Cosmological Models with Quintessence: Dynamical Properties and Observational Constraints

Shao-Chin Cindy Ng Department of Physics and Mathematical Physics Adelaide lJniversity South Australia, Australia

1-1- September 2OOL Contents

Abstract llI

Declaration iv

Acknowledgements v

1 Introduction 1 1.1 Quintessence as Missing Energy 1 L2 Cosmological Models with Quintessence. 4 Lz1 Inverse Power-Law Potential 4

L.2.2 Simple Exponential Potential . . 6 r.2.3 Double-Exponential Potential I

r.2.4 Pseudo Nambu-Goldstone Bosons (PNGB) . 7

1.3 Quintessential Difficulties . 9 1.4 Outlines of Chapters 9

2 Phase-Space Analysis 11 2.7 Simple Exponential Potential . . . 12 2.2 Inverse Power-Law Potential 13 2.2.1 Kinetic Energy Domination and Frozen Field 15 2.2.2 Tracker Solution 17 2.2.3 Inflationary Solution 18 2.3 Doubie-Exponential Potential 19 2.4 PNGB Potential 22 2.4.1 Field Variables 23 2.4.2 Energy Density Variables 28 3 Numericallntegration 32 3.1 Simple Exponential Potential 32

3.2 Inverse Power Law Potential , 35 .tù 3.3 Double-Exponential Potential .)f 3.4 PNGB Potential 39 3.5 Luminosity Distance 44

4 Supernovae Tests 46

4.I Constraints on Quintessential Parameters . 47 4.2 Evolution of Supernovae 54 4.3 Model Compartson bb 4.3.I Accelerating Universe versus Luminosity Evolution 6( 4.3.2 PNGB versus Other Potentials 70

5 Future Supernovae Probes 73 5.1 SNAP Data Simulation 74 5.2 Likelihood Function 76 5.3 Model Compartson 77 5.3.1 Accelerating Universe versus Luminosity Evolution 77 5.3.2 PNGB versus Other Potentials 80 5.4 Discussion 81

6 Gravitational Lensing Statistics Test 84 6.1 Likelihood Function 85 6.2 Constraints on Quintessence Models . 87

7 Conclusion 94 7.7 Discussion 94 7.2 Further Studies 97

Bibliography 100

Publications LO7

11 Abstract

Many recent astronomical observations indicate that ordinary gravitationally clumped matter constitutes of order' 113 of the total energy density of the universe, and that the remaining 213 should take the folm of a smooth component which does not clump significantly gravitationally and which is quite unlike any form of rnatter observed in the laboratory. One of the candidates for the "missing energy" is the self-interacting energy of a dynarnical scalar field with its potential, which is given the name "quintessence". This thesis aims to study different quintessence models, in particular, a quintessence arising from an ultra-light pseudo Nambu- Goldstone boson (PNGB). The dynamical properties of the different quintessence models are overvìewed by phase-space analyses, to study the attractor and "tracket" solutions in the phase space; the existence of these solutions could avoid fine tuning of the initial conditions. The high-redshift type Ia supernovae constraints upon these quintessence models are studied. With the PNGB models, the impact of existence of a simple phenomenological model for supernovae luminosity evolution to the supernovae constraints is also studied. The potentials of a future supernovae data set, as might be obtained by the proposed SuperNova Acceleration Probe (SNAP) satellite, to discriminate the PNGB models over the other quintessence models and to discriminate the existence of the supernovae luminosity evolution are studied. Finally, the gravitional lensing statistics of high luminosity quasars constraints upon the quintessence models are briefly discussed'

llr

Acknowledgements

I wish to thank the Adelaide University for providing me with an International Postgraduate Research Scholarship (IPRS), and a Research Abroad Scholarship to partiy support my study leave in United Kingdom. I wish to thank my supervisor, Dr. David Wiltshire, for inspiring my interest into this area of research, and for four years of guidance. I also wish to thank David for sparing a moment out of his busy schedule to see me everytime I needed an advice at work. I wish to thank the staff of the Centre for Theoretical Physics at University of Sussex, Unitecl Kingdom, for their kind hospitality during my stay there. I wish to thank Tiago Barreiro, Roger Clay, Ed Copeland, Elisa Di Pietro, Chris Kochanek, David Mota, Nelson Nunes, Ray Protheroe, Francesca Rosati, Luis Ureña-López, and Ioav Waga, for useful cliscussions' Finally, I wish to thank my family and friends, especially Jian Haur, Kwong, and Ling, for their love and encouragement. Chapter 1

Introduction

1.1 Quintessence as Missing Energy

The standard cosmological model is the h,ot big bang model which is a homogeneous and isotropic universe with evolution governed by the Friedmann equations obtainecl from General Relativity. Its main constituents can be described by matter and radiation fluids, and its kinematic properties match those we observe in the real universe. It has achieved the foliowing successes: the predictions of light element abundances produced during cosmological nucleosynthesis agree with observations; the cosmic microwave background is naturally explained as a relic of the initial hot thermal phase; it accounts naturally for the expansion of the universe; it provides a framework within which one can understand the formation of galaxies and other cosmic structures. The inflationary uniuerse scenario is designed to address several questions left unanswered by the standard hot big bang scenario. It is an early phase of exponential expansion of the universe that drastically changes the past light cone, removing the horizon problem, and diluting unwanted relics to such very low densities that they are close to unobservable. There exist many different versions of the inflationary universe. The first inflationary model was formulated by Guth [1] in 1981, although many of his ideas had been presented previously by Starobinsky [2] in 1979. Although Guth's model proved problematic, improved models with alternative in-

flationary mechanisms were subsequently proposed by Albrecht and Steinhardt [3] and by Linde [4] and there are no\n/ very many variants of inflationary cosmologies [5]. The inflationary universe scenario predicts that the universe is spatially flat and

1 that the total energy density of the universe, ptot, is very nearly equal to the critical clensity, p"=3Hll8nG - L7 xl0-2n g "--t, where Ils is the current value of the Hubble parameter and G is Newton's gravitational constant. The cosmic microwave background (CMB), first discovered by Penzias and Wil- son in 1965, is a background radiation whose spectrum is very nearly that of the blackbody radiation of temperature - 2.7 K. The angular power spectrum of temperature anisotropy in the CMB is a powerful probe of the content and na,ture of the universe. Current measurements of the CMB anisotropy include the balloon- borne experiments BOOMERanG [6] and MAXIMA [7], and the Degree Angular Scale Interferometer (DASI) [8]. These experiments have measured the CMB power spectrum ovel a wide range of spherical harmonic multipoles. New constraints have been set on a seven-climensional space of cosmological parameters within the class of inflationary adiabatic models using only weakly restrictive prior probabilities. Tlre total energy content of the universe is detelmined to be f)¿o¿ = puf P" - 1.0, consistent with the prediction of the inflationary universe scenario. At the same time, there is growing observational evidence that the total matter density p^ of the universe is significantly less than the critical density (see, e.g., refs. [9]-[14] and references therein). Constraints on the matter energy density of the universe can be derived, €.8., from galaxy cluster abundances [12, 13], and large- scale structure [14]. These tests are consistent with f]- = P,nlP. - 0.3. Primordial nucleosynthesis constraints require a significant ploportion of this matter density to be non-baryonic, which would indicate that the univelse contains significant "dark matter" of a form which is not readily found in the laboratory. If these results on the density of clumped matter hold and measurements of the CMB anisotropy establish that the universe is spatially flat, then there must be another contribution to the energy density of the universe, which cloes not clump gravitationally in the way that ordinary matter does. It is a "smooth" component, which is now often called a "dark energy" to contrast it from gravitationally clumped dark matter. Furthermore, the unexpected faintness of type Ia supernovae (SNe Ia) at recl- shift z - 0.5 provides the most direct evidence that the expansion of the Universe is accelerating [15, 16]. This conclusion is supported by measurements of the characteristic angular scale of fluctuations in the CMB which reveal a total energy densìty well in excess of the fraction attributed to gravitating mass 17,77,18]. Such an expansion is quite unexpected, violating Newtonian intuition and requiring some form

2 of universal repulsion to counteract the decelerating effects of normal matter and radiation. This suggests that the missing energy should possess negative pressure, : 7s 10,, and equation-of-state, u plp < 0. To date the evidence for a universe which is accelerating at the present epoch is based on renormalizing the peak luminosities of SNe Ia according to empirical relations, so-called "Phillips relations" [19] [25], which have been observed between the peak iuminosities and decay times of SNe Ia at low redshifts. Thus the evidence for acceleration is not fully conclusive, although the bulk of observational checks do seem to support the acceleration hypothesis [26]. It is one of the aims of this thesis to explore the extent to which current data and possible future satellite data can distinguish the signature of cosmological acceleration from other possible explanations, such as an evolution of the SNe Ia sources. Whatever doubts one might have about the evidence for cosmological acceleration, among theoretical cosmologists the possibility has seen a great resurgence of interest in models involving the cosmological constant,,A., a vacuum energy density wlrich is spatially uniform and time-independent, with wL : -r' The cosmological constant was first introduced by Einstein fol the purpose of allowing a static universe with the repulsive cosmological constant delicately balancing the gravitational attraction of rnatter [27]. Einstein's static universe model is theoretically unappealing since it is unstable, and it was abandoned by Einstein as soon as it was realised that the universe is expanding. The value of the cosmological constant envisaged today is orders of magnitude different from that envisaged by Einstein. Indeed, constraints on the expansion of the universe impose a very small upper value,

< 10-30 wù.p N 10-3eV, where m, is the Planck mass scale. This very unnatural fine^ tuning of parameters is refer-red to as the "cosmological constant problem", or more accurately the ttvacuum energy problem". Alternatively, a dynamical, time-dependent and spatially inhomogeneous component, which is hopefully well motivated by fundamental physics, can be introduced. As an extension of inflation, it is the enelgy associated with a scalar field / evolving down its potentialV(ó). The equation of state of the new component possesses a possibly time-varyìng adiabatic index lying in the range -1 < LU6 10,, which makes it different in nature to baryons, neutrinos, dark matter or radiation. This frfth component of the energy density, which has been studied extensively since

the pioneering work of Wetterich [2S] and Pebbles and Ratra [29] in the 1980s, has

3 been given the name "quintessence" in the more recent literature [30]. Quintessence is broadly deflned, allowing a spectrum of possibilities including a wq which is constant, unifolmly evolving or oscillating.

L.2 Cosmological Models \Mith Quintessence

In this thesis we focus on some well-studied examples of quintessence. For simplicity it is assumed that any couplings to other fields are negligibly small, so that the scalar field interacts with other matter only gravitationally, and is minimally coupled. A further step is to generalize lhe same mechanism in the framework of scalar-tensor theories of gravity. This is referred to as non-minimal quintessence [31], extended quintessence [32], coupled quintessence [33] or generalized quintessence [34]. The average energy density is pó: Ó'lZ + I/ and the average pïessure is p+ - ó, l2-V. As / evolves d,own its potential, the ratio of kinetic (ó' 12) to potential (V) energy can change; this would lead to a time-varying equation-of-state, wO: pOlpO. Many interesting candidates have been proposed in such a context. We wìll discuss these cases in turn.

1.2.I Inverse Power-Law Potential

Ratra and Peebles [29] first showed that if the potentialV($ of the scalar field that drove inflation had a power-law tail at Iarge S,

V(ó):Vtó-*, (1.1)

the mass density, pó, associated with / would act like a cosmological constant that decreases with time less rapidly than the mass densities of matter and radiation. A similar procedure was undertaken by Liddle and Scherrer [35]. They assumed the desired behaviour of p6 and the dominant component energy density p to scale exactly as a power of the scale factoî, pó x a-n and p x a-*. Substituting the required behaviour into the scalar field equation leads to the potential in eq. (1.1). If ihe exponent a is positive, then nx > n and the scalar fi,eld energy density grows compared to the matter. They showed that this solution is a stable attractor for all values of positive a. The inverse power law potential was then utilized by ZIatev et al.l36l as a form of cluintessence which they called a "tracker field" in view of the fact that its dynamical

4 behaviour decreased the amount of fine-tuning involved in the "cosmic coincidence probiem" [37]. This fi.ne-tuning problem may be understood as follows: Since the missing energy density and the matter density decrease at diffelent rates as the universe expands, it appears that their ratio must be set to a specific, infinitesimal value in the very early universe in order for the two densities to nearly coincide today, some 15 billion years later. The attractor like solutions of the equation-of- motion of tracker fields allow a very wide range of initial conditions and rapidly converge to a common, cosmic evolutionary track of p6(t). The initial value of p4 can vary by nearly 100 orders of magnitude without altering the cosmic history. The acceptable initial conditions include the theoretically natural possibility of an equipartition of energy after inflation - nearly equal energy density in / as in the other 100-1000 degrees of freedom (".g. flø t 10-3). The resulting cosmology has further desirable properties [36]. The tracker fields idea can be applìed to the standard cold dark matter component in cases where it is composed of oscillating fieids [38]: combining these ideas, a model in which quintessence, cold dark matter, and ordinary matter all contribute comparable amounts to the total energy density today irrespective of initial conditions can be constructed. Models of dynamical supersymmetry breaking readily lead to an inverse power- law potential model [39]. It is based on the gauge group Stl(¡/") and has l/¡ < l/" flavours: quarks Qi, i : 1 . . . lú¡ in fundamentals of ,9u(¡/") and antiquarks Ç¿, i : I.. . l/¡ in antifundamentals of .SU(^t). The potential for { has the form (1.1)' with o: z!!!!'N"- N¡ .' G.z) However, current observational data seemed to favour a small value of the parameter, a ( 5, and hence u6o I -0.5 [10, 40, 47]. This may be a problem for these models: Steinhardt et al. 142] showed that starting from the equipartition condition after inflation, it it necessary to have a ) 5 for the field to begin tracking before matter-radiation equality. Since the observational constraints indicate that tracking could only be achieved (if at all) at mole recent times, it is not cleal what the theoretical advantage, in terms of alleviating the cosmic coincidence problem, is gained by the tracker solution. This difficulty can be overcome, for example, by constructing scalar freld poten- : q-a It was argued that when the field is on tial of the form v (Ó) ¡y¡aia ""q'lz 143].

5 tracks today, one has ó = *, demonstrating that any realistic model of quintessence must be based on supergravity, and the above potential was constructed. T)ue to the presence of the exponential term, l.;u4o caî be pushed towards the value -1 in contrast to the usual case for which is it difficult to go beyond u6o = -0.5.

L.2.2 Simple Exponential Potential

The Liouville potential, with an exponential dependence upon the scalar field,

v(ó):v¡e-^"Ó, (1'3) is a common functional form for the self-interaction potential. It is to be found in higher-order or higher-dimensional gravity theories (see, e.g., refs. [ a]-[46] and references therein). In string or Kaluza Klein type models the moduli fields associated with the geometry of the extra dimensions may have effective exponential potentials due to the curvature of the internal spaces, or interaction of moduli with form fields on the internal spaces. Exponential potentials can also arise due to non- perturbative effects such as gaugino condensation [a7]. The possible cosmological roles of exponential potentials have been widely investigated before as a mean of driving a period of cosmological inflation [48]. In the standard cold dark matter model with the addition of a scalar field / with a simple exponential potential in eq. (1.3), there is a very special and interesting solution called the "scaling solution", in which the scalar field energy follows that of the dominant component, contributing a fixed fraction of the total energy density determined by ) [28]. The scaling solution is a global attractor solution [49], suggesting a possible solution to the cosmic coincidence problem that the missing energy density and the rratter density nearly coincide today' The main problem with this scenario is that if the scalar field has a significant contribution to the eneïgy density today, it should also have had a significant effect during the radiation-dominated era. Primordial nucleosynthesis bounds put a tight constraint on the fraction of energy in a smooth component in the radiation- dominated era. The current upper bound on f)6 at nucleosynthesis is estimated to

be in the range 0.13 < f¿d < 0.2 [45]. Recently, it has been emphasized that universes that accelerate without end have no well-defined asymptotic states from which a physical S-matrix can be built; thus on the basis of current theoretical ideas it wouid seem that endless acceleration is in

f) contradiction with the axioms of string theory [50]. A cosmological constant gives just such an eternal acceleration. Cline [51] and Kolda and Lahneman [52] separately showed that quintessence models described by an exponential potential can account for the present observed acceleration of the universe while also predicting only a finite period of acceleration, consistent with theoretical paradigms, thereby giving another motivation to these models.

L.2.3 Double-Exponential Potential

A possible extension of the exponential potential (1.3) is a double-exponential dependence upon the scalar field,

v :vt"xp(-A"ø"Ó) . (1 4)

In a class of modeis, reminiscent of many superstrìng models where supersymmetry is broken through gaugino condensation along the flat direction corresponding to the dilaton field, one recovers, in the limit of small stling coupling, a leading behaviour of the potential which is a double-exponential function of the scalar field [39]. De la Macorra [53] has also studied the cosmological evolution of string/M moduli fields T. At T > 1 it was found that the potential is given by a double exponential function of the field Z. A problem with these models arises from the fact that after renormalisation down to low energy the vacuum expectation value of the dilaton field provides a value for the fine structure constart If a. A'sliding'dilaton would make the fine structure constant vary with time at an unacceptable rate [54].

L.2.4 Pseudo Nambu-Goldstone Bosons (PNGB)

Frieman et al. 155) have explored the properties of cosmologies with an ultra-light pseudo-Nambu Goldstone boson (PNGB) field which satisfies the two conditions that: (i) it is currently relaxing to its vacuum state; and (ii) it dynamically dominates the energy density during the epoch in which it relaxes. PNGB models are characterized by two mass scales, a spontaneous scale (at which the effective La- gr.angian still retains the symmetry) and an explicit symmetry breaking scale (at which the effective Lagrangian contains the explicit symmetry breaking term). The two dynamical conditions above fix these two mass scales to values which are 'tea- sonable' from the view point of particle physics. Thus, we may have an explanation

I for the 'coincidence'that the vacuum energy is dynamically important at the present epoch. From the viewpoint of quantum field theory, PNGBs are the simplest way to have naturally ultra-low mass, spin-O particles. 'Technically' natural small mass scales are those that are protected by symmetries. An example of this phenomenon is the "schizon" model [56], based on a Z¡¡-\nvariant low-energy effective chiral Lagrangian for l/ fermions, e.g., neutrinos, with mass of order M, in which the small PNGB mass, Ìnó = M" I l, is protected by fermionic chiral symmetries. The potential for the light scalar field / is of the form

v(ó): Malcos($l/) + 1l (1.5)

Since / is extlemely light, we assume that it is the only classical field which has not yet reached its vacuum expectation value. The constant term in the PNGB potential has been chosen to ensur-e that the vacuum energy vanishes at the minimum of the / potential, in accord with our assurnption that the fundamental vacuum energy is zero owing to some as yet not understood mechanism. Vacuum energy is most simply stored in the potential energy V(Ó) - Ma of a scalar field, where M sets the characteristìc height of the potential, and we set V (ó*) : 0 at the minimum of the potential by the assumptions above. In order to generate a non zero l\ at the present epoch, / must initially be displaced from the minimum (ó¿ I ó* as an initial condition), and its kinetic energy must be small compared to its potential energy. This implies that the motion of the fi.eld is still overdamped,nó= fr,,@ùS 3110:5 x 10-33å eV. In addition, for fl¡ - 1, the potential eneïgy density shouid be of older the critical density, M4 - SHlMþ,18n,, ot M - 3 * 1g-z¡rtlz eV. Thus, the characteristic height and curvature of the potential are strongly constrained for a classical model of the cosmological constant. In terms of the mass scales introduced above, generally the PNGB mass takes values rmó - M'l f . Thus, the two dynamical conditions oL rnó and M above essentially fi.x these two mass scales to be M - I0-3 eV, interestingly close to the neutrino mass scale for the Mikheyev-smirnov-Wolfenstein (MSW) solution to the solar neutrino problem, and f - Mpt N 101s GeV, the Planck scale' Moreover, the small mass r??4 is technically naturall.

lAlternative mechanisms to generate the extremely small mass for the axion have been proposed by Nomura, Watari, and Yanagida [57].

