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Bubble free energy in cosmological phase transitions
Physics Letters B 309 (1993) 252-257 North-Holland
PHYSICS LETTERS B
Bubble free energy in cosmological phase transitions J. Ignatius 1 Department of Theoretical Physics, University of Helsinki, P.O. Box 9, Siltavuorenpenger 20 C, SF-O0014 Helsinki, Finland Received 2 January 1993; revised manuscript received 22 April 1993 Editor: P.V. Landshoff Free energy as a function of temperature and the bubble radius is determined for spherical bubbles created in cosmological first order phase transitions. The phase transition is assumed to be driven by an order parameter (e.g. a Higgs field) with quartic potential.
The purpose o f this Letter is to study the free energy o f bubbles which are created in cosmological first order phase transitions. Assuming that the phase transition can be described with an order parameter model with a quartic potential, we determine the radial dependence o f the free energy at given temperature. Expanded in powers o f 1/R, i.e. in the large R limit, the Helmholtz free energy o f a spherical bubble o f the low temperature phase nucleating in the high temperature phase can be written as
F ( R , T) = - ~ r t A p ( T ) R 3 + 4~ztrf(T)R 2 + 8nyf(T)R + 16n6f(T) + ....
(1)
where R is the bubble radius according to some specific definition and Ap ( T ) is the pressure difference between the two phases. It is expected that near the thermodynamical phase transition temperature Tc the relative change o f the functions cry (T), yf (T), 6f ( T) is slow, in contrast to the pressure difference Ap ( T ) , since properties o f the interface do not change dramatically with temperature. In cosmology, when we are discussing bubble formation near T¢, we can treat these functions as constants. In this approximation the free energy is for large R, in the vicinity o f T¢
I 252
Bitnet: IGNATIUS@FINUHCB
F (R, T) = - 4 z c A p ( T ) R s + 47rtrpR 2 + 8ttyR + 16zt6 + ...
(2)
-- _ 4 r t A p ( T ) R 3 + 47rtr(R)R 2,
(3)
where the latter form defines the generalized surface tension (see also [ 1 ] )
a(R) -
1
41tR 2
[ F ( R , T ) + ] z t A p ( T ) R 3]
(4)
Eq. (3) enables us to solve the general free energy F (R, T) for a given definition of the bubble radius from the critical free energy For(T) (to be defined later) and the pressure difference Ap ( T ) only: from eq. (4) we are able to determine a ( R c r ( T ) ) , where Rcr ( T ) is the radius of the critical bubble, and when the function Rcr ( T ) is known as well, we can solve for (R). Thus we know the general free energy in eq. (3). The quantities needed to solve the general free energy F (R, T) are known in the case we are interested in, namely when a cosmological phase transition can be described using an order parameter model. In other words, in the first place we know only the height o f the free energy wall for forming bubbles of the new phase, Fcr(T). But applying the procedure described above, we are able to determine also the radial dependence o f the free energy, F ( R , T). Naturally the height is the most important single feature of the wall - in the standard picture of homogenous nucleation it essentially determines the bubble nucle-
0370-2693/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
Volume 309, number 3,4
PHYSICS LETTERSB
ation rate. However, the free energy wall is of interest even more generally. For example, in the MIT bag model the free energy wall is such that small hadron bubbles exist already above the quark-hadron phase transition temperature [2]. We will discuss this observation later on. In order to determine the radial dependence of the free energy from eq. (3) we must first know Ap(T) and F e r ( T ) . Below, we will summarize the relevant physics of the order parameter model which enables us to calculate these quantities. The action for the bosonic order parameter field in the high temperature approximation is s = 7
d3x
+
v(¢r)],
(5)
where the effective potential is V(~,T)
= l~a(T2
- T2)t~ 2 -
laT@ 3 + 1~4.
(6)
This effective potential is commonly used to describe a first order electroweak phase transition [3-7]. It might be that the electroweak theory contains for example two Higgs doublets, but to a reasonable approximation this same finite temperature potential can be used for the combination of Higgs fields which first becomes massless [4]. For QCD, the choice of the order parameter is not so clear, since the theory has no classical potential. For example, in [8] the energy density with a quartic potential was used as the order parameter. The potential V (4, T) should be regarded as a phenomenological expansion for the effective order parameter, valid in the vicinity of T~. The latent heat coming from the decrease of the effective relativistic degrees of freedom is included in the potential. The constants To, ~)a, a and 2 are to be chosen so that the potential quantitatively correctly describes the phase transition. These four constants can be expressed in terms of the following four more physical quantities [ 9 ]: T¢, the length lc, the thermodynamical or planar surface tension ap and the latent heat of the transition. The first three ones are given by To
T¢-
v/l _
,
3x/2 1
15 July 1993
At To, the correlation lengths in both phases are equal to lc. It is convenient to express the temperature dependence of several quantities by using the function 2 (T), 92Ya ( l - r ° 2 ~ 2(T) - 2 a 2 T2J '
which satisfies 2(To) = 0, 2(To) = I. The pressure in the low temperature (broken symmetry) phase minus the pressure in the high temperature (symmetric) phase is [6] Ap(T) =
a4 T 4 { S 2 ( T ) 2 -
~2(T)
+1 + [1 - ~2(T)] 3/2} .
