Volume 207, number l PHYSICS LETTERS B 9 June 1988

O N T H E D Y N A M I C S O F T H E P O W E R LAW I N F L A T I O N DUE TO AN E X P O N E N T I A L P O T E N T I A L

Jun ' ichi Y O K O Y A M A Department of Physics, Faculty of Science, The University of Tokyo, Tokyo 113, Japan

and

Kei- ichi M A E D A NASA/Fermilab Astrophysics Center, Fermi National Accelerator Laboratory, Box 500, Batavia, IL 60510-0500, USA and Department of Physics, Faculty of Science, The University of Tokyo, Tokyo 113, Japan

Received 9 March 1988

The power law inflationary universe model induced by a scalar field with an exponential potential is studied. A dissipation term due to particle creation is introduced in the infiaton's classical equation of motion. It is shown that the power index of the inflation increases prominently with an adequate viscosity. Consequently, even in theories with a rather steep exponential such as some supergravity or superstring models, it turns out that a "realistic" power law inflation (with a power index p > 10) is possible.

The solution of the long-standing cosmological problems such as hor izon and flatness problems is now generally a t t r ibuted to inflation, or the exponen- tial expansion o f the cosmic scale in the early uni- verse [ 1 ]. Such exponential inflation may be realized i f a scalar field with a sufficiently flat potent ia l exists and i f its energy contr ibutes dominan t ly to the en- ergy densi ty of the early universe [2,3] . Unfor tu- nately, however, no scalar field with such a potent ia l is known yet to exist natural ly in high energy physics theories, such as superstr ing theory, which are ex- pected to describe the early his tory of the universe [4,5].

Exponent ial inflat ion, however, is by no means a necessary condi t ion to resolve the above-ment ioned cosmological problems. Models which may solve these problems without resorting to exponent ia l ex- pansion are also known as general ized inflat ion models [ 6 - 8 ] . They are based on the fact that any type of accelerated expansion may solve hor izon and flatness problems in principle [ 6 ].

Among various generalized inflat ionary models one

1 Permanent address.

with par t icular interest to us is the power law infla- t ion model [6,7], in which the cosmic scale factor a( t ) grows as

a ( t ) = a o t ~, p > l . (1)

Such a power law expansion may be relal ized i f a sca- lar field ~ with an exponent ial potent ia l V I i i = Vo exp ( - 20) dominates the energy densi ty of the universe. Indeed there is a set o f exact solutions in the spatial ly flat Rober t son-Walke r spacet ime in which a( t ) shows power law behavior , as will be seen below. The Einstein equations and the classical equa- t ion of mot ion o f the scalar field 0 are wri t ten as

( & / a ) 2 - - H 2 - ! f !~2-.t- - 3 t 2y - Vo exp ( -2¢~) ] - k / a 2 ,

(2)

} '+ 3 H } - 2 Vo exp( -2¢~) = 0 , (3)

where Vo and 2 are constants and 8riG is taken unity. Then these equations posses the following exact so- lution i f k = 0 [9,10]:

¢~c =#co + ( 2 / 2 ) l n ( t / t o ) , (4)

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V o l u m e 207, n u m b e r 1 P H Y S I C S L E T T E R S B 9 J u n e 1988

a(t)=aot p, p = 2 / 2 2 , (5)

where G (t) denotes a classical solution and ao, to and Go are integration constants. Thus power law infla- tion may take place if2 < x/2. In fact it has been shown by Halliwell [9 ] that the above solution is an attrac- tor for all the k~< 0 and some k> 0 initial conditions in the Friedmann-Robertson-Walker (FRW) universes.

Scalar fields with an effectively exponential poten- tial appear naturally in supergravity theories and su- pers•ring models [ 11-13 ]. However, their potential is usually so steep that the power index of the scale factor cannot be much larger than unity, which makes it difficult to construct an acceptable inflationary universe model. For example, we find 2=x/-2 and x/6 for two scalar fields in N = 2, six-dimensional su- pergravity model with S2-compactification [ 12 ], and the same is true for two scalar fields in N = 1, ten-di- mensional supergravity model with gaugino conden- sation [13]. What we intend to do here is to reconsider the dynamics of the classical field G ( t ) in the power law inflationary stage to show that the power index p increases sufficiently for inflation to occur with an adequate viscosity.

