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Vacuum tension effects on the evolution of domain walls in the early universe
Nuclear Physics B252 (1985) 523-537 © North-Holland Publishing Company
VACUUM TENSION EFFECTS ON THE EVOLUTION OF D O M A I N W A L L S IN T H E EARLY U N I V E R S E * A. AURILIA 1 Department of Physics, University of Toronto in Mississauga, Toronto, Ontario, Canada
G. DENARDO International Centre for Theoretical Physics, Trieste, Italy, Dipartimento di Fisica Teorica, Universita' di Trieste, Italy and Istitato Nazionale di Fisica Nucleare, Sezione di Trieste, Italy
F. LEGOVINI and E. SPALLUCCI Dipartimento di Fisica Teorica, Universita' di Trieste, Italy, and lstituto Nazionale di Fisica Nucleare, Sezione di Trieste, Italy
Received 4 July 1984 (Revised 25 October 1984) The "vacuum pressure" mechanism of the hadronic bag model is taken as a guide to formulate the dynamics of closed domain walls in the cosmological case. The effective action functional suggested by this analogy is a straightforward generalization of the Einstein-Maxwell action: it involves a 3-index antisymmetric potential whose coupling to matter generates two effective cosmological constants, one inside and one outside the domain wall. It is suggested that this mechanism, which is alternative to the introduction of a Higgs potential, is the source of the bubble nucleation process envisaged in the new inflationary cosmology. The dynamics of a spherical domain in a de Sitter phase is analyzed and is consistent with the geometrical formulation of shell dynamics proposed long ago by Israel.
I. Introduction The i n t e r p l a y b e t w e e n particle physics a n d cosmology has led to p r o f o u n d changes in o u r p e r c e p t i o n of the physics of the early universe. As o p p o s e d to the picture offered by the hot b i g - b a n g model, the g r o u n d state of the early universe resembles an e m u l s i o n o f different phases, The phase transitions p r o c e e d via the creation of v a c u u m b u b b l e s in one phase a n d their s u b s e q u e n t e v o l u t i o n in the b a c k g r o u n d of a different phase [1]. Such a characteristic d o m a i n structure is c o m m o n to other physical systems: ferromagnets a n d s u p e r c o n d u c t o r s are some of such systems whose properties have u n d o u b t e d l y inspired some of the c u r r e n t cosmological scenarios. However, what comes to our m i n d as a particularly fitting a n a l o g y with the case of the early u n i v e r s e is the h a d r o n i c v a c u u m pictured in the q u a r k - b a g m o d e l with * Work supported in part by the National Science and Engineering Research Council (NSERC) of Canada. On leave of absence from Istituto Nazionale di Fisica Nucleate, Sezione di Trieste, Italy. 523
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surface tension*. Our objective is to illustrate the precise sense of this analogy and to explore some of its physical consequences in cosmology and astrophysics. The hadronic vacuum envisaged in the bag model is also regarded as a medium consisting of two phases separated by a closed domain wall. The interior of the " b a g " represents the hadronic phase while the exterior region is regarded as a different phase of the vacuum inaccessible to the hadronic constituents. A key ingredient of the bag model is a sort of "cosmological constant" confined to the interior of the bag. This is tantamount to a non-vanishing vacuum energy density associated with the hadronic phase. Hopefully, this phenomenological scenario of the ground state of strong interactions will be substantiated by a deeper understanding of the hadronization phase in non-perturbative QCD. By the same reasoning, since at present the quantum properties of the cosmological ground state are poorly understood, it seems desirable to formulate an effective approach to the bubble nucleation process in the early universe. We suggest that the process of formation of vacuum bubbles in the cosmological case be interpreted in terms of a vacuum energy density which is Completely analogous to that of the hadronic phase in the physics of strong interactions. Once formed, the later evolution of the domain boundaries is governed by their surface tension and their interaction with matter. The problem of "bubble dynamics" is of interest in its own right and was considered recently by Berezin, Kuzmin and Tkachev [3] in connection with the vacuum state of the early universe and also by Maeda, and Sasaki and H. Sato I-4] in connection with a computer simulated evolution of intergalactic "voids" in an expanding universe. The relevant mathematical formalism for this type of problem in general relativity was developed some time ago by Israel [5] and later by Chase [6]. In particle physics, on the other hand, Dirac used his formalism of constrained hamiltonian system to develop an electrodynamic model of a conducting bubble in an attempt to resolve the electron-muon puzzle. Later, with the recognition that strongly interacting particles possess a finite spatial extension, the study of the evolution of extended structures in spacetime became a major line of research in particle physics. Our approach to bubble dynamics'is based on a general theory of relativistic extended objects formulated previously in ref. [7]; relevant points of that theory will be recalled in sect. 2. In order to discuss the cosmological and the astrophysical case we propose an effective action modelled on the hadronic bag. The resulting equation governing ~the dynamics of one vacuum in the medium of a different vacuum can be obtained either as a junction equation ~ la Israel, or from the generally covariant definition of the total mass energy of the system [8]. Particular solutions to our equation of motion fit into two classes: we find "oscillating bubbles", which expand from zero to a m a x i m u m radius and then collapse, and de Sitter-type bubbles which contract from infinity up to a minimum radius and then expand indefinitely. A special case * F o r a g e n e r a l r e v i e w see f o r i n s t a n c e [ 2 a ] ; f o r a c o n c i s e r e v i e w o f s o m e o f t h e a s p e c t s o f t h e b a g m o d e l w h i c h a r e r e l e v a n t to o u r d i s c u s s i o n see [2b].
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of this type of solution was advocated by Vilenkin [9] who pictures the birth of the universe as a tunneling effect from an initial "void" with total energy equal to zero into a de Sitter space. When the surface tension is zero we also find an isolated static solution representing a spherical vacuum domain of de Sitter type. This kind of bubble admits a limiting configuration where its radius, the Schwarzschild and the de Sitter radii are coincident. This solution was proposed some time ago as a model of hadronic matter in the framework of "strong" gravity [ 10]. The astrophysical meaning of such a solution, if any, remains an open question. Finally, when the action is formulated in "flat" Minkowski space, the equations of motion reproduce the confinement mechanism postulated in the quark-bag model. This paper is organized as follows: in sect. 2 we introduce and discuss the coupling of the membrane and its generalized gauge field in curved space. In sect. 3 the existence of vacuum domains is shown as a solution of the generalized Maxwell equations. In sect. 4 we find a spherical bubble solution of the Einstein equation and discuss its evolution in some particular cases. In sect. 5 we give a brief discussion of the results. 2. The action
In the current literature the vacuum state of the early universe is often modelled on the classical de Sitter solution to Einstein's equation with a cosmological term. This new type of "de Sitter vacuum" has been successfully exploited to get around some of the well-known problems of the standard big-bang model [1 1]. A similar artifact (the bag constant) is the key to the success of the quark-bag model in matching hadronic spectroscopy. Our approach is based on the recognition that the cosmological constant as well as the bag constant can be formulated, in a gauge invariant way, in terms of a 3-index antisymmetric potential A~,~ to which we associate a totally antisymmetric field strength tensor F~,~p =- O[~A~,vol [7, 12]. In geometric notation the field Fa,,vp is a 4-form F = dA which trivially satisfies the Bianchi identities d F = 0 and obeys the Maxwell-like equations d * F = *J. The source J will be specified shortly. For the moment we note that the field F remains invariant under the gauge transformation A ~ A + dh where the gauge function h is a differential 2-form. The field F is non-propagating and, in the absence of a source term, it represents a background field constant over the entire universe: indeed, since *F, the Hodge dual field strength, is a zero-form, the equation d * F = 0 implies * F = c, with c constant everywhere. This constant can be interpreted as the cosmological constant in general relativity. As a simple calculation (cf. eq. (2.14)), the energymomentum tensor of the F-field is given by T~F~= Ag,~, with A = 4zrc 2. Therefore the coupling of the F-field to the Einstein tensor G can be described by the equivalent sets of equations G=87rT,
d*F=O~G=Ag.
