Higgs inflation: A study of the phase diagrams in

the Einstein and the Jordan frames

Masterarbeit

zur Erlangung des Grades eines Master of Scienceder Fakultat Physik

der Universitat Bielefeld

vorgelegt von

Patrick Zok

Universitat BielefeldFakultat fur Physik

Juli 2013

Betreuer & 1. Gutachter: Dr. Dominika Konikowska2. Gutachter: Prof. Dr. Dominik Schwarz

Contents

1 Introduction 31.1 FLRW universe . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.1 Flatness and horizon problems . . . . . . . . . . . . . 51.2.2 Inflation and scalar field models . . . . . . . . . . . . 61.2.3 Slow-roll approximation . . . . . . . . . . . . . . . . . 71.2.4 Inflationary models . . . . . . . . . . . . . . . . . . . . 9

1.3 Conformal transformation . . . . . . . . . . . . . . . . . . . . 111.3.1 Conformal transformation: general case . . . . . . . . 111.3.2 Conformal transformation from the Jordan frame to

the Einstein frame . . . . . . . . . . . . . . . . . . . . 131.4 Higgs inflation . . . . . . . . . . . . . . . . . . . . . . . . . . 151.5 R2 inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2 Gravity and a scalar field: equations of motion 222.1 General case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.1.1 Tensor equation of motion . . . . . . . . . . . . . . . . 222.1.2 Scalar equation of motion . . . . . . . . . . . . . . . . 24

2.2 Equations of motion in the Einstein frame . . . . . . . . . . . 242.2.1 Equations of motion for the FLRW metric in the Ein-

stein frame . . . . . . . . . . . . . . . . . . . . . . . . 242.3 Equations of motion in the Jordan frame . . . . . . . . . . . 25

2.3.1 Equations of motion for the FLRW metric in the Jor-dan frame . . . . . . . . . . . . . . . . . . . . . . . . . 25

3 Initial conditions for the inflation 273.1 Initial conditions for the inflation in the Einstein frame . . . 273.2 Initial conditions for the inflation in the Jordan frame . . . . 283.3 Potential V (φ) = m2

2 φ2 in the Einstein frame . . . . . . . . . 29

3.4 Potential V (φ) = m2

2 φ2 in the Jordan frame . . . . . . . . . . 32

3.5 The Higgs inflation in the Einstein frame . . . . . . . . . . . 383.6 The Higgs inflation in the Jordan frame . . . . . . . . . . . . 403.7 Potential V (φ) = α

6φ6 in the Jordan frame . . . . . . . . . . . 43

4 Conclusions 46

5 Erklarung 50

2

1 Introduction

The introduction of inflation is a big success of modern cosmology. By as-suming an initial inflationary era, it is possible to solve some problems of thehot big bang cosmology, such as e.g. the horizon and the flatness problems.The Higgs inflation is a model which gives us a direct connection between theStandard Model (SM) of particle physics and the cosmic inflation, namely,that the Higgs boson acts as the inflaton.

Considering the Higgs inflation with a scalar field φ minimally coupledto gravity, the self-coupling of the Higgs field is too large to produce theright amplitude of the primordial density perturbations. However, if we

consider instead the Higgs inflation based on a non-minimal coupling ξφ2R2

in the Jordan frame, we can improve the viability of the model by a suitablechoice of the coupling constant ξ. Thanks to the new observations of thecosmic microwave background radiation from the Planck satellite [1], we canfurther constrain the possible inflationary models. Thus, the Higgs inflationwith the non-minimal coupling is fully consistent with the received data.

In this work we will study the Higgs inflation in the Einstein and theJordan frames. We will present the details of the conformal transforma-tion relating these frames and show its importance for realising the Higgs

inflation with a non-minimal coupling ξφ2R2 . As a next step, we will con-

sider a scalar-tensor theory of gravity and derive the equations of motion inthe Einstein and the Jordan frames, also for a spatially homogeneous andisotropic universe. Basing on these results, we will derive the initial con-ditions for the inflation in both the Einstein and the Jordan frames. Wewill use them for the phase diagram method to study the trajectories ofthe scalar field in both frames for the Higgs inflation, as well as some otherchaotic inflationary models with monomial potentials. We will show howthese trajectories coincide to an attractor solution (a separatrix), which canprovide an inflationary solution in both frames. We will study the featuresof the inflationary phase diagrams and compare them between the potentialsand the frames. We will show that contrary to the Einstein frame, in theJordan frame the general structure of the phase diagrams depend on theused potential. Moreover, we will study how the attractor solution dependson the value of the coupling constant ξ in the Jordan frame.

We will use natural units (~ = c = 1) throughout the work. Moreover,we will assume ξ > 0, unless otherwise specified.

1.1 FLRW universe

The cosmological principle states that the universe at sufficiently large dis-tances (> 100 Mpc) is nearly homogeneous and isotropic. The near homo-geneity of the cosmic microwave background radiation provides the mostpowerful evidence. Moreover, large-scale galaxy surveys such as the Sloan

3

Digital Sky Survey show its accuracy [2]. According to the cosmologi-cal principle, the universe should not have any preferred locations at agiven time. This requires that the spatial part of the metric used to de-scribe the geometry of the universe has a constant curvature. Therefore weadapt the most general maximally spatially symmetric Friedmann-Lemaıtre-Robertson-Walker (FLRW) metric(FLRW-metric)

ds2 = −dt2 + a2(t)

(dr2

1−Kr2+ r2

(dθ2 + sin2 θdϕ2

)), (1)

where a(t) is the scale factor describing the expansion of the universe. Itis normalized to have the present value a0 = 1. K = −1, 0, 1 is the spatialcurvature constant for an open, flat and closed universe, respectively.

The metric tensor evolves according to the Einstein equation

Rµν −1

2gµνR = 8πGTµν , (2)

where G is the gravitational constant, Rµν and R are the Ricci tensor andscalar, respectively. Tµν is the energy-momentum tensor which describes thedensity and the flux of the energy and the momentum. In general relativityit is the source of the gravitational field. For a perfect fluid the energy-momentum tensor reads

Tµν = (ρ+ P )uµuν + Pδµν , (3)

where P is the pressure, ρ is the energy density and uµ the fluid 4-velocity.In a local rest frame we obtain

ρ = T 00 and Tij = Pδij . (4)

Using the FLRW-metric (1), the Ricci scalar and the 00-component ofthe Ricci tensor read

R = 6

[a

a+

(a

a

)2

+K

a

2], (5)

R00 = −3a

a, (6)

where the dot denotes the derivative with respect to the time t. Calculatingthe 00-component of the Einstein equation (2), we obtain the Friedmannequation. For a perfect fluid it reads

H2 =

(a

a

)2

=ρ

3M2P

− K

a2(7)

4

where H is the Hubble parameter, and MP = 1√8πG

is the reduced Planck

mass. Taking the trace of the Einstein equation, we obtain the accelerationequation using Eqs. (5) and (7):

a

a= −ρ+ 3P

6M2P

. (8)

We can defineΩ =

ρ

ρc, (9)

where Ω is called the density parameter and ρc is the critical density corre-sponding to the flat universe. It is the density which we would obtain for aflat universe. From Eq. (7) it can be seen that for Ω = 1 the universe is flatand remains so. Otherwise it is curved and its curvature is evolving.

1.2 Inflation

Introducing an initial era, called inflation, we can solve some problems ofcosmology. In this part, we first point out some of the unsolved problems.Subsequently, we define the inflationary era and introduce some single-fieldinflationary models. Finally, we show how the inflation can solve the demon-strated problems by using the example of chaotic inflation.

1.2.1 Flatness and horizon problems

To understand the flatness problem, let us look at the density parameter Ωdefined in Eq. (9). Today the Ω = 1.000±0.004 [1], which means that Ω wasextremely close to 1 in the early universe. At nucleosynthesis for example weneed |Ω(tnuc)−1| ≤ 10−16 [3]. If Ω had been slightly larger at this time, theuniverse would have been closed and recollapsed a long time ago. If Ω hadbeen slightly smaller, the energy density would have been extremely smallat the present time. Consequently, we need finely tuned initial conditionsto realize our almost flat universe. Which is very unlikely. This is called theflatness problem.

The inverse of the Hubble parameter H−1 is called the horizon or Hubblelength. The horizon corresponds to distances at which the recession velocityof the universe v = c. If we go back in time, the horizon decreases. Andit is typically of the order of the particle horizon dph. The particle horizoncorresponds to the distance that light travelled since the beginning of theuniverse. This means that the present value of the particle horizon 14300Mpc defines the radius of the observable universe [4]. Regions separated bymore than twice the particle horizon are out of causal contact. Thus thequestion is how the observable universe can be so homogeneous, at leaston large scales, regarding that the observable part of the universe is muchlarger than the particle horizon at early times. This is called the horizonproblem.

5

1.2.2 Inflation and scalar field models

The inflation is defined as an era during which the rate of increase of thescale factor accelerates;

a > 0 . (10)

Using the Hubble parameter H = aa , it can be calculated that H

H2 = aaH2 −1.

