k-essence and Hybrid Inflation Ulrike Wißmeier

k-essence and Hybrid Inflation Ulrike Wißmeier München 2006

k-essence and Hybrid Inflation Ulrike Wißmeier Diplomarbeit an der Fakultät für Physik der Ludwig Maximilians Universität München vorgelegt von Ulrike Wißmeier aus München München, den 31.01.2006

Erstgutachter: Prof. Dr. Viatcheslav Mukhanov Zweitgutachter: Prof. Dr. Dieter Lüst

Contents Abstract 1 Introduction 3 1 Basics 7 1.1 Standard Cosmology.............................. 7 1.1.1 The Universe we see.......................... 7 1.1.2 The Friedmann Equations....................... 8 1.1.3 The Thermal History of the Universe................. 10 1.2 Inflation..................................... 11 1.2.1 Motivation for Inflation......................... 11 1.2.2 Inflation in the Abstract........................ 12 1.3 The Late Time Cosmic Acceleration...................... 13 2 Inflationary Models 15 2.1 Standard Slow-roll Inflation.......................... 15 2.2 Hybrid Inflation................................. 18 2.3 k-inflation.................................... 21 3 k-essence and Hybrid Inflation 29 3.1 General Model................................. 29 3.1.1 The Action Principle.......................... 29 3.1.2 Equations of Motion.......................... 31 3.2 Toy-model.................................... 32 3.2.1 Endless Inflation............................ 33 3.2.2 Inflation with Exit........................... 36 3.3 Power-law Hybrid-k-Inflation.......................... 37 3.3.1 Exit due to the Change of Sign.................... 38 3.3.2 Exit due to a Vanishing k-inflation-term............... 43 3.4 Slow-roll Hybrid-k-Inflation.......................... 45 4 Conclusions 51 A Power-law Inflation 53

vi Contents Danksagung 57

Abstract In this thesis we introduce a new inflationary model called hybrid-k-inflation. At the beginning inflation is driven by the non-quadratic kinetic terms of the first scalar field ϕ, analogously to k-inflation. The graceful exit from the inflationary stage occurs due to the interaction with the second scalar field σ, analogously to hybrid inflation. Depending on the model, the second field σ may have a comparable to ϕ contribution to the total energy density at the end of inflation. We study two different classes of models: power-law and slow-roll hybrid-k-inflation. We show that the initial conditions for which the inflation endures sufficiently long to solve the horizon and flatness problems, lie in a broad range. In addition we consider a toy model with two kinetically coupled ghost-like scalar fields. For this model it was shown that the de-sitter universe is a late time asymptotic for a broad range of initial data. We also suggest a way to implement a graceful exit from inflation in this scenario. The analysis is supported by numerical calculations.

2 Abstract

Introduction Inflation is the accelerated expansion of the universe. The most important property of inflation is that it can generate irregularities in the universe, which may lead to the formation of structure [23], [24]. Historically inflation was introduced to solve problems, which were thrown up by the hot big bang model. Successful inflation must possess a smooth graceful exit, which should take place after a long enough stage. There is a huge zoo of inflationary models, e.g. R 2 -Inflation [28],Old Inflation [13], New Inflation [17], Chaotic Inflation [18], Hybrid Inflation [19], k-inflation [6], DBI Inflation [2], Ghost Inflation [4], B-Inflation [3], N-flation [10], but at present there exists no preferred concrete inflationary scenario based on a convincing realistic particle physics model. It is therefore important to explore novel possibilities for implementing an inflationary evolution of the early universe. For the discussions in our thesis and for building the hybrid-k-inflation-model the following three models are important: Standard slow-roll chaotic inflation [18], where inflation is driven by the potential of the scalar field ϕ. In standard inflation the exit is brought by a slow rolling of the inflaton field, which gradually becomes faster and faster. Furthermore we consider hybrid inflation [19], where inflation ends due to a very rapid rolling ( waterfall ) of a scalar field σ triggered by another scalar field ϕ. Also k-inflation [6] plays an important role for our further discussions. In this model inflation is driven by non-standard kinetic-energy terms of the scalar field ϕ (called k-essence), while the potential is zero. In k-inflation the field varies from the strong-coupling region to the weak-coupling one, and the exit occurs. Summarizing we have: a one-field-model, where inflation is driven by the potential term (standard slow-roll inflation) a two-field-model, where inflation is also driven by the potential term and at its last stages by the vacuum energy density (hybrid inflation) a one-field-model, where inflation is driven by the non-linear kinetic energy terms (k-inflation)

4 Introduction Hybrid-k-inflation is a mixture of the previous two models: it is a two-field-model, where inflation is driven by non-linear kinetic-energy terms of the k-essence scalar field ϕ, whereas the exit is brought by the second field σ interacting with ϕ. Beside the interaction terms the field σ is described by the Lagrangian with the standard kinetic term and potential V (σ). We consider two different models: power-law and slow-roll hybrid-k-inflation. In the first model, if there were no interaction with σ, the universe would have a constant equation of state w < 1 during inflation, whereas in the second model we have w 1. 3 There are two classes of power-law models and for the first class two ways of exiting inflation will be discussed. In the first case the negative contribution to the k-inflation pressure changes its sign as the field σ rolls across a critical value σ. The last stages of inflation are driven by the potential of the second field σ and the exit from inflation occurs in the same way as in the normal slow-roll-inflation. The time-duration of the k-inflationary stage and the potential-driven inflationary stage can be varied by choosing different initial conditions and parameters. This feature of the model may be important for the formation of structure, because during these two inflationary stages the cosmological perturbations have different sound speeds and therefore, different amplitudes. This exit mechanism can lead to different reheating temperatures as compared to the usual cases of k-inflation and chaotic inflation. On the other hand one can realize the graceful exit as follows: When the Hubble parameter H becomes less than the mass m of the field σ the latter begins to oscillate, crosses a critical value σ and changes the sign of the pressure of the k-essence. In this case no potential-driven inflationary stage occurs. In the second class of power-law models the graceful exit is realized in the following way: When the Hubble parameter H becomes less than the mass m of the field σ the latter begins to oscillate near vacuum and suppresses the contribution of the k-essence to Friedmann equations. Slow-roll hybrid-k-inflation is also driven by the non-standard kinetic-energy terms of the k-essence ϕ, but here we have quasi exponential inflation. After a sufficiently long stage of inflation the second field σ intervenes in a similar way as in power-law case and the k-inflation term becomes negligible in comparison to the potential V (σ). The last stages of inflation are driven by V (σ) and the exit occurs again like in normal slow-rollinflation. In this model the time-duration of the k-inflationary stage and the potentialdriven inflationary stage can be varied as well. This model allows also to have no potentialdriven stage. Furthermore we consider a toy model without potential, but with two kinetically coupled ghost-like scalar fields. We show that de-sitter inflation will take place if the Lagrangian just depends on the field derivatives. This late time asymptotic corresponds to an attractive fixed point, which is of the type of a stable star. In the case of explicit dependence on the fields the exit occurs. The analysis for all models is supported by numerical calculations.

Introduction 5 This thesis is organized as follows: The first chapter gives a short introduction into standard cosmology and a brief motivation for inflationary models in general. In the second chapter we present different models, which are important for building the hybrid-k-inflation model. I.e. normal slow-roll inflation, hybrid inflation and k-inflation will be discussed. At the beginning of the third chapter we introduce a toy model without potential, but with two kinetically coupled ghost-like scalar fields. After that we introduce the two hybrid-kinflationary models: power-law and slow-roll hybrid-k-inflation with their different ways of exiting inflation.

6 Introduction

Chapter 1 Basics This chapter is divided into three parts. First we give a summary of the notions of standard cosmology needed in later chapters. Then, following [14], [16] and [22], we discuss the idea of inflation and give the basic principles for chapter 2, in which we discuss the different inflationary models. In the third part we give a short introduction to the idea of k-essence [7]. 1.1 Standard Cosmology 1.1.1 The Universe we see One of the most fundamental features of standard cosmology is the large-scale isotropy and homogeneity of the observed universe.the metric for a space with homogeneous and isotropic spatial sections is the Friedmann-Robertson-Walker metric (FRWM), which can be written in the form 1 [ ds 2 = dt 2 a 2 dr 2 ] (t) 1 kr + 2 r2 dθ 2 + r 2 sin 2 θ dφ 2, (1.1) where (t,r,θ,φ) are the coordinates, a(t) is the cosmic scale factor and 1 for spaces of constant positive spatial curvature k = 0 for spaces of zero spatial curvature 1 for spaces of constant negative spatial curvature Another feature of the standard cosmology is the expansion of the universe which was discovered in the 1920 s. The universality of the expansion is illustrated by the red shifted galaxy spectra. The expansion rate of the universe is given by the Hubble parameter H ȧ(t) a(t), 1 we use the same sign conventions as in [15], and [14]: ds 2 = dt 2 d l 2 ; R µν 1/2Rg µν G µν = +8πGT µν and R µ ναβ = αγ µ νβ βγ µ να + Γµ σα Γσ νβ Γµ σβ Γσ να and Γµ αβ = 1 2 gµσ ( β g σα + α g σβ σ g αβ ). Greek indices run from 0 to 3, Latin indices from 1 to 3. c = h = M P l = 1

8 1. Basics which is at present [27] H 0 = 100 h km s 1 Mpc 1 with h = 0.71 +0.04 0.03. Einstein had already assumed isotropy and homogeneity to simplify the mathematical analysis. The matter distribution of the universe seems to be homogeneous when averaged over large enough scales. The best evidence for the isotropy is the uniformity of the temperature of the Cosmic Microwave Background Radiation (CMBR). The CMBR consists of a gas of photons with Planckian spectrum with a temperature of around 2.7 K. Apart from the observed dipole anisotropy, which results of the earths movement relative to the cosmic rest frame, the temperature anisotropies are found to be of order 10 5, as shown in figure 1.1. If the expansion were not isotropic, the anisotropy would lead to a temperature anisotropy in the CMBR of similar magnitude. Figure 1.1: Following subtraction of the dipole anisotropy and components of the detected emission arising from dust, hot gas and charged particles interacting with magnetic fields in the Milky Way Galaxy, the CMBR anisotropy can be seen. CMBR anisotropy - tiny fluctuations in the sky brightness at a level of order 10 5 - was first detected by the COBE DMR instrument. The CMBR is a remnant of the Big Bang, and the fluctuations are the imprint of density contrast in the early universe. This picture was taken from NASAhomepage (http://lambda.gsfc.nasa.gov). Primordial nucleosynthesis is one of the earliest tests of the standard model. Nuclear reactions that took place from t 0.01 to 100 s resulted in the production of substantial amounts of D, 3 He, 4 He, 7 Li. At the moment, there is conformance between the predicted and observed abundances for these four isotopes. 1.1.2 The Friedmann Equations The Einstein equations are given by: R µν 1 2 R g µν = 8πG T µν, (1.2) where T µν is the stress-energy tensor for matter present in the universe, such as fields, radiation and dust.

