An inflationary cosmological model with induced gravity from higher dimensions

Volume 160B, number 1,2,3

PHYSICS LETTERS

3 October 1985

AN INFLATIONARY COSMOLOGICAL MODEL W I T H INDUCED GRAVITY F R O M H I G H E R DIMENSIONS M.D. P O L L O C K International Centre for Theoretical Physws, Trzeste, Italy ~ and Research Institute for Fundamental Phystcs, Kyoto Unwerstty, Kyoto, Japan Received 17 May 1985 It Is shown how dimensional reduction of a (4 + N)-dlmenslonal theory can lead to an effective four-dimensional, broken-symmetry theory of gravity, whose lagranglan density is of the form Aa= ~ 2 R + ~1 b ~¢ /, - V(¢), where R is the Raccl scalar, the field ~b is the reverse of the radius function of the internal space, and the potential contaans both a classical contrtbutton and a Caslmlr energy - ~ k1 ( ,~ 4 For small 4, the Casmur energy dominates, and q u a n t u m corrections automatically generate a C o l e m a n - W e m b e r g potential F ( ¢ ) = ¼kq4 [In (¢2//t2) - 12] + S. A generalized R u b a k o v Shaposhnikov ansatz is made in the (4 + N)-dimenslonal metric, by means of which the vacuum energy density can be made sufficiently small for compatiblhty w~th the cosmological nucrowave background radiation, for a gtven k It turns out that c ~ 2( N 2 + 2 N - 12)-l Cosmological inflation can be reahzed, provxded that N >_.20

The new inflationary universe proposed independently by Linde [1] and Albrecht and Steinhardt [2] appears capable, in principle, of solving a number of outstanding cosmological problems, associated with the horizon, the flatness and the abundance of magnetic monopoles, and at the same time explaming the origin of galaxies. The subject has recently been reviewed at length in ref. [3,4] where some of the remaining difficulties with the idea are discussed, together with the developments to which they have led. Within the standard SU(5) grand unified model, a serious problem is that the density fluctuations produced are much too large. For the Coleman-Weinberg potential [5], fluctuations are predicted to have the scale-free HarrisonZel'dovich spectrum [6] to good approximation, but the required amplitude [7] 8p/p < 3 × 10 -6 is only obtainable for an extremely small coupling parameter 2~_< 10 -13 [8]. Such a small value of h occurs naturally in supersymmetric theories, because of cancellations in the contributions from bosons and from fermions which determine 2~ in a gauge theory. 1 Present address.

Globally supersymmetric inflationary models were first constructed by Ellis et al. [9], but they do not reheat satisfactorily at the end of inflation [10]. It appears that this difficulty can be overcome by going to (N = 1) supergravity models [11], and the no-scale models [12] also solve the gravitino problem [13]. An alternative approach by Steinhardt [14] and Albrecht et al. [10,15] is to use the reverse-hierarchy supersymmetric models [16] based on the geometrical hierarchy model of Dimopoulos and Raby [17]. Here too reheating is a problem, but it is claimed [18] that correction terms in the potential due to local supersymmetry can produce a steep well near the absolute minimum, which is what is needed. And we mention also the chaotic inflationary cosmology of Linde [19], which can be implemented in the context of supergravity [20]. A common feature of these theories is the occurrence of a singlet scalar field, sometimes called the inflaton field. (Indeed, a separate inflaton sector can, perhaps rather artificially, be grafted onto the standard SU(5) grand unified model, in such a way as to give satisfactory inflation [21].) Now, some broken symmetry theories of gravity also contain a singlet scalar

0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

47

Volume 160B, number 1,2,3

PHYSICS LETTERS

field, 4,, the usual Hilbert lagrangian being replaced by one of the form [22,23] ~ = ½~4,2R + ±4, 2 .k4, k - v ( 4 , ) .

(1)

Here, R is the Ricci scalar, a semi-colon denotes a covariant derivative, and latin space-time radices run from 0 to 3. The gravitational constant is generated as the vacuum expectation value a -1 = 8 ,G (2) These theories are of the scalar-tensor type [24], with the parameter c related to the BransDacke parameter o~ by t~----(4c) -1. But the observatmnal constraint [25] ~< 5 x 10 -4

(3)

does not apply, because the field 4, is now effectively massive *1. The possibility of obtaining a successful inflationary model with such a theory has been studied recently [26] in the context of a kind of chaotic inflation [19], taking a Cole-Weinberg potential [5] V(4,) = 1X4,4[ln (4,2//t2) - ½] + A.

