A cosmological field theory

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A COSMOLOGICAL FIELD TH EO RY

A C C K P T E l )

T V Q C mK A L U A H . , : . T U D i L S b y

■rm

Steven P au l Starkovich B.A., Oregon S tate University, 1976

M.S . U niversity of Oregon, 1985 *r > y ..£ ________

A D issertation S u bm itted in P a rtia l Fulfillment of th e R equirem en > for th e Degree of

D O C T O R O F PH IL O SO PH Y

in the D ep artm en t of Physics and A stronom y

We accept this d issertatio n as conform ing to th e required sta n d a rd

D r. F. I. CooDerstock, Supervisor

^ .r>»- F n A Pnv:„Mr-k, D ep artm en tal M em ber

Dr. C. E. Picfaotto, D epartm ental M em ber

j_l p r . G. G. M iller, O utside M em ber

f—v—; -Dr. P. Hickson, E x tern al E xam iner

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Supervisor: Dr. F. I. Cooperstock

A B ST R ACT

Field theory is used to describe the m aterial content of th e universe th ro u g h

out its entire history, and an oscillating cosmological m odel w itho ut a singularity

is presented. In our theory, the “cosmological fluid” is described by a classical

scalar field th a t undergoes a series of phase transitions over th e lifetim e of th e

universe. Each tran sitio n corresponds to a discontinuous change in the eq uation

of s ta te of the field. In general, for an FRW universe a n d a given eq uation of

state, we show th a t th e field potential V{(j>) may be derived from th e solution of

P ic c a ti’s equation. The resulting expression for V(<j>) includes p aram eters whose

values are determ ined from the boundary conditions. In our theory, we em ploy th e

stan d ard cosmological m odel and th e fundam ental Planck quantities to provide

these bou nd ary conditions. We thereby determ ine th e scalar field L agrangian for

the entire history of th e universe. T h e resulting cosmological m odel is free of any

singularities, and includes an early inflationary epoch. Inflation arises in our th e

ory as a consequence of the initial conditions. The theory describes a universe th a t

is very cold at its m inim um radius, although it heats rapidly d uring th e in itial in

flationary era. This increase in the tem p eratu re of th e scalar field during inflation

is a direct consequence of applying classical therm odynam ics u n der th e assum ed

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m

param eters. Inflation continues un til a m axim um possible physical tem p eratu re

(th e P lanck te m p eratu re) is attain e d , at which point a phase transitio n occurs

and th e sta n d a rd m odel era begins. By relating the tem p eratu re of the scalar

field in o u r theory to th e rad ia tio n te m p eratu re in th e sta n d a rd model universe,

it is possible to establish a therm odynam ic con straint on a m ore com plete theory

of m a tte r for th e early universe. A lthough, in principle, inflation occurs for any

equ ation of sta te w here p < —( 1 /3 )p, we find th a t the initial equation of sta te

m ust be p ~ — p if th e la ter epochs of the universe are to resemble the stan d ard

model. In p articu la r, we find th a t Ho = 33 — 44 k m .s e e '1 M p c 1 is the value

of the H ubble p a ra m e te r a t th e cu rren t epoch th a t is least sensitive to the initial

equ ation of state.

Exam iners:

Dr. F. I. C ooperstock, Supervisor

Dr. F„ D . A. H artw ick, D ep artm en tal M em ber

DrT C. E . PicciotJ^>dt)epartm ental M em ber

£)r. Q ./G /' Miller, O utside M em b er, D epartm ent o f M athem atics

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List of Tables

Table 1 - Solutions of E quations (4.11) for Various Values of -yp ... 82

Table 2 - R esults for 7P = 1.85 x 10- 3 ... 84

T able 3 - R esults for 7p = 1.90 x 10~ 3 ... 85

Table 4 - R esults for 7p = 1.95 x 10- 3 ... 8 6

Table 5 - R esults for yp — 2.00 x 10- 3 ... 87

Table 6 - R esults for 7p = 2.01 x 10~ 3 ... 8 8

Table 7 - R esults for j p = 2.0153 x 10- 3 ... 89

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A C K NO W LEDG EM ENT

I wish to express my thanks to Professor Fred Cooperstock for giving m e th e

o p po rtu n ity to pursue this research. I am very grateful for his early ■ onfidence

and his continued support. In addition, I wish to express my app reciatio n for th e

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D E DICATION

To

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Chapter 1 Introduction

Cosmology is the stu d y of the origin a n d evolution of th e universe, an d m od

ern physical cosmology is inextricably bou:;d w ith g rav itatio n al theory. By one

account (B ekenstein 1992), m ore th a n forty gravitational theories have been p ro

posed, debated , anti accepted or rejected since the second decade of th e tw entieth

century, b u t it is the earliest of these theories — E in stein ’s theory of general

relativ ity - - th a t has been the m ost successful in describing th e universe at large.

T he physical and historical significance of general relativ ity lies p rim arily in

its fundam entally different view of space an d tim e from th a t afforded by New

tonian theory. E in stein ’s theory asserts th a t gravity is a m an ifestatio n of th e

geom etric stru c tu re of a four-dim ensional spacetim e m anifold, and th a t this ge

om etric stru ctu re is determ ined by the distribution of m a tte r an d energy w ithin

th a t spacetim e. Unlike th e o th er forces of n atu re, which are all form ulated w ithin

an assum ed underlying spacetim e structu re, the phenom enon we call “g rav ity” is

em bodied w ithin th e spacetim e stru ctu re itself. According to E in stein ’s theory,

g rav itation al a ttra c tio n is m anifested by th e curvature of spacetim e.

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stru c tu re and energy content of spacevime. W ith regard to cosmology, the most

em pirically successful theory is th a t which, as it happens, follows from the simplest

set of assum ptions, namely, th a t the universe’ is homogeneous and isotropic, and

th a t it is com prised of th e fam iliar forms of m a tte r and energy. T h e assum ptions of

hom ogeneity an d isotropy were m otivated initially by considerations of sim plicity

and a preconception of the way the universe likely appears when viewed in the

large. T h e result ng theory — the Friedm an-R obertson-W alker ( FRW ) model

has com e to be known as the “stan d ard cosmological m odel.”

T h e compelling n atu re of the stan d ard model arises from its ability to pre

dict several observed features of the universe in a consistent an d com prehensive

m anner. These features include the expansion of th e universe, th e characteris

tics of th e microwave background radiation, and the observed abundances of the

light elem ents. Several im p o rtan t issues w ithin the stan d ard m odel rem ain u n re

solved, m ost notably those pertain ing to a determ ination of w h eth er th e universe

is ‘o p en ” or “closed” (a question th a t only makes sense once; general relativity is

accep ted as th e underlying theory of gravity) and th e details of galaxy form ation.

However, during the p ast two decades, most of the criticism of th e sta n d a rd model

has stem m ed from w hat physicists have deem ed to be an u nacceptable dependence

on a precise ad ju stm en t (a “fi:.e-tuning” ) of initial conditions necessary to yield

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well-known horizon, sm oothness, and flatness “problem s.” The flatness problem is the

need to precisely adjust the conditions of th e early universe so th a t it resembles a

flat universe to a very high precision. T he horizon an d sm oothness problem s raise

the issue as to why causally disconnect,*3'! regions of the stand ard-m o del universe

arc so sim ilar in th eir physical characteristics. These issues are described in m ore

detail in A ppendix B.

