Nuclear Ph,,,,ic,. B226 (1993) 5 2 7 - 5 4 6 ' N o r t h - I l o l l a n d P u b h s h m g C o m p a n y
Q U A N T U M FI,UCTUATIONS IN THE NEW INFI,ATIONARY UNIVERSE
Ale,~andcr V I L F ~, K I N
Ph~ ~l~ ~ Department, Tu/t~ I m t er~tt~, l'ledf~rd. ~1,~ 02155. ~ ~ |
Recev, ed 1 r e b r u a r y 19~3 ( R e v i s e d 13 J u n e 19~3)
The e'~olutton of q u a n t u m fluctuation., of a s~.alar field ~n de Sitter space 1., a n a l y z e d m tile u ) n t c \ I of the nev, mflat~onar', sc_enar.) "I he d u r a t . m of the mflat.)nar~, p h a s e v, ,2,.tlllla.tcd and the p r o b l e m of denslt', p e r t u r b a h o n s resulting from q u a n t u m f l u c t u a t i o n s of the l h g g ' , field is ,.h so. us.,cd
I. Introduction
T h e i n f l a t i o n a r y c o s m o l o g i c a l m o d e l [1] a s s u m e s a f i r s t - o r d e r p h a s e t r a n s i t i o n in which the u m v e r s e s u p e r c o o l s a n d passes t h r o u g h a de Sitter p e r i o d o f e x p o n e n t i a l e x p a n s i o n . If the xnflatlonary p h a s e is sufficiently long, this m o d e l can give a n a t u r a l s o l u t i o n to the horizon, flatness a n d m o n o p o [ e p r o b l e m s Inltmll 3, it ~ a s not clear h o w to e n d i n f l a t i o n a n d get b a c k to a r a d l a U o n - d o m m a t e d universe. A s s u m i n g a
" ' n o r m a l " f i r s t - o r d e r p h a s e t r a n s i t i o n o c c u r r i n g t h r o u g h b u b b l e n u c l e a t i o n a n d coalescence, G u t h [1] a n d G u t h a n d W e l n b e r g [2] have shown that st leads to a strongls, i n h o m o g e n e o u s p i c t u r e o f the universe. Recently. L l n d e [3] and A l b r e c h t a n d S t e l n h a r d t [4] have suggested that this p r o b l e m can be a v o i d e d in C o l e m a n - W e l n b e r g t y p e models, in which the b a r r i e r m the effective p o t e n t i a l d i s a p p e a r s or b e c o m e s very small as the t e m p e r a t u r e goes to zero. A t s o m e t e m p e r a t u r e T , the false v a c u u m Is d e s t a b i l i z e d by t h e r m a l or q u a n t u m f l u c t u a t i o n s , a n d the Hlggs field q5 s t a r t s " r o l l i n g " d o w n the effective p o t e n t i a l while the unlver~' keeps e x p a n d i n g e x p o n e n t i a l l y . T h e C o l e m a n - W e l n b e r g p o t e n t i a l is very flat nca, the origin, a n d the r o l l o v e r time can be m u c h g r e a t e r than the e x p a n s i o n time, t i ~, where t l zs the H u b b l e c o n s t a n t o f the d e Sitter space. It IS e x p e c t e d that the initial scale o f s p a t i a l
• ,a r l a t i o n of the Higgs field is - I I ~, so that one can talk a b o u t f l u c t u a t i o n region,, o f initial size - I1 1. If the rollover time is g r e a t e r than ~ 50 H l then the whole visible universe is c o n t a i n e d m one f l u c t u a t i o n region. Thl,, ne',~, version of G u t h c o s m o l o g y ~s o f t e n called the new i n f l a t i o n a r y ~cenarlo
527
528 A VtlenAm / Quantumfluctuattons
Destabilization of the false vacuum in the new inflationary universe, with quan- tum and gravitational effects taken into account, has been discussed in refs. [5-8]*.
It has been shown that the symmetric state is destabilized much earlier than one would naively expect, due to anomalous behavior of fluctuations of a massless minimally coupled field in de Sitter space. For small values of ,#, the one-loop Coleman-Weinberg effective potential in de Sitter space has the form [11.10]
v ( ~ ) = ~ m 2 ( T ) q , 2 - {X,t, 4, (1.1)
where r e ( T ) = A g T is the temperature-dependent effective mass due to interaction with gauge bosons, g is the gauge coupling. X = Bg41n(o/H),q~ = o is the true minimum of the effective potential, A and B are model-dependent constants. At the one-loop level, the symmetric state ( 0 ) = 0 remains quasistable at all nonzero temperatures. However, when r e ( T ) becomes sufficiently small, hlgher-ordcr correc- tions become important.
The effective HIggs mass at ( ~ ) = 0 equals
rn~f, = ( V " ( eO)) = rn2( T ) - 3 X ( , 2 ) . (1.2) The last term in eq. (1.2) is the contribution of the Hlggs field fluctuations. The infinities in (~2) can be absorbed by renormahzing the mass. m = m ( T = 0), and the conformal parameter,/j. The renormahzed values of m and ( are assumed equal to zero. In fiat space-time (,~2) = 1 T 2 and X(q52) is always much smaller than m 2 ( T ) However, in de Sitter space the behavior of (~2) is quite different. As long as the fluctuations are not too large, one can use perturbation theory and calculate (~2) for a noninteractmg field (X = 0). Assuming that the de Sitter stage is preceded by a radiation-dominated Robertson-Walker expansion*" and representing (,#2) as a sum of thermal and vacuum contributions,
one obtains [5, 81
and [6.71
(.62) = ('/'-~) t + ( , 2 ) , .
(1.3)
T 2 H 2
(q52) r--- ~ - + 8~A~--g (1.4)
( ~ 2 ) ~ _ H 3 t / 4 , / r 2 + C H 2. (1.5)
* Hay, k i n g a n d M o s s [9] h a ' , e ¢.onsldered a m o d e l m w h i c h tile l h g g s field ha,, a small m a s s , m > 0, ,,o t h a l the false ~ a t . u u m r e m a i n s q u a , q s t a b l e d m s n to z e r o l e m p e r a m r e a n d d e c a , , s b', q u a n t t n n t u n n e h n g T h e rollo'ver t u n e r m the Hay, k i n g - M o s s , < e n a n o h a s b e e n e,,tmlated in ref [10] [ h e r e it p, ,,hown t h a t a l t h o u g h r d e p e n d s o n m . it c a n n o t be m a d e g r e a l e r t h a n - ( ~ I I ) i w h e i e c, ~ ~ 2 / 4 v , g is the g a u g e c o u p h n g
* * O t h e r b e g i n n i n g s o f the d e Sitter s t a g e h a ~ e also b e e n dis~.u,,,,ed in the l i t e r a t u r e [I5]
A Vtlenkm / Quantum fire tuatton~ 529 Here, C - 1 is a c o n s t a n t and time t is set equal to zero at the b e g l n m n g of the de Sitter stage. T h e temperature decreases like exp( - lit), so that (02)-1- a p p r o a c h e s a constant, while (q,2), grows (This unusual behavior of (q~2), is explained in sect.