(4)

It was found that inflation could continue for long enough to solve the horizon problem if c ~< 4 X 10 -3,

(5)

and that the constraint [27] on the anisotropy of the cosmological microwave background radiation would not be violated provided that X ~< 3 X 10 -12

(6)

A discussion of this possibility has been gwen by Accetta, Zoller and Turner [28] for the potential V(4,) = ~ ( 4 , 2 _ ~2)2. They list upper limits on and ;k, which a successful (ordinary or chaotic) inflationary model must satisfy, together with maximum reheating temperatures. In their model. too, the values of ~ are always very small, 3 X 10 - ~ < X ~<10 -~s, if we fix 8P/P ~ 3 × 10 -6, so that their parameter 3 - 3 × 10 -2. Maximum reheating temperatures are mostly in the range 10 n _< TR _< 1012 GeV. *~ Ref [26]contains an erroneousstatement on this point 48

3 October 1985

The purpose of the present paper is to show how the effective lagrangian (1) arises automatically as a result of dimensional reduction, starting from a theory in (D + N)-dimensions, with D = 4 space-time dimensions and N extra dimensions forming a compact internal space, whose length scale is 4,- 1. The parameter c and the potential V(4,) depend upon N, and the inequality (5) translates into the inequality N >_.20. The need to have a large number of extra dimensions in order for inflation to work in Kaluza-Klein type theories was first pointed out by Sahdev [29]. (See also ref. [30].) We make use of a generalization of the ansatz of Rubakov and Shaposhnikov [31], which provides an extra degree of freedom. This freedom was originally introduced in an attempt to explain why the cosmological constant today is essentially vanishing, since an arbitrary A D appears in the D-dimensional space-time components of the original ( D + N)-dimensional Einstein equations, which contain a fixed A. How the value A err = 0 might then be favoured was left as an open question in ref. [31]. Since that work appeared, there has been the suggestion by Hawking [32] and by Baum [33] that whenever Aeer is freely variable, then the path-integral mechanism automatically selects a space-time with vanishing cosmological constant, this being overwhelmingly the most probable of all de Sitter or anti-de Sitter space-times. Baum considered an arbitrary Aeff arising from some scalar field theory with potential V(4,), and showed that the mechanism also worked if the theory included a kinetic term 14,.k4,' k. Then the most probable configuration corresponds to a value of 4, which is not in general a solution to the classical field equation, 3S/34, ~ O, although one assumes that the metric g,j is a solution of the classical Einstein equations, since the contribution from this g,j is expected to dominate in the path integral [34]. Further, it has been shown [35] that the idea also works for a theory described by the lagrangian (1). But now the fields 4, and g,j are on the same footing, and self-consistency demands that we have both 3 S / 3 g 's = 0 and 3S/34, = 0. Conventionally, one says that the classical solution for 4' can always be sustained by addition of an external

Volume 160B, number 1,2,3

PHYSICS LETTERS

current, J, by including a term J ~ in .W, but in the present context, the meaning of "external" is unclear, and this term is better omitted. Actually, we shall find that whilst the equation of motion 8S/6ep = 0 m our theory is independent of the renormalization point/t, the potential V(~) is not. Therefore, we shall assume, as in ref. [26], that the renormahzation point is such as to make Aar(q~0) = 0 for the classical solution 3S/3q, l~=,~°

3 October 1985

Substituting expression (8) back into the action (7), we obtain

S=(16~d)-~f d O + U x ~ x U / ~ o ° / 2 × { ° - l k o - 2h + u°-i [x-lCTx + ~( N - 3)X- 2X,X k] + x-i[I{N

+ D o - i [ ] o "b

¼ D ( D - 3)o-20op~] }.

=0.

