Nonminimal coupling and cosmic no-hair theorem

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16 May 1994

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PHYSICS

LETTERS

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ELSEVIER

PltysicsLettersA 188 (1994) 130-136

Nonminimal coupling and cosmic no-hair theorem Salvatore Capozziello, Ruggiero de Ritis, Paolo Scudellaro Dipartimento di Scienze Fisiche, Universitridi Napoli, Istituto Nazionale di Fisica Nucleare, Sezione di Napoli, Mostra d’oltremare Pad. 19, 80125 Naples, Italy Received 1 November 1993; revised manuscript received 22 March 1994; accepted for publication 29 March 1994 Communicated by J.P. Vigier

Abstract

We give (suffkient) conditions for the extension of the cosmic no-hair conjecture to nonminimally coupled scalar-tensor theories in which the coupling is not given a priori. Furthermore, in a Friedman-Robertson-Walker spacetime, we get exact solutions for such theories using the so-called Noether symmetry approach. The results are interesting because we obtain behaviours able to avoid the initial singularity.

1. Introduction theories of gravity [ 1 ] is cosmology, especially due to inflationary models [ 21; also in theories of fundamental interactions, like superstrings, grand unified theories and quantum gravity, there is a lot of attention towards them [ 31. In Ref. [4] some of us have started to construct a new approach for finding exact cosmological solutions, or at least constants of motion in such theories. There a scalar field 4 minimally coupled to gravitation was introduced and, looking for Noether symmetries for the point-like Lagrangian describing the Friedman-Robertson-Walker (FRW ) flat cosmology, one could deduce a well defined expression for the potential I/ (4), also solving exactly the dynamical equations. This approach has the interesting feature that the form of the potential results directly from searching for a Noether symmetry in the theory. Such a program has also been performed introducing a nonminimal coupling F (4) between gravity and the scalar field C$leading to exact solutions when F (0) satisfies particular differential equations [ 61. Interest

widely

in scalar-tensor

spread in

Elsevicr Science B.V. SSDIO375-9601(94)00250-S

(The case F (4) equal to a constant and V = I/ (4) is also found in Ref. [ 61. In string cosmology, too, some interesting results have been obtained in such a way 171.) In this paper, we discuss some aspects of the socalled cosmic no-hair theorem in the framework introduced above. Such a conjecture has been developed in order to get the uniformity of the universe without imposing special initial conditions [ 9,10 1.It concerns the discussion whether forever expanding inflationary models asymptotically tend to a de Sitter behaviour or not. Actually, a de Sitter spacetime is a stable asymptotic solution of the Einstein equations [ 111. Supposing the existence of a positive cosmological n (or a &) in the presence of a potential V(4), all no-hair theorems also assume that dominant and strong energy conditions [ 121 are valid, even if such conditions, holding before inflation, must be violated afterwards, when pressure becomes negative. When there is inflation, we cannot expect to have a no-hair theorem with a potential possessing a minimum into which the scalar field 4 evolves [ 13 1. (Asymptotically, 4 begins to oscillate around such a minimum, and there

S. Capozzielloet al. /Physics LettersA 188 (1994) 130-136

cannot be a de Sitter behaviour at late times). While a no-hair theorem for a power-law inflation driven by some specific potential (such as, for instance, exponential [ 141) can be proved, also a sufficient condition for nonexistence of inflation can be given [ 131; looking at the decaying of the small perturbations of a de Sitter solution, this imposes limits on the kinds of potentials used in the theory. Usually, the no-hair theorems have been proved for Bianchi models with a nonpositive three-curvature scalar [ 10,15 1, using the key inequality Hz-&l.

