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Exact and asymptotic solutions to tensor-scalar cosmologies
ELSEVIER
Nuclear Physics B 486 (1997) 413-422
Exact and asymptotic solutions to tensor-scalar cosmologies A. O u k o u i s s 1 Facultd des Sciences, Universit~ Libre de Bruxelles, Campus de la Plaine, C.P 225, bd du Triomphe, B-I050 Brussels, Belgium Received 12 March 1996; revised 9 October 1996; accepted 18 October 1996
Abstract
In the tensor-scalar theory of gravity, we give a number of explicit solutions and examine analytically their asymptotic behavior. We consider homogeneous cosmologies with perfect fluid matter distribution satisfying the equation of state p = A p where A is a constant - 1 ~< A ~< 1. The convergence of our theory to general relativity is considered to be "good" if the scalar field ~b = const, is an attractor of the equations of motion. When p = - p new varieties of inflation arise in which the scale factor a ( t ) cx t"exp(K, t~). PACS: 98.80.Hw; 04.20.Jb
1. I n t r o d u c t i o n
Tensor-scalar theories of gravity [ 1-3] have raised new interest in gravitational physics and cosmology. From a theoretical point of view, they provide the simplest possible extension of Einstein's theory where, in addition to the usual gravitational forces, long-range scalar forces are present. Most unification models based on superstrings or supergravity naturally associate massless scalar fields to the tensor gravitational field. In these models, one expects the matter-scalar and matter-tensor couplings to be of the same order, namely 2 =1. Olun.th. 1E-mail: [email protected] 0550-3213/97/$17.00 Copyright (~ 1997 Elsevier Science B.V. All rights reserved PH S 0 5 5 0 - 3 2 1 3 ( 9 6 ) 0 0 5 9 3 - 7
(1)
414
A. Oukouiss/Nuclear Physics B 486 (1997) 413-422
From the experimental point of view, the most precise gravitational tests do not exclude the existence of a scalar admixture to gravity. The present solar system experiments however indicate that the maximum fractional contribution of scalars to the (Newtonian and post-Newtonian) gravitational interaction is 2
1
O~today< ~ .
(2)
Many authors [5-7] have tried to reconcile Eqs. (1) and (2) in exhibiting a mechanism driving the early unified tensor scalar theory towards general relativity. Recently, it has been shown by Gerard and Mahara [8] that a minimal tensor-scalar theory does not generically contain a natural attractor mechanism towards general relativity. Two classes of solutions are of particular interest: those for dust (~ -- 0) models, which describe the post-recombination history of the universe to a very good approximation, and those with A = - 1 , which describe an era of inflation dominated by a slowly evolving scalar field with p = - p . The purpose of this paper is to give some exact solutions and discuss their asymptotic forms. This paper is organised as follows: In Section 2 we introduce the field equations of the tensor-scalar gravity in both the Jordan-Fierz frame and in the Einstein conformal frame and we describe the model we are interested in. In Section 3 we give a number of explicit solutions and discuss their asymptotic forms.
2. Field equations The general form of the action in tensor-scalar theories expressed in the "physical" Jordan-Fierz frame reads [ 3,4,9]
-s= i - ~
]
~"~0.4~o~4,-u(~) + Sm[¢'.,.-~],
(3)
where R =-g~Ru~ denotes the curvature scalar of the physical metric g ~ , ~b is the Jordan-Fierz-Brans-Dicke scalar field and Sm [~bm,g ~ ] is the matter action which does not contain ~b. In this paper we shall limit ourselves to the simplest class of exactly massless tensor-scalar theories, where the potential U(~b) = 0. The Brans-Dicke parameter w is here treated as a function of ~b. More general tensor-scalar cosmologies have been proposed and discussed recently e.g. in Refs. [ 10,11]. In the Jordan-Fierz frame, since they do not couple to 4', the matter fields ~bm obey the same equations as in general relativity: the energy-momentum conservation equation e.g. reads --/ZP
T;~ = 0,
(4)
where
T~,p= 2 6Stn[Om,~u~]
(5)
A, Oukouiss/Nuclear Physics B 486 (1997) 413-422
415
denotes the physical stress energy tensor of the matter. The equation of motion for the gravitational and the scalar fields are straightforwardly derived from the variations of the action with respect to ~ and ~b, -
=
-
+
(6)
+87r~b-I Tu~, do9
~,
[]~b = 2to(~b) + 3 These equations are rather complicated and the presence of second order derivatives of ~b in the r.h.s, of Eq. (6) induces, as discussed in [7], several mathematical and physical unpleasant features. Hence, for many purposes, tensor-scalar theories are better described in the Einstein frame. This frame is obtained via the following conformal transformation: ~
1
= A 2 ( ~ ) g ~ = ~--~g~,
(8)
where G plays the role of the gravitational constant. With c~(~p) = A - l a__AA= [2~o(~b) + 3]-1/2, a~o
(9)
the action (3) becomes, up to a surface term which has been neglected,
S =
-6
11
;d4xx/-~(Rt.,J
2gUVOl, q~,9,,~o) + S,n[~b,,,,A2gu,,].
