Early Universe Evolution in Graviton-Dilaton Models

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MRI-PHY/96-35

hep-th/9611223

Early Universe Evolution in Graviton-Dilaton Mo dels

S. Kalyana Rama

Mehta Research Institute, 10 Kasturba Gandhi Marg,

Allahabad 211 002, India.

email: [email protected]

ABSTRACT. We present a class of graviton-dilaton mo dels which

leads to a singularity free evolution of the universe. We study

the evolution of a homogeneous isotropic universe. We follow an

approach which enables us to analyse the evolution and obtain its

generic features even in the absence of explicit solutions, which

are not p ossible in general. We describ e the generic evolution of

the universe and show, in particular, that it is singularity free in

the present class of mo dels. Such mo dels may stand on their own

as interesting mo dels for singularity free cosmology, and may be

studied accordingly. They may also arise from string theory. We

discuss critically a few such p ossibilities. 1

1. Intro duction

General Relativity (GR) is a b eautiful theory and, more imp ortantly,

has b een consistently successful in describing the observed universe. But,

GR leads inevitably to singularities. In particular, our universe according

to GR starts from a big bang singularity. We do not, however, understand

the physics near the singularity nor know its resolution. The inevitability

and the lack of understanding of the singularities p oint to a lacuna in our

fundamental understanding of gravity itself. By the same token, a successful

resolution of the singularities is likely to provide deep insights into gravity.

In order to resolve the singularities, it is necessary to go beyond GR,

p erhaps to a quantum theory of gravity. A leading candidate for such a

theory is string theory. However, despite enormous progress [1], string theory

has not yet resolved the singularities [2].

Perhaps, one may have to await further progress in string theory, or a

quantum theory of gravity, to b e able to resolve singularities. However, there

is still an avenue left op en - resolve the singularities, if p ossible, within the

context of a generalised Brans-Dicke mo del, and then inquire whether such

a mo del can arise from string theory, or a quantum theory of gravity.

Generalised Brans-Dicke mo del is sp ecial for more than one reason. It

app ears naturally in sup ergravity, Kaluza-Klein theories, and in all the known

e ective string actions. It is p erhaps the most natural extension of GR [3],

whichmay explain its ubiquitous app earance in fundamental theories: Dicke

had discovered long ago the lack of observational evidence for the principle

of strong equivalence in GR. Together with Brans, he then constructed the

Brans-Dicke theory [4] which ob eys all the principles of GR except that of

strong equivalence. This theory, generalised in [5], has b een applied in many

cosmological and astrophysical contexts [6]-[10].

Brans-Dicke theory contains a graviton, a scalar (dilaton) coupled non

minimally to the graviton, and a constant parameter ! . GR is obtained

when ! = 1. In the generalised Brans-Dicke theory, referred to as graviton-

dilaton theory, ! is an arbitrary function of the dilaton [3, 5]. Hence, it

includes an in nite number of mo dels, one for every function ! . Thus, very

likely, it also includes all e ective actions that may arise from string theory

or a quantum theory of gravity.

This p ersp ective logically suggests an avenue in resolving the singularities:

to nd, if p ossible, the class of graviton-dilaton mo dels in which the evolution 2

of universe is singularity free, and then study whether such mo dels can arise

from string theory or a quantum theory of gravity. On the other hand, even if

not deriveable from string theory or a quantum theory of gravity, such mo dels

may stand on their own as interesting mo dels for singularity free cosmology.

In either case, one may study further implications of these mo dels in other

cosmological and astrophysical contexts. Such studies are fruitful and are

likely to lead to novel phenomena, providing valuable insights.

Accordingly, in this pap er, we consider a class of graviton-dilaton mo dels

where the function ! satis es certain constraints. These constraints were

originally derived in [11] in a di erent context, and some of their generic

cosmological and astrophysical consequences were explored in [11, 12]. Here

we analyse the evolution of universe in these mo dels which, as we will show,

turns out to b e singularity free. The essential p oints of the analysis and the

results have b een given in a previous letter [13].

We consider a homogeneous isotropic universe, such as our observed one,

containing matter. The relevent equations of motion cannot, in general, be

solved explicitly except in sp ecial cases [10, 12]. Hence, a di erent approach

is needed for the general case which is valid for any matter and for any

arbitrary function ! , and which enables one to analyse the evolution and

obtain its generic features even in the absence of explicit solutions.

We will present such an approach b elow. We rst present a general anal-

ysis of the evolution and then apply it to describ e in detail the evolution of

universe in the present mo del. We show, in particular, that the constraints

on ! ensure that the evolution is singularity free. The universe evolves with

no big bang or any other singularity and the time continues inde nitely into

the past and the future.

An imp ortant question to ask, from our p ersp ective, is whether a function

! as required in the present mo del, can arise from a fundamental theory. We

discuss critically a few such p ossibilities in string theory. On the other hand,

the present mo del may stand on its own as an interesting mo del for singularity

free cosmology and may b e studied accordingly. We mention some issues for

future study.

This pap er is organised as follows. In section 2, we present our mo del. In

section 3, we write its equations of motion in various forms, so as to facilitate

our analysis. In section 4, we present the general analysis of the evolution.

In section 5, we illustrate our metho d by applying the results of section 4

to describ e the evolution of toy universes and show that their evolution is 3

singularity free in the present mo del. In section 6, we describ e the evolution

of our observed universe and show that its evolution is singularity free in the

present mo del. In section 7, we discuss further generalisations of our mo del.

In section 8, we give a brief summary, discuss critically a few p ossibilities

of our mo del arising from string theory, and mention some issues for future

study. In the App endix, we derive the suciency conditions for the absence

of singularities which are used in the pap er.

2. Graviton-Dilaton Mo del

We consider the following graviton-dilaton action in `Einstein frame':

1 1

4 2 ()

S = d x g R + (r) + S (M;e g ) ; (1)

16G 2

where G is the Newton's constant, is the dilaton and () is an arbitrary

function that charcterises the theory. S is the action for matter elds,

denoted collectively by M. They couple to the metric g minimally and to

the dilaton through the function (). This function cannot b e gotten rid

of by a rede nition of g except when the matter action S is conformally

()

invariant - in that case, by de nition, S (M;e g ) = S (M; g ) -

M M

which is assumed to b e not the case here.

We can de ne a metric

g = e g ; (2)

and write the action (1) in `Dicke frame':

1 1

2 2 4

S = R + + S (M;g ) ; (3) ge 3 1 (r) d x

16G 2

where . If has a nite upp er b ound < 1, then the

max

factor e can be absorb ed into G and the range of can be set to be

max

0. Also, we will work in the units where G = .

Both forms of the action in (1) and (3) are, however, equivalent [14].

In `Einstein frame', the matter elds feel, b esides the gravitational force,

the dilatonic force also which must be taken into account in obtaining the

@ g

In our notation, the signature of the metric is ( + ++) and R = + .

2 @x @x 4

physical quantities. Whereas, in `Dicke frame', the matter elds feel only the

gravitational force and, hence, the physical quantities are directly obtained

from the metric g . On the other hand, equations of motion are often easier

to solve if g is used.

De ning a new dilaton eld = e , the action (3) can be written as

1 ! ( )

4 2

d x S = g R + (r ) + S (M; g ) ; (4)

16G

where ! ( )isnow the arbitrary function that characterises the theory. This

is the form of the action used more commonly and, hence, we will also use

it in this pap er in most of what follows. Note that 0 since = e .

Furthermore, if has a nite upp er b ound < 1, then the factor

max

can b e absorb ed into G and the range of can b e set to b e 0 1.

max N

max

Also, we will work in the units where G = .

Let 2! +3. Then, and () are related to and ( ) as follows:

= or; equivalently ; = ( ) : (5)

Note that must be p ositive. If one starts from and (), then the rst

relation gives in terms of . To obtain in terms of ; () must be

inverted rst to obtain in terms of and then = ln must be used.

If one starts from and ( ) and uses = e , then the second relation

gives in terms of whichmust then b e inverted to obtain in terms of .

However, in the following, no inversion is necessary since no explicit forms

for ( ) and () will be used.

In the mo del we study here, the function () is required to satisfy the

following constraints:

jj 1

lim = ;

jj!1

= nite 8 n 1; 1 1

= 0 at =0 only

lim () = k ; n 1 ; (6)

!0 5

where > 0 and k > 0 are constants and n is an integer. The function

()is otherwise arbitrary.

The rst two constraints are imp ortant and were originally obtained in a

di erent context in [11, 12], where some of their generic consequences were

also explored. These constraints imply a nite upp er b ound on , i.e. ()

max

< 1. Hence, the factor e can be absorb ed into G and the range

max N

of can be set to be 0. Thus there is atleast one critical p oint where

= 0, and =0.

The third constraint is not imp ortant, but assumed here for the sake of

simplicity only. It implies that there is only one critical p oint, a maximum,

of (). As explained ab ove, we can take =0. Thus, the range of is

max

given by 1 0 and, hence, that of = e by 0 1. Also, by

adding a suitable constant to ,we can take the maximum of () to o ccur

at = 0 with no loss of generality.

