NON-MINIMAL DERIVATIVE COUPLING TO RICCI SCALAR GRAVITY

Transcription

1 NON-MINIMAL DERIVATIVE COUPLING TO RICCI SCALAR GRAVITY YUTTANA JAWRALEE A Thesis Submitted to Graduate School of Naresuan University in Partial Fulfillment of the Requirements for the Master of Science Degree in Theoretical Physics July 017 Copyright 016 by Naresuan University

2 Thesis entitled Non-Minimal Derivative Coupling to Ricci Scalar Gravity By Mr.Yuttana Jawralee has been approved by the Graduate School as partial fulfillment of the requirements for the Master Degree in Theoretical Physics of Naresuan University Oral Defense Committee... Chair Sekson Sukhasena, Ph.D.)... Advisor Associate Professor Burin Gumjudpai, Ph.D.)... Co-Advisor Pichet Vanichchapongjaroen, Ph.D.)... Co-Advisor Shingo Takeuchi, Ph.D.)... External Examiner Maneenate Wechakama, Ph.D.) Approved... Panu Putthawong, Ph.D.) Associate Dean for Administration and Planning For Dean of the Graduate School

3 ACKNOWLEDGMENTS I would like to thank my advisor, Associate Professor Burin Gumjudpai for giving me the motivations and push me to be a good physicist, and thank you for all additional knowledge. I thank all committees for spending their time to become my thesis examination s committees. I also thanks for all people at The Institute for Fundamental Study IF), Dr. Pichet Vanichchapongjareon, Dr. Seckson Sukhasena, Narakorn Kaewkhao, and Phongsaphat Rangdee U. of Phayao) for suggestions. I thank Klinbumrung s family for many assistances. At last I very very thank my MOM, my AUNT, and my SISTER for love and all supports. Yuttana Jawralee

4 LIST OF CONTENTS Chapter Page I INTRODUCTION... 1 Background and motivation... 1 Objectives... 3 Frameworks... 3 II STANDARD COSMOLOGY... 4 Cosmological Principle... 4 Hubble s law... 4 The Friedmann Robertson Walker Universe... 5 Power-Law expansion Super-acceleration expansion Scalar field model... 1 Slow-roll approximation III NON-MINIMAL DERIVATIVE COUPLING Literature review of the NMDC models Background Equations Non-Minimal Derivative Coupling model... 4 Non-Minimal Derivative Coupling and Cosmology IV RESULTS AND DISCUSSIONS Results... 41

5 LIST OF CONTENTS CONT.) Chapter Page Discussions V CONCLUSIONS REFERENCES APPENDIX BIOGRAPHY... 61

6 Title NON-MINIMAL DERIVATIVE COUPLING TO RICCI SCALAR GRAVITY Author Advisor Academic Paper Yuttana Jawralee Associate Professor Burin Gumjudpai, Ph.D. Thesis M.S. in Theoretical Physics, Naresuan University, 016 Keywords NMDC, Scalar Field, Dark Energy ABSTRACT In this work, we investigate a model of dark energy in which the scalar field kinetic term and a simple NMDC gravity with ξr a φ a φ term is considered a flat universe. The model is a trial model by transforming all the field to its logarithm mapping φ = ln φ. Equation of motion, scalar field solution and scalar potential are derived when we assume slow-roll approximation. The field solution and scalar potential are found for power-law, super acceleration and de-sitter expansions.

7 CHAPTER I INTRODUCTION 1.1 Background and motivation Recently, our universe is not only expanding but also accelerating due to some kind of unknown energy form called dark energy, which has been comfirmed by the cosmic acceleration reported in ,. This dark energy is usually in the form of either cosmological constant or scalar field 3, 4, 5, 6. There are many scalar field models proposed to describe the acceleration of the expansion of the universe, for example, quintessence models 7, 8, 9, 10, 11, 1, k-essence models 13, 14. Modifications of gravity, for instance, fr) theory 15, 16, the scalar field kinetic term is non-minimally coupled to the curvature 17, 18, 19, 0, 1,, 3, and others are as well possible answers of present acceleration. Accomplishing the acceleration needs the effective equation of state of matter species that a scalar field evolving under its potential to give rise to the negative pressure, p < ρc /3. In the scalar-tensor theories, we can extend the theories for non-minimal coupling NMC) between scalar fields and Ricci scalar in GR in form of gfφ)r, where fφ) is a function of scalar field φ and R is the Ricci scalar. The non-minimal coupling NMC) is motivated by scalar-tensor theories of the Jordan-Brans-Dicke models 4, 5, renormalizing term of quantum field theory in curved space 6 and in multidimensional theories like superstring theory 7, 8, 9, 30, 31. In the context of inflationary cosmology, first cosmology consideration of non-minimal derivative coupling NMDC) of scalar field was proposed by Amendola in Therein the function of coupling is in form fφ, a φ). In a derivative

8 coupling to Ricci tensor has been considered to study cosmological on the coupling parameter was proposed to study late time cosmological dynamics. In this thesis, we consider the simplest NMDC model with only term of ξr a φ a φ and free kinetic term. We propose modification to the model to improve the NMDC term in the small field value range. Therefore the NMDC model can be either considered as dark energy model or inflationary model when a field transformation introduced to the model. We will find solutions of the scalar field and scalar potential when the power-law, super-acceleration and de-sitter expansion were considered. We use units such that c = = 1, where c is the speed of light and is reduced Planck s constant. The gravitational constant G is related to the Planck mass m pl = GeV via G = 1/m pl and the reduced Planck mass M pl = GeV via 8πG = 1/Mpl, respectively. The signature of the metric is assumed to be, +, +, +). List of natural units mass = energy = M Length = time = L = M 1 G = L = M H = L 1 = M g ab = 1 R ab = R = Λ = L = M T ab = M L 3 = 1 L 4 = M4 φ = a = mass = M a φ = mass = M = mass = M = L 1 ξ = M = L

9 3 1. Objectives Our objectives are to study general aspects of FLRW cosmology of the NMDC coupling to Ricci curvature R model. Then we study cosmological field equation of this model. After that, we will obtain particular solutions of the scalar field and scalar potential corresponding to power-law, super-acceleration and de- Sitter expansion. 1.3 Frameworks In this thesis, the introduction and motivation of our works, objectives, and frameworks are shown in Chapter 1. In Chapter, we review basic cosmology and introduce the background equations, for example, the Friedmann equations, fluid equation and field equations. Then we review the non-minimal derivative coupling NMDC) of scalar field and determine the necessarily equations of the model in Chapter 3. Chapter 4 is results and discussions. The last Chapter is the conclusions.

10 CHAPTER II STANDARD COSMOLOGY.1 Cosmological Principle The fundamental of modern cosmology is belief that the place that we live in the Universe is not a special location. This is based on cosmological principle, and it follows observations that the Universe on large scales is homogeneous and isotropic. Homogeneity is the states that the Universe looks the same at each point, while isotropy states that the Universe looks the same in all directions. Note that homogeneity does not imply isotropy and vice versa, for example, a uniform electric field is homogeneous field, at all points are the same, but it is not isotropic at one point because directions of the field can be distinguished from those perpendicular to them. However, the universe looks approximately homogenous and isotropic on the large scales. One strong evidence to support this is the CMB observed in 1995 by the COBE mission, at least to one part in Hubble s law In 190, Edwin Hubble proved that the further galaxies are moving away from the Earth. Hubble realized a relation between the velocity of recession and distance of an object from us v = H 0 r,.1) where r is physical distance. This is known as Hubble s law, and the constant H 0 is known as the Hubble s constant at present time t 0. Let consider the velocity of recession is given by v = dr dt..)