3.) On the other hand, m 2 ( T ) decreases like exp( - 2Ht), at some point m~r f becomes negative and the s y m m e t r i c state is destabilized. For typical values of the parame- ters. this happens when ( ~ z ) is not much different from tl 2.
T h e p u r p o s e of this paper is to analyze the evolution of q u a n t u m fluctuations after the false v a c u u m is destabilized. Initially the fluctuations will grow a c c o r d i n g to eq. (1.5). W h e n (q~2) grows sufficiently large, self-interaction becomes important, and the behavior of (,~)2 is modified. Finally, at some point a classical description of the field 0 becomes appropriate, and the following evolution is described by the classical field equation.
Eq. (1.5) has been obtained in refs. [6,7]. T h e derivation of ref. [6] is rigorous but rather complicated, while that of ref. [7] is too sketchy. Since eq. (1.5) is i m p o r t a n t for the new inflationary scenario, I think It IS worthwhile to give a simple derivation of this equation, which will emphasize its physical meaning.
T o include the effect of self-interaction, one has to calculate ( 0 2 ) as a function of time for the self-interacting field q~ in de Sitter space. Needless to say, this problem is not tractable exactly even in flat space-time, and one has to resort to a p p r o x i m a t e methods. An expansion in powers of ~. is not very interesting, since it breaks d o w n when the fluctuations b e c o m e large. We shall do a little better than that and calculate ( ~ ? ) in a self-consistent a p p r o x l m a u o n , which a m o u n t s to a s u m m a t i o n of an infinite set of " c a c t u s " diagrams (cactus approximation).
The paper is organized as follows T h e self-consistent a p p r o x i m a t i o n is introduced in sect. 2. Sect. 3 reviews the k n o w n results c o n c e r n i n g q u a n t u m fluctuations for a free scalar field in de S m e r space and gives a new derivation of some re',ult~ using less rigorous but more intuitive methods These m e t h o d s are used in sect. 4 to analyze the evolution of q u a n t u m fluctuations in the new inflationary universe. In the same section, l estimate the rollo~,er time "r. T h e result is in a quahtatlve agreement with the estimation obtained by L m d e [7]. Sect. 5 discusses the problem of cosmological density fluctuations p r o d u c e d by the q u a n t u m fluctuations of the Hlggs field ¢, T h e results are summarized and discussed in sect 6.
2. Self-consistent approximation
Wc shall consider a q u a n t u m field 0 described by the lagrangian 2 -
(2 1) in de Sitter space. T h e absolute m i n i m u m of V(~) is at ~ = o >> H. T h e metric of the
530 4 ~'th'n~l~l / Quwltum flu(tuatlons
de Sitter s p a c e can be w r i t t e n as
d s 2 = d r 2 - e x p ( 2 1 1 t ) d x z (2 2) T h e r a d i u s of the d e Sitter h o r i z o n is I1 i. W e shall d i s r e g a r d t h e r m a l effects, ,,,ince they are u n i m p o r t a n t after the false v a c u u m is d e s t a b i l i z e d * T h e n the ( ' o l e m a n - W e l n b e r g p o t e n t i a l for ~ << o is
V ( 9 ) = - ' X , ¢ (2 3)
a n d the field ~ satisfies the o p e r a t o r e q u a t i o n
121,;5 - X4, ~ = 0 (2.4)
All i n f o r m a t i o n we need a b o u t the q u a n t u m f l u c t u a t i o n s is c o n t a i n e d in the G r e e n f u n c t i o n
G ( x , x') = ( r ~ ( ~ - ) ~ ( . ~ ' ) ) , (2 5)
which satisfies the D y s o n e q u a t i o n
riG(x, x') + f ~ ' ( ",;, x")(./(.~c", x ' ) ~ / - g d4x '' = - 1 6 ( .x, x ' ) (2.6)
Here,
27(x, x')
is the self-energy p a r t a n d the & f u n c t i o n is n o r m a l i z e d so thatf S ( x , U ) v / - g d a x ' = 1 . ( 2 V )
T o the first o r d e r in the s e l f - c o u p h n g X, s is given by the d i a g r a m s h o w n in fig l a : 221(x, x ' ) = - 37t(~52(x))8( v. x ' ) , (2 8) w h e r e
( , f - ( x ) ) = Go(x. x) (2.9)
a n d G o is the free field G r e e n function. ( ~ 2 ) and S i are, of course, d i v e r g e n t a n d s h o u l d be r e n o r m a h z e d As e x p l a i n e d in ref. [6], the infinities in S 1 can bc a b s o r b e d by r e n o r m a l i z l n g the m a s s m a n d the c o n f o r m a l p a r a m e t e r ,~ of the field ,~. (The
" The effect,, of "'I tawkmg temperature" are included m the ,,acuum part of (~2)
4 VtlenAtn / Quantum Hm tuatto,B 5t I
a) ~ b) c ~
I ~g 1 The self-energy {]lagrams (a) ul the fir,,/-order pcrturbatff}n thcor',, (b) ~n {.actu', appr(}xuTlat,on
r e n o r m a h z e d values of both ,,i and ,~ are assumed to be zero ) S u b s u t u t m g (2 8) in (2.6) we obtain, to first order m ~.,
[ZIG( ~, x") - 3tX<~2(x)>G(x, .~.") = -
la(x. x'), {2 lO)
T o make our a p p r o x i m a t i o n self-consistent, we can define ({p2/x in terms of the full Green function G rather than Go:
<~2(x)> = G ( x . x) (2.11)
Eqs. (2.10) and (2.11) with (¢2> properly renormallzed c o r r e s p o n d to the s u m m a t i o n of an Infinite n u m b e r of " c a c t u s " self-energy diagrams of the form shown in fig lb.
The " c a c t u s " a p p r o x i m a t i o n defined by eqs. (2 10), (2.11) is analogous to the Hartrce a p p r o x i m a t i o n used in m a n y - b o d y theory. Fhe same class of diagrams is included in the large-N a p p r o x i m a t i o n , which has been studied b~ a n u m b e r of authors [12] for a model of N scalar fields ¢,~ with a quartic self-interaction A({~,,4,,, )2.