8 l-.3 Quintessential Difficulties

These quintessence ideas might be difficult to implement in the context of realistic models. The main problem lies in the fact that the quintessence field must be ex- tremelyweakly coupled to ordinary matter (for a review see, e.g., ref. [58]). Carroll [59] pointed out that, in general, one should expect gravitational strength interactions of quintessence with ordinary matter and discussed how these interactions can give rise to interesting microscopic and macroscopic phenomena. Kolda and Lyth [60] r.emarked on the difficulty of keeping the mass of the quintessence field light. The results of [53]-[60] show that there is extreme fine tuning in the couplings of the quintessence field to matter, unless they are forbidden by some symmetry. This is somewhat reminiscent of the fine tuning associated with the cosmological constant. Despite these difficulties it is conceivable that new mechanisms might be found for overcoming these problems in a natural fashion. For example, some years ago Damour and Polyakov proposed a mechanism in the context of string cosmology for driving scalar fields to dynamical regimes in which they are very weakly coupled to other forms of matter [61]. Their arguments applied to the string dilaton, but could be generalizedto other scalar-tensor theories [62]' The quintessence fleld must be very light. This field must therefore be very weakly coupled to matter; otherwise its exchange would generate observable long ïange forces. Eötvös-type experiments put very severe constraints on such couplings. In order to alleviate this problem for the inverse power-law potential models reviewed in Section 12.7, Masiero, Pietroni, and Rosati [63] have proposed a solution much in the spirit of the least coupling principle of Damour and Polyakov [61] to ease the Eötvös experiments constraints. Thus although the status of quintessence is at the very best tentative at the theoretical level, it is conceivable that present objections could be overcome in future models, and given the observational evidence from supernovae and the CMB it is clear that such models warrant detailed study.

L.4 Outlines of Chapters

The aim of this thesis is in twofold. Firstly, we want to study the dynamical prop- er.ties of these quintessence models and their cosmological implications in more detail. Secondly, we want to explore the implications of observational data on these I quintessence modeis. We will focus on two sets of observational data, namely the high-redshift SNe Ia and the lensing statistics of high luminosity quasars. In Chapter 2, we will overview the dynamical properties of these quintessence models by a phase-space analysis of cosrnologies containing a barotropic fluid plus a scalar field / with the different potentials V(Ó)' In Chapter 3, we will discuss the numerical integration of the field equations of the quintessential cosrrologies forward fi-om the early matter-dominated ela. We will discuss the possible initial conditions, and will study the dependence of the scalar field energy density and Hubble expansion age on the quintessential parameters. We will also delive the luminosity distance - redshift relations numerically from the integration. In Chapter 4, we will study the constraints on the quintessential parameters as imposed by the luminosity distances of SNe Ia. With the PNGB models, we will explicitly allow for the possibility of evolution of the peak luminosities of the supernovae sources, using simple empirical models which have been recently discussed in the literature. By evaluating the "Bayes factor", we will study the ability of the SNe Ia data in discriminating between the PNGB models from the other quintessence models, and to discriminate between two possible explanations for the observed dimming of the SNe Ia data, namely either (i) a cosmological evolution for which the expansion of the universe has been accelerating for a substantial range of redshift s z - I; or (ii) an unexpected supernova luminosity evolution over such a redshift range. In Chapter 5, we will simulate future supernovae data sets as might be obtained by the proposed SuperNova Acceleration Probe (SNAP). We will study the degree of improvement of a future supernovae data set to discriminate between the two explanations for the observed dimming of the SNe Ia, and to discriminate the PNGB models from the other quintessence models. In Chapter 6, we will study the constraints on the quintessential parameters as imposed by the gravitational lensing statistics of high luminosity quasats. Finally, a discussion of this work, and some suggestions for further studies, are discussed in Chapter 7.

10 Chapter 2

Phase-Space Analysis

The classical action for gravity coupled to a scalar field / has the form

s : aarrE¡K# - * (2.1) I f,s,'0,óa,ó -v@)) t), where rc2 :8nG, r? is the Ricci scalat,, g :- det g¡,,, and L is the Lagrangian density of non-relativistic matter and radiation. For simplicity, we assume / is minimally coupled to the curvature, and we work in units in which fi, : c: I' Consider a spatially flat Friedmann-Robertson Walker (FRW) universe containing a fluid with barotropic equation of state P"r: ('l -l)p"r, where 7 is a constant, 0 < ,y ( 2, such as radiatior (? : 413) or dust (7 : 1). The governing equations are given by

H -tG"+P.,+ó"), (2.2) p^, P.,) (2.3) -3H(p.,I.dv , _JHó _ (2.4) ó dó , subject to the Friedmann constraint

rc2 1 H2 ó'+v (2.5) 3 h* 2 where H : à,la is the Hubble parameter, and an overdot denotes ordinary differen- tiation with respect to time Ú. In this chapter, we wiÌl perform a phase-space analysis for the governing equations (2.2)-(2.5), for the different potential functions I/(/) discussed in Section 1.2. We will discuss the phase-space analyses in turn.

11 2.! Simple Exponential Potential

The phase-space analysis of cosmologies containing a barotropic fluid plus a scalar fietd with an exponential potential,V ($) - Vas-\"$, has been presented by Copeland et al.146]" We briefly review their work in this section. Following ref. [46] we defrne new variables ot/V *=ffi;"ó a=ßE- (2.6)

The evolution equations (2.2)-(2.4) can be written as a plane-autonomous system:

dr y')l dN -3r*) +|,lz*+r(r -*'- , da EÐ (2.7) + r (t - - a")) , dN ^rlï", )a lz"'+ "' where lr/ is the logarithm of the scale factor, ly' : ln(ø), and the constraint equation (2.5) becomes rc2 pt J*r+r2+y2:\. (2.s) In terms of these new variables the scalar field energy density is given by f)ø= #:*'+a', (2.9) and its efiective equation of state u, un, point is given by

P_! i J-!- Ó', ,^:tQ: : ?r' . (2.10) pó - v+ó'lz- 12 lu2

Note that r and y are bounded, 0 1 12 +y2 1I,for p-, ) 0, and so the evolution of this system is completely described by trajectories within the unit disc. The lower half-disc, A I 0, corresponds to contracting univetses. As the system is symmetric under the reflection (r, y) -+ (*,, -y) and time reversal f -+ -t,we only consider the upper half-disc, y 2 0 in the following dìscussion. Depending on the values of 7 and ), there are up to flve fixed points (critical points) where drldN:0 and dyldN:0 which are listed in Table 2.1. There are only two possible late-time attractor solutions. One of these is the scalar field dominated solution (flø : 1) which exists for sufficiently flat potentials, )2 < 6. It is a late-time attractor when we have \' < 31. For À2 ) 37 there is a different late-time attractor where neither the scalar-freld nor the barotropic fluid

72 n a Existence Stability

0 0 AII À and 7 Saddlepointfor0

1 0 All ) and 7 Unstable node for < \/6 ^ Saddle point for > t/6 ^ -1 0 All À and 7 Unstable node for > -\/6 ^ Saddle point for < -\/6 ^ lt - l6)rt2 <6 Stable node for À' < 3'y ^2 ^2 ^1,/6 Saddle point for 37 < À2 < 6 Ql\tt21l), l3(2 - 1)1lD'2lt/2 \r>31 Stable node for 3l < À' < 2a12 lQy - 2) Stable spiral for > 24^t2 l(h - 2) ^2 Table 2.1: The properties of the critical points for the plane autonomous system (2.7).

entirely dominates the evolution. Instead we have a "scaling solution" where the energy density of the scalar field remains proportional to that of the barotropic fluid with f)¿ : 3t lÀ2. This solution was first found by Wetterich [28] and shown to be the global attractor solution for )2 > 3"y by Wands et al.la9l.

2.2 Inverse Power-Law Potential

In this section we will performed a phase-space analysis similar to that in the

previous section for V ($) : VtÓ-o . In the case of the simple exponential potential (1.3) analysed in the previous section, one of the differential equations decoupled, and the dynamics were effectively described by a phase plane with trajectories bounded by the circle *'+ A': 1. In the present case, and the following cases, however, no such simplification arises.

Hence, we introduce a third variable [64] V,A ) :-oV (2.rr) ó

In terms of the new variables æ, y , and ), the governing equations (2.2)-(2.5) become:

dr -rr * +|,lz.'+ r (1 - y')l , dN ^rE' -,' dy +!rulz*+ (r *' y')) , dN -^18., r - -

13 d^ (2.r2) dN -.Æ(r -r¡À2n , where [42] VttV a*1 n-| (2.13) -- v,2 d Note that in the case of the simple exponential potential (1.3), À(/) reduces to a constant and l: 1. For a, ó > 0, the system is confined to ) ) 0. We can identify critical points at finite ) from the evolution equations (2.12). They are three discrete points at (r.,U",À.): (+1,0,0), (0,1,0), and a one-parameter family at (0,0,)") for )" arbitrary. The stability of the critical points can be obtained by linearizing (2.I2) about these points and solving for the eigenvalues of small perturbations. The critical points and the eigenvalues are listed in Table 2.2. The kinetic energy dominated solution (1,0,0) is the global repellor. The potential energy dominated solution (0,1,0), and the barotropic fluid dominated solution are degenerate critical points, in particular with regard to perturbations orthogonal to the (r,,y) plane, for which the eigenvalues are zeto.

a ) Eigenvalues +1 0 0 6 - 3'y, 3, +2\/3 0 1 0 -3 * 3112, -37, 0 0 0 0(À"

Table 2.2: The critical points at finite À, and their eigenvalues of small perturbations for eqs. (2.I2).

In Fig. 2.I, we project some three-dimensional trajectories onto the (r, g) plane. Assuming the scaiar field / is evolving down the potential slope and / is positive, we only examine the r ) 0 solutions. The trajectories begin to evolve from the y-axis. As can been seen in Fig. 2.I,lhe potential energy dominated solution (0,1,0) is the global attractor where all the trajectories end up at late times. The behaviour of the trajectories will be explained in more detail when we discuss the behavioul of the solutions near the critical points latter of this section.

t4 1.0

0.8

0.6

o.4

0.2

0.0 0.0 0.2 0.4 0.6 0.8 1.0 X

Figure 2.1: The projection of the space trajectories onto the (r,y) plane for system (2.I2) with 7 : 1 and a: 4. The dashed line is the ï : A solution. 2.2.L Kinetic Energy Domination and Frozen Field Initially, the scalar field is high up the potential slope with / æ 0, and hence À : al ó ) 0. The pattern of trajectories near the ) -+ oo surface can be ascertained by continuity to the À -ì oo solutions, even though the latter are not physical. We bring the plane À -+ oo to a finite distance from the ) : 0 plane by the transformation \- e ^-I_€, 0(e1L, (2-I4) so that the e : 0 sulface corresponds to the ) : 0 surface and the e : 1 surface corresponds to À -ì oo. We define a new time coordinate z by dr : )dl/. From eqs. (2.12) we find

& 1)re(1 ,) (2.15) dr-= À(l+)'),o,lloy ,: -\Æ(f - - ,

On the plane €.: l, we find deldr: 0, and the evolution equations for r and y become dr d, d'E t¡ (2.t6) dr U2

15 Solving equations (2.16), we obtain t;

A tanlr[r l!(, ,,\] 1 Ly 2t' - 'trr 6 v / '"cht/; ? -,¿)l ,, (2.17) where A and Ti ate arbitrary constants. As shown in Fig. 2.1, before reaching the z-axis the trajectories are a series of near concentric circles well-described by equations (2.I7). For ) very large, the scalar fleld potential energy turns into kinetic energy very rapidly. Near the r-axis where the scalar field energy density is kinetic energy dominated (y < 1), The evolution equation fol r in (2.72) reduces to dr3 : ,'). (2.18) d,N -r(2 - r)r(1 - For r <0, drf dN is always positive and r is always increasinglfor r > 0, drf dN is always negative and r is decreasing towards the À-axis. As mention in the previous section, there are a one-parameter family of fixed points lying along the )-axis. Linear perturbation analysis shows that near the À-axis, all orbits are gathered along the r direction with a dependence

* o o-3(2-t)/2 (2.19)

They emanate from the y direction with a dependence

axa31/2. (2.20)

Near the )-axis wher-e dló : *' + y' : 0, we have a barotropic fluid dominated universe, i.e., matter or radiation dominated according to the value of 1. Solving eq. (2.2) by neglect irg ó" , we obtain

a -ztl2 H:H¿ (2.2r) úi where a¿ is the value of a at a particular instant H: H¿. Substituting H x o,-3tlz into the definition of r and y in ec1. (2.6), it means near the )-axis, the kinetic energy density redshifts u" 6' o ø-6, while the potential energy with the scalar field 'ftozett'.

16 2.2.2 Tlacker Solution

To examine the behaviour of the solutions near (*,y) : (0,0) as ) -l co, by the Poincaré method we consider the transformation

(2.22)

From eqs. (2.12) we fi.nd

de d^l dN t)xe . (2.23) dN- JaG - ^2 When e :0 (À - -), we find deldN:0 and the evolution equations for X and Y become

dX J + +t1x \/6(t r)x' , dN -3X - - dY 3 +11Y \/6(t i)xY . (2.24) dN - -

We can identify three frxed points from equations (2.24). The critical poìnts and their eigenvalues of small perturbations are listed in Table 2.3.

X Y Eigenvalues 0 0 3112, -3(2 - t)lz 0 3(2 3(4I 2 1) '/6(t -2)l4F - I) - 1) 12, - - 7)a(r - tßlzto +ffilt-3[2+t-2to

Table 2.3: The clitical points and their eigenvalues of small perturbations for equations (2.24).

Among the three fixed points in Table 2.3, only the third one on the list is an attractor solution, which is the well-known "tracker solution" '36, 42]. The eigenvalues are equivalent to eq. (15) of ref. [42]. For 0 1 lø S 1,2*1-2lo 2 0 and the real part of the eigenvalues is always negative. As shown in Fig. 2.I,traiectories leaving the À-axis are spiralling towards a single trajectory. Near the À-axis the trajectory is linear, corresponding to a constant 74 given by 'y a'l ^tó: erj:fr,' (2'25) The tracker solution approaches the u - y solution when a -+ oo.

17 Substituting the tracker solution into eq. (2.9), we find 3lo f^) ó (2.26) ^2 which increases steadily with decreasing ). Combining the Y value for the tracker solution with the deflnition of y in eq. (2.6) and the 11 solution (2.21), we can derive the exact solution for the tracker field / to be ó:lffiurf,'*'(i)""'"*' (2'zT)

The tracker solution is an attractor in the sense that a very wide range of initial conditions for / and / rapidly approach a common evolutionary track, so that the cosmology is insensitive to the initial conditions. The term "tracker" was introduced [36] because there is a subtle but important difference from attractor solutions in clynamical systems. Unlike a standard attractor, the tracker solution is not a fixed point: Í)¿ changes steadily as the scalar field proceeds down its track. This is desirable because one is interested in having the scalar- field energy ultimately overtake the background density and drive the universe towalds an accelerating phase. The general conditions for the existence of tracker fields were addressed in ref.1421.

2.2.3 Inflationary Solution

The tracker solution is important only when ) >> 1 and, r,,y K l. As shown in Fig. 2.1, eventually the trajectories will reach the critical point (r,y,À): (0,1,0), which is the exact and global attractor in the phase-space. To study the behaviour of the solutions near (0, 1,0), inspired by the scalar field dominated solution of the simple exponential potential (see Section 1) we consider the transformation

x:!.-r I y:JT=Ta e.28) - À ) In terms of X and Y, the evolution equations (2.12) become

dX -t(r" (1 +|x'x¡zx' - -Y')l t , dN ^'Y') "x' dY dN (2.2e)

i8 where 1)À, : (2.30) ": ,/6(r - *^, . When ) : 0 (c : 0) we find d^ldN : 0, and eqs. (2.29) are reduced to dx or, , ts ,rN : -zx+\fi'

Y# sz'3x2 +!.u(x, y',), (2.31) dN : ,E*ly'z" - t,t\" - From eqs. (2.31), we can identify a fixed point at

x": Y".c : . (2.J2) - - ^[l=! 6 ' with eigenvalues of small perturbations ffi+:-3, rr¿-:-37. (2'33)

The eigenvalues rrl+ a;te always negative, and hence this solution is an attractor solution. As shown in Fig. 2.I,trajectories are approaching the (0,1,0) point along the solution (2.32), which corresponds to the circular boundary c2 + A2 : I' Interestingly, ) and c are related to the slow-rol/ parameters €s¿ and q"¡ in the formulation of inflation [65]: rc2v' €sl : (2.34) 2V ^22' rc2v" : :c-l^, r2 (2.35) T"l V ) and the necessary conditions for the slow-roll approrimation to hold are 6"¿(1, ln",l

2.3 Double-Exponential Potential

The phase-space analysis lor V($) - V1,exp(-Aefr"ó) is similar to that for the inverse power-law potential discussed in the previous section. From the definition of À and I in eqs. (2.11) and (2.13), we obtain À: t/ZA"Jt"o, (Z.JT) f : I_(AeJî"ó)-t. (2.38)

19 Assuming that A is positive, the systemis confined to À > 0. substituting eq. (2.38) into the r, y, and À evolution equations (2.12), we can identify the critical points at finite ) to be three discrete points at (r",U",À.): (+1,0,0), (0,1,0), and a one-parameter family at (0,0,)") for )" arbitrary. The critical points and the eigenvaiues of small perturbations are listed in Table 2.4. The kinetic energy dorninated solution (1,0,0) is the global repellor. The potential energy dominated solution (0, 1,0), and the barotropic fluid dominated solution are degenerate critical points, in palticular with regard to perturbations orthogonal to the (r, y) plane, for which the eigenvalues ate zero.

lc U. ). Eigenvalues + 1 0 0 6 - 3'1,3, +2\/3 _3 _f _37, 0 1 0 \ 12, 0 0 0 0()"

Tal:'Ie 2.4: The critical points at finite ) and their eigenvalues for the double- exponential potential models.