11 5 and the relationship between physical distance r and the co-moving distance x can be written as r = at)x.3) The quantity at) is the scale factor. It is a function of time only, while x is a co-moving coordinate which is fixed by the definition. So v = ȧt) at) r.4) The Hubble s parameter defined as Ht) ȧt)/at), and therefore the Hubble s law can be written as v = Ht)r..5).3 The Friedmann Robertson Walker Universe The metric or line-element that describes a 4-dimensional homogeneous and isotropic space-time is called Friedmann-Lemaître-Robertson-Walker FLRW) space-time and is given by 33, 34 ds = g ab dx a dx b, = c dt + a t)dσ.6) where g ab is the metric tensor, at) is the scale factor with cosmic time t, dσ is the time-independent metric of the 3-dimensional space with a constant curvature K. dσ = dr 1 Kr + r dθ + sin θdφ )..7) Therefore, the line-element in equation.6) now become ds = c dt + a dr t) 1 Kr + r dθ + sin θdφ ),.8)

12 6 This is known as the Robertson-Walker metric. Where K is the curvature and takes the values 1, 0, or 1 depending on whether the spatial section has negative, zero or positive curvature respectively. Thus the covariant components g ab of the metric are g 00 = c, g 11 = a t) 1 Kr, g = a t)r, g 33 = a t)r sin θ. The Einstein s field equation has the following form G ab R ab 1 g abr = 8πG c 4 T ab..9) where G ab is Einstein tensor, R and R ab are Ricci scalar and the Ricci tensor of the metric g ab. G is the gravitational constant and T ab is the matter stress-energy tensor. From g ab we can obtain the connection or the Chritoffel symbol as Γ e ab = 1 gec b g ca + a g cb c g ab ).10) and we have the only non-zero coefficients are Γ 0 aȧ 11 = c1 Kr ), Γ0 = aȧr c, Γ0 33 = aȧr sin θ, c Γ 1 01 = ȧ ca, Γ1 11 = Kr 1 Kr, Γ1 = r1 Kr ), Γ 1 33 = r1 Kr )sin θ, Γ 0 = ȧ ca, Γ 1 = 1 r, Γ 33 = sinθcosθ, Γ 3 03 = ȧ ca, Γ3 13 = 1 r, Γ3 3 = cotθ. Next we need to find the components of the Ricci tensor which is defined by R d abc = c Γ d ab b Γ d ac + Γ e abγ d ec Γ e acγ d eb.11)

13 7 when the upper index is the same as the last index of the Reimann tensor R d abc we will obtain the Ricci tensor, R c abc = R ab, as R ab = c Γ c ab b Γ c ac + Γ e abγ c ec Γ e acγ c eb.1) We find that the off-diagonal components of the Ricci tensor are zero and the diagonal components are given by R 00 = 3 c ä a, R 11 = aä + ȧ + Kc 1 Kr )c, R = aä + ȧ + Kc )r c, R 33 = aä + ȧ + Kc )r sin θ c. and the Ricci tensor is then ä ȧ ) R = Ra a = g ab R ab = 6 a + + K..13) a a Let us now consider equation.9), R ab 1 g abr = 8πG c 4 T ab..14) It is convenient to express the field equations in the alternative form T ab 1 g abt R ab = 8πG c 4 )..15) In the FRW spacetime the energy-momentum tensor of the perfect fluid takes the form 35 T ab = ρ + Pc ) u a u b + P g ab.16) where u a is 4-velocity, ρ and P are mass density and pressure respectively. When we contracted the energy-momentum tensor by g ab we will obtain the trace of the energy-momentum tensor g ab T ab = g ab ρ + P c ) u a u b + P g ab g ab T = ρ + Pc ) c ) + 4P.17)

14 8 where u a u a = u a g ab u b = c, and therefore T = ρc + 3P..18) From equation.15) we can write the energy-momentum tensor as T ab = ρ + Pc ) u a u b + P g ab.19) and we find that T 00, T 11, T, and T 33 components are T 00 = ρc, a T 11 = 1 Kr )P, T = a r P, T 33 = a r sin θ)p. we substitute the 00-component of the energy-momentum tensor into equation.14), and given by ä a = 4πG ρ + 3P ) 3 c.0) This equation is known as the acceleration equation. Similarly, the ii-components are R ii = 8πG T c 4 ii 1 ) g iit ä ȧ a + a + Kc = 4πG ) P ρc a c.1) where i = 1,, 3, we see that the components 11,, and 33 are the same because the homogeneity and isotropy of the FLRW metric. Then we substitute equation.19) into equation.0), and we will obtain ȧ ) H = = 8πG a 3 ρ Kc.) a where H = ȧ/a. We finally arrive at the Friedmann equation.

15 9 Taking the time derivative of the Friedmann equation and substitute ä from the acceleration equation. We will obtain ρ + 3 Pc ) aȧ = ρa + ρaȧ.3) and dividing through by a, hence ρ + 3ȧ ρ + Pc ) = 0 a or ρ + 3H ρ + Pc ) = 0..4) This is called continuity equation. Another useful quantity is the density parameter, which is ratio of density and critical density, the quantity is defined as Ω = 8πG 3H ρ = ρ ρ c,.5) where the critical density is defined as density that is just enough for flat geometry. Hence, the Friedmann equation.) can be written as ρ c = 3H 8πG..6) Ω 1 = K H a..7) The sign of K is therefore determined if Ω is greater than, equal to, or less than, one. We have ρ < ρ c Ω < 1 K < 0 open ρ = ρ c Ω = 1 K = 0 flat ρ > ρ c Ω > 1 K > 0 closed

16 10 The density parameter tells us which one of the three FRW geometries describes our universe. Recent measurement of the cosmic microwave background found consistent with flat geometry, i.e. or Ω is very close to unity..4 Power-Law expansion In this section we introduce the power-law expansion. The power-law expansion were also studied in context of scalar field cosmology 36 and phantom scalar field cosmology 37. We use the power parameters p and q for two forms of power-law to separated between the canonical and phantom power-law, a t p and a t s t) q respectively. The power-law expansion used in the models under the assumptions that our flat FLRW universe is filled with dust matter and scalar fields, and dominated by dark energy. The power-law is defined as ) p at) t a 0 t 0 ) =,.8) t 0 where a 0 t 0 ) is the value of the scale factor at present time t 0 and p is a number which described the acceleration of the universe, where p > 1. The flat universe dominated by the dark energy and the Friedmann equation gives 1 < p <. We will consider the constant value of p in the range 0 < p < and in the short range of redshift z 0.45 to present, z = 0. Then we can write the power-law cosmology of the cosmic speed as t p 1 ȧ = a 0 p t p 0 t = a 0 p t 0 ), ) p 1 t, = a p t,.9)

17 11 and the cosmic acceleration t p ä = a 0 pp 1) t p 0 t = a 0 pp 1) ), ) p 1 t 0 t, pp 1) = a..30) t Hence, the Hubble parameter and its time derivative in the power-law expansion reads H = ȧ a, = p t,.31) Ḣ = p t..3) From equation.9), p is calculated at the present H 0, t 0 as p = H 0 t 0. Therefore,we can write the dust matter density in the power-law as ) 3p ) 3 t0 a0 ρ m = ρ m,0 = ρ m,0..33) t a where ρ m,0 is the dust matter density at present time t 0..5 Super-acceleration expansion In the case of super-acceleration expansion, the power-law is defined slightly different from previous and the scale factor becomes ) q at) a 0 t 0 ) = ts t,.34) t s t 0 where t s is called the big-rip time which is defined as 38 t s t 0 + q Ht 0 ),.35) In the same as previous, then we can written a cosmic speed and the cosmic acceleration as ȧ = a 0 q t s t) q 1 t s t 0 ), q a = q t s t),.36)

18 1 and ä = a 0 qq 1) t s t) q t s t 0 ), q qq 1) = a t s t)..37) The Hubble parameter and its time derivative in this case are q H = t s t,.38) q Ḣ = t s t)..39) At present, q = H 0 t 0 t s ). The dust matter density in the phantom power-law is ) 3 a0 ρ m = ρ m,0, a ) 3 a 0 = ρ m,0, a 0 t s t)/t s t 0 ) q ) 3q t s t 0 = ρ m,0..40) t s t where ρ m,0 is the dust matter density at present time t 0..6 Scalar field model In this section we will discuss about scalar field, φ = φx, t). Now the scalar field is a model of dark energy with the time dependence equation of state. It is assumed to be spatial homogeneous or invariance under transformation like φx ) = φx). Therefore, scalar field can be written as φ = φt), we roughly discuss about homogeneous scalar field, for example, quintessence field. Quintessence field Cosmological model of dark energy with a canonical scalar field φ is called quintessence. The action for quintessence is given by 3, 39 S q = d 4 x 1 g gab a φ b φ V φ),.41)

