time that means at an earlier time the Universe was much flatter. As the time grows on, will move away from. Hence, the Universe will become more and more curved, which is contradictory to the observed value at the present, since [4]. The problem is that why is the Universe still flat The horizon problem The horizon problem is the most crucial problem and is related to the communication between different regions of the Universe. We consider the light which is travelling in an observable part of the Universe that we are able to see. The light that we can see from every other parts of the sky has the same temperature of about K [1], which is described that if there are divergent regions of the sky which can interact and move close to thermal equilibrium. But the light we can notice from opposite sides of the sky are still travelling to us since decoupling (when the Universe cooled off. The Universe changed from is opaque to be transparent. The photons can travel and were unobstructed by the entire residue of the Universe s evolution.), which spends time around the time of the Big Bang. Because the light just arrived us, it cannot interact with the opposite side of the sky since it has no time enough

18 and cannot refer that the region used to interacted and created thermal equilibrium. Actually the problem is not well. Because the microwave still travelling since decoupling. Likely that region might use to interact and move to thermal equilibrium before that. in fact at early time, The observable Universe is much more smaller than the present so the light was able to travel less than the present. But it happens that regions occur around each other on the sky cannot to interact and create thermal equilibrium before decoupling The Problem with Inflation In 1981, Alan Guth suggested inflation that is a solution of various problems which inflation is defines to be an era of the Universe. Which a condition for inflation is We consider equation (2.8) and (2.9), we obtain (2.31) ( ) (2.32) Which we can see that equation (2.32) corresponds to (2.31) and we must have a negative pressure is (2.33) Which in cosmological constant have a fluid with Friedmann equation is so we have the new (2.34) But when the Universe is more and more expanded the first and second term are reduce. So we will have only cosmological constant term From, we obtain (2.35) ( ) (2.36) Which we see that when the Universe have cosmological constant dominated, have expansion rate of the Universe is much more than matter domination or radiation domination.

19 The flatness problem From the friedmann equation (2.37) In the Big Bang theory, the equation above accruing with time which constrain to get out from but inflation is opposite because (2.38) Since the condition for inflation constrain close to in exponential expansion ( ) (2.39) From the equation (2.39), which it correspond to observation in the present The horizon problem Inflation can solve the horizon problem that has a question refer to why any regions on the sky have the same temperature. Which explain that inflation expand the size of small part on the Universe that can converged thermal equilibrium. Inflation expand this small part to be much bigger than the our presently observable Universe. When we start to observe, we found that microwave have the same temperature to be equal to any part since these microwave used to be in thermal equilibrium before then.

20 2.12 Slow-roll approximation From section 2.11 we see that inflation can explain the problem of the big bang theory. By it add condition of inflation in idea of the big bang theory, which can explain these problem that correspond to observation of cosmology. Then in particle physics proposed scalar field given by (2.40) (2.41) Where is scalar field, dot is, is potential energy of scalar field, is energy density of scalar field and is pressure of scalar field. Since the scalar field is under inflation, we will call as inflaton field. From equation (2.40) and (2.41) we can rewrite the Fiedmann equation as follows: And from equation (2.9), (2.40) and (2.41) we obtain From condition of inflation [3], So we can rewrite equation (2.42) and (2.43) as ( ) (2.42) (2.43) (2.44) [ ] (2.45) (2.46) We will use equation (2.45) and (2.46) to find slow-roll approximation as and ( ) ( ) (2.47) (2.48) where and are slow-roll parameter and is Planck mass. So we see that if there is generation of inflation, inflation must be under condition of slow-roll approximation.

21 Chapter 3 Power Spectrum We shall consider the overdensity, which is defined as (3.1) where is an overdensity. is the density at any position in the Universe and is an average density. We shall study the fluctuation of density in the form of the Fourier expansion [5]. We shall adopt the convention of the Fourier series in 1-dimension as (3.2) where the period of is and the frequencies of the terms in the series are. If we consider the limit where. We use definition of the Fourier integral (3.3) Consider (3.3) in 3 dimension. We obtain (3.4) We use definition of Fourier integral in 3D to distribute the overdensity ( ) (3.5) where is the wave number which corresponds to. From equation (3.5), we consider the Universe that is homogeneous and isotropic. We calculate to find the average of the overdensity, where is the average of the over density and is the Dirac delta function.

