Volume 241, number 4 PHYSICS LETTERS B 24 May 1990
D O M A I N WALLS, W O R M H O L E S A N D T H E C O S M O L O G I C A L C O N S T A N T
Rober t MYERS Physics Department, McGill University, 3600 University Street, Montreal, Quebec, Canada H3A 2T8
Received 2 January 1990
We examine the Hawking-Baum-Coleman mechanism in the context of a universe containing a domain wall. We find new instantons with a negative cosmological constant in our model, but the above mechanism still selects zero cosmological constant. Unstable domain walls can produce a complex result, making unclear the interpretation of the euclidean gravitational path inte- gral as a probability distribution.
Recently there has been a great deal of interest in wormholes and the possibi l i ty that they may lead to a vanishing cosmological constant . This mechanism proceeds in two stages: First o f all, the low energy coupling constants in nature become quan tum vari- ables as a result o f wormhole interact ions [ 1 ]. These coupl ing constants include the vacuum energy den- sity (i.e., the cosmological cons tant ) . The eucl idean functional integral of gravity is supposed to provide a probabi l i ty d is t r ibut ion for the observed values o f these couplings. The four-sphere provides a saddle- point of the path integral when A > 0, and an essential s ingulari ty occurs at A - 0 +. This is then in terpreted as predic t ing that one will measure a vanishing cos- mological constant. The lat ter result was observed in- dependent ly by Hawking [2] and Baum [3] in the context of a single parent universe. Coleman [4] noted that the singulari ty is even more extreme with many large universes, which is more natural in the context of wormholes. Saddlepoints with A < 0 are not well explored, but in the case of pure gravity, one ar- gues that they make a negligible cont r ibut ion to the path integral [4 ].
In the present letter, a model is examined with gravity coupled to a scalar field. The scalar is re- quired to have discrete degenerate vacua, giving rise to the possibi l i ty of doma in walls. In this model, gravi ta t ional ins tantons are then constructed which describe a closed universe containing a single do-
main wall. The act ion of these instantons becomes negative and diverges for a par t icular negative value o f the cosmological constant. Applying the H a w k i n g - B a u m - C o l e m a n prescr ipt ion along with a volume regulator [5] , still favors the singulari ty at A ~ 0 + though.
Begin with the euclidean action for gravity coupled to a real scalar field,
IE : f d4x,~
× 1 U ( O ) ) . ( ( l-6-~nG ( - R + 2 A ) + ½ ( V O ) 2 + ) 1
The min ima of the scalar potent ia l U(O) are def ined to lie at zero. Any nonzero vacuum energy densi ty is defined to be a contr ibut ion to the cosmological con- stant, A. We also require that U(0 ) include at least two discrete degenerate minima, ~+ and ~_. Doma in walls will then occur at the boundary between regions where the scalar field lies at the dist inct vacuum val- ues. The precise form of the potent ial is un impor- tant, but a useful example is
2 rn 2 3/,~4 (2)
with ¢_+ = _+ (6rn2/2)l /2 .
Taking an SO (4) symmetr ic ansatz for the metric, one can write the line element as
Volume 241, number 4 PHYSICS LETTERS B 24 May 1990
ds2=f2(r) dr 2
+r2[dO2+sin20 (d02+ sin20 dx 2) ] . (3)
If we begin with O=0+ so that U ( 0 ) = 0 , it iw well known that solving Einstein's equations yields
f - 2 = l - ~ A r 2 . (4)
For A = 0, eq. ( 3 ) is then simply the line element for flat space in polar coordinates. For A > 0, eq. ( 3 ) ac- tually gives the metric on one hemisphere o f a four- sphere of radius (3/A) ~/2. We refer to the entire sphere as the de Sitter instanton. If A < 0, eq. (3) is the line element on an infinite hyperbolic sheet. One might refer to this solution as the anti-de Sitter in- stanton, except that strictly it is not an instanton since it has infinite (positive) action.