It can be shown that the cactus diagrams give a leading contribution in the hmlt of large N. F o r n o t a u o n a l s~mphclty 1 shall continue to d~scuss the model of one self-interacting scalar field, the whole analys~s can be easily reformulated for the case of an N - c o m p o n e n t field. For N = 1 we have no guarantee that the omitted diagrams are u n i m p o r t a n t . However, by including an infinite set of dmgrams, we go b e y o n d p e r t u r b a t i o n theory and one can hope that the results in cactus a p p r o x i m a - tion will reflect the qualitative b e h a w o r of the full nonlinear theory.
Instead of calculating the Green function
G(x. x')
from eqs. (2.10). (2.11), one can use an alternative p r o c e d u r e which I fred more illuminating. In the same a p p r o x i m a - tion the field o p e r a t o r q}(x} can be represented as~ ( x ) = ( 2 ~ ' )
Vnfd~k[a,~,k(t)e'* "+h.c.], (2.12)
where the m o d e functions ~ ( t ) satisfy the equation
~ + 3H~k + k2e 2that ` - 3~<~2>~. = 0, and (q~2) is given by
( ~ 2 > = ( 2 ~ )
~fl¢~12d3k.
(2.13)
(2.14)
532 A Vdenl~m / Quantum flu~tuatums
T h e p r o b l e m thus r e d u c e s to the s o l u u o n of the c o u p l e d e q u a t i o n s (2.13), (2.14) A s i m i l a r s y s t e m o f e q u a t i o n s has b e e n used xn ref. [10] to c a l c u l a t e the effective p o t e n t i a l in de S~tter s p a c e m the c a c t u s a p p r o x i m a t i o n .
Yet a n o t h e r v, ay o f f o r m u l a t i n g the c a c t u s a p p r o x i m a t i o n is in terms of an o p e r a t o r equatxon,
Ulq, - 3,X(q}'-)~ = 0. (2.15)
H e r e I have o m i t t e d the m , ~ a n d X r e n o r m a h z a u o n c o u n t c r t e r m s , which cancel out the d i v e r g e n c e s in (ca2).
In the next section we `shall dl`scuss the c a l c u l a t i o n o f (if2) for a free field Ca. This c o r r e s p o n d s to the f i r s t - o r d e r a p p r o x i m a t i o n for the ,self-energy part, eqs. (2.8). (2.9).
W e shall review the results o b t a i n e d m refs. [5-8], e m p h a s i z i n g the special case rn = ~2 = 0 a n d r e d e r l v e s o m e o f the results using le,ss r i g o r o u s b u t m o r e intuitive method,s. T h e s e m e t h o d s will then be used to a n a l y z e the f l u c t u a t i o n s in the c a c t u s a p p r o x i m a t i o n .
3. A free scalar field in de Sitter space
A free s c a l a r field o f m a s s m a n d c o n f o r m a l c o u p l i n g ,~ ts d e s c r i b e d by the e q u a t i o n
([] + m'- + ~R)q,= 0, (3 l)
v, here R = 12H-' is the c u r v a t u r e of the de S i t t e r space. T h e effccuve m a s s s q u a r e d of the field is m 2 + ~R. F o r m 2 + ~R > 0 the t h e o r y is s t a b l e a n d for m 2 + ~'R < 0 it is u n s t a b l e ( t h a t is, the field has g r o w i n g m o d e s ) W e ,shall p a y special a t t e n t i o n to the i n t e r m e d i a t e case, m 2 + ~R = 0. m which the b e h a v t o r of the t h e o r y is very interesting, a n d w h i c h is most relevant for the m f l a u o n a r y ,scenario. In the f o l l o w i n g I shall set ~ = 0 for s l m p h c t t y T h e d e p e n d e n c e on ~ can be recovered b~y r e p l a c i n g m'- ~ rn 2 + ,~R everywhere.
T o c a l c u l a t e v a c u u m averages like ( ~ 2 ) , in c u r v e d s p a c e - t i m e , o n e has to specify w h a t is m e a n t by " v a c u u m " . T h e c h o i c e o f a v a c u u m m curved s p a c e - t i m e ~s not unique, reflecting the fact that there is no u m q u e division into positive a n d negative f r e q u e n c y m o d e s . H o w e v e r , for a s t a b l e t h e o r y ( m 2 > 0) m a de Sitter space, there exists a u n i q u e d e S ~ t t e r - i n v a n a n t v a c u u m state, in which all the v a c u u m averages have the full s y m m e t r y of the de Sitter space. Bunch a n d Davies [13] (see also ref.
[14]) have c a l c u l a t e d the e x p e c t a t i o n values of q,2 a n d of the e n e r g y - m o m e n t u m t e n s o r T m this state T h e d i v e r g e n t p a r t of (,¢,2) has the form A + B R . where A a n d B are infinite eon`stants. T h e s e infinities can be a b s o r b e d b y the r e n o r m a h z a u o n of m a n d ,~. In the case o f small m a s s ( m << H ) , v, hich is of m o s t interest to us, the finite p a r t of (q,2) m the B u n c h - D a v i e s v a c u u m is [6,7]
3 H 4
(qb2)BD 8'/rzm 2 + O ( 1 1 2 ) . (3.2)
A Vdenkm / Quantumfluctuattons 533 T h e v a c u u m e n e r g y - m o m e n t u m tensor is given by [13]
= ¼ ,,, 2( , 2 ) g , , . ( 3 . 3 ) If the v a c u u m state is not de Sitter i n v a n a n t , it can be shown [6] that (@2) and (Tu,) a p p r o a c h the de Sitter-mvariant values (3.2). (3.3) on a time scale - t t / r n
2
and - H 1 respecuvely. In other words, all excitauons over the Bunch-Davies v a c u u m are redshifted away.In the lirmt m--* 0, (q~2)B D m eq. (3.2) dwerges. This infrared divergence artses even in the p o m t - s h t t e d expression (@(x)qS(X'))B D, indicating that a de Sltter- m v a n a n t v a c u u m state does not exist for a massless, m i m m a l l y coupled (~ = 0) field.
If we set t = 0 at the beginning of the de S~tter phase, then it can be shown that, for t > > H 1, ( 0 2 ) ls given by [6, 7]
(q?) = A + H3t/aTr 2,
(3.4)where A = const.
T o u n d e r s t a n d this unusual behavior of (~2), we have to analyze the behavior of the m o d e f u n c u o n s + k ( t ) m de S~tter space. F o r a massless, minimally coupled field, the m o d e functions are given by
~k~ ( t ) ( ~ r ) = 1/2
HG(k,7) ] , (3.5)
where rt = - H -~e- m 1c212 _ icll2 = 1 and
HJX/)2(x), HJZ/~(x)
are Hankel functions,tt~2(x)
= [H3~)~(x)] * =-(2/~rx)l/2e
'~(1 +1~ix).
