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The dynamics of an inflationary Bianchi-IX model in the presence of a rotating fluid
Physics Letters B 287 (1992) 61-68 North-Holland
P H YSI C $ / EI T ER S B
The dynamics of an inflationary Bianchi-IX model in the presence of a rotating fluid P.M. Sfi ~ and A.B. H e n r i q u e s Centro de Fisica da MaMria Condensada, Av. Prof. Gama Pinto, 2, P-1699 Lisbon Codex, Portugal and Departamento de Fisica, Instituto Superior T~cnico, Av. Rovisco Pais, P-1096 Lisbon Codex, Portugal
Received 25 February 1992
The role played by a rotating fluid is investigated in the context of an inflationary non-diagonal Bianchi-IX model. We find that rotation has not, in this model, a significant influence in the inflationary process, while its influence on the premature recollapse (either helping to prevent it or favouring it) depends strongly on the initial values taken by the dynamical variables. We also analyse a chaotic solution near the singularity, in the presence of matter.
1. Introduction
The idea of an early inflationary phase in the evolution of the universe has been proposed to provide a solution to cosmological problems (the most quoted being the homogeneity and the isotropy problems) without having to assume special initial conditions [ 1 ]. Several versions of the inflationary scenario exist (for a review see ref. [2 ] ), but in most of them the implementation of the model is made within the framework provided by the Friedmann-Lema~tre-Robertson-Walker ( F L R W ) metric, that is within a model already homogeneous and isotropic. To solve the homogeneity and isotropy problems one needs to work in a general inhomogeneous model. Due to the obvious difficulties in dealing with non-homogeneous metrics, we shall confine ourselves to a homogeneous anisotropic space. In recent years, m a n y general results have been obtained and numerical calculations performed, using the different Bianchi and Kantowsky-Sachs metrics [ 3-10 ]. In particular, the influence of the shear on the inflationary behaviour of the universe has been explored. Although astronomical limits on the present rotation of the universe are rather severe [ I 1 ], this might have not been necessarily the case near the singularity. Some investigations on anisotropic cosmologies have been generalized to include rotation [ 12-18 ]. In ref. [ 17 ] it has been shown that, under general conditions, rotation in type-IX models leads to a speedup of the expansion relative to the FLRW model, the exception corresponding to a curvature dominated universe. The influence of inflation on the rotation problem has also been analysed [19]. In this paper we introduce a Bianchi-IX model, with, for simplicity, only one non-diagonal term. The matter content of the model is given by a rotating perfect fluid plus a scalar field, playing the role of the inflaton. A comparison is explicitly made between two Bianchi-IX models, one with and the other without rotation, enabling us to analyse the influence of this dynamical variable on the inflationary behaviour of the model. Approximate, asymptotic, solutions near the singularity, in the spirit of ref. [ 20 ], are worked out, showing that chaos is maintained in the presence of matter.
~ Supponedbyagrant fromGTAE (GrupoTe6ricodeAltas Energias), Lisbon. 0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
61
Volume 287, number 1,2,3
PHYSICS LETTERS B
6 August 1992
2. The model
We now introduce our model based on a homogeneous Bianchi-IX cosmology with a rotating fluid and a scalar field. The energy-momentum tensor of the fluid is assumed to be of the hydrodynamical type: Tu~ = ( P + P ) u l , u~, +Pgl,~,,
uUu u = - 1 ,
( 1)
where uu is the fluid four-velocity, p the energy density and p the pressure. A non-zero rotation of the fluid is achieved by imposing the condition that uj, is not orthogonal to the homogeneous three-surfaces, i.e. that at least one of the ui be different from zero. The homogeneous Bianchi-IX metric reads ds2= -dt2+googifo t ,
din'= ~Cjko9 l i j ^co k ,
(2)
where the &j form a symmetric 3 × 3 matrix and the C')k are the structure constants of the Bianchi-IX isometry group. Consistency between the geometry (2) and the rotating perfect fluid (1) requires some of the nondiagonal components o f g o to be different from zero. For simplicity we assume u u = ( u o ( t ) , O, O, u3(t) ) = (Uo(t), O, O, c ( t ) x / u 2 ( t )
- 1) ,
(3)
and /a2(t) gij=~de~t)
d2(t) ) b2(t)~ . 0 C2(t)
(4)
Inflation in this model is driven by a spatially homogeneous scalar field, 0 = 0 (t), with an energy-momentum tensor given by SF = Ou¢ O,,(b- ~gu,, O,~(b O'~¢-gu,, • ½m2(b 2 , T g,,,
(5)
where we have chosen the potential to be of the chaotic type. The evolution of the dynamical variables are determined by the law of local conservation of energy-momentum of the fluid, 2(u2h/2h+b/c) /5=p Uo2(2_2)_2+ 1 '
(6)
and rio = -
U2o(--~7~)~-+1
(2-1)+
-c ( 2 - 2 )
(7)
,
by the Klein-Gordon equation,
(8) and by the Einstein equations, a-a + a x ~ + c J - Y - a 7 +2y~¢
6"
b Xa+
-y
+2y
a
-
+x
2a2b2c e
+Y
a2 + b 2 _ ¢ 2
-k-.v
62
a2+b2_c2 a2c 2
b2c2
- ½~¢m2OZ+ ½1¢p(2-2) ,
-½xrnZO2+½Kp(2-2),
(9)
(10)
Volume 287, number 1,2,3
b)
7g+x~(a-+ cka b
[{
d2
+ ~a2 -d\
-2y
PHYSICS LETTERS B
CCI, c4-(a2-b2) 2
-~±x
d-(xgt+x~
-2a~b2c2
~)
6 August 1992
2 - y ~ =½xm202+xp[l+2(u2-3)],
+2x~-~-+x
(a 2"Jffb2) 2-c4q d2[ 2a2b2c= j= ,½xmZO2+½x,o(2-,a.)]