In exponential inflation, the time variation of the inflaton field ~0 is generally negligibly small during the inflationary stage so that the dynamics of the field is well described by an equation of motion of the form [3]

3H~b+ V' [~0] = 0 , (6)

while in power law inflation, the inflaton G rolls down the exponential potential rather rapidly even in the inflationary stage. Indeed under the exact solution of (4) the ratio of the potential energy to the kinetic en- ergy takes a finite value determined by the power p, a s ,

V[G]I½6~ = 3 p - 1. (7)

Thus in power law inflation the time variation of the classical field G is not negligible.

This implies that the couplings of some field with become time dependent through G ( t ) and those

particles are produced due to G's variation [14]. Then the classical field G ( t ) loses it energy through these dissipation processes. In order to take into ac- count this energy dissipation, we have to introduce a

viscosity term in the inflaton's equation of motion [ 15 -17 ]. That is, eq. (3) should be modified to

~'+ 3 H ~ - 2 Vo e x p ( - 2 ¢ ) + C v b = 0 , (8)

where Cv is the phenomenologically introduced vis- cosity coefficient which generally depends on G, 0~, and the coupling strength between ~ and other fields.

Here let us investigate the effect of introducing a viscosity term as in (8) on the dynamics of the infla- tion. To do this - since we, unfortunately, have yet no method to calculate Cv properly - we assume that time scale of energy dissipation is proportional to the inflaton's "mass". That is, we assume that C~ takes the form

Cv - f M o - f V " [G] 1/2, (9)

where the constant f is treated as a free parameter. We further assume, for simplicity, that the inflaton energy released through the viscosity term is con- verted into that of radiation. Then energy density of q~ and that of radiation, which we write aspo~ andprad, respectively, evolve in terms of

dpoc= -3Hb2-C~b~, ( I 0 )

dprad/dt = - 4Hprad + Cv0~. ( I 1 )

Then the dynamical equations and a hamiltonian constraint equation of the above system are given as follows:

b'~ + 3H~c + f2x /V Oc-2V=O , (12)

/~/~ I " 2 - - ( i ¢ c -1- ] P r a d ) + k / a 2 , ( 1 3 )

HZ =31 (p~ +Prad) - k / a 2 • (14)

Defining a new time variable by

z - ~ V[G(t)]~/Rdt (15)

and

a ( r ) - l n a(z) , ~('C)=-Prad/V[~c], (16)

we obtain from ( 12 ) - ( 14 )

tt 1 t2 t • t q ~ c = ~ 2 ~ - f A ~ - 3 ~ a + 2 , (17)

a " ~ 1 d~t2 ..[_ 1 t r - g v c --~2~ca -2a ' 2+ -~ , (18)

Odt2 - - l ~ t 2 -1- - - 6 ~ c - - 1 "[- l a , ( 1 9 )

where we have set k = 0 for simplicity and where the prime denotes differentiation with respect to ~.

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Volume 207, number l PHYSICS LETTERS B 9 June 1988

In order to examine the dynamics of the above sys- tem, we draw a diagram of (O'c, c~' ). To do this, let us first draw the lines o f ~ ' = 0 and od' =0 , which are given by

~ ' = 0 = ½2 [0'c - (3 /2 ) (o~'+ -~f2) 12

- (9 /22) (o~'+ ~ f 2 ) 2 + 2 , (20)

I t 3 t ~20L )2...~ (2- 3 2 ,2 a " = 0 = ~ ( q $ c - ~2 )oz - ~ . (21)

As is seen above the inclusion of the viscosity term of the form of (9) implies that o~' is transported just to o~' +1 f 2 in eq. (20). Since 5 is non-negative, only the region satisfying

~2~ 1 ~ , 2 . . I_ ~- gv-~ --{ (22)

has physical significance. The resultant diagram is shown in fig. 1.

