Symmetry behavior in the Einstein universe: Effect of spacetime curvature and arbitrary field coupling

Volume 130B, number 1,2

PHYSICS LETTERS

13 October 1983

SYMMETRY BEHAVIOR IN THE EINSTEIN UNIVERSE: E F F E C T O F SPACETIME CURVATURE AND A R B I T R A R Y F I E L D COUPLING ~r D 2 . O'CONNOR, B.L. HU and T.C. SHEN Department of Physics and Astronomy, UniversTtyof Maryland, CollegePark, MD 20742, USA Recewed 14 April 1983 Revised manuscript received 23 June 1983

For the investigation of the effect of spacehme curvature and arbitrary field coupling on symmetry restoration, we have computed by zeta-function regulanzatxon the oneqoop effective potential for a self-interacting scalar field arbltrardy coupled to a static Robertson-Walker umverse. The classical contribution of curvature can restore broken symmetries for a wide range of parameters from conformal to near-minimal couphngs whereas the quantum contribution is Important only for a small range of values near the minimal eouphng point. The symmetry behavior of the two extremes is found to be very different

1. Introduction. Phase transition in the early unwerse is now believed to be a natural consequence o f Grand Unification (GU) and possibly quantum gravitational (QG) theories. At such early times, the background spacetime could have possessed strong curvature and m a t t e r could have existed in a state o f high temperature. To examine the s y m m e t r y behavior o f the early umverse more closely one should take into consideration the effects o f spacetime curvature and finite temperature corrections in their full rights * 1. As the first problem in a systematic investigation o f these issues we consider here the effect o f spatial curvature and different field couplings on the symmetry breaking in a static R o b e r t s o n - W a l k e r , or the Einstein universe. Later work will dwell on the effect o f curvature anlsotropy [2], spacetime dynamics [3] and finite temperature effects [4] ,2 Different aspects o f the present problem have been considered by a number o f authors: Ford [6] studied a X~b4 theory and derived the effective mass by calculating (~b2)0 . Toms [7], and Denardo and Spallucci [8] calculated the effective potential for massless and massive scalar fields by means o f the f-function regularization while Kennedy [9] studied this problem via dimensional regularization. Whereas most previous considerations have been confined to the calculation o f the critical curvature for conformally coupled fields, which happens to be an important but rather special case, they did not deal with how different field couplings could influence the symmetry behavior. We want to extend these previous calculations to encompass arbitrary field couplings. Specifically we have computed the one-loop effective potential o f a ~.~b4 theory by ~'-functlon regularization and derived the critical curvature for the near-conformal and near-minimal regions. The symmetry behavior o f non-conformal fields are found to differ greatly from the conformally coupled fields. The contribution o f quantum corrections is more important for the near-mlntmal (~ ~- 1) case, when the classical effect o f spacetime curvature is less pronounced. The infrared behavior exhibited by the non-conformal fields here in influencing the phase transition is similar to what happens m the inflationary scenario typical o f GU epoch [ 10].

Supported m part by the National Science Foundation under Grant PHY81-07387. ,1 Fpr a recent review of these issues, see ref [ 1] ,2 For a review on fimte temperature quantum processes m the early universe, see Hu [5]. 0 . 0 3 1 - 9 1 6 3 / 8 3 / 0 0 0 0 - 0 0 0 0 / $ 03.00 © 1983 North-Holland

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2. Effective potential. The metric o f the Einstein unvlerse can be written in the form ds 2 = dt 2 - a 2 [dx 2 + sm2x(d02 + sin20 dq52)] ,

(1)

where ×, 0, ¢ are the angular coordinates on a 3-sphere S 3 and a is its radius. We consider a massive (m) self-interacting (X) scalar field 4~coupled arbitrarily (~) to the curvature. The fluctuation field q~around the classical solution ~ obeys to linear order in ~ the wave equation (e g. refs. [ 4 - 7 ] ) , [[]+(1-~)R/6+m