(ll) (12)
where we have taken the equation of state of the rotating fluid to be p= ().-1)p,
1~<2~<2,
(13)
and have introduced the notation h - a 2b 2-- d 4' x -~a 2b 2/ h and y - d* / h. The overdot denotes differentiation with respect to time and x= 8~zG. The remaining two Einstein equations,
-~
+ ~ + ~)-y~2
c + a ;t +
-~ -~ +
~ + ~
-x
4~b~c ~ - y 2c ~
- ½ , ~ - ½~m~-Kp[,~(ug - 1) + 1 ] - - 0 ,
da
(14)
-~l)-b~-b-d)l-2KpCUox~o-l=O
(15)
appear as constraints to the system ( 6 )- ( 12 ). In the remaining of this section we take 2 = ~, corresponding to radiation. Let us introduce a mean scale factor defined b y / ~ - ( d e t gij)1/6. The Klein-Gordon equation (8) can be rewritten in terms of/~ as
(b+3(R/R)O+m ~ = 0 , 3R/R- (/t/2h + k/c) being interpreted as the mean
(16)
expansion rate of the three-surfaces of homogeneity. The behaviour of/~/R determines the evolution of the scalar field and, consequently, the duration of the inflationary period and the growth of the scale factor. We must investigate how much the behaviour o f / ~ is changed by the presence of rotation. Due to the complexity of the system of coupled non-linear differential equations (6)-( 12 ), one cannot hope to solve them analytically• Two important tools in our analysis will be the Raychaudhuri and the Friedmann equations. Raychaudhuri's equation follows from the definition of the Riemann tensor in terms of the commutator of the covariant differentiation: O . U ' + ½02=
-RuvuUuV-a2+o)2+F,
(17)
where a2=
_~(
~2 b2 b2 t~2 ab~bb_2xa~_8xya~l bll ~) 2X2aS+ 2x2-~ + 2 7-i + 2y( 4y+ 3 )-ds -2x(1-2y)-d-b-ZX-b-c ac -~-8xy-~ + 4y
•
2.2 (2~ +~Uo-2Uofto
-2
~)
+(U2-1)
(
)
2 X (a2-b2)2 2a2b2c2 +y~+-~xpu 2
(18)
is the shear of the fluid, c2
~o2= (u~ -
1 ) 2h
(19)
is the rotation of the fluid, and 63
Volume 287, number 1,2,3
PHYSICS LETTERS B
6 August 1992
(20)
+
its expansion. The matter term is given by
RuuuUuV= -
(21)
I lcm2(~2 + U2K~2.-}-K.p ,
and
F=(u~-
\2hc+C)+uofio(~
(22)
"
accounts for the acceleration of the fluid. Integrating the Raychaudhuri equation (17) we obtain the Friedmann equation: , 2 =~xm , 2 2• 2 Uo+~X¢ , - 2 2Uo+U2Xp+½a2+½m2( 1 - 2 a 2 + b 2 ~ + ~0 --p-]
+x
UoUo
h 2-h
a4+b4 +c4-2a2b2-2b2c2-2c2a 2 1 4a2b2c2 +y ~ .
(23)
Eqs. (17) and (23) are not useful, as they stand, if we wish to analyse the influence of the rotating fluid on the inflationary behaviour of the model. Indeed, as noticed after eq. (16), the evolution of the scalar field is governed by the expansion rate of the three-surfaces of homogeneity, 3/~//~, while O - u~':~= - i~o- 3uol~/R is the expansion rate of the fluid (in the orthogonal case the two quantities coincide). We further rewrite the Friedmann and Raychaudhuri equations in terms of the expansion rate of the threesurfaces of homogeneity:
(_~)2.=~tqm2(~z.4_~K.~2_t.]x~+{62x2a2b2-¥2b2c2-k-2c2a2-a4-b4-c412a2b2c 2
~
R
+ Y 3c21
,
(24)
~xp-~a
_
(25)
where f i = p + -~ (Uoz - 1 )p is the energy density of the radiation fluid, as seen by an observer in the frame of the normals to the three-surfaces of homogeneity (frame where the fluid appears tilted ); #2 is the shear of the threesurfaces of homogeneity 2/" 2 a2
b2
72
d2
-3 k x aS +X2bS + Y +y(4y+3)-~ - x ( 1 - Z Y ) ~
fib
b?
a~
-X bc-X--
ad
bd+2y ~
(26)
It is the comparison between (24) and (25) with, respectively, the Friedmann and the Raychaudhuri equations written for a non-rotating Bianchi-IX model (d 2(t) = 0, u0 (t) = 1 ) [ 10 ] that makes it possible to study the influence of the rotation on the inflationary process• Based on the fact that fi>p and #2> 0, it has been shown [ 16] that the expansion on the FLRW model is smaller than on the non-diagonal Bianchi-IX model (if either the curvature term is negative, or negligible at the temperature marking the onset of inflation). When comparing the expansion of rotating and non-rotating Bian• -2 chi-IX models the situation is much more complicated, smce O'rotating case -- O".2. . . . . rating case (see eq. (26)) has not a well defined sign, with the consequences that will be analysed in detail in section 3.2. In this numerical section the curvature terms will, obviously, be taken into account.