(2.1)
This is the basic property of the F-field; recently it has generated a lot of interesting
A. Aurilia et al. / Early universe
526
applications not only in connection with the long-standing question of the actual size of the cosmological constant [13] but also in connection with the problem of the compactification of extra dimensions in Kaluza-Klein supergravity [14]. From our vantage point the usefulness of the gauge invariant formulation of the cosmological constant is seen in the possibility of coupling the F-field to matter whereas the standard cosmological term is frozen in the action of general relativity. The gauge invariant coupling of the F-field to matter is easily understood by analogy with electrodynamics: as the standard electromagnetic field A~ mediates the interaction between the line elements along the world line of point particles, so the 3-index potential A ~ mediates the interaction between the hypersurface elements along the world-tube swept by the surface of a bubble in spacetime. This world-tube is the embedding ~:(~) of a connected, orientable, 3-dimensional manifold 2, in the 4-dimensional spacetime manifold M, described by x '~ = s~"(ta),
(2.2)
where t a = {t °, t l, t2},
x " = {x °, x l, x 2, x3},
(2.3)
are local coordinates on M and ,X respectively. In order to describe a bubble, which is usually thought .of as a spacetime 3-dimensional region bounded by a closed 2-dimensional spatial surface, we choose ,X-~ R x S where S is a 2-dimensional compact manifold. Then at any time the bubble is represented by the intersection between the x ° = const hyperplane and ~:(,X). Our model describes a surface layer, that is a m e m b r a n e of vanishing thickness to which we associate a 6-function. This is the limit where the thickness of the m e m b r a n e is very small with respect to the size of the bubble. The tangent timelike element to the world-tube ~(2;) is given by the 3-vector density ¢
__ e abc~_ ~ O~ Ot o A Ot--'-(A t3t----~= cgt b at c '
(2.4)
where e abe is the 3-dimensional Levi-Civita tensor*, and y is the determinant of * We adopt the following conventions: [ " * " ~ ] is the n-dimensional Levi-Civita tensor density, which is defined according to [~c "'~-] =
+!. -
cq, c~2. . . . o~. even permutation of reference sequence a l , a~, • • • c~. odd permutation of reference sequence s o m e indices e q u a l .
Consequently, the Levi-Civita tensor is ECrl'"an
=41___~[o,~q,
~ , . .... - - 4 - - ~ [ ~ , . ..... I.
We define the dual of a k-rank tensor T~,~.~+, as l *Ta,"'a. k=~.ea,'"-.
k+l'"ot.
TOtn-k+lCln "
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the induced metric rob = at o ~ - ~ g . ~
(2.5)
on ~:(~). ~ generalizes the velocity tangent vector to the case of an extended object like the 3-surface we are considering. The norm of such a 3-vector density, defined as I1~11= (-~a~'~a~,~/3 !)1/2 provides a suitable kinetic term for the surface. By taking into account (2.2), (2.4) and (2.5) one finds I1~112=-3, hence the kinetic term of the membrane can also be seen as its invariant "area". Finally we have to couple the membrane with the gauge potential. The photon field A , interacts with a pont-like charge through the current density J~'= d r ~ ' 6 ( x - ~:(r)) where ~ is the tangent element to the particle world-line x = ~:(r). Hence the gauge potential Aa,~ can interact with the bubble only through the current density
jx.~, = f~: ~x.,,8( x - ~:(t)) d3t.
(2.6)
Our effective action is therefore given by 1 I(g~,.,A~,~.£~)=~ fMd4X,/~-gR-PL d3t ll~
-I 2"4!
where we have used Gn = 1. Here the first term is the usual Hilbert-Einstein action and the fourth is a straightforward generalization of Maxwell's. The positive-definite constant p (which in classical electrodynamics would represent the mass of the pointlike charge) is here the surface energy density, or surface tension, and has dimension L -1 in geometric units. Finally the gauge coupling constant c has dimension L -1. The action (2.7) is manifestly invariant under general coordinate transformations on the spacetime manifold as well as on the embedded submanifold representing the world-track of the bubble. These invariance properties are reflected by the field equations: 6gI = 0 leads to Einstein's equation G = 8~-(TZ)+ TF~),
(2.8)
while 6AI = 0 leads to "Maxwell's equations" for the F-field: d F = O,
d*F = c'J,
(2.9)
where *J represents the 1-form current dual to JA~(x): 1
I Ata'v I * ' 8 = 3 ! pa.~-' , --
E
where epA~,~ is the Levi-Civita tensor in 4 dimensions.
(2.10)
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Finally 6~I = 0 leads to the "Lorentz force" equation governing the evolution of the bubble
F]vxO + ( I ' g ) ~ T
.bOx p Ox ~ 1 c Ot ~ Orb--3!
Yoga"
~f"~,;ll~l I
(2.11)
where (Fg)oQ is the connection of the metric g on the spacetime manifold and ~ v is the Laplace-Beltrami operator with metric Y~b. Just as in electrodynamics, the field equations imply the conservation of the Noether current (2.6) as well as the conservation of the symmetric energy-momentum tensor
2 61 T~" = -x/----g 8gA, = T ~ + T ~ ,
(2.12)
where
z)
f
2! 3
I111=
T ~ -- 31! Fa,,ov F ~'~ v
4I-
d3t'
1 2.4!ga~'F~t3V~F~t3v~=-lga~'(*F) 2"
(2.13)
(2.14)
Both in cosmology and in particle physics one is mainly concerned with spherical vacuum domains, so in sect. 4 we shall assume spherical symmetry for our closed membrane: in this case we shall be able to solve the whole set of coupled field equations derived from the action (2.7).
3. Exact solution of the "Maxwell" equations In principle, the field equations completely determine the motion of the bubble as well as the spacetime geometry. The general solution of Maxwell equations (2.9) in the coordinate free language was derived in ref. [7]. For the reader's convenience we report here the main steps, in the ordinary component form. Variation of I with respect to A~,v produces the covariant Maxwell equations ,9u ( ~/-----gF '~'°'-) = c-,/----gJ ~P'~.
(3.1)
By inserting in (3.1) the dual field strength *F: x ~ - g F ~'~°~= - [ ~ " ° " ] * F ,
(3.2)
and the dual current *J~,: (3.3) we find
O~,*F = c*J~,.
(3.4)
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By taking the dual of jx~,~ and considering that we get
529
~'~P is the tangent element to g(,~),
*J,. = n , , 8 ~ ( x ) ,
8~)(x)
is the surface 6-function
x/-~-~8(x -~(t)).
(3.6)
where n~, is the unit outward-normal to ~(,~), and
8~(x) 6~(x)
= I : d3t
(3.5)
satisfies the following relations:
; d4x 6~)(x)p(x)= I (d3x)~p(x) ,
(3.7)
where the integral on the r.h.s, is restricted to the volume enclosed by the membrane ~,Ou(x)
= -
n. ~ ( x ) ,
(3.8)
where 0o(x) is the characteristic function of the open subset U of the spacetime manifold associated with the interior of the bubble. Note that in connection with the quark-bag model this characteristic function is to be identified [2] with the volume step function defined by 0u(X)=~'l
inside the bubble outside the bubble.
Now, comparing (3.8) and (3.5) with (3.4), we find
*F = -cOo(x) + al ,
(3.9)
where a ' is a constant solution of the homogeneous Maxwell equations, representing an additional cosmological background. Indeed, when the physical parameters p and c are zero, the field equations reduce to Einstein equations with a cosmological term indicated previously (cf. eq. (2.1)). The general solution (3.1) represents the origin of the bubble nucleation process in the cosmic vacuum as well as in the hadronic vacuum. To appreciate the connection with the bag model it may help to gain a qualitative understanding of the solution in Minkowski space. In the absence of curvature, our action (2.2) represents the embedding in Minkowski space of a closed surface endowed with the energymomentum tensor (2.13) and coupled, in a gauge-invariant way, to the generalized Maxwell field F. A special solution of eq. (3.4) is therefore ( a ' = 0)
*F = -cOo(x).