First let us consider the case f = 0 (no viscosity). As is seen in fig. 1, there are two stationary points with o~' > 0. Solving (20) and (21) with f = 0, we find

(O', a ' ) = a o 6x/-6-L~_ 2 2 ,

B(2, ½2). (23)

The point B appears outside the physical region as long as 2 < 2 . On the other hand the point Ao is just

(z'

,\

, \\, / / /

\ \~ //

\xx~,. \\ l II

5=0 ~

%"=0 (f¢o)

. (%' / Fig. 1. Dynamical behavior of the universe in the diagram of (0;, a ' ) for the case 2 < 2. The power law solutions, A / (f# 0) and Ao (f=0) are attractors as shown by solid and broken ar- rows, respectively.

on the 5 = 0 line and furthermore it is an attractor for all the initial conditions (~'c, a ' ), provided 2 < 2. At Ao (~ co, a 3 ) the scale factor behaves as

a( t )=aot p, p=(2/2)(a'o/~)'co)=2/22 (24)

and 6= 0. That is, power law expansion with p = 2/22 is realized even in the initially radiation dominated case. Note that p = 2/22 is larger than 1/2 (the power index in the radiation dominated era) for 2 < 2. Ra- diation energy redshifts to 0 owing to this rapid ex- pansion caused by the exponential potential.

Next we consider the case with nonzero viscosity. The stationary point in the physical region, denoted by Ar(O'cy, a) ) , is also an attractor for all the physical initial conditions (O'c, ce' ) in this case. At Arthe scale factor shows asymptotically power law behavior with its power index given by

pf= (2 /2 ) (a'JO'f) , (25)

which is obviously larger than 2/22 as is seen in fig. 1. At this point 5 takes a finite value determined by (19). That is, if we take into account the effect of viscosity due to particle production, the power o f in- flation increases and finite radiation energy density remains. Fig. 2, in which 2 = 1, shows an example of the dependence of p / and 6 on the viscosity strength

P 16

14

12

10

8

6

P

4 r O . l O

2 0 . 0 5

0 ........ i ....... , ........ i ........ i ....... I0.00 10 3 10-2 10-1 10 o 101 10 2

f Fig. 2. Dependence of the power index p and 5 on the viscosity strength ffor the case 2 = 1.

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Volume 207, number 1 PHYSICS LETTERS B 9 June 1988

f. As is seen in fig. 2 the power index increases prom- inently if f is larger than ~ 3. This feature is common to all the cases with 2< 2, that is, the power becomes greater than ~ 10, if f > O (3-10).

There are several merits in power law inflation with larger power index. First the duration time of infla- tion which is necessary to explain the homogeneity and flatness of the present universe gets shorter. Sec- ond the spectrum of the density fluctuation after in- flation becomes closer to the Zel'dovich type, since the spetrum depends on p as

6p/pl,.(~) ock -1/(l+t') (26)

at the time tn(k) when fluctuation with wave num- ber k gets into the horizon [ 7 ]. It is desirable to have a power law inflation model of a larger power with the help of the effect of viscosity.

Hence it is important to examine if there is an ad- equate viscosity. In what follows, we investigate this briefly. As we mentioned earlier, we have no method to calculate Cv properly. However, it has been evalu- ated by perturbative expansion under the assump- tion that q~c(t) varies adiabatically and it has been shown to be related to the decay rate of the q~ particle through the imaginary part of its propagator [ 17]. We have performed a similar calculation in the pres- ent case with an exponential potential and yielded at the lowest order

C,~½F, (27)

where F stands for the decay rate of the ~ particle. In order to evaluate F, the coupling between ~ and ex- ternal fields rffust be specified.

For the typical coupling of a scalar field which ap- pears in supergravity theories with and exponential potential, we may consider the Yukawa type cou- pling ~ZiZi or the exponential coupling exp(-m2~)ZtZ~ with m a parameter and where Xi represents a fermion field such as the gaugino, the coupling strength may be of the order of unity. Then the decay rate F for each case is written as

F~ (n/4n)M~,

F~ (n/4n)mL~ 2 exp(-- 2rn~4~c) M~, (28)

respectively, for the lowest order of 2 where n stands for the number of decay modes, which can be

~O(100) . Putting (28) into (27) we find the di- mensionless parameter fas

f - Cv/Mo ~ n/8n,

f ~ (n/Sn) m222 exp ( - 2rn2Oc) . (29)

Since we are dealing with a strongly coupled case, formula (27), which has been evaluated perturba- tively, does not give a quantitatively correct viscos- ity. However, from (29), we may know qualitatively the effect of viscosity on the dynamics of the universe.