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Volume 287, number 1,2,3
PHYSICSLETTERSB
6 August 1992
3. Numerical analysis
3. I. Asymptotic solutions near the singularity We now derive some asymptotic solutions which approximately describe the model near the initial singularity. These solutions have been worked out, and their properties analysed in ref. [ 20 ]. We limit ourselves to emphasize some details concerning the behaviour of the matter terms (the energy density and the four-velocity) during a Kasner epoch, which we think have not been noticed. In this section, the influence of the scalar field will not be taken into account; the fluid will be given by the energy-momentum tensor ( 1 ), with the equation of state (13), assuming 2 < 2. From ref. [20] we know that, when approaching the singularity, the evolution of the model is characterized by an alternation of Kasner epochs, during which we can neglect the curvature and matter terms in the (0, 0) and (i, j ) Einstein equations. The following solutions then apply:
a(t)=aot ~, b(t)=bot ~, c(t)=Cot ~, d(t)=dot ~.
(27)
Substituting (27) into the Einstein equations ( 9 ) - ( 11 ), (14) we find the well known Kasner relations ot+fl+y=ot2+flz+72= 1 ,
(28)
while the Einstein equation (12) gives
(6-a)(a-fl)=O.
(29)
Considering the matter equations (6), (7) and (27), and the approximate power-law solutions
p=pot r, uo=l +½wot',
(30)
we deduce that r=-2,
~=2(2-~,-1),
(31)
where we assume ~> 0 (Uo--, 1 near t= 0). Finally, with these expressions, the Einstein equation (15 ) and the requirement a2bZ-d4> 0 (which means that 4~> 2 ( a + f l ) when t - , 0 ) , we find
a=a, fl=~-z,
forthechoiceu3=+c(t)
ux/-~o-1,
fl=~, a = 6 - z ,
forthechoiceu3=-c(t)x~o-1,
(32) (33)
where z=-2poa~Cox/-W-o/ dL From eq. (32) we get expressions for a, fl and ~ , = ~ -~(1--~z 2) l/z,
(34)
where y~ ( - 1, 1 ); the roles of ot and fl interchange when we use eq. (33). The relations (32) and (33 ) do not correspond to an independent constraint on the exponents, as z depends on ao, do, Po and Wowhich vary from Kasner epoch to Kasner epoch, allowing the exponents to follow the replacement law of Belinskii et al. [20]. The condition e > 0 means that ~,<2-1 and y is restricted to the interval --~ < ? < 2 - 1. But we know, from ref. [20], that the alternation of the Kasner epochs, and the corresponding law of replacement of the exponents a, fl and ~, will eventually bring 7 to take a value in the interval 2 - 1 < ~,< 1. The apparent contradiction is solved if we notice that we can have an alternative solution for Uo. We look for a solution of the form
p=~ot ~, Uo=~ot ~,
(35)
where this time we take g < 0 (Uo~ + ~ , when t ~ 0 ) . We find 65
Volume 287, number 1,2,3
e=-2(ce+fl),
~=
2-7-1 2-----~'
PHYSICS LETTERS B
6 August 1992
(36)
allowing us to write p =~oa2b2a -2b -2 and uo = #oCo (ao bo ) - ,/2 (ab) '/2c - ' (for 2 = 4 ), in agreement with eq. (2.30) in ref. [20]. We have, for 1 ~<2<2, that y > 2 - 1. The matter terms may thus have different kinds of behaviour in different Kasner epochs, and we must consider both solutions (30), ( 31 ) and ( 35 ), ( 36 ). Rotation is given by eq. (19). Near the origin o~~ t a+y-2 for the case of the solution (30), (31), and oJ ~ t ~2~- 3) ~, - y)/ (2 -~), for the alternative solution ( 35 ), ( 36 ). Near the singularity, in a radiation dominated universe, 09 never goes to infinity faster than p; it is never dynamically dominant. However, this is not necessarily so when 2 < 4; but even in this case we know, from the work of Ryan [ 13 ], that the character of the singularity changes very little compared to that of known solutions with only shear and expansion. For fluids rigid enough to have 2 > 3 the rotation, o~, increases during the corresponding Kasner epochs, when we go away from the origin. Such type of behaviour has been noticed before by Barrow [21 ], based on simple dimensional considerations. We have checked numerically that this critical value of,{ remains unchanged, even when the scalar field is taken into account. The presence of a scalar field, on the other hand, and in the diagonal case, alters the character of the chaotic solution (relations (28) are changed), bringing the oscillatory behaviour to an end after a few bounces, as has been demonstrated in ref. [22 ]. The same behaviour also occurs when we take the limiting value 2 = 2 in the equation of state (13). However, if non-diagonal terms are included, the chaotic behaviour is re-established (ref. [23 ] and Belinskii and Khalatnikov [22] ). 3.2. Numerical solutions
The numerical computations were carried out using a step-by-step fourth-order Runge-Kutta method. The accuracy of the numerical solutions was checked with the help of the constraint equations (14) and (15). In our simulations the scalar field was given an initial value of the order ( 3 - 4)mp, which turns out to be enough to obtain sufficient inflation (zX/~~ 103°) and the value of its mass m was limited by the upper bound m < 10-6mp, which follows from the requirement that the energy density fluctuations, 8p/p, must be of the order of 10 - 4 10- 5 at the horizon crossing time [ 2 ]. We also take, in what immediately follows, 2 = -~, a radiation fluid. In order to study the influence of a rotating fluid on the dynamics of the inflationary period, we computed the evolution of the model for a large variety of initial conditions, corresponding to different initial values of the fluid's rotation. This analysis shows that the rotation can either speed up or slow down the expansion, but such changes are always small (less than 1% of the total e-folding), when compared with the values obtained in the absence of rotation. Rotation does not seem to lead to any new qualitative feature in the inflationary period. We made sure that we included cases where the energy associated with rotation was of the order of, or even larger than, the energy contained in the potential term ½m 2q~2responsible for the inflationary process. The rotation 092 decreases approximately a s / ~ - 2 ( t ) and, after a short period of inflation, it is practically washed out and its effects on the evolution of the model can be neglected. It is well known that models with positive spatial curvature can recollapse before inflation takes place (premature recollapse problem). The presence of a rotating radiation field may, in some circumstances, help to overcome this problem. Indeed, if the initial conditions are such that #2 is small enough (see eq. (25)), the recollapse can be sufficiently delayed to allow for the scalar field to take over and dominate the dynamics of the universe, giving rise to an inflationary period of expansion. But the opposite effect is also observed, in which case the rotation, being responsible for an increase of #2, makes the premature recollapse to occur faster. The behaviour of the model is thus strongly dependent on the initial values of the dynamical variables and their subsequent evolution. In particular, small variations on the initial values of Uo can prevent inflation and lead to a premature recollapse, or lead to an increased expansion. The results so far were obtained assuming 2 = ~. We now take in our numerical simulations a value 2 > ~, the 66
Volume 287, number 1,2,3
PHYSICS LETTERS B
6 August 1992
60O
~4 5 0 - .