T h e wavelengths of the modes grow hke e m,h = k l e x p ( H t ) . (3 6)
In flat space-time we would define q,a(t) to be posluve frequency functions. In an e x p a n d i n g umverse, it makes sense to talk a b o u t p o s m v e and negative frequencies only for the m o d e s with wavelengths much shorter than the horizon, which go t h r o u g h m a n y osciilauons during one expansion time. In our case such are the modes with kl'q] >> 1 and rt3/2, " ' ( 2 ) ( k . q ) ' M O ) t ,.3/2~kr/) are the pos~twe and negative frequency functions, respectively. At the beginning of the de Sitter phase, t = 0, Ir/I = H ~, and we shall require that the coefficients c~(k) and c 2 ( k ) m eq. (3.5) are such that c 2---1 and c l---0 for k > > H . F o r k < H , c I and c 2 d e p e n d on one's a s s u m p t i o n s about the initial state of the universe and on the details of the t r a n s m o n to the de S~tter regime.
T h e crucml observation is that the m o d e functions (3.5) a p p r o a c h n o n z e r o c o n s t a n t s as t ---, oc. F o r k >> H,
Iq~12= ~ - S 1 + / ~ 5 e 2,, . (3.7)
We see that the h m m n g value
H Z / 2 k 3
is reached when ke u, becomes smaller than534 A Vtlenl~m / Quantumflu~tuattons
It, that is when the wavelength of the m o d e becomes greater than the de Sitter horizon, H - i. In the course of expansion, more and more modes c o m e out of the horizon, their c o n t r i b u t i o n s add up to (q,2), and (if2) grows u n b o u n d e d l y .
F r o m eqs. (3.2), (3.3) we see that, unhke (,~2), (T~,) BI~ has a finite hmlt at m ~ 0:
3 H 4
(L,,)BD --" 32~2 g . . (3.8)
As in the case of m 2 > 0, ( T ~ ) in an arbitrary state a p p r o a c h e s this limiting value on a nmescale - H 1. This difference between (~2) and (T~,,) is due to the fact that ( T ~ ) is not sensitive to the c o n t r i b u t i o n of very long wavelength modes.
A rigorous derivation of eq. (3.4) with p r o p e r regularlzanon and renormallzatlon of (q,2) has been given in ref. [6]. Here we shall rederlve this equation using simpler and more intuitive methods which, however, treat r e n o r m a l l z a u o n in a rather cavalier fashion.
T h e first a p p r o a c h is to calculate (if'-) from eq. (2.14) including only the c o n t r i b u t i o n of modes with wavelength greater than the horizon (since we know that these modes are responsible for the growth of (~,2)). Th~s means that the integration over k m (2.14) should be cut at k - He m. Then we can a p p r o x i m a t e l y replace ItPa I 2 u n d e r the integral by its limiting value. H Z / 2 k 3 :
(q~2) = A + 112 /;;cv, p(;;t) 1 H~t
4~
" ~ ~;t k d/, = A + - - (3 9)
4,n -2
Here A = const is the contribution of modes with k < t l (that is of the modes that had wavelengths greater than the horizon at thc beginning of the de Sitter phase).
T h e a n o m a l o u s growth of fluctuations is a specml feature of the de Sitter space. If the expansion was slower than exponential before t = 0. then the fluctuations at t = 0 have no reason to be large, and one can expect that A _< H 2
T h e u m e d e p e n d e n c e (q,2) cx t can be pictured as a b r o w n l a n motion of the field
¢;, As a result of q u a n t u m fluctuations, the m a g n i t u d e of q, on the horizon scale changes by +_ ( H / 2 1 r ) per expansion time ( / / 1 ) Then the a~.erage " d i s p l a c e m e n t "
squared is (~2) = ( H / 2 ~ r ) 2 N , where N = ltt is the n u m b e r of "'steps".
Eq. (3 4) can also be derived using the o p e r a t o r equation for the field o p e r a t o r ~.
Let us first consider a massive field.
Using this equation we find
(a +
m e ) ~ , ( x ) = o (3 lo)(17 + 2m2)(q) 2 ) = 2(q,.,ff " ) . F o r rn > 0, (~2) = const and eq. (3.11) gives
(~,,,q~ " ) = r/.12(q~2) .
(3 11)
(3.12)
In the hmlt m ~ 0
where I have used eq. (3.2).
4 l/llenkm / QuantunlflU~tuatmn~ 5"~5
(55 ,,55 ") = 3 t t 4 / 8 T r 2
o r
F o r a massless field 55 eq. (3.11) is
2 " 4 "~
~ 0 -~ )
=0,,55 5
= 3 H / 4 w -( d ~ 2 + 3 H d ( 5 5 2 ) = 3 H 4 / 4 7 r 2 .
(It is clear from the s y m m e t r y that ~7(55 2) = 0 ) T h e solutton of cq. (3.15) is (552) = A + B e - wn + H ~ t / 4 7 r 2 ,
(3.13)
(3.14)
(3.15)
(3 16)
A ( r . t) = (2w)
~fl4,~(t)lZe '*''
X " d ~ k . (3.18)F o r k >> t i , 1 ~ [ - ' is given b3, eq. (3.7). O m i t t i n g thc c o n t n b u n o n of m o d e s ',~,~th k < 1t, w e obtain
A ( r . t ) = ( 2 7 r ) 2 f S ( l l 2 + k Z e 2m)s'n/tr/,r dkk (3.19)
T h e physmal distance between the points x and x ' is / = r e x p ( l I t ) . For r << 1t i where r = I x - x'l. can be written as
I3.17)
A ( r, t ) = (55( x , t )55( x ' , t ) ) , in a g r e e m e n t w n h (3.4).
T h e two m e t h o d s just described wdl be used in the next section to analyze the g r o w t h of fluctuations m the cactus a p p r o x i m a t i o n .
It is interesting to study the spatial correlations m the field 55(x, t). T h e expecta- tion value of 55 ts (55) = 0, indicating that 55 can grow in the positive or n e g a n v e d l r e c t m n with equal p r o b a b l h t y . However, since the u m v e r s e e x p a n d s exponentmlly.
there are strong correlations between the values of 55 at points separated by distances smaller than H l e x p ( H t ) .
T h e correlation f u n c t m n
536
[that is, for 1 << H
A Vdenkm / Quantumfluctuattons
t e x p ( H t ) ] * :
1 47r 2H3----~t(1-\ n,1 )
A ( r , t ) = --14~r 2 - - - - ~ + - ~ , l n
HI .
