Quintessence from Large Extra Dimensions

2003:332 CIV

MASTER'S THESIS

Quintessence from Large Extra Dimensions

Daniel Sunhede

MSc Programmes in Engineering

Department of Kiruna Space and Environment Campus

M.Sc. Thesis

Quintessence from

Large Extra Dimensions

DANIEL SUNHEDE

daniel.sunhede@phys.jyu.fi

Corrected Version− 2004.03.12

Abstract

Recent observations indicate that the majority of all matter in the universe is in the form of a dark energy component, causing the universe to accelerate.

An attractrive solution to this scenario would be quintessence, i.e. a model in which the dark energy stems from a scalar field, causing an effective cosmological constant that accelerates the universe.

This master thesis begins with a review of the standard big bang sce- nario and the present observational situation. We continue with reviewing quintessence, in particular the natural quintessence scenario. Finally, we explore in detail a recent model by Albrecht, Burgess, Ravndal and Skordis, where the possible existence of two large extra dimensions give rise to a quintessence field in our 4-dimensional picture of the world [72, 73]. We find that when astrophysical constraints are correctly accounted for, they seem to spoil the perturbative expansion employed by Albrecht et al. However, if one improves on these approximations, it turns out that the scenario is salvageable with the sacrifice of fine-tuning parameters to an anomalously large one-loop correction, while two and higher order loop corrections can be small.

Cover illustrationdepicts the symbol for quinta essentia (the fifth element). The fifth element was introduced by the Pythagoreans, believed to be the substance composing all heavenly bodies. Alongside is a schematic illustration of the brane world scenario in chapter 6. The surface represents regular, four-dimensional space-time, whereas circles represent the two extra, torus shaped dimensions.

Contents

1 Introduction 1

2 Basic Cosmology 3

2.1 The Early Universe . . . . 3

2.1.1 The Planck Scale . . . . 5

2.1.2 The GUT Scale . . . . 6

2.1.3 The Electroweak Scale . . . . 6

2.1.4 The QCD Scale . . . . 6

2.1.5 Nucleosynthesis . . . . 7

2.1.6 Radiation Decouples / Recombination . . . . 7

2.2 Relativistic Cosmology . . . . 8

2.2.1 The Cosmological Principle . . . . 8

2.2.2 Matter As a Perfect Fluid . . . . 9

2.2.3 The Friedmann Equations . . . . 10

2.3 Inflation . . . . 12

2.3.1 Problems with the Standard Big Bang Scenario . . . . 12

2.3.2 The Basics of Inflationary Cosmology . . . . 14

2.3.3 The Inflaton Field . . . . 15

2.3.4 The Potential V (φ) . . . . 16

2.3.5 Solving the Problems . . . . 17

2.3.6 Concluding Remarks . . . . 18

3 Observational Situation 19 3.1 Type Ia Supernovae . . . . 19

3.2 CMBR . . . . 23

3.3 Summary . . . . 25

4 Quintessence 27 4.1 The Cosmological Constant Problem . . . . 27

4.2 The Quintessence Scenario . . . . 28

4.3 Tracker Solutions . . . . 29

4.4 Natural Quintessence . . . . 30

4.4.1 Exponential Potentials . . . . 30

CONTENTS

4.4.2 Natural Quintessence . . . . 31

4.5 Concluding Remarks . . . . 33

5 Extra Dimensions 35 5.1 Background . . . . 35

5.2 The Hierarchy Problem . . . . 36

5.3 Large Extra Dimensions and the Hierarchy Problem . . . . . 36

5.4 Observational Constraints . . . . 38

5.5 Summary . . . . 38

6 Natural Quintessence and Large Extra Dimensions 41 6.1 Background . . . . 41

6.1.1 The Brane World . . . . 41

6.1.2 Dimensional Reduction . . . . 42

6.1.3 Quantum Corrections . . . . 43

6.1.4 Toroidal Compactification and U (r) . . . . 43

6.2 4D Effective Brane Cosmology . . . . 44

6.2.1 The Einstein Frame . . . . 44

6.2.2 The Friedmann Equations . . . . 45

6.2.3 Equation of Motion for χ . . . . 46

6.3 The Scenario of Albrecht et al. . . . 47

6.3.1 Approximate Expressions . . . . 47

6.3.2 The Cosmological Evolution . . . . 48

6.4 The Analytical Solution for χ in Terms of M

_b

r . . . . 51

6.5 Observational Constraints . . . . 53

6.6 A Valid Cosmology . . . . 55

6.7 Summary . . . . 59

7 Summary and Discussion 61

Acknowledgements 63

A Dimensional Reduction 65

B Conformal Transformation 69

C Equation of motion for χ 73

D Approximate Expressions 75

E Numerical Issues 77

Bibliography 81

Chapter 1

Introduction

The detailed measurements of the cosmological microwave background ra- diation have in recent years proven to be a very valuable source for cosmo- logical parameters. In particular, we now know that the universe is flat and composed of 30% matter and 70% dark energy. This dark energy component causes the expansion of the universe to accelerate. Although the dark energy could be in the form of a “bare” cosmological constant, a more attractive scenario would be that of quintessence, where a scalar field acts to accelarete the present universe.

The quintessence scenario is the main concern of this thesis. In par- ticular, we will study a recent model by Albrecht et al. [72, 73], where the existence of two large extra dimensions acts to provide natural quintessence.

However, we first want to review the setting for such a scenario. Chapter 2 reviews the standard big bang scenario and chapter 3 gives an overview of the present observational situation. The quintessence scenario gets re- viewed in chapter 4, where we in particular discuss natural quintessence.

We continue by giving a brief background to the theory of extra dimensions in chapter 5. Finally, chapter 6 presents a detailed study of the Albrecht et al. scenario.

On notations: For general relativity we follow the sign conventions of

Landau and Lifshitz [4]. All equations are written in natural units (~ = c =

k

_B

= 1) unless the fundamental constants are explicitly displayed. With the

exception for chapter 2, we use the reduced planck mass M

_Pl

≡ (8πG)

^−1/2

.

Chapter 2

Basic Cosmology

In this chapter we will review some basic cosmological concepts as well as observations. This will be done to an extent suitable for the upcoming chapters and the review shall by no means be thought of as trying to cover the subject as a whole. For more complete discussions on cosmology, the reader is referred to Kolb & Turner [1], Peebles [2], and Weinberg [3].