(9)
This pressure difference is positive for temperatures smaller than T~. The critical action Scr (T) is the value of the action (5), with the appropriate boundary conditions, in the O(3)-symmetric extremum. More physically speaking, it is the critical free energy FCr(T), the free energy needed to form a critical bubble, divided by the temperature. The critical free energy is for To < T < Tc the maximum value of the general free energy as a function of the bubble radius at constant temperature, Fer(T) = F ( R c r ( T ) ,
T),
(10)
where Rer ( T ) is radius of the critical bubble.
The critical action can be analytically approximated in the small and large relative supercooling regimes [3], and in the full range it has been calculated numerically [6,7]. In terms of the function ~(T), Scr(T)
-
Fer(T) _ 29/27~ a f(2) T 35 23/2 (1 _ ~ ) 2 "
(11)
A fit to the function f ( 2 ) with an -0.1 ... +1.6% error was presented in [7]. Here we give for f (2) a four-parameter fit with an accuracy of ±0.1%: f ( ~ ) = ~3/2( a0 + a12 + a22--2 + a3 ~3 + a4~ 4) , a0 = 15.63628,
I~=---
(8)
Ear:' a2
=2.39731,
al = -18.03398, a3 = - 0 . 8 6 5 0 4 ,
2x/2 a s ap -
8----ff-2 5/-----iT 3 .
(7)
a4 = 1.86543.
(12) 253
Volume 309, number 3,4
PHYSICS LETTERS B
15 July 1993
3 0.2
g-
,~-
80.1 2 .o= 0.13
I,"<
e-O. 1 1
-0.2
/
/
/
/
!
i
i
-0.3
0 0.0
0.2
0.4
0.6
0.8
1.0
8
12
R'
Fig. 1. Non-divergent part of the critical action. Solid curve is f ( ~ ) , as defined in eq. (11), and dashed curve is
[f (2)/23/2]/6 . This fit is exact in the small relative supercooling (2 = 1 ) limit. The function f (2) is shown in fig. 1, together with f (~)/~3/2. It is interesting to note that the latter one, which is relevant in the large relative supercooling (2 = 0) limit [6], is an almost straight line over the full range. Now that we know both Ap ( T ) and For(T) we can proceed in determining the general free energy from eq. (3). Next, we must have a definition which gives the bubble radius for critical bubbles, Rcr ( T ) . Employing it we then obtain the corresponding curvature-dependent surface tension a ( R ) . In the rest o f this Letter we mainly study these two quantities. One should bear in mind, however, that when discussing cosmological bubble formation only the free energy has real physical importance, the surface tension as well as the bubble radius are only auxiliary quantities. But those quantities must be known before one is able to utilize eq. (3) which gives the radial dependence o f the free energy. We employ two different definitions for the bubble radius. The first one is Laplace radius RL defined by using Laplace's relation
RI.~(T) - 2aL(RLer(T) )
(13)
Ap(T) The second one is the tension radius RT, defined as the distance at which the gradient density term o f the critical action reaches its maximum. The definition o f RT is illustrated in fig. 2. F r o m Laplace's relation, it follows that the radius o f the critical bubble and the generalized surface tension 254
4
Fig. 2. Definition of RT. Solid curve is the gradient part of the action density of a critical (2 = 0.9)-bubble and dashed curve is the potential part, the first and the latter term in eq. (5), respectively. The dimensionless radius is R' = MR, where M stands for the bosonic mass in the potential, M 2 = 7a (T 2 - T2). The free energy density is shown in units of TM3a/23/2 (note that since 2 is fixed, so is T/Tc). From the figure, we can read that for 2 = 0.9, R~cr ~ 7.3 . in terms o f the critical free energy are RLcr(T) =
[ 3 F ( R L c r ( T ) , T ) ] 1/3 2~ Ap(T) '
(14)
3 F aL(RLer(T)) = [ T-~ ( R L e r ( T ) ' T ) A p ( T ) 2 I 1/3
(15) In the m a x i m u m tension definition, the radius o f the critical bubble, RTer(T), is obtained from the numerical solution o f the extremum differential equation for the action in eq. (5) for different values o f ~. The curvature-dependent surface tension aT (RTcr ( T ) ) is then calculated directly from eq. (4). The two definitions for the bubble radius are compared in fig. 3. One notes that always Rrcr >t RLcr. In the small relative supercooling limit, the two definitions coincide. On the other hand, in the large relative supercooling limit they differ even qualitatively: RLcr vanishes, whereas RTcr goes to infinity. However, this limit is beyond the validity range o f the expression (2, 3) for the free energy, in which all the temperature dependence comes from the pressure difference. Both o f the generalized surface tensions, aL (R) and aT(R), are plotted versus the inverse radius I¢/R in fig. 4. F o r comparison, we may note that in the theory o f fluid surfaces, the dependence o f the surface ten-
Volume 309, number 3,4
PHYSICS LETTERS B
15 July 1993
16 12
~z 0
8
0 0.0
//
-1
4
¢
0.2
0.4
0.6
0.8
-2
1.0
/ /.