For the Yukawa coupling case, (29) possesses the same form as (9) so that the power index may in- crease prominently owing to viscosity. For the expo- nentially coupled case, the effective coupling strength decreases with time so that the initially large power index of inflation eventually decreases to p -- 2/22. We may need more analysis to see whether we can find a sufficient power law inflation 2 ~ O ( 1 ).

So far we have concentrated on the inflationary stage in which the potential has effectively an expo- nential form V= Voexp (-2q~). In order to make a vi- able cosmological scenario, however, the universe must evolve into the radiation-dominated Fried- mann stage after enough inflation and successful re- heating. To implement this requirement, the potential must have a global minimum, where V= 0 to avoid nonzero cosmological constant, and ¢ should settle down there after inflation. Potentials with such a property appear in supergravity and superstring models [ 5,12 ].

Now let us describe an inflationary scenario based on an "inflaton" field with such a potential. To do so, since realistic potentials which appear in particle the- ories are too complicated, we make use of a toy potential:

V(q)) =2Vo [cosh (20) - 1 ]

-~ Voexp( -2¢) for -q~>> 1/2,

Vo2202 for I¢l << 1/2, (30)

where 2 < 2. At the Planck time when a classical description of

the evolution of the universe becomes possible, the universe is in a chaotic condition due to quantum and thermal fluctuations and the potential energy of 0 is typically of the order of the Planck scale [ 3 ]. Hence

may start at a large negative value. Then, as shown

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Volume 207, number 1 PHYSICS LETTERS B 9 June 1988

above, the in f la ton ' s energy soon becomes d o m i n a n t and the v i scos i ty -enhanced power law in f l a t ion is re- alized, which lasts un t i l ~ rolls d o w n the po ten t ia l to 10[ ~ 1/2 or V~ Vo where the po ten t ia l has no longer an exponen t ia l form. T h e n in f la ton starts to oscil late a r o u n d the global m i n i m u m an d the rehea t ing takes place to restore the r a d i a t i o n - d o m i n a t e d F r i e d m a n n universe.

For the above scenar io to be viable, it mus t satisfy several condi t ions , i.e. long enough in f l a t iona ry pe- r iod to solve the hor izon an d the f latness problem, the p roduc t ion of appropr ia te densi ty f luctuat ion, a n d good reheat ing. All these cond i t i ons t u rn ou t to be cons t ra in t s on Vo. If, for example , we assume the power index in the in f l a t ionary phase to b e p = 10 due to viscosity, the most s t r ingent cons t ra in t on Vo comes f rom the dens i ty pe r tu rba t i on r equ i remen t , which is Vo-- O ( 10 - ' 3_ 10 - i s) [ 7 ]. Once this c o n d i t i o n is satisfied, the un ive r se expands by a factor o f ~ e 2°° du r ing the in f l a t ionary stage which is cer ta in ly large enough. Fu r the rmore , since the present " i n f l a t o n " is strongly coupled with ma t t e r fields, the rehea t ing oc- curs rapidly in the field osci l la t ion phase so that we yield a reheat t empera tu re T = O ( 1 0 '4 G e V ) , which is high enough for baryogenesis [ 18 ]. No te that the above cons t ra in t on Vo can be sat isf ied in more real- istic potent ia l s which appear in some supers t r ing models by an appropr ia te choice o f mode l pa rame- ters [5 ]. In this choice the g rav i t ino mass is large enough for its o v e r p r o d u c t i o n in the rehea t ing phase to be avoided.

In conc lus ion we have shown that we m a y con- struct an acceptable in f l a t iona ry scenar io even in theories with a scalar field hav ing a ra ther steep ex- ponen t i a l po ten t ia l thanks to the f r ic t ion against the in f la ton ' s m o t i o n due to part ic le creat ion, i f the field is strongly coupled with o ther fields.

J.Y. is grateful to Professor K. Sato for his con t in -

uous encouragement . K.M. acknowledges Rocky Kolb and Theore t ica l Astrophysics G r o u p at Fe rmi l ab for the i r hospital i ty. This work was par t ia l ly supppor t ed by the D O E and the NASA at Fermi lab .

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