22 We obtain (3.6) where is power spectrum which refers to clustering of the density of matter in the Universe. From the theory of initial fluctuation, the relation between and is as follows: (3.7) where is the spectral index. From equation (3.7) we explain that the clustering of matter is equivalent to every scale for n = 1. In other words, there was no significant clustering of matter in any scale. We see Fig.1, in this case is independent scale. Fig.1: Graph of relation between and. In case and. We see that in case and depend on.

23 In case, it refers to the clustering of matter at larger scale. This case corresponds to the top-down theory. That means, there was initially a clustering of matter at larger scale. Then clustering of matter will split to smaller scales. In case, it refers to the clustering of matter happened at smaller scale. This case corresponds to the bottom-up theory. That means, there was initially a clustering of matter at smaller scale. Then clustering of matter will combine to be larger scale. However, from the observation in the present, we obtain the spectrum index as [6]. Which correspond to the case. That means, our presently observable Universe corresponds to the top-down theory.

24 Chapter 4 Calculation We consider 3 inflationary potentials to calculate the spectral index. Massive scalar field [3]: (4.1) where is mass of the inflaton field and is inflaton field. Self-interacting scalar field [3]: (4.2) where is the self-coupling constant. Hill-top potential [7]: ( ) ( ) (4.3) where is mass of inflaton field, is the some large mass scale such as the GUT or Planck scale, which, is the reduced Planck mass, kg and is a free parameter. 4.1 Calculation Step - the Massive Scalar Field and Self-interacting Scalar Field 1. Take each potential to calculate the inflaton field, which can make the slowroll approximation valid under various potentials. These are the slow-roll parameters defined as follows: where ( ) ( ) (4.4) (4.5) where is the Planck mass and prime indicates. From the calculation, we can work out the value of the inflaton field at the end of inflation. 2. Substitute the end inflaton field in the number of e-foldings [3], to find an initial inflaton field given by (4.6)

25 where is an end inflaton field, is an initial inflaton field, is wave number, is present scale factor and is present Hubble constant. 3. Substitute the value of the initial inflaton field in the amplitude of density perturbation. To find the amplitude of density perturbation that is function of wave number. The amplitude of density perturbation [3], given by (4.7) 4. Substitute in spectrum index equation, to find each spectrum index of massive scalar field and self-interacting scalar field. The spectrum index [3] given by (4.8) where is the spectral index. 4.2 Calculation Step - the Hill-top Potential We did the same method as for the massive scalar field and self-interacting scalar field for the hill-top potential. But we do a sampling of and in various values. We set. We obtain Table 1: Random sampling of and to find the spectrum index of hill-top potential

26 Table 1(continue): Random sampling of and to find the spectrum index of hill-top potential

27 Table 1(continue): Random sampling of and to find the spectrum index of hill-top potential

28 Table 1(continue): Random sampling of and to find the spectrum index of hill-top potential

29 Table 1(continue): Random sampling of and to find the spectrum index of hill-top potential

30 Chapter 5 Analysis 5.1 The Spectral Index of Massive Scalar Field - Analysis From the calculation, we obtained the spectral index of the massive scalar field as follows: (5.1) where is the spectral index that calculate form the massive scalar field. From the observation, we have between and as follows:. We will calculate the difference (5.2) where is the variance of the Gaussian distribution and is the deviation between and in term of. 5.2 The Spectral Index of Self-interacting Scalar Field - Analysis From the calculation, we obtained the spectral index of the self-interacting scalar field as follows: (5.3) where is the spectral index that calculate from the self-interacting scalar field. We will calculate difference between and as follows: (5.4) where is the deviation between and in term of.

