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Inflation in a Bianchi-IX cosmological model. The roles of primordial shear and gauge fields
Volume 256, number 3,4
PHYSICS LETTERS B
14 March 1991
Inflation in a Bianchi-IX cosmological model. The roles of primordial shear and gauge fields A.B. Henriques, J.M. M o u r ~ o a n d P . M . Sd lnstituto de Fisica e Matemdtica, Av. Prof. Gama Pinto 2, P-1699 Lisbon Codex, Portugal
Received 29 June 1990
The role played by primordial anisotropy and gauge field configurations is investigated in the context of a model with an inflationary stage driven by a self-interacting complex scalar field. We find that, in general, both the anisotropy and the gauge field aid inflation and we try to understand, qualitatively, the reasons for such behaviour. Asymptotic inflationary solutions and their stability are investigated.
I. Introduction Most o f m o d e r n cosmological models are based upon the F r i e d m a n n - R o b e r t s o n - W a l k e r ( F R W ) metric; this is not only due to the simplicity o f the resulting Einstein equations, but is also due to the fact that astronomical observations, in particular m e a s u r e m e n t s o f the cosmic microwave background radiation, seem to indicate an highly isotropic universe at the time o f decoupling. Assuming this isotropy to be verified at each point o f the universe, we m a y then conclude that it was also homogeneous at that time. The observations also tell us that the average density is close to the critical density, with 0.02 < g 2 - p / p c < 0.3, this being called the flatness problem. In fact, while inflationary models require that 12_~ 1, constraints from p r i m o r d i a l nucleosynthesis tell us that densities for which £2h 2>, 0.05 will have to be p r o v i d e d by some form of non-baryonic matter. As is well known, the idea o f an early inflationary phase was introduced in order to explain the observed state o f the universe (its isotropy, h o m o g e n e i t y and flatness) in a natural way, that is, without having to assume special initial conditions [ 1-6 ]. While in all the models the flatness p r o b l e m is u n d e r s t o o d and solved, the situation concerning the isotropy and h o m o g e n e i t y is far less clear [ 7 ]. Most o f the calculations have been m a d e in the f r a m e w o r k p r o v i d e d by the F R W metric and, indeed, what the F R W models can do is to solve the horizon problem, a necessary, but not sufficient condition for the solution of the isotropy and homogeneity problems. Due to the obvious difficulties in dealing with non-homogeneous metrics, we shall confine ourselves to the isotropy problem, assuming a homogeneous space. Even so, we can only claim to have a solution to this problem, if we work with anisotropic metrics and show that they can be inflated and isotropized under very general circumstances. In fact, no compelling reasons exist, apart from simplicity, to assume that the universe started in an isotropic c o n d i t i o n from its very beginning, or that it emerged from the Planck era already in an isotropic state. In recent years a series o f i m p o r t a n t general results have been obtained, and numerical calculations performed, using the different types o f Bianchi metrics [ 8-18 ]. In m a n y o f these works the inflationary stage was achieved with the help o f a cosmological constant or with a scalar field, the inflaton. Variations of these ideas have also been explored. F o r example, in ref. [ 19 ] the role o f the inflaton was taken by vector fields. In this p a p e r we investigate a m o d e l based on the Bianchi-IX metric, where m a t t e r is represented by a scalar Higgs field (playing the role o f the inflaton) and a gauge field. F o r purely illustrative and exploratory purposes we take the scalar field to be a charged complex singlet and the gauge group to be U ( 1 ); this allows us to fix uniquely their 0370-2693/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland )
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interaction via a minimal gauge coupling term and to investigate the interchange of energy taking place between the two fields. Though models with non-abelian gauge fields, coupled to multiplets of Higgs fields, are of course physically more interesting, we believe that the main features of their inflationary behaviour are already present in the abelian case. For a study of the dynamics of Friedmann-Robertson-Walker cosmologies in the presence of a gauge sector with a non-abelian gauge group see ref. [20]. We make here explicit use of the Maxwell (i.e. abelian Yang-Mills) equations for the U( 1 ) field and the Klein-Gordon equation for the scalar field, instead of the usual fluid apI=coximation. While our stress-energy tensor is that of a non-perfect fluid, such an approach allows us to derive the necessary equations of state from the lagrangian of the theory, rather than having to make an independent choice for them. We also assume the sell-interaction potential of the scalar field to be of the exponential form V= Vo e x p ( - R i l l ). Potentials of the exponential type may result from gravity theories in 4 + n dimensions undergoing dimensional reduction [18,21] or from supergravity models, like the SalamSezgin N = 2 supergravity model [22 ]. In fact, the precise form of the potential does not seem to be critical for the onset of an inflationary phase, as it has been discussed by Piran [23]. We study numerically the conditions under which inflation and isotropization are achieved, as well as the analytic form and stability of some of the asymptotic inflationary solutions. Particular attention is paid to the role of the primordial anisotropy and gauge field configurations. Although, after the inflationary regime, both the energy of anisotropy and the energy associated with the gauge fields become negligible, they are nevertheless important for the occurrence of inflation. We shall find cases where increasing the primordial anisotropy aids inflation and cases where it makes inflation more difficult, while in all cases studied we found that the gauge field helps inflation. We shall try to understand, at least qualitatively, the reasons for this behaviour. The present model is too simple to answer questions related to the post-inflationary era, like the reheating of the universe, baryogenesis and scalar density fluctuations (we hope to address such problems in a future work), but we give a simple example of how our model may naturally exit the inflationary era into an era with more moderate expansion. The paper is organized as follows. In section 2 we outline the essential features of the model and write down the corresponding system of coupled equations. The numerical methods used to solve the equations, the analysis of results and the study of the stability of the asymptotic inflationary solutions are described in section 3. The main conclusions of our work are summarized in the last section.
2. The coupled Einstein-scalar-vector field equations We investigate a cosmological model based on a homogeneous diagonal Bianchi-IX metric. ds2=
-dt2+gijo)ioY,
(2.1a)
where go is a time-dependent diagonal matrix,
g,j=diag(a2(t),
b2(t), c 2 ( t ) ) ,
(2.1b)
and the one-forms o) i satisfy the relations d~oi=
l('ijkO)J A (j)k,
(2.2)
with the relevant structure constants of the type IX models defined by C ~23 = C23~ = C3 ~2= 1. Here and in what follows latin indices run through the values 1, 2, 3 and greek ones through 0, 1, 2, 3; our units are such that 8 ~ G = c = 1. The stress-energy tensor will consist of threc parts, one describing the gauge field, another the complex scalar field and a third part representing the interaction between these fields. The lagrangian density for the U ( 1 ) gauge field is 360
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t e m = - l x ~ - g FuvF uu,
14 March 1991 (2.3)
where Fu, is the field tensor, F~,-A,,u-A~,,,+A,C ~, g = d e t g~,,, Cij~=C'jk and ('"~,~=0 if one or more indices are equal to zero. In a basis {X~} = {0o, ~}, dual to {o) ~} ={dt, 09'}, the vector potential has components {Ao, A~, A2, A3}. Given the homogeneous nature of our problem, these components are assumed to depend on time only, A~=Au(t). We further choose a gauge where Ao (t) = 0. In order to have a stress-energy tensor compatible with the diagonal metric, it can be seen that only one spatial component ofA u can be different from zero; we then take
A3=A3(t), A1 = A 2 = 0 .