2 + ~1 x~21 ~= 0

(2)

where R = 6/a 2 is the scalar curvature, ~ is the coupling constant (~ = 0 is conformal and ~ = 1 is minimal) and [] is the Laplace-Beltrami operator on S 3 × R 1 . On the classical level the curvature term is seen yielding a non-zero contribution to the effective mass ~ 2 =_M 2 + (1 - ~ ) R / 6 where M 2 - m 2 + i X~2 whenever the coupling is nonminimal (~ :~ 1). Suppose the system exists in a phase of a broken symmetry (m 2 < 0) then by Increasing the curvature of spacetime (or tracing the history of the universe backwards) beyond a certain critical value Re, symmetry can be restored. The effect o f quantum corrections and finite temperature corrections show up as additional terms in the effectwe m a s s ~-ffL2 In the form 5 X(~ ) = ~ X((~2)0 + (~2)T). In general the quantum correction term is smaller than the classxcal term by a factor ofXh/64rr 2, except for the case of minimal (or near minimal ~ 1) coupling, where quantum correction becomes the main source for symmetry restoration. An effective potential V can be defIned for constant background spacetlme g and field ~ from the effective action ['(~) by (see e.g. ref. [7]) V(~) = - ( V o l ) 11~(~)1~@, where Vol is the spacetime volume. In the functional integral perturbatlve approach, the effective action Y[~] is given by: r[~] = r(0)[~] + r(1)+ O(h2) ,

(3)

where Y(B°) ts the classical action

I'(BO)[g, ~] = -- f d4x x/~-(AB + ~B R + ½eBR 2 + ½~ [D + rn 2 + (1 - ~B)R/6] q] + (XB/4[)~4} ,

(4)

where A is the cosmological constant, x = (167rG)-t, the Newton constant, and e is the coupling constant for the quadratic curvature term* 3 Subscripts B under the quantities denote that they are bare. [,(1) is the l-loop quantum correction I "(1) = 71 1/~ ln(det/2-21A),

(5)

and A is the operator defined as A = [] + m 2 + (1 - ~)R/6 + ½Xq~2. A constant mass scale t2 is introduced to make the measure d [¢] of the functional integral for the generating functional dimensionless. The determinant of A is formally divergent and needs to be regularized. In the ~'-function regularization scheme [ 1 1 - 1 3 ] , one computes the elgenvalues o f the operator A on the euclideanized metric [obtained from ( 1 ) b y a Wick rotation to imaginary 2 time r = i t which, for finite temperature theory has, p e r i o d i c i t y / 3 ] given by XN _- k 02 + 6Onlm, where k 0 = 27rn0/3, and 6Onlrn are the elgenvalues of the operator A on the 3-sphere, N denotes the collective time and hyperspherlcal q u a n t u m n u m b e r s ( n o , n , l , m ) o n S 1 × S 3 , w i t h r a n g e s n o =O,+-l,+2,n=O, 1,2 , ;l---0,1, n , a n d m = - l , - l + 1, .. l. One then introduces the generalized ~'-function defined by ~'(v) = ~

(g-2XN)-V .

(7)

N

Using the regularization method of Dowker and Crltchley [ 11], one can express the one-loop effective potential from (5) as *3 Since for the Einstein universe in (4)

32

RgvRtaV and R#va3Rlava3are proportional

to R 2 , we Introduce JUSt one coupling constant e

Volume 130B, number 1,2

PHYSICS LETTERS

V(1)(@) -- - ½ h ( V o l ) -1 [~"(0) +

~(0)/v]

13 October 1983

.

(8)

The divergence in V(1) w111 be cancelled by the addition of counter terms. For the Einstein space, C°n21m = (q2 _ 1)/a 2 + c/~2,

where q - n + 1 ,

(9)

In the ltrnlt of lar:~,e ~3(the continum limit, or at low temperature) one can approximate £no b y / 3 f d k o / 2 n and write the generalized ~" function as

~(v)=U2"2~ f dk0 /~.m ( k 2 + _=

n, ,

_ /3 (ua) 2v P ( v - 1/2) Z ( v -

,~

~

r(~)

2 )-v 13 Conlm = p2v N / ~

lt(v - 1 / 2 ) ~ ,Co2 , - v + l / 2 r(v) n,l,m ~ nlm:

1/2,x)

(10)

where Z(v,x~

~-J q2 q=l(q2+x)V '

x=CT/{2a 2 - 1 =M2a 2 - ~ .