In Fig. 2.2, we project the three-dimensional trajectories onto the (r,g) plane. Assuming the scalar field / is evolving down the potential slope and / is positive, we only examine the z ) 0 solutions. As can been seen in Fig. 2.2, trajectories are approaching the barotropic fluid dominated solution (*,A,À): (0,0,À.), )" --) oo, at late times. This solution is the global attractor-for these models [39,53,64,66]. It means cosmologies with a scalar field with double-exponential potentials will be dominated by the barotlopic fluid eventually. The greatest distinction between the present case and the previolls case of the inverse power-law potential is that À increases as the scalar field rolls down the potential slope towards the positive / direction (see eq. (2.37), while in the previous case À is decreasing as $ increases (see eq. (2.11)). With the present situation, À is small initially and the pattern of the trajectories can be ascertained by continuity to the ) : 0 solutions, even though the solution is not physical. On the surface of À : 0, the potential energy dominated solution (*,A): (0,1) is the late time attractor and trajectories are driven towards a scalar field dominated univer-se (see Section 1). Therefore, beginning from a matter-dominated universe wiih ) sufficiently small, it is possible to have the universe dominated by the scalar field at present, consistent with current observa-

20 1.0

0.8

0.6 h a.4

0.2 0.0 0.0 0.2 a.4 0.6 0.8 1.0 x

Figure 2.2: The projection of the space trajectories onto the (r,y) plane for 7:1, for the double-exponential potential moclels. The dashed line is the r : y solution. tional results, before the matter dominates the universe again in the future. This might now seem to be a welcomingly situation, given that a scalar field dominated universe will lead to an eternal acceleration, thereby violating the prediction from

string theory [50]. At sufficiently late times when ) is large, the trajectories are approaching the r-axis, and they remain close to the r-axis when they approach the À-axis. The final evolution of the trajectories is magnified in Fig. 2.3. The behaviour of the trajectories near the r-axis and the À-axis can be explained as in Section 2.2.L The results in Section 2.2.2 apply to the general cases when f is a constant. In the present case, when ó > I, f æ 1 is nearly constant, and hence the lesults are applicable. Eventually, the trajectories aïe approaching the global attractor (*,a,À): (0,0, )"), À" -l oo, along the tracker solution (see Section2.2.2). The equation-of state of the tracker solution is .tó: ____f__ (2.39) 1 - ]e-Jzø ' and the exact solution for the tracker freld / can be derived from the following relation: (2.40) 0.10

0.08

0.06

o.o4

0.02 0.00 0.00 0.02 0.04 0.06 0.08 0.10 x

Figure 2.3: The magnification of the evolution near the )-axis in Fig. 2.2. The dashed line is the r : u solution.

For f æ 1, the trackel solution is well approximated by the r : y solution (see Fig. 2.3).

2.4 PNGB Potential

The dynamical properties for the PNGB models are different from those in the previous models. In particular, with the cosine potential V(Ó): Ma(cosÓlf -I), the scalar field is oscillating at the bottom of the potential well, in contrast to the previous cases when / is slowly evolving in a runaway potential which decreases monotonically to zero as / goes to infinity. Therefore, the phase-space analysis for the PNGB potential is slightly more complicated than that for the potentials discussed previously. Similar to the previous two cases, the simplest phase space for the ptesent case appears to be a three dimensional subspace of the full four-dimensional phase space. There are two alternative choices of variables which are useful to describe the dynamics, which we will discuss in turn.

22 2.4.L Field Variables

The first choice is to simply use the Hubble parameter, H, the scalar field and its first derivative as the elementary variables. By defining

)- rcó -v J- (2.41) .r F "ó we therefore obtain, from the governing equations (2.2)-(2.4), the system (2-l),'- H: -\n'+1-''2 2 (cos1+1) -lt-, (2.42) ìJ (2.43) ' -r m'' çr.nn¡ where ror notario:.r .*r:i;:-:;: ; 2 - R2M4, and F : rcr,so that .F is dimensionless, while m has dimensions of inverse time. The constraint equation (2.5) becomes

rc2 : m2(co"1 1) (2.45) pt 3H2 - + - il' . From eq. (2.3), it follows that þ"r: 0 if p",:0. Therefore trajectories do not cross the two-dimensional pt : 0 surface, which is a hyperboloid in the variables

H,, cos(I f 2), and .I. Physical trajectories with p., ) 0 are forced to lie within the volume of the H, I r,/ phase space bounded by the Pt : 0 surface. The only critical points of the system (2.42)-(2.44) at finite values or H, I, J occur at

1. Cr+ at H : LHt, I : 0 rnod 2tr, ./ : 0; and

2. Cz at, H :0, I : zr mod 2n, J :0, where ,, = \fi* . (2.46) Both of these points in fact lie on the p', : 0 surface' Furthermore, this surface intersects the fI :0 plane only at the isolated points Cz. The.tl > 0 and I{ < 0 subspaces are thus physically distinct, and the H < 0 subspace simply corresponds to the time reversal of the H ) 0 subspace. Therefore we can take 11 > 0 without loss of generality. The pattern of trajectories close to the pt : 0 surface can be ascertained by continuity to the pt : 0 solutions, even though the latter are not physical. The

23 p., : 0 subspace is obtained, for example, by regarding (2.45) as a quadratic equation for H, and using the solution to eliminate H, thereby obtaining a two-dimensional system for 1 and J given by Q.aJ) and (2.44). We plot the resulting ¡1 ) 0 pattern of trajectories in Fig. 2.4 for values of 1 e 10,2n). Since the potential,V($), is periodic the same pattern of trajectories repeats itself as \Me extend 1 to 1oo, with trajectories crossìng from one "cell" to another at the cell boundaries. The trajectories which occupy the lower half of Fig. 2.4 are obtained from those in the upper half by the symmefty I -+ 2tr - I, J -+ -J of the differential equations (2.42)-(2.45). Physically this simply corresponds to the scalar field rolling from the maximum to the right of the minimum as opposed to the one to the left.

0 I 2¡

Figure 2.4: The projection of the trajectories within lhe p",:0 subspace' onto the 1-./ plane for values of 1 e l0,2tr), for the system (2.42)-(2.45).

An analysis of small perturbations about the critical points Cr and C2 yield eigenvalues

1. ) : -r/6^^t, -T (16 + 6+41F2 at C11,

2. À: 0, +T at Cz.

Thus C11 attracts a two-dimensional bunch of trajectories but is a saddle point with respect to trajectories lying in the p^, : 0 surface, as is evident from Fig. 2.4. The

24 two-dimensional bunch of trajectories which approach C1.r- are found to correspond to inflationary solutionl with o o.tp(1þ,l3mt) as ú -+ oo and / -+ const - 2nn7, ntL. The point Cz is a clegenerate case, in particular with regard to perturbations orthogonal to the p., :0 surface, (i.e., into the surface p., ) 0 region), for which the eigenvalue is zero. It is a centle with respect to the trajectories lying in the pt : 0 surface, and when perturbations of higher order are considered it becomes a stable spiral point in the pt : 0 surface as can be seen in Fig. 2.4. Since there is a degeneracy, however, an alternative choice of phase-space variables is desirable. We will defer a discussion of the late time behaviour of the solution near C2 to Section 2.4.2. The points C1a correspond to models with a scalar field sitting at the maximum of the potential, whereas C2 corresponds to the scalar field sitting at the bottom of the potential well. The separatrices in Fig. 2.4 which joìn C1a to C2 correspond to the fleld rolling from the maximum to the minimum. It wouid appear from Fig. 2.4 trajectories which spiral into Cz become arbitrarily close to the separatrix at late times. The separatrices which join the points C11 to points at inflnity correspond to solutions for which the scalar field reaches the top of the potential hill as ú -+ too, (e.g., the rightmost trajectory in Fig.2.4). Finally, there are also straight line separatrices parallel to the I/-axis at each of the points C1a, extending from 11 :

-[ -FIr to infinity, which represent solutions with a static scalar field sitting on top of the potential hil1. To examine the critical points at infinity it is convenient to transform to spherical polar coordinates r,0, and / by defining H I - rsindsin/, (2.48) J : rsinîcos$, (2.49)

and to bring the sphere at infinity to a finite distance from the origin by the transformation, : pl(l - p), 0 I p < 1 [68]. Although the trajectories on the sphere at infinity do not represent physical cosmologies, it is useful to plot them since the form of the trajectories which lie just lThe possible role of scalar fields with PNGB potentials in driving an inflationary expansion of

the early universe has been discussed in ref. [67].

25 within the sphere will be similar. On the sphere p -- I we find

d0 L rr'vsin 0 lvucos2e fe rtn,t' óY +2+(Lan2t \"*" 0.or, 4 o;l ,) (2.50) ,tt = 4 - "'J Yde2 : l.o, e sinkþ, (2.51) where { is a new time coordinate defined bv dt : rdt' The resulting integral curves are plotted in Fig. 2.5. By (2.45) the plojection of the physical region p" ) 0 onto the sphere at infinity leads to the condition

cot2o>1r.o"'4. Q.52) 6 Values of g and / which violate this inequality lie in the shaded region.

þ 0 2¡

A A- 1+ 0

A!_

ît

Figure 2.5: The projection of trajectories within the sphere at infinity onto the plane of the spherical polar coordinates / and 0,for the system (2.42)-(2.45). The unphysical region is shaded.

The critical points on the sphere at infinity are

1. AË, Afl: four points at (0, ó) € {(+ tan-1 t/6,0), (t tan-1 t/6,r)} ot 11 : foo, I lH :0, and JIH : +t/6. 2. Bi: two points at

26 11 : too, I lH :0, and J lH : 0. Since the projection onto spherical polar coordinates Q.a7)-(2.49) is degenerate at the north and south poles d :0,T, these points are excluded from the chart (0,ó) but can be included using an alternative hemispherical projection (see Fig. 2.6).

3. Cfr: two points at

(0,ó) e {(tr 12,n l2),(n 12,3r l2)} or 1: *oo, HII :0, and JII : 0.

H I>0 B-. I<0

A-r*

,, f + J

A-1-

B- B-_

Figure 2.6: The projection of trajectories within the 1 ) 0 and I < 0 hemispheres at infinity onto the H-J pIane, for the system (2.42)-(2.44). The unphysical region is shaded.

An analysis of small perturbations shows that the points Añ,r* are repellors in ail directions of the phase space (cf., Table 2.5), while AÊ,r- are attractors. This therefore represents the most 'typical' early behaviour of soiutions. For 11 > 0, Añ,r+ correspond to the limit / -+ 0. we find that 11 -ll3t or a o( ¿tl3, while rcþ-+1þþlnú, for these solutions. The points Af,r- with fI ( 0 represent the time-reversed solutions. The point Bî (Bî) repels (attracts) a two dimensional bunch of trajectories travelling to (from) finite values of H, L, J, btú is a saddle point with respect to

27 Critical points Eigenvalues (with degeneracies) AN, Añ +3 (2), +3 (z -'v) Bi +T Q),+ 1(z - t) ci, 0 (3)

Table 2.5: The critical points on the sphere at infinity and their eigenvalues for the PNGB models. directions on the sphele at infinity. The points are found to correspond to ú + 0 with 1/ - fi¡ or a x t2l3' while nþ x t", n ) 0. The points Cf, are the projection of the points Cr+ and Cz into the sphere at inÊnity. The degenerate eigenvalues simply reflect the degeneracy of the projection. Bf acts as a repellor for trajectories with Ó = 0, / = const as f -+ 0. As shown in Fig. 2.6, trajectories are driven towards Bf before they reach Cf . This is consistent with the property that when 11 is large (3ø > rn6),the field evolution is over-damped by the expansion, and the field is effectively frozen to its initial value (d -+ o).

2.4.2 Energy Density Variables

In view of the eigenvalue degeneracy encounteted above, we can alternatively choose to represent the system by the variables H, r, and y, where similarly to eq. (2.6)

[46] rcsJ :L (2.53) Jan - ,/6H ' KJV 1/lmcos(I l2) a (2.54) ßH ",/zn As above, we will consider H > 0 only. The governing equations then take the form o ' r) H : (2.55) -|H'u,2

+_,my x: -F t-H: +|n*ç¡,-z¡ (2.56) Tnr T- a r +!Hru (2.57) -F -3Y'H.'2m'¿ 2 where p(r,a) = 7(1 - y') -t (2 - lr2 (2.58)

28 We note that in these variables *' + y' - 06 which is why we have adopted the terminology "energy density variables". The physical region of the phase space will be constrained to lie within the cylinder *'+ y' ( 1 since 0ø ( 1. The physical region of phase space is further restricted by the requirement that a' l3*'lH2 : Ht'lH', which is equivalent to cos2 (Il2) ( 1 in terms of the field variables. For values of H > H1 each f1 : const slice of the cylinder 12 + A2 < \ is cut off in the y-direction above and below the a : lHt 111 lines. Thus the "fundamental ceil" of the phase space can be considered tobeacylinderfor0 ( H 1H¡ cappedbyahornfot H ) Ht,whichtapersoff to a line segment -1 < r 3 7 on the r-axis as 11 -+ oo (see Fig. 2.7). In fact, the phase space consists of an infinite number of copies of the fundamental cell of tr\g. 2.7 as a result of the periodic structure of the potential. These celIs, Cn, can be labelled by an integer, n, with the variable 1 lying in the range Znr 1 I < 2(n * 1)zr for each n. For cells with even n the dynamics is described by (2.55)-(2.58) with the upper sign in (2.56) and (2.57), while for odd rl one must take the lower sign. H

I -'--¿- il \-

C Hl cr*

Figure 2.7: The "fundamental cell" of the phase space in terms of the energy density variables. On the y' : H?lH2 planes (The shaded planes), only one trajectory is possible as shown. It corresponds to the scalar field lying on top of the potential hill all along. On the H : 0 plane, the trajectories are concentric circles with centre at the origin.

Within the horn portion of each phase-space cell the motion of most trajectories

29 is roughly circular in H : const slices, in a clockwise sense in even ceIls, C2n, and anti-clockwise in odd cells Cz,+t. However, trajectories can cross from one cell to another along the g : LfulH boundaries of their horns, which correspond to the surfaces I :2nr in terms of the field variables. For even cells, C2n, trajectories join the cell C2n alongthe y : HtlH surface and the ceIIC2n.,r1 along the y : -HtlH surface. Below the .I1 - Hl plane solutions cannot cross from one cell to another, but lemain confined within the cylinder 12 + a2 < 7. When H :0 we see from (2.55)-(2.58) that å : malF and y : -mælT, so that trajectories which iie in the H :0 surface are purely concentric circles. Since à : 0 in the H : 0 plane, these do not represent physically interesting cosmologies, but by continuity the behaviour of the trajectories just above the plane will be of a spiral nature. The origin H : r: A:0 is in fact the critical point corresponding to Cz. The nature of the critical point is altered by the change of variables, however. In particular, whereas the eigenvalues for linear perturbations are unchanged, when higher order corrections are considered the point is no longer always an asymptotically stable spiral as rwas the case in Fig. 2.4. Asymptotically as I -) oo we have mt r: A(t) sinT , (2.5e)

: A(t) (2.60) a "orf , where the amplitud is governed by the equation ftø\ : BHA2 (, - r"t"'T) 0 - A'). (2 61)

The nature of Cz is I ow found to depend on 7:

1. If 1 ( 1 we ft d that over a cycle the average value of the right hand side (r.h.s.) of (2.61) is negat ve and A2 decreases o that C2 is an asymptotically stablespiral. Furthermoretoleadingorder H-fiast +ooor a

A(t) : s¡(,-,)t,""p rr.urt l- + I,* +."" (Y)f

wit B consta t. The la e-time attractor has 04 : 0 and 0", : 1

30 2. I1 1 ) 1 then over a cycle the average value of the r.h.s. of (2.61) is positive and. A2 increases until it reaches a limit cycle A2 : 1, i.e., *' + y' : 1 or f)d : 1, 0.y : 0. In this case

A(t):r - Bt2(,-r)exp l- l,* +"",(#)] (2.63)

3. If 7 : 1, which corresponds physically to an ordinary matter-dominated universe, then an intermediate situation obtains. Essentially any of the concentric circles in the H :0 plane of Fig. 2.7 can be approached asymptotically giving a universe for which 0ø -+ cv1 and 0ø -+ | - at where a1 is a constant in the range 0 ( ar < 1, which depends on the initial conditions and the parameters m and F.

The three different late-time behaviours can be understood as a consequence of the scalar field either decreasing more rapidly than the barotropic fluid (7 < 1), less rapidly ('y > 1), or at the same rate (f : 1). Since Pt x ct-31, we observe that for all values of 7, I pó o ; (2.64) at late times. This is because the effective barotropic index of the scalar field is .ló :2r2f n2 + A2 N sin2(mtf F) at late times, so that it varies from 7¿ : 0 to .yó:2 over each cycle, with an average value at 'lö - 1. The scalar field thus has a dependence on the scale factor identical to that of non-relativistic matter at late times.

31 Chapter 3

Numerical Integration

In Chapter 2, we performed a phase-space analysis to the governing equations (2.2)- (2.5) for the different quintessence models. In this chapter, we want to numerically evolve the governing equations forward from the time of onset of matter domination by choosing suitable initial conditions. This work extends the studies previously undertaken by various authors [40, 69, 70,71].

3.1 Simple Exponential Potential

Here we will extend the results of Frieman and Waga [71]. For the simple exponential potential models with V({) - V¡e-À"Ó we introduce the dimensionless variables:

1) : nlJ;O z) ru: )rcó (3.1) ,: H , I , -r"#, wher.e ,f)-¡ is the fractional energy density of mattel at the present epoch, lo. We will integrate the field equations forwarcl from the time of onset of matter domination, therefore we have 1 : l. In terms of the dimensionless variables defined above, the dynamical system becomes

,r' : )e-- (3.2) -ZIr*H' , H u (3.3) Ho Ii : Àu, (3.4) where the Hubble par-ameter is deflned implicitly according to

1 H /" 1" 1 2 -f -f -e-IU (3.5) H, = (r" Uu" ü

32 and a prime denotes a derivative with respect to the dimensionless time parameter Hot. Note that we have chosen the same dimensionless variables as Frieman and Waga

[71] except for the variable u. Our reason for making the new choice of u in eq. (3.1) is that it allows us to integrate the Friedman equation directly rather than a second order equation (2.2) which follows from the other equations by virtue of the Bianchi identity. This may possibly lead to better numerical stability since it is not necessary to implement the Friedmann constraint separately. We begin the integration at initial values of u, u, and tr,r chosen to correspond to initiai conditions expected in the early matter-dominated era. The integration proceeds then until the right-hand side (r.h.s.) of (3.5) is equal to 1, thereby determining the value of the present epoch, ts, to be the time at which H : Ho. We are then also able to determine Íì-s, since according to the definition of u in eq. (3.1)

0rno:rt(úo) . (3.6)

The choice of appropriate initiai conditions has been previously discussed [70, 71]. In particular, since the Hubble parameter is large at early times, it effectively acts as a damping term in eq. (2.2), driving the scalar field to a state with /:0 initially, i.e., z : 0. We take u : 1101 initially, which in view of the definition of u in eq. (3.1) and the fact that 0-0 -0.1 - 1 corresponds to the early matter dominated era 1100 I z 5 3000. Results of the integration do not change significantly if u is altered to values within the same order of magnitude. As eq. (3.1) shows, changing Vt is equivalent to rescaling the scalar field /. Without lost of generality we fix ó : 0 iniiially. Finally, we fix Hst :10-5 initially. The contours for Íìóo : 1-0-o are displayed in Figs 3.1 in theV¡, ) parameter space. In the far-left portion of the flgure, the fi,eld evolution is overdamped by the expansion, and the field is effectively'hozen' to its initial value. Thus, at early times, the fieid acts as an effective cosmological constant, with Q4o is determined by Vt,, independent of À, and the contours of constant f)¿o are nearly vertical. In the far-right portion of the figure, the field becomes dynamical before the present epoch and has already reached the attractor solution (see Section 2.1). For À2 ) 3, the late-time attlactor is the "scaling solution" and the field redshifts as non- lelativistic matter with 060 :31À2, independent of V¡. The contours of constant f)40 are nearly horizontal with the limit 040 : 1 at )t ( 3, where the scalar field dominated solution becomes the late-time attractor.

