Bubble Dynamics in Generalized Einstein Theories

1001 Progress of Theoretical Physics, Vol. 90, No.5, November 1993

Bubble Dynamics in Generalized Einstein Theories Nobuyuki SAKAI and Kei-ichi MAEDA

Department of Physics, Waseda University, Tokyo 169-50 (Received June 25, 1993) We present a formalism, encompassing an extension of Israel's thin-wall method, for studying the dynamics of bubbles in generalized Einstein theories. We use this formalism to show that in such theories, no bubble can be homogeneous both inside and out. The evolution equations are designed just in terms of variables in homogeneous outside. Only ordinary differential equations, including the initial constraints, arise from our formalism, which makes it useful for a variety of applications.

§ 1.

Introduction

Particle physics predicts that the universe experienced many phase transitions in its early history. If any of these phase transitions is first-order, the universe changes its phase from false vacuum to true vacuum through creation, expansion and collision of bubbles. It is very likely that this process has some influence on the evolution of the universe. Indeed, old inflation l) is based on a super-cooled first-order phase transition. In the inflationary scenario, the universe expands exponentially with time before the transition, thereby solving the horizon, flatness and monopole problems. However, it turns out that this exponential expansion is too rapid to permit a transition from false vacuum to true vacuum via percolation of true vacuum bubbles. 2) Extended inflation3) revived the idea of old inflation by using the Brans-Dicke theory4) instead of the Einstein theory. The Brans-Dicke field decelerates the expansion of the universe so that true vacuum bubbles can coalesce, thus ending the phase transition that drives inflation. Unfortunately, it was soon pointed out that extended inflation predicts anisotropies in the microwave background inconsistent with current observations. 5) So interest shifted to other extended models based on other generalized Einstein theories (GETs).6)-8) When discussing whether the inflationary phase transition ends, and if so, whether the presence of big bubbles is consistent with the isotropy of the cosmic microwave background, many authors have assumed that bubbles expand with the velocity of light, which has been proved only in the flat space-time. 9 ) (Strictly speaking, the asymptotic velocity of the wall in the Einstein theory differs from the velocity of light in generaF°)-l2) (see (5°6)), but in a realistic parameter range it almost agrees with the velocity of light, which does not change the analysis for inflation models.) Hence, it is important to study whether this assumption remains valid in the Brans-Dicke theory or other GETs as well. In the Einstein theory, using the thin-wall formalism devised first by Israel (Gauss-Codazzi formalism)/3) many authors undertook the study of bubble dynamics. lO ),1l),l4),l5) Suffern extended Israel's method to cover the Brans-Dicke theory/6) and Goldwirth and Zaglauer l7 ) and the present authors l2 ) applied Suffern's

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1002

N. Sakai and K. Maeda

formalism to bubbles in extended inflation. In this paper we generalize this thin-wall formalism to cover a larger class of GETs and derive the bubble equations of motion. The resulting formalism is useful not only for studying the phase transition that drives inflation, but also for other first-order phase transitions, such as the late time phase transition l8 ) in GETs. We organize our presentation as follows. In § 2 we briefly review the formalism of Berezin, Kuzumin and Tkachev, and discuss a method for analysing inhomogeneous bubbles. In § 3, we apply this method to derive the bubble equations of motion in those theories containing a scalar field ([J coupled to gravity through a term I( ([J)!R in the action. In § 4, we show how to set initial conditions for an inhomogeneous bubble. In § 5, as an example, we apply our formalism to hyperextended inflation.7) In Appendix A, we show how to use conformal field redefinitions to derive bubble equations of motion in GETs whose action includes a term non-linear in the scalar curvature. § 2.

A brief review of thin-wall formalism - - How to analyse inhomogeneous bubbles in an expanding universe--

As we will show below, bubbles in GETs become inhomogeneous when they are formed in an expanding universe. Hence, if we wish to analyse their dynamics, we must in general solve difficult partial differential equations unless we can somehow construct a formalism that avoids all spatial differentiation in its basic equations. Such a formalism is, of course, impossible for general inhomogeneous space-times. If, however, we assume that the bubble's exterior is homogeneous and that its interior matter fluid satisfies a condition to be discussed shortly, we can in fact circumvent the difficulties of the general case. The result is a framework for studying bubble dynamics in which only ordinary differential equations arise. Berezin, Kuzumin and Tkachev have presented such a framework in their systematic study of thin-wall bubbles/I) although they omitted complete explanation of how to obtain regular initial data and why additional conditions for the interior matter fluid are required. In this section, we briefly summarize their formalism for analysing a thin-wall bubble in the Einstein theory and discuss how to treat the problem when the spacetime of the new phase is inhomogeneous or its metric is unknown. Let a time-like hypersurface };, which represents the world-surface of a wall, divide space-time into two regions, V+ (outside) and V- (inside). We define a unit space-like vector N p , which is orthogonal to }; and points from V- to V+. It is convenient to introduce a Gaussian normal coordinate system (n, X i )19) such that the hypersurface n=O corresponds to};. According to the assumption that the wall is infinitely thin, its surface energy-momentum tensor is