(10)
In the Einstein frame, the matter field equations are more involved than the previous ones because matter now couples both to the tensor and to the scalar. The gravitational and scalar field equations however get a simpler form: they read
R~,,, - Rg~z_...~= 2Ou~oOvq~ _ guvO,~oO,~ ~p + KTu~ , 2 K
[q~p = - ~ a(go) T,
( 11) (12)
with
T~_
_ _ 2 6Sm[~bm,A2g~]
T.~ ~ = T a ( ~ ) O ~ .
= A6T~ ,
(13) (14)
T standing for the trace g~"T~, and K = 8~rG. t~ defined through Eq. (13) is real for w(~b) > - 3 , which we assume. In Eq. (12), one can interpret the function ce(~) as the ratio between the scalarmatter and tensor-matter couplings; as such, a plays a central role in discussing the observable consequences of tensor-scalar theories [3].
416
A. Oukouiss/Nuclear Physics B 486 (1997) 413-422
Homogeneous cosmological space-times can be represented both in the Einstein conformal frame, (15)
ds 2 = dt 2 - a2(t)dl 2
and in the Jordan-Fierz frame, ds 2 =dt 2 -
~2(7)d/2
(16)
describing homogeneous, isotropic and spatially flat cosmological models with a perfect fluid matter distribution. Note that the scalar field ~, depends only of the time variable. The matter distribution admits a perfect fluid description in both conformal frames: T~
= (p + p)u~'u ~ - pg~,
T u~ = ( ~ + ~ ) ~ ' ~
- p~'~
(17)
(18)
with positive matter densities p and fi and pressure p, p and g~z~u~u ~
= ~ u ~ u ~ ~ = +1.
(19)
From Eqs. (13) and (8) we have p = A4p,
(20)
p = A4p,
(21) (22)
dr = adt, ~(7) =aa(t)
(23)
.
Using Eqs. (15) and (17), the equations of motion written in (11) and (14) reduce to H+3H2_ 3 / ~ / + 5 (1 + A ) H 2 -
1 -A 2 Kp, I - h 2 2 '
,b + 3( 1 + a ) p H = ( 1 - 3A)pa4~, where an overdot denotes differentiation with respect to t and H ( t ) Eqs. (24) and (25) also imply 3H 2 = Kp + ~/,2.
(24) (25)
(26) = i~(t)/a(t).
(27)
In what tbllows, we shall restrict ourselves to the ranges - 1 ~< ,~ ~< 0. We will use the approach developed by Gurevich, Finkelstein and Ruban [ 12] for Brans-Dicke gravity (ce = const.) to the more complicated cases where a depends on ~o. We introduce the time coordinate r/defined by dt
=
a 3 a A ( 3 a - I ) x / ~ + 3dr/
and new dynamical variables defined as
(28)
A. Oukouiss/Nuclear Physics B 486 (1997) 413-422
417
fj (r/) = a3~b,
(29)
f2 (r/) =a3H.
(30)
The derivative of Eq. (29) with respect to r/reads f lt
K
= --~(1
3,~)pa3(l+a)A(3a_l) .
-
(31)
On the other hand, the energy-momentum conservation law expressed in the JordanFierz frame -~,~ d Tw = ~ - ~ + 3 ( l + a ) ~ n = 0
(32)
together with Eqs. (20), (22) and (23) gives p = const, x a-3(l+a)A -(3a-l).
(33)
Eq. (33) inserted in the r.h.s, of Eq. (31) leads to f l 07) = - B ( 1 - 3a)~7.
(34)
We have set f l (0) = 0. Hence, from Eqs. (24) and (30) one gets t ) (1 - a ) f~(r/ = B - - , o¢
(35)
where K~3( l +a) r ~,t a (3:t-1)
B = 7"
~qJ'~
p(r/) = const. > 0.
