Dilaton-Driven Inflation in String Cosmology

Home , Dilaton, String (physics)

〉 PostScript processed by the SLAC/DESY Libraries on 7 Jun 1995. HEP-TH-9506045 CERN-TH/95-147 June 1994 e v a It is , y yw vit ersit ork of string BGU PH-95/05 CERN-TH/95-147 TION en kinetic in ation. a 23, Switzerland vitational radiation is gen- ternational conference ) ; ysics, Ben-Gurion Univ CT olution in the framew hastic gra w its detection in future gra a 84105, Israel. tofPh ABSTRA y allo Ram Brustein Beer-Shev kground of sto c TON-DRIVEN INFLA IN STRING COSMOLOGY ebruary 2-5, 1995, Coral Gables , Florida. F ypical bac DILA Uni ed symmetry in the small and in the large, t address: Departmen tribution to the pro ceedings of the In ts. Theory Division, CERN CH-1211 Genev t an outline for cosmological ev Con ) Presen ) wn that a t exp erimen erated, with strength that ma I presen theory with emphasis on a phase of dilaton-driv sho

INTRODUCTION

I present an outline of cosmological evolution in the framework of string

theory. The main emphasis is on a phase of dilaton-driven kinetic in a-

tion and its p ossible observable consequences, in particular, a background of

sto chastic gravitational radiation. The results concerning the pro duced sp ec-

trum of gravitational radiation were obtained in [1, 2]. More details on vari-

ous asp ects of the suggested outline and additional references may b e found in

[3-11].

POTENTIAL-DRIVEN INFLATION

In ationary evolution of the universe requires a source of energy to drive

the expansion. The conventional exp ectation is that the energy source is

dominated by p otential energy of scalar elds, called in atons [12 ]. The in-

atons are exp ected to p osses non-vanishing p otential energy during some

phase in their evolution in which in ationary expansion takes place. Eventu-

ally, the in atons settle down to the true minimum of their p otential where

the p otential energy vanishes, thus depriving the universe of the necessary

source to drive its accelerated expansion. The in ationary phase ends and

the universe continues to expand sub-luminally until to day. If one tries to

implement similar ideas in the framework of string theory, an apparent prob-

lem is immediately encountered [13 , 14]. String theory do es indeed con-

tain many scalar elds, called mo duli, which seem particularly suitable for

the job of in atons [15, 16 ]. Among the mo duli the dilaton is an im-

p ortant and universal eld whose exp ectation value determines the string

coupling parameter g hexp()i. It couples to all other elds with gravi-

tational strength. If some scalar eld, for example, one of the mo duli elds

acquires a non-vanishing p otential so do es the dilaton. The typ e of gener-

ated dilaton p otential dep ends on the details of the mo del. Twotyp es are

distinguished, p erturbative V () exp( =M ), and non p erturbative

V () exp(exp( =M )), with particular numerical parameters , .

The equations of motion for the resulting string dilaton-gravity, assuming

isotropic and homogeneous universe

2 2 2 i

ds = dt + a (t)dx dx

= (t); (1) 1

are the following

8 1

2 2

H = + V ()

3M 2

+3H = (2)

The Hubble parameter, H , is related to the scale factor, a in the usual way,

H and V is the p otential. Consider, for example, the (unrealistic) case

of exp onential p otential V = V exp( =M ) for which can solve eqs. (2)

0 Pl

explicitly

16=

a(t)= a t : (3)

If the p otential is steep er than the critical steepness =4 , the dilaton

kinetic energy b ecomes dominantover p otential energy and the expansion

is subluminal. The generic situation in string theory is that the p otentials

in several mo dels are steep er than critical and therefore p otential-driven in-

ation requires sp ecial situations and is generally sp eaking hard to obtain.

Recently, some progress has b een made towards characterizing requirements

from mo dels in which p otential-driven in ation could b e supp orted [16, 17 ].

DILATON-DRIVEN KINETIC INFLATION

The outline for cosmological evolution that I present here relies heavily on

the fact that the kinetic energy of the dilaton tends to dominate the energy

density. Instead of trying to ght this tendency, one accepts it and turns this

feature into a virtue, using it to drive kinetic energy dominated in ationary

evolution. Kinetic in ation was also discussed in [18 ]. The evolution starts

when the dilaton is deep in the weak-coupling region ( 1) and Hubble

parameter, H , is small. The evolution in this ep o ch is shown b elowtobeac-

celerated expansion dominated by the dilaton kinetic energy and determined

by the vacuum solution of the string dilaton-gravity equations of motion [3].