~
300 150-
/
o~'~
X=1.5 2
3
, , , 410-7t/tp
Fig. 1. The graph represents the variation of 0)2 with time, for three different values of 2 (p= (2-1)p), as calculated numerically.
limiting case 2 = 2, corresponding to Zeldovich's stiff m a t t e r p=p. Increasing 2, rotation starts evolving slower t h a n / ~ - 2 (t) a n d for a value of,l = ~ it is a p p r o x i m a t e l y constant; for 2 larger than this critical value the rotation increases in t i m e (see fig. 1 ), in qualitative agreement with our considerations at the end o f the previous subsection, although the effects o f the presence o f the scalar field h a d not then been taken into account. This led us to expect that, in such cases, rotation might take energy away from the scalar, stopping inflation prematurely. However, as we checked numerically, rotation takes energy almost exclusively from the h y d r o d y n a m i c a l field, in the sense that the energy density p decreases m u c h faster than otherwise expected (in a diagonal model, p~R - 3;'(t) ). F o r the considered values o f 2 in the interval [-~, 2 ], we m a y conclude that the rotation has not, in our model, a significant influence on the evolution o f the inflationary expansion, while its influence on the recollapse problem is strongly d e p e n d e n t on the initial values o f the d y n a m i c a l variables, in particular on the value o f Uo. F o r 1 ~ 2 < ~ the influence o f rotation is identical to the cases just described.
Acknowledgement The authors would like to t h a n k J.M. Mour~o for useful discussions.
References [ 1] A.H. Guth, Phys. Rev. D 23 ( 1981 ) 347. [2] E. Kolb and M. Turner, The early universe, Lecture Note Series Frontiers in Physics, Vol. 69 (Addison-Wesley, Reading, MA, 1990). [ 3 ] R.M. Wald, Phys. Rev. D 28 ( 1983 ) 2118. [4] A. Zardecki, Phys. Rev. D 31 (1985) 718. [5] ~. Grin, Phys. Rev. D 32 (1985) 2522. [6] E. Martinez-Gonzalez and B.J.T. Jones, Phys. Lett. B 167 (1986) 37. [7] J.D. Barrow, Nucl. Phys. B 296 (1988) 697. [8] A.B. Burd and J.D. Barrow, Nucl. Phys. B 308 (1988) 929. [9] L.E. Mendes and A.B. Henriques, Phys. Lett. B 254 ( 1991 ) 44. [ 10] A.B. Henriques, J.M. Mourio and P.M. S~i,Phys. Len. B 256 ( 1991 ) 359. [ l 1] S.W. Hawking, Mon. Not. R. Astron. Soc. 142 (1969) 129; C.B. Collins and S.W. Hawking, Mon. Not. R. Astron. Soc. 162 (1973) 307. [ 12] R.A. Matzner, Ann. Phys. 65 ( 1971 ) 438. [ 13 ] M.P. Ryan Jr., Ann. Phys. 65 ( 1971 ) 506. 67
Volume 287, number 1,2,3
PHYSICS LETTERS B
[ 14] A.R. King and G.F.R. Ellis, Commun. Math. Phys. 31 (1973) 209. [15] T. Rothman and M.S. Madsen, Phys. Lett. B 159 (1985) 256. [ 16 ] T. Rothman and G.F.R. Ellis, Phys. Lett. B 180 ( 1986 ) 19. [ 17 ] R. Matzner, T. Rothman and G.F.R. Ellis, Phys. Rev. D 34 (1986) 2926. [ 18] A.K. Raychaudhuri and B. Modak, Class, Quantum Grav. 5 (1988) 225. [ 19] J. Ellis and K.A. Olive, Nature 303 (1983) 679. [20] V.A. Belinskii, I.M. Khalatnikov and E.M. Lifshitz, Adv. Phys. 31 (1982) 639. [21 ] J.D. Barrow, Mon. Not. R. Astron. Soc. 179 (1977) 47P. [ 22] V.A. Belinskii and I.M. Khalatnikov, Sov. Phys. JETP 36 ( 1973 ) 59; see also A.B. Henriques and R.G. Moorhouse, Phys. Lett. B 194 (1987) 353. [23] J.D. Barrow and F. Cotsakis, Phys. Lett. B 232 (1989) 172.