(36)
Eq. (27) becomes then 3fzZ(r/) = K p ( r l ) a 6 ( r l ) -t- f Z ( r l ) ,
(37)
namely f ( r / ) + (1 - 43A)Zr/2 = 4 @ f 22 0 7 ) ,
(38)
where we have put K
f(rl) = ~ffSp(rl)a
6
(r/).
(39)
Eq. (8) gives 1 dq~ d~ = - 2---d--~-"
(40)
On the other hand, from Eqs. (29) and (34) we get ~b = - B ( 1 - 3A)r/a -3 ,
(41)
418
A. Oukouiss/Nuclear Physics B 486 (1997) 413-422
SO that - - = 2B( I - 3,~) A (3a-l)r/a -3(I-a)
(42)
~b
which, by the use of Eqs. (36) and (39), still reduces to
4¢
r/
~- = ( 1 - 3A) f ( r / ) '
(43)
and whose solution reads
~b(r/)=~boexp
(1-3,~)
f(rl,) I
,
(44)
where ~bo and ~7o are positive constants. Eq. (42) then gives a('r/) 3(l-a) = 2B f ( r / ) A (3a-I)
(45)
and with Eq. (19) ~(r/) 3(l-a) = 2B f ( r / ) A 2.
(46)
3. Exact and asymptotic solutions In order to give exact solutions and to examine their asymptotic behaviour, we must specify p a 6 from which ~b(r/) follows from Eq. (44), ~(r/) from Eq. (46), f2(r/) from Eq. (38) and hence a ( r / ) from (35), and finally 7(r/) from Eq. (28). The choice of the function p a 6 is sufficient to produce an exact solution of the field equations and hence the behaviour of ce at late times. 3.1. The d u s t case
The importance of the dust models (,~ = 0) arises from the fact that they provide a very good description of the present thermodynamic state of matter. Recently Damour and Nordtvedt [ 16] have argued that the general relativistic limit acts as a cosmological attractor within parameter space of more general scalar-tensor gravity theories. This occurs when the Brans-Dicke parameter o) diverges. In the notation favored by Damour and Nordtvedt this corresponds to the Brans-Dicke field ~ as defined in Eq. (8) having a local maximum with respect to the field ~o with (~b 4= 0). To see how this emerges in our notation, we will consider the following example: p a 6 oc r//3,
/3 > 2.
Eqs. (44), (46) and (35) give
(47)
A. Oukouiss/Nuclear Physics B 486 (1997) 413-422
~b(r/) = ~boexp [K _2 - - ~ r / 2 - # ] ,
r
B3 = aor/t~ exp [ _ K z -/.~
419 (48)
j '
½,
(49)
(50)
K, ~bo and ao are constants. When r / ~ cx~, we obtain
7 oc r/~/2 .
(51)
Hence, at late times, we have ~(7) cx (7) 2/3 ,
(52)
05(7) cx exp [K 2-~72(2-n)/t~ ] ---~const, as 7 --* oo,
(53)
and we recover then the Friedmann-Robertson-Walker model based on general relativity
a(7) oc (7) <2-,~)/n,
(54)
a(7) ---~0 as 7--~ co.
(55)
This could be expected since the coupling to asymptotes to infinity at late times since ~b --~ const. This behavior naturally guarantees that these theories survive the scrutiny of weak-field solar system tests. In fact, to~to-3 ___,0 as 7 --~ c~. 3.2. Inflationary models
This corresponds to a matter source which is created by a potential-dominated scalar field. It is the standard matter source generating a wide variety of inflationary models. We recall that in Brans-Dicke models the ,~ = - l source produced an example of power-law inflation [13] in contrast to the exponential behavior obtained in general relativity with the same matter source. For p a 6 ~ 77fl, fl > 2, Eqs. (44), (46) and (35) give for ,~ = - 1 ~('r/) cx r/n/6 ,
(56)
or(r/) c< r/(2-/3)/2 ,
(57)
~b('q) c< exp [ _2 - ~ r / 2 - # ] ,
(58)
7 cx ln(r/). Then
(59)
420
A. Oukouiss/Nuclear Physics B 486 (1997) 413-422
~(~-) ocexp [~-7] ,
(60)
~b(7) c x e x P [ 2 _ ~ 4 f l e x p ( 2 - / 3 ) 7 ] ,
(61)
a(7) o ( e x p ] ~ - ~ '
(62)
I --~0
asT---~oo
we recover then the De Sitter model based on general relativity. 3.3. Brans-Dicke models Among the wider class of scalar-tensor gravity theories Brans-Dicke behavior looks atypical. It only occurs at late times where pa 6 (x rl 2 in the limit r / ~ oe, p a 6 = r/2 .