Sij=liml£ Tijdn. £-0

(2·1)

-£

Using the extrinsic curvature tensor, defined by Kij=Ni;j, and the Einstein equations, we can write the jump conditions at the wall as ll )


Bubble Dynamics in Generalized Einstein Theories

1003 (2·2)

(2'4)

where KZ =87fG, hij denotes the three metric of };, and I denotes the three dimensional covariant derivative. We have written 1Jf± for the value of any field variable 1Jf on }; on the ± side of V, and used a bracket to represent the difference of the ± values: [1Jf]±= 1Jf+-1Jf-. From (2·2) and (2'4) we eliminate K/- as follows: l l )

KrS/+

~z {S/S/- ~ (TrS)Z}=[Tn n]±.

(2'5)

Since (2·3) and (2'5) do not contain K/-, they may be useful when the inside is inhomogeneous. In fact, when the space-time is spherically symmetric, we can write down the equations of motion of the shell as follows. The world-interval on }; is given by (2·6)

where R(r) is the circumference radius of a wall and r is the proper time of the wall. Sij can be written in perfect fluid form: (2'7)

where 6, 111 and u i =(l, 0, 0) are the surface energy density, the surface pressure and a unit time-like vector tangent to };, respectively. We can then rewrite (2·3) and (2'5) as

d6 +2 dR 6+?l1 =[Tn]± dr dr R T,

(2'8) (2·9)

As is shown in Ref. 11), K/ contains d ZR/drz, while K/ contains only dR/dr. Hence, (2'8) and (2·9) determine the time evolution of Rand 6, even when we know nothing about the interior geometry. In Ref. 11), to analyse a bubble whose interior metric is unknown, Berezin et al. used only (2·3) and (2'5), or equivalently (2·8)' and (2·9), by giving the initial values of 5/ in advance. Here, however, we point out two serious problems with their method. If we give Kr and 5/ in advance, K/- is determined from (2· 2). Then two space-times V± could be connected mathematically. But we must give careful consideration to the boundary conditions at the center of the bubble. In the spherically symmetric case, if there is neither a black hole nor a naked singularity inside, regularity (i.e., smoothness) requires the flatness of space at the center of a bubble. The metric g;'v is constrained by this condition, and so is K/-. If we fix the values of 5/ and Kr in advance, and determine K/- from (2' 2) without taking this regularity


1004


condition into account, the bubble's center may turn out to be singular. Hence, S/ should not be given in advance. Rather it must be determined by solving the junction conditions and the regularity condition jointly. Here we show more precisely how to do this. When we know the metrics of both sides, we can use the angular component of (2·2):

(2'10) because it does not contain d 2R/dr2. Initial conditions for R, dR/dr and the matter density in V± fix the initial value of (J. Recall, however, that if we give an arbitrary value for (J and determine K/- from (2'10), it is not in general consistent with the regularity condition at the center. We have to solve the Einstein equations to get a g;;'v consistent with the regularity condition at the center. Once we find initial data which satisfy the regularity condition, however, their subsequent regularity is automatically guaranteed when we solve the above dynamical equations (2·8) and (2·9). The second point is this. Though (2·8) and (2·9) have no K/- terms, they do have Tn n- or T rn- terms, which include the density of the interior matter fluid and its velocity relative to the wall. Neither the density nor the velocity can in general be determined without knowing the inner metric. Only when Tp.-v takes the form of a cosmological constant, i.e., Tp.--;; = - pv - g;;'v (where pv is guaranteed to be constant because of the Bianchi identities), can we obtain the equations of motion using only field variables in V+. We summarize this section as follows: if and only if a bubble encloses pure vacuum energy, can we write down the equations of motion of a shell without knowing its interior metric. (We must, of course, specify the initial value of the surface density, (J.) We have to set up initial data that guarantee the bubble's regularity at its center. § 3.