1
0.4
0'.6
0'.8
1.0
To~To Fig. 3. Radius of the critical bubble for To/Tc = 0.58 as a function of the degree of relative supercooling. Dashed curve is Rtcr/lc, and solid curve RTcr/lc. sion on the definition o f the bubble radius is a subtle issue [ 10 ]. F o r large values o f the bubble radius the two surface tensions coincide. In this limit, depending on the value o f the constant To~To, the surface tensions can either decrease or increase with R, implying that the curvature coefficient can be positive or negative, respectively (see fig. 5). When the bubble radius gets smaller, the two surface tensions differ increasingly. The curves for ax (R) turn backwards, because with decreasing 2, R'r~ when expressed in physical units possesses a m i n i m u m , after which it begins to increase (see fig. 3). F o r small values o f the radius, aL (R) has to decrease with R, in order to compensate the faster decrease o f the term A p ( T ) R 3 in the free energy expansion in eq. (3). And vice versa,
i
.......i.......i........i........
Fig. 5. The coefficients 7 and $ in the free energy expansion in eq. (2), for different values of To~To. Dashed curve is 7/(aplc), coefficient of the curvature term, and solid curve is $/(apl2c ), coefficient o f the constant term. The width o f the curve for $ shows the uncertainty in eq. (19), due to numerical determination o f f " ( l ) (and to much less extent,
of f ' ( 1 ) ) . aa-(R) must increase with decreasing 2 in the large relative supercooling regime. The two-valuedness o f ax (R) does not imply that the distance o f m a x i m u m tension would be a b a d definition for the bubble radius. The radius R-r agrees well with the intuitive picture o f the bubble, and furthermore, as will be noted later, its m i n i m u m value can be used as an estimate for the break-down distance of the free energy expansion. Next, we expand a (R) as Taylor series in 1/R and thus determine the coefficients in the truncated free energy expansion in eq. (2): ap-
a(R)ll/R=0,
Y-
21 dd(a1(/RR))
l/R=0 '
1 1 dEa(R)
1
- 42! aTU -7 ll/R=0
(16)
""--.. ....... . ......
G
0.0
0.2
0.4
0.6
0.8
1.0
Ic/R Fig. 4. The generalized surface tension from both definitions, aL(R) and aT(R), plotted versus the inverse radius lc/R. Dashed curves show at (R)/ap, and solid curves aT (R)/ap. Upper curves are for To/Tc = 0.99, and lower curves for To~To = 0.58.
These coefficients were determined employing both definitions for the bubble radius. Within numerical accuracy, the results were equal. The reason for this is that both o f the radii, and both o f the generalized surface tensions as well, coincide in the limit of infinite bubble radius. Below, results for the coefficients are presented only for the case where the radius is defined by Laplace's relation. 255
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PHYSICS LETTERS B
We write trL(RLcr(T)) and lc/RLer(T) as Laurent expansions o f 2 around 2 = 1, which corresponds to infinite radius. We find that the coefficient o f the curvature term is
y=
1.49+0.01 -
2T~]aflc,
where
)
If T:
92ya -- 2or 2 -- 2 k, T0z - 1
(17) 0 0.2
6 = ~4{f"(1)+ 2u[5f'(l)
2f'(1)- 1] +
0.6
018
1.0
.
(18)
Fig. 6. The distances at which magnitudes o f the curvature term and the constant term equal the surface term in the truncated free energy expansion o f eq. (2), plotted versus To~To. Dashed curve shows the distance Rr/l¢ at which 4napR 2 = 8nlylRr, and solid curve the distance R6/lc at which 4napR~ = 16n[~].
6
27uE}apl2
(0.88 + 0.03- ^ 90 T~ o.
0.4
To~To
In eq. (17), f ' ( 1 ) denotes d f ( 2 ) / d 2 at 2 = 1, and its numerical value is c o m p u t e d from eq. (12). F o r the coefficient o f the constant term in eq. (2) we find
T4)apg.