1.1 Standard Cosmology 9 To be consistent with the symmetries of the metric, the total stress-energy tensor T µν must be diagonal, and by isotropy the spatial components must be equal. The simplest realization of such a stress-energy tensor is that of a perfect fluid characterized by a timedependent energy density ε(t) and pressure p(t): T µ ν = diag(ε, p, p, p). (1.3) The energy-momentum conservation (T µ0 ; µ = 0) gives the 1st law of thermodynamics in the form d(ε a 3 ) = p d(a 3 ). We can rewrite this equation into: ε = 3H(ε + p). (1.4) If the equation of state parameter w p ε (1.5) is constant, then the energy density evolves as ε a 3(1+w). Interesting examples are: Radiation p = ε 3 ε a 4 Dust p = 0 ε a 3 Vacuum energy p = ε ε = const (1.6) The dynamical equations that describe the evolution of the scale factor a(t) follow from the Einstein equations (1.2). The 0-0 component gives the so-called Friedmann equation while the i-i component gives ȧ 2 a + k 2 a = 8πG ε, (1.7) 2 3 2ä a + ȧ2 a + k = 8πGε. (1.8) 2 a2 The three field equations (1.4), (1.7) and (1.8) (sometimes all three equations are called the Friedmann equations) are related by the Bianchi identities and only two of them are independent. These two are usually taken as (1.4) and (1.7). The difference of (1.7) and (1.8) provides an equation for the acceleration alone: ä a = 4πG (ε + 3p). (1.9) 3 As already mentioned, the expansion rate of the universe is determined by the Hubble parameter H, which is not a constant. The Friedmann equation can be rewritten as k H 2 a 2 = ε ε c 1 Ω 1, (1.10)

10 1. Basics where ε c = 3H2 8πG (1.11) is the critical density and Ω is the cosmological parameter. As H 2 a 2 0, the following correspondence between the sign of k and the sign of Ω 1 can be found: k = +1 Ω > 1 closed universe k = 0 Ω = 1 flat universe k = 1 Ω < 1 open universe 1.1.3 The Thermal History of the Universe Einstein s equations can be used to track the evolution of the universe back in time and to describe the history of the universe. Our universe is expanding today and hence it was smaller, i.e. all cosmological scales were smaller, in the past. We can see this from the Friedmann equations. Because of the different scale factor dependencies of the various energy densities (1.6), their relative contributions to the total energy density change as one goes back in time. It is convenient to use the scale factor itself or the redshift z (a 0 /a) 1 to describe the evolution of the universe. In this description of the cosmic history, we proceed from small z to large z, which means from present to the past as the redshift grows when we proceed back in time. The universe is presently dominated by dark energy (see also section 1.3) and its energy density ε de soon becomes negligible, whereas the spatial curvature remains negligible all the way back from the present. Hence, during most of our recent past, the universe has been dust dominated. At z 1100, the photons decoupled from the electrons and were able to propagate freely. We observe this radiation today as the CMBR mentioned in subsection 1.1.1. The domination of dust ended at dust-radiation equipartition at z 4000, where ε rad = ε dust. For larger redshifts, radiation was the dominant component of the universe. As the redshift z increases the universe becomes smaller and hotter. When the temperature T was about 10 10 K, the light elements (D, 3 He, 4 He, 7 Li) were formed from a soup of baryons. When we continue further back in time, z, we are approaching the singularity. In this time-frame inflation takes place. It is unknown to the present day what happened prior to inflation. Without inflation, radiation still would have been the dominant component and if we follow Einsteins equations we will meet a point where the scale factor vanishes and the energy density becomes infinite. This point is called the Big Bang. Extrapolation past this singularity, a = 0, ε, is not possible in the framework of classical general relativity.

1.2 Inflation 11 1.2 Inflation 1.2.1 Motivation for Inflation Inflation is the accelerated expansion of the universe. In the modern view, by far the most important property of inflation, is that it can generate irregularities in the universe, which lead to the formation of structure [23], [24]. The historical motivation for inflation was rather different, and arose largely on philosophical grounds concerning the question of whether the initial conditions required for the Hot Big Bang seem likely or not. Now we briefly want to discuss the aspects of the historical motivation, geared to [16]. Flatness problem Equation (1.10) shows that if the universe is flat (Ω = 1), then it remains flat for all time. Otherwise, the cosmological parameter Ω evolves. The flatness problem is simply that during radiation or matter domination, the combination ah is a decreasing function of time. For instance, in a nearly flat, dust-dominated universe, we have 1 Ω t 2/3 (as a t 2/3 ) and in a nearly flat, radiation-dominated universe, we have 1 Ω t (as a t 1/2 ). We know from observations, see e.g. [8], that at the present Ω 0 is not hugely different from unity, which implies that at much earlier time it must have been extremely close to 1. The flatness problem states that such finely tuned initial conditions seem extremely unlikely. Almost all initial conditions lead either to a closed universe that recollapses immediately, or to an open universe that very quickly enters the curvature-dominated regime and cools to below 3K within the first second of its existence. Horizon Problem As mentioned in section 1.1.1, the universe seems to be homogeneous on large scales. Actually, we know that the universe was already nearly homogeneous at the time of recombination, since the temperature of the CMBR is nearly the same in all sky directions. It can be shown, however, that if cosmic evolution proceeded according to the old standard cosmology (i.e. without inflation), at the time of recombination the visible universe consisted of 10 6 causally disconnected regions. Within the old standard cosmology those regions never had a chance to interact and thermalize to a common temperature, and hence, there is no explanation for this initial homogeneity of the universe. Unwanted Relicts As it is written in [16] a Hot Big Bang at a very high temperature might have resulted in relics, that are forbidden by observation, surviving to the present. Among them is for example the gravitino, the spin- 3 - partner of the graviton. Depending on the theory, there 2

12 1. Basics might be also unwanted topological defects, such as magnetic monopoles, cosmic strings, domain walls etc. Homogeneity and Isotropy Finally, although the early universe was nearly homogeneous, it actually contained small inhomogeneities from which the structures we observe today emerged by gravitational instability. In the context of the old standard cosmology, the spectrum of these inhomogeneities belongs to the initial conditions. It can be neither explained nor predicted. Therefore we have to go beyond the standard Hot Big Bang model. 1.2.2 Inflation in the Abstract The problems mentioned above are naturally lifted by numerous inflationary theories, some of the first invented in the early 1980s, e.g. [28]. We cannot consider the inflationary cosmology as a replacement for the Hot Big Bang model. It is simply an add-on, occurring at very early stages of the evolution of the universe. In simple models inflation should be over at t 10 34... 10 36 s [22]. Inflation is the accelerated expansion of the universe, i.e. gravity acts as a repulsive force. INFLATION ä > 0 (1.12) This condition for inflation can be rewritten as a requirement on the matter driving the expansion. Directly from the acceleration equation (1.9) we get: INFLATION ε + 3p < 0 (1.13) Since we always assume ε > 0 (because in flat cosmology H 2 = 8πGε/3), it is necessary for the pressure p to be negative to satisfy the condition (1.13), which is independent of the curvature of the universe. Scalar Fields Matter with the unusual property of a negative pressure is required to obtain inflation. This matter may be a scalar field, describing scalar (spin-0) particles. Although there has been no direct observation of a fundamental scalar particle, such particles proliferate in modern particle theories. Inflationary cosmology is one field where they play an important role. The scalar field responsible for inflation is usually called the inflaton. Number of e-foldings Successful inflation must also possess a smooth graceful exit. Inflation can solve the initial conditions problem only if this exit takes place after 75 Hubble times (e-foldings) [22]. This

1.3 The Late Time Cosmic Acceleration 13 number of e-foldings is given by N ln ( ) a(tend ) tend = H(t)dt. (1.14) a(t in ) t in 1.3 The Late Time Cosmic Acceleration Till now we just mentioned the inflation, which occurred right after the big bang. But at the present the universe also undergoes a stage of accelerated expansion. A concordance of cosmological observations, see e.g. [8], of large scale structure, the CMBR anisotropy and type IA supernovae at large red shift suggest, that the matter density of the universe comprises about one-third of the critical value expected for a flat universe. The missing two-thirds correspond to a dark energy component with negative pressure, causing today s acceleration of the expansion. One candidate for this component is a cosmological constant Λ. Another candidate is a dynamical component whose energy density evolves with time as it is the case for quintessence [9], [11], [26] or for k-essence [7], which is specified in the following. k-essence The challenge for theoretical physics is to address the cosmic coincidence problem: Why does cosmic acceleration begin at such a late stage in the evolution of the universe? Most dark energy candidates need fine-tuning of the initial energy density to solve this problem. The advantage of the k-essence model, see [7], is that it provides a dynamical explanation which does not require the fine-tuning and is completely non-anthropic. In this model, cosmic acceleration and human evolution are related, since both phenomena are linked to the onset of matter-domination. The k-essence component has the property that it only behaves as a negative pressure component after matter-radiation equality, so that it can only overtake the matter density and induce cosmic acceleration after the matter has dominated the universe for a long period. As human evolution is also linked to matterdomination because the formation of planets, stars, galaxies and large-scale structure only occurs during this period, we obtain a relation between the two phenomena. A further advantage of k-essence is the insensitivity of cosmic evolution to initial conditions due to a dynamical attractor behavior. The distinctive feature of k-essence models is that k-essence tracks the equation of state during the radiation-dominated epoch. A tracking solution during the matter-dominated epoch is physically forbidden [7]. Instead, at the onset of matter-domination, the k-essence field energy density ε k drops several orders of magnitude as the field approaches a new attractor solution in which it acts as a cosmological constant, with p ε, i.e. w 1. The k-essence energy density overtakes the matter-density, typically several billions of years after matter-domination, driving the universe into a period of cosmic acceleration. As it overtakes the energy density of the universe, it begins to approach yet another attractor solution which might correspond to an accelerating universe with w < 1 or a decelerating 3

14 1. Basics or even dust-like solution with 1 < w < 0. In this scenario, we observe today s cosmic 3 acceleration, because the time for human evolution and the time for k-essence to overtake the matter density are both severals of billions of years due to independent but predictive dynamical reasons. The k-essence models rely on dynamical attractor properties of scalar fields with nonlinear kinetic energy terms in the action (k-field). The concept was first introduced to develop an alternative inflationary model, k-inflation, which is invented in [6] and will be discussed in section 2.3.

