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Quark-hadron phase transition and thermodynamics of anisotropic cosmology
Volume 112A, number 8
PHYSICS LETTERS
11 November 1985
QUARK-HADRON PHASE TRANSITION AND T H E R M O D Y N A M I C S OF A N I S O T R O P I C C O S M O L O G Y E. G I A N N E T T O Dipartimento di Fisica, Corso Italia 57, Catania 95129, Italy
and P. SALUCCI SISSA, Strada Costiera 11, Trieste 34014, Italy Received 12 February 1985; revised manuscript received 11 June 1985; accepted for publication 8 September 1985
A first-order quark-hadron phase transition is studied in the most general homogeneous cosmologies, where also curvature anisotropies and velocity fields are present. We have found an entropy conservation law that governs expansion thermodynamics and holds in every model.
The relevance of studying phase transitions in the very early universe has recently been [ 1] pointed out also for non-isotropic models; here very strict connections arise between the physics of phase transitions and the cosmological expansion. In a previous paper [1 ] we have analyzed the case of the quark-hadron "interface" [2] (the transition could not be singular) in Bianchi type I cosmologies, where no matter velocity field or curvature anisotropies are present, It is also well known that a phase transition can affect dramatically the evolution of the universe, i.e. driving it towards an inflationary [3] or recollapsing [4] stage. Moreover, it has been shown that, in FriedmannRobertson-Walker (FRW) models [5] as well as in a Bianchi type I framework [ 1], a possible q u a r k hadron phase transition induces the scale factor to follow an apparent oscillatory regime, although it is a monotone function of time but far away from a "radiative behaviour". In refs. [1,6], it has been shown that, though a first order phase transition is producing entropy, the expansion of the universe can take place at constant comoving entropy, providingthat supercooling is negligible. Indeed, the transition heath takes just account there is an entropy transfer from the quark-gluon into lepton-radiation form, for the 374
number of spin degrees of freedom of strongly interacting matter gets a sharp decrease. Then, it is this conservation law which allows us to compute the ratio between the "initial" and "final" scale factors in the phase transition and to find the actual dynamical solution: entropy conservation drives expansion thermodynamics. To continue with the study of the quark-hadron interface in general homogeneous cosmologies, we will show in this letter that the isentropy of the universe expansion is a very widespread characteristic of models that admits not only expansion anisotropies but also curvature ones. Let us consider the Bianchi type metrics: ds 2 = - d r 2 + e2a(e2~)i/rir] ,
(1)
where e~ is a scalar describing the isotropic scale factor, ~3is a 3 X 3 symmetric matrix showing spatial anisotropies and r i are time-independent one-forms such that dr i =--2'-ke" 1_r i ~k ^ r e ,
(2)
where C~e are the structure constants of the homogeneity group of the spatial geometry. The most relevant cases are [7] Bianchi type V, VII, and IX metrics, that include as particular subsets open and closed 0.375-9601/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 112A, number 8
PHYSICS LETTERS
Friedman models, respectively. Now, we have:
cj~ = ~j~ ~,~ + 8'[j~],
11 November 1985
v.s~>0. (3)
(11)
We will show that V "S -- 0 in analogy to V. T = 0, or
type V:
a = 1,
nie = O,
0 = (SFRwUU).,u = (SFRwU0)" + SFRwUUPVuv.
type VII:
a = 0,
nie = diagl0,1,11,
type IX:
a = 0,
n/e = diagl 1,1,11,
where the anisotropy of the expansion is described by the two tensors
In this case also entropy conservation holds, dictated by a Gauss theorem, and in a very deep sense from the principle that the boundary of a boundary is zero [8]: it is a machinery of such a particular coupling between expansion (spacetime gravitation) and fluid velocity (matter) that guarantees the automatic conservation of entropy. Entropy apparently produced in a unit spacetime volume is not else than that flows out of this volume, because fluid flux lines are not orthonormal to homogeneity hypersurfaces. Indeed, from (I0) we have
,,,j =
TOu ;u = Tku ;u = 0,
(4)
and the connections read
POj = &Sij + aij , 1 -a
Pra'=~e
A
FijO = vii, 3
3 C C ~ c { ( e i ) A ( e r-O )B(e s)
+(e~)A (e;~y~(~)c_ (e~)A (eJ)C(e~)C},
7ii
[(4)"
(5)
+
= } [(e~i)" (e-t3)~ _ (eg/)" (e-t3)ik ] .