2.1 The Early Universe

If cosmology was born together with general relativity [5], it left its infancy when Hubble discovered the expansion of the universe [6]. Since the universe is expanding today, becoming colder and less dense, reversing time will make it hotter and more dense. Extrapolated far enough back in time we will ultimately reach an extremely small, dense and hot state. The event which at this point set off the expansion is simply referred to as the big bang.

The expansion of the universe is only one of many observations that makes us believe in the big bang model. Indeed the main observations in favour of big bang are the following:

1. The universe is expanding. Or more specifically stated: Redshift of galaxies are proportional to their distance (Hubble’s law).

2. The sky is filled with electromagnetic radiation. This cosmic mi- crowave background is isotropic and has a black-body spectrum with temperature T ≈ 2.73 K.

3. Relative abundances of light elements; 99.9999% of all matter is made up of hydrogen and helium. He:H ≈ 0.25.

4. The universe is made of matter, with the mass ratio number of baryons to photons n

_b

/n

_γ

= 5 × 10

⁻¹⁰

.

Keeping these observations in mind, we will now review the standard big

bang model.

Basic Cosmology

Planck scale

GUTNuclear physicsAtomic physics Now

QCD Electroweak symmetry breaking

NucleosynthesisRadiation decouplesGalaxy formationQuark confinement

-13 10-9 10-6 10-3 100 103 106 109 1012 1015 1018 10PlM

5 1010 1015 1030 1020 1025 1032 102.73

T [K] E [ G e V]

17 1012 106 100 10-6 10-12 10-18 10-24 10-30 10-36 10-42 10 13.7 Gyr

t [ s]

TH E BI G BAN G

F ig u re 2. 1:

Overviewoftheevolutionoftheuniversefrombigbangtopresent,showingtheageoftheuniverset,thetemperatureT, andthecorrespondingenergyE=kT.

2.1 The Early Universe

The evolution of the universe can roughly be divided into four different eras, each one dominated by specific types of interactions:

• 0 - 1 min. – Electroweak and strong interactions.

• 1 - 30 min. – Nuclear interactions.

• 30 min. - 100 000 years – Electromagnetic interactions.

• 100 000 years - Present – Gravitational interactions.

Figure 2.1 outlines the evolution from big bang to present.

There is concrete evidence for the big bang model dating back to 1 − 100 s after the big bang. It his however possible to make educated speculations back to times as early as 10

⁻⁴³

s. Such speculations are possible via the theories of fundamental interactions at very short distances. Just as the weak and electromagnetic interactions are two different aspects of the same electroweak force, the strong interaction is believed to join them at even higher energies. Finally, this combined force will come together with gravity.

The assumption that all fundamental interactions are different aspects of one fundamental force is referred to as grand unification. Using this line of argument as a starting point we will now discuss the main events of the above figure.

2.1.1 The Planck Scale

T ∼ 10¹⁹ GeV

Since there is yet no theory which quantizes gravity, we expect the known laws of nature to break down at some point. To estimate when this happens one imagines a piece of matter/particle so dense that it is contained within its own Schwarzchild radius. For an observer residing outside this radius it is impossible to make any measurements on the particle. From quantum mechanics, the wavelength or “size” of a particle is given by its Compton wavelength. Setting the Schwarzchild radius equal to the Compton wave- length

2mG c

²

= h

mc (2.1)

yields the mass. Introducing a factor π, one defines this as the Planck mass

M

_Pl

≡

 ~ c G



1/2

= 2.18 × 10

⁻⁵

g = 1.22 × 10

¹⁹

GeV/c

²

. (2.2)

Working in natural units one easily obtains the corresponding Planck

length, l

_Pl

= 1.61 × 10

⁻³³

cm, and Planck time, t

_Pl

= 5.38 × 10

⁻⁴⁴

s. For

time intervals shorter than this, one end of the above piece of matter would

cease to be aware the other end. That is, the laws of causality would no

longer apply. Clearly, this is where the known laws of physics break down.

Basic Cosmology

There is no guarantee that they apply this early, but they will definitely not be valid at earlier times. Before the Planck time, the density was & 10

⁹⁴

g/cm

³

and all fundamental interactions are believed to have been unified as one fundamental force. At t ' t

Pl

, gravitation decoupled from the other forces.

¹

2.1.2 The GUT Scale

T∼ 10¹⁶ GeV

The known laws of nature break down at the Planck scale, but they have been confirmed to be valid at the electroweak scale. It is believed that somewhere inbetween these scales, the strong and the electroweak force were unified. This period is referred to as the GUT scale (Grand Unified Theory).

Either at the Planck or the GUT scale there was an inflationary period in the universe. That is, at some stage space-time underwent a super-luminal expansion. Except for making the universe flat this expansion also had some other effects. Section 2.3 will discuss inflation in greater detail.

2.1.3 The Electroweak Scale

T ∼ 100 GeV

Below the GUT scale the universe was filled with a relativistic plasma con- sisting of known particles: Bosons (γ, W

^±

, Z

⁰

, g, H

^±

, H

⁰

)

²

, leptons (e, ν

_e

, µ, ν

_µ

, τ , ν

_τ

), quarks (u, d, c, s, t, b), and all their antimatter counterparts.

All particles except H

⁰

were massless, and all of them were in thermal equi- librium. Simply put, the universe consisted of a primordial soup of particles, constantly being created, annihilated and transformed into each other.

At T ∼ 100 GeV electroweak symmetry breaking occurs. Below the phase transition temperature, the ground state of the theory breaks the symmetry

³

when Higgs scalar fields acquire nonzero expectation value. All particles coupling to these fields become massive. Quarks, leptons e, µ, τ and bosons W

^±

, Z

⁰

attained their masses at this point.

The electroweak scale is the latest time for creating a slight overproduc- tion of matter with respect to antimatter [7]. The main (vast) amount of matter and antimatter later annihilates, leaving just the overproduced part.

2.1.4 The QCD Scale

T ∼ 100 MeV

The thermal energy is now too low for keeping quarks free and they become bound into hadrons (quark confinement). With the continuing decrease in temperature many hadrons decay and almost all hadrons later annihi- late with antihadrons, leaving the observed predominance of baryons over

1Strictly speaking, the term grand unification was originally used when talking about a unification of the strong and the electroweak force. Hence the below headline.

2H^±, H⁰ – Higgs bosons.

3Spontaneous symmetry breaking.

2.1 The Early Universe

antibaryons. At this point the ratio between the number of quarks and pho- tons got frozen at about 10

⁻⁹

. Among baryons, protons and neutrons were in thermal equilibrium.