T h e past two decades also have borne witness to m ijo r advances in theo

retical particle physics. T he accepted m odel of particle physics, articu la ted in

the language of q uantum field theory, accom odates the existing experim ental ev

idence o f particle interactions. Nonetheless, there rem ains su b stan tial theoretical

discom fort w ith the num ber of free param eters w ithin the m odel for w hich there

are no com pelling restrictions. In addition, there is an aesth etic m otivation for

over-higher degrees of unification of the forces of n atu re. T h e effort to ex trap o late

quant m il field theory to energy realms exceeding those atta in a b le in th e labo

rato ry has led logically to a symbiosis between particle physics an d cosmology.

Consequently, th e universe — particularly the early universe — nas becom e a

“la b o ra to ry ” in which new field theories of m a tte r m ay be te sted for th e ir cosmo

logical consequences. O f course, th e present low-energy observable universe (w ith

photons a t an energy of approxim ately 10-13 G e V ) can provide a very-high-energy

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4 assum ption is m ade as to an underlying theory of cosmological e\o lu tio n . There-

fore, to a great extent, the viability of any given unified thee y of particles and

fields has come to depend upon the validity of the stan d ard theory of cosmology.

T he m ost notew orthy product of the symbiosis between fiold theory and cos

mology is th e inflationary universe theory that has served to alleviate' the tine-

tu n in g problem s of th e stan d ard model. Tcday, th e assum ption th a t th e n ' exists

an inflationary epoch in the early universe w herein the scale factor of the u n i

verse accelerates — is well-accepted. T h e recent literatu re contains num erous

com prehensive reviews of m odern cosmology and inflation theories, am ong them

being B orner 1988, B randenberger 1985, Collins et al. 1989, Kolb an d T urner

1990, Linde 1990, and Olive 1990. T he discussion in this In tro d u ctio n draws

up o n these reviews.

T h e field theory formalism for the description of m a tte r and the forces of n a

tu re has a long history (Pickering 1984). T he objective of a field theory (w hether

q u an tu m or classical) is the specification of a locally-invariant L agrangian th a t is

com prised of the various m a tte r and gauge fields ap p ro p riate to th e problem under

consideration, an d th a t yields equa ions whose solutions describe' the phenom ena

of interest. For our purposes here, it is useful to outline' the' basie- notions be1

hind th e m ost generally-accepted contem porary cosmeileigiral the-eine's se> th a t we

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of m a tte r and cosmological inflation.

In these theories, the Lagrangian — which m ay include a variety of fields cor

responding to different fundam ental particles — typically includes a scalar field

whose p o ten tial cm,, dies the energy content of th e vacuum . T h e pre-inflationary

ui.iverse is presum ed to be radiation-dom inated an d at a very high tem p eratu re, so

th a t as the universe emerges from a singularity, the scale facto r R ~ t 1/ 2 while th e

te m p eratu re T ~ 1 / R . W hen the tem p eratu re falls below a critical value, the vac

uum energy density dom inates the radiatio n density, and th e vacuum -dom inated

universe (w ith th e equation of state p = —p) inflates beyond th e particle horizon.

T he result is a causally-connected observable universe. Inflation ends w hen th e

sym m etry of th e scalar field p o te n t:"1, is broken, either by a tunnelling process

as in th e “old inflation” theory, or by a “slow-rcllover” process as in “new infla

tio n.” T h e sym m etry breaking serves to redefine, or restru ctu re, th e gro u n d sta te

(th e vacuum ) of th e theory. The radiatio n, having cooled exponentially d uring

inflation, m ust th e n be reheated.

A lthough set val contem porary inflation theories have offered solutions to th e

problem s of the stan d ard model, fundam ental problem s still rem ain w ith m ost

inflation m odels. P erhaps th e m ost serious problem is th e fact th a t a generally-

accepted field theory of m a tte r for th e early imiverse does n o t exist. T h e sim plest

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6 p lv sic s an d cosmological perspective, and most o th e r models are still extrem ely

speculative. F urth er, in those circum stances where a p articu lar theory of m a tte r

is being investigated for its cosmological consequences, it is usually necessary to

finely tu n e one or m ore free param eters in the m odel in ord er for th e end result

to be a universe resembling th e one th a t we observe. T he fine-tuning problem s

are m ost ap p aren t w ith regard to the specification of the details of th e transition

betw een the inflation era and the stan d ard model an issue th a t has plagued

inflation models since their inception. T hus, the fine-tuning problem s of the s ta n

dard cosmological m odel have, to some extent, been recast as fine-tuning problem s

w ithin th e field th eo ry th a t accounts for inflation. Finally, in m any inflation m o d

els, a scalar field is included in the theory for th e sole purpose of generating

inflation, an d is no t otherw ise integrated into a com prehensive theory of m a tter.

It is against th is background th a t we now describe th e m otivation for the*

present work. We w ish to develop a cosmological field theory th a t unifies the

early a n d late epochs of the universe w ithin a single theoretical fram ework th a t

employs field theory for the description of th e m aterial content of th e universe.

W hile q u an tu m field theory forms the basis of o th er contem porary m odels of the

early universe, this d issertatio n describes a cosmological theory rooted in the long

tra d itio n of classical field theory. W hile th e stan d ard m odel of particle physics

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theory of m a tte r we employ in this dissertation envisions th e m aterial content of

the universe as a continuum .

T h e ram ifications of our theory will become clear as we proceed th ro u g h this

dissertation. Even though the fundam ental notions about th e n a tu re of m a tte r

which underlie th e present work differ substantially from th e ideas underlying

oth er contem porary theories of the early universe, there are nonetheless some

com m on themes. In th e rem ainder of this Introduction we shall an ticip ate some

of our results a n d make a few com parisons betw een the two approaches. It shall

then b e easier to appreciate th e advantages, as well as the disadvantages, th a t our

theory holds over o th er models.

A n im p o rtan t concept th a t bears an im m ediate relevance to th e present work

is an id e a due to M arkov (1982) who suggested th a t, in m uch th e sam e way th a t

there exists a lim iting physical velocity, there m ight exist a lim iting density, and

lim its on other physical quantities as well. M arkov identifies th e P lanck density

as a n a tu ra l lim iting density. C ontem porary inflation m odels provide for an early

universe th a t emerges from a singularity into a rad iatio n -d o m in ated state, th en

becom es vacuum -dom inated, an d then evolves to th e radiatio n -d o m in ated epoch

th a t m arks rhe beginning of th e stan d ard m odel era. If, instead, th e universe

begins its expansion in a “vacuum-like” state, th en general relativistic cosmology

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s

the density is at its lim iting value. If the energy density is positive, the E instein

equ ations th e n require th e universe to be closed.