N o t e that these considerations necessarily apply at zero temperature. At any finite temperature, Aa will contain a term p~ = *r2,4 T4/90, m units such that c = h = k = 1, where A > 200 is the number of relativistic fields. One will only have a de Sitter or anti-de Sitter space-time if ,2 [3.¢fr[ >> ~r2A T4/90, and so the mechanism of refs. [32,33] will only work up to an uncertainty A A,f r ,r 2A T4/90 >_.20T 4. It is known from observations, however, that A¢er -< (Om~tt¢~)today, which means an equivalent maximum temperature so small as to be essentially vanishing. Let us consider the (D + N)-dimensional acUon

S = (16~r(~)-

lfd°+ xv (

(9) It is possible to integrate over the internal space, p r o w d e d that

f d ~Vxg~U ° t D-

2A).

fimte,

(10) f dNx ~-n~aD/2= finite. We assume that both these conditions are satisfied, so that (9) can be converted into the reduced action

S°= (16~rd)-l f d°x -~S~o x

-

2)/2 =

(7)

-

( ~xN/2(~ o - 4 ~ - ' ) ¼N(N

- 1 ) x ( N - - 4 ) / 2 X k x k "+ KX (N-2)/2 },

(11) /~ is the ( D + N)-dimensional Ricci scalar, G is the coupling constant. A is a cosmological constant and g is the determinant of the (D + N)dimensional metric, which we write as

where the constants G, g and r are defined by

~ & ' = f d"x g~/,,°, ~, f d Ux X/~--NN0 ~°- 2)/2 fdtCx ~ N o 0/2

=

o

goa(x")x(x*)'

(8)

g~

fd"~ ~vGo°~2[~N+ Do ,~o + ¼D(D- 3 ) ° ~o~°1 where gu(x k) is related to the metric of space-time, g~# describes the internal space, with greek suffices running from (D + 1) to (D + N), and the functions o(x") and X(x k) are to be determined from the theory. This is a modification of the R u b a k o v - S h a p o s h n i k o v ansatz proper, for which X = 1 [31].

,2 ~ denotes the energy density corresponding to the geometrical quantity A

fd'x ~

o o/2

(12)

At this point, it is conventional in Kaluza-Klein theories to make a conformal transformation of the s p a c e - t i m e metric gu ~ ~2(xk)gu' and to choose ~ so that the coefficient of R o is a constant, thus ensuring that the energy-moment u m tensor of the matter, represented by all other terms in the lagrangian, is conserved. This is done by Shaft and Wetterich [36], for example, who 49

Volume 160B, number 1,2,3

PHYSICS LETTERS

have also discussed inflation in Kaluza-Klein theories, including the effects of terms quadratic in /~As. But the scalar field X cannot be eliminated from the theory; one is always dealing with a scalar-tensor theory of some kind. Furthermore, it is known that theories described by a lagrangian of the type * ~ = l c * 2 R -J- 21* , k * ,k -- V ( * ) "l-*~mat (13) yield a matter energy-momentum tensor T,j = 2&£amat//Bg'j that is conserved identically provided that V does not interact with the matter and does not depend upon temperature. (This is proved explicitly by Zee, in the first of ref. [23].) Hence, an alternative conformal rescaling, in which the effective lagrangian function (11) is reduced to the form (13) leads to a viable physical theory (even at finite temperature, under certain conditions [37]). The f i e l d , has canonical dimensions (length)- 1, and hence the most natural identification would seem to be `/,2= X- 1. Choosing fa in accordance with this prescription, we derive the new reduced action

S,9= (16rrG)-l f dnx -~Z-~n{ ½~,2RD + ½~'[(D - 1)( O - 2)-1( N 2 _ 4)

-N(N-

1 ) ] , ; k , ' * - V j ( , ) },

(14)

3 October 1985

and Vc(, ) = o / ~ , N+2 -1.-B , N+4,

a = 2,/g,

B = (,/g)B'. (19) Usually, the action (16) is freely invented, with arbitrary Vc(,), but here a precise meaning attaches to the field , . Notice that both a and B are now arbitrary, apart from a question of sign, as a consequence of our making the ansatz (8), and that ~ is only positive for N >t 3. The constraints (3) and (5) both require the number of internal dimensions to be quite large, N >> 3, so that c > 0, which is necessary for stability, w h e n , is dynamical. The constraint (5) translates into the condition N >_ 20;

(20)

Since the internal space is compact, then the classical potential (18) must be augmented by a Casimir energy, which here will be of the form Vc~,m~r = - a"~'l X,.h4,