2.1.5 Nucleosynthesis

T' 0.1 − 1 MeV

When the temperature reaches about 1 MeV, only γ, e, neutrinos and a small amount of baryons, n

_b

/n

_γ

∼ 10

⁻⁹

, is left. The reaction rates keeping neutrinos in thermal equilibrium are no longer greater than the expansion rate of the universe, and neutrinos decouple from the rest of the plasma to form the neutrino background.

A little later, electrons and positrons annihilate and the thermal equi- librium between protons and neutrons comes to an end as well. Neutron decays continue, decreasing the ratio of neutrons to protons. Just before neutrinos froze out, the equilibrium value of this ratio was about

¹₆

, but when the universe has cooled down to about 0.1 MeV, neutron decays have resulted in a ratio of about

¹₇

.

The important new scale which enters during nucleosynthesis is the nu- clear binding energy (. 1 MeV). At about 0.1 MeV, light elements begin to form very rapidly. Almost all neutrons suddenly get caught inside nu- clei, freezing the ratio of neutrons to protons. Due to various reasons, no elements heavier than

⁴

He are produced in any significant amounts. When nuclear reactions stop, they have resulted in the following amounts of differ- ent elements:

⁴

He:H ≈ 0.25 (mass ratio),

²

H:H ' 10

⁻⁵

,

³

He:H ' 10

⁻⁵

and

7

Li:H ' 10

⁻¹⁰

(number density ratios).

In summary, different reaction rates during nucleosynthesis determine the produced amount of a species. By a detailed analysis one can explicitly show that all different reactions indeed end up in the observed ratio of helium to hydrogen, as well as the lack of heavier elements and other ratios. In this sense, nucleosynthesis is both the earliest and most convincing test of the hot big bang model. It shows that standard cosmology is valid back to temperatures of about 1 MeV and the laws of physics could not have changed much since these times. Indeed, this observation will turn out to be of great importance to us in chapter 6.

Finally, nucleosynthesis also provides a strong constraint on the baryon density parameter Ω

_b

. Values outside the interval 0.017 ≤ Ω

b

h

²

≤ 0.021 would not produce the observed ratios of different elements [8, 9].

2.1.6 Radiation Decouples / Recombination

T ∼ 1 eV

When the universe is about 10

¹¹

s old the matter density has become equal to

the radiation density and structure formation slowly begins. At 10

¹³

s after

the big bang the temperature has cooled down to T ' 1 eV, comparable to

atomic binding energies. Soon photons are no longer able to keep electrons

free and atoms form. The universe is now transparent for electromagnetic

Basic Cosmology

radiation and the decoupled photons form the cosmic microwave background.

This ends the near thermal equilibrium that existed in the early universe.

The surface of last scattering for the cosmic microwave background radiation (CMBR) provides us with a snapshot of the universe at decoupling. CMBR will be discussed further in chapter 3.

2.2 Relativistic Cosmology

2.2.1 The Cosmological Principle

The proper framework for describing the expansion of the universe after the Planck era is general relativity. Relativistic cosmology is based on two assumptions, where the first one is the cosmological principle. The cosmo- logical principle states that the universe is both homogeneous and isotropic.

This idea was at first introduced as a simplifying assumption, but observa- tions have now given very compelling evidence that this indeed is the case.

See figure 2.2.

For a homogeneous and isotropic space the most general space-time in- terval is the Friedmann-Robertson-Walker line element. Using spherical

 





 

 



 







 









Figure 2.2:

Coneplot of the full 2dF Galaxy Redshift Survey, illustrating the cosmological principle. The figure shows the distribution of galaxies across the sky, where the radial coordinate is distance from earth and the angular coordinate is angle on the sky. [10]

2.2 Relativistic Cosmology

coordinates (r, θ, φ) for space, it can be written in the form ds

²

≡ g

µν

dx

^µ

dx

^ν

= dt

²

− a(t)

²

 dr

²

1 − kr

²

+ r

²

(dθ

²

+ sin

²

θdφ

²

)



, (2.3) where g

_µν

is the metric tensor and the curvature k = −1, 0, 1, represents a spherical, Euclidean and hyperbolic geometry, respectively. More commonly referred to as the closed, flat and open universe. The most striking property of the Friedmann-Robertson-Walker line element is that the bracketed part is independent of time. These comoving coordinates remain constant for a specific point in space, while the expansion of the universe is encoded in the scale factor a(t).

⁴

The scale factor is the single parameter determining the evolution and it provides a measure on the relative size of the universe. t is usually referred to as cosmic time or world time, where each t = const.

hypersurface is a world map of simultaneous events (simultaneous in world time, that is).

2.2.2 Matter As a Perfect Fluid

The second assumption of relativistic cosmology is that the matter in the universe can be treated as a perfect fluid. The stress-energy tensor for a perfect fluid is given by

T

_µν

= (ρ + p)u

_µ

u

_ν

− pg

µν

, (2.4) where ρ represents its energy density, p is the pressure and u

_µ

is the nor- malized four-velocity of the fluid. Conservation of energy (i.e. the µ = 0 component from the conservation of stress-energy, T

^µν_;ν

= 0), yields the 1st law of thermodynamics

d(ρa

³

) = −pd(a

³

) , (2.5)

or, equivalently

dρ = − 1

a

³

(ρ + p)d(a

³

) . (2.6) The 1st law of thermodynamics states that the change in energy in a co- moving volume element is equal to minus the pressure times the change in volume.

The equation of state for a perfect fluid can in many cases be described by

p = wρ , (2.7)

where w is independent of time. Using (2.5) it is straightforward to show that this gives the following density evolution

ρ ∝ 1

a

^3(1+w)

. (2.8)

4Note that r is dimensionless, while it is a(t) that has the dimension of length.

Basic Cosmology

Obviously w is zero for pressureless matter (dust), corresponding to ρ ∝ a

⁻³

. For relativistic matter (radiation) w equals one third, corresponding to ρ ∝ a

⁻⁴

. Hence, even though the universe is matter dominated at present, it must have been radiation radiation dominated at sufficiently early times.

From equation (2.6) it follows that w = −1, corresponding to negative pressure, is an interesting special case. All states of matter with w > −1 gets diluted when the universe expands (d(a

³

) > 0), but for w = −1 the energy density remains constant. Below we will see that such a peculiar equation of state does indeed arise naturally.

Figure 2.3 illustrates the above special cases and shows the aforemen- tioned radiation domination in the early universe. Whether things will stay matter dominated is still an open question and we will partly address this when discussing quintessence. Note that if the universe is indeed flat due to a cosmological constant, one needs a tremendous coincidence for ρ

_Λ

to be on the same order as ρ

_m

at present. This is usually referred to as the cosmic coincidence problem.