-k>

(1)

According to (1), fi < 0, so that H z b/a 2 0 must be not increasing, a (t ) being the scale factor of the universe. In an expanding universe, the de Sitter asymptotic behaviour is shown to be reached quickly, while the energy density of matter and the anisotropy rapidly tend to zero [ IO,151. Here, we develop some considerations about the nohair theorem in a homogeneous spacetime with k d 0, filled with a scalar field 4 nonminimally coupled to gravity. Writing the curvature scalar (3)R in terms of the structure constants of the Lie algebra of the spatial symmetry group, we find [ lo,15 ] that all the models, except type IX, in the corresponding Bianchi classification surely have (3)R, i.e. k 6 0. The situation with k > 0 is not examined here. That is, we take into consideration the action A=

.I

d4xfi[F’(@R

+ ff’4;,&

- v(4) I,

(2) where V (4) is a generic potential for the field 4. The field equations are obtained by a variation with respect to gFy, F(4)Gpv

= -;T,v

- g,v Cl F(4)

+ F(d),v,

(3)

where G,,” is the Einstein tensor and qy = $;,&;Y- $g,&;&~

+ g,uv(+)

(4)

is the energy-momentum tensor of the scalar field (17 = j: is the box operator). By a variation with respect to 9, we obtain the generalized Klein-Gordon equation, 0 d-RF’(#)

+ v’(4)

= 0,

(5)

131

where the prime indicates the derivative with respect to 4. It is interesting to note that this equation is equivalent to the Bianchi identity [ 61. Let us consider now a homogeneous cosmology and introduce a mean scale factor of the expansion of the universe, a = a(t) = [x(t)y(t)z(t) I’/‘, where x(t), y(t), z(t) are the scale factors in the three spatial directions. Therefore, ‘3’R = 6k/a*. The Lagrangian density from which we get the Einstein and the Klein-Gordon equations generalizes the one present in Ref. [ 61, t = 6ati2F($)

+ 6ci$a2F’(g5) - 6kaF(qS)

+ a3[+j2 - V(4)

1 + $‘u,baab,

(6)

where the dot is the time derivative and cr2 = fo,#rab is the scalar shear. We could also note that a3u2 = Z2/3a3,

(7)

where .?Zis a constant, and rewrite C inserting Z2 /3a3 as the last term [ 5 ] (this Lagrangian is obtained from (2 ) imposing homogeneity). In Ref. [ 6 1, for u* = 0, we have already discussed the existence and the role played by such a symmetry when F (4) and V (4) are two generic functions. Choosing V (4) = n from the very beginning requires a different treatment, which we perform in this paper. Using our Noether symmetry approach in this case yields a peculiar seconddegree polynomial expression of the coupling function F (q5) . We discuss also the general cosmological solutions for this case, stressing also an interesting connection between the Hessian determinant of C (as given by Eq. (6) ) and the asymptotic value of F (4).

2. No-hair theorem with nonminimal coupling Following the standard way to prove the no-hair theorem [ lo,15 1, we have to start by considering the two Einstein equations, which in this case are the generalizations of the usual ones found in the literature (see, for example, Ref. [ 16]), Hz = -&

[H&F’(4)

+ k/a2

(8)

132

Ei=

S. Capozziello et al. /Physics Letters A 188 (1994) 130-l 36

WF’W - jv + p(4) -Hz- -*F:$)

+ @%$)+ pF”(q5) - 3= ‘6 bcFb] ’

(9)

where F’(4) E dF(4)/d4 (we use 8nG~ = 1 units and we see that if F (4) = - 4 we recover the standard Einstein equations of homogeneous cosmology). We rewrite (8) and (9), respectively, as V(4) H2+m=

--&

]Hp(4)

+ k/a*

(10)

+ &P’ + &%b~=bl,

_ *,m _

1 _4” ’ F(4) + -W

_--1 E(4) 2 F(4)

1 + -6F (4)

%bbab.

HZ-f/&~+_ 12lLl

1 2F(4)

ri=

= 7(X).

(iii) Finally we assume that lim HE

t--rot

F

= 0,

-(H*

(12)

that is we assume the standard Einstein regime

is restored more rapidly than the asymptotic H behaviour. Actually, there are also conditions on the first derivatives of F (4) in order to restore the general relativity and to allow that the scalar-tensor theory passes the solar system tests. In fact, recasting the action (2 ) in a Brans-Dicke form (see Ref. [ 17 ] ), we have that

-

;A,

‘2 -6,;ml

-

-

6,;m,~~b~Pb

(13)

6

0,

(14) from which we get -k=3(H2-$I)

2 (H*-f/i)

30,

(15)

where we have called

li = tk

+ -1.