The fourth constraint follows from the Taylor expansion of () near its

maximum at =0. Note, as follows from (5), that !1 as ! 0.

The constraints in (6), expressed equivalently in terms of and ( ),

b ecome

(0) =

= nite 8 n 1; 0 <1

! 1 at =1 only

lim = (1 ) ; <1; (7)

where > 0 and > 0 are constants, and the range of follows naturally

0 1

from (5) and (6). We further assume that ( ) is a strictly increasing func-

tion of . This constraint is not imp ortant, but assumed here for the sakeof

simplicity only. The function ( ) is otherwise arbitrary.

1 d 2

4 1

Note, for later reference, that !1 and = (1 ) in the

limit ! 1. Hence, for the range of given in (7), which follows naturally

1 d

from (6), we have that ! 0 in the limit ! 1. This ensures, as

will b ecome clear later, that the present mo del satis es the observational

constraints imp osed by solar system exp eriments.

A vast class of functions ( ), equivalently (), exists satisfying the

ab ove constraints. A simple example is = + ((1 ) 1). However,

0 1 6

no explicit form for ( ) will be used in the following.

3. Equations of Motion

In the following, we take the \matter" to be a p erfect uid with density

and pressure p, related by p = where 1 1. Note that the

value of indicates the nature of \matter". For example, = 0; , or 1

indicates that the \matter" is dust, radiation, or massless scalar eld resp ec-

tively. Hence, we enclose \matter" within inverted commas where necessary

in order to remind the reader that \matter" do es not mean dust only. In the

following, we study the evolution of a at homogeneous isotropic universe.

(The following analysis can, however, be extended to curved universes also

and the main results remain unchanged.) The line element is given by

2 2 2A(t) 2 2 2

ds = dt + e (dr + r dS ) ; (8)

where e is the scale factor and dS is the line element on an unit sphere,

and the elds dep end on the time co ordinate t only. The equation of motion

for A, following from (4), is given by

A _ ! _

A + = ; (9)

where upp er dots denote t-derivatives. Solving equation (9) for A,we obtain

_ _

A = + + ; (10)

2 6 12

where = 1 and we have used = 2!+3. The square ro ots in (10) and

in the following are to be taken with a p ositive sign always. The remaining

equations of motion, following from (4), are given by

_ (1 3 )

+3A _ + = (11)

2 2

3(1+ )A

= e ; (12)

where we have used p = and 0 is a constant. Equations of motion

following from (4) also contain the equation for A :

! _ p 2A _

+ + + =0 : (13) A 2A+3A +

2 2 7

In general, this equation follows from (9), (11), and (12) also. But there is

a subtlety p ointed out in [15]. What one obtains from (9), (11), and (12) is

the following:

A + A =0; (14)

where the expression for A is de ned in (13). Equation (13) then follows if

6= 0, i.e. if and only if e is not prop ortional to and only if A + .

This subtlety should be kept in mind. However, as checked explicitly, the

solutions presented in this pap er all satisfy equations (9) - (13).

Equations (10) and (11) can also b e written in di erent forms. Equation

(11) can be integrated once to obtain

_ (t) = ( (t)+c) (15)

3 A

(1 3 ) e

where (t) = (16) dt

and c = _ e j is a constant. An equation relating A and can now

t=t

be derived. Divide equation (10) by and substitute (15) for _ . We then

have

3(1 )A

dA 2 e

2 K ; K 1+ : (17) = 1+ sign ( _ )

d 3 ( (t)+c)

All the ab ove equations can be equivalently written in terms of and

() also. For example, equations (10) and (11) b ecome

_ _

A + = + (18)

2 6 12

(1 3 )

_ _ _

+3A+ = e ; (19)

Although the derivation of (14) is straightforward, an outline of the steps involved

is p erhaps helpful. In equation (11), restore the p-dep endence by replacing (1 3 ) by

( 3p). Now, di erentiate (9) and use _ = 3A( + p). Then substitute (9) for , and

(11) for !_ , equivalently . Again substitute (9) for . The result is equation (14) [15].

6 8

while equation (15) and (16) b ecome

(t) = e ( (t)+c) (20)

(1 3 )

3 A

dt e (21) where (t) =

3A+

and c = e j is a constant.

t=t

Wenow make a few remarks. Firstly, under time reversal t !t; !

in equations (10) and (18), whereas equations (11) and (19) are unchanged.

Thus, evolution for is same as that for , but with the direction of time

reversed.

Secondly, in general, there will app ear in the solutions p ositive non zero

A A

arbitrary constant factors e ; , and in frontofe ; , and resp ectively.

0 0

However, it follows from (10) and (11) that they can all b e set equal to 1 with

3(1+ )A

no loss of generality if time t is measured in units of . Therefore,

in the solutions b elow we often assume that t is measured in appropriate

units and, hence, set these constants equal to 1. Note, however, that the

mo del dep endent constants asso ciated with ( ), such as or in (7),

0 1

cannot b e set equal to 1.

Thirdly, if = 0 or = then (t) = 0 in (15). If it is also p ossible

to p erform the integrations given b elow then solutions to equations (10) and

(15) can, in principle, be obtained in a closed form. To see this, de ne two

new variables t and as follows:

p p

A A

dt dt ; e e : (22)

Note that these variables app ear naturally in `Einstein frame' given in (1)

and (2). After some algebra, equations (10) and (15) b ecome

dA 2 e ce

1+ (23) =

dt c

1 d ce

= ; (24)

from which it follows that

dA 2 e

= 1+ : (25)

d 12 c 9

In the ab ove equations = 1as in (10) and c is a constant as in (15).

It turns out that A-integrations in equations (23) and (25) can be p er-

formed explicitly. Omitting the details, which are straightforward, the result

is as follows. When and, hence, =0:

c 3

dt (26) e =

A = : (27)

When 6= 0, de ne y e . Then

q q

c 2

2 2

1+y ln ( 1+y +y) = dt (28) y

1+y +1) = : (29) lny ln(

One thus has an explicit relation between A and t , and also between A

and if one can obtain in a closed form. Thus, one has (t). If

one can obtain also in a closed form then one has an explicit relation

between t and t. Therefore, in principle, one has A(t) and (t) in a closed

form.

Evidently, solutions can b e obtained thus in a closed form only in sp ecial

cases. Even then the details of the required integrations and inversions of

functions are likely to obscure the general features of the solutions. Such

explicit solutions have b een studied in [10] for a few sp eci c functions ( ),

and in [12] for any arbitrary function ( ) but with = 0. In the present

pap er, however, we follow a di erent approach which is valid for any matter

and for any arbitrary function ( ).

4. Analysis of the Evolution

Our main goal in this pap er is to determine whether the evolution of the

universe in the present mo del is singular or not. The task is trivial if one

can solve the equations of motion (10) - (12) for arbitrary functions ( ) (or

()). However, explicit solutions can be obtained only in sp ecial cases as 10

describ ed ab ove. Hence, a di erent approach is needed for the general case

whichisvalid for any matter and for any arbitrary function ( ), and which

enables one to analyse the evolution and obtain its generic features even in

the absence of explicit solutions.

We will present such an approach b elow. We analyse the evolution and

obtain its generic features for the general case where matter and ( ) (or

()) are arbitrary. We will show, in particular, that when ( ) (or ())

satis es the constraints in (7) (or (6)), the evolution of the universe, such as

our observed one, is singularity free at all times.

To prove the presence of a singularity it suces to show that any one of

the curvature invariants, usually the curvature scalar, diverges. However, to

prove the absence of a singularity, it is necessary to show that al l curvature

invariants are nite. As proved in the App endix, a sucient condition for

al l curvature invariants to b e nite is that the quantities in (53) b e all nite.

Some or all of these quantities have a p otential divergence when, for example,

e and/or vanish or when ! 1. Hence, in the following analysis, we

pay particular attention to these quantities, determining their niteness or

otherwise. We p erform our analysis, as far as p ossible, in terms of and

( ). In the limit as ! 1, however, it is sometimes necessary to p erform

the analysis in terms of and (), as will b ecome clear b elow.

To pro ceed, we need initial conditions at an initial time, say t = t . That

is, we need A(t ); _ (t ), and (t ) or equivalently (t ). Here and in the

i i i i

following (t) means ( (t)), i.e. the function ( ) evaluated at time t.

Note that if _ (t ) = 0 and (t ) = 1, then (t ) = 1 and the evolution

i i i

is same as in Einstein's theory. In particular, there is a singularity at a

nite time in the past. Therefore, we assume that the initial conditions in

the following refer always to generic values of A(t ), _ (t ), and (t ) only,

i i i

i.e. A(t ) and _ (t ) are nite and non in nitesimal and (t ) < 1 in the

i i i

following.