-12 -10 -8 -6 -4 -2 0

-130 -120 -110 -100 -90 -80 -70

lg a

ρlg

w = -1, Λ w = 1/ 3, radiation

w = 0, dust

Figure 2.3:

The evolution of ρ for different types of matter. The universe is assumed to be flat at present, where the scale factor has been set to one.

2.2.3 The Friedmann Equations

The Friedmann-Robertson-Walker line element provides the form of the met-

ric g

_µν

for a homogeneous and isotropic universe, but the dynamical equa-

2.2 Relativistic Cosmology

tions for g

_µν

are given by the Einstein equations R

µν

− 1

2 g

µν

R = 8πGT

µν

+ Λg

µν

, (2.9) obtained from the Lagrangian

L = √

−g(R − 2Λ) + κL

m

, (2.10)

where L

m

is the matter Lagrangian and κ = 8πG is the coupling strength of the gravitational field to matter.

The phenomenology of (2.9) is quite simple; space-time (represented by the Ricci tensor R

_µν

and the Ricci scalar R) gets curved by matter (T

_µν

) and possibly the cosmological constant (Λ). The Einstein equations corresponding to the metric (2.3) and ideal fluid stress-energy (2.4) yield the Friedmann equations

 ˙a a



2

+ k

a

²

= 8πG 3 ρ + Λ

3 , (2.11)

¨ a

a = − 4πG

3 (ρ + 3p) + Λ

3 . (2.12)

From a classical point of view, the cosmological constant is not expected.

But it can be added and it indeed shows up in the most general formulation of the gravitational action (corresponding to (2.10)). When Einstein formu- lated the theory of general relativity, the universe was thought to be static.

Therefore, he did include the cosmological constant, making static solutions possible. In these solutions, the repulsive Λ exactly balances the gravita- tional attraction of matter. This idea along with the constant was however dropped with the discovery of the expanding universe. Nevertheless, since the cosmological constant shows up in the most general formulation, a com- plete cosmological theory must either explain its value or show why it should be exactly zero.

In the Friedmann equations, terms containing ρ and p correspond to ordinary matter states, but it is clear that Λ is equivalent to a matter state with w = −1. That is, matter with a negative pressure trying to expand the universe. The recent discovery that the expansion of the universe seems to be accelerating (for a recent update see [11, 12]), have thus once again put Λ in the limelight. The “bare” value of the cosmological constant is identified as the vacuum energy density.

⁵

The expansion rate of the universe is measured via the Hubble param- eter H ≡ ˙a/a. In general, H ∝

¹_t

and sets the timescale for the evolution

5The Einstein equations would in fact be somewhat unsatisfactory if there wasn’t a term corresponding to vacuum energies, since they show up over and over again in quantum field theory.

Basic Cosmology

of a. The universe roughly doubles its size during the time H

⁻¹

. Although the parameter is not constant in time, its present day value H

₀

is gen- erally referred to as the Hubble constant. The actual value of H

₀

isn’t known very well and it is usually written as H

₀

= 100h km/s Mpc

⁻¹

, where h = 0.72 ± 0.03 (statistical) ± 0.07 (systematic) [13] (for an estimate based on CMBR measurements, see [12]).

Incorporating Λ in ρ of the first Friedmann equation (2.11), one obtains the following expression for k

k

a

²

H

²

= ρ

3H

²

/8πG − 1 = Ω − 1 , (2.13) where

Ω ≡ ρ ρ

c

ρ

_c

≡ 3H

²

8πG . (2.14)

We know from observations that the present value of the density parameter Ω is very close to 1 [12]. If k = 0 then by definition Ω = 1. In other words, if the present universe is flat, its density is very close to the critical density ρ

_c

. Since (2.13) is valid for all times, it also has an interesting implication for the curvature of the very early universe. It turns out that the early universe has to be extremely flat in the beginning in order to be flat today as well.

This is a very undesirable feature of the big bang model and we will return to this problem when discussing the theory of inflation.

The acceleration/deceleration of the universe is measured via the decel- eration parameter q ≡ −¨a/(aH

²

). Putting this into (2.12) and using the fact that the present universe is dominated by matter (p ≈ 0), one obtains the following expression for the present deceleration:

q

₀

= 1

2 Ω

_m

− Ω

Λ

, (2.15)

where Ω

_m

is the present value of the matter density parameter and Ω

_Λ

≡ ρ

Λ

/ρ

c

≡ (Λ/8πG)/ρ

c

. Hence, wether the present universe is accelerating or decelerating is completely determined by Ω

_m

and Ω

_Λ

.

2.3 Inflation

2.3.1 Problems with the Standard Big Bang Scenario

The big bang model is very successful in predicting many of the phenomena

observed in the universe today. However, it does leave a number of problems

unsolved. In the context of inflation one usually discuss the flatness and

the horizon problem, large-scale structures and relics. We will see that

although inflation originally was introduced to solve the flatness and the

horizon problem, it also offers useful ways to get past the other obstacles.

2.3 Inflation

The flatness problem. It was previously mentioned that if the universe is flat today it must have been extremely flat in the beginning. We restate equation (2.13), slightly rewritten:

|Ω − 1| = |k|

a

²

H

²

. (2.16)

Using the first Friedmann equation (2.12) it is straightforward to show that for a matter dominated universe a ∝ t

^2/3

, and for a radiation dominated universe a ∝ t

^1/2

. By using the definition of H the above yields

|Ω − 1| ∝ t

^2/3

during matter domination,

|Ω − 1| ∝ t during radiation domination. (2.17) Hence, if the present universe is approximately flat one is forced to fine-tune Ω to a huge degree as t approaches zero. This would indicate that we live at a very special time in the universe, certainly not desirable. The alternative is that the universe indeed was almost identical to flat at early times, but the big bang model offers no mechanism which could accomplish this.

The horizon problem. Although the big bang gives the origin of the CMBR, it does not explain why it is isotropic. For the sky to have almost the exact same temperature everywhere, naturally one would expect that the whole universe must have been in thermal equilibrium at recombination.

But there was no time for reaching this. For the universe to be in thermal equilibrium it must also be in causal contact. A particle is in causal contact with all objects inside its horizon, that is, the comoving distance a light signal emitted by the particle travels during the time t. This radius is given

by Z

_t

0

dt

⁰

a(t

⁰

) . (2.18)

Consider the causally connected parts of the universe at the time of recom- bination compared to the ones at present;

Z

trec

0

dt a(t) 

Z

t0

trec

dt

a(t) . (2.19)

That is, a much larger part of the universe is visible today than at recombi- nation. Following the above line of argument it would thus be impossible for the CMBR to be isotropic. In fact, regions separated by more than roughly one degree in the sky today were disconnected at recombination [14].