19 13 where φ is the quintessence field with potential V φ), and g is determinant of g ab. Variation of the action.41) with respect to φ gives φ + 3H φ + d V φ) = 0,.4) dφ where dv φ)/dφ stands for the derivative of potential with respect to scalar field and over dot represents a derivative with respect to time. The energy momentum tensor of scalar field can be derive by varying the action.41) respect to the metric g ab, then the energy momentum tensor is taken form 1 T ab = a φ b φ g ab gcd c φ d φ + V φ)..43) Energy density and pressure for the scalar field are ρ φ = 1 φ + V φ),.44) P φ = 1 φ V φ)..45) The Friedmann equation.) and the acceleration equation.0) H = 8πG 1 3 φ + V φ),.46) ä a = 8πG φ V φ)..47) 3 Therefore we will see that accelerating expansion of the universe takes place when φ < V φ), that is ä > 0. The equation of state for the field φ is given by w φ P φ ρ φ = φ V φ) φ + V φ)..48) For the case of φ V φ) and φ 0, Eq..4) and.46) can be approximated as 3H φ d V φ),.49) dφ

20 14 and 3H 8πGV φ)..50) Hence the EoS parameter Eq..48) gives the approximation w φ 1 + ɛ,.51) 3 where ɛ dv/dφ)/v /16πG) is known as slow-roll parameter. Let us consider the fluid equation of the field is taken the form ρ φ + 3Hρ φ 1 + w φ ) = 0,.5) which can be written in an integrated form as ȧ ) ρ = ρ 0 exp 31 + w φ ) dt,.53) a or ρ = ρ 0 exp ) da 31 + w φ )..54) a where ρ 0 is an integration constant. Consider the Friedmann and the acceleration equation, Eqs..46). We can express the scalar potential V φ) and the field φ it follows V = 3H 1 + Ḣ,.55) 8πG 3H 1 φ = dt Ḣ..56) 8πG.7 Slow-roll approximation In cosmological inflation, the simplest way for inflation to happen is to approximate that φ V φ), so that the Friedmann equation is dominated by the potential term, i.e. H 8πG V φ)..57) 3

21 15 If we also assume that the second order time derivative of the field value can be negligible, i.e. φ 3H φ,.58) Then we have the slow-roll approximation. The Klein-Gordon equation under slow-roll approximation becomes φ 1 d V φ)..59) 3H dφ The 3H φ term is the friction term given by the expansion rate of the universe. This means that even when the potential is steep inflation could happen, satisfying the condition φ < V φ), due to the friction created from the rapid expansion. Another case is that inflation happens when the field moves slowly due to the almost flat d or slowly varying potential, i.e. V φ) is very small. dφ The approximation is governed by the slow-roll parameters. When the scalar field potential term dominates the Friedmann equation, the slow-roll parameters are defined as Ḣ H 1 1 V 16πG 1 d V 3H dφ 1 8πGV These parameters, in the slow-roll approximation, satisfy ) dv dφ εφ),.60) d V dφ ηφ)..61) εφ) 1,.6) ηφ) 1..63) These conditions are called the slow-roll conditions. The statements in the slowroll conditions are equivalent to slowly moving field and slowly-varying potential. Inflation comes to an end when the condition φ < V φ) is violated and hence φ becomes greater than V φ).

22 CHAPTER III NON-MINIMAL DERIVATIVE COUPLING We have known that our universe is under the accelerating expansion phase due to the dark energy 1,, 3, 4, 5, 6. In present, there are many scalar field models proposed to describe accelerating expansion of the universe, for example, quintessence 7, 8, 9, 10, 11, 1 and k-essence models 13, 14. We have seen that there are many models represent various modifications of scalar-tensor theories 4, 5. One of the scalar-tensor theories is the non-minimal couplings NMC) between the curvature and the scalar fields 17,. The NMC is motivated by scalar-tensor theories of the Jordan-Brans-Dicke models 4, 5. The non-minimal derivative coupling NMDC) is the extended model of the NMC. The coupling function fφ) is not only the function of the scalar field but it is also the function of the derivative of φ as well; f = fφ, a φ). The simplest form of the NMDC is the coupling between Ricci scalar and the derivative of scalar eld i.e. R a φ a φ. The NMDC term can phenomenologically be dominant either at late time for the quadratic or Higgs potential) or at inflation for runaway potential). Hence the model can be either considered as dark energy model or inflationary model of which the scalar field derivative is both non-minimally selfcoupling and coupling to Ricci scalar. 3.1 Literature review of the NMDC models In this section, we give a brief review of the recent NMDC gravity models Amendola s Model In , Amendola studied the scalar-tensor theory with the Lagrangian linear in the Ricci scalar R, quadratic in φ, and containing terms as

23 17 follows R a φ a φ, R ab a φ b φ, R φ φ, R ab φ a b φ, a R a φ, Rφ. Amendola showed that the non-minimal derivative couplings are an interesting source of new cosmological dynamics. He investigated a model with the only derivative coupling term R ab a φ b φ and presented some analytical inflationary solutions Capozziello, Lambiase and Schmidt s Result In , Capozziello, Lambiase and Schmidt studied theories of gravity where non-minimal derivative couplings of the form R a φ a φ and R ab a φ b φ are presented in the Lagrangian. They showed that the de Sitter space-time is an attractor solution. When considering only R a φ a φ with free Ricci scalar, free kinetic term, potential and matter terms, the equation of state parameter close to - 1. Assuming slow-roll condition and power law expansion, the scalar field potential is found Granda s Two Coupling constant Model In , Granda studied on scalar field with kinetic term coupled to itself and the curvature. The model consists of the Einstein Hilbert term, a kinetic term of scalar field, a potential term and two couplings term, ξ, η, in form of 1/)ξRφ a φ a φ and 1/)ηR ab φ a φ b φ. This model gives late time accelerated expansion even with no potential. In the case of scalar field dominated, the scalar field and potential are found and an expression are given by a power-law expansion. This implies that the NMDC gives an important role in the explanation of the dark energy or cosmological constant problem Granda s One Coupling constant Model In 010 0, Granda studied on scalar field with kinetic term coupled to itself and the curvature. From the model we can find solution for Friedmann

24 18 equations with the particular restriction on the scalar field φ = constant = φ 0. This model gives late time accelerated expansion even without potential and at limit t the system reproduces an effective cosmological constant, w 1. At last the scalar field potential is found by power-law expansion. 3. Background Equations In this work, we consider one coupling of Granda s model. We assumed the universe is flatted and filled with a perfect fluid and scalar field φ with the nonminimal derivative coupling NMDC) to the scalar curvature R. We considered the action 0 and transformed all the field value to its logarithm, φ φ = ln φ. The mass dimension of the field prior to the transformation is with the constant. Let start with the action with re-scaling field S = d 4 x 1 g 16πG R 1 ) φ a φ a φ 1 ) ξr φ a φ a φ V φ ) + S m, 3.1) where S m is the dark matter action which describes a fluid with barotropic equation of state, g = detg ab ), G is the gravitational constant, R is the scalar curvature, ξ is the coupling parameter, and V φ ) is the potential of scalar field. In the cosmological context, by using the Friedmann-Robertson-Walker metric ds = dt + a t)dx 3.) where dx is the metric in three space, at) is the scale factor. From the 00) and ii) components of the Einstein equations, the Friedmann equations and the acceleration equation respectively, are given by 3H = 8πGρ φ + ρ m ), 3.3) Ḣ + 3H = 8πGP φ + P m ). 3.4) where P m, P φ are the pressure of the matter and scalar field, ρ m and ρ φ are respectively the energy density of matter and scalar field. We consider dust matter here,