(6)
From the Einstein equation, with u the velocity field, we derive:
u i = (1/19u°) e - a [(e-#e0eO)AB CABc(eJ)C
ukTku Zu = O, with some algebraic manipulation (in a first order phase transition dp = d~" = 0) we get
rio = [6Uo - 3'(u2 - 1)]/Uo(O + P ) , (8)
where u is the four-velocity of the perfect fluid, and SFR W is the FRW entropy: SFRW = (19 + p ) e 3 ~ - - 1
(9)
with rt uv = diag[-1,1,1,1l. The motion of the fluid is governed by the curved space-time laws of thermodynamics plus the local law of e n e r g y - m o m e n t u m conservation,
"TVU ;u = O. The second law of thermodynamics tell us
(15) (16)
where - 7 = (O +P) [3~ + 4&Uk uk + OkeUk u e + ~eekUeuo ] , (17)
--8 = (O + P)[(ake + Uke)ukueuo + 4~UkUkUo + I~ekueumum
with p indicating pressure, 19 energy density, ~"temperature. The stress-energy tensor in the proper reference frame of a fluid element is identical to that in flat space-time (principle of equivalence), and it is given by (orthonormal frame)
TUV = (p + p) uUu v + p~UV ,
(14)
(7)
We define an entropy four-vector [8] as
S = SpRWU,
(13)
and substituting (9) in these equations and using
= [7(2u 2 - . 1 ) - 28uol/u20,
- oij(e-~)/CCAc].
(12)
(1 O)
+ r~kueukum ] •
(18)
Using (15) and (16) we obtain
b = (19 +p)(--rkekueu~l + OSceugue + &UkUk -- 36 -- 4&(UkUk ) 2 -- OkeUk ueuiui ) , ~l0 = [--OkeUkueuo + 4&(ukuk)2u 0 + Okeukueuiuiuo _ &uku k Uo + ~eekue(uiu i)2 ] .
(19)
Hence, recalling again dp = d~" = 0 for a first order phase transition, we can write
SFRW = e3a~'-I [3&(P +P) +/5] -~ v ' s = 0.
(20)
Thus, we have shown that, in the important subset of 375
Volume 112A, number 8
PHYSICS LETTERS
Bianchi types admitting asymptotically isotropic solutions, V, VII and IX (in the t y p e I such a conservation law is trivially secured), but allowing general anisotropic curvatures and velocity fields, the thermodynamics o f a first order phase transition is governed by an e n t r o p y conservation law. We want to thank Professor S. B o n o m e t t o for his e n c o u r a g e m e n t and help, and Dr. Dubal for useful discussions.
References [1] E. Giannetto and P. Salucci, Phys, Lett. 152B (1985) 171; G. Steigman and M. Turner, Ertrico Fermi preprint 83/24. [21 E. Giannetto and P. Salucci, Proc. General relativity, Italian Meeting (Florence, 1984).
376
11 November 1985
[3] A. Guth, Phys. Rev. D23 (1981) 347. [4] J. Lodenquai and V. Dixit, Phys. Lett. 124B (1983) 17. [5] S. Bonometto and O. Pantano, Astron. Astrophys. 130 (1984) 49; S. Bonometto and S. Matarrese, Phys. Lett. 133B (1983) 77; K.A. Olive, Nuck Phys. B190 (1981) 483; E. Witten, Phys. Rev. D30 (1984) 272. [6] S. Bonometto and M. Sakellariadou, Astrophys.,J. 282 (1984) 370; S. Bonometto and L. Sokolowski, Phys. Lett. 107A (1985) 210; S. Bonometto, P. Marchetti and S. Matarrese, DFPD preprint 8/85. [7] C. Collins and S. Hawking, Mon, Not. R. Astron. Soc. 162 (1973) 307; S. ttawking, Mon. Not. R. Astron. Soc. 142 (1969) 129. [8] C.W. Misner, K.S. Thorne and J.A. Wheeler, Gravitation (Freeman, San Francisco, 1973).