The analysis will be carried out for any arbitrary function ( ), satis-

fying only the constraints in (7). Hence, only the generic features of the

evolution can b e obtained but not the numerical details which require cho os-

ing a sp eci c function for ( ). Therefore, precise numerical values for A (t )

and _ (t ) are not needed. Their signs alone are sucient. We also need to

3, and also the value of in (10). Thus, there know whether or not (t )

are 16 sets of p ossible initial conditions: 2 each for the signs of A (t ); _ (t ),

i i

3 or not. and , and 2 for whether (t )

i 11

However, it is not necessary to analyse all of these 16 sets. The evolution

for is same as that for , but with the direction of time reversed. Hence,

only 8 sets need to be analysed. Now, let = 1. If _ < 0 then A can not

be negative for any value of , see (10). Similarly, if >_ 0 and > 3 then

_ _ _

also A cannot b e negative since + > in equation (10).

6 12 3 2 2

Thus, there remain only 5 sets of p ossible initial conditions. Wecho ose these

sets so as to b e of direct relevence to the evolution of observed universe and

analyse each of them for t > t . In the course of the analysis, equivalent

forms of the equations of motion given in section 3 will b ecome useful. We

then use these results in sections 5 and 6 to describ e the generic evolution of

the universe.

Two remarks are now in order. First, the amount and the nature of

the dominant \matter" in the universe are given by and where 1

A A

1. The later varies as the scale factor e of the universe evolves. If e is

increasing, i.e. if A> 0, then the universe is expanding, eventually b ecoming

dominated by \matter" with < , when such \matter" is present. For the

observed universe, whichisknown to contain dust for which = 0, one may

take 0. Thus, by cho osing t suitably we may assume, with no loss of

generality, that < ( 0 for observed universe) if A > 0. Similarly, if

A<0 then wemay assume, with no loss of generality, that when such

\matter" is present. This is valid for the observed universe which is known

to contain radiation for which = .

Second, consider (t) in (15) or (20). The integrand is p ositive and / .

(The dep endence on can b e absorb ed into the unit of time, as explained in

section 3.) Hence, the value of the integral is controlled by . For example,

in the limit ! 1, it is controlled by the constant in (7). We will use this

fact in the following. Also, since t > t , the sign of (t) is same as that of

(1 3 ). Moreover, (t)= 0 if =0 or = corresp onding to an universe

containing, resp ectively, no matter or radiation only.

4 a. A(t ) > 0; _ (t ) > 0, and (t ) > 3

i i i

These conditions imply that = +1 in (10) and the constant c > 0 in

(15). For t>t, the scale factor e increases and, eventually, can b e taken

when such \matter" is present. In fact, 0 for the ob eserved to be <

universe. Then, (1 3 ) > 0 and, hence, (t) increases. Thus, for t > t ,

_ > 0 and and b oth increase. Since > 3, it follows from equation 12

(10) that A>0 for t >t and, hence, e increases. Eventually, ! 1 and

! (1 ) .

In the limit as ! 1; ( (t)+c) > 0 is a constant to an excellent

approximation. Then, equation (17) can be solved relating and e :

2(1 )

3(1 )

e = (constant) (1 ) : (30)

Substituting this result in equation (15) then yields the unique solution in

the limit ! 1:

A A

3(1+ ) (1+ )(1 )

e = e t ; =1 t ; (31)

where A and > 0 are constants and t is measured in appropriate units.

0 0

Since ! 1, it follows from (31) that t ! 1. It can now be seen that

( (t)+ c) > 0 is indeed a constant to an excellent approximation. Also, the

quantities in (53) are all nite, implying that all the curvature invariants are

nite. Thus, there is no singularity as ! 1 and, thus, for t>t .

Note that the ab ove relation between and e and, hence, the solution

(31) is not valid if = 0 or = 1 i.e. if the universe contains no matter

or \matter" with = 1 only. Although these cases are not relevent for the

observed universe, we will now consider them for the sake of completeness.

The results, however, are useful in describing the evolution for toy universes

in section 5.

= 0 implies that = 0. In this case, (t) = 0 identically and the

analysis was rst given in [12], alb eit di erently. The dynamics turns out to

be clear in terms of and (). These variables, although equivalent to

and ( ), are particularly well suited in describing the evolution near =0,

or equivalently = 1, where !1. In fact, in this limit, it is not p ossible

sometimes to p erform the analysis in terms of and ( ).

At t , we have A>0; >0 and, with our normalisation, <0; <0

and > 0. Also, () has a maximum =0 at =0. The equations

max

of motion (18) and (19) are now given by

_ _

A =( 3 + sign()) ; = ce ; (32)

2 3 13

where c > 0. For t > t , the scale factor e and continue to increase.

Eventually, ! 0. In this limit, ! 0 and ! 0. The solution to (32) is

then given by

A A

e = e ; = c lnt; (33) t

where t is measured in appropriate units. From these expressions, it is clear

that the eld crosses the value 0 at t = 1 (in appropriate units) and

b ecomes p ositive, while the scale factor e remains nite and increasing. For

>0; () decreases and < 0. Equivalently, and ( ) b oth decrease.

Thus, we now have A>0; <_ 0, and =1. Further evolution is analysed

in section 4b.

Note the crossover at =0 where crosses 0 and b ecomes p ositive. In

terms of and ( ), this amounts to the following: reaches 1, corresp ond-

ingly reaches 1, and then \re ects" back and decreases, corresp ondingly

decreases. This crossover/re ection takes place in a nite time and with

e remaining nite during the pro cess. It is this phenomenon of crossover

that can be seen only in terms of and (), and not and ( ).

In this case, the universe contains \matter" with = 1 only. Hence, =

e and (t) is negative. However, as ! 1; (t) / , see equation

(7), and remains nite as can be veri ed a p osteriori. Also, the magnitude

of (t) can b e made as small as necessary by cho osing the mo del dep endent

constant suciently large. Hence, as ! 1, the factor ( (t)+c) can be

taken to remain p ositive and constant.

2 6A A

Now, _ = (constant) e and its dep endence on e is same as that of

. Therefore, in the limit ! 1 or equivalently !1, the solution to (18)

and (20) and, hence, the evolution turn out to b e identical to the = 0 case.

In particular, the dynamics is clear in terms of and () only. The eld

crosses the value 0 (at t = 1 in appropriate units) and b ecomes p ositive,

while the scale factor e remains nite and increasing. This crossover cannot

be seen in terms of and ( ).

For > 0; () decreases and < 0. Equivalently, and ( ) b oth

decrease. Thus, we now have A>0; <_ 0, and =1. Further evolution is

analysed in section 4b.

4 b. A(t ) > 0; _ (t ) < 0; (t ) > 3, and =1

i i i 14

The constant c < 0 in (15). For t > t ; and decrease and the scale

factor e increases. Equation (17) now b ecomes

2 = 1 K ;

where K (t) is given in (17).

Consider rst the case =0 or = 1, discussed in section 4a and which

led to the present phase. Then, (1 3 ) 0 and, therefore, (t) 0.

Thus, ( (t)+c) c< 0 and, hence, _ (t) is negative and non zero, implying

that (t) decreases for t >t. Also, K (t) remains nite and it follows from

equation (17) that < 0. Thus, A>0 since <_ 0 and, hence, e increases

for t>t .

3(1 )A

2 e

Eventually, as t increases, ! 0 and e !1. Also, 1 in

( (t)+c)

(17) since =0 or =1. Equation (17) can then be solved relating and

e :

3+ 3

e = (constant) ; (34)

where ! 0 and is given in (7). Substituting this result in equation (15)

then yields the unique solution in the limit ! 0:

n m A A

(t t ) ; = (t t ) ; (35) e = e

0 0 0

where A ; , and t >t are some constants and

0 0 0 i

3+ 3 2

p p

n = ; m = : (36)

3(1 + 3 ) 1+ 3

0 0

Note that m<0. Then, since ! 0, it follows from (35) that t !1.

Thus, as ! 0, t ! 1 and e ! 1. It can also be seen that the

quantities in (53) are all nite for t t 1, implying that all the curvature

invariants are nite. Thus, there is no singularity for t t 1.

Consider now the case where and are non zero and have generic values.

We have A(t ) > 0 and _ (t ) < 0. Hence, e increases and decreases. As

i i

e increases, can eventually be taken to be < , when such \matter" is

present. In fact, 0 for the ob eserved universe. In such cases where 0,

(t) grows faster than t as can be seen from (16). Then, as t increases, the

factor ( (t)+c) b ecomes p ositive. 15

This implies that _ , initially negative, passes through zero and b ecomes

p ositive. We then have A > 0 and _ > 0. As t increases further, and,

hence, ( ) increase. If > 3 then further evolution pro ceeds as describ ed in

section 4a. If < 3 then, up on making a time reversal, the initial conditions

b ecome identical to the one describ ed in section 4e. The evolution in the

present case then pro ceeds as in section 4d for the case which leads to the

initial conditions of section 4e.

4 c. A(t ) < 0; _ (t ) < 0, and (t ) > 3

i i i

These conditions imply that = 1 in (10) and the constant c < 0 in

(15). For t> t , the scale factor e decreases and, eventually, can b e taken

to be when such \matter" is present. In fact, this is the case for the

observed universe. Then (1 3 ) 0 and (t) decreases or remains constant.