Large-scale structures. In addition to the horizon problem, the detailed

form of the inhomogeneity of the universe is left unexplained as well. On

large scales, galaxies are observed to cluster, creating a “foam” filled with

voids on the order of 100 Mpc. Moreover, the CMBR spectrum show

Basic Cosmology

anisotropies at angular scales of up to one degree. There are some mech- anisms such as topological defects that could cause these large-scale struc- tures, but they are not compatible with the fine details of the observed CMBR spectrum.

Relics. During phase transitions in the very early universe, many gauge theories predict the creation of various topological defects such as magnetic monopoles, cosmic strings and domain walls. Magnetic monopoles are point- like defects carrying a magnetic net charge. These are in several models created in such extreme amounts that they easily overclose the universe. If magnetic monopoles were created in substantial amounts, how come they are not observed at present? Both domain walls and cosmic strings are in conflict with observations as well, see Kolb & Turner [1] for details.

2.3.2 The Basics of Inflationary Cosmology

Inflation was introduced by Alan Guth [15] and Andrei Linde [16] as a means to solve the horizon and flatness problem. Adding inflation may seem as somewhat ad hoc, but inflation does stem from the particle theories which are already an integral part of physics in the very early universe.

The basic idea of inflation is simple. One assumes that there was an epoch when the universe was dominated by the vacuum energy component.

That is, the very early universe underwent an accelerated expansion. De- noting the vacuum energy density by ρ

_φ

, the first Friedmann equation (2.11) then reduces to

 ˙a a



2

= 8πG

3 ρ

_φ

, (2.20)

where we have neglected the curvature and matter contributions. This can be motivated by the fact that when ρ

_φ

starts to accelerate the universe, we expect the large increase in a to make the curvature ( ∝ a

⁻²

), matter ( ∝ a

⁻³

) and radiation ( ∝ a

⁻⁴

) terms negligible. Since ρ

_φ

is constant, the solution to (2.20) is given by

a(t) ∝ e

^Ht

, t ∈ [t

i

, t

_f

], (2.21) where H =

q

8πG

3

ρ

_φ

, t

_i

is the time when ρ

_φ

begins to dominate and t

_f

is the time when inflation ends. The characteristic timescale for a typical inflationary model is H

⁻¹

∼ 10

⁻³⁴

s.

Models of the type (2.20) are referred to as de Sitter models. For true

vacuum, such a solution would continue to accelerate forever, redshifting

away both curvature and all forms of matter to the point where the universe

is virtually empty. This is obviously not the case, and that is why we demand

the expansion to end at some time t

_f

. Furthermore, the success of big bang

nucleosynthesis requires t

_f

 t

nucleos

.

2.3 Inflation

2.3.3 The Inflaton Field

The idea, or “definition” of inflation is indeed very general, making it possi- ble for a variety of sources to drive the exponential expansion. The crucial part is however to be able to “switch off” the expansion at some point. The majority of current models (and the original scenarios) realize the above by postulating a scalar matter field φ, usually referred to as the inflaton.

The action of the inflaton field is in general given by S =

Z d

⁴

x √

−gL , (2.22)

L = 1

2 (∂φ)

²

− V (φ) , (2.23)

where g is the determinant of the metric tensor and V (φ) is the potential of the field. The Lagrangian density L gives the stress-energy tensor and hence the density and pressure due to the field:

ρ

_φ

= 1

2 φ ˙

²

+ 1

2a

²

( ∇φ)

²

+ V (φ) , (2.24) p

_φ

= 1

2 φ ˙

²

− 1

6a

²

( ∇φ)

²

− V (φ) , (2.25) expressed in comoving coordinates. Assuming the field is homogeneous

⁶

one obtains

ρ

_φ

= 1

2 φ ˙

²

+ V (φ) , (2.26)

p

_φ

= 1

2 φ ˙

²

− V (φ) . (2.27)

From the equations above one sees that the requirement p = −ρ is ful- filled when ˙ φ

²

 V (φ). That is, when the kinetic energy of the field is negligible compared to the potential energy, the inflaton behaves as repul- sive matter causing the universe to accelerate. To determine the conditions for this to occur one needs the dynamical equation for φ. Varying the action with respect to φ results in the equation of motion for the inflaton field,

φ + 3H ˙ ¨ φ + V

⁰

(φ) = 0 , (2.28) where V

⁰

(φ) ≡ ∂V/∂φ. Terms containing ∇φ have been dropped since the field is assumed to be homogeneous. The above equation is similar to an equation describing the movement of a ball under a force V

⁰

with a friction term 3H ˙ φ. Within this context one refers to the requirement ˙ φ

²

 V (φ) as slow-rolling.

6Note that since the physical gradient is related to the comoving gradient via ∇physical= a⁻¹∇^comoving, inhomogeneities of the field are redshifted away when a increases. However, the homogeneity of the field is nothing but a simplifying assumption at this point.

Basic Cosmology

2.3.4 The Potential V (φ)

For the inflaton field to be acting as a cosmological constant, we want the field to be slowly rolling, ˙ φ

²

 V (φ), for a long time. This will be the case if the acceleration is also small, that is, if ¨ φ  V

⁰

(φ). Usually one imagines a potential roughly on the form of figure 2.4. The inflaton is assumed the to be in a false vacuum state at t

_i

, slowly rolling down towards the true vacuum. If the potential is flat enough, the time it takes for the inflaton to roll down is long compared to the expansion rate of the universe, resulting in the exponential expansion of (2.21). As φ comes closer to the minimum, the potential “steepens” and φ rolls down much faster. The field is no longer slow-rolling, the kinetic term begins to dominate, ˙ φ

²

 V (φ), and inflation ends.

Reheating tf

Slow-roll ti

φ V( )

φ

Figure 2.4:

Schematic illustration of an inflaton potential.

Due to the large kinetic term, φ will inevitably overshoot the minimum

and begin to oscillate around it. All excess energy of the false vacuum

state is now released in the process of reheating. This occurs since the above

oscillations are damped by the decay of inflatons into lighter particles. These

particles thermalize, increasing the temperature of the universe which has

cooled down significantly during the previous expansion. Note that the time

when full decay is accomplished, t

_r

, can be much later than when inflation

ends, i.e. t

_r

 t

f

. The detailed mechanism of how reheating occurs is still

being debated.