T herefore, these two considerations — an initial vacuum-like equation of state

in th e early universe and a lim iting physical density lead logically to a closed

universe w ith o u t an initial singularity. T he presence of an initial singularity sig

nifies a breakdow n of physics at th e beginning of th e universe. It is reasonable

to require th a t singularity avoidance be a criterion in any physical theory, and

the use of a vacuum -like equation of state ,o avoid a singularity in the early uni

verse has a long history th a t extends at least as far back as G liner (196G). More

recently, a singularity-free cosmological model w ith an early inflationary epoch

was proposed by Israelit an d Rosen (1989). In their m odel, which dot's no t em

ploy field theory for th e description of th e m aterial content of th e universe, they

p o stu la te th a t the early universe is com prised of a m aterial substance (they refer

to it as “p re m a tte r” ) whose equation of sta te is initially the sam e as th a t for

th e vacuum . T hey co nstruct a continuous function for th e equation of s ta te such

th a t, as th e universe evolves, the inflationary p re m a tte r epoch is followed by the

rad iatio n - a n d m atter-d o m in ated epochs of the stan d ard m odel. They do not,

however, provide a theory of m a tte r th a t accounts for this behavior.

Cosmological models which do not use field theory lack the; fu nd am en tal

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Oil tii<: o th er hand , as we have already noted, a suecessful q u an tu m field theory

for m a tte r in the early universe has yet 1 0 be discovered. In tois dissertation , we

develop a cosmological theory utilizing a single, classical scalar field to represent

the m a te rif' content of the universe. This approach is a logical extension of th a t

used by Coopt.’stock and Rosen (1989) to construct static particle solutions to

the field equations of a classical field theory. In a sense, changing th e bo u n d ary

conditions and th e independent variable cause th e static, scalar field solutions of

a par ticle theory to become cosmological models w herein th e m aterial content o f

the universe is a scalar field continuum .

T h e theory we present goes beyond a consideration of th e early universe alone,

and describes th e general behavior of the universt throu gho ut its en tire history.

In add ition , we consider a broader range of initial conditions th a n was considered

in th e work of Israelit and Rosen, although m any of th e physical resu lts they

o b tain ed are qualitatively sim ilar to the results we shall eventually o b ta in from

our theory.

T h e scalar field in our theory — the “cosmological fluid” — represents th e

to tal m aterial content of the universe an d we p o stu la te th a t th e field m ay a t

tain various equations of sta te in order th a t various cosmological epochs m ay b e

analyzed. Accordingly, th e “radiation ” an d “m a tte r” epochs are th o se eras in

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10

p — ( 1 /3 )p and p = 0, respectively. We shall use th e term “p re m a tte r” (from

Israelit an d Rosen) to describe the physical sta te of th e scalar field during the

inflation era where th e equation of state approaches p — —p.

D iscontinuous changes in th e equation of sta te of the scalar field in our theory

denote th e occurrence of phase transitions. These transitions occur a t critical field

values, a n d our theory p erm its fhe determ ination of these values. O ur use of the

phase tran sitio n concept removes the arbitrariness associated with co nstructing a

fu nctional form for th e equation of state, p articularly for th e tran sition s between

th e phases. However, we m ust make it very clear- th a t our use the phase tran si

tio n concept is m uch different from th a t which appears in th e literatu re on field

theories of th e early universe; so different, in fact, th a t only some term inology is

sim ilar. V acuum phase transitions are a staple of m ost contem porary cosmolog

ical theories, an d they are m anifested by a sym m etry-breaking of a scalar field

p o te n tia l in accordance w ith th e generalized theory of phase; tran sitio ns due to

L and au (L an dau and Lifshitz 1980).

These o th e r theories include the scalar field as only one of several compon. nts

of th e cosmological fluid. By contrast, in ou r theory, the; entire m aterial content

is em bodied w ithin a single com ponent — th e scalar field whose; value; e»volve‘s

sm oothly over tim e. A phase tran sitio n is th en nothing more; th a n a change; in

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value which we refer to here as a “critical value.” T h ere is no change in th e n a tu re

of th e po ten tial and no sudden change in th e field value to m ark th e occurrence

of th e tran sitio n this la tter feature being the central id ea behind th e L andau

theory. Consequently, our scalar field does not play th e role of an o rd er p aram eter

in a vacuum phase transition as it does in o th er theories.

From a practical point of view, our theory should b e able to speak to th e

im p o rtan t observable features of th e universe. A m ong these features is th e tem

p e ra tu re of th e microwave background radiation. T h e application of th e first an d

second laws of therm odynam ics to our theory shows th a t th e te m p e ra tu re of th e

scalar- field m ust increase during the inflationary era. T his is a consequence of

simplifying th e assum ptions ab o u t th e early universe — th a t is, by elim inating

a pre-inflationary rad iatio n era — an d of em bodying th e m aterial content solely

w ithin the scalar field. Thus, it is possible to reconcile inflation a n d “h eatin g ”

by appealing to th e laws of therm odynam ics w ithin the context of th e assum p

tions in our theory. Consequently, inflation and heating becom e com plem entary

phenom ena, unlike in oth er theories where a “reheating” o f a sep arate rad iative

com ponent m ust be introduced after inflation to com pensate for th e exponential

cooling of th e rad ia tio n during inflation. O f course, th e a c tu a l universe contains

a rad iativ e com ponent, and la ter in this p ap er we discuss how th e te m p e ra tu re

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12 observed universe. T he result of this discussion is w hat we shall refer to as a

“therm odynam ic con straint” on a more com plete theory of m a tte r for th e early

universe.

In fu rth e r accordance w ith the idea th a t there are finite lim its to physical

quantities, we assum e th a t th ere exists a m axim um possible physical te m p era

ture. T h is lim it is taken to be the Planck te m p eratu re (Tpi), whose value m ay

be o b ta in ed by setting Tpi = m pj, where, in n a tu ra l un its (see A ppendix A), the

B oltzm ann constant equals one. la our theory, inflation ends by a basic physical

consideration — th e attain m en t of this m axim um possible tem p eratu re and not

: ■; m eans of a finely-tim ed mechanism as required in other field theories of th e

early universe. W hen th is m axim um te m p eratu re is attain ed , a tran sitio n from

a vacuum -like equ ation of sta te to a radiative equation of s ta te occurs. At this

tran sitio n , inflation ends and th e era described by the stan d ard model begins.

T h e m ission of o th er field theories of the early universe is com pleted upon th e

tra n sitio n from a n inflationary epoch to the beginning of the s ta n d a rd m odel. In

co n trast, o u r theory unifies the early and late epochs w ithin th e t ame th eoretical

fram ew ork while still providing for a la ter universe th a t follows th e general outline

of th e ra d ia tio n and m a tte r epochs of the stan d ard model. As is the case during

th e early inflationary epoch, th e m aterial content of th e universe during th e la ter

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m axim um size, contract to a scale of the order of th e P lanck length, and th en

re-expand, retracing its steps cyclically ( th a t is, th e scale facto r of th e universe

will continue to “oscillate” betw een its m inim um and m axim um values). We

adopt th e fam iliar idealization th a t th e universe is homogeneous a n d isotropic,

and th is assum ption is im p o rtan t w ith regard to <dngularity-avoidance upo n the

co ntraction of th e universe. We discuss this issue near the en d of this dissertation.