Chapter 2 Inflationary Models As discussed in the first chapter, inflation is an add-on to the Hot Big Bang model and can explain the initial conditions of the standard scenario. In this chapter we discuss different inflationary models. We start with standard slowroll chaotic inflation [18], where inflation is driven by the potential of the scalar field ϕ. In the second section we consider hybrid inflation [19], a two-field-model, where inflation is also driven by the potential term and at its last stages by the vacuum energy density. At the end of this chapter k-inflation [6] will be presented. In this model inflation is driven by non-standard kinetic-energy terms of the scalar field ϕ, while the potential is zero. 2.1 Standard Slow-roll Inflation First of all, we want to discuss the standard slow-roll inflation, following the books [22] and [16]. The standard way to specify a theory is through its action from which the equations of motion can be derived. The action for a scalar field is given by S = d 4 x g ( 1 2 ( µϕ µ ϕ V (ϕ)), (2.1) where p is the Lagrangian density and an arbitrary function of ϕ and X 1 2 ( µϕ µ ϕ). Then the expressions for the energy density and pressure of a homogeneous scalar field ϕ ϕ(t) are: ε ϕ = 1 2 ϕ2 + V (ϕ), (2.2) p ϕ = 1 2 ϕ2 V (ϕ). (2.3) The term V (ϕ) is the potential, which should follow from high energy physics. The simplest examples are V (ϕ) = 1 2 m2 ϕ 2 and V (ϕ) = λϕ 4. The equation of motion can be found by substituting the relations (2.2) and (2.3) into the Friedmann and continuity equations ((1.4) and (1.7)). Assuming a spatially flat

16 2. Inflationary Models universe, i.e. k = 0, we obtain 1 : H 2 = 8π 3 ( 1 2 ϕ2 + V (ϕ)) (2.4) and the equation of motion ϕ + 3H ϕ = dv dϕ. (2.5) From the effective energy density and pressure we see that the condition for inflation is satisfied if ϕ 2 V (ϕ), i.e. if we have a slowly varying field. Notice that the second term in the equation (2.5) acts like a friction term for a harmonic oscillator (for a quadratic potential) with the friction determined by the Hubble parameter H. Once inflation gets under way, the curvature term in the Friedmann equations becomes less and less important. Normally, it is assumed negligible from the start; if it is not, then the beginning stages of inflation will render it so. The standard approximation technique for analyzing inflation is the slow-roll approximation. It throws away the first term of the right hand-side of equation (2.4) (because of ϕ 2 V (ϕ)) and the first term of the left hand-side of equation (2.5) (as ϕ V, ϕ ), leaving H 2 8π 3 V (ϕ) (2.6) and 3H ϕ dv dϕ. (2.7) For this approximation to be valid, it is necessary for the two following conditions to hold: ɛ sr (ϕ) = 1 16π η sr (ϕ) = 1 8π ( ) V 2 1, V (2.8) ( ) V 1. V (2.9) ɛ sr and η sr are the so-called slow-roll parameters. Here we have introduced the notation that primes are derivatives with respect to the scalar field ϕ. Notice that ɛ sr is positive by definition. The slow-roll parameters prove to be a very useful way of quantifying the predictions of inflation. If the first condition (2.8) is satisfied, the potential is flat enough to guarantee an exponential expansion. If the second condition (2.9) is satisfied the friction term in equation (2.5) dominates and therefore implies the slow rolling of the field on the potential, guaranteeing that the inflationary period lasts for sufficient time. The slow-roll parameters make it easy to see where inflation might occur on a given potential. For example, for V (ϕ) = 1 2 m2 ϕ 2, they are satisfied provided that ϕ 2 > 2. For 8π such a potential, inflation proceeds until the scalar field gets too close to the minimum for the slow-roll conditions to be maintained, and inflation comes to the end. 1 in this section we will use G 1

2.1 Standard Slow-roll Inflation 17 Example: V = 1 2 m2 ϕ 2 We consider the simple example with V = 1 2 m2 ϕ 2. A more detailed description of the evolution is given in [22]. Substituting H, given by (2.2) and H 2 = 8πε/3, into the equation of motion (2.5), we obtain a differential equation for the phase curves ϕ(ϕ): d ϕ 12π ϕ dϕ = ϕ2 + m 2 ϕ 2 + m 2 ϕ. (2.10) ϕ The behaviour of the solutions of (2.10) is shown in the phase diagram figure 2.1. ϕ m 12π frag replacements ϕ m 12π attractor Figure 2.1: The phase diagram for the simple example p = 1 2 ϕ2 1 2 m2 ϕ 2 is shown. The important feature of this diagram is the existence of an attractor solution to which all other solutions converge in time. The picture is taken from [21]. To show that the set of initial conditions is quite large we can consider a special region of the diagram, for example where ϕ > 0 and ϕ < 0. Assuming that initially the potential is small compared to the kinetic energy, i.e. ϕ mϕ, we have the case of the ultra-hard equation of state, where p ε. Neglecting mϕ compared to ϕ in (2.10) we get: By integration through separation of the variables we obtain: d ϕ dϕ = 12π ϕ. (2.11) ϕ exp ( 12πϕ), (2.12) and thus ϕ = const 1 12π ln t. (2.13) Substituting this result into H 2 = 8πε/3 and neglecting the potential term we obtain H 2 (9t 2 ) 1 and thus ε a 6. According to (2.12) the derivative of the scalar field decays exponentially more quickly than the value of the scalar field itself. Therefore, the large initial value of ϕ is damped within a short time interval before the field itself has

18 2. Inflationary Models changed significantly. The trajectory which begins at large ϕ goes up very sharply and meets the attractor, compare figure 2.1. This enlarges the set of initial conditions which lead to an inflationary stage. Now we study the inflationary solutions. If a trajectory joins the attractor where it is flat, at ϕ 1, then afterwards the solution describes a stage of accelerated expansion. To determine the attractor solution we assume that d ϕ/dϕ 0. It follows from (2.10) that ϕ attr m, (2.14) 12π and therefore ϕ attr (t) ϕ i m 12π (t t i ) m 12π (t f t), (2.15) where t i is the time when the trajectory joins the attractor and t f is the moment when ϕ formally vanishes. In reality, (2.15) fails well before the field ϕ vanishes. The corrections to the approximate attractor solution (2.15) can be found in [22]. Exit and Oscillation Inflation ends when the slow-roll conditions (2.8) and (2.9) are violated, as the field approaches a minimum of the potential with zero or negligible potential energy. In such cases, it can be assumed that inflation ends when ɛ sr (ϕ) reaches unity. This directly tells us the value of ϕ where inflation comes to an end. The field begins to oscillate with frequency w m, see [22], and the universe enters the stage of deceleration. In [22] the effective equation of state for a potential V ϕ n is calculated. The averaged w for an oscillating scalar field is then given by w = p ε n 2 n + 2. (2.16) I.e. for an oscillating massive field with V ϕ 2 we obtain w 0. In the case of a quartic potential we get w 1, i.e. the field mimics an ultra-relativistic fluid. 3 In models such as hybrid inflation, as we will see in the next chapter, extra physics intervenes to end inflation. 2.2 Hybrid Inflation Now we describe a model where inflation does not end by a slow rolling of the inflaton field, but by a very rapid rolling (waterfall) of a scalar field σ triggered by another scalar field ϕ. This model was invented in [19]. The effective potential of this model is given by V (σ, ϕ) = 1 4λ (M 2 λσ 2 ) 2 + m2 2 ϕ2 + g2 2 ϕ2 σ 2. (2.17)

2.2 Hybrid Inflation 19 V PSfrag replacements Figure 2.2: The figure above shows the potential V (σ, 0) of the theory of spontaneous symmetry breaking. V (0, 0) = M 4 /(4λ) is the vacuum energy density, which drives inflation at its last stages. σ Sfrag replacements σ ϕ Figure 2.3: The image above shows the three-dimensional plot of the Hybrid Inflation - potential V (σ, ϕ). The picture is taken from A. Linde s homepage www.stanford.edu/ alinde.

20 2. Inflationary Models This model looks like a hybrid of chaotic inflation with V (ϕ) = 1 2 m2 ϕ 2 and the theory with spontaneous symmetry breaking with V (σ) = 1 (M 2 λσ 2 ) 2, see figure 2.2. The 4λ specific choice of parameters allows the existence of the waterfall regime. Calculating the effective mass squared of the field σ we get Mσ 2 σ=0 = M 2 + g 2 ϕ 2. Therefore for ϕ > ϕ c = M/g the only minimum of the effective potential V (σ, ϕ) is at σ = 0. The curvature of the effective potential in the σ-direction is much greater than the curvature in the ϕ-direction. Thus we expect that at the first stages of expansion of the universe the field σ rolled down to σ = 0, its minimum, whereas the field ϕ could remain large for a longer time. Hence we consider the stage of inflation at large ϕ, with σ = 0 and therefore with the potential V (ϕ) = 1 4λ M 4 + m2 2 ϕ2, (2.18) which is composed of the slow-roll-potential (already mentioned in the previous section) and a constant vacuum energy density. When the inflaton field ϕ becomes smaller than ϕ c = M/g, the phase transition with the symmetry breaking occurs. If we assume m 2 ϕ 2 c = m2 M 2 /g 2 M 4 /λ, the Hubble constant H at the time of the phase transition is given by: H 2 = 8π 3M 2 P l ε = 2πM 4. (2.19) 3λMP 2 l We make a following additional assumption: m 2 H 2, what gives M 2 mm P l 3λ 2π. (2.20) One can verify, that, under the condition (2.20), the universe at ϕ > ϕ c undergoes a stage of inflation. Note that inflation at its last stages is driven not by the energy density of the inflaton field ϕ but by the vacuum energy density V (0, 0) = M 4 /(4λ), as in the new inflationary universe scenario [17]. Now we study the behavior of the fields ϕ and σ after the time t = H 1 from the moment t c when the field ϕ becomes equal to ϕ c. The equation of motion of the field ϕ during inflation is given by (2.7). Therefore during the time interval t = H 1 the field ϕ decreases from ϕ c by ϕ = m2 ϕ c 3H 2 = m2 λm 2 P l 2πgM 3. The absolute value of the negative effective mass squared M 2 σ σ=0 = M 2 + g 2 ϕ 2 of the field σ at that time becomes equal to M 2 σ (ϕ) = λm2 M 2 P l πm 2. (2.21) The value of M 2 σ (ϕ), the curvature of the potential, is much greater than H 2 for M 3 λmm 2 P l. (2.22)

