—journal of November 1999
physics pp. 833–841
Exact scalar field cosmologies in a higher derivative theory
B C PAUL
Physics Department, North Bengal University, Siliguri, District Darjeeling 734 430, India Email: bcpaul@nbu.ernet.in
MS received 27 May 1999; revised 1 October 1999
Abstract. We obtain exact cosmological solutions of a higher derivative theory described by the LagrangianL =R+2R2in the presence of interacting scalar field. The interacting scalar field potential required for a known evolution of the FRW universe in the framework of the theory is obtained using a technique different from the usual approach to solve the Einstein field equations.
We follow here a technique to determine potential similar to that used by Ellis and Madsen in Einstein gravity. Some new and interesting potentials are noted in the presence ofR2 term in the Einstein action for the known behaviours of the universe. These potentials in general do not obey the slow rollover approximation.
Keywords. Cosmology; higher derivative theory.
PACS Nos 04.20; 98.80
1. Introduction
In modern cosmology inflation is one of the essential ingredient for building cosmological models of the early universe. It is now understood that it can solve satisfactorily some of the outstanding problems of the standard big bang cosmology [1]. The basic idea of inflation is that there was an epoch in the early universe when the vacuum energy density of the universe dominated leading to an accelerated expansion of the universe. In this epoch universe may be described by either exponential expansion [2] or power law expansion [3]
(S(t)tn;n >>1). In cosmology the usual approach to build cosmological models is to use the Einstein’s equations for a given potentialV() or the field(t)to obtain the dynamical behaviour of the universe. However, a convincing and unambigous expression forV()is still lacking, a different view point may be useful to derive the potential. It was shown by Ram [4] that one can have power law or exponential inflation consistent with the model with an arbitrary potential which may drive inflation. Ellis and Madsen [5] considering a different approach obtain the expressions for the potential in an elegant way in the framework of Einstein gravity. They solved the field equations for the scalar field for a variety of given scale factor of a universe for which the required potential is then derived. In the early epoch curvature squared terms may play an important role in addition to the Einstein action. It is well known that with suitable counter terms viz.,
C Æ
C
Æ
;R
2andadded to the Einstein action, one gets a perturbation theory which
is well behaved. It is renormalizable and asymptotically free [6]. In this paper we explore scalar field potential for a known behaviour of the universe in the presence ofR2term in the Einstein action which is coupled with a scalar field. We derive exact expression for potentialV()and the behaviour of the scalar field(t)assuming a FRW-universe. The paper is organised as follows : Inx2 we set up the relevant field equations, inx3 different models are studied to derive the potential for a known behaviour of the universe in closed, open and flat cases. Finally inx4 we give a brief discussion.
2. Gravitational action
We consider a generalized gravitational action which is given by
I = Z
p
gd 4
x
1
2
R+R 2
+ 1
2
@
@
+V()
; (1)
we adopt the conventionc = 8G = 1, denotes the 4-dimensional indices,g is the 4-dimensional metric,R is the Ricci scalar, is inflaton field, V() is the scalar field potential andis a dimensional coupling constant. The Robertson–Walker line element is given by
ds 2
= dt 2
+S 2
(t)
dr 2
1 kr 2
+r 2
d 2
+sin 2
d 2
; (2)
whereS(t)is the scale factor of the universe,k = +1;0; 1represent theS3;R3and
H
3hypersurfaces respectively. The relevant field equations for the scale factorS(t)and homogeneous scalar field = (t) are obtained by varying the action (1). The field equations are given by
3 _
S 2
+k
S 2
!
= 1
2 _
2
+V()+2
1
2 R
2
6
H _
R _
HR H 2
R
; (3)
d
dt
1
2 _
2
+V()
= 3H _
2
; (4)
whereR= 6[(S=S) +((S_2+k)=S2)].
3. Cosmological models
The set of field equations (3) and (4) contain three unknownsS;andV() which can be solved at least in principle, for any particular choice of potential or the scale factor
S =S(t). In the following examples, we considerS =S(t)i.e, we choose known form of expansion of the universe, and then try to solve forandV().
From eqs (3) and (4), one can write the expressions for_2andV as
_
2
= 1
3H dX
dt
; (5)
V()=X 1
6H dX
dt
; (6)
whereX=3[H2+(k=S2)] 18[2HH+6H_H2 H_2+(2k=S2)H_ (2k=S2)H2
(2k=S 2
)+(k 2
=S 4
)]. For = 0, the above two equations reduce to the corresponding equations for a pure scalar field in Einstein gravity considered by Ellis and Madsen [5].