+ 8rg ]
(19)
The coefficients y and 8 are shown in fig. 5. If the temperature To is near To, the coefficient y is in units o f aplc close to unity, and 8 almost vanishes. If To/To is in the vicinity o f 0.58, the zero-point o f y, the constant term dominates over the curvature term in the free energy expansion even for rather large bubbles. When To~To decreases from ~ 0.4, y decreases and 8 increases rapidly. The magnitudes o f the different terms in the truncated free energy expansion are c o m p a r e d in fig. 6. F r o m the figure we can conclude that if To/To > 0.5, the truncated expansion for the free energy o f a spherical bubble in eq. (2) breaks down when the bubble radius is o f the order o f 2lc. If To/To is smaller, the truncated free energy expansion is valid only for much larger bubbles. The distance where the truncated free energy expansion breaks down is for different values o f To/Tc roughly equal to the smallest tension radius o f the critical bubbles. Finally, we will investigate if the low temperature phase bubbles can exist at temperatures higher than To. Here the full expression for the free energy, eq. (3), is used. Now the volume term in this equation is positive. Therefore the necessary condition for the existence o f stable low temperature phase bubbles is that 256
6
[~-~f'(1)-u]aflc
( U --
15 July 1993
the surface term (i.e. the latter term) not be a monotonically increasing function o f the radius R. The surface term is plotted in fig. 7 for two values o f To/Tc, and it behaves qualitatively in the same manner for all other values o f To~To. In spite o f the two-valuedness o f aT (R) it is clear that the behaviour o f the surface term 4 n a ( R ) R 2 shows no indication o f the existence o f low temperature bubbles above T¢. The result that the magnitude o f the generalized surface term grows with the bubble size is very natural (even though a(R) is a decreasing function o f
~'6 O
4
/
/"
2 //
o// 0
4
8
12
R/lc Fig. 7. Logarithm o f the surface part in the expression for the free energy in eq. (3), log[4na(R)R2/(apl~)], as a function o f the bubble radius R/l¢. In the same m a n n e r as in fig. 4, dashed curves represent 4naL(R)R 2, solid curves 4naT(R)R2; upper curves are for To~To = 0.99, lower for To~To = 0 . 5 8 .
Volume 309, number 3,4
PHYSICS LETTERS B
the bubble radius for large bubbles if To~To > 0.58, the combination tr ( R ) R 2 is an increasing one). However, in the M I T bag model calculation by M a r d o r and Svetitsky [2 ] it was found that the free energy as a function o f the bubble radius has a m i n i m u m for T > T¢. (In the context o f strangelet masses, the curvature energy in the M I T bag model has been recently studied in [ 11 ].) Clearly the conclusion must be that either the spherical M I T bag model, or any order parameter model with a potential as the one in eq. (6), describes the q u a r k - h a d r o n phase transition qualitatively incorrectly. To summarize, we have discussed bubble free energy in cosmological first order phase transitions which can be described by the quartic potential. We made the a p p r o x i m a t i o n that all the temperature dependence in the free energy comes from the pressure difference between the two phases. This approximation enabled us to solve, starting from knowledge o f the critical bubbles only, the general free energy F(R, T). W e studied in detail the quantities which using eq. (3) give the radial dependence o f the free energy: the bubble radius and the curvature-dependent surface tension. This research was supported by the Academy o f Finland. The author wishes to thank K. Kajantie for the inspiration to this work as well as for useful comments, and T. Ala-Nissil~i, A. Laaksonen and M. Laine for discussions.
15 July 1993
References [1] K. Kajantie, J. Potvin and K. Rummukainen, Institute for Theoretical Physics preprint NSF-ITP-9255 (1992), to be published in Phys. Rev. D. [2] I. Mardor and B. Svetitsky, Phys. Rev. D 44 (1991) 878; G. Lana and B. Svetitsky, Phys. Lett. B 285 (1992) 251. [3] A. Linde, Nucl. Phys. B 216 (1983) 421. [4] L. McLerran, M. Shaposhnikov, N. Turok and M. Voloshin, Phys. Lett. B 256 (1991) 451. [5] N. Turok, Phys. Rev. Lett. 68 (1992) 1803; G.W. Anderson and L.J. Hall, Phys. Rev. D 45 (1992) 2685; B.-H. Liu, L. McLerran and N. Turok, Phys. Rev. D 46 (1992) 2668; M. Kamionkowski and K. Freese, Phys. Rev. Left. 69 (1992) 2743. [6]K. Enqvist, J. Ignatius, K. Kajantie and K. Rummukainen, Phys. Rev. D 45 (1992) 3415. [7] M. Dine, R.G. Leigh, P. Huet, A. Linde and D. Linde, Phys. Rev. D 46 (1992) 550. [8] L.P. Csernai and J.I. Kapusta, Phys. Rev. D 46 (1992) 1379. [9] K. Kajantie, Phys. Lett. B 285 (1992) 331. [10] See e.g., G. Navascurs, Rep. Prog. Phys. 42 (1979) 1131; or J.S. Rowlinson and B. Widom, Molecular Theory of Capillarity, Ch. 2.4 (Oxford University Press, Oxford 1982). [11] J. Madsen, Phys. Rev. Lett. 70 (1993) 391; Aarhus University preprint AARHUS-ASTRO-1993-3, to be published in Phys. Rev. D.