2.3 k-inflation 21 In this case the field σ within the time t H 1 rolls down to its minimum at σ(ϕ) = M(ϕ)/ λ, rapidly oscillates near it and loses its energy due to the expansion of the universe. However, the field cannot simply relax near this minimum, since the effective potential V (σ, ϕ) at σ(ϕ) has a non vanishing partial derivative. One can check that the motion in this direction becomes very fast and the field ϕ rolls to the minimum of its effective potential within the time much smaller than H 1 if [19] M 3 λgmm 2 P l. (2.23) Thus, under the determined conditions, (2.22) and (2.23), inflation ends almost promptly, as soon as the field ϕ reaches its critical value ϕ c = M/g. 2.3 k-inflation Now we present a model where inflation is not driven by the potential V, but by nonstandard (i.e. non-quadratic) kinetic-energy terms for a scalar field ϕ. The motivation for including this terms is given by string theory. This model ( k-inflation for short) was invented in [6] (see also [5]). Basic Equations We consider a single scalar field ϕ minimally coupled to gravity. The action is given by 2 : S = d 4 x g ( R ) 6 + p(ϕ, ϕ), (2.24) where R is the Ricci scalar. Here only Lagrangians which just depend on the scalar X 1 2 ( ϕ)2 and on the field ϕ are considered. As we do not use potential terms in this model, we must impose that the function p(ϕ, X) vanishes when X 0. The Lagrangian p(ϕ, X) Taylor expanded around X = 0 gives: The energy-momentum tensor for p(ϕ, X) reads: p(ϕ, X) = K(ϕ)X + L(ϕ)X 2 +.... (2.25) T µν 2 δs ϕ p(ϕ, X) = g δg µν X µϕ ν ϕ p(ϕ, X)g µν. (2.26) This equation shows that, if µ ϕ is timelike, i.e. X > 0, the energy-momentum tensor has the perfect fluid form. Comparing (2.26) to the stress-energy tensor of a perfect fluid 2 Here we use: 8πG 3 = 1 and the signature ( +,,, ). T µν = (ε + p) u µ u ν p g µν, (2.27)

22 2. Inflationary Models we obtain: Pressure p = p(ϕ, X) (2.28) Energy density ε = ε(ϕ, X) 2X p(ϕ,x) X p(ϕ, X) (2.29) and a four-velocity u µ = σ µϕ 2X, where σ is the sign of ϕ = 0 ϕ. We consider, as usual, a flat background Friedmann model, (1.1) with k=0, and a homogeneous scalar field, i.e. X = 1 2 ϕ2. The equations of motions are given again by (1.4) and (1.7): ε = 3H(ε + p), (2.30) and what constrains ε > 0. Basic Idea H 2 = ε, (2.31) Now, we want to give the basic idea of k-inflation, starting with the ordinary case where the Lagrangian p depends only on X = 1 2 ( ϕ)2 and does not depend on ϕ. We have: and p = p(x) ε(x) = 2X p X p(x). The evolution equation can then be qualitatively solved by looking at the graph of the equation of state p = f(ε). The shape of this graph depends very much on the shape of the function p = p(x). We use ϕ ψ, and instead of (2.29) we obtain: ε(ψ) = ψ p p(ψ). (2.32) ψ Here we see that ε can be read geometrically off the graph p = p(ψ) as ε is the axis intercept. If the function p(ψ) = 1 2 Kψ2 + 1 4 Lψ4 +... is always convex, 2 p/ ψ 2 > 0, what can be the case if all the coefficients K,L,... are positive, p and ε will always be positive. Then, equation (2.30) shows that the energy density will monotonically decrease towards zero, and the evolution will be driven to an attracting solution with the equation of state given by p ε, valid near ε = 0 where the usual kinetic term dominates. But if the function p(ψ) is non-convex, the graph p = f(ε) could be more complicated and can allow exponential-type inflationary behavior. Looking at equation (2.32), we notice that the extrema of the function p = p(ψ) correspond to values where p = ε, i.e. w = 1. For a general function p = p(x) the graph of the equation of state might resemble figure 2.4.

2.3 k-inflation 23 p u p= ε s ε s u u u p= ε s Figure 2.4: Evolution of the equation of state w = p/ε for an hypothetical general Lagrangian. The region ε < 0 can be excluded because of flat cosmology, where H 2 = ε. Except for the origin and the point above it, the attractors of the evolution are inflationary fixed points with w = 1. The picture is taken from [6]. From (2.30) it follows that ε decreases above and increases below the line p = ε. Therefore, all the intersection points (in figure 2.4) with this line, are attractors of the evolution. As we can see by integration of H 2 = ε the fixed points, lying at this line, correspond to an exponential inflation (de-sitter inflation with w = 1): a attr (t) = a 0 exp ( ε t). (2.33) The region where the energy density is negative can be excluded, because it cannot be reached by flat cosmologies (see (2.31)). Till now we considered a Lagrangian, depending only on X. Unfortunately there is no graceful exit, no smooth transition to Friedmann universe and the cosmological perturbations are ill-filled in this model [12]. Therefore we should examine a Lagrangian, which depends explicitly on the scalar field ϕ. Slow-roll k-inflation As already mentioned before, we now consider a Lagrangian p = p(ϕ, X) and we want to find the conditions under which the influence of ϕ will represent only a relatively small perturbation of the attraction toward de-sitter-inflation. Therefore, we focus again on the simplest kinetic Lagrangian: p(ϕ, X) = K(ϕ)X + L(ϕ)X 2. (2.34)

24 2. Inflationary Models The energy density can be calculated with (2.29) and is given by: ε(ϕ, X) = K(ϕ)X + 3 L(ϕ)X 2. (2.35) Here we just want to assume that there are two regions: one where K > 0 in some range of values of ϕ(weak-coupling domain) and one where K < 0 (strong-coupling domain). p p=- ε p= ε weak coupling p=ε/3 ε strong coupling u s Figure 2.5: (ε-p)-trajectory for the model (2.34) as the field ϕ varies from the strongcoupling domain to the weak-coupling one. The picture is taken from [6]. Figure 2.5 shows the change in the form of the equation of state as ϕ varies from the strong-coupling region to the weak-coupling region. For large X the equation of state asymptotes the one of radiation (w = 1/3), but for X 0, w becomes unity (hard equation of state). Here we have again an inflationary fixed point, where w 1. As mentioned before we just want ϕ to bring a small perturbation onto the model discussed at the beginning. The question is: under what conditions on the functions K(ϕ) and L(ϕ) one can approximately solve the evolution equation (2.30)? By redefinition of the scalar field (see [6]) we get (in strong coupling K < 0 ) a simpler Lagrangian: p new (ϕ new, X) = K(ϕ new )X +X 2. Because of facility of inspection we omit the index new in further formulas. The zeroth-order slow-roll solution to equation (2.30), i.e. the solution of ε + p = 2X( p/ X) = 0, corresponds to: X sr = 1 2 K(ϕ sr), (2.36) ϕ sr = σ K(ϕ sr ), (2.37) ε sr = 1 4 K2 (ϕ sr ), (2.38)

2.3 k-inflation 25 H sr = 1 2 K(ϕ sr). (2.39) Notice, that K(ϕ sr ) > 0 in the slow-roll domain and σ denotes the sign of ϕ. The time evolution is given from this four equations through: t t in = ( ) a(t) N ln = a in ϕ ϕ in dϕ ϕ = σ ϕ tend ϕ in t in H(t)dt = σ 2 dϕ K(ϕ), ϕ ϕ in K(ϕ) dϕ, with N, the number of e-foldings of inflation (1.14). We obtain the post-slow-roll approximation, X = X sr + δx, by inserting (2.32) into (2.30): ε = 3H(ε + p) = 3H(2X) p X. Rewriting this equation we get: p X = K + 2X = 2 δx = ε 6X ε, and with the slow-roll approximations (2.36) - (2.39) we get: δx 1 X sr 12 ε ε σ 3/2 3 K/ ϕ ( K) 3/2 2σ 3 (( K) 1/2 ). (2.40) ϕ The criterion for the validity of the previous slow-roll solutions (2.36) - (2.39) is δx 1 (slow-roll-condition), (2.41) X which means 4 (( K) 1/2 ) 1. 9 ϕ There are different functions K(ϕ), which satisfy this condition (see [6] ). Summarizing the behaviour of the solutions, we are focusing on, we get the following: We start with some representative point in the strong coupling domain of the (ε, p)-plane. In a first evolution stage we can neglect the ϕ -dependence of the equation of state because there is a fast attraction toward the nearest inflationary attractor (where w 1). Then we can consider that our representative point follows the post-slow-roll motion X = X sr + δx, corresponding to a point near to the p = ε line. As the evolution continues, the slow-roll condition is less and less satisfied and the point straggles more away from the p = ε line. Then we reach a point where the slow-roll criterion (2.41) becomes violated and the exit occurs. After the transition to the weak-coupling domain, where K > 0, the cosmological evolution will be attracted toward the p ε equation of state, i.e. ε ϕ a 6. As this energy density decreases much faster than the energy density of radiation (ε rad a 4 ) and the energy density of dust (ε dust a 3 ), even small traces of ε rad or ε dust, present at the end of inflation, will ultimately dominate the expansion.

26 2. Inflationary Models weak coupling ϕ. strong coupling ϕ ε<0 ε+p=0 Figure 2.6: Schematic phase diagram of slow-roll k-inflation. Trajectories approach the attractor but do not reach the line ε + p = 0 where the speed of sound (2.45) vanishes. Around the point where the slow-roll condition is violated, the solutions leave the inflationary stage and approach then smoothly the vacuum ϕ = 0. The picture is taken from [6]. Power-law k-inflation If we consider the model of slow-roll inflation, i.e. the potential driven inflationary scenario, it is useful to have some exact solutions to the full equations of motion in order to study their properties. One exact solution is power-law inflation, where the potential is chosen to take the exponential form (see for instance [20]). Then there exists an attractor solution which describes a power-law inflating universe, but there is no graceful exit. In [6], it was shown that one can get power-law k-inflation, which is analogous to the power-law inflation for potential driven inflationary scenarios. Before, we used the Lagrangian (2.34), which can be rewritten through field redefinition into: p = F (ϕ)( X + X 2 ). (2.42) In [6] it is shown, that there exists a function F (ϕ) for which the master equation (2.30) has an exact solution which describes power-law inflation: F (ϕ) = 4 ( ) 4 3γ 1 9 γ 2 (ϕ ϕ ), (2.43) 2 where γ = w + 1 = (p/ε) + 1. For F (ϕ) given by (2.43) the model has an attractor solution with a fixed point X = X 0 = 2 γ 4 3γ, which describes power-law expansion a(t) t 2 3γ, (2.44)

2.3 k-inflation 27 if 0 < γ < 2. For negative values of γ one gets pole-like super-inflation, which has a 3 graceful exit problem [29]. In the model of power-law inflation this problem can be easily solved if in some range of ϕ the function F is modified in an obvious way. In appendix A the formulas for a more general Lagrangian are derived. Stability For the stability analysis we need the equation for the speed of sound [6], [12]: c 2 s = p, X ε, X = p, X p, X +2Xp, XX. (2.45) The comma denotes a partial derivative with respect to X. If c 2 s is negative then the model is absolutely unstable. The stability requirement c 2 s = p, X ε, X > 0 (2.46) is non trivial in the k-inflationary scenario, because, for instance, in slow-roll k-inflation in the zeroth-order slow-roll approximation the inflationary attractors are defined by p, X = 0, and therefore c 2 s = 0. As discussed in the section slow-roll k-inflation, in the post-slowroll approximation, with X = X sr + δx, p, X does not vanish. To first order in δx we can write p, X p, sr XX δx, and using (2.45) we obtain c 2 s δx 2X sr. (2.47) Thus stability requires that δx > 0, i.e. that on the equation of state graphs of figures 2.4 and 2.5, the ϕ-gradients of p be such that they displace the real, non-adiabatic, slow-roll attractor beyond the p = ε line. These stable stretches of the (ε-p)-graphs are labelled s in figure 2.4. These are also the stretches where the slope (dp/dε) is positive.