68
6 August 1992
P H YSI C $ / EI T ER S B
The dynamics of an inflationary Bianchi-IX model in the presence of a rotating fluid P.M. Sfi ~ and A.B. H e n r i q u e s Centro de Fisica da MaMria Condensada, Av. Prof. Gama Pinto, 2, P-1699 Lisbon Codex, Portugal and Departamento de Fisica, Instituto Superior T~cnico, Av. Rovisco Pais, P-1096 Lisbon Codex, Portugal
Received 25 February 1992
The role played by a rotating fluid is investigated in the context of an inflationary non-diagonal Bianchi-IX model. We find that rotation has not, in this model, a significant influence in the inflationary process, while its influence on the premature recollapse (either helping to prevent it or favouring it) depends strongly on the initial values taken by the dynamical variables. We also analyse a chaotic solution near the singularity, in the presence of matter.
1. Introduction
The idea of an early inflationary phase in the evolution of the universe has been proposed to provide a solution to cosmological problems (the most quoted being the homogeneity and the isotropy problems) without having to assume special initial conditions [ 1 ]. Several versions of the inflationary scenario exist (for a review see ref. [2 ] ), but in most of them the implementation of the model is made within the framework provided by the Friedmann-Lema~tre-Robertson-Walker ( F L R W ) metric, that is within a model already homogeneous and isotropic. To solve the homogeneity and isotropy problems one needs to work in a general inhomogeneous model. Due to the obvious difficulties in dealing with non-homogeneous metrics, we shall confine ourselves to a homogeneous anisotropic space. In recent years, m a n y general results have been obtained and numerical calculations performed, using the different Bianchi and Kantowsky-Sachs metrics [ 3-10 ]. In particular, the influence of the shear on the inflationary behaviour of the universe has been explored. Although astronomical limits on the present rotation of the universe are rather severe [ I 1 ], this might have not been necessarily the case near the singularity. Some investigations on anisotropic cosmologies have been generalized to include rotation [ 12-18 ]. In ref. [ 17 ] it has been shown that, under general conditions, rotation in type-IX models leads to a speedup of the expansion relative to the FLRW model, the exception corresponding to a curvature dominated universe. The influence of inflation on the rotation problem has also been analysed [19]. In this paper we introduce a Bianchi-IX model, with, for simplicity, only one non-diagonal term. The matter content of the model is given by a rotating perfect fluid plus a scalar field, playing the role of the inflaton. A comparison is explicitly made between two Bianchi-IX models, one with and the other without rotation, enabling us to analyse the influence of this dynamical variable on the inflationary behaviour of the model. Approximate, asymptotic, solutions near the singularity, in the spirit of ref. [ 20 ], are worked out, showing that chaos is maintained in the presence of matter.
~ Supponedbyagrant fromGTAE (GrupoTe6ricodeAltas Energias), Lisbon. 0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
61
Volume 287, number 1,2,3
PHYSICS LETTERS B
6 August 1992
2. The model
We now introduce our model based on a homogeneous Bianchi-IX cosmology with a rotating fluid and a scalar field. The energy-momentum tensor of the fluid is assumed to be of the hydrodynamical type: Tu~ = ( P + P ) u l , u~, +Pgl,~,,
uUu u = - 1 ,
( 1)
where uu is the fluid four-velocity, p the energy density and p the pressure. A non-zero rotation of the fluid is achieved by imposing the condition that uj, is not orthogonal to the homogeneous three-surfaces, i.e. that at least one of the ui be different from zero. The homogeneous Bianchi-IX metric reads ds2= -dt2+googifo t ,
din'= ~Cjko9 l i j ^co k ,
(2)
where the &j form a symmetric 3 × 3 matrix and the C')k are the structure constants of the Bianchi-IX isometry group. Consistency between the geometry (2) and the rotating perfect fluid (1) requires some of the nondiagonal components o f g o to be different from zero. For simplicity we assume u u = ( u o ( t ) , O, O, u3(t) ) = (Uo(t), O, O, c ( t ) x / u 2 ( t )
- 1) ,
(3)
and /a2(t) gij=~de~t)
d2(t) ) b2(t)~ . 0 C2(t)
(4)
Inflation in this model is driven by a spatially homogeneous scalar field, 0 = 0 (t), with an energy-momentum tensor given by SF = Ou¢ O,,(b- ~gu,, O,~(b O'~¢-gu,, • ½m2(b 2 , T g,,,
(5)
where we have chosen the potential to be of the chaotic type. The evolution of the dynamical variables are determined by the law of local conservation of energy-momentum of the fluid, 2(u2h/2h+b/c) /5=p Uo2(2_2)_2+ 1 '
(6)
and rio = -
U2o(--~7~)~-+1
(2-1)+
-c ( 2 - 2 )
(7)
,
by the Klein-Gordon equation,
(8) and by the Einstein equations, a-a + a x ~ + c J - Y - a 7 +2y~¢
6"
b Xa+
-y
+2y
a
-
+x
2a2b2c e
+Y
a2 + b 2 _ ¢ 2
-k-.v
62
a2+b2_c2 a2c 2
b2c2
- ½~¢m2OZ+ ½1¢p(2-2) ,
-½xrnZO2+½Kp(2-2),
(9)
(10)
Volume 287, number 1,2,3
b)
7g+x~(a-+ cka b
[{
d2
+ ~a2 -d\
-2y
PHYSICS LETTERS B
CCI, c4-(a2-b2) 2
-~±x
d-(xgt+x~
-2a~b2c2
~)
6 August 1992
2 - y ~ =½xm202+xp[l+2(u2-3)],
+2x~-~-+x
(a 2"Jffb2) 2-c4q d2[ 2a2b2c= j= ,½xmZO2+½x,o(2-,a.)]