33 T 0.1 03

0.9 0

Þ0

-1

qu? o\ Oi -2 o oo -1 0 2 3 tog(rczVn/u o2)

Figure 3.1: Contours of constant Opo in the parameter space for the simple exponential potential models, V(Ó) : V¿e-^"Ó.

0.?o 0

Þ0 o

-1.

o t- @ t -1 0 2 3 tog(rczvn/H o2)

Figure 3.2: Contours of constant /{ofo in the parameter space for the simple exponential potential models, V(Ó) : V¡e-^"Ó.

34 Fig. 3.2 shows contours of constant expansion age, Hoto, in the same parameter space. In the lower-left portion of the figure where the field acts as a cosmological constant at present, Hsts increases with increasing fl4o as expected. In the upper-right portion of the figure where )' ) 3, Ilsfs aslmptotically approaches lhe Ei,nstei,n-de Sitter value 213 as one moves to the right, since the "scaling" freld currently redshifts as the background matter ancl we have assumed a spatially flat unlverse

3.2 Inverse Power-Law Potential

The numerical integration for the inverse power-law potential models, V(/) : Vdó-o , is similar as above. Here we introduce a different u.t, w : nó. (3.7)

The dynamical system then becomes H 1.1' : 4?" I aVnw-@*r\ , (3.8) Hs H u' (3.e) Ho

1t) : U j (3.10) where Vn : rc'+2V¡f H!, and the Hubble parameter is (," +Hn:\" = -6*+!u' +lu*.,-')å'3'o* ) . (8.11) We take u : 1101 initially as discussed in the previous section. Since the inverse power-law potential gives rise to "tracker solutions", we assume the scalar field is on track early in the matter-dominated era. From the tracker solution given in Section 2.2.2, we derive the following relations among u, u, and wi u2 3w2 fr 6(H lHo)2 2(o+2)21 Vnw-o 3(a I4)w2 Y: ,@lÈe:ffi, (3'12) For the matter-dominated era, \¡re have (H I Hs)' = ,". Solving eq. (3.12) for u and

'¿-¿.' we fi.nd

u ÈN ) (3'13) l-g'"*'"]'t'fi'+z)'l

35 10

6 o õ 4

0 -2 0 2 4 6 I log(rc"*zv",/H o2)

Figure 3.3: Contours of constant 0¿o in the parameter space for the inverse power- law potential models , V (Ó) : VtÓ-o .

-.t 0 _L a 0 2 4 6 8 log(rc"*zV",/H oz)

Figure 3.4: Contours of constant Ilolo in the parameter space for the inverse power- law potential models , V (Ó) : VtÓ-o .

36 g(g!El-1/(a+2), w ã I (3.14) lr"@T*wl which fixes the initial values for u and u' given VB, a, and the initial value for r.,. Note that this approach is different from Waga and Frieman [40], who started the evolution of the scalar freld during the radiation dominated epoch, but also assumed that the scalar field is on track early in the evolution of the universe. The contours for f)40 are displayed in Figs 3.3 in theVp, o parameter space. As expected, the contours of constant Í)¿o increase with increasing V6. In the lower portion of the figure where a = 0, the scaiar field potentials are so flat that the field act effectively as a cosmological constant. In the upper portion of the figure, the scalar field is beginning to dominate only recently, which means that it ìs still well-approximated by the tracker field solution, with its energy density scaling as p6 x a-3t0, and 7¿ : al(a*2) (see Section 2.2.2). Hence, the larger a is, the more rapidly p4 decreases with respect to the scale factor a (and hence the redshift z I I), and the smaller f)40 is. Fig. 3.4 shows contours of constant Hsts in the same parameter space. As expected, flsús aslmptotically approaches the Einstein-de Sitter value 2/3 toward the upper-left corner of the figure where 0¿o -ì 0, and the current equation-of-state of the scalar field approaches that of the non-relativistic matter, 'uøo -+ I'

3.3 Double-Exponential Potential

For the double-exponential potential models, V(Ó) : V¡exp( - A.fr"Ó), we now redefine t¿' as lx: fing* lnÁ . (3.15)

The dynamical system then becomes H 'tt' : exp (-e-) (3.16) -t O" + t/lVee* , H u (3.1 7) Ho u) : t[2u (3. i8) ' where Vn: rc2VtlHl, and the Hubble parameter is H exp ( -)]' (3.1e) Hs þ'* å"' *I'" -e

o4 Jf 1

q7 0' q 0 6 o.7 o.9

Þ0 o

-1.

-2 -1 0 2 3 tog(rczv*,/a o2)

Figure 3.5: Contours of constant f}¿o in the parameter space for the rlouble- exponential potential models, V(Ó) : V¡exp(-A"Ø"Ó)'

o o.f 0.80 0 ogo S Þ.0 o -1

D 0 2 3 tog(rczv"/no2)

Figure 3.6: Contours of constant Hsts in the parameter space for the double- exponential potentiai models, V(Ó) : V¡exp(-A"Ø"Ó).

38 We choose the initial values u : 0 and u - 1101, as for the simple exponential potential models (see Section 3.1). Without lost of generality, we fix the scalar field ó:0 initially, as eq. (3.15) shows that changing A is equivalent to rescaiing /. The contour for f)¿o is displayed in Fig. 3.5 in the VB, A parameter space. In the far-left portion of the figure, the field acts as an effective cosmological constant, with 0¿o is determined by 76, independent of A, and the contours of constant f)60 are nearly vertical. In the far-right portion of the frgure, the fleld has become dynamical before the present epoch. For small values of Vn, the scalar field has become dynamical only relatively recently, whereas for large values of VB, the scalar field has been dynamical for some time. Therefore, f)60 decreases asymptotically since the universe is eventually matter-dominated, f,)¿o -+ 0 (see Section 2.3). Fig. 3.6 shows contours of constant Hsts in the same parameter space. As expected, llsfs aslmptotically approaches the Einstein-de Sitter value 2f3 as one moves to the right, where the univer-se is currently dominated by matter, 0¿o -+ 0.

3.4 PNGB Potential

F,or the pNGB models with Iz(/) : M4fcos(þlf) + 1l : (*"lor)fcos(rcglF) + tl, we now redefine the variable u as nó w F (3.20) The dynamical system then becomes -H m2 u * srn ?'., (3.21) -t ao" TR ' H (3.22) u Ho' u L1) : ,71 (3.23 ) wh.ere the Hubble patameter is

1 H 2 'u,' * + t)] (3.24) Ho f,'* ffir."", We choose the initial values u : 0 and u : 1101, as for the simple exponential potential models (see Section 3.1). The initial value of the scalar field variable, wi: u)(ti), can lead to some valiability in predicted cosmological parameters at the

39 present epoch. A few clifferent values of w(t¿) have been considered by different au- thor.s [40,69, 71], and the systematic stuclies of bounds in the M,f pararneter space have been performed. One must bear in mind that such bounds are also dependent oî,tzi) the value of which is not greatly restricted. Given that we are starting with u(t¿) x 0, so that the kinetic energy of the scalar field is initially negligible, the only physical restriction on the value of u¿ comes from the requirement that the scalar field should be sufficiently far from the minimum of the potential, V(Ó), that Cla(l¿) is smali. This will ensure that u¿ is consistent with a scalar field that has emerged from the radiation dominated era with Íì4 sufficiently small that it is consistent with bounds set by primordial nucleosynthesis, and by structure formation models. This still leaves considerable latitude for the choice of w¿, however. In Figs. 3.7 and 3.8 we display contour plots of f)¿o and ly'ofo in the M,f parameter space for two values u; : 1.5 and tl¿ : 0'5' Similar figures have been given by Frieman and Waga 140, 7l] in the u¿ : L5 case, although our resolution is somewhat better. As ru¿ decr-eases the contour plots do not change significantly in terms of their overall features, but contours with equivalent values shift to lower values of the / palameter. For example, for large values of M the f,)do : 0.7 contour lies at a value f x 2.05 x 1018 GeV \1 w¿ - 1.5, while the same contour lies at f x 0.94 x 1018 GeV if w¿: 0.5' The other principal feature of the plots 3.7 and 3.8, which was not commented on in ref. [71], is the waue-lilce properties of the contours at larger values of M. These features can be readily understood by considering the corresponding plots of the deceleration parameter, qs, which is defined in terms of u(fs), u(fo), and u.'(ls) by 3 1 .. : ". * I, eo zu"(to) ru"(to) - 1 (r+:clao (3.25) 2 -aao")

We display contour plots of qs in the M,,f parameter space in Fig. 3.9. Essentially, as M increases for roughly fixed / for sufficiently large M, the value of q6 oscillates over negative and positive values, from a minimum of qo - iG - 3Cløo) to a maximum of qo This corresponds to the scalar field, qo : T (t + 3fì40) about a mean - 0.5. /, having undergone mor-e and more oscillations by the time of the present epoch. The minimum value of qs is attained when Ó : O instantaneously, while the maximum value of q¡ is attained when / is instantaneously passing through the minimum of

40 6 U, 1.5 5

0) 4 æ

3 0.9 \ 0.9 o.7 2 o.? 0.6 .5 0.3 1 o.3 2 4 6 B 10 (u) tt (to-sht/zeY)

3.0

2.5 u¡ = A'5

q) f\ 2.0 €

v 1.5 \ o.9 .o.S

1.0 o.? o.z 9.5 0.5 0.5 2 4 6 B 10 (b) tt (to-3nL/zeY)

Figure 3.7: Contours of fl4o in the parameter space for the PNGB models for two choices of initial values: (u) .o : 1.5; (b) .o : 0.5

4t 6

c) 4 €

3 \

2 0.80 'U)t 1.5 I 2 + 6 8 10 (u) l,t (to-\nL/?eY)

3.0

2.5

c) ë 1.5 \ 1.0

0.5 u¡ = O'5 0.0 ? + 6 8 10 (b) l,t (to-}ttl/?eY)

Figure 3.8: Contours of Hstsin the parameter space for the PNGB models for two choices of initial values: (u) *n: 1.5; (b) ,, :0.5

42 6

0) 4 a @ O 3

1, 2 4 6 8 10 (") M (Io-3h1/2eY)

3.0

c) 2.0 O 1.5 o

1.0

0.5 2 4 6 8 10 (b) tt (to-"nL/zeY)

Figure 3.9: Contours of qs in the parameter space for the PNGB moclels for two choices of initial values: (.) ,, : 1.5; (b) .o :0.5. Values Ço ( 0, corresponding to a universe whose expansion is accelerating at the present epoch, are shaded.

43 its potential. For smaller values of M b the left of the piots the scalar field has only relatively recently become dynamical, whereas for larger values of M, the scalar can already have undergone several oscillations by the time of the present epoch, particularly if / is small. This variation can be understood in terms of the asymptotic period of oscillation of solutions which approach C1, which by eqs. (2.59), (2.60) is

t":2trF lm :2n I lM2. (3.26) The period lo is shorter for larger M, ot for smaller /. Since the final / values plotted in the w¿ :0.5 case are a factor of 2 smaller than the w¿: 1.5 case, this also explains why points with the same value of M have undergone more oscillations up to the present epoch for the smaller value of w¿. The value of Hús osciilates as M increases for roughly fixed /, according to whethel the universe has been accelerating or decelerating in the most recent past, with more rapid variation for parameter values with shorter asymptotic periods, fo. The "wiggles" in the ,ft¿o contours are a residual effect of the oscillation of the scalar freld as f,)¿ settles down to a constant value according to (2.61). The variation in the value of 040 with the value of w¿ can be understood from the fact that for smaller values of w¿ lhe scalar field begins its evolution in the matter-dominated era closer to the critical points, C1a, corresponding to the maximum of the potential, V(ó). For fixed M ancl / the period of quasi-inflationary expansion is therefore longer, and the present value of 040 larger.

3.5 Luminosity Distance

We have discussed numerical integration of govelning equations for the different quintessence models and determinecl f)¿0, Hsts. In this section we want to discuss an additional task of the numerical integration which is to determine numerically the luminosity distance - redshift relations. The luminosity distance of a light source, d¡,, is defi'ned by Hndt' (, : \z' +-r- t)r\ ['o \z +-r- r)d'r)rl /r, (J.27) " Jr with r - Hot, it is convenient to define an additional variable r: [' ,d, (3.28) - Jrn

44 which is proportional to the comoving coordinate distance

CT rcomoving : (3.2e) n*4to1*ç

We then adjoin a differential equation

Tl : -'u (3.30) to the differential equations of u, u, and u.r when performing the numerical integration. In terms of r and u, the lurninosity distance is then determined according to Hodr-'-o?-r(úo)) , (g.31) c a:Å; Empirical caliblation of the light curve - Iuminosity relationship of type Ia supernovae provides absolute magnitudes that can be used as distance indicators. The luminosity distance (3.31) can be used in the appropriate distance modulus to compare with the supernovae data. Glavitational lensing of distant light sources due to the accumulation of matter along the line of sight leads to a statistical test that can also put constraint on the luminosity distance - redshift relation. Basically, if the volume of space to a given redshift is larger then on average one can expect more lensing events. These observational tests will be the focus of the remaining chapters.

45 Chapter 4

Supernovae Tests

In the late 1980s type Ia supernovae (SNe Ia) were briefly considered promising candidates for standard candles, but observers quickly discovered that SNe Ia are not all identically brighi. The intrinsic disper-sion in the peak absolute magnitudes of SNe Ia, determined from studies of nearby events, is approximately 0.3-0.5 mag [72]. However, there is an apparent empirical correlation between the rate of de- ciine of the light cur-ve of a given type Ia supernova (SN Ia) and its luminosity at maximum brightness that was first quantified by Phillips et al.lL9l. Various techniques have been developed to take advantage of this correlation to determine the absolute magnitudes of individual supernovae using their light curves [19]-[25]; the relationships used in these analyses have come to be known generically as "Phillips relations". When applied to nearby SNe Ia, these methods reduce the dispersion of the distance moduli about the low-redshift Friedmann-Robertson-Walker (FRW) distance modulus versus redshift relation to approximately 0.75 .21,24]. Two experimental groups, the Supernova Cosmology project (SCP) [16, 25] and the Highredshift Supernovae Search (HZS) 115], have recently announced and published results from their independent programs to discover and study high-redshift supernovae. The r-esulting two data sets share many low-redshift supernovae discovered by previous surveys, but include different high-redshift supernovae, and differ in their analysis methods. SCP have published data on 60 SNe Ia. Of these, 18 were discovered and measured in the Calán-Tololo survey (all at low redshift) [73], and this group discovered 42 new SNe Ia at redshifts between 0.17 and 0.83. The distance moduli ) p) are inferred using the stretch factor (SF) light curve fitting method and are typically

46 uncertain to t0.2 mag at 1o uncertainties. The SF method [16, 25] is based on fitting a time stretched version of a single standard template to the observed light cur-ves. The stretch factor, s, is then used to estimate the absolute magnitude of the SNe Ia via a linear relationship that is determined jointiy with the cosmological parameters. The quoted ¡; values include a correction for extinction in the Galaxy based on the detailed model of Burstein and Heiles [74]' HZS have published results based on 50 SNe Ia. Of these, 37, including 27 at low redshift (z < 0.15) and 10 at high redshift (z > 0.15) have well-sampled light curves in addition to spectroscopic information; the quoted 1a uncertainties for ¡,r" for these SNe Ia ale typically smaller than f 0.2 mag at high redshift for determinations by either the multicolour light curve shape (MLCS) method or the M15/template fitting (Ttr) method. The data for 17 of the SNe Ia at low redshift come from the

Calán-Tololo survey [73].