(11)

To evaluate ~"(0), one can expand Z(u, x) m a Laurent series about their simple pole Z(v, x ) = Z and express lim [P(v - 1/2)/P(v)] (au)2/a = - ( x / ~ / a )

l(V, x ) / v + ZO,

Iv + (2 + in 4 a2p2)v 2 + O(p3)] .

v -+0

In this way, one obtains the l-loop effective potential (8) as V (1) = (h/4rr2a 4) [ Z 0 ( - 1 / 2 , x) + Z _ I ( - 1/2,x) (2 + In i

aZu2)] .

(12)

where p2 is redefined here (as used in ref. [11]) to include a divergent part ,a 3. Renormahzation. The divergences in (12) are contained in the term called VD, and removed by the renormalization o f t h e parameters, e g . , m 2 = m 2 + 6m 2, etc. Thus, V= V~ 0) + VtJ ) + VC(1) = V(0)ren+ V(1), where ~1) is the finite one-loop effective potential, and V(0)renis the classical potential with renormalized parameters V(C1) = 6A + (8•)R + (8e/Z)R 2 + (8m2/2)~ 2 - (8~/2)(R/6)q~2 + (6X/4!)~4 .

(13)

They are determined by the following conditions' m 2 = [Re 02V/0~2]~-0,R=0, 0 = IRe V]~;=O>37= 0 ,

X = [Re ~)4V/O~/ll~=~o,R= 0 ,

K = [ReOV/OR]g=O,R= 0 ,

e.= [Re~)2V/DR2]Oa=O,R=Ro ,

1(1--~)=

(14)

[ReO3V/ORO~Z]~,=O,R=n,.

The real part in the above equations should be taken, as the renormlizatlon terms are required to be real quantities. The coupling constants X, e, ~ are defined at the energy scales corresponding to the values q~= q~0 and R = R0, R 1 +4 E q u a t i o n s obtained using the regularlzatmn m e t h o d of D o w k e r and Cntchley [ 1 1 ] can be converted to those obtained by the m e t h o d ol Hawking [ 13] vm the t r a n s f o r m a t m n

In ~ t = In 4#~)C + 1v

2

F o r brevity, we have a p p h e d this t r a n s f o r m a t i o n to ~u2 ( = / z b C ). T h u s thereafter, one should regard ,,2 (=#~t) as carrying a hidden divergent part

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respectively, which may in general be all different as the energy scales at which they are measured need not be the same. How these coupling constants behave under the change of scales is determined by their associated renormalization group equations. To calculate the counter terms we need to compute Z 0 and Z _ 1 in V. From (11) by means of the Plana sum formula one gets

Z_I(-1/2,x ) = - ~x 2 ,

(lSa)

Z 0 ( - 1/2,x) = (1 + x ) 1/2 - ½x(l + ¼x) 2 + l x 2 {ln [½ (2 + ~ ) ]

2 + ½} + i1/2(x),

(15b)

where /a(x) = i f d t {(2 + it) 2 [(2 + it) 2 + x] a - (2 - it) 2 [(2 - it) 2 + x]a}/(e 2~rt - 1). 0 Since the values o f the coupling constants are physically determined at low energy or small curvature, and the properties of the theory are independent of the point of renormalizatlon, we therefore choose to renormalize V at the near-flat space hmlt (small R or large a ~ ~ 2 a 2 ,.~ M 2 a 2 >> 1 or large x). The expression for Z 0 in this limit (x >> 1) is given by,

Zo(-1/2,x ) = ~x2(ln¼x + ~)(x >>1).