257
PHYSICS LETTERS B
Bubble free energy in cosmological phase transitions J. Ignatius 1 Department of Theoretical Physics, University of Helsinki, P.O. Box 9, Siltavuorenpenger 20 C, SF-O0014 Helsinki, Finland Received 2 January 1993; revised manuscript received 22 April 1993 Editor: P.V. Landshoff Free energy as a function of temperature and the bubble radius is determined for spherical bubbles created in cosmological first order phase transitions. The phase transition is assumed to be driven by an order parameter (e.g. a Higgs field) with quartic potential.
The purpose o f this Letter is to study the free energy o f bubbles which are created in cosmological first order phase transitions. Assuming that the phase transition can be described with an order parameter model with a quartic potential, we determine the radial dependence o f the free energy at given temperature. Expanded in powers o f 1/R, i.e. in the large R limit, the Helmholtz free energy o f a spherical bubble o f the low temperature phase nucleating in the high temperature phase can be written as
F ( R , T) = - ~ r t A p ( T ) R 3 + 4~ztrf(T)R 2 + 8nyf(T)R + 16n6f(T) + ....
(1)
where R is the bubble radius according to some specific definition and Ap ( T ) is the pressure difference between the two phases. It is expected that near the thermodynamical phase transition temperature Tc the relative change o f the functions cry (T), yf (T), 6f ( T) is slow, in contrast to the pressure difference Ap ( T ) , since properties o f the interface do not change dramatically with temperature. In cosmology, when we are discussing bubble formation near T¢, we can treat these functions as constants. In this approximation the free energy is for large R, in the vicinity o f T¢
I 252
Bitnet: IGNATIUS@FINUHCB
F (R, T) = - 4 z c A p ( T ) R s + 47rtrpR 2 + 8ttyR + 16zt6 + ...
(2)
-- _ 4 r t A p ( T ) R 3 + 47rtr(R)R 2,
(3)
where the latter form defines the generalized surface tension (see also [ 1 ] )
a(R) -
1
41tR 2
[ F ( R , T ) + ] z t A p ( T ) R 3]
(4)
Eq. (3) enables us to solve the general free energy F (R, T) for a given definition of the bubble radius from the critical free energy For(T) (to be defined later) and the pressure difference Ap ( T ) only: from eq. (4) we are able to determine a ( R c r ( T ) ) , where Rcr ( T ) is the radius of the critical bubble, and when the function Rcr ( T ) is known as well, we can solve for (R). Thus we know the general free energy in eq. (3). The quantities needed to solve the general free energy F (R, T) are known in the case we are interested in, namely when a cosmological phase transition can be described using an order parameter model. In other words, in the first place we know only the height o f the free energy wall for forming bubbles of the new phase, Fcr(T). But applying the procedure described above, we are able to determine also the radial dependence o f the free energy, F ( R , T). Naturally the height is the most important single feature of the wall - in the standard picture of homogenous nucleation it essentially determines the bubble nucle-
0370-2693/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
Volume 309, number 3,4
PHYSICS LETTERSB
ation rate. However, the free energy wall is of interest even more generally. For example, in the MIT bag model the free energy wall is such that small hadron bubbles exist already above the quark-hadron phase transition temperature [2]. We will discuss this observation later on. In order to determine the radial dependence of the free energy from eq. (3) we must first know Ap(T) and F e r ( T ) . Below, we will summarize the relevant physics of the order parameter model which enables us to calculate these quantities. The action for the bosonic order parameter field in the high temperature approximation is s = 7
d3x
+
v(¢r)],
(5)
where the effective potential is V(~,T)
= l~a(T2
- T2)t~ 2 -
laT@ 3 + 1~4.
(6)
This effective potential is commonly used to describe a first order electroweak phase transition [3-7]. It might be that the electroweak theory contains for example two Higgs doublets, but to a reasonable approximation this same finite temperature potential can be used for the combination of Higgs fields which first becomes massless [4]. For QCD, the choice of the order parameter is not so clear, since the theory has no classical potential. For example, in [8] the energy density with a quartic potential was used as the order parameter. The potential V (4, T) should be regarded as a phenomenological expansion for the effective order parameter, valid in the vicinity of T~. The latent heat coming from the decrease of the effective relativistic degrees of freedom is included in the potential. The constants To, ~)a, a and 2 are to be chosen so that the potential quantitatively correctly describes the phase transition. These four constants can be expressed in terms of the following four more physical quantities [ 9 ]: T¢, the length lc, the thermodynamical or planar surface tension ap and the latent heat of the transition. The first three ones are given by To
T¢-
v/l _
,
3x/2 1
15 July 1993
At To, the correlation lengths in both phases are equal to lc. It is convenient to express the temperature dependence of several quantities by using the function 2 (T), 92Ya ( l - r ° 2 ~ 2(T) - 2 a 2 T2J '
which satisfies 2(To) = 0, 2(To) = I. The pressure in the low temperature (broken symmetry) phase minus the pressure in the high temperature (symmetric) phase is [6] Ap(T) =
a4 T 4 { S 2 ( T ) 2 -
~2(T)
+1 + [1 - ~2(T)] 3/2} .
