21 September 2000

Ž .Physics Letters B 489 2000 397–402www.elsevier.nlrlocaternpe

Static cosmological solutionsof the Einstein–Yang–Mills–Higgs equations

P. Breitenlohner a, P. Forgacs b, D. Maison a´a Max-Planck-Institut f ur Physik, Werner Heisenberg Institut, Fohringer Ring 6, 80805 Munich, Federal Republic of Germany¨ ¨

b Laboratoire de Mathemathiques et Physique Theorique, CNRS UPRES-A 6083, UniÕersite de Tours,´ ´ ´Parc de Grandmont, 37200 Tours, France

Received 21 July 2000; received in revised form 1 August 2000; accepted 14 August 2000Editor: P.V. Landshoff

Abstract

Numerical evidence is presented for the existence of a new family of static, globally regular ‘cosmological’ solutions ofthe spherically symmetric Einstein–Yang–Mills–Higgs equations. These solutions are characterized by two natural numbersŽ .mG1, nG0 , the number of nodes of the Yang–Mills and Higgs field respectively. The corresponding spacetimes arestatic with spatially compact sections with 3-sphere topology. q 2000 Published by Elsevier Science B.V.

There has been considerable progress in the studyof the spherically symmetric Einstein–Yang–MillsŽ . Ž .EYM and Einstein–Yang–Mills–Higgs EYMHequations, stimulated by the discovery of globally

w xregular solutions of the EYM eqs. 1 , for a revieww xsee e.g. 2 . Up to now most of the attention has been

focused on asymptotically flat, smooth particle-likeŽ .self-gravitating sphalerons and monopoles andblack hole solutions.

In this paper we present a new ‘cosmological’type of globally regular solutions of the EYMHequations describing static, spherically symmetricspatially compact space-times. Analogous solutionsŽ .‘static universes’ have already been found in EYM

w xtheory 3 in the presence of a cosmological constantŽ .L, EYMC . More precisely, numerical evidence has

w xbeen presented in Ref. 3 indicating the existence ofa discrete family of solutions indexed by the numberm of nodes of the YM field, for a special set of

� Ž .values of the cosmological constant, L m , ms

41, . . . . The ms1 solution is particularly simple asits energy density is constant and it has a simple

w xanalytic form 4,5 . It appears that the presence ofthe cosmological constant is essential for the veryexistence of such solutions. Therefore at first sight itmight appear surprising that such ‘static universe’solutions also exist in an EYMH theory without acosmological constant. In fact this should not cometotally unexpected as the self-interaction potential of

Ž .the scalar field, V F , can generate the necessaryenergy density to support a spatially compact space-

Ž .time. Intuitively V F can act as a ‘dynamical cos-mological constant’. This viewpoint has been partic-

w xularly stressed by Linde and Vilenkin 6 in theirwork on ‘topological inflation’.

The EYMH theory has three different mass scales,'the Planck mass M s1r G and the masses MPl W

and M of the YM resp. Higgs field, giving rise toHŽtwo dimensionless ratios asM rgM where g isW Pl

.the YM coupling and bsM rM . It turns out toH W

0370-2693r00r$ - see front matter q 2000 Published by Elsevier Science B.V.Ž .PII: S0370-2693 00 00931-X

( )P. Breitenlohner et al.rPhysics Letters B 489 2000 397–402398

be convenient to introduce an ‘effective cosmologi-4 2 Žcal constant’ Lsa b r4. Our ms1,ns0,1,2,

.. . . EYMH solutions bifurcate with the EYMC'w xsolution of 3 for as 3 , . . . at Ls3r4 with a

vanishing Higgs field. Varying a leads to 1-parame-Ž .ter families ns0,1, . . . with L determined as a

Ž .possibly multi-valued function of a . Similarly forsome discrete values a our solutions bifurcatem ,n

with the higher nodes cosmological EYMC solutionsw xof Ref. 3 , when the Higgs field tends to zero.

w xFollowing the notation of 7 , we write the spheri-cally symmetric line element as:

ds2 se2n ŽR.dt 2 ye2 lŽR.dR2 yr 2 R dV 2 . 1Ž . Ž .