28 2. Inflationary Models

Chapter 3 k-essence and Hybrid Inflation In the second chapter we have discussed three inflationary models: Standard slow-roll inflation, where the inflation is driven by the potential V (ϕ) of the scalar field ϕ. Hybrid inflation, a two-field-model, where inflation is also driven by the potential term and at its last stages by the vacuum energy density. k-inflation, a one-field model, where inflation is driven by the non-standard kineticenergy terms of the scalar field ϕ and where the potential vanishes. Now we want to discuss the combined model with two scalar fields ϕ and σ, where the potential V is small in comparison to the total energy density. First we present the general model and give the basic equations, which we will need for further discussions. Then we discuss a toy model without potential, but with two kinetically coupled ghosts. In section 3.3 the power-law model for hybrid-k-inflation will be introduced, while in section 3.4 the slow-roll hybrid-k-inflation model will be discussed. 3.1 General Model 3.1.1 The Action Principle We are going to consider two scalar fields ϕ and σ, with non-canonical kinetic terms, minimally coupled to gravity. Thus the total action describing our model is: S tot [g µν, ϕ, σ] = S EH [g µν ] + S k [g µν, ϕ, σ], (3.1) where g µν denotes the space-time metric. The total action is divided into different parts : S EH is the Einstein-Hilbert action of general relativity S EH = 1 d 4 x g R, 6

30 3. k-essence and Hybrid Inflation so we deal exclusively with Einstein gravity in four space-time dimensions. The factor 1/6 reflects our unit choice 8πG/3 1. The k-fields are described by the action S k = d 4 x g p(ϕ, σ, X, Y ), (3.2) where p depends on the field derivatives through the combinations X 1 2 ( µϕ) 2 and Y 1 2 ( µσ) 2. (3.3) The stress-energy tensor of the k-field is obtained by functional differentiation of the k-field action (3.2) with respect to the metric g µν : T µν 2 g δs k δg µν δs k = d 4 x g δg µν p(ϕ, σ, X, Y ) + g µν Furthermore we use the standard formula d 4 x g p(ϕ, σ, X, Y ) g µν δg µν. g = 1 g gµν, g µν 2 see e.g. [25] and [15], and p(ϕ, σ, X, Y ) = 1 p(ϕ, σ, X, Y ) g αβ 2 X α ϕ β ϕ + 1 p(ϕ, σ, X, Y ) α σ β σ. 2 Y Then we obtain ( g δs k = d 4 x 2 ) g µν p δg µν + ( g p d 4 x 2 X µϕ ν ϕ + p ) Y µσ ν σ δg µν. Therefore the stress-energy tensor is: T µν = p, X µ ϕ ν ϕ + p, Y µ σ ν σ pg µν, (3.4) with p = p(ϕ, σ, X, Y ) and p, X = p/ X, p, Y = p/ Y. The total pressure and the total energy density of the k-fields can be formally defined as p = 1 3 T i i (3.5) T 0 0 = ε = 2X p, X +2 Y p, Y p. (3.6)

3.1 General Model 31 3.1.2 Equations of Motion The equations of motion of our model are obtained by functional differentiation of the action with respect to the fields: δs δϕ = d 4 x g (p, ϕ δϕ + p, X ( µ ϕ) δ( µ ϕ)) = d 4 x g (p, ϕ δϕ µ (p, X µ ϕ)δϕ). µ is a covariant derivative. Thus the extremal principle δs δϕ = 0 reads with and where δs δϕ = p, ϕ p, X ϕ µ ϕ µ p, X = 0 µ p, X = p, Xϕ µ ϕ + p, Xσ µ σ + p, XX µ X + p, XY µ Y µ X = ( µ ν ϕ) ν ϕ, µ Y = ( µ ν σ) ν σ, ϕ = g µν µ ν ϕ. Analogously we vary the action with respect to the second field σ. following equations of motion: Then we get the K µν µ ν ϕ p, XY ( µ ν σ) ν σ µ ϕ + p, Xϕ µ ϕ µ ϕ p, Xσ µ σ µ ϕ p, ϕ = 0 (3.7) where and L µν µ ν σ p, XY ( µ ν ϕ) ν ϕ µ σ + p, Xσ µ σ µ σ p, Xϕ µ ϕ µ σ p, σ = 0 (3.8) K µν = p, X g µν + p, XX µ ϕ ν ϕ (3.9) L µν = p, Y g µν + p, Y Y µ σ ν σ. (3.10) As mentioned in the sections before, our universe is isotropic and homogeneous on large enough scales and there is firm evidence that its spatial sections are flat [8]. Therefore we will consider the motion of the k-fields in a homogeneous and isotropic spatially flat universe described by the metric (1.1) with k = 0. A homogeneous and isotropic flat universe is the consequence of a sufficiently long stage of inflation, i.e. the number of e-foldings N (compare (1.14)) should be larger than 75 [22]. As the universe is homogeneous, the spatial derivatives vanish j ϕ = 0 and with ϕ = ϕ + 3H ϕ we obtain and ϕ (p, X +p, XX ϕ 2 ) + σ (p, XY ϕ σ) +3Hp, X ϕ + p, Xϕ ϕ 2 + p, Xσ ϕ σ p, ϕ = 0 (3.11) } {{ } } {{ } α β σ (p, Y +p, Y Y σ 2 ) + ϕ (p, XY ϕ σ) +3Hp, Y σ + p, Y σ σ 2 + p, Y ϕ ϕ σ p, σ = 0. (3.12) } {{ } } {{ } γ δ

32 3. k-essence and Hybrid Inflation Rewriting this system of differential equation we get ϕ = γ(p, ϕ ϕ(3hp, X + ϕp, Xϕ + σp, Xσ )) + β( p, σ + σ(3hp, Y + ϕp, Y ϕ + σp, Y σ )) αγ βδ (3.13) and σ = δ(p, ϕ ϕ(3hp, X + ϕp, Xϕ + σp, Xσ )) + α( p, σ + σ(3hp, Y + ϕp, Y ϕ + σp, Y σ )). (3.14) αγ βδ The existence and uniqueness of solutions of this system can be guaranteed if The 00-component of the Einstein equations (1.2) gives again: where ε is given by (3.6). αγ βδ 0. (3.15) H 2 = ε, (3.16) 3.2 Toy-model As a warm-up we want to start with a toy model without potential, but with two kinetically coupled ghosts. The Lagrangian is given by p = A(ϕ, σ) ( µ ϕ) 2 + 2 C(ϕ, σ) ( µ σ) 2 D(ϕ, σ) + ( µ ϕ) 2 ( µ σ) 2. (3.17) 2 4 If we use our notations (3.3), then p reads p = A(ϕ, σ) X + C(ϕ, σ) Y + D(ϕ, σ) X Y. (3.18) With (3.6) we can calculate the energy density, and obtain: ε = A(ϕ, σ) X + C(ϕ, σ) Y + 3 D(ϕ, σ) X Y. (3.19) As we work within flat cosmology, where H 2 = ε, we want the energy density ε to be positive for X, Y. Thus we see from the two equations (3.18) and (3.19) that D(ϕ, σ) has to be positive. As we need a negative pressure to get inflation, A(ϕ, σ) and/or C(ϕ, σ) should be negative at the beginning. Then just for small X and Y the energy density is negative.

3.2 Toy-model 33 3.2.1 Endless Inflation First of all we consider a Lagrangian, which does not explicitly depend on the fields ϕ and σ. We choose A, C to be negative constants, while D is a positive constant. By redefinition of the fields we can rewrite (3.18) into with Λ 1 > 0. For the energy density we get p = X Y + Λ 1 X Y, (3.20) ε = X Y + 3Λ 1 X Y. (3.21) The energy density depends only on the field derivatives and thus ε > 0 if Y > X 3Λ 1 X 1, (3.22) compare figure 3.1. Using (3.11) and (3.12), we get the equations of motion: ( ) ( ) ( ) 1 1 1 ϕ 2Λ σ2 1 + σ Λ ϕ σ + 3H( ϕ, σ) 2Λ σ2 1 ϕ = 0, (3.23) ( ) ( ) ( ) 1 1 1 ϕ Λ ϕ σ + σ 2Λ ϕ2 1 + 3H( ϕ, σ) 2Λ ϕ2 1 σ = 0. (3.24) The existence and uniqueness of solutions of this system of ordinary differential equations can be guaranteed if ( 1 + Λ det 1 Y Λ 1 ) ϕ σ Λ 1 ϕ σ 1 + Λ 1 = 1 Λ 1 (X + Y ) 3Λ 2 XY 0. (3.25) X Substituting ϕ = v and σ = u, the equations of motion can be reduced to the system of two ordinary differential equations of first order where ( ) ( ) ( ) 1 1 1 v 2Λ u2 1 + u Λ uv + 3H(u, v) 2Λ u2 1 v = 0, (3.26) ( ) ( ) ( ) 1 1 1 v Λ uv + u 2Λ v2 1 + 3H(u, v) 2Λ v2 1 u = 0, (3.27) H(u, v) = v2 2 u2 2 + 3 4Λ u2 v 2. (3.28) Then the phase space is (u, v). Similarly to section 2.3, we can calculate a fixed point where w = 1 and X = 1 2 v2 = const, Y = 1 2 u2 = const. We can find a point (v, u ), where v = 0 = u: (X, Y ) = (Λ, Λ), (3.29)

34 3. k-essence and Hybrid Inflation Y 3 2.5 2 1.5 1 PSfrag replacements 0.5 0.5 1 1.5 2 2.5 3 X Figure 3.1: The numerically obtained phase trajectories are plotted in the XY-space for the model (3.20) with Λ 1 = 1. The fixed points are given by (3.29) (X, Y ) = (1, 1). The green graph is the (w = 1)-graph, where w > 1 above. The red graph shows where the determinant (3.25) changes its sign, while the (ε = 0)-graph is plotted in blue. Below the blue graph the energy density is negative. Thus this region can be excluded for flat cosmology, as there H 2 = ε. One can see that the fixed points are of the type of a stable star. i.e. (v, u ) = ( ± 2Λ, ± 2Λ ). (3.30) At all these four fixed points the energy density has a value of ε = Λ, i.e. H = Λ. We can calculate the stability matrix to classify the fixed point and to prove that the fixed point is an attractor. We have to rewrite (3.26) and (3.27) into a system of two ordinary differential equations, which are diagonal with respect to v and u. In our case this gives v = f(u, v) (3.31) u = g(u, v). (3.32) v = H(u, v) v2 u(u 2v) 2Λ(v u) 2 3v 2 u 2 + 2Λ(u 2 + v 2 2Λ) u = H(u, v) u2 v(v 2u) 2Λ(u v) 2 3v 2 u 2 + 2Λ(u 2 + v 2 2Λ) (3.33) (3.34)