(2.4)
The dynamics of the complex scalar field will be defined by the lagrangian density Lsc=- x/~
D,,0 D + " 0 * - ~ - - g V ( 0 ) ,
(2.5)
where the partial derivatives 0u were replaced by covariant derivatives D u = 0 , - i e A u , realizing the minimal gauge invariant coupling between the two fields, with e representing the coupling constant. The homogeneous ansatz for 0, corresponding to (2.1b) and (2.4) is 0=O(t).
(2.6)
The full matter lagrangian is thus L = Lem -t- Lsc. The stress-energy tensor, following from (2.3) and (2.5), is T ~ , = - [ 1 F ~ p F ~ + D , 0 D + ~ * + V(0) ] ~ u + F~,~F ~ + D~,0 D+~0* + D"0 D+ ~0".
(2.7)
Substituting (2.1), (2.4) and (2.6) in (2.7), we find that the components of the stress-energy tensor, in the {e) ~} frame, depend only on time and that there is just one off-diagonal component To3= T3o=ieA3(~O * 00" ), apparently leading to an inconsistency in the Einstein equations, once we have assumed a diagonal metric. The inconsistency is solved with the help of the Maxwell equation deduced from the matter lagrangian L:
F°"+F°"
+~+
-C~2a~Sb2 -ieg°"(~fb*-O~*)-2e2A3fbo*g3"=O,
(2.8)
where dot denotes d/dt. Taking c~=0, we obtain the constraint on the scalar field .00"-- 0~ * = 0 ,
(2.9)
which solves the inconsistency, as it implies that To3 = T3o = 0. The equations of motion of the fields 0 and ~* are
~'+~
+~+
+Oe2cT +
=0
(2.10)
and its complex conjugate for 0*From the constraint equation (2.9), we get
~(t)=OO1(t) ,
(2.11)
where ~ is real and 0 is a constant complex number which can be chosen such that 101 = 1 / , ~ . Eq. (2.10) then becomes an equation for ~ :
b'~+~' +b+
+0'e~+
d0--~--
(2.12)
We see that this equation differs from the usual Klein-Gordon equation by the appearance of a term representing the interaction between the scalar and vector fields. An analogous term also appears in the equation of 361
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motion of the vector field. The Maxwell equation for the only nonzero component of the vector potential is then
b
e2~)
=0
(2.13)
Finally we write the Einstein equations for the metric (2.1) and the stress-energy tensor (2.7):
+ - + abeb aT+
l(1 + 1 + 1) a4+b4+c4
ca
~ ~5
~5
75
a+a(b ~)aa-(b2-c2)
4aZbZc 2
2
a
[9"
a b +
+
b(~ ~) b4-(a2-c2) + ~ + + 2a2b2c 2
( + ? c c
+
+
A~
2a2b2c2
c4-(b2-a2)2 2a2b2c 2
2c 2 - e
~5c2~ 2
~C5C 2 +V(~,),
A] .;t 2 - 2a2b ~ + ~ + V(01), ----
2 -V(OI)=0,
(2.14)
+A~
- 2aZb 2
2
2aZb 2
A2 2a2b 2
ft2 + V ( q ~ l ) + e 2 c 2 v,, . 2c 2
(2.15a)
(2.15b) (2.15c)
Through the Bianchi identities only three of the Einstein equations are independent; hence we take eq. (2.14 ) as a constraint to the initial conditions. Once satisfied at an instant o f time t these same Bianchi identities ensure that it is satisfied at any other instant of time t.
3. Numerical analysis. Asymptotic solutions 3. I. The primordial anisotropy and gauge field configurations The system of eqs. (2.12)-(2.15 ) is rather complicated and was solved numerically, by a step-by-step method, for a variety of initial conditions. We used a fourth-order R u n g e - K u t t a method, using the constraint equation not only in the choice of the initial conditions, but also to check the numerical accuracy. Being interested in the role played by the primordial anisotropy, we obtain for the shear tensor
o"s =diag
- H , ~ - H , -c - H
)
,
(3. la)
where
H=g
+~+
(3.1b)
is the mean Hubble parameter; the shear is then
~r2= a , ; a ' J = ( ~ - H ) 2 + ( ~ - H ) 2 + ( ~ - H )
2.
(3.1c)
From (2.14), (2.15 ) and (3.1) we can write the Raychaudhuri equation = -
_~a2- _1( A 2 A2] -~2+~ 6 \ a 2 b 2 + cZJ
V(~)
(3.2)
where G is the mean scale factor G = (abc)1/3, V(O) = Vo e x p ( - 2 0 ) and Vo and 2 are positive constants (for convenience we redefined 2oid=2V/2); here and in the following we drop the index 1 in the scalar field. From
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the positive sign in front of V(O), we see why the potential is so important to achieve an inflationary regime, characterized by (~'> 0. The following condition must then be satisfied: 1(
A2
A 2~.q_ 1~2.
(3.3)
-~v(0)> ~ 2+ g \a-~b~ + c 21
If the condition (3.3) is not satisfied initially then two different cases can occur: (i) the universe recollapses after a finite time without undergoing any inflationary stage or (ii) the evolution of the fields brings the potential to dominate before recollapse occurs and the Universe begins to inflate. The latter case indeed happens for a large range of initial conditions, as we found numerically, and we may now try to understand it from the behaviour of the scalar and gauge fields. This typical behaviour is already present in the isotropic case studied in ref. [24]. Introducing the expressions for the energy density and the pressure of the scalar field, p0=½b2+g(g)),
and
po=½b2-V(O),
(3.4)
we can write the equation of motion of the scalar field as
d (;b~)_o~A2 ~tpo=-(po+po) + -b +
c2 ,
(3.5)
assuming a situaiton, quite usual in the present context, where 02 << V(~), we then have that dpJdt ~_0 neglecting also the last term due to the 1/c 2 factor. Such a situation does not occur for the gauge field and the anisotropy. This means that, in general, or at least for a large variety of initial conditions, the evolution of the scalar field potential is slower than the evolution of the terms on the RHS of (3.3); if V(O) is initially large, it will remain large for a time long when compared with the time of evolution of the gauge field and eventually, the condition expressed by the inequality (3.3) will be verified (see fig. 1 ). Hence, if we have a model where initially the energy of the matter fields is distributed among gauge and self-interacting scalar fields, we expect that the slower evolution of the scalar field can lead to an inflationary regime. Probably more surprising it is the fact that both the primordial anisotropy and gauge fields tend to enhance inflation instead of making it more difficult to occur, a fact that had been noticed before by Martinez-Gonzalez and Jones in the context of a Bianchi-V model [ 14 ]. The surprise comes from the fact that the shear ~2 and gauge field contribute with a negative sign to the RHS of the Raychaudhuri equation, thus making G more negative. The solution of the apparent paradox is in the Friedmann equation, which in our case is given by ~.0
0.8
,3 6
°2 t 0.0
l
O0
~ / 5 o
+ 1/6(
1.0
20
.