25 19 hence P m = 0. Let us derive the Einstein equations by varying for each term of the action 3.1), φ φ = ln φ, consider on the first term of the brace we obtain 1 16πG δ 1 gr) = δ g)r + gδr), 16πG 1 = 1 ggab δg 16πG ab )R = = + gr ab + g ab a a a b )δg, ab 1 g R ab 1 ) 16πG g abr δg ab + gg ab δg ab a a gδg ab a b, 1 g R ab 1 ) 16πG g abr δg ab. 3.5) The second term is δ 1 g a ln φ) a ln φ) = δ 1 g δ ab g)g φ aφ) φ a φ) = 1 φ aφ b φ) + gδg ab ) φ aφ b φ), = 1 1 ggcd δg cd ab )g + gδg ab ) φ aφ b φ), = 1 1 g g ab abg φ aφ b φ) φ aφ b φ) δg ab, φ aφ b φ) = 1 1 g φ g ab a φ a φ) φ aφ b φ) δg ab, = 1 1 g g ab φ aφ a φ) φ aφ b φ) δg ab. 3.6)

26 0 The third term is δ 1 ξr g φ gab a φ b φ = 1 ξδr) g φ gab a φ b φ) +ξrδ g) φ gab a φ b φ) +ξr g φ δgab ) a φ b φ), = 1 ξ R ab δg ab g φ gab a φ b φ) g ab a a δg ab g φ gab a φ b φ) + a b δg ab g φ gab a φ b φ) 1 R ab gg ab δg φ gab a φ b φ) +R g φ δgab a φ b φ), = 1 ξ R ab δg ab g φ gab a φ b φ) gg ab g ab δg ab a a φ aφ b φ) + gg ab δg ab a b φ aφ b φ) 1 R ab gg ab δg φ gab a φ b φ) +R ab gδg φ aφ b φ), = 1 ξ g R ab 1 g abr) φ aφ a φ) g ab a a φ aφ a φ) + R φ aφ b φ) + a b φ aφ a φ) δg ab. 3.7) and the last term we get δ gv φ )) = δ g)v φ ) + gδv φ )), = 1 ggab δg ab )V φ ), = 1 ggab δg ab V φ ). 3.8)

27 1 Now, by putting together for each term into the action δs = δ d 4 x 1 g 16πG R 1 ) φ a φ a φ 1 ξr φ 0 = 1 d 4 x 1 g R ab 1 ) 8πG g abr + 1 g ab ξ R ab 1 g abr) φ aφ a φ) g ab a a +R φ aφ b φ) + a b φ aφ a φ) ) a φ a φ V φ ) + δs m, φ aφ a φ) φ aφ b φ) φ aφ a φ) ) + g ab V φ ) δg ab. 3.9) Therefore, by the action principle, it can be written as 1 0 = R ab 1 ) 8πG g abr + 1 g ab φ aφ a φ) φ aφ b φ ξ R ab 1 g abr) φ aφ a φ) g ab a a φ aφ a φ) ) +R φ aφ b φ) + a b φ aφ a φ) + g ab V φ ), We can change index and rearrange as 1 R ab 1 ) 8πG g abr = φ aφ b φ 1 φ g ab d φ d φ g ab V φ ) +ξ R ab 1 ) g abr φ dφ d φ + ξr φ aφ b φ From Einstein s field equation, we get 1 R ab 1 ) 8πG g abr g ab ξ d d φ cφ c φ) + ξ a b φ dφ d φ). 3.10) = T ab. 3.11) Finally, after varying the action 3.1) with respect to the metric and from Eq. 3.10), the effective energy-momentum tensor for the scalar field is given by T ab = φ aφ b φ 1 ) g ab φ dφ d φ g ab V φ ) +ξ R ab 1 ) ) ) g abr φ dφ d φ + R φ aφ b φ ) ) g ab d d φ cφ c φ + a b φ dφ d φ. 3.1)

28 Assuming the spatially-flat Friedmann Robertson Walker FRW) metric, we can write the components of tensor T ab, as follows, See detail in appendix C) ρ φ = T 00 = 1 φ φ + V φ ) + 3ξ Ḣ + 3H ) φ φ + H φ φ φ H and P φ = T 11 a = 1 φ φ V φ ) + ξ Ḣ + 3H ) φ φ 4H φ 5 we get + φ φ + φ φ... φ 3 φ 3 φ 3 φ 3 φ φ φ + 4H φ 4 φ 4 φ φ φ ) ). 3.14) Let us consider the action 3.1) varying this action with respect to φ, and L φ φ = V φ ) φ. 3.15) and L φ a φ ) Lφ a a φ ) = 1 a φ ) bφ b φ 1 ξr bφ b φ, = 1 ) g bc a φ b φ c φ + ξrg bc b φ c φ, ) = 1 gbc c φ ) bφ ) a φ ) + bφ ) cφ ) a φ) + cφ ) ξr bφ ) a φ ) +ξr b φ ) cφ ), a φ ) = 1 gbc c φ )δa b + b φ )δa c + c φ )ξrδa b + ξr b φ )δa c, = 1 g ac c φ ) + g ab b φ ) + g ac ξr c φ ) + g ab ξr b φ ), = 1 g ac c φ ) + g ac ξr c φ ) = g ac c φ ) + g ac ξr c φ ), = a g ac c φ ) + g ac ξr c φ ), = g ac a c φ ) + g ac ξ a R) c φ ), +g ac ξr a c φ ) = a a φ ) + ξ a R) a φ ) + ξr a a φ ). 3.16)

29 3 Let us consider a a φ ) = a a φ ), = a a φ ) + Γ a ca c φ ), where the field is function of time only, and = 0 0 φ ) + Γ Γ 0 + Γ 3 03) 0 φ ), ȧ = 0 φ ) + a + ȧ a + ȧ ) a φ ), = φ + 3H φ ). 3.17) where R = g ab R ab = 6Ḣ + 1H, a R = 0 6Ḣ + 1H ), = 6Ḧ + 4HḢ. 3.18) We put Eq. 3.17) and Eq. 3.18) into Eq. 3.16) and then we will obtain Lφ a = a φ ) φ + 3H φ + ξ6ḧ + 4HḢ) φ + ξ6ḣ + 1H ) φ +3ξH6Ḣ + 1H ) φ, = φ + 3H φ + 6ξḦ φ + 4ξḢH φ + 6ξḢ φ + 1ξH φ +18ξḢH φ + 36ξH 3 φ, = φ + 3H φ + 6ξḦ φ + 6ξH7Ḣ + 6H ) φ +6ξḢ + H ) φ. 3.19) From the Euler-Lagrange equations Lφ a a φ ) L φ φ = 0 3.0) we replace Eq. 3.15) and Eq. 3.19) into Eq. 3.0). Therefore we can write the equation of motion EoM) of the system for the field φ as φ + 3H φ + 6ξḦ φ + 6ξH7Ḣ + 6H ) φ +6ξḢ + H ) φ + d dφ V φ ) = )

30 4 By transforming, φ = ln φ, we obtain φ = d ln φ) dt = φ φ, and φ = d dt φ φ ) φ = φ φ φ ). Finally, we substitute φ and φ into Eq. 3.1) and then we can write the equation of motion for the field φ as φ + 3H φ φ φ φ φ ξ6ḣ + 1H ) + 6ξḦ φ + 6ξH7Ḣ + 6H ) φ ) +ξ6ḣ + 1H ) φ φ d + V φ) = 0. 3.) dφ where dv φ )/dφ φ/)dv φ)/dφ, is derivative of potential with respect to scalar field φ, and over dot represents a derivative with respect to time t. 3.3 Non-Minimal Derivative Coupling model This section we will study about the field φ, the potential of the scalar field V φ), and the equation of state E.o.S) w φ, respectively Shape of the field φ Let us look for the shape of the field, φ, which compatible with late time or early time solution. Under slow-roll assumption of 0 < φ 1 and... φ φ φ, we neglect the terms with φ 3, φ..., φ φ, φ φ and φ4 hence the pressure and density now become ρ φ 1 φ φ + V φ ) + 3ξ Ḣ + 3H ) φ φ + H φ φ, 3.3) φ and P φ 1 φ φ V φ ) + ξ Ḣ + 3H ) φ φ 4H φ φ. 3.4) φ