PHYSICS LETTERS
11 November 1985
QUARK-HADRON PHASE TRANSITION AND T H E R M O D Y N A M I C S OF A N I S O T R O P I C C O S M O L O G Y E. G I A N N E T T O Dipartimento di Fisica, Corso Italia 57, Catania 95129, Italy
and P. SALUCCI SISSA, Strada Costiera 11, Trieste 34014, Italy Received 12 February 1985; revised manuscript received 11 June 1985; accepted for publication 8 September 1985
A first-order quark-hadron phase transition is studied in the most general homogeneous cosmologies, where also curvature anisotropies and velocity fields are present. We have found an entropy conservation law that governs expansion thermodynamics and holds in every model.
The relevance of studying phase transitions in the very early universe has recently been [ 1] pointed out also for non-isotropic models; here very strict connections arise between the physics of phase transitions and the cosmological expansion. In a previous paper [1 ] we have analyzed the case of the quark-hadron "interface" [2] (the transition could not be singular) in Bianchi type I cosmologies, where no matter velocity field or curvature anisotropies are present, It is also well known that a phase transition can affect dramatically the evolution of the universe, i.e. driving it towards an inflationary [3] or recollapsing [4] stage. Moreover, it has been shown that, in FriedmannRobertson-Walker (FRW) models [5] as well as in a Bianchi type I framework [ 1], a possible q u a r k hadron phase transition induces the scale factor to follow an apparent oscillatory regime, although it is a monotone function of time but far away from a "radiative behaviour". In refs. [1,6], it has been shown that, though a first order phase transition is producing entropy, the expansion of the universe can take place at constant comoving entropy, providingthat supercooling is negligible. Indeed, the transition heath takes just account there is an entropy transfer from the quark-gluon into lepton-radiation form, for the 374
number of spin degrees of freedom of strongly interacting matter gets a sharp decrease. Then, it is this conservation law which allows us to compute the ratio between the "initial" and "final" scale factors in the phase transition and to find the actual dynamical solution: entropy conservation drives expansion thermodynamics. To continue with the study of the quark-hadron interface in general homogeneous cosmologies, we will show in this letter that the isentropy of the universe expansion is a very widespread characteristic of models that admits not only expansion anisotropies but also curvature ones. Let us consider the Bianchi type metrics: ds 2 = - d r 2 + e2a(e2~)i/rir] ,
(1)
where e~ is a scalar describing the isotropic scale factor, ~3is a 3 X 3 symmetric matrix showing spatial anisotropies and r i are time-independent one-forms such that dr i =--2'-ke" 1_r i ~k ^ r e ,
(2)
where C~e are the structure constants of the homogeneity group of the spatial geometry. The most relevant cases are [7] Bianchi type V, VII, and IX metrics, that include as particular subsets open and closed 0.375-9601/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 112A, number 8
PHYSICS LETTERS
Friedman models, respectively. Now, we have:
cj~ = ~j~ ~,~ + 8'[j~],
11 November 1985
v.s~>0. (3)
(11)
We will show that V "S -- 0 in analogy to V. T = 0, or
type V:
a = 1,
nie = O,
0 = (SFRwUU).,u = (SFRwU0)" + SFRwUUPVuv.
type VII:
a = 0,
nie = diagl0,1,11,
type IX:
a = 0,
n/e = diagl 1,1,11,
where the anisotropy of the expansion is described by the two tensors
In this case also entropy conservation holds, dictated by a Gauss theorem, and in a very deep sense from the principle that the boundary of a boundary is zero [8]: it is a machinery of such a particular coupling between expansion (spacetime gravitation) and fluid velocity (matter) that guarantees the automatic conservation of entropy. Entropy apparently produced in a unit spacetime volume is not else than that flows out of this volume, because fluid flux lines are not orthonormal to homogeneity hypersurfaces. Indeed, from (I0) we have
,,,j =
TOu ;u = Tku ;u = 0,
(4)
and the connections read
POj = &Sij + aij , 1 -a
Pra'=~e
A
FijO = vii, 3
3 C C ~ c { ( e i ) A ( e r-O )B(e s)
+(e~)A (e;~y~(~)c_ (e~)A (eJ)C(e~)C},
7ii
[(4)"
(5)
+
= } [(e~i)" (e-t3)~ _ (eg/)" (e-t3)ik ] .