(ll) (12)
where we have taken the equation of state of the rotating fluid to be p= ().-1)p,
1~<2~<2,
(13)
and have introduced the notation h - a 2b 2-- d 4' x -~a 2b 2/ h and y - d* / h. The overdot denotes differentiation with respect to time and x= 8~zG. The remaining two Einstein equations,
-~
+ ~ + ~)-y~2
c + a ;t +
-~ -~ +
~ + ~
-x
4~b~c ~ - y 2c ~
- ½ , ~ - ½~m~-Kp[,~(ug - 1) + 1 ] - - 0 ,
da
(14)
-~l)-b~-b-d)l-2KpCUox~o-l=O
(15)
appear as constraints to the system ( 6 )- ( 12 ). In the remaining of this section we take 2 = ~, corresponding to radiation. Let us introduce a mean scale factor defined b y / ~ - ( d e t gij)1/6. The Klein-Gordon equation (8) can be rewritten in terms of/~ as
(b+3(R/R)O+m ~ = 0 , 3R/R- (/t/2h + k/c) being interpreted as the mean
(16)
expansion rate of the three-surfaces of homogeneity. The behaviour of/~/R determines the evolution of the scalar field and, consequently, the duration of the inflationary period and the growth of the scale factor. We must investigate how much the behaviour o f / ~ is changed by the presence of rotation. Due to the complexity of the system of coupled non-linear differential equations (6)-( 12 ), one cannot hope to solve them analytically• Two important tools in our analysis will be the Raychaudhuri and the Friedmann equations. Raychaudhuri's equation follows from the definition of the Riemann tensor in terms of the commutator of the covariant differentiation: O . U ' + ½02=
-RuvuUuV-a2+o)2+F,
(17)
where a2=
_~(
~2 b2 b2 t~2 ab~bb_2xa~_8xya~l bll ~) 2X2aS+ 2x2-~ + 2 7-i + 2y( 4y+ 3 )-ds -2x(1-2y)-d-b-ZX-b-c ac -~-8xy-~ + 4y
•
2.2 (2~ +~Uo-2Uofto
-2
~)
+(U2-1)
(
)
2 X (a2-b2)2 2a2b2c2 +y~+-~xpu 2
(18)
is the shear of the fluid, c2
~o2= (u~ -
1 ) 2h
(19)
is the rotation of the fluid, and 63
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6 August 1992
(20)
+
its expansion. The matter term is given by
RuuuUuV= -
(21)
I lcm2(~2 + U2K~2.-}-K.p ,
and
F=(u~-
\2hc+C)+uofio(~
(22)
"
accounts for the acceleration of the fluid. Integrating the Raychaudhuri equation (17) we obtain the Friedmann equation: , 2 =~xm , 2 2• 2 Uo+~X¢ , - 2 2Uo+U2Xp+½a2+½m2( 1 - 2 a 2 + b 2 ~ + ~0 --p-]
+x
UoUo
h 2-h
a4+b4 +c4-2a2b2-2b2c2-2c2a 2 1 4a2b2c2 +y ~ .
(23)
Eqs. (17) and (23) are not useful, as they stand, if we wish to analyse the influence of the rotating fluid on the inflationary behaviour of the model. Indeed, as noticed after eq. (16), the evolution of the scalar field is governed by the expansion rate of the three-surfaces of homogeneity, 3/~//~, while O - u~':~= - i~o- 3uol~/R is the expansion rate of the fluid (in the orthogonal case the two quantities coincide). We further rewrite the Friedmann and Raychaudhuri equations in terms of the expansion rate of the threesurfaces of homogeneity:
(_~)2.=~tqm2(~z.4_~K.~2_t.]x~+{62x2a2b2-¥2b2c2-k-2c2a2-a4-b4-c412a2b2c 2
~
R
+ Y 3c21
,
(24)
~xp-~a
_
(25)
where f i = p + -~ (Uoz - 1 )p is the energy density of the radiation fluid, as seen by an observer in the frame of the normals to the three-surfaces of homogeneity (frame where the fluid appears tilted ); #2 is the shear of the threesurfaces of homogeneity 2/" 2 a2
b2
72
d2
-3 k x aS +X2bS + Y +y(4y+3)-~ - x ( 1 - Z Y ) ~
fib
b?
a~
-X bc-X--
ad
bd+2y ~
(26)
It is the comparison between (24) and (25) with, respectively, the Friedmann and the Raychaudhuri equations written for a non-rotating Bianchi-IX model (d 2(t) = 0, u0 (t) = 1 ) [ 10 ] that makes it possible to study the influence of the rotation on the inflationary process• Based on the fact that fi>p and #2> 0, it has been shown [ 16] that the expansion on the FLRW model is smaller than on the non-diagonal Bianchi-IX model (if either the curvature term is negative, or negligible at the temperature marking the onset of inflation). When comparing the expansion of rotating and non-rotating Bian• -2 chi-IX models the situation is much more complicated, smce O'rotating case -- O".2. . . . . rating case (see eq. (26)) has not a well defined sign, with the consequences that will be analysed in detail in section 3.2. In this numerical section the curvature terms will, obviously, be taken into account.