31 5 Therefore, we start from Eq. 3.4) with the pressure and energy density Ḣ 8πG φ φ + 4ξḢ + 3H ) φ φ + ξh φ φ φ + ρ m, 3.5) by using linear-equation approximation of the NMDC field equation ) φ 3H φ φ d V φ). 3.6) dφ we substitute the equation of motion, Eq. 3.6), into Eq. 3.5), therefore Ḣ 8πG ) φ φ + 4ξḢ + 3H ) φ φ ξh 3H φ φ φ + φ d dφ V φ) + ρ m, 8πG φ φ + 8ξḢ φ φ + 6ξH φ φ ξh φ d dφ V φ) + ρ m, 8πG ) 1 + 6ξH + 8ξḢ φ φ ξh φ d dφ V φ) 8πG ρ m. From the Friedmann equation, Eq.3.3), we take time derivative of its HḢ = 8πG 3 3.7) ρ φ + ρ m ), 3.8) Let consider time derivative of the energy density, ρ φ, from equation 3.3) we will obtain ρ φ = d 1 dt φ φ + V φ ) + 3ξḢ + 3H ) φ φ + 6ξH φ φ φ 6ξH φ, φ 3 3 = φ φ φ φ 3 φ + φ d φ V φ) + 1ξḢ 3 dφ φ φ 1 ξḣ φ 3 φ 3 +6 ξḧ φ φ + 18 ξh φ φ φ 18 ξh φ 3 φ ξhḣ φ φ +6 ξḣ φ φ φ + 6 ξh φ φ 1 ξh φ φ φ + 6 ξh φ... φ 3 φ 18 ξh φ φ 3 φ + 6 ξḣ φ 3 φ ξh φ 4 φ 4, 3.9) By neglecting terms of order higher than the second one, the equation 3.9) reduces to ρ φ φ φ φ + φ d V φ) + 18ξḢ dφ φ φ φ + 6ξḦ φ φ +18ξH φ φ φ + 18ξHḢ φ φ. 3.30)

32 6 Similarly, we can use approximation from Eq. 3.6) into Eq. 3.30) to obtain ) ρ φ φ φ 3H φ φ d dφ V φ) + φ d dφ V φ) ) +18ξḢ φ φ 3H φ φ d dφ V φ) ) +6ξḦ φ φ + 18ξH φ φ 3H φ φ d dφ V φ) +18ξHḢ φ φ, 3H φ φ 18ξḢ)3H) φ φ 18ξḢ φ d V φ) + 6ξḦ dφ φ φ 18ξH )3H) φ φ 18ξH d φ V φ) + 18ξḢH dφ φ φ, 3 H φ φ 18ξHḢ φ φ + ξḧ φ φ 18ξH 3 φ φ +6ξHḢ φ φ 6ξḢ φ d dφ V φ) d 6ξH φ dφ V φ), 3 H φ φ 1ξHḢ φ φ + ξḧ φ φ 18ξH 3 φ φ 6ξḢ φ d dφ V φ) 6ξH φ d dφ V φ). 3.31) Therefore, substituting Eq. 3.31) into Eq. 3.8), we get HḢ = 8πG ρ φ + ρ m ), 3 ) 8πG H φ φ 1ξHḢ φ φ + ξḧ φ φ 18ξH 3 φ φ 68πG)ξḢ + H ) φ d 8πG V φ) + dφ 3 ρ m, 3.3) Finally, the equation 3.3) can be written as HḢ 8πG we can rearrange the Eq. 3.33), reads d dφ V φ) HḢ 8πG ) H 1ξHḢ + ξḧ 18ξH3 φ φ 8πGξ3Ḣ + 3H ) φ d 8πG V φ) + dφ 6 ρ m. 3.33) H 1ξHḢ + ξḧ 18ξH3 ) 8πGξ3Ḣ + 3H ) φ φ φ 8πG ρ 6 m, 3.34)

33 7 By replacing dv φ)/dφ from Eq. 3.34) into Eq. 3.7), and then Ḣ 8πG ) 1 + 8ξḢ + 6ξH φ φ 8πG ρ m HḢ 8πG H 1ξHḢ + ξḧ 18ξH3 ) φ φ 8πG 6 ρ m ) 8πGξH φ 8πGξ3Ḣ + 3H ) φ 8πG ) 1 + 8ξḢ + 6ξH φ φ H Ḣ 3Ḣ + 3H ) + 8πG H ρ m 6 3Ḣ + 3H ) + 8πG H 1ξH Ḣ + ξhḧ 18ξH4 ) φ φ 3Ḣ + 8πG 3H ) ρ m, Ḣ 8πG 1 + 8ξḢ + 6ξH ) φ φ H Ḣ 3Ḣ + 3H ) + 8πG H ρ m 3 3Ḣ + 3H ) + 8πG H 1ξH Ḣ + ξhḧ 18ξH4 ) φ φ 3Ḣ + 8πGρ m, 3H ) Multiply above equation by 3Ḣ + 3H ) ) Ḣ3Ḣ + 3H ) 8πG3Ḣ + 3H ) 1 + 8ξḢ + 6ξH φ φ +8πG H 1ξH Ḣ + ξhḧ 18ξH4 ) φ φ + 8πGH ρ m 3 8πG3Ḣ + 3H )ρ m H Ḣ,, Therefore, by using the continuity equation of matter, ρ m = 3Hρ m, then the kinetic term of scalar field in term of the Hubble parameter is φ φ 6Ḣ 8H Ḣ 8πGH ρ m 8πG3Ḣ + 3H )ρ m ). 3.35) 8πG 3Ḣ + 4H + ξ36h H Ḣ + 4Ḣ HḦ) In other word, we can write down dφ φ dt 6 Ḣ 8H Ḣ 8πGH ρ m 8πG3Ḣ + 1 3H )ρ m, 3.36) 8πG3Ḣ + 4H + ξ36h H Ḣ + 4Ḣ HḦ)) which can be integrated with respect to time for finding the shape of the field as following dln φ) 1 6 Ḣ 8H Ḣ 8πGH ρ m 8πG3Ḣ dt + 1 3H )ρ m, 8πG3Ḣ + 4H + ξ36h H Ḣ + 4Ḣ HḦ)) 3.37)

34 8 Therefore, we obtain the following expression for φ as function of time is ) 1 φt) exp 6Ḣ 8H Ḣ 8πGρ m 3Ḣ + 1/ 4H 8πG 3Ḣ + 4H + ξ 36H H Ḣ ) + 4Ḣ HḦ. This is an exact solution of the field in an integrated form for the Non-Minimal Derivative Coupling model The scalar potential V φ) Let us start from the Friedmann equation, Eq. 3.3), with Eq. 3.3) and substitute φ, Eq. 3.6), into the Friedmann equation H = 8πG ρ φ + ρ m, 3 8πG 1 3 φ φ + V φ ) + 3ξḢ + 3H ) φ φ + 6ξH φ φ φ + ρ m, 8πG 1 3 φ φ + V φ ) + 3ξḢ + 3H ) φ φ ) +6ξH φ φ 3H φ φ d dφ V φ) + ρ m, 8πG 6 φ φ + 8πG 3 V φ ) + 8πGξḢ + 3H ) φ φ 68πG)ξH φ φ 8πG)ξH φ d 8πG V φ) + dφ 3 ρ m, 8πG 6 φ φ + V φ ) + 6ξḢ + 3H ) φ φ 36ξH φ φ 1ξH φ d dφ V φ) + 8πG 3 ρ m, Finally, the Friedmann equation is H 8πG ) 1 + 1ξḢ 6 18ξH φ φ + V φ ) 1ξH φ d dφ V φ) Similarly as before, we can rearrange Eq. 3.7) as Ḣ + 8πG 1 + 6ξH + 8ξḢ d dφ V φ) 8πGξH φ ) φ + 8πG 3 ρ m. 3.39) φ + 8πGρ m. 3.40) 3.38)