3.2 Toy-model 35 w 1 0.5 20 40 60 80 N -0.5 PSfrag replacements -1 Figure 3.2: The figure above shows the equation of state w(n), where N is the number of e-foldings. When the system approaches the fixed point, the universe is in a quasi de-sitter inflationary stage with w 1. This de-sitter stage is a late-time asymptotic. Then we expand about the fixed point (v, u ). To first order this gives d dt ( ) δv = δu ( f,v (v, u ) f, u (v, u ) g, v (v, u ) g, u (v, u ) ) ( ) δv δu (3.35) where the 2 x 2 matrix is called the stability matrix S. In our case ( ) ( f,v (v S =, u ) f, u (v, u ) 3H 0 = g, v (v, u ) g, u (v, u ) 0 3H ). (3.36) The analysis of the eigenvalues of the stability matrix S characterizes the type of the fixed point. The eigenvalues are obviously negative and equal: λ 1 = λ 2 = 3H < 0. (3.37) The matrix has two different eigenvectors, because δu and δv decouple around (v, u ) in linear order. A fixed point for which the stability matrix has those eigenvalues is called a stable star and is a dynamical attractor. For a better understanding of the model we consider the numerically obtained solution in the XY -plane, see figure 3.1. The energy density (3.21) equals zero if Y = while the determinant (3.25) becomes zero if X 3Λ 1 X 1, Y = (1 Λ 1 X)Λ 1 + 3Λ 1 X.

36 3. k-essence and Hybrid Inflation At X Y = 2Λ 1 X 1 the equation of motion is w = 1. Numerical calculations prove the correctness of our analysis. As the system approaches the fixed point the universe is in a quasi de-sitter inflationary stage, compare figure 3.2. This de-sitter stage is a late-time asymptotic and thus the model might be interesting as a dark energy model. Unfortunately there is no graceful exit from de-sitter stage. As it already has been mentioned in section 2.3 we should consider a Lagrangian which depends explicitly on ϕ or/and σ to get a graceful exit. 3.2.2 Inflation with Exit In k-inflation we used the Lagrangian (2.34): p(ϕ, X) = K(ϕ)X + L(ϕ)X 2. There we have chosen the function K(ϕ) to be negative to get inflation and later to be positive to get a graceful exit as the pressure then becomes positive too. This means for our model, that we can choose the functions A(ϕ, σ) and C(ϕ, σ) to be negative for large negative ϕ and/or σ and to be positive for ϕ and/or σ being positive. The initial conditions, i.e. the value of ϕ(t = 0) and/or σ(t = 0), have to be selected in a way that the inflationary stage is long enough to solve the homogeneity problem, compare section 1.2 and [22]. Toy model with A = tanh σ and C = tanh ϕ We consider numerically a model, where the Lagrangian depends explicitly on both scalar fields ϕ and σ: p = tanh (σ)x + tanh (ϕ)y + DXY. (3.38) The energy density is p = tanh (σ)x + tanh (ϕ)y + 3DXY, (3.39) while the equations of motions are given by (3.11) and (3.12). The existence and uniqueness of solutions of this system of differential equations can be guaranteed if ( ) A + DY D ϕ σ det = AC + ADX + DCY 3D 2 XY = D ϕ σ C + DX tanh (σ) tanh (ϕ) + D (tanh (σ)x + tanh (ϕ)y ) 3D 2 XY 0. (3.40) Simulating the evolution with D = 1, ϕ(t = 0) = 110 = σ(t = 0) and ϕ(t = 0) = 6 = σ(t = 0) we receive a quasi de-sitter inflationary stage, where w 1, while σ and ϕ are negative. After a sufficiently long stage of inflation (N > 75) the exit occurs (see

3.3 Power-law Hybrid-k-Inflation 37 figure 3.3), as the functions A(σ) = tanh σ and C(ϕ) = tanh ϕ change their sign. The equation of state towards the hard equation of state w 1. The new attractor, with p ε, corresponds to a very fast decrease of the energy density: ε a 6. As this energy density decreases much faster than the energy density of radiation (ε rad a 4 ) and the one of dust (ε dust a 3 ), even small traces of ε rad or ε dust, present at the end of inflation, will ultimately dominate the expansion. Varying the initial conditions we see that these have to be extremely fine-tuned. The initial values for the fields should not be too different: ϕ(0) σ(0) < 2. w 1 0.5 20 40 60 80 N -0.5 PSfrag replacements -1 Figure 3.3: The figure above shows the equation of state w(n) of the model (3.38). The universe undergoes a stage of quasi de-sitter inflation till N > 75. Then the graceful exit occurs and w +1, where ε a 6. In this section 3.2 we considered a toy model without potential, but with two kinetically coupled ghost-like scalar fields. Recapitulating we can see that even the Lagrangian with the simple XY -coupling term and without further non-linear kinetic terms, such as X 2 and/or Y 2, can bring quasi de-sitter inflation. When the Lagrangian does not depend on the scalar fields ϕ and σ the de-sitter stage is a late-time asymptotic. The explicit ϕ- or σ-dependence is responsible for the graceful exit, but in this case the initial conditions have to be extremely fine-tuned. This model may be interesting as a dark energy model. 3.3 Power-law Hybrid-k-Inflation In the second chapter we discussed power-law inflation for the normal k-field. Now the power-law model for hybrid-k-inflation will be derived analogously. The basic idea is to make k-inflation (power-law) with the first field ϕ and after a sufficiently long stage of inflation the second field σ brings the graceful exit.

38 3. k-essence and Hybrid Inflation In appendix A the general formula for the Lagrangian which makes power-law-kinflation is derived (A.6): p = 4 Cϕ 2 ( X + αx2 ). (3.41) C and α are constants. To this Lagrangian we add the kinetic term and potential of the second field (Y V (σ)) and we could build in a further function which depends explicitly on the second field σ. Thus we obtain a Lagrangian, which depends on both fields and both kinetic energy terms: p = p(x, Y, ϕ, σ). In the following we discuss two different models: one where the graceful exit takes place due to the change of sign of a function g(σ) - similar to the idea in section 2.3 (p. 23). The other one where the exit is brought through a function f(σ), which makes the k-inflation-term negligible in comparison to the energy density of the second field. 3.3.1 Exit due to the Change of Sign Lagrangian and Initial Conditions Taking (3.41), where α = 1, we can build a new model: p = 4 Cϕ 2 ( g(σ)x + X 2 ) + Y V (σ), (3.42) with C = 9γ2 4 3γ and 0 < γ < 2 3. k-inflation (power-law) should be made by the first field. Thus the parameters and initial conditions have to be set as follows: m << H in 4 Cϕ 2 ( g(σ)x + 3X 2 ) and Y + V (σ) 4 Cϕ 2 ( g(σ)x + 3X 2 ) to guarantee a sufficiently long stage of inflation and g(σ) 1 to have normal k-inflation made by the field ϕ. Then normal k-inflation with p 4 ( X + X 2 ) takes place at the beginning, where the Cϕ 2 fixed point is given by (A.3) X 2 γ. As the function g(σ) changes its sign when σ 4 3γ reaches a certain value, the k-inflation-term becomes positive and thus the graceful exit occurs. A possible function for g(σ) is the hyperbolic tangent: g(σ) = tanh (σ σ ), with σ as a positive constant. Analytical Considerations For p = 4 Cϕ 2 ( tanh (σ σ )X + X 2 ) + Y 1 2 m2 σ 2 (3.43)

3.3 Power-law Hybrid-k-Inflation 39 we want to do the analytical calculations. Inserting (3.43) into the equations of motions ((3.11) and (3.12)), we obtain: and ϕ( tanh (σ σ ) + 6X) + 3H( tanh (σ σ ) + 2X) ϕ + + 2 ϕ (tanh (σ σ )X 3X 2 ) sech 2 (σ σ) ϕ σ = 0 (3.44) σ + 3H σ + m 2 σ + 4 Cϕ 2 Xsech2 (σ σ) = 0. (3.45) At the beginning we choose σ σ and H 4 Cϕ 2 ( X + 3X 2 ) to satisfy the initial conditions. Then we can simplify (3.44) and rewrite the obtained equation: ϕ = 6 C ϕ 3X2 X(2X 1) ϕ + 2 (X ϕ 3X2 ) 6X 1. (3.46) We set ϕ = 0 to get the fixed points, i.e. the possible constant values for X during the k-inflationary phase: X (3) = C + 24 + C 2 + 16C 48 X (1) = 0 and X (2) = 1 3, (3.47) and X (4) = C + 24 C 2 + 16C. (3.48) 48 The fixed points X (1) and X (2) can be excluded as ε X(3X 1) and thus the energy density would vanish. Which of the other two fixed points will be reached, depends on the initial value of ϕ. During this stage of inflation the second field σ decreases slowly.then, there is a point where σ σ = 0 and thus we have to consider different equations of motions: and ϕ = 3H ϕ + 1 2 ϕ 2 ϕ + 1 3 σ ϕ (3.49) σ + 3H σ + m 2 σ + 4 X = 0. (3.50) Cϕ2 The first and the last term of the right hand-side of (3.49) are negative (since σ < 0 and we assume ϕ > 0), the second term becomes very small as ϕ increases. Therefore ϕ < 0 and ϕ 0. The k-inflation-term becomes small in comparison to the energy density of the second field. Notice that in this model H m is necessary for k-inflation. But in the end m has to be larger than H. Then we get oscillation of the second field σ. As g(σ) can contribute to the effective mass, the function has to be chosen in a way that it does not reduce m 2 eff. For (3.42) with V = 1 2 m2 σ 2, m 2 eff is given by m 2 eff = m2 + 4X Cϕ 2 2 g(0) σ 2, (3.51)

40 3. k-essence and Hybrid Inflation g 1 0.5 PSfrag replacements -0.5 50 55 60 65 70 75 N -1 Figure 3.4: Illustration of the function g = tanh (σ σ ) with σ = 4 for the Lagrangian (3.43). At N 65, g(σ) changes its sign and the k-inflationary phase goes over into the normal slow-roll inflationary stage, driven by V (σ). w 1 0.5 50 55 60 65 70 75 N PSfrag replacements -0.5-1 Figure 3.5: Illustration of the equation of state w = w(n) for (3.43) with σ = 4. w 0.95 during the inflationary phase, as we have chosen γ = 1/20. Till N 65 k-inflation, made by the k-essence field ϕ, takes place. Then the function g(σ) changes its sign and the energy density of the second field σ dominates. The last stages of inflation are driven by V (σ). < w > 1 0.5 77.5 78 78.5 79 79.5 N PSfrag replacements -0.5-1 Figure 3.6: Illustration of the averaged equation of state (3.55) for (3.43) with V σ 2. As shown in section 2.1, < w > mimics a dust-like equation of state for a quadratic potential.