(3.10)
It is perhaps worth noticing here that in the literature on hadronic bag models, this step function is introduced as an "ad hoc" mathematical tool [2] in order to extend the space integrations over a finite domain to be over all space and in any Lorentz frame. In our approach the step function serves the same purpose-but appears in
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a natural way as a solution of Maxwell's equations of bubble dynamics. The same observation applies to the "bag constant" (or vacuum tension) itself: this physical parameter is introduced " b y h a n d " in the bag model in order to confine quarks and gluons. On our part we define the value of the field strength on the surface * F / b u b b l e = -½c as the arithmetic mean of the values inside and outside. The F-field evidently acts as an order parameter in spacetime. Eq. (3.4) with solution (3.10) represents the basic mechanism of confinement postulated in the bag model (3.11) which follows immediately from the solution (3.10) combined with eq. (2.14). Thus the bag constant is related to the coupling constant of our theory and the net result of the coupling of the F-field to the bubble's degree of freedom is a static effect: in the bubble's interior there exists a non-vanishing volume energy density which, in a minkowskian background, is precisely the "volume tension" advocated in the bag model in order to confine quarks and gluons. From our viewpoint the vacuum energy density of the bag is simply the self-energy density of the F-field. Note also that, unlike electrodynamics, the total self-energy of the F-field is finite and equal to ~c'2 V (when a ' = 0) where V is the volume enclosed by the membrane at a given time. Note, finally, that, unlike the original formulation of the M I T bag model [15], the bubble's surface in our theory is an independent dynamical system evolving according to the "Lorentz force" (2.9). When the metric is minkowskian, and the bubble is spherical, the Lorentz force equation can be solved analytically. The full details of this solution were given in ref. [16] and will not be reproduced here. The results which concern us here are as follows: the dynamical behaviour of the spherical solution depends on the combination A = a c / p of the physical parameters (with a = - ½c+ a ' ) and is dynamically unstable for all possible values of A. In particular, when a ' = 0 then A < 0 and there exist no static configuration whatever the value of the initial radius. Under the combined effect of surface tension and volume tension, the bag collapses to the central singularity unless an extra agent provides the necessary balancing pressure. In the quark model it is the quark-gluon system which stabilizes the bag into a physical hadron. It would cost, however, an infinite amount of energy to isolate the hadronic constituents as free non-interacting particles.
4. Cosmic voids
In the presence of a gravitational field the constant a ' cannot be safely taken equal to zero because a constant F-field provides a non-trivial energy-momentum tensor coupled to gravity through the Einstein equations. Even in this case the problem admits a "spherical bubble" solution which is represented in a locally static
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531
coordinate system by an interior de Sitter metric dS,2.)- -(1
-1A,r2) dt,2.+ (1-½A,r2) -' dr2+ r 2 d a 2 ,
(4.1)
and an exterior Schwarzschild-de Sitter geometry d S 2 ) = - ( 1 - ~ - ~ - ½ A o r2) dt2o+ (1-~-~-½Aor2) -1 d r 2 + r2 d~-22,
(4.2)
where Ai = 47r(c- a') 2 ,
Ao = 47r(a') 2 ,
and E is the total mass-energy of the system. Due to the spherical symmetry of the solution we have on the bubble surface the metric of a two-sphere: dS 2) = - d r 2 + R ( r ) 2 d/'2 2 ,
(4.3)
with a time-dependent radius R(r). Eqs. (4.1)-(4.3) completely fix the geometry of the problem. The dynamical equation for the bubble, i.e. the evolution equation for R(r), can be obtained either through the purely geometrical method of Israel [5, 6] or using the integral expression for the total mass-energy [8] =
T K~, dS~,
(4.4)
S
where K , is the unit-norm, timelike Killing vector. Both methods lead to the following equation for R(r):
EfA,-Ao
dRy2 =
dr]
)
E2
- l + n 2 R 2 + ~ \ ~ - ~ - ~ p 2 1 4 167r2p2R4,
(4.5)
where H = (247rp)-~[(A~ + Ao+ 487r2p2) 2 -
4A,Ao]1/2 •
Except for the last term, (4.5) is formally equivalent to the dynamical equation for a "matter-dominated", spatially-closed Friedman universe: (dR~ 2 = -1 dr]
+lAR2÷ 81rp~R3 3R
(4.6) '
with a positive cosmological constant A = 3 H 2 and an "exotic" matter density Pm= 3 E ((A~ - Ao)/48 rr2p 2 - 1)/87rR 3. Therefore we can treat the bubble evolution like a standard cosmological problem. To our knowledge there is no general way of solving eq. (4.5), other than by numerical methods. Presently we shall only consider some suitable approximation in order to obtain fully analytical solutions. (i) in the case H 2 ~ 0 , E2/167r2p2~O, eq. (4.5) reads dRy2 = E/Ai-Zo - l +-Z |T2"-3-'5 /~ \ q . ~ ¢ r - p dr/
)
1 ,
(4.7)
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A. Aurilia et al. / Early universe
and can be viewed as the classical equation describing the one-dimensional motion
/
- V (R )
of a unit-mass particle, moving with constant total energy e = -½ in the potential V (R ) = -
E-ff( A \ , - A oO
1) .
(4.8)
This problem admits a physically meaningful solution only if A~- Ao> 48"n'2p 2 (in the opposite case the particle always stays in a classically forbidden region), and the motion is confined to the interval O ~ R < ~ R M = E ( ( A i - A o ) / 4 8 ~ 2 p 2 - 1). A parametric solution of eq. (4.7) can easily be obtained in terms of the conformal time ~7: d r = R(~7) dr/, where
{Ai-Ao R(rl) = \487r2p2
I ) E sin 2 (½~/),
(4.9a)
and therefore
1(Ai-Ao )
r=~\4--~2-p---~ 1 E ( ~ ? - s i n q ) .
(4.9b)
Solution (4.9) represents an "oscillating bubble" which expands from a vanishing initial radius up to RM, and then collapses. The whole cycle occurs in the time
A~- = ~rE ( ( Ai - Ao) / 48 rrEp2 - 1). Near the origin the dominant contribution to (4.5) comes from the 1/R 4 term we have discarded, so we expect solution (4.9) to become unreliable as ~'-~ 0. (ii) The c a s e H E = 0, A i ~ Ao+487r2p 2 gives ( d R ~ 2 = -1
21
(4.10)
The classical meaning of (4.10) is qualitatively the same as in the case (i) and we expect again a motion confined to the interval 0 ~< R ~< RM = (E/4rrp) 1/2. Eqs. (4.9a) and (4.9b) are replaced by R(n)
= ( E ~ '/2 \4-~p] sin 1/2(2n),
(4.11a)
~-= x/~RM(21F(a, x/~) - F(ce,x/~))
(4.1 lb)
a = arcsin (~/2 sin (lzr - r/)).
(4.1 lc)
where
F(a, x/~-~)and IF(a, x/~) are the elliptic integrals of the first and second kind respec-
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533
tively [l?]. An expansion-recontraction cycle requires a time
\4"rrp/
F(½-rr,x/~2 !)
4 2x/~\4"rrp/
"
(4.12)
We notice that (4.11) represent a good approximation for a general solution to (4.5) near the origin, where the 1 / R 4 term dominates over the others. (iii) From E - - 0 , it follows ( d~--~/ R~ 2 =
-l+
H2R 2 .
(4.13)
This is the case of "vacuum bubbles" expanding according to the de Sitter law R(7) = H -I cosh (H~').
(4.14)
This kind of solution acquires a particular relevance in the framework of the new inflationary cosmology because it provides the exponentially expanding phase needed to solve the inconsistency of the standard cosmological model [l l, 18]. Moreover, it has been suggested by Tryon that a closed universe with total vanishing energy could emerge as a spontaneous vacuum fluctuation [19]. This approach to "cosmogenesis" was recently reconsidered by Vilenkin who interpreted the solution (4.14) as describing a quantum tunneling from "nothing" to de Sitter space [9]. We have already argued elsewhere that in our classical approach "nothing" is rather replaced by the exterior de Sitter phase corresponding to a vanishing total energy for the system [8]. (iv) p = 0, and for the sake of simplicity A0 = 0, gives E = ~AiR 3 .
(4.15)
In this case the vanishing of the surface tension p turns a dynamical problem into a static one: the membrane no longer has a physical meaning, it is simply the geometrical boundary of the bubble. The dynamical equation (4.5) becomes the condition (4.15) and the total energy E is completely saturated by the volume energy. One can read eq. (4.15) also from a geometrical point of view: once the Schwarzschild and de Sitter radii, R, = 2E and RA = (3/Ai)1/2 respectively, are fixed, the radius of the bubble is R = ( R s R ~ ) ~/3. As an extreme case we can choose R=Rs=R~.
In view of the peculiar equation of state P, = - p , associated with the de Sitter vacuum, it is somewhat surprising that if one takes for A~ the typical energy density of the nuclear matter, i.e. p, ~ l0 4 g c m -3, then the resulting value of R is close to l0 kin, the correct size of neutron star. The physical meaning, if any, of such a solution is presently unclear. Similar objects, but with R - - 1 0 - 1 3 cm, have also been obtained in the framework of strong gravity, where they provide a geometrical model for the hadron structure and dynamics [10].