31 5.3 The Spectral Index of the Hill-top Potential - Analysis The Spectral Index at From the calculation, we obtained the spectral index of the hill-top potential when we did a sampling of and in various values. We set and obtained the graph of spectrum index as follows: Fig. 2: Graph of spectrum index at where. From Fig. 2, we obtained the maximum value of the spectral index which is The value of the spectral index is constant when lies within to. We will consider the spectral index to calculate the difference between and, where is spectrum index at as follows: (5.5) where is the deviation between and in term of The Spectral Index at From the calculation, we obtained the spectral index of hill-top potential when we did a sampling of and in various values. We set, and obtained the graph of spectral index as follows:

32 Fig. 3: Graph of spectrum index at. Where. From Fig. 3, we obtained the maximum value of the spectral index which is The value of the spectral index is constant when lies within to. We will consider the spectral index to calculate the difference between and, where is spectrum index at as follows: (5.6) where is the deviation between and in term of. Then we will consider,, and with Graph of Gaussian Distribution that show in Fig. 4,

33 Fig. 4: The graph of the Gaussian distribution with confidence level at different multiple of the standard deviation. Potential Deviation Deviation in term of Massive scalar field Self-interacting scalar field Hill-top potential, Hill-top potential, Table 2: Deviation between observable index and calculated values.

34 From Fig. 4 and Table 2, we see that, which value at, lies within to. That means the massive scalar field is in a good agreement with the observed value. Then we consider, which value at 2.076, lies within and. That means, in comparison with the observed value of the spectral index, the probability that the self-interacting scalar field being consistent with the observation is less than. Finally, we consider and, which value at and respectively. The value lie within and. That means, in comparison with the observed value of the spectral index, the possibility that the hill-top potential with and begin consistent with observation is less than. Thus, they do not agree with the observed value. If we increase the value of, the value of the spectral index is lower and less consistent with observation.

35 Chapter 6 Conclusion This project studies the effect of various inflationary potentials which affect the power spectrum under the slow-roll approximation in order to find the index of the power spectrum. From the calculation, we obtain the values of the inflaton field which are valid under the slow-roll approximation. The inflaton field gives spectral index of various potentials as follows: Potentials Massive scalar field Self-interacting scalar field Hill-top potential Spectral indices Table 3: A list of spectral indices. Then we will calculate the deviation between the observed spectral index and these spectral indices as follows: Potentials Massive scalar field Self-interacting scalar field Hill-top potential Deviation in term of Table 4: List of the deviation between observable index and calculated values.

36 We will compare the value of,, and with the graph of Gaussian distribution in Fig. 4. We can see that, whose value is, lies within to. That means the massive scalar field is in a good agreement with the observed value of the spectral index. Next, we have shown that, whose value is, lies within and. That means, in comparison with the observed value of the spectral index, the probability that the self-interacting scalar field being consistent with the observation is less than. Finally, we have shown that and, whose value is and respectively. The value lie within and. That means, in comparison with the observed value of the spectral index, the possibility that the hill-top potential with and begin consistent with observation is less than. Thus, they do not agree with the observed value. If we increase the value of, the value of the spectral index is lower and less consistent with observation. From the analysis, we conclude that only the spectral index of massive scalar field is in a good agreement with the observed value. That means the inflaton potentials, which are valid under the slow-roll approximation, gives the spectral index less than 1. The overall effect is to make scale-dependent.