3.3 Power-law Hybrid-k-Inflation 41 and thus 2 g(0) has to be positive to guarantee m > H. As we have chosen g = tanh (σ σ ) σ 2 with 2 g(0) = 2 sech 2 (σ σ) tanh (σ σ), we get a positive contribution to the mass for σ 2 σ < σ. As X 0, from (3.50) only the normal Klein-Gordon equation is left: σ + 3H σ + m 2 σ = 0. (3.52) From now on the evolution is determined by the field σ and oscillation of σ, mentioned in section 2.1 and calculated in [22], takes place. For V σ 2 we get an averaged equation of state < w > 0, while for V σ 4 we get < w > 1/3. The number of e-foldings for the k-inflationary phase is given by ( ) a(tend ) tend N ln = H(t)dt. (3.53) a(t) t in t end is the time, where σ = σ, i.e. where the k-inflation term changes its sign. Numerical Investigations for V σ 2 The properties of the model are investigated numerically for the Lagrangian (3.43). The parameters and initial conditions are set by: m = 0.01, γ = 1/20, ϕ(0) = 1.5, σ(0) = 6.91, ϕ(0) = 1, σ(0) = 0.001. As γ = 1/20 we get a t 40/3 (see (A.2)). Figure 3.5 shows that k-inflation, made by the first field ϕ, takes place with an equation of state given by w = γ 1 0.95. After a large number of e-foldings, N 65, the function g changes its sign (see figure 3.4) and the energy density of the second field σ becomes larger than the energy density of the ϕ-field. Thus the last stages of the inflation are driven by the potential V (σ). As X 0, lim w gx + X 2 ϕ lim X 0 X 0 gx + 3X = +1. 2 I.e. ε ϕ a 6 and the field ϕ vanishes very quickly. The exit occurs after N 77. As V σ 2 the averaged equation of state < w > 0. This is the equation for the density of non-relativistic matter, i.e. the energy density of the field σ falls as a 3. As 2 a t 3(w+1), we get I.e. we obtain an averaged equation of state which can be plotted (compare figure 3.6). H = ȧ a = 2 3(w + 1)t. (3.54) < w >= 2 1, (3.55) 3Ht

42 3. k-essence and Hybrid Inflation The peak in figure 3.5 separates the k-inflationary phase from normal slow-roll inflation, driven by V (σ). By changing the initial condition σ(0) and the parameter σ we can shorten the k-inflationary and thus enlarge the potential-driven inflationary phase. This feature of the model may be important for the formation of structure, because during these two inflationary stages the cosmological perturbations have different speeds of sound and therefore, different amplitudes. We can imagine the following situations: The peak lies at 60... 20 e-foldings before end. Then we get a change of structure and a change in the fluctuations. The peak lies at more than 70 e-foldings before end. Then we just have standard slow-roll inflation, what is not interesting for our model. If g(σ) changes its sign when H < m, then slow-roll inflation is not possible anymore and we just have k-inflation. The squared speed of sound during the k-inflationary phase is given by (2.45) c 2 s = p, X ε, X c 2 s for the slow-roll inflationary phase is unity, compare figure 3.7. c 2 s 1 + 2X 1 + 6X. (3.56) 1 0.8 0.6 PSfrag replacements 0.4 0.2 10 20 30 40 50 60 70 80 N Figure 3.7: Illustration of the squared speed of sound c 2 s. During the k-inflationary phase c 2 s 0.005, while for N > 65 slow-roll inflation takes place with c 2 s = 1. By variation of the initial conditions and parameters we see that these can lie in a wide range, the main objective is that the above mentioned initial conditions are fulfilled. Numerical Investigations for V σ 4 We use the same initial conditions and do the same calculations for a quartic potential. The evolution is entirely the same: The first field ϕ vanishes very quickly after the change of sign of the function g(σ), while the second field σ starts to oscillate. As shown in figure 3.8 and as calculated in section 2.1, < w > 1. I.e. the oscillating field mimics an 3 ultra-relativistic fluid with a decreasing energy density ε a 4.

3.3 Power-law Hybrid-k-Inflation 43 < w > 1 0.5 79.5 80 80.5 81 81.5 N PSfrag replacements -0.5-1 Figure 3.8: Illustration of the averaged equation of state for a quartic potential V σ 4. < w > 1, i.e. the oscillating field σ mimics an ultra-relativistic fluid. 3 3.3.2 Exit due to a Vanishing k-inflation-term Lagrangian and Initial Conditions In this model the exit occurs due to a function f(σ), which forces the k-inflation-term to vanish. We use a similar Lagrangian to the aforementioned model: 4 p = Cϕ 2 (1 + f(σ)) ( X + X2 ) + Y V (σ), (3.57) with C = 9γ2 4 3γ and 0 < γ < 2 3. k-inflation (power-law) should be made again by the first field ϕ and thus the parameters and initial conditions have to be set in a certain way: m << H in and Y + V (σ) << stage of inflation and 4 Cϕ 2 (1+f(σ)) ( X + 3X 2 ) to guarantee a sufficiently long 1 >> f(σ) to have normal k-inflation through the field ϕ, at first. If at the beginning σ > 0 and σ < 0, the function f(σ) could be chosen as f(σ) = 1. σ 2 Then the rough evolution is the following: We start with power-law k-inflation made by field ϕ, where p 4 ( X + X 2 ). As Cϕ 2 σ becomes smaller, the function f(σ) gets more and more important and when σ 0 the graceful exit occurs. σ rolls down to its minimum of the potential V (σ) and starts oscillating. Analytical Considerations Like in section 3.3.1, we want to derive the properties of the model analytically using the following toy Lagrangian: 4 p = Cϕ 2 (1 + 1 ) ( X + X2 ) + Y 1 2 m2 σ 2. (3.58) σ 2

44 3. k-essence and Hybrid Inflation w 1 0.75 0.5 0.25-0.25-0.5 PSfrag replacements -0.75-1 72 74 76 78 80 N Figure 3.9: Illustration of the equation of state w = w(n) for (3.58). Power-law k-inflation with w 0.95 takes place till N 77. Then the exit occurs. w 1 0.5 77.5 78 78.5 79 79.5 80 80.5 N PSfrag replacements -0.5-1 Figure 3.10: This figure shows the averaged equation of state < w > for a quadratic potential, i.e. < w > 0.

3.4 Slow-roll Hybrid-k-Inflation 45 The equations of motion for this Lagrangian are given by: ϕ(6x 1) + 3H(2X 1) ϕ 2 2 ϕ (3X2 X) + (2X 1) ϕ σ = 0 (3.59) σ(σ 2 + 1) and σ + 3H σ + m 2 σ 8σ Cϕ 2 (σ 2 + 1) 2 (X2 X) = 0 (3.60) To get k-inflation with the first field we have to claim 1 1/σ 2 and H 4 ( X + 3X Cϕ 2 ). 2 Inserting these two conditions into (3.59), we obtain the same equation of motion as in section 3.3.1: 6 C ϕ 3X2 X(2X 1) ϕ + 2 ϕ ϕ = (X 3X2 ), (3.61) 6X 1 with the fixed points given by (3.48). During this stage of inflation the second field σ decreases slowly, while ϕ increases. I.e. the k-inflation-term vanishes. The normal Klein- Gordon equation is left and from now on the evolution is determined by the second field σ. σ oscillates and as we have chosen a quadratic potential, the averaged equation of motion of the oscillating field σ is zero (compare section 2.1 or [22]) The number of e-foldings is given by ( ) a(tend ) tend N ln = H(t)dt, (3.62) a(t) t in where t end is given through the time, when the condition 1 >> f(σ) is violated. Numerical Calculations We investigate the properties of the full model (3.58) numerically. The parameters and initial conditions are set by: m = 0.001, γ = 1/20, ϕ(0) = 1.5, σ(0) = 3.63, ϕ(0) = 1, σ(0) = 0.5. I.e. a t 40/3. Figure 3.9 shows that w = γ 1 0.95 and thus power-law inflation takes place. After a sufficiently large number of e-foldings, N 77, the graceful exit occurs and the k-inflation-term vanishes. As we have chosen a quadratic potential, < w > 0 (see figure 3.10). We vary again the initial conditions and parameters in the program and it follows that these can lie in a wide range, but the above mentioned initial conditions have to be fulfilled. 3.4 Slow-roll Hybrid-k-Inflation In the section before we considered power-law hybrid-k-inflation, where a t 3(w+1). Now we want to come back to the exponential inflation with a exp ( ε t). Analogously to the slow-roll k-inflationary model, mentioned in section 2.3 and [6], we want to derive a model where slow-roll inflation is made by the first field ϕ (called k-essence). The graceful exit is brought by the interaction with a second field σ, i.e. we add the kinetic and potential term (Y V (σ)) and build in a further function K(ϕ, σ) = s(σ)k(ϕ). 2

46 3. k-essence and Hybrid Inflation Lagrangian and Initial Conditions We consider the following Lagrangian: p = s(σ)k(ϕ)x + X 2 + Y V (σ). (3.63) s is a function of the second field and initially should be negative due to the desired inflationary behaviour. k = k(ϕ) is a positive function of the first field and has to be introduced to have a non-zero squared speed of sound for the k-inflationary phase. Possible functions for k(ϕ) are given in [6] and [5].The idea is that if s(σ) changes its sign, the k- inflation-term s(σ)k(ϕ)x + X 2 becomes positive and the exit occurs. The energy density can be calculated, using (3.6) ε = s(σ)k(ϕ)x + 3X 2 + Y + V (σ). (3.64) To get an analogous model to the one of section 2.3, we set the initial conditions as follows: s(σ) 1 (Y + V (σ)) (s(σ)k(ϕ)x + 3X 2 ), i.e. m H with H in Equations of Motion and Analysis k(ϕ)x + 3X 2. The equations of motion are given by (3.11) and (3.12).In first order k/ ϕ and s/ σ are small. First of all we consider the k-inflationary stage. Therefore we can set s(σ) = 1 and thus we can do the same analysis as in section 2.3 with K(ϕ) = k(ϕ) < 0. The zeroth-order slow-roll solution corresponds to: X sr = 1 2 k(ϕ sr), (3.65) ϕ sr = σ k(ϕ sr ), (3.66) ε sr = 1 4 k2 (ϕ sr ), (3.67) H sr = 1 2 k(ϕ sr). (3.68) The number of e-foldings for the k-inflationary phase is given by N = tend t in H(t)dt = σend σ in H dσ, (3.69) σ with σ end = σ. σ is the critical value where the initial condition for k-inflation (s(σ) 1) is not fulfilled anymore and the function s(σ) changes its sign.