(21)

with ~' > 0. (Remember that , - x is the radius function of the internal space.) Now because N >> 1, for sufficiently small values of , the classical part ~ ( , ) can be ignored. In this regime, we are dealing with a pure 2~,4 theory, and quantum corrections will automatically give a Coleman-Weinberg potential Vq = ¼X,4 [ln (,/,=//,2)_ ½] + A,

where the classical potential V¢'(,) is given by V c ' ( , ) --~ Ol,fik,(D+2N)/(D -2) + B,,(3D + 2N-4)/(D- 2),

(15)

in which we have set a' = 2 and the constant B' is arbitrary, ultimately because the space ~ p must be independent of x k. (In ref. [31], an arbitrary A n takes the place of B'.) Finally, in order to have a lagrangian in the form (13), we set the coefficient of the kinetic term equal to ½. Specializing to the case D = 4, and setting 16~rG = g, we obtain the action $4 =

where/, is a renormalization point and A is a constant. Adding expressions (18) and (22), we have, as an approximation to the full potential, the expression V ( , ) = otJ'~k*N+2 + B * N+4

+ ¼x,' [in

_ 1] +

(23)

Let us assume a Friedmann-Robertson-Walker space-time ds 2 = a t 2 - a2(t)d2x, (24)

(16) (17)

in which a dot denotes differentiation with respect to comoving time t, the Hubble parameter is

where

50

(22)

and ignore spatial variations of , . Then we write the equation of motion of the f i e l d , as [37] //; + 3H,~, + ,-1,~,2 = - ( l + 6 , ) a u / a , , (25)

fd4x_~g7411,,2R4+ ½ , , * * , k V¢(,)],

, = 2 / ( N 2 + 2N + 12),

(18)

with

Volume 160B, number 1,2,3

PHYSICS LETTERS

H = a - lfi, and we have introduced the potential U, which is related to the potential V by

epOU/ Oep = epOV/ cgep- 4V.

(26)

The existence of two distinct potentials, one determining the energy and the other governing the evolution of the field q~, is a well-known feature of this kind of theory - see Smolin [23]. (The same feature occurs in the work of Shaft and Wettench [36], but there the two potentials are connected in a different way.) F r o m expression (23), we deduce that U(q~) = [ ( N -

2 ) / ( N + 2)] aAep u+2

+ [ N / ( N + 4)] B , N+4 + ~X~4 - 4A In (~/t~,),

(27)

where /xa is arbitrary. [If we want to have U(q~o) = 0, then condition (29) below will imply that t~ - q~o.] For small values of q~, the first three terms in expression (27) can be neglected, and we than have U(q,) -- - 4.& In (ff//~x)-

(28)

Notice that this is the same potential as one finds in the reverse-hierarchy supersymmetric models [16,17]. It is claimed [14,15,18] that such a potential can successfully give rise to inflation, provided that local supersymmetry correction terms are included at large values of ~, in order to generate a steep well around the minimum [18]. In our model, extra terms of just the reqmred types are also present, and we shall see that they can act in the same way. We suppose that the universe passes through a phase when expansion of the four-dimensmnal space-time is accompanied by contraction of the internal space, whose inverse radius funcuon 4~ evolves from a value ~a to its final stable value q'0, at which the net cosmological constant V(~0) must vanish. We suggest that this cancellation can be justified in the path-integral approach [3.2,33,35] discussed above, and impose the condition ~Xq~o + A.

0,

(29)

with the newtonian gravitational constant given by G~ 1 = 8~r,* 2.

(30)

3 October 1985

[In writing eq. (29), we have set bt2 = ~ . Note that this lifts the degeneracy.] We assume that q~ << q,2, so that V ( ~ ) = A. Higher-order terms m R u have been ignored, and self-consistency then demands that V(~)GZec <_%1, which means that .&G~(epo/~) 4 _< 1. From the constraint (35) below, we see that this inequality is satisfied for values of @ such that (q~/q~0)2 > 10- 5, which does permit ~2 << ~2. Similar considerations allow one to ~gnore any initial quantum growth of q~, which in the free-field approximation should be q~2= (2qr)-2Hat [38].) Then, during much of the expansion ~2 << ~ , and eq. (25) reduces to

eO~ + (3~/c)1/2+ = _+2 + 4(1 + 6¢)-1A.