4.L Constraints on Quintessential Parameters

In this section, we will study the SNe Ia constraints upon the quintessence models discussed previously. The SNe Ia constlaints upon quintessence models have been pleviously studied by a number of authors [40, 77, 75,,76]. It is our aim here to extend these results in a number of ways: both in terms of regions of parameter space considered, inclusion of the possibility of source evolution, and the statistical tests used. In this section, we will begin by performing a likelihood analysis similar to previous analyses, but will use a somewhat different approach to calculating the likelihood function, by using analyti,c marginalization in a fashion similar to that which has been appiied to standard FRW models by Drell, Loredo and Wasserman l( (). We will make use of the larger available data set of the two, namely the 60 SNe Ia published by SCP. The distance modulus for supernovae number i is estimated to be pi : r'rL.i - Ms, (4.1)

where we use the stretch-luminosity corrected effective B-band peak magnitude as tlre apparent magnitude m¿, arrd Mo is the fiducial absolute magnitude. In the quintessence models with parameters Q, when the (relative) peculiar velocity of the source is negligible, the distance modulus is determined by the source's

47 redshift, z, according to

ttQ; Q,flo) : \Iogd;(z; Q, Ho) + 25 , (4.2) if the luminosity distance, cl¡,, as defi.ned by eqs. (3.27) or (3.31), is given in units of Megaparsecs. As in the analyses of HZS and SCP, we adopt the Bayesian approach for inferring the quintessential parameters Q. The additional parameters, such as f10 and Ms, are dealt with by marginalizing. We follow the statistical procedures adopted by Drell eú al. l77l and marginalize over' //s and Mo analytically' Rigorous calculation of the iikelihood for Q, IIs, and Ms is very complicated, requiring the introduction and estimation of many additional parameters, including parameters from the lightcurve model and parameters for characteristics of the individual SNe. With several simplifying assumptions, the finai result is relatively simple. The result is

lp?¿; Q, Ho) tt¿l' L(Q, Ho, Mo) xff - (4 3) ""0 l- 2 i:t L where ol: ol,r*lp,'(z¿))2ol,n. (4.4)

Here l/¿ is the number of data points, p¿ is given by (4.1), or,iis its uncertainty, "q. z¿ is the redshift, and o",¿ is its uncertainty (mostly due to the source's peculiar velocity). The total variance øf depends on Q through p'(r¿).However, this dependence is weak in general, and p,'(z) is actually independent of Q at low redshift in the pure cosmology model, with t,Q):ryi (4.5) We follow the practice of HZS and simply use this formula for all redshifts. We can separate p into a part g that depends implicitly only on Q,

p(z; Q, Ho) : g(r; Q) - n , (4.6) and the I1s dependence is contained in 17,

\ : Slog llo . Ø.7)

Let us define the quadratic form X2 according to ll¿ l^n Ms g(z¿; Q) + nlz x'(Q, Ho,, Mo): - - (4 8) Ð,i=l ol

48 This is the y2 statistic used in HZS; the joint likelihood for Q, Ho, and Ms is simply proportional lo e-x'12. Note here there is a degeneracy between r7 and ffi, since they play identical roles (up to a sign) in the model for the distance moduli. We can analytically marginalize over the quantity

u:Mo-\ (4 e) to find the marginal likelihood for Q. We use a prior that is flat in u. We bound this prior over some range A¡u. The prior range has negligible effect on all our results (so long as it contains the peak of the likelihood). Since the u parameter is common to all models the prior range cancels out of all probability ratios. Thus we could let it become infinite. Using the flat prior, the marginal likelihood for Q is

a 1 L L" I clue-"12. (4.10) To perform the integral we complete the square in y2 as a function of z, writing

x'(Q,¿:LP+q(Q), (4.11) where N¿ 1 1 Ð (4.r2) J^2 i=l ol2', À/¿ m¿ g(z¿; Q) ù(Q) : Ð - (4.13) "t i=l ol and the Q-dependence is isolated in î/(A.), g(r¿;Q)l' q(Q\ : _-l*;,? - a\-/ -= 32 Ol

:,\tr3[-n -g(z¿;q-r@)]' Ø14)

The integral in (4.10) is thus simply an integral over a Gaussian in ¿z located at / with standard deviation s; ù is the best-fit (most probable) value of z given Q, and s is its conditional uncertainty. As long as ¡u is inside the prior range and the value of this integral is well approximated by so that s K L,u, "t/?n, Le) - "F (4.15) A,U "-nt,.

49 This is the "marginal likeiihood" one would use to infer Q in the absence of any systematic error terms. Note from (4.14) that the quadratic form is just what one would obtain by calculating the "profile likelihood" for the density parameters (the Iikelihood maximized over the nuisance parameters, a frequentist method sometimes used to approximately treat nuisance parameters). Since the uncertainty s is independent of Q, it follows from (4.15) that the marginal likelihood is proportional to the profile likelihood in this problem. ufit In ref. [tO] Perlmutter eú ¿/. consider-ed a subset C" of 54 SNe Ia which excluded six supernovae events: two that are the most significant outliers from the average light-curve width, two that have the largest differences between observed and expected magnitudes (or fluxes), and two that are likely to be reddened. It was discovered that excluding these six supernovae produced a more robust flt of the cosmological parameters 0-o and ,Q¡s. Since we are investigating models with different cosmological parameters from those fitted in ref. [16] and we do not wish to prejúdge matters, we will do the analysis for both the entire data set of ali 60 SNe Ia and the "fit C" data set of SCP. With the PNGB models, the 68.3% and 95.470 joint credible regions for M and / with the initial value u¿ : I.5 (see Section 3.a) are displayed in Fig. 4.1, which is a direct analogue of Fig. 1 of ref. [40] where a similar analysis was performed on 37 supernovae given in HZS. The position of the region of parameters which are included at both the 68.3% and 95.4% confidence levels is broadly similar to that obtained from the HZS data. Although Frieman and Waga did not include the parameter region M > 0.004h eY , we have redone their analysis on the HZS data and fincl that the parameter values to the right of trig. 4.1 which are admitted at the 95.470IeveI but excluded at the 683% level for the SCP data set, (labelled region II in Fig. 4.1), are in fact excluded also at the 95.4% confrdence level if the 37 supernovae of the HZS data set are used. It is possible that this discrepancy between the outcomes of the analysis of the two data sets has its origin in the different techniques used by Riess et al.115) to determine the distance moduli. Possible systematic discrepancies in the SF method of SCP versus the MLCS and TF methods of HSZ have been discussed in some detail in ref. [77]. The importance of the 2ø included parameter region to right of Fig. 4.1 di- minishes, however, if other observations are taken into account, since it largely corresponds to parameter values with flao ) 0.9. The overwhelming evidence of

50 6

0) 4 oC,

3 \ 2

1 2 4 6 B 10 (") ttt (tO-1nt/zeY)

c) 4 €

3 \ I

I 2 4 6 a 10 (b) ttt (to-3nt/2eY)

Figure 4.1: Confidence limits on M,f parameter values of the PNGB models with w¿ : 1.5: (a) for the 60 SNe Ia in the SCP data set; (b) for the 54 SNe Ia in the reduced "fit C" SCP data set. Parameter values excluded at the 95.4% level are darkly shaded, while those excluded at the 68.3% level are lightly shaded. Overplotted are the contours for Q4o (dashed) and Hsts (dotted).

51 many astronomical observations over the past two decades [9]-[11] would tend to indicate that 0-o ry 0.2 +.0.1, indicating that a vacuum energy fraction of f)óo : 1 - 0-o - 0.7-0.8 is desirable, and fl¿o 5 0.9 in any case. Furthermore, in view of recent estimates [84], a lower bound of 10 Gyr for the age of the Universe appears to currently indicated. With h - 0.7 this would require Hotoà0.72. The few 2o allowed values below the Íì¿o : 0.9 contour in this part of the parameter space (which disappears entirely for "fit C") have unacceptably small values for the age of the Universe, ,[/sús. The parameter region which flom Figs. 4.1 is admitted at both the 68.3% and 95.4% levels (labelled region I), by contrast, corresponds to acceptable values of both 060 and 1/61s. Comparing with Fig. 3.9, we see that this region has -0.1 S qo 5 -0.6, corresponding to a Universe with a scalar field still in an early stage of rolling down the potentialV($). The label I is thus indicative of the fact that the scalar field is rolling down the potential for the first time (from left to right), while in region II the scalar field is rolling down the potential for the second time (from right to left). In region II qs is positive - however, it corresponds to parameter values for which there would have been a cosmological acceleration at modest redshifts in the past, e.g., al, z - 0.2 [78], well within the range of the current supernovae data set. The tension between observably acceptable values of 040 and I1sls arising from parameter values of M and / in region II is more easily mitigated by choosing a lower value of w¿, for which the scalar field spends more of its early dynamical history higher up the potential hill. Confidence lìmits on the M, f parameter space obtained from the SCP data set with the initial value w¿:0.2 are shown in Fig.4.2. Note that although there \s no 2o allowed area left below the f)¿o : 0.9 contours to the right of Fig 4.2,, the conclusions legarding statistically preferred regions of the parameter space can change if evolution of the sources occurs. We think it is important that this legion of the parameter space is also examined, as we will now do. For the PNGB model with initial value w; : 0.2 we find that in the best-fit case the normalized y2 parameter is reduced from 101.6, i.e., 1.75 per degree of freedom, ufit for the full data set to 58.65, or 1.13 per degree of freedom, for the reduced C" data set. For the PNGB models with u;¿ : 1'5 and the other quintessence models the best-fit y2 pararneters are similarly greatly reduced. Thus the "fit C" data

52 2.O

1.5 (,0) 3 O c 1.0 \

0.5

2 4 6 10 (") trt (to-"ht/?eY)

2.O

1.5 c) € 1.0 \

0.5

2 4 6 I 10 (b) M (Ll-slnt/zeY)

Figure 4.2: Confr.dence limits on M,f parameter values of the PNGB models, with wi : 0.2i (a) for the 60 SNe Ia in the SCP data set; (b) for the 54 SNe Ia in the reduced "fit C' SCP data set. Parameter values excluded at the 95.4% level are darkly shaded, while those exciuded at the 68.37a level are lightly shaded. Overplotted are the contours for fl4o (dashed) and 110úo (dotted).

53 provides a moïe robust fit in all the quintessence models we have studied. The marginal likelihood for the quintessential parameters for the SCP data ar-e displayed as 68.3% and 95.4% confidence level limits in the parameter spaces of the simple exponential potential models (see Fig. 4.3), the inverse power-law potential models (see Fig. 4.4), and the double-exponential potential models (see Fig. 4.5). We see that for these models the differences between the parameter values allowed by fitting all 60 Sne Ia in the full data set and the 54 SNe Ia in the reduced "fit C" data set are much smaller than for the PNGB model. Indeed, only in the case of the double-exponential potential is any difference realiy discernible, and then it is only a broadening of the 1o allowed region. This difference between the PNGB model and the other models no doubt has its origin in the fact that the PNGB model encompasses a greater variety of possible outcomes for the cosmological evolution. In particular, it can lead to solutions for which the expansion of the universe is decelerating at the present epoch just as easily as it can lead to solutions for which the expansion ìs presently accelerating, whereas the other models are more restricted to behave similarly to models with a cosmological constant. Of the quintessence models which we have considered in this thesis the PNGB model is therefore the one which is most naturally suited to testing the hypothesis that the Sne Ia luminosity distance observations are due to an unaccounted intrinsic evolution of the sources, rather than to the cosmological model. We will discuss testing this hypothesis in the next section.

4.2 Evolution of Supernovae

In allowing for source evolution we are acknowledging that the peak luminosities of distant SNe Ia have been normalisecl according to empirical Phillips relations 127,24,79] between observed peak luminosity and supernova decay time, which have been found to be valid at low redshifts. Although we would hope that such relations remain applicable at high redshifts, until the Phillips relations can be modelled and accounted for physically some doubt will always remain about applying these lelations at higher redshifts. In particular, a recent analysis by Riess et al.180) indicates that the sample of SNe Ia shows a possible evolution in rise times from moderate (z -0.3) to large (z-7) redshifts. Although the statistical signifi.cance of this result has been diminished -

54 o.4

0.2

0.0

Þ¡

-o.4

-0.6

-0.8 - 1.0 0 2 3 4 (") tog(rczv"/Ho2)

0.4

0.2

0.0

-0.4

-0.6

- 0.8 - 1.0 0 2 3 +

(b) tog(rczvr,/a o2)

Figule 4.3: Confidence limits on the palameter space for the simple exponential potential models V : Vte-À"Ó; (a) for the 60 SNe Ia in the SCP data set; (b) for the 54 SNe Ia in the reduced ufit C" SCP data set. Parameter values excluded at the 95.470level are darkly shaded, while those excluded at the 683% level are lightly shaded. Overplotted are the contours for f-l4o (solid) and flsús (dotted).

55 1.4

1.2

10 I u

0 0 5 10 t:) (") rog(rc"*zv"/H o2)

1.2

0 0 5 10 15 (b) rog(rc"*z v*/H o2)

Figure 4.4: Confidence limits on the parameter space for the inverse power-law potentiai models V : Vtó-", (.) for the 60 SNe Ia in the SCP data set; (b) for the 54 SNe Ia in the reduced "fit C" SCP data set. Parameter values excluded at the 95.4% level are darkly shaded, while those excluded at the 68.3% Ievel are lightly shaded. Overplotted ai-e the contours for fl¿o (solid) and 116ús (dotted).

56 0.0

-0.5 f b¡ o - 1.0

- 1.5

-2.0 0.0 0.5 1.0 1.5 2.O (^) tog(rczv"/tt o2)

0.0

-0.5

Þ0 o - 1.0

- 1.5

-2.0 0.0 0.5 1.0 1.5 2.O

(b) Iog(rczv",/H o'z)

Figure 4.5: Confi.dence limits on the parameter space for the double exponential potentiai models V : Vtexp(-Aeø"d)' (") for the 60 SNe Ia in the SCP data set; ((fit (b) for the 54 SNe Ia in the reduced C" SCP data set. Parameter values excluded at the 95.4% level are darkly shaded, while those excluded at the 683% ievel are lightty shaded. Overplotted are the contours for Íì¿o (solid) and floúo (dotted). The lower right-hand region is excluded from the plot due to computational difficulties.

{\( from the 5.8a level [80] to the 1.5o level [81] - upon a more rigorous treatment of the uncertainties in the data [81], it remains true that while a systematic evolution in the rise times of the supernovae is not conclusively ruled in, neither is it conclusively ruled out. Given that an evolution in the shape of the light curves of the supernovae measured in their rest frame remains a real possibility, it would not be surprising if the peak luminosity - which is the effective standard candle used - were also to evolve. Riess et al. 180) conclude that the SNe Ia data could conceivably be explained entirely within the context of an open FRW universe together with a rea- sonable astrophysical evolution model, e.8., a consequence of a time variation of the abundances of relevant heavy elements in the environment of the white dwarf supernovae progenitors. Detailed astrophysical modelling - see, e.g., ref. [82] - should hopefully eventually resolve the issue, although at this stage the difference between our theoretical understanding and the observations remains quite substantial [83]. In the absence of a detailed physical model to explain precisely how the source peak luminosities vary with redshift, one approach is to assume some particular 'empirical'form for the source evolution, and to examine the consequences. Such an analysis has been recently preformed by Drell, Loredo and Wasserman l77l in the case of Friedmann-Lemaître models with constant vacuum energy. We will undertake an equivalent analysis for the case of PNGB quintessential cosmologies. We would argue that PNGB cosmologies in fact represent a much more natural set of models in which to test the evolution hypothesis than the open FRW models studied by Drell et al.177l in view of the recent cosmic microwave background results t6]-tS] which now provide overwhelming evidence that the universe is very close to spatially flat. This makes the choice of a spatially open FRW model uncompelling. On the other hand, the PNGB cosmologies have the virtue that while they are spatially flat, thele is no ø ytri,ori preference for either accelerated or decelerated expansion. Both possibilities are available at modest redshifts, depending on parameter values. Ultimately, the scalar field will undergo coherent oscillations about its minimum, and the resulting luminosity distance will become indistinguishable from that of an Einstein cle Sitter model at late times, although the fraction of energy density in clumped matter, f)-, and the fraction of energy density in quintessence, f)6, can take any values consistent with 0- * flø : 1. Whether the scalar field is currently rolling down the potential for the first time in the history of the universe

58 - leading to a luminosity distance relation similar to that produced by a cosmological constant - or whether it has already undergone one or more oscillations by the present epoch, is a matter of a choice of initial conditions. Beyond some bounds set by primordial nucleosynthesis these initial conditions are not very much constrained by our present knowledge of the models, resulting in diverse possibilities for cosmological evolution. The current and final values of ,f)- and 0¿ likewise depend on initial conditions for the scalar field, and on the parameters M and f . The particular luminosity function we will choose here is simply taken as an illustrative example, and is not singled out by any particular physical model. Nat- urally, one can ctiticize it on these grounds. However, the Phillips relations are also purely empirical, and the purpose of our study as with that of ref . l77l is simply to test the extent to which any form of source evolution is able to account for the observed data as compared to an accelerated expansion. Following Drell, Loredo and Wasserman [77] we will assume that the intrinsic luminosities of SNe Ia scale as a power of 1 f z as a result of evolution. This model introduces a continuous magnitude shift of the forrrr B ln(1* z) to the SNe Ia sample. Eq. (4.8) then becomes lÍ¿ l-o g(z¿; Q) u - þln(I + z¿)12 x'(Q, Ho, Mo, Ð :Ð - - (4.16) o?-x

The parameter B will be assumed to have a Gaussian prior distribution with mean ps and standard deviation ó. Physically the parameter Bs represents a ledshift- dependent evolution of the peak luminosity of the supernovae sources, which might be expected to arise as a result of the chemical evolution of the environment of the supernovae progenitors, as abundances of heavier elements increase with cosmic time. Ultimately, one should hope to account for this evolution with astrophysical modelling of the supernovae explosions [82]. The parameter ó would then account for a local variability in the supernovae environments between regions of individual galaxies at the same redshift which are richer or poorer in metals, or with progenitor populations of different ages and masses etc. We now have two parameters to marginalize over) u and B. As in the case of models with no evolution, we will marginalize over z using a flat prior. We use a Gaussian prior for B with mean þo, o, equivalently a Gaussian prior for the new variable þt : þ - Bs with zero mean, and standard deviation ô so that t= P(þ'): þ?lzt'z Ø'r7) bt/2r' "-

59 The marginal likelihood is calculated by multiplying the prior (4.I7) by the likelihood and integrating over B¡ Le): * I oU, I d, p(8,)s-x2/2. (4.1s) The u malginalization is similar as in Section 4.1. The resulting likelihood is given by (4.15), withm¡-g(r¿;Q) replaced withm¿-g(ro;Q)-(þo+0r) ln(1+z¿). "q. To do the marginalization over 81, we complete the square in þt,, identifying the p1 uncertainty, o, given by

1 1 z¿) ,Ç, +ft"'(t ! , (4.1e) or: b, - " T=t oí and the conditional best-fit B1,

-lú¿ h("¡; Q)In(I -l z¿) p(Q): a' -"çu(e) +Ð (4.20) i=L ol where

h(,¿; Q) (4.21)

ç (4.22)

7rQ) (4.23)

Integrating over the Gaussian dependence on B1 gives a factor of aJî, and the final likelihood for Q is Le):'2u9;nt, (4.24)

where

z¿l Q s2?t(Q))' q@) lh - ol Bfqf , 3 [¡,(,0; Q) - s'?uçq - P9)rn(r ¡ z¿)]2 - -r-- (4.25) v % Although ø is independent of Q, the marginal likelihood is no longer proportional to the profile likelihood because q is now given lry (a.25) rather than by the chi square type statistic @.Ia). We have performed a detailed numerical analysis on the entire SCP data set of 60 SNe Ia varying B¡, b and u.'¿. As a result we find a best-fit value of þo: B* - 0.4L4,

60 +lo

390=õoF H gñ¿Ë-ocrW \J ¿ *õ-9 É.ã- + è È xÈxË ã +sFú x!rr-è (U ¿J Þnia-¡r EË 5ôtll- I I I a-dã'+ 3 o a aÐaf,v UP^ N O) N O') N o)

o-Þ-&2.i'OJ.irP E'ã'G0c + a)oo-:n.D oYd;ì

JivLt Fó1q o0 æ @ 8ãqdFÞ g. :þ o o) --' ñÂþO L!Y :r fuqoPF F2 il 8, FÈ '+P5JA O O OO.CA -^-L- ãX¡-.rØfoo 6

o 4 €

I 2 4 6 I 10 (u)É.=o,b=o l,t (lo-3h.t/zev)

q) o 4 @

1 2 4 6 B 10 (b)p,=0,b=o.25 l,t (lo-3tt"t/2ev)

c) 4 @

3 \ 2

1 2 4 6 8 10 (") É.=o,b=0.5 tvt (lo-\ttt/?ev)

Figure 4.7: Confidence limits on the PNGB models in the þo : 0 slice of the (M,f ,Éo) puru.treter space relative to a best-fit value þ* - 0.414, arìsing from the 60 SNe Ia in the SCP data set, with w¿ : I'5' Parameter values exciuded at the 95.4% level are darkly shaded, while those excluded at the 68.3% level are lightly slraded. For reference, contours of f,16¡ and Hsts are superposed as dashed and dotted lines respectively. 62 2.O

^ 1.5 c) L) I 3 t.o

0.5

2 4 6 I 10 (")É,=0.435,b=o tt (to-\nt/?ev)

2.O

r.5

L) @ 3 t.o \

0.5

2 4 6 8 10 (b) p.=0.435,b=0.25 M (lo-3nt/?ev)

2.O

^ 1.5 0) LJ @ 3 t.o \

0.5

2 4 6 a 10 (")p,=0.435,b=0.5 M (to-3ttr/2ev)

Figure 4.8: Confidence limits on the PNGB models in the best-fit Éo : 0'435 slice of the (M,f ,Éo) putu-eter space, arising from the 60 SNe Ia in the SCP data set, with tl¿ : 0.2. Parameter values excluded at the 95.4%o level are darkly shaded, while those excluded at the 68.3% level are lightly shaded. For reference, contouts of 060 and flolo are superposed as dashed and dotted lines respectively.