Therefore, the factor ( (t)+c) c<0 and, since e decreases, wehave that

_ (t) < _ (t ) < 0 for t> t. Thus and, hence, decrease for t> t. Also,

i i i

> 0 since <_ 0 and A<0.

In (17), K (t ) > 1 since > 3. From the b ehaviour of e ; ; ( ), and

( (t)+c) describ ed ab ove, it follows that K decreases monotonically. The

lowest value of K is ,achievable when vanishes. This, together with

3 9

K (t ) > 1 and the monotonic b ehaviour of K (t), then implies that there

exists a time, say t = t > t where K (t ) = 1 with (t ) > 0. Hence,

m i m m

(t )=0. Therefore, A(t )=0 since _ (t ) is non zero. This is a critical

m m m

p oint of e and is a minimum. Also, equation (17) can be written as

(t)

(1+ K) = nite ; (37) A(t ) A(t )=

i m

(t )

where the last equality follows b ecause b oth the integrand and the interval

of integration are nite. Therefore, A(t ) is nite and > 1 and, hence,

A(t ) 3

e is nite and non vanishing.

continues to decrease and reaches a Thus, for t > t , the scale factor e

non zero minimum at t = t . The precise values of t ; A(t ); (t ), and

m m m m

(t ) are mo del dep endent. Hence, nothing further can b e said ab out them

3 A

The existence of such a non vanishing minimum of e has also b een deduced in the

variable mass theories of [6]. However, singularities may subsequently arise in these theo-

ries. See section 7 also. 16

except, as follows from K (t )=1, that (t ) 3 in general and =3 when

m m

=0.

However, the ab ove information suces for our purp oses. It can now be

seen that the quantities in (53) are all nite, implying that all the curvature

invariants are nite. Thus, there is no singularity for t t t .

i m

For t>t , one has A(t) > 0; _ (t) < 0, and ( (t )) 3 by continuity.

m m

Further evolution is analysed b elow.

4 d. A (t ) > 0; _ (t ) < 0; (t ) 3, and = 1

i i i

This implies that the constant c<0 in (15) and that (t ) < 0 in (17).

Hence, K (t ) < 1. For t>t , the scale factor e increases and, eventually,

i i

can b e taken to b e < when such \matter" is present. In fact, 0 for the

observed universe. Then, (1 3 ) > 0 and, hence, (t) increases. Thus, e

is increases, decreases, and ( (t)+c) increases. Now, dep ending on , ,

and the details of ( ), K (t) in (17) may or may not remain < 1 for t>t.

Consider rst the case where K (t) remains < 1 for t>t . It then follows

that ( (t)+ c) cannot have a zero for t > t and, since c < 0, must remain

negative and non in nitesimal. Note that must be > 0 for otherwise (t)

grows as fast as or faster than t, making ( (t)+c)vanish at some nite time.

Let

lim ( (t)+c)=c (38)

t!1

where c is a negative, non in nitesimal constant. We then have for t t

e i

1; c ( (t)+c) c < 0 and, hence, _ (t) < 0. Therefore, eventually

(t) ! 0. Since K (t) < 1 and _ (t) < 0, it also follows from (17) that < 0

and, hence, A(t) > 0. We will now consider the limit ! 0.

K (t) < 1 implies that as ! 0, K ! < 1 where

3(1 )A

(39) lim e 1+

e 0

is a constant and is given in (7). Now, as ! 0, equation (17) can be

solved relating and e :

3 3

e = (constant) : (40) 17

Note that (3 3 ) > 0 since < 3. Hence, e ! 1 as ! 0.

e e

Substituting (40) in equation (15) then yields the unique solution in the

limit ! 0: For 6= ,

n m

A A

e = e (t sign(m)t) ; = (t sign (m)t) ; (41)

0 0 0

where t is measured in appropriate units, A ; , and t > t are some

0 0 0 i

constants, and

3 2 3

p p

: (42) ; m = n =

3(1 3 ) 1 3

e e

For = ,

c (tt )

A A c (tt )

0 0

e = e e ; = e ; (43)

where c is de ned in (38).

Let > . Hence, m>0. Also, n<0 b ecause < 3. Since ! 0, it

e e

follows from (41) that t ! t >t, which implies that vanishes at a nite

0 i

time t . In this limit, the scale factor e !1 since n<0. The quantities

in (53), for example , also diverge, implying that the curvature invariants,

including the Ricci scalar, diverge. Thus, for > , there is a singularity

at a nite time t .

1 1

Let , which is the case in our mo del, see (7). When < , m<0.

e e

3 3

Also, n>0 b ecause < 3. Since ! 0, it follows from (41) that t !1.

In this limit, the scale factor e ! 1 since n > 0 now. The same result

holds also for the solution in (43) with = .

For K (t) to be < 1, ( (t)+c) must not be to o small. Since c < 0

and (t) > 0, it necessarily implies that (t) must remain nite. It follows

from the ab ove solutions that (t) can remain nite only if 3 > ,

3 1

. Under this condition, it can b e checked easily that equivalently, >

3 3

3(1 )A

lim e =0. Hence, = as follows from (39).

!0 e 0

It can now be seen, for = , that the quantities in (53) are all

e 0

nite for t t 1, implying that all the curvature invariants are nite.

. Thus, there is no singularity for t t 1 when

i 0

Consider the second case where K (t) in (17) may not remain < 1 for all

t > t . This is the case for the observed universe where 0 and, hence,

(t) grows as fast as or faster than t. Then there is a time, say t = t , when

1 18

the value of K =1. It follows that c< ((t )+c) < 0 and, hence, _ (t) < 0

for t t t . Also, (t ) is non vanishing since the case of (t ) ! 0 is

i 1 1 1

same as the case describ ed ab ove. Therefore, wehave that (t ) = 0 which

implies, since _ (t ) 6=0, that A(t )=0. This is a critical p oint of e and is

1 1

a maximum.

For t > t ;K(t) b ecomes > 1 and, hence, A (t) < 0. Also, _ < 0 and

(t) < 3. Then = 1 necessarily. Further evolution is analysed b elow.

4 e. A (t ) < 0; _ (t ) < 0, and (t ) < 3

i i i

The ab ove conditions imply that = 1, the constant c<0 in (15), and

(t ) > 0. Hence, K (t ) > 1. For t > t , ( (t)+c) which is negative at t

i i i i

may or may not vanish for non zero .

Consider the case where ( (t)+c) do es not vanish and remains negative

for non zero . Therefore, _ (t) < 0 and, hence, (t) decreases. For t > t ,

the scale factor e decreases and, eventually, can b e taken to b e when

such \matter" is present. In fact, this is the case for the observed universe.

Then, (1 3 ) < 0 and (t) decreases. Hence, ( (t)+c) also decreases.

It follows from (17) that K , initially > 1, is now decreasing since and

A 2

e are decreasing and ( (t)+c) is increasing. Its lowest value = ,

3 9

achieveable at = 0. Therefore, there exists a time, say t = t , where

K = 1 and, hence, = 0. Clearly, (t ) and _ (t ) are non zero for the

M M

same reasons as in section 4c, implying that A(t ) = 0. This is a critical

p oint of e and is a minimum.

The existence of this critical p ointofe can b e seen in another way also.

As ( (t)+ c) < 0 grows in magnitude, it follows from equation (15) that

2 6A

_ / e , whereas is given by (12). Note that _ (t) < 0 and, hence, (t)

decreases and eventually ! 0. Then, in the limit ! 0, the term in

equation (10) dominates term. Hence, in this limit where ! < 3,

_ _

wehave that A>0. Since A<0 initially, this implies the existence of a zero

of A. This is a critical p oint of e and is a minimum.

Note that all quantities remain nite for t t t . In particular, the

i M

quantities in (53) are all nite, implying that all the curvature invariants are

nite. Thus, there is no singularity for t t t .

i M

For t > t , we have A > 0; _ < 0; < 3, and = 1. Further

evolution then pro ceeds as describ ed in section 4d. The evolution can thus

b ecome rep etetive and oscillatory. 19

Consider now the case where ( (t)+c) vanishes, say at t = t , for

non zero . Hence, _ (t ) vanishes and this is a minimum of . For t >

t ; ( (t)+c) and _ are p ositiveby continuity and, hence, increases. Also,

A < 0 and the scale factor e decreases. Eventually, can be taken to be

when such \matter" is present. In fact, this is the case for the observed

universe. Then, (1 3 ) < 0 and (t) decreases. Hence, ( (t)+ c) also

decreases.

Now, as increases and ! 1, ( (t)+c) may or may not vanish with

<1. If ( (t)+c) vanishes with <1, then _ vanishes. This is a critical

p oint of and is a maximum, beyond which we have A < 0; _ < 0, and

= 1. Further evolution then pro ceeds as describ ed in section 4c if > 3,

and as in section 4e if < 3. The evolution can thus b ecome rep etetive and

oscillatory.

If ( (t)+c) do es not vanish and remain p ositive for 1 then, eventually,

2 6A

! 1. It follows from equation (15) that _ / e , whereas is given

by (12). Note that A<0 and e is decreasing. Then, in this limit, the

term in equation (10) dominates or, if = 1, is of the same order as the

term.