T h ere is a t least one other im p o rtan t difference betw een o u r th e o ry and m ost

others, and it is a difference of approach tow ard solving th e field e q u a tio n ;. In

other m odels, a p articu la r scalar field p o ten tial is m otivated b y some independent

consideration an d then tested for its cosmological consequences. In co n trast, our

approach is to determ ine th e scalar field p o te n tial which yields reasonable cosm o

logical results throughout the entire history of th e universe. In o th e r words, we

shall app eal to th e stan d ard cosmological m odel in order to establish th e p o ten tial

function corresponding to th a t behavior. Toward this end, we show th a t for an

FRW universe and a given equation of state, th e field p o te n tial m ay be derived

from th e solution of R iccati’s eq ation. A lternatively, our developm ent m ay be

altered slightly so th a t th e equation of sta te m ay b e determ ined if th e p o te n tia l is

known. T his could be of interest in those studies th a t exam ine th e cosmological

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14 W e now sum m arize th e im p o rtan t features of th e cosmological field theory

presented in th is dissertation.

1) T he universe is closed and oscillates between a m inim um a n d m axim um

size. A rgum ents against oscillating models th a t are founded upon com pletely

different theoretical premises from those in the present work have no bearing on

this resu lt (for exam ple, see G u th an d Sher 1983);

2) The early universe includes an inflationary epoch th a t arises as a con

sequence of th e im posed b oundary conditions which, in tu rn , are expressed in

term s of th e fu nd am en tal constants of n a tu re (th e Planck qu an tities). Inflation is

independent of th e existence of an earlier radiation-dom inated universe;

3) T he universe is initially very cold, b u t its tem p eratu re increases rapidly.

This increase in th e te m p eratu re of th e scalar ^eld is com plem entary w ith inflation,

it follows from basic therm odynam ic considerations, and it does not require a fine-

tu n in g of free param eters;

4) R elating th e te m p eratu re of th e scalar field to a rad iativ e te m p e ra tu re

in th e sta n d a rd m odel yields a therm odynam ic constraint on a m ore complete;

theory of m a tte r th a t m ay b e superim posed on our mode) while reta in in g our

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5) T he a ttain m en t of a physical state (a m axim um possible physical tem p er

atu re) m arks th e end of inflation. There is no need to co nstru ct a finely tu n e d

m echanism for m aking a tran sition from inflation to th e sta n d a rd model;

6) The m aterial content of th e universe is described by a field th eo ry th a t is

relevant for the entire history of th e universe, ra th e r th a n only for an early epoch.

T he scalar field p o ten tial (and thereby th e Lagrangian function) th a t describes

the evolution of the universe throughout its entire h istory is established; and

7) A fter utilizing all the b oundary conditions, th e only rem aining free p a

ram e ter in th e entire theory is th e initial equation of sta te . It is found th a t —

w ithout excessive fine-tuning — th e initial equation of s ta te m ust be p » - p if

the la te r epochs are to resemble th e stan d ard model. We are able to predict th e

value of the H ubble p aram eter a t th e present epoch th a t is least sensitive to th e

choice of the initial eq uation of state.

In addition, th e theory also p e r n r ts th e d eterm ination of th e critical values

of th e scalar field where th e phase transitions occur. Finally, we discover th a t th e

scalar field possesses a negative specific h eat during th e p re m a tte r p hase ( th a t is,

its te m p e ra tu re increases while its density decreases).

In C h ap ter 2 we develop th e basic equations an d specify th e b ou n d ary condi

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16 o th er sources in the literature, including the review m onographs cited earlier. We

include this m aterial here tor th e sake of completeness. In C h ap ter 3, we derive

th e differential eq uation — whose form is th a t of R iccati’s equation from which

we th e n determ ine th e field potential. In C hapter 4 we develop th e cosmological

theory in detail, an d in C h ap ter 5 we present th e cosmological resu lts and offer

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Chapter 2 The Field Equations

and Boundary Conditions

T h e E instein equations of general relativity are

G ki = 8 n T k (2.1)

where G k an d T k are th e Einstein and energy-m om entum tensors, respectively,

and th e indices * an d k take on th e values 0, 1, 2, a n d 3. We ado pt th e n a tu ra l

system of u n its (ft = c = l ) s o th a t th e gravitational con stan t G = 1 / m 2t ,

where m pi is th e Planck mass. The equations are fu rth e r simplified if all physical

quantities, including the gravitational constant, are m easured in P lanck u n its (see

A ppendix A).

Because we assum e th e universe to be homogeneous a n d isotropic, we adopt

the Friedm ann-R obertson-W alker (FRW ) spacetim e whose interval is given by

d r2

d s2 = dV - R l + r (d0 + s in 9d<j> )

1 — hr2

w here, generally, d s2 = g ik d x 'd x k a n d </,•*. is th e m etric tensor.

(2.2)

In equation (2.2) t is the co-moving (proper) tim e as m easured in Planck

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18 facto r of th e universe. The param eter k denotes th e spatial cu rvature am i takes on

th e values k = +1, — 1 or 0 for positive, negative, or zero curvature, respectively.

If k = +1, R is th e radius of the universe in Planck lengths; otherwise' R is only

a m easure of th e relative pro per separation of given points at different, times. For

th e reasons est iblished in C hapter 1, we shall be considering a closed (k — +1)

m odel in this dissertation.

T h e energy-m om entum tensor in equation (2.1) is th a t of th e m a tte r p rod uc

ing th e g rav itatio n al field. In order to describe th e m aterial content of th e universe

w ith in th e field theory formalism, it is sufficient to specify the L agranginn of the

theory. T he Lagrangian in our theory is th a t of a real scalar field <j>, a n d may be

w ritte n in the sta n d a rd form

L = ^ W ) ( 3 V ) - V (4 ) (2.3) w here V{<j>) is th e potential.

T h ere are two expressions for T k :

T? = (% < /> )(& 4)-6*L t (2.4) w hich follows from equation (2.3) and the definition of T k from field theory, an d

T? = (p + p )u iu k - p 6 t , (2.5) w hich applies to any macroscopic perfect fluid (L andau and Lifshitz 1975). In

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four-velocity of th e cosmological fluid, where a takes on th e values 1, 2, a n d 3. T he

spatial com ponents, it", vanish under th e constraints of hom ogeneity and isotropy.

Com parison of these expressions for T k yields expressions for p a n d p in te rm s of

the field quantities:

P = + V ( 4 . )

(2.6)

P = \ * 2 -

V (j).

(2.7)

A n additional term ( r k) could have been included in eq uation (2.5) for the

purpose of representing a contribution to th e energy-m om entum te n so r from dis

sipative processes w ithin th e cosmological fluid. T he m ost general form for r* ,

given by

Ti = + u ’? ~ ~ u lu kUi.t] + (C - | » / ) ( ^ - UiUk )u \t (2 .8 )

(L an d au and Lifshitz 1987), would be reduced to

r? = C(«? - «.-«*)«!, (2-9)

under th e co nstraints of homogeneity and isotropy. In these eq uations, r) is a

sh ,‘ar viscosity coefficient an d ( is a bulk viscosity coefficient for th e cosmological

fluid. In the FRW m etric u?f = 3H where H = R / R is th e H ubble p aram eter.