The scale factor a is always positive. Thus, we obtain an equivalent condition

for the inflation − HH2 < 1. If

|H| H2 , (11)

then H will not change much in the characteristic time ∆t = H−1. Andtherefore we have an almost exponential expansion [5]:

a(t) ∝ eHt . (12)

For an almost exponential expansion we can neglect the curvature term Ka2

in the Friedmann equation (7):

H2 ∼=ρ

3M2P

. (13)

Using aa = H + H2, and the Eqs. (11) and (13), the acceleration equation

(8) can be approximated as

0 ∼=ρ+ P

2M2P

. (14)

Consequently, ρ ∼= −P is required to have an almost exponential expansion.We will show that this scenario can be realized with scalar fields. Let us

assume that the universe is dominated by scalar fields during the inflation.And that just one scalar field φ varies, which we call the inflaton. TheLagrangian of a scalar field minimally coupled to gravity is

L = −1

2∂µφ∂µφ− V (φ) , (15)

where V (φ) is the potential. In the presence of a gravitational field theaction takes the form

S =

∫d4x√−gM2P

2R− 1

2∂µφ∂µφ− V (φ)

, (16)

where g is the determinant of the metric tensor gµν .Varying the action with respect to the scalar field φ we obtain the Klein-

Gordon equationφ+ V ′(φ) = 0 , (17)

6

with = gµν∇µ∇ν , and the prime indicating the derivative with respect toφ.

To obtain a homogeneous and isotropic expanding universe we use theFLRW metric (1). For a homogeneous field φ we obtain the following scalarequation of motion:

φ+ 3Hφ+ V ′(φ) = 0 . (18)

For a homogeneous scalar field the energy density reads

ρ =1

2φ2 + V . (19)

Using Eq. (19), the Friedmann equation (7) reads

3M2PH

2 = V (φ) +1

2φ2 . (20)

Taking the derivative of Eq. (20) with respect to time and using Eq. (18), weobtain a useful relationship:

2M2P H = −φ2 . (21)

1.2.3 Slow-roll approximation

Assuming that the scalar field varies slowly, we can use the approximation

1

2φ2 V . (22)

Then the Friedmann equation(20) reads

3M2PH

2 ∼= V . (23)

Taking the derivative of this equation and using Eq. (21), we obtain

3Hφ ∼= −V ′ , (24)

what is equivalent to|φ| 3H|φ| , (25)

as we can see from Eq. (18). Taking the derivative of Eq. (24), we obtain

φ ∼= −H

Hφ− V ′′φ

3H. (26)

By using Eq. (21) and Eq. (23), we see that the slow-roll inequality of Eq. (11)is fulfilled. Hence, we obtain an almost exponentially expanding universe.For such a universe the condition ρ ∼= −P has to be fulfilled, as said inSection 1.2.2. For a homogeneous scalar field the pressure reads

P =1

2φ2 − V . (27)

7

Hence, if we use the slow-roll approximation (22) for Eqs. (19) and (27), thecondition ρ ∼= −P is fulfilled.

Now we can define the slow-roll parameters

ε =M2P

2

(V ′

V

)2

and η = M2P

V ′′

V. (28)

It follows from Eqs. (11), (23), and (26) that for the slow-roll approximation

ε(φ) 1 and |η(φ)| 1 , (29)

the so-called slow-roll conditions. We obtain a slow-roll inflationary period,as long as these conditions are fulfilled. The advantage of the slow-rollparameters is that they just depend on the used potential.

However, we can still obtain an inflationary period if the slow-roll con-ditions are not fulfilled, as the inequality (11) is just a condition for analmost-exponential inflation, not a condition for the inflation in general:

we just need − HH2 < 1 to be met. Let us therefore rewrite the slow-roll

parameters with the Hamilton-Jacobi formulation.Using H = dH

dφdφdt , Eq. (21) reads

φ = −2M2PH

′(φ) . (30)

By inserting Eq. (30) into Eq. (20), we obtain

(H ′)2 − 3

2M2P

H2 = − 1

2M4P

V , (31)

which we call the Hamilton-Jacobi equation [6]. Thus, we can use H(φ) andH ′(φ) instead of the potential V (φ). Consequently, let us define a differentversion of the slow-roll parameters:

εH = 2M2P

(H ′

H

)2

and ηH = 2M2P

H ′′

H. (32)

Taking the first and second derivative of Eq. (23), we obtain

H ′ ∼=V ′

6M2PH

and H ′′ ∼=V ′′

6M2PH− (H ′)2

H. (33)

Using Eqs. (23) and (33), the slow-roll parameters (32) can be approximatedas

εH ∼=M2P

2

(V ′

V

)2

= ε , (34)

and

ηH ∼= 2M2P

V ′′

V−M2P

2

(V ′

V

)2

= η − ε . (35)

8

Consequently, εH −→ ε and ηH −→ η− ε in the slow-roll limit. The inequal-

ity εH < 1 is equivalent to − HH2 < 1, as we can see by inserting Eq. (30) into

Eq. (11). Hence, εH < 1 is an exact condition for the inflation.The duration of inflation is given by the number of e-foldings

N = lna(tend)

a(t)=

∫ tend

tHdt = − 1

2M2P

∫ φend

φ

H

H ′dφ , (36)

where tend and φend are the time and the value of φ at the end of inflation.In the slow-roll approximation we obtain

N ∼=1

2M2P

∫ φ

φend

V

V ′dφ . (37)

After inflation ends, the inflaton oscillates around the minimum of the po-tential. The slow-roll approximation is not valid anymore. According toEq. (18), the scalar field starts to oscillate around the minimum of the po-tential, with 3Hφ corresponding to a friction term [3]. Consequently, theinflaton decays in other particles and these particles become eventually ther-malized. This era after inflation is called reheating.

1.2.4 Inflationary models

Chaotic inflation

Let us consider potentials V (φ) growing slower than exp( 6φMP

). Thiscondition is fulfilled for every potential that grows like a power law forφ ≥MP [5]:

V (φ) ∝ φn . (38)

We will assume in the following sections that n is an even integer. Suchmonomial potentials are just one of the possible classes of potentials torealize the chaotic inflation.

Taking the square Eq. (24) and using Eq. (23), we obtain

1

2φ2 ∼=

M2P

6V(V ′)2 . (39)

For a monomial potential (38), Eq. (39) reads

1

2φ2 ∼=

n2M2P

6φ2V . (40)

As shown in Section 1.2.3, 12 φ

2 V is required for an almost-exponentialinflation. This condition is fulfilled for

φ nMP√6

. (41)

9

During the chaotic inflation the scale factor reads

a(t) ∼= ai exp4π

nM2P

(φ2i − φ2(t)) , (42)

where ai and φi are the values of the scale factor and the scalar field at thebeginning of slow-roll inflation, respectively [5].

The potential is constrained by the upper bound

V (φ) ≤M4P . (43)

To be more specific, due to the Heisenberg uncertainty principle we candetermine the energy density just with an accuracy of O(M4

P ) at the Plancktime tP ≈ M−1P . During the inflation the energy density is dominated bythe potential V (φ), thus we obtain the upper bound from Eq. (43) [5]. Wecan realize an inflationary period, as long as the conditions from Eq. (41)and Eq. (43) are fulfilled for the initial value φi.

Now we will show how we can solve the problems from Section 1.2.1, ifwe assume chaotic inflationary period during the evolution of the universe.As an example, we will use the V (φ) = λ

4φ4 theory from [5]. So that such a

theory is consistent with the observations of the universe, therefore the self-coupling constant has to be λ ≈ 10−12−10−14 . Let us assume λ ≈ 10−14. Asa consequence a domain of the size of the Planck length lp = M−1P grows to

the size l ≈ 10105

during the inflation. Such a domain is significantly largerthan the observable universe. This means that even domains larger thanthe observable universe were in causal contact at the beginning of inflation,which solves the horizon problem.

To solve the flatness problem, let us consider the Friedmann equation(7). During the inflation the term K

a2becomes exponentially small, since we

assume an almost exponential evolution of the scale factor. During inflationthe universe had grown to a many times larger size than the observableuniverse. Consequently, the term K

a2is significantly smaller than the term

ρ3M2

P, and the density parameter (9) is driven close to 1 during the inflation.

Thus the universe is not distinguishable locally from a flat universe.

Hybrid inflation

In hybrid inflation models we use two interacting scalar fields instead ofone. Namely, the inflaton φ and the waterfall field σ. If in these modelsthe field φ becomes smaller than a critical value φc, the waterfield becomesunstable, a phase transition occurs, and the inflation ends. As an examplelet us consider the potential

V (φ, σ) =α

4

(σ2 − M2

α

)+

1

2m2φ2 +

1

2g2φ2σ2 , (44)

10

which was studied in [7]. If we assume m2 << H2 << M2, we can realize astage of inflation for large values of φ and σ = 0. After the inflaton becomessmaller than φc = M

g , the fields roll down to the minimum σ = ± M√α

, φ = 0

and the inflation ends. Thus contrary chaotic inflation, hybrid inflation canend although the flatness conditions Eq. (29) are still fulfilled.

Hilltop inflation

In hilltop inflationary models the inflation takes place near the maximumof the potential. And if we have a maximum for φ = 0, the potential takesthe general form [8]:

V (φ) = V0 −1

2m2φ2 + ... = V0

(1− 1

2|η0|

(φ

MP

)2

+ ...

), (45)

where the dots denote additional terms. The slow-roll inflation can be real-ized near the maximum of the potential, if |η0| << 1. The inflation ends inthe same way as in the chaotic inflation.

1.3 Conformal transformation

Scalar-tensor theories of gravity can be formulated in the Jordan frame orin the Einstein frame. If we have a non-minimal coupling of a scalar fieldto gravity, we are calling the corresponding frame the Jordan frame. If wehave a minimal coupling, we are calling the corresponding frame the Einsteinframe. If there is a non-minimal coupling of the scalar field to gravity, whichis linear in the Ricci scalar R, we can switch from the Jordan frame to theEinstein frame by making a conformal transformation.