The ‘minimal’ spherically symmetric Ansatz for theYM field is

W aT dx mm a

sW R T duqT sinu dw qT cosu dw , 2Ž . Ž . Ž .1 2 3

Ž .where T denote the generators of SU 2 and for thea

Higgs field

F a sH R na , 3Ž . Ž .

na denoting the unit vector in the radial direction.The reduced EYMH action can be expressed as

X2X X1Žnql. y2 l 2Ssy dRe 1qe r qn rŽ Ž . Ž .Ž .H 2

y2 l 2ye r V yV , 4Ž .1 2

with

2XWŽ . 2X1V s q H , 5Ž . Ž .1 22r

and

22 2 21yW b rŽ . 22 2 2 2V s q H ya qW H .Ž .2 2 82 r6Ž .

Ž .Varying the reduced action 4 one obtains the EYMHequations:

XX 2 Xy2 l 2 y2 l 21ye r qn r q2e r V y2V s0 ,Ž .Ž . 1 2

7aŽ .

XX 2 Xy2 l yl yl y2 l 21qe r y2e e rr y2e r V y2VŽ . 1 2

s0 , 7bŽ .

X XX Xyl yl ynyl nyle r e qe e rnŽ . Ž .

E r 2V E VŽ .1 2y2 lqe qr s0 , 7cŽ .E r E r

W 2 y1XXnyl nql 2e W ye W qH s0 , 7dŽ . Ž .2ž /r

b 2 r 2XX2 nyl nql 2 2 2r e H ye H 2W q H yaŽ . Ž .ž /2

s0 . 7eŽ .

Introducing the combinations

N'eyl rX , k'eyl rX qrn X , 8Ž . Ž .

Ž .the field Eqs. 7aa-e take the form:

2 ˙ 2 2 ˙ 22kNs1qN q2W qr H y2V , 9aŽ .2

rsN , 9bŽ .˙

˙ 2N W2˙ ˙Ns kyN y2 yrH , 9cŽ . Ž .

r r

2 2 2W b r 22 2 2ks 1yk q2 y H yaŽ .˙ 2 2r

2 2y2 H W rr , 9dŽ .

2 ˙W y1 WŽ .2WsW qH y kyN , 9eŽ . Ž .2ž / rr

2 2 ˙W b H2 2Hs2 H q H ya y kqN .Ž . Ž .2ž /4 rr

9fŽ .

˙ yl X Ž .where f[dfrdsse f . Note that in Eqs. 9a theŽremaining gauge freedom i.e. diffeomorphisms in

.the variable R is implicit in the choice of the

( )P. Breitenlohner et al.rPhysics Letters B 489 2000 397–402 399

Žindependent variable s . A convenient choice also. lfor the numerical integration is e s1.

Next we recall that solutions with a regular originw xat rs0 7 have the expansion

W r s1ybr 2 qO r 4 , 10aŽ . Ž . Ž .H r sarqO r 3 , 10bŽ . Ž . Ž .

a2 a 4b 22 2 4N r s1y 2b q q r qO r ,Ž . Ž .ž /2 24

10cŽ .

a2 a 4b 22 2 4k r s1q 2b y y r qO r ,Ž . Ž .ž /2 8

10dŽ .

where a, b are free parameters. Solutions satisfyingŽ .the conditions 10a which also stay globally regular

Ž .contain among others asymptotically flat self-Ž .gravitating monopoles. Integrating the field Eqs. 9a

one finds that for the generic solution, however,Ž .N s becomes zero at some finite sss with0

finite values of all the other dependent variables. AsŽ . Ž .it is immediately seen from Eq. 9ab N s s00

Ž .implies stationarity of r at r s sr . Then in0 0Ž .general r s decreases on some interval for s)s .0

We refer to such a point as an ‘equator’. In the casew xof the EYM equations it has been proven in Ref. 8

Ž .that for all solutions with an equator r s decreasesfrom r all the way to rs0 where a curvature0

Ž .singularity develops ‘bag of gold’ . Adding a cos-mological constant radically changes this conclusionand there exists an infinite family of regular solu-

w xtions with 3-sphere topology 3 .Ž .Since for H r '0 the EYMH system reduces to

an EYMC theory where the cosmological constant isgiven as Lsa 4b 2r4, it is natural to search for