(3.20)T h e first t e r m m (3.20) is the usual, f l a t - s p a c e , z e r o - p o m t f l u c t u a t i o n term. W e see f r o m eq. (3.20) t h a t for the & s t a n c e s
H - l << l<< H - t e x p ( H t ) (3.21) a n d for
Ht
>> 1 the c o r r e l a u o n f u n c t i o n is a slowly v a r y i n g f u n c u o n o f l. T h i s m e a n s t h a t if we d o a m e a s u r e m e n t of ,~ at two p o i n t s s e p a r a t e d b y a & s t a n c e in the r a n g e (3.21), the results will b e the s a m e with an a c c u r a c y - ( H t ) - t / 2 ( l n ( H l ) )1/2.
T h e c o r r e l a t i o n s g r a d u a l l y d i e o u t as we go to d i s t a n c e s - H t e x p ( H t ) . In the b r o w n l a n m o t i o n picture, the values of q5 at p o i n t s s e p a r a t e d by 1 << H - l e x p ( l I t ) m a d e m a n y b r o w n l a n s t e p s t o g e t h e r a n d s t a r t e d w a n d e r i n g a w a y f r o m o n e a n o t h e r o n l y a f t e r the c o - m o v i n g scale I c a m e o u t of the horizon.4. Growth of quantum fluctuations
N o w we shall use eqs. ( 2 . 1 3 ) - ( 2 . 1 5 ) to a n a l y z e the g r o w t h of q u a n t u m f l u c t u a t i o n s m o u r m o d e l of a s e l f - i n t e r a c t i n g s c a l a r field. S u p p o s e the false v a c u u m is destab~- h z e d at t = 0 a n d let us a s s u m e for s i m p l i c i t y that ( ~ 2 ) = 0 at that m o m e n t . (This c o r r e s p o n d s to n e g l e c t i n g the c o n s t a n t A m eq. (3.4).) I m t i a l l y ( ~ 2 ) wdi grow a c c o r d i n g to eq. (3.4). T h e n its g r o w t h will be a c c e l e r a t e d b y the negative effectwe m a s s s q u a r e d mc2ff = _ 3 ~ ( ~ 2 ) . Let us first c o n s i d e r the imttal stage of the e v o l u t t o n w h e n
( ~ 2 ) <<
H2/X,
(4.1)so that Im~Zffl << H 2. Comparing the last two terms in eq. (2.13) we see that only the modes with wavelengths greater than the hortzon ( k <<
He n')
are destabdtzed d u r i n g this stage, whtle the h i g h e r m o m e n t u m m o d e s are p r a c t i c a l l y u n a f f e c t e d . Like m the p r e v i o u s section, we shall d i s r e g a r d the c o n t r i b u t t o n o f w a v e l e n g t h s s h o r t e r t h a n the h o r i z o n a n d writeIt can be c h e c k e d t h a t
(dp2) = 2 ~ flilexp(th)l~b~]Zk2 dk.
t~a ( t ) = ~b~°~( t )exp{-~ fi(dp2) dt ) •
(4.2)
(4.3)
* Note that for r << H I the contribution of modes with k, <
II
to A (r,t)
is thc same constant 4 that appcars m cq (3 9) This contribution is omitted m eqs (3 19), (3 20),4 Vth'nktn / Quantum flu{tuattom 537
a p p r o x i m a t e l y solves the mode equation (2.13), provided that the conditions (4.1) and
( d / d / ) ( q ) 2 )
<< l_t((p2)
(4.4) are sausfled*. Here ,/,~)) is the u n p e r t u r b e d (~. = 0) m o d e function and t, is the time at which the wavelength of the m o d e becomes greater than the horizon,t a = n t l n ( k / H ) .
(4.5)Finally. the integration in (4.2) is taken over modes with X > H ~, and we can a p p r o x i m a t e l y replace Iq~°)l z by ItS h m m n g value,
HZ/2k 3.
T h e n eq. (4.2) gives
(~2) = 41r---S'l/
--~-exp ~ - d t 'H 2 f l i t t ,
- 4 ~ 2 , ( , d z e x p { ~ - f H , ( ~ 2 ( " ) ) dr'} (4.6)
where z =
ln(k/H).
Differentiating (4.6) with respect to t, we obtain a differenual equation for (~2):d 2 H 3 2~ ~ 2
~--] ( ~ ) = 4~r~ + ~ - ((#-)
.(4 7)
T h e solution of this equation with the initial condition ( ~ 2 ) I t = t ) = 0 is H 2
(~2( t )) - - - tan X t , (4.8)
where
F o r
Xt
<< 1,X = ( ~ ) 1/2'/T - ' t t . ( 4 . 9 )
(dp 2) = H 3 t / 4 1 r 2 , ( 4 . 1 0 )
* T h e lower h m l t of m t e g r a U o n m eq (4 3) 1s c h o s e n so that ~b~ -- ~b~ °) for t - t~, reflecting the fact that the & f f e r e n c e between G a n d ~b~ °~ is small for m o d e s with '*avelcngth,, smaller t h a n the horizon Eq (4 3) apphe,, for t > t~
53N 4 I, denAm / Quantum flu~tuatums as expected. The solution (4.8) can also be written as
where
H2 c o t [ x ( ,)] (4.11)
( 4 ' 2 ) - 27r¢-5~ ' ~ ' - "
t o = r r / 2 x = r r 2 ( 2 X ) - ' " " H ' (4 12) At t = t 0 eq. (4.11) diverges, but our a p p r o x i m a t i o n breaks d o w n earher For X ( t o - t)<< 1 we can write
I t
( 4 , : ) = 22~(r o - r) " (4 13)
N o w It lS easll) seen that both conditions (4.1) and (4.4) are sausfled if H ( t o - t ) >> 1.
F r o m eqs. (4.3) and (4.8) we find the behavior of the m o d e functions i+~l 2 _ H2 c o s x t ~
2A ~ c o s x t ( t > t ~ ) . (4.14)
Eq. (4.7) for (4,2) can also be derived using the o p e r a t o r equation m e t h o d F r o m the o p e r a t o r equation (2.15) we obtain
E3(4, z ) = 2(4, ,,ep ") + 6~.(4,2 ) 2 .