(63)
In this case we obtain the solutions (~(7) O( 711-3t,
(64)
a ( I ) O( 7] (l+3A)/3(I-3A)
(65)
When r/--~ oc, we obtain from Eq. (24) O( 7 (I+3A2)/(I--A)
(66)
Hence, at late times, we have ~(7) O((7) (1 +3,~)/3(1+3a 2) ,
(67)
~ ( 7 ) O ( ( t ) (I-A)(I-3A)/(I+3"h?) ,
(68)
o~(~) oc const.
(69)
and we recover then the Brans-Dicke solution [ 1]. 3.4. New superinflation The only known solution to the horizon, flatness and monopole problems of the standard cosmological is a period of expansion in a De Sitter phase, where the cosmic scale factor a(t) increases by a factor 1028, at least. The first scenario, called "old" inflation [ 17], was based on a supercooled first-order phase transition where a(t) grew exponentially with time. The old inflationary model failed [ 18] since the false vacuum expanded too rapidly for this problem, known as the "graceful exit" problem, gave rise to the so-called "new" inflationary scenarios [ 19]. New inflation is based on a slow-rollover phase transition and requires extreme fine-tuning of the effective potential parameters in order to become extremely flat near the false vacuum. "Extended" inflation [20] was of new inflation. Recently Steinhardt and Accetta [21] present a simple scenario called "hyperextended inflation".
A. Oukouiss/Nuclear Physics B 486 (1997) 413-422
In order to give new model, we choose p a 6 cx 7721n(77) when 77 ~ from Eqs. ( 2 8 ) , ( 3 5 ) , (44) and (46)
421 ~ . We obtain
a ( t ) o( 72(1 --K)/(2+K) exp[ (2 + K)7] 2/(2+K) ,
(70)
~ ( 7 ) cx [t] 2K/(z+x) ,
(71)
Ol(t) O< ( 3 ) - ] / ( 2 + X ) ,
(72)
where K is a constant positive. These solutions display new varieties of inflationary universe. When K = 1, we have K(7) oc exp[37] 2/3. This is the particular example ot' the "intermediate inflation" proposed in [ 14]. It is especially interesting that this form o f the scale factor, which arises when K = 1, is precisely that which generates the exact Z e l ' d o v i c h - H a r r i s o n spectrum for the density and gravitational wave fluctuations produced during inflation to first order in perturbation theory [ 15]. In this paper we have shown how to derive exact cosmological solutions for the Friedmann IK = 0 models with a perfect fluid characterised by the equation of state p = Ap, where ,~ is a constant and 0 ~< A <~ I, in general scalar-tensor gravity theories. By an appropriate choice of variables we have shown that the solutions can be defined in terms o f a function p(77)a6(77). We have provided a number o f specific examples of dust solutions and inflationary universe theories. The availability o f new exact solutions allows cosmological tests o f gravitation theories to be combined with weak-field and laboratory limits on allowed deviations from general relativity to constrain the permitted departures o f 0,) - 1 and w l w - 3 from zero.
Acknowledgements We gratefully acknowledge enlightening discussions with Prof. Marc Henneaux, Prof. Frangois Anglert, L. Musongla, M. Belmir, R. Argurio, C. Iannuzzo and C. Schomblond.
References [II c. Brans and R.H. Dicke, Phys. Rev 124 (1961) 925. [2] D. La and P.J. Steinhardt, Phys. Rev. Lctt. 62 (1989) 376. 13] R.V. Wagoner, Phys. Rev. D 1 (1970) 3209. [4] K.N. Nordtvedt, Astrophys. J. 161 (1970) 1059. [5] P.J. Steinhardt and ES. Accetta, Phys. Rev. Lctt. 64 (1990) 2740. [6] J.Garcia Bellido and M. Quiros, Phys. Len. B 243 (1990) 45. [7] T. Damour and K. Nordtvedt, Phys. Rev D. 48 (1993) 3436. [8] J.M. G6rard and I. Mahara, Phys. Lett. B 346 (1995) 35. [9] P.G. Bergman, Int. J. Th¢or. Phys. 1 (1968) 25. 10] T. Damour and K. Nordtvedt, Phys. Rev. Lett. 70 (1993) 2217. 11] J.D. Barrow, Phys. Rev. D 47 (1993) 5329. 12] LE.Gurevich,A.M.Finkelstein and V.A.Ruban, Astrophys. Space Sci. 22 (1973) 231. 13] P.J. Steinhardt and F.S. Accetta, Phys, Rev. Lett. 64 (1990) 2470. 14] J.D. Barrow, Phys. Lett. B 235 (1989) 40. 151 J.D. Barrow and A.R. Liddle, Phys. Rev. D 47 (1993) R5219.