Now we wish to introduce a domain wall at a cer- tain radius, r = r~,~. We work in the thin wall approx- imation as described in ref. [6] . (The necessary junction conditions were first discussed in a more general context by ref. [7] . ) One assumes 0 = 0 + up to the domain wall, and then the domain wall pro- duces a discontinuity in the extrinsic curvature
y%= ~+ % - ~_ %. (5)
For r>r . . . . one then sets 0 = 0 , - Einstein's equa- tions relate 7% to the stress-energy of the domain wall in the limit o f zero thickness,
' f - 8rrG ( 7 % - h % 7 ~ c ) = S % = l i m T%dn, (6)
where hat, is the induced metric on the domain wall surface. 7r± % is the extrinsic curvature of the surface approached from rma x__ e. In this approximation for the domain wall, S % = - X h % , where X is some (positive) constant determined by the precise details of the scalar theory. An explicit formula will be pre- sented below.
To construct a finite volume universe (in order to have finite action), one must smoothly close off the space in the region r> rm~x. This can be done with a second copy of the original manifold extending from r = 0 to rma x. rma x is chosen then to match the discon- tinuity in the extrinsic curvature of these two spaces with the stress-energy of the domain wall in the thin wall approximation. For A > 0, the resulting space is two segments of a round four-sphere glued together.
I f A = 0, the result is two balls in N4 glued together at their surface. The somewhat surprising result is that finite action instantons are also constructed for A < 0.
To make the matching condition precise, intro- duce a new parameter related to the domain wall stress-energy: Y= 2zrGX. It is straightforward to cal- culate the extrinsic curvature of the surface r = rmax as z~ij = ~go,k~ k, where ~k Ok = f - ~ 0~ is the radial unit vec- tor for the metric in (3). One finds that g % = h a b ( l _ t 2 jArn~a~ )~/2/r,na~. By the symmetry of our construction g+ab= - - ~ % . Then from eqs. (5) and (6), one finds
- xArmax) ( 7 ) y~__ ( l 1 2 1/2
rmax
Therefore given Y and A, r ~ is determined as
I r ~ - y2+] A . (8)
Note that this is a positive quantity and hence this construction is only possible when A ~> - 3 y2.
Now the action o f these new gravitational instan- tons is evaluated. The trace of Einstein's equations yield
R (A ) 1 6 z r G + ~ ( V 0 ) 2 = - 2 8 - ~ + U ( 0 ) , (9)
which when substituted into ( 1 ) yields
,___f (10)
The first term in this expression is a familiar term proportional to the volume of the instanton, - ( A / 8~G) V. The volume is easily evaluated using (8)
rmax
24 2( I+½A/Y 2 ) _ l - ; . ( l l )
The scalar potential term in (10) makes a contribu- tion localized at the domain wall. To evaluate this term, consider the stress-energy of the domain wall. First the normal component vanishes, yielding
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Volume 241, number 4 PHYSICS LETTERS B 24 May 1990
o = S r r = f T'rfdr --.2
= f F l (O¢~2-U(~) f ] dr. (12) _ L 2 f \ 0 r J
This is a useful result for evaluat ing the tangential components of S%,
+Jl-' °0r_ 0,qdr - X = S ° ° = - 2fkOrJ
- e
= - 2 j U(O)fdr . (13)
The second term in (10) then becomes
f U((~)fr 3 )Xr3max(2~r 2) . (14) dr d3£2=
Combin ing these results as well as eq. (8) , the total act ion becomes
A / ~ _ _ 2 3 - - V-- XTr rmax 8zrG
[ A 247r2( I + I A / y 2 "~ = - 8~G A 2 1- ( l+IA/y2)3/2 J
27rG \ y2 + ~A J d
3~(l - ' ) - - GA ( l+ lA /y2 )~ /2 . (15)
One check on this result is that in the case Y= 0, one recovers the famil iar result for the de Sit ter instan- ton: I ( Y= 0) = - 3zr/GA = - 3/8G2p, where p = 8~rGA is the vacuum energy density.
In ( 15 ) then, one recovers the expected divergence as A - , 0 at Y= 0. For Y4= 0, the action is finite at A = 0, but diverges 1-+ - Go at A = - yz < 0. F rom eq. (8) , one sees that rma x and hence the volume diverges at this point. The term - ( A / 8 7 r G ) Vcont r ibu tes + o e to the action. The area of the domain wall also di- verges here, and the U(0) term makes an infini te negative contribution. To leading order in fl=A + 3 y2, these two divergences cancel in eq. ( 15 ), but a nega- t ive subleading divergence remain, as can be seen in the final result. A pr ior i then, these saddlepoints with
A < 0 cannot be ignored in evaluating the path integral.