3.1 de Sitter model
For a pure de Sitter expansion of the universe, one has
S(t)=S
0 e
!t
; (7)
whereS0and! =H(t =0)are constants andH = S=S_ . In this caseH_ = 0and we obtain fork 6=0and6= 0, the evolution of the scalar field using (5) and (6) which is given by
=
0 k
p
12
!
e
!t
S 1
0 q
A 2
S 2
0 e
2!t
+A 2
sin 1
(AS
0 e
!t
) 2
;
(8) whereA2=(1+24!2)=(24k). The corresponding potential is
V(t)=3!
2
+
2k(24!
2
+1)
S 2
0 e
2!t
12k 2
S 4
0 e
4!t
: (9)
We note that for a real inflaton field one requires > 0 for a closed universe and > (1=24!2) for an open universe. This model is valid for a time t
(1=2!)ln(24k=(S 2
0
(1+24!
2
))). As it is difficult to invert eq. (8) to gett = t(), it is not possible to write V explicitly as a function of. However, for a simple case say= (1=24!2), one obtains from (5), the evolutionary behaviour of the scalar field, which is given by
=
0
k
p
2!
2
S 2
0 e
2!t
: (10)
The required potential in this case can be expressed in terms ofwhich is
V()=3!
2
+! 2
(
0 )
2
: (11)
The potential has a minimum at=0. For=0similar potential can be obtained [5].
We note that the decay of the kinetic energy of the scalar field is faster inR2-theory than in pure Einstein gravity. For a closed universe, plus sign in eq. (10) implies a decreasing and minus sign an increasing mode. For an open universe (k= 1), plus sign represents an increasing and minus sign a decreasing mode respectively. For a flat universe (k=0) one gets a solution with constant scalar field in a potentialV =3!2. The exponentially expanding universe gives rise to sufficient inflation for
R
t
f
ti
Hdt65.
3.2 de Sitter expansion from singularity
For de Sitter expansion from singularity, one has
S(t)=S
0
sinh!t; (12)
whereS0and!are constants andH =! coth!t. From eqs (5) and (6) we get
_
2
=
sinh 2
!t
sinh 4
!t
; (13)
where=24!2(1 + 24!2)1 + (k=S2
0
! 2
))and=48!4(1 + (k=S2
0
! 2
))((k=S 2
0
! 2
) 3). The corresponding potential is
V(t)=3!
2
+
sinh 2
!t
4sinh 4
!t
: (14)
On integrating eq.(13) for>0and>0, one gets
=
0
p
+
!
[F(;r) E(;r)]; (15)
where0is an integration constant,F(;r)andE(;r)are the elliptic integrals of first and second kind respectively,=sin 1(2 (=)sinh 2 !t)1=2andr=
p
=(+). As it is difficult to invert (15) to gett =t(), it is not possible to writeV explicitly as a function of. But in principle, the potentialV can be expressed in terms offor various values ofandwhich are determined from,!andS0.
The following simple cases are noted:
Case (i). When = we get(k=S2
0
! 2
)=2 (1=24!
2
)and eq. (13) now can be integrated easily, which yields
=
0
p
!sinh!t
; (16)
where=((1+24!2)(72!2 1))=12and0is a constant of integration. It is now simple to express the potential in terms ofwhich is given by
V()=3!
2
+! 2
(
0 )
2
1+ 1
4
! 2
(
0 )
2
: (17)
The minimum of the potential is found to exist at =0. It has maxima at =0
(2 p
=!
2
). The solution can be accommodated fork = 0; 1under some constraints satisfied by!and. In a flat universe,!2=1=48which leads to=
p
3!. In a closed universe!2 >1=72for positiveand!2 > (1=24)for negative. Open universe permits only positivewhich satisfies the limit1=72<!2<1=48.
Case (ii). When =0one obtainsS0
! =1=
p
3, in a closed universe. In this case eq.
(13) can be integrated to yield
=
0
p
! ln
tanh
!t
2
(18)
and the scalar field potential can be expressed as
V()=3!
2
+ sinh 2
!(
0 )
p
; (19)
where=8!2(1+24!2). The potential has a minimum at=0. The solution admits a negativefor !2 < (1=24). It is interesting to note that for = 0one obtains the solution in closed, open and flat models of the universe [5]. But withR2-term in the Einstein action one gets only a closed universe scenario.
Case (iii). When = 0one gets = (1=24!2), and = 2!2(1+(k=!2S2
0 ))
((k=!
2
S 2
0
) 3). In this case the solution of the scalar field is
=
0
p
!
coth!t: (20)
A real scalar field results for0<S0
!<1=
p
3in a closed universe and for0<S0
!<1
in an open universe. Thus there exist overlapping region0<S0
! <1=
p
3for which the behaviour of the inflaton field is independent of the models of the universe. This solution is not found in a flat case. The scalar field potential in this case is obtained as
V()=3!