64
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3. Numerical analysis
3. I. Asymptotic solutions near the singularity We now derive some asymptotic solutions which approximately describe the model near the initial singularity. These solutions have been worked out, and their properties analysed in ref. [ 20 ]. We limit ourselves to emphasize some details concerning the behaviour of the matter terms (the energy density and the four-velocity) during a Kasner epoch, which we think have not been noticed. In this section, the influence of the scalar field will not be taken into account; the fluid will be given by the energy-momentum tensor ( 1 ), with the equation of state (13), assuming 2 < 2. From ref. [20] we know that, when approaching the singularity, the evolution of the model is characterized by an alternation of Kasner epochs, during which we can neglect the curvature and matter terms in the (0, 0) and (i, j ) Einstein equations. The following solutions then apply:
a(t)=aot ~, b(t)=bot ~, c(t)=Cot ~, d(t)=dot ~.
(27)
Substituting (27) into the Einstein equations ( 9 ) - ( 11 ), (14) we find the well known Kasner relations ot+fl+y=ot2+flz+72= 1 ,
(28)
while the Einstein equation (12) gives
(6-a)(a-fl)=O.
(29)
Considering the matter equations (6), (7) and (27), and the approximate power-law solutions
p=pot r, uo=l +½wot',
(30)
we deduce that r=-2,
~=2(2-~,-1),
(31)
where we assume ~> 0 (Uo--, 1 near t= 0). Finally, with these expressions, the Einstein equation (15 ) and the requirement a2bZ-d4> 0 (which means that 4~> 2 ( a + f l ) when t - , 0 ) , we find
a=a, fl=~-z,
forthechoiceu3=+c(t)
ux/-~o-1,
fl=~, a = 6 - z ,
forthechoiceu3=-c(t)x~o-1,
(32) (33)
where z=-2poa~Cox/-W-o/ dL From eq. (32) we get expressions for a, fl and ~ , = ~ -~(1--~z 2) l/z,
(34)
where y~ ( - 1, 1 ); the roles of ot and fl interchange when we use eq. (33). The relations (32) and (33 ) do not correspond to an independent constraint on the exponents, as z depends on ao, do, Po and Wowhich vary from Kasner epoch to Kasner epoch, allowing the exponents to follow the replacement law of Belinskii et al. [20]. The condition e > 0 means that ~,<2-1 and y is restricted to the interval --~ < ? < 2 - 1. But we know, from ref. [20], that the alternation of the Kasner epochs, and the corresponding law of replacement of the exponents a, fl and ~, will eventually bring 7 to take a value in the interval 2 - 1 < ~,< 1. The apparent contradiction is solved if we notice that we can have an alternative solution for Uo. We look for a solution of the form
p=~ot ~, Uo=~ot ~,
(35)
where this time we take g < 0 (Uo~ + ~ , when t ~ 0 ) . We find 65
Volume 287, number 1,2,3
e=-2(ce+fl),
~=
2-7-1 2-----~'
PHYSICS LETTERS B
6 August 1992
(36)
allowing us to write p =~oa2b2a -2b -2 and uo = #oCo (ao bo ) - ,/2 (ab) '/2c - ' (for 2 = 4 ), in agreement with eq. (2.30) in ref. [20]. We have, for 1 ~<2<2, that y > 2 - 1. The matter terms may thus have different kinds of behaviour in different Kasner epochs, and we must consider both solutions (30), ( 31 ) and ( 35 ), ( 36 ). Rotation is given by eq. (19). Near the origin o~~ t a+y-2 for the case of the solution (30), (31), and oJ ~ t ~2~- 3) ~, - y)/ (2 -~), for the alternative solution ( 35 ), ( 36 ). Near the singularity, in a radiation dominated universe, 09 never goes to infinity faster than p; it is never dynamically dominant. However, this is not necessarily so when 2 < 4; but even in this case we know, from the work of Ryan [ 13 ], that the character of the singularity changes very little compared to that of known solutions with only shear and expansion. For fluids rigid enough to have 2 > 3 the rotation, o~, increases during the corresponding Kasner epochs, when we go away from the origin. Such type of behaviour has been noticed before by Barrow [21 ], based on simple dimensional considerations. We have checked numerically that this critical value of,{ remains unchanged, even when the scalar field is taken into account. The presence of a scalar field, on the other hand, and in the diagonal case, alters the character of the chaotic solution (relations (28) are changed), bringing the oscillatory behaviour to an end after a few bounces, as has been demonstrated in ref. [22 ]. The same behaviour also occurs when we take the limiting value 2 = 2 in the equation of state (13). However, if non-diagonal terms are included, the chaotic behaviour is re-established (ref. [23 ] and Belinskii and Khalatnikov [22] ). 3.2. Numerical solutions
The numerical computations were carried out using a step-by-step fourth-order Runge-Kutta method. The accuracy of the numerical solutions was checked with the help of the constraint equations (14) and (15). In our simulations the scalar field was given an initial value of the order ( 3 - 4)mp, which turns out to be enough to obtain sufficient inflation (zX/~~ 103°) and the value of its mass m was limited by the upper bound m < 10-6mp, which follows from the requirement that the energy density fluctuations, 8p/p, must be of the order of 10 - 4 10- 5 at the horizon crossing time [ 2 ]. We also take, in what immediately follows, 2 = -~, a radiation fluid. In order to study the influence of a rotating fluid on the dynamics of the inflationary period, we computed the evolution of the model for a large variety of initial conditions, corresponding to different initial values of the fluid's rotation. This analysis shows that the rotation can either speed up or slow down the expansion, but such changes are always small (less than 1% of the total e-folding), when compared with the values obtained in the absence of rotation. Rotation does not seem to lead to any new qualitative feature in the inflationary period. We made sure that we included cases where the energy associated with rotation was of the order of, or even larger than, the energy contained in the potential term ½m 2q~2responsible for the inflationary process. The rotation 092 decreases approximately a s / ~ - 2 ( t ) and, after a short period of inflation, it is practically washed out and its effects on the evolution of the model can be neglected. It is well known that models with positive spatial curvature can recollapse before inflation takes place (premature recollapse problem). The presence of a rotating radiation field may, in some circumstances, help to overcome this problem. Indeed, if the initial conditions are such that #2 is small enough (see eq. (25)), the recollapse can be sufficiently delayed to allow for the scalar field to take over and dominate the dynamics of the universe, giving rise to an inflationary period of expansion. But the opposite effect is also observed, in which case the rotation, being responsible for an increase of #2, makes the premature recollapse to occur faster. The behaviour of the model is thus strongly dependent on the initial values of the dynamical variables and their subsequent evolution. In particular, small variations on the initial values of Uo can prevent inflation and lead to a premature recollapse, or lead to an increased expansion. The results so far were obtained assuming 2 = ~. We now take in our numerical simulations a value 2 > ~, the 66
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60O
~4 5 0 - .