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On the basis of the former particular cases, and by analogy with the motion of a classical, unit-mass particle with total energy e = - I, in the potential ,
v(r)
2 2
r
E [
A~-ao~
E2
48~2p2]
1
(4.16)
32~r2p 2 R 4,
we can deduce some features about the general solutions of (4.5). We expect all the bubbles for E > 0 to exhibit either an oscillating or an asymptotically de Sitterlike behaviour. In the first case R ~ , ~ o 0 , and (4.5) can be approximated in this region by
(dR] 2 --d-~T]
(4.17)
= \ 4 7rp ,] R 4'
so that near the origin we have the following asymptotic behaviour: [ 3E'~'/3 1/3 R(~')'~o~-~p) ~" .
(4.18)
It is easy to check that (4.18) agrees with the asymptotic form (4.11 ) in the same limit. In the second case R ( r ) ~ . . . . oo, so in the remote past and in the distant future (4.5) approaches (4.13). As a consequence, also for E > 0 , there exist solutions whose asymptotic form is (4.14). Both kind of bubbles have a rest point (the particle's turning point) corresponding to the m a x i m u m expansion or the minimum contraction radius. The mass of the bubble at the rest point is given by y = ax3+x2(1
(4.19)
- - X 2 ) 1/2 ,
where we have introduced the adimensional quantities y = ( E a f f 12 ~rp),
(4.20a)
x = R ( 3 / a i ) -1/2 ,
(4.20b)
a =~
\aft
( a i - Ao - 487r2p2) •
(4.20c)
The function y ( x ) exhibits a single m a x i m u m in the interval 0 ~
x=" k
" ~ + ~-5-~
'/2
j
if
a>0,
(4.21)
or
4 + 3 a 2 - a (9a 2 + 8)1/2] 1/2 ~+~ j
=L
when
a < 0.
In the limiting case a = 0 we have simply ~ = ~/~ and )7 = 2x/3.
(4.22)
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535
Finally we notice that for a <~0 the curve (4.9) intersects the x-axis. The intercept represents the minimum radius of a zero-total-energy bubble: Ro = H -1 in agreement with the exact solution (4.14).
5. Summary and discussion The particular solutions of eq. (4.5) we found in sect. 4 can be distinguished into two classes according to E > 0 or E = 0. In the first case the mass energy of the bubble must be provided by some external environment. Tfierefore we can use these massive bubbles as models for the cosmic voids in the large-scale galaxy distribution [4]. In the second case, E = 0, bubbles can form spontaneously in vacuum without violating the energy conservation principle. The gravitational effects on the vacuum decay process have been studied by Coleman and De Luccia (from now on CDeL) in the framework of a semiclassical scalar field theory [20]. In this approach the decay of the false vacuum to a lowest energy density state, i.e. the true vacuum, proceeds through a quantum tunnelling mediated by a classical, euclidean, field configuration minimizing the action, the so-called bounce. Let us verify the consistency of our model with the C D e L theory. As far as the notation is concerned the following identities hold: p=Sj, (5.1) A, = 87rGN U(th_),
(5.2)
A0 = 8~rGN U ( 6 + ) ,
(5.3)
where the quantities on the r.h.s, are defined in ref. [20], and we have explicitly written the Newton constant GN. If we assume, with CDeL, that the bubble is materialized at rest, then from eq. (4.14) we find immediately for the materialization radius R=H-~=~. (5.4) Again the last symbol on the right in (5.4) refers to the notation of ref. [20]. We notice that the value of R has been derived by us directly from the equation of motion for the bubble, while C D e L obtained it from a stationarity condition for the action of the bounce. If we specialize H to the case of vanishing U(~b_) or U(~b+), then we get (3.15) and (3.16) of C D e L respectively. In the case A~ = --8'rrGNe <0, where e = - U(~b_) and A o = 0 , H vanishes for 67rS 2= GNe and the radius of the bubble becomes infinite. This is precisely the condition for the gravitational vacuum stabilization given by (3.21) in ref. [20]. In our general-covariant approach the total mass-energy of the system is expressed through the integral expression (4.4); an explicit calculation [8] for the case Ao = 0, Ai =--8"rrGN e, e = - U(~b_), R : 0 gives
E=-4"n'eR3+2"n'SIR2[(1-2E~GRY)I/E+(I+S'n-GNeR2)'/2
] .
(5.5)
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In the limiting case of a weak gravitational field we may expand equation (5.5) in powers of GN: E~---4~reR3(I-27rSIGNR)+4ZrSlR2(I+2zrGNeR2)-8~'2S2GNR3. By taking into account the relation R = 3 S t / e
(5.6)
we find
E =8rr2GNe2gS=- E 8.....
(5.7)
Hence, in the classical limit our general formula (5.5) gives the total gravitational energy as computed in ref. [20]. Following C D e L we define a classical volume energy as Ev = --4ZrGN e fo R dx x2(l - %rex 2) = - ~ZrGNeg3(1 - ~ G N e R 2 ) .
(5.8)
Then, inserting (5.8) into (5.5), taking into account (5.7) and the definitions ENewton ------ 8 7 r 2 e 2 G N R 5 ,
(5.9)
Egeo m.=i3"n" 16 2 e 2,.-, n5 , t.tNlX
(5.10)
we obtain the C D e L energy balance condition Eg .... = ENewton"l'-Eg. . . . .
(5.11)
Concerning the kinematics of the system it is worth mentioning that as far as one considers only the case with E = 0, and with either A~ or Ao vanishing, it is unnecessary to resort explicitly to an equation of motion. The evolution of the bubble can be simply inferred from the matching condition between the Minkowski and de Sitter (or anti-de Sitter) metrics. On the other hand, the case E > 0 is more easily tackled in the framework of the Israel approach [5] to the surface layer dynamics. Besides this quite general, purely geometrical, method of investigation we have at our disposal, in the action functional approach we adopted, the Lorentz force equation and the first integral of the motion (4.4). Both eq. (2.11) and (4.4) lead to the radial equation (4.5). Let us conclude summarizing what we consider the distinguishing features of our model: (i) the form of the action functional which defines the physical system we are investigating is dictated by two fundamental symmetries: invariance under general coordinates transformations and local gauge invariance; (ii) all the basic fields involved in our model, i.e ga~,, AA~,~, ~:a, have a precise geometrical meaning; (iii) a two-phase ground state for the system is obtained as an exact solution of the coupled field equations; (iv) the same effective action functional provides us with an effective description for vastly different physical systems, as are the hadronic vacuum, the very early universe, and the intergalactic voids.