5.0
Fig. 1. It is shown how condition(3.3) is indeed verified after a short time. We took V0=3, 2~0.7, and fixed (ii/a)o,,~ 1.4, (b/b)o ~ 1.4, (/'/C)o,~ 0.7, (A3)o = 1, (]t3)o=2, q}o~0.7, ~0 ~ 1.4.
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31a4q-b4q-c4-(a2-b2)2-4a2b2c 2(b2-c2)2-
(c2-a2)2
14 March 1991
(3.6)
p being the total energy density due to the scalar and gauge fields and their interaction. We see that both p and cr2 contribute with a positive sign and thus a large initial anisotropy will give rise to a large initial value o f ( G / G) 2. But, as both a2 and the gauge field tend to decrease faster than the scalar field, eventually condition (3.3) can be satisfied and, when this happens, the larger the initial anisotropy (as well as the gauge field) the larger the velocity of expansion will be, as translated by ( G / G ) 2, making it easier for the potential V(0) to expand the system. Incidentally, we can also understand now why, without very special assumptions, a self-interacting vector field cannot by itself sufficiently inflate a system [ 19 ]. Its contributions for the equations of motion are proportional to A 2 for example, and we have that A 2 =gU,AuA, ' giving rise to combinations like A 2/c2 or similar ones; when the scales inflate these terms quickly become negligible and their influence disappears. It is, nevertheless, important to realize that the influence of the anisotropy is not always the same. We found, in particular, that in cases where a small change o f the initial conditions may bring a premature recollapse, an increase in the value of the primordial anisotropy may either lead to an inflationary situation or it may contribute to the quick collapse of the system. On the other hand, the increase of the initial value of the energy density associated with the gauge field always seems to help inflation. 3.2. The asymptotic solutions We verified numerically that, for initial conditions such that the system inflates, the scale factors a, b, c approach a behaviour proportional to t p, where p is given by 2/22, although the absolute values of the scales may be different; the system also isotropizes in the sense that the Hubble factors, corresponding to the three directions, become equal. The scalar field has then a logarithmic variation with t. This is not difficult to understand once we realize that, in these circumstances, the gauge field and the curvature terms in eq. (2.15) quickly approach zero and can be neglected. The following asymptotic solution (analogue to the solutions found for the isotropic case in refs. [ 18,24 ] ) then applies: a = a o t p,
b=bo tp,
C=Cot p and
O=kln(#t),
(3.7)
where ao, bo, and Co are usually different, p = 2/22, k = 2/2 and # = 22x/Vo/( 12 - 2)~2), requiring that 22 < 6, this behaviour can be seen in fig. 2. It is possible now to make more precise our statement that the anisotropy and the gauge field aid inflation. What we mean is that, although the solution asymptotically approaches the same power-law behaviour t p, with p fixed and given by 2/22, the actual scales, defined by ao, bo and Co increase when we increase the initial value of a 2 and the energy of the gauge field. For small values of 2, p can be large enough to solve the horizon and flatness problems [ 6 ]. For 22 > 2, the curvature terms in eq. (2.15 ) can be seen to decrease more slowly than the kinetic terms. Then, even if we reach a stage where such terms appear negligible, with the universe in an almost isotropic condition, this is only temporary; as the system evolves, those terms will again be important and the universe may go back to an anisotropic regime. O f course, for such large values of 2, our model is not even able to inflate. Unfortunately, the interacting terms, appearing in the equations of motion (2.12) and (2.13 ) of the scalar and gauge fields, do not seem to lead to a steady transfer of energy away from the scalar field into the gauge field, as we hoped; not only do they become very small, they give rise instead to a mutual interchange of energy between the two fields. We could try to remedy this situation by introducing a viscosity term in the equation of motion o f the scalar field; this has been used by several authors [24], and allows for large inflation, even with values of 22 > 2. From dimensional and field theoretic arguments, such terms are expected to have the general form Cv0d~ 5-2d, where d is a n u m b e r usually taken to be one, but it is not known how to calculate the viscosity coefficient Cv. We have not explored this possibility here. 364
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12o] (b)
(a)
25
14 March 1991
P asyrnptoiic 20
~
f f/
//
num~ric
1.5
50 @asymptotic
0
50
100
150
t
200
0.0
0 ....
sb ....
1;0 ....
1;0 ....
t
2~0
Fig. 2. (a) We see that the numerical exponent p quickly approaches the theoretical asymptotic value 2/22; (b) the same happens with the scalar field $ with respect to the asymptotic solution ¢ = k In (,ut).
We can easily check that the solutions we have just described are asymptotically stable. For this it is convenient to introduce the redefinitions a = ¢ ' , b = e a and c=¢", together with a new time variable r through d / dt = ~ d/dz. We then perturb the system with the help of the small quantities x, y, z and w and the replacements o~-.ce+x, fl--,fl+y, 7--+7+z and O--,O+w; we neglect the curvature and gauge field terms. Finally, after linearizing the equations in x, y, z and w and making the redefinitionsf=x', g=y', h=z' and rn=w', where primes denote derivatives with respect to r, we find f'=
+ ( ½ 2 0 ' - 4 a ' ) f -c~' g-o~' h + 12o~'m,
h'= -o~'f-o~'g+ (½2c~'-4o~' ) h + ½2o~'m,
g' = -o~' f + (½20'-4c~' )g-oC h + ½2o~'m ,
(3.8a,b)
m'= -O'f-O'g-O'h+ (20'-3c~' )m,
(3.8c,d)
we have that ½ 2 0 ' - 4 c e ' = ~ / ~ / V o ( 1 - 8 / 2 2 ) , ½2o~'=(1/2)x/lz2/Vo, o ~ ' = ( 2 / 2 2 ) ~v@~V0, 0 ' = ( 2 / 2 2 ) × x//z2/Vo and 20' - 3o~' = x//fl/Vo (2 - 6/22), where all the terms have a c o m m o n factor x//~2/Vo. After these replacements the system takes the autonomous form x' =Ax (notice that the solution whose stability is being investigated has o~' = fl' = 7' ). After some algebra, we find that the eigenvalues of the matrix A are the roots of the equation (4 + E3(27/,12-- 4) -- e213( 1 - - 8 / , 1 2 ) ( 1 - 3/22 ) + 3( 1 -- 8/2 2 ) 2+ 1 / 2 2 ( 3 _ 4 / 2 2 )
]
+ e [ - (1 - 8 / 2 2) 3_ 3( 1 - 8 / 2 2 ) 2 ( 1 - 3 / , l 2 ) - 6 / ) . 2 ( 1 - 8 / , 1 2 ) - 12/,16+4/,14(1 - 8 / , 1 2 ) - 4 / , 1 4 ] + [ ( 1 -- 8/2 2 ) 3( 1 -- 3/,12 ) + 3/,12( 1 -- 8/). 2 ) 2-31-12/26( 1 -- 8/,12+4/,16 ) -t-4/,14( 1 -- 8/,12 ) ] = 0 .