Thus we have A<0; >_ 0; !1, equivalently !1, and = 1.

Also, term in equation (10) dominates or, if =1, is of the same order

as the term. But this is precisely the time reversed version of the evolution

analysed in section 4b and in section 4a for the case which led to the initial

conditions of section 4a, where the dynamics is clear in terms of and ().

Applying the results of sections 4a and 4b, it follows that will cross the

value 0 after which e and, hence, b egins to decrease. We then have A< 0,

<_ 0, and > 3. Also, = 1 necessarily. Further evolution then pro ceeds

as describ ed in section 4c. The evolution can thus b ecome rep etetive and

oscillatory.

5. Evolution of toy Universes

Wenow use the results of the analysis in section 4 to describ e the generic

evolution of universe in the present mo del. For the purp ose of illustration we

rst describ e in this section the generic evolution of three toy universes: One,

where the universe contains no matter, i.e. =0. Two, where the universe

contains radiation only, i.e. 6= 0 but = . And three, where the universe

contains massless scalar eld only, i.e. 6= 0 but = 1. Clearly, however, 20

none of these toy mo dels are realistic. The generic evolution of a realistic

universe, such as our observed one, will be describ ed in the next section.

Note that (t)=0 in the rst two mo dels. As describ ed in section 3, it

may then b e p ossible to obtain closed form solutions, although only in sp ecial

cases [10, 12]. Moreover, the details of the sp eci c calculations obscure the

general features of the solutions. Hence, and also for the purp ose of illustra-

tion, we describ e the evolution of the toy universes within the framework of

our analysis. However, the explicit evolutions describ ed in [10, 12] for sp eci c

cases all conform to the ones describ ed by the present general analysis.

We rst describ e the evolution for t > t taking, as initial conditions

A(t ) > 0; _ (t ) > 0, and (t ) > 3. Then, = 1 necessarily. To describ e

i i i

the evolution for t

conditions, A(t ) < 0; _ (t ) < 0, and (t ) > 3. Then, = 1 necessarily.

i i i

The required evolution is that for t> t in terms of the reversed time variable,

which is also denoted as t. The results of section 4 can then be applied

directly.

When the initial conditions are di erent, the evolution can again b e anal-

ysed along similar lines as follows. However, the main result that the evolu-

tion is singularity free remains unchanged.

5 a. =0; t>t

The evolution for this case was rst describ ed in [12] where explicit so-

lutions to the equations of motion were obtained for any arbitrary function

( ) in terms of the variables de ned in (22). We will now describ e this

evolution within the framework of the analysis presented in section 4 which

is valid for any matter and for any arbitrary function ( ).

Initially, we have A(t ) > 0; _ (t ) > 0, and (t ) > 3. Then, = 1

i i i

necessarily. For t > t , the evolution pro ceeds as describ ed in section 4a.

Both e and increase. Hence, increases. Eventually, ! 1 and !1.

The dynamics near = 1, equivalently = 0 in our normalisation, is clear in

terms of and (). It follows from the analysis of section 4a that crosses

0 and continues to increase. Then, () and, hence, decrease.

We then have A>0, but <_ 0. The evolution then pro ceeds as describ ed

A A

in section 4b. e increases whereas and, hence, decrease. As t !1; e

and evolve as given in (35) and (36). 21

In particular, it can be seen that the quantities in (53) are all nite for

t t 1, implying that all the curvature invariants are nite. Hence, the

evolution is singularity free.

t (t )

i i

To describ e the evolution for t < t , we reverse the direction of time.

Then, in terms of the reversed time variable also denoted as t, we have

A(t ) < 0, _ (t ) < 0, and (t ) > 3. Then, = 1 necessarily. For t > t ,

i i i i

the evolution pro ceeds as describ ed in section 4c. Both e and decrease.

Hence, decreases. Then, as shown in section 4c, there exists a time, say

A(t )

t = t > t where K (t ) = 1 in equation (17), e > 0, (t ) > 0, and

m i m m

_ (t ) < 0. Note that, in this case where and, hence, vanish, we have

m 0

(t )=3.

As shown in section 4c, K (t ) = 1 implies that A(t ) = 0. Hence, the

m m

scale factor e reaches a minimum. For t>t , we then have A>0, <_ 0,

< 3, and = 1. The evolution pro ceeds as describ ed in section 4d.

It follows that (t) = 0 since and, hence, vanishes. Hence, in equation

(38), c = c which is negative and non in nitesimal. Therefore, as describ ed

A A

in section 4d, e increases whereas and, hence, decrease. As t !1, e

and evolve as given in (41) - (43).

In particular, it can be seen that, for , the quantities in (53) are

all nite for t t 1, implying that all the curvature invariants are nite.

Hence, the evolution is singularity free.

Thus, the evolution of the toy universe with = 0 and with the initial

conditions A (t ) > 0, _ (t ) > 0, and (t ) > 3 pro ceeds in the present mo del

i i i

as follows. For t>t, the scale factor e increases continuously to 1. The

eld increases, reaches the value 1, and then decreases continuously to 0.

Corresp ondingly, increases, reaches 1, and then decreases continuously to

, given in (7).

For t < t , the scale factor e decreases, reaches a non zero minimum,

and then increases continuously to 1. The eld decreases continuously to

0. Corresp ondingly, decreases continuously to , given in (7).

Also, the curvature invariants all remain nite for 1 t 1. Hence,

in the present mo del, the evolution of the toy universe with =0 is singu-

larity free. 22

When the initial conditions are di erent, the evolution can again b e anal-

ysed along similar lines. However, the main result that the evolution is

singularity free remains unchanged.

; t>t 5 b. =

The evolution for this case was rst describ ed in [10] where explicit so-

lutions to the equations of motion were obtained for a few sp eci c functions

( ). We will now describ e this evolution within the framework of the anal-

ysis presented in section 4 whichisvalid for any matter and for any arbitrary

function ( ).

Initially, we have A(t ) > 0; _ (t ) > 0, and (t ) > 3. Then, = 1

i i i

necessarily. For t > t , the evolution pro ceeds as describ ed in section 4a.

A A

Both e and increase. Hence, increases. As t !1, e and evolve as

given in (31) where = now.

In particular, it can be seen that the quantities in (53) are all nite for

t t 1, implying that all the curvature invariants are nite. Hence, the

evolution is singularity free.

t (t )

i i

To describ e the evolution for t < t , we reverse the direction of time.

Then, in terms of the reversed time variable also denoted as t, we have

A(t ) < 0, _ (t ) < 0, and (t ) > 3. Then, = 1 necessarily. For t > t ,

i i i i

the evolution pro ceeds as describ ed in section 4c. Both e and decrease.

Hence, decreases. Then, as shown in section 4c, there exists a time, say

A(t )

t = t > t where K (t ) = 1 in equation (17), e > 0, (t ) > 0, and

m i m m

_ (t ) < 0. Note that (t ) < 3.

m m

As shown in section 4c, K (t ) = 1 implies that A(t ) = 0. Hence, the

m m

scale factor e reaches a minimum. For t>t , we then have A>0, <_ 0,

< 3, and = 1. The evolution pro ceeds as describ ed in section 4d.

It follows that (t) = 0 since = . Hence, in equation (38), c = c

which is negative and non in nitesimal. Therefore, as describ ed in section

A A

4d, e increases whereas and, hence, decrease. As t ! 1; e and

evolve as given in (41) - (43).

, the quantities in (53) are In particular, it can be seen that, for

all nite for t t 1, implying that all the curvature invariants are nite.

Hence, the evolution is singularity free. 23

Thus, the evolution of the toy universe containing only radiation, i.e.

6= 0 but = , and with the initial conditions A(t ) > 0; _ (t ) > 0, and

i i

(t ) > 3 pro ceeds in the present mo del as follows. For t > t , the scale

i i

factor e increases continuously to 1. The eld increases continuously to

1. Corresp ondingly, increases continuously to 1.

For t < t , the scale factor e decreases, reaches a non zero minimum,

and then increases continuously to 1. The eld decreases continuously to

0. Corresp ondingly, decreases continuosly to , given in (7).

Also, the curvature invariants all remain nite for 1 t 1. Hence,

in the present mo del, the evolution of the toy universe with 6= 0 but =

is singularity free.

When the initial conditions are di erent, the evolution can again b e anal-

ysed along similar lines. However, the main result that the evolution is

singularity free remains unchanged.

5 c. =1; t> t

Initially, we have A(t ) > 0; _ (t ) > 0, and (t ) > 3. Then, = 1

i i i

necessarily. For t > t , the evolution pro ceeds as describ ed in section 4a.

Both e and increase. Hence, increases.

The constant c> 0in (15). However, (1 3 ) < 0 and, hence, (t) < 0.

Therefore, ( (t)+c) can b ecome negative for < 1, or remain p ositive as

! 1. In the case where ( (t)+c) b ecomes negative for < 1, it must pass

through a zero where _ vanishes. This critical p oint is a maximum of (t)

beyond which we have A>0 and <_ 0, and > 3.