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References [1] The very early universe, ed. G.W. Gibbons, S.W. Hawking and S.T.C. Siklos (Cambridge University Press, 1983) [2a] P. Hasenfratz and J. Kuti, Phys. Reports 40C (1978) 75; [2b] K. Johnson, in Proc. 17th Scottish University Summer School, 1976, ed. Barbour and Davies [3] V.A. Berezin, V.A. Kuzmin and I.I. Tkachev, Phys. Lett. BI20 (1983) 91 [4] H. Sato, Progr. Theor. Phys. 68 (1982) 236; K. Maeda, M, Sasaki and H. Sato, Progr. Theor. Phys. 69 (1983) 89 [5] W. Israel, Nuovo Cim. 44B (1966) 1; Phys. Rev. 153 (1966) 1388; V. de la Cruz and W. Israel, Nuovo Cim. 51A (1967) 744 [6] J.E. Chase, Nuovo Cim. B67 (1970) 136 [7] A. Aurilia, D. Christodoulou and F. Legovini, Phys. Lett. 73B (1978) 429; A. Aurilia and F. Legovini, Phys. Lett. 76B (1977) 299; A. Aurilia and D. Christodoulou, J. Math. Phys. 20 (1979) 1446; 20 (1979) 1642; A. Aurilia and Y. Takahashi, Progr. Theor. Phys. 66 (1981) 693 [8] A. Aurilia, G. Denardo, F. Legovini and E. Spallucci,'Phys. Lett. 147B (1984) 258 [9] A. Vilenkin, Phys. Rev. D27 (1983) 2848 [10] C. Sivaram and K.P. Sinha, Phys. Reports 51 (1979) 111 [11] A.H. Guth, Phys. Rev. D23 (1981) 347 [12] A. Aurilia, H. Nicolai and P.K. Townsend, Nucl. Phys. B176 (1980) 509; M.J. Dutt and P. van Nieuwenhuizen, Phys. Lett. B94 (1980) 179 [13] S.W. Hawking, Phys. Lett. 134B (1984) 403; T. Banks, Phys. Lett. 52 (1984) 1461 [14] P.G.O. Freund and M.A. Rubin, Phys. Lett. 97B (1980) 233; M.J. Duff, B.E.W. Nilsson and C.N. Pope, in Fourth Workshop on Grand unification, Univ. of Pennsylvania, 1983; ed. H.A. Weldon, P. Langaker and P.J. Steinhardt; L. Castellani, R.d.'Auria and P. Fr6, in Supersymmetry and supergravity 1983, Karpacz School, ed. B. Milewski (World Scientific, Singapore); F. Englert and H. Nicolai, preprint TH-3711-CERN; P. van Nieuwenhuizen, lectures at Les Houches Summer School 1983, to appear [15] A. Chodos, R.L. Jafie, K. Johnson, C.B. Thorn and V.S. Weisskopf, Phys. Rev. D9 (1974) 3471 [16] A. Aurilia and D. Christodoulou, Phys. Lett. 78B (1978) 589 [17] I.S. Gradshteyn and I.M. Ryzhik, Tables of integrals, series and products (Academic Press, London, 1980) 181 [18] A.D. Linde, Phys. Lett. 108B (1982) 389, A. Albrecht and P.J. Steinhardt, Phys. Lett. 48 (1982) 1220 [19] E.P. Tryon, Nature 246 (1973) 396 [20] S. Coleman and F. De Luccia, Phys. Rev. D21 (1980) 3305
VACUUM TENSION EFFECTS ON THE EVOLUTION OF D O M A I N W A L L S IN T H E EARLY U N I V E R S E * A. AURILIA 1 Department of Physics, University of Toronto in Mississauga, Toronto, Ontario, Canada
G. DENARDO International Centre for Theoretical Physics, Trieste, Italy, Dipartimento di Fisica Teorica, Universita' di Trieste, Italy and Istitato Nazionale di Fisica Nucleare, Sezione di Trieste, Italy
F. LEGOVINI and E. SPALLUCCI Dipartimento di Fisica Teorica, Universita' di Trieste, Italy, and lstituto Nazionale di Fisica Nucleare, Sezione di Trieste, Italy
Received 4 July 1984 (Revised 25 October 1984) The "vacuum pressure" mechanism of the hadronic bag model is taken as a guide to formulate the dynamics of closed domain walls in the cosmological case. The effective action functional suggested by this analogy is a straightforward generalization of the Einstein-Maxwell action: it involves a 3-index antisymmetric potential whose coupling to matter generates two effective cosmological constants, one inside and one outside the domain wall. It is suggested that this mechanism, which is alternative to the introduction of a Higgs potential, is the source of the bubble nucleation process envisaged in the new inflationary cosmology. The dynamics of a spherical domain in a de Sitter phase is analyzed and is consistent with the geometrical formulation of shell dynamics proposed long ago by Israel.
I. Introduction The i n t e r p l a y b e t w e e n particle physics a n d cosmology has led to p r o f o u n d changes in o u r p e r c e p t i o n of the physics of the early universe. As o p p o s e d to the picture offered by the hot b i g - b a n g model, the g r o u n d state of the early universe resembles an e m u l s i o n o f different phases, The phase transitions p r o c e e d via the creation of v a c u u m b u b b l e s in one phase a n d their s u b s e q u e n t e v o l u t i o n in the b a c k g r o u n d of a different phase [1]. Such a characteristic d o m a i n structure is c o m m o n to other physical systems: ferromagnets a n d s u p e r c o n d u c t o r s are some of such systems whose properties have u n d o u b t e d l y inspired some of the c u r r e n t cosmological scenarios. However, what comes to our m i n d as a particularly fitting a n a l o g y with the case of the early u n i v e r s e is the h a d r o n i c v a c u u m pictured in the q u a r k - b a g m o d e l with * Work supported in part by the National Science and Engineering Research Council (NSERC) of Canada. On leave of absence from Istituto Nazionale di Fisica Nucleate, Sezione di Trieste, Italy. 523
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surface tension*. Our objective is to illustrate the precise sense of this analogy and to explore some of its physical consequences in cosmology and astrophysics. The hadronic vacuum envisaged in the bag model is also regarded as a medium consisting of two phases separated by a closed domain wall. The interior of the " b a g " represents the hadronic phase while the exterior region is regarded as a different phase of the vacuum inaccessible to the hadronic constituents. A key ingredient of the bag model is a sort of "cosmological constant" confined to the interior of the bag. This is tantamount to a non-vanishing vacuum energy density associated with the hadronic phase. Hopefully, this phenomenological scenario of the ground state of strong interactions will be substantiated by a deeper understanding of the hadronization phase in non-perturbative QCD. By the same reasoning, since at present the quantum properties of the cosmological ground state are poorly understood, it seems desirable to formulate an effective approach to the bubble nucleation process in the early universe. We suggest that the process of formation of vacuum bubbles in the cosmological case be interpreted in terms of a vacuum energy density which is Completely analogous to that of the hadronic phase in the physics of strong interactions. Once formed, the later evolution of the domain boundaries is governed by their surface tension and their interaction with matter. The problem of "bubble dynamics" is of interest in its own right and was considered recently by Berezin, Kuzmin and Tkachev [3] in connection with the vacuum state of the early universe and also by Maeda, and Sasaki and H. Sato I-4] in connection with a computer simulated evolution of intergalactic "voids" in an expanding universe. The relevant mathematical formalism for this type of problem in general relativity was developed some time ago by Israel [5] and later by Chase [6]. In particle physics, on the other hand, Dirac used his formalism of constrained hamiltonian system to develop an electrodynamic model of a conducting bubble in an attempt to resolve the electron-muon puzzle. Later, with the recognition that strongly interacting particles possess a finite spatial extension, the study of the evolution of extended structures in spacetime became a major line of research in particle physics. Our approach to bubble dynamics'is based on a general theory of relativistic extended objects formulated previously in ref. [7]; relevant points of that theory will be recalled in sect. 2. In order to discuss the cosmological and the astrophysical case we propose an effective action modelled on the hadronic bag. The resulting equation governing ~the dynamics of one vacuum in the medium of a different vacuum can be obtained either as a junction equation ~ la Israel, or from the generally covariant definition of the total mass energy of the system [8]. Particular solutions to our equation of motion fit into two classes: we find "oscillating bubbles", which expand from zero to a m a x i m u m radius and then collapse, and de Sitter-type bubbles which contract from infinity up to a minimum radius and then expand indefinitely. A special case * F o r a g e n e r a l r e v i e w see f o r i n s t a n c e [ 2 a ] ; f o r a c o n c i s e r e v i e w o f s o m e o f t h e a s p e c t s o f t h e b a g m o d e l w h i c h a r e r e l e v a n t to o u r d i s c u s s i o n see [2b].
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of this type of solution was advocated by Vilenkin [9] who pictures the birth of the universe as a tunneling effect from an initial "void" with total energy equal to zero into a de Sitter space. When the surface tension is zero we also find an isolated static solution representing a spherical vacuum domain of de Sitter type. This kind of bubble admits a limiting configuration where its radius, the Schwarzschild and the de Sitter radii are coincident. This solution was proposed some time ago as a model of hadronic matter in the framework of "strong" gravity [ 10]. The astrophysical meaning of such a solution, if any, remains an open question. Finally, when the action is formulated in "flat" Minkowski space, the equations of motion reproduce the confinement mechanism postulated in the quark-bag model. This paper is organized as follows: in sect. 2 we introduce and discuss the coupling of the membrane and its generalized gauge field in curved space. In sect. 3 the existence of vacuum domains is shown as a solution of the generalized Maxwell equations. In sect. 4 we find a spherical bubble solution of the Einstein equation and discuss its evolution in some particular cases. In sect. 5 we give a brief discussion of the results. 2. The action
In the current literature the vacuum state of the early universe is often modelled on the classical de Sitter solution to Einstein's equation with a cosmological term. This new type of "de Sitter vacuum" has been successfully exploited to get around some of the well-known problems of the standard big-bang model [1 1]. A similar artifact (the bag constant) is the key to the success of the quark-bag model in matching hadronic spectroscopy. Our approach is based on the recognition that the cosmological constant as well as the bag constant can be formulated, in a gauge invariant way, in terms of a 3-index antisymmetric potential A~,~ to which we associate a totally antisymmetric field strength tensor F~,~p =- O[~A~,vol [7, 12]. In geometric notation the field Fa,,vp is a 4-form F = dA which trivially satisfies the Bianchi identities d F = 0 and obeys the Maxwell-like equations d * F = *J. The source J will be specified shortly. For the moment we note that the field F remains invariant under the gauge transformation A ~ A + dh where the gauge function h is a differential 2-form. The field F is non-propagating and, in the absence of a source term, it represents a background field constant over the entire universe: indeed, since *F, the Hodge dual field strength, is a zero-form, the equation d * F = 0 implies * F = c, with c constant everywhere. This constant can be interpreted as the cosmological constant in general relativity. As a simple calculation (cf. eq. (2.14)), the energymomentum tensor of the F-field is given by T~F~= Ag,~, with A = 4zrc 2. Therefore the coupling of the F-field to the Einstein tensor G can be described by the equivalent sets of equations G=87rT,
d*F=O~G=Ag.