37 References [1] A. Liddle. (2003). An introduction to modern cosmology. (2nd Ed.). Chichester: John Wiley Sons Ltd. [2] D. Lyth and A. Liddle. (2009). The primordial density perturbation (Cosmology, inflation and the Origin of Structure). (1st Ed.). New York: Cambridge University Press. [3] A. Liddle. (1999). An introduction to cosmological inflation. Proceeding of ICTP summer school in high-energy physics [4] K. Lake. (2005). The flatness problem and. Phys. Rev. Lett. 94(201102). 1-4 [5] M. Boas. (2006). Mathematical Methods in the Physical Sciences. (3rd ed.). New Jersey: John Wiley Sons, Inc. [6] G. Hinshaw, D. Larson, E. Komatsu, D. N. Spergel, C. L. Bennett, J. Dunkley, M. R. Nolta, M. Halpern, R. S. Hill, N. Odegard, L. Page, K. M. Smith, J. L. Weiland, B. Gold, N. Jarosik, A. Kogut, M. Limon, S. S. Meyer, G. S. Tucker, E. Wollack and E. L. Wright. Nine-year Wilkinson Microwave AnisotropyProbe (WMAP) Observations: Cosmological parameter results. [7] D. Lyth. (2007). MSSM inflation. JCAP. 0704(006).

38 APPENDIX

39 APPENDIX MATHEMATICA CODE 1 Code for calculate spectrum index of Massive scalar field In[1] : = ( ) ( Planck mass ) ( Massive scalar field ) ( ) ( ) ( ) In[7] : = Reduce [ ] Out[7]= In[8] : = Reduce [ ] Out[8]= ( ) ( ) ( Slow-roll parameter ) ( ) ( Slow-roll parameter ) In[9] : = Print [ [ { }]] In[10] : = Print * [ [ ] ]+ { [ ] } { [ ]} ( ) In[11] : = [ ] ( ) [ ] ( ) [ ]

40 In[14] : = Print[ * [ ] [ ] +] [ ] ( ) In[15] : = Print [ ] [ ] ( ) In[16] : = Print [ [ [ ] [ ]] ] ( ) In[17] : = [ ] Print [ [ ]] (* End of Code *)

41 2 Code for calculate spectrum index of Self-interacting scalar field In[1] : = ( ) ( Self-interacting scalar field ) ( ) ( ) ( ) ( Slow-roll parameter ) ( ) ( Slow-roll parameter ) ( ) In[7] : = Reduce [ ] Out[7]= In[8] : = Reduce [ ] Out[8]= ( ) In[9] : = Print [ [ { }]] In[10] : = Print * [ [ ] ]+ { [ ] } { [ ]} ( ) In[11] : = [ ] ( ) [ ] ( ) [ ] ( )

42 In[14] : = Print[ * [ ] [ ] +] ( ) In[15] : = Print [ ] (( [ ]) ) [ ] [ ] ( ) In[16] : = Print [ [ [ ] [ ]] ] ( ) In[17] : = [ ] Print [ [ ] (* End of Code *)

43 3 Example code for calculate spectrum index of Hill-top potential In[1] := ( ) ( ) [ ] ( ) [ ] ( ) ( ) ( Slow-roll parameter ) ( ) ( Slow-roll parameter ) In[6]:= In[7]:= [ [ ] ] [ ][[ ]] [ [ [[ ]]] ] [ [ ] ] [ ][[ ]] [ [ [[ ]]] ] [ [ ] [ ]] In[16]:= [ ] [ ] [ ] In[17]:= [ ] [ [ [ ] ] [ ] ][[ ]] In[18]:= [ ] [ [ ]] [ [ ]] In[19]:= [ ] [ [ [ ]] [ ]] (* End of Code *)

44 4 Code for plotting in Fig. 2 In[1]:= ( Spectral index at k=1 ) In[2]:= [ [ ] In[3]:= [ ] In[4]:= [ ] ]

45 Out[4]:= (* End of Code *)

46 5 Code for plotting in Fig. 3 In[1]:= ( Spectral index at k=0.01 ) In[2]:= [ [ ] ] In[3]:= [ ] In[4]:= [ ]

47 Out[4]:= (* End of Code *)

48 BIOGRAPHY

49 BIOGRAPHY Name-Surname Chiraphan Chincho Date of Birth March 30, 1991 Place of Birth Address Uttaradit, Thailand 13 Moo3 Tambon Lo, Chun District, Phayao Province, Thailand Educational Background 2002 Wattaitalad School (primary level) 2009 Chunwittayakom School (secondary level)