(4 15)
F o r (4,2) << it2/2~, one can expect that (4,,,4,'") is a p p r o x i m a t e l y given by the free field expression (3.13), since we know that this quantity is not sensitive to the c o n t r i b u t i o n of long wavelength modes. Then
(42
- - + 3 H It(4, 2) = 3 H 4 + 6~.(4,2) 2 . (4.16)d t 2 , 4 ~ r 2
F o r (4,2),<,<
H2/~,
d / d t << H, the second derivative can be neglected, and we obtain eq. (4.7).A l t h o u g h eq. (4.8) becomes inaccurate for large (4,2), it allows us to estimate the duration of the inflationary phase. The exponential expansion ends when (4,2) b e c o m e s - 0 2 . where 4, = o is the true m i n i m u m of the effective potential. The a p p r o x i m a t i o n leading to eq. (4.8) breaks d o w n for (4,2) > H 2 / k , when the growth of fluctuations is faster than the expansion, and " r o l h n g d o w n " from (4,2) _ t12/)x to ( 4 , 2 ) _ o2 takes no longer than - H l (it can be shown that, in the cactus a p p r o x i m a t i o n , (4,2) grows like ( r - t) 2 for (4,2) >> H 2 / ) k ) . This means that the rollover time r can be different from t o by no more than H ~, and thus
" r = , n ' 2 ( 2 ~ . ) - l / 2 H ] (4 17)
4 Ptlenl~m / Quantum tim tuatmn~ 539
T o have e n o u g h inflation, we need Hr >__ 50, and so X must satisfy the constraint
X Z 0 02. (4.18)
T h e accuracy of eqs. (4.17). (4 18) depends on the accuracy of the cactus a p p r o x i m a t i o n . The cactus a p p r o x H n a u o n ~s slmdar to the r a n d o m phase approxi- mation, since it neglects correlations between different modes. O n e can expect, however, that at some point correlations b e c o m e i m p o r t a n t and the behavior of the field q5 becomes close to that of a classical field, ~ . A classical behavior of the fluctuatxons has been assumed b~ several authors [16 18] who haxe &scussed the density fluctuations in the new mflaUonar~ scenario. As long as ~ << H 2 / X , the evolution of the classical field is described by the equation
d , 2X 4
~ = ~f]q,2, (4.19)
which has the solution
ea~ = 311/2X( t , ; - t ), (4.20)
where to = const
T h e u m e t , at which the classmal evolutkm takes over will be estimated in the next secuon. Here we note that eq (4.8) ~s certainly correct up to first order in ~, a n d one expects that the Ume t , should be after the quadratic and higher terms in the expansion of (4.8) in powers of )~ b e c o m e important. For Xt ~ 1. the first two terms of th~s expansmn,
H ~ t ( 1 XllZt 2 ')
(qS2)=7--~\47r- + - - + 6 ~ 2 "-- (4.21)
give an accuracy of better than 15%. This m & c a t e s that
t , ) X 1 (4 22)
If we match the s o l u u o n s (4.8) and (4 20) at t = t , , we find
Xt0 = X t , + ( 3 / ~ ' ) t a n X t , . (4.23) T h e rollover time r equals t~;, and we obtain from eqs. (4.22). (4.23)
1.57 < X r < 1.61. (4.24)
This differs from eq (4.17) by no more than 3%, i n & c a t m g that eq. (4 17) gives a reasonable e s u m a t e of the rollover time.
540 4 l/denkm / Quantum flu<tuatton~
5. Generation of cosmological density fluctuations
Eq. (4.17) estimates only the m e a n value of the rollover time. The actual value of -r can be different from (4.17) and can fluctuate in space. This is precisely the effect that gives rise to density fluctuations in the new inflationary universe [16-18]. If 6r~
is the f l u c t u a u o n of -r on a scale characterized by w a v e n u m b e r / , , then at the time when this scale re-enters the horizon [16-18]
8 p i p - H6"r~ . (5.1)
8r, can be found from
6"i" k - 8ep~/+ (5.2)
and [16]
8 d & ( t ) = [ k 3 A ~ ( t ) ] t/2.
(5.3)
Here, A , ( t ) is the Fourier t r a n s f o r m of the two-point correlation function (3.17),
,a,(t)= (2,,) 3f~a(x,t)e
" Xd3x.(5.4)
If the evolution of the field 4' is adequately described by classical field equations, then it can be shown that [16] 8 + k ( t ) e c ~ ( t ) and r k IS i n d e p e n d e n t of t. Estimating 6"r k from eqs. (5.2), (5.3) at the time when the galactic scale comes out of the horizon.
one finds for that scale 8 0 / p - 50. This is five or six orders of m a g n i t u d e too large.
Things look more encouraging in the cactus a p p r o x i m a t i o n . T h e fluctuation of in this a p p r o x l m a u o n can be written as
8eOk ( t ) = ( k / 2 1 r )3/21q~ ( t )l .
(5.5)
F r o m eqs. (4.8) a n d (4.14) we see that as t--* t o, the relative fluctuation of q~
a p p r o a c h e s a small c o n s t a n t value.
(6q~,) 2 ¢'2X
• - - cos Xta. (5.6)
(~,2) 8~r2
It can be shown that this relation is a p p r o x i m a t e l y preserved at late stages of the evolution when (qfl) > H 2 / X (see the Appendix). For ~qfl) - o 2, + m eq. (4 26) is
-- ~ 1 / 2 0 2 and
8~" k - cos X t ~ / 2 r r x l / a o . (5.7)
A Vdenktn / Quantumfluttuatton~ 541 F o r sufficiently large scales, c o s Xtk ~ 1 and
6p~, H
p 2 ,n-~l/4o •
(5.8)
With reasonable values of the parameters, eq. (5.8) gives ~pl,/p ~ 10 - 4. This is just the value needed for the galaxy formation!
T h e trouble with this a r g u m e n t is that the cactus a p p r o x i m a u o n neglects correla- tions between different modes and can hardly be trusted m the analysts of the fluctuattons of ~'. T h e time t , at which the classical description takes over can be estimated in the following way. T h e field q, fluctuates with a m p h t u d e - H on a timescale - H - i and thus the fluctuation of ,~ is 8,~ - H 2. (This follows also from eq. (3.13)). O n the other hand, from eq. (4.19), the classical " v e l o c i t y " is given by
~k = X q ~ / 3 H . O n e can expect that the transition to the classical regime occurs when ,~
becomes greater than 8~, that is when
(alp2)(/,) - )k- 2/3H2. (5.9)
A s s u m i n g that t k < t , and evaluating 8"r k at t - t , , we fred* using eqs. (5.2). (5.6), (5.9) and (4.8)
H S $ t - 8 e p k ( t , ) / H >_. ( 2 ~ ' ) - s / 2 _ 0.1. (5.10) T h e denstty fluctuations are still too large. It is possible to obtain ~ p / p - - 10 4 for t~ > t , , but this requires ridiculously small values of }, [16-18].