(2.1)
This is the basic property of the F-field; recently it has generated a lot of interesting
A. Aurilia et al. / Early universe
526
applications not only in connection with the long-standing question of the actual size of the cosmological constant [13] but also in connection with the problem of the compactification of extra dimensions in Kaluza-Klein supergravity [14]. From our vantage point the usefulness of the gauge invariant formulation of the cosmological constant is seen in the possibility of coupling the F-field to matter whereas the standard cosmological term is frozen in the action of general relativity. The gauge invariant coupling of the F-field to matter is easily understood by analogy with electrodynamics: as the standard electromagnetic field A~ mediates the interaction between the line elements along the world line of point particles, so the 3-index potential A ~ mediates the interaction between the hypersurface elements along the world-tube swept by the surface of a bubble in spacetime. This world-tube is the embedding ~:(~) of a connected, orientable, 3-dimensional manifold 2, in the 4-dimensional spacetime manifold M, described by x '~ = s~"(ta),
(2.2)
where t a = {t °, t l, t2},
x " = {x °, x l, x 2, x3},
(2.3)
are local coordinates on M and ,X respectively. In order to describe a bubble, which is usually thought .of as a spacetime 3-dimensional region bounded by a closed 2-dimensional spatial surface, we choose ,X-~ R x S where S is a 2-dimensional compact manifold. Then at any time the bubble is represented by the intersection between the x ° = const hyperplane and ~:(,X). Our model describes a surface layer, that is a m e m b r a n e of vanishing thickness to which we associate a 6-function. This is the limit where the thickness of the m e m b r a n e is very small with respect to the size of the bubble. The tangent timelike element to the world-tube ~(2;) is given by the 3-vector density ¢
__ e abc~_ ~ O~ Ot o A Ot--'-(A t3t----~= cgt b at c '
(2.4)
where e abe is the 3-dimensional Levi-Civita tensor*, and y is the determinant of * We adopt the following conventions: [ " * " ~ ] is the n-dimensional Levi-Civita tensor density, which is defined according to [~c "'~-] =
+!. -
cq, c~2. . . . o~. even permutation of reference sequence a l , a~, • • • c~. odd permutation of reference sequence s o m e indices e q u a l .
Consequently, the Levi-Civita tensor is ECrl'"an
=41___~[o,~q,
~ , . .... - - 4 - - ~ [ ~ , . ..... I.
We define the dual of a k-rank tensor T~,~.~+, as l *Ta,"'a. k=~.ea,'"-.
k+l'"ot.
TOtn-k+lCln "
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the induced metric rob = at o ~ - ~ g . ~
(2.5)
on ~:(~). ~ generalizes the velocity tangent vector to the case of an extended object like the 3-surface we are considering. The norm of such a 3-vector density, defined as I1~11= (-~a~'~a~,~/3 !)1/2 provides a suitable kinetic term for the surface. By taking into account (2.2), (2.4) and (2.5) one finds I1~112=-3, hence the kinetic term of the membrane can also be seen as its invariant "area". Finally we have to couple the membrane with the gauge potential. The photon field A , interacts with a pont-like charge through the current density J~'= d r ~ ' 6 ( x - ~:(r)) where ~ is the tangent element to the particle world-line x = ~:(r). Hence the gauge potential Aa,~ can interact with the bubble only through the current density
jx.~, = f~: ~x.,,8( x - ~:(t)) d3t.
(2.6)
Our effective action is therefore given by 1 I(g~,.,A~,~.£~)=~ fMd4X,/~-gR-PL d3t ll~
-I 2"4!
where we have used Gn = 1. Here the first term is the usual Hilbert-Einstein action and the fourth is a straightforward generalization of Maxwell's. The positive-definite constant p (which in classical electrodynamics would represent the mass of the pointlike charge) is here the surface energy density, or surface tension, and has dimension L -1 in geometric units. Finally the gauge coupling constant c has dimension L -1. The action (2.7) is manifestly invariant under general coordinate transformations on the spacetime manifold as well as on the embedded submanifold representing the world-track of the bubble. These invariance properties are reflected by the field equations: 6gI = 0 leads to Einstein's equation G = 8~-(TZ)+ TF~),
(2.8)
while 6AI = 0 leads to "Maxwell's equations" for the F-field: d F = O,
d*F = c'J,
(2.9)
where *J represents the 1-form current dual to JA~(x): 1
I Ata'v I * ' 8 = 3 ! pa.~-' , --
E
where epA~,~ is the Levi-Civita tensor in 4 dimensions.
(2.10)
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Finally 6~I = 0 leads to the "Lorentz force" equation governing the evolution of the bubble
F]vxO + ( I ' g ) ~ T
.bOx p Ox ~ 1 c Ot ~ Orb--3!
Yoga"
~f"~,;ll~l I
(2.11)
where (Fg)oQ is the connection of the metric g on the spacetime manifold and ~ v is the Laplace-Beltrami operator with metric Y~b. Just as in electrodynamics, the field equations imply the conservation of the Noether current (2.6) as well as the conservation of the symmetric energy-momentum tensor
2 61 T~" = -x/----g 8gA, = T ~ + T ~ ,
(2.12)
where
z)
f
2! 3
I111=
T ~ -- 31! Fa,,ov F ~'~ v
4I-
d3t'
1 2.4!ga~'F~t3V~F~t3v~=-lga~'(*F) 2"
(2.13)
(2.14)
Both in cosmology and in particle physics one is mainly concerned with spherical vacuum domains, so in sect. 4 we shall assume spherical symmetry for our closed membrane: in this case we shall be able to solve the whole set of coupled field equations derived from the action (2.7).
3. Exact solution of the "Maxwell" equations In principle, the field equations completely determine the motion of the bubble as well as the spacetime geometry. The general solution of Maxwell equations (2.9) in the coordinate free language was derived in ref. [7]. For the reader's convenience we report here the main steps, in the ordinary component form. Variation of I with respect to A~,v produces the covariant Maxwell equations ,9u ( ~/-----gF '~'°'-) = c-,/----gJ ~P'~.
(3.1)
By inserting in (3.1) the dual field strength *F: x ~ - g F ~'~°~= - [ ~ " ° " ] * F ,
(3.2)
and the dual current *J~,: (3.3) we find
O~,*F = c*J~,.
(3.4)
A. Aurilia et aL / Early universe
By taking the dual of jx~,~ and considering that we get
529
~'~P is the tangent element to g(,~),
*J,. = n , , 8 ~ ( x ) ,
8~)(x)
is the surface 6-function
x/-~-~8(x -~(t)).
(3.6)
where n~, is the unit outward-normal to ~(,~), and
8~(x) 6~(x)
= I : d3t
(3.5)
satisfies the following relations:
; d4x 6~)(x)p(x)= I (d3x)~p(x) ,
(3.7)
where the integral on the r.h.s, is restricted to the volume enclosed by the membrane ~,Ou(x)
= -
n. ~ ( x ) ,
(3.8)
where 0o(x) is the characteristic function of the open subset U of the spacetime manifold associated with the interior of the bubble. Note that in connection with the quark-bag model this characteristic function is to be identified [2] with the volume step function defined by 0u(X)=~'l
inside the bubble outside the bubble.