63 for w¿ : 1.5 in the PNGB models. This would correspond to supernovae being intrinsically dimmer by 0.17 magnitudes at a redshift of z:0.5, which is an effect of the typical order of magnitude being addressed in current attempts to better' model the supernova explosions [82]. Furthermore, we find that inclusion of a non- zero variance, b2, does not alter the prediction of the best-fit value of þo, although it naturally does lead to a broadening of the areas of parameter space included at the 2o level. There is relatively little broadening of the region of parameter values included at the 1ø level in the 0o : 0* plane, however, which is no doubt a consequence of the steepness of the contouls of deceleration parameter, qs, in Fig. 3.9(a) in the area corresponding to region II. In Figs. 4.6 and 4.7 we display the joint credible regions for M and /, for two slices through the three-dimensional (M,f ,þo) parameter space for w¿: 1.5: (u) the best-fit case 0o : 0*; and (b) þo : 0. Analogously, the best-fit case is shown in Fig. 4.8 for w¿ :0.2. We see from Fig. 4.7 that once the likelihood is normalized relative lo þ* no regions remain in the þ : 0 parameter plane which are admitted at the 1o level when ó : 0. Furthermore, even when a non-zero standard deviation, b, is included, region II of the (M, f) parameter plane is favoured at the la level, in contrast to Fig. 4.1.

103 102 0

101 - o lar "Þ)) f{ o- ?" 100 Y" 99 9B 0.0 0.2 0.4 0.6 0.8 b

Figure 4.9: Variation of the least value of Q : -2ln(LLu) as a function of the standard deviation ó of the prior distribution for Bs for values 0o : 0 and Bs : þ*.

We have also undertaken an analysis of the models with B6 : 0 but variable ó, similarly to the study of ref. [77]. In that case, we once again find that region II of

64 the (M,/) parameter plane is admitted at the 1ø level if ó : 0.25 or ó : 0.5. The dependence of the value of

Q = -2rn(LLu):8 - rr"(+u) rn.rul on the value of ó is displayed in Fig. 4.9, for l3o : 0 as compared with the best- fit case þo : 0*. The quantity Q is analogous to the chi-square statistic of the maximum likelihood method. We see that the 0o : 0 models favour a non-zero value of b - 0.36 by a very small margin as compared to the ó : 0 case. For b : 0.36 the points of greatest likelihood lie mainly in region II, in contrast to the b : 0 case in trig. 4.1. Varying the initial condition ur¿ does not appear to affect the best-fit value of ps significantly. Folu¿ : 0.2 (c.f., Fig. 4.8), for example, the best fit value was þ* :0.435, a difference of 5To from the us¿ : I.5 case. Furthermote, the numerical va,lue of the least value of Q was only 0.270 greater in the w¿ : 0.2 case. We have not attempted to find a best-fit value lor w¿. There is some cause for concern, however, if we consider the favoured values of 0¿o and 11sl6. We always found that the best-fit parameter values occurred at the 040 -+ 1 boundary of the (M,,/) parameter space, which can be discounted by dynamical measurements of 0-0 : 1- f)po [9]-[1i]. Although acceptable parameter values with ,Q4o < 0.8 certainly fall within both the 2o and 1a portions of region II of Fig. 4.6, for all values of b, they generally lead to unacceptably low values of Hús. As we have already remarked in Sec. 4.1 the tension between the values of 1161¡ and 040 is somewhat mitigated for lower values of rn¿. For u.'¿ - 0.2, for example, we see from Fig. 4.8 that the f^ldo : 0.7 and llsls meet in region II, and there is a small region of parameters there with 0.7 5 fløo S 0.9 and 11s16 I 0.8, which is also consistent with the other cosmological tests. The most significant aspect of these results is the fact that the mere introduction of an additional dispersion, b > 0.I7, in the peak luminosities while leaving their- mean value fixed (cf., Fig.4.9), gives rise to a change in the best-fit region of parameter space from region I to region II (see ref. [78] for further details). One would imagine that an increased dispersion is likely to be a feature of many models of source evolution, even if evolutionary effects are of secondary importance. Thus even if the empirical models with non-zero B ar-e somewhat artificial, more sophisticated scenarios could well lead to similar changes in regard to the fitting of cosmological parameters in the PNGB model.

65 Ostensibly the source evolution models appear to provide a slightly better fit. However, this may simply be due to the fact that we have an extra free parameter (B) to fit. Naturally we could repeat the analysis of the present section for the reduced ((fit C" data set. However, it is much more important to attempt to statistically quantify the fit of the SCP data for the case of source evolution relative to the case of no source evolution. We will do this in the next section for both the full data set of SCP and for their "fit C" reduced data set. As we will show, it turns out that in neither case is the difference in the fit of the clata to the source evolution model significant enough to be considered positive evidence for the source evolution hypothesis over the hypothesis that the inferred SNe Ia luminosity distances are purely cosmological in origin. This being the case, we have not consiclered it worthwhile to repeat the analysis of the present section for the reduced "fit C" data set. While we would expect some small changes to the areas of the 1ø and 2o allowed regions in Figs. 4.6-4.8, overall the qualitative features of the various plots would remain broadly similar as Bs and b are varied. We will now turn our attention to the question of statistically comparing different models.

4.3 Model Comparison

One relatively straightforward way to compa e the relative strengths of rival theoretical models in fitting a common set of data, even when the models involve different numbers of parameters, is by calculation of "Bayes factors" [77, 85]. Given more parameters, one may always find a better fit to the data - however, the Bayes factor approach includes an effective "Ockham's razot factor" which adds a weighting against the inclusion of extra parameters. The Bayes factor method may be readily applied to the results of Section 4.1 and 4.2 to give explicit statistics to compare the relative strengths of the PNGB model versus all the other quintessence models, and a cosmological explanation of the SNe Ia data versus an explanation in terms of source evolution. In Bayesian inference, when comparing rival models M¿, each with parameters Q¿, the likelihood for a model conditional on the data (D) in a model comparison calculation is equal to the average likelihood for its parameters,

p(DlMi): dQ¿ p(Q,lM¿)L(Qr) (4.27) I ,

bb where p(Q,lM¡) is the prior probability for Q¿, and L(Q¿) is the sampling probability for D presuming M¿ to be true. The ratio of model likelihoods,

p( D M¿ B¡j : ) (4.28) p(DlM¡) is called the Bayes factor. When the prior odds does not strongly favour one model over another, the Bayes factor can be interpreted just as one would interpret an odds in betting. Kass and Raftery [85] provide a comprehensive review of Bayes factors, and the Lecommended interpretation is summarized in Table 4.1.

B¿j Strength of evidence for H¿ over H¡ 1to3 Not worth more than a bare mention 3to20 Positive 20 to 150 Strong > 150 Very Strong

Table 4.1: Interpretation of Bayes Factors

4.3.t Accelerating IJniverse versus Luminosity Evolution

In this section we will calculate the Bayes factor of the SCP data set to compare the model without souïce evolution to the model with source evolution in the context of the PNGB model. For simplicity, we will take a somewhat different approach to Section 4.2 in our treatment of the case with source evolution. We will assume that the parameter p has a flat prior bounded over some range L0 - 2.3, with limits corresponding to -0.6 < P < 1.7, and marginalize the likelihood function over this prior. The marginalization can be performed analytically as in section 4.2 by replacing the Gaussian prior (4.17) with llLP. The question of what bounds to place on the prior range for the parameter B is an interesting one, since we are dealing with a purely empirical model of source evolution with no a priori restrictions on þ, other than that very large values of B would not be physically plausible. We will therefore seek the narrowest range of values of B consistent with our numerical results. One condition on the prior range for B is that B also lies inside the prior range, where B it tto* the best-fit value of B obtained by marginalizing over a flat prior, analogously to eq. (4.20). Setting

67 too tight a bound on LB would count against parameter values of M and / with best-fit values B outside the range, which might otherwise be included. By explicit numerical integration we have found that for the initial conditions and the range of parameter values of M and / we consider in this chapter (and in Chapter f) ¡î ües in the range -0.6 < P a l.Z. We will therefore take these bounds to be the prior range for B. Despite the differences from the approach used in Section 4.2 the conclusions of the analysis are broadly similar and are shown ìn Fig. 4.10. In particular, in the absence of any evolution the SCP data set favours region I, whereas if evolution is allowed for then region II appears to be favoured slightly more than region L As observed in Section 4.I, the goodness of fit of the data as measured by the normalized y2 valte in the best-fit case was dramatically improved by using the reduced "fit C" SCP data set, rather than all of their 60 SNe Ia. It is therefore important to check whether the Bayes factor and other statistical results are changed by using the "fit C" data set. We find that in the best-fit case the normalized X2 parameter is reduced from 101.1, or L74 per degree of freedom, for the full data set, to 58.21, or I.I2 per clegree of freedom, for "fit C". Thus the "fit C" data provides a more robust fit even for the model with source evolution. Note that over the entire parameter space considered, we find the value of B with the best fit of all is þ x 0.622 using the entire SCP data set, which occuïs at parameter values M :4.55 * 1g-z¡rrlzev, -f : 0.884 x 1018GeV, while for the "fit C" data set it is B x 0.720, which occurs at M : 4.64 * 1g-zlrtlzev, / : 0.901 x 1018GeV. Averaged over the (M, f) parameter space we find (Ð = 0.286 for ufit the full SCP data set, and (B) = 0.348 for the C' data set. Thus in both cases the value of B is peaked in region II, but for the "fit C" data the overall values of B are some\ryhat larger. For the full SCP data set the Bayes factor for the model without luminosity evolution veïsus the model with luminosity evolution, which is the ratio of the ufit average parameter likelihoods, is B - 2.6. For the reduced C' data set we obtain a Bayes factor of 2.5, which is the same as far as its interpretation is concerned. Thus the data alone cannot discriminate between the two hypotheses.

68 2.0

1.5 O

@ O a 1.0 \

0.5

2 4 6 B 10 (u) M (rc-1hL/?eY)

2.O

1.5 q) € c 1.0 \

0.5

2 + 6 8 10 (b) ttt (to-}nt/?eY)

Figure 4.10: Confidence limits on M,f parameter values of the PNGB model, with u¡ : 0.2, marginalized over a flat prior for the luminosity evolution parameter þ, (") for the 60 SNe Ia in the SCP data set; (b) for the 54 supernovae Ia in the reduced "fit C" SCP data set. Parameter values excluded at the 95.4% level are darkly shaded, while those excluded at the 68.3% level are lightly shaded. Overplotted are the contours for 040 (dashed) and Hsls (dotted).

69 4.3.2 PNGB versus Other Potentials

In this section we will calculate the Bayes factor of the SCP data set to compare the PNGB model with all the other quintessence models. We will use the initial value w¿:0.2, for which the tension between the favoured values of Í^16o and .Flolo in region II is somewhat mitigated relative to the results for some larger values of tr.r¿. Confidence limits on the M, f pararneter space obtained from the SCP data set, assuming u¿ -- 0.2, are shown in Fig. 4.2. ln order to calculate Bayes factors which compa e different theoretical models we must naturally make choices of the range of prior parameters integrated over-, and we are dealing with different parameters in the different models. Different choices of priors for these parameters would leacl to some variatìon in the resulting Bayes factors. We will make a choice of the range of prior parameters by using the resulting values of 060 and ËIsls that they give rise to. In particular, we will choose a conservative bound of 0.5 ( 0¿o < 0.9. For the expansion age of the universe we choose 0.72 < Hsts 11.15: for Ho : 70 km s-lMpc-l it corresponds to a conservative bouncl of ús - 13 + 3 Gyr, consistent with recent estimates [84]. The parameter space of the PNGB model bounded by these two constraints is indicated by the contours of 0¿o and .flsls in Figs. 4.2. In the large / region in region I where the contouls appear to be parallel to the /-axis, a cut-off at / - lgts GeV, the Planck scale, is chosen. We calculate the average parameter likelihood by integrating L(M,/) over the prior ranges for the parameter and dividing the integral by the relevant area. The parameter spaces of the remaining quintessence models are not completely bounded by the f^l¿o and 116ús constraints, however. Lacking a physical cut-off for the parameters, such as a cut-off for / at Planck scale for the PNGB model, we will investigate in each case the dependence of the Bayes factor comparison on the prior ranges of the parameters. We will show that in some cases, the choice of the prior parameter ranges do not affect the conclusion from the Bayes factor analysis. For the simple exponential potential model, as shown in Fig. 4.3, the f,)60 and .FIsls contours diverge at large 7¿, putting a bound on the parameter space. In the small ) region, where the potential is effectively a cosmological constant model whose value depends on V¡ only, the Í)¿o and fI¡fs contours remain near parallel. For the full SCP data set the average parameter likelihood over the region bounded by the 0¿o and /1sfs constraints, with a cut-off at À : 0.1, is Lu," a 0.67L¡1,

70 where L¡7 is the average parameter likelihood for the PNGB model. In the small À region the likelihood only changes slightly with decreasing ) and tends to a constant, corresponding to an average parameter likelihood which we estimate to be Luu. a 7.\Ln. Since the small ) region extends to an infinite range in terms of the parameterization shown in Fig. 4.3, if we enlarge the prior range of ) to include smaller values, then it has the effect of blinging the average parameter likelihood closer to the average parameter likelihood in the small ) region, which is effectively an upper bound. Thus the Bayes factor for the simple exponential potential model versus the PNGB model lies in the range 0.67 < B < L5 for the full SCP data set. Similarly 0.65 < B < L5 for the reduced "flt C' data set. In each case the larger value is the one appropriate to including the entire small parameter range for À. Such a Bayes factor is too small to confidently discriminate between the two models. For the double-exponentiai potential model, the average parameter likelihood over the region bounded by the Íì40 and l1olo constraints, with a cut-off at A : 0.01, is .4,.'," - 0.92Lp for- the full SCP data set. The average parameter likelihood in the small A legion, where the potential is effectively a cosmological constant model whose value depends on V¡ only, is L^u" 1 L.\Ln. By a similar argument to above, the Bayes factor for the double-exponential potential model versus the PNGB model varies over a range 0.92 < B < 1.5 for the full SCP data set. Similarly, 0.38 < B < L.5 for the reduced "fit C" data set. Again, the Bayes factor is too small to confidently discriminate between the two models. For the inverse power-law potential model, as shown in Fig. 4.4, the 040 and lloúo contours remain nearly parallel towards the upper right-hand region where both I/¿ and o are large. The average parameter likelihood over the region bounded by the fl6o and f/¡ús constraints with arbitrary cut-offs at rcor2WlH3: 1015 and a : 15 is .C..," - 0.l4Ln for the full SCP data set, and .4.," - O.7lLnc for the reduced "fit C"data set, where Lnc is the average parameter likelihood for the PNGB model based on the "frt C" data set. This gives a Bayes factor for the PNGB model versus the inverse power-law potential model of B : 7.2 for the full SCP data set, ot B :9.2 for the ((fit C" data set. Both values are large enough to provide slightly positive evidence that the SCP data set favours the PNGB model over the inverse power-law potential model. Since the Bayes factor is only weakly positive, however, we cannot not place strong confidence in this conclusion. Note that the 040 and fIoúo

7r contouts both extend towards the smaller likelihoods region. Therefore, increasing the prior ranges lor Vt and a will decrease the average parameter likelihood, and increase the slightly positive preference for the PNGB potential as compared to the inverse power-law potential in the Bayes factor test. Given that the existing data is not sufficiently abundant to allow one to confidently distinguish between competing quintessence models, nor to distinguish source evolution from a cosmological explanation of SNe Ia luminosity distances, it is interesting to ask the question as to what extent we would expect this situation to be improved with future data. We will address this question in the next chapter.

72 Chapter 5

Future S.tpernovae Probes

In Chapter 4 we have studied the observational constraints arising from type Ia su- peïnovae (SNe Ia) from 60 supernovae data from the Supernova Csomology project (SCP) of Perlmutter et a/. [16] on the quintessence models. By evaluating Bayes factors, we have shown that the SNe Ia data cannot discriminate between the quintessence models. In the context of the PNGB model (1.5), we have also shown that the SNe Ia data cannot discriminate between the models with and without an empirical supernovae evolution. Very recently the identification of an SN Ia event at redshift z-L.7 [86] has provided tantalizing evidence that the expansion of the universe could have been decelerating at that epoch. Naturally, no film conclusions can be drawn from a single event. However, if this result is supported by future data ìt would rule out the simplest models of source evolution. Furthermore, supernovae events at such redshifts are the type of observations which should enable a discrimination to be made between some of the different quintessence models. Much more data is needed to enable an accurate estimation of the nature of the dark energy. This might be accomplished by a dedicated space telescope, the SuperNova Acceleration Probe (SNAP) [87, 88], which aims to collect a large number of supernovae with z 12. In this chapter, we assess the ability of the SNAP mission to determine various properties of the "dark energy". By analysing a simulated data set, as might be obtained by the proposed SNAP satellite, we can test the ability of such experiments to distinguish among currently attractive quintessence models. The feasibility of determining the properties of the dark energy component by using simulated data sets has already been considered by several authors [39]-[93].

73 One common approach is to assume that the quintessence field is described by a perfect fluid with equation of state P : . p, where tu is approximately constant over epochs of interest, or else slowly varying. For example, various authors [90, 92, 93] consìder models with an equation of state linear in redshift, .(") - u)o I wtz. However, many realistic cosmological models could fall outside the confines of these approximations. Our approach will be rather different here since, as in the previous chapters, we will deal with quintessence fielcls given directly by a Lagrangian without any prior assumptions about -(t), which has the freedom to vary widely over measurable redshifts. We will flt simulated SNAP data sets to the exact dr(r) of different quintessence models derived from numerical integration of the coupled Einstein-scalar field equations.