1.3.1 Conformal transformation: general case

Let M be an n-dimensional manifold with the metric gµν . If Ω is a non-vanishing regular function, then the metric gµν arises from a conformal trans-formation:

gµν = Ω−2gµν . (46)

We say that we conformally transformed the metric gµν .If we assume that gµν is a Lorentz metric, then a timelike, null, or space-

like vector with respect to gµν has to satisfy the same property with respectto gµν . A conformal transformation changes the length of time(space)-likeintervals and the norm of time(space)-like vectors, but it leaves the causalstructure of the space-time manifolds (M, gµν) and (M, gµν) unchanged.Conversely, if (M, gµν) and (M, gµν) have identical causal structure, thengµν and gµν are related by a conformal transformation [9].

11

Let us transform conformally the quantities which we will need in thefollowing sections. The conformal transformation of the metric tensor de-terminant in a n-dimensional spacetime reads

g = Ω2ng . (47)

We will now derive the conformal transformation of the Ricci scalar R,starting from the Christoffel symbol Γρµν . It is related to the metric tensorgµν by

Γρµν =1

2gρλ(∂µgνλ + ∂νgµλ − ∂λgµν) . (48)

Then the conformal transformation of the Christoffel symbol can be obtainedas

Γρµν =1

2Ω2gρλ(∂µ(Ω2gνλ) + ∂ν(Ω2gµλ)− ∂λ(Ω2gµν)

)= Γρµν + Ω−1(δρν∇µΩ + δρµ∇νΩ− gµν∇ρΩ) ,

(49)

where we used gµν = Ω−2gµν and ∂µΩ = ∇µΩ.The Ricci tensor is related to the Christoffel symbol by

Rµν = ∂ρΓρνµ − ∂νΓρρµ + ΓρρλΓλνµ − ΓρνλΓλρµ . (50)

Thus, the conformally transformed Ricci tensor in a n-dimensional spacetimecan be calculated as

Rµν =Rµν + Ω−2(

2(n− 2)∇µΩ∇νΩ− (n− 3)gµν∇αΩ∇αΩ)

+ Ω−1(−(n− 1)∂µ∇νΩ + gρν∂µ∇ρΩ− gµν∂ρ∇ρΩ

+ (n− 1)Γρµν∇ρΩ + gανΓαρµ∇ρΩ− gµνΓραρ∇αΩ).

(51)

To remove the explicit dependence on the remaining Christoffel symbols, weuse the definitions of the covariant derivative for a contravariant vector V µ

and a covariant vector Vµ:

∇µV ν = ∂µVν + ΓνµρV

ρ ,

∇µVν = ∂µVν − ΓρµνVρ .

(52)

Hence, we obtain the following relations:

∂µ∇νΩ = ∇µ∇νΩ− Γνµα∇αΩ ,

∂µ∇νΩ = ∇µ∇νΩ + Γαµν∇αΩ .

(53)

12

Using these relations in Eq. (51), we obtain the conformal transformation ofthe Ricci tensor:

Rµν = Rµν + Ω−2(

2(n− 2)∇µΩ∇νΩ− (n− 3)gµν∇αΩ∇αΩ)

+ Ω−1(−(n− 2)∇µ∇νΩ− gµν∇α∇αΩ

),

(54)

and in 4 dimensions

Rµν =Rµν + Ω−2[4∇µΩ∇νΩ− gµν∇αΩ∇αΩ

]

− Ω−1[2∇µ∇νΩ− gµν∇α∇αΩ

].

(55)

The Ricci scalar is related to the Ricci tensor by R = gµνRµν . Thus, theconformal transformation of the Ricci scalar in 4 dimensions reads

R = Ω−2

[R− 6Ω

Ω

]. (56)

1.3.2 Conformal transformation from the Jordan frame to theEinstein frame

The starting point is the action (16) for the canonical Einstein gravity witha scalar field in the Einstein frame. If we now include the non-minimalcoupling ξφ2R

2 , we obtain the following action in the Jordan frame:

SJ =

∫d4x√−gM2P + ξφ2

2R− 1

2∂µφ∂

µφ− V (φ)

, (57)

where ξ is a dimensionless coupling constant, and J indicates that we areusing the Jordan frame. Inserting the relations from Eqs. (47) and (56) intothe action (57), we obtain

S =

∫d4x√−gM2

P + ξφ2

2

(Ω2R− 6ΩΩ

)− Ω2

2∂µφ∂

µφ− Ω4V (φ),

(58)

where the tilde denotes quantities and contractions calculated with the met-ric gµν .

We find the function Ω by requiring the coupling of the scalar field togravity to be indeed minimal in the Einstein frame, which yields

Ω2 =M2P

M2P + ξφ2

. (59)

13

Inserting Eq. (59) into Eq. (58), we obtain

S =

∫d4x√−gM2

P

2R−

(3ξ2φ2

M2P

Ω4 +1

2Ω2)∂µφ∂

µφ

− Ω4V (φ).

(60)

It is convenient to define a new scalar field χ with

∂χ

∂φ= Ω

√6ξ2φ2

M2P

Ω2 + 1 , (61)

for which the kinetic term in the action (60) will be minimal. Thus, weobtain the action in the Einstein frame as

SE =

∫d4x√−gM2P

2R− 1

2∂µχ∂

µχ− U(χ)

, (62)

where U(χ) = Ω4V (φ(χ)), and E indicates that we are using the Einsteinframe.

The equations of motion in the Einstein frame are usually easier tosolve than the corresponding equations in the Jordan frame. The energy-momentum tensor Tµν for a scalar field minimally coupled to gravity, forexample, reduces to that of a perfect fluid. Instead, Tµν for a non-minimallycoupled scalar field to gravity is considerably more complicated [10]. Theequations of motions in both frames are mathematically equivalent to eachother. However, it is not clear if the conformally transformed theory doesnot lead to a different, physically inequivalent theory. Therefore, it is notclear which frame represents the physical one. The physicists working inclassical gravitational physics either identify one of the frames as the physi-cal one, ignore this problem, or they are aware of physical inequivalence butdo not identify one of the frames as the physical one[10].

There are many reasons to believe that a non-minimal coupling of thescalar field to gravity emerges, as shown in the Refs.[10], [11] and [12]. Ininflationary scenarios such a non-minimal coupling can improve the viabilityof the used theory. Let us look, for example, at a V (φ) = λ

4φ4 theory. As

stated in Section 1.2.4, we need λ ≈ 10−12 − 10−14 for a minimally coupledscalar field to gravity. It seems unnatural that the self-coupling constantλ has to be tuned to such an excessively small value. However, we obtaina larger value of λ for a scalar field non-minimally coupled to gravity, asshown in Ref. [13]. The amplitude of the density contrast δρ

ρ is proportional

to√λ in the Einstein frame, but to

√λξ in the Jordan frame. Hence we

obtain e.g. λ ≈ 10−3 for ξ ≈ 103.

14

1.4 Higgs inflation

We will now present how the results of an inflationary model can improveby a scalar field non-minimal coupled to gravity(through the example ofHiggs inflation). In the Higgs inflation [14] we assume, that there might bea direct connection between the Higgs field and the cosmic inflation, namely,that the Higgs field itself acts as the inflaton. Let us follow the discussionpresented in [14].

The tree-level Lagrangian for the Higgs field, including the Einstein-Hilbert term, reads

S =

∫d4x√−g[M2P

2R+ |DµH|2 − λ(|H|2 − v2)2

], (63)

where Dµ is the covariant derivative with respect to the Standard Model(SM) gauge symmetry, H is the SM Higgs boson field, v = 246 GeV itsvacuum expectation value, and λ is the self-coupling constant [15]. We candetermine the value of λ at tree level by

mH =√

2λv , (64)

where mH is the mass of the Higgs boson. We obtain λ ∼= 0.13 for mH∼= 125

GeV [16].Taking the unitary gauge H = (0, ν + φ)/

√2, with φ being a real scalar

field, the action (63) reduces to

S =

∫d4x√−gM2P

2R− 1

2∂µφ∂

µφ− λ

4

(φ2 − v2

)2. (65)

However, due to the relatively large value of λ, the primordial density pertur-bations become many orders of magnitude larger than the observed densityperturbations and we can not realize an inflationary scenario within thismodel [14].

As stated in Section 1.3.2, a non-minimal coupling of the scalar field togravity can improve the viability of an inflationary model. Hence, let us

consider a model with a non-minimal coupling ξHH†R2 . Including into the

action (63) we obtain the action in the Jordan frame:

SJ =

∫d4x√−gM2 + ξHH†

2R+ |DµH|2 − λ(|H|2 − v2)2

, (66)

where M is a mass parameter. The action (66) corresponds to a fundamentaltheory. Consequently, the corresponding energy scale M needs not be thePlanck scale MP . Using the unitary gauge we obtain

SJ =

∫d4x√−gM2 + ξ(φ2 + 2φv + v2)

2R− 1

2∂µφ∂

µφ− λ

4

(φ2 − v2

)2.

(67)

15

For high energies φ v holds and we can neglect the terms φvR and v2R2 .

Then the action reads

SJ =

∫d4x√−gM2 + ξφ2

2R− 1

2∂µφ∂

µφ− λ

4

(φ2 − v2

)2. (68)

Using

Ω2 =M2P

M2 + ξφ2, (69)

and the results of the Section 1.3.2, we conformally transform the action(68) from the Jordan frame to the Einstein frame:

SE =

∫d4x√−gM2P

2R− 1

2∂µχ∂

µχ− Ω4λ

4

(φ2(χ)− v2

)2. (70)

For low energies (φ MP ) the Einstein and the Jordan frames should beundistinguishable, i.e. we need Ω2 ∼= 1 for low energies. Hence, we treat themass parameter M and the reduced Planck mass MP as approximately thesame.