For a given scalar theory, there are two relevant cases. The first is s imply the de Sitter instanton with the scalar sitting at ei ther o f the two possible vacuum values, and the other is the instanton with a domain wall whose construct ion was discussed above ~1. To discuss the predicted value of A, the divergences of the act ion in these two cases must be compared. The accepted procedure is to regulate the divergences by restricting the volume of these parent universes to be less than some max imum value lv~ [5] . Although more intr icate schemes are possible, it will then suf- fice for us to compare the act ions o f the instantons at V= Vm as Vm--'~. TO implement this, we parame- trize I(A, Y), the result in (15 ), in terms of the vol- ume V, and a dimensionless paramete r z = A / 3 Y 2. Using eq. ( 11 ), one may write
( l+ z [247r 2 3 1 A=+_ L Vm 1 ( f ~ Z ) 3 / 2 j j , (16)
where the + ( - ) sign applies for posi t ive (nega- t ive) z. At V= Vm, the action eq. (15) becomes
, f 3 V~m/2 I = - 2 , ~ 2 G h ( z ) = - B o h ( z ) , (17)
where
1 - 1 / ( 1 + z ) 1/2 h ( z ) = + (18) - [ l _ ( l + 3 z ) / ( l + z ) 3 / 2 1 1 / 2 .
The range of z is - 1 to 0 where A < 0, and then 0 to + ~ where A > 0. One can show that h (z) is a mono- tonical ly increasing function beginning at zero at z = - l , and going to +1 at z - . + ~ . The action o f the de Sitter instanton is evaluated with 1"2=0 or z = +oe , and hence Ildsl =Bo. For finite Y, z is finite and Llr, I =Boh(z)<Bo. Therefore the de Sitter in- s tanton has a larger divergence than any instanton with a domain wall. Of course there is a great deal of ambigui ty in regulating these singularit ies [ 10], and
~ A third possibility would be the de Sitter instanton with the scalar at the unstable equilibrium point between the two vacua. It is difficult to determine the contribution of this unstable saddlepoint to the gravity path integral because evaluating the determinant of quadratic fluctuations is problematic (com- pare the results of refs. [8,91 ). In our discussion, contribu- tions of this third solution are ignored, and a positive deter- minant is assumed for the other de Sitter instantons.
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it may be possible to choose a regulating procedure which favors the instanton with a domain wall. Un- doubtedly such a procedure would seem less natural, and we do not consider such a possibility further here.
Following the usual wormhole calculations, one now determines the relative probability that the ob- served universe has A = 0 or A < 0 by comparing the contributions to the expectation value of some oper- ator [ 4,11 ]. In a semiclassical limit, one finds the un- normalized expectation value of an operator M takes the form
( M ) ~ f dN(o~) ~ / M e x p ( - l ~ )
× e x p ( E e x p ( - l , ) ) . (19) J
Here dN is some measure for integrating over the coupling constants adjusted by wormhole interac- tions. The two discrete sums indicate summing over district parent universe saddlepoints (i.e., the de Sitter and the domain wall instantons). The domi- nant double exponential arises from summing over the number of disconnected parent universes, while the first exponential is the contribution of the uni- verse distinguished by the operator insertion M. In the simplest case thatA is the only low energy param- eter adjusted by the wormholes, the ratio of the de Sitter contribution to that of the domain wall case with some finite z is (comparing only the double ex- ponential) exp{exp(Bo)-exp[Boh(z)]}-+oc as Vm-+m. Thus the de Sitter instanton overwelms the other contribution. In the case that wormholes cou- ple to all of the parameters in U(0) as well as A, there is an integral over domain walls with different stress- energies X. The case of interest will be very light do- main walls, but one can show Oh(z)/OXfx=o~-Ov as V,~oo, indicating that the de Sitter instanton will overwelm the contributions of all saddlepoints with domain walls with any finite stress-energy. Thus in either case, one recovers the desired prediction that the observed value of A is zero.