(9)
This pressure difference is positive for temperatures smaller than T~. The critical action Scr (T) is the value of the action (5), with the appropriate boundary conditions, in the O(3)-symmetric extremum. More physically speaking, it is the critical free energy FCr(T), the free energy needed to form a critical bubble, divided by the temperature. The critical free energy is for To < T < Tc the maximum value of the general free energy as a function of the bubble radius at constant temperature, Fer(T) = F ( R c r ( T ) ,
T),
(10)
where Rer ( T ) is radius of the critical bubble.
The critical action can be analytically approximated in the small and large relative supercooling regimes [3], and in the full range it has been calculated numerically [6,7]. In terms of the function ~(T), Scr(T)
-
Fer(T) _ 29/27~ a f(2) T 35 23/2 (1 _ ~ ) 2 "
(11)
A fit to the function f ( 2 ) with an -0.1 ... +1.6% error was presented in [7]. Here we give for f (2) a four-parameter fit with an accuracy of ±0.1%: f ( ~ ) = ~3/2( a0 + a12 + a22--2 + a3 ~3 + a4~ 4) , a0 = 15.63628,
I~=---
(8)
Ear:' a2
=2.39731,
al = -18.03398, a3 = - 0 . 8 6 5 0 4 ,
2x/2 a s ap -
8----ff-2 5/-----iT 3 .
(7)
a4 = 1.86543.
(12) 253
Volume 309, number 3,4
PHYSICS LETTERS B
15 July 1993
3 0.2
g-
,~-
80.1 2 .o= 0.13
I,"<
e-O. 1 1
-0.2
/
/
/
/
!
i
i
-0.3
0 0.0
0.2
0.4
0.6
0.8
1.0
8
12
R'
Fig. 1. Non-divergent part of the critical action. Solid curve is f ( ~ ) , as defined in eq. (11), and dashed curve is
[f (2)/23/2]/6 . This fit is exact in the small relative supercooling (2 = 1 ) limit. The function f (2) is shown in fig. 1, together with f (~)/~3/2. It is interesting to note that the latter one, which is relevant in the large relative supercooling (2 = 0) limit [6], is an almost straight line over the full range. Now that we know both Ap ( T ) and For(T) we can proceed in determining the general free energy from eq. (3). Next, we must have a definition which gives the bubble radius for critical bubbles, Rcr ( T ) . Employing it we then obtain the corresponding curvature-dependent surface tension a ( R ) . In the rest o f this Letter we mainly study these two quantities. One should bear in mind, however, that when discussing cosmological bubble formation only the free energy has real physical importance, the surface tension as well as the bubble radius are only auxiliary quantities. But those quantities must be known before one is able to utilize eq. (3) which gives the radial dependence o f the free energy. We employ two different definitions for the bubble radius. The first one is Laplace radius RL defined by using Laplace's relation
RI.~(T) - 2aL(RLer(T) )
(13)
Ap(T) The second one is the tension radius RT, defined as the distance at which the gradient density term o f the critical action reaches its maximum. The definition o f RT is illustrated in fig. 2. F r o m Laplace's relation, it follows that the radius o f the critical bubble and the generalized surface tension 254
4
Fig. 2. Definition of RT. Solid curve is the gradient part of the action density of a critical (2 = 0.9)-bubble and dashed curve is the potential part, the first and the latter term in eq. (5), respectively. The dimensionless radius is R' = MR, where M stands for the bosonic mass in the potential, M 2 = 7a (T 2 - T2). The free energy density is shown in units of TM3a/23/2 (note that since 2 is fixed, so is T/Tc). From the figure, we can read that for 2 = 0.9, R~cr ~ 7.3 . in terms o f the critical free energy are RLcr(T) =
[ 3 F ( R L c r ( T ) , T ) ] 1/3 2~ Ap(T) '
(14)
3 F aL(RLer(T)) = [ T-~ ( R L e r ( T ) ' T ) A p ( T ) 2 I 1/3
(15) In the m a x i m u m tension definition, the radius o f the critical bubble, RTer(T), is obtained from the numerical solution o f the extremum differential equation for the action in eq. (5) for different values o f ~. The curvature-dependent surface tension aT (RTcr ( T ) ) is then calculated directly from eq. (4). The two definitions for the bubble radius are compared in fig. 3. One notes that always Rrcr >t RLcr. In the small relative supercooling limit, the two definitions coincide. On the other hand, in the large relative supercooling limit they differ even qualitatively: RLcr vanishes, whereas RTcr goes to infinity. However, this limit is beyond the validity range o f the expression (2, 3) for the free energy, in which all the temperature dependence comes from the pressure difference. Both o f the generalized surface tensions, aL (R) and aT(R), are plotted versus the inverse radius I¢/R in fig. 4. F o r comparison, we may note that in the theory o f fluid surfaces, the dependence o f the surface ten-
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PHYSICS LETTERS B
15 July 1993
16 12
~z 0
8
0 0.0
//
-1
4
¢
0.2
0.4
0.6
0.8
-2
1.0
/ /.