(3.9)
We solved this equation numerically for a large range of values of,1; we either have four real negative roots, or two real negative plus two complex conjugate roots with their real parts negative, indicating that we are in the presence o f asymptotically stable solutions. This is still valid for a Z as small as 1 0 - 6. It is interesting to notice that for a small 2 the system may evolve towards a different temporary attractor corresponding to an exponential solution, depending on our choice of initial conditions. This solution arises as follows. For values of 2-~ 0 and 6 2 << 1, we have that V(0) ~- Vo( 1 -,10) = Vo; then, and again neglecting asymptotically the curvature terms and the gauge field, the exponential solution applies, with
a=aoexp(pt),
b=boexp(pt),
C=Coexp(pt)
and
O~-&+q,
(3.10) 365
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where 6-~ 2p, p _ ( ] Vo)l/2 and q is a constant; thus, for very small 2 6 is small and ~ changes slowly. Except for 2 identically zero, the exponential solution will act only as a temporary attractor. From V~ V o ( 1 - 2 ~ ) we expect the system to inflate for a period of time of the order of ( 6 2 ) - 1 ~ 2 - 2 ( ~ Vo)-l/Z, till V(O)~-O, giving Gi/ G f - exp ( 1/2 2 ); after this period of time the system will evolve towards the attractor corresponding to the powerlaw solution. To have a realistic model, the universe must naturally evolve towards a radiation dominated phase. As an illustration of how this can happen, we consider a simple potential with a local minimum. For simplicity, we take the following combination of exponential potentials: V(¢) = Vo(exp (20) + e x p ( - 2 ¢ ) - 2 ) ,
(3.11)
V ( 0 = 0 ) = 0 and near this minimum V(¢) - Vo2202. Assuming that the initial model has been isotropized and flattened, after a more or less long period of inflation, the equations are
3(G/G)~=½q)2+ V(¢), ~'+ 3((7,/G)~+~V(O)~-~)'+3((7,/G)~+ZVo220=O.
(3.12a,b)
If we look for an approximate solution, taking only into account the dominant terms for large t, we have that ¢-~0ot"COSO)t
(a=-l)
G~-Gotp,
and
(3.13)
o)_~2 ( 2 Vo) 1/2 and p-~ 2 ( ~ Vo) 1/2. In the inflationary phase this form of the potential does not alter our previous conclusions and p ~- 2/22. Hence, for a small value of 2 we have a large p during the inflationary regime, evolving then to a small p ~ 2 x / -~Vo, when ¢ reaches the minimum of the potential and inflation comes to an end. Such behaviour can be verified numerically (see fig. 3). In our model the energy lost by the scalar field goes mainly to the expansion of the Universe; in a more realistic model, with the right coupling between the scalar field and other fields taken in account, this energy would, to a large extent, be used for reheating and particle creation [ 14,24]. As we said in the introduction, we leave the study of these problems to a future work.
(a)
300
10
Iog~oG
(b)
O0
20.0 !0
-20 100 30
0.0
o
2b
40
60
80
L
I00
--4"020
!O
60
80
t
I00
Fig. 3. (a) Plot of the mean scale factor, clearly showing G = (abc) U3 evolving away from the inflationary phase; (b) the damped oscillations of the scalar field around the m i n i m u m of the potential (3.11 ) are here clearly seen.
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4. Conclusions In this work we investigated the role played by the anisotropy and gauge field configurations in the very early Universe, in the context of an inflationary scenario driven by a self-interacting charged complex scalar. We used directly the Yang-Mills and the K l e i n - G o r d o n equations to describe the evolution of the gauge and of the scalar fields, rather than going to a fluid description. The m i n i m a l gauge coupling allowed the interaction between the gauge and the scalar fields to be fixed in a u n i q u e way. For most of the initial conditions, we found that an increase in the initial value of the anisotropy aided inflation, the exceptions referring to those cases where the system is near a situation of recollapse; then an increase of the anisotropy may, sometimes, lead to a still faster recollapse. On the contrary, in all the situations studied, an increase in the energy associated with the gauge field always favoured inflation. The different roles played by the anisotropy, and the scalar and gauge fields can, in part, be qualitatively understood with the help of the Raychaudhuri and F r i e d m a n n equations and the equations of m o t i o n of the two fields. We also studied the asymptotic behaviour of the inflationary solutions. We were able to identify an attractor corresponding to a power-law solution, a n d another one to an exponential solution for 2 = 0 . This last one, Ibr 2 # 0 and small, acted only as a temporary attractor. In both cases the system quickly isotropized and the gauge fields became negligible. A small value of the parameter, appearing in the exponential potential, is required to solve the horizon and flatness problems, a limitntion considered to be a negative feature of these models. The interacting terms between the scalar and gauge fields did not lead to a steady transfer of energy away from the scalar field, but rather to a mutual transfer of energy between the two fields; they were not then able to perform the role of a viscosity, transferring energy away from the scalar field and allowing enough inflation to take place, even for larger values of 2.
Acknowledgement The authors would like to thank their colleagues O. Bertolami and A. Liddlc for useful discussions and suggestions.