(ii) Second, we suppose the existence of a relation between the two functions F(4) and V(4), that is V(4) = F(F(4)),Fbeingsuchthat F(-X)

1

lab3 0, 12(Ii,( aubO

(11)

Let us make the following three (sufficient) hypotheses: (i) First, we assume that, for large t, F ( C#J ( t ) ) tends to a constant (in our conventions this constant has to be less than zero). This is a natural hypothesis since we are requiring that, after a nonminimal coupling regime, the standard Einstein gravity is restored: G,=

the Brans-Dicke field is Q, = F (4) and the BransDicke coupling function is o(Q) = F($)/2F’($)*. To recover the Einstein regime for t + co, the condition 0 + const must hold (in our case, as said above, F(4) + -3). This implies F’(4) + 0 and then w + 00. Furthermore, the solar-system tests require [18] (do/d@)cc-3+0,whichis (dcc/d0)mm3 = 4F’ (q5j4/F (q5J3 + 0. This condition is satisfied when F (4) approaches a constant without asymptotic variations in the first derivative. In what follows, we see that this situation can be achieved in our context. Using all such hypotheses, Eqs. ( 10) and ( 11) become, respectively, for large t,

F‘(F(+)) 6F($(t))

F(Fcm) = aF,’

(16)

that is, we get again the key inequality ( 1). It is relevant to stress that the condition on the function 7 (it has to be an even function) is satisfied if the Lagrangian (6) has a Noether symmetry which does occur when k = 0 and IJ* = 0 (in fact, it is possible to demonstrate the nonexistence of a Noether symmetry in nonminimal coupling when the number of spatial dimensions is n = 3 and k # 0 [ 61). Unfortunately, not all the functions F which can be obtained from the existence of such a symmetry give rise to models (i.e. to specific potentials) which, once integrated, satisfy also the other two hypotheses we have made. This is the case, for example, for the stringdilaton cosmology (see Ref. [ 71). Essentially, when k = u* = 0, the existence of such a symmetry is a sufficient condition to obtain the relation we require between F (4) and V(4). In what follows, we analyze the model in which k = u* = 0 (i.e. a FRW spacetime) and the potential V (4) is constant but F (4) is a priori generic. From one side this is a particular case of condition (i), and

S. Capozzielloet al. /PhysicsLettersA 188(1994)130-136 we do not have any control on the validity of the other two assumptions we have made. On the other side, this situation is very peculiar in the approach we have developed by using the Noether symmetry. We will show that also in the case l’(4) = const and F (4) generic, the analysis of the existence of this symmetry is relevant for fixing the form of the function F (4)as well as for finding the exact cosmological solutions.

Eq. (2 1) is equivalent to the system

=

(22)

0,

3a + 12F’(4)~

@F"(4) + 3. The ease of a constant potential

[2; + (;)‘]F($J)

+

J+ [

25j

aa

1

+ cj2F"(4)4(&i"-A) =o,

aP = 0,

(23)

+ 2as

F'(4)

Za+a~+a~

>

(

+2eF(@+

Let us suppose that V(4) = A (in the philosophy adopted in Ref. [ 61, this case corresponds to choosing the parameter s = - 1) . The dynamical equations derived from (6 ) are

133

;g

= 0,

(24)

3cYA= 0.

(25)

Supposing F'(4)# 0,it is easy to show that the vector X which is solution of Eq. (21) (or, equivalently, of the system (22)-(25)) is

F'(4) (17) if the function F (4)is given by

d+39+6[;+

($2]F'c+0.