2.3 Inflation

2.3.5 Solving the Problems

With the very basic parts of inflation at hand it is easy to see how it solves the problems mentioned above. We revisit them in the previous order.

The flatness problem. Since Ω ≈ 1 at present, the standard big bang scenario implies that Ω must be extremely close to one at very early times.

In the inflationary scenario Ω is instead driven towards one, independently of initial conditions. When vacuum density dominates, the expansion of the universe is accelerating, ¨ a > 0 ⇔

_dt^d

(aH) > 0. Since |Ω − 1| = |k|/(aH)

²

, the total energy density is thus forced closer and closer to one during the acceleration. In the case of exponential expansion,

|Ω − 1| = |k|

H

²

e

^−2Ht

, (2.29)

and Ω is driven towards one so rapidly that the universe becomes indistin- guishable from being flat.

The horizon problem. We previously saw that the observable universe could not have been in thermal equilibrium at recombination. Hence, there must have been some other course of events responsible for the isotropic tem- perature distribution observed today. Consider the universe before inflation.

Inevitably, there will exist some regions that are in thermal equilibrium.

That is, if expansion is large enough, such a region could blow up to the size of the observable universe. The amount of expansion needed is measured via the number e-foldings N ≡ log

^a(t_a(t^f_i)⁾

. One can show that N & 60 solves the horizon problem [17], a number easily realized within the majority of inflationary models. In summary, as long as inflation takes place during a certain number of e-foldings it will smoothen out all inhomogeneities.

Large-scale structures. Although the huge expansion during inflation acts to smoothen out all pre-inflationary inhomogeneities, it will also provide the “seeds” that later turn out as large-scale structures. Due to the tremen- dous expansion of space-time, the inevitable vacuum quantum fluctuations will be enlarged to macroscopic scale. These fluctuations will later evolve into the present structure of the universe. The so-called scale-invariant spec- trum of such fluctuations seem to fit nicely with CMBR data (for a review see [18]).

Relics. The solution to the problems of primordial relics is trivial. In-

flation causes the universe to increase its size tremendously. Hence, the

number density of topological defects will be diluted to the point when they

might be no longer observable. However, it is important to note that for

any inflationary model, one must make sure that reheating does not raise

the temperature to levels where these defects are created once again.

Basic Cosmology

2.3.6 Concluding Remarks

In summary, inflation is the idea that the very early universe underwent an accelerating expansion. This expansion lasted long enough to make the universe flat and resulted in the observed universe being homogeneous and isotropic. In addition to solving these problems, inflation also offers a useful way to provide “seeds” for large-scale structures and to get rid of unwanted relics.

The above discussion was carried out with only small assumptions on the form of the potential. That is, a remarkable feature of inflation is the rela- tively large freedom it gives for choosing V (φ). In the overall picture one can however distinguish for major models, namely old inflation (Guth’s original scenario) [15], new inflation (discussed above) [16], chaotic inflation [19] and hybrid inflation (involves several scalar fields) [20]. Chaotic inflation does not rely on phase transitions, instead the value of φ is displaced from the vacuum state by means of quantum fluctuations. Due to a large damping term in the equation of motion, the scalar field will roll down slowly and the precise form of V (φ) is irrelevant to the mechanism.

Finally, one shall remember that inflation is not constrained to arise

from scalar fields. The basic requirement on the mechanism is just that the

universe shall go through an accelerating expansion which will stop at times

much earlier than nucleosynthesis. Extensive reviews on inflation can be

found in [21, 22].

Chapter 3

Observational Situation

This chapter reviews the present observational situation. We will treat ob- servations concerning the energy density parameters, primarily since these makes us able to determine the curvature of the universe and the decelera- tion parameter q

₀

. Although there are several methods in determining Ω

_m

and Ω

_Λ

, we will focus on the two most important ones: Type Ia supernovae and CMBR observations.

3.1 Type Ia Supernovae

In astronomy, one usually measures distance in terms of the distance mod- ulus m − M, where m is the apparent magnitude and M is the absolute magnitude. The distance modulus is related to the luminosity distance d

L

via

m − M = 5 lg d

L

+ 25 , (3.1)

where d

_L

is measured in units of Mpc. Additionally, the luminosity distance depends on cosmological parameters, d

_L

= d

_L

(z, Ω

_m

, Ω

_Λ

, H

₀

).

For obvious reasons it is not possible to directly measure the absolute magnitude M . Instead one needs to identify a population of “standard can- dles”, i.e. a certain type of object easily identifiable and abundant, where all members have the same absolute magnitude and therefore can be calibrated by observations nearby. For distance measurements to nearby galaxies one can use cepheids as such standard candles. Classical cepheids are pulsat- ing, population I supergiants belonging to spectral class F-K (prototype δ-cephei), whose period of pulsation correlates well with their luminosity.

By using parallax to measure the distance to cepheids in the Milky Way one can then use their period to determine the distance to cepheids in nearby galaxies. With the Hubble space telescope cepheids can be used to measure distances up to the Virgo cluster some 16 Mpc away.

Having established distances to nearby galaxies one needs another stan-

dard candle which is detectable at much larger distance scales. By com-

Observational Situation

paring the apparent magnitude of such an object in a nearby galaxy to the one in a remote galaxy, one can hence determine the distance. Type Ia supernovae (or at least a large subsample of these) are good candidates for this since they seem to have nearly uniform and large intrinsic luminosity, M ∼ −19.5 [23]. This extreme brightness allow them to be detectable at high redshifts, up to z ∼ 1, which is needed for observing cosmological ef- fects. A type Ia supernova is generally believed to be a white dwarf, accreting mass from a companion star until it reaches the Oppenheimer-Volker mass of 1.4M



. After this the white dwarf collapses and thermonuclear pressure in the core rips it apart in the observed explosion. Since an accreting white dwarf always explode at 1.4M



, all type Ia supernovae have close to similar characteristics.

Calan/Tololo (Hamuy et al, A.J. 1996)

Supernova Cosmology Project

effective mBmag residualstandard deviation

(a)

(b)

(c)

(0.5,0.5) (0, 0) ( 1, 0 ) (1, 0) (1.5,–0.5) (2, 0) (Ω_Μ,Ω_Λ) = ( 0, 1 )

Flat

(0.28, 0.72)

(0.75, 0.25 ) (1, 0) (0.5, 0.5 ) (0, 0) (0, 1 ) (Ω_{Μ ,}Ω_Λ) =

Λ = 0

redshift z

14 16 18 20 22 24

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

0.0 0.2 0.4 0.6 0.8 1.0

-6 -4 -2 0 2 4 6

Figure 3.1:

Hubble diagram from the Supernova Cosmology Team [25] with curves representing different choices of Ωmand ΩΛ. The bottom figure shows the number of standard deviations of each point from the best-fit curve.