35 9 Substituting Eqs 3.35) and 3.40) into Eq. 3.39) the Friedmann equation can be written as H { 8πG ) 1 + 1ξḢ 6 8H Ḣ 8πGρ m 3Ḣ + ) 4H ) 8πG3Ḣ + 4H + ξ36h H Ḣ + 4Ḣ HḦ) 1ξH φ 1 + 6ξH + 8ξḢ Ḣ + 8πG } ) ) 6Ḣ 8H Ḣ 8πGρm3Ḣ+4H ) + 8πGρ 8πG3Ḣ+4H +ξ36h 4 +54H Ḣ+4Ḣ HḦ) m 8πGξH φ +V φ ) + 8πG 3 ρ m, { 8πG ) 1 + 1ξḢ 6 18ξH 6Ḣ 8H Ḣ 8πGρ m 3Ḣ + ) 4H ) 8πG3Ḣ + 4H + ξ36h H Ḣ + 4Ḣ ) HḦ) ξḢ + 6ξH 8πG 6Ḣ 8H Ḣ 8πGρ m 3Ḣ + ) 4H ) 3Ḣ + 4H + ξ36h H Ḣ + 4Ḣ HḦ) 6ρ m 1Ḣ 8πG + V φ ) } + 8πG 3 ρ m, ) 6H 8πG 5 54ξH 36ξḢ 6Ḣ 8H Ḣ 8πGρ m 3Ḣ + ) 4H ) 8πG3Ḣ + 4H + ξ36h H Ḣ + 4Ḣ HḦ) 4ρ m 1Ḣ 8πG + V φ ). Now we can write down the potential of the scalar field in terms of the Hubble parameter, H, Ḣ, Ḧ, and ρ m as ) 6Ḣ V a, H, Ḣ, Ḧ) 8πG + 3H 8πG + 4ρ m ξH + 36ξḢ ) 6Ḣ 8H Ḣ 8πGρ m 3Ḣ + 4H 8πG 3Ḣ + 4H + ξ 36H H Ḣ + 4Ḣ HḦ ). 3.41)

36 The equation of state of scalar field w φ This subsection we study the equation of state of the model. Consider equations 3.3) and 3.30) ρ φ 1 φ φ + V φ ) + 3ξ Ḣ + 3H ) ρ φ φ φ φ + φ d V φ) + 18ξḢ dφ φ φ φ + 6ξḦ φ φ φ φ + H φ φ, 3.4) φ +18ξH φ φ φ + 18ξHḢ φ φ, 3.43) Consider the standard continuity equation of the scalar field ρ φ + 3Hρ φ 1 + w φ ) = ) Substituting Eq. 3.4) into the continuity equation 3.44) gives ) 1 ρ φ + 3H φ φ + V φ ) + 3ξ Ḣ + 3H ) φ φ + H φ φ 1 + w φ φ ) = 0, and we obtain ) 1 ρ φ 3H φ φ + V φ ) + 3ξ Ḣ + 3H ) φ φ + H φ φ 1 + w φ φ ). compare equations between 3.43) and 3.45), we obtain φ φ φ + φ d V φ) + 18ξḢ dφ φ φ φ + 6ξḦ φ φ + 18ξH φ φ φ + 18ξHḢ φ φ ) 1 3H φ φ + V φ ) + 3ξ Ḣ + 3H ) φ φ + H φ φ 1 + w φ φ ), or we can write as 1 + w φ φ 3.45) φ φ + φ d V φ) + 18ξḢ φ φ + 6ξḦ φ + 18ξH φ φ + 18ξHḢ φ dφ φ φ φ φ ), 1 3H φ φ + V φ ) + 3ξ Ḣ + 3H ) φ φ + H φ φ φ Therefore, the equation of state parameter E.o.S) of the model is w φ φ φ φ + φ d V φ) + 18ξḢ φ φ + 6ξḦ φ + 18ξH φ φ + 18ξHḢ φ dφ φ φ φ φ ) H φ φ + V φ ) + 3ξ Ḣ + 3H ) φ φ + H φ φ φ 3.46)

37 31 Substituting φ from Eq. 3.6) into Eq. 3.46), the equation of state parameter E.o.S) of the model now becomes 3H 54ξH 3 36ξḢH + 6ξḦ w φ 3 H + 7ξH3 18ξḢH φ φ φ 18ξḢ + H ) φ d dφ V φ) φ 3HV φ ) + 18ξH φ d dφ V φ) ) In addition, we can find the equation of state in the general kinematical form by using the Friedmann equation, Eq. 3.3). H 8πG 1 3 φ φ + V φ ) + 3ξḢ + 3H ) φ φ + 6ξH φ φ φ + ρ m, 3H 8πG φ φ + 8πGV φ ) + 38πG)ξḢ + 3H ) φ φ Therefore, the potential is +68πG)ξH φ φ φ + 8πGρ m, V φ ) 3H 8πG 1 φ φ 3ξḢ + 3H ) φ φ 6ξH φ φ φ ρ m. 3.48) we substitute Eq. 3.48) into the energy density, ρ φ. Therefore, the equation 3.3) now becomes ρ φ 1 φ φ + V φ ) + 3ξḢ + 3H ) φ φ + 6ξH φ φ, φ 1 φ φ + 3H 8πG 1 φ φ 3ξḢ + 3H ) φ φ 6ξH φ φ φ ρ m + 3ξḢ + 3H ) φ φ + 6ξH φ φ, φ ρ φ + ρ m 3H 8πG. 3.49) Then we put Eq. 3.49) into Eq. 3.4) Ḣ 8πG ) 3H 8πG + P φ + P m, Ḣ 8πG 3H 8πG + P φ + P m,

38 3 Hence, the pressure is P φ 3H 8πG Ḣ 8πG P m. 3.50) From the relation of the equation of state, w φ = P φ /ρ φ, and in the absence of matter, w m = 0. We obtain the equation of state as w φ P φ ρ φ, 3H Ḣ 8πG 8πG 3H ρ, 8πG m w φ H, Ḣ, ρ 3H + Ḣ m). 3.51) 3H 8πGρ m From all information above, we can write the effective equation of state parameter, w eff, with dust-matter content as or simplify to w eff w φρ φ, ρ φ + ρ m 3H w φ ρ 8πG m w φ 3H 8πG 1 8πGρ m 3H 3H + Ḣ w eff 1 8πGρ m, 3H 8πGρ m 3H 3H + Ḣ 3H 8πGρ m, 3H 8πGρ m 3H 3H + Ḣ 3H,,, 3.5) 1 Ḣ 3H. 3.53) In the case of power-law expansion, we can write the equation of state as w eff 1 p/t ) 3p /t ), 1 + 3p. 3.54) and the accelerated expansion occurs for p > 1 which contributes to w eff < 1/3.