(18)

F(b)

(27)

= I$d + F;4 + Fo,

Eq. (17) is the second-order Einstein equation, whereas ( 18) is the Klein-Gordon equation. The energyjimction associated with 15gives the flrstorder Einstein equation [4,6]

where F,' and FOare integration constants. To avoid the degeneracy of Lagrangian (6)) we assume

(;)2FO) +(;) $F'(@ + i&b2 +A) = 0.

which means that there is no suitable value of the constant 40 such that the coupling can be written as F(4) = A($ + 40)~. In fact, if we compute the Hessian determinant connected to the Lagrangian (6), we find

(19) Such equations are not easy to solve exactly, and we turn to search for Noether symmetries for ,C, giving a straightforward way to derive a change of variables such that our equations become easier. We look for a (lift) vector field defined on the tangent of the minisuperspace manifold Q (a, 4))

where (Yand f3 are unknown functions, such that the Lie derivative of fZalong X is zero [4,6],

LXL =o.

(21)

Fo # 3F;‘,

(28)

l-l = 12a4[F(+) - 3F’@)‘] = 12a2R,

(29)

where we have called R = F(4) - 3F’(+)2. Using (27), 7i is given by 12a4 (Fo - 3Fd’); then, H is different from zero if (28) holds (of course, we are not considering the big-bang singularity). Having found the symmetry described by X, it is known that it is possible to find two new variables w = w (a, C#J)and z = ~(a, 4) such that L results independent of z, giving easier equations to deal with [ 81. To this purpose, we pose [ 4,6]

ixdw=~fi=O, a a4

ix&

-

ifi

-a&j-

-

1

’

(30)

134

S. Capozziello et al. /Physics titters A 188 (I 994) 130-l 36

Particular solutions of ( 30 ) are w = a, z = ad,

Integrating (38 ), we get (31)

4(t)

=

0

so that Lagrangian (6) is transformed into I? = 6Fowti2+ 6F;wtii

+ ;wi2 - Aw3,

Aa:

+

Zo e-“’ a0 >

- 6F& (32)

not depending on z, as expected. Eqs. ( 17), ( 18 ) and ( 19) become

(41)

where zo is another arbitrary constant. Therefore we have lim 4(t) E c& = -6Fd,

t-02

(42)

so that we find

12Fswzij + 6F;Zw + 6F0ti2 - ii’ + 3w2A =o

(

-$eear + L + 6F4

(33)

F~=F(+.,,)=%.

6Fdwti + 6Flti? + wz + iti = 0,

(34)

6Feti2 + 6Fltbi+ fi2+Aw2 = 0.

(35)

Thus, the situations leading to the standard de Sitter behaviour are (i)A>OandRrFm 0. This means that situation (ii) is not physically relevant, being necessary to get Fm < 0 to recover minimally coupled theory at late times. This fact, in general, forbids also the case with A and R having the same sign. Before analyzing some relevant special cases of the general solutions (40) and (41), we want to stress some features of such solutions. Concerning the formula (40) giving the behaviour of a(t), let us assume that CI = cz = a0 > 0.With this particular Cauchy data,wehavea(t) = @[cosh(,/mt)]1/2;that isa = a(t).Thescalefactorisnotonlywelldelined on the whole range of variability of the cosmic time (this is true in general as given by (40) ) but it is an even function of time. This time reversal symmetry is present only in the scale factor: from (41) we seethat for t < 0 the scalar field 40) diverges. Furthermore the leading term in F (q%), asgiven by (27 ) , diverges implying Gd is practically zero at that epoch. what is then described by (40) and (41) is a model of the universe in which there is no singularity in the geometry, but we still find a sort of gravitational singularity because at the very early stage of the universe (that is when t + -00) there is no gravity at all; the standard gravity is then restored with a de Sitter expansion. This peculiar behaviour is essentially due to the nonminimal coupling, that is to the time variation of the gravitational coupling constant: in fact we can compare such a model with the one we get posing F = - 4, that is with a model in which there is only a

Using the vector field (26) defining Noether’s symmetry for L, it is also possible [ 81 to get a constant of motion, 3 = L&cc,

(36)

where 6.~ is the Cartan one-form associated with C

PI,

er =gda+

z d&

SothefirstintegralofEq.