3.1 Type Ia Supernovae

Two independent groups have made searches for distant supernovae in order to measure cosmological parameters. Figure 3.1 shows the results in terms of m − M vs. z for the Supernova Cosmology Project [24, 25], and figure 3.2 shows the corresponding results for the High-Z Supernovae Search Team [26, 27, 28]. What makes the above data interesting is that, by comparing observations with theoretical predictions, one can use it to put limits on Ω

_m

and Ω

_Λ

. The theoretical predictions are sensitive on model parameters, in particular the deceleration q

₀

, and hence supernovae data put limits on the linear combination

¹₂

Ω

_m

− Ω

Λ

. The resulting bounds can be seen in figure 3.3 and 3.4. It is clear that the independent results do agree, but what is more surprising is that both teams favour a positive cosmological constant and strongly rule out the traditional flat universe filled with ordinary matter, (Ω

m

, Ω

Λ

) = (1, 0).

34 36 38 40 42 44

ΩM=0.24, Ω_Λ=0.76 Ω_M=0.20, Ω_Λ=0.00 ΩM=1.00, Ω_Λ=0.00

m-M (mag)

MLCS

0.01 0.10 1.00

z -0.5

0.0 0.5

∆(m-M) (mag)

Figure 3.2:

Hubble diagram from the High-Z Supernovae Search Team [27] with curves representing different choices of Ωm and ΩΛ. The lower figure shows the difference between the observed distance modulus and that of an open universe.

Observational Situation

Ω

Μ No Big Bang

1 2

0 1 2 3

expands foreve r Ω

Λ

Λ = 0Flat Universe

-1 0 1 2 3

2 3

closed ope

n 90%

68%

99%

95%

recollapses eventu ally

flat

Figure 3.3:

Constraints in the ΩmΩΛ-plane from the Supernova Cosm. Team [25].

0.0 0.5 1.0 1.5 2.0 2.5

ΩM

-1 0 1 2 3

ΩΛ

68.3%

95.4%

95.4%

99.7%

99 .7%

99.7%

No Big Bang

Ωtot=1 Expands to Infinity

Recollapses Ω_Λ=0 Open

Close d Accele

rating Decele

rating q0=0 q0=-0.5

q0=0.5

^

MLCS

Figure 3.4:

Constraints in the Ω^mΩ^Λ-plane from the High-Z SNe Search [27].

3.2 CMBR

In contrast (as we will see in the following section), CMBR points to- wards a flat universe. For this geometry Perlmutter et al. [25] find the best fit value of Ω

_m

to be

Ω

m

= 0.28

^+.09_−.08

(statistical)

^+.05_−.04

(systematic) , (3.2) so that

Ω

_Λ

≈ 0.7 , (3.3)

where Λ > 0 with a confidence level of 99%. The data is also found to be inconsistent with an open universe with zero cosmological constant. The High-Z Supernovae Search independently come to the same conclusions and their measurements are consistent with an accelerating expansion, q

₀

< 0, with a confidence level of 99.5% [27].

3.2 CMBR

Along with the expansion of the universe, the observation of the cosmic microwave background radiation [29] is the most important evidence for the standard big bang model. The CMBR is almost perfectly isotropic, but small anisotropies were detected in 1992 by the COBE satellite [30], later by several balloon experiments and most recently by the WMAP satel- lite [11, 12]. These small temperature variations are caused by primordial density fluctuations and acoustic oscillations between baryons and photons at recombination. Moreover, the detailed form of these fluctuations de- pend intricately on the cosmological parameters. By measuring the CMBR anisotropies to a high accuracy it is thus possible to obtain very detailed information about almost all of the fundamental cosmological parameters.

A map of the temperature fluctuations across the sky is shown in figure 3.5. This picture still has a lot of foreground contamination coming in particular from the Galaxy (the horizontal stroke in the the middle). The spectrum of foreground sources differs from the CMBR signal, and can be removed by use of several independent maps from measurements at different frequencies. The resulting map of temperature fluctuations is then expanded in spherical harmonics Y

_lm

(naturally geocentric coordinates):

∆T

T (θ, φ) = X

lm

a

_lm

Y

_lm

(θ, φ) . (3.4)

The studied quantity is however the angular two-point correlation function:

C(θ) =

 ∆T ( ˆ m) T

∆T (ˆ n) T



= X

l

2l + 1

4π C

_l

P

_l

(cos θ) , (3.5)

where θ is the angle between the unit vectors ˆ m and ˆ n, and P

_l

are Legen-

dre polynomials. That is, specification of the correlation function C(θ) is

Observational Situation

Figure 3.5:

Temperature fluctuations across the sky as measured by WMAP (W-band), shown in galactic coordinates [11].

equivalent to specification of the power spectrum C

_l

. Note that physics is independent of m beacuse of the fact that there is no preferred direction in the universe. The CMBR power spectrum is usually plotted as l(l + 1)C

_l

vs. l, where l roughly corresponds to an angular scale ∆θ ' π/l.

The angular scale for the largest anistropies are determined by the size of the largest disconnected regions at recombination, i.e. the horizon. This roughly corresponds to an angle of one degree in the sky today. Thus, at l ' 10

²

the power spectrum starts to show Doppler peaks correspond- ing to the density fluctuations of the early universe (for a more detailed account concerning the underlying mechanisms, see Lineweaver [31] and Tegmark [32]). However, the apparent angular size of the above regions is naturally affected by the geometry of the universe. Compared to a flat uni- verse, the angle will appear larger for a closed geometry and smaller for an open geometry. Hence, the first peak of the spectrum will appear at different values of l depending on the curvature of the universe. Figure 3.6 shows the power spectra for one flat and one open universe, with otherwise identical parameters. It is seen that the first Doppler peak is situated around l ' 200 for a flat geometry, while it is shifted towards larger values for an open uni- verse. Data from the Boomerang [33], MAXIMA [34], and WMAP [11, 12]

experiments have in this manner shown that the universe is very close to being flat. Using WMAP data, a best-fit cosmological model to the CMBR and other measures of large scale structure yields [12]

Ω = 1.02 ± .02 , (3.6)

essentially confirming the standard inflationary big bang model.