39 Non-Minimal Derivative Coupling and Cosmology NMDC with power-law expansion In this subsection, we present the NMDC cosmology under power-law expansion, a t p. We use the information of the power-law expansion from sec..4) and the continuity equation of matter, ρ m = 3Hρ m. Since we consider on small red-shifts, the radiation sector is ignored. Therefore, the kinetic term of the model from Eq. 3.35) can be written as and then φ φ 6Ḣ 8H Ḣ 8πGρ m H 8πGρ m 3Ḣ + 3H ) 8πG 3Ḣ + 4H + ξ36h H Ḣ ), + 4Ḣ HḦ) ) 8πGρ m p ) 8πGρ t m 3p 3p ) t, 8πG 4p 3p ) + ξ 36p4 54p 3 +0p ) t t 4 3p +4p3 t 4 3p +4p 3 t 4 8πG ) 8πGρ m 4p 3p t ) 4p 3p t ) + ξ 36p4 54p 3 +0p t 4 ) p4p 3) 8πGρ m t 4p 3) 8πG 4p 3)t + pξ36p 54p + 0). 3.55), φ φ 1 p4p 3) 8πGρm t 4p 3) 8πG 4p 3)t + pξ36p 54p + 0) 1 8πG p 8πGρ m t t + pξ36p 54p+0) 4p 3) 1, ) Let α = pξ36p 54p + 0)/4p 3), In the case of scalar field dominated or the matter term can be neglected, we get dln φ) p 8πG 1 dt t + α, 3.57) Integration of above equation, Eq. 3.57), which gives the scalar field as a function of time ) φ ln φ 0 ) p t 8πG sinh 1, 3.58) α

40 34 Finally, we obtain the field as ) ) p t φt) φ 0 exp sinh ) 8πG α where φ 0 is an integration constant. Inverting of Eq. 3.58) and define χ = 8πG/ p we can obtain the time parameter as a function of φ as ) φ t α sinh χ ln φ ) Consider the scalar potential of the model, Eq. 3.41). We use the powerlaw expansion, a t p, and write the scalar potential as the function of time. Hence, the scalar potential is given by V t) 6p 8πGt + 3p 8πGt 4p 3 t 4 3pp ) 8πGt 3p 8πGt 3pp ) 8πGt + 3pp ) 8πGt + 8πG ξ p ) + 54ξ p ) t 4 t t ) t 4 36p 4 54p 3 +0p, + 3p 4p 3p t + ξ t 4 4p 3 3p 4p 8πG 3p + ξ t 4p 3 3p )5t +54ξp 36ξp) t 6 p4p 3)t +ξp 36p 54p+0) 5t +54ξp 36ξp t 36p 4 54p 3 +0p t 4 ),, 8πG t 4 p 4p 3) 5t + 54ξp 36ξp. 3.61) 8πG p4p 3)t 4 + ξp t 36p 54p + 0) The corresponding the scalar potential in term of φ, which is obtained after substituting t from Eq. 3.60) into Eq. 3.61), is given by V φ) 3pp ) + 8πGα sinh χ ln φ φ 0 ) } p 4p 3) {5α sinh χ ln φ φ 0 ) + 54ξp 36ξp }. 8πG {p4p 3)α 4 sinh 4 χ ln φ φ 0 ) + p ξα sinh χ ln φ φ 0 ) 36p 54p + 0) 3.6)

41 35 Next we want to derive the equation of state E.o.S) of the model by using the power-law expansion. The Hubble parameter is Ht) = ȧ/a = p/t with Ḣ = p/t and Ḧ = p/t3. Hence the E.o.S parameter, Eq. 3.47), is 3H 54ξH 3 36ξḢH + 6ξḦ φ ) 18ξ Ḣ φ + H φ d V φ) dφ w φ 1, 3 H + 7ξH3 18ξḢH φ φ 3HV φ ) + 18ξH φ d V φ) dφ ) ) p 18ξ 3 p 1ξ t 3 t 3 ) ) p 9ξ 3 p 6ξ + 1 p ) t 3 t 3 t φ ) p 18ξ 1ξ ) p t t 4ξ 1 ) t + 1 ) p 9ξ 6ξ ) p t t + 1 φ 4ξ ) p t + p ) 3 t φ φ + 6ξ φ + p t ) V φ ) 6ξ φ ) p p t t φ d V φ) dφ φ + 6ξ p ) 1 t t φ d V φ) dφ p t ) φ d dφ V φ) 1, φ + V φ ) 6ξ 1, ) p t φ d V φ) dφ 3.63) Substituting the scalar field kinetic term, φ /φ ), from Eq. 3.55) to above equation. For the case of potential is not equal to zero, we get w φ 18ξp 1ξp 4ξ+t t 9ξp 6ξp+t / t p4p 3) 8πG4p 3)t +pξ36p 54p+0) p4p 3) 8πG4p 3)t +pξ36p 54p+0) 18ξp 1ξp 4ξ+t )p4p 3) 8πG4p 3)t 4 +pξt 36p 54p+0) + 6ξ p 1) t 9ξp 6ξp+t /)p4p 3) + V 8πG4p 3)t 4 +pξt 36p 54p+0) φ ) 6ξ p t + 6ξ p 1) t φ d dφ V φ) + V φ ) 6ξ 1, ) p t φ d V φ) dφ φ d V φ) dφ ) 1, 3.64) φ d V φ) dφ In the case of with out the potential, V φ ) = 0, the consequence of its derivative is dv φ)/dφ = 0. Therefore, the EoS parameter reduces to 18ξp 1ξp 4ξ+t p4p 3) t 8πG4p 3)t +pξ36p 54p+0) w φ 1 +, w φ 1 9ξp 6ξp+t / t 18ξp 1ξp 4ξ + t 9ξp + 6ξp t / p4p 3) 8πG4p 3)t +pξ36p 54p+0). 3.65)

42 NMDC with de-sitter expansion Let us now consider the kinetic term of the scalar field in term of the de-sitter expansion, a exp H 0 t). The Hubble parameter is H = a 0H 0 exp H 0 t) a 0 exp H 0 t), H = H ) where H 0 is the Hubble constant at the present time, and we use ρ m = ρ m,0 a 3, then we get ρ m = ρ m,0 exp 3H 0 t) 3.67) Therefore the kinetic term of the model, Eq. 3.35), is φ φ 6Ḣ 8H Ḣ 8πGρ m 3Ḣ + 4H ) 8πG 3Ḣ + 4H + ξ 36H H Ḣ + 4Ḣ HḦ ) ), 3πGρ m H 8πG 4H + 36ξH 4 ), ρ m 1 + 9ξH ). 3.68) Substituting Eq. 3.67) into Eq. 3.68) and integrating these equation with respect to time is given by dln φ) 1 ) φ ln φ 0 3 ρm,0 exp 3H 0 t) dt 1 + 9ξH0) 1 ρm,0 exp 3H 0 t) H01 + 9ξH0) ) Finally, we obtain the field φt) φ 0 exp { 1 ρm,0 exp 3H 0 t) 3 H01 + 9ξH0) } 3.70) Inverting Eq. 3.69), we can write the time parameter as function of field φ as { t 1 ) } H ln ξH0) 3 φ 3H 0 ρ m,0 ln. 3.71) φ 0

43 37 Next we will find solutions giving rise to accelerated expansion and look for the scalar potential, Eq. 3.41). The scalar potential of the model is given by V t) 3H 0 8πG + 4ρ m,0 exp 3H 0 t) ξH0) 4H0) ρ 4H0 + 36ξH0) 4 m,0 exp 3H 0 t), 3H ξH 8πG ξH0) 4 ρ m,0 exp 3H 0 t). 3.7) To find the scalar potential in term of φ, which is obtained after substituting t from Eq. 3.71) into Eq. 3.7) V φ) 3H ξH 8πG ξH0 3H 0 8πG H ξH 0) 3H 0 8πG H ξH 0) { 4 ρ m,0 exp ln H 0 + 9ξH 4 0) ρ m,0 ) } 3 φ ln, φ 0 ) ξH 0 3 φ ξH0 ln, φ 0 ) φ ln. 3.73) φ 0 Now we will derive the equation of state of the model with function of the de-sitter expansion. Let us consider Eq. 3.47) 3H 54ξH 3 36ξḢH + 6ξḦ φ w φ 3 H + 7ξH3 18ξḢH φ φ 18ξḢ + H ) φ d dφ V φ) φ 3HV φ ) + 18ξH φ d dφ V φ) 1, By using Eq. 3.68), and H = ȧ/a = H 0, in the case of non zero potential and zero potential, respectively. we obtain 3H 0 54ξH0 3 w φ 3H 0 + 7ξH0 3 ρ m,0 exp 3H 0 t) 18ξH φ 1+9ξH0 ) 0 d V φ) dφ 1, ρ m,0 exp 3H 0 t) 3H 1+9ξH0 ) 0 V φ ) + 18ξH0 φ d V φ) dφ 3.74) and w φ 1 1 3H 0 54ξH0 3 3 H, 0 7ξH ξH ) 9ξH 0