Equations of motion in generalized Einstein theories

Here we derive the equations of motion for a bubble in GETs. consider here is

The action we

(3'1) where FU/), IR) and E( (J) are arbitrary functions. Unfortunately, we find that it is very complicated to write down the equations of motion in the physical frame for general forms of F( (J), IR). In this section, instead, we restrict our theories such that F is a linear function of IR, i.e.,

(3·2) where I( (J) and V( (J) are arbitrary functions. When we wish to consider bubble dynamics in a theory to which (3'2) is inapplicable, such as IR 2 -theory, we may be able to recover the Einstein gravity by introducing unphysical, conformally transformed field variables. 21 ) We show in detail how this works in Appendix A.


1005

Bubble Dynamics in Generalized Einstein Theories The variations of (3·1) with respect to gp.v and ([J yield the field equations

(3·3)

where Tp.v == {E( ([J) + 2!"( ([J)}l7 p.([J[7 v([J -{

E~([J) +2!"(([J)}gp.v([7([J)2+2/'(([J)([7p.[7v([J-gp.vD([J) + Tp.v,

Y( ([J)== 2f(

([J)E(~tJ-6f'( ([J)2 ,

(3·5)

(3·6)

and a prime denotes derivatives with respect to ([J. Besides Sij defined in (2 ·1), we introduce another surface energy-momentum tensor as

(3.7) where the second equality is obtained from (3·4) and (3·5). Replacing Tp.v, Sij and K2 in the Einstein theory with Tp.v, Sij and 1/2/( ([J), we can use the formalism described in § 2. An additional constraint is the junction condition on ([J. As Suffern showed in the Brans-Dicke theory/6) we can obtain this condition of ([J directly from (3·4): (3·8) From this we see that V+ and V- cannot both be homogeneous. (Strictly speaking, we need more than (3·8) to see this. We must also explain how the Gaussian normal coordinate n relates to space-time coordinates in V±. This we do in § 4 just below (4·10).) Anyway, this is why we write the equations using only the field variables on the homogeneous side of space-time, where we may know the solution exactly, or at worst have only simple decoupled ordinary differential equations to solve for the space-time metric. We can formally rewrite (2·2), (2·3) and (2·5) in GETs as (3·9)

-

{ f(5/([J) }Ij

K j i+S- i

j+

1 (Tr S-)2}-[T4f(1([J) {S-iS-j j i -2 nn]+-.

(3·10) (3·11)

Though the calculation is cumbersome, with the help of (3·5) and (3·7), we can rewrite these equations as


1006

N. Sakai and K. Maeda (3 ·12) (3·13) (3·14)

which are the basic equations in GETs. Equation (3·13) is the same as that in the Einstein theory. Henceforth we restrict the space-time V+ to be Friedmann-Robertson-Walker (FRW): (3·15)

where

r(x+)=

sinx+ x+ sinhx+

1

(k+= + 1, closed universe) , (k+=O, flat universe) , (k+= -1, open universe) .

Since V- is inhomogeneous, (3·13) and (3·14) become useful. Substituting (2·7) in (3 ·13) and (3 ·14), we get

- 6K/+ +2?JJ Kr +

8A

([J) 6(6+4?JJ) +21'( ([J) Y( ([J)( - 6+2?JJ)2=[ Tnn]± , (3 ·16)

d6 + dR 6+?JJ =[Tn]± dr dr R r,.

(3 ·17)

where K/+ and Kr are given by (B·I0) and (B·8). The conditions of metric continuity on }; are obtained as R(r)=a(t+(r»r(x+(r»I1.' , dr2=dt+2-a 2dx+2I1.' from (2·6) and (3·15).

(3 ·18)

Here we consider perfect fluid matter, (3·19)

where p, p and Up. are the pressure, the energy density and the four velocity of the matter, respectively. Then we can rewrite (3 ·16) and (3 ·17) as

- r+ {(1-2 ?JJ)v H _2k ?JJ} 6 + R dx+ 6 6

(3·20) (3·21)

where

_ dx+1 v+=a dt+ 1."

and


H- da/dt+ . a

(3·22)

Bubble Dynamics in Generalized Einstein Theories

1007

v+ denotes the peculiar velocity of the wall relative to the background expansion. We get the relation between dR/dt+ and v+ from (3'18) and (3·22): dR dr dt+ = dx+ v++ HR.