References [ l ] A.H. Guth, Phys. Rev. D 23 ( 1981 ) 347. [2] A.D. Linde, Phys. Lett. B 108 (1982) 389. [3] A. Albrecht and P.J. Steinhardt, Phys. Rev. Lett. 48 (1982) 1220. [4] A.D. Linde, Phys. Lett. B 129 (1983) 177; Mod. Phys. Lett. A 1 (1986) 81. [5] L.F. Abbott and M.B. Wise, Nucl. Phys. B 244 (1984) 541. [6] S. Matarrese and P. Lucchin, Phys. Rev. D 32 (1985) 1316. [ 7 ] T. Rothman and G.F.R. Ellis, The metaphysics of inflation, Institute of Theoretical Physics and Astrophysics (University of Cape Town ) preprint 86/10. [8] R.M. Wald, Phys. Rev. D 28 ( 1983 ) 2118; X.-F. Lin and R.M. Wald, Phys. Rev. D 40 ( 1989 ) 3280. [ 9 ] J.D. Barrow and M.S. Turner, Nature 292 ( 1981 ) 35. [ 10] G. Steigman and M.S. Turner, Phys. Lett. B 128 (1983) 295. [ 11 ] A. Zardecki, Phys. Rev. D 28 (1983) 1235; D 31 (1985) 718. [ 12] M. Demianski, Nature 307 (1984) 140. [ 13 ] T. Rothman and M.S. Madsen, Phys. Lett. B 159 ( 1985 ) 256. [ 14] E. Martinez-Gonzalez and B.J.T. Jones, Phys. Lett. B 167 (1986) 37. [ 15] L.G. Jensen and J.A. Stein-Schabes,Phys. Rev. D 34 ( 1986 ) 931. [ 16] T. Rothman and G.F.R. Ellis, Phys. Lett. B 180 (1986) 19. [ 17] J.D. Barrow, Nucl. Phys. B 296 (1988) 697. 367
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[ 18 ] A.B. Burd and J.D. Barrow, Nucl. Phys. B 308 ( 1988 ) 929. [ 19] L.H. Ford, Phys. Rev. D 40 (1989) 967. [20 ] P.V. Moniz and J.M. Mourn.o, Einstein-Ynag-Mills-Higgs homogeneous and isotropic cosmologies, Lisbon preprint IFM-11/90. [21 ] J.J. Halliwell, Phys. Lett. B 185 (1987) 341; Yu.A. Kubyshin, V.A. Rubakov and I.I. Tkachev, Intern. J. Mod. Phys. A 4 (1989) 1409. [22] A. Salam and E. Sezgin, Phys. Lett. B 147 (1984) 47. [23] T. Piran, Phys. Lett. B 181 (1986) 238. [24] A. Albrecht, P.J. Steinhardt, M.S. Turner and F. Wilczek, Phys. Rev. Lett. 48 (1982) 1437; J.J. Halliwell, Nucl. Phys. B 286 (1987) 729; J. Yokoyama and K. Maeda, Phys. Lett. B 207 (1988) 31; A.R. Liddle, Phys. Lett. B 220 (1989) 502.
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Inflation in a Bianchi-IX cosmological model. The roles of primordial shear and gauge fields A.B. Henriques, J.M. M o u r ~ o a n d P . M . Sd lnstituto de Fisica e Matemdtica, Av. Prof. Gama Pinto 2, P-1699 Lisbon Codex, Portugal
Received 29 June 1990
The role played by primordial anisotropy and gauge field configurations is investigated in the context of a model with an inflationary stage driven by a self-interacting complex scalar field. We find that, in general, both the anisotropy and the gauge field aid inflation and we try to understand, qualitatively, the reasons for such behaviour. Asymptotic inflationary solutions and their stability are investigated.
I. Introduction Most o f m o d e r n cosmological models are based upon the F r i e d m a n n - R o b e r t s o n - W a l k e r ( F R W ) metric; this is not only due to the simplicity o f the resulting Einstein equations, but is also due to the fact that astronomical observations, in particular m e a s u r e m e n t s o f the cosmic microwave background radiation, seem to indicate an highly isotropic universe at the time o f decoupling. Assuming this isotropy to be verified at each point o f the universe, we m a y then conclude that it was also homogeneous at that time. The observations also tell us that the average density is close to the critical density, with 0.02 < g 2 - p / p c < 0.3, this being called the flatness problem. In fact, while inflationary models require that 12_~ 1, constraints from p r i m o r d i a l nucleosynthesis tell us that densities for which £2h 2>, 0.05 will have to be p r o v i d e d by some form of non-baryonic matter. As is well known, the idea o f an early inflationary phase was introduced in order to explain the observed state o f the universe (its isotropy, h o m o g e n e i t y and flatness) in a natural way, that is, without having to assume special initial conditions [ 1-6 ]. While in all the models the flatness p r o b l e m is u n d e r s t o o d and solved, the situation concerning the isotropy and h o m o g e n e i t y is far less clear [ 7 ]. Most o f the calculations have been m a d e in the f r a m e w o r k p r o v i d e d by the F R W metric and, indeed, what the F R W models can do is to solve the horizon problem, a necessary, but not sufficient condition for the solution of the isotropy and homogeneity problems. Due to the obvious difficulties in dealing with non-homogeneous metrics, we shall confine ourselves to the isotropy problem, assuming a homogeneous space. Even so, we can only claim to have a solution to this problem, if we work with anisotropic metrics and show that they can be inflated and isotropized under very general circumstances. In fact, no compelling reasons exist, apart from simplicity, to assume that the universe started in an isotropic c o n d i t i o n from its very beginning, or that it emerged from the Planck era already in an isotropic state. In recent years a series o f i m p o r t a n t general results have been obtained, and numerical calculations performed, using the different types o f Bianchi metrics [ 8-18 ]. In m a n y o f these works the inflationary stage was achieved with the help o f a cosmological constant or with a scalar field, the inflaton. Variations of these ideas have also been explored. F o r example, in ref. [ 19 ] the role o f the inflaton was taken by vector fields. In this p a p e r we investigate a m o d e l based on the Bianchi-IX metric, where m a t t e r is represented by a scalar Higgs field (playing the role o f the inflaton) and a gauge field. F o r purely illustrative and exploratory purposes we take the scalar field to be a charged complex singlet and the gauge group to be U ( 1 ); this allows us to fix uniquely their 0370-2693/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland )
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interaction via a minimal gauge coupling term and to investigate the interchange of energy taking place between the two fields. Though models with non-abelian gauge fields, coupled to multiplets of Higgs fields, are of course physically more interesting, we believe that the main features of their inflationary behaviour are already present in the abelian case. For a study of the dynamics of Friedmann-Robertson-Walker cosmologies in the presence of a gauge sector with a non-abelian gauge group see ref. [20]. We make here explicit use of the Maxwell (i.e. abelian Yang-Mills) equations for the U( 1 ) field and the Klein-Gordon equation for the scalar field, instead of the usual fluid apI=coximation. While our stress-energy tensor is that of a non-perfect fluid, such an approach allows us to derive the necessary equations of state from the lagrangian of the theory, rather than having to make an independent choice for them. We also assume the sell-interaction potential of the scalar field to be of the exponential form V= Vo e x p ( - R i l l ). Potentials of the exponential type may result from gravity theories in 4 + n dimensions undergoing dimensional reduction [18,21] or from supergravity models, like the SalamSezgin N = 2 supergravity model [22 ]. In fact, the precise form of the potential does not seem to be critical for the onset of an inflationary phase, as it has been discussed by Piran [23]. We study numerically the conditions under which inflation and isotropization are achieved, as well as the analytic form and stability of some of the asymptotic inflationary solutions. Particular attention is paid to the role of the primordial anisotropy and gauge field configurations. Although, after the inflationary regime, both the energy of anisotropy and the energy associated with the gauge fields become negligible, they are nevertheless important for the occurrence of inflation. We shall find cases where increasing the primordial anisotropy aids inflation and cases where it makes inflation more difficult, while in all cases studied we found that the gauge field helps inflation. We shall try to understand, at least qualitatively, the reasons for this behaviour. The present model is too simple to answer questions related to the post-inflationary era, like the reheating of the universe, baryogenesis and scalar density fluctuations (we hope to address such problems in a future work), but we give a simple example of how our model may naturally exit the inflationary era into an era with more moderate expansion. The paper is organized as follows. In section 2 we outline the essential features of the model and write down the corresponding system of coupled equations. The numerical methods used to solve the equations, the analysis of results and the study of the stability of the asymptotic inflationary solutions are described in section 3. The main conclusions of our work are summarized in the last section.