T hus, th e whole effect of r* in our theory would be to replace th e pressure p

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2 0 cosmological fluid is w ell-approxim ated by a perfect fluid, and therefore r* = 0.

As a resu lt, no entropy pro du ctio n can be forthcom ing from th e dynam ics of the

scalar field.

E valuating th e 0-0 a n d a — a com ponents of eq uation (2.1), and using eq ua

tions (2.2) and (2.5), we ob tain two equations which describe th e tim e evolution

of a hom ogeneous, isotropic universe:

( R \ \ k (21())

\ R I R2

en d

3

We have defined th e p aram eter 7 as

R + ^ r ( 3 7 - 2)pR = 0. (2.1 1)

7 = - + 1 = . -2—- (2.1 2a)

P <f>2 + 2 V

so th a t

v = p (2 ^ T ) t { 2 l 2 b )

w here we have used equations (2.6) and (2.7). If 7 = 4 /3 th e universe is radiatio

n-dom in ated, while if 7 = 1 the universe is m atter-d o m in ated (p = 0). T h e case

o f 7 —- 0 corresponds to th e vacuum equation o f s ta te (p — —p). From equ ation

(2.11), we see th a t inflation (R > 0) arises for any 7 < 2/3 .

E quations (2.10) a n d (2.11) are related to each o th e r by th e conservation

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conservation laws = 0. The result m ay be w ritte n in at least th ree equivalent

forms:

<jt + ZH(j> + = 0 (2.13)

(which can also be obtained by varying th e Lagrangian w ith respect to <f>),

P + 3JTt/> = 0, (2.14)

or

d

d R (pB?) = - S p R 2. (2.15)

T h e two independent field equations we shall solve are eq u atio n (2.13) a n d

R + k(4>2 - V ) R = 0, K = (2.16)

O

which follows from equations (2.6), (2.1 1), and (2.1 2).

T h e first a n d second laws of therm odynam ics com bine to give the G ibbs

relatio n

T d S = d E + pd(2rr2R 3) (2.17)

w here S an d E are th e to tal entropy a n d energy, respectively, an d 2 n 2R 3 is th e

volum e of a A: = + 1 imiverse of radius R . L etting s and p b e th e entropy a n d

energy densities, and applying th e integrability conditions of eq u atio n (2.17), we

o b ta in (W einberg 1972)

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*

22 w here we have used equation (2.1 2a).

D ifferentiating equation (2.17) w ith respect to R , and using equation (2.15),

we o b ta in an eq u atio n th a t reflects the perfect fluid assum ption:

T ^ - (sR 3) = 0. (2.19)

at

C arryin g out th e differentiation in equation (2.19), an d using equations (2.14) and

(2.18), we o b ta in

7 - 7 ^ - 3 f f7(7 - l ) = 0. (2.20)

Therefore, once th e field equations are solved, the te m p eratu re evolution of th e

cosmological fluid m ay be determ ined from equations (2.12) a n d (2.20). T he

evolution of th e density a n d pressure may also be found from eq uations (2.1 2).

In order to establish the in itial conditions for th e field equations, we assum e

th a t th e m axim um attain a b le density in any physical system is th e Planck density

(M arkov 1982). W hen this constraint is applied in conjunction w ith th e initial

condition i?(0) — 0, th e com plete set of in itial conditions becom es

p( 0) = 1 (2.2 1a)

p{0) - 0 (2.216)

# ( ° ) = Q ) ? (2.2 1c)

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w here p is given in term s of th e field quantities by equation (2.6), an d w here we

have used equation (2.1 0) w ith k — +1.

In C h ap ter 4 it shall become apparent th a t we need tw o ad d itio n al b o u n d

ary conditions in order to com plete the cosmological theory. T hese b o u n d ary

conditions are established by considering th e rad ia tio n epoch of th e stan d ar d cos

mological m odel. At th e end of th e radiation epoch, th e rad ia tio n te m p e ra tu re is

approxim ately 2900 ° K (K olb and Turner 1990). T his corresponds to a rad ia tio n

energy density a t decoupling of

A 7r2

pdec = (2.2 2a)

15

Planck densities, where A = 1.7539 x 10~115. As one of th e b o u n d ary conditions,

we shall assum e th a t th e energy density in equ ation (2.2 2a) equals th e energy

density of th e scalar field at th e corresponding tim e in our theory.

In o u r theory, th e radiatio n epoch begins w hen th e te m p e ra tu re has a tta in e d

a m axim um value. T his is a consequence of equ ation (2.20) an d a featu re of ou r

m odel which will be specified later in this p ap er afte r a m ore general developm ent

in C h a p te r 3; nam ely th a t 7 will b e a specified constant for a given epoch. T hen ,

since 7 < 2 /3 during th e inflation era while 7 = 4 /3 during th e rad ia tio n era,

eq uation (2.20) shows th a t T > 0 during inflation a n d T < 0 d uring th e rad ia tio n

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24 For th e purpose of establishing the value of this m axim um tem p eratu re, we

assum e th a t th e energy density at the beginning of the rad iatio n e ra of th e s ta n

d ard m odel is given by

ir2 7T2

p r i = — T*i = — , (2.226)

H ' 15 r,‘ 15 K

w here th e te m p eratu re has been scaled in such a way th a t T r,i = 1. T his n a tu ra l

scaling of th e te m p e ra tu re, in conjunction w ith equation (2.2 2a), yields the Planck

te m p eratu re

Tpi = Tr<i = 1.42 x 1032 °Jv. (2.23)

T his is th e sam e tem p eratu re th a t one would o btain from settin g Tvi = rnpi. We

shall assum e th a t the energy density of the scalar field a t th e end of th e inflation

era in o ur theory is given by equation (2.22b). T his is the second bo un dary

condition we seek.

These two b o u n d ary conditions allow us to com plete our cosmological theory,

while appealing to th e stan d ard m odel for the determ in atio n of tw o free; p a ra m

eters. We should n o te th a t th e values for p in equations (2.22) are calculated

u n d er th e simplifying assum ption th a t only photons co n trib u te to th e energy d en

sity du rin g th e rad ia tio n era of th e stan d ard model. T h e q u alitativ e behavior of

our m odel w ould not b e affected were we to include o th e r relativistic species w hen

specifying th e values in equations (2.2 2).

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and m ay be solved if either 7 o'- V is specified. T h e approach in this p ap er will

be to specify an equation of s ta te in term s of th e field <j>. T he field equations are

then solved and, as we show in C hapter 3, the scalar field p o te n tial is determ ined.

We thereby o b ta in th e Lagrangian of the field theory for th e entire history of

the universe. By specifying 7, ou r approach more closely resem bles th a t used in

s ta n d a rd cosmology th a n the approach used in q u an tu m field theories of th e early

universe. In th e la tte r, the p o ten tial V(<f>) is p o stu la ted a n d th e n te ste d for its

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Chapter 3 The Determination

of the Scalar Field Potential

If <f> is a constant, equations (2.12) - (2.16) show how th e behavior of th e scale

facto r depends on th e value of dV/d<j>. In particular, if dV/d(f> — 0 w hen <p = 0,

th e scale factor undergoes eternal exponential inflation, and B.(t) ~ c x p ( x t) for

all tim e, w here x = (8 7T/0/3 ) 1 / 2 an d the density is constant. On th e oth er hand , if

dV/d<f> ^ 0 for co nstant <j>, then <f> evolves away from zero and etern al inflation is

avoided.