(31)

This is eq. (10) of ref. [26], and it admits the solution = [16cA/3(1 + 6c)211/2(t- to),

(32)

for which ~ = 0, so there really is slow rolhng. Meanwhile, the universe expands according to the law

a( t ) -~ ao( t - to) 1/'~

(33)

(which is also given in ref. [28]). This is essentially a power-law expansion, with coefficient n = (16~) -1/2 = [ ( N 2 + 2N 12)/32] 1/2 ~ N / 3 ~ >_ 4, for N>_ 20. Such an expansion can normally be brought about by matter with energy-density P and pressure p related by p = ( - / - 1)0, with 7 = 2/3n = 4 / N < 0.2, so that ( p + p ) / p ~<0.2. One would naively expect the usual results of inflationary cosmology to hold in this situation, when appropriate allowance has been made for the fact that the Hubble parameter H now varies with time. It has been shown that this is so for n >_.10 [39]. Thus, for example, density perturbations created during inflation, via fluctuations of ~, are expected to have an approximately scale-invariant spectrum [39]. Their amplitude must satisfy 80/0 ~< 3 × 10 -6 at the time of horizon crossing, in order not to produce too much anisotropy in the dipole component of the cosmological microwave background radiation [7]. This inequality in turn implies an upper limit/kl upon the constant A. 51

Volume 160B, number 1,2,3

PHYSICS LETTERS

F r o m expression (28), we would expect density perturbations of amplitude 8p//p - A1/2G N, which would mean that ~IG~-

10 -11.

(34~

Another upper hmat A2 follows from the requirement [27] that primordial gravitational waves amplified during inflation do not give rise to too great an anisotropy in the quadrupole and higher-order components of the microwave background. For exponential expansion, the dipole component furnishes the stronger constraint [40], so that .g,~ < Az, whilst for the expansion law of the chaotic inflationary universe, the quadrupolecomponent constraint appears to be the stronger [41], so that Jkz < A1. The situation regarding the present model is not immediately clear, and this aspect requires further investigation. The reasoning of ref. [26] leads to the estimate A2G 2 - 4 X 10 -11,

(35)

if we take the observational constraint on the temperature anisotropy of the microwave background ( A T / T ) z _< 3 × 10 -8 [42], which does not differ appreciably from (34). In ref. [26], this constraint was translated into the conditmn h .%<3 × 10-12 on the coupling constant. Such a small value of X is not admissible in the present model, because the Caslmir parameter is always many orders of magmtude larger. Now, however, the potential V(q,) contains the two additional terms in a and B, both freely adjustable, so that a solution to eq. (29) can be found for the given 3,, such that the constraint (34) or (25) is not violated. Because N > 20, these extra terms have a very strong dependence upon ¢, and by taking aA < 0 and B > 0, we construct a very steep potential well around the minimum at = 4'0, which ought in principle to permit reheating with almost 100% efficiency. Then, we may estimate the maximum reheating temperature T R via the relation ~rZAT 4 - A (36) which leads to the result T R - 2 X 10 I5 GeV.

(37)

Of course, what actually happens will depend upon the precise way in which other matter fields 52

3 October 1985

couple to ~. The original purpose of Kaluza-Klem type theories was to create these fields out of the geometry. In the present model, we have not investigated this aspect, for which off-diagonal terms in the metric (8) would have to be included. A1 least one can say that the problematic decoupiing theorems which afflict the reverse-hierarchy super-symmetric models [10,15,17] will not cause trouble here. Further work is needed on this point. Both in the present paper and in ref. [26], it is supposed that cp evolves towards the equilibrium point ~0 from some small initial value ~1 << ~0, but the motivation ts different in the two cases. Here, we imagine that the internal space is in a state of contraction from some initial radius ¢ - 1 which is large relative to the final radius epo i ( 8 ~ r ~ ) l / 2 / p -- lp/10, where I v = G~( 2 is the Planck radius. Notice, however, that ¢ = 0 is not a solution to the field equation for d? in the broken symmetry theory of gravity [35], unlike in the usual theory, this being traceable to the presence of the term 4V in eq. (25). (When ~2 << ~ , we have 4V = 4A.) This means that an evolution of #, from ~x << ~o to ~o may be regarded as "chaotic" in the sense that 4, = 0 is not an equilibrium position, and this was our point of view in ref. [26]. In this original works on the subject, Linde [19] considered a chaotic evolution of the scalar field starting from some initial e~ > e?0, and so the picture is not quite the same. This idea is based upon his observation [4,19] that above some critical temperature T . , there is insufficient time for the field to reach its equilibrium state, so that the notion of high-temperature symmetry-restoration becomes inoperative. For the broken symmetry theory of gravity, however, the symmetry is not restored at any finite temperature, even if thermal equilibrium is a meaningful concept [37]. This is a further consequence of the replacement of the potential V by the potential U in the equation of motion (25). We have seen that the self-consistency condition V(,h)G 2 _< 1 is satisfied for all ~ in the range 3 × 103 ~<~/~bo < 1, which enabled us to analyse a model starting from d? ~ 0. On the other hand, if we examine values of ~ > q~0, than we find that the potential very rapidly becomes dominated by =