2. The coupled Einstein-scalar-vector field equations We investigate a cosmological model based on a homogeneous diagonal Bianchi-IX metric. ds2=
-dt2+gijo)ioY,
(2.1a)
where go is a time-dependent diagonal matrix,
g,j=diag(a2(t),
b2(t), c 2 ( t ) ) ,
(2.1b)
and the one-forms o) i satisfy the relations d~oi=
l('ijkO)J A (j)k,
(2.2)
with the relevant structure constants of the type IX models defined by C ~23 = C23~ = C3 ~2= 1. Here and in what follows latin indices run through the values 1, 2, 3 and greek ones through 0, 1, 2, 3; our units are such that 8 ~ G = c = 1. The stress-energy tensor will consist of threc parts, one describing the gauge field, another the complex scalar field and a third part representing the interaction between these fields. The lagrangian density for the U ( 1 ) gauge field is 360
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t e m = - l x ~ - g FuvF uu,
14 March 1991 (2.3)
where Fu, is the field tensor, F~,-A,,u-A~,,,+A,C ~, g = d e t g~,,, Cij~=C'jk and ('"~,~=0 if one or more indices are equal to zero. In a basis {X~} = {0o, ~}, dual to {o) ~} ={dt, 09'}, the vector potential has components {Ao, A~, A2, A3}. Given the homogeneous nature of our problem, these components are assumed to depend on time only, A~=Au(t). We further choose a gauge where Ao (t) = 0. In order to have a stress-energy tensor compatible with the diagonal metric, it can be seen that only one spatial component ofA u can be different from zero; we then take
A3=A3(t), A1 = A 2 = 0 .
(2.4)
The dynamics of the complex scalar field will be defined by the lagrangian density Lsc=- x/~
D,,0 D + " 0 * - ~ - - g V ( 0 ) ,
(2.5)
where the partial derivatives 0u were replaced by covariant derivatives D u = 0 , - i e A u , realizing the minimal gauge invariant coupling between the two fields, with e representing the coupling constant. The homogeneous ansatz for 0, corresponding to (2.1b) and (2.4) is 0=O(t).
(2.6)
The full matter lagrangian is thus L = Lem -t- Lsc. The stress-energy tensor, following from (2.3) and (2.5), is T ~ , = - [ 1 F ~ p F ~ + D , 0 D + ~ * + V(0) ] ~ u + F~,~F ~ + D~,0 D+~0* + D"0 D+ ~0".
(2.7)
Substituting (2.1), (2.4) and (2.6) in (2.7), we find that the components of the stress-energy tensor, in the {e) ~} frame, depend only on time and that there is just one off-diagonal component To3= T3o=ieA3(~O * 00" ), apparently leading to an inconsistency in the Einstein equations, once we have assumed a diagonal metric. The inconsistency is solved with the help of the Maxwell equation deduced from the matter lagrangian L:
F°"+F°"
+~+
-C~2a~Sb2 -ieg°"(~fb*-O~*)-2e2A3fbo*g3"=O,
(2.8)
where dot denotes d/dt. Taking c~=0, we obtain the constraint on the scalar field .00"-- 0~ * = 0 ,
(2.9)
which solves the inconsistency, as it implies that To3 = T3o = 0. The equations of motion of the fields 0 and ~* are
~'+~
+~+
+Oe2cT +
=0
(2.10)
and its complex conjugate for 0*From the constraint equation (2.9), we get
~(t)=OO1(t) ,
(2.11)
where ~ is real and 0 is a constant complex number which can be chosen such that 101 = 1 / , ~ . Eq. (2.10) then becomes an equation for ~ :
b'~+~' +b+
+0'e~+
d0--~--
(2.12)
We see that this equation differs from the usual Klein-Gordon equation by the appearance of a term representing the interaction between the scalar and vector fields. An analogous term also appears in the equation of 361
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motion of the vector field. The Maxwell equation for the only nonzero component of the vector potential is then
b
e2~)
=0
(2.13)
Finally we write the Einstein equations for the metric (2.1) and the stress-energy tensor (2.7):
+ - + abeb aT+
l(1 + 1 + 1) a4+b4+c4
ca
~ ~5
~5
75
a+a(b ~)aa-(b2-c2)
4aZbZc 2
2
a
[9"
a b +
+
b(~ ~) b4-(a2-c2) + ~ + + 2a2b2c 2
( + ? c c
+
+
A~
2a2b2c2
c4-(b2-a2)2 2a2b2c 2
2c 2 - e
~5c2~ 2
~C5C 2 +V(~,),
A] .;t 2 - 2a2b ~ + ~ + V(01), ----
2 -V(OI)=0,
(2.14)
+A~
- 2aZb 2
2
2aZb 2
A2 2a2b 2
ft2 + V ( q ~ l ) + e 2 c 2 v,, . 2c 2
(2.15a)
(2.15b) (2.15c)
Through the Bianchi identities only three of the Einstein equations are independent; hence we take eq. (2.14 ) as a constraint to the initial conditions. Once satisfied at an instant o f time t these same Bianchi identities ensure that it is satisfied at any other instant of time t.
3. Numerical analysis. Asymptotic solutions 3. I. The primordial anisotropy and gauge field configurations The system of eqs. (2.12)-(2.15 ) is rather complicated and was solved numerically, by a step-by-step method, for a variety of initial conditions. We used a fourth-order R u n g e - K u t t a method, using the constraint equation not only in the choice of the initial conditions, but also to check the numerical accuracy. Being interested in the role played by the primordial anisotropy, we obtain for the shear tensor
o"s =diag
- H , ~ - H , -c - H
)
,
(3. la)
where
H=g
+~+
(3.1b)
is the mean Hubble parameter; the shear is then
~r2= a , ; a ' J = ( ~ - H ) 2 + ( ~ - H ) 2 + ( ~ - H )
2.
(3.1c)
From (2.14), (2.15 ) and (3.1) we can write the Raychaudhuri equation = -
_~a2- _1( A 2 A2] -~2+~ 6 \ a 2 b 2 + cZJ
V(~)
(3.2)
where G is the mean scale factor G = (abc)1/3, V(O) = Vo e x p ( - 2 0 ) and Vo and 2 are positive constants (for convenience we redefined 2oid=2V/2); here and in the following we drop the index 1 in the scalar field. From
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the positive sign in front of V(O), we see why the potential is so important to achieve an inflationary regime, characterized by (~'> 0. The following condition must then be satisfied: 1(
A2
A 2~.q_ 1~2.