(3·23)

As we mentioned in § 2, p- and v_ cannot be determined without knowing the metric in V-. Only when p-= - p-=const does v- disappear in (3·20) and (3·21), allowing us to integrate them without solving the field equations (3'3) in V-. Once we have acceptable initial data, we can find the evolution of the wall using (3·20), (3'21) and (3'23). As we mentioned in § 2, the initial values are constrained by the angular component of (3 '12): (3'24)

which contains no second derivative terms like dv+/dt+ or d 2 R/dr2 • Since in the present inhomogeneous case we have no exact solution for K/-, we must calculate it numerically by solving the constraint equations of (3'3). In the next section, we describe in detail how to get the required initial data. § 4.

Initial conditions for an inhomogeneous bubble

In this section we present a numerical method to determine initial data when V+ is the FRW space-time and the metric in V- is unknown. The spherically symmetrical metric in V- is generally given by (4 '1)

where A, Band C are functions of Land x-. Using this metric, we write down the Hamiltonian and momentum constraint equations following from (3'3) as C"

1- C 2C2

2

C( 3 C) 1 +c 8- 2C + 4/(cP) T t

t

,

(Cy=C( 8- 2&)+ 4/((1») ytr,

(4·2) (4'3)

where (4·4)

From the definition of yp.].J, we find

y/ = - c(i) dJ2- c(i) +2f"( (1») (1)'2+2/'( (1»)( 8dJ - (1)" - 2g' (1)') + T/ ,

(4· 5)

ytr= -{c( (1»)+2f"( (1»)}dJ(1)' +2/'( (1») { -(dJ)'+( 8- 2& )(1)'} + T tr .

(4'6)

Giving appropriate initial configurations of (1)(x-) and dJ(x-), we can integrate numeri-


1008


cally (4'2) and (4·3) with boundary conditions at the center of a bubble and at the wall. We now discuss these boundary conditions.

en

Regularity conditions at the center of a bubble. The curvature must vanish at the center, which implies (4'7) (ii) Continuity at the wall (x-=%.,1;.). We impose junction conditions for (jJ, (jJ,r, (jJ,n, R, R,r and R,n, where R(r, n) is the circumference radius of the n=const, r=const, (J =7r/2 curves in the Gaussian normal coordinates introduced in (B·2). Since (jJ and R must be continuous across the wall, we have (4·8) (a) The continuity of (jJ,r and R,r: d(jJ+ (m'- + d.-) r+ dt+ =r- V-w. w

(4'9)

,

(4'10) where 1

(4'11)

and we have assumed that (jJ is homogeneous in V+, i.e., 0(jJ+ /ox+=O . (b) The jump conditions for (jJ,n and R,n: As for the behaviour of (jJ,n and R,n, we have the jump conditions (3·8) and (3·24). Because R,n=RK/ from (B·3), the condition on R,n implies that on K/. K/+ in the FRW metric can be described by two expressions: one is our (B'8) and the other is that of Berezin et aI., (B ·11). Similarly, K/- in a spherical symmetric space-time has the following two expressions: (4 '12) and (4'13) In deriving (4 '12), we have chosen the interior coordinates so as to satisfy condition that the sign of (ox-/on)=the sign of (oL/or) = + L Just the same as the case of K/, (jJ,n can be described in terms of the coordinates in V± as d(jJ+ (jJ,t = r + v+ dt+'

(jJ,-.. = r -( (jJ'- + v- dr) .

The jump conditions for (jJ,n and

la,

(3'8) and (3'24), can be written as


(4·14)

1009

Bubble Dynamics in Generalized Einstein Theories r+v+

~++ -(([)'-+v-aJ-)= Y(([»)( -6+2?11) ,

(4 ·15)

r+(;;+ +v+HR)-r-(C+v-C)=- 4A([») {6+2j'(([»)Y(([»)(-6+2?11)}R.