5.1- SNAP Data Simulation

For the fiducial cosmological model used in the simulation of the supernova data set, we choose the PNGB model (1.5). Our reasons for this choice of quintessence model are twofold. Firstly, from the viewpoint of quantum field theory PNGB models are the simplest way to have naturally ultra-low mass, spin-0 particles and hence perhaps the most natural candidate for a presentiy-existing minimally- coupled scalar field. The second motivation for choosing a PNGB model rather than other forms of quintessence since it provides a natural framework for studying the question of the possibility of source evolution, as was discussed in section 4.2. In particular, unlike the open FRW models which have been previously used to study source evolution

177) fhe PNGB models ale all spatially flat, as is consistent with current observations t6]-tS], while allowing both accelerated or decelerated expansion depending on the parameter values of M, f and w¿. In this chapter we will use the initial condition w¿ = ó¿l I - 0.2, as was used in Sec 4.3.1. Here we will again use the more simplified approach of Sec 4.3.1 to treat the case with source evolution. As was the case there, we will add a term B ln(1 + z) to the distance modulus, and assume that B has a flat prior bounded over some range Lþ :2.3, with limits corresponding to -0.6 < P < I.7, and marginalize the likelihood function over this prior. We will simulate two data sets from fiducial cosmologies chosen with parameters

74 in each of regions I and II. In each case we will introduce a random error to the exact distance moduli to simulate a future supernova data set that has been con- verted to a table of effective magnitudes ?7?¿ and redshifts z¿ of objects with a single fiducial absolute magnitude ffi. We will consider both statistical and systematic uncertainties in the magnitudes. Typically the redshifts are known to sufficiently high precision that their uncertainties can be ignored. We assume the supernovae are observed uniformly within four different redshift ranges with the following different sampling rates, which are the same as those chosen by Weller and Albrecht [92]: In the first range frorn z:0 0.2 we assume that there are 50 observations, in tlre second and largest redshift range from z : 0.2-I.2 there are 1800 SNe and in the two high redshift bins, z : 1..2-I.4 and z : I.4-L.7, there are 50 SNe and 15 SNe observations respectively. The statistical error in magnitude is assumed to be orna':0.15, including both measurement error and any residual intrinsic dispersion after calibration. For data set A, we assume there is no luminosity evolution. We choose the fiducial parameters M :2.53 x |g-z¡rtlz"y and / : 0.58 x 1018GeV; these parameters give (0-0, f-lao) t (0.33,0.67) and f/6ú¡ t 0.9. For data set B, we assume there is a luminosity evolution and we add a terrn B ln(1 * z) to the distance moduli. We choose the fiducial parameters M : 4 x 1g-z¡rtlz"y and / : 0.676 x 1018GeV. These parameters give (0-0,0¿o) t (0.25,0.75) and Hsts x 0.8. The best-fit value of the parameter B for this particular choice of M and / is þ :0.659 for the full data set, and, þ : 0.727 for the "fit C" reduced data set. We will take B t 0.659 as the fiducial parameter for data set B, as it is the somewhat more conservative value, representing less luminosity evolution. The frducial parameters for the two data sets have been deliberately chosen so as to give similar values of Í)-s,f)¿o and Hsts or one hand, so as to be consistent with values favoured by other observational tests. On the other hand, the ficlucial parameters fol data set A are centred in region I, which corresponds to a universe in which the scalar field is rolling down the potential for the first time at the present epoch, whereas the fiducial parameters for data set B are centred in region II, which corresponds to a universe in which the scalar field is rolling down the potential for the second time at the present epoch. As was discussed in Sections 4.2 and 4.3.2 rcgion I is favoured if the SNe Ia luminosity distances are true cosmological distances, whereas region II becomes significantly favoured if the data hides a simple

/Ð luminosity evolution.

5.2 Likelihood F\rnction

The calculation of the marginal likelihood for quintessential parameters Q has been discussed in Section 4,1, where Ms and Hubble parameter,É/s are treated as additional parameters and are dealt with by marginalizing. In this section, to simplify the computation we assume that Ilo and Ms are known to sufficiently high precision that their uncertainty can be ignored, and we frx Ho:70 km/s Mp.-t. One then finds the likelihood lor Q, LQ) - e-x212, (5.1) where 1,, ._ pe¿;e)], x2(e): t Lt's'N (5.2) o o'r,¿ In the above equation,

p(z¡; Q): S log dt (r¿; Q) + zs (5.3) is the distance modulus predicted by each model with parameters Q, while þ",i : rn¿ - Mo is the simulated distance modulus, and its uncertainty is o,",t: 0.15. The model with an unexpected luminosity evolution corresponds to replacing eq. (5.2) with

Itt",n BIn(I * ,o) ttQ¿; Q)l' X,(Q,P):\ - - (5.4) olt,x

As in the analysis of the SCP data in Section 4.3, we will marginalize over B with a flat prior, to obtain the marginal likelihood L(Q): il t dBe-x'/z (5.5) The above integration can be performed by an analytic marginalization [77]. We separate q from ¡2 where B'@) , ,-h'(ro;Q) q@) - -o, -+ "? (5.6)

76 is independent of B. The integral is thus an integral over a Gaussian in B located at B with standard deviation a. þ is the best-fit value of B given Q, and. o is its conditional uncertainty. They are given by

i: ¡[ln(1 tz¿)]r. (57) o2 ? orr,n ' t h(r,; Q)ln(1 f z¿) pQ) o- (5.8) t o i lf, ¡x where h(r¿; Q) : þs,i - ttQ¿; Q) . (5.9)

As long as B is inside the prior range and o < Lp, the vaiue of the integral is well approximated by a\Æ;, so that

oY"^n L@) : . (5.10) L{J "-ol,

5.3 Model Comparison

We will compare the reiative strengths of rival theoretical models by the calculation of Bayes factors [85] in the manner already described in Section 4.3. The recom- mended interpretation of Bayes factors is summarized in Table 4.1.

5.3.1 Accelerating lJniverse versus Luminosity Evolution

If we frt the data set obtained from fiducial model A assuming that there is no luminosity evolution, the parameterc M and / are well constrained by the simulated data. The 95.4% confldence level contour bounds a very small region around the best-fit values of M x 2.55x10-3hLl2eV and / = 0.592 x 1018GeV, with normalized X2 : 1910. These best-fit parameters are very close to the fiducial parameters that generate the data set. The average parameter likelihood is .C.u. - 7.I x l\-aLs, where Lo : exp(-1910/2). If luminosity evolution is considered to exist (see Fig. 5.1), fitting the data set from fiducial model A we find that the 95.4% confidence ievel contour includes a larger region in region I of the parameter space around the fiducial parameters, and a region in region II. The average parameter likelihood is Luu" r: 1.9 x 10-3,Cs. Hence, the Bayes factor for the model with luminosity evolution versus the model without luminosity evolution is B - 2.7. We conclude

77 2.O

1.5

a) €

FI 1.0 \

0.5

2 4 6 8 10 M(lo-s h1/2eY)

Figure 5.1: Confidence limits on M,f parameter values, with u.'¿ - 0.2, marginalized over a flat prior for the luminosity evolution parameter B, arising from the 1915 SNAP supernovae simulated assuming M : 2.53 ,19-s¡t/zeV , f : 0.58 x 1018GeV, and B : 0. Parameter values excluded at the 95.4% level are darkly shaded, while those excluded at the 68.370level are lightly shaded. Overplotted are the contours for 0¿o (dashed) and 1{oúo (dotted).

that it is not possible to discriminate between the two hypotheses even though the underlying simulated data does not have any luminosity evolution. If we fit the data set obtained from the fiducial model B by assuming that there is no luminosity evolution, then the 95/% level bounds a very small region around the best-fit values of M x 2.27 x 1g-z¡rrlz"y and / x 0.764x 1018GeV. The average parameter likelihood is Luu. - 9.4 x 10-1816, where ,Co : exP(-192412). If luminosity evolution is considered to exist (see Fig. 5.2), on the other hand, then the 95.4% level bounds a small region around the best-fit values of M = 4.00 x 1g-z¡rt'lz"y and / = 0.695 x 1018GeV, with þ x 0.626 and normalized, y2 -- Lg24. The average parameter likelihood is.C".,," - 8.3 x1.0-aLo. This time, the Bayes factor for the model with luminosity evolution lelative to the model without luminosity evoiution is much greater than one, B = 9.I x 1013. Thus if the true data were derived from fiducial model B, it would show up clearly as giving a d;(z) which can be discriminated from the model without luminosity evolution. This result in no way contradicts the inconclusive result obtained from data set

78 2.0

1.5 q)

@ 1.0 \

0.5

'10 2 4 6 B M(rc-3 hL/zeY)

Figure 5.2: Confrdence limits on M,f parameter values, with u.'¿ : 0.2, marginalized over a flat prior for the luminosity evolution parameter B, arising from the 1915 SNAP supernovae simulated assuming M : 4 * 1g-s¡rtlzev, .f : 0.676 x 10l8GeV, and B : 0.659. Parameter values excluded at the 95.4% level are darkly shaded, while those excluded at the 68.3% level are lightly shaded. Overplotted ate the contours for 040 (dashed) and l1olo (dotted).

A. If the underlying data were to truly follow a simple luminosity evolution, then the analysis above shows that this would stand out as a very strong positive signal in the Bayes factor test. If the underlying data has no luminosity evolution on the other hand, then it may be possible to obtain a good fit with an extra luminosity evolution parameter-, simply because one has an extra parameter to fit. Thus a simple luminosity evolution is easy to rule in, but difficult to rule out. Combined together the two results demonstrate the efficacy of the Bayes factor approach. With data set B we would obtain a decisive result, but by Ockham's tazor we should reject the more complicated evolutionary hypothesis if we were to obtain an inconclusive result such as that pertaining to data set A. In fact, we have found that by narrowing the range of the prior AB that the Bayes factor for data set A can even be made slightly greater than the cut-off value of 3 listed in Table 4.1. Clearly any weakly positive results should therefore also be treated with some caution.

79 5.3.2 PNGB versus Other Potentials

In Section 4.3.1, we showed that the SCP data set does not particularly favour any of the PNGB, simple exponential potential, and double-exponential potential model relative to the others. The SCP data set does slightly disfavour the inverse power- law potential model as compared with the PNGB model. This might be considered to result from the fact that the 2o-confidence r-egion only overlaps with a small region (the low a region) of the whole parameter space allowed by the priors set on f)60 and IIsÍs. Howevet, even for the model comparison between the PNGB and inverse-power law potentials the Bayes factor determined from the SCP data is not large enough to give a strong statistical conciusion. In this section we want to compare the typical results we should expect when a SNAP data set is used. We will use data set A, simulated from a fiducial PNGB model assuming no luminosity evolution. We will study the potential of this data set to discriminate the PNGB model from other quintessence models. We will only compare the quintessence models on the basis of assuming that the true data follows the PNGB model, with the fiducial parameters of data set A. Naturally, we could assume any of the other potentials as our fiducial model, and then compare the other models on the basis of a new fiducial data set. This would allow us, for example, to compare the inverse power-law potentiai with the simple- and double exponential potentials, a test which we have not performed here. We will restrict our attention to the fiducial data set A relevant to the PNGB potential, however, since the amount of computer- time involved in computing the fiducial data sets is large, and an analysis based on a fiducial PNGB model will prove to be sufficient to illustrate the dramatically increased discriminatory power of a SNAP data set as opposed to presently available SNe Ia data. Naturally, one could extend the discussion to other fiducial data sets, but we will leave that to future work. As discussed in Section 4.3.7, we will make a choice of the range of prior parameters by using the resulting values of 040 and //sús that they give rise to. The parameter spaces are not completely bounded by the 0.5 < fl¿o ( 0.9 and 0.72 < Hoto 1 1.15 constraints, however. Without a physical cut-off for the parameters, we will investigate in each case the dependence of the Bayes factor comparison on the prior ranges of the parameters. Just as in Section 4.3.2, we will show that in some cases, the choice of the prior parameter ranges do not affect the conclusion from the Bayes factor analysis.

80 For the simple exponential potential model with y(d) : VAe-\^Ô, the average parameter likelihood over the region bounded by the f,)40 and fleús constraints, with a cut-off at ) : 0.1, is Luu" a 2.I x I0-5 Ls, where .Co : exp(-I9I0 12). In the small ) region, we estimate the average palameter likelihood to be Luu. - 6.6 x 10-31,C0. Comparing with the average parameter likelihood for the PNGB model, Luu" a 7.1 x 10-aLs,the Bayes factor for the PNGB model versus the simple exponential potential model takes values in the range 34-1027, depending on the prior range for ). This would provide a very strong conclusion that data set A favours the PNGB model over the simple exponential potential model' For the double-exponential potential model with Iz(/) : Vtex¡(-¡""/2"Ó), the average parameter likelihood over the region bounded by the f,)¿o and llsls constraints, with a cut off at A:0.01, is Lu,"t 1.6 x 10-3,Cs. The average parameter likelihood in the small A region, is .C,.,," - 2.I x I0-2s Lo. The Bayes factor for tþe PNGB model ver-sus the double-exponential potential model lies in the range 0.44-1025, depending on the the prior range fot A. The lower bound of the Bayes factor, B:0.44, is too small to confrdently discriminate between the two models. However, provided small values of the parameter A are included in the prior range, the Bayes factor would provide strong evidence that data set A favours the PNGB model over the double-exponential potential model. For the inverse power-law potential model with y(d) -- v¡Ó-", the average parameter likelihood over the region bounded by the f)¿o and llolo constraints with arbitrary cut-offs at rcoI2VtlH3 :1015 and a: 15 is Luu. - 1.9 X 10-10¿0. This gives a Bayes factor for the PNGB model versus the inverse power-law potential model of B - 106. This provides very strong evidence that data set A would favour the PNGB model over the inverse power law potential model.

5.4 Discusslon

Now, let us consider our results in this chapter in reiation to previous work concern- ing the feasibility of determining the properties of the dark energy from future supernovae surveys [39]-[93]. One approach in past studies of scalar field quintessence has been to assume a potential7(ó) over which the scalar field, /, would have slowly varied during cosmological time scales, and then test the efficacy of reconstlucting the potential [S9]. Another approach has been to assume that the quintessence field

81 can be described by a perfect fluid with slowiy valying equation of state P : w(z)p, expand w(z) in a power series, and then test the effi,cacy of determining the coeffi- cients in the power selies [90, 92, 93]. The conclusions of investigations to date are mixed. Weller and Albrecht [92] find that many models can be distinguished with a fit to a linear equation of state for the dark energy, P : *(r)p with ru(z) - u)s I wtz, but only if the current mass density,{'L*, is known to a high precision. Barger and Marfatia [93] find that even by putting Q* :0.3 exactly, that there is still a possibility of obtaining data sets which might not discriminate between quintessence and "k-essence", namely an alternative form of dark energy with a scalar field characterized by non lineal kinetic terms [94, 95]. Wang and Garnavich [96] consider two classes of functions us(z), corresponding respectively to a linear variation and to k-essence. Using somewhat different techniques to other authors they are more optimistic about prospects for determininSw(z) from future SNe Ia data. In this work we have taken a different approach to those above, by considering a class of models - the PNGB models which are very well motivated from the point of view of particle physics, but for which the above methods will not be satisfactory in the case of all plausible parameter ranges, given the potentially oscillatory nature of uQ) and the corresponding fact that the scalar field may have varied over a widerangeof values of V(þ) overobservabletimescales. Incontrasttothevarious reconstruction approaches we have fitted the simulated SNAP data sets to the exact dt (r) of different models obtained by numerical integration, and compared them to othel models. One real drawback of all approaches is that as yet there is no preferred physical model for the dark energy. On one hand this means that any approximations made in potential reconstruction methods may be too restrictive, since many different potential energy functions V (þ) are conceivable, and many of these may give results degenerate with each other. On the other hand, using a given Lagrangian for the quintessence field, as we have done, limits us to a model by model test. Nevertheless, we find that data sets such as those that would be produced by SNAP promise to be very successful on some tests, even if they will probably be less successful on others. In particular, while existing data is not yet sufficiently large to discriminate between various quintessence models or models with evolution, we have shown that the much larger size and smaller error bars of the simulated SNAP SNe

82 Ia data sets provide much tighter constraints on the parameters for quintessence models such as those corresponding to pseudo Nambu-Goldstone bosons. In conclusion, we find that future supernova measurements such as those that would be afforded by the SNAP satellite, will have the power to significantly increase our knowledge of the properties of the dark energy in the universe. To be completely confident, however, we will require a deeper theoretical understanding of the nature of the dark energy and hopefully new input from fundamental physics.

83 Chapter 6

Gravitational Lensittg Statistics Test

Gravitational lensing of distant light sources due to the accumulation of matter along the line of sight provides another relatively sensitive constraint on the cosmological models of interest. For cosmology the situation of most interest is the lensing of high luminosity quasars by intervening galaxies. The abundance of multiply imaged quasars and the observed separation of the images to the source puts constraints on the number count-redshift relation and hence the model parameters. This leads to a statistical test, which has been used to put bounds on.A. [97]-[99]. In principle, the gravitational lensing statistics test is a sensitive probe of the cosmology; however, it is susceptible to a number of systematic errors. Uncertaintìes in the luminosity function for source and lens, lens evolution, lensing cross section, and dust extinction for optical lenses threaten to render the constraints compatible with or even favour a low density open universe over a spatially flat one. Gravitational lensing statistics are thus useful since they provide a test which potentially provides opposing constraints to those obtained from supernovae magnitude-redshift tests. In particular, in the case of models with a vacuum energy provided by a cosmological constant, the high redshift supernovae have been interpreted as favouring relatively large values of Í-)¡ - Perlmutter et al.116l give a value of 0¡ : 0.72 at lo level - whereas the gravitational lensing data leads to upper bounds on fl¡: Kochanek [97] quotes Qn ( 0.66 at the 2o level. A combined likelihood analysis has been performed by various authors on models with a cosmological constant or quintessence [40, 99, 102].