(3.3)
-~v(0)> ~ 2+ g \a-~b~ + c 21
If the condition (3.3) is not satisfied initially then two different cases can occur: (i) the universe recollapses after a finite time without undergoing any inflationary stage or (ii) the evolution of the fields brings the potential to dominate before recollapse occurs and the Universe begins to inflate. The latter case indeed happens for a large range of initial conditions, as we found numerically, and we may now try to understand it from the behaviour of the scalar and gauge fields. This typical behaviour is already present in the isotropic case studied in ref. [24]. Introducing the expressions for the energy density and the pressure of the scalar field, p0=½b2+g(g)),
and
po=½b2-V(O),
(3.4)
we can write the equation of motion of the scalar field as
d (;b~)_o~A2 ~tpo=-(po+po) + -b +
c2 ,
(3.5)
assuming a situaiton, quite usual in the present context, where 02 << V(~), we then have that dpJdt ~_0 neglecting also the last term due to the 1/c 2 factor. Such a situation does not occur for the gauge field and the anisotropy. This means that, in general, or at least for a large variety of initial conditions, the evolution of the scalar field potential is slower than the evolution of the terms on the RHS of (3.3); if V(O) is initially large, it will remain large for a time long when compared with the time of evolution of the gauge field and eventually, the condition expressed by the inequality (3.3) will be verified (see fig. 1 ). Hence, if we have a model where initially the energy of the matter fields is distributed among gauge and self-interacting scalar fields, we expect that the slower evolution of the scalar field can lead to an inflationary regime. Probably more surprising it is the fact that both the primordial anisotropy and gauge fields tend to enhance inflation instead of making it more difficult to occur, a fact that had been noticed before by Martinez-Gonzalez and Jones in the context of a Bianchi-V model [ 14 ]. The surprise comes from the fact that the shear ~2 and gauge field contribute with a negative sign to the RHS of the Raychaudhuri equation, thus making G more negative. The solution of the apparent paradox is in the Friedmann equation, which in our case is given by ~.0
0.8
,3 6
°2 t 0.0
l
O0
~ / 5 o
+ 1/6(
1.0
20
.
5.0
Fig. 1. It is shown how condition(3.3) is indeed verified after a short time. We took V0=3, 2~0.7, and fixed (ii/a)o,,~ 1.4, (b/b)o ~ 1.4, (/'/C)o,~ 0.7, (A3)o = 1, (]t3)o=2, q}o~0.7, ~0 ~ 1.4.
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31a4q-b4q-c4-(a2-b2)2-4a2b2c 2(b2-c2)2-
(c2-a2)2
14 March 1991
(3.6)
p being the total energy density due to the scalar and gauge fields and their interaction. We see that both p and cr2 contribute with a positive sign and thus a large initial anisotropy will give rise to a large initial value o f ( G / G) 2. But, as both a2 and the gauge field tend to decrease faster than the scalar field, eventually condition (3.3) can be satisfied and, when this happens, the larger the initial anisotropy (as well as the gauge field) the larger the velocity of expansion will be, as translated by ( G / G ) 2, making it easier for the potential V(0) to expand the system. Incidentally, we can also understand now why, without very special assumptions, a self-interacting vector field cannot by itself sufficiently inflate a system [ 19 ]. Its contributions for the equations of motion are proportional to A 2 for example, and we have that A 2 =gU,AuA, ' giving rise to combinations like A 2/c2 or similar ones; when the scales inflate these terms quickly become negligible and their influence disappears. It is, nevertheless, important to realize that the influence of the anisotropy is not always the same. We found, in particular, that in cases where a small change o f the initial conditions may bring a premature recollapse, an increase in the value of the primordial anisotropy may either lead to an inflationary situation or it may contribute to the quick collapse of the system. On the other hand, the increase of the initial value of the energy density associated with the gauge field always seems to help inflation. 3.2. The asymptotic solutions We verified numerically that, for initial conditions such that the system inflates, the scale factors a, b, c approach a behaviour proportional to t p, where p is given by 2/22, although the absolute values of the scales may be different; the system also isotropizes in the sense that the Hubble factors, corresponding to the three directions, become equal. The scalar field has then a logarithmic variation with t. This is not difficult to understand once we realize that, in these circumstances, the gauge field and the curvature terms in eq. (2.15) quickly approach zero and can be neglected. The following asymptotic solution (analogue to the solutions found for the isotropic case in refs. [ 18,24 ] ) then applies: a = a o t p,
b=bo tp,
C=Cot p and
O=kln(#t),
(3.7)
where ao, bo, and Co are usually different, p = 2/22, k = 2/2 and # = 22x/Vo/( 12 - 2)~2), requiring that 22 < 6, this behaviour can be seen in fig. 2. It is possible now to make more precise our statement that the anisotropy and the gauge field aid inflation. What we mean is that, although the solution asymptotically approaches the same power-law behaviour t p, with p fixed and given by 2/22, the actual scales, defined by ao, bo and Co increase when we increase the initial value of a 2 and the energy of the gauge field. For small values of 2, p can be large enough to solve the horizon and flatness problems [ 6 ]. For 22 > 2, the curvature terms in eq. (2.15 ) can be seen to decrease more slowly than the kinetic terms. Then, even if we reach a stage where such terms appear negligible, with the universe in an almost isotropic condition, this is only temporary; as the system evolves, those terms will again be important and the universe may go back to an anisotropic regime. O f course, for such large values of 2, our model is not even able to inflate. Unfortunately, the interacting terms, appearing in the equations of motion (2.12) and (2.13 ) of the scalar and gauge fields, do not seem to lead to a steady transfer of energy away from the scalar field into the gauge field, as we hoped; not only do they become very small, they give rise instead to a mutual interchange of energy between the two fields. We could try to remedy this situation by introducing a viscosity term in the equation of motion o f the scalar field; this has been used by several authors [24], and allows for large inflation, even with values of 22 > 2. From dimensional and field theoretic arguments, such terms are expected to have the general form Cv0d~ 5-2d, where d is a n u m b e r usually taken to be one, but it is not known how to calculate the viscosity coefficient Cv. We have not explored this possibility here. 364
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12o] (b)
(a)
25
14 March 1991
P asyrnptoiic 20
~
f f/
//
num~ric
1.5
50 @asymptotic
0
50
100
150
t
200
0.0
0 ....
sb ....
1;0 ....
1;0 ....
t
2~0
Fig. 2. (a) We see that the numerical exponent p quickly approaches the theoretical asymptotic value 2/22; (b) the same happens with the scalar field $ with respect to the asymptotic solution ¢ = k In (,ut).