(16)

Using this for V and the conditions (14), the counter terms are calculated to be

6m 2 = (hXm2/32rr 2) [ln (m2//a 2) - 1] --- ~ m 2 ( r

- r2) ,

(17a)

89t = (3hX2/32rr2){ln [(m 2 + 1 )t~b2)/bt2] + 5a _ g8(m 2 + ¼X(p2)m2/(m2 +l$ Xq~2)2) -- -~XX(r -- r0) ,

(17b)

8A = (hmn/64rr 2 ) / I n (m2/g 2) - ~] + 6A((@) = (hm2/64rr 2) (r - r5) + 8A ((q~)) ,

(17c)

6K = - ( h m 2 / 3 2 r r 2) (~/6)/In (m2/# 2) - 1] = - ( h m 2 / 3 2 r r 2) (~/6) (r - r 4 ) ,

(17d)

8e = (h/32rr 2) (~2/36) (In [(m 2 - ~Ro/6)/la2]} -= (/~/32rr 2) (~2/36) (r - r 3 ) ,

(17e)

8~ = (h/32rr2)~X {ln [(m 2 - ~R1/6)/la2]} - (X/4)~(r - r l ) ,

(17f)

where r = In (m2/tz2), ~ - hX/8rr 2. From these counter terms one can obtain the renormalization group equations * s for each o f the parameters: d X / d r = a R 2,

d m 2 / d r = ¼ X m 2,

dA/dr =/~mn/64rr 2 ,

d~/dr=¼~t~,

dr~dr = -(~/m2/32rr 2) ~/6,

de~dr = (~/32rr 2) (~/6) 2 .

(18)

Solutions to these equations can be obtained with the appropriate boundary conditions which preserve eq. (17) at the specific renormalization point:

~(r) = ~,(ro)/L3(r) , ~(r) = ~i(rl)[L(rl)/L(r)] , m2(r) = m 2 ( r 2) [L(r2)/L(r)] ,

(19a) (19b,c)

e(r) = e(r3) + (P//32rr 2) [~2(rl)/9X(ro)]L2(rl)[L(r 3) - L ( r ) ] ,

(19d)

K(r) = K(r4) + (~/32rr 2) [2~(rl)/3 ] [m2(r2)/~(ro)] { L ( r 2 ) L ( r l ) [L(r) - L ( r 4 ) ] } ,

(19c)

A(r) = A(rs) - [hm4/167r2R(ro)]L2(r2) [L(r) - L(rs)] ,

(19f)

,s For a background discussion on renormaliatzion group equation in curved spacetlme, see, e.g. [ 14]. For an example of its application, see, e.g. [ 15] 34

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where L(r) = [ 1 - ~ ~(ro) (r - r0) ] 1/3 and r 0 to r 5 are defined m eq. (17). Henceforth, whenever the expficit functional dependance of the parameters on r is not displayed they should be regarded as assuming the value at their respective renormalization points. The renormalized effectwe potential is given by V = A + K R + 71 eR2 + (1 - ~)(R/6)~2/2 + VCW(~ ) + ( h / 6 4 g 2 ) { - ~a m2~R [In IM2/m21 - ½] + ~ 2 R 2 [ 1 n [M2/(m 2 _ ~R0/6) I _ 3] _ 6~X~2R {In IM2/(m 2 - fiR1/6)[ - 3 }

- (x2/a 4) [In [M2a2/4[+ ½]} + (l~/47r2a4)Zo + 6A((~b)),

(20)

where 8A is determined by the vacuum energy in flat space. Here VCW denotes the Coleman-Weinberg effective potential for k¢ 4 theories m flat space gwen by VCW = ~1m2q~2 + (x/a!)q] 4 + (iv//64rr2){M 4 In [g2/m21 - lm2Xq$2 + ¼Xq~4 {In Im2/(m 2 + ~1X~bO) 12 + ~m2( m2 + ¼X¢2)/(m 2 + ½7tgb2)2}}

(21)