44 NMDC with super-acceleration expansion In this subsection, let us consider a model with function of the superacceleration expansion. The Hubble in this case is H = ȧ a = q t s t, 3.76) time derivative of Hubble parameter is q Ḣ = t s t), 3.77) and Ḧ = q t s t). 3.78) 3 where q < 0 and the dust matter density, ρ m = ρ m,0 a 3 0/a 3 ) is then ) 3q ts t 0 ρ m = ρ m,0 3.79) t s t Therefore, the kinetic term of the model, Eq. 3.35), can be written as φ φ 6Ḣ 8H Ḣ 8πGρ m H 8πGρ m 3Ḣ + 3H ) 8πG 3Ḣ + 4H + ξ36h H Ḣ ), + 4Ḣ HḦ) ) ) 3q +4q 3 q t s t) 8πGρ 4 m t s t) 8πGρ m 3q 3q t s t) ) ) ), 4q 8πG 3q 36q t s t) + ξ 4 54q 3 +0q t s t) 4 3q +4q 3 4q 8πGρ 3q m 8πG t s t) 4 ) 4q 3q t s t) ) + ξ t s t) ) 36q 4 54q 3 +0q t s t) 4 ), q4q 3) 8πGρ m t s t) 4q 3) 8πG 4q 3)t s t) + qξ36q 54q + 0). 3.80) and φ φ 1 q4q 3) 8πGρm t s t) 4q 3) 8πG 4q 3)t s t) + qξ36q 54q + 0) 1 8πG q 8πGρ m t s t) t s t) + qξ36q 54q+0) 4q 3) 1, )

45 39 Let β = 36q 54q + 0)/4q 3) and q = i q, in this case we assume no the matter contribution, ρ m = 0, so that dln φ) i q 8πG 1 dt s t) ts t) + qξβ, 3.8) Integration of these equation which gives the scalar field as function of time is ) ) φ ln i q φ 0 t s t 8πG sinh 1 i 3.83) q ξβ Finally, the field is found ) ) q φt) φ 0 exp sin 1 t s t. 3.84) 8πG q ξβ Similarly, we define χ = 8πG/ q and we can write the time parameter as function of field as or ) φ t s t β sinh χ ln. 3.85) φ 0 ) φ t t s β sinh χ ln. 3.86) φ 0 As the scalar potential of the model, Eq. 3.61) we have the scalar potential as function of time V t) 3q q ) 8πG t s t) q 4q 3) 5t s t) + 54ξq 36ξq +, 3.87) 8πG q4q 3)t s t) 4 + ξq t s t) 36q 54q + 0) Substituting Eq. 3.85) into Eq. 3.87) and we have used relations, sinh ix) = i sinx) and sin 1 x) = i sinh 1 ix). Hence we obtain the scalar potential is V φ) 3q ) 5β sin χ ln φ φ 0 ) + 54q πGξβ sin χ ln φ φ 0 ) 8πGξβ 4 sin χ ln φ φ 0 ) cos χ ln φ φ 0 ) 3.88)

46 40 Similar method to the power-law expansion, we want to derive the equation of state E.o.S) from the case of super-acceleration expansion. By using Eqs. 3.76), 3.77), and 3.78) into Eq. 3.47). Therefore the E.o.S parameter for the simple case, without potential, is given by w φ 1 + w φ 1 t s t) 8πG4q 3)t s t) +qξ36q 54q+0) 18ξq 1ξq 4ξ+t s t) q4q 3) t 8πG4p 3)t s t) +qξ36q 54q+0) 9ξq 6ξq+t s t) / q4q 3) 18ξq 1ξq 4ξ + t s t) 9ξq + 6ξq t s t) /,. 3.89)

47 CHAPTER IV RESULTS AND DISCUSSIONS In the previous chapter, we have presented the derived essential solutions, e.g. the scalar field and the scalar potential, from our investigation in the NMDC model. In the present chapter we show the results and discussions. The power-law function has been widely considered in astrophysical observation, see detail 36, 40. Previously, Granda has studied simplest ξr a φ a φ model and has found scalar potential by assuming power-law expansion 0. In our study, we have considered a result of a field transformation introduced to the model. After that we have found the scalar potential when power-law, super-acceleration, and de-sitter expansion were considered. 4.1 Results As can be seen from the previous chapter, if we know the exact form of scale factor a = at) we can derive the field solution and the scalar potential. We show cosmological solutions where the ingredients are dark energy and dark matter could be viewed as the presence of the NMDC field with dust matter fluid. In this model, we assume the known expansions form are power-law, de- Sitter and phantom power-law or super- acceleration. Hence the scalar field and the scalar potential are found Power-law expansion We consider an expansion function, a t p where p > 0. To find the exact scalar field solution of this case, we need to disregard small contribution of ρ m in order to obtain exact solution, ) ) p t φt) φ 0 exp sinh 1, 4.1) 8πG α

48 4 where α = pξ 36p 54p + 0) / 4p 3) and χ = 8πG/ p. Then the scalar potential is V φ) 3pp ) + 8πGα sinh χ ln φ φ 0 ) } p 4p 3) {5α sinh χ ln φ φ 0 ) + 54ξp 36ξp }. 8πG {p4p 3)α 4 sinh 4 χ ln φ φ 0 ) + p ξα sinh χ ln φ φ 0 ) 36p 54p + 0) 4.) 4.1. de-sitter expansion Assuming the de-sitter expansion, a e H 0t and we keep the dust matter contribution in the solution here. The scalar solution is { 1 ρm,0 exp 3H 0 t) φt) φ 0 exp 3 H01 + 9ξH0) with the potential }, 4.3) V φ) 3H 0 8πG + 9 ) φ 4 H ξH0) ln. 4.4) φ Super-acceleration expansion The super-acceleration expansion is assumed, a t s t) q where q < 0 and t s is the future singularity, t s > t. The solution is ) ) q φt) φ 0 exp sin 1 t s t. 4.5) 8πG q ξβ where β = 36q 54q + 0)/4q 3) with χ = 8πG/ q. Therefore we obtain the potential 3q ) 5β sin χ ln φ φ 0 ) + 54q 36 V φ) +. 8πGξβ sin χ ln φ φ 0 ) 8πGξβ 4 sin χ ln φ φ 0 ) cos χ ln φ φ 0 ) 4.6)

49 43 4. Discussions According to this results, our results give slightly different from the result given in 0. The complicated structure of the FLRW equations in this model is not easy to analysis of their solutions and in general we cannot find an explicit of the potential. Nevertheless, when power-law, super-acceleration, and de-sitter expansion are assumed with under slow-roll assumption of 0 < φ 1 and... φ φ φ, an explicit of potential can be derived in 4.), 4.4) and 4.6), respectively. However, all cases of this studies have only considered in the case of without the matter contribution, ρ m = 0, except for a de-sitter case. Including a matter term might completely changes the evolution of the scale factor, possibly giving rise a past deceleration followed by the present acceleration. In the future work, we will use employ the skills learn from this work to approach NMDC palatini model and find scalar field exact solution and scalar field potential. The stability analysis of the model will be investigated.

50 CHAPTER V CONCLUSIONS In this thesis, we study general aspect of FLRW cosmology of the NMDC model for explanation the accelerated expansion of the universe. In our work, we are interested in and starting with the action of Granda s model 0. We propose a transformation φ = ln φ which enhance domination of the NMDC terms. We assumed a flat FLRW universe filled with scalar field and pressureless matter. Our scalar fields are in the non-minimal derivative coupling to Ricci scalar with the coupling constant ξ. After constructing the scenario, we derived field equations and equation of motion for this model under slow-roll approximation. Then we find cosmological solutions of the scalar field and scalar potential as function of a, H, Ḣ and Ḧ when considering power-law, super-acceleration, and de-sitter expansion. In conclusion, the field solution, φt) and the potential, V φ) can be found for an explicit forms by using three types of expansion functions: power-law, de- Sitter and super-acceleration expansions 41.

51 REFERENCES

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