422 1161 [17] [ 18] [191 [201 [211
A. Oukouiss/Nuclear Physics B 486 (1997) 413-422 T. Damour and K. Nordtvedt, Phys. Rev. Lett. 70 (1993) 2217. A.H. Guth, Phys. Rev. D 23 (1981) 347. A.H. Guth and EJ. Weinberg, Nucl. Phys. B 212 (1983) 321. A.D. Linde, Phys. Lett. B 108 (1982) 398. C. Mathiazhagan and V.B. Johri, Class. Quant. Grav. 1 (1984) L 29. P.J. Steinhardt and F.S. Accetta, Phys. Rev. Lett. 64 (1990) 2740.
Nuclear Physics B 486 (1997) 413-422
Exact and asymptotic solutions to tensor-scalar cosmologies A. O u k o u i s s 1 Facultd des Sciences, Universit~ Libre de Bruxelles, Campus de la Plaine, C.P 225, bd du Triomphe, B-I050 Brussels, Belgium Received 12 March 1996; revised 9 October 1996; accepted 18 October 1996
Abstract
In the tensor-scalar theory of gravity, we give a number of explicit solutions and examine analytically their asymptotic behavior. We consider homogeneous cosmologies with perfect fluid matter distribution satisfying the equation of state p = A p where A is a constant - 1 ~< A ~< 1. The convergence of our theory to general relativity is considered to be "good" if the scalar field ~b = const, is an attractor of the equations of motion. When p = - p new varieties of inflation arise in which the scale factor a ( t ) cx t"exp(K, t~). PACS: 98.80.Hw; 04.20.Jb
1. I n t r o d u c t i o n
Tensor-scalar theories of gravity [ 1-3] have raised new interest in gravitational physics and cosmology. From a theoretical point of view, they provide the simplest possible extension of Einstein's theory where, in addition to the usual gravitational forces, long-range scalar forces are present. Most unification models based on superstrings or supergravity naturally associate massless scalar fields to the tensor gravitational field. In these models, one expects the matter-scalar and matter-tensor couplings to be of the same order, namely 2 =1. Olun.th. 1E-mail: [email protected] 0550-3213/97/$17.00 Copyright (~ 1997 Elsevier Science B.V. All rights reserved PH S 0 5 5 0 - 3 2 1 3 ( 9 6 ) 0 0 5 9 3 - 7
(1)
414
A. Oukouiss/Nuclear Physics B 486 (1997) 413-422
From the experimental point of view, the most precise gravitational tests do not exclude the existence of a scalar admixture to gravity. The present solar system experiments however indicate that the maximum fractional contribution of scalars to the (Newtonian and post-Newtonian) gravitational interaction is 2
1
O~today< ~ .
(2)
Many authors [5-7] have tried to reconcile Eqs. (1) and (2) in exhibiting a mechanism driving the early unified tensor scalar theory towards general relativity. Recently, it has been shown by Gerard and Mahara [8] that a minimal tensor-scalar theory does not generically contain a natural attractor mechanism towards general relativity. Two classes of solutions are of particular interest: those for dust (~ -- 0) models, which describe the post-recombination history of the universe to a very good approximation, and those with A = - 1 , which describe an era of inflation dominated by a slowly evolving scalar field with p = - p . The purpose of this paper is to give some exact solutions and discuss their asymptotic forms. This paper is organised as follows: In Section 2 we introduce the field equations of the tensor-scalar gravity in both the Jordan-Fierz frame and in the Einstein conformal frame and we describe the model we are interested in. In Section 3 we give a number of explicit solutions and discuss their asymptotic forms.