Now, comparing (3.8) and (3.5) with (3.4), we find
*F = -cOo(x) + al ,
(3.9)
where a ' is a constant solution of the homogeneous Maxwell equations, representing an additional cosmological background. Indeed, when the physical parameters p and c are zero, the field equations reduce to Einstein equations with a cosmological term indicated previously (cf. eq. (2.1)). The general solution (3.1) represents the origin of the bubble nucleation process in the cosmic vacuum as well as in the hadronic vacuum. To appreciate the connection with the bag model it may help to gain a qualitative understanding of the solution in Minkowski space. In the absence of curvature, our action (2.2) represents the embedding in Minkowski space of a closed surface endowed with the energymomentum tensor (2.13) and coupled, in a gauge-invariant way, to the generalized Maxwell field F. A special solution of eq. (3.4) is therefore ( a ' = 0)
*F = -cOo(x).
(3.10)
It is perhaps worth noticing here that in the literature on hadronic bag models, this step function is introduced as an "ad hoc" mathematical tool [2] in order to extend the space integrations over a finite domain to be over all space and in any Lorentz frame. In our approach the step function serves the same purpose-but appears in
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A. Aurilia et al. / Early universe
a natural way as a solution of Maxwell's equations of bubble dynamics. The same observation applies to the "bag constant" (or vacuum tension) itself: this physical parameter is introduced " b y h a n d " in the bag model in order to confine quarks and gluons. On our part we define the value of the field strength on the surface * F / b u b b l e = -½c as the arithmetic mean of the values inside and outside. The F-field evidently acts as an order parameter in spacetime. Eq. (3.4) with solution (3.10) represents the basic mechanism of confinement postulated in the bag model (3.11) which follows immediately from the solution (3.10) combined with eq. (2.14). Thus the bag constant is related to the coupling constant of our theory and the net result of the coupling of the F-field to the bubble's degree of freedom is a static effect: in the bubble's interior there exists a non-vanishing volume energy density which, in a minkowskian background, is precisely the "volume tension" advocated in the bag model in order to confine quarks and gluons. From our viewpoint the vacuum energy density of the bag is simply the self-energy density of the F-field. Note also that, unlike electrodynamics, the total self-energy of the F-field is finite and equal to ~c'2 V (when a ' = 0) where V is the volume enclosed by the membrane at a given time. Note, finally, that, unlike the original formulation of the M I T bag model [15], the bubble's surface in our theory is an independent dynamical system evolving according to the "Lorentz force" (2.9). When the metric is minkowskian, and the bubble is spherical, the Lorentz force equation can be solved analytically. The full details of this solution were given in ref. [16] and will not be reproduced here. The results which concern us here are as follows: the dynamical behaviour of the spherical solution depends on the combination A = a c / p of the physical parameters (with a = - ½c+ a ' ) and is dynamically unstable for all possible values of A. In particular, when a ' = 0 then A < 0 and there exist no static configuration whatever the value of the initial radius. Under the combined effect of surface tension and volume tension, the bag collapses to the central singularity unless an extra agent provides the necessary balancing pressure. In the quark model it is the quark-gluon system which stabilizes the bag into a physical hadron. It would cost, however, an infinite amount of energy to isolate the hadronic constituents as free non-interacting particles.
4. Cosmic voids
In the presence of a gravitational field the constant a ' cannot be safely taken equal to zero because a constant F-field provides a non-trivial energy-momentum tensor coupled to gravity through the Einstein equations. Even in this case the problem admits a "spherical bubble" solution which is represented in a locally static
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531
coordinate system by an interior de Sitter metric dS,2.)- -(1
-1A,r2) dt,2.+ (1-½A,r2) -' dr2+ r 2 d a 2 ,
(4.1)
and an exterior Schwarzschild-de Sitter geometry d S 2 ) = - ( 1 - ~ - ~ - ½ A o r2) dt2o+ (1-~-~-½Aor2) -1 d r 2 + r2 d~-22,
(4.2)
where Ai = 47r(c- a') 2 ,
Ao = 47r(a') 2 ,
and E is the total mass-energy of the system. Due to the spherical symmetry of the solution we have on the bubble surface the metric of a two-sphere: dS 2) = - d r 2 + R ( r ) 2 d/'2 2 ,
(4.3)
with a time-dependent radius R(r). Eqs. (4.1)-(4.3) completely fix the geometry of the problem. The dynamical equation for the bubble, i.e. the evolution equation for R(r), can be obtained either through the purely geometrical method of Israel [5, 6] or using the integral expression for the total mass-energy [8] =
T K~, dS~,
(4.4)
S
where K , is the unit-norm, timelike Killing vector. Both methods lead to the following equation for R(r):
EfA,-Ao
dRy2 =
dr]
)
E2
- l + n 2 R 2 + ~ \ ~ - ~ - ~ p 2 1 4 167r2p2R4,
(4.5)
where H = (247rp)-~[(A~ + Ao+ 487r2p2) 2 -
4A,Ao]1/2 •
Except for the last term, (4.5) is formally equivalent to the dynamical equation for a "matter-dominated", spatially-closed Friedman universe: (dR~ 2 = -1 dr]
+lAR2÷ 81rp~R3 3R
(4.6) '
with a positive cosmological constant A = 3 H 2 and an "exotic" matter density Pm= 3 E ((A~ - Ao)/48 rr2p 2 - 1)/87rR 3. Therefore we can treat the bubble evolution like a standard cosmological problem. To our knowledge there is no general way of solving eq. (4.5), other than by numerical methods. Presently we shall only consider some suitable approximation in order to obtain fully analytical solutions. (i) in the case H 2 ~ 0 , E2/167r2p2~O, eq. (4.5) reads dRy2 = E/Ai-Zo - l +-Z |T2"-3-'5 /~ \ q . ~ ¢ r - p dr/
)
1 ,
(4.7)
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and can be viewed as the classical equation describing the one-dimensional motion
/
- V (R )
of a unit-mass particle, moving with constant total energy e = -½ in the potential V (R ) = -
E-ff( A \ , - A oO
1) .
(4.8)
This problem admits a physically meaningful solution only if A~- Ao> 48"n'2p 2 (in the opposite case the particle always stays in a classically forbidden region), and the motion is confined to the interval O ~ R < ~ R M = E ( ( A i - A o ) / 4 8 ~ 2 p 2 - 1). A parametric solution of eq. (4.7) can easily be obtained in terms of the conformal time ~7: d r = R(~7) dr/, where
{Ai-Ao R(rl) = \487r2p2
I ) E sin 2 (½~/),
(4.9a)
and therefore
1(Ai-Ao )
r=~\4--~2-p---~ 1 E ( ~ ? - s i n q ) .
(4.9b)
Solution (4.9) represents an "oscillating bubble" which expands from a vanishing initial radius up to RM, and then collapses. The whole cycle occurs in the time
A~- = ~rE ( ( Ai - Ao) / 48 rrEp2 - 1). Near the origin the dominant contribution to (4.5) comes from the 1/R 4 term we have discarded, so we expect solution (4.9) to become unreliable as ~'-~ 0. (ii) The c a s e H E = 0, A i ~ Ao+487r2p 2 gives ( d R ~ 2 = -1
21
(4.10)
The classical meaning of (4.10) is qualitatively the same as in the case (i) and we expect again a motion confined to the interval 0 ~< R ~< RM = (E/4rrp) 1/2. Eqs. (4.9a) and (4.9b) are replaced by R(n)
= ( E ~ '/2 \4-~p] sin 1/2(2n),
(4.11a)
~-= x/~RM(21F(a, x/~) - F(ce,x/~))
(4.1 lb)
a = arcsin (~/2 sin (lzr - r/)).
(4.1 lc)
where
F(a, x/~-~)and IF(a, x/~) are the elliptic integrals of the first and second kind respec-
A. Aurilia et al. / Early universe
533
tively [l?]. An expansion-recontraction cycle requires a time
\4"rrp/
F(½-rr,x/~2 !)
4 2x/~\4"rrp/
"
(4.12)
We notice that (4.11) represent a good approximation for a general solution to (4.5) near the origin, where the 1 / R 4 term dominates over the others. (iii) From E - - 0 , it follows ( d~--~/ R~ 2 =
-l+
H2R 2 .
(4.13)
This is the case of "vacuum bubbles" expanding according to the de Sitter law R(7) = H -I cosh (H~').