In th e rem ainder of this dissertation, we consider only the case of <f> ^ 0 or,

equivalently, 7 0. We proceed on the assum ption th a t an effective eq uatio n of

s ta te 7 = 7 (<f)) is specified. In this C hapter, the field equations are transform ed

into eq uation s for In V an d In R as functions of <f>, an d these new equations are,

in tu rn , com bined to yield a R iccati equation whose solution perm its us to find

v(4).

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E qu atio n (2.13) is transform ed using equation (2.12a), equ ation (3.1), an d

the H ubble p aram eter ( H ) to obtain

din 72 1

d<j>

1 cfln V 1 din 7 7 d<j> 2 — 7 d(j>

(3.2)

In order to transform equation (2.16), we first differentiate th e H ubble p a

ram eter w ith respect to time:

R rV t t2 7 din 72 •oCplni? •2 f d ] x i R \ 2

E quations (2.12a), (3.1) and (3.3) are com bined to yield

R R ^ 1 din 7 1 din V 2 — 7 d<j> ^ * 2 d<j> d l n R c P ln R ( d l n 7 2 \2 . . - ^d(j> + - ^ + ( - 5 r ) ’ 3' 4) an expression th a t arises only from the definitions of H and 7 .

In addition, equations (2.12) and (2.16) are com bined to yield

— ~ = t t" ( 2 — 3 7).

R<f>2 27

C om parison of equations (3.4) a n d (3.5) yields

din R

1 din 7 1 din V

2 — 7 d<f> 2 d<f> d(f>

(3.5)

d2ln72 / d l n R \ k

E quations (2.13) an d (2.16) have thu s been transform ed into eq uations (3.2)

and (3.6). In a sim ilar way, equation (2.20) is transform ed in to

din 7 d ln T .d in 72

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2 8 D ifferentiating equation (3.2) w ith respect to <f> yields

1 d2\a.V 1 din 7 din V

d2l n i ? 1

d<f>2 ~ 3 7 d(f>2 7 d(j) d<j>

1 d2l n7 7 / dln ~ )\

+ 2 - 7 d(jy + ( 2 - 7 ) 2 V d(f> ) (3.8)

S u b stitu tin g equations (3.2) an d (3.8) into equ ation (3.3) and collecting term s

yields ^ + a(<j>)u + b(<f>)u2 + c{<j>) = 0, a<p (3.9) where u = din V / d<j>, m = 9 7 — 1 0 din 7 6 ( 2 — 7) d<f> ’ 3 7 - 2 6 7 (3.10a) (3.10&) and 7 d2l n7 7 ( 3 7 + 2 ) / d i n7 \ 2 3k

+

2 - 7 d(f>2 3(2 - 7)

U

₎ + t ( 2 - 3 7 ). (3.10c)

E q u atio n (3.9) is of the general form of R iccati’s eq uation for u a n d is

solved by converting it to a second-order, linear, homogeneous differential eq u a

tion (D avis 1962). A function w((f>) may be defined such th a t

1 w'(<j))

U^ b(<j>) w(<j>)

where th e ' denotes differentiation w ith respect to <f>.

(3.11)

S u b stitu tin g equation (3.11) into equation (3.9) yields

b ' m

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T h e task of solving th e field equations has thus b een reduced to solving

equation (3.12). The m ost im p ortant special case of eq uation (3.12), an d th e one

which we shall consider for the rem ainder of this dissertation, is for 7 equal to a

constant. In this circum stance, equations (3.10) become

a = 0; b = ( 3 7 — 2 ) /6-y; c = 3k(2 - 37 )/2 . (3.13)

S u b stitu tin g eq uation (3.13) into equation (3.12), we o btain a differential eq uatio n

for w(<f>),

w " — a 2w = 0, (3.14a)

where

a = 36(/«7)1/ 2. (3.146)

E qu atio n (3.14a) has the general solution

u>(<f>) = 0 1e - a+ + lS2ea+, (3.15)

where fi\ and /? 2 are constants of integration.

E qu atio n (3.11) then yields

din V a

d<j> b

which, upon in teg ration , yields

1 - /3e2a^

1 + /?e2“ ^ (3.16)

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3 0 w here ft = /i2//^i and C is a constant of integration.

E q uation (3.17) is th e g c n e i;. solution for the scalar field p o te n tia l u nder the

condition th a t 7 is a nonvanishing constant. We em phasize again th a t, a t this

stage of th e developm ent, no o th e r boundary conditions have been im posed. In

p articu la r, this result is applicable to all inflation models w hich can accom odate

th is condition on 7, a n d this includes a wide range of m odels currently in the

literatu re. T he constants ft and C in equation (3.17) are determ ined from w hat

ever ad d itio n al b o u n dary conditions are im posed in conjunction w ith a p articu la r

app licatio n of this general result.

Finally, equations (3.2) and (3.7) become

din R 1 din V d<j> 3 7 d(j) an d (3.18) din T _ ^ 7 — 1 \ din V (3.19) d(j> = K ~ r ) - d(f> a n d eq u atio n (2.1 2a) yields ^ = ± (2? 7 ) # VX' 2^ ) - (3.20)

T h e choice of eith er th e plus or minus sign in equation (3.20) will depen d upon

o th e r conditions im posed on th e problem.

It should be n o ted th a t it is possible to combine the field equ ations in a

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One such approach was developed by Ellis an d M adsen (1991) w here an explicit

dependence on th e H ubble p aram eter is retained in th e ir equations. O u r m eth od

pursues a different line of development th a t eventually leads to an equ atio n of the

form of R iccati’s equation from which we can then determ ine an expression for

V{<j>). A cosmological theory m ay now be constructed by applying these results

to consecutive cosmological epochs, each w ith its own characteristic value of 7 .

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32

Chapter 4 The Cosmological Theory

here are m any possible applications of equation (3.17), each corresponding

to a different set of bo undary conditions. The theory we offer in this dissertation

is am ong the m ost comprehensive of th e possible applications in th e sense th a t

we em body in th e scalar field the entire m aterial content of th e universe. As

an altern ative, eq uation (3.17) also could be used (w ith correspondingly different

b o u n d ary conditions) in a m odel where th e scalar- field was only one of several

com ponents to th e to tal energy.

In b ro ad outline, th e model presented her. resembles th e model of Israelit and

Rosen (1989) in th a t it consists of p rem atter-, radiation-, and m atter-d o m in ated

epochs. In our n o ta tio n below, the subscripts p, r , and m refer to these epochs,

respectively. As we have already noted, 7„, = 1 end 7r = 4 /3 . We shall consider

th e physical consequences of several different values for 7p.

T h e L agrangian of the field theory for th e en tire history of the universe* is

determ ined once we find V(<f>) for each epoch. T he general solution or V ( <j>) with

7 equal to a co nstant has already been determ ined in equation (3.17). It rem ains

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the constants b, a, ft, a n d C for each epoch. The constants b and a are obtained

directly from their definitions in term s of 7 (see eq. [3.10b] and eq. [3.14b]).