(6)
From the Einstein equation, with u the velocity field, we derive:
u i = (1/19u°) e - a [(e-#e0eO)AB CABc(eJ)C
ukTku Zu = O, with some algebraic manipulation (in a first order phase transition dp = d~" = 0) we get
rio = [6Uo - 3'(u2 - 1)]/Uo(O + P ) , (8)
where u is the four-velocity of the perfect fluid, and SFR W is the FRW entropy: SFRW = (19 + p ) e 3 ~ - - 1
(9)
with rt uv = diag[-1,1,1,1l. The motion of the fluid is governed by the curved space-time laws of thermodynamics plus the local law of e n e r g y - m o m e n t u m conservation,
"TVU ;u = O. The second law of thermodynamics tell us
(15) (16)
where - 7 = (O +P) [3~ + 4&Uk uk + OkeUk u e + ~eekUeuo ] , (17)
--8 = (O + P)[(ake + Uke)ukueuo + 4~UkUkUo + I~ekueumum
with p indicating pressure, 19 energy density, ~"temperature. The stress-energy tensor in the proper reference frame of a fluid element is identical to that in flat space-time (principle of equivalence), and it is given by (orthonormal frame)
TUV = (p + p) uUu v + p~UV ,
(14)
(7)
We define an entropy four-vector [8] as
S = SpRWU,
(13)
and substituting (9) in these equations and using
= [7(2u 2 - . 1 ) - 28uol/u20,
- oij(e-~)/CCAc].
(12)
(1 O)
+ r~kueukum ] •
(18)
Using (15) and (16) we obtain
b = (19 +p)(--rkekueu~l + OSceugue + &UkUk -- 36 -- 4&(UkUk ) 2 -- OkeUk ueuiui ) , ~l0 = [--OkeUkueuo + 4&(ukuk)2u 0 + Okeukueuiuiuo _ &uku k Uo + ~eekue(uiu i)2 ] .
(19)
Hence, recalling again dp = d~" = 0 for a first order phase transition, we can write
SFRW = e3a~'-I [3&(P +P) +/5] -~ v ' s = 0.
(20)
Thus, we have shown that, in the important subset of 375
Volume 112A, number 8
PHYSICS LETTERS
Bianchi types admitting asymptotically isotropic solutions, V, VII and IX (in the t y p e I such a conservation law is trivially secured), but allowing general anisotropic curvatures and velocity fields, the thermodynamics o f a first order phase transition is governed by an e n t r o p y conservation law. We want to thank Professor S. B o n o m e t t o for his e n c o u r a g e m e n t and help, and Dr. Dubal for useful discussions.
References [1] E. Giannetto and P. Salucci, Phys, Lett. 152B (1985) 171; G. Steigman and M. Turner, Ertrico Fermi preprint 83/24. [21 E. Giannetto and P. Salucci, Proc. General relativity, Italian Meeting (Florence, 1984).
376
11 November 1985
[3] A. Guth, Phys. Rev. D23 (1981) 347. [4] J. Lodenquai and V. Dixit, Phys. Lett. 124B (1983) 17. [5] S. Bonometto and O. Pantano, Astron. Astrophys. 130 (1984) 49; S. Bonometto and S. Matarrese, Phys. Lett. 133B (1983) 77; K.A. Olive, Nuck Phys. B190 (1981) 483; E. Witten, Phys. Rev. D30 (1984) 272. [6] S. Bonometto and M. Sakellariadou, Astrophys.,J. 282 (1984) 370; S. Bonometto and L. Sokolowski, Phys. Lett. 107A (1985) 210; S. Bonometto, P. Marchetti and S. Matarrese, DFPD preprint 8/85. [7] C. Collins and S. Hawking, Mon, Not. R. Astron. Soc. 162 (1973) 307; S. ttawking, Mon. Not. R. Astron. Soc. 142 (1969) 129. [8] C.W. Misner, K.S. Thorne and J.A. Wheeler, Gravitation (Freeman, San Francisco, 1973).