(37) (34) is

J = a2 ifs + 6Flt + 4 = 6Fdww + wi, (* . >

(38)

which is nothing else but a ~?//ai (in fact a ,?Ja z = 0, then d/dt (az/ai)= 0). Let us now consider the coupling ( 27 ) . Taking ( 33 ) and (3 5 ) into account, and differentiating the constant of the motion (38) (or, equivalently, consider (34)), it is easy to get ti + (ZA/R)w = 0.

(39)

The general solution of (39) is a(t)= (cle" +c2e-"')1'2,

(40)

where cl and c2 are integration constants and A = J_.

(43)

S. Capozziello et al. /Physics Letters A 188 (1994) 130-136

&kinematical term and a cosmological constant. Eqs. (17) and (19) become ..

2% + HZ = -f$‘+

A,

135

and

214 x=-t+B. 3l’)2l

(44)

When Fd = 0, R 3 F. results and 3H2 = $$’ + A.

(45)

Summing up these two equations we get ..

.

%+z

0

2

f

3 4(t) = --me

-2.U + P e-“‘Y

(49)

infinitesimal at infinity, so that

-A=O.

(46) Fm E F(&)

Then the question is to see if a = (cl # +CZema )” can be a solution of the equation where (Yis a nonzero is a solution, real number (a = Qexp[(fAt)‘/2] but in this case a (-t ) # a (t ) 1. A simple calculation shows that if (Y# 1, the above function cannot be a solution; in the case (Y= 1 we get the condition CICZ= 0, that is a de Sitter behaviour. Then we conclude that (40) is strictly connected with the form of the coupling F (4) given by (27). Finally it is noteworthy to stress that the three hypotheses we have made on the no-hair theorem are easily satisfied for t >>0.

= Fo.

(50)

This forces FO to be negative (and, therefore, A to be

positive), if we want to recover the minimally coupled theory late in time. As we have said, the situation with FO > 0 and A < 0 must be disregarded as it is unphysical. Let us just notice that the coupling (27) with F; = 0 reduces, for F. = - f , to the case F (4) = i(C;+’ - 1) (when { = a, the Noether symmetry exists, see Ref. [4] ).

4. Conclusions 3. I. Special cases

First of all, let us notice that the situation determined by Fd = FO = 0 is the one giving the degeneracy of Lagrangian (6), being N = 0. We do not consider it here. When FO = 0, it is ‘7%= -3Fl’ c 0, i.e. the asymptotic de Sitter behaviour can be obtained again only for /i > 0 ( (43) is still valid). Putting /i < 0 into the equations leads to the solution a(t)

= Asin’/2(~~t

+ B),

(47)

where A > 0 and B are integration constants. Furthermore, we get 4(t) = g

$_

TF(n,l/&)

x sin-“‘(dmt

+ B)

+ $- sin-‘/2( dmt

where F((Y, l/d) where (Ys arcsin

+ B) - 6Fd,

(48)

(51)

is a first kind elliptic integral [19],

t/2 2sin(fx) 1 + sin(+) + cos(fx) >

(

In this paper, we have clarified two aspects of the so called scalar-tensor theories of gravity. First, we have given some (sufficient) conditions concerning the cosmic no-hair conjecture revisited in the context of a general nonminimal coupling between a scalar field, usually introduced in the inflationary theories, and gravity. Then we have improved the method we have developed in Ref. [ 61 to a case which is singular in that general treatment, that is when the potential is V(4) = n. As a concluding remark we want to stress the interesting behaviour shown by the scale factor, in this nonminimal coupling case, as given by (40). First of all, we do not have the big bang singularity and it is also possible to find special conditions which give rise to an even function for a (t ) . Finally, we can compute the Hubble parameter H = h/a, which is

We immediately get lim H = fL,

t--rat

lim HZ-$,

t--m

(52)

136

S. Capozzielloetal. /PhysicsLettersA

which are two cosmological constants giving asymptotic de Sitter evolutions for any initial conditions (that is for any value of cl,2).

161S. Capozziello

Acknowledgement The authors want to acknowledge the referee for the useful suggestions, which have allowed us to improve the results of this paper.

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