2. Field equations The general form of the action in tensor-scalar theories expressed in the "physical" Jordan-Fierz frame reads [ 3,4,9]
-s= i - ~
]
~"~0.4~o~4,-u(~) + Sm[¢'.,.-~],
(3)
where R =-g~Ru~ denotes the curvature scalar of the physical metric g ~ , ~b is the Jordan-Fierz-Brans-Dicke scalar field and Sm [~bm,g ~ ] is the matter action which does not contain ~b. In this paper we shall limit ourselves to the simplest class of exactly massless tensor-scalar theories, where the potential U(~b) = 0. The Brans-Dicke parameter w is here treated as a function of ~b. More general tensor-scalar cosmologies have been proposed and discussed recently e.g. in Refs. [ 10,11]. In the Jordan-Fierz frame, since they do not couple to 4', the matter fields ~bm obey the same equations as in general relativity: the energy-momentum conservation equation e.g. reads --/ZP
T;~ = 0,
(4)
where
T~,p= 2 6Stn[Om,~u~]
(5)
A, Oukouiss/Nuclear Physics B 486 (1997) 413-422
415
denotes the physical stress energy tensor of the matter. The equation of motion for the gravitational and the scalar fields are straightforwardly derived from the variations of the action with respect to ~ and ~b, -
=
-
+
(6)
+87r~b-I Tu~, do9
~,
[]~b = 2to(~b) + 3 These equations are rather complicated and the presence of second order derivatives of ~b in the r.h.s, of Eq. (6) induces, as discussed in [7], several mathematical and physical unpleasant features. Hence, for many purposes, tensor-scalar theories are better described in the Einstein frame. This frame is obtained via the following conformal transformation: ~
1
= A 2 ( ~ ) g ~ = ~--~g~,
(8)
where G plays the role of the gravitational constant. With c~(~p) = A - l a__AA= [2~o(~b) + 3]-1/2, a~o
(9)
the action (3) becomes, up to a surface term which has been neglected,
S =
-6
11
;d4xx/-~(Rt.,J
2gUVOl, q~,9,,~o) + S,n[~b,,,,A2gu,,].
(10)
In the Einstein frame, the matter field equations are more involved than the previous ones because matter now couples both to the tensor and to the scalar. The gravitational and scalar field equations however get a simpler form: they read
R~,,, - Rg~z_...~= 2Ou~oOvq~ _ guvO,~oO,~ ~p + KTu~ , 2 K
[q~p = - ~ a(go) T,
( 11) (12)
with
T~_
_ _ 2 6Sm[~bm,A2g~]
T.~ ~ = T a ( ~ ) O ~ .
= A6T~ ,
(13) (14)
T standing for the trace g~"T~, and K = 8~rG. t~ defined through Eq. (13) is real for w(~b) > - 3 , which we assume. In Eq. (12), one can interpret the function ce(~) as the ratio between the scalarmatter and tensor-matter couplings; as such, a plays a central role in discussing the observable consequences of tensor-scalar theories [3].
416
A. Oukouiss/Nuclear Physics B 486 (1997) 413-422
Homogeneous cosmological space-times can be represented both in the Einstein conformal frame, (15)
ds 2 = dt 2 - a2(t)dl 2
and in the Jordan-Fierz frame, ds 2 =dt 2 -
~2(7)d/2
(16)
describing homogeneous, isotropic and spatially flat cosmological models with a perfect fluid matter distribution. Note that the scalar field ~, depends only of the time variable. The matter distribution admits a perfect fluid description in both conformal frames: T~
= (p + p)u~'u ~ - pg~,
T u~ = ( ~ + ~ ) ~ ' ~
- p~'~
(17)
(18)
with positive matter densities p and fi and pressure p, p and g~z~u~u ~
= ~ u ~ u ~ ~ = +1.
(19)
From Eqs. (13) and (8) we have p = A4p,
(20)
p = A4p,
(21) (22)
dr = adt, ~(7) =aa(t)
(23)
.
Using Eqs. (15) and (17), the equations of motion written in (11) and (14) reduce to H+3H2_ 3 / ~ / + 5 (1 + A ) H 2 -
1 -A 2 Kp, I - h 2 2 '
,b + 3( 1 + a ) p H = ( 1 - 3A)pa4~, where an overdot denotes differentiation with respect to t and H ( t ) Eqs. (24) and (25) also imply 3H 2 = Kp + ~/,2.
(24) (25)
(26) = i~(t)/a(t).
(27)
In what tbllows, we shall restrict ourselves to the ranges - 1 ~< ,~ ~< 0. We will use the approach developed by Gurevich, Finkelstein and Ruban [ 12] for Brans-Dicke gravity (ce = const.) to the more complicated cases where a depends on ~o. We introduce the time coordinate r/defined by dt
=
a 3 a A ( 3 a - I ) x / ~ + 3dr/
and new dynamical variables defined as
(28)
A. Oukouiss/Nuclear Physics B 486 (1997) 413-422
417
fj (r/) = a3~b,
(29)
f2 (r/) =a3H.