If we compare action (70) with the action (65), we see that thanks to theconformal transformation the potential is modified by a factor Ω4, reading

U(χ) = Ω4λ

4

(φ2(χ)− v2

)2. (71)

For small values of the new scalar field χ (χ . 10−5MP ), we obtain χ ∼= φand Ω ∼= 1.

Thus, the new potential (71) is approximately as the form of the Higgspotential, as shown in Figure 1. For large values χ

√6MP (or φ(χ)

MP /√ξ v), Eq. (61) becomes approximately

dφ

dχ∼=

√ξφ√

6ξ + 1MP. (72)

Solving this differential equation, we get

φ ∼= C exp

( √ξχ√

6ξ + 1MP

), (73)

where C is an integration constant. We put C = MP√ξ

, so that φ(χ) from

Eq. (73) coincides approximately with the exact solution for the large fieldvalues. Inserting this result into Eq. (71), we can see that the potential(71) is exponentially flattened with respect to the Higgs potential V (φ) =λ4

(φ2 − v2

)2:

U(χ) ∼=λM4

P

4ξ2

(1 +

M2P

ξφ2

)−2∼=λM4

P

4ξ2

(1 + exp

(− 2

√ξχ√

6ξ + 1MP

))−2,

(74)

16

10-55´10-610-6ΧHMPL

3 Λ´10-13

Ξ2

6 Λ´10-13

Ξ2

9 Λ´10-13

Ξ2

UHMP4L

Figure 1: The blue line shows the potential U(χ) from Eq. (71). The dashed

line shows the Higgs potential V (χ) = λ4

(χ2 − v2

)2with λ = 0.13 and v =

246 GeV .

1 5 9ΧHMPL

Λ

8 Ξ2

Λ

4 Ξ2

UHMP4L

Figure 2: The blue line shows the exact potential U(χ) from Eq. (71). Thered line shows the approximated potential (74).

17

as shown in Figure 2.The inflation becomes possible due to the flatness of the potential [14].

Let us address it within the slow-roll approximation introduced in Section1.2.3. Using Eq. (72) we calculate the first and the second derivative of thepotential (74) with respect to the scalar field χ:

dU

dχ∼=

λM5P√

6ξ3φ(χ)2

(1 +

M2P

ξφ(χ)2

)−3 ∼= λM5P√

6ξ3φ(χ)2, (75)

d2U

dχ2∼= −

λM4P

3ξ3φ(χ)2. (76)

With these results we can express the slow-roll parameters (28) as functionsof φ(χ):

ε ∼=4M4

P

3ξ2φ(χ)4and η ∼= −

4M2P

3ξφ(χ)2. (77)

The slow-roll parameters depend on the coupling constant ξ, which can be sochosen that our model matches up with the observations of the universe. Ifwe assume that the inflation lasted until the COBE scale enters the horizon,we need 62 e-foldings [14]. We need for 62 e-foldings

ξ ∼= 49000√λ ∼= 18000 , (78)

with λ = 0.13. However, as mentioned above, the potential (74) is not avalid approximation for small field values. Hence, if we use the approximatedslow-roll parameters (77), we should obtain an incorrect duration of inflation,since the slow-roll conditions (29) will be violated for the wrong field values.

To investigate this issue, we calculate numerically the slow-roll parame-ters (28) with the use of the exact potential (71), and compare them withthe approximated slow-roll parameters (77). The results are shown in Figure3 and 4. The slow roll conditions are violated for χ ≈ 0.94MP and 0.35MP

in case of of the exact and the approximated slow-roll parameters, respec-tively. Using these field values, we can calculate the number of e-foldingsN with Eq. (37) and compare the results. The inflation lasts for ∆N ≈ 3.7e-foldings longer in case of the approximated calculation.

We calculate numerically the slow-roll parameter εH (32) from the Hamilton-Jacobi formulation. To do this, first we solved numerically the differentialequation (31) to obtain H(φ). The result for εH is shown in Figure 3. Forthis case the slow roll conditions are violated for χ ≈ 0.61MP . Finally, wecalculate the number of e-foldings with Eq. (36). Comparing the result withthe results from above with the exact or the approximated potential, the in-flation lasts for ∆N ≈ 0.8 e-foldings longer or ∆N ≈ 2.9 e-foldings shorter,respectively.

Figure 5 shows the running of the self-coupling constant λ in dependenceof the energy scale µ. The shown results were obtained by calculating the

18

1 3 5ΧHMPL

0.2

0.4

0.6

0.8

1.0

ΕH ΧL

Figure 3: The slow-roll parameter ε(χ) calculated with the exact potentialU(χ) from Eq. (71) (continuous curve), and with the approximated potentialfrom Eq. (74)(dashed). The dotted line shows the slow-roll parameter εH(χ).

1 4 7ΧHMPL

0.2

0.4

0.6

0.8

1.0

ÈΗH ΧLÈ

Figure 4: The slow-roll parameter |η(χ)| calculated with the exact potentialU(χ) from Eq. (71)(continuous curve), and calculated with the approximatedpotential from Eq. (74) (dashed).

19

Mt ! 172.9 GeV"s!MZ" ! 0.1184

Mt !174.4 GeV

Mt !171.4 GeV

"s !MZ " !0.1191"s !MZ " !0.1177

4 6 8 10 12 14 16 18

! 0.02

0.00

0.02

0.04

0.06

Log10!""GeV #

#$"%

MH$126GeV

Figure 5: Evolution of λ with the energy scale µ: 2 loop (dashed, blue) and3 loop (red, continuous) results [17].

β-function of the renormalisation group equation at the two and three looplevel [17]. The reheating temperature TR for the Higgs inflation reads

TR ∼= 6× 1013GeV , (79)

as shown in [19]. If we use this temperature as the energy scale µ, we obtainthe value λ ∼= 0.01. And as a consequence we get ξ ∼= 5000. However, if werepeat the e-folding calculations from above for ξ = 5000 and λ = 0.01, wesee that we obtain approximately the same differences in e-foldings ∆N ≈3.7, 0.8 and 2.9, respectively.

1.5 R2 inflation

The action for the R2 inflation reads

S =MP

2

∫d4x√−g(R− R2

6µ2

). (80)

It is convenient to make the conformal transformation

gµν = χgµν , χ = exp

√

23φ

MP

. (81)

20

Consequently, the R2-term is replaced by the scalar field terms. The newaction reads

S =

∫d4x√−g

[MP

2R− 1

2gµν∂µφ∂νφ−

3µ2M2P

4

(1− 1

χ(φ)

)2], (82)

where χ serves as the inflaton during the inflation [18].Let us follow the discussion presented in [19]. For large field values the

potential can be approximated as

V (χ) =A4

4

(1− exp

(− 2χ√

6MP

))2

, (83)

where A is some constant dependent on the parameter µ. The potential hasfor ξ 1 approximately the same form as the potential (74) for the Higgsinflation during the inflationary stage. If we choose for both potentials the

same normalization, the constant A4

4 has the same value as prefactorλM4

P4ξ2

of the potential (74) for the Higgs inflation. Hence, the models have samepredictions for the spectral index ns and the tensor-to-scalar ratio r [19].The spectral index ns is defined by

ns − 1 =d lnPζ(k)

d ln k, (84)

where Pζ(k) is the spectrum of the curvature perturbation. The scale-invariant spectrum corresponds to ns = 1. The tensor-to-scalar ratio isdefined by

r =Pg(k)

Pζ(k), (85)

where Pg(k) is the spectrum of the tensor perturbation.However, both models have a different behaviour during reheating. In

the Higgs inflation the inflaton field produces bosons of weak interactions.These decay eventually into other SM particles. These particles interactwith each other and eventually become thermalized with the reheating tem-perature

TR ∼= 6× 1013GeV . (86)

In the R2 inflation the scalaron first decays into the Higgs boson. The Higgsbosons rescatter into other SM particles. This leads to a different reheatingtemperature

TR ∼= 3.1× 109GeV . (87)

The needed number of e-foldings of inflation depends on TR . Hence, weobtain different numbers for both models:

NH = 57.66 , NR2 = 54.37 . (88)

21

The spectral index ns and the tensor-to-scalar ratio r can be expressed independence of N :

ns ∼=1− 8(4N + 9)

(4N + 3)2, r ∼=

192

(4N + 3)2. (89)

Thus, we obtain for the Higgs inflation

ns ∼= 0.967 , r ∼= 0.0032 , (90)

and for the R2 inflation

ns ∼= 0.965 , r ∼= 0.0036 . (91)

For the Higgs inflation ns is slightly larger and r slightly smaller. However,both models are fully consistent with the Planck data [1].

2 Gravity and a scalar field: equations of motion

2.1 General case

We will now study the following action with an arbitrary f(φ) coupling ofthe scalar field to gravity, and an arbitrary potential V (φ):

S =

∫d4x√−g

f(φ)R︸ ︷︷ ︸A

− 1

2∂µφ∂

µφ︸ ︷︷ ︸B

−V (φ)︸ ︷︷ ︸C

. (92)

Afterwards, we will insert for f(φ) the minimal coupling MP2 and the non-

minimal couplingM2

P+ξφ2

2 to obtain the actions in the Einstein (62) and theJordan (57) frames.