The validity of the thin wall approximation will be considered now. An essential restriction was that S~r=0. In fact for a domain wall of finite thickness, the normal components of the stress-energy are non- vanishing [ 12 ]. The requirement for the validity of our approximation is that the ratio of the normal
components to the tangential components of the stress-energy be much less than one, (Srr/S°o) << 1. It was shown in ref. [13] that this requirement is equivalent to
= GXT<< 1, (20)
where T is the thickness of the wall (see also ref. [ 14 ] ). T is typically the order of inverse scalar mass 1 / ms, and hence (20) should be valid for a wide range of scalar field theories.
A word of caution is required though, since in fact both X and T will be affected by the background cos- mological constant. This is most easily seen by re- casting the metric ( 3 ) into the following form:
dsZ=dT2+r(-c) 2 [d02Wsin20 (d¢2+sin20 dz 2) ] •
The primary case of interest is A < 0 for which r= cosh(x/½ [A[ r) when the scalar is fixed at
¢ = ¢ +_. To linear order, the scalar field equation then becomes
0 = g i 0 - - m 2 0
=¢,,+ 3r '¢ , m e 0 r
~--0"+ 3 X / ~ ¢ ' - - m 2 0 , (21)
where' denotes O/Or, and the last equation is approx- imate for large r. Searching for solutions of the form
= 0± + q e ~, (21 ) is solved by
oz= + ( 31Al + m 2) 1/2(~ IAI)W2.
The plus sign gives the relevant exponentially grow- ing solution, and for small I A[ one recovers c~_~ m, but for large [AI, c~-~ m2/x/3 IAI. One expects that T= O (c~ - ~ ) since c~ - ~ is the characteristic proper distance scale on which # varies. So that requiring the domain wall to be thin puts restrictions not only on the scalar mass but also on the cosmological con- stant. Similarly X is also not only a function of the couplings of the scalar field, but also of the cosmolog- ical constant. From ( 13 ) one may approximately de- termine this dependence with
X=2 g (0 ) d ~ - 2 U(¢) de 1O+-¢~- I
In this approximation then, X has the same depen- dence on A as T.
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V o l u m e 2 4 1 , n u m b e r 4 P H Y S I C S L E T T E R S B 2 4 M a y 1 9 9 0
One may easily extend the above calculation to un- stable domain walls. Such instantons were consid- ered by refs. [ 15,16 ]. This is the case for which the scalar potential has two local minima, hut one of these has a nonvanishing vacuum energy density. To be specific let U(0+ ) = 0 and U(0_ ) =Po> 0. Our con- struction of the instanton can be adapted essentially unchanged except that there are asymmetric match- ing conditions to be satisfied at the domain wall at rma~. The final result for the euclidean action is
[=½[I(A+, Y + ) + I ( A _ , Y _ ) ] , (23)
where I(A, Y) is the expression presented in (15), and A+=A, A_=A+8xGpo. Y+ are related to the extrinsic curvatures on the domain wall surface, and must satisfy Y + + Y _ = 4 r c G X and r2m.x= ( Y~ + ~A ± ) - t. One finds that
Po Po ( 24 ) Y+ =2JrGX+ ~ , Y_ = 2 ~ G X - - ~ .
Eq. (23) applies when both Y+>0 , and so when Po < 6 ~zGX2. This action diverges at rmax--* O% but again with a volume regulator, one finds that this diver- gence is overwhelmed by that of the de Sitter instan- ton at A = 0. Note that in this case the two de Sitter instantons with ~ = ~+ are distinct, since ~+ is a sta- ble vacuum but 0_ is not.