Consider the case where ( (t)+c) remains p ositive as ! 1. Further-

more, as describ ed in section 4, it can be taken to be non in nitesimal by

cho osing the mo del dep endent constant in (7) large enough. The dynam-

ics near =1, equivalently =0 in our normalisation, is clear in terms of

and (). It follows from the analysis of section 4a that crosses 0 and

continues to increase. Then, () and, hence, decrease. We then have

A>0, but <_ 0.

Thus, in b oth of the ab ove cases, we now have A > 0 and _ < 0. The

evolution then pro ceeds as describ ed in section 4b. e increases whereas

and, hence, decrease. As t ! 1, e and evolve as given in (35) and

(36). 24

In particular, it can be seen that the quantities in (53) are all nite for

t t 1, implying that all the curvature invariants are nite. Hence, the

evolution is singularity free.

t (t )

i i

To describ e the evolution for t < t , we reverse the direction of time.

Then, in terms of the reversed time variable which is also denoted as t, we

have A (t ) < 0, _ (t ) < 0, and (t ) > 3. Then, = 1 necessarily. For

i i i

t > t , the evolution pro ceeds as describ ed in section 4c. Both e and

decrease. Hence, decreases. Then, as shown in section 4c, there exists

A(t )

a time, say t = t > t where K (t ) = 1 in equation (17), e > 0,

m i m

(t ) > 0, and _ (t ) < 0. Note that (t ) < 3.

m m m

As shown in section 4c, K (t ) = 1 implies that A(t ) = 0. Hence, the

m m

scale factor e reaches a minimum. For t>t , we then have A>0, <_ 0,

< 3, and = 1. The evolution pro ceeds as describ ed in section 4d.

It follows that (t) = 0 since = 1. Note that the constant c < 0 in

(15). Thus, ( (t)+c) < c < 0 and, hence, _ < 0 for t > t . Therefore,

decreases. Eventually, ! 0. Moreover, and as can be veri ed a p osteriori,

(t) ! (constant) as ! 0. Hence, in equation (38), the constant c is

negative and non in nitesimal. Therefore, as describ ed in section 4d, e

increases whereas and, hence, decrease. As t !1, e and evolve as

given in (41) - (43).

In particular, it can be seen that, for , the quantities in (53) are

all nite for t t 1, implying that all the curvature invariants are nite.

Hence, the evolution is singularity free.

Thus, the evolution of the toy universe containing only massless scalar

eld, i.e. 6= 0 but = 1, and with the initial conditions A(t ) > 0,

_ (t ) > 0, and (t ) > 3 pro ceeds in the present mo del as follows. For t>t ,

i i i

the scale factor e increases continuously to 1. The eld increases, reaches

a maximum 1, and then decreases continuously to 0. Corresp ondingly,

increases, reaches a maximum 1, and then decreases continuously to ,

given in (7).

For t < t , the scale factor e decreases, reaches a non zero minimum,

and then increases continuously to 1. The eld decreases continuously to

0. Corresp ondingly, decreases continuously to , given in (7).

0 25

Also, the curvature invariants all remain nite for 1 t 1. Hence,

in the present mo del, the evolution of the toy universe with 6= 0 but =1

is singularity free.

When the initial conditions are di erent, the evolution can again b e anal-

ysed along similar lines. However, the main result that the evolution is

singularity free remains unchanged.

6. Evolution of observed Universe

Wenow use the results of the analysis in section 4 to describ e the generic

evolution of a realistic universe, such as our observed one, in the present

mo del. A realistic universe may contain di erenttyp es of \matter" describ ed

by di erent values of , but with 1 1. For example, = 1; 0; ,

and 1 resp ectively for vacuum energy, dust, radiation, and massless scalar

eld. Our observed universe is certainly known to contain dust and radiation.

However, in avariety of mo dels, vacuum energy and/or massless scalar eld

are often assumed to be present. Our assumption that 1 1 then

covers all these cases. As follows from (12), \matter" with larger dominates

A A

the evolution for smaller value of e and vice versa. Therefore, when e is

decreasing we take eventually, and when e is increasing we take 0

eventually.

In considering the evolution of the universe, we start with an initial time

t , corresp onding to a temp erature, say 10 GeV, such that GUT symmetry

breaking, in ation, and other (matter) mo del dep endent phenomena may

o ccur for t > t only. Our observed universe is expanding at t and, hence,

i i

A(t ) > 0. For the sake of de niteness, we take _ (t ) > 0 and (t ) > 3,

i i i

equivalently ! (t ) > 0, as commonly assumed.

We rst describ e the evolution for t > t taking, as initial conditions

A(t ) > 0; _ (t ) > 0, and (t ) > 3. Then, = 1 necessarily. To describ e

i i i

the evolution for t

conditions, A(t ) < 0; _ (t ) < 0, and (t ) > 3. Then, = 1 necessarily.

i i i

The required evolution is that for t> t in terms of the reversed time variable,

which is also denoted as t. The results of section 4 can then be applied

directly.

When the initial conditions are di erent, the evolution can again b e anal-

ysed along similar lines as follows. However, the main result that the evolu-

tion is singularity free remains unchanged. 26

t>t

Initially, we have A(t ) > 0; _ (t ) > 0, and (t ) > 3. Then, = 1

i i i

necessarily. For t > t , the evolution pro ceeds as describ ed in section 4a.

A A

Both e and increase. Hence, increases. Eventually, as e increases, we

can take 0. Then, as t !1, e and evolve as given in (31).

In particular, it can be seen that the quantities in (53) are all nite for

t t 1, implying that all the curvature invariants are nite. Hence, the

evolution is singularity free.

Thus, in the present day universe in our mo del, (today ) ! 1 and

1 d 2

4 1

(today ) !1. Also, as follows from (7), (today )= (1 ) !

0. Therefore, our mo del satis es the observational constraints imp osed by so-

1 d

lar system exp eriments, namely (today ) > 2000 and (today ) < 0:0002

(see [3] pg. 117, 124-5, and 339).

t< t , equivalently (t) > (t )

i i

To describ e the evolution for t < t , we reverse the direction of time.

Then, in terms of the reversed time variable, also denoted as t, we have

A(t ) < 0, _ (t ) < 0, and (t ) > 3. Then, = 1 necessarily. For t > t ,

i i i i

the evolution pro ceeds as describ ed in section 4c. Both e and decrease.

Hence, decreases. Then, as shown in section 4c, there exists a time, say

A(t )

t = t > t where K (t ) = 1 in equation (17), e > 0, (t ) > 0, and

m i m m

_ (t ) < 0. Note that (t ) < 3.

m m

As shown in section 4c, K (t ) = 1 implies that A(t ) = 0. Hence, the

m m

scale factor e reaches a minimum. For t>t , we then have A>0, <_ 0,

< 3, and = 1. The evolution pro ceeds as describ ed in section 4d. Now,

however, the evolution for t>t is complicated since the universe is known

to contain \matter" with =0. Nevertheless, all qualitative features of its

evolution can b e obtained using the results of section 4.

For t t , A > 0, _ < 0, and ( (t)+c) < 0. Hence, e increases and

decreases and, eventually, \matter" with 0 dominates the evolution.

Then (1 3 ) > 0 and (t) b egins to increase. Hence ( (t)+c), which is

negative, also b egins to increase. For 0, (t) grows atleast as fast as t,

as follows from (16). Then eventually, as describ ed in section 4d, e reaches

a maximum eventually at time, say t = t . Also, ( (t )+c) < 0 and, hence,

1 1

_ (t ) < 0. Note that this critical p ointofe exists indep endent of the details

1 27

of the mo del, as long as \matter" with 0 is present. Further evolution

then pro ceeds as describ ed in section 4e.

For t t , we then have A<0; <_ 0, and < 3. Also, ( (t)+c) < 0

but (t) remains increasing. Hence, dep ending now on the details of the

mo del, ( (t)+c) may remain negative for all t>t , or it may reach a zero

and b ecome p ositive.

In the rst case, ( (t)+c) remains negative for all t>t . Hence, <_ 0

and continues to decrease. The scale factor e , as shown in section 4e,

reaches a minimum at time, say t = t >t , with (t ) > 0 non vanishing.

M 1 M

For t > t ; e increases, and we have A > 0; _ < 0, and < 3. Further

evolution then pro ceeds as describ ed in section 4d.

Note that this means that the scale factor increases and reaches a maxi-

mum, then decreases and reaches a minimum, then increases and so on. The

existence of maxima of e is mo del indep endent as long as \matter" with

0 is present, which is the case for the observed universe. The minima of

e are all non zero for the reasons describ ed in section 4c. Their existence de-

p ends on whether ( (t)+c) remains negative or not, but is otherwise mo del

indep endent as long as \matter" with is present, which is the case

for the observed universe. The evolution can thus b ecome rep etetive and

oscillatory.