84 In this chapter we will study the gravitational lensing statistics constraints on the quintessence models compared with the type Ia supernovae (SNe Ia) constraints obtained in Chapter 4. Gravitational lensing statistics test has been used to test properties of some quintessence models [40, 100, 101, 102], such as the inverse power- law potential models [100] and the PNGB models [a0]. We will update the results by performing calculations similar to those recently described by Waga and Miceli, who undertook a statistical lensing analysis of optical sources described ear-lier by Kochanek [97]. Waga and Miceli [102] studied cosmological models with an exotic fluid described by an equation of state p, : (ml3 - t)p,. They used a total of 862 (z > 1) high luminosity quasars (HLQs) plus 5 lenses from seven major optical surveys [103]. Another alternative not considered here is to analyse data from radio surveys as discussed, e.8., in refs. [98, 99]

6.1- Likelihood Function

We are interested in determining the likelihood function defined as [104]

Ng N1 N7 Lt"n,: II(t - p:)Ilpl¡np'"t. (6.1) i=l j=l It=\

Here ly'¡ is the number of quasars that have multiple images, l/y is the number of quasars that don't have, p'¿ K I is the probability that quasar i is lensed and p'"¡* is the configuration plobability, which can be considered to be the probability that quasar k is lensed with the observed image separation. The lens galaxies are modelled as singular isothermal spheres (SIS), and only lensing by early-type galaxies is considered. The total optical depth (r), obtained by integrating the differential probability, dr, that a line of sight intersects a galaxy at redshift z7 in the interval dz;, along the line of sight from 0 to the quasar's redshift zs) can be expressed analytically [102], ,(,s)= fi [++4':i1' (62) where F = 0.026 measures the effectiveness of the lens in producing multiple images [104]. Since lensing increases the apparent brightness of a quasar and since there are more faint quasars than bright ones, thele will be an over representation of lensed

85 quasars in a flux limited sample. The magnification corrected probabilities are

p¿ : r(z¿)B(*0, (6.3) "¿) where B(*, z) : M3 ß(*, z, Ms, M2) (6.4) is the bias factor [97, 105], and -r dN, fM' clM clNq ß(*, z, Mt, Mz) : 2 (m t 2.5los M, z) . (6.5) dm J*, W d*

The lenses are modelled by a SIS profile where Mo :2, and Mz : 104 is used in the numerical computation. The quasar luminosity function is expressed as follows [97]:

(6.6) +*lro-,{*-^)11g-a(---)]dmL , where mo*(z+l) for z1I, rrùs for Lt (6.10) J-L and l:f(Ð-1go'+anz(a) (6.1 1)

86 The configuration probability that a lens in a spatially flat universe has critical radius b (and image separation L,0 :2ó) is [105]

b2_ p.(b) 30;tr(1 t a - 2-y-',i''t') -zâr1r t a - 41-1,ûtz¡ +ô'?r(l I a - 61-',þ/\llr0* a * 4t-') (6.12) where b : blb.. The characteristic length scale ô* : t".8(oill250 km r-t)', the one component velocity dispersion oil:225 Km/s, a: -1.0, and 7: {. To simplify computation only two selection functions a e used, one for the HST observations and another one for all the glound based surveys [105]. For the ground based surveys, the Yee et al. selection function is used, but is integrated up to 7 arc- seconds. Using more accurate selection functions for each ground based observation separately has little statistical effect.

Falco et al. 198] observed that statistical lensing analysis based on optical and radio observations can be reconciled if the existence of dust in E/SO galaxies is considered. As suggested by theìr estimates a mean extinction of L,m:0.5 mag is assumed in the computation. By expressing L¡"," as a function of the quintessential parameters we obtain the maximum of the likelihood function (LT:ä) and form the ratio I : L¡"n"f LT:i,i. h can be shown that with two parameters, the distribution of -2ln I tends to a y2 distribution with two degrees of freedom [105].

6.2 Constraints on Quintessence Models

In Fig. 6.1 we show the 68.3% and 95.4% joint credible regions from lensing statistics in the two parameter space (V¡,À) of the simple exponential potential models, V(ó) : V¡e-^"Ó. In the upper portion of the figure, the 2o level is consistent with the lower bound of the expansion age, Hsts S 1.15. If we adopt the constraint f)60 I 0.9, the 2o levels from both lensing statistics and the 60 SNe Ia from the Supernova Cosmology Project (SCP) (see Fig. 4.3) constrain the allowed region of the parameter space to the small À region only, where the scalar field effectively acts like a cosmological constant and becomes dynamical only relatively recently. In this region, as expected, the lensing statistics give 040 S 0.7 at 2o, which is lower than that given by the SNe Ia data.

87 0.4

o.2

0 0

0.2 Þ0 o -0.4

- 0.6

- 0.8

1.0

0 1 2 7 4 log (rcz vn/H ot)

Figure 6.1: Confidence limits on the pa-,-ameter space for the simple exponential potential models, V (ó) : Vte-^"Ó , alising from the gravitational lensing statistics of the high luminosity quasars. Parameter values excluded at the 95/% level are darkly shaded, while those excluded at the 68.3% level ale lightly shaded. For reference, contours of 0¿o and llsls are superposed as solid and dotted lines respectively.

88 In Fig. 6.2 we show the 63.3% and 95.4% joint credible regions from lensing statistics in the two parameter space (Vt,o) of the inverse power-law potential models, V(ó) : V¡ó-'. The 2o level from lensing statistics rules out the lower right-hand region of the parameter space, consistent with the constraint from the SNe Ia data (see Fig. .a). However, the upper left-hand region of the parameter space which was ruled out by the the 2o Ievelof the SNe Ia data is now 1o allowed by the lensing statistics. The 2o levels from both the SNe Ia data and lensing statistics, together with the constraint fl6o 5 0.9, bound a narrow band in the parameter space extending from the lower left-hand corner to the upper right-hand corner of the figure. In Fig. 6.3 we show the 63.3% and 95.470 joint credible regions from lensing statistics in the two parameter space (Vt,A) of the double-exponential potential models, V(ó) : V¡exp(_ A.Õ"Ó). Similarly to the previous cases, a large region in the parameter space which was excluded at 2o by the SNe Ia data (see Fig. 4.5) is now 1a allowed by the lensing statistics. The 2o levels from both lensing statistics and the SNe Ia data constrain the allowed region of the parameter space to the small À region where the scalar field effectively acts like a cosmological constant and becomes dynamical only relatively recently. In this region, as expected, the lensing statistics gives 0¿o S 0.7 at 2o, which is lower than that given by the SNe Ia, data. Gravitational lensing constraints on the PNGB models have been very recently given by Waga and Frieman [40] for u¡: 1.5. However, Waga and Frieman considered a more restricted range of parameter space, M < 0.005h eV , since they did not consider the possibility of source evolution in the case of the type Ia supernovae and therefore took values of M > 0.005å, eV to be ruled out. We wish to extend the range of M for the gravitational lensing statistics to consider parameter values corresponding to region II (see Chapter 4) in the supernovae constraint graphs so as to compare the constraints from different tests. Undertaking a similar analysis for the increased parameter range, we arrive at Fig. 6.4, which shows the 68.3% and95.4%joint credible regions lor M and /, for two values of w¿. The only regions of parameter space excluded at the 2o level turn out to be areas of parameter space for which the deceleration is presently negative (cf., Fig. 3.9), with the scalar freld still commencing its first oscillation at the present epoch. Furthermore, region II of Figs. 4.6, 4.8, and 4.10, which is favoured by the SNe Ia data if models with supernovae source evolution are considered, coincides

89 I4

B õ

0 E 0 J 10 15 log (rc"*'v,/a ot)

Figure 6.2: Confidence limits on the parametel space for the inverse power-law potential models, V(Ó) : V.¿,Ó-', arising from the gravitational lensing statistics of the high luminosity quasars. Parameter values excluded at the 95.4% Ievel are darkly shaded, while those excluded at the 68.3% level are lightly shaded. For reference, contours of 040 and Ilsls ale superposed as solid and dotted lines respectiveiy.

90 0.0

- 0.5 f Þ0 o - 1.0

-1.5

-2.0 0.0 0.5 1.0 1.5 2.O log (rcz v,/H ot)

Figure 6.3: Confidence iimits on the parameter space for the double-exponential potential models, V(ó) : V¿exp(- A.fr"Ó), arising from the gravitational lensing statistics of the high luminosity quasars. Parameter values excluded at the 95.4% level are darkly shaded, while those excluded at the 68.3% level are lightly shaded. For reference, contours of 060 and 110úo are superposed as solid and dotted lines respectively. The lower right-hand region is excluded from the plot due to computational difficulties.

91 with a region included at even the 1o level in Fig. 6.4. Clearly models with supernovae source evolution would probably be more easily reconciled with the gravitational lensing statistics. However, it is still quite possible that one might find small regions of parameter space for quintessence models such as the PNGB model which satisfy both competing constraints. On the basis of the regions of parameter space allowed by the SCP data it is still too early to draw any firm conclusions as to whether gravitational lensing bounds are in conflict with luminosity distances. However, the data which would be collected by the SNAP mission should allow us to refine a comparison of the two tests in the near future.

92 6

0) 4 o

3 \ 2

L 2 4 6 B 10 M(to-3nt/2ev) (.) -" -- 1.5

2.0

1.5

O (J € 1.0 \ 0.5

2 4 6 B '10 (b) -, = 0.2 lt(to-3nt/2ev)

Figure 6.4: Confidence iimits arising from gravitational lensing statistics: (a) w¿ : t.6; (b) w¿:0.2. Parameter values excluded at the 95.470level are darkly shaded, while those excluded at the 68.3% level are lightly shaded. For reference, contours of 0¿o and llolo are superposed as dashed and dotted lines respectively.

93 Chapter 7

Conclusion

7 .1, Discussron

In Chapters 4-6, we have considered the observational constraints arising from supernovae Ia (SNe Ia) data and gravitational lensing data on cosmological models based on Einstein gravity minimally coupled to a scalar quintessence field with an inverse power-law potential (1.1), a simple exponential potential (1.3), a double- exponential potential (1.4) and a effective potential of an ultra-light pseudo Nambu- Goldstone boson (PNGB) (1.5). We numerically evolved the coupled Einstein-scalar field equations of motion forward from the epoch of matter-radiation equality to obtain the luminosity distance - redshift relation. The numerical integrations were cliscussed in Chapter 3. In Chapter' 4, we have studied empirical models with evolution of SNe Ia in the context of the PNGB quintessence models: following Drell, Loredo, and Wassermann l77lwe analysed the observational constraints when a term Bln(l f z) was added to the distance modulus. Although this particular empirical contribution is not favoured by any particular theoretical model, other types of source evolution might be expected to affect the analysis in similar ways. An obvious avenue for further research would be to investigate the extent to which other types of source evolution change the results. It might also be interesting to investigate the possibility of source evolution in the case of other quintessence models or a cosmological constant. As discussed in Section 2.4, the PNGB models are qualitatively different from cosmological constant models since their final state corresponds to one in which the ultimate destiny of

94 the universe is to expand at the same rate as a spatialiy flat Friedmann-Robertson- Walker (FRW) model, rather than to undergo an accelerated expansion. This is of course precisely why we chose the PNGB models as the basis of our investigation, rather than a model in which a late-time accelerated expansion had been built in by hand. If we wish to test the hypothesis that the faintness of the SNe Ia is at least partly due to an intrinsic variation of their peak luminosities then a quintessence rnodel which possesses a variety of possibilities for the present-day variation of the scale factor is probably the best type of model to investigate. Nevertheless source evolution studies in models with other for"ms of dark energy could be checked to confirm this. If only supernovae luminosity distances (cf., Figs.4.6,4.8, and 4.10) and gravìta- tional lensing statistics (cf., Fig.6.4) are compared then we see that for the PNGB models with empirical supernovae evolution there is a remarkable concordance between the two tests - region II of Figs. 4.6,4.8, and 4.10 coincides with a region included at even the 1ø level in Fig. 6.4. This is pelhaps not surprising, since in vìew of the results of phase-space analyses in Section 2.4 and the plots of the deceleration parameter, Ç0, in Fig. 3.9, region II corresponds to parameter values for which the present day universe has already undergone almost one complete oscillation of the scalar field about the frnal critical point C2 of the dynamial systems. It is thus already well on the way towards its asymptotic behaviour, which closely resembles that of a standard spatially flat FRW model. Due to the oscillatoly behaviour, parameter values in region II correspond to models in which there has been a recent cosmological acceleration, (e.g., at z - 0.2), but with a q(z) which changes sign three times over the larger range of redshifts, 0 < z ( 4, in the quasar lensing sample, and therefore differing significantly from Friedmann Lemaître models over this larger redshift range. As shown in Chapter 5, extending the SNe Ia sample to include objects at redshifts 0.1 < z 1 0.4 and z > 0.85 in substantial numbers, as proposed by the future SuperNova Accelela- tion Probe (SNAP), would greatly improve the ability to decide between models in legions I and II. Even if the SNe Ia sources undergo evolution it is clear that parameter values in region I which are favoured in the absence of evolution of peak SN Ia luminosities, are still included at the 2o level in the models with evolution, in view of Figs. 4.6 - 4.8 and 4.10.

95 What we wish to emphasize, however, is that an effective vacuum energy which is cosmologically significant at the present epoch should not simply be thought of in terms of a "cosmic acceleration". A dynamical vacuum energy with a varying effective equation of state allows for many possibilities for the evolution of the universe, and overly restrictive assumptions, such as equating quintessence to models with a late period of continuous cosmological acceleration, should be avoided. If detailed astrophysical modelling of SNe Ia explosions ultimately shows that the dimness of distant supernova events is largely due to evolutionary effects, it does not spell the end for cosmologies with dynamical scalar' fields. In Chapter 4, our analysis has considered both the entire data set of SCP and ufit a reduced C" data set for which six supernovae were excluded [16]. We have found that in the best-fit case the y2 parameters are greatly reduced, showing that the "fit C" data provided a more robust fit in all cases we have studied, whatever the model assumptions. We have also shown that for the models with supernovae evoiution, the overall values of best-fit B are somewhat larger for "fit C". In Chapter 5, we have shown that much tighter bounds on the parameter space of quintessence models will be obtained over the next decade as more SNe Ia data is collected. By evaluating Bayes factors in the context of the PNGB models, we have shown that future satellite SNe Ia data sets should have greater success in detecting whether the observed luminosity distance - redshift relation is purely cosmological in origin, or is significantly contaminated by evolutionary effects of the sources. The results of section 5.3.1 showed that although it may be difficult to completely rule out luminosity evolution, if the true data were from a population with luminosity evolution then this would provide a strong distinctive signal. If there is no such signal, then our experience with the data simulations would indicate that by the principle of Ockham's razor the evolutionary hypothesis should be rejected. We have only studied one simple illustrative supernova luminosity evolution function, but we expect that similar conclusions would apply to other simple luminosity evolution rnodels. We have further shown that with the future data it should be possible to discriminate PNGB models from some other particular types of quintessence; in particular, it gives a very different signature to that of simple inverse power-law potentials or simple exponential potentials. The case of a double-exponential potential gives a lower Bayes factor, and may therefore be more difficult to distinguish from the

96 PNGB models. However, even in this case some distinction between the two models is possible if one allows a suitably large prior range of the parameter A, to include smail values. A number of obvious extensions of our analysis are possible. In the case of testing source evolution versus the case with no evolution, for example, it would be interesting to determine by how much we can reduce the parameter þ for the simulated SNAP data set B while still obtaining a very strong result for the discriminatory power of the relevant Bayes factor. Given the magnitude of the value obtained in section 5.3.1, we suspect that the fiducial value of B could be significantly reduced. Likewise many other tests could be performed with fiducial data sets based on other quintessence potentials.

7.2 Further Studies

Finally, we want to mention a few other interesting recent developments in quintessence models. One appealing idea is the possibility that the quintessence scalar field / has driven inflation. This picture was called "Quintessential Inflation" by Peebles and

Vilenkin [107] who assumed that the entropy and matter of the familiar universe arise from gravitational particle production at the end of inflation. While this eliminates the problem of finding a satisfactory coupling of the inflaton and matter fields, the model of Peebles and Vilenkin involves a very unnatural potential obtained by cutting and pasting two other potentials to get one with the desired behaviour. If quintessential inflation could be obtained with a more "natural" potential it would be a significant advance. Another interesting idea is to considel multiple scalar fields with different potentials. Scalar fields with exponential potentials evolve so as to act cooperatively to "assist" inflation, as has been shown by Liddle et al. [108]. Barreira et al. 11091 considered potentials containing two exponentiai terms and one field. They showed that the universe can evolve through a barotropic fluid dominated regime and at some recent epoch evolve into the scalar field dominated regime. When deriving the inverse power-law potentiai in eq. (1.1) with a given by (L.2), Binetruy "q. [39] chose the same initial conditions for all the l/7 vacuum expectation values (v.e.v.) and their time derivatives. In the more general case, different initial conditions are

97 assigned to different v.e.v.'s, and the system is desclibed by l/¡ coupled different ecluations [63]. Recently Balbi et al. lII0l have performed a Bayesian likelihood analysis, using the MAXIMA-I and BOOMERanG-98 published bandpowers, to explore the space of quintessence parameters of the inverse power-law potential models. Cos- mic microwave background (CMB) anisotropy angular power spectra in the inverse power-law potential models have been obtained by suitably modifying CMBFAST [111] to include / perturbations in the synchronous gauge, as detailed by Perrotta and Baccigalupi [112], and numerically evolving the background quantities, account- ing for the tracker scalar field trajectories, as discussed by Baccigalupi et al. l1l3l, where results were presented for some specific values of the quintessence parameters. The CMB anisotropy test can of course be applied to other quintessence models. In the case of the PNGB models it would be interesting to compare the resulting constraints on the parameter space with the constraints from SNe Ia luminosity distances and gravitational lensing statistics. There are a few other interesting quintessence models to which it might be interesting to apply an analysis similar to that undertaken in this thesis. We list some examples:

1. Caldwell [114] constructed a toy model of a "phantom" energy component which possesses an equation of state w I -I, in violation of the dominant energy condition.

2. Quintessence models in which late times oscillations of scalar field give tise to an effective equation of state which can be negative; wø 1-I13, in contrast to the PNGB models where uó N 0, have been proposed by Sahni and Wang

[115].

3. Recently, a novel class of models for inflation has been found in which the inflationary dynamics is driven solely by (non-canonical) kinetic terms rather than by potential terms. As an obvious extension, Chiba et al.194] have showed that a scalar field with non-canonical kinetic terms alone behaves like an energy component which is time-varying and has negative pressure presently, i.e.,

quintessence. This scalar field is called "k-essence' [95].

4. Dodelson et al. [116] proposed that the dark energy has periodically dominated in the past so that its preponderance today is natural; they illustrate this

98 paradigm with a model potential.

5. Quintessence in "brane cosmology" has recently been studied by various authors [117].

This brief and by no means exhaustive list of currently active areas of study in quintessential cosmologies shows that much remains to be done, and the field is still in its infancy. Perhaps the greatest problem is that fundamental scalar fields are not observed in the laboratory, and we have no direct experimental guidance as to the nature of the dark energy. Hopefully the coming decades will provide us with a wealth of observational data, including much refined CMB anisotropy measurements from the Microwave Anisotropy Probe (MAP) [118] and Planck [119] satellites, SNe Ia measurements from the SNAP satellite, and also improved gravitational lensing measurements. In this way we might hope to discover the nature of what appears to be the dominant component of the energy density of the universe today, even if we cannot directly measure it in the laboratory.

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106 Publication

1. S.C.C. Ng and D.L. Wiltshire, "Propert,ies of cosmologies with dynamical pseud"o Nambu-Galdstone bosons", Phys. Rev. D63, 023503 (2001).

2. S.C.C. Ng, " Obseruational constrai,nts upon qui,ntessence models arising from moduli fi,elds," Phys. Lett. 8485, 1 (2000).

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107