(30)
The derivative of Eq. (29) with respect to r/reads f lt
K
= --~(1
3,~)pa3(l+a)A(3a_l) .
-
(31)
On the other hand, the energy-momentum conservation law expressed in the JordanFierz frame -~,~ d Tw = ~ - ~ + 3 ( l + a ) ~ n = 0
(32)
together with Eqs. (20), (22) and (23) gives p = const, x a-3(l+a)A -(3a-l).
(33)
Eq. (33) inserted in the r.h.s, of Eq. (31) leads to f l 07) = - B ( 1 - 3a)~7.
(34)
We have set f l (0) = 0. Hence, from Eqs. (24) and (30) one gets t ) (1 - a ) f~(r/ = B - - , o¢
(35)
where K~3( l +a) r ~,t a (3:t-1)
B = 7"
~qJ'~
p(r/) = const. > 0.
(36)
Eq. (27) becomes then 3fzZ(r/) = K p ( r l ) a 6 ( r l ) -t- f Z ( r l ) ,
(37)
namely f ( r / ) + (1 - 43A)Zr/2 = 4 @ f 22 0 7 ) ,
(38)
where we have put K
f(rl) = ~ffSp(rl)a
6
(r/).
(39)
Eq. (8) gives 1 dq~ d~ = - 2---d--~-"
(40)
On the other hand, from Eqs. (29) and (34) we get ~b = - B ( 1 - 3A)r/a -3 ,
(41)
418
A. Oukouiss/Nuclear Physics B 486 (1997) 413-422
SO that - - = 2B( I - 3,~) A (3a-l)r/a -3(I-a)
(42)
~b
which, by the use of Eqs. (36) and (39), still reduces to
4¢
r/
~- = ( 1 - 3A) f ( r / ) '
(43)
and whose solution reads
~b(r/)=~boexp
(1-3,~)
f(rl,) I
,
(44)
where ~bo and ~7o are positive constants. Eq. (42) then gives a('r/) 3(l-a) = 2B f ( r / ) A (3a-I)
(45)
and with Eq. (19) ~(r/) 3(l-a) = 2B f ( r / ) A 2.
(46)
3. Exact and asymptotic solutions In order to give exact solutions and to examine their asymptotic behaviour, we must specify p a 6 from which ~b(r/) follows from Eq. (44), ~(r/) from Eq. (46), f2(r/) from Eq. (38) and hence a ( r / ) from (35), and finally 7(r/) from Eq. (28). The choice of the function p a 6 is sufficient to produce an exact solution of the field equations and hence the behaviour of ce at late times. 3.1. The d u s t case
The importance of the dust models (,~ = 0) arises from the fact that they provide a very good description of the present thermodynamic state of matter. Recently Damour and Nordtvedt [ 16] have argued that the general relativistic limit acts as a cosmological attractor within parameter space of more general scalar-tensor gravity theories. This occurs when the Brans-Dicke parameter o) diverges. In the notation favored by Damour and Nordtvedt this corresponds to the Brans-Dicke field ~ as defined in Eq. (8) having a local maximum with respect to the field ~o with (~b 4= 0). To see how this emerges in our notation, we will consider the following example: p a 6 oc r//3,
/3 > 2.
Eqs. (44), (46) and (35) give
(47)
A. Oukouiss/Nuclear Physics B 486 (1997) 413-422
~b(r/) = ~boexp [K _2 - - ~ r / 2 - # ] ,
r
B3 = aor/t~ exp [ _ K z -/.~
419 (48)
j '
½,
(49)
(50)
K, ~bo and ao are constants. When r / ~ cx~, we obtain
7 oc r/~/2 .
(51)
Hence, at late times, we have ~(7) cx (7) 2/3 ,
(52)
05(7) cx exp [K 2-~72(2-n)/t~ ] ---~const, as 7 --* oo,
(53)
and we recover then the Friedmann-Robertson-Walker model based on general relativity
a(7) oc (7) <2-,~)/n,
(54)
a(7) ---~0 as 7--~ co.