(4.14)
This kind of solution acquires a particular relevance in the framework of the new inflationary cosmology because it provides the exponentially expanding phase needed to solve the inconsistency of the standard cosmological model [l l, 18]. Moreover, it has been suggested by Tryon that a closed universe with total vanishing energy could emerge as a spontaneous vacuum fluctuation [19]. This approach to "cosmogenesis" was recently reconsidered by Vilenkin who interpreted the solution (4.14) as describing a quantum tunneling from "nothing" to de Sitter space [9]. We have already argued elsewhere that in our classical approach "nothing" is rather replaced by the exterior de Sitter phase corresponding to a vanishing total energy for the system [8]. (iv) p = 0, and for the sake of simplicity A0 = 0, gives E = ~AiR 3 .
(4.15)
In this case the vanishing of the surface tension p turns a dynamical problem into a static one: the membrane no longer has a physical meaning, it is simply the geometrical boundary of the bubble. The dynamical equation (4.5) becomes the condition (4.15) and the total energy E is completely saturated by the volume energy. One can read eq. (4.15) also from a geometrical point of view: once the Schwarzschild and de Sitter radii, R, = 2E and RA = (3/Ai)1/2 respectively, are fixed, the radius of the bubble is R = ( R s R ~ ) ~/3. As an extreme case we can choose R=Rs=R~.
In view of the peculiar equation of state P, = - p , associated with the de Sitter vacuum, it is somewhat surprising that if one takes for A~ the typical energy density of the nuclear matter, i.e. p, ~ l0 4 g c m -3, then the resulting value of R is close to l0 kin, the correct size of neutron star. The physical meaning, if any, of such a solution is presently unclear. Similar objects, but with R - - 1 0 - 1 3 cm, have also been obtained in the framework of strong gravity, where they provide a geometrical model for the hadron structure and dynamics [10].
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On the basis of the former particular cases, and by analogy with the motion of a classical, unit-mass particle with total energy e = - I, in the potential ,
v(r)
2 2
r
E [
A~-ao~
E2
48~2p2]
1
(4.16)
32~r2p 2 R 4,
we can deduce some features about the general solutions of (4.5). We expect all the bubbles for E > 0 to exhibit either an oscillating or an asymptotically de Sitterlike behaviour. In the first case R ~ , ~ o 0 , and (4.5) can be approximated in this region by
(dR] 2 --d-~T]
(4.17)
= \ 4 7rp ,] R 4'
so that near the origin we have the following asymptotic behaviour: [ 3E'~'/3 1/3 R(~')'~o~-~p) ~" .
(4.18)
It is easy to check that (4.18) agrees with the asymptotic form (4.11 ) in the same limit. In the second case R ( r ) ~ . . . . oo, so in the remote past and in the distant future (4.5) approaches (4.13). As a consequence, also for E > 0 , there exist solutions whose asymptotic form is (4.14). Both kind of bubbles have a rest point (the particle's turning point) corresponding to the m a x i m u m expansion or the minimum contraction radius. The mass of the bubble at the rest point is given by y = ax3+x2(1
(4.19)
- - X 2 ) 1/2 ,
where we have introduced the adimensional quantities y = ( E a f f 12 ~rp),
(4.20a)
x = R ( 3 / a i ) -1/2 ,
(4.20b)
a =~
\aft
( a i - Ao - 487r2p2) •
(4.20c)
The function y ( x ) exhibits a single m a x i m u m in the interval 0 ~
x=" k
" ~ + ~-5-~
'/2
j
if
a>0,
(4.21)
or
4 + 3 a 2 - a (9a 2 + 8)1/2] 1/2 ~+~ j
=L
when
a < 0.
In the limiting case a = 0 we have simply ~ = ~/~ and )7 = 2x/3.
(4.22)
A. Aurilia et al. / Early universe
535
Finally we notice that for a <~0 the curve (4.9) intersects the x-axis. The intercept represents the minimum radius of a zero-total-energy bubble: Ro = H -1 in agreement with the exact solution (4.14).
5. Summary and discussion The particular solutions of eq. (4.5) we found in sect. 4 can be distinguished into two classes according to E > 0 or E = 0. In the first case the mass energy of the bubble must be provided by some external environment. Tfierefore we can use these massive bubbles as models for the cosmic voids in the large-scale galaxy distribution [4]. In the second case, E = 0, bubbles can form spontaneously in vacuum without violating the energy conservation principle. The gravitational effects on the vacuum decay process have been studied by Coleman and De Luccia (from now on CDeL) in the framework of a semiclassical scalar field theory [20]. In this approach the decay of the false vacuum to a lowest energy density state, i.e. the true vacuum, proceeds through a quantum tunnelling mediated by a classical, euclidean, field configuration minimizing the action, the so-called bounce. Let us verify the consistency of our model with the C D e L theory. As far as the notation is concerned the following identities hold: p=Sj, (5.1) A, = 87rGN U(th_),
(5.2)
A0 = 8~rGN U ( 6 + ) ,
(5.3)
where the quantities on the r.h.s, are defined in ref. [20], and we have explicitly written the Newton constant GN. If we assume, with CDeL, that the bubble is materialized at rest, then from eq. (4.14) we find immediately for the materialization radius R=H-~=~. (5.4) Again the last symbol on the right in (5.4) refers to the notation of ref. [20]. We notice that the value of R has been derived by us directly from the equation of motion for the bubble, while C D e L obtained it from a stationarity condition for the action of the bounce. If we specialize H to the case of vanishing U(~b_) or U(~b+), then we get (3.15) and (3.16) of C D e L respectively. In the case A~ = --8'rrGNe <0, where e = - U(~b_) and A o = 0 , H vanishes for 67rS 2= GNe and the radius of the bubble becomes infinite. This is precisely the condition for the gravitational vacuum stabilization given by (3.21) in ref. [20]. In our general-covariant approach the total mass-energy of the system is expressed through the integral expression (4.4); an explicit calculation [8] for the case Ao = 0, Ai =--8"rrGN e, e = - U(~b_), R : 0 gives
E=-4"n'eR3+2"n'SIR2[(1-2E~GRY)I/E+(I+S'n-GNeR2)'/2
] .
(5.5)
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A. Aurilia et al. / Early universe
In the limiting case of a weak gravitational field we may expand equation (5.5) in powers of GN: E~---4~reR3(I-27rSIGNR)+4ZrSlR2(I+2zrGNeR2)-8~'2S2GNR3. By taking into account the relation R = 3 S t / e
(5.6)
we find
E =8rr2GNe2gS=- E 8.....
(5.7)
Hence, in the classical limit our general formula (5.5) gives the total gravitational energy as computed in ref. [20]. Following C D e L we define a classical volume energy as Ev = --4ZrGN e fo R dx x2(l - %rex 2) = - ~ZrGNeg3(1 - ~ G N e R 2 ) .
(5.8)
Then, inserting (5.8) into (5.5), taking into account (5.7) and the definitions ENewton ------ 8 7 r 2 e 2 G N R 5 ,
(5.9)
Egeo m.=i3"n" 16 2 e 2,.-, n5 , t.tNlX
(5.10)
we obtain the C D e L energy balance condition Eg .... = ENewton"l'-Eg. . . . .
(5.11)
Concerning the kinematics of the system it is worth mentioning that as far as one considers only the case with E = 0, and with either A~ or Ao vanishing, it is unnecessary to resort explicitly to an equation of motion. The evolution of the bubble can be simply inferred from the matching condition between the Minkowski and de Sitter (or anti-de Sitter) metrics. On the other hand, the case E > 0 is more easily tackled in the framework of the Israel approach [5] to the surface layer dynamics. Besides this quite general, purely geometrical, method of investigation we have at our disposal, in the action functional approach we adopted, the Lorentz force equation and the first integral of the motion (4.4). Both eq. (2.11) and (4.4) lead to the radial equation (4.5). Let us conclude summarizing what we consider the distinguishing features of our model: (i) the form of the action functional which defines the physical system we are investigating is dictated by two fundamental symmetries: invariance under general coordinates transformations and local gauge invariance; (ii) all the basic fields involved in our model, i.e ga~,, AA~,~, ~:a, have a precise geometrical meaning; (iii) a two-phase ground state for the system is obtained as an exact solution of the coupled field equations; (iv) the same effective action functional provides us with an effective description for vastly different physical systems, as are the hadronic vacuum, the very early universe, and the intergalactic voids.
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