H o w can we save the inflationary universe? O n e possibility is to consider N - c o m p o n e n t Higgs fields with N >> 1, in which case the cactus a p p r o x t m a t l o n m a y be rehable [12]. Perhaps a more attractive alternative is to consider particle models which give a different shape of the effective potentml at small ,~. The authors of rcfs.
[17,18] have pointed out that the density fluctuations can be m a d e small if the effective potential is sufficiently flat somewhere m the range H << ~ << o.
T o illustrate this possibility, suppose that for q> >__ q'l << o the effective potential has the form
V(q,) = - lp2qfl (5.11)
with #2 << H 2 and ~ > 0. A model with V(q~) like (5.1l) has been recently discussed by D i m o p o u l o s and Georgi [19]. We first note that V(4~) c a n n o t keep the form (5.11)
*Eq (5 10) assumes that eq (48) apphes up to t - t , It Is po%lble, however, that there r, an intermediate regime when (~2) change,, from - H2/2~r~ 1/2 to - ,~ 2/~H2 In that case H6"r~ can be greater than given by eq (5 10)
542
A l/tlenl~m / Quantum fluctuatton~up to ea - o, since otherwise the false v a c u u m (ff = 0) energy is 0, - 1/2202 and H 2 = ~wGp, 4 # 202
a "tr ---g- <<
~2.m y
H e r e mp is the Planck mass and I have a s s u m e d that o << rap. In order to have
~2 << H 2, V(q,) should have a big dip at q~ > ~&. (The model of Ref. [19] does not s a u s f y this r e q m r e m e n t . ) It will also be necessary to r e q m r e that
~1 >>
I I 3 / t "t2" (5.12)A race thing a b o u t the model (5.11) ~s that it describes a free field wtth a t a c h y o m c m a s s ~p,,
(151- p 2 ) , = 0 , (5 13)
and the m o d e f u n c u o n s can be found exactly [13,6]:
q'h = ( ~Tr )'/~ HrlV2H,!2)( kTI). (5.14)
H e r e v/= - H - Xexp( - H t ) and
(9 p.2 ) 1/2 ~ -- I I ~
= " 3 . = (5.15)
T h e a s y m p t o t i c f o r m of q't at t >> tt is
I2p'2 t~)] (5 16)
Iq~(t)l 2--- ( l l 2 / 2 k 3 ) e x p ~ - ~ ( t - , and analysis s l m d a r to that of the p r e w o u s secuon g~ves
3 H a
[ ]
(q52) 80r2p, 2 exp ~ t ) - - ! - 1 . (5.17) W e shall use eq (5.17) a s s u m i n g that t is sufficiently large, so that the e x p o n c n u a l m the square brackets ,s much greater than one.
T h e f l u c t u a u o n of the rollover Ume, 8"r~. can be e s u m a t e d from
8"r~, - 3O~/cb, (5.18)
where 6e& ls given by eq. (5.5) and q~ = ((q~,))1/2.
" 3 ,,1/2 ~ ,
l l e x p ( - ~a-q I (5 19)
4 Vth,nl~m / Quantumflu~tuanon~ 543 We see that 80~/0 is small if the galactic scale comes out of the horizon at sufficiently late nine (t~ >> 3 H / ~ 2 ) . If this happens when {4, 2) - 4,], then
~ p / p -- 0 I H 3//x24,1 ,
(5.20)
and the fluctuations have the desired magnitude if 4,1 - 10~11~/# 2
If V(~) becomes much steeper than (5.11) at 4, > 4,1. the rollover time is the time it takes to get from (4,2) = 0 to (4,2) _ 4,]:
H'r ~ ( 3 H 2 / # 2 )ln( l ~ l / n 2 ) > ( 3 H 2 / p f l ) I n ( H / l ~ ) ,
(5.21)
where the last inequahty follows from the condition (5.12). The inflation is suffi- ciently large provided that H > 4p,.
The conclusion is that, depending on the parameters of the model (5.11), 6 0 / 0 can take practically any value, including the right one. Other forms of V(~) have been suggested in refs. [17, 18,20]. The effective potential used in refs. [18,20] is based on the geometric hierarchy model of Dlmopoulos and Raby [21]. This model can glve small density fluctuations, but probably falls to produce sufficient reheating [20]. In general, to produce small density fluctuations and efficient reheating, the effecm, e potential must have a rather special shape, and it is not clear whether such effecnve potential,, can be naturally obtained in realistic models.
6. Conclusions
In this paper we have discussed the physics of q u a n t u m fluctuations in de Sitter space for a simple model of a self-interacting scalar field, eqs. (2.1), (2.3). We have found the average "rollover" time in the new inflationary scenario:
,r = q7.2(2~, ) i / 2 H 1 (6.1)
This is in a quahtatlve agreement with the estimation obtained by Llnde [7J. (The difference is only in the numerical coefficient )
Requiring that the expansion factor, e x p ( H r ) , be sufficiently large ( H r > 50). we obtain a constraint on X:
~ . < 2 × 1 0 - 2 ( 6 2 )
A warning should be issued that, in general, the results of this paper are not dlrectl3 applicable to reahstlC gauge theories, even if the effective potential at small has the form (2.3). The reason is that the Hlggs fields in reahstlc theories are multi-component fields, and a potential like (2.3) is obtained by choosing some direction in the group space and suppressing other degrees of freedom This is all
544 A Vdenkm / Quantum fluttuattons
right if one is mterested in the classical evolution of the field; however, for the analysis of q u a n t u m fluctuations one has to include all degrees of freedom.
As an example, let us consider an O( N )-symmetric model
L = ~3~,~.O~'ep. + l~k(q~aq~a) 2 , (6.3)
where a = 1 . . . N. If we pick some direcUon in the N-&menslonal q,-space, say,
qa. = q~(X,0 . . . 0), (6.4)
then the Lagrangian for ~ takes the form of eqs. (2.1), (2.3). However, q u a n t u m fluctuations occur in all N components of %, and the equation corresponding to eq.
(4.7) is
d H 3
2A
d-t
(if2>
= 4~r ---S + 3-H ( N + 2)(q~2> 2 . (6.5) Here (¢2) = N 1(%q~ ). The rollover time for this model is= -H- 2 ( N + 2 ) X " (6.6)
We note that if N is not too large, eq. (6.1) stdl gives a reasonably good order-of-magnitude estimate of r. For N = 10, ~" changes only by a factor of 0.5 Eq.