2.1.1 Tensor equation of motion

Varying the action (92) with respect to the inverse metric tensor gµν wecan obtain the tensor equation of motion. For a better overview, we splitthe action in the three parts A, B, and C, and we vary them separately.Varying the part A of the action we obtain:

δSA =

∫d4xδ√−gf(φ)R+

√−gf(φ)δgµνRµν︸ ︷︷ ︸

A1

+√−gf(φ)gµνδRµν︸ ︷︷ ︸

A2

,

(93)

22

where we split δSA in the two parts A1 and A2 . Using

δ√−g = −1

2

√−gδgµνgµν (94)

we obtain for the part A1:

δSA1 =

∫d4x√−gf(φ)

Rµν −

1

2gµνR

δgµν . (95)

UsingδRµν = ∇α

(δΓαµν

)−∇ν

(δΓαµα

), (96)

the part A2 reads

δSA2 =

∫d4x√−ggµν[δΓααµ (∇νf(φ))− δΓανµ (∇αf(φ))

]+∇α

(f(φ)gµνδΓαµν

)−∇ν

(f(φ)gµνδΓαµα

).

(97)

And using

δΓαµν =1

2gαρ (∇ν(δgµρ) +∇µ(δgνρ)−∇ρ(δgµν)) , (98)

Eq. (97) reads

δSA2 =

∫d4x√−g−δgµν∇µ∇νf(φ) + gµνδg

µνf(φ)

+∇α(f(φ)gµνδΓαµν)−∇ν(f(φ)gµνδΓαµα)

+1

2

[∇µ(δgµα∇αf(φ)) +∇ν(δgνα∇αf(φ))

−∇µ(δgαρgαρgµν∇αf(φ))−∇ρ

(δgµνgµνg

αρ∇αf(φ))]

.

(99)

The boundary terms do not contribute, if the variation δgµν vanishes on thesurface of the 4-volume. Hence, Eq. (99) simplifies to

δSA2 =

∫d4x√−g−∇µ∇νf(φ) + gµνf(φ)

δgµν . (100)

Varying the parts B and C of the action (92) we obtain

δSB =

∫d4x√−g1

2

(−∂µφ∂νφ+

1

2gµν∂µφ∂

µφ

)δgµν (101)

and

δSC =

∫d4x√−g(

1

2gµνV

)δgµν . (102)

23

With the results of Eqs. (95), (100), (101), and (102) we find finally:

δS =

∫d4x√−gf(φ)(Rµν −

1

2gµνR)−∇µ∇νf(φ) + gµνf(φ)

− 1

2

(∂µφ∂νφ−

1

2gµν∂µφ∂

µφ

)+

1

2gµνV

δgµν .

(103)

The action principle tells us that the variation of action has to vanish, i.e.δS = 0. Hence, the tensor equation of motion reads

Rµν −1

2gµνR =f−1(φ)

[−1

4gµν∂αφ∂

αφ+1

2∂µφ∂νφ−

1

2gµνV (φ)

+∇µ∇νf(φ)− gµνf(φ)].

(104)

2.1.2 Scalar equation of motion

Varying the action (92) with respect to the scalar field φ we obtain

δS =

∫d4x√−gRf ′(φ)δφ− V ′(φ)δφ+ (φ) δφ−∇µ (δφ∂µφ)

. (105)

If we neglect the boundary term, the scalar equation of motion reads

Rf ′(φ)− V ′(φ) +φ = 0 . (106)

2.2 Equations of motion in the Einstein frame

Substituting f(φ) = MP2 in the action (92), we have the action in the Einstein

frame (62). For such a substitution the equations of motion (104) and (106)read

Rµν −1

2gµνR =

1

M2P

[−1

2gµν∂αχ∂

αχ+ ∂µχ∂νχ− gµνU(χ)

], (107)

dU

dχ− χ = 0 . (108)

2.2.1 Equations of motion for the FLRW metric in the Einsteinframe

We now insert the FLRW-metric (1) into the Eqs. (107) and (108) in orderto obtain the equations of motion for a homogeneous and isotropic universein the Einstein frame. Using the tensor equation of motion (107), we cando the same calculations as in Section 1.1 to obtain the Friedmann and theacceleration equations. The 00-component of (107) gives us the Friedmannequation:

H2 +K

a2=

1

3M2P

[1

2χ2 + U

], (109)

24

where H =˙aa . Taking the trace of Eq. (107), we obtain

R =1

3M2P

(−1

2χ2 + 2U

). (110)

And by using Eqs. (5) and (109), we obtain the acceleration equation:

¨a

a=

1

3M2P

(−χ2 + U

). (111)

The scalar equation of motion (108) for the FLRW-metric reads

χ+ 3Hχ+ U ′ = 0 , (112)

where we used χ = −χ− 3Hχ.

2.3 Equations of motion in the Jordan frame

Substituting f(φ) =M2

P+ξφ2

2 to the action (92), we obtain the action in theJordan frame (57). From Eq. (104) we get the respective tensor equation ofmotion in the Jordan frame:

Rµν −1

2gµνR =

−2

M2P + ξφ2

[1

4gµν∂αφ∂

αφ− 1

2∂µφ∂νφ+

1

2gµνV (φ)

−∇µ∇ν(M2

P + ξφ2

2

)+ gµν

(M2P + ξφ2

2

)],

(113)

which can be simplified to

Rµν −1

2gµνR =

1

M2P + ξφ2

[(1 + 2ξ)∂µφ∂νφ−

(1

2+ 2ξ

)gµν∂αφ∂

αφ

+ 2ξφ(∇µ∇ν − gµν)φ− gµνV].

(114)

From Eq. (106) we obtain the scalar equation of motion

−ξφR+ V ′ −φ = 0 . (115)

2.3.1 Equations of motion for the FLRW metric in the Jordanframe

From the conformal transformation of the line element

ds2 = gµνdxµdxν = Ω2gµνdx

µdxν , (116)

for the FLRW metric (1)

ds2 = −dt2 + a2(t)dx2 = Ω2(−dt2 + a2(t)dx2) , (117)

25

with dx2 = dr2

1−Kr + r2dΘ2 + r2 sin2 Θdϕ2, we can see that Ω−1dt = dt and

Ω−1a(t) = a(t) [20]. Carrying out analogous calculations as in Section 2.2.1,we obtain the modified Friedmann equation, the acceleration equation andthe scalar equation of motion in the Jordan frame:

H2 +K

a2=

1

3(M2P + ξφ2

) [1

2φ2 − 6ξφHφ+ V

], (118)

a

a=

1

M2P + ξφ2

[−(

1

3+ ξ

)φ2 − ξφ

(φ+Hφ

)+

1

3V

], (119)

6ξφ

(a

a+H2 +

K

a2

)= φ+ 3Hφ+ V ′ . (120)

It can be shown that the equations of motion in the Jordan frame(Eqs. (118), (119) and (120)), just as in the Einstein frame, do not con-stitute an independent set. Let us rewrite Eqs. (118), (119), (120) in theform A = 0, B = 0, and C = 0, respectively:

3(M2P + ξφ2)

(H2 +

K

a2

)− φ2

2+ 6ξφHφ− V = 0 , (121)

3(M2P + ξφ2

) aa

+ (1 + 3ξ) φ2 + ξφ(φ+Hφ

)− 1

3V = 0 , (122)

φ+ 3Hφ+ V ′ − 6ξφ

(H2 +

a

a+K

a2

)= 0 . (123)

Taking the time derivative of Eq. (121) and dividing it by φ, we obtain:

A

φ= 6ξφ

(H2 +

K

a2+ H

)+ 6ξφH − φ− V ′

− 6H

φ

(M2P + ξφ2

)(Ka2− H

)+

6ξφφH

φ.

(124)

This equation can be rewritten in the form of

A

φ− 3Hφ = −C +

2H

φ

[3(M2

P + ξφ2)

(a

a−H2

)− 3ξHφφ+ 3ξφ2 + 3ξφφ

],

(125)

where we used H = aa −H

2. We rewrite this equation using the Eqs. (121)and (122) as

A

φ− 3Hφ = −C +

2H

φ

(B −A− 3

2φ2). (126)

26

This equation takes the form of

C =2H

φ(B −A)− A

φ, (127)

which means that, Eqs. (118), (119) and (120) constitute a dependent setof equations.

3 Initial conditions for the inflation

3.1 Initial conditions for the inflation in the Einstein frame

In this part we will show how an inflationary period arises for a wide rangeof initial conditions in the Einstein frame. As shown in Section 2.2.1, weobtain for the action (62)

S =

∫d4x√−gM2P

2R− 1

2∂µφ∂µφ− V (φ)

, (128)

the following equations of motion:

φ+ 3Hφ+ V ′(φ) = 0 , (129)

H2 +K

a2=

1

3M2P

[1

2φ2 + V

]. (130)

We introduce a variable

N = ln

(a

ai

)with

d

dt= H

d

dN, (131)

and will use the number of e-foldings N instead of the time t. At first weinsert the new variable into the Friedmann equation (130) and the scalarequation of motion (129), respectively:

H2 =V − 3M2

PKe−2N

3M2P −

12

(dφ

dN

)2 , (132)

H2

[d2φ

dN2+

(1

H

dH

dN+ 3

)dφ

dN

]+ V ′ = 0 . (133)

Using Eq. (132) and the relation

1

H

dH

dN= − 1

2M2P

(dφ

dN

)2

, (134)

27

Eq. (133) can be rewritten as

V

3M2P −

12

(dφ

dN

)2[d2φ

dN2+ 3

(1− 1

6M2P

(dφ

dN

)2)dφ

dN

]+ V ′ = 0 , (135)

where we assumed a flat universe (K = 0). Introducing now a variable

y =dφ

dN, (136)

we can reduce Eq. (135) to a first-order equation

dy

dφ=dN

dφ

dy

dN= −

(1− y2

6M2P

)(3 +

3M2PV′

yV

). (137)

3.2 Initial conditions for the inflation in the Jordan frame

We will now perform similar calculations as in Section 3.1, but for the equa-tions of motion in the Jordan frame from Section 2.3.1. We start with themodified Friedman equation (118) for a flat universe (K = 0), and rewriteit in terms of the new variable N , defined as in Eq. (131):

H2 = V

[3(M2

P + ξφ2)− 1

2

(dφ

dN

)2

+ 6ξφdφ

dN

]−1. (138)

Using the relation H = aa −H

2, we subtract Eq. (118) from Eq. (119) to get

H =1

M2P + ξφ2

[−(

1

2+ ξ

)φ2 − ξφφ+ ξφφH

], (139)

and by inserting the relations (131) we obtain

HdH

dN

[(M2

P + ξφ2) + ξφdφ

dN

]= H2

[−(

1

2+ ξ

)(dφ

dN

)2

− ξφ d2φ

dN2+ ξφ

dφ

dN

].