•

For conformal fields, all terms contaIning ~ vanish, eq. (20) then reduces to the result of refs. [7] and [8] ,6. Note that for Z 0 in (20), the exact non-flat form (15) should be used. In contrast to the near-fiat space regime approximation (C~2a2 >~ 1) used earlier f o r Z 0 m the determination of the counter-terms o f renormalization, here for the study o f symmetry restoration due to spacetime curvature, we are interested in the high curvature regime corresponding to small conformally-related effective mass cl?~2a2 ~ 1. In this regime ( y ~ 1, wherey - x + 1 -= C~2a2)Z0 can be approximated by

Z o ( - 1 / 2 , y - 1) = - 0 . 4 1 1 5 0 + y l / 2 _ 0.60223y - 0.0031778y 2 oo

+ ~ ] ( - 1 ) n F ( n - 1/2) n=3 n! I'(--1/2) r l ( n - 1 / 2 ) y n = b l / 2 y l / 2 +

~

bnyn

(y'~l),

(22)

n=0

where r~(v)= Zs=2S2/(s2 - 1) v. Notice the appearance o f the y 1/2 term. Symmetry restoration occurs when the effective potential assumes a minimum at ~ = 0, 1.e., d 2 Vide) 2 I~=0 = 0, or dV/dq~ 2 I~,=0= 0. This yields an algebraic equation f o r y 0 -=-y(~= 0) for the determination o f the critical curvature R c-

YO + XC(a, ~) + XYo 112 = 0 ,

(23)

1 C(a, ~) = 2b 1 + ¼m2a2(r2 - 2 - lnl:} m2a2 0 - g~(r 1 - 2 - i n 1¼m2a21)

= 2b 1 - ¼m2a2(1 + in 1¼m2a21) + ¼~(2+ In 1¼(m 2 - ~R1/6)a21).

(24)

4. Higher order contribution. The equation as it stands is however not invariant under the renormalization group (RG) transformation. The problem arises from the ~y-1/2 term and could be rectified by adding higher order contributions to the effective potential. Consider the so-called daisy diagrams (see, e.g. fig. 5 of ref. [16]), with k - 1 one-vertex bubbles on a k-vertex bubble which are the most dominant k t h order contributions in an O (N) invariant theory under the large N approximation. It gives the effective mass c-~ 2ff a contribution (we use C~e2ff to denote the classical effective mass c/g2 plus quantum corrections) (_l)k[(hX/2 V)tr(l/XN)]k(l~X/2V)tr(1/XN)k+l = [X~ V(1)/~c~2

[~=o]k(1/k!)X(~/~ c~2)k+1 V(1)[~=O .

(25)

Summing up all such diagrams, one obtains 3 *6 There is an error m eq (3.12) and (3 18) m [8] the signs of the terms with 3" (m4/642) should be reversed. Also, m eq (3.19) the logarithmic term should have been multlphed by M4 rather than M 2 , a misprint probably.

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C~eff2 _- c-~ 2B + [exp ()k 0 V (1)/0c'~2 i~=0 ~/~ Q,~ 2)] )k ~ V (1)/~ c ~ 2 i03=0 ,

13 October 1983 (26)

or,

= ~ 2 B + Xc~V(1)/~O'/~2i~g+xaVO)la~ ~1~,

(27)

where 02B;

= 0) = m g + (1 -

B)R/6 •

(28)

2 fa 2 After renormalization m the same manner as described above, this becomes for Yeff = c ~ el Yeff =Y0 + XC' + 2XdZo/dy[yo+XC,+2XdZo/dyly ° ,

(29)

where Y0 - c ~ 2a2 and C' = C - 2b l" Taking the appropriate limit as discussed above, while dropping subdominate terms in ~, one obtains instead o f (23) the following expression, Yeff = Y0 + ~C + X/Cv0 + x c ) 1/2 .

(30)

This equation is RG invariant to order Xs/3, as can be seen by differentiating with respect to r. Setting Yeff = 0 gives the critical value for Yoc = - h C +X2/3, which is equivalent to the critical radius, ac2 = _ m _ 2 ( 1 _ ~ _ ~,2/3 + ~C) ,

(31)

or the critical curvature Rc = _ 6 m 2 / ( 1 _ ~ _ ~ 2 / 3 + X C ) .