In the second case, ( (t)+c) reaches a zero at time, say t = t >t and

m 1

b ecomes p ositive. It then follows that _ (t ) = 0 and that this is a minimum

of . For t > t , we have A < 0, _ > 0, and < 3. Hence, e decreases

. and increases. Eventually, we can take

Now, (1 3 ) 0 and (t) b egins to decrease or remains constant. Hence

( (t)+c), which is p ositive, also b egins to decrease or remains constant.

Thus, dep ending on the details of and ,((t)+c)mayormay not vanish

with <1. If ( (t)+ c)vanishes with <1, then _ vanishes. This critical

p oint is a maximum of , beyond which we have A<0, <_ 0, and = 1.

Further evolution then pro ceeds as describ ed in section 4c if > 3 and as in

section 4e if < 3. The evolution can thus b ecome rep etetive and oscillatory.

If ( (t)+c) do es not vanish and remain p ositive for <1 then, eventually,

2 6A

! 1. It follows from equation (15) that _ / e , whereas is given

by (12). Note that A<0 and e is decreasing. Then, in this limit, the

term. term in equation (10) dominates or, if =1,is of the order of the

Thus we have A < 0, _ > 0, ! 1, equivalently ! 1, and = 1. 28

term. Also, term in equation (10) dominates or is of the order of the

But this is precisely the time reversed version of the evolution analysed in

section 4b, where the dynamics is clear in terms of and (). Applying the

results of section 4b, it then follows that will cross the value 0 after which

e and, hence, will b egin to decrease. We then have A<0, <_ 0; >3.

The evolution then pro ceeds as describ ed in section 4c. The evolution can

thus b ecome rep etetive and oscillatory.

Thus it is clear that dep ending on the details of the mo del, the universe

undergo es oscillations, p erhaps in nitely many. As follows from the ab ove

description, the oscillations can stop, if at all, only in the limit ! 0. The

solutions then are given by (41) - (43). In particular, however, the quantities

in (53) all remain nite during the oscillations, implying that all the curvature

invariants remain nite. They also remain nite in the solutions (41) - (43)

if in (7).

Thus, it follows that if then the quantities in (53) are all nite

for t t 1, implying that all the curvature invariants are nite. Hence,

the evolution is singularity free.

In summary, the evolution of a realistic universe, such as our observed

one, with the initial conditions A(t ) > 0, _ (t ) > 0, and (t ) > 3 pro ceeds

i i i

in the present mo del as follows. For t > t , the scale factor e increases

continuously to 1. The eld increases continuously to 1. Corresp ondingly,

increases continuously to 1.

For t < t , the scale factor e decreases and reaches a non zero mini-

mum. It then increases and reaches a maximum, then decreases and reaches

a minimum, then increases and so on, p erhaps ad in ntum. The oscillations

can stop, if at all, only in the limit ! 0. The solutions then are given by

(41) - (43). The eld may, dep ending on the details of mo del, continuously

decrease to 0, or undergo oscillations, its maxima always b eing 1.

Also, the curvature invariants all remain nite for 1 t 1. Hence,

the evolution, although complicated and mo del dep endent in details, is com-

pletely singularity free. Thus wehave that a homogeneous isotropic universe,

such as our observed one, evolves in the present mo del with no big bang or

any other singularity. The time continues inde nitely into the past and the

future, without encountering any singularity.

When the initial conditions are di erent, the evolution can again b e anal-

ysed along similar lines. However, the main result that the evolution is

singularity free remains unchanged. 29

7. Generalisations of the mo del

The analysis and the results of sections 4 - 6 are generic and valid for any

function satisfying only the constraints (7). However, these constraints

can b e relaxed further thus generalising the mo del, but still maintaining the

evolution singularity free. We now study these generalisations.

We rst summarise the essential features of the constraints (7) on ( )by

noting the following p oints. First, as evident from the results of the previous

sections, any singularity is likely to arise, if at all, only in the limit ! 0.

It then follows from the relevent solutions (41) - (43) that this p otential

singularity is avoided by the constraint in (7).

Second, we used only the suciency condition given in the App endix to

determine the absence of singularity. Hence, the requirement of niteness of

; 8n 1.

Third, there is no singularity in the limit ! 1, although !1. The

evolution of the present day universe corresp onds to this limit. Also, the

b ehaviour of ( ) in this limit, in particular the constrainton given in (7)

which follows naturally from (6), ensures that our mo del satis es the obser-

vational constraints imp osed by solar system exp eriments. Note, however,

that the observational constraints are satis ed for any > .

Fourth, the monotonicity of ( ) for 0 < < 1 is not required either

to avoid the singularity or to conform with the observations. It was invoked

only to keep the analysis simple and transparent. Without this requirement,

the evolution would be more complicated, its history p ossibly b earing the

imprints of the ups and downs of ( ). The analysis in section 4 would then

involve further sub classes not directly relevent for the issue of singularity.

It is now clear how the present mo del, equivalently the constraints (7) on

( ), can b e generalised further. In the generalised mo del also we take to

have a nite upp er b ound < 1 which was naturally the case in

max

the previous mo del. Then, as in section 2, the factor can be absorb ed

max

into the Newton's constant G and the range of can set to be 0 1.

The generalised mo del is now given by the function ( ) > 0 satisfying the

following more general constraints:

(0)

= nite 8 n 1; 0 < 1

d 30

! 1 at =1 only

lim = (1 ) ; > ; (44)

where > 0is a constant. The function ( )is arbitrary otherwise.

Thus, in the generalised mo del, ( ) in the limit ! 0 need not nec-

essarily be a constant, but can be a function of . For example, one may

0 0

have ( )= as ! 0 where > 0 and > 0 are constants. Such

0 0

functions do arise in the variable mass theories of [6] also. Note that the

p osition of the p ole, namely = 1, is not a constraint but only a choice

of normalisation of . However, the order of the p ole can now be, for

example, 1.

We have taken , 8n 1 for the sake of simplicity. This requirement

can p erhaps b e relaxed if one uses only the appropraite necessary conditions

to determine the absence of singularity instead of the suciency conditions

given in the App endix. However, we will not pursue it further in this pap er.

The evolution of the universe in the generalised mo del can be analysed

along the same lines as in sections 4 - 6. It is clear that the only imp ortant

limits are ! 0 and, p erhaps, ! 1. Hence, we now analysis only these

two limits in the generalised mo del. To b e de nite, we consider the following

example:

lim = ; > 0

lim = (1 ) ; 1 ; (45)

where > 0 and > 0 are constants. Note that the analysis in section

0 1

4 applies directly with no mo di cation in the limit ! 1 if < 1.

Hence, we have taken 1 in (45). Using (5), the constraints (45) can be

expressed equivalently in terms of and () as follows:

lim = ln( )

lim = e ; if =1

= ; if >1; (46)

where and are constants. The range of =ln is given by 1

0 0

0. Hence, the constant < 0 in (46). Also, the limit ! 0 corresp onds

0 31

to !1 and, hence, to ! whereas the limit ! 1 corresp onds to

! 0 and, hence, to !1.

! 0

In this limit, the analysis is similar to that given in section 4d. Under

the conditions of section 4b, equation (17) can be solved in the limit ! 0

relating and e :

e = (constant) : (47)

Hence, e !1 as ! 0. Substituting (47) in equation (15) then yields the

unique solution in the limit ! 0: For 6=1,

A A

e =e ; = (t sign (m)t) ; (48) (t sign(m)t)

0 0 0

where t is measured in appropriate units, A , , and t > t are constants

0 0 0 i

and m = . For =1,

c (tt )

A A c (tt )

0 0

e e =e ; = e ; (49)

where c < 0 is a constant. Note, as p ointed out in [15] and as discussed in

section 3, that when and e satis es equation (47) the eld equation (13)

is not guaranteed to follow from (10) - (12). Nevertheless, the ab ove solution

satis es equation (13) also, as checked explicitly.

Let > 1. Then, m > 0. Since ! 0, it follows from (48) that

t ! t > t , implying that vanishes at a nite time t . In this limit, the

0 i 0

scale factor e !1as follows from (48). The quantities in (53), for example

, also diverge implying that the curvature invariants, including the Ricci

scalar, diverge. Thus, for > 1, there is a singularity at a nite time t . This

implies, for example, that the variables mass theories of [6] which exhibit a

minimum of e can nevertheless have singularities b ecause can be > 1 in

these theories.

Let 1. When < 1; m<0. Since ! 0, it follows from (48) that

t !1. In this limit, the scale factor e !1as follows now from (48). The

same result holds when = 1 also, as follows from (49).

If 6=0 then the consistency of the ab ove solutions requires that .

This is analogous to the condition > 0 required in section 4d.

It can now be seen, for 1, that the quantities in (53) are all -

nite implying that all the curvature invariants are nite. Thus, there is no

singularity in the ab ove solutions when 1. 32

! 1

In this limit, the analysis is similar to that given in section 4a. For 6=0

and 6= 1, the solution is given uniquely by

A A

3(1+ )

e = e t

= 1 e ; if =1

= 1 (1 kT ) ; if >1; (50)

where > 0 and k > 0 are constants, t is measured in appropriate units,

and we have de ned T t which ! 0 in the limit t ! 1. The ab ove

solution then implies that e ! 1 and ! (1 ). The precise value

of is mo del dep endent. In particular, it also dep ends on and can be

0 1

made as small as necessary bycho osing a suciently large . Equivalently,

as t !1, can be made as large as necessary.