(55)
This could be expected since the coupling to asymptotes to infinity at late times since ~b --~ const. This behavior naturally guarantees that these theories survive the scrutiny of weak-field solar system tests. In fact, to~to-3 ___,0 as 7 --~ c~. 3.2. Inflationary models
This corresponds to a matter source which is created by a potential-dominated scalar field. It is the standard matter source generating a wide variety of inflationary models. We recall that in Brans-Dicke models the ,~ = - l source produced an example of power-law inflation [13] in contrast to the exponential behavior obtained in general relativity with the same matter source. For p a 6 ~ 77fl, fl > 2, Eqs. (44), (46) and (35) give for ,~ = - 1 ~('r/) cx r/n/6 ,
(56)
or(r/) c< r/(2-/3)/2 ,
(57)
~b('q) c< exp [ _2 - ~ r / 2 - # ] ,
(58)
7 cx ln(r/). Then
(59)
420
A. Oukouiss/Nuclear Physics B 486 (1997) 413-422
~(~-) ocexp [~-7] ,
(60)
~b(7) c x e x P [ 2 _ ~ 4 f l e x p ( 2 - / 3 ) 7 ] ,
(61)
a(7) o ( e x p ] ~ - ~ '
(62)
I --~0
asT---~oo
we recover then the De Sitter model based on general relativity. 3.3. Brans-Dicke models Among the wider class of scalar-tensor gravity theories Brans-Dicke behavior looks atypical. It only occurs at late times where pa 6 (x rl 2 in the limit r / ~ oe, p a 6 = r/2 .
(63)
In this case we obtain the solutions (~(7) O( 711-3t,
(64)
a ( I ) O( 7] (l+3A)/3(I-3A)
(65)
When r/--~ oc, we obtain from Eq. (24) O( 7 (I+3A2)/(I--A)
(66)
Hence, at late times, we have ~(7) O((7) (1 +3,~)/3(1+3a 2) ,
(67)
~ ( 7 ) O ( ( t ) (I-A)(I-3A)/(I+3"h?) ,
(68)
o~(~) oc const.
(69)
and we recover then the Brans-Dicke solution [ 1]. 3.4. New superinflation The only known solution to the horizon, flatness and monopole problems of the standard cosmological is a period of expansion in a De Sitter phase, where the cosmic scale factor a(t) increases by a factor 1028, at least. The first scenario, called "old" inflation [ 17], was based on a supercooled first-order phase transition where a(t) grew exponentially with time. The old inflationary model failed [ 18] since the false vacuum expanded too rapidly for this problem, known as the "graceful exit" problem, gave rise to the so-called "new" inflationary scenarios [ 19]. New inflation is based on a slow-rollover phase transition and requires extreme fine-tuning of the effective potential parameters in order to become extremely flat near the false vacuum. "Extended" inflation [20] was of new inflation. Recently Steinhardt and Accetta [21] present a simple scenario called "hyperextended inflation".
A. Oukouiss/Nuclear Physics B 486 (1997) 413-422
In order to give new model, we choose p a 6 cx 7721n(77) when 77 ~ from Eqs. ( 2 8 ) , ( 3 5 ) , (44) and (46)
421 ~ . We obtain
a ( t ) o( 72(1 --K)/(2+K) exp[ (2 + K)7] 2/(2+K) ,
(70)
~ ( 7 ) cx [t] 2K/(z+x) ,
(71)
Ol(t) O< ( 3 ) - ] / ( 2 + X ) ,
(72)
where K is a constant positive. These solutions display new varieties of inflationary universe. When K = 1, we have K(7) oc exp[37] 2/3. This is the particular example ot' the "intermediate inflation" proposed in [ 14]. It is especially interesting that this form o f the scale factor, which arises when K = 1, is precisely that which generates the exact Z e l ' d o v i c h - H a r r i s o n spectrum for the density and gravitational wave fluctuations produced during inflation to first order in perturbation theory [ 15]. In this paper we have shown how to derive exact cosmological solutions for the Friedmann IK = 0 models with a perfect fluid characterised by the equation of state p = Ap, where ,~ is a constant and 0 ~< A <~ I, in general scalar-tensor gravity theories. By an appropriate choice of variables we have shown that the solutions can be defined in terms o f a function p(77)a6(77). We have provided a number o f specific examples of dust solutions and inflationary universe theories. The availability o f new exact solutions allows cosmological tests o f gravitation theories to be combined with weak-field and laboratory limits on allowed deviations from general relativity to constrain the permitted departures o f 0,) - 1 and w l w - 3 from zero.
Acknowledgements We gratefully acknowledge enlightening discussions with Prof. Marc Henneaux, Prof. Frangois Anglert, L. Musongla, M. Belmir, R. Argurio, C. Iannuzzo and C. Schomblond.
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