(32)

F o r conformal coupling (~ = 0) R c ~- - 6 m 2 / ( 1 - ~ t 2/3 + h C )

(m2 < 0 ) ,

(33a)

whereas for minimal couphng (~ = 1), R c ~- 6m2(1 + C~1/3)/X2/3

(rn 2 > 0 ) .

(33b)

For the massless case

R 2 = 4 ' ~R 1 exp [4(1 -

~)/x~ + (2 -

4/xl/3~)1 .

(34)

This differs from the result o f Ford and Toms [ 15] b y the ~ - 1 / 3 term which, however, can be absorbed into a rede fined R 1. Interating (29) by including super daisy diagrams we obtain (see fig. 6 o f ref. [ 16])

Yeff = YO + k C ' + 2~ d Z o / d Y lyeff .

(35)

In the neighborhood of the critical point we find Yelf =Yo + ~tC + X/'ref !1/2 ,

(36)

or equivalently, c ~ 3 f f = [m 2 +

(1 -

~ + ~tC)/a2]C~ eff + ~/a3 •

(37)

To determine how the effective mass behaves as the critical pohat as approached from the symmetric state (a < ac), one can expand c ~ elf m a Taylor series about the critical value ac, i.e., c'~ elf = C'~effla=ac + dC'fl~eff/da[a=ac (a - ac) + "'" Using d C ~ e f f / d a b c = -370/3/a2c one gets. C~ef f = --(3~1/3/a 2) (a - ac) ~ 3xl/3(1/a -- 1/ac) . 36

(38)

Volume 130B, number 1,2

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It IS interesting to note that unlike the finite temperature flat space result (15), X does enter into this expression. For small radii but away from the critical value a ~ ac, cff~ef f = [m2(a 2 _ a 2) + ~:2/3] l / 2 a - I ~ imlac/a,

(39)

which behaves similar to the finite temperature case [16].

4. Discussion. The result we have obtained for the critical curvature for conformally coupled fields (33) differs slightly from that o f Denardo and Spalluci [8]. These authors obtained R c = - 6 m 2 / ( 1 - hX/96n2). This discrepancy arises from the fact that the previous work assumed x ~ 1 in the expansion o f Z 0 in (20) for conformal coupling, which corresponds to working under the condition that M2a 2 ~ 1 instead of c'/K2a2 ~ I. Had one assumed M2a 2 ~ 1, then for near-conformal coupling (~ ~- 0) one would have used the expansion on Z 0 for x ~ 1 : v4x-

,

vx 2 +

--

(2n-3)x n

(4o) where "y is the Euler constant and ~"is the Rlemann zeta function. However, M2a 2 ~ 1 corresponds to the region where symmetry restoration could not possibly have occurred, because of the presence o f curvature contribution in the effective mass for all non-minimally coupled fields. Indeed, the result (33a) indicates that im2a21 ~- 1, which is inconsistent with the assumption behind the use of (40): 1.e., m2a 2 ~ 1. (Near the symmetry restoration point ~b= 0, M 2 -~ m2). Our result for both cases are consistent with the approximation assumed, c ~ 2a2 ~ 1, i.e., for conformal coupling rn2a 2 ~ - 1 , and for mmtmal coupling m2a 2 ~ 1. For conformally coupled fields, the dominant contribution to symmetry restoration comes from the classical effect of curvature, i.e., when R c > - 6 m 2 / ( 1 - ~:2/3). For near minimally-coupled fields, while the classical effect of curvature is small, the quantum correction can induce symmetry restoration (usually at a large critical curvature R c ~- - 3 6 m 2 / h l n ~ due to the smallness o f X) for those values o f ~ ~< ~0 such that the denomenator in (31) is positive (as m 2 < 0, m the phase of broken symmetry). The range of the couphng constant ~ E [~0, 1] wherein no symmetry restoration occurs can thus be determined by the condition 1 - ~0 - ?t2/3 + C(~0)X > 0 .