This will imply, as in section 6, that in the presentday universe in the gen-

1 d

eralised mo del, (today ) ! 1 , (today ) ! , and (today ) !

0 1 3

4 1

. Therefore, by cho osing suciently large one can ensure that

the generalised mo del also satis es the observational constraints imp osed

1 d

by solar system exp eriments, namely (today ) > 2000 and (today ) <

0:0002 (see [3] pg. 117, 124-5, and 339).

For =0, the solution is obtained easily in terms of and (), and is

given by

A A

t e = e

e = (constant) t ; (51)

where A is a constant. () and, hence, = e are then obtained from

(46). Note that, as in section 4a, these solutions also apply to the case when

6= 0, but = 1. As t ! 1 in the ab ove solution, we have e ! 1 and

!1. Thus, as follows from (46), ! 0. Hence, ! 1 and !1.

It can be seen that the quantities in (53) are all nite implying that all

the curvature invariants are nite. Thus, there is no singularity in the ab ove

solutions.

Now the evolution of the universe in the generalised mo del can be anal-

ysed as in sections 4 - 6. For the three toy universes of section 5, the evolution 33

pro ceeds as describ ed in section 5. For a realistic universe, such as our ob-

served one, the evolution is complicated but pro ceeds as describ ed in section

6. In particular, it follows that the evolution of the universe is singularity

free in the generalised mo del when the function ( ) satis es the constraints

(45). Although not proved here, the evolution is likely to b e singularity free

when ( ) satis es the constraints (44) only.

Before concluding this section, we would like to p oint out an interesting

asp ect of the generalised mo dels, where ( ) ! 0, equivalently ! ! , in

the limit ! 0. When ! = , the graviton-dilaton part of the action in

(4) acquires a conformal symmetry under which

(x) (x)

g ! e g and ! e ; (52)

where (x) is an arbitrary function of space time co ordinates [16]. In general,

the matter action S breaks this symmetry. However, it can be seen from

the solutions presented in section 4 and the present one that precisely in

the limit ! 0 do the matter terms b ecome negligible compared to the

dilatonic terms. This suggests that the matter action S may be neglected

and that the conformal symmetry of the action (4) may b ecome exact as

! 0. A p ossible implication of the app earance of this conformal symmetry

is mentioned in the conclusion.

8. Summary and Conclusion

To summarise, we considered the evolution of a at homogeneous isotropic

universe in a graviton-dilaton mo del. The mo del is sp eci ed by the function

( ) which satis es the constraints given in (7), or more generally (44), but

is arbitrary otherwise. We studied the evolution of universe in this mo del

with the main purp ose of determining whether it is singular or not.

Assuming generic initial conditions where A(t ) and _ (t ) are nite and

i i

non in nitesimal and (t ) < 1, we presented a general analysis which is

valid for any matter and for any arbitrary function ( ), and which en-

ables one to analyse the evolution and obtain its generic features even in the

absence of explicit solutions.

We illustrated our metho d by applying it to describ e the evolution of three

toy universes. We showed, in particular, that their evolution in the present

mo del is singularity free. We then describ ed the evolution of a universe, such

as our observed one. We showed, in particular, that its evolution, although 34

rich and complicated, is singularity free. This is a generic result valid for any

( ) satisfying only the constraints given in (7) or (44).

An imp ortant question to ask now, from our p ersp ective, is whether a

function ( ) as required in the present mo del can arise from a fundamental

theory. We now discuss critically a few such p ossibilities in string theory.

In the string theoretic context, Einstein metric formulation given in (1)

is more natural where is the dilaton and () the arbitrary function. The

strong coupling limit in string theory, relevant for singularities, corresp onds

to diverging e ective Newton's constant (= ) and, thus, to the limit

() ! 0. In this limit ln ! in our mo del (see equation (7) and

(1)).

(1) Note that, in string theory, ln( ()) can be thought of as Kahler

p otential for , which is exp ected to be mo di ed at strong coupling by non

p erturbative e ects [17]. But, such mo di cations are usually exp onential in

nature and, hence, are unlikely to lead to a ()as required here.

(2) However, at strong coupling, here may not be the stringy dilaton

directly, but instead a combination of the stringy dilaton and other compact-

i cation dep endent mo duli. In that case, after writing the relevant e ective

action as given in (1), the Kahler p otential ln for this `e ective dilaton'

may well turn out to b e of the required form.

(3) Another, p erhaps more promising, p ossibility is the following which

arises when = (see equation (7) and (1)). This case corresp onds to a ve

dimensional space time compacti ed to four dimensions on a circle [18]. In

recent developments in string theory at strong coupling, analogous phenom-

ena relating a d-dimensional theory and a (d + 1)-dimensional theory on a

circle are found to o ccur: Using S-duality symmetries, Witten has discovered

that the ten dimensional string at strong coupling is an eleven dimensional

(M-)theory compacti ed on a circle [19, 1]. A similar phenomenon, that the

three dimensional string at strong coupling is a four dimensional theory com-

pacti ed on a circle, is at the heart of Witten's novel prop osal for solving

the cosmological constant problem [20]; for its p ossible stringy realisation see

[21]. A similar phenomenon in four/ ve dimensions, if exists, may p ossibly

lead to a () as required here.

(4) As discussed in section 7, the graviton-dilaton action in (4) acquires

a conformal symmetry in the limit ! 0 where ( ) ! 0, equivalently

! ( ) ! . Note that this is also the limit where singularities can p oten-

2 35

tially arise, but are avoided in the present mo del. The app erance of this

conformal symmetry is, p erhaps, a hint of a connection between the present

mo del and string theory, known to lead generically to conformal eld theories,

but which must now be in the strong coupling limit since ! 0.

Admittedly, at present, these are plausibility arguments only. Neverthe-

less, given the elegantway the present mo del resolves the big bang singularity

it is worthwhile to derive it, p erhaps along the ab ove lines, from a funda-

mental theory such as string theory.

On the other hand, however, such a mo del, even if not derivable from

string theory,may stand on its own as an interesting mo del for singularity free

cosmology. One may then study its implications in other cosmological and

astrophysical contexts. There are many issues that can be studied further.

We mention some of them here.

(1) One can study the evolution of a more complicated universe, e.g. an

inhomgeneous anisotropic universe, and determine whether it is singularity

free or not.

(2) In section 6 we found that the evolution of a realistic universe can b e

oscillatory. Such early universe oscillations have signi cant implications and

may also lead to observable predictions. It is therefore imp ortant to study

this issue in more detail.

(3) In the astrophysical context, one may study the evolution and/or col-

lapse of stars. In fact, in [11] where the present mo del was originally derived

in a di erent context, we found that our mo del suggests anovel scenario for

stellar collapse in which a black hole is unlikely to form. Such a phenomenon,

if established rigorously, is likely to have far reaching consequences.

(4) The present mo del is characterised by one arbitrary function ( ),

required to satisfy the constraints given in (7) or (44). These constraints,

however, do not x ( ) uniquely. The consequent arbitrariness in the mo del

greatly diminishes its predictive power. It is therefore desirable to nd the

criteria which can x ( ) uniquely.

In this regard, note that the present mo del naturally incorp orates, as

describ ed in [12], an ingredient crucial for the success of hyp erextended in-

ation [8]. Therefore, with a view to x ( ) uniquely, one may require the

present mo del to lead to a successful in ation also. However, our preliminary

calculations [22] indicate that this requirement is not strong enough to x

( ) uniquely. Hence, more criteria are needed.

Considering the simplicity of the present mo del and the novel conse- 36

quences that follow from it generically, we b elieve that its further study

is fruitful and is likely to lead to interesting phenomena.

Acknowledgement: We thank T. R. Seshadri for discussions.

APPENDIX

Finiteness of Curvature Invariants

In this App endix, we derive a sucient condition for al l curvature invari-

ants to be nite. Curvature invariants are constructed using metric tensor,

Riemann tensor, and covariant derivatives. When the metric is diagonal,

every term in any curvature invariant can be group ed into three typ es of

p p

factors: (A) g g g g R , (B) g g g , and (C) g @

acting multiply on (A) and (B) typ e factors (no summation over rep eated

indices).

Using the metric given by (8) and evaluating the ab ove factors explicitly,

one obtains that (A) and (B) typ e factors are functions of A; A, and e .

The action of (C) pro duces extra time derivatives. Thus, it follows that the

d A

curvature invariants are functions of e and ; n 1.

d A

By rep eated use of equations (10)-(12), for any n 1 and, hence, any

curvature invariant, can be expressed in terms of the following quantities:

2 n n

_ _ d _

e ; ; ; ; ; and ; 8n 1 : (53)

2 n

The resultant expressions are nite if the ab ove quantities are nite. The

algebra is straightforward and, hence, we omit the details here.

Hence, a sucient condition for al l curvature invariants to be nite is

that the quantities in (53) be al l nite.

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