To describ e the rst phase in more detail, lo ok for solutions of the e ective

string equations of motion in which the metric is of the isotropic, FRWtyp e

with vanishing spatial curvature and the dilaton dep ends only on time. One 2

nds three indep endent rst order equations for the dilaton and H

1 1

H = H 3H + U + e U + e p (4a)

2 2

=3H 3H +U +e (4b)

_ +3H( +p)=0 (4c)

where U = e V . Some sources in the form of an ideal uid were included [3]

as well. The () signi es that either (+) or ()ischosen for b oth equations

simultaneously. The solutions of equations (4a-4c) b elong to two branches,

according to which sign is chosen. In the absence of any p otential or sources

the (+) branch solution for fH; g is given by

1 1

(+)

H =

t t

(+)

= +( 3 1) ln (t t) ; t

0 0 0

This solution describ es either accelerated contraction and evolution towards

weak coupling or accelerated in ationary expansion and evolution from a

cold, at and weakly coupled universe towards a hot, curved and strongly

coupled one. I assume that the initial conditions are such that the latter

is chosen. In general, the e ects of a p otential and sources on this branch

are quite mild. After a p erio d of time, of length determined by the initial

conditions, a \Branch Change" event from the dilaton-driven accelerated

expansion era into what will eventually b ecome a phase of decelerated ex-

pansion has to o ccur. It o ccurs either when curvatures and kinetic energies

reach the string curvature or when quantum e ects b ecome strong enough.

The correct dynamical description of this phase should, therefore, b e stringy

in nature. If the value of the dilaton is small throughout this stage of evo-

lution, dynamics can b e describ ed by classical string theory in terms of a

two-dimensional conformal eld theory. This stage is not yet well under-

stood. At the moment, the only existing examples are not quite realistic

[19, 20 ]. More ideas ab out this stage may b e found in [21, 22 ]. The value

which the dilaton takes at the end of this ep o ch is an imp ortant param-

end

eter. After the \Branch Change" event, the universe co ols down and may

b e describ ed accurately, again, by means of string dilaton-gravity e ective 3

theory. Now, however, radiation and matter are imp ortant factors. The

dilaton remains approximately at the value . The universe evolves as

end

a regular Friedman-Rob ertson-Walker (FRW) radiation-dominated universe.

TENSOR PERTURBATIONS AND RELIC GRAVITATIONAL

WAVES

The phase of accelerated evolution, describ ed in the previous section,

pro duces a typical and unique sp ectrum of gravitationl radiation. The basic

mechanism is bynowwell known [23] (see [24, 25 ] for recent reviews). Quan-

tum mechanical p erturbations exist as tiny wrinkles on top of the classically

homogeneous and isotropic background. These wrinkles are then magni ed

by the accelerated evolution and b ecome classical sto chastic inhomogeneities.

BelowI sketch the derivation of the sp ectrum of tensor p erturbations. Many

technical elements are omitted here and can b e found in gory details in [1].

The classical solution (5) in conformal time , where dt ad ,isgiven by

2 2

g = diag (a ( ); a ( ) ) i; j =1;2;3 (6)

where

2 1=2

a ( ) jj ; () 3lnjj + (7)

for ! 0 . One expands the metric around the classical solution g =

g + g where g is given in the previous equation and g = a ( )h (; ~x).

cs cs ij ij

The resulting equation of motion for each of the two indep endent tensor

p erturbation comp onents is given in Fourier space by

00 0 2

h +2 h +k h =0 (8)

k k

and has the general solution h = A + B ln jkj. Initial conditions corre-

k k k

sp onding to quantum uctuations at short scales h 1=(a k ) exp[i(k ~x k)],

determine h

ln jkj

jh j' ' ln jkj: (9)

The amplitude of sto chastic tensor p erturbations in x space is characterized

3=2

by jh jk jh j.From eq.(9) we obtain

k k

max

3 2

jk j (ln jkj) : (10) jh j ()

max k

Pl 4

The end of the dilaton-driven ep o ch is assumed [5] to take place when the

curvature scale H reaches the string scale M . In the Einstein frame, in

which M is constant, the string scale dep ends up on the dilaton as M =

Pl s

exp(=2)M .Thus we assume the dilaton-driven era to end at conformal

time j j = where H ' ( a( )) = M ( ) = exp(( )=2)M .At the

1 1 1 1 s 1 1 Pl

end of the dilaton-driven era wethus have

3=2

jh ( )j (k ) ln(k ) (11)

k 1 1 1

This is the nal result for the primordial sp ectrum of tensor p erturbations.