~
300 150-
/
o~'~
X=1.5 2
3
, , , 410-7t/tp
Fig. 1. The graph represents the variation of 0)2 with time, for three different values of 2 (p= (2-1)p), as calculated numerically.
limiting case 2 = 2, corresponding to Zeldovich's stiff m a t t e r p=p. Increasing 2, rotation starts evolving slower t h a n / ~ - 2 (t) a n d for a value of,l = ~ it is a p p r o x i m a t e l y constant; for 2 larger than this critical value the rotation increases in t i m e (see fig. 1 ), in qualitative agreement with our considerations at the end o f the previous subsection, although the effects o f the presence o f the scalar field h a d not then been taken into account. This led us to expect that, in such cases, rotation might take energy away from the scalar, stopping inflation prematurely. However, as we checked numerically, rotation takes energy almost exclusively from the h y d r o d y n a m i c a l field, in the sense that the energy density p decreases m u c h faster than otherwise expected (in a diagonal model, p~R - 3;'(t) ). F o r the considered values o f 2 in the interval [-~, 2 ], we m a y conclude that the rotation has not, in our model, a significant influence on the evolution o f the inflationary expansion, while its influence on the recollapse problem is strongly d e p e n d e n t on the initial values o f the d y n a m i c a l variables, in particular on the value o f Uo. F o r 1 ~ 2 < ~ the influence o f rotation is identical to the cases just described.
Acknowledgement The authors would like to t h a n k J.M. Mour~o for useful discussions.
References [ 1] A.H. Guth, Phys. Rev. D 23 ( 1981 ) 347. [2] E. Kolb and M. Turner, The early universe, Lecture Note Series Frontiers in Physics, Vol. 69 (Addison-Wesley, Reading, MA, 1990). [ 3 ] R.M. Wald, Phys. Rev. D 28 ( 1983 ) 2118. [4] A. Zardecki, Phys. Rev. D 31 (1985) 718. [5] ~. Grin, Phys. Rev. D 32 (1985) 2522. [6] E. Martinez-Gonzalez and B.J.T. Jones, Phys. Lett. B 167 (1986) 37. [7] J.D. Barrow, Nucl. Phys. B 296 (1988) 697. [8] A.B. Burd and J.D. Barrow, Nucl. Phys. B 308 (1988) 929. [9] L.E. Mendes and A.B. Henriques, Phys. Lett. B 254 ( 1991 ) 44. [ 10] A.B. Henriques, J.M. Mourio and P.M. S~i,Phys. Len. B 256 ( 1991 ) 359. [ l 1] S.W. Hawking, Mon. Not. R. Astron. Soc. 142 (1969) 129; C.B. Collins and S.W. Hawking, Mon. Not. R. Astron. Soc. 162 (1973) 307. [ 12] R.A. Matzner, Ann. Phys. 65 ( 1971 ) 438. [ 13 ] M.P. Ryan Jr., Ann. Phys. 65 ( 1971 ) 506. 67
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[ 14] A.R. King and G.F.R. Ellis, Commun. Math. Phys. 31 (1973) 209. [15] T. Rothman and M.S. Madsen, Phys. Lett. B 159 (1985) 256. [ 16 ] T. Rothman and G.F.R. Ellis, Phys. Lett. B 180 ( 1986 ) 19. [ 17 ] R. Matzner, T. Rothman and G.F.R. Ellis, Phys. Rev. D 34 (1986) 2926. [ 18] A.K. Raychaudhuri and B. Modak, Class, Quantum Grav. 5 (1988) 225. [ 19] J. Ellis and K.A. Olive, Nature 303 (1983) 679. [20] V.A. Belinskii, I.M. Khalatnikov and E.M. Lifshitz, Adv. Phys. 31 (1982) 639. [21 ] J.D. Barrow, Mon. Not. R. Astron. Soc. 179 (1977) 47P. [ 22] V.A. Belinskii and I.M. Khalatnikov, Sov. Phys. JETP 36 ( 1973 ) 59; see also A.B. Henriques and R.G. Moorhouse, Phys. Lett. B 194 (1987) 353. [23] J.D. Barrow and F. Cotsakis, Phys. Lett. B 232 (1989) 172.
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