Ž .solutions of Eqs. 7a with a nontrivial Higgs fieldbifurcating with the EYMC ones. To see if such abifurcating class really exists one should first estab-lish the existence of regular solutions of the lin-

Ž .earized Higgs-field Eq. 7ae in the background of a3-sphere type solution of the EYMC system sincethe bifurcation occurs for H™0. To discuss thelinearization of the Higgs-field equation around solu-tions of the EYMC equations, it is more convenientto choose a different gauge from el s1, namelynsl and use the variable r defined by drrdrsrX

selN. Then the linearization of the Higgs-field Eq.Ž . Ž7ae , around an EYMC background solution Ls

Ž . Ž . Ž . .L , rsr r , nsn r , WsW r , HsH '0b b b b b4 2 . Ž .with L sa b r4 and Hsh r , reads as:b b b

LX bX2 2n 2 2br h s2e W y r h . 11Ž .Ž .b b b2ž /a

Let us consider first the simplest, ms1 EYMCw xsolution 4 :

'W scos x , H s0 , r s 2 sin x , 12aŽ .b b b

N sk scos x , n s0 , L s3r4 , 12bŽ .b b b b

' Ž .where xsrr 2 . Then Eq. 11 in the backgroundŽ .12a , reduces to the following simple equation:

3XX2 2 2sin x h x s 2cos x y sin x h x ,Ž . Ž . Ž . Ž . Ž .Ž . 2ž /a

13Ž .

where now X stands for the derivative with respect toŽ .x. We note that Eq. 13 can be transformed to a

hypergeometric equation, however, the solutions ofinterest, i.e. regular on the background geometry and

Ž .satisfying the condition 10ab , can be directly foundby the following trigonometric polynomial Ansatz

Ž .for h xn

2 kq1h x s c sin x . 14Ž . Ž .Ýn kks0

One then easily obtains the recursion relation for thecoefficients c :k

2ky1 2kq1 y2y3ra 2Ž . Ž .c s c ,k ky12kq2 2kq1 y2Ž . Ž .

ks1 . . . n , 15Ž .Ž .together with the termination condition c s0nq1

32a s , ns0,1, . . .1,n 2nq1 2nq3 y2Ž . Ž .

16Ž .

yielding a 2 s3, a 2 s3r13, a 2 s1r11, . . .1,0 1,1 1,2

That is we have found an infinite family of regu-lar solutions of the linearized Higgs field equationŽ .bounded ‘zero modes’ in the ms1 EYMC back-ground. These ‘zero modes’, indexed by the number

Ž . Ž .of zeros of h x ns0,1, . . . , indicate the existence

( )P. Breitenlohner et al.rPhysics Letters B 489 2000 397–402400

of a family of globally regular solutions of theŽ Ž .EYMH equations with H x having the same num-

.ber of zeros bifurcating with the ms1 EYMCsolution for as0 and asa .1, n

Ž .From the a-dependence of the r.h.s. of Eq. 13 itŽ .follows that the solutions h x vanishing at xs0w .have nq1 zeros for all ag a ,a . Accord-1, nq1 1,n

ing to standard theorems on Sturm–Liouville opera-tors this number gives also the number of bound

Ž .states of the differential operator in Eq. 13 . Thuswhenever a crosses a from above a new bound1, n

state appears, indicated by the existence of a zeromode for asa .1, n

In contrast to the case ms1, the higher nodeŽ .EYMC solutions mG2 are not known analytically.

Nevertheless one can still conclude that for any ofthese EYMC backgrounds an infinite family of regu-lar zero modes exists. To see this, it is sufficient tonote that for a sufficiently small the r.h.s. of Eq.Ž .11 becomes arbitrarily negative over a finite inter-val of r. This implies that the number of boundstates tends to infinity for a™0. According to theprevious argument the same holds for the number ofzero modes accumulating at as0. Numerical val-ues for the first five bifurcation points with ms1, . . . ,5,10,20 are given in Table 1.