3.3 Summary

Other cosmological parameters influence the height and positions of var- ious peaks in characteristic ways. By making a fit to the observed power spectrum one obtains the preferred values for a specific set of parameters.

The WMAP best-fit cosmological model gives [12] for example:

Ω

m

h

²

= 0.135

^+.008_−.009

,

Ω

_b

h

²

= 0.0224 ± .0009 , (3.7) where h = 0.71

^+.04_−.03

. That is, in summary, the universe is flat and composed of 4.4% baryons, 22% non-baryonic matter (i.e. dark matter) and 73% dark energy.

0 500 1000 1500

2 4 6 8

Figure 3.6:

Power spectra for a flat universe, Ωm = 0.3 and ΩΛ = 0.7, and an open universe, Ωm = 0.3 and ΩΛ = 0. We see that the open universe with first Doppler peak around l' 400 is conclusively ruled out by current data [35]. Adapted from [18].

3.3 Summary

Early CMBR balloon experiments were only able to constrain Ω

_m

+ Ω

_Λ

≈ 1.

However, SNe Ia observations effectively measures the acceleration of the

Observational Situation

universe, proportional to

¹₂

Ω

_m

− Ω

Λ

. This makes these methods highly com- plementary in the (Ω

_m

, Ω

_Λ

)-plane. In this manner measurements pointed towards a non-zero cosmological constants. Recent data from the WMAP satellite has remarkably increased the accuracy in determining cosmological parameters from CMBR. It measures independently all parameters and in particular finds Ω

_Λ

= 0.73 ± .04 [11] which is in good agreement with SNe Ia observations. These determinations are also in concert with measurements separately constraining Ω

_m

(for a short review see [39]).

In summary, observations has led to a new cosmological concordance model, according to which:

Ω

m

≈ 0.27 , (3.8)

Ω

_b

≈ 0.044 , Ω

_Λ

≈ 0.73 ,

and h ≈ 0.71. That is, the universe is flat, it contains a lot of non-baryonic matter

¹

and it is dominated by a dark energy component, causing the uni- verse to accelerate

q

0

≈ 1

2 0.27 − 0.73 < 0 . (3.9) This dark energy is either in the form of a “bare” cosmological constant or as some exotic matter with an equivalent equation of state. In particular, it could be in the form of a scalar field causing an effective cosmological constant; the quintessence scenario.

1Note that nucleosynthesis constrain baryonic matter to Ωb ≈ 0.04, clearly in agree- ment with observations.

Chapter 4

Quintessence

This chapter reviews the quintessence scenario. We begin by discussing why a true vacuum energy is not an attractive way to explain the acceleration of the universe. The review continues with the basic theory of quintessence, followed by a section on tracker solutions. The discussion comes to an end with natural quintessence.

4.1 The Cosmological Constant Problem

The present observations leave us with an unsolved problem. Why and how is Ω

_Λ

> 0 ? The simplest solution would be a “bare” cosmological constant Λ

0

corresponding to true vacuum,

ρ

_Λ0

≈ 0.73ρ

c

∼ (10

⁻³

eV)

⁴

. (4.1) This approach has two serious problems. First of all, estimating the vacuum energy expected from quantum field theory yields

ρ

_vac

∼ (10

¹⁸

GeV)

⁴

. (4.2)

Comparing this density to equation (4.1) result in the famous discrepancy of about 120 orders of magnitude between the theoretical and the observed value of the cosmological constant. Although this discrepancy more con- servatively becomes of 45 order of magnitude (corresponding to replacing M

_Pl

by the QCD energy scale ∼ 150 MeV), it is needless to say that this is unacceptable.

The theoretical vacuum density is a sum of contributions originating in

the phase transitions of the early universe. One could imagine these terms

to cancel and just leave the tiny observed value, but this seems, to put it

mildly, unlikely. Even if one did believe this to be the case, one still has

to face the second problem. While ordinary matter density have decrease

many orders of magnitude since early times, a “bare” cosmological constant

Λ

₀

is truly constant. For them to be of the same order today is an amazing

coincidence that would require an extreme amount of fine-tuning.

Quintessence

4.2 The Quintessence Scenario

Since a true cosmological constant is haunted by serious problems, it is likely that we have to follow a different path. A very attractive idea is that some (unknown) symmetry argument would set Λ

₀

precisely to zero. One could then model an effective cosmological constant.

The present observations point towards an accelerating universe. By analogy with inflation, a tempting solution would thus be a scalar field causing an effective cosmological constant. There are three possibilities for this to occur. The scalar field could be at an absolute minimum of non-zero potential energy. It could be in a metastable false vacuum state. Or it could be slowly rolling down a potential. The first two of the these alternatives are indistinguishable from a true cosmological constant. The third is what we usually refer to as quintessence [40, 41, 42]. That is, the quintessence scenario assumes that the “bare” cosmological constant is set to zero (or at least negligible magnitude) and then use a scalar field to model the effective cosmological constant.

A spatially homogeneous scalar field ϕ with minimal coupling to gravity

¹

has the familiar equation of motion

¨

ϕ + 3H ˙ ϕ + V

⁰

(ϕ) = 0 , (4.3) where primes denote derivatives with respect to ϕ. The friction caused by the Hubble term makes the field approximately constant (i.e. overdamped) when H > p

V

⁰⁰

(ϕ), and free to roll (i.e. underdamped) when H < p V

⁰⁰

(ϕ).

The equation of state for the quintessence field is naturally given by

w

_ϕ

≡ p

_ϕ

ρ

_ϕ

=

1

2

ϕ ˙

²

− V (ϕ)

1

2

ϕ ˙

²

+ V (ϕ) , (4.4)

with the field acting as an effective cosmological constant when ˙ ϕ

²

 V (ϕ).

However, note that all states of matter with w < −

¹₃

, i.e. densities that dilute slower than curvature (ρ

_curv

∝ a

⁻²

), act to accelerate the universe.

In summary, the quintessence scenario use the idea of inflation to obtain a moderate acceleration in the present universe. While the matter density ρ

M

is steadily decreasing, the quintessence field ϕ is slowly rolling down its potential. Eventually ρ

ϕ

will begin to dominate and start to accelerate the universe. In its basic form, this scenario does not avoid the problem of fine-tuning. We still have to adjust the initial conditions so that ρ

_ϕ

and ρ

M

end up in the present values. Additionally, the origin of the field has to be explained as well.

1That is, the Lagrangian of the field ϕ does not contain any terms of the form f (ϕ)R.