(140)Using Eq. (138), we get

H = HdH

dN=

V

M2P + ξφ2 + ξφ dφ

dN

−(12 + ξ

) ( dφdN

)2− ξφ d2φ

dN2 + ξφ dφdN

3(M2P + ξφ2)− 1

2

(dφdN

)2+ 6ξφ dφ

dN

.

(141)We also rewrite the scalar equation of motion (120) in terms of the variableN as

H2

[d2φ

dN2+ 3

dφ

dN− 12ξφ

]+H

dH

dN

[dφ

dN− 6ξφ

]+ V ′ = 0 , (142)

28

and use Eqs. (138) and (141) to obtain

V

d2φdN2 + 3 dφ

dN − 12ξφ

3(M2P + ξφ2)− 1

2( dφdN )2 + 6ξφ dφdN

+dφdN − 6ξφ

M2P + ξφ2 + ξφ dφ

dN

−(12 + ξ)( dφdN )2 − ξφ d2φdN2 + ξφ dφ

dN

3(M2P + ξφ2)− 1

2( dφdN )2 + 6ξφ dφdN

+ V ′ = 0 .

(143)

As in Section 3.1, we reduce this to a first-order equation by introducing anew variable y defined as in Eq. (136):

V

yy′ + 3y − 12ξφ

3(M2P + ξφ2)− 1

2y2 + 6ξφy

+

y − 6ξφ

M2P + ξφ2 + ξφy

[ −(12 + ξ)y2 − ξφyy′ + ξφy

3(M2P + ξφ2)− 1

2y2 + 6ξφy

]+ V ′ = 0 ,

(144)

which is equivalent to

dy

dφ=

1

2[M2P + ξ(1 + 6ξ)φ2]V y

V [24ξφ(M2

P + ξφ2)

− 6(M2P + (1− 6ξ)ξφ2)y − 2ξ(7 + 6ξ)φy2 + (1 + 2ξ)y3]

+ [−6(M2P + ξφ2)2 − 18ξφ(M2

P + ξφ2)y

+ (M2P + (1− 12ξ)ξφ2)y2 + ξφy3]V ′

.

(145)

3.3 Potential V (φ) = m2

2φ2 in the Einstein frame

We follow in this section the discussion presented in [21]. We will nowanalyze Eq. (137) using the phase diagram method. The phase diagramis symmetric about the origin, and the general structure of the diagramdoes not depend on the specific form of the potential. Since we want toinvestigate the chaotic inflation, we assume that the potential is symmetric,and for |φ| → ∞ the potential grows slower than an exponential.

Figure 6 shows the phase diagram for the potential V (φ) = m2

2 φ2. We

use m = 3 × 10−6MP from [22]. First of all, we see that we can separatethe trajectories into the Lorentzian (physical real-time) and the Euclideansolutions. For a positive potential V (φ), we obtain the Lorentzian solutionsif |y| ≤

√6MP . If y becomes bigger, we have the Euclidean solutions,

since the Hubble parameter H is imaginary in this case, as we can see fromEq. (132). If the potential is negative, we get the Lorentzian solutions for|y| ≥

√6MP . If y becomes smaller, we have the Euclidean solutions.

29

y(MP)

Φ(MP)

-5 5

-3

-2

-1

1

2

3

Figure 6: Phase diagram for a minimally coupled system with potentialV (φ) = m2

2 φ2 and m = 3 × 10−6MP . The blue (red) trajectories show

the Lorentzian (Euclidean) solutions.

We will now concentrate on the Lorentzian solutions for a positive po-tential V (φ), and begin with the solutions near |y| =

√6MP . Therefore we

writey = ±

√6MP (1−∆y) (146)

with 0 < ∆y 1. Using this in Eq. (137) and linearizing it, we get anequation for ∆y:

d∆y

dφ= ±∆y

( √6

MP± V ′

V

). (147)

This equation can be integrated, yielding

∆y = ∆yi

(V

Vi

)exp

[±√

6

MP(φ− φi)

], (148)

where ∆yi = ∆y(φi) and Vi = V (φi).

For the potential V = m2

2 φ2 , Eq. (147) reads

d∆y

dφ= ±

( √6

MP± 2

φ

)∆y. (149)

For large negative values of φ, the bracket in Eq. (149) is positive, and ifwe increase φ, y moves away from

√6MP . Nevertheless, for small negative

values of φ, the bracket reaches√

6MP again, as it can be seen in Figure

30

y(MP)

Φ(MP)

-0.2 -0.1 0.1 0.2

-3

-2

-1

1

2

3

Figure 7: After the inflation: Oscillations of the scalar field φ around theminimum of the potential V = m2

2 φ2 with m = 3× 10−6MP .

6. If we move away from the line y =√

6MP , we eventually reach anothercurve, which is an attractor for solutions, a separatrix. For large values ofφ (|φ|

√3MP ), the attractor is close to the line dy

dφ = 0, and |y| becomessmall (|y| MP ). Hence, we use these approximations in Eq. (137). Thenthe attractor takes the form

y ∼= −M2P

V ′

V. (150)

We rewrite Eq. (21) in terms of the new variable y from Eq. (136) as

−˙H

H2=

1

2M2P

y2 1 , (151)

which is a condition for an almost exponential inflation, as explained inSection 1.2.2. Hence, these solutions can describe an inflationary period.

Substituting Eq. (136) in Eq. (150), we obtain:

dN ∼= −1

M2P

V

V ′dφ . (152)

Integrating this equation and employing N = ln( aai ), we get

a(t) ∼= ai exp

(− 1

M2P

∫V

V ′dφ

). (153)

31

The quasi de-Sitter solution will be reached if ∆y = O(1) and |φ| √

3MP ,where ∆y is defined as in Eq. (146). Consequently, we can rewrite Eq. (148)as

φ2i ≤ Vi exp

( √6

MP|φi|

), (154)

which is a constraint on the initial conditions of inflation.After the inflation ends, the approximation from Eq. (25) is not valid

anymore. According to Eq. (18), the scalar field starts to oscillate aroundthe minimum of the potential, with 3Hφ corresponding to a friction term[3]. For the potential V (φ) = m2

2 φ2, the minimum will be reached for φ = 0.

We can see these damped oscillations in Figure 7.

3.4 Potential V (φ) = m2

2φ2 in the Jordan frame

We will now analyze Eq. (145) with the phase diagram method for the po-

tential V = m2

2 φ2 in the Jordan frame. The phase diagram for ξ = 18000

is shown in Figure 8. We can see that the Lorentzian and the Euclideansolutions are not separated by a constant line anymore. Since we are justinterested in the Lorentzian solutions, we have to find the condition for ob-taining them. The line which separates the Euclidean and the Lorentziansolutions grows with |φ|. Hence, we consider |φ| MP and |y| MP tosimplify Eq. (145), which can be then approximated as

y′ ∼= 12ξ(1+6ξ)yφ2

24ξ2φ3 − 6ξ(1− 6ξ)φ2y − 2ξ(7 + 6ξ)φy2 + (1 + 2ξ)y3

+[−6ξ2φ4 − 18ξ2φ3y + (1− 12ξ)ξφ2y2 + ξφy3

]V ′

V

.

(155)Usually we need ξ 1 for the inflationary cosmology, as shown already inSection 1.4. Hence, let us consider ξ 1 for Eq. (155). We obtain

y′ ∼= 2φ

y+ 3− y

φ+

1

6ξ

y2

φ2− V ′

V

[1

2

φ2

y+

3

2φ+ y − 1

12ξ

y2

φ2

]. (156)

For the potential V = m2

2 φ2 this differential equation reads

y′ ∼=φ

y− 3y

φ+

y2

3ξφ2. (157)

For the line, which separates the Lorentzian and the Euclidean solutions, it

holds that y is of the same order as φ. Hence, the terms of the order O( y2

ξφ2)

are negligible, and Eq. (157) can be approximated as

y′ ∼=φ

y− 3y

φ, (158)

32

y(MP)

Φ(MP)

-10 10 20

-4

-2

2

4

Figure 8: Phase diagram for V (φ) = m2

2 φ2 with m = 3 × 10−6MP and ξ =

18000 in the Jordan frame. The blue (red) trajectories show the Lorentzian(Euclidean) solutions.

and is analytically solvable, yielding

y ∼= ±√φ8 + C

2φ3, (159)

where C is an integration constant. Hence, the Lorentzian and the Euclideansolutions are separated by the line y = ±φ

2 . We obtain the Lorentziansolutions with

|y(φi)| <|φi|2

(160)

as a constraint on the initial conditions.As shown in Figure 8, we do no see an attractor solution. However,

we can still realize a short period of inflation. To show this, we use theapproximations |φ| MP , |y| MP , and ξ 1 for Eq. (145), in order tostudy the behaviour of the trajectories near y ∼= 0. We obtain

y′ ∼=φ

y. (161)

For the inflation the condition

− H

H2=

(12 + ξ

)y2 + ξφyy′ − ξφy

M2P + ξφ2 + ξφy

< 1 (162)

33

y(MP)

Φ(MP)

-15 -10 -5 5 10 15

-4

-2

2

4

Figure 9: Phase diagram for the potential V = m2

2 φ2 with m = 3× 10−6MP

and ξ = 0.01 in the Jordan frame. The blue trajectories show the Lorentzianinflationary solutions. The green trajectories show the Lorentzian non-inflationary solutions. The red trajectories show the Euclidean solutions.

has to be fulfilled, where we used Eqs. (138) and (141). Applying Eq. (161),we get (

12 + ξ

)y2 − 2ξφy

M2< 1 . (163)

The trajectories go towards φ→ −∞ and y → −∞ for growing times. Thus,if we choose y(φi) > 0, the condition (163) will be fulfilled in any case forsome period of time, and we can realize the inflation. Nevertheless, afterthe inflation, the scalar field never reaches the minimum of the potential.Instead it diverges to infinity.