Volume 160B, number 1,2,3

PHYSICS LETTERS

t h e t e r m in B , so t h a t V ( ~ ) -- B~bu+4 ~ B~b24, a n d w e a r e r e s t r i c t e d to v a l u e s o f ff s u c h t h a t ~ / ~ < 1 + 6, w i t h 6 << 1. F o r l a r g e r v a l u e s o f ~, t h e a n a l y s e s o f ref. [36] will b e r e l e v a n t . I s h o u l d like to t h a n k P r o f e s s o r Z. M a k i f o r hospitality at the Research Institute for Fundam e n t a l P h y s i c s , K y o t o U n i v e r s i t y , w h e r e this w o r k w a s b e g u n . I s h o u l d also like to t h a n k P r o f e s s o r Abdus Salam, the International Atomic Energy A g e n c y a n d U N E S C O f o r h o s p i t a l i t y at t h e I n t e r n a t i o n a l C e n t r e for T h e o r e t i c a l Physics, T r i e s t e , a n d a l s o P r o f e s s o r H.-y. G u o , Dr. K. M a e d a a n d P r o f e s s o r H. S a t o f o r discussions. F i n a n c i a l s u p p o r t is a c k n o w l e d g e d f r o m K y o t o U n i v e r s i t y , J a p a n , a n d f r o m the J a p a n S o c i e t y for t h e P r o m o t i o n o f S c i e n c e s for a n E x c h a n g e F e l l o w s h i p a w a r d j o i n t l y w i t h t h e R o y a l Society.

References [1] A D Lmde, Phys Lett 108B (1982) 389 [2] A Albrecht and P J Stemhardt, Plays Rev Lett 48 (1982) 1220 [3] A D Llnde, Rep. Prog Phys 47 (1983) 925 [4] R H Brandenburger, Rev. Mod Phys 57 (1985) 1 [5] S Coleman and E J Wemberg, Phys Rev D7 (1973) 1888, B Allen, Nucl Phys B226 (1983) 228; A Vilenlon, Nucl Phys B226 (1983)504 [6] E,R Harnson, Phys Rev. D1 (1970) 2726, Ya.B Zel'dovlch, Mon Not R Astr Soc 160 (1972)1P [7] L F Abbott and M B Wise, Astrophys J 272 (1984) L47 [8] S W Hawkang, Phys Lett 115B (1982)295, A H Guth and S-Y P1, Phys Rev Lett 49 (1982) 1110, A A Staroblnsky, Phys Lett 117B (1982) 175, J Bardeen, P J Stelnhardt and M S Turner, Phys Rev D28 (1983) 679 [9] J Elhs, D V Nanopoulos, K A Ohve and K Tamvakas, Phys Lett 118B (1982) 335, Nucl Phys B221 (1983) 524, Phys Lett 120B (1983) 331 [10] A Albrecht, S Dlmopoulos, W Flschler, E W Kolb, S Raby and P J Stemhardt, Nucl Phys B229 (1983) 528, B A. Ovrut and P J Stemhardt, Plays Lett 133B (1983) 161 [11] D V Nanopoulos, K A Ohve, M Sredmclo and K Tamvakls, Phys Lett 123B (1983) 41, D V Nanopoulos, K A Ohve and M Sredmclo, Phys Lett 127B (1983) 30 [12] J Elhs, C Kounnas and D V Nanopoulos, Plays Lett 143B (1984) 410, J Ellis, J Enqwst, G Gelrmm, C Kounnas, A Maslero,