We can easily check that the solutions we have just described are asymptotically stable. For this it is convenient to introduce the redefinitions a = ¢ ' , b = e a and c=¢", together with a new time variable r through d / dt = ~ d/dz. We then perturb the system with the help of the small quantities x, y, z and w and the replacements o~-.ce+x, fl--,fl+y, 7--+7+z and O--,O+w; we neglect the curvature and gauge field terms. Finally, after linearizing the equations in x, y, z and w and making the redefinitionsf=x', g=y', h=z' and rn=w', where primes denote derivatives with respect to r, we find f'=
+ ( ½ 2 0 ' - 4 a ' ) f -c~' g-o~' h + 12o~'m,
h'= -o~'f-o~'g+ (½2c~'-4o~' ) h + ½2o~'m,
g' = -o~' f + (½20'-4c~' )g-oC h + ½2o~'m ,
(3.8a,b)
m'= -O'f-O'g-O'h+ (20'-3c~' )m,
(3.8c,d)
we have that ½ 2 0 ' - 4 c e ' = ~ / ~ / V o ( 1 - 8 / 2 2 ) , ½2o~'=(1/2)x/lz2/Vo, o ~ ' = ( 2 / 2 2 ) ~v@~V0, 0 ' = ( 2 / 2 2 ) × x//z2/Vo and 20' - 3o~' = x//fl/Vo (2 - 6/22), where all the terms have a c o m m o n factor x//~2/Vo. After these replacements the system takes the autonomous form x' =Ax (notice that the solution whose stability is being investigated has o~' = fl' = 7' ). After some algebra, we find that the eigenvalues of the matrix A are the roots of the equation (4 + E3(27/,12-- 4) -- e213( 1 - - 8 / , 1 2 ) ( 1 - 3/22 ) + 3( 1 -- 8/2 2 ) 2+ 1 / 2 2 ( 3 _ 4 / 2 2 )
]
+ e [ - (1 - 8 / 2 2) 3_ 3( 1 - 8 / 2 2 ) 2 ( 1 - 3 / , l 2 ) - 6 / ) . 2 ( 1 - 8 / , 1 2 ) - 12/,16+4/,14(1 - 8 / , 1 2 ) - 4 / , 1 4 ] + [ ( 1 -- 8/2 2 ) 3( 1 -- 3/,12 ) + 3/,12( 1 -- 8/). 2 ) 2-31-12/26( 1 -- 8/,12+4/,16 ) -t-4/,14( 1 -- 8/,12 ) ] = 0 .
(3.9)
We solved this equation numerically for a large range of values of,1; we either have four real negative roots, or two real negative plus two complex conjugate roots with their real parts negative, indicating that we are in the presence o f asymptotically stable solutions. This is still valid for a Z as small as 1 0 - 6. It is interesting to notice that for a small 2 the system may evolve towards a different temporary attractor corresponding to an exponential solution, depending on our choice of initial conditions. This solution arises as follows. For values of 2-~ 0 and 6 2 << 1, we have that V(0) ~- Vo( 1 -,10) = Vo; then, and again neglecting asymptotically the curvature terms and the gauge field, the exponential solution applies, with
a=aoexp(pt),
b=boexp(pt),
C=Coexp(pt)
and
O~-&+q,
(3.10) 365
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where 6-~ 2p, p _ ( ] Vo)l/2 and q is a constant; thus, for very small 2 6 is small and ~ changes slowly. Except for 2 identically zero, the exponential solution will act only as a temporary attractor. From V~ V o ( 1 - 2 ~ ) we expect the system to inflate for a period of time of the order of ( 6 2 ) - 1 ~ 2 - 2 ( ~ Vo)-l/Z, till V(O)~-O, giving Gi/ G f - exp ( 1/2 2 ); after this period of time the system will evolve towards the attractor corresponding to the powerlaw solution. To have a realistic model, the universe must naturally evolve towards a radiation dominated phase. As an illustration of how this can happen, we consider a simple potential with a local minimum. For simplicity, we take the following combination of exponential potentials: V(¢) = Vo(exp (20) + e x p ( - 2 ¢ ) - 2 ) ,
(3.11)
V ( 0 = 0 ) = 0 and near this minimum V(¢) - Vo2202. Assuming that the initial model has been isotropized and flattened, after a more or less long period of inflation, the equations are
3(G/G)~=½q)2+ V(¢), ~'+ 3((7,/G)~+~V(O)~-~)'+3((7,/G)~+ZVo220=O.
(3.12a,b)
If we look for an approximate solution, taking only into account the dominant terms for large t, we have that ¢-~0ot"COSO)t
(a=-l)
G~-Gotp,
and
(3.13)
o)_~2 ( 2 Vo) 1/2 and p-~ 2 ( ~ Vo) 1/2. In the inflationary phase this form of the potential does not alter our previous conclusions and p ~- 2/22. Hence, for a small value of 2 we have a large p during the inflationary regime, evolving then to a small p ~ 2 x / -~Vo, when ¢ reaches the minimum of the potential and inflation comes to an end. Such behaviour can be verified numerically (see fig. 3). In our model the energy lost by the scalar field goes mainly to the expansion of the Universe; in a more realistic model, with the right coupling between the scalar field and other fields taken in account, this energy would, to a large extent, be used for reheating and particle creation [ 14,24]. As we said in the introduction, we leave the study of these problems to a future work.
(a)
300
10
Iog~oG
(b)
O0
20.0 !0
-20 100 30
0.0
o
2b
40
60
80
L
I00
--4"020
!O
60
80
t
I00
Fig. 3. (a) Plot of the mean scale factor, clearly showing G = (abc) U3 evolving away from the inflationary phase; (b) the damped oscillations of the scalar field around the m i n i m u m of the potential (3.11 ) are here clearly seen.
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4. Conclusions In this work we investigated the role played by the anisotropy and gauge field configurations in the very early Universe, in the context of an inflationary scenario driven by a self-interacting charged complex scalar. We used directly the Yang-Mills and the K l e i n - G o r d o n equations to describe the evolution of the gauge and of the scalar fields, rather than going to a fluid description. The m i n i m a l gauge coupling allowed the interaction between the gauge and the scalar fields to be fixed in a u n i q u e way. For most of the initial conditions, we found that an increase in the initial value of the anisotropy aided inflation, the exceptions referring to those cases where the system is near a situation of recollapse; then an increase of the anisotropy may, sometimes, lead to a still faster recollapse. On the contrary, in all the situations studied, an increase in the energy associated with the gauge field always favoured inflation. The different roles played by the anisotropy, and the scalar and gauge fields can, in part, be qualitatively understood with the help of the Raychaudhuri and F r i e d m a n n equations and the equations of m o t i o n of the two fields. We also studied the asymptotic behaviour of the inflationary solutions. We were able to identify an attractor corresponding to a power-law solution, a n d another one to an exponential solution for 2 = 0 . This last one, Ibr 2 # 0 and small, acted only as a temporary attractor. In both cases the system quickly isotropized and the gauge fields became negligible. A small value of the parameter, appearing in the exponential potential, is required to solve the horizon and flatness problems, a limitntion considered to be a negative feature of these models. The interacting terms between the scalar and gauge fields did not lead to a steady transfer of energy away from the scalar field, but rather to a mutual transfer of energy between the two fields; they were not then able to perform the role of a viscosity, transferring energy away from the scalar field and allowing enough inflation to take place, even for larger values of 2.
Acknowledgement The authors would like to thank their colleagues O. Bertolami and A. Liddlc for useful discussions and suggestions.
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