(4 ·16) In the Einstein theory, only (4 ·10) and (4 ·16) remain. When the metrics in both 6, v+ and v-. If one assumes v+ (or dR/dr) as an initial value, v- is determined from (4·10). Then (4·16) provides the relation between Rand 6. Therefore, in GETs, (4·9) and (4·15) are interpreted as additional conditions which determine ([)'-(xz) and dr(xz). We summarize the procedure of setting initial data. We must first solve the constraint equations (4·2) and (4·3) with boundary conditions at the ~enter, (4·7), and at the wall, (4·9), (4·10), (4·15) and (4·16). This entails the following steps. First, we fix the values of Xz, v+ and e. Integrating (4·2) and (4·3) from x-=O, with the boundary conditions (4·7), we obtain R=C(xz), C(Xz) and C(Xz). The boundary conditions (4·9), (4·10), (4·15) and (4·16) determine ([)'-(xz), aJ-(Xz), v- and 6. Because in the integration, we must give the distributions of ([)-(x-) and aJ-(X-) in advance, we have to iterate the above procedure until ([)-(x-) and aJ-(x-) become consistent with the boundary conditions at the wall. In an example of the next section, we use the 4th-order Runge-Kutta method in solving the initial equations as well as the equations of motion. Now, we can show inhomogeneity of the interior space-time as follows. As mentioned above, two jump conditions, (4·9) and (4 ·15), fix ([)'-(Xz) and aJ-(Xz) uniquely. While, if the metric in V- were homogeneous, ([)'-(xz) would vanish, and hence this assumption would be inconsistent with the too many boundary conditions. Therefore, the inside is inhomogeneous in general. Finally we shall give some comments on the initial velocity of the wall. This problem is related to the validity of the 0(4) symmetric solution of Coleman and De Luccia. 20 ) When V+ is the de Sitter spacetime and V- is the Minkowski spacetime, they derived a formula for the initial, radius of a bubble: V± are given, the unknown variables are R,

4A/3 R 0-H-l (2A/3)2+ 1 '

(4 ·17)

where k=-(Pv/H6) with pv being the vacuum energy in V+. This is consistent with the junction condition (B·12) with P+=Pv and p-=O, only if dR/dr=O. However, we have shown in Ref. 12) that when the self-gravity of the wall is dominant, i.e., A~1.5, (4 ·17) is inapplicable to the spatially fiat universe even as an approximation. Hence, it is not clear whether the 0(4) symmetric solution of Coleman and De Luccia for non-de Sitter background is valid even if the gravitational effect is small. This may force the initial velocity to be uncertain. § 5.

Application to hyper extended inflation

As an example of our present formulation, we shall discuss a bubble motion in


1010


hyperexterrded inflation. The action is given by (3 ·1), with setting (S·I) where we have normalized the dimensionless field (]J to be unity at the present epoch. For simplicity, we consider only vacuum energy outside and no vacuum energy inside the bubble: (S·2) and assume the outer 3·space is homogeneous and flat. dr/dx+=1 into (3·20) and (3·21), we obtain

Inserting (S·I), (S·2) and

2w( (]J) 3K2 3+2w( (]J) 4(]J (f,

(S·3)

and (f=const. Once we determine the initial data as in § 4, we can find the evolution of the wall from (3·23) and (S· 3). Here we write down the junction condition (3 ·12) following Berezin et al.: (S·4)

where 3{1+2w((]J)}2 + {3+2w( (]J)}{S+6w( (]J)} pv

(S·S)

and Ll and E are defined just below (B ·1). In our analysis, E- is always + 1. In the Einstein theory, since the inner region is simply the Minkowski space-time, Ll-=1. We may regard E =- (Ll- -1) /H2 R2 as the measure of inhomogeneity inside a bubble. In order to determine initial data and solve the equations of motion, we must know the functions aCt) and (]JCt), which we can get only from a specific model for w( (]J). Qualitatively, however, if w( (]J) varies slowly, aCt) and (]JCt) can be approximated by those corresponding to a constant w. Here we use this approximate solution, and regard w not as a dynamical variable but as a parameter. In our previous paper,I2) we showed the results only for the case W>S, and we mainly discussed the problems of Ref. 17). Here we also include the case W0, because our present universe is in the region of aF jaPe >0 and a spacetime region in which aFjaPe=O becomes singular, so that no spacetime can evolve into a region in which aFjaPe < O. W( ¢, (fJ, ¢) is also explicitly given, but here we need only know that it is an effective potential of the three scalar fields. The variations of (A'4) with respect to fjpu and ¢ yield the field equations:' (A'5) (A·6)

where

- - p¢{7- u¢+ e -.f2i3K¢ {c{7- p(fJ{7- u(fJ +{7-u¢} T- pu ={7 p¢{7 - fjpu[

~

(£7¢)2+

~ e-.f2i3K¢{c(£7(fJ)2+(£7¢)2} + we¢, (fJ,

¢)].

(A'7)

The variations with respect to (fJ and ¢ yield

c£7(e-.f2i3K¢£7(fJ) = ~~ ,


(A·S)


1014

We assume here that