The con stan ts ft and C will be determ ined by th e application of th e b ou ndary

conditions.

T h e boundary conditions also allow determ in ation of ‘‘he in itial an d critical

values of the scalar field. The initial value of <b is denoted as (j>0, the q u an tity <f)pr

is the critical value of <j> a t the p rem atter-to -rad iatio n phase tran sitio n , an d <j>rm is

the critical value of <j> a t the rad iatio n -to -m atter tran sition. We re iterate a point

m entioned in th e Introduction concerning our use of th e te rm “phase tra n sitio n .”

In this work, a phase tran sitio n is a discontinuous change in th e equation of sta te

of th e scalar field. In o th e r models, the phase tran sitio n concept is used to describe

a sym m etry breaking o f th e vacuum state of a scalar field p o ten tial in accordance

w ith th e generalized th eo ry of phase transitions due to L andau (see Linde 1979

for a com prehensive review). T his la tte r in terp reta tio n of “phase tra n sitio n ” has

110 relevance to the present work.

As a convenient choice for one of th e bo undary conditions in o u r theory, we

set <f> = 0 at th e point o f m axim um expansion in th e scale factor, a t which point

the density a tta in s a m inim um value. From eq uation (2.12b), we th e n see th

<f> = 0 corresponds to a m inim um in the p oten tial (F mm) whose value m ay be

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of th e universe is sym m etric w ith its expansion. Therefore wo shall only consider

th e case of expansion.

In our m odel, 7 is discontinuous at the critical values of tin' scalar field.

However, th e energy density (p) and the scale factor (R ) m ust he continuous across

each tran sitio n if the to ta l energy is to he a continuous function of </>. E quatio n

(2.12b) th en yields discontinuities in V(<f>) at <f>rm and <f>pr. In p articular, for two

phases (which we denote in general as I and I I ) , the po tentials 011 ('it,her side of

th e tran sitio n are related according to

Vi 2 - 7 /

.2 - H i .

Vn . (4.1) In th e equations below, we shall denote the values of the po tential 011 the p re m a tte r

a n d rad iatio n sides of <t>pr as (Vpr)r and ( V),r )r , respectively. Similarly, the values

of th e p o ten tial on th e rad iation an d m a tte r sides of <j>rm are denoted as (V r m )r

a n d (Vrm )mt respectively.

We m ust next consider the behavior of (Ih iV /d fi a t the critical values of <f>.

Since j p = 02 ^ 0 (see eqns. [2.6] an d [2.7]), equation (2.14) m ay be w ritten as

p' = - 3 /7 0 . (4.2) T h e continuity of p and R implies th a t the H ubble p aram eter is continuous across

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which, w ith equation (2.1 2b), becomes /<fl n V \ V d<j> ) , I I . I I I 1/2/ <flnV\ \ d(f> ) jj (4.4)

E quations (2.21), (3.16)-(3.18), (4.1), and (4.4) then provide the b ou n d ary

conditions. At 0 = 0 and t = t m : Vmin = C m [ l +f 3

m}1/bm,

(4.5a) and 1 + f i n = 0; (4.56) a t <j> = (j>rm and t = t r :

(

Vrm)m = Cm

+

f)m ea”'+r”']

**1/6m = | c r [e"0- ^ + /9r e°"*” " ] 1/6r,**

(4.5c) and 1 / 3 n 2 Q m ^ r m « / v 1 __ / ? / > 2 ® r ^ r m (4.5d) ' 1 - /3me2Qm^rm' \ / 3 o . 1 — /?r e2" r ^rm 6„, 1 + A’me2am^rm 2 br 1 + /?r e2a,r^rm at 0 = </>pr and / = t p: (Vpr) r = C r [ c - ° - ^ ' + ^ re « " ^ ] 1/6r = i ( ^ _ ) c „ [ t— ap<)>pr

_|_ ^ ^g^p^pr j

^

(4.5e) and O r 1 - /3re2ar<t,PT 1 + /dr e2Qr‘b,r = 2(37p) - 1 / 2 Q P ' 1 — /?pe2Q>^',r 1 + /?pe2afp^>r (4 -5 /)

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36 at <j) = (j>Q and t = 0: (4.5#/) and a p 1 — /Jpe2 a ',,,!>0 bp i + flpe^pto

T h e q u an tities t p an d t r denote the tim es at which th e two phase tran sition s occur,

while th e q u an tity t rn denotes the tim e of m axim um expansion in 72. We have*

used 7m = 1 a n d j r = 4 /3 , b u t we have not specified a (co n stan t) value for 7,,. If

Tpi Vrmi an d VPr are specified, equations (4.5) comprise a to ta l of 1 0 equations in

10 unknow ns (/?ni* flr? ftp 1 C Tll. C r , C*p, 0rm. . and V7,,,,,,).

T h e lifetim e ( A/ ) of each epoch may be determ ined once equations (4.5) are

solved. W ith 7 equal to a constant, equations (3.17) and (3.20) yield

w here x = exp [2a0], and th e subscripts i and / refer to the in ital an d final

reaches zero a t the p oint of m axim um expansion in 72, we; have; use;d the; minus

sign in equation (3.20). The tim es which are noteel w ith regard to ecpiations (4.5)

are t m = A t m -f- A t r + A t p; t r = A t r + A /p; and t p .= A t p. In ae;cordance

w ith th e general outline of th e stan d ard cosmological me>de;l, we; would e;xpe;e;t,

A t m » A t T » A t v .

(4.6)

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It is convenient to adopt the following notation:

Xtnrm = eXp [2o(m^rm]) (4.7a)

Xrrm — eXp [2GV<^rm]) (4.76)

x rpr = exp [2otrtf>pr ], (4.7 c)

x PVr = exp [2ap<^pr], (4.7d)

x 0 = exp [2ap(j>o], (4.7e)

and

0 < y = — - < 1. (4.8)

X p p f

T h e n p p er lim it in equation (4.8) arises because 0 < 7P < 2 /3 im plies a p < 0,

while <j>o > (f)pr from th e boundary conditions.

Using equations (3.10b) and (3.14b), we ob tain

1 \ /k = gi ®m = 2 > (4.9a) K = <*r = (4.96) and , _ 3 tp - 2 67P , ctp — 3 bp yj K-yp. (4.9c)

T h e qu antities F rm and Vpr now may b e specified on eith er side of th e cor

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3S p articu la r, an d r t r \ — r< f lmx m r m ) ' _ A7r'2 ^ \ V * rm )m — ^mrm ^vJ — ( V \ - C ( 1 + fir*rpr) _ _ __ 2 z 2rpr 45 3 ( 2 - 7;,)'

A fter several calculations and substitutions, equations (4.5) yield the

ing results: (1 + y ) y ~ 1^2 = 2[3(Vpr )r ]ftp 1 / 2 y (1 + % m rm ) ( l + y) 3 ( V p r ) r [ 2( VTm)in J firm — h i Tn rIn 2 a m a _ X ^~V3 r r J m r m 2 r 2v^ C r = - ( V r m ) 3 ' ( 1 + * m r fttn = f c m = (Frm) \4 mrrn / ( 1 4" ^ m r m ) (j)pr — 111 Xrpr 2 a, 2-0 — y^ppr (j) o = —— lnxo ( 4 .1 0 a ) (4.10/.) follow-(4.11a) (4.11/.) ( 4 .l i e ) (4. l i d ) (4. H r ) (4 .1 1 /) (4. llr/) (4.11/t) (4.11*) (4. H i ) (4.1 l k )

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0

* X(j and , _ ( 2 - l p \ x \ l2b* ' p - \ 2 ) 2 ' / b» Vmin ~ 64C m . (4.11/) (4.11m ) (4.11n)

Since (Vpr )r an d (Vrm)m are specifed in equations (4.10), equations (4.11) m ay

be solved if -yp is specified. T h a t is, j p is the only rem aining free p aram eter in th e

theory.