3.4 Slow-roll Hybrid-k-Inflation 47 The squared speed of sound during the k-inflationary stage is given by (2.45) c 2 s = p, X ε, X k + 2X k + 6X. (3.70) A possible function k(ϕ), which satisfies the stability requirement (2.46) and the slow roll condition (2.41) δx 2 ( ) 1 k 1 (3.71) X sr 3 ϕ is k(ϕ) = 1/ϕ. Initially the scalar field σ is chosen to be large and during the k-inflationary stage it rolls down very slowly, as H m. As a possible function for s(σ) one can chose s(σ) = tanh (σ σ ). Then the change of sign occurs when σ crosses the critical value σ. Like in the power-law model 3.3.1, there are two ways of exiting inflation: In the first case σ crosses the critical value σ when the Hubble parameter H is still larger than the mass m of the field σ. Then the k-inflation term becomes negligible in comparison to the potential V (σ). The last stages of inflation are driven by V (σ) and the exit occurs like in normal slow-roll-inflation. The time-duration of the k-inflationary stage and the potential-driven inflationary stage can be varied. In the other case the Hubble parameter H is smaller than m of the field σ. The latter begins to oscillate and then crosses the critical value σ. Then we do not have a potential-driven stage. After the change of sign of the function s(σ), the energy density of the second field σ dominates the energy density of the first field. X towards zero and thus lim w skx + X 2 ϕ lim X 0 X 0 skx + 3X = +1, 2 i.e. ε ϕ a 6. As this energy density decreases much faster than the energy density of radiation (ε rad a 4 ) and the energy density of dust (ε dust a 3 ), the first field ϕ vanishes very quickly. From now on the evolution is determined by the second field σ and the equation of motion for a potential V (σ) = 1 2 m2 σ 2 is given by: σ + 3H σ + m 2 σ = 0. (3.72) In [22] the approximate solution for this equation is given. For the case that the Hubble parameter H becomes smaller than the mass m, the WKB solution gives ( ) σ m 1/2 a 3/2 sin mdt, (3.73) i.e. σ starts to oscillate. As we haven chosen a potential V σ 2, we obtain with (2.16) an averaged equation of state of < w > 0. This is the equation for the density of non-relativistic matter (dust)

48 3. k-essence and Hybrid Inflation and thus the energy density falls as a 3. In the case of a quartic potential we would obtain < w > 1. I.e. the oscillating field σ mimics an ultra-relativistic fluid with a decreasing 3 energy density a 4. Numerical considerations with V (σ) = 1 2 m2 σ 2 We want to consider the model (3.63) numerically with V (σ) = 1 2 m2 σ 2, k(ϕ) = 1/ϕ and s(σ) = tanh (σ σ ). First of all we choose the following parameters and initial conditions: σ = 4, m = 0.0003, ϕ(0) = 100, σ(0) = 9.15, ϕ(0) = 1, σ(0) = 0.001. With this parameters and initial conditions we obtain the figures 3.12 till 3.14. Figure 3.13 shows that w 1 and thus quasi-exponential inflation takes place. During this k-inflationary stage the squared speed of sound is 0.005. After a large number of e-foldings, N > 65, the function s(σ) changes its sign and the energy density of the second field σ dominates. The k-inflationary stage ends and standard slow-roll inflation, driven by the potential V (σ), takes place. Then c 2 s = 1, compare figure 3.14. After the graceful exit at N > 75, σ starts oscillating and < w > 0, as we have chosen V σ 2. For a quartic potential we would get < w > 1. 3 We change the initial conditions and take σ = 8.5, m = 0.0003, ϕ(0) = 100, σ(0) = 10.5, ϕ(0) = 1, σ(0) = 0.001. Then the function s(σ) changes its sign at N 20 and we have a longer potential-driven inflationary stage. The squared speed of sound for these initial conditions is illustrated in figure 3.11. Like in the power-law model we see, that the time of the transit from k-inflation to the potential-driven inflation, can be varied by changing the initial value of σ and the parameter σ. This feature of the model may be important for the formation of structure, because during these two inflationary stages the cosmological perturbations have different sound speeds and therefore, different amplitudes. c 2 s 1 0.8 0.6 PSfrag replacements 0.4 0.2 10 20 30 40 50 60 70 80 Figure 3.11: Illustration of the squared speed of sound c 2 s for the initial condition σ(0) = 10.5 and the parameter σ = 8.5. During the k-inflationary phase c 2 s 0. At N 20 the transit to the potential-driven inflationary phase occurs and from now on c 2 s = 1. N

3.4 Slow-roll Hybrid-k-Inflation 49 s 1 PSfrag replacements 0.5-0.5 10 20 30 40 50 60 70 80 N -1 Figure 3.12: Illustration of the function s(σ) = tanh (σ 4) for (3.63). When s 0 the energy density of the k-essence field ϕ becomes negligible in comparison to the one of the second field. w 1 0.5 PSfrag replacements -0.5 10 20 30 40 50 60 70 80 N -1 Figure 3.13: Illustration of the equation of state w = w(n) for (3.63) with V = 1 2 m2 σ 2. Till N = 65 we have k-inflation, driven by the first field ϕ. The last e-foldings are made by the inflation with the potential V (σ). The exit occurs when we reach N 77. c 2 s 1 0.8 0.6 PSfrag replacements 0.4 0.2 10 20 30 40 50 60 70 80 N Figure 3.14: Illustration of the squared speed of sound c 2 s. During the k-inflationary phase c 2 s 0.005, while c2 s = 1 for the potential driven inflationary stage.

50 3. k-essence and Hybrid Inflation

Chapter 4 Conclusions The inclusion of non-canonical kinetic terms in the Lagrangian of a scalar field can have quite non-trivial and unexpected cosmological effects. Non-linear terms of this type are expected to appear in any effective field theory and do indeed arise in most models unifying gravity with other particle forces, including supergravity and superstring models. Using two scalar fields σ,ϕ, we have shown that inflation can be even driven by the mixterm of both kinetic-energy terms XY. For this case of two kinetically coupled massless ghost-like scalar fields we have shown, that de-sitter inflation is a late time asymptotic. This solution corresponds to an attractive fixed point, which is of the type of a stable star. This toy model might be a candidate for a dynamical dark energy. Another interesting feature of this model is a dynamical crossing of the so-called phantom divide w = 1. This crossing is not possible for general one-scalar-field models of dark energy [29]. However the observations of dark energy even slightly prefer this transition through w = 1 [1]. In the case when the Lagrangian depends not only on the field derivatives, but also on the fields, the exit from inflation can be implemented. We have introduced a further model called hybrid-k-inflation, in which the k-essence field ϕ drives inflation, while the second field σ brings the exit due to an interaction with ϕ. This hybrid-k-inflationary model can produce a power-law accelerating expansion, where a t 2/(3(w+1)), or a quasi-exponentially inflationary stage (slow-roll model), where a exp (Ht). We presented two classes of power-law models. In the first case the pressure changed its sign as the field σ rolled down and crossed a critical value σ. The last stages of inflation were driven by the potential of the second field σ and the exit from inflation occurred in the same way as in standard slow-roll-inflation. The time-duration of the k-inflationary stage and the potential-driven inflationary stage can be varied by choosing different initial conditions and parameters. Furthermore we realized a graceful exit in this model as follows: When the Hubble parameter H became less than the mass m of the field σ the latter began to oscillate, crossed a critical value σ and then changed the sign of the pressure of the k-essence. In this case no potential-driven inflationary stage occurred. In the second class of power-law models the graceful exit was realized in a different way. When the Hubble parameter H became less than the mass m of the field σ the latter began

52 4. Conclusions to oscillate near vacuum and suppressed due to a specific interaction the contribution of the k-essence ϕ to the Friedmann equations. In this model we only had k-essence and no potential-driven inflation. For the slow-roll hybrid-k-inflationary model we have shown that after a sufficiently long stage of inflation the second field σ intervened and the graceful exit from inflation can be implemented in the same way as in the power-law model. In this model the time-duration of the k-inflationary stage and the potential-driven inflationary stage can be varied as well. As in the power-law case, this model allows also to have no potential-driven stage. We have shown that the cosmological dynamics governed by two non-trivially coupled scalar fields is very rich. The models of this type can lead to two observationally different inflationary stages, occurring one after another. The transition to the Friedmann decelerating universe can occur with different energy scales, which can be independent on the generated spectrum of fluctuations. Thus the presence of an additional degree(s) of freedom makes it difficult to distinguish the various inflationary models by observations.

Appendix A Power-law Inflation Analogously to power-law inflation in [6] the formulas for a more general Lagrangian p = F (ϕ)( X + αx 2 ) (A.1) will be derived. α is a positive constant (normally taken = 1) and 0 < γ < 2. Note that 3 γ = w + 1 = (p/ε) + 1. I.e. by selection of a certain γ the power of t is set, since the scale factor evolves as a t 2 3γ. (A.2) The energy density can be calculated with (3.6) and gives p = F (ϕ)( X+3αX 2 ). Inserting p and ε into ε + p = γε one gets the fixed point X = 2 γ α(4 3γ). (A.3) Expressing ε in terms of p and substituting (A.1) with X given by (A.3) into one gets the differential equation for F (ϕ): ε tot = 3H(ε tot + p tot ), ( ) 2 F f(ϕ) 3 = 9 ϕ 4 C (A.4) with C = 4 3γ ( 2 γ (γ 1) 2 4 3γ (γ 1 + 1 ) 2 α ) γ. (A.5) Note, that if α = 1 we obtain the same result as in [6]: C = 4γ2. The differential equation 4 3γ has the following solutions: 4 F 1 (ϕ) = ( Cϕ ϕ ), 2

54 A. Power-law Inflation F 2 (ϕ) = 4 ( Cϕ + ϕ ) 2, with ϕ =constant. Thus the Lagrangian for power-law k-inflation is: p = 4 Cϕ 2 ( X + αx2 ), (A.6) with ϕ = 0 and C given by (A.5).

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Danksagung Zunächst möchte ich meinem Betreuer Prof. Dr. Viatcheslav Mukhanov danken, der mir diese Arbeit ermöglicht und mit Anregungen meine Arbeit gefördert hat. Weiterhin gilt mein Dank dem Zweitkorrektor Prof. Dr. Dieter Lüst. Ein großer Dank geht an Alexander Vikman, der immer zu Diskussionen bereit war und mir mit Rat und Tat zur Seite stand. Desweiteren möchte ich mich bei Dr. Serge Winitzki, meinen Zimmerkollegen, Kommilitonen, Freunden und Eltern für die stete Hilfsbereitschaft und Unterstützung bedanken. Erklärung Ich versichere, diese Arbeit selbständig angefertigt und dazu nur die im Literaturverzeichnis angegebenen Quellen benutzt zu haben. München, den 31.01.2006 Ulrike Wißmeier