One may also consider the case Y _ < 0 or po> 6gGX 2. In this case r . . . . the radius o f the domain wall is no longer the maximum radius of the instan- ton. Rather this solution may be described qualita- tively as a de Sitter instanton with radius (3 /A_)J /2 containing a bubble of radius rmax with reduced en- ergy density. In the bubble, the cosmological con- stant is A+ = A _ - 81rGpo. The total action of this so- lution is
].= 3zC ( 1 ) 2GA_ 1+ ( I + ~ A /y2_)i7~
3rt (1 - l ) 2GA+ (1+~ 2 J/2 - 25) ~A+/Y+)
Now I becomes negative and diverges as A_-~0 + where the volume diverges. In this limit, the domain wall remains finite in size though, and hence the dif- ference between the action of the de Sitter instanton with A_ and that including the bubble is finite. In the limit that A _ - , 0 , one finds that
K = I - I a s ( A _ ) - 3zt k(£) , 2GIA+ [
where
1 2 1 k ( 2 ) = l 2 1 + 5 ( 1 + 2 ) '/2
(26)
(27)
and ~=A+/3Y2+ which takes values in the range - 1 < 2 < 0 . One can show that in this range k is al- ways positive, and diverges at 5 ~ - 1, corresponding to an infinite bubble. In the limit where the volume of the de Sitter instanton is diverging (i.e., A_ -, 0 +, but that of the bubble remains finite, one can con- struct solutions with any number o f bubbles. Evalu- ating the contribution of these solutions in (19) in- volves summing over all numbers of bubbles, as well as taking into account zero modes and symmetry fac- tors. The final result is that the action of the de Sitter instanton with ~ = ~ _ is shifted,
37r 37r / dS = - - - - __, __ - - + H V e - ~ ,
GA _ GA _
where K is given in (26). The instanton volume V arises due to translational zero modes of the relative positions of bubbles, and H represents the modifica- tions to the quadratic fluctuation determinant in the presence of a bubble. The presence of a bubble should introduce one new negative mode in the scalar field [ 17 ], and hence H is imaginary. In the context where these instantons were originally considered, that is decay of the false vacuum [15,16 ] this leads to an interpretation of [HI e -~ as the decay probability per unit volume. In the present context, one has two dis- tinct de Sitter instantons for which the real part of the action is equally divergent at A + --, 0 +. Unfortunately the action for the unstable instanton has an imagi- nary phase as well, creating a problem of interpreta- tion It is difficult to reconcile understanding the eu- clidean path integral as a probability distribution for the low energy coupling constants with the fact that in general it yields complex results [ 8 ].
It is simple to further extend the constructions pre- sented here to include more complicated networks of domain walls as well as other topological defects such as strings, but again the singularities encountered at infinite regulator volume are still overwhelmingly dominated by that of the de Sitter instanton. This
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Volume 241, number 4 PHYSICS LETTERS B 24 May 1990
seems to ind ica te a cer ta in robus tness o f the la t te r
singulari ty. The or ig ina l a rgumen t s for A = 0 only
cons ide red the s implest , m o s t s y m m e t r i c pa ren t uni-
verse saddlepoin ts . H e r e we have e x t e n d e d the anal-
ysis to inc lude less s y m m e t r i c solut ions , bu t the de
Si t ter i n s t an ton still d o m i n a t e s the pa th integral pro-
duc ing the p red i c t i on A = 0.
The d iscuss ion here has been phrased ent i re ly in
the con tex t o f a rgumen t s due to C o l e m a n , H a w k i n g
and Baum. This a p p r o a c h has recent ly r ece ived s o m e
cr i t i c i sm [8,18 ]. In fact, the uns tab le d o m a i n wall
case reveals one o f the sho r t comings o f these argu-
ments . The singulari t ies i nvo lved our analysis though
are pe rvas ive to any semiclass ical t r e a t m e n t o f grav-
ity, and so wou ld r eappea r in lo ren tz i an tunne l l ing
ca lcu la t ions [ 19] ~2, or th i rd q u a n t i z a t i o n o f the
W h e e l e r - D e W i t t e q u a t i o n [ 18,20 ]. The conc lus ion
r ema ins that the s ingular con t r i bu t i on o f the de Si t ter
i n s t an ton at A = 0 o v e r w e l m s that o f the d o m a i n wall
ins tantons .
I wou ld like to thank C. Burgess for useful conver -
sations. Th i s research was suppor t ed in par t by funds
p r o v i d e d by the Physics D e p a r t m e n t o f M c G i l l
Unive r s i ty .
~2 It is interesting to note that all of the instantons discussed here, even those with A < 0, may be continued to solutions describ- ing expanding lorentzian space-times.
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