Figure 9 shows the phase diagram for the potential V (φ) = m2

2 φ2 and

ξ = 0.01. We see that we obtain now an attractor solution. In contrastto the Einstein frame, the inflationary attractor solution is bounded by amaximum value of |φ|, which depends on ξ. To determine this value, weuse the slow-roll approximation (22) for the equations of motion for a flatuniverse (K = 0). The modified Friedmann equation (118) reads

H2 ∼=1

3(M2 + ξφ2)

(V (φ)− 6ξHφφ

). (164)

The respective scalar equation of motion (120) reads

3Hφ ∼= 12ξH2φ− V ′(φ) , (165)

34

where we used H2 ∼= aa additionally. This is a valid approximation, since

H = aa − H2 and H ∼= 0 during the inflation. Inserting Eq. (165) into

Eq. (164) we obtain

H2 =1

3(M2 + ξφ2)

[V (φ)− 2ξφ

(12ξH2φ− V ′(φ)

)]. (166)

Thus, for a monomial potential

V (φ) =λM4−n

P

nφn (167)

Eq. (166) reads

H2 =φ2

3(M2 + ξφ2)

[λM4−n

P

(1

n+ 2ξ

)φn−2 − 24ξ2H2

]. (168)

It follows for the quadratic potential V (φ) = m2

2 φ2 that

H2 =φ2

3(M2 + ξφ2)

[m2

(1

2+ 2ξ

)− 24ξ2H2

]. (169)

From Eq. (165) we obtain the differential equation

φ = 4ξHφ− V ′(φ)

3H, (170)

which for the potential (167) reads

φ = φ

(4ξH − λφn−2

3H

). (171)

By solving this differential equation for V (φ) = m2

2 φ2 we obtain

φ = φi exp

[(4ξH − m2

3H

)(t− ti)

]. (172)

In order for the scalar field to reach the minimum of the potential (167),its absolute value has to decrease with time. Otherwise the inflation wouldnever end. Therefore we need

ξ <m2

12H2. (173)

We can rewrite Eq. (169) as

H2 =m2(12 + 2ξ

)3(M2

P+ξφ2)φ2

+ 24ξ2, (174)

35

-8 -6 -4 -2 0ΦHMPL

0.5

1.0

1.5

2.0

2.5

yHMPL

Figure 10: Attractor solution for the potential V = m2

2 φ2 with m = 3 ×

10−6MP in the Jordan frame, and: ξ = 0 (blue curve), ξ = 0.001 (green),ξ = 0.01 (purple), ξ = 0.05 (red), ξ = 0.1 (black).

and using the inequality (173) we obtain

1

12ξ>

(12 + 2ξ

)3(M2

P+ξφ2)φ2

+ 24ξ2, (175)

which leads to the condition:

ξφ2 < M2P . (176)

Hence, we get an inflationary solution for a specific, ξ-dependent range ofthe scalar field φ only.

Figure 9 shows the phase diagram for ξ = 0.01. For this value of ξEq. (176) yields |φ| < 10MP . Figure 10 shows the attractors for differentvalues of ξ. From Eq. (176) we obtain |φ| <

√10MP , |φ| < 2

√5MP and

|φ| < 10√

10MP for ξ = 0.1, ξ = 0.05 and ξ = 0.001, respectively. So thesmaller the value of ξ, the larger the range of |φ| for an inflationary solution.

These calculations explain why we did not see an attractor for the poten-tial V (φ) = m2

2 φ2 in Figure 8. There we assumed ξ = 18000. So Eq. (176)

yields |φ| . 7 × 10−3MP . But the chaotic inflation usually ends alreadyfor φ = O(MP ), as shown 1.2.4. So, even if we obtain an attractor for|φ| . 7× 10−3MP , we do not obtain an inflationary solution.

36

Let us now consider ξ < 0. We obtain from Eq. (174) the inequality

m2(12 + 2ξ

)3(M2

P+ξφ2)

φ2+ 24ξ2

> 0 , (177)

since we are just interested in Lorentzian solutions (H2 > 0). This conditioncan be fulfilled for two different cases.

In the first case the inequalities

1

2+ 2ξ < 0 , (178)

and3(M2

P + ξφ2) + 24ξ2φ2 < 0 (179)

have to be fulfilled. We get ξ < −14 from Eq. (178). Eq. (179) can be

rewritten in the form of

M2P

φ2< −ξ(1 + 8ξ) . (180)

This condition is fulfilled for ξ > −18 . Hence, the first case can never be

realized.In the second case the inequalities

1

2+ 2ξ > 0 , (181)

and3(M2

P + ξφ2) + 24ξ2φ2 > 0 (182)

have to be fulfilled. We obtain ξ > −14 from Eq. (181). We rewrite Eq. (182)

in the form ofM2P

φ2> −ξ(1 + 8ξ) . (183)

We can distinguish the solutions in two cases again. For −18 ≥ ξ > −1

4 theinequalities (181) and (182) are always fulfilled independent of the value ofthe scalar field. For 0 > ξ > −1

8 we obtain Lorentzian solutions as long as

M2P > −ξ(1 + 8ξ)φ2 (184)

is fulfilled.To summarise, we can distinguish the inflationary solutions in the Jordan

frame for the potential V = m2

2 φ2 in 4 cases. For ξ > 0 we can realize infla-

tionary solutions for a specific range of φ. We obtain inflationary solutionsfor ξφ2 < M2

P . For 0 > ξ > −18 the range of inflationary solutions depends

on φ as well, but with the condition M2P > −ξ(1 + 8ξ)φ2. For −1

8 ≥ ξ > −14

we obtain inflationary solutions independent of φ. Finally, for ξ ≤ −14 we

can never realize the inflation, because all solutions are Euclidean.

37

3.5 The Higgs inflation in the Einstein frame

We will employ now the phase diagram method for the Higgs potentialV (φ) = λ

4

(φ2 − v2

)2in the Einstein frame by inserting it into Eq. (137).

We adapt for the constants λ = 0.13 and v = 246 GeV, as explained inSection 1.4. The result can be seen in Figure 11. The general structure ofthe phase diagram stays the same as the phase diagram for the potentialV (φ) = m2

2 φ2 in the Einstein frame, as expected. For a wide range of initial

conditions the trajectories reach an attractor solution. It should be men-tioned again that this inflationary model is inconsistent with observationsfor this potential, even though we get an attractor which provides an infla-tionary period. Though the reason is the too big primordial perturbationsgenerated by the Higgs potential [14].

This changes if we include a non-minimal coupling of the scalar fieldto gravity, and transform the action conformally from the Jordan to theEinstein frame, as discussed in Section 1.4. The phase diagrams for the po-tential U(χ) = Ω4 λ

4

(φ2(χ)− v2

)2from the action (62) are shown in Figure

12. According to the discussion leading to Eq. (78), we used ξ = 18000. Tobe more specific, from Eq. (61) we obtain the differential equation

dφ

dχ=

[(M2

M2 + ξφ(χ)2

)(6ξ2φ(χ)2

M2 + ξφ(χ)2+ 1

)]− 12

. (185)

We calculate φ(χ) by solving numerically this differential equation and useit in Eq. (137):

dy

dχ= −

(1− y2

6M2

)(3 +

3M2U ′

yU

). (186)

As we can see in Figure 13, the attractor depends now on the couplingconstant ξ. The bigger ξ, the steeper is the attractor for 0 < ξ . 100.For ξ & 100 the attractor converges to an approximately common solution.Changing the self-coupling constant λ has no effect on the phase diagram,since λ cancels in Eq. (186).

38

y(MP)

Φ(MP)

-5 5

-3

-2

-1

1

2

3

Figure 11: Phase diagram for a minimally coupled system with the potentialV (φ) = λ

4

(φ2 − v2

)2, λ = 0.13 and v = 246 GeV. The blue (red) trajectories

show the Lorentzian (Euclidean) solutions. The green line separates theEuclidean and Lorentzian solutions.

y(MP)

Χ(MP)

-6 -4 -2 2 4 6

-3

-2

-1

1

2

3

Figure 12: Phase diagram for the Higgs potential conformally transformedto the Einstein frame i.e. U(χ) = Ω4 λ

4

(φ2(χ)− v2

)2, with λ = 0.13, v =

246 GeV, and ξ = 18000. The blue (red) trajectories show the Lorentzian(Euclidean) solutions.

39

-1-4-7ΧHMPL

1

2

yHMPL

Figure 13: Attractor solution for the potential V (χ) = λ4

(χ2 − v2

)2(pur-

ple curve) and U(χ) = Ω4 λ4

(φ2(χ)− v2

)2for ξ = 0.1 (green) and ξ = 1

(red). The attractor solutions for ξ = 1000 and ξ = 18000 coincide to anapproximately common solution (blue). We used λ = 0.13 and v = 246 GeV.