3 October 1985

D V Nanopoulos and A Yu Sm~rnow, Phys Lett 147B (1984) 27 [13] H Pagels and J R Pnmack, Phys Rev Lett. 48 (1982) 223, S Welnberg, Phys Rev, Lett 48 (1982) 1303, J Elhs, J.E Kam and D,V Nanopoulos, Phys Lett 145B (1984) 181 [14] P.J Stemhardt, in. The very early umverse, Proc. Nuffield Workshop, eds G W Gibbons, S W Hawking and S.T C Slklos (Cambridge U P, London, 1982) p 251 [15] A Albrecht, S Dlmopoulos, W Flschler, E.W Kolb, S Raby and P J Stemhardt, in Proc Thard Marcel Grossman meeting on the Recent development m general relativity, ed H Nmg (North-Holland, Amsterdam, 1982) p 511 [16] E W~tten, Phys Lett 105B (1981)267, Nucl Phys B177 (1981) 477 [17] S Dlmopoulos and S Raby, Nucl. Phys B219 (1983) 479 [18] A Albrecht and P J Stemhardt, Phys Lett 131B (1983) 45 [19] A D Lmde, Pls'maZh Eksp Teor Flz 38 (1983)149 [Sov Phys JETP Lett 38 (1983)176], Phys Lett 129B (1983) 177 [20] A S Goncharov and A D Llnde, Phys Lett 139B (1984) 27, A D Lmde, Lett Nuovo Clmento 39 (1984) 401, Zh Eksp, Teor Flz 87 (1984) 369 [Sov Phys JETP (1985)] [21] Q Shaft and A Vflenkan, Phys Rev Lett 52 (1984)691 [22] Y Fujn, Phys. Key D9 (1974) 874, D26 (1982) 2580 [23] P Mmkowska, Plays Lett 71B (1977)419, L Smohn, Nucl Phys B160 (1979)253, Phys Lett 93B (1980) 95; A. Zee, Plays Rev Lett. 42 (1979) 417, 44 (1980) 703, Phys Rev D23 (1981) 858 [24] P Jordan, Z Plays 157 (1959)112, C Brans and R H Dlcke, Phys Rev 124 (1961) 925 [25] R D Reasenberg, I I Shapiro, P E. MacNeil, R B Goldstein, J C Breldenthal, J P Brenkle, D L Caan, T M Kaufman, T A. Komarek and A I Zygielbaum, Astrophys J Lett 234 (1979) L219 [26] M D Pollock, Phys. Lett 156B (1985) 301 [27] V A Rubakov, M V Sazhm and A V Veryaskan, Phys Lett 115B (1982) 189, R Fabbn and M D Pollock, Phys Lett 125B (1983) 445 [28] F S Accetta, D J Zoller and M S Turner, Phys Rev D31 (1985) 3046 [29] D Sahdev, Phys Lett 137B (1984) 155 [30] R B Abbott, S M Barr and S D Elhs, Phys Rev D30 (1984) 720; D31 (1985) 673, E W Kolb, D Lmdley and D Seckel, Phys Rev, D30 (1984) 1205, K Maeda, prepnnt Quantum effect on Kaluza-Klem cosmologles- b_xghtemperature case, 15AS59/84/A(1984) [31] V A Rubakov and M E Shaposhmkov, Phys. Lett 125B (1983) 139 [32] S W Hawlong, Phys Lett 134B (1984) 430. [33] E Baum, Phys Lett 133B (1983)185 53

Volume 160B, number 1,2,3

PHYSICS LETTERS

[34] S W Hawking, Nucl Phys B144 (1978)349 [35] M.D Pollock, Plays Lett. 148B (1984) 287 [36] W Shaft and C. Wetterich, Phys Lett. 129B (1983) 387; 152B (1985) 51; C Wettench, Nucl Phys. B252 (1985) 309. [37] M D Pollock, Phys. Lett 132B (1983) 61 [38] A D Lmde, Phys. Lett l16B (1982) 335, A Vdenkin and L H Ford, Plays. Rev D26 (1982) 1231 [39] L H Abbott and M B Wise, Nud. Plays B244 (1984) 541

54

3 October 1985

[40] S W. Hawking, Phys Lett 150B (1985) 339 [41] R. Kahn and R.H Brandenburger, Phys Lett 141B (1984) 317 [42] R Fabbri, I Gmch and V Natale, Phil Trans R Soc Lond A307 (1982) 67; P.M Lubm, G L Epstein and G.F Smoot, Phys Rev Lett 50 (1983) 616; D T Wdkanson, Phys Trans R. Soc Lond. A307 (1982) 55