2
+ jj
4
! 2
2
(
0 )
2
+
! 4
4jj
(
0 )
4
: (21)
It is easy to see that=0does not lead to the above solution. Thus the evolution of the scalar field and the potential obtained above is because of the presence ofR2term in the Einstein action.
Case (iv). ForS0
!=1one gets ==0in an open universe. In this case one obtains a potential which is constant with an invariant scalar field for6=0.
3.3 de Sitter expansion without singularity
For a de Sitter expansion without singularity, one has
S(t)=S
0
cosh!t; (22)
whereS0and!are constant andH =! tanh!t. Using eqs (5) and (6) we get
_
2
=
cosh 2
!t +
cosh 4
!t
; (23)
V(t)=3!
2
cosh 2
!t +
4cosh 4
!t
; (24)
where=2!2(1+24!2)(1 (k=S2
0
! 2
))and=48!4(1 (k=S2
0
! 2
))((k=S 2
0
! 2
)+3). On integrating eq. (23) for>0and>0, we get
=
0
p
!
E(;r); (25)
where0 is an integration constant,E(;r)is the elliptic integrals of second kind with
= sin 1
( tanh!t)andr =
p
=( ). Here it is difficult to invert eq. (25) to get
t=t(), it is not possible to writeV explicitly as a function of. But one can expressV in terms offor the following simple cases:
Case (i). When = we get(k=S02
! 2
) = (1=24!
2
) 2. Equation (23) can be integrated easily, which yields
=
0
p
! cosh!t
; (26)
where0 is a constant of integration. Using the evolution of the scalar field we get the potential which is
V()=3!
2
+! 2
(
0 )
2
1+ 1
4
! 2
(
0 )
2
; (27)
where=((1+24!2)(72!2 1))=12. The potential has a minimum at=0. The solution can be accommodated fork=0;1under constraints satisfied betweenand!. In a flat universe (k=0) for!2=(1=48)which makes=
p
3!. In a closed universe for!2<(1=72)and in an open universe!2< (1=24)for negative.
Case (ii). When = 0one obtainsS0
! = 1=
p
3, in a closed universe. In this case eq. (23) can be integrated to yield
=
0
2 p
! tan
1
(e
!t
) (28)
which is real for <0i.e., with a negativesatisfying the limit < (1=24!2). The scalar field potential in this case is
V()=3!
2
32!
2
(1+24!
2
) sin
!(
0 )
p
8(1+24!
2
)
!
2
(29) which is an oscillatory potential. Similar potential was obtained by Ellis and Madsen [5] in Einstein gravity forS0
! 1in a closed universe. Thus,R2 theory also permits inflationary scenario in a closed universe with an oscillatory potential forS0
!=1=
p
3. Case (iii). When = 0 we get!2 = (1=24) and = 2!2(1 (k=!2S2
0 ))
((k=!
2
S 2
0
)+3). In this case the solution of the scalar field is obtained for a negative, which is
=
0
p
!
tanh!t (30)
for a closed universe with0<S0
!<1and for an open universe with0<S0
!<1=
p
3. Thus there exist overlapping region0 < S0
! < 1=
p
3for which the behaviour of the inflaton field is independent of the models of the universe. A flat universe does not permit such a solution. The scalar field potential in this case is obtained as
V()=3!
2
+
4
! 2
2
(
0 )
2
+
! 4
4
(
0 )
4
: (31)
In Einstein gravity this solution cannot be obtained as = 0. Thus the scalar field be- haviour and the potential obtained above is because of the presence ofR2 term in the Einstein action which is new.
Case (iv). ForS0
!=1one gets==0in a closed universe. In this case one obtains a potential which is constant and a constant scalar field for6=0.
3.4 Power law inflation
For power law expansion, one has
S(t)=S
0 t
b
; (32)
whereS0andbare arbitrary constants. It represents power law inflation forb >1. The Hubble parameter isH =b=t. From (5) we obtain
_
2
= 2b
t 2
+ 2k
t 2b
+
144(2b 1)
t 4
+
24k(1+b) 2
t 3+2b
+ 24k
t 2b+1
+ 6k
2
t 4b+1
: (33) It is difficult to integrate eq. (33) to express(t), hence it is not possible to invert it to get
t=t(). The case=0was discussed by Ellis and Madsen [5]. It is interesting to look for solution with6= 0. It is difficult to integrate eq. (33) fork6=0. For simplicity we consider a flat universe (k=0) and on integrating eq. (33), one gets
=
0 ln
p
2bt+ p
2bt 2
+144b(2b 1)
p
2bt 2
+144b(2b 1)
t
:
(34) We now discuss our solution for different values ofb.
Case (i). It is evident that forb =1=2one recovers the scalar field behaviour similar to Einstein gravity. Here theR2term in the action does not effect the scalar field and the potential respectively. In this case the behaviour of the scalar field is given by
=
0 lnt
with an exponential potentialV() = 1=4 e2( 0). The scale factor of the universe evolves asS(t)
p
twhich is similar to a radiation dominated universe.