In order to verify the correctness of the aboveŽ .scenario we have integrated the field Eqs. 9a nu-

merically. To simplify the singular boundary valueproblem, we have looked only for solutions, which

Ž .are anti symmetric about the equator. This meanswe impose boundary conditions at rs0 and at the

Ž .equator a regular point . In addition to the vanishingof k at the equator, we require the vanishing of the

˙ ˙functions W resp. W and H resp. H depending if m

and n are even or odd. Thus 3 functions must have acommon zero with N forcing us to tune 3 of the 4available parameters a , b , a, and b. Fig. 1 showssome solutions with ms1,2 zeros of W and ns0,1zeros of H.

In Fig. 2 we have plotted the values of theparameters a and L for the solutions with m,nF2.While a runs to large values with decreasing L forthe ns0 solutions, i.e. with a nodeless Higgs field,the parameters for the solutions with nG1 show theopposite behaviour. This structure remains true forhigher values of m, although the change occurs in

Žgeneral for some n )1 increasing with m e.g.0. Ž .n s6 for ms10 . Similar to the case 2,0 some of0

the graphs in the a-L plane for m)2 show aminimum of a , before they tend to large values of a

Ž .e.g. ns1,2,4,5 for ms10 .The families of numerical solutions for which a

becomes large seem to have a smooth limit fora™` with L™1r4. In this limit, assuming that H

Ž .stays finite, Eqs. 7a reduce to an EYMH systemwith a cosmological constant Ls1r4 instead of a

Ž .Higgs potential bs0 . The spatial sections of thecorresponding space-times are no longer compact

'Ž .s™` and r ™ 2 . We have also numerically0

integrated the limiting field equations and our resultsfully confirm the existence of the limit a™`.

w xIt was argued in 11 that the solutions of thew xEYMC system of 3 are unstable. As usual the

criterion for instability is the existence of imaginarymodes of the linearized time-dependent field equa-tions. Although we do not quite approve of the

w xmethods of 11 , we nevertheless believe that theirresult is correct. For the case ms1 the instability

w xhas already been shown in 5 . We expect the insta-

Table 1Bifurcation points.

m L a a a a am m ,0 m ,1 m ,2 m ,3 m ,4

1 0.75 1.732051 0.707107 0.480384 0.369274 0.3015112 0.3642442 1.929400 1.080086 0.692265 0.521313 0.4205863 0.2932176 2.694371 1.331930 0.957678 0.712995 0.5695244 0.2703275 3.629807 1.686984 1.170945 0.922451 0.7397885 0.2608951 4.730633 2.061886 1.424943 1.102191 0.912326

10 0.2512791 12.441074 4.080242 2.777478 2.137747 1.74542820 0.2501165 37.982205 8.256034 5.578608 4.284836 3.494772

( )P. Breitenlohner et al.rPhysics Letters B 489 2000 397–402 401

Ž . Ž . Ž .Fig. 1. Solutions with ms1,2 zeros of W and ns0,1 zeros of H. The figures display W, H solid , r, k dashed , and N dotted asfunctions of s .

bility of the YM system to persist in our case withthe Higgs field with the same number of unstablemodes at least for small values of a. In view of the

Ž .discussion of the solutions of Eq. 11 given abovethere are additional instabilities of the EYMC solu-

Žtions viewed as solutions of the EYMH theory with

Fig. 2. The effective cosmological constant Ls a 4b 2r4 as afunction of a for the solutions with m,nF2.

.as0 in the Higgs sector. More precisely, there arew .nq1 unstable modes for ag a ,a . Turn-m ,nq1 m ,n

ing to solutions with a/0 but small the only ques-tions is, if the number of the unstable Higgs modes

Ž .of the m,n solutions is equal to n or nq1. Theanswer depends on the behaviour of the zero mode atthe bifurcation point a when a deviates fromm ,n

zero. If it turns into a bound state one gets nq1, ifit moves into the continuum n unstable modes.

Similar to the globally regular solutions discussedin this paper, there are families of solutions with ahorizon branching off from the corresponding solu-

w xtions of 3 without a Higgs field. In fact there areeven more general solutions having two horizons,one of them replacing the regular origin of theforementioned class. We plan to give a detailedaccount of these solutions in a forthcoming publica-tion.1

1 w xUncited references: 9,10

( )P. Breitenlohner et al.rPhysics Letters B 489 2000 397–402402

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