1
0.4
0'.6
0'.8
1.0
To~To Fig. 3. Radius of the critical bubble for To/Tc = 0.58 as a function of the degree of relative supercooling. Dashed curve is Rtcr/lc, and solid curve RTcr/lc. sion on the definition o f the bubble radius is a subtle issue [ 10 ]. F o r large values o f the bubble radius the two surface tensions coincide. In this limit, depending on the value o f the constant To~To, the surface tensions can either decrease or increase with R, implying that the curvature coefficient can be positive or negative, respectively (see fig. 5). When the bubble radius gets smaller, the two surface tensions differ increasingly. The curves for ax (R) turn backwards, because with decreasing 2, R'r~ when expressed in physical units possesses a m i n i m u m , after which it begins to increase (see fig. 3). F o r small values o f the radius, aL (R) has to decrease with R, in order to compensate the faster decrease o f the term A p ( T ) R 3 in the free energy expansion in eq. (3). And vice versa,
i
.......i.......i........i........
Fig. 5. The coefficients 7 and $ in the free energy expansion in eq. (2), for different values of To~To. Dashed curve is 7/(aplc), coefficient of the curvature term, and solid curve is $/(apl2c ), coefficient o f the constant term. The width o f the curve for $ shows the uncertainty in eq. (19), due to numerical determination o f f " ( l ) (and to much less extent,
of f ' ( 1 ) ) . aa-(R) must increase with decreasing 2 in the large relative supercooling regime. The two-valuedness o f ax (R) does not imply that the distance o f m a x i m u m tension would be a b a d definition for the bubble radius. The radius R-r agrees well with the intuitive picture o f the bubble, and furthermore, as will be noted later, its m i n i m u m value can be used as an estimate for the break-down distance of the free energy expansion. Next, we expand a (R) as Taylor series in 1/R and thus determine the coefficients in the truncated free energy expansion in eq. (2): ap-
a(R)ll/R=0,
Y-
21 dd(a1(/RR))
l/R=0 '
1 1 dEa(R)
1
- 42! aTU -7 ll/R=0
(16)
""--.. ....... . ......
G
0.0
0.2
0.4
0.6
0.8
1.0
Ic/R Fig. 4. The generalized surface tension from both definitions, aL(R) and aT(R), plotted versus the inverse radius lc/R. Dashed curves show at (R)/ap, and solid curves aT (R)/ap. Upper curves are for To/Tc = 0.99, and lower curves for To~To = 0.58.
These coefficients were determined employing both definitions for the bubble radius. Within numerical accuracy, the results were equal. The reason for this is that both o f the radii, and both o f the generalized surface tensions as well, coincide in the limit of infinite bubble radius. Below, results for the coefficients are presented only for the case where the radius is defined by Laplace's relation. 255
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We write trL(RLcr(T)) and lc/RLer(T) as Laurent expansions o f 2 around 2 = 1, which corresponds to infinite radius. We find that the coefficient o f the curvature term is
y=
1.49+0.01 -
2T~]aflc,
where
)
If T:
92ya -- 2or 2 -- 2 k, T0z - 1
(17) 0 0.2
6 = ~4{f"(1)+ 2u[5f'(l)
2f'(1)- 1] +
0.6
018
1.0
.
(18)
Fig. 6. The distances at which magnitudes o f the curvature term and the constant term equal the surface term in the truncated free energy expansion o f eq. (2), plotted versus To~To. Dashed curve shows the distance Rr/l¢ at which 4napR 2 = 8nlylRr, and solid curve the distance R6/lc at which 4napR~ = 16n[~].
6
27uE}apl2
(0.88 + 0.03- ^ 90 T~ o.
0.4
To~To
In eq. (17), f ' ( 1 ) denotes d f ( 2 ) / d 2 at 2 = 1, and its numerical value is c o m p u t e d from eq. (12). F o r the coefficient o f the constant term in eq. (2) we find
T4)apg.