Case (ii). Forb>1=3and=0one gets=0
p
2b lntandV() =b(3b 1)
e (
p
2(
0
)=b). For large value ofb (i.e.,>1) this solution corresponds to power law inflation. The solutions represents a non-inflationary universe for1=3<b<1.
4. Discussion
In this work we obtain a set of cosmological solutions of a higher derivative theory in the presence of a scalar field. The different types of scalar field potential required for the solutions are obtained. However, the potentials obtained here are not derived from the field theoretic model. The parameterin the action when set equal to zero one gets the results
obtained by Ellis and Madsen (in short EM) [5]. We, in this paper explored the other types of potentials that permit cosmological solutions in the presence ofR2term in the action in addition to that found by EM. Our results are given below :
Inxx3.1 we obtain scalar field potentials for a pure de Sitter expansion of the uni- verse. The potential given by (11) is similar to that obtained by EM which is however obtained here when = (1=24!2) ( < 0). The rate of decrease of the scalar field is however found less than that of=0case.
Inxx3.2 we obtain de Sitter expansion of the universe from a singularity. The general potential in this case is given by (14) which can be reduced to a simple form assum- ing a relation forS0
;!;. For example, whenk=S02
=2!
2
(1=24), we obtain a cosmological solution with a polynomial potentialV() =4
i=0
i
i, the above solution is not allowed for=0case. One gets a flat universe for=1=48!2, a closed universe for>1=72!2and an open universe for1=72!2<<1=48!2. We note here another quartic potential (21) for = (1=24!2). A realistic cos- mological scenario may be obtained in these cases fort > 0. The sine hyperbolic type scalar field potential (19) obtained by EM for closed, open and flat models of universe permits de Sitter expansion from singularity for a closed model of universe only in the presence of higher derivative term (6=0).
Inxx3.3 we discuss a de Sitter expansion without singularity. We note that the poly- nomial potential (27) given by V() =
P
4
i=0
i
i admits solution fork=S2
0
=
1=24 2!
2in theR2-theory. The above potential do not give solution for=0. It is interesting to note that one gets a flat universe for =1=48!2, a closed uni- verse for<1=72!2, and an open universe for< (1=24!2). We further note that a quartic potential (31) admits singularity free solution for!2 = (1=24) and S0 satisfying the constraint 0 < S0
! < 1=
p
3 in a closed model and for
0 < S
0
! = 1=
p
3in an open model of universe respectively. EM also obtained an oscillatory scalar field potential in a closed model forS0
! 1but we note that the presence of higher derivative term admits similar scenario forS0
!=1=
p
3.
Inxx3.4 we discuss power law inflation. We discuss the results corresponding to the solutions obtained by EM. It may be mentioned here that one gets an exponential potential required for PLI similar to that obtained in superstrings and supergravity models [7].
To conclude, we note in the present exercise that the value of the parameterin the action plays an important role to determine the behaviour of the early universe.
Acknowledgement
I would like to thank Inter-University Centre for Astronomy and Astrophysics, Pune for the warm hospitality where a part of the work was done. I also like to thank the referee for constructive criticism that has resulted in an improvement in the presentation of this work.
References
[1] A H Guth, Phys. Rev. D23, 347 (1981)
[2] A D Linde, Particle physics and inflationary cosmology (Harwood Academic Publishers, 1990) R H Brandenberger, Rev. Mod. Phys. 57, 1 (1985)
E Kolb and M S Turner, The early universe (New York, Addison Wesley, 1990) D La and P J Steinhardt, Phys. Rev. Lett. 62, 376 (1989)
A D Linde, Phys. Lett. B108, 389 (1982); Phys. Lett. B129, 177 (1983); Phys. Lett. B162, 281 (1985); Mod. Phys. Lett. A1, 81 (1986); Phys. Lett. B175, 395 (1986); Phys. Today September issue 61 (1987)
A Albrecht and P J Steinhardt, Phys. Rev. Lett. 48, 1220 (1982) [3] L F Abbot and M B Wise, Nucl. Phys. B244, 541 (1984)
F Luccin and S Matarrese, Phys. Rev. D32, 1316 (1985) Ya B Zelovich, Mon. Not. R. Astron. Soc. 160, 1P (1972) E Harrison, Phys. Rev. D1, 2726 (1970)
[4] B Ram, Phys. Lett. A172, 404 (1993)
[5] G F R Ellis and M S Madsen, Class. Quantum Gravit. 8, 667 (1991) [6] K S Stelle, Phys. Rev. D16, 953 (1977)
[7] A Salam and E Sezgin, Phys. Lett. B147, 47 (1984)