From the primordial sp ectrum one wishes to compute the observable sp ec-

trum to day. A nice feature of gravitational waves is that gravitons are af-

fected practically only by the evolution of the background curvature since

right after the \Branch Change" era. Thus the sp ectrum that should b e seen

to day should mainly re ect what happ ened in the very early universe pro-

cessed through presumed known background evolution. While frequencies

shift according to the evolution of the background scale factor throughout

the evolution, amplitudes of tensor p erturbations freeze while outside the

horizon and evolve only when inside the horizon. If the dilaton-driven era

is followed by a stringy phase characterized by an almost constantvalue of

H ,we exp ect scales whichwent out of the horizon during the dilaton-driven

era to keep moving further outside and to reenter only much later, during

the radiation, or p ossibly even matter dominated era. If we assume this to

b e the case for all (comoving) scales larger than ,wemust also assume

that h remains frozen, for all these scales, at the value given in eq.(11) until

reentry.

The result is [2] that the part of the pro cessed sp ectrum which lies b elow

a certain maximal frequency ! , the highest frequency ampli ed during

max

the dilaton-driven era, is presently given, in the string frame, by

1 ! 1 !

1=2

1=4

jh j = H =M z z exp ( )( ) ln( ) (12)

! 0 s end

out

2 ! !

max max

where z (k )=a (k )=a (k ) is the red-shift while the scale k was outside

out re ex

the horizon, z is the red-shift from the matter-radiation equality epochuntil

to day, M is the presentvalue of string scale (usually estimated to b e ab out

17 18

2 5 10 GeV ), H 10 Hz is the presentvalue of the Hubble parameter,

0 5

= ( ), and

end 1

1=2

1=4

! = H M z z : (13)

max 0 s

out

The fraction of energy in gravitational waves in units of the critical density

is given by

! d !

3 1

) ln ( ): (14) = z exp( )(

end

d ln ! ! !

max max

Equations (12-14) were derived assuming reentry during the radiation dom-

inated era and should b e taken as go o d estimates and not as numerically

accurate expressions. The pro cessed sp ectrum of gravitational radiation is

presented graphically in Figure 1,

δ hω Ω

GW = 10 Ω −22 −4 10 GW = 10 Ω −10 GW =10 δ −16 h ω max

ω1/2 10−26

1/2 zout

ω1/2

10−30

ω1/2 φ e end

1 102 106 1010

Frequency in Hertz

Figure 1. The characteristic sp ectral amplitude of gravitational waves jh j. The

solid lines show several individual sp ectra for di erentvalues of z and =0.

out end

max

The thick dashed line shows the maximum amplitude jh j as a function of z

out

for = 0. The dashed lines are lines of xed and therefore lines of constant

end end

energy density. is the maximal amount of gravitational energy densityata

given . Also shown in the gure is a triangular shap e marking the sensitivity

end

goals for detection of sto chastic background h , of the \Advanced LIGO".

3=y r 6

Two p ossible devices may b e able to detect the predicted sto chastic grav-

itational wave background, in the lower frequency region 1 10 Hz, large

interferometers, such as the planned LIGO[26 ] and VIRGO[27] and in the

6 9

higher range of frequencies 10 10 Hz, ro om-size microwavecavities. For

1=2

a given set of parameters the amplitude grows as jh j! and therefore

it may seem that the b est sensitivity for detection is at the high end of the

sp ectrum ! = ! .However, the noise in a given interferometer grows as

max

5=4

h ! [28]. Therefore for a given interferometer the b est sensitivity ac-

tually is in the lowest frequency range available. Microwavecavities maybe

6 9

op erated as gravitywave detectors [29] for the high frequency range 10 10

Hz. For the MHz range sp eci c suggestions [30 , 31 ] have b een implemented

[32], but not op erated as gravitational radiation detector. As can b e seen

from Figure 1, the required sensitivity for detection at the MHz region is

h 10 corresp onding to h of the same order and therefore to a noise

c 3=y r

level of h 10 [28], assuming a bandwidth of MHz. With attainable Q

factors of the order of 10 , this sensitivity goal do es not seem out of reach.

For the GHz region the required sensitivityis h 10 corresp onding to

h 10 .

ACKNOWLEDGMENT

Research supp orted in part by an Alon Grant. I would like to thank M.

Gasp erini, M. Giovannini, V. Mukhanov and G. Veneziano for enjoyable and

fruitful collab oration and S. Finn, P. Michelson P. Saulson and N. Rob ertson

for discussions ab out gravitywave detectors.

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