(41)

For ~ ~ 1, to the first approximation, 1 - ~0 = ~2/3. However, as ~ -7 1, the lny term In C [eq. (24)] can assume very large negative value (whence C-* ¼In X2/3), thus preventing the symmetry restoration from occurrmg for all values of the mass parameter m 2 < 0. This term arises from the zero-mode contribution and is important to the symmetry restoration behavior only for near minimally-coupled fields. It is the same infrared behavior that plays an outstanding role m the phase transition of inflationary cosmologaes. There, as the otherwise short-wavelength modes are redshifted rapidly to long-wavelength modes, this infrared contribution becomes (for non-conformal fields only) more pronounced. To summarize our findings: the effect o f spacetime curvature can influence the symmetry behavior o f a system on two levels: the classical contribution in general enhances restoration, while the quantum contribution restrains it. Considering the effect of field couphngs m a background with fixed (posmve curvature), classical contribution outweights the quantum contribution for almost the full range of coupling parameter ~, from the conformal to the near-minimal 0 ~< ~ < ~0 ~- 1. This means that if the universe had started out in a phase o f broken symmetry, then decreasing the value of ~ from ~0 to 0 (conformal coupling) will lead to symmetry restoration. For a small neighborhood o f ~ around the value for minimal coupling ~ c [~0, 1], the value of~o depending on the self-interaction parameter X via (38), quantum contributions of curvature dominate over the classical contribution. According to our present result,if the universe had started out m a symmetric phase (with m 2 >t 0), for ~ E [~0,1 ], increasing the curvature above the critical value (34) will lead to symmetry breaking. For minnnal coupling, no symmetry restoration can take place for the normal range of values of X. Details of the present calculation are contained in ref. [2].

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One o f u s ( D J . O ' C o n n o r ) a c k n o w l e d g e s the s u p p o r t o f a R e s e a r c h A s s l s t a n t s h i p in t h e s u m m e r o f 1982 u n d e r NSF grant PHY82-05659.

References [ 1 ] G. Gibbons, S. Hawking and S Slklos, eds, Proc Nufheld Workshop on the Very early unwerse (Cambridge U.P., London, 1983). [ 2] T C. Shen, B.L. Hu and D J O'Connor, Symmetry behavior m the static Taub unwerse : Effect of curvature amsotropy, Unw of Maryland report (1983) [3] B.L. Hu, D.J. O'Connor and T.C. Shen, m preparation. [4] B L. Hu, Phys. Lett. 123B (1983) 189, B.L. Hu, Symmetry behavior at finite temperature m dynamic spacetlmes, m Proc Nuffleld Workshop on the Very early umverse, eds. G. Gibbons, S Hawking and S Slklos (Cambridge U.P., London, 1983). [5] B.L Hu, m Proc Third Marcel Grossmann Meeting m General relatwity, ed Hu Nmg (Smence Press, Bmjmg, 1983) [6] L.H. Ford, Phys Rev. D22 (1980) 3003. [7] D.J. Toms, Phys. Rev D21 (1980) 2805. [8] G. Denardo and E. Spallucci, Nuovo Cunento 64A (1981) 27. [9] G Kennedy,Phys Rev D23 (1981)2884 [10] See the talks given by B. Allen, L. Ford, B.L. Hu, A.D. Lmde, I. Moss and A.A Starobmsky In Princeton Umv preprmt (1982) [11] J.S Dowker and R Critchley,Phys. Rev. D13 (1976) 3224. [12] E Baum, m ' Proc. Nuffield Workshop on the very early umverse, eds G. Gibbons, S Hawking and S Slktos (Cambridge U.P, London, 1983). [13] S W. Hawking, Comm. Math Phys 55 (1977)133 [14] B.L. Nelson and P. Panangaden, Phys. Rev. D25 (1982) 1019 [15] L.H. Ford and D.J. Toms, Phys Rev. D25 (1982) 1510. [16] L. Dolan and R. Jackiw, Phys. Rev. D9 (1974) 3320, Sec. IIID

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