+ 8rg ]
(19)
The coefficients y and 8 are shown in fig. 5. If the temperature To is near To, the coefficient y is in units o f aplc close to unity, and 8 almost vanishes. If To/To is in the vicinity o f 0.58, the zero-point o f y, the constant term dominates over the curvature term in the free energy expansion even for rather large bubbles. When To~To decreases from ~ 0.4, y decreases and 8 increases rapidly. The magnitudes o f the different terms in the truncated free energy expansion are c o m p a r e d in fig. 6. F r o m the figure we can conclude that if To/To > 0.5, the truncated expansion for the free energy o f a spherical bubble in eq. (2) breaks down when the bubble radius is o f the order o f 2lc. If To/To is smaller, the truncated free energy expansion is valid only for much larger bubbles. The distance where the truncated free energy expansion breaks down is for different values o f To/Tc roughly equal to the smallest tension radius o f the critical bubbles. Finally, we will investigate if the low temperature phase bubbles can exist at temperatures higher than To. Here the full expression for the free energy, eq. (3), is used. Now the volume term in this equation is positive. Therefore the necessary condition for the existence o f stable low temperature phase bubbles is that 256
6
[~-~f'(1)-u]aflc
( U --
15 July 1993
the surface term (i.e. the latter term) not be a monotonically increasing function o f the radius R. The surface term is plotted in fig. 7 for two values o f To/Tc, and it behaves qualitatively in the same manner for all other values o f To~To. In spite o f the two-valuedness o f aT (R) it is clear that the behaviour o f the surface term 4 n a ( R ) R 2 shows no indication o f the existence o f low temperature bubbles above T¢. The result that the magnitude o f the generalized surface term grows with the bubble size is very natural (even though a(R) is a decreasing function o f
~'6 O
4
/
/"
2 //
o// 0
4
8
12
R/lc Fig. 7. Logarithm o f the surface part in the expression for the free energy in eq. (3), log[4na(R)R2/(apl~)], as a function o f the bubble radius R/l¢. In the same m a n n e r as in fig. 4, dashed curves represent 4naL(R)R 2, solid curves 4naT(R)R2; upper curves are for To~To = 0.99, lower for To~To = 0 . 5 8 .
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PHYSICS LETTERS B
the bubble radius for large bubbles if To~To > 0.58, the combination tr ( R ) R 2 is an increasing one). However, in the M I T bag model calculation by M a r d o r and Svetitsky [2 ] it was found that the free energy as a function o f the bubble radius has a m i n i m u m for T > T¢. (In the context o f strangelet masses, the curvature energy in the M I T bag model has been recently studied in [ 11 ].) Clearly the conclusion must be that either the spherical M I T bag model, or any order parameter model with a potential as the one in eq. (6), describes the q u a r k - h a d r o n phase transition qualitatively incorrectly. To summarize, we have discussed bubble free energy in cosmological first order phase transitions which can be described by the quartic potential. We made the a p p r o x i m a t i o n that all the temperature dependence in the free energy comes from the pressure difference between the two phases. This approximation enabled us to solve, starting from knowledge o f the critical bubbles only, the general free energy F(R, T). W e studied in detail the quantities which using eq. (3) give the radial dependence o f the free energy: the bubble radius and the curvature-dependent surface tension. This research was supported by the Academy o f Finland. The author wishes to thank K. Kajantie for the inspiration to this work as well as for useful comments, and T. Ala-Nissil~i, A. Laaksonen and M. Laine for discussions.
15 July 1993
References [1] K. Kajantie, J. Potvin and K. Rummukainen, Institute for Theoretical Physics preprint NSF-ITP-9255 (1992), to be published in Phys. Rev. D. [2] I. Mardor and B. Svetitsky, Phys. Rev. D 44 (1991) 878; G. Lana and B. Svetitsky, Phys. Lett. B 285 (1992) 251. [3] A. Linde, Nucl. Phys. B 216 (1983) 421. [4] L. McLerran, M. Shaposhnikov, N. Turok and M. Voloshin, Phys. Lett. B 256 (1991) 451. [5] N. Turok, Phys. Rev. Lett. 68 (1992) 1803; G.W. Anderson and L.J. Hall, Phys. Rev. D 45 (1992) 2685; B.-H. Liu, L. McLerran and N. Turok, Phys. Rev. D 46 (1992) 2668; M. Kamionkowski and K. Freese, Phys. Rev. Left. 69 (1992) 2743. [6]K. Enqvist, J. Ignatius, K. Kajantie and K. Rummukainen, Phys. Rev. D 45 (1992) 3415. [7] M. Dine, R.G. Leigh, P. Huet, A. Linde and D. Linde, Phys. Rev. D 46 (1992) 550. [8] L.P. Csernai and J.I. Kapusta, Phys. Rev. D 46 (1992) 1379. [9] K. Kajantie, Phys. Lett. B 285 (1992) 331. [10] See e.g., G. Navascurs, Rep. Prog. Phys. 42 (1979) 1131; or J.S. Rowlinson and B. Widom, Molecular Theory of Capillarity, Ch. 2.4 (Oxford University Press, Oxford 1982). [11] J. Madsen, Phys. Rev. Lett. 70 (1993) 391; Aarhus University preprint AARHUS-ASTRO-1993-3, to be published in Phys. Rev. D.
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