(6.6) also suggests that taking into account the suppressed degrees of freedom wdl tend to decrease the magnitude of r. The single-degree-of-freedom effecuve potential for the standard SU(5) model has ~ - 0.5 [16], and we expect H~" < ~r2(2X) 1/2 - 10, which Is not enough.
The analysis of the density perturbations resulting from the q u a n t u m fluctuauons of q~ gives, m the cactus approxlmaUon, 6 0 / 0 - 1 0 - a _ an answer one may be tempted to believe. However, the cactus approximation neglects important correla- uons between the modes and probably becomes unrehable for large values of (q52).
The argument of sect. 5 suggests that the classical evolution takes over no later than when (,~) - )~- 2/~H2. This gives 8 p / p >_. 0.1, and thus the density fluctuaUons are too large. To get around this difficulty, one can consider effectwe potentials of a
&fferent shape. W h a t one needs [17,18] is an effecuve potenual which is sufficiently flat somewhere in the range H << ,b << o and has a big dip near q~ = o. The latter is required for efficient reheating [20].
I am grateful to Larry Ford for stimulating dtscussions. This work was supported in part by the National Soence F o u n d a u o n under G r a n t # PHY-8206202.
A Vtlen~tn / Quantumflu(tuatlon~ 545
A p p e n d i x
This appendix discusses the evolution of (q)2) when ( 0 2 ) >> H2/2t m the cactus a p p r o x i m a t i o n . In ttus regime the effect of space-time c u r v a t u r e is negligible, and we can use the M l n k o w s k y space d ' a l a m b e r t i a n in eq. (4.15). The growth of (,~2) is due _..,2 ~1/2 where m~f t = - 37x(~2). We to unstable modes with wavelengths X > ( '"off* ,
shall assume that the d o m i n a n t c o n t r i b u u o n to (q,2) is gwen by the modes with
>> ( _ mcrt2 )1/2; then
( + 2 ) >> ( ( V ~ ) 2 ) . ( A . I ) With this assumpUon, eq. (4.15) takes the form
d z
d t 2 (q5 2) = 2 ( ¢ 2) + 6 ~ ( 9 ~ 2 ) 2 . (A.2)
T a k i n g the e x p e c t a u o n value of the energy conservation law. T~., = 0. we obtain also
( ¢ 2 ) _ ~ . ( ~ 2 ) 2 = 2 E = c o n s t . (A.3) Eqs. (A.2) and (A.3) gwe the following equation for (qfl)"
d 2
dt 2 (q~2) = 92~(~2>2 + 4 E . (A.4)
At large values of (,~2) we can neglect the constant 4 E ; then eq. (A 4) has a s o l u u o n
(~2> = 2 (A.5)
3 a ( / 0 -- t) 2 "
In the same a p p r o x i m a t i o n , the m o d e functmns satisfy the equation
4'k = 32'(q'2) ¢ , = 2(to - t ) - 2 ¢ ~ , (A.6) which has solutmns with ~k cx (t o - t) 2 and q'k cc (t o - t) -i. Thus,
~b, = A k ( t o - t) -I + B k ( t o - t) 2. (A.7) a n d
1~,[ 2
- - -~ const as t ~ t o . (A.8)
546 4 l/tlenl~m / Quantum flu~tuatton~
T h e w e a k p o m t o f t h i s a n a l y s i s is t h e a s s u m p t i o n ( A . I ) w h i c h 1 c a n n o t n g o r o u ' , l ~ justtf3,. I h a v e s t u d i e d t h e s a m e p r o b l e m t a k i n g eq. (2.13), (2 14) as a s t a r t i n g p o m t , w i t h a n a p p r o p r i a t e c u t o f f in t h e i n t e g r a l o v e r / ~ T h e r e s u l t ts t h e s a m e , eqs. ( A , 5 ) a n d ( A . 8 ) It s h o u l d b e e m p h a s i z e d t h a t eqs. ( A . 5 ) a n d ( A . 8 ) a p p l y o n l y m t h e c a c t u s a p p r o x i m a t i o n . A s e x p l a m e d in sect. 5. t h i s a p p r o x i m a t i o n b e c o m e s u n r e h a b l e a t l a r g e v a l u e s o f (q52).
References [1] A l l (ruth, Ph3,, Re', D23 (1981) 347
[21 A ti Guth and E Wemberg. Nucl Phi,, B212 (1982) 321 ['q A D I rode, Phi,, Lett B108 (1982) 38'9
[4] A Albrecht and P J Stcmhardt, Phv,, Re', Lctt 48 (1982) 1220 [5] A Vdenkm, Ph,,,, Lclt B115(1982) 91
16] A Vflcnkln and I H Ford. Phys Rcv D26 (1982)1231 [7] A I) Lmde, Ph',,, I ett lI6B (1982) 335
I8] A Starobm,,k), Ph,~.~ Lctl I17B (1982) 175
19] S W I-lay, king and I G Mo,,,,. Phv,, Left BllO (1982) 35 [10] A ~¢llenkm. Nucl Phi', B226 (1983) 504
[111 (i Shore, Ann Ph',,, 128 (1980)376.
B Allen, ( a m b n d g c Um`*er,,lt> prcprmt (1982) [12] I Dolan and R Jacklv, Ph',', Re`* I)9 (1974) 3320.
H J S~.hmtzcr, Phi', Re', I)10 (1974) 1800, 2042,
S (oleman. R Jacklv, and H I) Pohtzer. Phv', Re`* I)10 (1974) 2491.
I t- Abbott. J S Kang and 1t J Schmtzcr, Ph>', Re', D13 (1976) 2212 [13] T S Bun~_h and P C W Da`*le,,, t>ro~- Rov Soc (London) A360 (1978) 117 [14] J S Dowker and R ('ntchlc'~, Phy', Re', I)13 (1976) 224
[15] R Brout, 1"" Englcrt and P Spmdcl. Ph',', Rcv Lctt 43 (1979) 417, A Starobm,,k~. Ph'¢,~ Left 91B (1980) 99.
J R (Jott, Nature 295 (1982) a04, A Vdcnkm. Phss Iett 117B(1982) 25
[16] A H G u t h a n d S-~t Pl. Phvs Re,, l.ctt 49(1982) 111{) [171 S W Ilav, kmg, Ph~s l e t t Bl15(1982) 295
[18] J P, ardeen, P J Stcmhardt and M '¢, Turner, Umvcrs~t`* of Pcnns~,l~ama prcprmt UPR-0202T (1982) [19] S l)lmopoulos and H Gcorgl. Ham'ard Umvcrslt,, prcprmt, IIUTP-82/A046 (1982)
[20] P J Stemhardt, Um`*crslt~ of Pennsvl`*ama prcprmt. UPR-01981 (1982)