Wi t h regard to th e lifetimes of the three epochs, eq uation (4.6) yields

A t , 47T 1 / 2 l/~ V2 9 mm (•Tmrm 1 )x mrm + t a n ~ H x 1l 2 ( T + v mrm/ 4 mrm \ x) (4.12)

foi th e lifetime of the m a tte r epoch, and

A t , ' 1 ■

1 / 2 j

1 1

32tt _{P r C l ' 2} “I- $r*£rrm 1 “1“ ftr'^rpr. (4.13) for th e life; ime of the rad iatio n epoch.

In considering the lifetime of th e p re m a tte r epoch, we first n o te th a t a com

bin atio n of equations (4.10), (4.11a), and (4.11b) yields

15 2bv

(4.14) ( l + xm r m ) 2 4A1 / 2

Using A = 1.7539 x 10~ 115 (see eqn. [2.22a]) an d equ atio n (4.9c) for bp, it is

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4 0

7P = 1 /3 , we would find th a t x mrm is a complex num ber a result th a t clearly

is an u nphysical solution for x mr,» given its definition in eq uation (4.7a). F u rth er

investigation shows th a t a physically meaningful solution for x , „ i s found only

if 7p < < 1. It should be em phasized th a t this lim itation on 7,, is a consequence of

the b o u n d a ry conditions in equations (2.22) and (4.10) which, in tu rn , arose from

our ap p ea l to th e stan d ard cosmological model. Therefore, while any 7,, < 2 /3

may, in principle, provide inflation, only 0 < 7P < < 1 (p « —p in the early

universe) yields a universe whose la ter epochs resemble the sta n d a rd cosmological

model.

Consequently, when evaluating equation (4.6) for the lifetime of th e p rem a tte r

epoch, it suffices to consider only the case where 7p < < 1. We th en ob tain

k- 1/ 2

A tp ~ ---2— lnJ/ (4-13)

for th e lifetim e of th e p re m a tte r epoch.

W e have now developed the theory in sufficient detail to b e able to consider

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Chapter 5 Results and Discussion

W ith Vrm and Vpr specified by equations (4.10), th e only rem aining free p a 

ram e ter in th e entire theory is -yp. For a given 7p, th e first ta sk is to solve equations

(4.11) for th e o th e r param eters in th e theory, and th e results corresponding to sev

eral different values of j p are presented in Table 1. T he cosmological consequences

then follow from equations (2.12), (2.21), (2.23), (3.17) - (3.19), (4.12), (4.13), an d

(4.15), and these results are presented in Tables 2 - 8 .

In these tables, <f> is given in un its of th e Planck mass ( m pi), th e scale factor

(R ) is expresed in cm , th e tem p eratu re (T ) is expressed in degrees Kelvin, an d

the density is given in g m / c m 3. T h e quantities Cm , C r , Cp, an d Vmin are given in

u n its of the P lanck density, and th e present value of the H ubble p a ra m e te r (Ho)

is given in Arm sec- 1 M p c ~ l . T h e conversion factors betw een P lanck u n its an d

cgs u n its are given in A ppendix A.

Tables 2 - 7 give results for t = t p, t = t r, and t = t m as well as for six values

of t in th e range of th e estim ated present age of th e universe (15 — 20 G y r ). T he

results for t p, t r , and t m follow from the above-m entioned equations. T h e results

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4 2 (4.6) w ith A t set equal to the present age. We also set .<•, = .r a n d the o th e r

q u an tities in equation (4.6) assum e their respective values for the m a tte r epoch.

Once we solve for X f — which here corresponds to th e value o f .r at th e present age

— th e present value of 4> then follows directly, and th e above-m entioned equations

are th e n used to solve for the o th er cosmological quantities.

T h e value for H a is obtained from equation (2.10) using th e present value of

th e density as given in Tables 2 - 7 , and using Ho = 1 0 0/# kin .see- 1 M p c ~ l =

1.7470ft X 10- 6 1 . T he results for H 0 are given in Table 8. T h e lower and

u p p er lim its for Hq correspond to the u p p er and lower lim its of th e present, age

of th e universe (th a t is, 20 and 15 Gyr), respectively.

O nce the p o te n tia l is determ ined for a given 7 , , , the in itial te m p e ra tu re is

found from equation (3.19) using the fact th a t T = Tpi at t = it,,, w here T,,/ is

given by equ ation (2.23). T he initial tem peratures are given in T able 8 for each

value o f 7p.

T h e fact th a t the initial equation of sta te affects the overall dynam ical b eh av

ior of th e later universe m ay at first seem puzzling in light of the fam iliar result,

th a t inflation “erases” any “m em ory” of o th er initial conditions, such as those

in itial conditions p ertaining to th e p ertu rb a tio n sp ectru m (th e so-called cosmic

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broad range of initial conditions for th e p ertu rb a tio n sp ectru m a n d yield a seem

ingly sm ooth, homogeneous, and isotropic universe. However, th e dynam ics of the

scale factor and, consequently, the overall density a n d the H ubble p aram eter, are

driven by th e equation of state, and th e in itial value of 7 has lasting effects th a t

are unique to a closed cosmological model.

If a closed universe began in a radiation-dom inated sta te at th e P lanck den

sity, it would quickly collapse. In the model presented here, however, th e universe

begins w ith 0 < j p < < 1 a n d inflates by an am ount which is reg ulated by th e

values of j p and Tpi. Since Tpi is fixed in term s of th e fund am en tal co nstants, th e

value of 7p determ ines the subsequent behavior of th e universe.

Q ualitatively, th e early universe in our theory m ust inflate (i? > 0) from

its in itial size (on the order of a P lanck length) to a large enough size (R is an

absolute m easure in a k = + 1 model) in order th a t once th e phase tran sitio n to

rad ia tio n occurs (an d R becomes negative), b o th R an d R will b e large enough

( H = R / R « (8 7t /3) ' / 2) to insure th a t the universe will n ot quickly co ntract, b u t

will instead continue to expand for a t least 15 — 20 Gyr. A con tractio n ph ase is

a characteristic of a closed cosmological model th a t is not shared by open o r flat

models. T h a t is, for th e closed, non-singular m odel th a t we describe in our theory,

the ex ten t of inflation determ ines how long th e universe will continue to ex pan d

A cosmological field theory