3.6 The Higgs inflation in the Jordan frame

For the Higgs potential V (h) = λ4

(h2 − v2

)2in the Jordan frame based on

Eq. (145), the phase diagram is shown in Figure 14, where we used ξ = 1for a better visualization, since the attractor for ξ = 18000 is much steeper,and the general structure of the phase diagram is the same. Similarly tothe Einstein frame, we get an attractor solution for a wide range of initialconditions. In Figure 15 we compare the attractor solution for differentvalues of ξ. The bigger ξ, the steeper the attractor for 0 < ξ . 1000. Forξ & 1000 the attractor converges to an approximately common solution.

For φ MP v, the Higgs potential V = λ4 (φ2 − v2)2 can be approx-

imated as V ∼= λ4φ

4. Assuming additionally y MP and ξ 1, Eq. (145)reduces to

y′ ∼=−5y

φ− 3 +

y2

2ξφ2. (187)

For the line, which separates the Lorentzian and the Euclidean solutions, y is

of the same order as φ. Hence, the terms of the order O( y2

ξφ2) are negligible,

and Eq. (187) can be approximated as

y′ ∼=−5y

φ− 3 . (188)

40

y(MP)

Φ(MP)

-5 5

-3

-2

-1

1

2

3

Figure 14: Phase diagram for the Higgs potential V (φ) = λ4

(φ2 − v2

)2with

λ = 0.13, v = 246 GeV, and ξ = 1 in the Jordan frame. The blue (red)trajectories show the Lorentzian (Euclidean) solutions.

This differential equation is analytical solvable, yielding

y ∼= −φ

2+C

φ5. (189)

Hence, the Lorentzian and the Euclidean solutions are for the Higgs potentialV = λ

4 (φ2 − v2)2 separated by the line y = −φ2 . We obtain the Lorentzian

solutions for

|y(φi)| <|φi|2

(190)

as a constraint on the initial conditions. It is the same constraint as for thepotential V (φ) = m2

2 φ2 in the Jordan frame from Section 3.4. Inserting the

approximated solution y = −φ2 to Eq. (156), we see that the term in brackets

vanishes. Thus, this approximation is independent of the specific form ofthe potential.

As we have seen in Figures 14 and 15, for the Higgs potential V =λ4 (φ2− v2)2 we obtain an attractor solution, which we now want to analyze.

For large values of φ (φMP ) the attractor is close to the line dydφ∼= 0, and

y becomes small (y MP ), as shown in the Figures 14 and 15. Hence, let

41

-1-2ΦHMPL

0.05

0.1

yHMPL

Figure 15: Attractor solution for the Higgs potential V (φ) = λ4

(φ2 − v2

)2with λ = 0.13 and v = 246 GeV in the Jordan frame: ξ = 10 (green curve),ξ = 100 (red), ξ = 1000 (blue).

us consider φMP y, dydφ∼= 0, and additionally ξ 1. Eq. (145) can be

approximated as

y ∼=V ′(2M2

Pφ+ ξφ3)− V

(4M2

P + ξφ2)

3ξφ(2V − φV ′). (191)

For the Higgs potential V = λ4 (φ2 − v2)2, Eq. (191) reads

y = −2(ξv2φ2 +M2

P

(v2 + φ2

))3ξφ(v2 + φ2)

. (192)

In Figure 16 we can see that Eq. (192) is a good approximation for theattractor solution. To verify whether the condition for inflation (11) is ful-filled, we use the approximations φ MP y and dy

dφ∼= 0 for Eqs. (138)

and (141), respectively:

H2 ∼=V

3(M2 + ξφ2) + 6ξφy, (193)

H ∼=V

M2 + ξφ2 + ξφy

(ξφy

3(M2 + ξφ2) + 6ξφy

). (194)

Thus, we obtain

− H

H2=

−ξφyM2 + ξφ2 + ξφy

< 1 . (195)

42

-1-2ΦHMPL

0.05

0.1

yHMPL

Figure 16: Approximated attractor (dashed curve) and the exact attractor

for the Higgs potential V (φ) = λ4

(φ2 − v2

)2with λ = 0.13 and v = 246 GeV

in the Jordan frame.

This condition is definitely fulfilled for large field values (φ >> MP ). Hencethe attractor provides an inflationary solution.

3.7 Potential V (φ) = α6φ6 in the Jordan frame

We skip the phase diagram for the potential V (φ) = α6φ

6 in the Einsteinframe. The general structure of the phase diagram is the same as for thepotential V (φ) = m2

2 φ2 (Figure 6) and the Higgs potential (Figure 11 and

12) in the Einstein frame. We employ now the phase diagram method forthe potential V (φ) = α

6φ6 with α = 10−6M−2P in the Jordan frame.

In Figure 17 we see the corresponding phase diagram for ξ = 1. Weobtain a relatively steep attractor solution compared to the attractor of theHiggs potential (Figure 14 and 15). And in contrast to the Higgs potential,|y(φ)| goes towards infinity when the scalar field approaches the minimum ofthe potential. Consequently it never starts to oscillate around the minimum.Hence, if we obtain an inflationary solution, we have no graceful exit for theinflation.

In Figure 18 we compare the attractor solutions for different values ofξ. The bigger ξ, the steeper is the attractor solution. And for ξ & 10 the

43

attractors coincide to a common solution, which can be approximated as

y(φ) ∼= −φ

4. (196)

For inflationary solutions the condition − HH2 < 1 has to be fulfilled.

From Eq. (138) and (141) we obtain

− H

H2=

(12 + ξ

)y2 + ξφyy′ − ξφy

M2P + ξφ2 + ξφy

< 1 . (197)

Using Eq. (196) and y′ ∼= −14 from Eq. (196) we obtain

(1 + 12ξ)φ2

32M2 + 24ξφ2< 1 . (198)

We can rewrite this condition as

φ2

32(1− 12ξ) < M2

P . (199)

This is always fulfilled for at least ξ & 112 . Thus the attractor provides an

inflationary solution.To calculate the attractor solution for small ξ 1 we assume y(φ) φ.

Hence, for ξ 1 and φ y &MP Eq. (145) can be approximated as

y′(φ) ∼= −6ξφ

y− 18

φy− 3(1 + 18ξ) . (200)

For ξ 1 the attractor is close to the line dydφ∼= 0. Assuming additionally

y′(φ) ∼= 0 the differential equation (200) is analytically solvable:

y(φ) ∼= −2(3 + ξφ2)

(1 + 18ξ)φ. (201)

In Figure 19 we compare the approximated solution from Eq. (201) with theexact attractor solution from Eq. (145) for ξ = 0.01. The exact solution wascaculated numerically.

44

y (MPL

Φ(MP)

-25 -20 -15 -10 -5

1

2

3

4

5

6

Figure 17: Phase diagram for the potential V = α6φ

6 with α = 10−6M−2P withξ = 1 in the Jordan frame. The blue (red) trajectories show the Lorentzian(Euclidean) solutions.

-12 -10 -8 -6 -4 -2ΦHMPL

0.5

1.0

1.5

2.0

2.5

3.0

yHMPL

Figure 18: Attractor solution for the potential V = α6φ

6 with α = 10−6M−2Pin the Jordan frame: ξ = 1000 (blue curve), ξ = 1 (dotted), ξ = 0.1 (green),ξ = 0.01 (red), ξ = 0 (dashed).

45

-20 -15 -10 -5 0ΦHMPL

0.5

1.0

1.5

2.0

2.5

3.0

yHMPL

Figure 19: The approximated (dashed curve) and the exact attractors (solidcurve) for the potential V = α

6φ6 with α = 10−6M−2P for ξ = 0.01 in the

Jordan frame.

4 Conclusions

In this thesis we studied the Higgs inflation and some chaotic inflationarymodels with monomial potentials in the Jordan and the Einstein frames. Wepresented the details of the conformal transformation relating these frames,

and how we can realise the Higgs inflation with a non-minimal coupling ξφ2R2

in the Jordan frame. We derived the equations of motion in both frames,and showed that the equations of motion for a scalar-tensor theory of gravityin the Jordan frame do not constitute an independent set. Subsequently, weobtained the initial conditions for the inflation in both the Einstein and theJordan frames. We used them for the phase diagram method to study thescalar field trajectories and compare the results between the potentials andthe conformal frames. We showed that these trajectories converge to anattractor solution (a separatrix), which can provide an inflationary periodin both frames.

In contrast to the solutions in the Einstein frame, we showed that theresults depend qualitatively on the power of the used potential. Moreover,for a quadratic potential the inflationary solutions in the Jordan frame de-pend on the coupling constant ξ. The smaller ξ, the larger the range ofthe scalar field φ for an inflationary solution. For a quadratic potential westudied additionally the inflationary solutions for ξ < 0. We showed that

46

we can realize the inflation for ξ > −14 only.

For a potential of the power of 6 or higher, we always obtain an infla-tionary solution in both frames. However, in contrast to the solutions inthe Einstein frame, we have no graceful exit for the inflation in the Jordanframe when the scalar field approaches the minimum of the potential.

The inflationary solutions for the Higgs potential in the Jordan frame,or the quartic potentials in general, have unique features. In contrast to thequadratic potentials, we can realize inflationary solutions regardless of thevalue of ξ. Additionally, we have a graceful exit at the end of inflation, incontrast to the models with a potential of the power 6 or higher.

47

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5 Erklarung

Hiermit versichere ich, die vorliegende Masterarbeit selbststandig und nurunter Benutzung der im Literaturverzeichnis angegebenen Quellen erstelltzu haben.

Bielefeld, den 30.07.2013 (Patrick Zok)

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